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abstract: 'We show that any finite monoid or semigroup presentation satisfying the small overlap condition $C(4)$ has word problem which is a deterministic rational relation. It follows that the set of lexicographically minimal words forms a regular language of normal forms, and that these normal forms can be computed in linear time. We also deduce that $C(4)$ monoids and semigroups are rational (in the sense of Sakarovitch), asynchronous automatic, and word hyperbolic (in the sense of Duncan and Gilman). From this it follows that $C(4)$ monoids satisfy analogues of Kleene’s theorem, and admit decision algorithms for the rational subset and finitely generated submonoid membership problems. We also prove some automata-theoretic results which may be of independent interest.'
title: 'Small Overlap Monoids II: Automatic Structures and Normal Forms'
---
MARK KAMBITES
School of Mathematics, University of Manchester,\
Manchester M13 9PL, England.
Introduction
============
Small overlap conditions are natural combinatorial conditions on monoid and semigroup presentations, which serve to limit the complexity of derivation sequences between equivalent words. They are the natural semigroup-theoretic analogues of the *small cancellation conditions* extensively employed in combinatorial and geometric group theory [@Lyndon77]. It has long been known that monoids with presentations satisfying the condition $C(3)$ have decidable word problem [@Higgins92; @Remmers71; @Remmers80]; recent research of the author [@K_smallover1] has shown that the slightly stronger condition $C(4)$ implies that the word problem is solvable in linear time on a 2-tape Turing machine.
In this paper, we take an automatic-theoretic approach to the study of small overlap semigroups and monoids. Our main result is that the word problem for any $C(4)$ monoid or semigroup presentation is a deterministic rational relation (and moreover, effectively computable as such). It follows from results of automata theory [@Johnson85; @Johnson86] that the set of all words which are lexicographically minimal in their equivalence classes forms a regular language of normal forms, and that a normal form for any element can be computed in linear time. We are also able to deduce that every monoid or semigroup admitting a presentation satisfying the condition $C(4)$ is rational (in the sense of Sakarovitch [@Sakarovitch87]) and hence also asynchronous automatic, and word hyperbolic (in the sense of Duncan and Gilman [@Duncan04]). Another consequence is that $C(4)$ monoids satisfy an analogue of Kleene’s theorem (see for example [@Hopcroft69]): their rational subsets coincide with their recognisable subsets. It follows also that membership is uniformly decidable for rational subsets, and hence also for finitely generated submonoids, of such monoids.
In addition to this introduction, this article comprises four sections. Section \[sec\_background\] briefly reviews the definitions of monoid and semigroup presentations, and of small overlap conditions. Section \[sec\_prefmod\] contains some purely automata-theoretic results which will be used to establish our main results, and may be of some independent interest. In Section \[sec\_smalloverlap\] we combine the results of the previous section with those of [@K_smallover1] to prove our main theorem. Finally, in Section \[sec\_consequences\] we deduce some consequences.
Preliminaries {#sec_background}
=============
In this section we briefly recall the key definitions of semigroup and monoid presentations and of small overlap conditions, which will be used in the rest of this paper.
Let $A$ be a finite alphabet (set of symbols). A *word* over $A$ is a finite sequence of zero or more elements from $A$. The set of all words over $A$ is denoted $A^*$; under the operation of *concatenation* it forms a monoid, called the *free monoid* on $A$. The length of a word $w \in A^*$ is denoted $|w|$. The unique *empty word* of length $0$ is denoted $\epsilon$; it forms the identity element of the monoid $A^*$. The set $A^*\setminus \lbrace \epsilon \rbrace$ of non-empty words forms a subsemigroup of $A^*$, called the *free semigroup on $A$* and denoted $A^+$. For $k \in \mathbb{N}$ we write $A^k$, $A^{\leq k}$ and $A^{< k}$ to denote the set of words in $A^*$ of length respectively exactly $k$, less than or equal to $k$, and strictly less than $k$. If $w \in A^*$ is a word, we write $w^R$ to denote the *reverse* of $w$, that is, the word composed of the letters of $w$ written in reverse order.
A finite *monoid presentation* $\langle A \mid R \rangle$ consists of a finite alphabet $A$ (the letters of which are called *generators*), together with a finite set $R \subseteq A^* \times A^*$ of pairs of words (called *relations*). We say that $u, v \in A^*$ are *one-step equivalent* if $u = axb$ and $v = ayb$ for some possibly empty words $a, b \in A^*$ and relation $(x,y) \in R$ or $(y,x) \in R$. We say that $u$ and $v$ are *equivalent*, and write $u \equiv_R v$ or just $u \equiv v$, if there is a finite sequence of words beginning with $u$ and ending with $v$, each term of which but the last is one-step equivalent to its successor. Equivalence is clearly an equivalence relation; in fact it is the least equivalence relation containing $R$ and compatible with the multiplication in $A^*$. We write ${\overline{u}}$ for the equivalence class of a word $u \in A^*$. The equivalence classes form a monoid with multiplication well-defined by ${\overline{u}} \ {\overline{v}} = {\overline{uv}}$; this is called the *monoid presented* by the presentation.
The *word problem* for a (fixed) monoid presentation $\langle A \mid R \rangle$ is the algorithmic problem of, given as input two words $u, v \in A^*$, deciding whether $u \equiv_R v$.
Definitions corresponding to all of those above can also be made for semigroups (without necessarily an identity element), by taking $A^+$ in place of $A^*$ (in all places except the definition of one-step equivalence, where $a$ and $b$ must still be allowed to be empty).
Now suppose we have a fixed monoid or semigroup presentation $\langle A \mid R \rangle$. We begin by recalling some basic definitions from the theory of small overlap conditions [@Higgins92; @Remmers71]. A *relation word* is a word which appears as one side of a relation in $R$. A *piece* is a word which appears more than once as a factor in the relations, either as a factor of two different relation words, or as a factor of the same relation word in two different (but possibly overlapping) places. Let $m \in \mathbb{N}$ be a positive integer. The presentation is said to *satisfy $C(m)$* if no relation word can be written as a product of *strictly fewer than $m$* pieces. Thus $C(1)$ says that no relation word is empty (which in the semigroup case is a trivial requirement); $C(2)$ says that no relation word is a factor of another.
Retaining our fixed presentation, we now recall some more specialist terminology from [@K_smallover1]. For each relation word $R$, let $X_R$ and $Z_R$ denote respectively the longest prefix of $R$ which is a piece, and the longest suffix of $R$ which is a piece. If the presentation satisfies $C(3)$ then $R$ cannot be written as a product of two pieces, so this prefix and suffix cannot meet; thus, $R$ admits a factorisation $X_R Y_R Z_R$ for some non-empty word $Y_R$. If moreover the presentation satisfies the stronger condition $C(4)$ then $R$ cannot be written as a product of three pieces, so $Y_R$ is not a piece. The converse also holds: a $C(3)$ presentation such that no $Y_R$ is a piece is a $C(4)$ presentation. We call $X_R$, $Y_R$ and $Z_R$ the *maximal piece prefix*, the *middle word* and the *maximal piece suffix* respectively of $R$.
If $R$ is a relation word we write ${\overline{R}}$ for the (necessarily unique, as a result of the small overlap condition) word such that $(R, {\overline{R}})$ or $({\overline{R}}, R)$ is a relation in the presentation. We write ${\overline{X_R}}$, ${\overline{Y_R}}$ and ${\overline{Z_R}}$ for $X_{{\overline{R}}}$, $Y_{{\overline{R}}}$ and $Z_{{\overline{R}}}$ respectively. (This is an abuse of notation since, for example, the word $X_R$ may be a maximal piece prefix of two distinct relation words, but we shall be careful to ensure that the meaning is clear from the context.)
A *relation prefix* of a word is a prefix which admits a (necessarily unique, as a consequence of the small overlap condition) factorisation of the form $a X Y$ where $X$ and $Y$ are the maximal piece prefix and middle word respectively of some relation word $XYZ$. An *overlap prefix (of length $n$)* of a word $u$ is a relation prefix which admits an (again necessarily unique) factorisation of the form $b X_1 Y_1' X_2 Y_2' \dots X_n Y_n$ where
- $n \geq 1$;
- $b X_1 Y_1' X_2 Y_2' \dots X_n Y_n$ has no factor of the form $X_0Y_0$, where $X_0$ and $Y_0$ are the maximal piece prefix and middle word respectively of some relation word, beginning before the end of the prefix $b$;
- for each $1 \leq i \leq n$, $R_i = X_i Y_i Z_i$ is a relation word with $X_i$ and $Z_i$ the maximal piece prefix and suffix respectively; and
- for each $1 \leq i < n$, $Y_i'$ is a proper, non-empty prefix of $Y_i$.
Let $u \in A^*$ be a word and let $p$ be a piece. We say that $u$ is *$p$-active* if $p u$ has a relation prefix $a XY$ with $|a| < |p|$, and *$p$-inactive* otherwise.
We now recall some basic definitions from automata theory. If $A$ is an alphabet, we denote by $A^\$ $ the alphabet $A \cup \lbrace \$ \rbrace$ where $\$ $ is a new symbol not in $A$. The symbol $\$ $ will be used as an *end-marker* for certain types of automata. If $R \subseteq A_1^* \times A_2^*$ is a relation, we denote by $R^\$ $ the set $$R^\$ \ = \ R \ (\$, \$) \ = \ \lbrace (u \$, v \$) \mid (u,v) \in R \rbrace \ \subseteq \ A_1^* \$ \times A_2^* \$ \ \subseteq \ (A_1^\$)^* \times (A_2^\$)^*.$$
A *rational transducer* from an alphabet $A_1$ to an alphabet $A_2$ is a finite directed graph with edges labelled by elements of $A_1^* \times A_2^*$, together with a distinguished initial vertex and a set of distinguished terminal vertices. The labelling of edges extends to a labelling of paths via the multiplication in the direct product monoid $A_1^* \times A_2^*$. A pair $(u,v) \in A_1^* \times A_2^*$ is *accepted* by the transducer if it labels some path from the initial vertex to a terminal vertex. The *relation accepted* by the transducer is the set of all pairs accepted. A relation accepted by some transducer is called a *rational relation* or *rational transduction*. Transductions, which were introduced in [@Elgot65], are of fundamental importance in the theory of formal languages and automata; a detailed study can be found in [@Berstel79].
A *deterministic 2-tape finite automaton* consists of two alphabets $A_1$ and $A_2$, a finite state set $Q$ partitioned into two disjoint subsets $Q_1$ and $Q_2$ with a distinguished initial state and set of distinguished terminal states, and for each $i = 1,2$ a partial function $$\delta_i : Q_i \times A_i^\$ \to Q.$$ Let $\mapsto$ be the smallest binary relation on $A_1^* \$ \times A_2^* \$ \times Q$ such that
- $(au,v, p) \mapsto (u, v, q)$ for all $a \in A_1$, $u \in A_1^* \$ $, $v \in A_2^* \$ $, $p \in Q_1$, $q \in Q$ such that $\delta_1(p, a)$ is defined and equal to $q$; and
- $(u,bv, p) \mapsto (u, v, q)$ for all $b \in A_2$, $u \in A_1^* \$ $, $v \in A_2^* \$ $, $p \in Q_2$, $q \in Q$ such that $\delta_2(p, b)$ is defined and equal to $q$;
and let $\mapsto^*$ be the reflexive, transitive closure of $\mapsto$. We say that a pair $(u,v) \in A_1 \times A_2$ is *accepted* by the automaton if there exists an initial state $q_0$ and a terminal state $q_1$ such that that $(u \$, v \$, q_0) \mapsto^* (\epsilon, \epsilon, q_1)$. Once again, the *relation accepted* by the automaton is the set of all pairs accepted.
A relation is called a *deterministic rational relation* if it is accepted by a deterministic 2-tape automaton, and a *reverse deterministic rational relation* if the relation $$\lbrace (u^R, v^R) \mid (u, v) \in R \rbrace$$ is accepted by a deterministic 2-tape automaton. In general, a deterministic rational relation need not be reverse deterministic rational [@Fischer68 Theorem 1]. Every \[reverse\] deterministic rational relation is accepted by a transducer [@Fischer68] and so is indeed a rational relation. The following elementary proposition gives a partial converse to this statement; the general idea is well known but the precise formulation we need does not seem to have appeared in the literature, so for completeness we give an outline proof.
\[prop\_dettransducer\_condition\] Let $R \subseteq A_1^* \times A_2^*$ be a relation and suppose $R^\$ $ is accepted by a transducer with the property that for every state $q$, one of the following (mutually exclusive) conditions holds:
- $q$ has an edge leaving it, and every edge leaving $q$ has the form $(a, \epsilon)$ for some $a \in A_1^\$ $, and there is at most one such edge for each $a \in A_1^\$ $;
- $q$ has an edge leaving it, and every edge leaving $q$ has the form $(\epsilon, a)$ for some $a \in A_2^\$ $, and there is at most one such edge for each $a \in A_2^\$ $;
- there are no edges leaving $q$;
- there is exactly one edge leaving $q$, and that edge has label $(\epsilon, \epsilon)$;
Then $R$ is accepted by a deterministic 2-tape automaton.
Let $M$ be the transducer accepting $R^\$ $ with the given property, and let $Q$ be the state set of $M$. Notice that for each state $q$, there is at most one state, which we call ${\overline{q}}$, with the property that there is a path from $q$ to ${\overline{q}}$ labelled $(\epsilon, \epsilon)$ and ${\overline{q}}$ satisfies condition (i) or (ii) in the statement of the proposition. Since (i) and (ii) are mutually exclusive, we may choose a partition $Q = Q_1 \cup Q_2$ of $Q$ into disjoint subsets such that for every $q \in Q$ with ${\overline{q}}$ defined we have that ${\overline{q}}$ satisfies condition (i) if and only if $q \in Q_1$, and similarly ${\overline{q}}$ satisfies condition (ii) if and only if $q \in Q_2$. (States $q$ for which ${\overline{q}}$ is not defined may be assigned arbitrarily to either $Q_1$ or $Q_2$).
We now define a new deterministic 2-tape automaton $N$ as follows. The two tape alphabets of $N$ are $A_1$ and $A_2$. The state set of $N$ is the state set $Q$ of $M$ partitioned into the subsets $Q_1$ and $Q_2$ constructed above. The initial state of $N$ is the initial state of $M$. The terminal states of $N$ consist of all states $p \in Q$ such that $M$ has a path from $p$ to a terminal state with label $(\epsilon, \epsilon)$. For each $a \in A_1^\$ $, $p \in Q_1$ and $q \in Q$ we set $\delta_1(p, a) = q$ if and only if ${\overline{p}}$ is defined and $M$ has an edge from ${\overline{p}}$ to $q$ with label $(a, \epsilon)$. Similarly, for each $a \in A_2^\$ $, $p \in Q_2$ and $q \in Q$ we set $\delta_2(p, a) = q$ if and only if ${\overline{p}}$ is defined and $M$ has an edge from ${\overline{p}}$ to $q$ with label $(\epsilon, a)$. It follows directly from the criteria on the automata that each $\delta_i$ is a well-defined partial function from $Q_i \times A_i^\$ $ to $Q$.
It is now a routine matter to verify that the deterministic 2-tape automaton $N$ accepts a pair $(u,v)$ if and only if $M$ accepts $(u \$, v \$)$.
Prefix-Rewriting Automata {#sec_prefmod}
=========================
In this section, we study a type of automaton called a *2-tape prefix-rewriting automaton*. We show that any relation accepted by a \[deterministic\] 2-tape prefix-rewriting automaton with a certain property called *bounded expansion* is a \[deterministic\] rational relation. In Section \[sec\_smalloverlap\] we shall apply this result to show that the word problem for a $C(4)$ monoid presentation is a deterministic rational relation.
Let $k \in {\mathbb{N}}$ and $A_1$ and $A_2$ be finite alphabets. A *$k$-prefix-rewriting automaton from $A_1$ to $A_2$* is a finite directed graph with edges labelled by elements of $$\left( (A_1^{\leq k} \times A_1^{\leq k}) \cup (A_1^{< k} \$ \times A_1^{< k} \$) \right) \times \left( (A_2^{\leq k} \times A_2^{\leq k}) \cup (A_2^{< k} \$ \times A_2^{< k} \$) \right),$$ together with a distinguished initial vertex and a set of distinguished terminal vertices. Given such an automaton with vertex set $Q$, we define a binary relation $\to$ on $A_1^* \$ \times A_2^* \$ \times Q$ by $$(u_1 \$, v_1 \$, q_1) \to (u_2 \$, v_2 \$, q_2)$$ if and only if there exist words $x_1$, $x_2$, $y_1$, $y_2$, $u'$ and $v'$ in the appropriate alphabets such that $$u_1 = x_1 u', \ u_2 = x_2 u', \ v_1 = y_1 v', \ v_2 = y_2 v'$$ and $(x_1, x_2, y_1, y_2)$ labels an edge from $q_1$ to $q_2$. If this holds we say that *the edge $e$ is applicable in the configuration $(u_1 \$, v_1 \$, q_1)$*. We call the automaton *deterministic* if in each configuration $(u,v,q) \in A_1^* \$ \times A_2^* \$ \times Q$ there is at most one edge applicable.
Let $\to^*$ denote the reflexive, transitive closure of the relation $\to$. We say that a pair $(u,v) \in A_1^* \times A_2^*$ is *accepted* by the automaton if there exists a terminal state $q_1$ such that $$(u \$, v \$, q_0) \to^* (\$, \$, q_1)$$ where $q_0$ is the initial state. As usual, the *relation accepted* by the automaton is the set of all pairs in $A_1^* \times A_2^*$ which are accepted by the automaton.
Intuitively, a 2-tape prefix-rewriting automaton is very similar to a 2-pushdown automaton; the only essential difference is that we allow both stacks to be initialised with non-empty words, and view the automaton as accepting pairs of words and defining a relation instead of a language. As one might expect, such automata are extremely powerful, being easily seen to accept in particular any relation of the form $L \times \lbrace \epsilon \rbrace$ where $L$ is a recursively enumerable language. However, we shall be interested in a more restricted class of such automata. We say that a prefix-rewriting automaton has *bounded expansion* if there exists a constant $b \in {\mathbb{N}}$ such that whenever $$(u_1, v_1, q_1) \to^* (u_2, v_2, q_2)$$ we have $|u_2| \leq |u_1| + b$ and $|v_2| \leq |v_2| + b$. We call such a value of $b$ an *expansion bound* for the automaton.
Note that the bounded expansion condition places a requirement on the contents of each store independently. This contrasts with the *shrinking* and *length-reducing* conditions on 2-pushdown automata, used to describe growing context-sensitive and Church-Rosser languages [@Buntrock98], where a restriction is applied to the total size of the 2 stores considered together. It transpires that our condition is a very strong one, in that a relation accepted by a prefix-rewriting automaton with bounded expansion is necessarily rational.
\[thm\_bpm\_imp\_rational\] Any relation accepted by a \[deterministic\] 2-tape prefix-rewriting automaton with bounded expansion is a \[deterministic\] rational transduction. Moreover, given a \[deterministic\] 2-tape prefix-rewriting automaton and an expansion bound for it, one can effectively construct a \[deterministic\] transducer recognising the same relation.
Let $M$ be a $2$-tape $k$-prefix-rewriting automaton with bounded expansion accepting a relation $R \subseteq A_1^* \times A_2^*$, and let $b \in {\mathbb{N}}$ be an expansion bound for $M$. We construct from $M$ a finite transducer $N$ which simulates $M$ and so accepts $R^\$ $. Intuitively, the new transducer will read $u$ and $v$, buffering at least the first $k$ characters of each in the finite state control. Prefix-modification can thus be simulated by modifying only the contents of the finite state control. Since a prefix-rewriting automaton can replace a prefix with a longer one, it may be necessary to store more than $k$ characters of each word in the finite state control, but the expansion bound serves to ensure that a buffer of some fixed size (namely $k+b$) will always suffice.
Formally, for $i = 1,2$ we let $C_i = A_i^{\leq k+b} \cup A_i^{< k+b} \$ $ and let $B_i$ be the set of all words $x \in C_i$ such that either $|x| \geq k$ or the final letter of $x$ is $\$ $. (Intuitively, $C_i$ will be the set of all possible states for the buffer on tape $i$, while $B_i$ will be the set of “adequately populated” buffer states in which it is not immediately necessary to read any more of the input word.)
We construct a transducer $N$ as follows. The state set of $N$ is $C_1 \times C_2 \times Q$ where $Q$ is the state set of $M$. The initial state is $(\epsilon, \epsilon, q_0)$ where $q_0$ is the initial state of $M$. The terminal states are those of the form $(\$, \$, q)$ with $q$ a terminal state of $M$. The edges are as follows:
- for every $x \in C_1$, $y \in C_2$ with $x \notin B_1$, every $a \in A_1^\$ $ such that $xa \in C_1$ and every state $q$, there is an edge from $(x,y,q)$ to $(xa,y,q)$ with label $(a, \epsilon)$;
- for every $x \in C_1$, $y \in C_2$ with $x \in B_1$ but $y \notin B_2$, every $a \in A_2^\$ $ such that $ya \in C_2$ and every state $q$, there is an edge from $(x,y,q)$ to $(x,ya,q)$ with label $(\epsilon, a)$;
- for each edge in $M$ from $p$ to $q$ with label $(u_1, u_2, v_1, v_2)$ and each $x', y'$ such that $u_1 x' \in B_1$ and $v_1 y' \in B_2$, there is an edge from $(u_1 x', v_1 y', p)$ to $(u_2 x', v_2 y', q)$ with label $(\epsilon, \epsilon)$ provided $u_2 x' \in C_1$ and $v_2 u' \in C_2$.
Edges of types (1) and (2) serve simply to read the input words into the buffers until each contains sufficient data (at least $k$ letters or the entire of the input if this is less), while edges of type (3) simulate the transitions of the prefix-rewriting automaton $M$ by operating only on the buffers.
Notice that once the transducer reaches a state in $A_1^{<k+b} \$ \times C_2 \times Q$ (that is, one where the first buffer content contains the symbol $\$ $), it will always remain in such a state, and will never again read from the first input word. Similarly, once it reaches a state in $C_1 \times A_2^{<k+b} \$ \times Q$ it will always remain in such a state and will never again read from the second input word. Noting also that all the terminal states lie in both of these sets, it follows that all pairs accepted by the transducer lie in $A_1^* \$ \times A_2^* \$ $.
We say that a configuration $(u_1,v_1,q_1)$ has expansion bound $(c, d) \in \mathbb{N} \times \mathbb{N}$ if whenever $(u_1, v_1, q_1) \to^* (u_2, v_2, q_2)$ we have $|u_2| \leq |u_1| + c$ and $|v_2| \leq |u_1| + d$. Note that the expansion bound condition on the automaton means that $(b,b)$ is an expansion bound for every configuration. We shall need the following lemma.
\[lemma\_prefmodimptransducer\] Suppose $(u_1,v_1,q_1) \to^* (u_2,v_2,q_2)$ in the prefix-rewriting automaton $M$. Suppose further than $(u_1, v_1, q_1)$ has expansion bound $(c_1, d_1)$ and that $u_1 = s_1 s_1'$, $v_1 = t_1 t_1'$ where $|s_1| \leq k+b-c_1$ and $|t_1| \leq k+b-d_1$. Then there exist factorisations $u_2 = s_2 s_2'$ and $v_2 = t_2 t_2'$ and an expansion bound $(c_2, d_2)$ for $(u_2, v_2, q_2)$ such that $|s_2| \leq k+b-c_2$, $|t_2| \leq k+b-d_2$ and the transducer $N$ has a path from $(s_1, t_1, q_1)$ to $(s_2, t_2, q_2)$ with label $(g, h)$ where $s_1' = g s_2'$ and $t_1' = h t_2'$.
We use induction on the number of steps in the transition sequence from from $(u_1, v_1, q_1)$ to $(u_2, v_2, q_2)$. Certainly if $(u_1, v_1, q_1) = (u_2, v_2, q_2)$ it suffices to take $s_2 = s_1$, $s_2' = s_1'$, $t_2 = t_1$, $t_2' = t_1'$, $c_2 = c_1$, $d_2 = d_1$ and $g = h = \epsilon$.
Next we consider one-step case, that is, the case in which $(u_1,v_1,q_1) \to (u_2,v_2,q_2)$. Let $g$ be the shortest prefix of $s_1'$ such that $s_1 g \in B_1$; similarly, let $h$ be the shortest prefix of $t_1'$ such that $t_1 h \in B_2$. It follows easily from the definition that our transducer $N$ has a path from $(s_1, t_1, q_1)$ to $(s_1 g, t_1 h, q_1)$ with label $(g, h)$.
Now since $(u_1, v_1, q_1) \to (u_2, v_2, q_2)$, by definition there exist words $x_1$, $x_2$, $y_1$, $y_2$, $u'$ and $v'$ such that $u_1 = x_1 u'$, $u_2 = x_2 u'$, $v_1 = y_1 v'$, $v_2 = y_2 v'$ and $(x_1, x_2, y_1, y_2)$ labels an edge from $q_1$ to $q_2$. Since $|x_1|, |y_1| \leq k$ we have that $x_1$ and $y_1$ are prefixes of $s_1 g$ and $t_1 h$ respectively, say $s_1 g = x_1 x'$ and $t_1 h = y_1 y'$. But now by the definition of our transducer, there is an edge from $(s_1 g = x_1 x', t_1 h = y_1 y', q_1)$ to $(x_2 x', y_2 y', q_2)$ with label $(\epsilon, \epsilon)$. Thus, setting $s_2 = x_2 x'$ and $t_2 = y_2 y'$ and defining $s_2'$ and $t_2'$ accordingly, we obtain a path from $(s_1, t_1, q_1)$ to $(s_2, t_2, q_2)$ with label $(g,h)$.
Now we have $$x_2 x' s_2' = s_2 s_2' = u_2 = x_2 u'$$ so cancelling on the left we obtain $u' = x' s_2'$. But now $$s_1 s_1' = u_1 = x_1 u' = x_1 x' s_2' = s_1 g s_2'$$ so cancelling again yields $s_1' = g s_2'$ as claimed. An entirely similar argument shows that $t_1' = h t_2'$.
Next, notice that we have $|u_1| - |u_2| = |s_1| - |s_2|$ and similarly $|v_1| - |v_2| = |s_1| - |s_2|$. Set $c_2 = c_1 + |s_1| - |s_2|$ and $d_2 = d_1 + |t_1| - |t_2|$. Clearly since any state derivable from $(u_2, v_2, q_2)$ is also derivable from $(u_1, v_1, q_1)$, it is readily verified that $(c_2, d_2)$ is an expansion bound for $(u_2, v_2, q_2)$. But now we have $$|s_2| \ = \ |s_1| + c_1 - c_2 \ \leq \ (k + b - c_1) + c_1 - c_2 \ = \ k + b - c_2$$ and similarly $|t_2| \leq k + b - d_2$ as required to complete the proof of the lemma in the one-step case.
The inductive argument for the general case is now straightforward.
Now if $(u,v)$ is accepted by the prefix-rewriting automaton then by definition we have $(u \$,v \$, q_0) \to^* (\$, \$, q_t)$ where $q_0$ is the initial state and $q_t$ is some terminal state. Since the automaton has expansion bound $b$, the state $(u \$,v \$, q_0)$ has expansion bound $(b,b)$. So taking $u_1 = u$, $v_1 = v$, $q_1 = q_0$, $q_2 = q_t$ $c_1 = d_1 = b$, $s_1 = t_1 = \epsilon$, $s_1' = u$ and $s_2' = v$ and applying Lemma \[lemma\_prefmodimptransducer\], our transducer has a path from $(\epsilon, \epsilon, q_0)$ to $(s_2, t_2, q_t)$ with label $(g,h)$ where $s_2 s_2' = t_2 t_2' = \$ $, $u = s_1' = g s_2'$ and $v = t_1' = h t_2'$.
Now either $s_2 = \epsilon$ and $s_2' = \$ $, or $s_2 = \$ $ and $s_2' = \epsilon$. In the latter case we have $g = u \$ $. In the former case we have $g = u$ and there is clearly an edge from $(s_2, t_2, q_t)$ to $(s_2 \$ = \$, t_2, q_t)$ labelled $(\$, \epsilon)$, so in either case there is a path from $(\epsilon, \epsilon, q_0)$ to $(\$, t_2, q_t)$ with label $(u \$, h)$. A similar argument deals with the case that $h = v$, showing that in all cases there is a path from the start state $(\epsilon, \epsilon, q_0)$ to the terminal state $(\$, \$, q_t)$ with label $(u \$,v \$)$. Thus, the transducer $N$ accepts $(u \$,v \$)$ as required.
Conversely, suppose $(u \$, v \$)$ is accepted by our transducer. Then there must be a path $\pi$ from $(\epsilon, \epsilon, q_0)$ to $(\$, \$, q_t)$ for some initial state $q_0$ and terminal state $q_t$. Now clearly $\pi$ admits a unique decomposition of the form $$\pi \ = \ \lambda_0 \rho_1 \lambda_1 \rho_2 \dots \rho_n \lambda_n$$ where each $\rho_i$ is a single edge of type (3) and each $\lambda_i$ is a (possibly empty) path consisting entirely of edges of types (1) and (2). Clearly each $\rho_i$ has label $(\epsilon, \epsilon)$. Suppose each $\lambda_i$ has label $(u_i, v_i)$; then clearly $u \$ = u_0 u_1 \dots u_n$ and $v \$ = v_0 v_1 \dots v_n$. Suppose that for $0 \leq i \leq n$, after traversing the initial segment of the path $\pi$ up to and including $\lambda_i$, the automaton is in configuration $(x_i, y_i, q_i)$. Notice that, since the paths $\lambda$ do not change the state component, $q_0$ is consistent with its use above, and in particular is an initial state in the prefix-rewriting automaton $M$. Similarly, $q_n = q_t$ is a terminal state of $M$. Now for $0 \leq i \leq n$ define $$c_i = x_i u_{i+1} u_{i+2} \dots u_n \text{ and } d_i = y_i v_{i+1} v_{i+2} \dots v_n.$$ Clearly we have that $x_0 = u_0$ and $y_0 = v_0$, from which it follows that $c_0 = u \$ $ and $d_0 = v \$ $. We also have $x_n = y_n = \$ $ so that $c_n = d_n = \$ $.
Now it is straightforward to see that for $1 \leq i \leq n$ we have $$(c_{i-1}, d_{i-1}, q_{i-1}) \to (c_i, d_i, q_i)$$ so that $$(u \$, v \$,q_0) = (c_0, d_0, q_0) \ \to^* \ (c_n, d_n, q_n) = (\$, \$, q_t).$$ which by definition means that $(u,v)$ is accepted by the 2-tape prefix-rewriting automaton $M$. This completes the proof that the transducer $N$ accepts the relation $R^\$ $. It is easy to show that for any relation $T$, $T$ is a rational relation if and only if $T^\$ $ is a rational relation, so this suffices to prove that $R$ is a rational relation.
Finally, suppose that the original prefix-rewriting automaton $M$ is deterministic. We claim that the transducer $N$ which we have constructed to accept $R^\$ $ satisfies the conditions of Proposition \[prop\_dettransducer\_condition\], from which it will follow that $R$ is a deterministic rational relation, as required.
To this end, consider a state $(x,y,q)$ in $N$. If $x \notin B_1$ then it follows immediately from the definition that all out-edges have labels of the form $(a, \epsilon)$ with $a \in A_1$ and that there is exactly one such for each $a \in A$, so that condition (i) holds. Similarly, if $x \in B_1$ but $y \notin B_2$ then all out-edges have labels of the form $(\epsilon, a)$ and there is exactly one such for each $a \in A_2$ so condition (ii) holds.
Finally, suppose $x \in B_1$ and $y \in B_2$. From the definition of $N$, any edge leaving $(x,y,p)$ must have label $(\epsilon, \epsilon)$. If there were more than one such edge, then each would correspond to a different possible transition in $M$ from the state $(x,y,p)$; but by the determinism assumption on $M$ there can only be one such transition, so this would give a contradiction. Thus we deduce that there is at most one such edge, so that either condition (iii) or condition (iv) holds. This completes the proof.
We emphasise that Theorem \[thm\_bpm\_imp\_rational\] does *not* give a means to effectively construct a transducer for a relation $R$ starting only from a 2-tape prefix-rewriting automaton with bounded expansion which accept $R$. The construction in the proof makes explicit use of the expansion bound for the prefix-rewriting automaton, and it is not clear that one can effectively compute an expansion bound from the automaton, even given the knowledge that such a bound exists.
Automata for the Word Problem in Small Overlap Monoids {#sec_smalloverlap}
======================================================
The aim of this section is to show that the word problem for any $C(4)$ monoid must be a deterministic rational relation. Throughout this section, we fix a monoid presentation $\langle A \mid R \rangle$ satisfying the condition $C(4)$.
In [@K_smallover1] we presented an efficient recursive algorithm which can be used to solve the word problem for such a presentation. For ease of reference the algorithm is reproduced in Figure 1.
$u = \epsilon$ or $v = \epsilon$ \[li\_start\_a\] $u = \epsilon$ and $v = \epsilon$ and $p = \epsilon$ \[li\_allepsilon\] \[li\_someepsilon\] \[li\_end\_a\] $u$ does not have the form $XYu'$ with $XY$ a clean overlap prefix $u$ and $v$ begin with different letters \[li\_start\_b\] \[li\_uvdifferentstart\] $p \neq \epsilon$ and $u$ and $p$ begin with different letters \[li\_updifferentstart\] $u \gets u$ with first letter deleted $v \gets v$ with first letter deleted $p \neq \epsilon$ $p \gets p$ with first letter deleted $\proc{WP-Prefix}(u,v,p)$ \[li\_rec\_nomop\] \[li\_end\_b\]
$\kw{let}\ X, Y, u'$ be such that $u = XY u'$ \[li\_start\_c\]
$p$ is a prefix of neither $X$ nor ${\overline{X}}$ \[li\_pnotprefix\]
$v$ does not begin either with $XY$ or with ${\overline{XY}}$ \[li\_vstartswrong\]
$u = XYZ u''$ and $v = XYZ v''$ $u''$ is $Z$-active $\proc{WP-Prefix}(Z u'', Z v'', \epsilon)$ \[li\_rec\_case1a\] $\proc{WP-Prefix}({\overline{Z}} u'', {\overline{Z}} v'', \epsilon)$ \[li\_rec\_case1b\]
$u = XY u'$ and $v = XY v'$ $p$ is a prefix of $X$ $\proc{WP-Prefix}(u',v', \epsilon)$ \[li\_rec\_case2a\] $\proc{WP-Prefix}(u',v', Z)$ \[li\_rec\_case2b\]
$u = XYZ u''$ and $v = {\overline{XYZ}} v''$ $u''$ is $Z$-active $\proc{WP-Prefix}(Z u'', Z v'', \epsilon)$ \[li\_rec\_case3a\] $\proc{WP-Prefix}({\overline{Z}} u'', {\overline{Z}} v'', \epsilon)$ \[li\_rec\_case3b\]
$u = XY u'$ and $v = {\overline{XYZ}} v''$ $\proc{WP-Prefix}(u', Z v'', \epsilon)$ \[li\_rec\_case4\]
$u = XYZ u''$ and $v = {\overline{XY}} v'$ $\proc{WP-Prefix}({\overline{Z}} u'', v', \epsilon)$ \[li\_rec\_case5\]
$u = XY u'$ and $v = {\overline{XY}} v'$ $z$ be the maximal common suffix of $Z$ and ${\overline{Z}}$ $z_1$ be such that $Z = z_1 z$ $z_2$ be such that ${\overline{Z}} = z_2 z$ $u'$ does not begin with $z_1$ or $v'$ does not begin with $z_2$; \[li\_case6no\] $u''$ be such that $u' := z_1 u''$ $v''$ be such that $v' := z_2 v''$; $\proc{WP-Prefix}(u'', v'', z)$ \[li\_rec\_case6\] \[li\_end\_c\]
It takes as input a piece of the presentation $p \in A^*$ and two words $u, v \in A^*$ and outputs **YES** if $u \equiv v$ and $p$ is a possible prefix of $u$ (and hence also of $v$). Otherwise it outputs **NO**. In particular, if $p = \epsilon$ then the algorithm outputs **YES** if $u \equiv v$ and **NO** if $u \not\equiv v$, thus solving the word problem for the presentation. See [@K_smallover1 Lemma 5] and [@K_smallover1 Lemma 6] for proofs of correctness and termination respectively.
The proof strategy for our main result is to show that this algorithm can be implemented on a deterministic 2-tape prefix-rewriting automaton with bounded expansion. The results of Section \[sec\_prefmod\] then allow us to conclude that the word problem is a deterministic rational relation.
\[thm\_main\] Let $\langle A \mid R \rangle$ be a finite monoid presentation satisfying the small overlap condition $C(4)$. Then the relation $$\lbrace (u, v) \in A^* \times A^* \mid u \equiv v \rbrace$$ is deterministic rational and reverse deterministic rational. Moreover, one can, starting from the presentation, effectively compute 2-tape deterministic automata recognising this relation and its reverse.
Let $k$ be twice the maximum length of a relation word in the presentation. We construct a deterministic 2-tape $k$-prefix-rewriting automaton recognising the desired relation, and an expansion bound for this automaton. By Theorem \[thm\_bpm\_imp\_rational\], this suffices to show that the given relation is deterministic rational and that a 2-tape deterministic automaton for it can be effectively constructed. Since the $C(4)$ condition on the presentation is entirely left-right symmetric, the claim regarding the reverse relation also follows.
Let $P$ be the set of all pieces of the presentation $\langle A \mid R \rangle$, and let $+$ be a new symbol not in $P$. Recall that $\epsilon$ is by definition a piece of every presentation, so certainly $\epsilon \in P$. Let $W = A^k \cup A^{< k} \$ $. We define a 2-tape prefix-rewriting automaton with
- state set $P \cup \lbrace + \rbrace$;
- initial state $\epsilon$,
- unique terminal state $+$;
and edges defined as follows.
- an edge from $\epsilon$ to $+$ labelled $(\$, \$, \$, \$)$.
- for every $u \in W$ with $u \neq \$ $ and such that $u$ has no clean overlap prefix of the form $XY$, and every $v \in W$ such that $v \neq \$ $ and $u$ and $v$ begin with the same letter, a transition from $p$ to $p'$ labelled $(u, u', v, v')$ where $u'$, $v'$ and $p'$ are obtained from $u$, $v$ and $p$ respectively by deleting the first letter.
In addition for every $p \in P$ and $u,v \in W$ such that $u$ has a clean overlap prefix (say $XY$) and $p$ is a prefix of either $X$ or ${\overline{X}}$ or both, the automaton may have an edge from $p$ to another state in $P$ as follows:
- If $u = XYZ u''$, $v = XYZv''$ and $u''$ is $Z$-active, the automaton has an edge from $p$ to $\epsilon$ labelled $(u, Zu'', v, Zv'')$.
- If $u = XYZ u''$, $v = XYZv''$ and $u''$ is **not** $Z$-active, the automaton has an edge from $p$ to $\epsilon$ labelled $(u, {\overline{Z}}u'', v, {\overline{Z}} v'')$.
- If $u = XY u'$, $v = XY v'$, $u$ and $v$ do not both have $XYZ$ as a prefix, and $p$ is a prefix of $X$, the automaton has an edge from $p$ to $\epsilon$ labelled $(u, u', v, v')$.
- If $u = XY u'$, $v = XY v'$, $u$ and $v$ do not both have $XYZ$ as a prefix, and $p$ is **not** a prefix of $X$, the automaton has an edge from $p$ to $Z$ with label $(u, u', v, v')$.
- If $u = XYZ u''$, $v = {\overline{XYZ}}v''$ and $u''$ is $Z$-active, the automaton has an edge from $p$ to $\epsilon$ labelled $(u, Zu'', v, Zv'')$.
- If $u = XYZ u''$, $v = {\overline{XYZ}}v''$ and $u''$ is **not** $Z$-active, the automaton has an edge from $p$ to $\epsilon$ labelled $(u, {\overline{Z}}u'', v, {\overline{Z}} v'')$.
- If $u = XY u'$, $v = {\overline{XYZ}} v''$ and $u$ does not have $XYZ$ as a prefix, the automaton has an edge from $p$ to $\epsilon$ labelled $(u, u', v, Z v'')$.
- If $u = XYZ u''$, $v = {\overline{XY}} u'$ and $v$ does not have ${\overline{XYZ}}$ as a prefix, the automaton has an edge from $p$ to $\epsilon$ labelled $(u, {\overline{Z}} u'', v, v')$.
- If $u = XY u'$, $v = {\overline{XY}} v'$, $u$ does not begin with $XYZ$, $v$ does not begin with ${\overline{XYZ}}$, $z$ is the maximum common suffix of $Z$ and ${\overline{Z}}$, $Z = z_1 z$, ${\overline{Z}} = z_2 z$, $u' = z_1 u''$, $v' = z_2 v''$, the automaton has an edge from $p$ to $z$ labelled $(u, u'', v, v'')$.
First, notice that this automaton is deterministic. Indeed, all edges leaving a given vertex $p \in P$ have labels of the form $(u, x, v, y)$ with $u, v \in W$. Notice that no member of the set $W$ is a prefix of another; it follows that no word has two distinct words in $W$ as prefixes, which means that the choice of prefixes $u$ and $v$ to act on is uniquely determined by the configuation in which the action is to be applied. Now it can be verified by examination that the various conditions on $u$, $v$ and $p$ which result in the inclusion of an edge from $p$ with label of the form $(u, x, v, y)$ are mutually exclusive, so that there is at most one such edge, and hence at most one transition applicable in any given configuration.
It is now an entirely routine matter to prove by induction that for every piece $p \in A^*$ and words $u, v \in A^*$ we have $$(u \$, v \$, p) \ \to^* \ (\$, \$, +)$$ if and only if the algorithm outputs **YES**, that is, if and only if $u \equiv v$ and $p$ is a possible prefix of $u$. Transitions of types B, C1, C2, C3, C4, C5, C6, C7, C8 and C9 correspond to the recursive calls at lines 15, 24, 25, 28, 29, 32, 33, 35, 37, 46 respectively, while transition of type A corresponds to termination with the answer **YES** at line 3 of the algorithm. The conditions under which the algorithm terminates with the answer **NO** (at lines 4, 7, 9, 19, 21 and 43) all correspond to non-terminal configurations of the automaton in which no transitions are applicable. It follows from [@K_smallover1 Lemma 7] that the tests for clean overlap prefixes and $Z$-activity on the buffer contents are equivalent to performing the corresponding tests on the whole of the remaining input, as demanded by the algorithm.
In particular, we have $$(u \$, v \$, \epsilon) \ \to^* \ (\$, \$, +)$$ if and only if $u \equiv v$, as required to show that our prefix-rewriting automaton solves the word problem. It remains only to find an expansion bound for the automaton. Let $b$ be the length of the longest relation word in the presentation $\langle A \mid R \rangle$.
Suppose $(u_0, v_0, q_0) \to^* (u_1, v_1, q_1)$ and suppose that $u_0 = z_0 u_0'$ and $v_0 = z_0 v_0'$ where $z_0$ is either a proper suffix of a relation word or the empty word. We claim that there are factorisations $u_1 = z_1 u_1'$ and $v_1 = z_1 v_1'$ where $z_1$ is a proper suffix of relation word or the empty word, $|u_1'| \leq |u_0'|$ and $|v_1'| \leq |v_0'|$.
We consider first the one-step case, that is, where $(u_0, v_0, q_0) \to (u_1, v_1, q_1)$. If the transition from $(u_0, v_0, q_0)$ to $(u_1, v_1, q_1)$ is of type A or B then the claim is clear, so suppose the transition is of type C1-C9. Then from the definitions of these transitions, we must have $u_0 = XY u'$ for some maximum piece prefix $X$ and middle word $Y$ of a relation word $XYZ$. Now $XY$ cannot be a piece, so it cannot be a prefix of $z_0$, which is a proper suffix of a relation word. Thus, we must have $|XY| > |z_0|$ and hence $|u'| < |u_0'|$. Looking again at the definitions of the transitions, we see that $u_1$ and $v_1$ either
- are (not necessarily proper) suffixes of $u'$ and $v'$ respectively; or
- have the form $u_1 = Z u''$ and $v_1 = Z v''$ where $u''$ and $v''$ are (not necessarily proper) suffixes of $u'$ and $v'$ respectively; or
- have the form $u_1 = {\overline{Z}} u''$ and $v_1 = {\overline{Z}} v''$ where $u''$ and $v''$ are (not necessarily proper) suffixes of $u'$ and $v'$ respectively.
In case (i) it suffices to set $z_1 = \epsilon$ and $u_1' = u_1$. In case (ii) \[respectively, case (iii)\] it suffices to set $z_1 = Z$ \[respectively, $z_1 = {\overline{Z}}$\] and $u_1' = u''$, noting that $Z$ \[respectively, ${\overline{Z}}$\] must be a proper suffix of a relation word since is a maximal piece suffix of $XYZ$ \[${\overline{XYZ}}$\] and no relation word can be a piece.
It now follows easily by induction that the claim also holds when $$(u_0, v_0, q_0) \to^* (u_1, v_1, q_1).$$ In particular, taking $z_0 = \epsilon$ and $u_0' = u_0$ and then writing $u_1 = z_1 u_1'$ as above we have $$|u_1| = |z_1| + |u_1'| \ \leq \ |z_1| + |u_0'| \ = \ |z_1| + |u_0| \ \leq \ |u_0| + b$$ and similarly $|v_1| \leq |v_0| + b$, as required to show that the automaton has expansion bound $b$.
As an immediate corollary we obtain a corresponding statement for semigroups.
\[cor\_mainsemi\] Let $\langle A \mid R \rangle$ be a finite semigroup presentation satisfying the small overlap condition $C(4)$. Then the relation $$\lbrace (u, v) \in A^+ \times A^+ \mid u \equiv v \rbrace$$ is deterministic rational and reverse deterministic rational. Moreover, one can, starting from the presentation, effectively compute 2-tape deterministic automata recognising this relation and its reverse.
Since the presentation has no empty relation words, the semigroup with presentation $\langle A \mid R \rangle$ arises as the subsemigroup of non-identity elements in the monoid with presentation $\langle A \mid R \rangle$. It follows that $$\lbrace (u, v) \in A^+ \times A^+ \mid u \equiv v \rbrace
\ = \ \lbrace (u, v) \in A^* \times A^* \mid u \equiv v \rbrace \setminus \lbrace (\epsilon, \epsilon) \rbrace.$$ Now it is easy to verify that a relation $R$ between free monoids is a deterministic rational relation only if $R \setminus \lbrace ( \epsilon, \epsilon) \rbrace$ is a deterministic rational relation between free semigroups, so the result follows from Theorem \[thm\_main\].
Consequences {#sec_consequences}
============
In this section we consider a number of interesting consequences and corollaries of Theorem \[thm\_main\]. We begin with some terminology from language theory.
Let $A$ be a finite alphabet, and choose some arbitrary total order $\leq$ on the letters of $A$. Recall that the corresponding *lexicographic order* is an extension of this order to a total order $\leq_L$ on the free monoid $A^*$, defined inductively by $\epsilon \leq_L w$ for all $w$, and for all $x, y \in A$ and $u, v \in A^*$ we have $xu \leq_L yv$ if either $x \neq y$ and $x \leq y$, or $x = y$ and $u \leq_L v$. Lexicographic order is a total order but not (unless $|A|=1$) a well-order, since it contains infinite descending chains such as $$b, \ ab, \ aab, \ aaab, \ \dots, \ a^i b, \ \dots$$ Hence, if $R$ is an equivalence relation on $A^*$ (even a rational one) there is no guarantee that every equivalence class of $R$ will contain a lexicographically minimal element. In the case that $R$ is *locally finite* (that is, each equivalence class is finite), however, every class must clearly contain a unique lexicographically minimal element, and the set of elements which are minimal in their class forms a *cross-section* of the relation, that is, a language of unique representatives for the equivalence classes of the relation; we shall call these representatives *lexicographic normal forms*. Remmers showed that if $\langle A \mid R \rangle$ is a $C(3)$ monoid \[semigroup\] presentation then the corresponding equivalence relation on $A^*$ \[respectively, $A^+$\] is locally finite [@Higgins92; @Remmers71]; it follows that every element of a $C(3)$ monoid has a lexicographic normal form. Johnson [@Johnson85; @Johnson86] showed that if $R$ is a deterministic rational locally finite equivalence relation then the function which maps each word to the corresponding lexicographic normal form can be computed by a deterministic transducer. Thus, we obtain the following corollary to Theorem \[thm\_main\].
Let $\langle A \mid R \rangle$ be a monoid presentation satisfying $C(4)$ and suppose $A$ is equipped with a total order. Then the relation $$\lbrace (u, v) \in A^* \times A^* \mid u \equiv v \text{ and } v \text{ is a lexicographic normal form} \rbrace$$ is a deterministic rational function.
The image of a rational function is always a regular language [@Berstel79 Corollary II.4.2]) and deterministic rational functions can be computed in linear time Johnson [@Johnson86 Theorem 5.1] so we have:
\[cor\_lexmin\] Let $\langle A \mid R \rangle$ be a monoid presentation satisfying $C(4)$ and suppose $A$ is equipped with a total order. Then the lexicographic normal forms comprise a regular language of unique representatives for elements of the monoid. Moreover, there is an algorithm which, given a word $w$ in $A^*$, computes in linear time the corresponding lexicographic normal form.
A monoid $M$ is called *rational* [@Sakarovitch87; @Sakarovitch90] if there exists a finite generating set $A$ for $M$ and a regular cross-section $L \subseteq A^*$ for $M$ such that the normal forms in $L$ are computed by a transducer.
Every monoid admitting a $C(4)$ presentation is *rational*.
Recall that the *rational subsets* of a monoid $M$ are those which can be obtained from finite subsets by the operations of union, product and submonoid generation (the “Kleene star” operation). If $M$ is generated by a finite subset $A$ then the rational subsets of $M$ are exactly the images in $M$ of regular languages over $A$, which means they have natural finite representations as finite automata over $A$. The *recognisable subsets* of $M$ are the homomorphic pre-images in $M$ of subsets of finite monoids. In the case that $M$ is a free monoid, the rational subsets are just the regular languages. Kleene’s Theorem asserts that the rational subsets of a free monoid (that is, the regular languages) coincide with the recognisable subsets [@Hopcroft69]. More generally, a monoid in which the rational and recognisable subsets coincide is called a *Kleene monoid*, or sometimes is said to *satisfy Kleene’s Theorem*. Rational monoids were originally introduced in an attempt to obtain a concrete characterisation of Kleene monoids [@Sakarovitch87], and indeed every rational monoid is a Kleene monoid (although it transpires that the converse does not hold). Thus, we obtain:
Let $M$ be a monoid or semigroup admitting a $C(4)$ presentation, and $S$ a subset of $M$. Then $S$ is rational if and only if $S$ is recognisable.
Recall that a collection of subsets of some given base set is called a *boolean algebra* if it contains the empty set and is closed under union, intersection and complement. As another corollary of the rationality of $M$ we obtain the following fact about rational subsets of $M$.
\[cor\_booleanalgebra\] Let $M$ be a monoid admitting a $C(4)$ presentation $\langle A \mid R \rangle$. Then the rational subsets of $M$ form a boolean algebra. Moreover, if rational subsets of $M$ are represented by automata over $A$, then the operations of union, intersection and complement are effectively computable.
Let $\sigma : A^* \to M$ be the canonical morphism mapping $A^*$ onto $M$, and let $$\rho = \lbrace (u, v) \in A^* \times A^* \mid u \equiv v \text{ and } v \text{ is a lexicographic normal form} \rbrace.$$ Suppose $X, Y \in A^*$ are rational subsets, with say $X = \hat{X} \sigma$ and $Y = \hat{Y} \sigma$ where $\hat{X}, \hat{Y} \subseteq A^*$ are regular languages. Then using the facts that $A^* \rho$ contains a unique representative for every element and that $\rho \sigma = \sigma$, it is readily verified that $M \setminus X = (A^* \rho \setminus \hat{X} \rho) \sigma$, $X \cap Y = (\hat{X} \rho \cap \hat{Y} \rho) \sigma$ and $X \cup Y = (\hat{X} \rho \cup \hat{Y} \rho) \sigma$. The result now follows from the fact that regular languages in a free monoid form a boolean algebra with effectively computable operations.
Recall that the *rational subset membership problem* for a finitely generated monoid $M$ is the problem of deciding, given a rational subset of $M$ (represented by a finite automaton over some fixed generating set for $M$) and an element of $M$ (represented as a word over the same generating set), whether the given element belongs to the given subset. The decidability of this problem is independent of the chosen generating set [@KambitesGraphRat Corollary 3.4].
Any monoid admitting a $C(4)$ presentation has decidable rational subset membership problem (and hence decidable submonoid membership problem).
Suppose $M$ has $C(4)$ presentation $\langle A \mid R \rangle$, and let $\sigma : A^* \to M$ be once again the canonical morphism. Suppose we are given a finite automaton recognising a language $\hat{X} \subseteq A^*$ (representing the rational subset $\hat{X} \sigma \subseteq M$) and a $w \in A^*$ (representing the element $w \sigma \in M$). Certainly we can compute from the latter a finite automaton recognising the singleton language $\lbrace w \rbrace$. Hence, by Corollary \[cor\_booleanalgebra\] we can compute a finite automaton recognising a language $\hat{Y} \subseteq A^*$ such that $\hat{Y} \sigma = \hat{X} \sigma \cap \lbrace w \rbrace \sigma$. But $w \sigma \in \hat{X} \sigma$ if and only if $\hat{X} \sigma \cap \lbrace w \rbrace \sigma$ is non-empty, so this reduces the problem to deciding emptiness of the regular language $\hat{Y}$; the latter is well known to be decidable.
A monoid $M$ is called *asynchronous automatic* (see, for example, [@Hoffmann01]) if there exists a finite generating set $A$ and a regular language $L \subseteq A^*$ such that $L$ contains a representative for every element of $M$, and the relation $$\lbrace (u,v) \in A^* \times A^* \mid ua \equiv v \rbrace$$ is a rational transduction for each $a \in A$ and for $a = \epsilon$. It has been shown [@Hoffmann01 Theorem 6.2] that rational monoids are asynchronous automatic, so we also obtain the following.
Every monoid admitting a $C(4)$ presentation is asynchronous automatic.
We have already remarked that small overlap conditions are the natural semigroup-theoretic analogue of the small cancellation conditions extensively used in combinatorial group theory (see, for example, [@Lyndon77]). It is well known that a group admitting a finite presentation satisfying sufficiently strong small cancellation conditions is *word hyperbolic* in the sense of Gromov [@Gromov87]. The usual geometric definition of a word hyperbolic group has no obvious counterpart for more general monoids or semigroups; however, Gilman [@Gilman02] has given a language-theoretic characterisation of word hyperbolic groups. Specifically, he showed that a group is word hyperbolic if and only if it admits a finite generating set $A$ and a regular language $L \subseteq A^*$ containing a representative for every element of $M$ such that the *multiplication table* $$\lbrace u \# v \# w^R \mid uv \equiv w \rbrace$$ is a context-free language, where $\#$ is a new symbol not in $A$. Motivated by this result, Duncan and Gilman [@Duncan04] have suggested calling a monoid *word hyperbolic* if it satisfies this language-theoretic condition. Since every rational monoid is word hyperbolic [@Hoffmann01 Theorem 6.3] we can deduce that every $C(4)$ monoid is word hyperbolic in this sense.
Every monoid admitting a $C(4)$ presentation is word hyperbolic in the sense of Duncan and Gilman (and furthermore admits a hyperbolic structure with unique representatives).
Acknowledgements {#acknowledgements .unnumbered}
================
This research was supported by an RCUK Academic Fellowship. The author would like to thank A. V. Borovik and V. N. Remeslennikov for a number of helpful conversations.
[10]{}
J. Berstel. . Informatik. Teubner, 1979.
G. Buntrock and F. Otto. Growing context-sensitive languages and [C]{}hurch-[R]{}osser languages. , 141(1):1–36, 1998.
A. Duncan and R. H. Gilman. Word hyperbolic semigroups. , 136(3):513–524, 2004.
C. C. Elgot and J. E. Mezei. On relations defined by generalized finite automata. , 9:47–68, 1965.
P. C. Fischer and A. L. Rosenberg. Multitape one-way nonwriting automata. , 2:88–101, 1968.
R. H. Gilman. On the definition of word hyperbolic groups. , 242(3):529–541, 2002.
M. Gromov. Hyperbolic groups. In [*Essays in Group Theory*]{}, volume 8 of [*Math. Sci. Res. Inst. Publ.*]{}, pages 75–263. Springer, New York, 1987.
P. M. Higgins. . Oxford Science Publications. The Clarendon Press Oxford University Press, New York, 1992. With a foreword by G. B. Preston.
M. Hoffmann, D. Kuske, F. Otto, and R. M. Thomas. Some relatives of automatic and hyperbolic groups. In G. M. S. Gomes, J.-E. Pin, and P. V. Silva, editors, [ *Semigroups, Algorithms, Automata and Languages*]{}, pages 379–406, 2003.
J. E. Hopcroft and J. D. Ullman. . Addison-Wesley, 1969.
J. H. Johnson. Do rational equivalence relations have regular cross sections? In [*Automata, languages and programming (Nafplion, 1985)*]{}, volume 194 of [*Lecture Notes in Comput. Sci.*]{}, pages 300–309. Springer, Berlin, 1985.
J. H. Johnson. Rational equivalence relations. , 47(1):39–60, 1986.
M. Kambites. Small overlap monoids [I]{}: the word problem. , 2007. (Preprint available at [arXiv:0712.0250 \[math.RA\]]{}).
M. Kambites, P. V. Silva, and B. Steinberg. On the rational subset problem for groups. , 309:622–639, 2007.
R. C. Lyndon and P. E. Schupp. . Springer-Verlag, 1977.
M. Pelletier and J. Sakarovitch. Easy multiplications [I]{}[I]{}. [E]{}xtensions of rational semigroups. , 88:18–59, 1990.
J. H. Remmers. . PhD thesis, University of Michigan, 1971.
J. H. Remmers. On the geometry of semigroup presentations. , 36(3):283–296, 1980.
J. Sakarovitch. Easy multiplications [I]{}. [T]{}he realm of [K]{}leene’s theorem. , 74:173–197, 1987.
|
---
abstract: '[Widespread interest in the diffusion of information through social networks has produced a large number of Social Dynamics models. A majority of them use theoretical hypothesis to explain their diffusion mechanisms while the few empirically based ones average out their measures over many messages of different content. Our empirical research tracking the step-by-step email propagation of an invariable viral marketing message delves into the content impact and has discovered new and striking features. The topology and dynamics of the propagation cascades display patterns not inherited from the email networks carrying the message. Their disconnected, low transitivity, tree-like cascades present positive correlation between their nodes probability to forward the message and the average number of neighbors they target and show increased participants’ involvement as the propagation paths length grows. Such patterns not described before, nor replicated by any of the existing models of information diffusion, can be explained if participants make their pass-along decisions based uniquely on local knowledge of their network neighbors affinity with the message content. We prove the plausibility of such mechanism through a stylized, agent-based model that replicates the *Affinity Paths* observed in real information diffusion cascades.]{}'
address:
- 'Instituto de Ingeniería del Conocimiento, Universidad Autónoma de Madrid, 28049 Madrid, Spain '
- |
Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM,\
Departamento de Matemáticas & GISC, Universidad Carlos III de Madrid, 28911, Leganés (Madrid),\
Instituto de Ingeniería del Conocimiento, Universidad Autónoma de Madrid, 28049 Madrid, Spain
author:
- José Luis Iribarren
- Esteban Moro
bibliography:
- 'Master.bib'
date: 'October 25, 2010'
title: Affinity Paths and Information Diffusion in Social Networks
---
Word-of-Mouth, Viral Marketing, Information Diffusion, Social Networks, Complex Systems
Introduction and Background
===========================
The discovery of quantitative laws in the collective properties of large numbers of people, for example the birth and death rates or crime frequencies, was one of the factors pushing the development of statistics and led scientists and philosophers to call for some quantitative understanding on how such precise regularities stem from the apparently erratic behavior of individuals. Hobbes, Laplace, Comte, Stuart Mill and many others shared, to a different extent, this line of thought [@Ball2004]. The question to investigate was how the interactions between social agents create order in their behavior from an initially disordered state. The basic premise was that agents’ repeated interactions should make people more similar since the information exchanges involved led to higher degrees of homogeneity in values, thoughts or preferences. The dynamic nature of the information diffusion, the poor understanding of the human behavior causes and the fact that the agents interactions take place in the thick of complex social networks, made the Social Dynamics problem largely untractable for a long time.
The appearance of new social phenomena related to the Internet (Social Media, Collaborative Filtering, Social Tagging...) whose interactions can be captured in large databases and the tendency of social scientists to move toward the formulation of simplified models and their quantitative analysis, have ushered in an era of scientific research in the field of Social Dynamics [@Lazer2009]. Several key questions have been posed: What favors the homogenization process? What hinders it? What are the fundamental interaction mechanisms fostering the adoption of innovations, the spreading of rumors, the evolution towards a dominant opinion or the emergence of trends and fashions?
Initially, the difficulty in obtaining micro-level data on the diffusion of information between individuals, the absence of suitable mathematical algorithms to rigorously analyze the phenomena and the calculation complexity involved in simulations with large real networks limited theoretical advancements to the construction of population average diffusion models based on master or differential equations. Those models were in general borrowed from mathematical epidemiology [@Hethcote2000] since it was assumed that information would propagate just like diseases do. However information diffusion research has deeply evolved since step-by-step tracking of interactions through electronic media made detailed diffusion data plentiful (although not necessarily accessible or easy to gather).
The development of the science of complex systems and advancements in the computerized treatment of Social Network Analysis methods have spurred the emergence of a “new” science of networks [@Watts2004] which provides more robust tools for the scientific treatment of social dynamics processes. As a result scientists realized that information spreading mechanisms vary with the type of information which spawned a rush to develop the appropriate model for each. According to their algorithmic approach those models can be categorized as population-average or network-based. The population-average models assume fully-mixed or homogeneous substrate networks and describe the agents’ social dynamic behavior at the aggregate level through differential or master equations. Examples of those are the seminal “two-step influence model” of information diffusion by @Katz1955, the rumor diffusion model of @Daley1965, the innovations adoption model of @Bass1969, its stochastic version by @Niu2002, the minority spreading opinion formation model of @Galam2002, the innovation diffusion model with influentials and imitators of @Bulte2007 or the percolation-based product lifecycle model of @Frenken2008. On the other hand, network-based models include the influence of the underlying social network topology by way of agent-based stochastic algorithms. Some examples of them are the classic innovation adoption “threshold model” of @Granovetter1978, the model of diffusion of technological innovations with upgrading costs of @Guardiola2002, the fads and fashion formation model of @bettencourt2002, models on the impact of the structural characteristics of a network on innovations diffusion [@Jackson2005; @Liu2005], the stochastic model for opinion formation of @Sznajd-Weron2005 or the network variant of the Daley-Kendall rumor model by @Nekovee2007.
However, this profusion of theoretical models was mainly justified by plausibility arguments and Social Dynamics models based on empirical data are still scarce. A few examples are the referral networks study of @Vilpponen2006 which found that the structure of electronic communication networks is different from that of the traditional interpersonal communication ones, the chain-letter diffusion research of @Liben-Nowell2008 whose strikingly long and narrow spreading chains were attributed to a new mechanism involving asynchronous response times of the forwarders or the study on information diffusion through blogs of @Gomez-Rodriguez2010 which found a core periphery structure in the blogosphere news diffusion network. Nevertheless, all these studies could only trace the propagation of messages with varying content and are unable to discriminate the propagation of individual content items. As a result, none of them could study the impact of the information content on the diffusion processes. While the lack of insight into the content impact would be expected of past century information diffusion research, its absence in more recent literature can only be explained because propagation data at the individual level, being usually proprietary because of its economic value or usage restrictions, is kept under tight wraps and results very hard to obtain.
Our research addresses such shortcoming. Unlike the works cited that study information propagation through the aggregate effect of propagating messages of varying content, ours tracked the precise paths of a viral marketing campaign fixed and invariable message as it spread through an email social network. The message content remained identical through the propagation. This allowed us to scrutinize the individuals’ reactions to a particular message instead of just averaged out behavior over diverse information items. By discriminating all factors impacting the participants’ spreading patterns from the message content we were able to detect the effects produced by the latter. We found that the message diffusion cascades evolve through a branching process that presents some characteristic and unique patterns unexamined until now although some literature [@Leskovec2007; @Watts2007a] has shown an inkling of them. We noticed a steady increase in the spreaders’ activity parameters as the message gets deeper in the propagation cascades. This surprising pattern can not be observed in empirical experiments collecting propagation data of varying content messages. It can be explained if the cascades growth stems from a mechanism based on the affinity between the message content and the preferences of those receiving it and not on the receiving node neighbors’ status or on the underlying social network structure used in many of the current models. We test and validate that hypothesis through a stylized agent-based propagation model. The rest of the article is organized as follows: First we describe the data obtained from our empirical research on real viral marketing campaigns and the control parameters of their messages propagation. Second, we present our findings on the structure and growth patterns of the information cascades. Third we introduce the message affinity propagation model and compare its predictions with the empirical results. The article ends with our conclusions.
Word-Of-Mouth diffusion research {#viral}
================================
We tracked and measured the “word-of-mouth” diffusion of viral marketing campaigns ran in eleven European markets which offered incentives to current subscribers of an IT company online newsletter to promote new subscriptions through recommendation emails to friends and colleagues. The campaigns were entirely web based: banner ads, emails, search engines and the company homepage drove participants into the campaign site. In it, participants accessed a referral form to register themselves and enter the addresses of those to whom they recommended subscribing the newsletter. The submission of this form triggered a personalized, but otherwise identical, recommendation message with a link to the campaign registration form. The link customized URL was appended with codes allowing to uniquely trace clicks on it to sender and addressee of the corresponding email. The form checked email addresses for syntax correctness and to prevent self recommendations. Cookies in the participants’ email client prevented sending multiple recommendations to the same address[^1] and improved the user experience by pre-filling the sender’s profile in subsequent sessions. Additionally, the campaign web server registered a time stamp for each of the process steps (subscription, recommendations, referral link clicks) and removed from records referrals to undeliverable email addresses.
[p[1.6cm]{}p[1cm]{}p[1cm]{}p[1cm]{}p[1cm]{}p[1cm]{}p[0.9cm]{}r]{} **Market** & $N$ & $N_s$ & $N_v$ & $N_p$ & $Arcs$ & $Casc.$ & $s_{max}$\
France & 11,758 & 3,247 & 524 & 7,987 & 8,593 & 3,248 & 139\
DE+AT & 7,943 & 1,760 & 567 & 5,616 & 6,239 & 1,750 & 146\
Spain & 5,260 & 855 & 505 & 3,900 & 4,454 & 843 & 122\
Nordic & 2,509 & 530 & 176 & 1,803 & 2,004 & 524 & 34\
UK+NL & 2,111 & 521 & 107 & 1,483 & 1,618 & 518 & 25\
Italy & 1,602 & 323 & 108 & 1,171 & 1,324 & 319 & 41\
**All markets** & **31,183** & **7,225** & **2,002** & **21,956** & **24,207** & **7,188** & **146**\
The incentive offered to recommenders was the possibility of winning laptop computers in a lottery to be held at the end of the campaign period. Aside from the obvious goal of increasing participation, the incentive mission was twofold: Firstly, discourage indiscriminate referrals to prevent spamming-like behavior and, secondly, ensure legal cover for the tracking of sender-receiver data. To accomplish such requirement, participation in the lottery was limited to the so-called successful referrals defined as the recommendation emails whose recipients clicked on the coded URL included on them. Thus, the more referral emails sent to recipients opening them and visiting the campaign site, the higher the sender’s winning odds. More importantly, both sender and receiver of any successful referral drawn in the lottery were entitled to receive the lottery prize. Terms and conditions, accessible from all web site pages and referral emails, specified that participation in the lottery implied the sender’s and receiver’s approval of the campaign registration of their email transaction details as this was necessary to ensure that both parties could receive the prize if their referral email was the winning one. Subscription to the newsletter was not required to participate in the prize draw. Campaigns ran in each country local language but were identical otherwise: Identical message, incentive, eligibility rules, lottery mechanism, campaign duration, web user interface and tracking processes. This homogeneity of data ensured that behavioral differences between countries were not caused by the campaigns execution but due to the market specifics. It also validates the analysis of country aggregated results.
Campaigns propagation data set {#data}
------------------------------
Spurred by the campaign sponsor web site and exogenous online advertising, a total of 7,225 individuals initiated message diffusion cascades which grew through viral pass-along driven by 2,002 secondary spreaders. Thus, the viral offering touched another 21,956 passive nodes who did not forward it further. All in all, 31,183 individuals of whom 9,227 were spreaders, received the viral message. Thus 77% of the individuals received the message through the endogenous viral propagation mechanism. The *Cascades Network* resulting of the message diffusion constitutes a directed graph with 7,188 independent cascades whose nodes represent participants linked by 24,207 directed arcs representing the recommendation emails. We call *Seed Nodes* ($N_s$) the individuals who spontaneously initiate recommendation cascades from the campaign site without having received a recommendation message from others and *Viral Nodes* ($N_v$) those who forward a previously received message. Table \[tab:1\] presents the summary data set of the campaigns message propagation[^2]. Unsuccessful emails, disconnected nodes, nodes with invalid or undeliverable email addresses, loops and multiple referrals between same nodes were discarded. In compliance with the sponsor rigorous policy, all personal information was codified and masked to guarantee the participants’ privacy protection.
Cascades Network structural metrics
-----------------------------------
[p[1.5cm]{}p[0.9cm]{}p[1cm]{}p[1cm]{}p[1.1cm]{}p[1.1cm]{}p[0.9cm]{}c]{} **Market** & $\overline{k}$ & $\sigma_{k}$ & $\overline{k}_{nn}$ & $C$ & $C_{rand}$ & $\overline{\ell}$ & $g_{max}$\
France & 1.46 & 1.594 & 3.99 & 0.0000 & 0.00012 & 2.164 & 8\
DE+AT & 1.57 & 2.027 & 5.59 & 0.0049 & 0.00020 & 2.671 & 7\
Spain & 1.69 & 2.383 & 7.17 & 0.0054 & 0.00032 & 3.287 & 9\
Nordic & 1.60 & 1.575 & 4.07 & 0.0077 & 0.00064 & 2.243 & 5\
UK+NL & 1.53 & 1.364 & 3.43 & 0.0112 & 0.00073 & 2.026 & 5\
Italy & 1.65 & 1.918 & 5.22 & 0.0234 & 0.00103 & 2.229 & 6\
**All markets** & **1.55** & **1.868** & **4.97** & **0.0048** & **0.00005** & **2.671** & **9**\
Here we examine differences and similarities between the *Cascades Network* topology and that of the reported email networks through which they propagate. Table \[metric\] shows the *Cascades Network* structural parameters measured without considering links direction. The cumulative distribution function (c.d.f) of the undirected network total degree $k$ is a power-law $P(k)\sim k^{-2.8}$ whose significant probability of very connected nodes evidences higher heterogeneity than the exponential degree distributions found in some email networks [@Guimera2003; @Newman2002]. However, their heterogeneity is less marked than that of the email network studied by @Ebel2002b whose power-law degree distribution (p.d.f.) of exponent $\gamma_{k} = 1.81$ is fatter tailed. Additionally, email networks present positive correlations between the nodes degree at either end of an edge, a property called degree assortativity and measured, according to @Newman2002b, by the Pearson correlation coefficient. For example, the degree correlation coefficient in the email network of @Guimera2003 is $\rho_k=+0.188$, indication of a correlated network. The equivalent for the *Cascades Network* $\rho_{k}=-0.001$ shows total uncorrelation. Besides, in networks with skewed node degree distributions and degree correlations, such as the email networks, the average connectivity of the network $\overline{k}$ is typically lower than that of the nearest neighbors of a node $\overline{k}_{nn}$. For example in the @Guimera2003 email network, the ratio $\overline{k}_{nn}/\overline{k}$ is approximately 2. Such phenomenon is responsible for the first neighbors of a node having in average more contacts than such node or, quoting @Feld1991, for the fact that “your friends always have more friends than you do.” Interestingly, this feature is more marked in the *Cascades Network* whose $\overline{k}_{nn}$ to $\overline{k}$ ratio ranges from 2.24 in UK+NL to 4.24 in Spain.
Another difference between the *Cascades Network* and the email networks through which they propagate lies in their transitivity, a property typical of acquaintance networks whereby two individuals with a common friend are more likely than average to know each other. The *Clustering coefficient* $C$, defined as the fraction of all triangles found in the network relative to the total number of triads[^3] measures the transitivity. Table \[metric\] shows that our *Cascades Networks* with a *Clustering coefficient* $C=4.8\times10^{-3}$ for the graph of **All markets** are highly intransitive yet ten times more transitive than an equivalent random network of the same size and connectivity. In any case, a very low value compared to the range $C
$ \[0.15 - 0.60\] found in social or email networks [@Newman2003c]. Probabilistic considerations show the logic of such feature: since the *Cascades Network* percolates its underlying email network only partially, the dyadic closure that builds clustering in the former must be just a fraction of the one in the latter. As a result our campaigns viral diffusion cascades, like the one in Fig. \[cascade\], are almost pure trees.
The last distinctive property of email networks, the *Small World* or low average shortest path length [@Boccaletti2006], although seemingly present since $\overline\ell=2.67$ (Table \[metric\]) and lower than that of email networks $\overline{\ell}_{email}\sim3.5$ [@Eckmann2004; @Guimera2003] is not comparable with those due to the nature of the *Cascades Network* that, split in many disconnected components, limits paths calculation to reachable pairs of nodes which necessarily yields lower values. The distribution of those cascades size ($s$), like the total degree, is a very skewed power-law whose c.d.f. exponent is $\gamma_s=1.35$. With largest cascade size $s_{max}=146$ nodes, mean size $\overline{s}=4.33$, and $\sigma_s=5.27$, the cascade in Fig. \[cascade\] is 25 times more likely to appear in our campaigns than in percolation through a random network[^4].
In consequence, the viral *Cascades Network* topology lacks all the four key features of email networks (fat tailed node degree distribution, nodes degree correlations, high clustering and the *Small World* property) and can not be formally characterized as a social network. This is quite logical since the viral propagation cascades of diffusion processes far from saturation, such as ours, overlay just sections of the underlying email network and, as a result, can only unveil a small portion of it. Paraphrasing @Liben-Nowell2008 in their study of chain-letters propagation, it is as if “the progress of the viral messages had a type of stroboscopic effect serving to briefly light up the structure of the global email network.” Unfortunately, not having any details on the topology of the email network substrate, we can not judge the extent of its influence on the *Cascades Network* topology.
Cascades Network Dynamic Parameters {#parameters}
-----------------------------------
While the structure of the undirected cascades is weakly related to that of the email network substrate, the flow of messages in the *Cascades Network* is not (except for the substrate network setting the boundary conditions) and fully depends on the recommendation mechanism. To study it we will consider the distribution of recommendation emails sent by spreaders of the viral message which offers a better picture of the cascades dynamics than the total node degree $k$ considered so far because 70% of the network nodes are inactive. This new variable, equivalent to the out-degree of the network nodes, is measured separately for *Seed Nodes* and *Viral Nodes* and designated as $r_s$ and $r_v$ respectively. While most *Viral Nodes* sent just a few recommendations a significant fraction displayed a very intense activity: thus for the ensemble of **All markets** in our dataset, the mean of the number of recommendations sent by *Viral Nodes*, the so-called *Fanout Coefficient*, was $\overline{r}_v=2.96$ (see Table \[dynamical\]), its standard deviation $\sigma_v=7.47$ and the highest number of recommendations sent by a single individual $r_v(max)=72$. Its distribution can be fitted to a fat tailed power-law of the form $$\label{harris}
PL_{\alpha\beta}(r_v)=\frac{H_{\alpha\,\beta}}{\beta+r_v^{\alpha}}$$ whose parameters for the **All markets** network take the values $H_{\alpha\,\beta}=11.6$, $\alpha=2.83$ and $\beta=10.96$ using Maximum Likelihood Estimation.
[p[1.5cm]{}p[0.9cm]{}p[0.75cm]{}p[0.75cm]{}p[1.05cm]{}p[0.9cm]{}p[0.75cm]{}p[0.75cm]{}c]{} **Market** & $\lambda$ & $\overline{r}_s$ & $\overline{r}_v$ & $\overline{r}_v\textrm{ SEM}$ & $R_0$ & $\overline{s}$ & $\overline{s^*}$ & $\%\,Dev.$\
France & 0.062 & 2.21 & 2.50 & 0.1023 & 0.154 & 3.62 & 3.61 & -0.22\
DE+AT & 0.092 & 2.48 & 3.06 & 0.1155 & 0.281 & 4.54 & 4.45 & -2.04\
Spain & 0.115 & 3.16 & 3.45 & 0.1909 & 0.400 & 6.24 & 6.23 & -0.20\
Nordic & 0.089 & 2.82 & 2.91 & 0.1836 & 0.259 & 4.79 & 4.81 & +0.31\
UK+NL & 0.067 & 2.49 & 2.87 & 0.2398 & 0.236 & 4.08 & 4.09 & +0.15\
Italy & 0.084 & 2.87 & 2.80 & 0.2301 & 0.236 & 5.02 & 4.76 & -5.20\
**All markets** & **0.083** & **2.51** & **2.96** & **0.065** & **0.246** & **4.34** & **4.33** & **-0.30**\
We can visualize the cascades of a viral propagation process growing through successive layers, or generations, as nodes reached in one generation resend the message to nodes in the next generation. The latter nodes constitute the off-spring of the earlier ones in an evolution of the propagation trees whose node-level dynamics is well described by the Galton-Watson Branching model[^5] [@Harris2002]. Two parameters fully describe this growth process at the population level: the aforementioned *Fanout Coefficient* $\overline{r}_v$ and the message *Transmissibility* $\lambda$ defined as the fraction of the touched nodes that become secondary spreaders. The *Transmissibility* results from data in Table \[tab:1\] as
$$\lambda=\frac{N_v}{N-N_s}$$
and both parameters combine to yield the Basic Reproductive Number $R_0$ or average number of secondary recommendations produced by reached nodes as
$$R_0=\lambda\overline{r}_v$$
This number is widely used in mathematical epidemiology [@Hethcote2000] to determine the moment when a disease outbreak becomes a self-sustaining epidemic. Thus, if $R_0\geq 1$ the spreading process reaches the *Tipping-point*[^6] an elusive goal that none of our campaigns attained. Table \[dynamical\] presents the propagation dynamic parameters and cascades average size $\overline{s}$ of our campaigns and their predicted value $\overline{s*}$ for the infinite propagation limit given by the Galton-Watson Branching model as
$$\label{avgcascade}
\overline{s*}=1+\frac{\overline{r}_s}{1-R_0},\qquad R_0\leq1$$
where $\overline{r}_s$ is the average number of messages sent by *Seed Nodes* and $R_0$ the viral propagation Basic Reproductive Number. The last column in Table \[dynamical\] shows the remarkable accuracy of the cascades average size predicted by the Galton-Watson Branching model versus the empirical values.
Patterns of the information cascades growth {#patterns}
===========================================
Despite the Galton-Watson model statistically accurate description of the distribution of cascades at a global level, a detailed study of the *Cascades Network* growth, reveals patterns indicating that viral messages spreading dynamics is quite peculiar. Firstly, we present a node level analysis showing the correlation in the spreading activity of a node with that of its active offspring down the message propagation tree. Secondly, we conduct a generation level analysis on the probability of the nodes becoming active as a function of their ordinal position in the message diffusion path which shows that viral messages diffusion propensity increases with distance from the *Seed Node*. Both findings lead to a striking prediction corroborated by the measurements on our viral campaigns: The viral messages diffusion dynamic parameters at the population level are correlated, a fact that has not been observed in other social dynamics processes such as innovations adoption, rumors spreading or opinions propagation. Note that both findings are incompatible with the assumptions in the Galton-Watson model in which the branching mechanism is homogeneous both at the social network level and within the cascades.
Correlated spreading of active nodes {#correlated}
------------------------------------
The first distinctive pattern of the viral messages *Cascades Network* growth is the marked positive correlation of the spreading activity between *Viral Nodes* and their active off-spring. In undirected networks, the nodes total degree correlation is given by the conditional probability $P(k\mid{k'})$ of a node of degree $k$ pointing to a node of degree $k'$. This function is very noisy in finite networks and is usually replaced by the average degree of the nearest neighbors of $k$-degree nodes $\overline{k}_{nn}(k)=\sum_k' k' P(k\mid{k'})$ [@Boccaletti2006]. When $\overline{k}_{nn}(k)$ is an increasing function of the degree $k$ the nodes tend to connect to others of similar connectivity and such network, called assortative, displays positive node total-degree correlations.
However the active nodes network is directed and instead one should study its out-degree correlation defined as the tendency of nodes to connect with others that have similar out-degrees to themselves. Its formal metric is the *out-assortativity coefficient*[^7] but considering throughout only the active nodes throughout a simplified analysis of the average out-degree of the *active nearest neighbors* $(\overline{r}_v)_{ann}$ of nodes of out-degree $r_v\geq1$ presented in Fig. \[assortativity\] suffices to prove that, in terms of the number of recommendations sent in our campaigns, the more active a node is the more prolific in average its progeny is. We studied the out-degree spreading pattern of active nodes in our campaigns (*Seed Nodes* excluded) and found that the activity of a node ($r_v$) correlates with that of its active nearest neighbors. Such correlation implies that the average number of recommendations sent by the active nearest neighbors of a node $(\overline{r}_v)_{ann}$ grows with the number of recommendations $r_v$ that it has sent. The slope of the linear regression of $(\overline{r}_v)_{ann}(r_v)$ is +0.69 indicating strong out-degree correlation. The actual values of $(\overline{r}_v)_{ann}$ range between 1 and 31.33, the mean of their distribution is 2.48 and its standard deviation 2.08.
This very peculiar feature of viral messages diffusion has not been observed on any other type of propagation processes in social networks. We can hypothesize two different explanations of it. One, the increased spreading activity of the active children of a node is a reflection of the out-degree correlation present in the substrate email network. Lacking any data on such network for our campaigns this hypothesis is impossible to verify. Besides, the out-degree positive correlation in the substrate email network merely means that its nodes tend to link to others of similar out-degree but does not by any means indicate that the number of recommendations made by active participants, hence the interest in participating in the campaign, should be a growing function of the number of recommendations made by their parent in the cascade. The other possible explanation, which we adopt, is that the intrinsic mechanism whereby participants in viral marketing campaigns forward the messages involves the sender selecting targets among those of her contacts perceived to be the most receptive to the content of the message being passed-along. The iteration of these target filtering decisions through several generations of senders would lead, in a process akin to targeted search, to focusing the message on groups of individuals genuinely interested on it. Those, in turn, would also be in average more active than their ancestors. The fact that this mechanism has not been observed in other types of information diffusion, such as referral networks [@Vilpponen2006], e-commerce recommendations [@Leskovec2007] or email chain-letters [@Liben-Nowell2008] may indicate either that the phenomenon is specific of viral marketing messages or that those authors analysis did not isolate the content factor.
Diffusion acceleration with path length
---------------------------------------
The second characteristic of viral spreading dynamics appears when measuring the probability of the nodes becoming active spreaders as a function of their position in the propagation tree. Thus, the *Transmissibility* by generation $\lambda_g$ in our campaigns grows in correlation with the ordinal $g$ representing the individuals’ location in the message propagation path. As shown in Table \[generations\] for the **All markets** data, $\lambda_g$ increases steadily with the generation ($\rho(g|\lambda_g)=0.908$) with parallel growth of the Reproductive Number by generation
$$R_g=\lambda_g(\overline{r}_v)_g=\frac{N_{g+1}}{N_g}$$
where $N_g$ is the total number of individuals reached at generation $g$. Besides, there is a growth trend for $(\overline{r}_v)_g$, the *Fanout* by generation which is visible in our campaigns (Table \[generations\]) whose *Fanout* ratio through generations $(\overline{r}_v)_{g+1}/(\overline{r}_v)_g$ positively correlates with the generation number ($\rho=0.4$). Those properties of messages diffusion were detected, but not studied, by @Watts2007a or @Leskovec2007 as shown in Fig. \[utility\] along with our campaigns measurements. As before, we posit that such pattern is due to “preferential forwarding,” defined as the spreaders’ propensity of passing a message preferentially to neighbors they presume to have more interest, or affinity, for it. Such mechanism results in an increase of the recipients propensity to pass the message along. As a consequence, the message follows network paths such that the *Transmissibility* by generation $\lambda_g$ increases as the propagation progresses. We denominate *Affinity Paths* to the chains of individuals with similar or increasing affinity for the message. They imply some knowledge by message spreaders of their immediate neighbors interests, a local awareness with global impact that leads to a different class of propagation than that of other Social Dynamics processes. Its consciously driven spreading mechanism causes messages to progress through paths presenting the homophily[^8] properties typical of social networks [@Mcpherson2001]. This phenomenon has been observed in the web where, according to @Singla2008 “there is correlation between preferences and behavior of an individual and those of others in its immediate circle”.
Dynamic Parameters correlation
------------------------------
As a result of the previous two properties the parameters $\lambda$ and $\overline{r}_v$ are correlated. Let us consider the relationship between the *Fanout Coefficient* and the generation parameters in Table \[generations\]
$$\overline{r}_v=\frac{\sum_{g=2}N_g}{\sum_{g=1}\lambda_gN_g}=\frac{1-P_g(1)}{\sum_{g=1}\lambda_gP_g}$$
where $P_g(1)=N_1/\sum_{g=1}N_g=N_s\overline{r}_s/(N-N_s)$ is the probability of an individual to have received the message from a *Seed Node*. Since $\sum_{g=1}\lambda_gP_g=\lambda$ one obtains the important expression $\lambda\overline{r}_v=1-P_g(1)$ which means that for $\lambda$ and $\overline{r}_v$ to increase simultaneously one must reduce the probability $P_g(1)$ of finding nodes in the first generation or, equivalently, grow longer cascades. Thus, a growing $\lambda_g$ yields longer paths and causes a parallel growth of $\overline{r}_v$. Our campaigns show that the average shortest path length $(\overline{\ell})$ of the diffusion cascades and the dynamic parameters are strongly correlated: $\rho(\overline{\ell}|\overline{r}_v)=0.88$ and $\rho(\overline{\ell}|\lambda)=0.89$. An increase of the *Transmissibility* $\lambda$ grows the paths length and the average number of recommendations made $\overline{r}_v$ as well. Plotting the dynamic parameters for various markets (Fig. \[2nd\_law\]) their correlation was found to be very strong with a Pearson coefficient $\rho(\lambda|\overline{r}_v)=0.92$. The values of $\lambda$ and $\overline{r}_v$ by country from Table \[dynamical\] fit to the decreasing exponential[^9]
$$\label{model}
\overline{r}_v = 1+b(1-e^{-c\lambda})$$
which for $\lambda\ll1$, and through a MacLaurin series expansion of $e^{-c\lambda}$, turns into $\overline{r}_v=1+a\lambda$ ($a=bc$). One can consider the slope $a$ of this “response line” as the message “fitness” with respect to each market. The exponential decrease for large $\lambda$ in Eq. (\[model\]) is due to the substrate network nodes clustering which limits propagation through saturation and finite size effects.
In principle this correlation between Fanout Coefficient and Transmissibility should invalidate the Galton-Watson model used in Section \[parameters\], because that model assumes that those parameters are uncorrelated. However, this is not the case since most of the participants in the campaign appear at very low generation numbers and thus the phenomena observed here is only a significant correction affecting a small fraction of participants.
The Message Affinity Model (MAM) {#hypo}
================================
The correlation between the messages propagation dynamic parameters $\lambda$ and $\overline{r}_v$ and the independence of the nodes spreading activity from the substrate email network they run upon are intriguing properties of the viral marketing diffusion processes. @Watts2007 built a model proving that information propagation can happen independently of the underlying social network structure and concluded that “large cascades of influence are driven not by the influentials but by a critical mass of easily influenced individuals.” However, their model does not explain the dynamic parameters correlation nor the increase with the generation of the nodes propensity of becoming spreaders. We posit that both features are due to the fact that the decisions of forwarding a viral message and of the number of neighbors to send it to, typically made in a single act by each forwarding individual, are correlated and that such correlation emerges as a function only of their affinity with the content of the message being spread.
The agent-based Message Affinity Model (MAM) incorporates that mechanism by assigning to the substrate network nodes a propensity value representing their affinity with the message being forwarded. Furthermore, the model propagation rules combine a variant of the states transition steps of the SIR epidemic model on networks [@Pastor-Satorras2001] with the stochastic evolution of a pseudo-markovian[^10] Galton-Watson Branching model. At any step, the network nodes are in one of the following three states:
- *Susceptible (S)*: Node has not received the message
- *Informed (I)*: Node is propagating the message
- *Refractory (R)*: Node does not spread the message anymore
Unlike the SIR model, MAM does not use a global probability for the nodes states transitions. Instead, they stem from the aggregate decisions that result from the interplay between the nodes pass-along propensity and the message “fitness” to diffuse. Drawn from a continuous probability density function $p(a)$, the *Affinity* $a_n\in[0,1]$ of a node represents its propensity to engage in spreading the message. The message fitness to trigger the node activations is represented by their *Affinity Threshold* $A_T\in[0,1]$, the lowest $a_n$ value for which such message can push the node into the *Informed* state: low threshold messages are capable of activating more nodes and are, as a result, forwarded more often than high threshold ones. The process starts by turning a random fraction of the substrate network nodes into the *Informed* state while leaving all others *Susceptible*. From that point onwards the following rules govern the stochastic propagation:
1. \[step4\] *Susceptible* nodes touched by the message become *Informed* if their *Affinity* is higher than the message threshold ($a_n>A_T$) and *Refractory* otherwise while, if touched, *Informed* or *Refractory* nodes stay unchanged.
2. \[step2\] An *Informed* node $n$ forwards a number of messages $(r_v)_n=(a_n-A_T)\times r $, with $r$ drawn from a PL distribution. The neighbors receiving those messages are
1. those with highest $a_n$ with probability $(a_n-A_T)$
2. chosen randomly with probability $1-(a_n-A_T)$
3. \[step5\] *Informed* nodes become *Refractory* immediately after spreading the message and the process ends when no *Informed* nodes are left
The quantity $a_n-A_T$ embodies the interplay between the participants interests and the message content. The choice in Rule (\[step2\]) of the neighbors that will receive the message represents the evaluation *Informed* nodes make, based on their local knowledge, of their neighbors’ affinity. It implies that local knowledge grows with the *Affinity*: nodes of high $a_n$ are more likely to choose targets with the highest propensity to pass the message while those with low $a_n$ will mostly choose their targets randomly. $A_T$ may vary by individual but, without loss of generality, we take it constant including all variations in $p(a)$.
MAM Simulation Results {#validation}
----------------------
Here we present the result of Monte Carlo simulations of viral messages propagation ran with the MAM model and show that they replicate the patterns observed in real processes. The simulations ran on two substrate networks with the same degree distribution but different structure: the real email network of a Spanish university [@Guimera2003] and a synthetic configuration model network built with the Molloy and Reed method [@Callaway2001]. They differ in their *Clustering Coefficient* ($C_{email}=0.22$ vs. $C_{conf}=0.014$) and in the fact that the email network node degrees are correlated while the configuration network ones are not. Their nodes *Affinity*, with correlation between nearest neighbors, was drawn from a uniform distribution. The *Cascades Network* resulting from the propagation of messages with *Affinity Threshold* between 0.6 and 0.97 were averaged over 15K cascades with 500 different allocations of the substrate nodes *Affinity*.
The simulations generate graphs with a large number of disconnected components that, like those in the real campaigns, feature distributions of Eq. (\[harris\]) type for both their viral nodes activity $P(r_v)$ and cascades size $P(s)$. The exponents $\gamma_k$ and $\gamma_s$ of their power-laws are in the range 1 - 3 depending on the values of the model parameters nodes *Affinity* ($a_n$) and message *Affinity Threshold* ($A_T$) used. Besides, the average cluster size of the graphs obtained in the simulations follows closely the branching model predictions as shown in Fig. \[cascades\]. It plots the average size $\overline{s}$ of the propagation network components obtained with different values of the message Affinity Threshold versus their reproductive number $R_0$ for each. The lines are not a fit to the data but the prediction $\overline{s*}$ given by Eq. (\[avgcascade\]). Notice their remarkable agreement and the fact, shown in the inset, that when the effect of *Seed Nodes* is removed by plotting $(\overline{s}-1)/\overline{r}_s$ the results for the simulations on both substrate networks match exactly. This indicates that as our model predicts, for processes running well below the *Tipping-point* the impact of the substrate network in the cascades average size or the dynamic parameters of the propagation is very low.
The plot of the *Cascades Network* dynamic parameters in the main panel of Fig. \[Affinitymod\] and their fit to Eq. (\[model\]) shows how MAM accurately replicates their correlation pattern. This proves that the viral messages propagation patterns are independent of the substrate network structure for low $\lambda$. However their $\overline{r}_v$ values diverge as $\lambda$ grows because the email network clustering and degree correlations accelerate saturation effects and curtail propagation. The diffusion acceleration with path length presented in Fig. \[utility\] and typical of viral messages propagation is also properly replicated with MAM. The inset of Fig. \[model\] presents the evolution of $\lambda_g$ with $g$ for simulations on the real email network (dotted lines) alongside that of our empirical results. The striking similarity of both up until $g=5$ is quite significant. The low number of active nodes left in the substrate network beyond that point, renders the statistics of the results unreliable. The same pattern (not shown) appears for simulations on the configuration model network. The growth of $\lambda_g$ can not prevent the propagation process ending. In fact for $R_g<1$, $\lambda_g$ is a probability below unity applied at each subsequent generation to an ever shrinking cohort of nodes. As proved by the Branching Process theory, the cascades inevitably reach a point where there is no new offspring and they die off. Actually, even for $R_g>1$ the cascades extinction has a non-zero probability that increases with the heterogeneity of the participants’ activity distribution [@Harris2002].
Conclusions and Discussion {#conclusions}
==========================
We tracked and analyzed the structure and growth dynamics of the propagation network created by the diffusion of a content-controlled message in real viral marketing campaigns driven through email forwarding. The resulting *Cascades Network*, formed by almost pure trees of very low clustering, shows two striking dynamical patterns not observed so far in other Social Dynamics processes like rumor spreading, innovations adoption or email chain-letters. First, there is positive correlation between the spreading nodes activity level as measured by their out-degree and that of their active off-spring and, second, the propensity of nodes reached by the message to becoming spreaders, the *Transmissibility* $\lambda$, grows with those nodes depth in the propagation path. These novel properties can only be detected by scrutinizing the propagation of messages of fixed and identical content. The scarcity of such type of data may explain why they have remained unobserved until now. The discovered patterns have two remarkable consequences. On the one hand, the dynamic parameters *Transmissibility* and *Fanout Coefficient* for a given message across different markets are correlated. On the other, the topology of the email network underlying the propagation has limited influence on the *Cascades Network* although its features are compatible with the structure of the substrate email network that conditions their formation.
Our explanation of all those peculiarities stems from the mechanism driving the messages propagation which involves the affinity of the campaign participants with the content of the message. Participants would make a simultaneous and conscious decision of spreading it or not and to whom which leads to a the positive correlation between the probability of becoming a spreader after receiving the message and the average number of messages forwarded. This decision would result from a single intrinsic property of the nodes in the substrate network, their affinity with the message being passed-along. Besides, the dynamic parameters by generation $\lambda_g$ and $(\overline{r}_g)$ tend to grow with $g$ since the choice of targets to forward the message to is based on the participants’ awareness of their neighbors’ affinity with it. Such mechanism steers the message through paths of increased affinity termed *Affinity Paths*.
This hypothesis is tested through an agent-based model (MAM) that replicates the patterns discovered and validates the proposed Affinity-driven information diffusion mechanism. It combines a stochastic branching process with propagation rules that create cascades of touched nodes by taking the substrate network nodes message awareness through a sequence of *Susceptible*, *Informed* and *Reluctant* states. The MAM uses just two control parameters: the *Affinity* distribution $p(a)$ of the substrate network nodes to assign them an affinity value between 0 (message is not sent) and 1 (message will certainly be forwarded) and the *Affinity Threshold* $A_T$ representing the message fitness to be passed-around. As the model runs through a substrate network list of edges, the interplay between $A_T$ and the nodes *Affinity* generates cascades with all the expected features while providing a glimpse into the substrate network topology. The empirical analysis and the theoretical model validate our conclusion that the mechanism driving viral marketing messages propagation results from the affinity between the campaign participants’ preferences and the messages content. In fact, the viral cascades features depend more on the individuals’ reaction to the message than on the substrate network topology. However, we could not verify this conclusion empirically since the structure of our campaigns substrate network being unknown, a comparison between the *Cascades Network* and the substrate email network was impossible. Also, MAM does not replicate the merging of cascades that occurs near the *Tipping-point* as it assumes that *Seed Nodes* are planted in a boundless network and far apart of each other to avoid propagation clashing. Finally, MAM only runs on undirected and fully connected networks.
[^1]: However, participants with cookies disabled could send multiple referrals to the same person. Thus 183 referrals (0.76% of total) were discarded
[^2]: The time dynamics of the message diffusion is covered on a separate paper
[^3]: A triad is a group of three nodes connected by two links
[^4]: The tail of the cascade size distribution in large random networks near the transition to the giant component goes as $n^c_s\sim s^{-5/2}$ [@Albert2002] and the probability of a cascade of size 122 is $\sim 6.1\times10^{-6}$.
[^5]: A markovian model of a population where each individual in generation $g$ produces in generation $g+1$ a random number of individuals extracted from the same probability distribution.
[^6]: Defined by analogy to phase transitions in Physics as the process inflection point where propagation speed accelerates drastically and becomes unstopped so that the message propagation reaches a very large fraction of the audience.
[^7]: A convoluted combination of the probability distributions of a link going *out of* a node of out-degree $r_v$, of a link going *into* a node of out-degree $r'_v$ and the joint probability of links to go from a node of out-degree $r_v$ to another of out-degree $r'_v$ [@Piraveenan2009]
[^8]: The tendency of individuals to associate and bond with similar others.
[^9]: Y intercept set to 1 since $\overline{r}_v\rightarrow 1$ as $\lambda\rightarrow 0$ because fit is on active nodes.
[^10]: The Galton-Watson Branching model used in Section \[parameters\] explains well the growth of the cascades at the average level but fails to predict the activity correlations that appear in the evolution through generations. This is because the Galton-Watson model stochastic process is markovian while, in reality, one node’s activity depends on that of its parent.
|
---
abstract: |
We describe typical degenerations of quadratic differentials thus describing “generic cusps” of the moduli space of meromorphic quadratic differentials with at most simple poles. The part of the boundary of the moduli space which does not arise from “generic” degenerations is often negligible in problems involving information on compactification of the moduli space.
However, even for a typical degeneration one may have several short loops on the Riemann surface which shrink simultaneously. We explain this phenomenon, describe all rigid configurations of short loops, present a detailed description of analogs of desingularized stable curves arising here, and show how one can reconstruct a Riemann surface endowed with a quadratic differential which is close to a “cusp” by the corresponding point at the principal boundary.
address:
- ' Department of Mathematics, UIC, Chicago, IL 60607-7045, USA\'
- ' IRMAR, Université Rennes-1, Campus de Beaulieu, 35042 Rennes, cedex, France '
author:
- Howard Masur
- Anton Zorich
title: Multiple Saddle Connections on Flat Surfaces and the Principal Boundary of the Moduli Spaces of Quadratic Differentials
---
[^1]
Introduction {#introduction .unnumbered}
============
Saddle connections on flat surfaces {#saddle-connections-on-flat-surfaces .unnumbered}
-----------------------------------
We study flat metrics on a closed orientable surface of genus $g$, which have isolated conical singularities and linear holonomy restricted to $\{Id,-Id\}$. If the linear holonomy group is trivial, then the surface is referred to as a [*translation surface*]{}, such a flat surface corresponds to an Abelian differential $\omega$ on a Riemann surface. If the holonomy group is nontrivial, then such a flat surface arises from a meromorphic quadratic differential $q$ with at most simple poles on a Riemann surface. In this paper, unless otherwise stated, a quadratic differential is [*not*]{} the square of an Abelian differential and a [*flat surface*]{} is the Riemann surface with the flat metric corresponding to an Abelian or to a quadratic differential.
It is natural to consider families of flat surfaces sharing the same combinatorial geometry: the genus, the number of singularities and the cone angles at singularities. Such families correspond to the [*strata*]{} ${{\mathcal Q}}(d_1,\dots,d_m)$ in the moduli space of quadratic differentials, where $d_i\in\{-1,0,1,2,3,\dots\}$ stands for the orders of singularities (simple poles, marked points, zeroes) of quadratic differentials. The collection $\alpha=\{d_1,\dots,d_m\}$ is called the [*singularity data*]{} of the stratum.
A [*saddle connection*]{} is a geodesic segment joining a pair of conical singularities or a conical singularity to itself without any singularities in its interior. For the flat metrics as described above, regular closed geodesics always appear in families; any such family fills a maximal cylinder bounded on each side by a closed saddle connection or by a chain of parallel saddle connections. Thus, when some regular closed geodesic becomes short the corresponding saddle connection(s) become short as well. More generally, a degeneration of an Abelian or of a quadratic differential corresponds to collapse of some saddle connections.
Any saddle connection on a flat surface $S\in{{\mathcal Q}}(\alpha)$ persists under small deformations of $S$ inside ${{\mathcal Q}}(\alpha)$. It might happen that any deformation of a given flat surface which shortens some specific saddle connection necessarily shortens some other saddle connections. We say that a collection $\{\gamma_1, \dots, \gamma_n\}$ of saddle connections is *rigid* if any sufficiently small deformation of the flat surface inside the stratum preserves the proportions $|\gamma_1|:|\gamma_2|: \dots:|\gamma_n|$ of the lengths of all saddle connections in the collection.
Degeneration of Abelian differentials {#degeneration-of-abelian-differentials .unnumbered}
-------------------------------------
In the case of Abelian differentials $\omega$, rigid collections of saddle connections were studied in the paper [@Eskin:Masur:Zorich]. It was shown that all saddle connections in any rigid collection are [*homologous*]{}. In particular, they are all parallel and have equal length and either all of them join the same pair of distinct singular points, or they are all closed.
This implies that when the saddle connections in a rigid collection are contracted by a continuous deformation, the limiting flat surface generically decomposes into several components represented by nondegenerate flat surfaces $S'_1,
\dots, S'_k$, where $k$ might vary from one to the genus of the initial surface. Let ${{\mathcal H}}(\beta'_j)$ be the stratum ambient for $S'_j$. The stratum ${{\mathcal H}}(\beta')={{\mathcal H}}(\beta'_1)\sqcup \dots \sqcup
{{\mathcal H}}(\beta'_k)$ of disconnected flat surfaces $S'_1\sqcup
\dots\sqcup S'_k$ is referred to as a [*principal boundary stratum*]{} of the stratum ${{\mathcal H}}(\beta)$. For any connected component of any stratum ${{\mathcal H}}(\beta)$ the paper [@Eskin:Masur:Zorich] describes all principal boundary strata; their union is called the *principal boundary* of the corresponding connected component of ${{\mathcal H}}(\beta)$.
The paper [@Eskin:Masur:Zorich] also presents the inverse construction. Consider any flat surface $S'_1\sqcup \dots \sqcup
S'_k\in {{\mathcal H}}(\beta')$ in the [*principal boundary*]{} of ${{\mathcal H}}(\beta)$; consider a sufficiently small value of a complex parameter $\delta\in{{\mathbb C}^{}}$. One can reconstruct the flat surface $S\in{{\mathcal H}}(\beta)$ endowed with a collection of homologous saddle connections $\gamma_1, \dots, \gamma_n$ such that $\int_{\gamma_i}\omega=\delta$, and such that the degeneration of $S$ that consists of contracting the saddle connections $\gamma_i$ in the collection gives the surface $S'_1\sqcup \dots
\sqcup S'_k$. This inverse construction involves several [*basic surgeries*]{} of the flat structure. Given a disconnected flat surface $S'_1\sqcup \dots \sqcup S'_k$ one applies an appropriate surgery to each $S'_j$ producing a surface $S_j$ with boundary. The surgery depends on the parameter $\delta$: the boundary of each $S_j$ is composed of two geodesic segments of lengths $|\delta|$; moreover, the boundary components of $S_j$ and $S_{j+1}$ are compatible, which allows one to glue the compound surface $S$ from the collection of surfaces with boundary.
A collection $\gamma=\{\gamma_1, \dots, \gamma_n\}$ of homologous saddle connections determines the following data on combinatorial geometry of the decomposition $S\setminus \gamma$: the number of components, their boundary structure, the singularity data for each component, the cyclic order in which the components are glued to each other. These data are referred to as *configuration* of homologous saddle connections. A configuration ${{\mathcal C}}$ uniquely determines the corresponding boundary stratum ${{\mathcal H}}(\beta'_{{{\mathcal C}}})$.
The constructions above explain how configurations of homologous saddle connections on flat surfaces $S\in{{\mathcal H}}(\beta)$ determine the “cusps” of the stratum ${{\mathcal H}}(\beta)$. Consider a subset ${{\mathcal H}}_1^{\varepsilon}(\beta)\subset{{\mathcal H}}(\beta)$ of surfaces of area one having a saddle connection shorter than $\varepsilon$. Up to a subset ${{\mathcal H}}_1^{\varepsilon,thin}(\beta)$ of negligibly small measure the set ${{\mathcal H}}_1^{\varepsilon}(\beta)$ can be represented as a disjoint union over all admissible configurations ${{\mathcal C}}$ (i.e. as a union over different “cusps”) of neighborhoods of the corresponding “cusps”. When a configuration ${{\mathcal C}}$ is composed from homologous saddle connections joining distinct zeroes, the neighborhood of the corresponding cusp has the structure of a fiber bundle over the corresponding boundary stratum ${{\mathcal H}}(\beta'_{{{\mathcal C}}})$ with the fiber represented by an appropriate ramified cover over the Euclidean $\varepsilon$-disc. Moreover, the canonical measure in the corresponding connected component of ${{\mathcal H}}_1^{\varepsilon,thick}(\beta) =
{{\mathcal H}}_1^{\varepsilon}(\beta) \setminus
{{\mathcal H}}_1^{\varepsilon,thin}(\beta)$ decomposes as a product measure of the canonical measure in the boundary stratum and the Euclidean measure in the fiber, see [@Eskin:Masur:Zorich].
We warn the reader that the correspondence between the compactification of the moduli space of Abelian differentials and the Deligne—Mumford compactification of the underlying moduli space of curves is not straightforward. In particular, the desingularized stable curve corresponding to the limiting flat surface generically *is not* represented as the union of corresponding Riemann surfaces $S'_1, \dots, S'_k$: the stable curve might contain more components.
Structure of the paper and statements of theorems {#s:structure:of:the:paper}
=================================================
This paper concerns the study of similar phenomena in the case of quadratic differentials that are not squares of Abelian differentials.
Ĥomologous saddle connections
-----------------------------
A meromorphic quadratic differential $q$ with at most simple poles on a Riemann surface $S$ defines a canonical (ramified) double cover $p:\hat S\to S$ such that $p^* q = \omega^2$ is a square of an Abelian differential $\omega$ on $\hat S$. Let $P=\{P_1, \dots, P_m\}\subset S$ be the collection of singularities (zeroes and simple poles) of $q$; let $\hat
P=p^{-1}(P)$ be the set of their preimages under the projection $p:\hat S \to S$.
Given an oriented saddle connection $\gamma$ on $S$ let $\gamma',\gamma''$ be its lifts to the double cover. If $[\gamma']=-[\gamma'']$ as cycles in $H_1(\hat S,\hat P;\,{\mathbb Z})$ we let $[\hat\gamma]:=[\gamma']$, otherwise we define $[\hat\gamma]$ as $[\hat\gamma]:=[\gamma']-[\gamma'']$.
\[def:homologous\] The saddle connections $\gamma_1,\gamma_2$ on a flat surface $S$ defined by a quadratic differential $q$ are ĥomologous[^2] if $[\hat\gamma_1]=[\hat\gamma_2]$ in $H_1(\hat S,\hat P;\,{\mathbb Z})$ under an appropriate choice of orientations of $\gamma_1,
\gamma_2$.
(0,0)(0,0) (140,360)
(0,-152)(0,-152) (-310,-526)[$\gamma_3$]{} (-332,-533)[$P_0$]{} (-286,-533)[$P_0$]{} (-318,-550)[$\alpha$]{} (-318,-583)[$\beta$]{} (-318,-615)[$\beta$]{} (-310,-637)[$\gamma_3$]{} (-332,-632)[$P_0$]{} (-286,-632)[$P_0$]{} (-297,-550)[$\alpha$]{} (-296,-583)[$\delta$]{} (-296,-615)[$\delta$]{} (-310,-559)[$\gamma_2$]{} (-332,-566)[$P_0$]{} (-286,-566)[$P_0$]{} (-310,-592)[$\gamma_1$]{} (-332,-599)[$P_1$]{} (-286,-599)[$P_2$]{}
(4,0)(2,0) (-310,-526)[$\gamma'_3$]{} (-320,-550)[$\alpha'$]{} (-320,-583)[$\beta'$]{} (-321,-615)[$-\beta''$]{} (-310,-637)[$\gamma_3'$]{} (-298,-550)[$\alpha'$]{} (-298,-583)[$\delta'$]{} (-304,-615)[$-\delta''$]{} (-310,-559)[$\gamma'_2$]{} (-310,-592)[$\gamma'_1$]{}
(-29,0)(-31,0) (-314,-526)[$-\gamma''_3$]{} (-321,-550)[$-\alpha''$]{} (-321,-583)[$-\beta''$]{} (-320,-615)[$\beta'$]{} (-314,-637)[$-\gamma_3''$]{} (-304,-550)[$-\alpha''$]{} (-304,-583)[$-\delta''$]{} (-297,-615)[$\delta'$]{} (-314,-559)[$-\gamma''_2$]{} (-314,-592)[$-\gamma''_1$]{}
(-111,0)(-115,0) (-310,-526)[$\gamma'_3$]{} (-320,-550)[$\alpha'$]{} (-320,-583)[$\beta'$]{} (-323,-615)[$-\beta''$]{} (-310,-637)[$\gamma_3'$]{} (-298,-550)[$\alpha'$]{} (-298,-583)[$\delta'$]{} (-305,-615)[$-\delta''$]{} (-310,-559)[$\gamma'_2$]{} (-310,-592)[$\gamma'_1$]{}
(-249,33)(-253,33) (-314,-526)[$-\gamma''_3$]{} (-321,-615)[$-\alpha''$]{} (-314,-637)[$-\gamma_3''$]{} (-297,-550)[$\beta'$]{} (-304,-583)[$-\beta''$]{} (-304,-615)[$-\alpha''$]{} (-314,-559)[$-\gamma''_1$]{} (-314,-592)[$-\gamma''_2$]{}
(0,0)(-3,350) (252,-102)[$\gamma_1$]{} (229,-97)[$P_1$]{} (275,-97)[$P_2$]{} (229,-64)[$P_0$]{} (275,-64)[$P_0$]{} (258,-53)[$P_0$]{} (258,-20)[$P_0$]{} (257,-36)[$\alpha$]{} (244,-25)[$\gamma_3$]{} (244,-46)[$\gamma_2$]{} (252,-63)[$\gamma_2$]{} (252,-72)[$\gamma_3$]{} (241,-83)[$\beta$]{} (266,-83)[$\delta$]{}
(0,0)(-97,350) (252,-102)[$\gamma_1$]{} (229,-97)[$P_1$]{} (275,-97)[$P_2$]{} (253,-39)[$\alpha$]{} (231,-59)[$\gamma_2$]{} (272,-60)[$\gamma_3$]{} (252,-71)[$P_0$]{} (241,-83)[$\beta$]{} (266,-83)[$\delta$]{}
(0,0)(5,500) (244,-15)[$\gamma'_3$]{} (244,-48)[$\gamma'_2$]{} (258,-55)[$-\gamma''_3$]{} (244,-72)[$\gamma'_2$]{} (258,-88)[$-\gamma''_1$]{} (244,-108)[$\gamma'_1$]{} (258,-124)[$-\gamma''_2$]{} (244,-141)[$\gamma'_3$]{} (258,-147)[$-\gamma''_2$]{} (258,-180)[$-\gamma''_3$]{}
(0,0)(-77,500) (253,-48)[$\gamma'_2$]{} (270,-48)[$-\gamma''_3$]{} (253,-72)[$\gamma'_2$]{} (270,-72)[$-\gamma''_3$]{} (261,-87)[$-\gamma''_1$]{} (261,-110)[$\gamma'_1$]{} (253,-125)[$\gamma'_3$]{} (270,-125)[$-\gamma''_2$]{} (253,-148)[$\gamma'_3$]{} (270,-148)[$-\gamma''_2$]{}
We begin with the following example which illustrates many of the main ideas.
\[ex:homologous:sad:connections\] Consider three unit squares, or rather a rectangle $1\times 3$ and glue a torus from it as indicated at the top left corner of Figure \[fig:examples:of:homologous:sad:connections:1\]. Identifying the three corresponding sides $\beta$, $\gamma_1$ and $\delta$ of the two bottom squares we obtain a “pocket” with two “corners” $P_1$ and $P_2$ at the bottom and with two “corners” $P_0$ at the boundary on top. Identifying the points $P_0$ we obtain a “pocket” with a “figure-eight” boundary (the bottom fragment of the top right picture at Figure \[fig:examples:of:homologous:sad:connections:1\]). Identifying the sides $\alpha$ of the remaining square we obtain a cylinder which we glue to the previous fragment. Topologically the surface thus obtained is a torus. Metrically this torus has three conical singularities. Two of them (“the corners $P_1,
P_2$ of the pocket”) have cone angle $\pi$; the third conical singularity $P_0$ has cone angle $4\pi$. Such a flat torus gives us a point in the stratum ${{\mathcal Q}}(2,-1,-1)$.
The bottom picture illustrates the canonical double covering over the above torus. The cycle $\gamma'_2$ is homologous to $\gamma'_3$ on the double cover and the cycle $\gamma''_2$ is homologous to $\gamma''_3$. This implies that the cycles $\hat\gamma_1$, $\hat\gamma_2$ and $\hat\gamma_3$ on the double cover are homologous to the waist curve of the thick cylinder fragment of the right bottom picture. Thus, the saddle connections $\gamma_1$, $\gamma_2$ and $\gamma_3$ are ĥomologous, though $\gamma_1$ is a segment joining distinct points $P_1$ and $P_2$, and $\gamma_2, \gamma_3$ are the closed loops with the base point $P_0$.
It essentially follows from the definition that ĥomologous saddle connections are parallel on $S$ and that their lengths either coincide or differ by a factor of two. The following simple statement proved in appendix \[ap:Long:saddle:connections\] characterizes rigid collections of saddle connections on a flat surface with nontrivial linear holonomy.
\[pr:rigid:configurations:hat:homologous\] Let $S$ be a flat surface corresponding to a meromorphic quadratic differential $q$ with at most simple poles. A collection $\gamma_1, \dots, \gamma_n$ of saddle connections on $S$ is rigid if and only if all saddle connections $\gamma_1,
\dots, \gamma_n$ are ĥomologous.
There is an obvious geometric test for deciding when saddle connections $\gamma_1, \gamma_2$ on a translation surface $S$ are homologous: it is sufficient to check whether $S\setminus
(\gamma_1\cup\gamma_2)$ is connected or not (provided $S\setminus\gamma_1$ and $S\setminus\gamma_2$ are connected). It is slightly less obvious to check whether saddle connections $\gamma_1, \gamma_2$ on a flat surface $S$ with nontrivial linear holonomy are ĥomologous or not. In particular, a pair of closed saddle connections might be homologous in the usual sense, but not ĥomologous; a pair of closed saddle connections might be ĥomologous even if one of them represents a loop homologous to zero, and the other does not; finally, a saddle connection joining a pair of [*distinct*]{} singularities might be ĥomologous to a saddle connection joining a singularity to itself.
Section \[s:hat:homologous:saddle:connections\] describes geometric criteria for deciding when two saddle connections are ĥomologous and what is the structure of the complement $S\setminus(\gamma_1\cup\gamma_2)$. These criteria are intensively used in the remaining part of the paper. In particular, we prove in section \[s:hat:homologous:saddle:connections\] the following statement.
\[th:unique:trivial:holonomy\] Let $S$ be a flat surface corresponding to a meromorphic quadratic differential $q$ with at most simple poles. Two saddle connections $\gamma_1, \gamma_2$ on $S$ are ĥomologous if and only if they have no interior intersections and one of the connected components of the complement $S\setminus
(\gamma_1\cup\gamma_2)$ has trivial linear holonomy. Moreover, if such a component exists, it is unique.
Graph of connected components
-----------------------------
A collection $\gamma$ of ĥomologous saddle connections $\gamma=\{\gamma_1,\ldots,\gamma_n\}$ divides $S$ into simpler surfaces $S_j$ with boundary. We associate to any such decomposition a graph $\Gamma(S,\gamma)$. The vertices of the graph correspond to the connected components $S_j$ of $S\setminus
(\gamma_1\cup\dots\cup\gamma_n)$. We denote the vertices corresponding to cylinders (if any) by small circles “$\circ$”. The remaining vertices are labelled with a “$+$” sign if the corresponding surface $S_j$ has trivial linear holonomy and with a “$-$” sign if it does not. We do not label the vertices of “$\circ$”-type: it is easy to see that the cylinders always have trivial linear holonomy.
The edges of the graph are in the one-to-one correspondence with the saddle connections $\gamma_i$. Each saddle connection $\gamma_i$ is on the boundary of either one or two surfaces. If $\gamma_i$ is on the boundary of pair of surfaces, it corresponds to an edge joining the corresponding vertices. If $\gamma_i$ is on the boundary of only one surface, then it corresponds to an edge of the graph which joins the vertex to itself; such an edge contributes $2$ to the valence of the vertex.
\[rm:dual:graph\] The union $\gamma=\gamma_1\cup\dots\cup\gamma_n$ of saddle connections can be considered as a graph $\gamma$ embedded into the surface $S$. By definition $\Gamma(S,\gamma)$ is a graph *dual* to $\gamma$. Namely, $\Gamma(S,\gamma)$ can be realized as graph embedded into the surface $S$ in the following way. A vertex of $\Gamma(S,\gamma)$ corresponding to a connected component $S_j$ of $S\setminus\gamma$ is mapped to a point $v_j$ located in the interior of the corresponding surface with boundary $S_j$. The line representing the image of an edge of $\Gamma(S,\gamma)$ corresponding to a saddle connection $\gamma_i$ has a single transversal intersection with $\gamma_i$ in some interior point; this line does not intersect itself nor any other such line nor some other saddle connection $\gamma_{i'}$, where $i'\neq i$, in an interior point.
(0,0)(197,0) (88,-35)[$S_2$]{} (90,-80)[$S_1$]{} (131,-14)[$v_2$]{} (138,-84)[$v_1$]{} (113,-61)[$\gamma_2$]{} (147,-61)[$\gamma_3$]{} (130,-118)[$\gamma_1$]{}
(0,0)(66,0) (131,-16)[$v_2$]{} (131,-75)[$v_1$]{} (100,-45)[$\gamma_2$]{} (159,-45)[$\gamma_3$]{} (131,-118)[$\gamma_1$]{}
\[ex:graph:Gamma:for:fig:1\] Consider the surface $S$ and the collection $\gamma$ of ĥomologous saddle connections $\{\gamma_1, \gamma_2,
\gamma_3\}$ as in example \[ex:homologous:sad:connections\] above (see figure \[fig:examples:of:homologous:sad:connections:1\]). The complement $S\setminus\gamma$ has two connected components; both represented by flat cylinders. The graph $\Gamma(S,\gamma)$ contains two vertices, both of the “$\circ$”-type, and three edges. The graph $\Gamma(S,\gamma)\subset S$ is dual to the graph $\gamma\subset S$, see figure \[fig:example:of:a:graph\].
It follows from the definition of ĥomologous saddle connections that their lengths are either the same or differ by a factor of two. Having a collection $\gamma$ of ĥomologous saddle connections $\gamma_1, \dots, \gamma_n$ we can normalize the length of the shortest one to $1$. Then the other saddle connections have lengths either $1$ or $2$, which endows the edges of the graph $\Gamma$ with the weights $1$ or $2$.
The theorem below classifies all possible graphs corresponding to nonempty collections of ĥomologous saddle connections.
(0,0)(0,0) (-140,470.5)
(0,0)(0,0) (38,-29)[a)]{} (285,-53)[b)]{} (132,-130)[c)]{} (132,-362)[d)]{} (132,-450)[e)]{}
\[th:graphs\] Let $S$ be a flat surface corresponding to a meromorphic quadratic differential $q$ with at most simple poles; let $\gamma$ be a collection of ĥomologous saddle connections $\{\gamma_1, \dots, \gamma_n\}$, and let $\Gamma(S,\gamma)$ be the graph of connected components encoding the decomposition $S\setminus (\gamma_1\cup\dots\cup\gamma_n)$.
The graph $\Gamma(S,\gamma)$ either has one of the basic types listed below or can be obtained from one of these graphs by placing additional [“$\circ$”]{}-vertices of valence two at any subcollection of edges subject to the following restrictions. At most one [“$\circ$”]{}-vertex may be placed at the same edge; a [“$\circ$”]{}-vertex cannot be placed at an edge adjacent to a [“$\circ$”]{}-vertex of valence $3$ if this is the edge separating the graph.
The graphs of basic types, presented in Figure \[fig:classification:of:graphs\], are given by the following list:
- An arbitrary (possibly empty) chain of “$+$”-vertices of valence two bounded by a pair of “$-$”-vertices of valence one;
- A single loop of vertices of valence two having exactly one “$-$”-vertex and arbitrary number of “$+$”-vertices (possibly no “$+$”-vertices at all);
- A single chain and a single loop joined at a vertex of valence three. The graph has exactly one “$-$”-vertex of valence one; it is located at the end of the chain. The vertex of valence three is either a “$+$”-vertex, or a [“$\circ$”]{}-vertex (vertex of the cylinder type). Both the chain, and the cycle may have in addition an arbitrary number of “$+$”-vertices of valence two (possibly no “$+$”-vertices at all);
- Two nonintersecting cycles joined by a chain. The graph has no “$-$”-vertices. Each of the two cycles has a single vertex of valence three (the one where the chain is attached to the cycle); this vertex is either a “$+$”-vertex or a [“$\circ$”]{}-vertex. If both vertices of valence three are [“$\circ$”]{}-vertices, the chain joining two cycles is nonempty: it has at least one “$+$”-vertex. Otherwise, each of the cycles and the chain may have arbitrary number of “$+$”-vertices of valence two (possibly no “$+$”-vertices of valence two at all);
- “Figure-eight” graph: two cycles joined at a vertex of valence four, which is either a “$+$”-vertex or a [“$\circ$”]{}-vertex. All the other vertices (if any) are the “$+$”-vertices of valence two. Each of the two cycles may have arbitrary number of such “$+$”-vertices of valence two (possibly no “$+$”-vertices of valence two at all).
Every graph listed above corresponds to some flat surface $S$ and to some collection of saddle connections $\gamma$.
Theorem \[th:graphs\] is proved in section \[s:graph\] with exception of the final statement on realizability, which is proved in sections \[s:Local:Constructions\]–\[s:Nonlocal:constructions\].
Parities of boundary singularities {#ss:Anatomy:of:decomposition:of:a:flat:surface}
----------------------------------
In section \[s:Anatomy\] we give a detailed analysis of each connected component $S_j$ of the decomposition $S\setminus\gamma$.
It is convenient to consider a closed surface with boundary $S^{comp}_j$ canonically associated to $S_j$ by taking the natural [*compactification*]{} of $S_j$. Note, that $S^{comp}_j$ need not be the same as the [*closure*]{} of $S_j$ in $S$. For example, if we cut a surface $S$ along a single saddle connection $\gamma$ joining a pair of distinct singularities we obtain a surface $S_1$ whose compactification is a surface with boundary composed of a pair of parallel distinct geodesics of the same length, while the closure of $S_1=S\setminus\gamma_1$ in $S$ coincides with $S$. The closure of $S_j$ in $S$ is obtained from the compactification $S^{comp}_j$ of $S_j$ by identification of some boundary points (if necessary), or by identification of some boundary saddle connections (if necessary).
Ribbon graph {#ribbon-graph .unnumbered}
------------
Given a vertex $v$ of a finite graph $\Gamma$ consider a tree $\Gamma_v$ obtained as a small neighborhood of $v$ in $\Gamma$ in the natural topology of a one-dimensional cell complex. The tree $\Gamma_v$ together with the canonical mapping of the graphs $\Gamma_v\to\Gamma$ will be referred to as the *boundary* of $v$. The number of edges of $\Gamma_v$ is exactly the valence of $v$ (and hence is at most $4$ for the graphs from figure \[fig:classification:of:graphs\]).
Suppose that the boundary of $S^{comp}_j$ has $r=r(j)$ connected components (called for brevity *boundary components*). Every boundary component is composed of a closed chain of saddle connections $\gamma_{j_{i,1}}, \dots, \gamma_{j_{i,p(i)}}$, where $1\le i \le r$. The case $p(i)=1$ is not excluded: a boundary component might be composed of a single saddle connection. The canonical orientation of $S^{comp}_j$ determines the orientation of every boundary component ${{\mathcal B}}_i$ of $\partial S^{comp}_j$ and hence determines the cyclic order $$\label{eq:chain:i}
\to\gamma_{j_{i,1}}\to \dots \to\gamma_{j_{i,p(i)}}\to$$ on every such chain; by convention we let $j_{i,p(i)+1}:=j_{i,1}$. Thus, we get a natural decomposition of the set of edges of $\Gamma_{v_j}$ into a disjoint union of subsets, each endowed with a cyclic order, $$\label{eq:all:chains}
\{\to\gamma_{j_{1,1}}\to \gamma_{j_{1,2}}\to \dots \to\gamma_{j_{1,p(1)}}\to\}\
\sqcup\ \dots\ \sqcup\
\{\to\gamma_{j_{r,1}}\to \dots \to\gamma_{j_{r,p(r)}}\to\}$$ It is convenient to encode such combinatorial structure by a *local ribbon graph* ${\mathbb G}_{v_j}$ which is defined in the following way.
Consider a realization of $\Gamma(S,\gamma)$ by an embedded graph dual to the graph $\gamma$ in $S$ (see remark \[rm:dual:graph\] above). For every vertex $v_j$ of $\Gamma(S,\gamma)$ we get an induced embedding $\Gamma_{v_j}\hookrightarrow S^{comp}_j$. Let a connected component ${{\mathcal B}}_i$ of $\partial S^{comp}_j$ be represented by a chain of saddle connections. A tubular neighborhood in $S^{comp}_j$ of the union of the corresponding edges $\{\gamma_{j_{i,1}}\cup \dots
\cup\gamma_{j_{i,p(i)}}\}$ of $\Gamma_{v_j}\subset S^{comp}_j$ (as in the left picture of figure \[fig:example:of:ribbon:graph\]) inherits the canonical orientation of $S$. This orientation induces a natural cyclic order on the edges $\gamma_{j_{i,1}},\dots,\gamma_{j_{i,p(i)}}$ of $\Gamma_{v_j}$. We choose the embedding $\Gamma_{v_j}\hookrightarrow S^{comp}_j$ in such way that turning counterclockwise around $v_j$ (considered as a point of $S_j^{comp}$) we see the edges $\gamma_{j_{i,1}},\dots,\gamma_{j_{i,p(i)}}$ appear in the cyclic order .
When the boundary $\partial S_j^{comp}$ contains several connected components, the ribbon graphs corresponding to different components overlap at $v_j$ (as in the left picture of figure \[fig:example:of:ribbon:graph\]). However, it is easy to make them disjoint by a small deformation, subject to an appropriate choice of the initial embedding $\Gamma_{v_j}\hookrightarrow S^{comp}_j$. From now on we shall always assume that the embedding is chosen appropriately.
(0,0)(250,0) (80,-35)[$S^{comp}_2$]{} (80,-105)[$S^{comp}_1$]{} (133,-53)[$\gamma_2$]{} (159,-53)[$\gamma_3$]{} (138,-127)[$\gamma''_1$]{} (148,-143)[$\gamma'_1$]{} (118,-54)[$1$]{} (109,-67)[$2$]{} (166,-100)[$2$]{} (174,-107)[$1$]{} (166,-150)[$1$]{} (174,-141)[$2$]{}
(0,0)(130,0) (133,-25)[${\mathbb G}_{v_2}$]{} (111,-58)[$\gamma_2$]{} (154,-58)[$\gamma_3$]{} (109,-110)[$\gamma''_1$]{} (154,-110)[$\gamma'_1$]{} (133,-120)[${\mathbb G}_{v_1}$]{}
(0,0)(200,0) (303,-15)[${\mathbb G}(S,\gamma)$]{}
\[ex:ribbon:graph:Gamma:for:fig:1\] Consider once again the surface $S$ and the collection $\gamma$ of ĥomologous saddle connections $\{\gamma_1, \gamma_2,
\gamma_3\}$ as in example \[ex:homologous:sad:connections\], see figure \[fig:examples:of:homologous:sad:connections:1\]. In example \[ex:graph:Gamma:for:fig:1\] we have constructed the associated graph $\Gamma(S,\gamma)$, see figure \[fig:example:of:a:graph\].
The complement $S\setminus\gamma$ has two connected components; their compactifications $S_1^{comp}, S_2^{comp}$ are represented by a pair of flat cylinders. Each of the two connected components of the boundary of $S_2^{comp}$ (the top cylinder in figure \[fig:example:of:ribbon:graph\]) is formed by a single saddle connection, so we get $\partial
S_2^{comp}=\{\gamma_2\}\sqcup\{\gamma_3\}$. Each of the two connected components of the boundary of $S_1^{comp}$ (the bottom cylinder in figure \[fig:example:of:ribbon:graph\]) is formed by a pair of saddle connections, so we get $\partial
S_2^{comp}=\{\gamma_2\to \gamma_3\}\sqcup\{\gamma'_1\to
\gamma''_1\}$. The orientation of the boundary components induced by the canonical orientation of $S$ is indicated in the left picture.
The picture in the center of figure \[fig:example:of:ribbon:graph\] shows the corresponding local ribbon graphs and the picture on the right shows the global ribbon graph ${\mathbb G}(S,\gamma)$ for this example.
For vertices $v$ of valence $1,2,3,4$ figure \[fig:local:ribbon:graphs\] gives a complete list of all possible partitions of the edges of $\Gamma_v$ into a disjoint union of subsets endowed with a cyclic order and of the corresponding local ribbon graphs ${\mathbb G}_v$. Note that the canonical orientation of $S$ induces the counterclockwise ordering of the edges of $\Gamma_v$.
(0,0)(0,-120) (-140,362)
(0,0)(0,10) ( 0, 4)[$\gamma_{j_{11}}$]{} ( 72, 4)[$\gamma_{j_{11}}$]{} (163, 4)[$\gamma_{j_{12}}$]{} (219, 4)[$\gamma_{j_{11}}$]{} (308, 4)[$\gamma_{j_{12}}$]{}
(0,-75)(0,-35) (-58, -115)[$\gamma_{j_{13}}$]{} ( 20, -115)[$\gamma_{j_{12}}$]{} ( -19, -175)[$\gamma_{j_{11}}$]{}
(-102,-75)(-102,-35) (-28, -115)[$\gamma_{j_{21}}$]{} ( 50, -115)[$\gamma_{j_{22}}$]{} ( 11, -175)[$\gamma_{j_{11}}$]{}
(-377,-75)(-377,-35) (-58, -115)[$\gamma_{j_{31}}$]{} ( 20, -115)[$\gamma_{j_{21}}$]{} ( -19, -175)[$\gamma_{j_{11}}$]{}
(0,0)(0,80) (407, -175)[$\gamma_{j_{14}}$]{} (450, -135)[$\gamma_{j_{13}}$]{} (495, -175)[$\gamma_{j_{12}}$]{} (450, -216)[$\gamma_{j_{11}}$]{}
(0,0)(0,80) (537, -175)[$\gamma_{j_{22}}$]{} (580, -135)[$\gamma_{j_{21}}$]{} (625, -175)[$\gamma_{j_{12}}$]{} (580, -216)[$\gamma_{j_{11}}$]{}
(0,0)(0,80) (681, -175)[$\gamma_{j_{23}}$]{} (722, -135)[$\gamma_{j_{11}}$]{} (768, -175)[$\gamma_{j_{22}}$]{} (722, -216)[$\gamma_{j_{21}}$]{}
(9,0)(9,115) (407, -295)[$\gamma_{j_{32}}$]{} (450, -255)[$\gamma_{j_{11}}$]{} (495, -295)[$\gamma_{j_{21}}$]{} (450, -336)[$\gamma_{j_{31}}$]{}
(16,0)(16,115) (681, -295)[$\gamma_{j_{41}}$]{} (722, -255)[$\gamma_{j_{31}}$]{} (768, -295)[$\gamma_{j_{21}}$]{} (722, -336)[$\gamma_{j_{11}}$]{}
(-420,-55)(-420,-40) (-40, -70)[$\{\to \gamma_{j_{11}} \to\}$]{} ( 80, -70)[$\{\to \gamma_{j_{11}} \to \gamma_{j_{12}} \to \}$]{} (215, -70)[$\{\to \gamma_{j_{11}} \to\} \sqcup \{\to \gamma_{j_{21}} \to\}$]{} (-45,-205)[$\{\gamma_{j_{11}}\!\to\!\gamma_{j_{12}}\!\to\!\gamma_{j_{13}}\}$]{} ( 80,-205)[$\{\gamma_{j_{11}}\!\to\} \sqcup \{\gamma_{j_{21}}\!\to\gamma_{j_{22}}\}$]{} (225,-205)[$\{\gamma_{j_{11}}\} \sqcup \{\gamma_{j_{21}}\} \sqcup \{\gamma_{j_{31}}\}$]{} (-45,-360)[$\{\gamma_{j_{11}}\!\!\to\! \gamma_{j_{12}}\!\!\to\!\gamma_{j_{13}}\!\!\to\!\gamma_{j_{14}}\}$]{} ( 80,-360)[$\{\gamma_{j_{11}}\!\!\to\!\gamma_{j_{12}}\}\!\sqcup\!\{\gamma_{j_{21}}\!\!\to\gamma_{j_{22}}\}$]{} (210,-360)[$\{\gamma_{j_{11}}\}\!\sqcup\!\{\gamma_{j_{21}}\!\!\to\!\gamma_{j_{22}}\!\!\to\gamma_{j_{23}}\}$]{} (-45,-515)[$\{\gamma_{j_{11}}\}\sqcup\{\gamma_{j_{21}}\}\sqcup\{\gamma_{j_{31}}\!\to\!\gamma_{j_{32}}\}$]{} (185,-515)[$\{\gamma_{j_{11}}\}\sqcup\{\gamma_{j_{21}}\}\sqcup\{\gamma_{j_{31}}\}\sqcup\{\gamma_{j_{41}}\}$]{}
Boundary singularities {#boundary-singularities .unnumbered}
----------------------
Let $S_j$ be a connected component of the decomposition $S\setminus (\gamma_1\cup\dots\cup\gamma_n)$; let $S_j^{comp}$ be its compactification, and let a connected component ${{\mathcal B}}_i$ of the boundary $\partial S_j^{comp}$ be represented by a chain of saddle connections. The common endpoint of $\gamma_{j_i}$ and $\gamma_{j_{i+1}}$ is called the [*boundary singularity*]{} of $S_j^{comp}$. Since all saddle connections $\gamma_1, \dots, \gamma_n$ are parallel, the corresponding angle between $\gamma_{j_i}$ and $\gamma_{j_{i+1}}$ is an integer multiple of $\pi$. There might be also several conical singularities in the interior of $S_j^{comp}$; they are called [*interior singularities*]{}.
\[def:order:of:boundary:singularity\] If the total angle at a boundary singularity is $(k+1)\pi$ the [*order of the boundary singularity*]{} is defined to be $k$, and the [*parity of the boundary singularity*]{} is defined to be the parity of $k$. If the total angle at an interior singularity is $(d+2)\pi$ the [*order of the interior singularity*]{} is defined to be $d$.
The order of the interior singularity coincides with the order of the zero (simple pole) of the corresponding germ of a quadratic differential. By convention, boundary singularities, and their orders will always refer to the compactification $S^{comp}_j$.
When $S_j$ is represented by a “$+$”-vertex of the graph $\Gamma(S,\gamma)$, we include the parities of the boundary singularities in our combinatorial structure represented by the embedded local ribbon graph ${\mathbb G}_{v_j}$. Let ${{\mathcal B}}_i$ be a connected component of the boundary $\partial S^{comp}_j$ constituted by a chain of saddle connections. The edges $\gamma_{j_{i,1}},\dots,\gamma_{j_{i,p(i)}}$ of the embedded graph $\Gamma_{v_j}\hookrightarrow S^{comp}_j$ subdivide a neighborhood of $v_j$ in $S_j$ into $p(i)$ sectors. To each sector bounded by a pair of consecutive edges $\gamma_{j_{i,l}}$ and $\gamma_{j_{i,l+1}}$ we associate the parity of the order $k_{j_{i,l}}$ of the corresponding boundary singularity of $S^{comp}_j$: of the common endpoint of the consecutive saddle connections $\gamma_{j_{i,l}}\to\gamma_{j_{i,l+1}}$ in ${{\mathcal B}}_i$.
Any connected component $S_j$ of the decomposition $S\setminus
\{\gamma_1, \dots, \gamma_n\}$ determines the following combinatorial data which we refer to as the *boundary type* of $S_j$: the structure of the local ribbon graph ${\mathbb G}_{v_j}$ as in figure \[fig:local:ribbon:graphs\]; an embedding $\Gamma_{v_j}\hookrightarrow\Gamma(S,\gamma)$ and a collection of parities of boundary singularities of $S_j$.
\[th:all:local:ribon:graphs\] Consider a decomposition of a flat surface $S$ as in theorem \[th:graphs\]. Every connected component $S_j$ of the decomposition has one of the boundary types presented in figure \[fig:embedded:local:ribbon:graphs\] and all indicated boundary types are realizable.
The dotted lines in figure \[fig:embedded:local:ribbon:graphs\] indicate pairs of edges of a vertex $v\in\Gamma(S,\gamma)$ of valence $3$ or $4$, which are joined by a loop in the graph $\Gamma(S,\gamma)$ (see figure \[fig:classification:of:graphs\]) and encode in this way the embedding $\Gamma_{v_j}\hookrightarrow\Gamma(S,\gamma)$.
\[rm:indexation\] We use the following convention on indexation of local ribbon graphs in figure \[fig:embedded:local:ribbon:graphs\]: the first symbol represents the type (“$+$”, “$-$”, or “$\circ$”) of the vertex $v_j$ in the graph $\Gamma(S,\gamma)$; the second symbol is the valence of $v_j$; the number after a dot is the number of boundary components of $S_j$. An extra letter “$a,b,c$” is employed when it is necessary to distinguish different embedded local ribbon graphs sharing the same vertex type, valence and number of boundary components.
The first part of the statement of theorem \[th:all:local:ribon:graphs\] which claims that every connected component of the decomposition has one of the boundary types in figure \[fig:embedded:local:ribbon:graphs\] is quite elementary; it is proved at the end of section \[s:Anatomy\]. The statement about the realizability of all boundary types presented in figure \[fig:embedded:local:ribbon:graphs\] is much less trivial; it follows from theorem \[th:from:boundary:to:neighborhood:of:the:cusp\] which is proved in sections \[s:Local:Constructions\] and \[s:Nonlocal:constructions\].
Configurations of ĥomologous saddle connections
-----------------------------------------------
We formalize the data on combinatorial geometry of the decomposition $S\setminus \gamma$ in definition \[def:configuration\] below.
\[def:configuration\] The following combinatorial structure is called a *configuration of ĥomologous saddle connections*.
1. A finite graph $\Gamma$ endowed with a labelling of each vertex by one of the symbols “$+$”, “$-$, or “$\circ$”, of one of one of the types described in theorem \[th:graphs\] (see figure \[fig:classification:of:graphs\]).
2. For any vertex $v$ of the graph $\Gamma$ an embedded ribbon graph ${\mathbb G}_v$ (encoding the decomposition of $\Gamma_v$ into a disjoint union of subsets, called *boundary components*, each endowed with a cyclic order; see equation ) of one of the types described in theorem \[th:all:local:ribon:graphs\] (see figure \[fig:embedded:local:ribbon:graphs\]).
3. For every “$+$”-vertex $v$ of $\Gamma$ and for every pair of consecutive elements $\gamma_{i,l}\to\gamma_{i,l+1}$ of ${\mathbb G}_v$ (called *boundary singularities*) an associated parity (even or odd) as in figure \[fig:embedded:local:ribbon:graphs\].
4. For every vertex $v$ of $\Gamma$ and for every boundary singularity of ${\mathbb G}_v$ a nonnegative integer $k_{i,l}$ (referred to as the *order of the boundary singularity*) satisfying the following conditions. The order of the boundary singularity has the same parity as the parity of the corresponding boundary singularity when $v$ is of the “$+$”-type; the order of any boundary singularity of any vertex of the “$\circ$”-type is equal to zero. The sum $D_i+2=k_{i,1}+\dots+k_{i,p(i)}$ of orders of boundary singularities along any boundary component ${{\mathcal B}}_i$ of $v$ satisfies $ D_i\ge 0$ for a vertex of “$+$”-type and $
D_i\ge -1$ for a vertex of “$-$”-type.
5. For every vertex $v$ of $\Gamma$ of “$-$”-type an unordered (possibly empty) collection of integers $\{d_1, \dots,
d_{s(v)}\}$, where $d_l\in\{-1,1,2,3,\dots\}$; for every vertex $v$ of $\Gamma$ of “$+$”-type an unordered (possibly empty) collection of positive even integers $\{d_1, \dots, d_{s(v)}\}$, where $d_l\in\{2,4,\dots\}$. In both cases these collections of integers (called *orders of interior singularities*) satisfy the following compatibility conditions with the collection of boundary singularities of ${\mathbb G}_v$: $$-4\ \le\ \big( \sum d_l + \sum D_i \big) \ \equiv\ 0\mod 4\,,$$ where the first sum is taken over all interior singularities and the second sum is taken over all boundary components ${{\mathcal B}}_i$ of ${\mathbb G}_v$.
6. When the vertex $v$ is of the “$-$”-type the couple \[unordered collection of interior singularities, unordered collection of boundary singularities\] is in addition not allowed to belong to the following exceptional list:
$$
[|c|c|]{}
& \[,{2}\];;;\
& \[{1},{5}\];;; \[,{6}\];\
& \[,{2, 0}\];;;\
& \[{1},{0,1}\];;\
& \[{3,1},{2,0}\]; \[{3,1},{1,1}\]; \[{3},{3,0}\]; \[{3},{2,1}\]\
& \[{1},{5,0}\]; \[{1},{4,1}\]; \[{1},{3,2}\];; \[{4},{1,1}\]\
& \[,{6,0}\]; \[,{5,1}\];,{4,2}\]; \[,{3,3}\]\
& \[,{2,2}\]; \[,{1,3}\];;;\
& \[,{3,5}\];;;\
& \[,{2,6}\];\
$$
Singularity data corresponding to a configuration {#singularity-data-corresponding-to-a-configuration .unnumbered}
-------------------------------------------------
Any two flat surfaces realizing the same configuration ${{\mathcal C}}$ of ĥomologous saddle connections belong to the same stratum ${{\mathcal Q}}(\alpha)$ of quadratic differentials. The singularity data $\alpha$ are defined by the configuration ${{\mathcal C}}$ as follows.
First note that any configuration ${{\mathcal C}}$ determines a natural *global ribbon graph* ${\mathbb G}$ in the following way. We have defined a structure of a local ribbon graph for a small neighborhood $\Gamma_v$ of every vertex $v\in\Gamma$. For every vertex $v$ of $\Gamma$ we have a ribbon going along a germ of every edge of $\Gamma_v\subset\Gamma$ in direction from $v$ to the center of the edge. Note that all local ribbon graphs carry the canonical orientation induced from the canonical orientation of the embodying plane. For every edge of $\Gamma$ we can extend the ribbons from the endpoints towards the center of the edge and glue them together respecting the canonical orientation. Applying this procedure to all edges of $\Gamma$ we get a global ribbon graph endowed with the canonical orientation.
Consider the global ribbon graph ${\mathbb G}$ as a surface with boundary. The boundary components of this surface are in a one-to-one correspondence with the subset of those conical points of $S$ which serve as the endpoints of the saddle connections $\gamma_i$ in the collection $\gamma_1, \dots, \gamma_n$. The orders of the corresponding singularities are calculated as follows. For any connected component $(\partial{\mathbb G})_m$ of its boundary define an integer $b_m$ as $$\label{eq:order:of:a:newborn:singularity}
b_m = \sum_{\substack{\text{boundary singularities}\\
\text{which belong to } (\partial{\mathbb G})_m}} (k_{i,l}+1)\quad-2$$ The set with multiplicities $\alpha$ can be defined now as $$\label{eq:singularity:data:defined:by:configuration}
\alpha = \bigg(\bigcup_{\substack{\pm\text{-vertices}\\
v_j\in\Gamma({{\mathcal C}})}} \text{interior singularities of }v_j\bigg) \
\bigcup\ \bigg( \bigcup_{\substack{\text{components
}(\partial{\mathbb G})_m
\\\text{of the boundary}\\\text{of } {\mathbb G}({{\mathcal C}})}} b_m\bigg)$$
\[ex:example:of:a:configuration\] The configurations ${{\mathcal C}}$ presented in the left picture of figure \[fig:global:ribbon:graph\] has $8$ saddle connections $\gamma=\{\gamma_1\cup\dots\cup\gamma_{8}\}$; the surface $S\setminus\gamma$ decomposes into $7$ connected components $S_1\sqcup\dots\sqcup S_7$. Two components are represented by cylinders and thus have no interior singularities. Among the remaining five components three have no interior singularities and are denoted with ${\varnothing}$, one has one interior singularity of order $2$, and one has two interior singularities of orders $4$. Thus, we get $$\bigcup_{\substack{\pm\text{-vertices}\\
v_j\in\Gamma({{\mathcal C}})}}\text{interior singularities of }v_j =
\{2,4,4\}$$
The boundary of the global ribbon graph ${\mathbb G}$ has two components $(\partial{\mathbb G})_1$ and $(\partial{\mathbb G})_2$ which correspond to two conical singularities $P_1$ and $P_2$ of $S$. The saddle connections $\gamma_5, \gamma_6, \gamma_7$ join $P_2$ to itself; the other saddle connections join $P_1$ to itself. Turning counterclockwise around the point $P_l$, $l=1,2$, we see geodesic rays parallel to $\gamma_i$ appear in the same order as they appear when we follow the corresponding component ${{\mathcal B}}_l$ in the positive direction. Denoting by “$x$” the geodesic rays which do not belong to the configuration we get the following (cyclically ordered) list for the zero $P_2$: $$\gamma_5 x \gamma_6 x \gamma_7 \gamma_7 x \gamma_6 x \gamma_5$$ We have $10$ geodesic rays; this corresponds to the cone angle $10\pi$ matching our formula for the order $b_2$ of the zero $P_2$: $$(0+1)+(1+1)+(1+1)+(1+1)+(1+1)+(0+1)-2 =8\,,$$ The analogous list for $P_1$ is as follows $$xx\gamma_1 x \gamma_2 x \gamma_3 xxxxx \gamma_4 \gamma_4
xxxxxxxxx \gamma_3 \gamma_8 \gamma_8 x \gamma_2 xxx \gamma_1$$ The number of consecutive “$x$” coincides with the order of the corresponding boundary singularity (see definition \[def:order:of:boundary:singularity\]). Thus, at $P_1$ we find $32$ geodesic rays parallel to $\gamma_i$, which corresponds to the cone angle $32\pi$, and the order $b_1$ of $P_1$ equals to $$(2+1)+(1+1)+(1+1)+(5+1)+(0+1)+(9+1)+
(0+1)+(0+1)+(1+1)+(3+1)-2\!=\!30$$ Finally, we get the following set with multiplicities: $$\alpha=(2,4,4,8,30).$$ The surface $S$ has genus $g=13$; the configuration ${{\mathcal C}}$ represents the stratum ${{\mathcal Q}}(2,4,4,8,30)$. Note, that the picture on the right represents the same configuration as the picture of the left.
The example above gives an idea of how can one construct all configurations (in the sense of definition \[def:configuration\]) for a given stratum ${{\mathcal Q}}(\alpha)$ of meromorphic quadratic differentials with at most simple poles. This algorithm is discussed in more details in appendix \[a:List:of:configurations:in:genus:2\], where as an illustration we present a complete list of all configurations of ĥomologous saddle connections for holomorphic quadratic differentials in genus two.
(0,0)(197,-20) (24,-32)[$\scriptstyle 1$]{} (34,-43)[$\scriptstyle 3$]{} (85,-70)[$\scriptstyle 2$]{} (92,-74)[$\scriptstyle {\varnothing}$]{} (72,-78)[$\scriptstyle 1$]{} (68,-97)[$\scriptstyle 1$]{} (87,-97)[$\scriptstyle 0$]{} (10,-40)[$\scriptstyle \{2\}$]{} (75,-30)[${{\mathcal B}}_1$]{} (60,-45)[$\gamma_1$]{} (40,-70)[$\gamma_2$]{} (85,-116)[$\gamma_3$]{} (119,-95)[$\gamma_8$]{} (100,-152)[$\gamma_4$]{} (132,-157)[$\gamma_5$]{} (145,-140)[$\gamma_6$]{} (148,-121)[$\gamma_7$]{} (57,-135)[$\scriptstyle \{4,4\}$]{} (75,-145)[$\scriptstyle 5$]{} (88,-140)[$\scriptstyle 9$]{} (154,-107)[$\scriptstyle 0$]{} (163,-114)[$\scriptstyle 0$]{} (120,-165)[$\scriptstyle 0$]{} (132,-172)[$\scriptstyle 0$]{} (154,-132)[$\scriptstyle 1$]{} (169,-134)[$\scriptstyle 1$]{} (148,-151)[$\scriptstyle 1$]{} (156,-164)[$\scriptstyle 1$]{} (171,-125)[$\scriptstyle {\varnothing}$]{} (163,-156)[$\scriptstyle {\varnothing}$]{} (183,-142)[${{\mathcal B}}_2$]{}
(0,0)(0,-20) (26,-32)[$\scriptstyle 3$]{} (34,-43)[$\scriptstyle 1$]{} (87,-73)[$\scriptstyle 1$]{} (87,-97)[$\scriptstyle 0$]{} (72,-92)[$\scriptstyle 1$]{} (64,-78)[$\scriptstyle 2$]{} (95,-74)[$\scriptstyle {\varnothing}$]{} (10,-40)[$\scriptstyle \{2\}$]{} (60,-45)[$\gamma_2$]{} (40,-70)[$\gamma_1$]{} (85,-116)[$\gamma_3$]{} (119,-95)[$\gamma_8$]{} (96,-149)[$\gamma_4$]{} (115,-156)[$\gamma_5$]{} (132,-154)[$\gamma_6$]{} (144,-139)[$\gamma_7$]{} (57,-135)[$\scriptstyle \{4,4\}$]{} (75,-145)[$\scriptstyle 5$]{} (88,-140)[$\scriptstyle 9$]{} (159,-129)[$\scriptstyle 0$]{} (165,-139)[$\scriptstyle 0$]{} (98,-170)[$\scriptstyle 0$]{} (108,-163)[$\scriptstyle 0$]{} (146,-153)[$\scriptstyle 1$]{} (155,-165)[$\scriptstyle 1$]{} (127,-161)[$\scriptstyle 1$]{} (128,-176)[$\scriptstyle 1$]{} (162,-160)[$\scriptstyle {\varnothing}$]{} (133,-178)[$\scriptstyle {\varnothing}$]{}
Principal boundary {#principal-boundary .unnumbered}
------------------
Analogously to the case of Abelian differentials a configuration ${{\mathcal C}}$ of ĥomologous saddle connections determines the corresponding principal boundary stratum ${{\mathcal Q}}(\alpha'_{{{\mathcal C}}})$ or ${{\mathcal H}}(\beta'_{{{\mathcal C}}})$. Namely, to each boundary component ${{\mathcal B}}_i$ $$\{\to\gamma_{j_{i,1}}\to \dots \to\gamma_{j_{i,p(i)}}\to\}$$ of every “$+$” or “$-$”-vertex $v_j$ of the graph $\Gamma({{\mathcal C}})$ (i.e. to each connected component of the corresponding local ribbon graph ${\mathbb G}_j$) we assign a number $$\label{eq:Dji}
D_{j_i}=k_{j_{i,1}}+\dots+k_{j_{i,p(i)}}-2,$$ where $k_{j_{i,1}}, \dots, k_{j_{i,p(i)}}$ are the orders of the boundary singularities corresponding to this boundary component. By lemma \[lm:sum:of:boundary:singularities:is:even\] proved in the beginning of section \[s:hat:homologous:saddle:connections\] the number $D_{j_i}$ is always a nonnegative even integer whenever $v_j$ is a “$+$”-vertex. To every “$+$”-vertex $v_j$ of the graph $\Gamma({{\mathcal C}})$ we assign the stratum $$\label{eq:H:boundary:stratum}
{{\mathcal H}}(\beta'_j)={{\mathcal H}}\left(\frac{d_1}{2}, \dots, \frac{d_{s(j)}}{2}, \frac{D_1}{2}, \dots, \frac{D_{r(j)}}{2}\right)$$ of holomorphic Abelian differentials, where $d_1, \dots,
d_{s(j)}$ are the orders of interior singularities of $v_j$. Note that conditions (4) and (5) in definition \[def:configuration\] of a configuration of ĥomologous saddle connections imply that the entries of $\beta'_j$ are integers and that their sum is even, so the stratum ${{\mathcal H}}(\beta'_j)$ is nonempty.
We assign to a “$-$”-vertex $v_j$ the stratum $$\label{eq:Q:boundary:stratum}
{{\mathcal Q}}(\alpha'_j)={{\mathcal Q}}(d_1, \dots, d_{s(j)}, D_1, \dots, D_{r(j)})$$ of meromorphic quadratic differentials with at most simple poles, where $d_1, \dots, d_{s(j)}$ are the orders of interior singularities of $v_j$. Note that condition (5) in definition \[def:configuration\] of a configuration of ĥomologous saddle connections guarantees that the sum of entries of $\alpha'_j$ defined above equals $0$ modulo $4$, while condition (6) guarantees that $\alpha'\not\in\big\{({\varnothing},\{-1,1\},\{3,1\},\{4\}\big\}$, which implies that the stratum ${{\mathcal Q}}(\alpha'_j)$ is nonempty.
Given a configuration ${{\mathcal C}}$ we assign to every “$\pm$”-vertex of the graph $\Gamma$ the corresponding stratum. When $\Gamma$ does not contain “$-$” vertices we get a stratum ${{\mathcal H}}(\beta'_{{\mathcal C}})$ of disconnected translation surfaces $S'_1\sqcup\dots\sqcup S'_k$, where $S'_j\in{{\mathcal H}}(\beta'_j)$, $j=1,\dots,k$. Otherwise we get a stratum ${{\mathcal Q}}(\alpha'_{{\mathcal C}})$ of disconnected flat surfaces $S'_1\sqcup\dots\sqcup S'_k$, where $S'_j\in{{\mathcal H}}(\beta'_j)$ when $S'_j$ is represented by a “$+$”-vertex and $S'_j\in{{\mathcal Q}}(\alpha'_j)$ when $S'_j$ is represented by a “$-$”-vertex. The resulting stratum is called the [*principal boundary stratum*]{} corresponding to the admissible configuration ${{\mathcal C}}$.
\[ex:principal:boundary\] Let us compute the principal boundary stratum corresponding to the configuration from example \[ex:example:of:a:configuration\], see figure \[fig:global:ribbon:graph\]. The components represented by cylinders, encoded by $\circ$-vertices do not contribute to the principal boundary: they shrink and disappear. The vertex $v_1$ of valence four has type $+4.2c$, see figure \[fig:embedded:local:ribbon:graphs\]; the corresponding local ribbon graph ${\mathbb G}_{v_1}$ has two connected components, $r(1)=2$, which correspond to two connected components ${{\mathcal B}}_1,
{{\mathcal B}}_2$ of the boundary $\partial S^{comp}_1$. The corresponding zeroes of the induced Abelian differential on $S'_1$ are calculated in terms of $D_1=2-2=0$ and $D_2=1+0+1-2=0$, see . Since $S^{comp}_1$ does not have interior singularities, the corresponding closed flat surface $S'_1$ is a torus with two marked points, $S'_1\in{{\mathcal H}}(0,0)$, see .
The remaining four vertices of $\Gamma(S,\gamma)$ have type $+2.1$; the boundary of each of the corresponding components $S_2,\dots,S_4$ is connected. Applying formulae and we get the following list of surfaces $S'_j$: $$S'_2\in{{\mathcal H}}\left(\frac{2}{2},\ \frac{3+1-2}{2},\right)\quad
S'_3\in{{\mathcal H}}\left(\frac{4}{2},\ \frac{4}{2}, \frac{5+9-2}{2},\right)\quad
S'_4,
S'_5\in{{\mathcal H}}\left(\frac{1+1-2}{2}\right)$$ The corresponding principal boundary stratum is $${{\mathcal H}}(0,0)\sqcup{{\mathcal H}}(1,1)\sqcup{{\mathcal H}}(6,2,2)\sqcup{{\mathcal H}}(0)\sqcup{{\mathcal H}}(0)$$
Main Theorems {#main-theorems .unnumbered}
-------------
In sections \[s:Local:Constructions\] and \[s:Nonlocal:constructions\] we describe some basic surgeries which depend continuously on a small complex parameter $\delta\in{{\mathbb C}^{}}$ (responsible for the length and direction of the saddle connections which form the boundary) and on an additional discrete parameter having finitely many values. The theorem below makes a bridge between the formal combinatorial constructions discussed above and the geometry of the moduli spaces of quadratic differentials and is proved in those sections.
We denote by ${{\mathcal Q}}_1^{\varepsilon}(\alpha)\subset{{\mathcal Q}}_1(\alpha)$ the subset of those flat surfaces of area one, which have at least one saddle connection of length at most $\varepsilon$.
\[th:from:boundary:to:neighborhood:of:the:cusp\] For each configuration ${{\mathcal C}}$ of ĥomologous saddle connections as in definition \[def:configuration\], let $\Gamma$ be the graph of connected components corresponding to this configuration. Let ${{\mathcal Q}}(\alpha'_{{\mathcal C}})$ (or ${{\mathcal H}}(\beta'_{{\mathcal C}})$) be the boundary stratum corresponding to the configuration ${{\mathcal C}}$.
For any flat surface $S'\in {{\mathcal Q}}(\alpha'_{{\mathcal C}})$ (correspondingly in ${{\mathcal H}}(\beta'_{{\mathcal C}})$), and any sufficiently small value of the complex parameter $\delta$, if one applies the basic surgeries to the connected components of $S'$ and assembles a closed surface $S$ from the resulting surfaces with boundary according to the structure of the graph $\Gamma({{\mathcal C}})$, then the result is a surface in ${{\mathcal Q}}^{\varepsilon}(\alpha)$.
Similar to the case of Abelian differentials, we denote by ${{\mathcal Q}}_1^{\varepsilon,thick}(\alpha)\subset{{\mathcal Q}}_1(\alpha)$ the subset of those flat surfaces of area one, which have a collection of ĥomologous saddle connections of length at most $\varepsilon$ and no other short saddle connection. Here “short” means, of length less than $\lambda {\varepsilon}^r$ for some parameters $\lambda\geq 1$ and $0< r\leq 1$, where the values of the parameters depend on the stratum. Then one can show that any surface in ${{\mathcal Q}}_1^{\varepsilon,thick}(\alpha)$ can be obtained by this construction. We will not prove this statement in order not to overload this paper.
We put theorem \[th:graphs\], theorem \[th:all:local:ribon:graphs\] and theorem \[th:from:boundary:to:neighborhood:of:the:cusp\] together in one statement which may be considered as our main theorem.
We say that a collection $\gamma$ of ĥomologous saddle connections $\{\gamma_1, \dots, \gamma_n\}$ on a flat surface $S\in{{\mathcal Q}}(\alpha)$ is *in general position* if there are no other saddle connections on $S$ parallel to saddle connections in the collection $\gamma$. It follows from proposition \[pr:homologous:equiv:parallel\] stated in the end of section \[s:hat:homologous:saddle:connections\] that for almost all flat surfaces in any stratum any collection of ĥomologous saddle connections is in general position. This implies, that we can always put a collection of ĥomologous saddle connections in general position by an arbitrary small deformation of the flat surface inside the stratum.
Any collection $\gamma$ of ĥomologous saddle connections $\{\gamma_1, \dots, \gamma_n\}$ in general position on a flat surface $S\in{{\mathcal Q}}(\alpha)$ naturally defines a corresponding configuration ${{\mathcal C}}(S,\gamma)$.
Any “formal” configuration of ĥomologous saddle connections as in definition \[def:configuration\] corresponds to some actual collection of ĥomologous saddle connections on an appropriate flat surface.
By theorem \[th:graphs\] any collection $\gamma$ of ĥomologous saddle connections $\{\gamma_1, \dots,\!\gamma_n\}$ on a flat surface $S\in{{\mathcal Q}}(\alpha)$ naturally defines a graph of connected components $\Gamma(S,\gamma)$ (structure 1 of a configuration). According to theorem \[th:all:local:ribon:graphs\], for every vertex $v$ of $\Gamma(S,\gamma)$ the collection $\gamma$ also defines a local ribbon graph (structure 2 of a configuration) as well as the orders $d_l$ and $k_{i,l}$ of all interior and boundary singularities. By theorem \[th:all:local:ribon:graphs\], for vertices of “$+$”-type, the orders $k_{i,l}$ of the boundary singularities are compatible with the corresponding parities (structures 3 and 4 of a configuration). The lower bounds for the sums $D_i$ of orders of boundary singularities follow from lemma \[lm:lower:bounds:for:D:i\]. The necessary condition of the compatibility of the orders of interior singularities with the orders of boundary singularities formalized as structure 5 is proved in lemma \[lm:sum:of:boundary:singularities:minus:vertex\]. The list of nonrealizable singularity data for the vertices of the “$-$”-types presented in structure 6 of a configuration is justified in lemma \[lm:nonrealizable:data\] at the end of section \[s:Nonlocal:constructions\]. This completes the proof of the first part of the statement.
The realizability of all formal configurations immediately follows from theorem \[th:from:boundary:to:neighborhood:of:the:cusp\].
Appendices. Long saddle connections {#appendices.-long-saddle-connections .unnumbered}
-----------------------------------
In appendix \[ap:Long:saddle:connections\] we study collections of ĥomologous saddle connections when they are not necessarily short.
The next proposition follows immediately from definition \[def:homologous\] and the notion of configuration.
Let $\gamma(S_0)=\{\gamma_1, \dots, \gamma_n\}$ be a collection of ĥomologous saddle connections on a flat surface $S_0$ in ${{\mathcal Q}}(\alpha)$. Let a flat surface $S$ be obtained by a sufficiently small continuous deformation of $S_0$ in ${{\mathcal Q}}(\alpha)$ and $\gamma(S)$ the corresponding collection of saddle connections. Then all saddle connections in the collection $\gamma(S)$ are ĥomologous. The configuration ${{\mathcal C}}(S,\gamma(S))$ defined by the collection $\gamma(S)$ of ĥomologous saddle connections on $S$ coincides with the initial configuration ${{\mathcal C}}(S_0,\gamma(S_0))$.
By definition, a configuration ${{\mathcal C}}$ of ĥomologous saddle connections is admissible for a given connected component ${{\mathcal Q}}^{c}(\alpha)$ of the stratum ${{\mathcal Q}}(\alpha)$ if there is *at least one* flat surface $S_0\in{{\mathcal Q}}^{c}(\alpha)$ and at least one collection $\gamma$ of ĥomologous saddle connections $\gamma=\{\gamma_1,\dots,\gamma_n\}$ on $S_0$ realizing ${{\mathcal C}}$. Consider any surface $S$ in the same connected component ${{\mathcal Q}}^{c}(\alpha)$. By $N_{{\mathcal C}}(S,L)$ denote the number of collections $\gamma=\{\gamma_1,\dots,\gamma_n\}$ of ĥomologous saddle connections on $S$ defining the same configuration ${{\mathcal C}}(S,\gamma)={{\mathcal C}}$ and such that $\max_{1\le i\le n}
|\gamma_i|\le L$. The results in [@Eskin:Masur] imply the following statement proved in the appendix.
\[pr:counting\] For almost every flat surface $S$ in the connected component ${{\mathcal Q}}^{c}(\alpha)$ containing $S_0$ the following limit exists $$\lim_{L\to +\infty} \cfrac{N_{{\mathcal C}}(S,L)}{L^2}=c_{{\mathcal C}}(S)$$ and is strictly positive. Moreover, for almost all surfaces $S$ in ${{\mathcal Q}}^{c}(\alpha)$ this limit is the same, $c_{{\mathcal C}}(S)=const_{{\mathcal C}}$. (This limit is called Siegel–Veech constant.)
In particular, any admissible configuration is presented on almost every flat surface in the corresponding connected component of the stratum by numerous collections of ĥomologous saddle connections.
Final comments, open problems, applications {#final-comments-open-problems-applications .unnumbered}
-------------------------------------------
The thick part ${{\mathcal Q}}_1^{\varepsilon,thick}(\alpha)$ decomposes into a disjoint union $${{\mathcal Q}}_1^{\varepsilon,thick}(\alpha)=
\bigsqcup_{{\text{configurations }{{\mathcal C}}}} {{\mathcal Q}}_1^{\varepsilon}(\alpha,{{\mathcal C}})$$ of (not necessarily connected) components corresponding to admissible configurations; the surfaces in any such component of ${{\mathcal Q}}_1^{\varepsilon}(\alpha,{{\mathcal C}})$ share the same configuration ${{\mathcal C}}$ of ĥomologous saddle connections. Following the lines of the paper [@Eskin:Masur:Zorich] one could extend theorem \[th:from:boundary:to:neighborhood:of:the:cusp\] and prove that up to a defect of a very small measure, for every configuration ${{\mathcal C}}$ there is an integer $M({{\mathcal C}})$ such that ${{\mathcal Q}}_1^{\varepsilon}(\alpha,{{\mathcal C}})$ is a (ramified) covering of order $M({{\mathcal C}})$ over the following space. The space is a fiber bundle over the boundary stratum ${{\mathcal Q}}_1(\alpha'_{{\mathcal C}})$ (correspondingly ${{\mathcal H}}_1(\beta'_{{\mathcal C}})$). It has a Euclidean $\varepsilon$-disc as a fiber when ${{\mathcal C}}$ does not contain cylinders, and the space ${{\mathcal H}}_1^\varepsilon(0,\dots,0)$ when ${{\mathcal C}}$ contains cylinders (number of marked points on the torus equals the number of cylinders). In both cases it is easy to express the measure on ${{\mathcal Q}}_1^{\varepsilon}(\alpha,{{\mathcal C}})$ in terms of the product measure on the fiber bundle, and compute the volume of ${{\mathcal Q}}_1^{\varepsilon}(\alpha,{{\mathcal C}})$ in terms of volumes of the strata, and using the Siegel—Veech formula compute the constants $c_{{\mathcal C}}$.
However, the evaluation of the constants $M$ (which depend on the configuration ${{\mathcal C}}$) requires some additional work. In particular, if the corresponding surgeries (see theorem \[th:from:boundary:to:neighborhood:of:the:cusp\]) are nonlocal (i.e. those, which use a path on a surface, see section \[s:Nonlocal:constructions\]) one needs to study the dependence of the resulting surface on the homotopy type of the path. These and related issues will be discussed in the forthcoming paper [@Boissy:in:progress].
Another subject which we do not discuss in this paper is the individual study of the connected components of the strata of quadratic differentials: different connected components of the same stratum ${{\mathcal Q}}(\alpha)$ have their individual lists of admissible configurations, graphs, boundary strata, etc. In particular, one can use the lists of admissible configurations to determine the connected component to which a given flat surface belongs. For example, a saddle connection joining the zero and the simple pole on any flat surface from the component ${{\mathcal Q}}^{ir}(9,-1)$ has a ĥomologous saddle connection joining the zero to itself, while analogous saddle connections on surfaces from the complementary connected component ${{\mathcal Q}}^{reg}(9,-1)$ may have multiplicity one. The existing invariant called the *Rauzy class* used to distinguish these components is rather complicated, see [@Lanneau]. Configurations of ĥomologous saddle connections for nonconnected strata will be studied in the papers [@Boissy:to:appear] and [@Boissy:in:progress].
Given a billiard in a rational polygon $\Pi$, one can build a translation surface $\hat S$ from an appropriate number $2N$ of copies of $\Pi$ such that geodesics on $S$ will project to the billiard trajectories in $\Pi$. Taking $N$ copies instead of $2N$ one obtains a flat surface with ${{\mathbb Z}/2{\mathbb Z}}$-holonomy with the same properties of geodesics. In some cases this latter construction is more advantageous. In the paper [@Athreya:Eskin:Zorich] there is the study of billiards in polygons whose angles are multiples of $\pi/2$. Identifying two copies of such polygons by their boundaries one obtains a flat surface corresponding to a meromorphic quadratic differential on ${{\mathbb C}^{}}P^1$ with at most simple poles. The results of this paper are used to classify closed billiard trajectories and generalized diagonals in the paper [@Athreya:Eskin:Zorich], see also [@Boissy:to:appear].
Acknowledgements
Conceptually this paper is a continuation of the paper [@Eskin:Masur:Zorich]. We want to thank A. Eskin for his participation in the early stage of this project. We thank C. Boissy and E. Lanneau for several helpful conversations concerning nonconnected strata. The first author thanks the University of Rennes for its support and hospitality during the preparation of this paper. The second author thanks the University of Chicago, UIC, Max-Planck-Institut für Mathematik at Bonn and IHES for hospitality and support during the preparation of this paper.
Preliminaries on flat surfaces and on ĥomologous saddle connections {#s:hat:homologous:saddle:connections}
===================================================================
In this section of preliminary results we describe geometric criteria for deciding when two saddle connections are ĥomologous and describe the structure of the complement $S\setminus(\gamma_1\cup\gamma_2)$. The key result in this section is proposition \[pr:homologous:saddle:connections\].
In the case of a translation surface $S$ it is obvious that two saddle connections $\gamma_1,\gamma_2$ are homologous if and only if $S\setminus (\gamma_1\cup\gamma_2)$ is disconnected (provided $S\setminus\gamma_1$ and $S\setminus\gamma_2$ are connected). It is less obvious to check whether saddle connections $\gamma_1,
\gamma_2$ on a flat surface $S$ with nontrivial linear holonomy are ĥomologous or not. In particular, a pair of closed saddle connections might be homologous in the usual sense, but not ĥomologous; a pair of closed saddle connections might be ĥomologous even if one of them represents a loop homologous to zero, and the other does not; finally, a saddle connection joining a pair of [*distinct*]{} singularities might be ĥomologous to a saddle connection joining a singularity to itself. The flat torus described in in the introduction gives an example of these phenomena (see example \[ex:homologous:sad:connections\] and figure \[fig:examples:of:homologous:sad:connections:1\]).
We start this section with several lemmas establishing some restrictions on the orders of singularities of a flat surface with boundary. By convention we consider only those flat structures which have linear holonomy in $\{Id, -Id\}$. Throughout this paper we assume that the boundary components of any flat surface with boundary are made up of parallel saddle connections, unless otherwise noted. We also assume that a sufficiently small collar neighborhood of any boundary component is a topological annulus (or, in the other words, that the natural compactification of $S\setminus \partial S$ coincides with $S$).
\[lm:sum:of:boundary:singularities:is:even\] If a flat surface $S_j$ with boundary has trivial linear holonomy, then the sum of the orders of the boundary singularities along each boundary component is even: $$k_{j_{i,1}}+ \dots + k_{j_{i,p(i)}} \equiv 0 \mod 2$$
Take a loop following the $i$-th boundary component $$\{\to\gamma_{j_{i,1}}\to \dots \to\gamma_{j_{i,p(i)}}\to\}$$ at a sufficiently small constant distance. Recall that by definition \[def:order:of:boundary:singularity\] of the order of a boundary singularity, the angle between the saddle connection $\gamma_{j_{i,l}}$ and the saddle connection $\gamma_{j_{i,l+1}}$ at the boundary singularity $\gamma_{j_{i,l}}\to\gamma_{j_{i,l+1}}$ equals $(k_{j_{i,l}}+1)\pi$. Thus, the linear holonomy around the loop is trivial if and only if the total sum of the angles $k_{j_{i,1}}\pi+\dots+ k_{j_{i,p(i)}}\pi$ is an integer multiple of $2\pi$, or, equivalently, if and only if the sum $k_{j_{i,1}}+\dots+ k_{j_{i,p(i)}}$ of the orders of the boundary singularities is even.
\[lm:sum:of:boundary:singularities:minus:vertex\] Let $d_{j_l},k_{j_{i,l}}$ denote the orders of correspondingly interior singularities and boundary singularities of a flat surface with boundary $S_j$. Then $$2 r(S_j)-4\le\
\sum d_{j_l} + \sum k_{j_{i,l}} \ \equiv\ 2 r(S_j)\mod 4,$$ where $r(S_j)$ is the number of boundary components, the first sum is taken over all interior singularities and the second sum is taken over all boundary singularities.
Consider one more copy of the surface $S_j$ taken with the opposite orientation. We can naturally identify these two copies along the common boundary. It follows from our assumptions on $S_j$ that the resulting surface $S$ is a nonsingular oriented closed flat surface without boundary. In other words, the closed flat surface $S$ corresponds to a meromorphic quadratic differential on a Riemann surface.
Each interior singularity of $S_j$ of order $d_{j_l}$ produces two distinct singularities of $S$ of order $d_i$. Each boundary singularity of $S_j$ of order $k_{j_{i,l}}$ gives rise to an interior singularity of $S$ of order $2k_{j_{i,l}}$. The surface $S$ has genus $\hat{g}=2g+r(S_j)-1$. Now recall that for any quadratic differential on a closed Riemann surface $S$ of genus $\hat{g}$ the sum of orders of singularities equals $4\hat{g}-4$. Hence, $$2\big(\sum_{\substack{interior\\singularities\\of\ S_j}} d_l +
\sum_{\substack{boundary\\singularities\\of\ S_j}} k_{j_{i,l}} \big) =
4\hat{g}-4=4(2g+r(S_j)-1)-4=8(g-1)+4r(S_j)\,$$ which implies the desired relation.
\[lm:lower:bounds:for:D:i\] The sum $D_i+2=k_{i,1}+\dots+k_{i,p(i)}$ of orders of boundary singularities along some boundary component ${{\mathcal B}}_i$ of a flat surface $S$ is equal to zero if and only if a sufficiently narrow collar neighborhood of ${{\mathcal B}}_i$ in $S$ is isometric to a flat cylinder.
When $S$ has trivial linear holonomy and the sum $D_i+2$ of orders of boundary singularities along a boundary component ${{\mathcal B}}_i$ is strictly positive, then $D_i\ge 0$.
By definition \[def:order:of:boundary:singularity\] the order $k_{i,l}$ of any boundary singularity is nonnegative. Thus, $D_i+2$ is equal to zero if and only if the orders of all boundary singularities along the boundary component ${{\mathcal B}}_i$ are equal to zero, which implies the first part of the statement.
The second statement is an obvious corollary of the first one combined with lemma \[lm:sum:of:boundary:singularities:is:even\].
\[lm:total:boundary:holonomy:is:trivial\] Let $\beta$ denote the total boundary of a translation surface defined by a holomorphic 1-form $\omega$. Then
- $\int_\beta\omega=0$.
- $\beta$ cannot consist of a single saddle connection.
- If $\beta$ is composed of exactly two saddle connections $\gamma_1,\gamma_2$ then $\gamma_1,\gamma_2$ are parallel and have equal length. Moreover, the oriented surface obtained by isometric identification of $\gamma_1$ and $\gamma_2$ is a translation surface (i.e. it is a closed flat surface with trivial linear holonomy).
Note that the canonical orientation of the surface induces a canonical orientation of the boundary $\beta$. Thus, the first statement is an immediate consequence of Stokes formula. The second statement follows from the first since the holonomy $\int_\gamma\omega$ along a saddle connection $\gamma$ cannot be $0$.
For the third let $\beta=\gamma_1-\gamma_2$. Then $\int_{\gamma_1}\omega=\int_{\gamma_2}\omega$. This implies that $\gamma_1,\gamma_2$ are parallel, have equal length and that their directions defined by the chosen orientations are compatible with linear holonomy. We can isometrically identify $\gamma_1$ either with $\gamma_2$ or with $-\gamma_2$. However, the second identification produces a nonorientable surface, so $\gamma_1$ must be identified with $\gamma_2$ which implies that the resulting surface is a translation surface.
Let $S$ be a flat surface with boundary; let $\gamma_1, \gamma_2$ be a pair of parallel saddle connections $\gamma_1, \gamma_2$ of equal length at the boundary of $S$. By convention, throughout this paper we always identify $\gamma_1$ and $\gamma_2$ in such way that the resulting flat surface is orientable.
Suppose that, moreover, $S$ has trivial linear holonomy.
\[def:flip\] We say that $\gamma_1$ and $\gamma_2$ are identified by a *translation* if the resulting flat surface has trivial linear holonomy; otherwise we say that $\gamma_1$ and $\gamma_2$ are identified by a *flip*.
\[lm:trivial:plus:trivial:is:trivial\] Assume that a flat surface $S$ with nontrivial linear holonomy is divided by a pair of parallel saddle connections $\gamma_1,\gamma_2$ into two connected components $S_1, S_2$. Then at least one of the components must have nontrivial linear holonomy.
If one of the $\gamma_1, \gamma_2$ is a closed curve homologous to zero, say $\gamma_1$, then $\gamma_2$ lies in one of the two components of the complement $S\setminus\gamma_1$. Then, the boundary of the other component, say, $S_1$ consists solely of $\gamma_1$, $\partial S_1=\gamma_1$. By property 2 of lemma \[lm:total:boundary:holonomy:is:trivial\] the component $S_1$ has nontrivial linear holonomy.
Therefore, we may assume that $\gamma_1$ and $\gamma_2$ are not homologous to zero so they are homologous to each other. Choosing appropriate orientations of $\gamma_1$ and $\gamma_2$ we get $$\partial S_1 = \gamma_1 - \gamma_2 \qquad
\partial S_2 = -\gamma_1 + \gamma_2$$ where the orientations of $S_1, S_2$ are induced from the canonical orientation of $S$. If both $S_1$ and $S_2$ have trivial linear holonomy we can choose the defining holomorphic 1-forms $\omega_1,\omega_2$ on $S_1$ and $S_2$ in such way that $$\int_{\gamma_1}\omega_1 = \int_{\gamma_1}\omega_2=
\int_{\gamma_2}\omega_1 = \int_{\gamma_2}\omega_2.$$ The latter relations imply the compatibility of $\omega_1$ and $\omega_2$ on $S$. Thus, the flat structure on $S$ can be defined by a holomorphic 1-form $\omega$ such that $\omega|_{S_1}=\omega_1, \omega|_{S_2}=\omega_2$, and $S$ has trivial linear holonomy contrary to the initial assumption.
\[lm:gamma:2gamma\] Any two ĥomologous saddle connections $\gamma_1, \gamma_2$ on a flat surface $S$ are parallel. When both relations $[\gamma'_1]=-[\gamma''_1]$ and $[\gamma'_2]=-[\gamma''_2]$ are simultaneously valid or simultaneously not valid the saddle connections $\gamma_1, \gamma_2$ have the same length, $|\gamma_1|=|\gamma_2|$. When one of the relations, say, $[\gamma'_1]=-[\gamma''_1]$, is valid while the other one is not, $[\gamma'_2]\neq-[\gamma''_2]$, the lengths differ by the factor of two, $|\gamma_1|=2|\gamma_2|$.
The proof is a straightforward corollary of definition \[def:homologous\] and the fact that the length of a saddle connection $\delta$ on the translation surface $\hat
S$ is defined as $|\delta|=|\int_\delta \omega|$ and its direction is defined by the argument of $\int_\delta \omega$.
\[lm:antiinvariant:cycle\] Let $\gamma$ be a saddle connection on a flat surface $S$ having nontrivial linear holonomy. The following properties are equivalent
- $[\gamma']=-[\gamma'']$ in $H_1(\hat{S},\hat P;\,{\mathbb Z})$;
- $\hat S\setminus (\gamma'\cup\gamma'')$ contains two connected components;
- $[\gamma]=0$ in $H_1(S,P;\,{\mathbb Z})$.
(a)$\Rightarrow$(c). Consider the map $p_\ast:
H_1(\hat{S},\hat P;\,{\mathbb Z})\to H_1(S,P;\,{\mathbb Z})$ induced by the covering $p$. By definition of $\gamma',\gamma''$ we have $[\gamma]=p_\ast[\gamma']=p_\ast[\gamma'']$. Thus, when $[\gamma']=-[\gamma'']$ we get $[\gamma]=-[\gamma]$, so $[\gamma]=0$.
(c)$\Rightarrow$(b). Since $[\gamma]=0$, $S\setminus\gamma$ contains two connected components $S_1, S_2$, such that $\partial
S_1=\gamma$, $\partial S_2 = -\gamma$. By property 2 of lemma \[lm:total:boundary:holonomy:is:trivial\] both $S_1$ and $S_2$ have nontrivial linear holonomy, which implies that both $\hat S_1=p^{-1}(S_1)$, $\hat S_2=p^{-1}(S_2)$, are connected. Thus, $\hat S\setminus (\gamma'\cup\gamma'')=\hat S_1\sqcup \hat
S_2$ contains two connected components.
(b)$\Rightarrow$(a). Since $\gamma'$ and $\gamma''$ are symmetric, the two connected components $\hat S', \hat S'$ of $\hat S\setminus (\gamma'\cup\gamma'')$ are also symmetric with respect to the involution. This restricts the possible situations to the following three (up to an interchange of the superscripts of $\hat S_1', \hat S_1''$ if necessary):
— either $\partial \hat S'$ is composed of two copies of $\gamma'$ and $\partial \hat S''$ from two copies of $\gamma''$;
— or $\partial \hat S' = \gamma'-\gamma''$ and $\partial \hat
S'' = \gamma''-\gamma'$;
— or $\partial \hat S' = \gamma'+\gamma''$ and $\partial \hat
S'' = -\gamma''-\gamma'$.
The first situation implies that $\hat S$ contains two connected components which contradicts the assumptions that $S$ has nontrivial linear holonomy. Hence, the first situation is excluded. The second situation implies that isometrically identifying the boundary components $\gamma'$ and $\gamma''$ of $S'$ we obtain a flat surface isometric to $S$. By property 3 of lemma \[lm:total:boundary:holonomy:is:trivial\] this again implies that $S$ has trivial linear holonomy which contradicts the assumptions. This case is also excluded. In the only remaining case we have $\partial \hat S' = \gamma'+\gamma''$ which implies $[\gamma']=-[\gamma'']$.
The next proposition is the key to the proofs of theorems \[th:unique:trivial:holonomy\] and \[th:graphs\]. We do not assume that the saddle connections in the proposition below are parallel.
\[pr:homologous:saddle:connections\] Two saddle connections $\gamma_1,\gamma_2$ on a flat surface $S$ are ĥomologous if and only if they have no interior intersections and one of the following holds
1. The union $\gamma_1\cup\gamma_2$ does not separate the surface $S$ and the complement $S\setminus\{\gamma_1\cup\gamma_2\}$ has trivial linear holonomy. (In this case $|\gamma_1|=|\gamma_2|$; all combinations: loop — loop, loop — segment, segment — segment are possible.)
2. The union $\gamma_1\cup\gamma_2$ separates $S$; neither $\gamma_1$ nor $\gamma_2$ by itself separates; the complement $S\setminus\{\gamma_1\cup\gamma_2\}$ has two connected components, one of them has trivial linear holonomy, the other — nontrivial. (In this case $|\gamma_1|=|\gamma_2|$; the saddle connections are either two segments or two loops.)
3. One of $\gamma_1, \gamma_2$, say, $\gamma_1$ separates $S$, the other one does not; the complement $S\setminus\{\gamma_1\cup\gamma_2\}$ has two connected components, one of them has trivial linear holonomy, the other one, whose boundary is $\gamma_1$, has nontrivial holonomy. (In this case $|\gamma_1|=2|\gamma_2|$; the separating saddle connection $\gamma_1$ is a loop, $\gamma_2$ might be a segment or a loop.)
4. Both $\gamma_1$ and $\gamma_2$ separate $S$; the complement $S\setminus\{\gamma_1\cup\gamma_2\}$ has three connected components; two of which have nontrivial linear holonomy, while the remaining one, whose boundary is $\gamma_1\cup\gamma_2$, has trivial linear holonomy. (In this case $|\gamma_1|=|\gamma_2|$; both $\gamma_1$ and $\gamma_2$ are loops.)
According to lemma \[lm:gamma:2gamma\] ĥomologous saddle connections are parallel. If two ĥomologous saddle connections $\gamma_1$ and $\gamma_2$ have a common point, this point is an endpoint for both $\gamma_1$ and $\gamma_2$. Thus, from now on we can assume that $\gamma_1,\gamma_2$ have no interior intersections.
Two saddle connections $\gamma_1,\gamma_2$ without interior intersections subdivide a flat surface $S$ in one of the following ways:
- The union $\gamma_1\cup\gamma_2$ does not separate the surface $S$.
- The union $\gamma_1\cup\gamma_2$ separates $S$; neither $\gamma_1$ nor $\gamma_2$ by itself separates.
- One of $\gamma_1, \gamma_2$, say, $\gamma_1$ separates $S$, the other one does not.
- Both $\gamma_1$ and $\gamma_2$ separate $S$.
For each of these cases we prove that the additional assumption that $\gamma_1$ and $\gamma_2$ are ĥomologous is equivalent to the corresponding additional assumptions (1)–(4) on triviality of linear holonomy of the corresponding components. In each case we use lemmas \[lm:gamma:2gamma\] and \[lm:antiinvariant:cycle\] to determine the corresponding relation between the lengths $|\gamma_1|$ and $|\gamma_2|$. We combine this information with lemma \[lm:antiinvariant:cycle\] (when appropriate) to prove that one of $\gamma_1, \gamma_2$ (or both $\gamma_1$ and $\gamma_2$) is a closed cycle. The proof of realizability of combinations loop — loop, loop — segment, segment — segment indicated in proposition \[pr:homologous:saddle:connections\] is left to the reader as an exercise. Note that example \[ex:homologous:sad:connections\] already proves realizability of combinations loop—segment in (1) and loop—loop in (2). The remaining combinations can be found in sections \[s:Local:Constructions\] and \[s:Nonlocal:constructions\].
Let $X\subseteq S$ be a subset of $S$. By $\hat X$ we denote the preimage $\hat X= p^{-1}(X)$. Let $S_j$ be a connected component of $S\setminus (\gamma_1\cup\gamma_2)$. We use the following obvious criterion: $S_j$ has nontrivial linear holonomy if and only if the preimage $\hat S_j$ is connected. Now let us pass to consideration of cases (i)–(iv).
\(i) In this case $S\setminus (\gamma_1\cup\gamma_2)$ is connected; denote it by $S_1$. Neither of $\gamma_1,\gamma_2$ is homologous to zero, so $[\hat\gamma_1]=[\gamma_1']-[\gamma_1'']$, and $[\hat\gamma_2]=[\gamma_2']-[\gamma_2'']$ (see lemma \[lm:antiinvariant:cycle\]). By lemma \[lm:gamma:2gamma\] when such $\gamma_1$ and $\gamma_2$ are ĥomologous, we have $|\gamma_1|=|\gamma_2|$.
If the saddle connections $\gamma_1$ and $\gamma_2$ are ĥomologous then the cycle $[\gamma_1']-[\gamma_1'']$ is homologous (in the usual sense) to one of the $\pm([\gamma_2']-[\gamma_2''])$ which means that $\hat
S_1=\hat{S}\setminus
(\gamma_1'\cup\gamma''_1\cup\gamma'_2\cup\gamma''_2)$ is not connected. Hence, by the above criterion $S_1$ has trivial linear holonomy.
Suppose now that $S_1=S\setminus (\gamma_1\cup\gamma_2)$ has trivial linear holonomy. Then $\hat S_1$ has two connected components $\hat S_1'$ and $\hat S_1''$. Note, that the flat surface $S$ has nontrivial linear holonomy. Hence, it follows from property 3 in lemma \[lm:total:boundary:holonomy:is:trivial\] that both $S\setminus\gamma_1$ and $S\setminus\gamma_2$ have nontrivial linear holonomy. This implies that there exist a pair of loops $\rho_1, \rho_2$ on $S$ such that $\rho_i$ and $\gamma_i$ have single transversal intersection, $i=1,2$; such that $\rho_1\cap\gamma_2={\varnothing}$, $\rho_2\cap\gamma_1={\varnothing}$; and such that holonomy along each $\rho_i$, $i=1,2$, is nontrivial. Interchanging the superscripts of $\hat S_1', \hat
S_1''$ if necessary, we may assume that $\gamma'_1$ is on the boundary of $\hat S_1'$. Since $\rho_1$ has nontrivial linear holonomy, the lift $\rho'_1\subset\hat S_1'$ of $\rho$ starting at $\gamma'_1$ is not closed and hence it ends on $-\gamma''_1$. This implies that both $\gamma'_1$ and $-\gamma''_1$ belong to the boundary of $\hat S_1'$. Since $S_1=S\setminus
(\gamma_1\cup\gamma_2)$ is connected, at least one of both $\pm\gamma'_2$ and $\pm\gamma''_2$ belongs to the boundary of $\hat S_1'$. Applying the same argument as above and using the obvious symmetry between $\hat S_1'$ and $\hat S_1''$ we conclude that under an appropriate choice of orientations of $\gamma_1$ and $\gamma_2$ one has $$\partial \hat S_1' = \gamma'_1 - \gamma''_1 - \gamma'_2 + \gamma''_2$$ which is equivalent to $$[\gamma'_1] - [\gamma''_1] = [\gamma'_2] - [\gamma''_2]$$ and hence, $\gamma_1$ and $\gamma_2$ are ĥomologous.
\(ii) In this case $\gamma_1$ and $\gamma_2$ are homologous in the usual sense, and not homologous to zero; the complement $S\setminus\{\gamma_1\cup\gamma_2\}$ has two connected components $S_1, S_2$. This implies that either both of $\gamma_1$ and $\gamma_2$ are segments, or both are closed cycles. Since neither of $\gamma_1, \gamma_2$ is homologous to zero, lemma \[lm:antiinvariant:cycle\] claims that $[\gamma'_i]\neq-[\gamma''_i]$ for $i=1,2$. Thus, if such $\gamma_1, \gamma_2$ are ĥomologous we get $|\gamma_1|=|\gamma_2|$ by lemma \[lm:gamma:2gamma\].
By lemma \[lm:trivial:plus:trivial:is:trivial\] at least one of two components, say, $S_1$ has nontrivial linear holonomy. Under an appropriate choice of orientations of $\gamma_1,\gamma_2$ we have $\partial S_1 =\gamma_1-\gamma_2$, which implies $$\partial \hat S_1 =(\gamma'_1+\gamma''_1)-(\gamma'_2+\gamma''_2).$$
Since $[\gamma'_i]\neq-[\gamma''_i]$, for $i=1,2$, the condition that $\gamma_1$ and $\gamma_2$ are ĥomologous is equivalent to one of the relations $([\gamma'_1]-[\gamma''_1])=\pm([\gamma'_2]-[\gamma''_2])$. Together with the above equation on $\partial \hat S_1$ it is equivalent to one of the following systems $$\begin{cases}
[\gamma'_1]=[\gamma'_2]\\
[\gamma''_1]=[\gamma''_2]
\end{cases}
\qquad
\begin{cases}
[\gamma'_1]=[\gamma''_2]\\
[\gamma''_1]=[\gamma'_2]
\end{cases}$$ The systems might be identified by interchange of superscripts of, say, $\gamma_2'$ and $\gamma_2''$, thus we can consider just the first one.
Since by the second property of lemma \[lm:total:boundary:holonomy:is:trivial\] neither of $[\gamma'_i], [\gamma''_i]$, $i=1,2$, is homologous to zero, the latter system is valid if and only if cutting $\partial \hat S$ by any of two pairs $[\gamma'_1],[\gamma'_2]$ or $[\gamma''_1],[\gamma''_2]$ of saddle connections we get two connected components. Since $\hat S_1$ is connected the latter is true if and only if $\hat S_2$ contains two connected components. By the criterion formulated above this is true if and only if $S_2$ has trivial linear holonomy. The equivalence is proved.
\(iii) In this case $\gamma_1$ is a closed cycle homologous to zero, while $\gamma_2$ is not homologous to zero. This implies that the complement $S\setminus\{\gamma_1\cup\gamma_2\}$ has two connected components $S_1, S_2$. Combining lemma \[lm:antiinvariant:cycle\] with lemma \[lm:gamma:2gamma\] we see that if such $\gamma_1$ and $\gamma_2$ are ĥomologous, we have $|\gamma_1|=2|\gamma_2|$.
Choose the orientation of $\gamma_1$ and enumeration of the components in such way that $$\partial S_1 = \gamma_1 \qquad
\partial S_2 = -\gamma_1 + \gamma_2 - \gamma_2$$ Then $$\partial \hat S_1 = \gamma'_1 + \gamma''_1\qquad
\partial \hat S_2 = -\gamma'_1 -\gamma''_1 + \gamma'_2 - \gamma'_2 + \gamma''_2 - \gamma''_2$$ Note that $S_1$ has nontrivial linear holonomy (see property 2 of lemma \[lm:total:boundary:holonomy:is:trivial\]) so $\hat S_1$ is connected. If $\gamma_1$ and $\gamma_2$ are ĥomologous, then $[\gamma'_1]=\pm ([\gamma'_2]-[\gamma''_2])$. This implies that $\hat S\setminus(\gamma'_1\cup\gamma'_2\cup\gamma''_2)$ contains at least two connected components. Since $\hat
S\setminus(\gamma'_1\cup\gamma'_2\cup\gamma''_2)= \hat
S_1\cup\gamma''_1\cup \hat S_2$ where $\hat S_1$ is connected and $\gamma_1''$ connects $\hat S_1$ and $\hat S_2$, this implies that $\hat S_2$ is not be connected. Hence $S_2$ has trivial linear holonomy.
Conversely, consider the connected component of $S\setminus\gamma_1$ containing $\gamma_2$; denote it by $\tilde
S_2$. Property 2 of lemma \[lm:total:boundary:holonomy:is:trivial\] implies that $\tilde S_2$ has nontrivial linear holonomy. Note that $S_2=\tilde S_2\setminus\gamma_2$. Thus, when $S_2$ has trivial linear holonomy, there exist a a closed path $\rho$ on $\tilde
S_2$ transversally intersecting $\gamma_2$ such that holonomy along $\rho$ is nontrivial. Since $S_2$ has trivial linear holonomy, $\hat S_2$ has two connected components $\hat S_2',
\hat S_2''$. Changing if necessary the superscripts of $\hat
S_2', \hat S_2''$ we may assume that $\gamma_2'$ is on the boundary of $\hat S'_2$. Since the holonomy along $\rho$ is nontrivial, following the lift of $\rho$ which starts at $\gamma_2'$ and goes inside $\hat S'_2$ the path $\rho$ ends at $-\gamma''_2$, which shows that $\gamma_2'$ and $-\gamma''_2$ make part of the boundary of the same component $\hat S'_2$. Taking into consideration the symmetry between components $\hat
S_2', \hat S_2''$ and choosing an appropriate orientation of $\gamma_1$ we get $$\partial \hat S'_2 = -\gamma'_1 + \gamma'_2 - \gamma''_2
\qquad
\partial \hat S''_2 = -\gamma''_1 - \gamma'_2 + \gamma''_2$$ which implies that $\gamma_1$ and $\gamma_2$ are ĥomologous.
\(iv) In this case the complement $S\setminus\{\gamma_1\cup\gamma_2\}$ has three connected components. Both $\gamma_1$ and $\gamma_2$ are homologous to zero, so they are represented by closed cycles. This also implies that $[\hat \gamma_i]=\gamma'_i$, $i=1,2$. If such $\gamma_1$ and $\gamma_2$ are ĥomologous, we have $|\gamma_1|=|\gamma_2|$ (see lemma \[lm:gamma:2gamma\]).
Denote the connected components of $S\setminus(\gamma_1\cup\gamma_2)$ in such way that under an appropriate choice of orientations of $\gamma_1,\gamma_2$ one gets $$\partial S_1 = \gamma_1 \qquad
\partial S_2 = -\gamma_2 \qquad
\partial S_3 = -\gamma_1+\gamma_2$$ By property 2 of lemma \[lm:total:boundary:holonomy:is:trivial\] the components $S_1$ and $S_2$ have nontrivial linear holonomy, so $\hat S_1$ and $\hat S_2$ are connected. We get $$\partial \hat S_1 = \gamma'_1 + \gamma''_1 \qquad
\partial \hat S_2 = -\gamma'_2 - \gamma''_2\qquad
\partial \hat S_3 = -\gamma'_1-\gamma''_1 + \gamma'_2 + \gamma''_2$$
If $\gamma_1$ and $\gamma_2$ are ĥomologous then $[\gamma'_1]=\pm[\gamma'_2]$ which implies that cutting $\hat S$ by $\gamma'_1, \gamma'_2$ we get two connected components, which means that $\hat S_3$ is not connected and hence $S_3$ has trivial linear holonomy.
Conversely, if $S_3$ has trivial linear holonomy then $\hat
S_3$ contains two connected components $\hat S'_3, \hat S''_3$ which (under appropriate enumeration) have boundaries $$\partial \hat S'_3 = -\gamma'_1 + \gamma'_2 \qquad
\partial \hat S''_3 = -\gamma''_1+ \gamma''_2$$ which implies that $\gamma_1$ and $\gamma_2$ are homologous. Proposition \[pr:homologous:saddle:connections\] is proved.
Theorem \[th:unique:trivial:holonomy\] follows from proposition \[pr:homologous:saddle:connections\].
Cutting $S$ by $\gamma_1, \gamma_2$ we get one of the decompositions (i)–(iv). According to proposition \[pr:homologous:saddle:connections\], the additional assumptions (1)–(4) on the triviality of the linear holonomy of the corresponding component are necessary and sufficient conditions for $\gamma_1, \gamma_2$ to be ĥomologous. It remains to note that in each of the cases (1)–(4) there is a unique component with trivial linear holonomy.
The following criterion is analogous to the corresponding statement in [@Eskin:Masur:Zorich]. It is proved in appendix \[ap:Long:saddle:connections\], where the notion of a measure on each stratum is discussed.
\[pr:homologous:equiv:parallel\] For almost every flat surface in any stratum, two saddle connections are parallel if and only if they are ĥomologous.
Graph of connected components {#s:graph}
=============================
In this section we give the proof that every graph is given by the list in theorem \[th:graphs\]. Denote by $\dot{S}$ the surface $S$ punctured at the singularities. Any closed loop $\rho$ on $\dot{S}$ can be homotoped to have a finite number of transverse intersections with the saddle connections from the collection $\gamma=\{\gamma_1\ldots\gamma_n\}$. It naturally induces a path $\rho_\ast$ on the graph $\Gamma(S,\gamma)$ by recording the surfaces $S_j$ intersected by $\rho$. It is easy to see that paths $\rho\sim\rho'$ homotopic on the punctured surface $\dot{S}$ define paths $\rho_\ast\sim \rho'_\ast$ homotopic on the graph. Mark a point $x\in\dot{S}\setminus\{\gamma_i\} $; let $v(x)$ be the corresponding vertex of the graph $\Gamma(S,\gamma)$. We get a natural homomorphism $\pi_1(\dot{S},x)\to\pi_1\big(\Gamma(S,\gamma),v(x)\big)$.
Any finite connected graph can be retracted by a deformation to a bouquet of circles (possibly to a point). We can choose the retraction in such way that $v(x)$ retracts to the base point of the bouquet of circles. We can consider the bouquet of circles $\operatorname{B}$ as a graph obtained from the graph $\Gamma(S,\gamma)$ by identifying some subtree of $\Gamma(S,\gamma)$ to a single vertex of $\operatorname{B}$. Thus, some edges of $\Gamma(S,\gamma)$ remain nondegenerate under the retraction, and some edges collapse to a point.
Now we can prove the lemma which is the main technical tool in the proof of theorem \[th:graphs\].
\[lm:nontriv:lin:holon:along:loops:of:the:graph\] Let $\alpha\subset\Gamma(S,\gamma)$ be a closed path on $\Gamma(S,\gamma)$ realized as a subgraph of $\Gamma(S,\gamma)$. If under some retraction of $\Gamma(S,\gamma)$ to a bouquet of circles, $\alpha$ retracts to one of the circles, then there exists a closed path $\rho$ on the punctured surface $\dot{S}$ such that $\rho_\ast=\alpha$ and the linear holonomy along $\rho$ is nontrivial.
We suppose that a retraction of $\Gamma(S,\gamma)$ to a bouquet of circles is fixed. We start with consideration of the general case, when the bouquet of circles contains at least two circles.
Let $\gamma_1$ and $\gamma_2$ be a pair of edges of $\Gamma(S,\gamma)$, which remain nondegenerate under retraction, such that $\gamma_1\in\alpha$ and $\gamma_2\not\in\alpha$ (since the bouquet contains at least two circles, such $\gamma_2$ exists). Cutting $\Gamma(S,\gamma)$ by these edges we get a connected graph. Equivalently, cutting the surface $S$ by a pair of ĥomologous saddle connections $\gamma_1, \gamma_2$ we get a connected surface $S_{(1,2)}=S\setminus(\gamma_\cup\gamma_2$ which, by proposition \[pr:homologous:saddle:connections\], has trivial linear holonomy. By construction $\partial S_{(1,2)}=
\gamma_1\cup-\gamma_1\cup\gamma_2\cup-\gamma_2$. Gluing back the boundary components $\gamma_1$ and $-\gamma_1$ of $S_{(1,2)}$ we get a surface $S_{(2)}=S\setminus\gamma_2$ which has nontrivial linear holonomy by lemma \[lm:total:boundary:holonomy:is:trivial\]. Thus, the boundary components $\gamma_1$ and $-\gamma_1$ of the [*translation*]{} surface $S_{(1,2)}=S\setminus(\gamma_1\cup\gamma_2)$ are identified by a flip (see definition \[def:flip\] in the previous section).
Consider any path $\rho$ in $S$ such that $\rho_\ast=\alpha$ and such that $\rho$ has unique transversal intersection with $\gamma_1$. By construction $\rho$ it gives a nonclosed connected path $\rho'$ on $S_{(1,2)}$ joining a pair of points on $P_+\in\gamma_1$ and $P_-\in-\gamma_1$ corresponding to the same point $P\in\gamma_1$ on $S$ upon gluing of $\gamma_1$ with $-\gamma_1$. Since $\gamma_1$ and $-\gamma_1$ are identified by a flip, we see that the linear holonomy along $\rho$ is nontrivial.
It remains to consider the case, when the bouquet of circles corresponding to the graph $\Gamma(S,\gamma)$ has a single circle. It follows from proposition \[pr:homologous:saddle:connections\] that the graph cannot be just a single loop composed of “$+$”-vertices of valence two and of “cylinder vertices” of valence two. Thus, either $\Gamma(S,\gamma)$ is a loop composed of vertices of valence two with some “$-$”-vertices, or there is at least one nontrivial subtree with a vertex on the base loop.
In the first case choose any path $\rho'$ on $S$ such that $\rho'_\ast=\alpha$. If the linear holonomy along the path $\rho'$ is nontrivial, we choose $\rho:=\rho'$ and the lemma is proved. If the holonomy is trivial, we can compose $\rho'$ with a closed path $\rho''$, such that $\rho''$ is contained entirely inside some $S^-_j$, and such that the linear holonomy along $\rho''$ is nontrivial. Since $\rho''\subset S^-_j$ the projection $\rho''_\ast$ is a trivial path. Thus, $(\rho'\cdot\rho'')_\ast = \rho'_\ast = \alpha$, and by construction the linear holonomy along $(\rho'\cdot\rho'')$ is nontrivial. The required path $\rho$ is given by $\rho'\cdot\rho''$.
In the second case the subtree necessarily has a vertex $S_j$ of valence one, which by Lemma \[lm:total:boundary:holonomy:is:trivial\] is a “$-$”-vertex. Denote by $\gamma_1$ the edge adjacent to this vertex of valence one; we denote by the same symbol $\gamma_1$ the corresponding saddle connection in $S$. Consider any path $\rho'$ on $S$ such that $\rho'_\ast=\alpha$. If $\alpha\subset\Gamma(S,\gamma)$ passes through $S^-_j$, we apply the same construction as in the previous case. If $\alpha$ does not pass through $S^-_j$ then $\gamma_1\cap \alpha={\varnothing}$, and any path $\rho\in S$ such that $\rho_\ast=\alpha$ has trivial intersection with the saddle connection $\gamma_1$. Choose some edge $\gamma_2\in\alpha$ which is nondegenerate under retraction. Cutting $S$ by the pair of ĥomologous saddle connections $\gamma_1, \gamma_2$ we get two connected components: a connected surface $S_{(1,2)}$ and a surface $S^-_j$ (corresponding to the vertex of valence one). The closed path $\rho$ on $S$ becomes a nonclosed connected path on $S_{(1,2)}$ joining the boundary components $\gamma_2'$ and $\gamma_2''$. By proposition \[pr:homologous:saddle:connections\] the surface $S_{(1,2)}$ has trivial linear holonomy. By construction $\partial S_{(1,2)}= \gamma_1\cup\gamma_2\cup-\gamma_2$. Glue back the boundary components $\gamma_2$ and $-\gamma_2$ of $S_{(1,2)}$. We get a surface $S_{(1)}$ which coincides with one of the two components of the initial surface $S$ cut by a single saddle connection $\gamma_1$. Since $\partial S_{(1)} = \gamma_1$ , by lemma \[lm:total:boundary:holonomy:is:trivial\] the surface $S_{(1)}$ has nontrivial linear holonomy. Thus, the boundary components $\gamma_2$ and $-\gamma_2$ of the [*translation*]{} surface $S_{(1,2)}$ were identified by a flip which implies that the linear holonomy along $\rho$ is nontrivial.
\[lm:subgraph:with:minus:vertex:has:nontrivial:holonomy\] Consider a connected subgraph $\Upsilon$ of the initial graph $\Gamma(S,\gamma)$. If $\Upsilon$ has a vertex labelled with “$-$” or if it is not a tree, the surface with boundary $S_\Upsilon$ corresponding to this subgraph has nontrivial linear holonomy.
If the subgraph has some vertex labelled with “$-$”, the corresponding surface $S^-_j$ has a closed path with nontrivial linear holonomy. The bigger surface $S_\Upsilon$ has the same path, so it also has nontrivial linear holonomy. If the subgraph is not a tree, then it has a loop which is not homotopically trivial. By Lemma \[lm:nontriv:lin:holon:along:loops:of:the:graph\] there is a closed path $\rho$ on $S_\Upsilon$ corresponding to this loop such that $\rho$ has nontrivial linear holonomy.
First we note that a cylinder has trivial linear holonomy, so by lemma \[lm:total:boundary:holonomy:is:trivial\] a [“$\circ$”]{}-vertex cannot have valence $1$.
If the valence of a [“$\circ$”]{}-vertex is two, then each of the two boundary components of the corresponding cylinder represents a single saddle connection. Hence, two [“$\circ$”]{}-vertices of valence two cannot have a common edge, otherwise the pair of corresponding cylinders would be identified along a boundary component of each which would result in a longer cylinder contradicting the assumption that each cylinder is maximal.
Now note that the bouquet of circles to which the graph $\Gamma(S,\gamma)$ is retracted contains at most two circles. Otherwise there would be edges $\gamma_1$ and $\gamma_2$ such that $\Gamma(S,\gamma)\setminus(\gamma_1\cup\gamma_2)$ would be connected but not simply connected. Thus, according to lemma \[lm:subgraph:with:minus:vertex:has:nontrivial:holonomy\] the surface $S_{(1,2)}=S\setminus(\gamma_1\cup\gamma_2)$ would have nontrivial linear holonomy, which contradicts proposition \[pr:homologous:saddle:connections\].
[**Two loops.**]{} Suppose that the bouquet of circles contains exactly two circles. Cut them by some edges $\gamma_1$ and $\gamma_2$ which correspond to different circles of the bouquet. By proposition \[pr:homologous:saddle:connections\] the resulting surface has trivial linear holonomy. Lemma \[lm:subgraph:with:minus:vertex:has:nontrivial:holonomy\] implies that the surface and therefore the graph does not have any “$-$”-vertices, in particular, no vertices of valence $1$. Since the Euler characteristic of $S^1\vee S^1$ equals to $-1$ we get $$\begin{gathered}
-1=\chi(S^1\vee S^1)=\chi(\Gamma(S,\gamma))=
-\frac{1}{2}\cdot(\text{number of vertices of valence 3})-\\
-\frac{2}{2}\cdot(\text{number of vertices of valence 4})
-\frac{3}{2}\cdot(\text{number of vertices of valence 5})-\dots\end{gathered}$$ which means that either $\Gamma(S,\gamma)$ has two vertices of valence $3$ while all the other vertices have valence $2$, or $\Gamma(S,\gamma)$ has a single vertex of valence $4$ while all the other vertices have valence $2$. All graphs of this type except one are in the list of theorem \[th:graphs\], see types d and e. The type which we have to rule out is schematically presented at figure \[fig:banned:graph\].
We prove by contradiction that this graph is not realizable as $\Gamma(S,\gamma)$. Let $S^+_n$ be a vertex of valence three; let $\gamma_1,\gamma_2,\gamma_3$ be the edges adjacent to it. Cutting $\Gamma(S,\gamma)$ by any pair of distinct edges $\gamma_i,
\gamma_j$, $i=1,2,3$, we still get a connected graph. This means that no pair of ĥomologous saddle connections $\gamma_i{\cup}\gamma_j$, $i=1,2,3$, separates $S$. Hence, by proposition \[pr:homologous:saddle:connections\] the lengths $|\gamma_i|$, $i=1,2,3$ are equal and all $\gamma_i$ are parallel.
Let $\omega$ be the holomorphic $1$-form representing the translation structure on $S^+_n$. Under an appropriate choice of orientations of $\gamma_1,\gamma_2,\gamma_3$ we get $\partial
S^+_n = \gamma_1\cup\gamma_2\cup\gamma_3$, and hence $\int_{\gamma_1}\omega + \int_{\gamma_2}\omega +
\int_{\gamma_3}\omega = 0$. On the other hand the fact that all the lengths $|\gamma_i|$, $i=1,2,3$ are equal and all $\gamma_i$ are parallel implies that $\int_{\gamma_1}\omega =
\pm\int_{\gamma_2}\omega = \pm\int_{\gamma_3}\omega$. These two relations are incompatible, which is a contradiction.
[**One loop.**]{} If $\Gamma(S,\gamma)$ is a loop of vertices of valence two then by proposition \[pr:homologous:saddle:connections\] it has at least one “$-$”-vertex $S_i^-$. Let $\gamma_1$ and $\gamma_2$ be the edges of $\Gamma(S,\gamma)$ adjacent to $S_i^-$. The complement $S\setminus(\gamma_1\cup\gamma_2)$ has two connected components: $S_i^-$ and $S\setminus S_i^-$. Since $S_i^-$ has nontrivial linear holonomy, by proposition \[pr:homologous:saddle:connections\] the flat surface $S\setminus S^-_i$ has trivial linear holonomy. It follows now from Lemma \[lm:subgraph:with:minus:vertex:has:nontrivial:holonomy\] that $S\setminus S^-_i$ is a chain of “$+$”-vertices of valence two and of [“$\circ$”]{}-vertices of valence two. Thus, in this case the graph $\Gamma(S,\gamma)$ is of the type b, see theorem \[th:graphs\] and figure \[fig:classification:of:graphs\].
If the graph has a nontrivial subtree attached to the base loop then any such subtree necessarily has a vertex of valence one, which by lemma \[lm:total:boundary:holonomy:is:trivial\] is a “$-$”-vertex. Let us show that $\Gamma(S,\gamma)$ can have only one “$-$”-vertex of valence one. Suppose that there are two vertices $S^-_i$ and $S^-_j$ of valence one; denote by $\gamma_1$ and $\gamma_2$ the edges of $\Gamma(S,\gamma)$ adjacent to these vertices. Cutting $S$ by $\gamma_1$ and $\gamma_2$ we obtain three connected components: $S^-_i, S^-_j$ and $S_{(1,2)}:=S\setminus(S^-_i{\cup}S^-_j)$. Since the first two surfaces have nontrivial linear holonomy, it follows from proposition \[pr:homologous:saddle:connections\] that $S_{(1,2)}$ has trivial linear holonomy. But by assumption the graph $\Upsilon$ corresponding to $S_{(1,2)}$ has a nontrivial loop, so by Lemma \[lm:nontriv:lin:holon:along:loops:of:the:graph\] the flat surface $S_{(1,2)}$ has nontrivial linear homology, which is a contradiction.
Thus, the graph has the structure of a union of a circle with a segment attached to a circle. The graph has single vertex of valence three, a single vertex of valence one and arbitrary number of vertices of valence two. Choosing an appropriate pair of edges $\gamma_1, \gamma_2$ and combining proposition \[pr:homologous:saddle:connections\] with Lemma \[lm:nontriv:lin:holon:along:loops:of:the:graph\] and Lemma \[lm:subgraph:with:minus:vertex:has:nontrivial:holonomy\] we see that the only “$-$”-vertex of the graph is the vertex of valence one located at the free end of the segment. This is the graph of the type c in the list of graphs in theorem \[th:graphs\].
[**A tree.**]{} In this case $\Gamma(S,\gamma)$ has at least two vertices of valence one which are therefore of “$-$”-type. Let $\gamma_1$ and $\gamma_2$ be the edges of $\Gamma(S,\gamma)$ adjacent to this pair of vertices $S^-_i$, $S^-_j$. Cutting the surface $S$ by $\gamma_1,\gamma_2$ we get three connected components $S^-_i$, $S^-_j$, and $S\setminus(S^-_i\cup S^-_j)$. By proposition \[pr:homologous:saddle:connections\] the component $S\setminus(S^-_i\cup S^-_j)$ has trivial linear holonomy. Thus, by Lemma \[lm:subgraph:with:minus:vertex:has:nontrivial:holonomy\] it does not have any “$-$”-vertices. Since $\Gamma(S,\gamma)$ is a tree it means that $\Gamma(S,\gamma)$ is a chain of “$+$”-vertices of valence two bounded at the ends by a pair of “$-$”-vertices of valence one. This is the graph a from theorem \[th:graphs\] (see also figure \[fig:classification:of:graphs\]).
Two [“$\circ$”]{}-vertices of valence two cannot be neighbors. It remains to prove that a [“$\circ$”]{}-vertex of valence $3$ cannot be joined by a separating edge to a [“$\circ$”]{}-vertex of valence $2$. If that were the case then on one boundary component of the cylinder there would be a marked point. If this boundary component were joined to a [“$\circ$”]{}-vertex of valence $2$ it would produce a “fake singularity” on $S$.
We have proved that all graphs must be of the type in theorem \[th:graphs\]. The fact that the weights are as described follows from the next lemma.
\[lm:lengths:of:saddle:connections\] The type of the graph $\Gamma({{\mathcal C}})$ uniquely determines the distribution of unsigned weights $1$ and $2$ on the edges of the graph; the corresponding weights are presented in Figure \[fig:classification:of:graphs\].
For every vertex representing a component with trivial linear holonomy we can choose signs for the weights $1$ and $2$ on the edges adjacent to the vertex. The sum of these signed weights is zero. This immediately implies that the globally defined unsigned weights on both edges adjacent to a valence two “$+$”-vertex or to a valence two [“$\circ$”]{}-vertex are the same. This in turn implies that all the weights on the graphs of types a) and b) coincide, and hence are marked by $1$.
The remaining graphs do not have “$-$”-vertices. The edges of any vertex of valence three are weighted by $1, 1$ and $2$. This implies that the weights of the graphs of types c) and d) are as in figure \[fig:classification:of:graphs\].
Let $\gamma_1, \gamma_2$ be a pair of edges adjacent to a valence four vertex, and belonging to two different loops. The surface cut along these saddle connections is connected. By proposition \[pr:homologous:saddle:connections\], $|\gamma_1|=|\gamma_2|$, and hence the corresponding edges have the same weight. Since all the edges in a chain of “$+$”-vertices or [“$\circ$”]{}-vertices of valence $2$ have the same weight, we see that all edges of a graph of type e) are weighted by $1$.
This completes the proof of the necessity part of theorem \[th:graphs\].
Parities of boundary singularities {#s:Anatomy}
==================================
In this section we prove the necessity part of theorem \[th:all:local:ribon:graphs\] which says that for any decomposition of a flat surface $S$ as in theorem \[th:graphs\] every connected component $S_j$ has one of the boundary types presented in figure \[fig:embedded:local:ribbon:graphs\].
theorem \[th:graphs\] and figure \[fig:classification:of:graphs\] give the types of graphs $\Gamma$; figure \[fig:local:ribbon:graphs\] gives the list of all abstract local ribbon graphs of valences from one to four. Basically, what remains to check is that for any “$+$”-vertex $v$ of $\Gamma$ an embedding ${\mathbb G}_v\hookrightarrow
\Gamma$ of the local ribbon graph ${\mathbb G}_v$ into the graph $\Gamma$ uniquely determines the parities of the boundary singularities, and that these parities are exactly as in figure \[fig:embedded:local:ribbon:graphs\].
Signs of the weights {#signs-of-the-weights .unnumbered}
--------------------
Given a collection $\gamma$ of ĥomologous saddle connections $\gamma_1, \dots, \gamma_n$ on a flat surface $S$ we have assigned weights $1$ and $2$ to saddle connections $\gamma_i$ (see the paragraph preceding theorem \[th:graphs\] for the definition of the weights and figure \[fig:classification:of:graphs\] for the distribution of the weights in $\Gamma$). If a connected component $S_j$ of $S\setminus\gamma$ has trivial linear holonomy (i.e. if it is represented by a “$+$” or by a [“$\circ$”]{}-vertex of $\Gamma$) we may assign signs $\pm$ to the weights of the saddle connections on the boundary of $S_j$. The canonical orientation of $S_j$ induces the canonical orientation of the boundary $\partial S_j$. Let $\omega$ be a holomorphic $1$-form representing the flat structure on $S_j$ normalized so that $$\int_{\gamma_i}\omega =\text{weight of }\gamma_i,$$ for some saddle connection $\gamma_i$ on the boundary of $S_j$. Then for the other saddle connections on $\partial S_j$ we get $\int_{\gamma_{i'}}\omega =\pm 1$ or $\int_{\gamma_{i'}}\omega =
\pm 2$ (see also the tables in section \[ss:tables\]).
There is an ambiguity in the choice of signs: we may simultaneously change the signs of all weights to the opposite ones. This corresponds to choosing $-\omega$ instead of $\omega$.
\[lm:weights:define:parity\] Consider two consecutive saddle connections $\gamma_{j_{i,l}}\to\gamma_{j_{i,l+1}}$ on the same boundary component ${{\mathcal B}}_i$ of $\partial S_j$. The parity of the corresponding boundary singularity is even if the weights of $\gamma_{j_{i,l}}$ and $\gamma_{j_{i,l+1}}$ have the same signs, and odd if the weights of $\gamma_{j_{i,l}}$ and $\gamma_{j_{i,l+1}}$ have opposite signs.
The holomorphic 1-form $\omega$ chosen above defines an oriented horizontal foliation on $S_j$: the kernel foliation of ${\operatorname{Im}}(\omega)$. The above normalization of $\omega$ implies that any saddle connection at the boundary $\partial S_j$ is horizontal. The weight of a saddle connection $\gamma_{j_{i,l}}$ on the boundary of $S_j$ is positive if the orientation of $\gamma_{j_{i,l}}$ induced from the orientation of the boundary matches the orientation of the foliation and negative if it does not.
The cone angle between two incoming or two outgoing separatrix rays (in the sense of the orientation of the foliation) is an even multiple of $\pi$ and the cone angle between an incoming and an outgoing separatrix ray (in the sense of the orientation of the foliation) is an odd multiple of $\pi$. The statement of the lemma now follows from definition \[def:order:of:boundary:singularity\] of the order of a boundary singularity.
Consider now a particular case when $S_j$ is represented by a vertex $v_j$ of valence four of the graph $\Gamma(S,\gamma)$. Four edges of $\Gamma_{v_j}$ are distributed into two pairs: each pair bounds one of the two loops of the graph $\Gamma(S,\gamma)$, see figure \[fig:classification:of:graphs\].
\[lm:signs:of:weights:for:valence:4\] The weights of saddle connections on the boundary of a component $S_j$ represented by a vertex of valence four have same signs if they bound the same loop in $\Gamma(S,\gamma)$ and opposite signs otherwise.
From lemma \[lm:lengths:of:saddle:connections\] we know that the absolute values of weights of all edges of $\Gamma_{v_j}$ for a vertex $v_j$ of valence four are equal to one (see also figure \[fig:classification:of:graphs\]). Hence, it follows from Stokes theorem that we have two edges of weight $+1$ and two edges of weight $-1$ in $\Gamma_{v_j}$. We want to show that the weights of a pair of edges of $\Gamma_{v_j}$ bounding the same loop in $\Gamma$ have the same signs.
Let $\gamma_1, \gamma_2\in \Gamma_{v_j}$ bound *different* loops in $\Gamma$. Cutting $S$ by $\gamma_1, \gamma_2$ we get a connected flat surface $S_{12}$. Using the same notation as in the proof of proposition \[pr:homologous:saddle:connections\] we get $$\partial S_{12}=\gamma'_1\cup -\gamma''_1\cup\gamma_2'\cup-\gamma''_2$$ By theorem \[th:unique:trivial:holonomy\] the surface $S_{12}$ has trivial linear holonomy. Hence, we can extend the form $\omega$ to $S_{12}$ which enables us to assign signs to the weights of saddle connections $\gamma'_1, \gamma''_1, \gamma'_2,
\gamma''_2$ on the boundary $\partial S_{12}$ of $S_{12}$.
The last statement of lemma \[lm:total:boundary:holonomy:is:trivial\] implies that gluing the initial closed surface $S$ from $S_{12}$ the boundary component $\gamma'_1$ is glued to $-\gamma''_1$ by a flip (see definition \[def:flip\] in section \[s:hat:homologous:saddle:connections\]). Similarly $\gamma'_2$ is glued to $-\gamma''_2$ by a flip. Hence, the weights of $\gamma'_1$ and of $\gamma''_1$ have the same signs, and the weights of $\gamma'_2$ and of $\gamma''_2$ have the same signs.
This completes the proof of the lemma in the case when the corresponding loop contains no vertices at all. An induction on the number of vertices in the loop completes the proof in general case.
\[lm:signed:lengths:of:saddle:connections\] For any “$+$”-vertex or [“$\circ$”]{}-vertex $v$ of the graph $\Gamma(S,\gamma)$ the type of the graph uniquely determines the distribution of signed weights $\pm 1$ and $\pm 2$ on the edges of $\Gamma_v$ (up to simultaneous interchange of all signes to the opposite ones).
By Stokes theorem the sum of weights of all saddle connections of $\Gamma_v$ is equal to zero. Taking into consideration lemma \[lm:lengths:of:saddle:connections\] (see also figure \[fig:classification:of:graphs\]) this implies that when the vertex $v$ has valence $2$, the weights of the edges of $\Gamma_v$ are $+1,-1$; when $v$ has valence $3$, the weights are $+1,+1,-2$; when $v$ has valence $4$, the weights are $+1,+1,-1,-1$. Moreover, according to lemma \[lm:signs:of:weights:for:valence:4\] the weights of edges of $\Gamma_v$ which bound the same loop in $\Gamma(S,\gamma)$ coincide.
Now we are ready to prove the following proposition, which corresponds to the necessity part of theorem \[th:all:local:ribon:graphs\]. (The sufficiency part of theorem \[th:all:local:ribon:graphs\] immediately follows from theorem \[th:from:boundary:to:neighborhood:of:the:cusp\] proved in the next section).
For any decomposition of a flat surface $S$ as in theorem \[th:graphs\] every connected component $S_j$ has one of the boundary types presented in figure \[fig:embedded:local:ribbon:graphs\].
The necessity part of theorem \[th:graphs\] proved in section \[s:graph\] claims that the graph $\Gamma(S,\gamma)$ of the decomposition has one of the types presented in figure \[fig:classification:of:graphs\]. Note that for “$+$” and “$-$”-vertices, figure \[fig:embedded:local:ribbon:graphs\] describes *all* possible embeddings of abstract local ribbon graphs ${\mathbb G}_v$ that are given in figure \[fig:local:ribbon:graphs\] into graphs $\Gamma$ as in figure \[fig:classification:of:graphs\]. We use dotted lines to indicate the pairs of edges bounding cycles in the graphs in figure \[fig:embedded:local:ribbon:graphs\]; dotted lines are not indicated in symmetric situations. Since there are no restrictions on the parities of boundary singularities of “$-$”-vertices this complets the proof for “$-$”-vertices.
Any [“$\circ$”]{}-vertex $S^{comp}_j$ corresponds to a flat cylinder. Hence, it has exactly two distinct boundary components. The boundary singularities on each of the components correspond to marked points, so the order of any boundary singularity of a [“$\circ$”]{}-vertex is zero. By lemma \[lm:weights:define:parity\] this implies that all edges of $\Gamma_{v_j}$ which correspond to the same boundary component of the cylinder $S_j$ have weights of the same sign. Taking into consideration lemma \[lm:signed:lengths:of:saddle:connections\] these two conditions restrict the possible structures of an embedded local ribbon graph ${\mathbb G}_v\hookrightarrow\Gamma(S,\gamma)$ for [“$\circ$”]{}-vertices to structures $\circ 2.2$, $\circ 3.2$ and $\circ
4.2$ in figure \[fig:embedded:local:ribbon:graphs\].
By lemma \[lm:signed:lengths:of:saddle:connections\] for any “$+$”-vertex of $\Gamma(S,\gamma)$ we know the signed weights of the edges of $\Gamma_v$ (up to simultaneous interchange of all signs to the opposite ones). For “$+$”-vertices of valence two and three this distribution follows immediately from figure \[fig:classification:of:graphs\] and from Stokes theorem; for “$+$”-vertices of valence four this distribution is described by lemma \[lm:signs:of:weights:for:valence:4\]. Hence, using lemma \[lm:weights:define:parity\] we can determine the parities of all boundary singularities for any embedded local ribbon graph ${\mathbb G}_v\hookrightarrow \Gamma$. It remains to check that for all possible embeddings listed in figure \[fig:embedded:local:ribbon:graphs\] the parities are the ones listed. This is an easy exercise.
\[cr:unique:choice:of:parities\] Given any abstract graph $\Gamma$ as in theorem \[th:graphs\] (see figure \[fig:classification:of:graphs\]), any “$+$” or “$-$”-vertex $v_j$ of $\Gamma$, any choice of the structure of a local ribbon graph ${\mathbb G}_{v_j}$ on $\Gamma_{v_j}$ and any embedding ${\mathbb G}_{v_j}\hookrightarrow \Gamma$, one can find a flat surface $S$ and a collection $\gamma$ of ĥomologous saddle connections on it such that $\Gamma(S,\gamma)=\Gamma$ and such that the boundary type of the component $S_j$ is represented by the chosen embedded ribbon graph. Moreover, if $v_j$ is represented by a “$+$”-vertex of $\Gamma$, then the parities of boundary singularities of $S_j$ are completely determined by the choice of the embedded ribbon graph.
Conversely, given an abstract graph $\Gamma$ as in theorem \[th:graphs\] (see figure \[fig:classification:of:graphs\]), a “$+$”-vertex $v_j$ of $\Gamma$, an abstract local ribbon graph ${\mathbb G}_{v_j}$, and a choice of the parities of boundary singularities as given in figure \[fig:embedded:local:ribbon:graphs\] there is a unique way (up to a symmetry of the ribbon graph ${\mathbb G}_{v_j}$) to embed the local ribbon graph with marked parities into the graph $\Gamma$. This unique way is expressed by the dotted lines in figure \[fig:embedded:local:ribbon:graphs\].
For “$+$” and “$-$”-vertices $v_j$ all possible embeddings of local ribbon graphs as in figure \[fig:local:ribbon:graphs\] into the graphs $\Gamma$ as in figure \[fig:classification:of:graphs\] are represented in figure \[fig:embedded:local:ribbon:graphs\]. Thus, the first statement follows from theorem \[th:all:local:ribon:graphs\]. The second statement immediately follows from theorem \[th:all:local:ribon:graphs\] combined with lemmas \[lm:weights:define:parity\] and \[lm:signed:lengths:of:saddle:connections\].
Neighborhood of the principal boundary: local constructions {#s:Local:Constructions}
===========================================================
In this section and in the next one we construct surfaces with boundaries representing all boundary types listed in figure \[fig:embedded:local:ribbon:graphs\]. We first prove the key proposition below. Combining it with some elementary extra arguments we prove theorem \[th:from:boundary:to:neighborhood:of:the:cusp\] (and, hence, the missing realizibility parts of theorems \[th:graphs\] and \[th:all:local:ribon:graphs\]).
\[pr:realizability:of:all:vertices\] Consider any configuration ${{\mathcal C}}$ as in definition \[def:configuration\], and any vertex $v_j$ of the graph $\Gamma({{\mathcal C}})$. Let $S'_j$ be any flat surface from the component ${{\mathcal Q}}(\alpha'_j)$ (or ${{\mathcal H}}(\beta'_j)$) of the principal boundary stratum ${{\mathcal Q}}(\alpha')$ (or ${{\mathcal H}}(\beta')$) corresponding to $v_j$. Choose any sufficiently small value of a complex parameter $\delta$ (depending on $S'_j$).
Applying to $S'_j$ an appropriate basic surgery (depending on $\delta$) as described below one gets a surface $S_j$ with boundary, such that the boundary type of $S_j$ and the collections of interior singularities and of boundary singularities of $S_j$ are represented by the local ribbon graph ${\mathbb G}_{v_j}$ and by the corresponding structures $\{d_l\}_j,
\{k_{i,l}\}_j$ of the configuration ${{\mathcal C}}$.
Recall that the principal boundary stratum corresponding to a “$+$”-vertex is of type ${{\mathcal H}}(\beta'_j)$; the principal boundary stratum corresponding to a “$-$”-vertex is of type ${{\mathcal Q}}(\alpha'_j)$. The singularity data $\beta'_j, \alpha'_j$ are defined by equations and correspondingly. Unlike the initial singularity data $\alpha$ the collections $\beta'_j$ and $\alpha'_j$ might contain entries “$0$” representing marked points of the surface $S'_j$.
Though the principal boundary stratum corresponding to a [“$\circ$”]{}-vertex is empty, proposition \[pr:realizability:of:all:vertices\] is not meaningless (though very simple) even for such vertices. We leave the construction of surfaces $S_j$ with boundary realizing each of $\circ 2.2, \circ 3.2, \circ 4.2$-boundary types to the reader as an elementary exercise.
We split proposition \[pr:realizability:of:all:vertices\] into a collection of propositions \[pr:local:constructions\], \[pr:parallelogram:constructions\] and \[pr:minus:2:2\]. To avoid excessive repetitions we abbreviate the statements of the corresponding propositions; they should be read as the statement of proposition \[pr:realizability:of:all:vertices\] applied to vertices of specified types.
Part of the surgeries (namely, “breaking up a zero by a local construction” and a “parallelogram construction”) are taken from the paper [@Eskin:Masur:Zorich]. For the sake of completeness we present their outline in the current paper. For more details we address the reader to the original paper [@Eskin:Masur:Zorich].
Local constructions
-------------------
We reserve the word “degree” for the zeroes of [*Abelian*]{} differentials. A zero of degree $l$ has cone angle $\pi(2l+2)$. We reserve the word “order” for the zeroes of [*quadratic*]{} differentials. A zero of order $m$ has cone angle $\pi(m+2)$. Recall that a boundary singularity of order $k$ has cone angle $\pi(k+1)$.
We distinguish two kinds of surgeries. The surgeries of the first type are purely local: they do not change the flat metric on $S'_j$ outside a small neighborhood of one or two points on $S'_j$. The surgeries of the second type depend on a nonlocal construction. In the remaining part of this section we describe local surgeries.
\[pr:local:constructions\] Every surface with boundary type $+2.1$, $+3.1$, $+4.1a$, $+4.1b$, $+4.2a$ is realizable by a local construction.
We use the indexation of the boundary types as in figure \[fig:embedded:local:ribbon:graphs\] and in remark \[rm:indexation\] in section \[ss:Anatomy:of:decomposition:of:a:flat:surface\].
The principle boundary stratum corresponding to a vertex $v_j$ of a “$+$”-type has type ${{\mathcal H}}(\beta'_j)$. The singularity data $\beta'_j$ is given by equation , namely $$\beta'_j=\{d_1/2, \dots, d_{s(j)}/2,\ D_1/2, \dots, D_{r(j)}/2\},$$ where $d_1, \dots, d_{s(j)}$ are the orders of interior singularities, and $D_1, \dots, D_{r(j)}$ are expressed in terms of the orders of boundary singularities by formula . Conditions 4 and 5 in definition \[def:configuration\] of a configuration guarantee that all the entries of $\beta'_j$ are nonnegative integer numbers, and that the total sum of these numbers is even. According to [@Masur:Smillie:realizability] this implies that the stratum ${{\mathcal H}}(\beta'_j)$ is nonempty.
Consider any surface $S'_j$ in ${{\mathcal H}}(\beta'_j)$. Denote the length of the shortest saddle connection on $S'_j$ by $4\varepsilon$. We shall apply a surgery to $S'_j$, which would continuously depend on a small complex parameter considered as a vector $\vec v$ in ${{\mathbb R}^{2}}\simeq{{\mathbb C}^{}}$. It is convenient to change slightly the notations and to denote by $\delta$ the *norm* of $\vec
v$. We always assume that $\delta<\varepsilon$. Our surgery would not affect interior singularities of $S'_j$.
We provide all the details of the proof in the case of the boundary type $+2.1$ and we point out the differences in the other cases.
Boundary type +2.1 {#boundary-type-2.1 .unnumbered}
------------------
In this case the boundary has single component, $r(j)=1$, and $D_1=k_{1,1}+k_{1,2}-2$, where $k_{1,1}, k_{1,2}$ are the orders of the two boundary singularities of ${\mathbb G}_{v_j}$. Both $k_{1,1},
k_{1,2}$ are odd positive integers, see figure \[fig:embedded:local:ribbon:graphs\].
Let $P$ be the zero of $S'_j$ of degree $m$, where $m=D_1/2$. We can represent $m$ as the sum $m=m'+m''$, where $m'=(k_{1,1}-1)/2$ and $m''=(k_{1,2}-1)/2$. Consider a metric disc of radius $\varepsilon$ centered at $P$. By the choice of $\varepsilon$ the disc does not contain any other singularities and is isometrically embedded into $S'_j$. It can be glued from $2(m+1)$ copies of standard metric half-discs of radius $\varepsilon$; see the picture at the top of figure \[fig:breaking:up:a:zero\]. Let $\vec{v}\in{{\mathbb R}^{2}}$ be a vector of length $\delta<\varepsilon$. Following [@Eskin:Masur:Zorich] we may [*break up*]{} the zero $P$ of degree $m$ into a pair of zeroes of degrees $m'$ and $m''$ joined by a single saddle connection with affine holonomy $\vec{v}$.
(0,0)(0,-20) (10,-10)
(0,0)(0,0) (-154,-55)[$\scriptstyle \varepsilon$]{} (-125,-55)[$\scriptstyle \varepsilon$]{} (-38,-66)[$\scriptstyle 6\pi$]{}
(10,-30)
(0,0)(0,0) (-145,-148)[$\scriptstyle 2\delta$]{} (-160,-197)[$\scriptstyle \varepsilon+\delta$]{} (-130,-197)[$\scriptstyle \varepsilon-\delta$]{} (-160,-247)[$\scriptstyle \varepsilon-\delta$]{} (-130,-247)[$\scriptstyle \varepsilon+\delta$]{} (-39,-197)[$\scriptstyle 2\delta$]{} (-65,-197)[$\scriptstyle \varepsilon+\delta$]{} (-19,-197)[$\scriptstyle \varepsilon+\delta$]{} (-64,-173)[$\scriptstyle \varepsilon-\delta$]{} (-19,-173)[$\scriptstyle \varepsilon-\delta$]{} (-65,-217)[$\scriptstyle \varepsilon-\delta$]{} (-17,-217)[$\scriptstyle \varepsilon-\delta$]{}
We do this by changing the way of gluing the half-discs as indicated on the bottom picture of figure \[fig:breaking:up:a:zero\]. As patterns we still use the standard metric half-discs, but move the marked points on their diameters. Two special half-discs have two marked points on the diameter at distance $\delta$ from the center. Each of the remaining $2m$ half-discs has a single marked point at distance $\delta$ from the center. We alternate the half-discs with the marked point moved to the right and to the left of the center. The picture shows that all the lengths along identifications match; gluing the half-discs we obtain a topological disc with a flat metric. Now the flat metric has two cone-type singularities with cone angles $2\pi(m'+1)$ and $2\pi(m''+1)$. Here $2m'$ and $2m''$ are the numbers of half-discs with one marked point glued in between the distinguished pair of half-discs with two marked points.
The case when one of $m', m''$ (or both of them) is equal to zero is not excluded, in this case the corresponding “newborn” singularity is just a marked point.
Note that a small annular neighborhood of the boundary of the initial disc is isometric to the corresponding annular neighborhood of the boundary of the deformed disc. Thus, we can glue the deformed disc back into the surface. Gluing back we can turn it by any angle $\varphi$, where $0\le\varphi< 2\pi(m+1)$ in such way that the newborn saddle connection will have the prescribed affine holonomy $\vec v$.
Making a slit along the resulting saddle connection we get a surface $S_j$ with boundary having prescribed boundary type $+2.1$, a pair of boundary singularities of prescribed orders $k_{1,1}, k_{1,2}$, and a collection of interior singularities of prescribed orders $d_1, \dots, d_{s(j)}$ (see the corresponding entry in the table in section \[ss:tables\]). We have completed the proof of proposition \[pr:local:constructions\] for the boundary type $+2.1$.
Boundary types +3.1 and +4.1a {#boundary-types-3.1-and-4.1a .unnumbered}
-----------------------------
Boundary type $+3.1$ can be considered as a particular case of boundary type $+4.1a$. To see this compare the surfaces with boundary representing the corresponding ribbon graphs (see the appropriate entries in the table in section \[ss:tables\]). Marking a point in the middle of the saddle connection labelled by “$+2$” on the boundary of the surface of type $+3.1$ we get a surface with boundary type $+4.1a$, where the boundary singularity joining the pair of edges labelled by “$+1$” has order $0$.
Consider a local ribbon graph of type $+4.1a$, a collection $\{2m_1, \dots, 2m_n\}$ of orders of interior singularities and a collection $\{2a_1+1,2a_2,2a_3+1,2a_4\}$ of orders of four boundary singularities (see figure \[fig:embedded:local:ribbon:graphs\] for their parities). The singularity data $\beta'_j$ of the corresponding component ${{\mathcal H}}(\beta'_j)$ of the principal boundary stratum has the form $\beta'=\{m_1, \dots, m_n, a_1+a_2+a_3+a_4\}$, see equations and .
(0,0)(0,-150) (10,-6)
(0,0)(0,0) (-98,-199)[$\scriptstyle \varepsilon$]{} (20,-199)[$\scriptstyle \varepsilon-\delta$]{} (28,-185)[$\scriptstyle \varepsilon+\delta$]{} (3,-160)[$\scriptstyle \varepsilon-\delta$]{} (3,-209)[$\scriptstyle P_3$]{} (-9,-199)[$\scriptstyle P$]{} (-8,-173)[$\scriptstyle \varepsilon$]{} (-29,-178)[$\scriptstyle \varepsilon$]{} (-52,-199)[$\scriptstyle \varepsilon-\delta$]{} (-4,-209)[$\scriptstyle \delta$]{} (-20,-209)[$\scriptstyle \delta$]{} (-28,-199)[$\scriptstyle P_1$]{} (-29,-229)[$\scriptstyle \varepsilon-\delta$]{} (-58,-217)[$\scriptstyle \varepsilon+\delta$]{} (-8,-236)[$\scriptstyle \varepsilon$]{} (10,-232)[$\scriptstyle \varepsilon$]{} (125,-209)[$\scriptstyle P_3$]{} (89,-198)[$\scriptstyle P_1$]{} (112,-185)[$\scriptstyle P_4$]{} (112,-221)[$\scriptstyle P_2$]{}
Choose an Abelian differential $S'_j\in{{\mathcal H}}(\beta')$. As before denote the length of the shortest saddle connection on $S'_j$ by $4\varepsilon$. This time we split the distinguished zero $P$ of degree $a_1+a_2+a_3+a_4$ into [*three*]{} zeroes $P_1,P,P_3$ such that the zero $P_1$ of degree $a_1$ is joined to the zero $P$ of degree $a_2+a_4$ by a saddle connection, and the zero $P$ of degree $a_2+a_4$ is joined to the zero $P_3$ of degree $a_3$ by a saddle connection, see figure \[fig:4:1a\]. The two saddle connections have the same holonomy vector $\vec{v}$. We assume as before that $\|\vec{v}\|=\delta<{\varepsilon}$. We then cut along both saddle connections and detach the zero $P$ into two boundary singularities $P_2, P_4$ of orders $2a_2$ and $2a_4$ correspondingly, getting a surface $S_j$ with boundary of desired geometric combinatorial type (see the corresponding entry in the table in section \[ss:tables\]).
Boundary type +4.1b {#boundary-type-4.1b .unnumbered}
-------------------
Consider a local ribbon graphs of type $+4.1b$ and a corresponding collection $\{2m_1, \dots, 2m_n\}$ of orders of interior singularities and a collection $\{2a_1+1,2a_2+1,2a_3+1,2a_4+1\}$ of orders of four boundary singularities.
The singularity data $\beta'_j$ of the corresponding component ${{\mathcal H}}(\beta'_j)$ of the principal boundary stratum has the form $\beta'=\{m_1, \dots, m_n, a_1+a_2+a_3+a_4+1\}$, see equations and .
Choose an Abelian differential $S'_j\in {{\mathcal H}}(\beta'_j)$; let $P$ be a zero of $S'_j$ of degree $(a_1+a_2+a_3+a_4+1)$. We split $P$ into three zeroes $P_1,P,P_3$ such that the zero $P_1$ of degree $a_1$ is joined to the zero $P$ of degree $a_2+a_4+1$ by a saddle connection with a holonomy vector $\vec{v}$ and the zero $P_3$ of degree $a_3$ is also joined to $P$ by a saddle connection with the same holonomy vector $\vec{v}$, see figure \[fig:4:1b\]. Note that the new saddle connections are oriented differently than in the previous case. We then cut along both saddle connections and detach the zero $P$ into two boundary singularities $P_2, P_4$ of orders $2a_2+1$ and $2a_4+1$ correspondingly.
(0,0)(0,-150) (10,-6)
(0,0)(0,0) (-104,-199)[$\scriptstyle \varepsilon$]{} (-48,-199)[$\scriptstyle \varepsilon$]{} (-2,-198)[$\scriptstyle \delta$]{} (-20,-209)[$\scriptstyle \delta$]{} (20,-199)[$\scriptstyle \varepsilon$]{} (-50,-176)[$\scriptstyle \varepsilon-\delta$]{} (-31,-162)[$\scriptstyle \varepsilon+\delta$]{} (-9,-160)[$\scriptstyle \varepsilon-\delta$]{} (17,-162)[$\scriptstyle \varepsilon$]{} (32,-182)[$\scriptstyle \varepsilon$]{} (-48,-217)[$\scriptstyle \varepsilon$]{} (-30,-229)[$\scriptstyle \varepsilon$]{} (-9,-242)[$\scriptstyle \varepsilon-\delta$]{} (7,-229)[$\scriptstyle \varepsilon+\delta$]{} (21,-218)[$\scriptstyle \varepsilon-\delta$]{} (125,-209)[$\scriptstyle P_3$]{} (89,-198)[$\scriptstyle P_1$]{} (112,-185)[$\scriptstyle P_4$]{} (102,-221)[$\scriptstyle P_2$]{}
By construction the resulting surface $S_j$ with boundary has the desired boundary type “$+4.1b$” (see the corresponding entry in the table in section \[ss:tables\]), and collections of interior and boundary singularities of prescribed orders.
Boundary type +4.2a {#boundary-type-4.2a .unnumbered}
-------------------
Consider a local ribbon graphs of type $+4.2a$ and corresponding collection $\{2m_1, \dots, 2m_n\}$ of orders of interior singularities and collections $\{2a'+1,2a''+1\}$, $\{2b'+1,
2b''+1\}$ of orders of two pairs of boundary singularities (see figure \[fig:embedded:local:ribbon:graphs\] for their parities). The singularity data $\beta'_j$ of the corresponding component ${{\mathcal H}}(\beta'_j)$ of the principal boundary stratum has the form $\beta'=\{m_1, \dots, m_n, a'+a'',b'+b''\}$, see equations and .
Choose an Abelian differential $S'_j\in {{\mathcal H}}(\beta'_j)$; let $P_1$ be a zero of $S'_j$ of degree $a'+a''$; let $P_2$ be a zero of degree $b'+b''$. As in the case $+2.1$ we break each of the distinguished zeroes $P_1, P_2$ into a pair of zeroes of degrees $a',a''$ and $b',b''$ correspondingly. We apply the surgery in such way that each of the two corresponding pairs of zeroes is joined by a saddle connection with a holonomy vector $\vec{v}$. We then cut open the modified flat surface along the saddle connections. As a result we get a surface $S_j$ with boundary of the desired boundary type “$+4.2a$” (see the corresponding entry in the table in section \[ss:tables\]), and with collections of interior and boundary singularities of prescribed orders.
Proposition \[pr:local:constructions\] is proved.
Neighborhood of the principal boundary: nonlocal constructions {#s:Nonlocal:constructions}
==============================================================
Recall that a direction $\pm v\in{{\mathbb R}^{2}}\setminus\{0\}$ determines a corresponding line field on the flat surface and a foliation in direction $v$. The foliation is orientable if and only if $S$ has trivial linear holonomy. Such an auxiliary direction $v$ is an element of all our constructions. We are creating surfaces with boundary from closed flat surfaces; the direction $v$ is the direction of parallel geodesic segments which form the boundary components.
An interior singularity $P$ of order $d$ has $d+2$ adjacent separatrix rays (or just separatrices) of the foliation in direction $v$. They divide a disc of small radius ${\varepsilon}$ centered at $P$ into $d+2$ sectors, each with cone angle $\pi$ (see the top part of figure \[fig:breaking:up:a:zero\] which represents a singularity of order $4$). When $P$ is a regular point (a marked point) we still have two such adjacent sectors, each having cone angle $\pi$. When $P$ is a simple pole, we have a single separatrix adjacent to $P$; cutting an $\varepsilon$-neighborhood of $P$ by this separatrix we get a single sector. When we speak about “sectors” adjacent to a singularity we always mean the sectors bounded by a pair of neighboring separatrices of the foliation in direction $v$.
When a flat surface has trivial linear holonomy, the foliation parallel to $v$ is oriented by the choice of direction $\vec{v}$. The separatrix rays adjacent to any point $P$ inherit the natural orientation: incoming and outgoing rays alternate with respect to the natural cyclic order on the collection of rays adjacent to $P$. The sectors adjacent to any singularity $P$ are also naturally divided into two classes: the ones which are located to the right of the corresponding oriented separatrix rays and the ones which are located to the left. We shall refer to them as to the “right” and to the “left” sectors correspondingly.
In all nonlocal constructions we shall use a surgery along a smooth path without self-intersections joining a pair of singularities of a compact flat surface (sometimes joining a singularity to itself). This path $\rho$ (two paths in some constructions) will be always chosen to be transverse to the direction $v$ (and hence, transverse to the foliation in direction $v$); in particular, $\rho$ never passes through singularities. We shall often call such path a “transversal”.
The following theorem from [@Hubbard:Masur] gives us a key instrument for all nonlocal constructions:
Consider a closed flat surface $S$ with nontrivial linear holonomy, a pair of points $P_1,P_2$ on $S$, a direction $\pm
v\in{{\mathbb R}^{2}}\setminus\{0\}$ and a pair of sectors $\Sigma_i$ adjacent to the corresponding points $P_i$, $i=1,2$.
For any such data there exist a transversal $\rho$ with the endpoints at $P_1$ and $P_2$ which leaves $P_1$ in $\Sigma_1$ and arrives at $P_2$ in $\Sigma_2$. The case when $P_1$ and $P_2$ coincide, or even when $\Sigma_1$ and $\Sigma_2$ coincide is not excluded.
If $S$ has trivial linear holonomy the statement above is valid under additional assumption that one of the sectors is a “right” sector, and the other one is a “left” sector.
Parallelogram construction {#ss:Parallelogram construction}
--------------------------
In this section we extend the “parallelogram construction” from [@Eskin:Masur:Zorich] to flat surfaces with nontrivial linear holonomy. For more details (including restrictions on the choice of parameter $\delta$ in terms of the length $4\varepsilon$ of the shortest saddle connection on $S$, and generalization of the “parallelogram construction” to piecewise-transverse paths) we address the reader to the original paper [@Eskin:Masur:Zorich] and to the forthcoming paper [@Boissy:in:progress].
Consider a transversal $\rho$ as in the theorem above. In the construction below, we assume that if $S$ has trivial linear holonomy, then $P_1\neq P_2$. If $S$ has nontrivial linear holonomy, then we allow $P_1= P_2$ unless $P_1$ is a singularity of order $-1$. If $P_1=P_2$ we allow $\Sigma_1=\Sigma_2$.
Fix the orientation of $\rho$ from $P_1$ to $P_2$. Since the path is smooth, it has well-defined tangent directions $\vec{u}_1=\dot\rho|_{P_1}$ and $\vec{u}_2=\dot\rho|_{P_2}$ at the endpoints. If the surface has trivial linear holonomy we assume that the frame $\{\vec{u_i},\vec{v}\}$ represents the canonical orientation (upon interchanging, if necessary, the ordering of $P_1, P_2$).
For some interior point $P\in\rho$ let $\vec{u}=\dot\rho|_P$ be the vector tangent to $\rho$. Chose a vector $\vec{v}\in T_P(S)$ at $P$ parallel to $v$ such that the frame $\vec{u},\vec{v}$ represents the canonical orientation of the surface. Perform a parallel transport of $\vec{v}$ along $\rho$ to all points of $\rho$.
For a sufficiently small $\delta>0$ and any positive $s\le\delta$ we can construct a [*parallel shift*]{} $\rho_s$ of $\rho$ in direction $\vec{v}$ at the distance $s$. Suppose that $\Sigma_1,\Sigma_2$ do not coincide nor are adjacent. Then for any $0\le s_1< s_2\le \delta$ the corresponding shifts $\rho_{s_1}$ and $\rho_{s_2}$ do not intersect and do not have self-intersections. If $\Sigma_1$ and $\Sigma_2$ coincide or are adjacent, the same is true upon an appropriate choice of orientation of $\rho$. .
Let $\gamma_i$ be a segment of the separatrix ray in direction $\vec{v}$ at $P_i$ of length $\delta$. Even when $P_1=P_2$ (in particular, when $\Sigma_1=\Sigma_2$) the segments $\gamma_1\neq\gamma_2$ are well-defined. The interior of the domain $\Omega$ bounded by $\rho,\rho',\gamma_1,\gamma_2$ is homeomorphic to an open disc and can be thought of as a “curvilinear parallelogram”, see figure \[fig:paral\].
Remove $\Omega$ from $S$ and identify $\rho$ and $\rho'$ by a parallel translation. When $P_1\neq P_2$, as a result of this surgery we get a surface with two boundary components each with a single singular point, see figure \[fig:paral\]. When $P_1=P_2$ we detach the resulting boundary singularity into two getting a surface with a single boundary component with two boundary singularities, see figure \[fig:m:2:1:nonlocal\]. We refer to this surgery as to the [*parallelogram construction.*]{}
(0,0)(0,-150) (10,-6)
(0,0)(0,0) (-114,-152)[$\scriptstyle P_2'$]{} (-114,-183)[$\scriptstyle P_2$]{} (-173,-206)[$\scriptstyle P_1$]{} (-173,-162)[$\scriptstyle P_1'$]{} (-142,-159)[$\scriptstyle \rho'$]{} (-142,-198)[$\scriptstyle \rho$]{} (-142,-180)[$\scriptstyle \Omega$]{} (11,-171)[$\scriptstyle P_2'$]{} (12,-195)[$\scriptstyle P_2$]{} (-49,-206)[$\scriptstyle P_1$]{} (-49,-181)[$\scriptstyle P_1'$]{} (-20,-177)[$\scriptstyle \rho'$]{} (-20,-198)[$\scriptstyle \rho$]{} (133,-182)[$\scriptstyle P_2$]{} (74,-205)[$\scriptstyle P_1$]{} (105,-198)[$\scriptstyle \rho$]{}
If the parallelogram construction is applied to a pair of distinct points $P_1\neq P_2$, let $D_i$ be the order of the corresponding singularity $P_i\in S$, $i=1,2$. In the case when $P_1= P_2$, let $\pi(a_1+1)$ be the angle between $\gamma_1$ and $\gamma_2$ counted in the positive direction, and let $\pi(a_2+1)$ be the angle between $\gamma_1$ and $\gamma_2$ counted in the negative direction. By construction $a_i\ge 0$, $i=1,2$. The order of the singularity $P$ in this case is $a_1+a_2$.
In this notation, the orders of the boundary singularities of a surface obtained by a parallelogram construction are equal to $\{D_1+2\}, \{D_2+2\}$, when $P_1\neq P_2$ and to $\{a_1,a_2+2\}$ when $P_1=P_2$. To see this, when $P_1\neq P_2$ it is sufficient to observe figure \[fig:paral\]; in the remaining case it is sufficient to observe Figure \[fig:m:2:1:nonlocal\].
Nonlocal surgeries {#ss:nonlocal:surgeries}
------------------
The remaining constructions are a combination of one of the local constructions described in the previous section with a parallelogram construction. The parameters ${\varepsilon}, \delta$ are chosen as before.
\[pr:parallelogram:constructions\] Every surface of any of boundary types $+2.2$, $+3.2a$, $+4.2b$, $+3.2b$, $+4.2c$, $+4.3a$, $+3.3$, $+4.3b$, $+4.4$ is realizable by a combination of a local construction with a parallelogram construction.
Applying the same arguments as in the beginning of the proof of proposition \[pr:local:constructions\] we check that the singularity data $\beta'$ defined by equation from formal combinatorial data $\big({\mathbb G}_{v_j}$, $\{d_1, \dots, d_s\}$, $\{k_{1,1}, \dots,
k_{r,p(r)}\big)$ as in proposition \[pr:parallelogram:constructions\] represents a nonempty stratum ${{\mathcal H}}(\beta'_j)$. Having a closed flat surface $S'_j\in{{\mathcal H}}(\beta'_j)$ we now need to construct a surface $S_j$ with boundary realizing the initial combinatorial data $\big({\mathbb G}_{v_j}$, $\{d_1, \dots, d_s\}$, $\{k_{1,1}, \dots,
k_{r,p(r)}\big)$.
Boundary type +2.2 {#boundary-type-2.2 .unnumbered}
------------------
We begin with boundary type $+2.2$. All interior singularities $\{2m_1, \dots, 2m_s\}$ have positive even orders; each of the two boundary components contains a single boundary singularity. The boundary singularities also have positive even orders $2m',
2m''$ (see figure \[fig:embedded:local:ribbon:graphs\] and condition (4) of definition \[def:configuration\] of a configuration), so in this case $\beta'=\{m_1,\dots,m_s,
m'-1,m''-1\}$.
Choosing an Abelian differential $S'_j\in{{\mathcal H}}(\beta')$ and performing the parallelogram construction at the zeroes of degrees $m'-1,m''-1$ (see figure \[fig:paral\]) we get a flat surface $S_j$ with boundary of type “$+2.2$” (see the corresponding entry in the table in section \[ss:tables\]), having collections of interior and of boundary singularities of prescribed orders.
Boundary types +3.2a and +4.2b {#boundary-types-3.2a-and-4.2b .unnumbered}
------------------------------
Boundary type $+3.2a$ can be considered as a particular case of $+4.2b$ when one of the boundary singularities has order $0$ (see the appropriate entries in the table in section \[ss:tables\]).
Consider a ribbon graph of type $+4.2b$. Let $\{2m_1, \dots,
2m_s\}$ be a collection of orders of interior singularities. According to figure \[fig:embedded:local:ribbon:graphs\] the orders of all boundary singularities are even for boundary type $+4.2b$; denote by $2a_1,2a_2+2$ the orders of boundary singularities corresponding to the first boundary component and by $2a_3, 2a_4+2$ the orders of boundary singularities corresponding to the second component. By condition (4) of definition \[def:configuration\] of a configuration the numbers $a_i$ are nonnegative integers for $i=1,\dots,4$. We get $\beta'=\{m_1, \dots, m_s, a_1+a_2, a_3+a_4\}$.
Choose a flat surface $S'_j\in{{\mathcal H}}(\beta'_j)$. Choose a pair of separatrices $\gamma_1,\gamma_2$ in direction $\vec{v}$ adjacent to the first zero. Choose $\gamma_1$ to be an outgoing separatrix and $\gamma_2$ to be incoming separatrix in such way that the angle from the separatrix ray $\gamma_1$ to the separatrix ray $\gamma_2$ in the clockwise direction is $(2a_1+1)\pi$. Let $\Sigma_1$ be the sector adjacent to $\gamma_1$ counterclockwise; let $\Sigma_2$ be the sector adjacent to $\gamma_2$ clockwise.
Similarly, choose a pair of separatrices $\gamma_3,\gamma_3$ in direction $\vec{v}$ adjacent to the zero of degree $a_3+a_4$ in such way that $\gamma_3$ is outgoing, $\gamma_4$ is incoming; the counterclockwise angle from $\gamma_3$ to $\gamma_4$ is $(2a_3+1)\pi$. Let $\Sigma_3$ be the sector adjacent to $\gamma_3$ clockwise; let $\Sigma_4$ be the sector adjacent to $\gamma_4$ counterclockwise.
Join $\Sigma_3$ to $\Sigma_1$ by a transversal $\rho_1$; join $\Sigma_2$ to $\Sigma_4$ by a transversal $\rho_2$. If $\rho_1$ intersects $\rho_2$ we can resolve the intersections to achieve nonintersecting transversals.
Suppose that in resolving the intersections (if any) we did not change the correspondence between the sectors and $\rho_1$ still joins $\Sigma_3$ to $\Sigma_1$ and $\rho_2$ joins $\Sigma_2$ to $\Sigma_4$. Choosing some small $\delta$ we can apply parallelogram construction to the transversal $\rho_1$ and the direction $\vec{v}$ and to the transversal $\rho_2$ and the direction $-\vec{v}$, see Figure \[fig:42b\].
In the remaining case after the resolution of intersections, the correspondence between the sectors changed and the transversal $\rho_1$ joins sector $\Sigma_4$ to $\Sigma_1$ while the transversal $\rho_2$ joins $\Sigma_2$ to $\Sigma_3$. In this case we deform the transversals slightly in such way that they still do not intersect and $\rho_2$ lands on the ray $\gamma_3$ at a distance $\delta$ from the zero (in the same sector $\Sigma_3$) and $\rho_1$ starts at a point on the ray $\gamma_4$ at a distance $\delta$ from the zero (in the same sector $\Sigma_4$). We can construct two “curvilinear parallelograms” $\Omega_1,
\Omega_2$ (see figure \[fig:42b\]) which do not intersect, so we can proceed as above.
(0,0)(0,-165) (10,-6)
(0,0)(0,0) (-114,-186)[$\scriptstyle P'$]{} (-114,-200)[$\scriptstyle \gamma_1$]{} (-126,-210)[$\scriptstyle \Sigma_1$]{} (-114,-218)[$\scriptstyle P$]{} (-126,-227)[$\scriptstyle \Sigma_2$]{} (-114,-231)[$\scriptstyle \gamma_2$]{} (-114,-245)[$\scriptstyle P''$]{} (-165,-187)[$\scriptstyle \rho_1'$]{} (-165,-205)[$\scriptstyle \rho_1$]{} (-165,-228)[$\scriptstyle \rho_2$]{} (-165,-247)[$\scriptstyle \rho_2'$]{} (7,-186)[$\scriptstyle P'$]{} (1,-223)[$\scriptstyle P_1$]{} (12,-213)[$\scriptstyle P_2$]{} (7,-245)[$\scriptstyle P''$]{} (-45,-187)[$\scriptstyle \rho_1'$]{} (-45,-205)[$\scriptstyle \rho_1$]{} (-45,-228)[$\scriptstyle \rho_2$]{} (-45,-247)[$\scriptstyle \rho_2'$]{} (110,-218)[$\scriptstyle P_1$]{} (130,-218)[$\scriptstyle P_2$]{} (60,-204)[$\scriptstyle \rho_1$]{} (60,-229)[$\scriptstyle \rho_2$]{}
Detaching each of the resulting singularities into pairs $P_1,
P_2$ and $P_3, P_4$ (see figure \[fig:42b\]) we get the desired surface $S_j$ with boundary of type “$+4.2b$” (see the corresponding entry in the table in section \[ss:tables\]), and with prescribe collections of interior and of boundary singularities.
Recall that if we identify the opposite sides of each hole of a surface constructed above we obtain a closed surface with a pair of even order zeroes simultaneously broken up into a pair of odd order zeroes.
Boundary types +3.2b and +4.2c {#boundary-types-3.2b-and-4.2c .unnumbered}
------------------------------
Boundary type $+3.2b$ can be considered as a particular case of boundary type $+4.2c$. To see this compare the surfaces with boundary representing the corresponding ribbon graphs (see the appropriate entries in the table in section \[ss:tables\]). Marking a point in the middle of the saddle connection labelled by “$+2$” on the boundary of the surface of type $+3.2b$ we get a surface of boundary type $+4.2c$ with the corresponding boundary singularity of order $0$.
Consider a ribbon graph of type $+4.2c$. Let $\{2m_1, \dots,
2m_s\}$ be a collection of orders of interior singularities. Let $2a_1+1,2a_2,2a_3+1$ be the orders of the boundary singularities on the boundary component composed from three saddle connections; let $2a_4+2$ be the order of the single boundary singularity on the complementary boundary component, see figure \[fig:embedded:local:ribbon:graphs\]. By condition (4) of definition \[def:configuration\] of a configuration the numbers $a_i$ are nonnegative integers for $i=1,\dots,4$. We get $\beta'=\{m_1, \dots, m_s, a_1+a_2+a_3, a_4\}$.
Choose a flat surface $S'_j\in{{\mathcal H}}(\beta'_j)$. Let $P$ be the zero of degree $a_1+a_2+a_3$, and $P_4$ be the zero of degree $a_4$. Choose a separatrix $\gamma_1$ in direction $\vec{v}$ adjacent to $P_4$ and a separatrix $\gamma_2$ in direction $\vec{v}$ adjacent to $P$. Let $\Sigma_1$ be the “right” sector adjacent to $\gamma_1$; let $\Sigma_2$ be the “left” sector adjacent to $\gamma_2$, see figure \[fig:42c\]. Join $\Sigma_1$ to $\Sigma_2$ by a separatrix $\rho$. Choose $\delta$ small enough, so that the intersection of $\rho$ with an $\delta$-neighborhood of $P$ contains a single connected component contained in $\Sigma_2$.
(0,0)(0,-165) (10,-6)
(0,0)(0,0) (-173,-236)[$\scriptstyle P_4$]{} (-113,-187)[$\scriptstyle P'$]{} (-112,-216)[$\scriptstyle P$]{} (-113,-249)[$\scriptstyle P_1$]{} (-152,-190)[$\scriptstyle \rho'$]{} (-152,-228)[$\scriptstyle \rho$]{} (-53,-236)[$\scriptstyle P_4$]{} (7,-187)[$\scriptstyle P'$]{} (8,-216)[$\scriptstyle P''$]{} (26,-216)[$\scriptstyle P_2$]{} (7,-249)[$\scriptstyle P_1$]{} (-32,-190)[$\scriptstyle \rho'$]{} (-32,-228)[$\scriptstyle \rho$]{} (67,-236)[$\scriptstyle P_4$]{} (127,-214)[$\scriptstyle P_3$]{} (157,-227)[$\scriptstyle P_2$]{} (127,-249)[$\scriptstyle P_1$]{} (88,-228)[$\scriptstyle \rho$]{}
Choose a separatrix $\gamma_3$ at $P$ such that the angle from $\gamma_2$ to $\gamma_3$ (in the counterclockwise direction) equals $\pi(2a_3+1)$. Break the zero $P$ along $\gamma_3$ into two zeroes $P$ and $P_1$ of degrees $a_2+a_3$ and $a_1$ correspondingly joined by a saddle connection in direction $\vec{v}$ of length $\delta$ and perform the parallelogram construction along $\rho$ (strictly speaking to a transversal naturally corresponding to $\rho$), see figure \[fig:42c\]. Detaching $P$ into two points we obtain a surface of type “$+4.2c$” with the desired singularity data.
Boundary type +4.3a {#boundary-type-4.3a .unnumbered}
-------------------
Let $\{2a_1+2\}$, $\{2a_2+2\}$, $\{2a_3+1, 2a_4+1\}$ be the orders of the boundary singularities naturally distributed into the corresponding boundary components, see figure \[fig:embedded:local:ribbon:graphs\]. By condition (4) of definition \[def:configuration\] of a configuration the numbers $a_i$ are nonnegative integers for $i=1,\dots,4$. We get $\beta'=\{m_1, \dots, m_s, a_1, a_2, a_3+a_4\}$.
Surfaces with boundary of this type are obtained by a trivial combination of a parallelogram construction applied to a pair of distinct zeroes of degrees $a_1$ and $a_2$ and by breaking up a zero of degree $a_3+a_4$ into two zeroes of degrees $a_3, a_4$ with a subsequent slit along the resulting saddle connection.
Boundary types +3.3 and +4.3b {#boundary-types-3.3-and-4.3b .unnumbered}
-----------------------------
The boundary type $+3.3$ can be considered as a particular case of the boundary type $+4.3b$ when the appropriate boundary singularity has order $0$.
Surfaces of type $+4.3b$ can be constructed in complete analogy with surfaces of type $+4.2b$ (see figure \[fig:42b\]) with the only difference that now we choose sectors $\Sigma_3$ and $\Sigma_4$ at two distinct points.
Boundary type +4.4 {#boundary-type-4.4 .unnumbered}
------------------
In this case the orders of boundary singularities have the form $\{2a_1+2, 2a_2+2,2a_3+2,2a_4+2\}$; according to condition (4) of definition \[def:configuration\] of a configuration all numbers $a_i$ are nonnegative integers. Thus, in this case we get $\beta'=\{m_1, \dots, m_s, a_1, a_2, a_3, a_4\}$. To construct a desired surface with boundary of type $+4.4$ it is sufficient to apply a pair of independent parallelogram constructions.
Proposition \[pr:parallelogram:constructions\] is proved.
Surfaces with boundary of “–” type {#ss:minus:type}
----------------------------------
To complete the proof of proposition \[pr:realizability:of:all:vertices\] it remains to construct surfaces with boundary realizing any combinatorial data $\big({\mathbb G}_{v_j}$, $\{d_1, \dots, d_s\}$, $\{k_{1,1}, \dots,
k_{r,p(r)}\big)$ satisfying conditions 2–6 of definition \[def:configuration\] for local ribbon graphs ${\mathbb G}_{v_j}$ of “$-$”-types.
\[pr:minus:2:2\] Combinatorial data representing boundary types $-2.2, -1.1$ and $-2.1$ are realizable by appropriate surfaces with boundaries.
(See initial proposition \[pr:realizability:of:all:vertices\] for the detailed formulation.)
The component of the principle boundary stratum corresponding to a vertex $v_j$ of “$-$”-type has type ${{\mathcal Q}}(\alpha'_j)$. The singularity data $\alpha'_j$ is given by equation , namely $$\alpha'_j=\{d_1, \dots, d_{s(j)},\ D_1, \dots, D_{r(j)}\},$$ where $d_1, \dots, d_{s(j)}$ are the orders of interior singularities, and $D_1, \dots, D_{r(j)}$ are expressed in terms of the orders of boundary singularities by formula . Conditions 4 and 5 in definition \[def:configuration\] of a configuration guarantee that all the entries of $\alpha'_j$ are from the set $\{-1,0,1,2,\dots \}$, that the total sum of the entries of $\alpha'_j$ is divisible by $4$ and that this sum is greater than or equal to $-4$. Moreover, condition 6 in definition implies that $\alpha'_j$ neither belongs to the exceptional list given by equation below, nor can be obtained from an entry of this list by adding additional elements “$0$” (see lemma \[lm:nonrealizable:data:combinatrorics\] in the next section). According to the results of the paper [@Masur:Smillie:realizability] this implies that the stratum ${{\mathcal Q}}(\alpha'_j)$ is a nonempty.
Consider any flat surface $S'_j$ in ${{\mathcal Q}}(\alpha'_j)$. We use the same conventions on parameters $\delta, {\varepsilon}$, and $v$ as in the proof of proposition \[pr:local:constructions\]. Applying an appropriate surgery to the closed surface $S'_j$ we are going to construct a surface $S_j$ with boundary realizing the initial combinatorial data $\big({\mathbb G}_{v_j}$, $\{d_1, \dots, d_s\}$, $\{k_{1,1}, \dots, k_{r,p(r)}\big)$.
Boundary type –2.2 {#boundary-type-2.2-1 .unnumbered}
------------------
Boundary type $-2.2$ is constructed in complete analogy to $+2.2$ by a parallelogram construction. Each of the two boundary components contains a single boundary singularity. The boundary singularities have strictly positive orders $k_{1,1}, k_{2,1}$ (see inequality on $D_i$ in condition (4) of definition \[def:configuration\] of a configuration), so in this case $\alpha'=\{d_1,\dots,d_s, k_{1,1}-2, k_{2,1}-2\}$.
Choosing a quadratic differential $S'\in{{\mathcal Q}}(\alpha')$ and performing the parallelogram construction at the zeroes of orders $k_{1,1}-2, k_{2,1}-2$ (see figure \[fig:paral\]) we get a flat surface $S_j$ with boundary of type “$-2.2$”, having collections of interior and of boundary singularities of prescribed orders.
Boundary types –1.1 and –2.1 {#boundary-types-1.1-and-2.1 .unnumbered}
----------------------------
Note next that boundary type $-1.1$ can be considered as a particular case of boundary type $-2.1$ when one of the two boundary singularities has order $0$.
Consider a ribbon graph of type $-2.1$. Let $\{d_1, \dots, d_s\}$ be the orders of interior singularities, let $\{k_{1,1},
k_{1,2}\}$ be the orders of boundary singularities. By condition (4) of definition \[def:configuration\] we have $D_1\ge -1$, where $D_1=k_{1,1}+k_{1,2}-2$, which implies that nonnegative integers $k_{1,1}, k_{1,2}$ cannot be simultaneously equal to zero. Thus, we may assume that $k_{1,1}\ge 1$. We get $\alpha'_j=\{d_1,\dots,d_s, k_{1,1}+k_{1,2}-2\}$.
Consider a flat surface $S'_j$ in ${{\mathcal Q}}(\alpha'_j)$. When both $k_{1,1},k_{1,2}$ are odd we can break up the zero of order $k_{1,1}+k_{1,1}-2$ into a pair of zeroes of orders $k_{1,1}-1$ and $k_{1,2}-1$ as in figure \[fig:breaking:up:a:zero\].
When one of $k_{1,1},k_{1,2}$ is odd and another one is even we can break up the zero of order $k_{1,1}+k_{1,1}-2$ into a pair of zeroes of orders $k_{1,1}-1$ and $k_{1,2}-1$ by a similar construction, see figure \[fig:minus:2:1:local\]. (Recall that by convention a “zero of order $-1$” is a simple pole of the corresponding meromorphic quadratic differential.) Cutting along the saddle connection we obtain the desired surface of type $-2.1$ with prescribed orders of interior and boundary singularities.
(0,0)(0,-140) (10,-6)
(0,0)(0,0) (-160,-199)[$\scriptstyle \varepsilon$]{} (-5,-199)[$\scriptstyle P_2$]{} (-25,-175)[$\scriptstyle \varepsilon-\delta$]{} (-52,-199)[$\scriptstyle \varepsilon+\delta$]{} (-10,-209)[$\scriptstyle \delta$]{} (-23,-199)[$\scriptstyle P_1$]{} (-25,-229)[$\scriptstyle \varepsilon-\delta$]{} (20,-189)[$\scriptstyle \varepsilon$]{} (20,-216)[$\scriptstyle \varepsilon$]{} (126,-199)[$\scriptstyle P_2$]{} (87,-199)[$\scriptstyle P_1$]{}
When both $k_{1,1},k_{1,2}$ are even, in fact $k_{1,1}\geq 2$ and $k_{1,1}+k_{1,2}-2\ge 0$. Let $P$ be the zero of order $k_{1,1}+k_{1,2}-2$ of the quadratic differential representing the flat surface $S'_j$. Choose a pair of separatrices $\gamma_1,
\gamma_2$ in such way that the angle from $\gamma_1$ to $\gamma_2$ counted counterclockwise is $\pi(k_{1,2}+1)$. Let $\Sigma_1$ be the sector adjacent to $\gamma_1$ in the clockwise direction and $\Sigma_2$ be the sector adjacent to $\gamma_2$ in the counterclockwise direction. Apply the parallelogram construction to $\Sigma_1, \Sigma_2$ and detach $P$ into two singularities $P_1, P_2$ (see figure \[fig:m:2:1:nonlocal\]). The orders of the boundary singularities of the resulting surface $S_j$ are $k_{1,1}$ and $k_{1,2}$.
(0,0)(0,-150) (28,12)
(0,0)(0,0) (-114,-200)[$\scriptstyle \gamma_2$]{} (-127,-208)[$\scriptstyle \Sigma_2$]{} (-127,-227)[$\scriptstyle \Sigma_1$]{} (-114,-231)[$\scriptstyle \gamma_1$]{}
We have completed the proof of proposition \[pr:minus:2:2\] and, thus, the proof of proposition \[pr:realizability:of:all:vertices\].
Now we are ready to prove theorem \[th:from:boundary:to:neighborhood:of:the:cusp\]. Note that theorem \[th:from:boundary:to:neighborhood:of:the:cusp\] immediately implies the missing realizability parts of theorems \[th:graphs\] and \[th:all:local:ribon:graphs\].
Consider a configuration ${{\mathcal C}}$ (in the sense of the formal combinatorial definition \[def:configuration\]). Let ${{\mathcal Q}}(\alpha'_{{{\mathcal C}}})$ (resp. ${{\mathcal H}}(\beta'_{{{\mathcal C}}}$) be the principal boundary stratum corresponding to the configuration ${{\mathcal C}}$. Let $S'$ be a (possibly nonconnected) flat surface in ${{\mathcal Q}}(\alpha'_{{{\mathcal C}}})$ (resp. ${{\mathcal H}}(\beta'_{{{\mathcal C}}}$). To every connected component of $S'$ apply the appropriate surgeries as in sections \[s:Local:Constructions\] and \[s:Nonlocal:constructions\] realizing the corresponding local ribbon graphs. We apply the surgeries in such a way that the saddle connections on the boundary of each surface $S_j$ are, say, horizontal, and have length proportional to their weight in $\Gamma$ with coefficient $\delta$. For every [“$\circ$”]{}-vertex of $\Gamma$ consider an appropriate flat cylinder, with the same requirement for the boundary. Now we glue a compound surface from the components $S_j$ as prescribed by the graph $\Gamma$. By construction the result is a closed surface $S$ endowed with a flat metric with linear holonomy restricted to $\{Id, -Id\}$.
By construction each flat surface $S_j$ with boundary is endowed with the canonical orientation. By definition the global ribbon graph ${\mathbb G}({{\mathcal C}})$ is endowed with the canonical orientation compatible with the canonical orientation of the embedded local ribbon graphs. This implies that the resulting closed surface $S$ inherits the canonical orientation. By construction $S$ has a collection of saddle connections $\gamma_1, \dots, \gamma_n$ realizing the configuration ${{\mathcal C}}$.
It remains to prove that $S$ is nonsingular, i.e. that it does not have any double (triple, ...) points. Suppose it does. Detaching them we get a nonsingular closed flat surface $\tilde
S$. By construction $\tilde S$ still has a collection of saddle connections $\gamma_1, \dots, \gamma_n$ realizing the configuration ${{\mathcal C}}$, which means that assembling the initial surface $S$ we have performed some superfluous identifications of several points of $\tilde S$.
Nonrealizable collections of singularities {#ss:nonrealizable:configurations}
------------------------------------------
It was proved in [@Masur:Smillie:realizability] that for the following exceptional list $\{\alpha''_1, \dots, \alpha''_4\}$ of singularity data $$\label{eq:empty:strata}
\big\{{\varnothing},\ \{1,-1\},\ \{3,1\},\ \{4\}\big\}$$ the four corresponding strata ${{\mathcal Q}}(\alpha''_j)$ are empty. It is clear, that completing any of these lists with entries “$0$” (which stand for marked points) we also get an empty stratum. This gives rise to restriction 6 in definition \[def:configuration\] of a configuration which we justify in this section.
Let ${\mathbb G}_{v_j}$ be a local ribbon graph of one of types $-1.1$, $-2.1$, $-2.2$ and let $\{d_1, \dots, d_s\}$, $\{k_{1,1}, \dots,
k_{r,p(r)}\}$ be a couple of unordered collections of integers satisfying conditions 4 and 5 of definition \[def:configuration\] of a configuration. (In our formal combinatorial definition they represent orders of interior and of boundary singularities of a virtual flat surface with boundary.) Applying formally equations and (which evaluate the singularity data of the corresponding component of the virtual principal boundary stratum) to our combinatorial data we obtain an unordered collection $\alpha'_j$ of integers. Consider a collection $\alpha''_j$ obtained from $\alpha'_j$ by omitting all entries “$0$” (if any).
\[lm:nonrealizable:data:combinatrorics\] The collection $\alpha''_j$ belongs to the exceptional list if and only if the combinatorial data $\big({\mathbb G}_{v_j}$, $\{d_1, \dots, d_s\}$, $\{k_{1,1}, \dots,
k_{r,p(r)}\big)$ as above belongs to the list (6) in definition \[def:configuration\] of a configuration.
The proof of the lemma is an exercise in elementary combinatorics.
Haven justified the combinatorial obstructions we complete this section with the corresponding geometric lemma.
\[lm:nonrealizable:data\] Let $S_j$ be a flat surface with boundary of one of types $-1.1$, $-2.1$, $-2.2$. Assume that $S_j$ does not have any saddle connections parallel to the boundary different from those which belong to the boundary. Then the corresponding combinatorial data $\big({\mathbb G}_{v_j}$, $\{$unordered collection of interior singularities$\}$, $\{$unordered collection of boundary singularities$\}\big)$ does not belong to the exceptional list (6) in definition \[def:configuration\] of a configuration.
We use the following strategy to prove the lemma. If some surface $S_j$ with boundary would define an entry from the list (6) in definition \[def:configuration\] we would shrink the boundary of $S_j$ to get as a limit a nondegenerate surface $S'_j$ from the corresponding component ${{\mathcal Q}}(\alpha'_j)$ of the principal boundary stratum. However, lemma \[lm:nonrealizable:data:combinatrorics\] implies that such ${{\mathcal Q}}(\alpha'_j)$ is empty, which leads to a contradiction.
To complete the proof we need to describe how can one “shrink the boundary” of a flat surface. First note, that boundary type “$-1.1$” can be considered as a particular case of boundary type “$-2.1$” when the order of one of the boundary singularities is equal to zero (see the corresponding surfaces with boundary in the table in section \[ss:tables\]).
Having a surface $S_j$ of type $-2.1$ we can isometrically identify the pair of boundary components to get a closed flat surface $S$. The corresponding singularity data $\alpha$ of $S\in{{\mathcal Q}}(\alpha)$ is expressed in terms of the singularity data of $S_j$ as follows: $$\alpha=\{d_1, \dots, d_{s(j)}, k_{1,1}-1, k_{1,2}-1\}$$ This implies that the couples $[\{d_1, \dots, d_{s(j)}\}$, $\{k_{1,1}, k_{1,2}\}]$ of collections of orders of interior and of boundary singularities in the list below $$\begin{array}{c}
{\varnothing}, \{1, 1\}\qquad
{\varnothing}, \{2, 0\}\\
\{1,-1\}, \{1,1\};\qquad
\{1\}, \{1,0\};\qquad
\{-1\},\{2,1\}\\
\{3,1\},\{1,1\};\qquad
\{3\},\{2,1\};\qquad
\{1\}, \{4,1\}\\
\{4\}, \{1,1\};\qquad
{\varnothing}, \{5,1\};\qquad
{\varnothing}, \{4,2\}.
\end{array}$$ are not realizable by any surface $S_j$ with boundary of type $-2.1$, for in these cases we would get a flat surface $S$ from an empty stratum, see equation .
In the remaining cases we get a closed surface $S\in{{\mathcal Q}}(\alpha)$ with a distinguished pair of singularities $P_0, P_1$ joined by a distinguished saddle connection $\gamma$. By assumptions of the lemma this saddle connection is not parallel to any other saddle connection on $S$. This implies that deforming, if necessary, $S$ and then applying an appropriate element of $SL(2,{{\mathbb R}})$ the surface $S$ can be continuously deformed inside ${{\mathcal Q}}(\alpha)$ to a surface $\tilde S$ with a single [*short*]{} saddle connection $\tilde\gamma$ and with no other short saddle connections. The deformation might be performed in such a way that the conical singularities $\tilde P_0, \tilde P_1$ serving as endpoints of $\tilde\gamma$ would have the same cone angles as $P_0$ and $P_1$ correspondingly. But then we would apply an appropriate surgery inverse to the one presented in figures \[fig:breaking:up:a:zero\], \[fig:minus:2:1:local\] or \[fig:m:2:1:nonlocal\] to coalesce the corresponding pair of zeroes into one. This would give a nondegenerate flat surface $S'$. Forgetting, if necessary, the resulting marked points on $S'$ we get $S'\in{{\mathcal Q}}(\alpha'')$, where $\alpha''$ is in the list (see lemma \[lm:nonrealizable:data:combinatrorics\]). The latter leads to a contradiction since these strata are empty.
The proof in the case of boundary type $-2.2$ is completely analogous. Lemma \[lm:nonrealizable:data\] is proved.
Long saddle connections {#ap:Long:saddle:connections}
=======================
We recall the definition of the natural $GL(2;{{\mathbb R}^{}})$-invariant measure in the stratum ${{\mathcal Q}}(\alpha)$. Let $\hat P = p^{-1}(P)$ be the collection of preimages of the singularities of a flat surface $S\in{{\mathcal Q}}(\alpha)$. Let $H_1^-(\hat S,\hat P;\,{\mathbb Z})$ be the subgroup in the relative homology group of $\hat S$, odd with respect to the involution $\tau$. Similarly, let $H^1_-(\hat
S,\hat P;\,{{\mathbb C}^{}})$ be the subspace in the relative cohomology odd with respect to the involution $\tau$ (i.e. the invariant subspace corresponding to the eigenvalue $-1$ of the induced linear involution $\tau^\ast: H^1(\hat S,\hat P;\,{{\mathbb C}^{}})\to
H^1(\hat S,\hat P;\,{{\mathbb C}^{}})$). We can choose a basis in $H_1^-(\hat
S,\hat P; {\mathbb Z})$ obtained as lifts $\hat\gamma_i$, $i=1, \dots,
\dim_{{{\mathbb C}^{}}}{{\mathcal Q}}(\alpha)$, of a collection of saddle connections on $S$. For any surface near $S$ the affine holonomy vectors $\int_{\hat\gamma}\omega$ serve as local coordinates for ${{\mathcal Q}}(\alpha)$. We define a measure $d\nu(S)$ on ${{\mathcal Q}}(\alpha)$ as Lebesgue measure defined by these coordinates, normalized so that the volume of a fundamental domain of the integer lattice in $$H^1_{-}(\hat S,\hat P;{{\mathbb Z}}\oplus i{{\mathbb Z}})
\subset H_-^1(\hat S,\hat P;{{\mathbb C}})$$ is equal to one.
Note that the Abelian differential $\omega$ on $\hat S$ has a regular point at the preimage $P_i'\in p^{-1}(P_i)$ of a simple pole $P_i$ of the quadratic differential $q$ on $S$. Consider the set $\tilde P\subseteq \hat P$ obtained by removing these regular points. It is easy to see that the canonical homomorphism $H_-^1(\hat S,\hat P;{{\mathbb C}})\to H_-^1(\hat S,\tilde P;{{\mathbb C}})$ induced by the inclusion $\tilde P\subseteq \hat P$ is actually an isomorphism. Thus, it does not matter which of two sets $\hat P,
\tilde P$ is used to define the coordinate charts.
Let ${{\mathcal C}}$ be an admissible configuration of ĥomologous saddle connections. Let $\gamma=\{\gamma_1, \dots, \gamma_n\}$ be a collection of ĥomologous saddle connections on the flat surface $S_0$ representing configuration ${{\mathcal C}}$. Choose some saddle connection $\gamma_i$ corresponding to an edge of weight $1$ of the graph $\Gamma(S,\gamma)$; such edge always exists, see figure \[fig:classification:of:graphs\]. We associate to the collection $\gamma$ a pair of vectors $\pm \vec{v}(\gamma) \in
{{\mathbb R}^{2}}$ setting $v=\int_{\gamma_i}\omega\in{{\mathbb C}}\cong{{\mathbb R}^{2}}$. For every surface $S$ in the same connected component we consider the discrete subset $V_{{\mathcal C}}(S)$ by taking the union $V_{{\mathcal C}}(S)=\cup \pm
v(\gamma)$ over all collections of ĥomologous saddle connections $\gamma$ realizing ${{\mathcal C}}$.
It is easy to see that the set $V_{{\mathcal C}}(S)$ satisfies axioms $(A)$, $(B)$, $(C_\mu)$ in [@Eskin:Masur]. Proposition \[pr:counting\] now follows from the general results in [@Eskin:Masur] and from theorem \[th:from:boundary:to:neighborhood:of:the:cusp\] which implies that the Siegel–Veech constant $const_{{\mathcal C}}$ is nonzero.
If saddle connections $\gamma_1$ and $\gamma_2$ are parallel, then $\int_{\hat\gamma_1}\omega= r \int_{\hat\gamma_2} \omega$ for $r$ real. If $\gamma_1$ and $\gamma_2$ are not ĥomologous then the homology classes of the lifts $\hat\gamma_1$ and $\hat\gamma_2$ are independent in $H_1^{-}(\hat S,\hat P;\,{\mathbb Z})$. Then the above equation holds only for a set of measure zero in $H^1_{-}(\hat S,\hat P;\,{{\mathbb C}})$. Taking a countable union of sets of measure zero corresponding to possible pairs of cycles and different coordinate charts, we see that two nonĥomologous saddle connections on $S$ are parallel only for a set of $S$ of measure zero.
Suppose that there are two saddle connections $\gamma_1,
\gamma_2$ in the collection which are not ĥomologous. Then the corresponding periods $\int_{\hat\gamma_1}\omega$ and $\int_{\hat\gamma_1}\omega$ correspond to two independent coordinates in a small neighborhood of the initial flat surface, and hence they can be deformed independently. Since the length $|\gamma|$ equals $|\int_{\hat\gamma}\omega|$ or $1/2|\int_{\hat\gamma}\omega|$ (depending on whether $\gamma$ is homologous to zero or not), we conclude that a collection containing two nonĥomologous saddle connections cannot be rigid.
The necessity of the condition in proposition \[pr:rigid:configurations:hat:homologous\] is proved. Sufficiency immediately follows from lemma \[lm:gamma:2gamma\] which says that the lengths of ĥomologous saddle connections are either the same or differ by a factor of two.
List of configurations in genus 2 {#a:List:of:configurations:in:genus:2}
=================================
Using definition \[def:configuration\], theorem \[th:from:boundary:to:neighborhood:of:the:cusp\] and corollary \[cr:unique:choice:of:parities\], and following examples \[ex:example:of:a:configuration\] and \[ex:principal:boundary\] in section \[s:structure:of:the:paper\] one can construct a complete list of configurations for any given stratum ${{\mathcal Q}}(\alpha)$. In this section we present an outline of the algorithm and list all configurations for holomorphic quadratic differentials in genus 2.
There are two natural parameters measuring “complexity” of singularity data $\alpha=\{d_1, \dots, d_m\}$: the genus $g$ of a flat surface $S$ in ${{\mathcal Q}}(\alpha)$ and the number $N$ of simple poles on $S$ (i.e. the number of conical points with the cone angle $\pi$). Having a configuration ${{\mathcal C}}$ denote by $N'$ the number of interior singularities of order $-1$ corresponding to this configuration and by $g'_1, \dots, g'_k$ the genera of surfaces $S'_1, \dots, S'_k$ corresponding to the principal boundary ${{\mathcal Q}}(\alpha'_{{\mathcal C}})$ (correspondingly ${{\mathcal H}}(\beta'_{{\mathcal C}})$ when ${{\mathcal C}}$ does not have “$-$”-vertices). It is easy to see that the number of simple poles on $S$ (i.e. the number of entries “$-1$” of $\alpha$) might vary from $N'$ to $N'+4$, and that the genus $g$ might vary from $\sum_{j=1}^k g'_j$ to $\sum_{j=1}^k g'_j + 2$ (see [@Boissy:to:appear] for an explicit expression of $g(S)$ in terms of genera $g'_j$ of components and of a structure of the global ribbon graph). Thus, having fixed the upper bounds for $g$ and $N$, we confine the list of corresponding configurations to a finite one.
A naive algorithm of enumeration of all configurations for a given stratum ${{\mathcal Q}}(\alpha)$ can be represented as follows. Let $g=g(\alpha)$ be the genus corresponding to the singularity data $\alpha$, $$d_1+\dots+d_m=4g(\alpha)-4$$ Consider complete lists of (possibly disconnected) strata ${{\mathcal H}}(\beta')$ of genera $g-2$, $g-1$, $g$. These lists are finite and can be easily constructed. Consider complete lists of (possibly disconnected) strata ${{\mathcal Q}}(\alpha')$ of genera $g-2,
g-1$, $g$ such that $\alpha'$ contains from $N-4$ to $N$ entries “$-1$” and at most two connected components $\alpha'_i,
\alpha'_j$ representing strata of quadratic differentials ${{\mathcal Q}}(\alpha'_i)$, ${{\mathcal Q}}(\alpha'_j)$ (the remaining connected components are represented by strata of holomorphic differentials ${{\mathcal H}}(\alpha'_l)$). These lists are also finite and can be easily constructed. Add the empty set to these lists when $0\le g \le
2$.
For every entry $\alpha'=\alpha'_1\sqcup\dots\sqcup\alpha'_k$ (correspondingly $\beta'$) as above consider all possible ways to organize the set $\{\alpha'_1,\dots,\alpha'_k\}$ into one of the graphs as in figure \[fig:classification:of:graphs\], in such way that vertices corresponding to the strata ${{\mathcal H}}(\alpha'_j)$, ${{\mathcal H}}(\beta'_j)$ have “$+$”-type, and vertices corresponding to the strata ${{\mathcal Q}}(\alpha'_j)$ have “$-$”-type. Using these basic graphs, construct all possible “extended” graphs adding vertices of the [“$\circ$”]{}-type as described in theorem \[th:graphs\].
For every vertex of every graph as above consider all possible structures of an embedded local ribbon graph as in figure \[fig:embedded:local:ribbon:graphs\].
At the current stage we have already chosen $\alpha'=\{\alpha'_1,\dots,\alpha'_k\}$ (correspondingly $\beta'$), the graph $\Gamma$, the bijection of $\{\alpha'_1,\dots,\alpha'_k\}$ (correspondingly $\{\beta'_1,
\dots, \beta'_k\}$) with the set of vertices of $\Gamma$ compatible with the structure of “$+$” and “$-$”-vertices, and the structure of a local ribbon graph for every vertex of $\Gamma$. Now for every local ribbon graph ${\mathbb G}_j$ representing a “$+$” or “$-$”-vertex $S_j$ consider all possible ways to arrange orders of interior singularities and of boundary singularities of $S_j$ in a way compatible with conditions (3)–(6) of definition \[def:configuration\] and with equation for the corresponding singularity data $\beta'_j$ (correspondingly equation for the singularity data $\alpha'_j$). By “compatibility” with equations – we mean that singularity data computed by these equations should produce $\beta'_j$ (correspondingly $\alpha'_j$) possibly completed with several (from $1$ to $r_j$) entries “$0$” (where $r_j$ is the number of connected components of the local ribbon graph ${\mathbb G}_j$).
From the resulting lists of configurations extract those which correspond to the required singularity data $\alpha$.
Certainly this algorithm is not very efficient for large values of $g$ or $N$. Nevertheless, for strata in small genera having reasonable number of simple poles, it works quite well (especially being slightly optimized using specific properties of given data $\alpha$).
As an example we present a complete list of configurations of ĥomologous saddle connections for holomorphic quadratic differentials in genus 2. We are grateful to Alex Eskin, who helped us to test completeness of this list.
$$\begin{array}{|c|c|c|}
\hline
&&\\
[-\halfbls]
{{\mathcal Q}}(2,2) & {{\mathcal Q}}(2,1,1) & {{\mathcal Q}}(1,1,1,1) \\
[-\halfbls]&&\\
\hline&&\\
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[KaZm]{}
J. Athreya, A. Eskin, A. Zorich, [*Rectangular billiards and volumes of spaces of quadratic differentials*]{}, in progress.
C. Boissy, [*Degenerations of meromorphic quadratic differentials on ${{\mathbb C}^{}}P^1$ and on hyperelliptic Riemann surfaces*]{}, to appear.
C. Boissy, [*Configurations of ĥomologous saddle connections for quadratic differentials in exceptional strata*]{}, in progress.
A. Eskin, H. Masur, [*Asymptotic formulas on flat surfaces*]{}, Ergodic Theory and Dynamical Systems, [**21**]{} (2) (2001), 443–478.
A. Eskin, H. Masur, A. Zorich, [*Moduli spaces of Abelian differentials: the principal boundary, counting problems and the Siegel–Veech constants*]{}, Publications Mathématiques de l’IHÉS, [**97**]{} (1) (2003), 61–179.
J. Hubbard, H. Masur, [*Quadratic differentials and foliations*]{}, Acta Math., [**142**]{} (1979), 221–274.
M.Kontsevich, Lyapunov exponents and Hodge theory. “The mathematical beauty of physics” (Saclay, 1996), (in Honor of C. Itzykson) 318–332, Adv. Ser. Math. Phys., 24, World Sci. Publishing, River Edge, NJ, 1997.
E. Lanneau, [*Connected components of the moduli spaces of quadratic differentials*]{}, Ph.D. Thesis.
H. Masur, [*Interval exchange transformations and measured foliations*]{}. Ann. of Math., [**115**]{} (1982), 169-200.
H. Masur, J. Smillie, [*Hausdorff dimension of sets of nonergodic foliations*]{}, Ann. of Math., [**134**]{} (1991), 455-543.
H. Masur, J. Smillie, [*Quadratic differentials with prescribed singularities and pseudo-Anosov diffeomorphisms*]{}, Comment. Math. Helvetici, [**68**]{} (1993), 289–307.
W. A. Veech, [*Moduli spaces of quadratic differentials*]{}, Journal d’Analyse Math. [**55**]{} (1990), 117–171.
[^1]: Research of the first author is partially supported by NSF grant 0244472
[^2]: The notion “homologous in the relative homology with local coefficients defined by the canonical double cover induced by a quadratic differential” is unbearably bulky, so we introduced an abbreviation “ĥomologous”. We stress that the circumflex over the “h” is quite meaningful: as it is indicated in the definition, the corresponding cycles are homologous [*on the double cover*]{}.
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abstract: 'Recently, it has become apparent that the thermal stability of topologically ordered systems at finite temperature, as discussed in condensed matter physics, can be studied by addressing the feasibility of self-correcting quantum memory, as discussed in quantum information science. Here, with this correspondence in mind, we propose a model of quantum codes that may cover a large class of physically realizable quantum memory. The model is supported by a certain class of gapped spin Hamiltonians, called stabilizer Hamiltonians, with translation symmetries and a small number of ground states that does not grow with the system size. We show that the model does not work as self-correcting quantum memory due to a certain topological constraint on geometric shapes of its logical operators. This quantum coding theoretical result implies that systems covered or approximated by the model cannot have thermally stable topological order, meaning that no system can be stable against both thermal fluctuations and local perturbations simultaneously in two and three spatial dimensions.'
address: 'Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA'
author:
- Beni Yoshida
title: 'Feasibility of self-correcting quantum memory and thermal stability of topological order'
---
self-correcting quantum memory ,thermal stability of topological order ,quantum coding theory ,topological phase ,stabilizer formalism ,topological quantum field theory
Introduction {#sec:introduction}
============
In recent years, ideas from quantum information science have become increasingly useful in condensed matter physics where quantum information theoretical viewpoints give an important insight on studies of many-body entanglement arising in ground states or quasi-particle excitations of correlated spin systems [@Bravyi06; @Vidal03; @Fradkin06; @Osborne02; @Li08; @Kitaev06; @Levin06; @Kitaev03; @Eisert10; @Vidal07; @Gu08]. In particular, it has been realized that many interesting physical systems in condensed matter physics may be described in the language of quantum codes such as stabilizer codes and its generalization [@Kitaev97; @Raussendorf03; @Hein04; @Bombin09b; @Bacon06]. The emerging closeness between two fields has made it possible to study several problems concerning many-body correlated spin systems through quantum coding theoretical tools [@Beni10b].
In the present paper, we shall explore such a capacity of quantum coding theory for studying problems in condensed matter physics further, by making use of the correspondence between the following two open problems:
(a) Feasibility of self-correcting quantum memory.
(b) Thermal stability of topological order.
Feasibility of self-correcting quantum memory is an important open problem in quantum information science concerning reliable storage of qubits. Thermal stability of topological order at finite temperature is an open problem of fundamental importance in condensed matter physics which concerns whether topological order may survive at finite temperature or not. While these two problems may look very different from each other, they are fundamentally akin to each other, in a sense that by searching for self-correcting quantum memory, one can search for topological ordered spin systems which are stable at finite temperature [@Dennis02; @Nussinov08; @Bravyi09; @Alicki09; @Alicki10]. This surprising correspondence enable us to address a condensed matter theoretical question, thermal stability of topological order, by analyzing a quantum information theoretical question, feasibility of self-correcting quantum memory.
![The correspondence between the feasibility of self-correcting quantum memory and the thermal stability of topological order, and the plan of the paper. []{data-label="fig_plan"}](fig_plan.pdf){width="0.65\linewidth"}
The main contributions of the present paper toward these two open problems are the followings;
- We propose a certain model of quantum codes which may cover a large class of physically realizable quantum codes, which is called a stabilizer code with translation and scale symmetries.
- We solve the $D$-dimensional model exactly by determining its coding properties completely and analyze the feasibility of self-correcting quantum memory for $D=1,2,3$.
- We establish the connection between self-correcting quantum memory and stable topological order at finite temperature, and analyze thermal stability of topological order arising in the model.
The plan of the paper is schematically summarized in Fig. \[fig\_plan\]. We begin the introduction of the paper by reminding why these two problems are important in each field. Then, we give brief summaries of our results in the rest of the introduction.
**Feasibility of self-correcting quantum memory:** Quantum entanglement decays easily. This underlying difficulty in quantum information science gave birth to the beautiful art of protecting qubits from decoherence; *quantum coding theory*. The central idea of quantum error- correcting codes is to encode a qubit in many-body entangled states, and perform error-corrections so that encoded qubits are not lost. After discoveries of first examples of quantum codes [@Shor95; @Laflamme96; @Bennett96; @Calderbank96; @Kitaev97] which culminated in stabilizer codes [@Gottesman96], a large number of quantum codes have been found. Now, quantum coding theory constitutes one of the most important building blocks for realizing fault-tolerant quantum computation [@Shor96].
Yet, there still remain important gaps between theoretical constructions of quantum codes and their physical realizations as quantum memory devices. First of all, most quantum coding schemes encode qubits *dynamically* by applying a large number of logical gates, and encoding is discussed only in terms of the Hilbert space. However, since encoded qubits will be eventually lost in the presence of interactions with the external environment, it is desirable to store logical qubits *statically* in some physical subspaces which are naturally protected from decoherence.
One plausible approach may be to store logical qubits in the gapped ground space of some quantum many-body system. In this light, stabilizer codes constructed with geometrically local generators are promising candidates for physical realizations of quantum codes since such *local stabilizer codes* can be realized as the ground space of gapped Hamiltonians by using their local generators as interaction terms. Indeed, from purely theoretical viewpoint, it is known that sufficiently accurate and frequent error-corrections can store logical qubits reliably [@Dennis02]. However, one may still dream of having a quantum memory device which would work without active error-corrections, given the difficulties and inefficiencies of performing fast and accurate error-corrections in reality[^1].
Self-correcting quantum memory is an ideal memory device which corrects errors by itself [@Dennis02; @Bacon06; @Nussinov08; @Bravyi09; @Alicki09; @Pastawski09]. Due to the large energy barrier separating degenerate ground states, natural thermal dissipation processes restore the system into the original encoded states by correcting errors automatically without any active error-correction. If such a memory device could exist, it will be a perfect quantum information storage device which may be used commercially in the future. Also, the reliable storage of qubits seems to be the starting point for building scalable quantum computers.
There has been significant progress toward construction of self-correcting quantum memory. It has been pointed out that the Toric code defined on a four-dimensional system ($D=4$) serves as self-correcting quantum memory [@Dennis02; @Takeda04]. Yet, such a four-dimensional code cannot be embedded in a three-dimensional system. While there have been several proposals for three-dimensional self-correcting memory [@Bacon06; @Hamma09; @Haah11], validities of none of these proposals have been verified yet[^2]. In addition, it has been shown that a self-correcting local stabilizer code cannot exist in two-dimensional systems [@Bravyi09; @Kay08]. It seems that two-dimensional spin systems are not promising as resource for self-correcting quantum memory. Now, the question we want to address can be summarized as follows:
- Is it possible to have three-dimensional self-correcting quantum memory?
**Thermal stability of topological order:** The feasibility of self-correcting quantum memory is closely related to another important open problem in condensed matter physics; the thermal stability of topological order.
Studies of topologically ordered systems [@Laughlin83; @Wen90; @Kitaev97; @Misguich02; @Kitaev03; @Kane05; @Levin06; @Kitaev06b; @Fu07] have been frontiers of researches in condensed matter physics community, as systems with topological order are beyond the description of the Landau’s symmetry-breaking paradigm which was once considered almost as “theory of everything” for studies of many-body systems. Topologically ordered systems are also of practical importance in quantum information science since many-body entanglement arising in ground states and quasi-particle excitations of topologically ordered systems is a primary resource for realizing various quantum information processing tasks [@Kitaev97; @Kitaev03].
The notion of topological order was originally introduced in order to characterize the stability of ground states of many-body quantum systems against local perturbations [@Wen90]. Loosely speaking, a system is said to have topological order when its ground state properties do not change significantly under any types of small, but finite local perturbations. This stability of ground states against local perturbations is also valuable for quantum information processing since topologically ordered spin systems can be used as good quantum codes with macroscopic code distances [@Bravyi10].
However, the situation changes completely when one considers the effect of thermal fluctuations on topologically ordered systems. In fact, it is known that topological order in a two-dimensional Toric code is not stable at any finite temperature which may be quantitatively seen from the fact that topological order parameters such as topological entanglement entropy vanish at any non-zero temperature at the thermodynamic limit [@Castelnovo07]. A similar result is obtained in a recent numerical work on topological entanglement entropy in a spin liquid model at finite temperature [@Isakov11]. It seems that topological order in a two-dimensional system is not stable at finite temperature according to general studies on the ground state properties of two-dimensional frustration-free Hamiltonians [@Bravyi09; @Bravyi10]. Now, the question concerning the stability of topological order can be summarized as follows[^3]:
- Is there any system with thermally stable topological order, which is stable against both thermal fluctuations and local perturbations?
**Main results:** Interestingly, this condensed matter theoretical question on the stability of topological order can be addressed through quantum coding theory. In fact, it has been pointed out that when topological order in correlated spin systems is stable at finite temperature, such a system can be used as self-correcting quantum memory [@Bravyi09; @Castelnovo08; @Nussinov09]. This *correspondence* between self-correcting quantum memory and the thermal stability of topological order may be better understood by identifying thermal fluctuations as random errors acting on a quantum memory. Therefore, one may take a slight liberty and say that *a search for self-correcting quantum memory is equivalent to a search for a novel quantum phase with topological order which is stable at finite temperature.*
In this paper, with this correspondence in mind, we analyze coding and physical properties of a certain model of local stabilizer codes with physically reasonable constraints [@Beni10b]. The model, which is called Stabilizer code with Translation and Scale symmetries (STS model), is constrained to the following physical conditions.
- Qubits are defined on a $D$-dimensional square (hypercubic) lattice with periodic boundary conditions.
- The Hamiltonian consists only of geometrically local interaction terms with translation symmetries.
- The number of logical qubits does not grow with the system size (scale symmetries).
We show that the model has a certain dimensional duality on geometric shapes of logical operators, as summarized in the following informal theorem:
In a $D$-dimensional STS model ($D\leq 3$), $m$-dimensional and $(D-m)$-dimensional logical operators always form anti-commuting pairs where $m$ is an integer.
Based on this dimensional duality on logical operators, we give answers to open questions **(a)** and **(b)**, as summarized below:
(a) The three-dimensional STS model does not work as self-correcting quantum memory since the energy barrier is finite for the encoding with respect to a two-dimensional logical operator.
(b) Systems covered or approximated by the model cannot have thermally stable topological order, meaning that systems cannot be stable against both thermal fluctuations and local perturbations simultaneously in two and three spatial dimensions.
**Code distance:** Our results on STS models also provide a partial answer to a long-standing open problem in quantum coding theory concerning the upper bound on the code distance of *local stabilizer codes*. The code distance $d$ is a measure of the robustness of quantum codes against errors, and one of the ultimate goals in quantum coding theory is to find a quantum code with a large code distance for a fixed system size $N$ (the total number of qubits). While an upper bound on the code distance of stabilizer codes is roughly known [@Knill97], the upper bound for local stabilizer codes is currently not known yet. In other words, despite the fact that the stabilizer formalism is a canonical framework in quantum coding theory, we still do not know how robust the best local stabilizer code can be !
For a long time, it had been believed that the code distance of local stabilizer codes with $N$ qubits is upper bounded by $O(\sqrt{N})$: $d \leq O(\sqrt{N})$ at $N \rightarrow \infty$ since all the examples of local stabilizer codes ever found satisfied this upper bound [@Kitaev97; @Dennis02]. Later, an example of a local stabilizer code whose code distance scales as $O(\sqrt{N}\log N)$ was found [@Freedman02]. While the code distance of this local stabilizer code exceeds the previously believed upper bound $O(\sqrt{N})$ *logarithmically*, an example of a local stabilizer code whose code distance exceeds $O(\sqrt{N})$ *polynomially* has not been found yet. Therefore, the code distance of local stabilizer codes seemed to be upper bounded by $O(N^{\frac{1}{2}+\epsilon})$ at $N \rightarrow \infty$ where $\epsilon$ is an arbitrary small positive number, although no analytical result was known on the upper bound.
It was recently proven that the code distance of local stabilizer codes is upper bounded by $O(L^{D-1})$ where $L$ is a linear length of the system and $D$ is the spatial dimension ($N \sim O(L^{D})$) [@Bravyi09]. While this work does not rule out the possibilities for the existence of a local stabilizer code whose code distance exceeds $O(N^{\frac{1}{2}})$ polynomially, this bound was proven to be tight only for $D = 1,2$. Thus, whether the tight upper bound is $$\begin{aligned}
d \ \leq \ O(N^{\frac{1}{2}+\epsilon}) \quad \mbox{or} \quad d \ \leq \ O(L^{D-1}) \qquad \mbox{at} \quad N \ \rightarrow \ \infty\end{aligned}$$ for $D >2$ seems to be one of the most important open questions concerning coding properties of local stabilizer codes.
Our analysis on STS models provides the following tight upper bound on the code distance of STS models:
- A three-dimensional STS model has a code distance which is tightly upper bounded by $O(L)$ where $L$ is the linear length of the system. Thus, in a three-dimensional system, the upper bound on code distances turns out to be more strict than not only $O(L^{2})$, but also $O(\sqrt{N})$.
**Organization:** The paper is organized as follows. In section \[sec:model\], we give a brief review of stabilizer codes and introduce STS models. In section \[sec:result\], we discuss the feasibility of self-correcting quantum memory. In section \[sec:topo\], we discuss the thermal stability of topological order at finite temperature. In section \[sec:physics\], we present a certain universal property of logical operators shared among all the STS models. In section \[sec:summary\], we discuss possible future problems. The definition of topological order at finite temperature is discussed further in \[sec:topo\_ap\]. A possible relevance to topological quantum field theory is discussed in \[sec:topology\]. The proof of the dimensional duality of logical operators is presented in \[sec:decomposition\] and \[sec:construction\].
The main part of the paper is self-consistent and accessible to readers without previous knowledge on quantum coding theory and topological order. However, the proof part is rather technical, and relies heavily on theoretical tools developed previously in [@Beni10b; @Beni10] by the author. In particular, we owe a lot of arguments to [@Beni10b] which introduced and solved the two-dimensional STS model originally.
Stabilizer code with physical constraints {#sec:model}
=========================================
While local stabilizer codes are physically realizable as quantum memory devices in principle, realistic physical systems are often constrained to not only the locality of interaction terms, but also various physical symmetries. In this section, we give the definition of the STS model which are local stabilizer codes with translation and scale symmetries [@Beni10b].
In section \[sec:model1\], we give a brief review of stabilizer codes. In section \[sec:model2\], we describe the definition of the STS model.
Stabilizer code {#sec:model1}
---------------
Here, we give a brief review of stabilizer codes which are quantum codes possessing Hamiltonians to support logical qubits in the ground space with a finite energy gap [@Gottesman96]. Some notations which will be used throughout this paper are also fixed here. Note that we shall use the notations $\{\}$ for a *set* and $\langle \rangle$ for a *group*.
**Stabilizer formalism:** The main idea of stabilizer codes is to encode $k$ logical qubits into $N$ physical qubits ($N>k$) by using a subspace $V_{\mathcal{S}}$ spanned by states $|\psi\rangle$ that are invariant under the action of the *stabilizer group* $\mathcal{S}$: $$\begin{aligned}
V_{\mathcal{S}} \ = \ \Big\{ \ |\psi\rangle \ \in \ (\mathbb{C}^{2})^{\otimes N} \ : \ U|\psi\rangle \ = \ |\psi\rangle, \ \forall U \ \in \ \mathcal{S} \ \Big\}.\end{aligned}$$ Here, the stabilizer group $\mathcal{S}$ is an arbitrary Abelian subgroup of the Pauli group $$\begin{aligned}
\mathcal{S} \ \subset \ \mathcal{P} \ = \ \Big\langle \ iI, X_{1},Z_{1},\dots , X_{N}, Z_{N} \ \Big\rangle\end{aligned}$$ such that $-I \not\in \mathcal{S}$. The elements in $\mathcal{S}$ are called *stabilizers*. The logical subspace $V_{\mathcal{S}}$ can be realized as the ground space of the following Hamiltonian (see Fig. \[fig\_stabilizer\_summary\]) $$\begin{aligned}
H \ = \ - \sum_{j} S_{j}, \qquad \mathcal{S} \ = \ \left\langle \ S_{1}, S_{2}, \cdots \ \right\rangle\end{aligned}$$ since the energy eigenvalue is minimized for states satisfying $S_{j}|\psi\rangle = |\psi\rangle$ for all $j$. There are $k$ logical qubits encoded in $V_{\mathcal{S}}$ where $k \equiv N - G(\mathcal{S})$. Here, $G(\mathcal{S})$ represents the number of independent generators in $\mathcal{S}$. The ground space is separated from excited states by a finite energy gap since eigenstates are simultaneously diagonalized with respect to eigenvalues $\pm 1$ of $S_{j}$.
![Energy spectrum of a stabilizer Hamiltonian. Logical qubits are encoded in the degenerate ground space. []{data-label="fig_stabilizer_summary"}](fig_stabilizer_summary.pdf){width="0.35\linewidth"}
**Logical operators:** In analyzing properties of logical qubits stored in the ground space, operators called *logical operators* play central roles. Logical operators are Pauli operators which commute with the Hamiltonian, but not inside the stabilizer group $\mathcal{S}$. Logical operators can be found inside the centralizer group: $$\begin{aligned}
\mathcal{C} \ = \ \Big\langle \ \Big\{ \ U \ \in \ \mathcal{P} \ : \ [U,S_{j}] \ = \ 0,\hspace{1ex} \mbox{for all}\ j \ \Big\} \ \Big\rangle\end{aligned}$$ which is a group of Pauli operators commuting with all the stabilizers. Then, a set of logical operators is $$\begin{aligned}
\textbf{L} \ = \ \Big\{ \ U \ \in \ \mathcal{C} \ : \ U^{2} \ = \ I, \hspace{1ex} U \ \not \in \ \mathcal{S} \ \Big\}.\end{aligned}$$ Logical operators may transform encoded qubits since they act non-trivially inside the ground space $V_{\mathcal{S}}$.
**Equivalence relation:** One may introduce an equivalence relation between logical operators by seeing how they act inside the ground space. Two logical operators $\ell$ and $\ell'$ are said to be *equivalent* if and only if $\ell$ and $\ell'$ act in the same way inside the ground space: $$\begin{aligned}
\ell \ \sim \ \ell' \ &\Leftrightarrow \ \ell|\psi\rangle \ = \ \ell' |\psi\rangle, \qquad \forall |\psi\rangle \ \in \ V_{\mathcal{S}}\\
&\Leftrightarrow \ \ell \ell' \ \in \ \mathcal{S}.\end{aligned}$$ Therefore, logical operators remain equivalent under multiplications of stabilizers.
**Canonical form:** It is often convenient to represent a set of $2k$ independent logical operators in the following *canonical* form [@Beni10]: $$\begin{aligned}
\left\{
\begin{array}{cccccc}
\ell_{1}, & \cdots , & \ell_{k} \\
r_{1}, & \cdots , & r_{k}
\end{array}
\right\}.\end{aligned}$$ Here, $\ell_{p}$ and $r_{p}$ are independent logical operators whose commutation relations are $\{ \ell_{p},r_{p}\}=0$, $[\ell_{p},r_{q}]=0$ for $p \not=q$, $[\ell_{p},\ell_{q}]=0$ and $[r_{p},r_{q}]=0$. Thus, only the operators in the same column anti-commute with each other. Note that choices of logical operators are not unique.
**Code distance:** The code distance is a measure of the robustness of a quantum code, which is quantified by the minimal weight of logical operators: $$\begin{aligned}
d \ = \ \min(w(U)) \qquad \mbox{where} \quad U \ \in \ \textbf{L}.\end{aligned}$$ Here, $w(U)$ denotes the number of non-trivial Pauli operators constituting $U$. The code distance corresponds to a minimal number of single Pauli errors necessary to destroy an encoded qubit. Roughly speaking, a quantum code with a large code distance can securely protect logical qubits.
**Bi-partition:** It is often convenient to split the entire system of qubits into two complementary subsets of qubits in studying coding properties of stabilizer codes [@Fattal04; @Wilde10]. Let us recall a useful formula to study geometric shapes of logical operators in stabilizer codes through a bi-partition. For a stabilizer code in a bi-partition, the following theorem is known to hold [@Beni10] (Fig. \[fig\_stabilizer\_summary\_bipartition\]).
\[theorem\_partition\] For a stabilizer code with $k$ logical qubits, let the number of independent logical operators supported by a subset of qubits $R$ be $g_{R}$. Then, for an arbitrary bi-partition into two complementary subsets of qubits $R$ and $\bar{R}$, the numbers of logical operators supported by $R$ and $\bar{R}$ obey the following constraint: $$\begin{aligned}
g_{R}+g_{\bar{R}} \ = \ 2k.\end{aligned}$$
![A bi-partition of a stabilizer code. Each dot represents a qubit. []{data-label="fig_stabilizer_summary_bipartition"}](fig_stabilizer_summary_bipartition.pdf){width="0.20\linewidth"}
This bi-partition theorem is useful for analyzing geometric sizes and geometric shapes of logical operators. For example, if we find a region $R$ where there is no logical operator: $g_{R}=0$, we immediately know that all the logical operators can be supported inside $\bar{R}$ since $g_{\bar{R}}=2k$ [@Bravyi09]. Thus, one can restrict geometric regions of qubits where logical operators are supported. For an extension of the theorem to subsystem codes, see [@Bravyi11; @Haah10].
Stabilizer code with Translation and Scale symmetries {#sec:model2}
-----------------------------------------------------
Here, we describe the definition of STS models, which are local stabilizer codes with translation and scale symmetries.
**(1) Locality of interaction:** Physically realistic systems must have only geometrically local interaction terms. To introduce the notion of locality to stabilizer codes, we consider a system of qubits defined on a $D$-dimensional square lattice (hypercubic lattice) which consists of $N = L_{1} \times \cdots \times L_{D}$ qubits where $L_{m}$ is the total number of qubits in the $\hat{m}$ direction for $m = 1 ,\cdots ,D$. Therefore, qubits are distributed in the physical space with a *metric*.
Here, the entire system is separated into a collection of hypercubes which consists of $v = v_{1} \times \cdots \times v_{D}$ qubits without overlaps by assuming that $n_{m}\equiv L_{m}/ v_{m}$ are integer values (see Fig. \[fig\_STS\]). We consider a block of $v = v_{1} \times \cdots \times v_{D}$ qubits as the single unit block which constitutes the entire system. In particular, we consider these unit blocks as single *composite particles* with a larger Hilbert space $(\mathbb{C}^{2})^{\otimes v}$ (Fig. \[fig\_STS\]). Thus, the entire system is viewed as a hypercubic lattice of $n_{1}\times \cdots \times n_{D}$ composite particles.
Now, we assume that interaction terms of the stabilizer Hamiltonian are defined *locally*: $$\begin{aligned}
H \ = \ - \sum_{j} S_{j} \end{aligned}$$ where $S_{j}$ are supported inside some regions with $2 \times \cdots \times 2$ composite particles (Fig. \[fig\_STS\]). (Otherwise, we coarse-grain the system). In this paper, instead of qubits, we consider composite particles as the smallest building blocks of the system.
![An illustration of the STS model. A two-dimensional example is shown where a unit block of $3 \times 2$ qubits is considered as a composite particle with a larger Hilbert space. Interaction terms $S_{j}$ are defined locally inside a region of $2 \times 2$ composite particles. The Hamiltonian is invariant under unit translations of composite particles. []{data-label="fig_STS"}](fig_STS.pdf){width="0.70\linewidth"}
**(2) Translation symmetries:** Physically realistic systems often have not only local interactions, but also some physical symmetries. Here, we assume that the stabilizer Hamiltonian possesses *translation symmetries*: $$\begin{aligned}
T_{m}(H)\ =\ H \qquad (m = 1, \cdots, D)\end{aligned}$$ where $T_{m}$ represent *unit translations of composite particles* in the $\hat{m}$ direction (Fig. \[fig\_STS\]).
For simplicity of discussion and in order to accommodate translation symmetries, we set the periodic boundary conditions. Then, the entire system may be viewed as a $D$-dimensional torus: $\textbf{T}^{D} = \textbf{S}^{1} \times \cdots \times \textbf{S}^{1}$ where $\textbf{S}^{1}$ is a circle. Thus, the entire system has a topologically non-trivial geometric shape *a priori*.
**(3) Scale symmetries:** In this paper, we are interested in coding properties at the limit where the system size goes to the infinity (in other words, at the *thermodynamic limit*). So far, we have considered the cases where the system size $\vec{n} \equiv (n_{1}, \cdots, n_{D})$ is fixed. Here, we consider changes of the number of composite particles $n_{m}$ while keeping interaction terms $S_{j}$ the same.
It is commonly believed that there is a tradeoff between the number of logical qubits $k$ and the code distance [@Bravyi10; @Haah10] where the code distance $d$ decreases as the number of logical qubits $k$ increases for a fixed $N$. Then, it may be legitimate to limit our considerations to the cases where the number of logical qubits $k$ remains small when the system size increases.
We assume that stabilizer codes have *scale symmetries* by requiring that the number of logical qubits $k_{\vec{n}}$ is independent of the system size $\vec{n}$: $$\begin{aligned}
k_{\vec{n}} \ = \ k, \qquad \forall \vec{n}.\end{aligned}$$ Here, we emphasize that, in a system with scale symmetries, the number of logical qubits $k$ remains constant under not only global scale transformations: $\vec{n} \rightarrow c \vec{n}$ where $c$ is some positive integer, but also arbitrary changes of $n_{m}$.
One might think that scale symmetries are too strong as physical constraints. However, through appropriate coarse-graining, a fairly large class of local stabilizer codes with translation symmetries can be considered as the STS model. For example, let us consider the cases where the number of logical qubits $k_{\vec{n}}$ is small: $$\begin{aligned}
k_{\vec{n}} \ \leq \ k_{0}, \qquad \forall \vec{n} \label{eq:small}\end{aligned}$$ where $k_{\vec{n}}$ does not grow with the system size $\vec{n}$. Then, there always exists some finite coarse-graining such that a coarse-grained system has scale symmetries, as proven in [@Beni10b]. As a result, by analyzing coding properties of stabilizer codes in the presence of scale symmetries, one can easily deduce coding properties of stabilizer codes which satisfy Eq. (\[eq:small\]). Therefore, solutions of STS models are sufficient to discuss coding properties of translation symmetric stabilizer codes with a small number of logical qubits. Note that STS models cover the Toric code and its generalizations to $D$-dimensional systems.
**Translation equivalence of logical operators:** There is a certain property of logical operators which emerges naturally as a result of translation and scale symmetries. For STS models, the following theorem holds [@Beni10b] (Fig. \[fig\_the\_tools\]).
\[theorem\_TE\] For each and every logical operator $\ell$ in an STS model, a unit translation of $\ell$ with respect to composite particles in any direction is always equivalent to the original logical operator $\ell$: $$\begin{aligned}
T_{m}(\ell) \ \sim \ \ell, \qquad \forall \ell \ \in \ \textbf{L}_{\vec{n}} \qquad (m \ = \ 1, \cdots, D) \end{aligned}$$ where $\textbf{L}_{\vec{n}}$ is a set of all the logical operators for an STS model defined with the system size ${\vec{n}}$.
![The translation equivalence of logical operators. Each square represents a composite particle. Shaded regions represent translated logical operators which are equivalent to each other. []{data-label="fig_the_tools"}](fig_the_tools.pdf){width="0.55\linewidth"}
Here, we only give an intuition on why this theorem holds. Let us consider the case where the system size $\vec{n}$ is large. Then, since the number of logical qubits $k$ does not depend on the system size, $k$ is relatively small compared with the system size $\vec{n}$. Now, due to the translation symmetries of the system Hamiltonian, translations of a given logical operator $\ell$ are also logical operators. However, there are only $2k$ independent logical operators. Then, there must be a finite integer $a_{m}$ such that $\ell \sim T_{m}^{a_{m}}(\ell)$ for all the logical operators $\ell$. (Otherwise, there would be so many independent logical operators). It turns out that $a_{m}=1$ for any $\ell$ and $m$. While we have used only the condition that the number of logical operators $k$ is small, due to scale symmetries ($k$ is constant), one can prove the above theorem by showing $a_{m}=1$ for any $\ell$, $m$ and $\vec{n}$.
Feasibility of self-correcting quantum memory {#sec:result}
=============================================
In this section, we present one of our main results in this paper, concerning coding the feasibility of self-correcting quantum memory. In section \[sec:result1\], we begin by giving brief discussion on the relation between the qubit storage time and the energy barrier. In section \[sec:result2\], we describe geometric shapes of logical operators in STS models, and discuss whether a three-dimensional STS model works as self-correcting quantum memory or not.
Self-correction, energy barrier and storage time {#sec:result1}
------------------------------------------------
In subsection, we review how self-correcting quantum memory works by establishing the connection between the energy barrier and the qubit storage time.
**Self-correcting classical memory:** For simplicity of discussion, let us start by analyzing an example of self-correcting *classical* memory. Consider two-dimensional Ising model: $$\begin{aligned}
H\ =\ - \sum_{i,j}Z_{i,j}Z_{i+1,j} - \sum_{i,j}Z_{i,j}Z_{i,j+1} \end{aligned}$$ which consists of $L \times L$ qubits with periodic boundary conditions. The model works as a classical code since one can encode a classical bit in the ground space by labeling $|0\cdots0\rangle$ as $0$ and $|1\cdots1\rangle$ as $1$.
Now, let us see why this model works as self-correcting classical memory. Assume that the system is originally $|0\cdots0\rangle$. Then, in order for errors to change a ground state $|0\cdots0\rangle$ into another ground state $|1\cdots1\rangle$, errors must flip all the spins from $|0\rangle$ to $|1\rangle$. However, during these spin flips, the excitation energy becomes at least $O(L)$ because there is a domain wall separating the regions with $|0\rangle$s and $|1\rangle$s (Fig. \[fig\_self\]). In other words, ground states $|0\cdots0\rangle$ and $|1\cdots1\rangle$ are separated by a *large energy barrier*. Then, before errors accumulate, natural thermal dissipation processes restore the system into the original encoded state[^4]. Therefore, the system corrects errors by itself.
One may estimate the bit storage time of two-dimensional Ising model by using the so-called Arrhenius law: $$\begin{aligned}
\tau \ \sim \ \exp(\Delta E /T) \end{aligned}$$ up to some polynomial corrections where $\tau$ is the storage time, $\Delta E$ is an energy barrier and $T$ is the temperature. According to the Arrhenius law, the bit storage time can be estimated as $\tau \sim \mbox{EXP}(L)$ since the energy barrier is $\Delta E \sim O(L)$. While it should be noted that this is an empirical law, it is commonly believed that the law correctly estimates the bit or qubit storage time of spin systems (for systems with well defined energy barrier) below a critical temperature. Indeed, the law is rigorously verified in various models of classical and quantum memory [@Brey96; @Nussinov08; @Alicki09; @Alicki10; @Chesi10]. However, it should be noted that verification of the Arrhenius law for classical and quantum memory is usually a very difficult task, involving an evaluation of time evolution of the system in a presence of interactions with an external environment.
Next, let us consider an example of a classical code which is not self-correcting; one-dimensional Ising model: $$\begin{aligned}
H\ =\ - \sum_{j}Z_{j}Z_{j+1} .\end{aligned}$$ One may see that this model does not work as self-correcting classical memory since one can change from $|0\cdots0\rangle$ to $|1\cdots1\rangle$ by costing only a finite energy. In other words, one can create a kink by costing only a finite energy: $$\begin{aligned}
|0 0 0 0 \cdots 0 \rangle \ \rightarrow \ |0 1 1 1 \cdots 1 \rangle \ \rightarrow \ |0 0 1 1 \cdots 1 \rangle \ \rightarrow \ \cdots \rightarrow \ |1 1 1 1 \cdots 1 \rangle\end{aligned}$$ which may propagate the lattice freely without costing any extra energy. Then, according to the Arrhenius law, the bit storage time is $\tau \sim O(1)$ which is independent of the system size since the energy barrier is $\Delta E \sim O(1)$.
![How self-correcting *classical* memory works in two-dimensional Ising model. []{data-label="fig_self"}](fig_self.pdf){width="0.60\linewidth"}
**Energy barrier and logical operators:** One can associate the self-correcting property of two-dimensional Ising model with geometric shapes of logical operators by viewing the model as a stabilizer code. Note that $D$-dimensional Ising model satisfies the definition of STS models since interaction terms are local and translation symmetric, and there is a single logical qubit ($k=1$ and $d=1$) regardless of the system size. Two-dimensional Ising model has the following pair of zero-dimensional and two-dimensional logical operators: $$\begin{aligned}
\ell\ = \ Z_{1,1},\qquad r\ =\ \prod_{i,j}X_{i,j}.\end{aligned}$$ Then, a classical bit is encoded in eigenstates of a zero-dimensional logical operator $\ell$. In order to change the encoded bit, one needs to apply a two-dimensional logical operator $r$ since $|1\cdots1\rangle = r|0\cdots0\rangle$. Then, an intermediate state during the change from $|0\cdots0\rangle$ to $|1\cdots1\rangle$ may be represented as $r^{*}|0\cdots0\rangle$ where $r^{*}$ is some “subpart” of the original two-dimensional logical operator $r$ (see Fig \[fig\_self2\](a)). Since interaction terms anti-commute with Pauli operators at the boundary of $r^{*}$, the excitation energy associated with $r^{*}|0\cdots0\rangle$ is proportional to the perimeter of $r^{*}$. Thus, during the change from $|0\cdots0\rangle$ to $|1\cdots1\rangle$, the excitation energy must become $O(L)$ since the perimeters of subparts of a two-dimensional logical operator $r$ are always one-dimensional.
One can also understand why one-dimensional Ising model does not work as self-correcting classical memory through geometric shapes of logical operators. One-dimensional Ising model has the following pair of zero-dimensional and one-dimensional logical operators: $\ell = Z_{1}$ and $r = \prod_{i}X_{i}$. Since a subpart $r^{*}$ of a one-dimensional logical operator $r$ is always zero-dimensional, the energy barrier is $\Delta E \sim O(1)$ (see Fig \[fig\_self2\](b)).
![(a) A subpart of a two-dimensional logical operator. The excitation energy is $O(L)$. (b) A subpart of a one-dimensional logical operator. The excitation energy is $O(1)$. []{data-label="fig_self2"}](fig_self2.pdf){width="0.55\linewidth"}
**Self-correcting quantum memory:** Next, let us discuss how self-correcting quantum memory works. It is known that many of good local stabilizer codes with macroscopic code distances do not have self-correcting properties since they have string-like logical operators which lead to $O(1)$ energy barrier. As an example, let us consider two-dimensional Toric code: $$\begin{aligned}
H_{Toric} \ = \ - \sum_{s} A_{s} - \sum_{p} B_{p} \end{aligned}$$ where qubits live on edges of $L \times L$ square lattice. Interaction terms $A_{s}$ and $B_{p}$ are shown in Fig. \[fig\_Toric\_code\]. Two-dimensional Toric code can be viewed as an STS model since the number of logical qubits is constant: $k=2$ regardless of the system size.
![(a) A subpart of a two-dimensional logical operator. The excitation energy is $O(L)$. (b) A subpart of a one-dimensional logical operator. The excitation energy is $O(1)$. []{data-label="fig_Toric_code"}](fig_Toric_code.pdf){width="0.55\linewidth"}
One can understand why two-dimensional Toric code does not work as self-correcting quantum memory through geometric shapes of logical operators. Since the Toric code has pairs of one-dimensional logical operators as shown in Fig. \[fig\_Toric\_code\], the energy barrier is $\Delta E \sim O(1)$. As a result, the qubit storage time is $\tau \sim O(1)$ according to the Arrhenius law. A similar discussion holds for three-dimensional Toric code too.
As two-dimensional and three-dimensional Toric codes have $O(1)$ qubit storage time, they are not reliable quantum memories. Then, what kinds of quantum codes may serve as reliable quantum memory devices? It is worth noting that two-dimensional Toric code works as a reliable quantum memory device if one perform sufficiently frequent and accurate error-corrections. In particular, it has been shown [@Dennis02] that the qubit storage time can be exponentially long: $\tau\sim \mbox{EXP($L$)}$ in the presence of active error-corrections; however, it seems very difficult to reach such accuracy and frequency which are necessary for reliably storing qubits.
In the same paper [@Dennis02], it has been also pointed out that four-dimensional Toric code may have exponentially long storage time $\tau \sim \mbox{EXP($L$)}$ below the critical temperature since the model has a large energy barrier which scales as $O(L)$. This remarkable insight has been further investigated in [@Takeda04], and later, rigorously verified in [@Alicki10]. One can associate the self-correcting property of four-dimensional Toric code with geometric shapes of logical operators, as the model has pairs of two-dimensional logical operators which lead to $O(L)$ energy barrier. Yet, since one cannot embed four-dimensional Toric code in a three-dimensional space, one hopes to have three-dimensional self-correcting quantum memory whose qubit storage time grows as the system size increases.
**Previous proposals:** So far, there have been several interesting proposals of three-dimensional self-correcting quantum memory. First, a certain model of three-dimensional subsystem codes [@Bacon06], called three-dimensional quantum compass model, was proposed as a candidate of self-correcting quantum memory. Subsystem codes may be considered as generalizations of stabilizer codes which are supported by Hamiltonians whose interaction terms are Pauli operators, but may anti-commute with each other. While the model opened a new possibility of quantum memory devices supported by frustrated Hamiltonians and initiated studies of Hamiltonian realizations of subsystem codes, its validity as self-correcting quantum memory has not been verified since it is difficult to solve the frustrated Hamiltonian, and its properties are hard to determine. Also, it seems difficult for such frustrated systems to have the thermal stability since the model supports gapless excitations. Furthermore, its validity has been denied in several literature including [@Nussinov08].
Later, an interesting model of the mixture of two-dimensional Toric code and three-dimensional Bosonic gas, called the Toric-Boson model, was proposed [@Hamma09] where Bosonic gas induces confining potential between anyonic excitations. While the model opened new capacities of quantum codes constructed in the so called mixed-dimensional configurations, which are of particular interest in ultracold atom physics community, the model itself has two serious drawbacks as a candidate of self-correcting quantum memory. First, its storage time is only polynomial in $L$: $\tau \sim \mbox{POLY}(L)$ since an effective energy barrier is: $\Delta E \sim \mbox{LOG($L$)}$. However, it is known that it takes at least $O(d)$ gate operations, which cannot be implemented simultaneously, to read out the encoded qubit, where $d$ is the code distance of the system. Then, since the code distance $d$ scales polynomially in $L$ in the model, polynomially long storage time is not sufficient for a model to work as efficient quantum memory device [@Alicki09]. Second, the model needs a very strong coupling between the Toric code and Bosonic gas whose strength scales polynomially in $L$. Therefore, the model has a problem in the scalability.
Recently, an interesting proposal of three-dimensional spin glass models [@Haah11] has been made whose relaxation dynamics is very slow due to the existence of a large number of energy local minima. The model seems to have a polynomially long storage time: $\tau \sim \mbox{POLY($L$)}$ with logarithmically large energy barrier: $\Delta E \sim \mbox{LOG($L$)}$, though it needs verifications (so, it could be longer or shorter). The system undergoes a phase transition at $T=0$, which mat imply a potential thermal instability of the system properties. Finally, it seems difficult o find an efficient decoding scheme for the model. Here, it should be noted that the model does not have scale symmetries since the number of logical qubits is highly sensitive to the system size, and there is no finite upper bound on it. See section \[sec:summary\] for discussion on stabilizer codes without scale symmetries too.
At this moment, validities of none of these proposals of three-dimensional self-correcting quantum memory has been verified yet, and the feasibility of self-correcting quantum memory remains open. Below, we summarize the qubit storage time and the energy barrier in these proposals of self-correcting quantum memory.
-- -- -- --
-- -- -- --
Dimensional duality of logical operators {#sec:result2}
----------------------------------------
In this subsection, we describe geometric shapes of logical operators in STS models, and discuss the feasibility of self-correcting quantum memory.
**One-dimension:** We begin by reviewing geometric shapes of logical operators in one-dimensional STS models. In a one-dimensional chain of $n_{1}$ composite particles, we denote $j$th composite particle as $P_{j}$. Then, the following theorem holds [@Beni10b].
\[theorem\_1dim\] For a one-dimensional STS model, there exists a canonical set of logical operators: $$\begin{aligned}
\left\{
\begin{array}{cccccc}
\ell_{1}, & \cdots , & \ell_{k} \\
r_{1}, & \cdots , & r_{k}
\end{array}
\right\}\end{aligned}$$ whose pair of anti-commuting operators $\ell_{j}$ and $r_{j}$ has one of the following property ($j=1,\cdots,k$).
- $\ell_{j}$ is a zero-dimensional logical operator defined inside $P_{1}$, while $r_{j}$ is a one-dimensional logical operator defined in a periodic way: $T_{1}(r_{j})=r_{j}$.
It is worth presenting geometric shapes of logical operators graphically (Fig. \[fig\_1D\_logical\]). The code distance of one-dimensional models is upper bounded by $v$ which is the inner dimension of a composite particle. Since there always exist zero-dimensional logical operators, such systems cannot have topological order.
![Logical operators in a one-dimensional system.[]{data-label="fig_1D_logical"}](fig_1D_logical.pdf){width="0.30\linewidth"}
**Two-dimension:** Next, we analyze geometric shapes of logical operators for two-dimensional STS models. Let us first introduce some regions of composite particles in order to define geometric shapes of logical operators concisely. A square region of $x_{1}\times x_{2}$ composite particles is denoted as $P(x_{1},x_{2})$ (Fig. \[fig\_2D\_logical\](a)): $$\begin{aligned}
P(x_{1},x_{2}) \ \equiv \ \Big\{ \ P_{r_{1},r_{2}} \ : \ 1 \ \leq \ r_{1} \ \leq \ x_{1}, \ 1 \ \leq \ r_{2} \ \leq \ x_{2} \ \Big\}\end{aligned}$$ where $1 \leq x_{1} \leq n_{1}$ and $1 \leq x_{2} \leq n_{2}$. Note that a composite particle at the position $(r_{1},r_{2})$ is denoted as $P_{r_{1},r_{2}}$. Then, the following theorem holds [@Beni10b].
\[theorem\_2dim\] For a two-dimensional STS model, there exists a canonical set of logical operators: $$\begin{aligned}
\left\{
\begin{array}{cccccc}
\ell_{1}, & \cdots , & \ell_{k} \\
r_{1}, & \cdots , & r_{k}
\end{array}
\right\}\end{aligned}$$ whose pair of anti-commuting operators $\ell_{j}$ and $r_{j}$ has one of the following two properties ($j=1,\cdots,k$).
- $\ell_{j}$ is a zero-dimensional logical operator defined inside $P(1,2v)$, while $r_{j}$ is a two-dimensional logical operator defined in a periodic way: $T_{1}(r_{j})=r_{j}$ and $T_{2}(r_{j})=r_{j}$.
- $\ell_{j}$ is a one-dimensional logical operator defined inside $P(1,n_{2})$ in a periodic way: $T_{2}(\ell_{j})=\ell_{j}$, while $r_{j}$ is a one-dimensional logical operator defined inside $P(n_{1},1)$ in a periodic way: $T_{1}(r_{j})=r_{j}$.
It is worth presenting geometric shapes of logical operators graphically (Fig. \[fig\_2D\_logical\](b)). There is a dimensional duality on geometric shapes of logical operators as follows: $$\begin{aligned}
\left\{
\begin{array}{cc}
\mbox{0 dim}, & \mbox{1 dim} \\
\mbox{2 dim}, & \mbox{1 dim}
\end{array}
\right\}.\end{aligned}$$ Logical operators are periodic in the directions in which they are stretched. The code distance is upper bounded by $O(L)$ which is consistent with the upper bound $O(L^{D-1})$ obtained in [@Bravyi09]. The existence of one-dimensional logical operators gives rise to topological order.
![Dimensional duality in a two-dimensional system. (a) A region of $x_{1}\times x_{2}$ composite particles is denoted as $P(x_{1},x_{2})$. (b) Geometric shapes of logical operators in a two-dimensional STS model. []{data-label="fig_2D_logical"}](fig_2D_logical.pdf){width="0.65\linewidth"}
**Three-dimension:** Finally, let us proceed to coding properties of a three-dimensional STS model. A region with $x_{1}\times x_{2} \times x_{3}$ composite particles is denoted as $P(x_{1},x_{2},x_{3})$: $$\begin{aligned}
P(x_{1},x_{2},x_{3}) \ \equiv \ \Big\{ \ P_{r_{1},r_{2},r_{3}} \ : \ 1 \ \leq \ r_{m} \ \leq \ x_{m},\ \ m \ = \ 1,2,3 \ \Big\}\end{aligned}$$ where $P_{r_{1},r_{2},r_{3}}$ represents a composite particle at $(r_{1},r_{2},r_{3})$. Then, for logical operators in a three-dimensional STS model, the following theorem holds.
\[theorem\_3dim\] For a three-dimensional STS model, there exists a canonical set of logical operators: $$\begin{aligned}
\left\{
\begin{array}{cccccc}
\ell_{1}, & \cdots , & \ell_{k} \\
r_{1}, & \cdots , & r_{k}
\end{array}
\right\}\end{aligned}$$ whose pair of anti-commuting operators $\ell_{j}$ and $r_{j}$ has one of the following four properties.
- $\ell_{j}$ is a zero-dimensional logical operator defined inside $P(1,2v,(2v)^{2})$, while $r_{j}$ is a three-dimensional logical operator defined in a periodic way: $T_{1}(r_{j})=r_{j}$, $T_{2}(r_{j})=r_{j}$ and $T_{3}(r_{j})=r_{j}$.
- $\ell_{j}$ is a one-dimensional logical operator defined inside $P(n_{1},2v,1)$ in a periodic way: $T_{1}(\ell_{j})=\ell_{j}$, while $r_{j}$ is a two-dimensional logical operator defined inside $P(1,n_{2},n_{3})$ in a periodic way: $T_{2}(r_{j})=r_{j}$ and $T_{3}(r_{j})=r_{j}$.
- $\ell_{j}$ is a one-dimensional logical operator defined inside $P(1,n_{2},2v)$ in a periodic way: $T_{2}(\ell_{j})=\ell_{j}$, while $r_{j}$ is a two-dimensional logical operator defined inside $P(n_{1},1,n_{3})$ in a periodic way: $T_{1}(r_{j})=r_{j}$ and $T_{3}(r_{j})=r_{j}$.
- $\ell_{j}$ is a one-dimensional logical operator defined inside $P(2v,1,n_{3})$ in a periodic way: $T_{3}(\ell_{j})=\ell_{j}$, while $r_{j}$ is a two-dimensional logical operator defined inside $P(n_{1},n_{2},1)$ in a periodic way: $T_{1}(r_{j})=r_{j}$ and $T_{2}(r_{j})=r_{j}$.
This is the main technical result of the present paper. We present the proof of the theorem in appendices \[sec:decomposition\] and \[sec:construction\]. It is worth representing geometric shapes of logical operators graphically (Fig. \[fig\_3D\_logical\]). Note that logical operators are periodic in the directions in which they are stretched. There is a dimensional duality on geometric shapes of logical operators as follows: $$\begin{aligned}
\left\{
\begin{array}{cc}
\mbox{0 dim}, & \mbox{1 dim} \\
\mbox{3 dim}, & \mbox{2 dim}
\end{array}\right\}\end{aligned}$$
![Dimensional duality in a three-dimensional system. []{data-label="fig_3D_logical"}](fig_3D_logical.pdf){width="0.90\linewidth"}
As a result of this dimensional duality, one may find the upper bound on the code distance. When $n_{1}=n_{2}=n_{3}=L$, the code distance of a three-dimensional STS model is upper bounded as follows: $$\begin{aligned}
d \ \leq \ 2vL \ \sim \ O(L).\end{aligned}$$ Note that this bound is tight for the three-dimensional Toric code. The system may have topological order when it is free from zero-dimensional logical operators.
**Higher-dimensions:** Though our primary interests are in coding properties of three-dimensional STS models, it is worth extending the analysis to higher dimensions. For a $D$-dimensional STS model ($D \geq 4$), we make the following *conjectures*:
- In a $D$-dimensional system, $m$-dimensional and $(D-m)$-dimensional logical operators form anti-commuting pairs where $m$ is an integer.
- The code distance is tightly upper bounded by $O(L^{\frac{D}{2}})$ when $D$ is even and by $O(L^{\frac{D-1}{2}})$ when $D$ is odd.
Note that generalizations of the Toric code to $D$-dimensional systems have the above dimensional duality for arbitrary integer $m$. See \[sec:topology2\] for constructions of $D$-dimensional Toric code. Such a dimensional duality arising in geometric shapes of logical operators may be viewed as a manifestation of Poincaré duality on coding properties of stabilizer codes supported on a $D$-torus, as very briefly discussed in \[sec:topology\].
However, we report that straightforward generalizations of analysis tools presented in this paper did not work for STS models with $D \geq 4$, and the conjecture above needs to be verified.
**Feasibility for a three-dimensional STS model:** Finally, let us show that three-dimensional STS models do not work as self-correcting quantum memory. First, in order for the system to work as a quantum code, there must be an anti-commuting pair of one-dimensional and two-dimensional logical operators. We denote them as $\ell$ and $r$ respectively. Let us assume that the system is initially in the eigenstate of $\ell=1$ denoted as $|\psi (\ell=1)\rangle$. Then, $|\psi (\ell=-1)\rangle = r |\psi (\ell=1)\rangle$ and $r$ is a two-dimensional logical operator. Since the boundary of a subpart of $r$ is one-dimensional, the excitation energies associated with intermediate states are $O(L)$. Thus, the encoding with respect to $\ell$ is self-correcting.
Next, let us assume that the system is initially in the eigenstate of $r=1$ denoted as $|\psi (r=1)\rangle$. Then, $|\psi (r=-1)\rangle = \ell |\psi (r=1)\rangle$ and $\ell$ is a two-dimensional logical operator. Since the boundary of a subpart of $\ell$ is zero-dimensional, the excitation energies associated with intermediate states are $O(1)$. Thus, the encoding with respect to $r$ is not self-correcting, and the qubit storage time is $\tau \sim O(1)$, independent of the system size. Therefore, while such a system is a good quantum code with the code distance $O(L)$, it works only as self-correcting *classical* memory, with the bit storage time $\tau \sim \mbox{EXP}(L)$. However, in order for the system to work as self-correcting quantum memory, there must be a pair of anti-commuting two-dimensional logical operators.
**Summary and discussion:** We summarize coding properties of STS models based on dimensions of pairs of logical operators: $$\begin{aligned}
\begin{array}{ccccccc}
\mbox{Spatial dim} & \mbox{Logical operators} & \mbox{Code distance} & \mbox{Bit storage time} & \mbox{Qubit storage time} & \mbox{Self-correction} \\ \hline
\mbox{1 dim} & \mbox{0 dim + 1 dim} & O(1) & O(1) & O(1)& \\
\mbox{2 dim} & \mbox{0 dim + 2 dim} & O(1) &\mbox{EXP}(L) & O(1)&\mbox{classical} \\
\mbox{2 dim} & \mbox{1 dim + 1 dim} & O(L) & O(1) & O(1)& \\
\mbox{3 dim} & \mbox{0 dim + 3 dim} & O(1) &\mbox{EXP}(L) & O(1)& \mbox{classical} \\
\mbox{3 dim} & \mbox{1 dim + 2 dim} & O(L) &\mbox{EXP}(L) & O(1)&\mbox{classical} \\
\mbox{4 dim} & \mbox{2 dim + 2 dim} & O(L^{2}) &\mbox{EXP}(L) & \mbox{EXP}(L)&\mbox{quantum}
\end{array}\notag\end{aligned}$$ where, for $D=4$, we presented coding properties of four-dimensional Toric code.
Thermal stability of topological order {#sec:topo}
======================================
In this section, we present another main result of the present paper, concerning the thermal stability of topological order at finite temperature by establishing the connection between the feasibility of self-correcting quantum memory and the stability of topological order.
In section \[sec:topo1\], we begin by establishing the connection between self-correcting classical memory and the thermal stability of ferromagnetic order at finite temperature. In section \[sec:topo2\], we establish the connection between quantum codes and topological order at zero temperature. In section \[sec:topo3\], we establish the connection between self-correcting quantum memory and topological order at finite temperature, and analyze the thermal stability of topological order arising in STS models. Note that discussion in this section is rather heuristic, and more rigorous treatment follows in \[sec:topo\_ap\]
Classical equivalence {#sec:topo1}
---------------------
In this subsection, we establish the connection between self-correcting classical memory and thermal stability of ferromagnetic order at finite temperature: $$\begin{aligned}
\begin{array}{ccc}
\mbox{Classical code} & \leftrightarrow & \mbox{Ferromagnetic order at $T=0$}\\
\mbox{Self-correcting classical memory} & \leftrightarrow & \mbox{Ferromagnetic order at $T>0$}
\end{array}\notag\end{aligned}$$
**Two-dimensional ferromagnet:** We have seen that two-dimensional Ising model works as self-correcting classical memory since the energy barrier separating two degenerate ground states is $O(L)$. This self-correcting property of two-dimensional Ising model is closely related to the thermal stability of ferromagnetic order as seen from the following thermodynamic argument.
Consider two-dimensional Ising model: $$\begin{aligned}
H(\epsilon) \ = \ - \sum_{i,j}Z_{i,j}Z_{i+1,j} - \sum_{i,j}Z_{i,j}Z_{i,j+1} - \epsilon \sum_{i,j} Z_{i,j}\end{aligned}$$ with an initial bias (symmetry-breaking field); $-\epsilon \sum_{i,j} Z_{i,j}$ for $\epsilon \geq 0$. As a result of an initial bias, the system is not degenerate anymore, and the ground state is $|0\cdots0\rangle$. The thermal stability of ferromagnetic order can be analyzed through the expectation value of the total magnetization $m$: $$\begin{aligned}
m \ = \ \frac{1}{N} \sum_{i,j} Z_{i,j}\end{aligned}$$ at finite temperature, which can be computed as follows: $$\begin{aligned}
\langle m \rangle \ = \ \lim_{\epsilon \rightarrow 0} \frac{1}{\beta}\frac{\partial \log Z }{\partial \epsilon}\end{aligned}$$ where the partition function is: $Z(\beta, \epsilon) = \text{Tr} e^{- \beta H(\epsilon)}$. Here, we evaluate the expectation value of $m$ at the limit where $\epsilon \rightarrow 0$ after we take the limit where $N$ goes to infinity. (Otherwise, there is no use of introducing an initial bias). Then, we have: $$\begin{aligned}
\left\langle \ \frac{1}{N} \sum_{i,j} Z_{i,j} \ \right\rangle_{\epsilon \rightarrow 0} \ &= \ 1 \qquad (T \ = \ 0) \\
1 \ > \ \left\langle \ \frac{1}{N} \sum_{i,j} Z_{i,j} \ \right\rangle_{\epsilon \rightarrow 0} \ &> \ 0 \qquad ( T_{c} \ > T \ > \ 0) \\
\left\langle \ \frac{1}{N} \sum_{i,j} Z_{i,j} \ \right\rangle_{\epsilon \rightarrow 0} \ &= \ 0 \qquad ( T \ > \ T_{c} ) \end{aligned}$$ where $T_{c}$ is some finite transition temperature, as plotted in Fig. \[fig\_topo\_2\](a).
One may notice that the total magnetization has some non-zero value as long as the temperature is below the transition temperature $T_{c}$. This indicates that the system is stable against thermal fluctuations for $T<T_{c}$. In particular, the system properties for $T_{c} > T > 0$ are close to the ground state properties at $T=0$. However, the system properties change drastically at $T=T_{c}$, and for $T > T_{c}$, the total magnetization vanishes. This indicates that the system undergoes a phase transition at $T=T_{c}$, and is unstable against thermal fluctuations for $T > T_{c}$.
![Total magnetizations. (a) Two-dimensional Ising model. (b) One-dimensional Ising model. []{data-label="fig_topo_2"}](fig_topo_2.pdf){width="0.70\linewidth"}
Now, one can establish the connection between self-correcting properties and the thermal stability. In a coding theoretical language, the computation of the magnetization at finite temperature may be interpreted as follows. We first apply an initial bias and encode a bit $0$ to a ground state $|0\cdots\rangle$. Then, we let the system interact with the external environment at finite temperature, and wait until the system reaches the equilibrium while slowly turning off the initial bias. Finally, we decode the initially encoded bit by measuring the total magnetization. Therefore, self-correcting property of two-dimensional Ising model with exponentially long bit storage time is a direct consequence of the thermal stability of ferromagnetic order at finite temperature.
In general, the total magnetization we have used above is called an *order parameter* in condensed matter physics since it can be used to distinguish different phases of matters. On the other hand, quantum coding theoretically, one may view the total magnetization as a symmetric summation of a zero-dimensional logical operator $Z_{i,j}$ with which a classical bit is encoded. Later, we shall see that expectation values of logical operators can be used as order parameters to characterize the stability of topological order.
**One-dimensional ferromagnet:** Next, let us see that one-dimensional Ising model, which does not work as self-correcting classical memory, is not stable against thermal fluctuations. Consider one-dimensional Ising model: $$\begin{aligned}
H(\epsilon) \ = \ - \sum Z_{j}Z_{j+1} - \epsilon \sum Z_{j}\end{aligned}$$ with an initial bias, and compute the expectation values of the total magnetization. Then, we have $$\begin{aligned}
\left\langle \ \frac{1}{N} \sum_{j} Z_{j} \ \right\rangle_{\epsilon \rightarrow 0} \ = \ 1 \qquad ( T \ = \ 0) \\
\left\langle \ \frac{1}{N} \sum_{j} Z_{j} \ \right\rangle_{\epsilon \rightarrow 0} \ = \ 0 \qquad ( T \ > \ 0)\end{aligned}$$ as plotted in Fig. \[fig\_topo\_2\](b). One may notice that the total magnetization becomes zero at any finite temperature. This indicates that the system is not stable against thermal fluctuations at any finite temperature, and the system undergoes a phase transition at $T=0$ as seen from the sudden change of the magnetization at $T=0$. The system properties for $T=0$ and for $T>0$ are significantly different, and the ground state properties are not stable against thermal fluctuations. This is the underlying reason why one-dimensional Ising model has $O(1)$ bit storage time, and does not work as self-correcting classical memory.
**Summary of the equivalence:** With observations above, one may notice that a large energy barrier, which is essential to self-correcting properties, is the key to the thermal stability of ferromagnetic order. In Fig. \[fig\_topo\_3\], we give a summary of the equivalence concerning classical memories. However, it is not clear whether a large energy barrier is sufficient for the thermal stability of the system or not. We address this issue briefly in \[sec:topo\_ap3\].
![A summary of the classical equivalence. []{data-label="fig_topo_3"}](fig_topo_3.pdf){width="0.70\linewidth"}
Here, it is worth noting that ferromagnetic systems, such as one-dimensional and two-dimensional Ising model are not stable against perturbations, as described in Fig. \[fig\_topo\_3\]. If one chooses $V$ as the initial bias, one can easily break the ground state degeneracy, and thus, the ground state property is not stable against perturbations. This indicates that ferromagnetic systems are not topologically ordered (see section \[sec:topo2\] too).
Existence of topological order and quantum codes {#sec:topo2}
------------------------------------------------
Now, we return to the main discussion in this section, concerning the stability of topological order at finite temperature. In this subsection and the next, we establish the connection between self-correcting quantum memory and the thermal stability of topological order: $$\begin{aligned}
\begin{array}{ccc}
\mbox{Quantum code} & \leftrightarrow & \mbox{Topological order at $T=0$}\\
\mbox{Self-correcting quantum memory} & \leftrightarrow & \mbox{Topological order at $T>0$}
\end{array}\notag\end{aligned}$$ In this subsection, we describe the definition of topological order in spin systems on a lattice, and argue that quantum codes have topological order at zero temperature. Note that our discussion closely follows pioneering works [@Nussinov09; @Bravyi10b].
**Phenomenological definition:** Let us first begin by describing the phenomenological definition of topological order which is commonly used in physics community. A system is said to have topological order when its ground state properties do not change significantly under any types of small perturbations [@Wen90].
\[def:topo\] Consider a degenerate spin system defined on some closed geometric manifold governed by a Hamiltonian $H$. The system is said to be topologically ordered at $T=0$ if and only if the system satisfies the following conditions:
- There exists some finite positive number $\delta$ such that, for any perturbations $V$: $$\begin{aligned}
H' \ = \ H + V, \qquad V \ = \ \sum_{j} V_{j}\end{aligned}$$ where $V_{j}$ are locally defined and $|V_{j}|\leq \delta$, the ground state degeneracy is not broken at the thermodynamic limit (Fig. \[fig\_topo\_def\]).
- The perturbed ground space $G$ can be approximated by the original ground space $G_{0}$ through some local unitary transformation: $$\begin{aligned}
UG_{0}U^{\dagger} \ \approx \ G\end{aligned}$$ which can be represented as $$\begin{aligned}
U \ = \ \int_{0}^{1} \exp(- i \hat{h} t)dt\end{aligned}$$ where $\hat{h}$ is some hermitian operator which is a summation of local terms with finite amplitudes.
![The stability of the ground state properties against local perturbations. []{data-label="fig_topo_def"}](fig_topo_def.pdf){width="0.70\linewidth"}
It is worth noting that a classical ferromagnet (Ising model) is not topologically ordered while it works as self-correcting classical memory. First of all, the ground state degeneracy of a classical ferromagnet is broken if a local magnetic field term $Z_{j}$ is added as a perturbation: $$\begin{aligned}
V \ = \ - \epsilon \sum_{j} Z_{j}.\end{aligned}$$ While the original Hamiltonian has $|0\cdots0\rangle$ and $|1\cdots1\rangle$ as ground states, the perturbed Hamiltonian has $|0\cdots0\rangle$ as a single ground state for positive $\epsilon$. Second, the ground state property significantly changes as a result of perturbations. While a classical ferromagnet has $\frac{1}{\sqrt{2}}(|0\cdots0\rangle + |1\cdots1\rangle)$ as a ground state, the perturbed ground state is $|0\cdots0\rangle$, and there is no local unitary transformation which transforms $\frac{1}{\sqrt{2}}(|0\cdots0\rangle + |1\cdots1\rangle)$ into $|0\cdots0\rangle$ [@Bravyi06]. As a straightforward extension of this discussion, one may notice that STS models with zero-dimensional logical operators are not topologically ordered, and topological order may exist only in two or higher-dimensional systems due to the dimensional duality of logical operators.
**Coding theoretical definition:** While the definition above captures the stability of ground state properties in topologically ordered systems, one may wonder what kinds of systems are actually topologically ordered. Also, one may wonder if there is a universal feature commonly shared among all the topologically ordered systems.
Somewhat surprisingly, a quantum coding theoretical viewpoint gives a hint to answer these questions by providing an alternative definition of topological order which may capture universal properties of topologically ordered systems [@Bravyi06]. In fact, a system of spins on a lattice may be said to be topologically ordered when the ground space is separated from excited states by a finite energy gap at the thermodynamic limit and the ground space realizes a quantum code whose code distance $d$ is comparable to the system size [@Bravyi10b]. One may rewrite this characterization of topological order through coding properties more explicitly in the following way.[^5]
\[def:red\] Consider a degenerate spin system defined on some closed geometric manifold. The system is said to be topologically ordered when it satisfies the following condition:
- Let the degenerate ground state space be $G$, which is protected by a finite energy gap. Consider reduced density matrices of ground states for a region $R$ which is contractable to a single point: $$\begin{aligned}
\rho_{R}(|\psi\rangle) \ \equiv \ Tr_{\bar{R}} (|\psi\rangle\langle\psi|).\end{aligned}$$ So, $R$ is a topologically trivial *zero-dimensional region* (Fig. \[fig\_reduce\]). Then, all the reduced density matrices of degenerate ground states are the same at the thermodynamic limit $$\begin{aligned}
\rho_{R}(|\psi\rangle) \ = \ \rho_{R}(|\psi'\rangle) \qquad \mbox{for all} \quad |\psi\rangle, \ |\psi'\rangle \ \in \ G \quad \mbox{at}\quad N \ \rightarrow \ \infty.\end{aligned}$$
![A coding theoretical definition of topological order. []{data-label="fig_reduce"}](fig_reduce.pdf){width="0.60\linewidth"}
Let us see why the condition above ensures that the system serves as a quantum code with a macroscopic code distance. When all the reduced density matrices are the same for all the topologically trivial regions, any errors acting inside such regions cannot change the encoded logical qubits. In order to change logical qubits, some global operator acting on a region with a topologically non-trivial geometric shape is necessary. Such a global operator has a weight equal to the code distance. Therefore, the ground space of such a system works as a good quantum code with a macroscopic code distance.[^6]
**Equivalence of two definitions:** Now, let us see that definition \[def:red\] implies definition \[def:topo\]. Under a small perturbation $V$ which is a summation of local terms, we perform a perturbative analysis. Since the ground states are separated by excited states by a finite energy gap, only the coupling between degenerate ground states is dominant. Since there is no local operator to couple two ground states, one needs to consider $O(L)$th order perturbation to have a coupling between degenerate ground states. Since such a coupling will be exponentially suppressed by a factor $\exp(- L / L_{0})$ where $L_{0}$ is some finite length scale, and the energy splitting is exponentially suppressed and goes to zero as $L$ goes to infinity. Thus, the ground state degeneracy is protected at the thermodynamic limit. When the ground state degeneracy is protected and the ground space is separated from excited states by a finite energy gap, one can approximate the ground space from the original unperturbed ground space according to the theory of adiabatic continuation [@Bravyi10b; @Hastings05]. A complete mathematical proof was presented in [@Bravyi10b]. However, it is not known if definition \[def:topo\] implies definition \[def:red\] or not.
Here, we briefly mention the usage of the expression “topological order” since it is sometimes used in a fuzzy and elusive way. In the present paper, by topological order, we mean the ground state properties of topologically ordered systems. By its definition, topological order is stable against any types of small local perturbations, meaning that the ground state degeneracy is protected and perturbed ground states can be approximated through local unitary transformations. Therefore, by the ground state properties, we mean any global features of ground states which are free from length scales and are not affected by local unitary transformations. We will return to discussion on the meaning of “global features” later.
**Topological order in STS models:** Let us conclude this subsection on the definition of topological order by seeing that two-dimensional and three-dimensional STS models can have topological order according to definition \[def:red\]. Consider the case when two-dimensional STS model has $k$ pairs of one-dimensional logical operators. Let us split the entire system into two complementary regions $P(n_{1}-1,n_{2}-1)$ and $\overline{P(n_{1}-1,n_{2}-1)}$ where $P(n_{1}-1,n_{2}-1)$ is a region with $n_{1}-1 \times n_{2}-1$ composite particles. Note that $P(n_{1}-1,n_{2}-1)$ is topologically trivial zero-dimensional region since one can shrink it into a single point continuously. Since there are $2k$ logical operators defined inside $\overline{P(n_{1}-1,n_{2}-1)}$, we have $g_{\overline{P(n_{1}-1,n_{2}-1)}}=2k$. However, this means that $g_{P(n_{1}-1,n_{2}-1)}=0$ from theorem \[theorem\_partition\], and there is no logical operator defined inside $P(n_{1}-1,n_{2}-1)$. Then, all the reduced density matrices are the same, and such a two-dimensional STS model is topologically ordered since it satisfies definition \[def:red\], and thus, definition \[def:topo\]. A similar discussion holds for a three-dimensional STS model with pairs of one-dimensional and two-dimensional logical operators.
Quantum equivalence {#sec:topo3}
-------------------
In this subsection, we establish the connection between self-correcting quantum memory and topological order at finite temperature, and analyze the thermal stability of topological order arising in STS models.
**Two-dimensional Toric code:** Two-dimensional Toric code is topologically ordered since it has a macroscopic code distance: $d \sim O(1)$. Yet, it does not work as self-correcting quantum memory since the energy barrier is $\Delta E \sim O(1)$. This is closely related to the thermal instability of topological order as we shall see below. Let us perform a thermodynamic analysis on two-dimensional Toric code in a way similar to Ising model by adding an initial bias: $$\begin{aligned}
H_{\ell}(\epsilon) \ = \ H_{Toric} - \epsilon \sum_{j=1}^{L} T_{2}^{j}(\ell) \\
H_{r}(\epsilon) \ = \ H_{Toric} - \epsilon \sum_{i=1}^{L} T_{1}^{i}(r)\end{aligned}$$ where $\ell$ and $r$ are one-dimensional logical operators extending in the $\hat{1}$ and $\hat{2}$ directions respectively. Here, we define the following “normalized logical operators”: $$\begin{aligned}
m_{\ell} \ &= \ \frac{1}{L}\sum_{j=1}^{L} T_{2}^{j}(\ell) \\
m_{r} \ &= \ \frac{1}{L}\sum_{i=1}^{L} T_{1}^{i}(r)\end{aligned}$$ by taking symmetric summations of logical operators, as we did for Ising model. Then, we have $$\begin{aligned}
\left\langle m_{\ell} \right\rangle_{\epsilon \rightarrow 0} \ = \ \left\langle m_{r} \right\rangle_{\epsilon \rightarrow 0} \ = \ 1 \qquad ( T \ = \ 0) \\
\left\langle m_{\ell} \right\rangle_{\epsilon \rightarrow 0} \ = \ \left\langle m_{r} \right\rangle_{\epsilon \rightarrow 0} \ = \ 0 \qquad ( T \ > \ 0)\end{aligned}$$ as plotted in Fig. \[fig\_topo\_4\](a) where $\langle m_{\ell}\rangle$ is evaluated for $H_{\ell}(\epsilon)$, and $\langle m_{r}\rangle$ is evaluated for $H_{r}(\epsilon)$. This indicates that the system is not stable against thermal fluctuations at any finite temperature, meaning that topological order arising in two-dimensional Toric code is thermally unstable. This implies that one cannot read out initially encoded qubit by measuring $m_{\ell}$ and $m_{r}$.
**Three-dimensional Toric code:** Next, let us consider thermodynamic properties of three-dimensional Toric code. While three-dimensional Toric code does not work as self-correcting quantum memory, it works as self-correcting *classical* memory. This is because it has pairs of one-dimensional and two-dimensional logical operators, and as a result, the bit storage time is $\tau \sim \mbox{EXP($L$)}$ while the qubit storage time is $\tau \sim O(1)$. These coding properties are closely related to thermodynamic properties of three-dimensional Toric code, as seen from expectation values of logical operators: $$\begin{aligned}
m_{\ell} \ &= \ \frac{1}{L}\sum_{z=1}^{L} T_{3}^{z}(\ell) \\
m_{r} \ &= \ \frac{1}{L^{2}}\sum_{x,y=1}^{L} T_{1}^{x}T_{2}^{y}(r)\end{aligned}$$ where $\ell$ is a two-dimensional logical operator extending in the $\hat{1}$ and $\hat{2}$ directions, while $r$ is a one-dimensional logical operator extending in the $\hat{3}$ direction. Their expectation values behave as follows: $$\begin{aligned}
\left\langle m_{\ell} \right\rangle_{\epsilon \rightarrow 0} \ = \ 1 \qquad ( T \ = \ 0) \\
\left\langle m_{\ell} \right\rangle_{\epsilon \rightarrow 0} \ = \ 0 \qquad ( T \ > \ 0)\end{aligned}$$ and $$\begin{aligned}
\left\langle m_{r} \right\rangle_{\epsilon \rightarrow 0} \ &= \ 1 \qquad (T \ = \ 0) \\
1 \ > \ \left\langle m_{r} \right\rangle_{\epsilon \rightarrow 0} \ &> \ 0 \qquad ( T_{c} \ > T \ > \ 0) \\
\left\langle m_{r} \right\rangle_{\epsilon \rightarrow 0} \ &= \ 0 \qquad ( T \ > \ T_{c} ) \end{aligned}$$ where $T_{c}$ is some finite transition temperature, as plotted in Fig. \[fig\_topo\_4\](b). This implies that three-dimensional Toric code undergoes phase transitions both at $T=0$ and $T=T_{c}$, and the ground state properties are not completely stable against thermal fluctuations at any finite temperature. Yet, the ground state properties partially survive at finite temperature as a direct consequence of being self-correcting classical memory.
![Expectation values of logical operators. (a) Two-dimensional Toric code. (b) Three-dimensional Toric code. (c) Four-dimensional Toric code. []{data-label="fig_topo_4"}](fig_topo_4.pdf){width="0.70\linewidth"}
**Four-dimensional Toric code:** Finally, let us see that four-dimensional Toric code, which works as self-correcting quantum memory, is stable against thermal fluctuations. Here, we define $$\begin{aligned}
m_{\ell} \ = \ \frac{1}{L^{2}}\sum_{z,w=1}^{L} T_{3}^{z}T_{4}^{w}(\ell) \\
m_{r} \ = \ \frac{1}{L^{2}}\sum_{x,y=1}^{L} T_{1}^{x}T_{2}^{y}(r)\end{aligned}$$ where $\ell$ is a two-dimensional logical operator extending in the $\hat{1}$ and $\hat{2}$ directions, and $r$ is a two-dimensional logical operator extending in the $\hat{3}$ and $\hat{4}$ directions. Then, we have $$\begin{aligned}
\left\langle m_{\ell} \right\rangle_{\epsilon \rightarrow 0} \ &= \ \left\langle m_{r} \right\rangle_{\epsilon \rightarrow 0} \ = \ 1 \qquad (T \ = \ 0) \\
1 \ > \ \left\langle m_{\ell} \right\rangle_{\epsilon \rightarrow 0} \ &= \ \left\langle m_{r} \right\rangle_{\epsilon \rightarrow 0} \ > \ 0 \qquad ( T_{c} \ > T \ > \ 0) \\
\left\langle m_{\ell} \right\rangle_{\epsilon \rightarrow 0} \ &= \ \left\langle m_{r} \right\rangle_{\epsilon \rightarrow 0} \ = \ 0 \qquad ( T \ > \ T_{c} ) \end{aligned}$$ where $T_{c}$ is some finite transition temperature, as plotted in Fig. \[fig\_topo\_4\](c). This implies that the ground state properties are stable against thermal fluctuations and topological order arising in four-dimensional Toric code is stable at finite temperature.
**Summary of the equivalence:** With these observations, one may notice that large energy barrier inside the ground space, which is essential to self-correcting properties, is the key to the thermal stability of topological order. In Fig. \[fig\_topo\_7\], we give a summary of the equivalence concerning quantum memory.
![The quantum equivalence. []{data-label="fig_topo_7"}](fig_topo_7.pdf){width="0.70\linewidth"}
With this connection between the feasibility of self-correcting quantum memory and the thermal stability of topological order, we conclude that topological order arising in STS models is not stable against thermal fluctuations. In other words, for $D \leq 3$, there is no system which is stable against both thermal fluctuations and local perturbations simultaneously. While discussion here is rather heuristic, we give a more rigorous treatment on the definition of the thermal stability of topological order in \[sec:topo\_ap\]. We summarize physical properties of STS models based on dimensions of pairs of logical operators: $$\begin{aligned}
\begin{array}{ccccccc}
\mbox{Spatial dim} & \mbox{Logical operators} & \mbox{Local perturbations} & \mbox{Thermal fluctuations} & \mbox{Memory property} \\ \hline
\mbox{1 dim} & \mbox{0 dim + 1 dim} & & & \mbox{Classical code} \\
\mbox{2 dim} & \mbox{0 dim + 2 dim} & & \mbox{stable} & \mbox{Classical self-correction} \\
\mbox{2 dim} & \mbox{1 dim + 1 dim} & \mbox{stable} & & \mbox{Quantum code} \\
\mbox{3 dim} & \mbox{0 dim + 3 dim} & & \mbox{stable} & \mbox{Classical self-correction} \\
\mbox{3 dim} & \mbox{1 dim + 2 dim} & \mbox{stable} & & \mbox{Quantum code} \\
\mbox{4 dim} & \mbox{2 dim + 2 dim} & \mbox{stable} & \mbox{stable} & \mbox{Quantum self-correction}
\end{array}\notag\end{aligned}$$ where, for $D=4$, we presented coding properties of four-dimensional Toric code.
Emergence of topology in logical operators {#sec:physics}
==========================================
So far, we have addressed specific questions concerning coding and physical properties of gapped spin systems on a lattice. While these two questions are of particular importance in quantum information science and condensed matter physics, the ultimate goal is to find universal properties of arbitrary gapped spin systems on a lattice and develop a unified theoretical framework to discuss their coding and physical properties. In particular, it would be very beautiful if arbitrary gapped spin systems can be analyzed universally through some unified theoretical tool which is yet to be discovered.
In this section, as a necessary first step toward this goal, we make an attempt to find universal properties of logical operators which are commonly shared among all the STS models. Logical operators in STS models have a certain interesting topological property which is a direct consequence of physical constraints we have imposed on stabilizer Hamiltonians. Here, we show that one can always deform geometric shapes of logical operators continuously while keeping them equivalent by applying appropriate stabilizers as long as one does not change geometric shapes in a topologically non-trivial way. We prove this continuous deformability of logical operators in STS models for $D=1,2,3$ with various examples of continuous deformations of logical operators. Some of these results were also presented in our previous papers [@Beni10; @Beni10b], especially for $D=2$.
The program of finding a universal theory of gapped spin systems is continued in \[sec:topology\] where the role of topology in analyzing coding and physical properties of stabilizer codes is further examined, while we concentrate on demonstrating topological properties of logical operators in this section.
**One-dimension:** We begin by illustrating a topological property of logical operators for one-dimensional STS models. Recall that $P(x)$ represents a region of $x$ composite particles, and $g_{P(1)} = k$ in one-dimensional systems regardless of the system size since there always exist $k$ zero-dimensional logical operators. Then, as a result of the bi-partition theorem (theorem \[theorem\_partition\]), we have $$\begin{aligned}
g_{P(1)} \ = \ g_{\overline{P(1)}} \ = \ k.\end{aligned}$$
![A deformation of a zero-dimensional logical operator. []{data-label="deform_1dim"}](deform_1dim.pdf){width="0.45\linewidth"}
Now, let us interpret the equation above from a geometric viewpoint. Consider some logical operator $\ell$ defined inside $\overline{P(1)}$. Then, since $g_{P(1)}=g_{\overline{P(1)}}=k$, there must exist some equivalent logical operator $\ell' \sim \ell$ which is defined inside $P(1)$. In other words, if there exists a logical operator defined inside $\overline{P(1)}$, one can shrink its geometric shape into $P(1)$ by applying some stabilizer (Fig. \[deform\_1dim\]): $$\begin{aligned}
\overline{P(1)} \ \rightarrow \ P(1).\end{aligned}$$ Here, one may notice that $\overline{P(1)}$ and $P(1)$ are topologically equivalent regions since one can deform $\overline{P(1)}$ into $P(1)$ continuously. On the other hand, if a logical operator is a one-dimensional logical operator defined all over the lattice (i.e. defined inside $P(n)$ where $n$ is the linear length of the system), one may not be able to deform the logical operator into $P(1)$.
**Two-dimensions:** Next, we analyze a topological property of logical operators for two-dimensional STS models. For the convenience of presentation, we assume that zero-dimensional logical operators defined inside $P(1,2v)$ can be actually defined inside $P(1,1)$ in a two-dimensional STS model. This may be done through some appropriate coarse-graining.
We first list regions which serve as references to classify geometric shapes of logical operators in a two-dimensional system. We define *topological unit regions* as follows (Fig. \[fig\_2D\_unit\](a)): $$\begin{aligned}
Q(0,0) \equiv P(1,1), \quad Q(1,0) \equiv P(n_{1},1), \quad Q(0,1) \equiv P(1,n_{2}), \quad Q(1,1) \equiv P(n_{1},n_{2}).\end{aligned}$$ “$1$” and “$0$” represent whether a region extends in the corresponding direction or not. For example, $Q(1,0)$ and $Q(0,1)$ are one-dimensional unit regions which extend in the directions of $\hat{1}$ and $\hat{2}$ respectively. $Q(0,0)$ is a zero-dimensional unit region with a single composite particle. $Q(1,1)$ is a two-dimensional unit region which consists of all the composite particles in the system. These topological unit regions are shown graphically in Fig. \[fig\_2D\_unit\](a). We also denote a union of all the $m$-dimensional topological unit regions as $R_{m}$: $$\begin{aligned}
R_{0} \equiv Q(0,0), \quad R_{1} \equiv Q(1,0) \cup Q(0,1), \quad R_{2} \equiv Q(1,1).\end{aligned}$$ We call $R_{m}$ *$m$-dimensional concatenated topological unit regions*. All the concatenated unit regions are graphically shown in Fig. \[fig\_2D\_unit\](b).
![Reference regions. (a) Topological unit regions. (b) Concatenated unit regions. []{data-label="fig_2D_unit"}](fig_2D_unit.pdf){width="0.70\linewidth"}
In a two-dimensional system, there are five different unions of topological unit regions: $R_{0}$, $Q(1,0)$, $Q(0,1)$, $R_{1}$ and $R_{2}$. We call these regions, except $R_{2}$, *reference regions*, whose set is denoted as $R_{ref}$: $$\begin{aligned}
R_{ref} = \{ R_{0}, Q(1,0), Q(0,1), R_{1} \}.\end{aligned}$$ Then, one can introduce equivalence relations between these reference regions and their complements in terms of continuous deformations. For example, as shown in Fig. \[fig\_deformation\](a), $\overline{R_{0}}$ can be continuously deformed into $R_{1}$ by enlarging the hole of $\overline{R_{0}}$ gradually. Also, as shown in Fig. \[fig\_deformation\](b), $\overline{R_{1}}$ can be continuously deformed into $R_{0}$ since both $\overline{R_{1}}$ and $R_{0}$ are zero-dimensional regions without any winding around the torus. Finally, as shown in Fig. \[fig\_deformation\](c), $\overline{Q(1,0)}$ can be deformed into $Q(1,0)$ since both regions have a winding in the $\hat{1}$ direction. In summary, we have the following equivalence relations between reference regions and their complements: $$\begin{aligned}
\overline{R_{0}} \simeq R_{1}, \qquad \overline{R_{1}}\simeq R_{0}, \qquad \overline{Q(1,0)}\simeq Q(1,0), \qquad \overline{Q(0,1)} \simeq Q(0,1).\end{aligned}$$
![The topological deformations of logical operators.[]{data-label="fig_deformation"}](fig_deformation.pdf){width="0.70\linewidth"}
Now, we discuss how geometric shapes of logical operators can be determined. A useful observation regarding geometric shapes of logical operators can be obtained by considering the number of independent logical operators defined inside a region $R$. Let the number of independent logical operators inside $R$ be $g_{R}$. Here, we consider the case where we have two regions $R$ and $R'$ where $R$ is larger than $R'$, meaning that $R$ includes all the composite particles inside $R'$ (Fig. \[fig\_deform\_intuition\]). Then, if $g_{R}=g_{R'}$, $R$ and $R'$ support the same logical operators since all the logical operators defined inside $R$ have equivalent representations which are supported inside $R'$. This means that, for a given logical operators $\ell$ defined inside $R$, one can always find another equivalent logical operator $\ell'$ defined inside $R'$. In other words, one can *deform* the geometric shape of $\ell$ into $\ell'$ by applying some appropriate stabilizer (Fig. \[fig\_deform\_intuition\]). Thus, by finding two connected regions $R$ and $R'$ where $R$ is larger than $R'$ and $g_{R}=g_{R'}$, one can conclude that logical operators defined inside $R$ can be deformed into $R'$.
![A shrinkage from $R$ to $R'$ when $g_{R}=g_{R'}$.[]{data-label="fig_deform_intuition"}](fig_deform_intuition.pdf){width="0.40\linewidth"}
With the above observation on geometric shapes of logical operators and deformations in mind, let us describe a topological property of geometric shapes of logical operators. The following lemma summarizes the topological property of logical operators in two-dimensional STS models.
\[lemma\_2dim\] In two-dimensional STS models, the following equations hold: $$\begin{aligned}
g_{\overline{R_{0}}} = g_{R_{1}}, \quad g_{\overline{R_{1}}} = g_{R_{0}}, \quad g_{Q(1,0)} = g_{\overline{Q(1,0)}} = k, \quad g_{Q(0,1)} = g_{\overline{Q(0,1)}} = k.\end{aligned}$$
In other words, one can shrink geometric shapes of logical operators by applying some appropriate stabilizers in the following ways: $$\begin{aligned}
\overline{R_{0}} \rightarrow R_{1}, \qquad \overline{R_{1}}\rightarrow \ R_{0}, \qquad \overline{Q(1,0)}\rightarrow \ Q(1,0), \qquad \overline{Q(0,1)} \rightarrow \ Q(0,1).\end{aligned}$$ Note that these shrinkages preserve topological properties of geometric shapes of logical operators.
Let us begin with the proof of $g_{\overline{R_{0}}} = g_{R_{1}}$. Let $k_{0}$ be the numbers of pairs of anti-commuting zero-dimensional and two-dimensional logical operators. Let $k_{1}$ be the number of pairs of anti-commuting one-dimensional logical operators. Then, we notice that $g_{R_{1}}=2k_{1}+k_{0}$ since $R_{1}$ supports both zero-dimensional and one-dimensional logical operators. On the other hand, since $g_{\overline{R_{0}}} = 2k - g_{R_{0}}$ from theorem \[theorem\_partition\] and $g_{R_{0}}=k_{0}$, we have $g_{\overline{R_{0}}} = g_{R_{1}}$. If we use theorem \[theorem\_partition\] to $g_{\overline{R_{0}}} = g_{R_{1}}$, we readily obtain $g_{R_{0}} = g_{\overline{R_{1}}}$ too. Next, let us show $g_{\overline{Q(1)}} = g_{Q(1)}$. Note that $g_{Q(1)}=k$. Then, we have $g_{\overline{Q(1)}}=k$, and have $g_{\overline{Q(1)}} = g_{Q(1)}$.
**Three-dimensions:** Let us continue our analysis on STS models for higher-dimensional cases ($D>2$). Below, we analyze the topological property of logical operators in three-dimensional STS models. We assume that zero-dimensional logical operators and one-dimensional logical operators in a three-dimensional STS model can be defined inside $P(1,1,1)$, $P(n_{1},1,1)$, $P(1,n_{2},1)$ and $P(1,1,n_{3})$.
We begin by finding reference regions for $D$-dimensional systems. Reference regions for $D>2$ can be defined from topological unit regions in a way similar to two-dimensional cases. Let $\vec{d}$ be an arbitrary binary $D$ component vector $\vec{d} = (d_{1},\cdots,d_{D})$ with $d_{m}=0,1$. Then, topological unit regions are: $$\begin{aligned}
Q(\vec{d}) \equiv P(\vec{x}), \qquad \mbox{where} \quad x_{m}=n_{m}^{d_{m}}.\end{aligned}$$ For example, $Q(1,1,0)=P(n_{1},n_{2},1)$ for $D=3$. We denote the weight of the binary vector $\vec{d}$ as $w(\vec{d}) \equiv \sum_{m=1}^{D}d_{m}$, which represents the dimension of $Q(\vec{d})$. Then, concatenated unit regions are defined as follows: $$\begin{aligned}
R_{m} \equiv \bigcup_{w(\vec{d})=m} Q(\vec{d}).\end{aligned}$$ A set of reference regions can be obtained by considering all the possible unions of $Q(\vec{d})$, which is denoted as $R_{ref}$. One may notice that topological unit regions $Q(\vec{d})$ are like independent generators for $m$th homology group for a $D$-torus: $H_{m}(T^{D})$.
![Topological unit regions in a three-dimensional system. Recall that we set the periodic boundary conditions. []{data-label="fig_3D_unit"}](fig_3D_unit.pdf){width="0.55\linewidth"}
It is worth presenting some examples here. In a three-dimensional system ($D=3$), we have the following topological unit regions. $$\begin{split}
\mbox{0 dim:} \qquad &Q(0,0,0)\\
\mbox{1 dim:} \qquad &Q(1,0,0),\ Q(0,1,0),\ Q(0,0,1)\\
\mbox{2 dim:} \qquad &Q(1,1,0),\ Q(0,1,1),\ Q(1,0,1)\\
\mbox{3 dim:} \qquad &Q(1,1,1).
\end{split}$$ Some examples are shown in Fig. \[fig\_3D\_unit\]. Also, concatenated topological unit regions are: $$\begin{split}
R_{0} &= Q(0,0,0)\\
R_{1} &= Q(1,0,0) \cup Q(0,1,0) \cup Q(0,0,1)\\
R_{2} &= Q(1,1,0)\cup Q(0,1,1)\cup Q(1,0,1)\\
R_{3} &= Q(1,1,1)
\end{split}$$ which are described in Fig. \[fig\_3D\_R\].
![Concatenated unit regions in a three-dimensional system. []{data-label="fig_3D_R"}](fig_3D_R.pdf){width="0.45\linewidth"}
One can introduce the equivalence relations in terms of reference regions. Equivalence relations among them are shown as follows: $$\begin{split}\label{eq:list}
R_{0} &\simeq \overline{R_{2}} \\
Q(1,0,0) &\simeq \overline{Q(1,1,0)\cup Q(1,0,1)}\\
Q(0,1,0) &\simeq \overline{Q(1,1,0)\cup Q(0,1,1)}\\
Q(0,0,1) &\simeq \overline{Q(1,0,1)\cup Q(0,1,1)}\\
Q(1,0,0) \cup Q(0,1,0)&\simeq \overline{Q(1,1,0)\cup Q(0,0,1)}\\
Q(0,1,0) \cup Q(0,0,1)&\simeq \overline{Q(0,1,1)\cup Q(1,0,0)}\\
Q(0,0,1) \cup Q(1,0,0)&\simeq \overline{Q(1,0,1)\cup Q(0,1,0)}\\
R_{1} &\simeq \overline{R_{1}}\\
Q(1,1,0)&\simeq \overline{Q(1,1,0)}\\
Q(0,1,1) &\simeq \overline{Q(0,1,1)}\\
Q(1,0,1)&\simeq \overline{Q(1,0,1)}.
\end{split}$$ Then, we have the following theorem.
\[theorem\_Ddim\] For $D$-dimensional STS models ($D=1,2,3$), let $R$ and $R'$ be reference regions: $R,R' \in R_{ref}$. When $\overline{R'}\simeq R$, one can deform geometric shapes of logical operators continuously from $\overline{R'}$ to $R$: $$\begin{aligned}
g_{R} = g_{\overline{R'}} \qquad \mbox{for}\quad R \simeq \overline{R'}.\end{aligned}$$
The proof of the lemma is straightforward from theorem \[theorem\_partition\] and theorem \[theorem\_3dim\], so we present a proof only for $R_{1} \simeq \overline{R_{1}}$. Since all the zero-dimensional and one-dimensional logical operators can be supported inside $R_{1}$, we have $g_{R_{1}}\geq k$. Similarly, we have $g_{\overline{R_{1}}}\geq k$. Then, since $g_{R_{1}} + g_{\overline{R_{1}}}=2k$, we have $g_{R_{1}} = g_{\overline{R_{1}}}=k$.
**Topological deformations of logical operators:** So far, our discussion on a topological property of logical operators has been limited only to regions generated by $m$-dimensional unit regions while some connected regions (such as examples presented in Fig. \[topo\_deform\]) cannot be generated by taking unions of unit regions. Then, a naturally arising question is whether one can perform similar continuous deformations for arbitrary connected regions of composite particles or not. Here, we provide a complete description on topological properties of logical operators in STS models by extending the notion of continuous deformations to arbitrary connected regions.
It turns out that one can continuously deform a geometric shapes of a logical operator defined on any connected region of composite particles continuously for STS models with $D=1,2,3$. Here, we summarize topological properties of logical operators in STS models as follows[^7]:
\[theorem\_continuous\_deformation\] Consider STS models for $D=1,2,3$. Consider two connected regions of composite particles $R$ and $R'$ which are topologically equivalent: $R \ \simeq \ R'$. Then, we have $$\begin{aligned}
g_{R} \ = \ g_{R'} \qquad \mbox{for}\quad R \ \simeq \ R'.\end{aligned}$$ Therefore, for any given logical operator $\ell$ defined inside $R$, one can always find an equivalent logical operator defined inside $R'$ as long as $R$ can be continuously deformed into $R'$.
![Examples of continuous deformations. Note that systems have periodic boundary conditions.[]{data-label="topo_deform"}](topo_deform.pdf){width="0.65\linewidth"}
For clarity of presentation, we skip the proof of theorem \[theorem\_continuous\_deformation\]. One may verify it by using scale symmetries, a bi-partition theorem (theorem \[theorem\_partition\]), and the translation equivalence of logical operators (theorem \[theorem\_TE\]). We present some examples of continuous deformations in Fig. \[topo\_deform\].
Summary and open questions {#sec:summary}
==========================
In this paper, we have established the connection between the feasibility of self-correcting quantum memory and the thermal stability of topological order, and provided partial answers to these two open problems by solving a model that may cover a large class of physically realizable quantum codes. Our discussion is limited to stabilizer codes with translation and scale symmetries, and these two problems still remain open for an even larger class of gapped spin systems on a lattice. Yet, we hope that our analysis will provide an important insight and a useful guidance on studies of coding and physical properties of gapped spin systems on a lattice. We also hope that our work will contribute to stimulating the use of quantum coding theoretical concepts in studying many-body quantum systems further.
Below, we discuss possible future problems and make some comments on them.
**Frustration-free systems with scale symmetries:** While our discussion is limited to stabilizer codes, any gapped spin systems with degenerate ground states can be used as quantum memory devices in principle. It seems that, for an arbitrary gapped spin system defined on a lattice, there exists some frustration-free Hamiltonian which approximates the original system and serve as its low energy effective Hamiltonian (except some subtle properties such as chirality). In fact, this claim has been rigorously proven for one-dimensional gapped spin systems [@Hastings06] by showing that any ground state of one-dimensional gapped spin Hamiltonians can be efficiently simulated through the matrix product state formalism. Now, this claim seems to be widely believed among the condensed matter physics community, and it is at the heart of recent progress in classifications of quantum phases in gapped spin systems [@Chen10; @Beni10b; @Chen11]. Therefore, the analyses on coding properties of arbitrary frustration-free Hamiltonians with translation and scale symmetries may provide a useful insight on questions concerning coding properties and quantum phases arising in arbitrary gapped spin systems on a lattice.
To the best of our knowledge, all the examples of frustration-free Hamiltonians with translation and scale symmetries, such as the quantum double model [@Kitaev03] and the string-net model [@Levin05], have the dimensional duality on geometric shapes of certain operators which may be considered as generalizations of logical operators. Also, these operators seem to have the continuous deformability in a way similar to logical operators in STS models. With these observations, we feel that our results are universal for all the frustration-free Hamiltonians with translation and scale symmetries, and thus, effectively true for arbitrary gapped spin systems defined on a lattice with a small number of ground states. Also, these observations imply that any gapped spin systems with scale symmetries (or a small number of ground states) may be described through TQFT, as discussed in \[sec:topology\]. Yet, the connection between TQFT and coding properties of gapped spin systems on a lattice with scale symmetries must be further established.
**Beyond scale symmetries:** While our treatments in the present paper are limited to quantum codes with scale symmetries, there are several interesting models of quantum codes which do not have scale symmetries [@Chamon05; @Newman99; @Bravyi11b; @Haah11]. Of particular interests are models proposed in [@Newman99] and in [@Haah11] which are classical and quantum memories respectively with partially self-correcting properties. These models do not have scale symmetries since the number of logical qubits $k_{\vec{n}}$ is highly sensitive to the system size $\vec{n}$, and there is no constant upper bound on $k_{\vec{n}}$. These models are known to have logarithmically large energy barrier $\Delta E \sim \mbox{LOG($L$)}$ with a large number of energy local minima. As a result, these models seem to have POLY($L$) relaxation time with slow dynamics, which may result in POLY($L$) bit or qubit storage time.
Whether these models with broken scale symmetries are useful as classical and quantum memories is a complicated problem. First of all, for a system to be a efficient quantum memory device, it is desirable to have exponentially long qubit storage time: $\tau \sim \mbox{EXP($L$)}$, since it needs at least $d$ gate operations to write or readout a logical qubit, and these encoding and decoding processes takes at least polynomially long time $\tau \sim \mbox{POLY($L$)}$. Also, it seems difficult to find an efficient decoding algorithm with an efficient error-correction for these models. Finally, these models undergo phase transitions at $T=0$, which may imply their potential thermal instability. At this moment, there are many issues to be analyzed concerning implementability of quantum codes with broken scales symmetries from engineering viewpoints.
**Beyond TQFT:** Stabilizer codes with broken scale symmetries are remarkable examples which may be beyond the description of the standard TQFT. Such models do not have continuously deformable logical operators, and are not expected to be characterized by topological properties of logical operators as in STS models. This observation is consistent with the fact that systems described by TQFT, in a sense of the axiomatic formulation developed by Atiyah, are allowed to have only a finite number of degenerate ground states. Thus, finding an effective field theoretical description for stabilizer codes with broken scale symmetries may be an interesting future problem which may lead to discoveries of novel quantum phases that are currently unknown and are governed by some deeper mathematical formalism than topology.
To the best of our knowledge, all the examples of stabilizer codes with broken scale symmetries still possess certain discrete scale symmetries. STS models have continuous scale symmetries since $k_{\vec{n}}=k$ for all the system sizes $\vec{n}$, while stabilizer codes with broken scale symmetries seem to have discrete scale symmetries since the number of logical qubits $k_{\vec{n}}$ has good scaling properties under global scale transformations: $(n_{1},n_{2},\cdots)\rightarrow (cn_{1},cn_{2},\cdots)$ where $c$ is some appropriate integer with behaviors such as $k_{c\vec{n}}=c k_{\vec{n}}$. The distinction between continuous and discrete scale symmetries becomes particularly important when one considers effects of RG transformations. As is pointed out in [@Wilson71], systems with continuous scale symmetries correspond to fixed points of RG transformations, while systems with discrete scale symmetries correspond to limit cycles of RG transformations. Physical realizations of systems with discrete scale symmetries include the Effimov effect which is recently of particular interest in the ultracold atom physics community. This observation implies that stabilizer codes with discrete scale symmetries correspond to limit cycles of RG transformations. Yet, the connection between limit cycles and discrete scale symmetries for lattice systems must be further established since discussion in [@Wilson71] is given primary for continuum systems.
**Spin glass behaviors and translation symmetry breaking:** Another interesting property of stabilizer codes with broken scale symmetries is a glassy behavior with slow relaxation dynamics. This glassy behavior may be understood as a direct consequence of broken scale symmetries, and associated broken translation symmetries in the ground space.
In general, even if a Hamiltonian possesses translation symmetries, its ground states may not possess translation symmetries. Let us assume that a stabilizer Hamiltonian is translation invariant under unit translations of composite particles: $T_{m}(H)=H$ for all $m$. Then, ground states of this stabilizer Hamiltonian may break translation symmetries: $T_{m}(|\psi\rangle)\not=|\psi\rangle$. Some examples of systems with broken translation symmetries are presented in [@Wen02; @Kitaev06b; @Beni10b], and translation symmetry breaking of ground states arising in topologically ordered systems are studied in [@Wen02; @Kitaev06b].
An interesting connection between scale symmetries and translation symmetries is that when a stabilizer code has scale symmetries: $k_{\vec{n}}=k$, all the ground states are invariant under unit translations of composite particles: $T_{m}(|\psi\rangle)=|\psi\rangle$ regardless of the system size, as proven in [@Beni10b]. In other words, if a system is coarse-grained such that scale symmetries are satisfied, translation symmetries are protected inside the ground space. On the other hand, if a system does not have scale symmetries, translation symmetries may be broken inside the ground space. In particular, when the system has a large number of logical qubits, translation symmetries are strongly broken such that there is no finite translation which keep ground states equivalent.
This strong breaking of translation symmetries seems to be the key to the glassy behavior observed in stabilizer codes with broken scale symmetries. In conventional spin glass models, a Hamiltonian consists of mutually commuting terms whose signs are random, and translations symmetries of the Hamiltonian are initially broken due to the randomness. This randomness in the Hamiltonian gives rise to spin configurations of ground states which are not uniform over real space, and leads to a complicated and slow relaxation dynamics in spin glasses. On the other hand, in stabilizer codes with broken scale symmetries, stabilizer Hamiltonians possess translation symmetries initially. However, spin configurations are not uniform since translation symmetries are strongly broken inside the ground space due to broken scale symmetries. Thus, systems with broken scale symmetries are expected to exhibit spin glass behaviors with slow relaxation dynamics and a large number of local minima. Yet, the connection between the glassy behavior and broken scale symmetries must be further established since the observation given here is very heuristic.
Topological order at finite temperature {#sec:topo_ap}
=======================================
In the main part of the paper, we have discussed the thermal stability of topological order by using expectation values of logical operators. In this appendix, we verify the use of expectation values of logical operators as topological order parameters, and define the thermal stability of topological order more rigorously.
In \[sec:topo\_ap1\], we begin by verifying that expectation values of logical operators can be used for topological order parameter at $T=0$. In \[sec:topo\_ap2\], we define topological order at finite temperature by extending the definition of topological order at zero temperature to finite temperature. In \[sec:topo\_ap3\], we discuss whether the existence of a large energy barrier is sufficient for the thermal stability of the system or not.
Topological order parameters and logical operators {#sec:topo_ap1}
--------------------------------------------------
While the ground state properties change only slightly under small perturbations in topologically ordered systems, the original ground state properties will be eventually lost under large perturbations. One may see whether topological order is lost by checking whether a new ground state can be approximated by the original ground state through a local unitary transformation. However, this naive approach will require a substantial amount of computations and cannot capture changes of coding and physical properties of the ground state under perturbations.
Fortunately, the loss of the ground state properties under perturbations can be quantitatively characterized by some physical quantities, called topological order parameters. Here, we characterize the stability of topological order against perturbations through topological order parameters.
**Topological order parameter:** Topological order parameters are physical quantities with which one can judge if topological order is lost under perturbations or not. In this light, one may deduce the necessary properties of topological order parameters. By its definition, a topological order parameter must be a global function of the system which does not change under local unitary transformations. Also, a topological order parameter must change only slightly when topological order is protected, while it undergoes some non-analytic change when topological order is lost as a result of perturbations.
There have been several proposals for such topological order parameters. The number of ground states serves as a topological order parameter since the ground state degeneracy is protected for topologically ordered systems at the thermodynamic limit [@Wen90]. Another interesting proposal is to use a certain entanglement measure, called topological entanglement entropy [@Kitaev06; @Levin06], which corresponds to a constant correction to the entanglement area law. Since topological entanglement entropy is a non-local quantity which involves a large number of spins, it does not change under local unitary transformations.
Below, we analyze the stability of topological order thorough expectation values of logical operators in order to further build the connection between topological order and quantum codes.
**Expectation values of logical operators:** Let us first consider two-dimensional Toric code and denote a pair of anti-commuting logical operators as $\ell$ and $r$ where $\ell$ extends in the $\hat{1}$ direction and $r$ extends in the $\hat{2}$ direction. Then, one may use the expectation values of the summations of translations of logical operators as order parameters: $$\begin{aligned}
U(\ell) \ = \ \frac{1}{L} \sum_{y} T_{2}^{y}(\ell), \qquad U(r) \ = \ \frac{1}{L} \sum_{x} T_{1}^{x}(r)\end{aligned}$$ where $L$ is the linear length of the system. Here, we took the summation of logical operators in a symmetric way.
Now, let us discuss if topological order is stable under a perturbation $V$ by considering the following perturbed Hamiltonian with an initial bias: $$\begin{aligned}
H_{\ell}(\epsilon) \ = \ H_{STS} - \epsilon U(\ell) + V.\end{aligned}$$ Note that, when $V=0$, the small bias breaks the ground state degeneracy and $$\begin{aligned}
\langle U(\ell)\rangle_{\epsilon \rightarrow 0} \ = \ 1.\end{aligned}$$ Recall that, in evaluating the expectation value of $U(\ell)$ in a presence of $V$, one must take the limit where $\epsilon$ goes to zero after taking the limit where $L$ goes to infinity. Since the ground space is separated from excited states by a finite energy gap, one may neglect the effect of excited states. So, let us examine the effect of $\epsilon U(\ell)$ and $V$ on the ground space. As a result of $\epsilon U(\ell)$, the ground state space is split into two sectors with $\ell = 1$ and $\ell = -1$ where the energy splitting between two sectors is $\epsilon$. On the other hand, the energy splitting induced by a perturbation $V$ is exponentially suppressed by a factor $\exp(- L / L_{0})$. Then, in considering the effect of $V$ on a sector with $\ell =1$, one may neglect the effect of a sector with $\ell = -1$ when $\epsilon$ is $O(1)$. Now, we take the limit of $L \rightarrow \infty$. Then, the ground space corresponds to a sector with $\ell=1$, and we have $$\begin{aligned}
\langle U(\ell)\rangle_{\epsilon \rightarrow 0} \ > \ 0\end{aligned}$$ even when $\epsilon$ goes to zero. However, if $V$ is large enough, the ground state properties will be lost and $\langle U(\ell)\rangle_{\epsilon \rightarrow 0}$ will be close to the value for the ground state of $V$. In a similar way, one may consider the following initial bias: $$\begin{aligned}
H_{r}(\epsilon) \ = \ H_{STS} - \epsilon U(r) + V\end{aligned}$$ and have $$\begin{aligned}
\langle U(r)\rangle_{\epsilon \rightarrow 0} \ > \ 0.\end{aligned}$$ Therefore, the stability of topological order can be characterized by expectation values of $U(\ell)$ and $U(r)$.
This argument implies that the loss of the ground state properties may be captured through the stability of logical qubits encoded in the ground space. The initial bias of $\epsilon U(\ell)$ represents the initial encoding of a logical qubit. The expectation value of $U(\ell)$ under a perturbation $V$ represents how much information is protected in a presence of perturbations.
**Stability of topological order:** Based on these observations, let us formulate the stability of topological order in terms of logical operators. Let us first generalize the definition of logical operators. We call operators $U$ which satisfy the following conditions *generalized logical operators*: $$\begin{aligned}
&[H_{0},U] \ = \ 0 \\
&U \ \not= \ e^{i\theta}I \quad \mbox{and} \quad |U|_{\mbox{max}} \ = \ 1 \qquad \mbox{in the ground space}\end{aligned}$$ where the action of $U$ is defined inside the ground space of the original unperturbed Hamiltonian $H_{0}$. In other words, generalized logical operators are any operators which does not change the energy of the system, but acts non-trivially inside the ground space. Generalized logical operators are not $c$-numbers inside the ground space. For example, a scaled summation of translations of logical operators is a generalized logical operator, while projectors onto the ground space are not generalized logical operators.
Based on generalized logical operators, we define the stability of topological order in the following way.
\[def:order\] The system has topological order when there exists a pair of generalized logical operators $U(\ell)$ and $U(r)$ which satisfy the following conditions:
- $U(\ell)$ and $U(r)$ do not commute with each other: $[U(\ell),U(r)]\not =0$ inside the ground space. This implies the existence of the ground state degeneracy, and $U(\ell)$ and $U(r)$ may characterize a logical qubit or qudit.
- For perturbed Hamiltonians: $$\begin{aligned}
H_{\ell} \ = \ H_{0} - \epsilon U(\ell) + V, \qquad H_{r} \ = \ H_{0} - \epsilon U(r) + V\end{aligned}$$ with any types of local perturbations $V$, $$\begin{aligned}
\langle U(\ell) \rangle_{\epsilon \rightarrow 0, \ V \rightarrow 0} \ &= \ 1 \qquad \mbox{for $H_{\ell}$} \\
\langle U(r) \rangle_{\epsilon \rightarrow 0, \ V \rightarrow 0} \ &= \ 1 \qquad \mbox{for $H_{r}$}.\end{aligned}$$
In evaluating $\langle U(\ell) \rangle_{\epsilon \rightarrow 0, V \rightarrow 0}$, we first take the thermodynamic limit, and then, take the limit of $V \rightarrow 0$, and finally, take the limit of $\epsilon \rightarrow 0$. It should be emphasized that we consider any types of perturbations $V$ in analyzing the stability of topological order[^8].
Let us look at some examples here. For a one-dimensional classical ferromagnet, let $\ell$ and $r$ be a zero-dimensional and a one-dimensional logical operators: $$\begin{aligned}
\ell\ = \ Z_{1},\qquad r\ =\ \prod_{j}X_{j}.\end{aligned}$$ Then, for $V = t \sum_{j}Z_{j}$, we have $$\begin{split}
H_{\ell} \ &= \ - \sum Z_{j}Z_{j+1}- \epsilon \Big(\frac{1}{L}\sum_{j}Z_{j} \Big) + t \sum_{j} Z_{j} \\
&= \ - \sum Z_{j}Z_{j+1} + \Big(t - \frac{\epsilon}{L} \Big) \sum_{j} Z_{j}.
\end{split}$$ At the thermodynamic limit, we have $$\begin{aligned}
\langle U(\ell) \rangle \ &= \ -1 \qquad \mbox{for $H_{\ell}$}\end{aligned}$$ regardless of $t>0$ and $\epsilon>0$. Then, taking the limit of $t \rightarrow 0$ and $\epsilon \rightarrow 0$, we have $$\begin{aligned}
\langle U(\ell) \rangle_{\epsilon \rightarrow 0, \ V \rightarrow 0} \ &= \ -1 \qquad \mbox{for $H_{\ell}$}.\end{aligned}$$ Therefore, the system is not topologically ordered.
**Topological order parameters and local unitary:** Strictly speaking, expectation values of logical operators cannot be used as topological order parameters. In fact, while logical operators are non-locally defined for topologically ordered systems, their expectation values may change under local unitary transformations. However, topological order must be characterized by some physical quantities or objects which are not affected by local unitary transformations. In fact, both the ground state degeneracy and topological entanglement entropy do not change under local unitary transformations. In this light, one might think that the above definition of the stability of topological order is not legitimate. Here, we make some comments on this issue briefly.
Despite the fact that expectation values of logical operators are not topological order parameters, we can formulate the stability of topological order through them. This is because we considered not only a specific type of local perturbations but all the types of local perturbations. Here, we demonstrate that, if we consider only one type of perturbations, expectation values of logical operators may fail to capture quantum phase transitions. In two-dimensional Toric code, let us consider a one-dimensional logical operator $\ell$ which consists only of Pauli $X$ operators. Then, consider the following perturbation: $$\begin{aligned}
V \ = \ - t \sum_{r} X_{r},\end{aligned}$$ where $t$ is some positive parameter and $r$ represents the position of each qubit. Then, when we increase $t$, there must be a quantum phase transition and the ground state properties will be close to the ones for $V$. However, such a perturbation may not change the value of $\langle U(\ell) \rangle$ since $V$ commutes with the logical operator $\ell$. In general, if one carefully choose the types of perturbations, expectation values of logical operators remain unchanged. This implies that if one tries to study a quantum phase transition only through $\langle U(\ell) \rangle$, one may fail to detect the transition.[^9]
Topological order at finite temperature {#sec:topo_ap2}
---------------------------------------
So far, we have focused on the ground state properties of topologically ordered systems at zero temperature and seen that topological order characterizes the ground state properties which are stable against any types of small perturbations. Here, we address the thermal stability of topological order.
The stability of topological order against perturbations can be discussed by seeing if the perturbed ground state can be reached by local unitary transformations or not. However, the stability of topological order against thermal fluctuations cannot be formulated in a similar way since there is no unitary transformation which connects a pure state and a statistical ensemble. Here, we define the stability of topological order against thermal fluctuations by the changes of topological order parameters.
**Stability against thermal fluctuations:** Below, we give the definition of the stability of topological order through expectation values of logical operators.
The system is said to have topological order which is stable at finite temperature if and only if there exists a pair of non-commuting generalized logical operators $U(\ell)$ and $U(r)$, and some finite transition temperature $T_{c}$ which satisfy the following conditions:
- Consider perturbed Hamiltonians with initial biases $$\begin{aligned}
H_{\ell} \ = \ H_{0} - \epsilon U(\ell) + V, \qquad H_{r} \ = \ H_{0} - \epsilon U(r) + V.\end{aligned}$$ For $T=0$, $$\begin{aligned}
\langle U(\ell) \rangle_{\epsilon \rightarrow 0, \ V \rightarrow 0} \ &= \ 1 \qquad \mbox{for $H_{\ell}$} \\
\langle U(r) \rangle_{\epsilon \rightarrow 0, \ V \rightarrow 0} \ &= \ 1 \qquad \mbox{for $H_{r}$},\end{aligned}$$ and for $T_{c}>T>0$, $$\begin{aligned}
\langle U(\ell) \rangle_{\epsilon \rightarrow 0, \ V \rightarrow 0} \ &> \ 0 \qquad \mbox{for $H_{\ell}$} \\
\langle U(r) \rangle_{\epsilon \rightarrow 0, \ V \rightarrow 0} \ &> \ 0 \qquad \mbox{for $H_{r}$}\end{aligned}$$ for any types of $V$.
One may see the connection between self-correcting quantum memory and the stability of topological order. In particular, if topological order in a spin system is stable at finite temperature, such a system works as self-correcting quantum memory. A logical qubit encoded with respect to $U(\ell)$ and $U(r)$ can be read out by measuring $U(\ell)$ and $U(r)$ even at finite temperature. This implies that encoded logical qubits are not lost in a presence of the interaction with the external environment, and there must be some self-correcting thermal dissipation processes.
It should be noted that the expectation values of logical operators $\ell$ and $r$ usually vanishes at any finite temperature as one may see from a direct calculation. Only the symmetric summations of logical operators may give rise to non-vanishing expectation values of generalized logical operators.
**Comment on the definition of stability:** Our definition of the stability of topological order relies on topological order parameters associated with a pair of non-commuting logical operators $\ell$ and $r$. The definition of the stability of topological order in the present paper is motivated purely from quantum information theoretical viewpoints. However, the definition we have used in the present paper may be too strict. For example, if one is interested only in the ground state properties associated to the expectation value of $\ell$, it makes perfect sense to claim that a three-dimensional STS model has topological order which is stable at finite temperature. In fact, once the system is held at finite temperature, the system properties are stable against any types of small perturbations.
Similarly, the stability of the ground state properties depends crucially on types of perturbations. For example, while there is no topological order in a one-dimensional system in a strict sense, the ground state properties of a one-dimensional classical ferromagnet are stale against small perturbations if one declares to consider only perturbations which are products of $X$ operators. Similarly, the ground state properties of the one-dimensional AKLT model is stable against small perturbations which do not break the time-reversal symmetry. Therefore, in discussing the stability of topological order, one needs to specify types of symmetries which are of interest.
Thermal stability of topological order and energy barrier {#sec:topo_ap3}
---------------------------------------------------------
We have seen that a large energy barrier in classical or quantum memory is the key to the thermal stability of ferromagnetic order or topological order at finite temperature. Yet, it is not clear if the existence of a large energy barrier is sufficient for the thermal stability or not.
While STS models always have energy barrier $O(L^{a})$ where $a$ is an integer ($a \geq 0$) due to the dimensional duality of logical operators, there are several examples of stabilizer codes with broken scale symmetries which have logarithmic energy barrier: $\Delta E \sim \mbox{LOG($L$)}$. Let us look at an example of classical memory with broken scale symmetries [@Newman99]. The model is constructed on a square lattice with $L\times L$ qubits: $$\begin{aligned}
H \ = \ - \sum_{i,j}Z_{i,j}Z_{i+1,j}Z_{i,j+1}.\end{aligned}$$ The model does not have scale symmetries, and there is no upper bound on the number of logical qubits (bits) $k$ since $k$ is highly sensitive to $L$. The model has logarithmic energy barrier $\Delta E \sim \mbox{LOG($L$)}$, and as a result, the bit memory time is expected to be $\tau \sim \mbox{POLY($L$)}$ if one trusts the Arrhenius law.
Despite a large energy barrier which scales with the system size, the model is known to be thermally unstable. This may be easily verified by computing the partition function: $Z(\beta)=\text{Tr}e^{-\beta H}$. Since $N-k$ stabilizers are independent, we have $$\begin{aligned}
(e^{-\beta} + e^{\beta})^{N-k}(e^{-\beta})^{k} \ \leq \ Z(\beta) \ \leq \ (e^{-\beta} + e^{\beta})^{N-k} (e^{\beta})^{k}\end{aligned}$$ and, we have $$\begin{aligned}
\lim_{N \rightarrow \infty} \frac{1}{N} \log Z(\beta)\ = \ \log (e^{-\beta} + e^{\beta}).\end{aligned}$$ Therefore, the thermodynamic property of this model is equivalent to a single qubit in a magnetic field, and the model does not have the thermal stability. A similar discussion holds for stabilizer codes with $N$ qubits, supported by $N$ interaction terms, which include one-dimensional Ising model, two-dimensional Toric code, models proposed in [@Chamon05; @Haah11]. Note that $M > N$ in two-dimensional Ising model and four-dimensional Toric code where $M$ is the number of interaction terms.
It is not clear whether polynomial energy barrier is necessary for thermal stability of ferromagnetic order or not. Another important characteristic of models with logarithmic energy barrier is the existence of a logarithmically large number of local minima. As a result, at finite temperature, the entropic term easily dominates the free energy function. So, it seems that the thermal instability results from both the logarithmic energy barrier and the logarithmically large number of local minima.
It is worth mentioning that there is an interacting spin model with a logarithmic energy barrier, but is not defined on a lattice. The model is called two-dimensional XY model, which is known to undergo a thermal phase transition at finite temperature. The model is of particular interest since it was not expected to possess any thermal phase transitions as a result of the Mermin-Wagner theorem which states that continuous symmetries cannot be spontaneously broken at finite temperature for $D \leq 2$. The reason why two-dimensional XY model undergoes a thermal phase transition at finite temperature is because the transition, known as Kosterlitz-Thouless transition, is induced by topological defects which are not directly related to continuous symmetries of the model. In two-dimensional XY model, it takes logarithmic energy to create a vortex: $\Delta E \sim \log r$ where $r$ is the size of a vortex. As a result, at low temperature, a configuration without vortices is favored, while at high temperature, a configuration with a large number of vortices is favored. Between these configurations, a phase transition occurs.
From the observations above, it seems that the logarithmic energy barrier may lead to the thermal stability, while the existence of local minima may break the thermal stability. Yet, at this moment, the connection between the energy barrier and the thermal stability has not been completely established.
Topology and quantum codes {#sec:topology}
==========================
In section \[sec:physics\], we have shown that one can deform geometric shapes of logical operators continuously while keeping them equivalent. This continuous deformability of logical operators implies that one can introduce the notion of topology in analyzing and classifying coding and physical properties of STS models. In this appendix, we further discuss the role of topology in analyzing stabilizer codes.
Many interesting properties of STS models, such as the dimensional duality of logical operators, naturally appear as corollaries of the continuous deformability of logical operators. In \[sec:topology1\], we begin by showing that the dimensional duality of logical operators can be derived only by assuming the continuous deformability of logical operators. In particular, we show that, for stabilizer codes with continuously deformable logical operators, $m$-dimensional and $D-m$-dimensional logical operators always form anti-commuting pairs where $m$ is an arbitrary positive integer ($m\leq D$). As an example of $D$-dimensional stabilizer codes with continuously deformable logical operators, we present generalizations of the Toric code to $D$-dimensional systems with anti-commuting pairs of $m$-dimensional and $D-m$-dimensional logical operators for arbitrary positive integers $D$ and $m$.
The topological property of logical operators naturally leads us to consider a possible relevance to another well-celebrated theoretical framework equipped with the notion of topology; called topological quantum field theory (TQFT) [@Witten89; @Birmingham91; @Nayak08]. In \[sec:topology2\], we establish the connection between quantum codes with the continuous deformability and TQFT further by demonstrating that the braiding of anyonic excitations in a $D$-dimensional stabilizer code is characterized by a topological invariant in a $D+1$-dimensional system. This implies that such a $D$-dimensional stabilizer code can be effectively described by $D+1$-dimensional TQFT.
Dimensional duality as a corollary of continuous deformability {#sec:topology1}
--------------------------------------------------------------
The continuous deformability of logical operators implies that topology is the essential notion in analyzing coding and physical properties of STS models. Then, a naturally arising question concerns the role of topology in determining coding and physical properties of stabilizer codes. Here, we show that the dimensional duality of logical operators is a universal property for all the stabilizer codes with continuously deformable logical operators. We also give concrete examples of $D$-dimensional stabilizer codes which have continuous deformability of logical operators and the dimensional duality by generalizing the Toric code to $D$-dimensional systems.
**Dimensional duality from topological deformation:** Let us begin by counting the number of independent logical operators defined inside $m$-dimensional regions. Recall that $m$-dimensional concatenated unit regions are obtained by taking unions of all the $m$-dimensional topological unit regions. Then, one may call logical operators which can be defined inside $R_{m}$, but cannot be defined inside $R_{m-1}$, *$m$-dimensional logical operators*.
$m$-dimensional logical operators are logical operators which have representations supported inside $R_{m}$, but do not have representations supported inside $R_{m-1}$.
Now, let us denote the number of independent $m$-dimensional logical operators as $g_{m} \equiv g_{R_{m}} - g_{R_{m-1}}$ where $g_{0} \equiv g_{R_{0}}$ by setting $g_{R_{-1}}\equiv0$. Then, there exists an interesting relation among the numbers of $m$-dimensional logical operators. In particular, the following lemma holds.
\[lemma\_duality\] There are the same number of $m$-dimensional and $D-m$-dimensional logical operators: $$\begin{aligned}
g_{m} = g_{D-m} \qquad \mbox{for} \quad m = 0,\cdots,D.\end{aligned}$$
The proof of this lemma can be obtained through a simple algebra by combining theorem \[theorem\_partition\] and the deformability of logical operators.
Consider a bi-partition of the entire system into $R_{m}$ and $\overline{R_{m}}$. From the topological deformation of logical operators, we have $$\begin{aligned}
g_{\overline{R_{m}}} = g_{R_{D-m-1}}\end{aligned}$$ since $\overline{R_{m}} \simeq R_{D-m-1}$. Thus, $R_{m}$ and $\overline{R_{m}}$ support the following logical operators: $$\begin{aligned}
R_{m} &: \quad \mbox{$0$-dim}, \ \mbox{$1$-dim}, \ \cdots, \ \mbox{$m$-dim} \notag \\
\overline{R_{m}} &: \quad \mbox{$0$-dim}, \ \mbox{$1$-dim}, \ \cdots, \ \mbox{$D-m-1$-dim}. \notag\end{aligned}$$ Therefore, we have $$\begin{aligned}
g_{R_{m}} = \sum_{j=0}^{m} g_{j}, \qquad g_{\overline{R_{m}}} = \sum_{j=0}^{D-m-1}g_{j}.\end{aligned}$$
Recall that $g_{R}+g_{\bar{R}}=2k$ as presented in theorem \[theorem\_partition\]. Using this formula for $R = R_{m}$, we have $$\begin{aligned}
g_{R_{m}} + g_{\overline{R_{m}}} = 2k.\end{aligned}$$ Then, we have $$\begin{aligned}
\sum_{j=0}^{m} g_{j} + \sum_{j=0}^{D-m-1}g_{j} = 2k, \qquad \mbox{for} \quad m = 0,\cdots,D.\end{aligned}$$ Since the total number of independent logical operators is $\sum_{j=0}^{D}g_{j}=2k$, we have $g_{m} = g_{D-m}$ for all $m$.
This lemma implies the existence of a dimensional duality in geometric shapes of logical operators. One may notice that this lemma is just a manifestation of Poincaré duality in a $D$-torus where the $m$th and $D-m$th Betti numbers are equal.
To completely establish relations between each logical operator with different dimensions, let us analyze their commutation relations. We have the following theorem.
\[theorem\_duality\] One can choose a set of $2k$ independent logical operators of $D$-dimensional systems with the deformability of logical operators in the following way: $$\begin{aligned}
\left\{
\begin{array}{cccccc}
\ell_{1}, & \cdots , & \ell_{k} \\
r_{1}, & \cdots , & r_{k}
\end{array}
\right\}.\end{aligned}$$ where $\ell_{p}$ are $m_{p}$-dimensional logical operators and $r_{p}$ are $D-m_{p}$-dimensional logical operators for some integer $m_{p}$ ($0 \leq m_{p} \leq D$) for any $p=1,\cdots,k$.
In other words, one can choose logical operators such that the summation of dimensions of pairs of anti-commuting logical operators is always $D$. Theorem \[theorem\_duality\] follows immediately from the following lemma.
$m$-dimensional and $m'$-dimensional logical operators commute with each other if $m + m' <D$.
Consider a $m$-dimensional logical operator $\ell$ and a $m'$-dimensional logical operator $\ell'$ which is defined inside $R_{m}$ and $R_{m'}$ respectively. For $m + m'<D$, there exists a translation of $R_{m}$ such that $R_{m}$ and $R_{m'}$ have no overlap. Then, due to the translation equivalence of logical operators, some translation of $\ell$ do not have an overlap with $\ell'$, which leads to $[\ell,\ell']=0$.
With this lemma, the proof of theorem \[theorem\_duality\] is immediate by using lemma \[lemma\_duality\]. For example, from the lemma, zero-dimensional logical operators may anti-commute only with $D$-dimensional logical operators. Since there are the same number of zero-dimensional and $D$-dimensional logical operators, there exists a canonical set of logical operators where $D$-dimensional logical operators can anti-commutes only with zero-dimensional logical operators. Similarly, one can show that there exists a set of $2k$ independent logical operators such that $m$-dimensional logical operators anti-commute only with $D-m$-dimensional logical operators for all $m$.
**Generalization of the Toric code:** Now, we give concrete examples of $D$-dimensional stabilizer codes which have continuously deformable logical operators. The model we present here is a straightforward generalization of two-dimensional Toric code to $D$-dimensional settings for arbitrary positive integer $D$. In particular, we illustrate the construction of $D$-dimensional Toric code with anti-commuting pairs of $m$-dimensional and $D-m$ dimensional logical operators for arbitrary positive integers $D$ and $m$ ($m \leq D$).
![An example of the construction for $D=3$ and $m=2$. []{data-label="D_dim"}](D_dim.pdf){width="0.50\linewidth"}
We first consider a $D$-dimensional hypercubic lattice which consists of $N=L\times \cdots \times L$ of $D$-dimensional unit hypercubes with periodic boundary conditions. We denote a set of $p$-dimensional unit hypercubes in this lattice as $h_{p}$. Note that $|h_{p}| = {}_D C_{p} \cdot N$ since one needs to specify $p$ directions from $D$ directions in defining $p$-dimensional unit hypercubes. We put qubits at the centers of $m$-dimensional hypercubes. Then, the total number of qubits is ${}_D C_{m} \cdot N$, and qubits are labeled by $m$-dimensional unit hypercubes in $h_{m}$. Fig. \[D\_dim\] shows a construction of the model for $D=3$ and $m=2$ where qubits live at centers of two-dimensional unit squares.
The Hamiltonian consists of plaquette terms and star terms as in the conventional two-dimensional Toric code: $$\begin{aligned}
H \ = \ H_{plaquette} + H_{star}\end{aligned}$$ Here, we call $(m+1)$-dimensional unit hypercubes in $h_{m+1}$ “plaquettes”. Then, plaquette terms consist of Pauli $Z$ operators which act on qubits included in plaquettes: $$\begin{aligned}
H_{plaquette} \ = \ - \sum_{p \in h_{m+1}} B_{p}, \qquad B_{p}\ = \ \prod_{r \subset p} Z_{r}\end{aligned}$$ where $r$ represent $m$-dimensional unit hypercubes included inside a $m+1$-dimensional unit hypercube $p$. Note that there are ${}_D C_{m+1}\cdot N$ plaquette terms. Next, we call $(m-1)$-dimensional unit hypercubes in $h_{m-1}$ “stars”. Then, star terms consist of Pauli $X$ operators which act on qubits neighboring to a star: $$\begin{aligned}
H_{star} \ = \ - \sum_{s \in h_{m-1}}A_{s}, \qquad A_{s} \ = \ \prod_{s \subset r} X_{r}\end{aligned}$$ where $s \subset r$ means that a star $s$ is included inside $m$-dimensional unit hypercube $r$. Note that there are ${}_D C_{m-1}\cdot N$ star terms. Noting that $[A_{s},B_{p}]=0$ since $A_{s}$ and $B_{p}$ share either zero or two qubits in common, the model is a stabilizer code. Fig. \[D\_dim\] shows constructions of star terms and plaquette terms for $D=3$ and $m=2$.
One can verify that the model has $k = {}_D C_{m}$ logical qubits with $k$ anti-commuting pairs of $m$-dimensional and $(D-m)$-dimensional logical operators where a $m$-dimensional logical operator consists of Pauli $Z$ operators supported on a $m$-dimensional hyperplane, while a $(D-m)$-dimensional logical operator consists of Pauli $X$ operators supported on a $(D-m)$-dimensional hyperplane. Since they may share either zero or one qubit, they may commute or anti-commute with each other. One may easily see that the construction above reproduces two-dimensional, three-dimensional and four-dimensional Toric code for choices of $(D,m)=(2,1), (3,1), (4,2)$. For $m=0$ and $m=D$, the model is reduced to the $D$-dimensional Ising model. One can verify that logical operators arising in this model can be deformed continuously by using the bi-partition theorem (theorem \[theorem\_partition\]). The construction above can be easily generalized to arbitrary $D$-dimensional graph embedded in a $D$-dimensional geometric manifold in a way similar to two-dimensional Toric code.
$D$-dimensional STS model and $D+1$-dimensional TQFT {#sec:topology2}
----------------------------------------------------
The emergence of the notion of topology in geometric shapes of logical operators leads us to consider a possible relevance to topological quantum field theory (TQFT), which also deals with systems whose physical properties depend heavily on topological characteristics of the systems. Here, we make an attempt to establish the connection between stabilizer codes with continuously deformable logical operators and TQFT further.
Roughly speaking, TQFT is a field theory which is invariant under diffeomorphism (continuous deformation), and particularly suited for describing topologically ordered systems. The most important characteristic in systems described by TQFT is the invariance of all the correlation functions under diffeomorphism. Consider an arbitrary diffeomorphism through a continuous change of space-time coordinates $x \rightarrow x'$. Then, the correlation function of a scalar operator $\phi(x)$ satisfies $$\begin{aligned}
\langle 0_{i} | \phi(x_{1}) \phi(x_{2})\cdots \phi(x_{n}) | 0_{j} \rangle = \langle 0_{i} | \phi(x_{1}') \phi(x_{2}')\cdots \phi(x_{n}') | 0_{j} \rangle\end{aligned}$$ where $|0_{i}\rangle$ represent degenerate ground states. Therefore, the vacuum expectation value of any products of scalar operators is invariant under differmorphism, and only the topological properties of products may characterize their expectation values.
Below, in order to establish the connection between stabilizer codes and TQFT, we show that the braiding of anyonic excitations in $D$-dimensional stabilizer codes can be characterized by some topological invariant in a $D+1$-dimensional system. In particular, by characterizing propagations of anyonic excitations in a $D+1$-dimensional system, we demonstrate that the braiding of anyonic excitations corresponds to a configuration of $m$-dimensional and $D-m$-dimensional closed objects with a non-zero linking number.
We begin by analyzing propagations of anyonic excitations in two-dimensional Toric code. Anyonic excitations can be created by applying a segment of a one-dimensional logical operator to a ground state of the Toric code Hamiltonian since endpoints of a segment of a logical operator $\ell^{seg}$ may not commute with interaction terms and create localized excitations (Fig. \[fig\_anyon1\](a)): $\ell^{seg}|\psi_{gs}\rangle$, and one may make anyons propagate along a geometric shape of a one-dimensional logical operator. Since one can deform a geometric shape of a one-dimensional logical operator continuously in the Toric code, anyons can propagate freely on the lattice by applying a segment of a deformed one-dimensional logical operator (Fig \[fig\_anyon1\](b)). Therefore, the continuous deformability of logical operators is the key to propagations of anyonic excitations.
![Anyonic excitations created from segments of logical operators. (a) A one-dimensional logical operator. (b) A deformed one-dimensional logical operator. []{data-label="fig_anyon1"}](fig_anyon1.pdf){width="0.40\linewidth"}
Propagations of anyonic excitations can be characterized by a one-dimensional closed loop drawn in a three-dimensional space. To illustrate this point, let us consider a process of creation, propagation, and annihilation of anyons, as described in Fig. \[fig\_anyon2\](a). By drawing propagations of anyonic excitations in a three-dimensional system by adding the time axis, the entire process can be represented as a one-dimensional closed loop as shown in Fig. \[fig\_anyon2\](b). In general, arbitrary one-dimensional closed loop in a three-dimensional system may characterize some propagations of anyonic excitations. See an example in Fig. \[fig\_anyon3\] which involves creations of two pairs of anyonic excitations.
![The correspondence between anyonic excitations in two-dimensional Toric code and a closed loop in a three-dimensional system. (a) Creation, propagation and annihilation of anyonic excitations. (b) Anyonic excitations described in a three-dimensional system. []{data-label="fig_anyon2"}](fig_anyon2.pdf){width="0.60\linewidth"}
![Propagations of anyonic excitations and an associated closed loop in a three-dimensional system. []{data-label="fig_anyon3"}](fig_anyon3.pdf){width="0.35\linewidth"}
Next, let us consider the braiding between anyonic excitations in the Toric code. One interesting property of topologically ordered spin systems is the non-trivial braiding property between anyonic excitations. There are two distinct type of anyons in two-dimensional Toric code which are created by a pair of anti-commuting logical operators respectively, and the braiding of different types of anyonic excitations may give rise to non-trivial change inside the ground space. In two-dimensional Toric code, the braiding of different anyons give rise to an additional phase $-1$ to the original ground state $|\psi_{gs}\rangle \rightarrow - |\psi_{gs}\rangle$, which is a direct consequence of the anti-commutation between logical operators.
One may understand this non-trivial braiding arising in two-dimensional Toric code as a topological invariant in a three-dimensional system. Let us consider the braiding of anyons described in Fig. \[fig\_anyon4\](a) where two types of pairs of anyonic excitations are created. One can characterize this braiding process as two closed loops in a three-dimensional system where loops are linked as described in Fig. \[fig\_anyon4\](b). In a more technical language, the braiding of anyons occurs only if the linking number between two loops has non-zero value. In particular, the final state is $|\psi_{gs}\rangle \rightarrow (-1)^{N_{link}} |\psi_{gs}\rangle$ where $N_{link}$ is the linking number of a given configuration of one-dimensional closed loops. This observation indicates that the braiding of anyonic excitations can be characterized by topological invariants, such as the linking number, in a ($2+1$)-dimensional system.
![The braiding as a topological invariant in a three-dimensional system. (a) A braiding of anyonic excitations. (b) Loops with non-zero linking number. []{data-label="fig_anyon4"}](fig_anyon4.pdf){width="0.70\linewidth"}
A similar observation holds for $D$-dimensional Toric code with a pair of $m$-dimensional and $D-m$-dimensional logical operators. The first type of anyonic excitations can be created by a segment of $m$-dimensional logical operator, while the second type of anyonic excitations can be created by a segment of $D-m$-dimensional logical operator. The propagation of anyonic excitations of the first type can be characterized by a $m$-dimensional object in a $D+1$-dimensional systems, while the propagation of anyonic excitations of the second type can be characterized by a $D-m$-dimensional object. These anyonic excitations are braided when the linking number between $m$-dimensional and $D-m$-dimensional objects is non-zero. While we have discussed the braiding for $D$-dimensional Toric code, similar discussion holds for any stabilizer codes with continuously deformable logical operators. Therefore, one may expect that $D$-dimensional stabilizer codes with continuous deformability can be effectively described by $D+1$-dimensional TQFT.
Decomposition of logical operators {#sec:decomposition}
==================================
We present the proof of theorem \[theorem\_3dim\] in this and the next appendices. The goal of this appendix is to prove the following theorem which will be the key to the proof of theorem \[theorem\_3dim\].
\[theorem\_decomposition\] Consider a three-dimensional STS model with the system size $n_{1}= 2\cdot2^{2n_{2}v}!$, $n_{2}=2^{m}$ and arbitrary $n_{3}$ where $m$ is an arbitrary positive integer. For a given logical operator $\ell$ supported inside $P(n_{1},n_{2},1)$, one can decompose $\ell$ as a product of the following centralizer operators $$\begin{aligned}
\ell \ \sim \ \ell_{a}\ell_{b}, \qquad \ell_{a},\ell_{b}\ \in \ \mathcal{C}_{P(n_{1},n_{2},1)}\end{aligned}$$ where $$\begin{aligned}
T_{1}^{\beta}(\ell_{b})\ = \ \ell_{b}, \qquad \beta \ \leq \ 2^{2n_{2}v}\end{aligned}$$ and $\ell_{a}$ is defined inside $P(2v, n_{2},1)$.
Here, $\mathcal{C}_{R}$ represents the restriction of the centralizer group $\mathcal{C}$ onto a region of composite particles $R$, meaning that $\mathcal{C}_{R}$ is a subgroup of centralizer operators defined inside $R$. Therefore, $\ell_{a},\ell_{b}\in \mathcal{C}_{P(n_{1},n_{2},1)}$ means that $\ell_{a}$ and $\ell_{b}$ are centralizer operators defined inside $P(n_{1},n_{2},1)$. We show the claim of the theorem graphically in Fig. \[fig\_decomposition3D\]. The theorem claims that a two-dimensional logical operator defined inside $P(n_{1},n_{2},1)$ can be decomposed as a product of a one-dimensional centralizer operator $\ell_{a}$ and a two-dimensional centralizer operator $\ell_{b}$ which is periodic in the $\hat{1}$ direction. As we shall see later, “$2v$” comes from the number of independent generators for the Pauli group acting on a single composite particle.
Before starting the proof of theorem \[theorem\_decomposition\], let us describe the entire sketch of the proof of theorem \[theorem\_3dim\]. As a simple extension of theorem \[theorem\_decomposition\], one can show that a one-dimensional logical operator defined inside $P(2v, n_{2},1)$ can be further decomposed as a product of a one-dimensional and a zero-dimensional centralizer operators. After these decompositions, one can classify geometric shapes of logical operators according to their dimensions and can find commutation relations between them.
Although theorem \[theorem\_decomposition\] is limited to some specially chosen system sizes: $n_{1}=2\cdot2^{2n_{2}v}!$ and $n_{2}=2^{m}$, one can construct logical operators for arbitrary system sizes from theorem \[theorem\_decomposition\]. For example, due to scale symmetries, one can show that one-dimensional logical operators found in theorem \[theorem\_decomposition\] are also logical operators for the systems with arbitrary $n_{1}$. In fact, one can find logical operators in the forms described in theorem \[theorem\_3dim\]. These arguments will be presented in \[sec:construction\].
![The claim of theorem \[theorem\_decomposition\]. One can decompose a two-dimensional logical operator as a product of a one-dimensional centralizer operator $\ell_{a}$ and a two-dimensional centralizer operator $\ell_{b}$. []{data-label="fig_decomposition3D"}](fig_decomposition3D.pdf){width="0.65\linewidth"}
Sketch of proof of theorem \[theorem\_decomposition\]
-----------------------------------------------------
First, we note that theorem \[theorem\_decomposition\] was proven for $n_{2}=1$ ($m=0$) in [@Beni10b] since such a system with $n_{2}=1$ can be considered as a two-dimensional system which extends only in the $\hat{1}$ and $\hat{3}$ directions. For a two-dimensional STS model, we have the following lemma.
\[lemma\_decomposition\] Consider a two-dimensional STS model where $n_{1}= 2\cdot2^{2v}!$ and arbitrary $n_{2}$. For a given logical operator $\ell$ supported inside $P(n_{1},1)$, one can decompose $\ell$ as a product of the following centralizer operators $$\begin{aligned}
\ell \ \sim \ \ell_{a}\ell_{b}, \qquad \ell_{a},\ell_{b}\ \in \ \mathcal{C}_{P(n_{1},1)}\end{aligned}$$ where $$\begin{aligned}
T_{1}^{\beta}(\ell_{b})\ = \ \ell_{b}, \qquad \beta \ \leq \ 2^{2v}\end{aligned}$$ and $\ell_{a}$ is defined inside $P(2v,1)$.
We present the claim of the lemma graphically in Fig. \[fig\_decomposition2D\]. The lemma claims that a one-dimensional logical operator defined inside $P(n_{1},1)$ can be decomposed as a product of a zero-dimensional centralizer operator $\ell_{a}$ and a one-dimensional centralizer operator $\ell_{b}$ which is periodic.
![The claim of lemma \[lemma\_decomposition\]. One can decompose a one-dimensional logical operator as a product of a zero-dimensional centralizer operator $\ell_{a}$ and a one-dimensional centralizer operator $\ell_{b}$. []{data-label="fig_decomposition2D"}](fig_decomposition2D.pdf){width="0.65\linewidth"}
A three-dimensional STS model may be viewed as a two-dimensional system if one considers $1\times n_{2} \times 1$ composite particles as a single composite particle which consists of $vn_{2}$ qubits (see Fig. \[fig\_decomposition3Dsub\]). In other words, we view the entire system as a two-dimensional lattice of one-dimensional tubes. Then, as a direct consequence of the lemma above, we notice the following corollary.
\[corollary\_decomposition\] A logical operator $\ell$ considered in theorem \[theorem\_decomposition\] can be decomposed as a product of the following centralizer operators $$\begin{aligned}
\ell \ \sim \ \ell_{a}\ell_{b}, \qquad \ell_{a},\ell_{b} \ \in \ \mathcal{C}_{P(n_{1},n_{2},1)}\end{aligned}$$ where $$\begin{aligned}
T_{1}^{\beta}(\ell_{b}) \ = \ \ell_{b}, \qquad \mbox{where}\quad \beta \ \leq \ 2^{2n_{2}v}\end{aligned}$$ and $\ell_{a}$ is defined inside $P(2vn_{2}, n_{2},1)$.
We present the claim of the corollary graphically in Fig. \[fig\_decomposition3Dsub\]. The corollary claims that a two-dimensional logical operator defined inside $P(n_{1},n_{2},1)$ can be decomposed as a product of a one-dimensional logical operator $\ell_{a}$ and a two-dimensional logical operator $\ell_{b}$ which is periodic in the $\hat{1}$ direction.
However, a one-dimensional logical operator $\ell_{a}$ described in corollary \[corollary\_decomposition\] is *not one-dimensional in a strict sense* since it is defined inside $P(2^{m}\cdot 2v,2^{m},1)$, and its “width” $2^{m}\cdot 2v$ grows as $m$ increases. On the other hand, $\ell_{a}$ described in theorem \[theorem\_decomposition\] is truly one-dimensional since its width is at most $2v$. Therefore, we need to show that a logical operator defined inside $P(2^{m}\cdot 2v,2^{m},1)$ have an equivalent logical operator defined inside $P(2v,2^{m},1)$.
![The claim of corollary \[corollary\_decomposition\]. The width of a one-dimensional logical operator $\ell_{a}$ increases as $m$ increases. []{data-label="fig_decomposition3Dsub"}](fig_decomposition3Dsub.pdf){width="0.65\linewidth"}
The rest of this appendix is dedicated to the proof of the following lemma.
\[lemma\_decomposition\_final\] For system sizes considered in theorem \[theorem\_decomposition\], a logical operator operator $\ell$ defined inside $P(x,2^{m},1)$ always has an equivalent logical operator $\ell'$ which is defined inside $P(x-1,2^{m},1)$ when $2v < x < n_{1}$.
By using this lemma, one can shorten the width of $\ell_{a}$ from $2^{m}\cdot 2v$ to $2v$.
Identity generating matrix
--------------------------
Here, we discuss how to shorten the width of $\ell_{a}$. In particular, we introduce a certain binary matrix which is essential in reducing the width of $\ell_{a}$.
**Shrinkage in two dimensions:** To give an intuition on how to shorten the width of $\ell_{a}$, let us first consider the case where $m=0$. (So, this is a two-dimensional system with $n_{2}=1$, and instead of the “width”, we use the “length”). As an example, consider the case when $m=0$, $x=4$ and $\ell$ is given by $$\begin{aligned}
\ell \ = \ \begin{bmatrix}
A, & B,& C ,& AB
\end{bmatrix}\end{aligned}$$ where $\ell$ is defined inside $P(4,1,1)$, and $A$, $B$ and $C$ are some Pauli operators. Then, consider the following logical operator: $$\begin{aligned}
\ell'' \ &\equiv \ \ell T_{1}^{2}(\ell) T_{1}^{3}(\ell) \\
&= \
\begin{bmatrix}
A, & B,& AC ,& I ,& BC ,& ABC ,& AB
\end{bmatrix}.\end{aligned}$$ Note that $\ell'' \sim \ell$ since $\ell''$ is a product of three logical operators which are equivalent to each other due to the translation equivalence of logical operators. Then, we notice that following operators are centralizer operators: $$\begin{aligned}
\ell_{1} \ = \ \begin{bmatrix}
A, & B,& AC
\end{bmatrix},\qquad
\ell_{2} \ = \ \begin{bmatrix}
BC ,& ABC ,& AB
\end{bmatrix}\end{aligned}$$ where $\ell'' = \ell_{1}T_{1}^{4}(\ell_{2})$ since stabilizers in STS models are defined inside $2\times 2$ composite particles and cannot overlap with $\ell_{1}$ and $T_{1}^{4}(\ell_{2})$ simultaneously. Now, due to the translation equivalence of logical operators, we have $$\begin{aligned}
\ell \ \sim \ \ell' \ \equiv \ \ell_{1}\ell_{2} \ = \
\begin{bmatrix}
A, & B,& AC
\end{bmatrix}
\ \times \
\begin{bmatrix}
BC ,& ABC ,& AB
\end{bmatrix}
\ = \
\begin{bmatrix}
ABC, & AC,& BC
\end{bmatrix}.\end{aligned}$$ Thus, a logical operator $\ell$, with the length $4$, is shrunk into an equivalent logical operator $\ell'$, with the length $3$.
An important observation is that one can form an identity operator $I$ by taking a product of Pauli operators in $\ell$. In general, if the length $x$ of $\ell$ is larger than $2v$, one can always form an identity operator $I$ by taking a product of some Pauli operators in $\ell$ since there are $2v$ independent generators for single Pauli operators acting on a single composite particle. Now, let us consider $\ell$ represented as $$\begin{aligned}
\ell \ =\ \begin{bmatrix}
U_{1}, & U_{2},& \cdots ,& U_{x}
\end{bmatrix}.\end{aligned}$$ which is defined inside $P(x,1,1)$ where $x > 2v$. Then, there always exists a binary vector $B=(B_{1},\cdots,B_{x}) \not= (0,\cdots,0)$ which satisfies the following condition: $$\begin{aligned}
\prod_{j=1}^{x} U_{j}^{B_{j}} \ = \ I.\end{aligned}$$ Now, let us take the following product of translations of $\ell$: $$\begin{aligned}
\prod_{j=1}^{x}T_{1}^{x-j}(\ell^{B_{j}}).\end{aligned}$$ Then, one may readily know that the $x$th entry of the above operator is $I$. From this operator, one can find two centralizer operators. By using them, one can readily shrink the length of $\ell$ from $x$ to $x-1$. This trick is the key to the proof of lemma \[lemma\_decomposition\]. Although the argument above works only when $B$ has an odd number of $1$ entries, a slight modification makes the shrinkage of $\ell$ possible when $B$ has an even number of $1$ entries.
**Identity generating matrix:** Now, we consider more general cases with $m>0$. Let us represent a logical operator $\ell$ defined inside a region $P(x, 2^{m},1)$ as a $x\times 2^{m}$ matrix whose entries are single Pauli operators: $$\begin{aligned}
\ell \ = \ \begin{bmatrix}
U_{1,1},& \cdots& U_{x,1} \\
\vdots & \ddots& \vdots \\
U_{1,2^{m}},& \cdots& U_{x,2^{m}}
\end{bmatrix}\end{aligned}$$ where each Pauli operator $U_{i,j}$ acts on each composite particle. We also represent each column of $\ell$ as follows: $$\begin{aligned}
U_{j} \ =\ \begin{bmatrix}
U_{j,1} \\
\vdots \\
U_{j,2^{m}}
\end{bmatrix}\qquad ( j\ =\ 1,\cdots,x ).\end{aligned}$$ Here, we denote a group of Pauli operators supported by a single column $P(1,2^{m},1)$ as $\mathcal{P}^{m}_{col}$ and call it the *column operator group* and its elements *column operators*. Note that $U_{j}\in \mathcal{P}^{m}_{col}$, and $\mathcal{P}^{m}_{col}$ has $2^{m}\cdot 2v$ independent generators: $G(\mathcal{P}^{m}_{col}) = 2^{m}\cdot 2v$.
When $m=0$ and $x >2v$, we found a binary vector $B$ with $x$ components which characterizes how to form an identity operator from $\ell$. When $m>0$ and $x >2v$, we can find an $x \times 2^{m}$ binary matrix $B$ which characterizes how to form an identity operator from $\ell$. Here, we introduce the *identity generating matrices* as follows.
Consider a logical operator $\ell$ defined inside $P(x,2^{m},1)$.
- For a $x\times 2^{m}$ binary matrix $B$ $$\begin{aligned}
B \ = \ \begin{bmatrix}
B_{1,1} ,& \cdots & B_{x,1} \\
\vdots & \ddots & \vdots \\
B_{1,2^{m}} ,& \cdots & B_{x,2^{m}}
\end{bmatrix}, \qquad B_{i,j} \ = \ 0,1, \end{aligned}$$ we define the following operations: $$\begin{aligned}
\ell(B) \ &\equiv \ \prod_{i,j} T_{1}^{x -i}T_{2}^{j-1} ( \ell^{B_{i,j}} ) \\
\ell(B)_{x}\ &\equiv \ \prod_{i=1}^{x}\prod_{j=1}^{2^{m}}T_{2}^{j-1}(U_{i}^{B_{i,j}})\ \in \ \mathcal{P}^{m}_{col}\end{aligned}$$ where $\ell(B)$ is a product of translations of $\ell$ taken according to $B$ while $\ell(B)_{x}$ is the $x$th column of $\ell(B)$.
- We call a binary matrix $B$ *identity generating matrix* if and only if $$\begin{aligned}
\ell(B)_{x} \ = \ I \quad \mbox{and}\quad B \ \not= \ \begin{bmatrix}
0, & \cdots & 0 \\
\vdots & \ddots & \vdots \\
0, & \cdots & 0
\end{bmatrix}.\end{aligned}$$
- We assign parities to each column of a binary matrix $B$ as follows: $$\begin{aligned}
Par(B)_{i}\ \equiv \ \sum_{j} B_{i,j} \qquad ( \mbox{mod 2})\end{aligned}$$ where $i = 1,\cdots,x$. We call a binary matrix $B$ *odd* if and only if $$\begin{aligned}
\exists i \quad \mbox{s.t} \quad Par(B)_{i} \ =\ 1.\end{aligned}$$
Therefore, when we form an identity operator, we considered translations of $\ell$ both in the $\hat{1}$ and $\hat{2}$ directions. The identity generating matrix is said to be odd when there exists a column with an odd parity.
**Shrinkage through odd matrices:** Note that there always exists an identity generating matrix when $x >2v$. Then, with the existence of identity generating matrices, one might hope that the width of $\ell_{a}$ can be reduced until it becomes $2v$ in a way similar to the cases where $m=0$. However, there is a caveat. In fact, only identity generating matrices with some special properties can be used for shrinking. In particular, we have the following lemma.
\[lemma\_odd1\] If there exists an odd identity generating matrix for $\ell$ defined inside $P(x, 2^{m}, 1)$, $\ell$ has an equivalent logical operator $\ell'$ defined inside $P(x-1, 2^{m}, 1)$.
Therefore, if there exists an odd identity generating matrix for any $x$ with $n_{1}>x>2v$, one can complete the proof of lemma \[lemma\_decomposition\_final\]. Below, we present the proof of lemma \[lemma\_odd1\]. The existence of an odd identity generating matrix will be proven later.
Assume that $B$ is an odd identity generating matrix for $\ell$. Assume that for some $i'$ ($1 \leq i' \leq x$), we have $$\begin{aligned}
Par(B)_{i'} \ = \ 1 \quad \mbox{and} \quad Par(B)_{i} \ = \ 0 \quad \mbox{for} \ i \ < \ i'.\end{aligned}$$ So, $i'$ is the smallest integer such that $i'$th column has an odd parity. Here, we define the following binary matrix $B'$ (see Fig. \[fig\_proof\_aid\]): $$\begin{aligned}
B'_{i,j} \ &\equiv \ B_{i,j} \qquad & \ (\ i \ \leq \ i'\ )\\
&\equiv \ 0 \qquad & \ (\ i \ > \ i' \ ). \end{aligned}$$ Note that $B'$ consists of $i$th columns of $B$ with $i\leq i'$. Based on $B'$, we consider the following logical operator $\ell'$: $$\begin{aligned}
\ell' \ \equiv \ \ell(B') \ \sim \ \ell.\end{aligned}$$ Note that $\ell'$ is equivalent to $\ell$ since $\ell'$ is a product of an odd number of translations of $\ell$. (Note that $\ell(B)$ may not be equivalent to $\ell$ since the number of $1$ entries in $B$ may be even).
See Fig \[fig\_proof\_aid\] for graphical representations of $B$, $B'$, $\ell(B)$ and $\ell(B')$. Note that $\ell(B)$ has an identity operator at $x$th column. Note that $\ell(B')$ has identity operators in the first $x - i'$ columns since $B_{i,j}=0$ for $i >i'$.
![Constructions of $B$, $B'$, $\ell(B)$ and $\ell(B')$. []{data-label="fig_proof_aid"}](fig_proof_aid.pdf){width="0.75\linewidth"}
Since $\ell(B)$ has an identity operator at $x$th column, we can decompose it as a product of two centralizer operators whose lengths are at most $x-1$. Let us denote these centralizer operators as $U_{a}$ and $U_{b}$: $$\begin{aligned}
\ell(B) \ = \ U_{a}U_{b}\end{aligned}$$ where $U_{a}$ is the centralizer on the left hand side and $U_{b}$ is the centralizer on the right hand side (See Fig. \[fig\_proof\_aid\]). $U_{a}$ is defined from $1$st column to $x-1$th column, and $U_{b}$ is defined from $x+1$th column to $2x-1$th column.
Since $B$ and $B'$ have the same entries from $1$st column to $i'$th column, we notice that $\ell(B')$ and $U_{b}$ have the same Pauli operators from $2x-2-i'$st column to $2x-1$th column as shown in Fig. \[fig\_proof\_aid\]. Then, by applying $U_{b}$ to $\ell(B')$, one can shrink the size of $\ell(B')$. In particular, $U_{b} \ell(B')$ has the length at most $x-1$. Although $U_{b}\ell(B')$ may not be equivalent to $\ell(B')$ as $U_{b}$ may not be a stabilizer, one may consider the following logical operator: $$\begin{aligned}
U_{b} \ \times \ \ell(B') \ \times \ T_{1}^{- i'}(U_{b}) \ \sim \ \ell(B') \ \sim \ \ell\end{aligned}$$ which is equivalent to $\ell(B')$ due to the translation equivalence of logical operators, and is defined inside a region with $x-1 \times 2^{m} \times 1$ composite particles. Then, due to the translation equivalence of logical operators, there exists a logical operator $\ell'' \sim \ell$ which is defined inside $P(x-1, 2^{m}, 1)$. This completes the proof.
Existence of an odd matrix for $m=1$
------------------------------------
Next, we present a proof of the existence of an odd identity generating matrix for $x > 2v$, in order to complete the proof of lemma \[lemma\_decomposition\_final\] and theorem \[theorem\_decomposition\]. In particular, we shall prove the following lemma.
\[lemma\_existence\] When $x > 2v$, there always exist an odd identity generating matrix.
We start by discussing cases with $m=0$ and $m=1$ before presenting general discussion. First of all, when $m=0$, identity generating matrices are always odd since all the binary matrices are odd except $B=(0,\cdots,0)$. Therefore, we consider the cases where $m=1$ below.
**Characteristic value:** Recall that we represented $\ell$ as a $x \times 2$ binary matrix: $$\begin{aligned}
\ell \ =\ \begin{bmatrix}
U_{1,1},& \cdots& U_{x,1} \\
U_{1,2},& \cdots& U_{x,2}
\end{bmatrix}\end{aligned}$$ and each column of $\ell$ as follows: $$\begin{aligned}
U_{j} \ = \ \begin{bmatrix}
U_{j,1} \\
U_{j,2}
\end{bmatrix}\qquad ( j \ = \ 1,\cdots,x ).\end{aligned}$$ Now, *we suppose that there is no odd identity generating matrix* for $\ell$, in order to use the contradiction for the proof of lemma \[lemma\_existence\].
First, it is convenient to classify column operators in $\mathcal{P}_{col}^{1}$ into two types as follows. For a column operator $U$ represented as $$\begin{aligned}
U = \begin{bmatrix}
U_{1,1} \\
U_{1,2}
\end{bmatrix},\end{aligned}$$ we assign a *characteristic value* $b$ and *characteristic operator* $V$ as follows:
- If $U_{1,1}=U_{1,2}$, we assign a characteristic value $b = 1$ and a characteristic operator $V=U_{1,1}$.
- If $U_{1,1}\not=U_{1,2}$, we assign a characteristic value $b= 0$ and a characteristic operator $V=U_{1,1}U_{1,2}$.
One may easily understand this classification by representing $U$ explicitly. A column operator $U$ with $b=1$ is $$\begin{aligned}
U \ = \ \begin{bmatrix}
V \\
V
\end{bmatrix}\end{aligned}$$ and a column operator $U$ with $b=0$ is $$\begin{aligned}
U \ = \ \begin{bmatrix}
VU_{1,2} \\
U_{1,2}
\end{bmatrix}.\end{aligned}$$ So, a column operator with $b=1$ is symmetric while a column operator with $b=0$ is not. Note that a characteristic operator is not an identity operator $I$ except when $U=I$.
Next, we introduce the following $x \times 2$ binary matrices: $$\begin{aligned}
E(i';0) \ \equiv \ \begin{bmatrix}
E_{1,1} ,& \cdots ,& E_{x,1} \\
E_{1,2} ,& \cdots ,& E_{x,2}
\end{bmatrix}\end{aligned}$$ such that $$\begin{aligned}
E_{i,1} \ &= \ 1 \qquad ( i \ = \ i')\\
E_{i,j} \ &= \ 0 \qquad \mbox{otherwise}\end{aligned}$$ and $$\begin{aligned}
E(i';1) \ \equiv \ \begin{bmatrix}
E_{1,1} ,& \cdots ,& E_{x,1} \\
E_{1,2} ,& \cdots ,& E_{x,2}
\end{bmatrix}\end{aligned}$$ such that $$\begin{aligned}
E_{i,1} \ &= \ E_{i,2} \ = \ 1 \qquad ( i \ = \ i')\\
E_{i,j} \ &= \ 0 \qquad \ \ \quad \qquad \mbox{otherwise}\end{aligned}$$ For example, $$\begin{aligned}
E(2;0) \ = \ \begin{bmatrix}
0, & 1 ,& 0 ,& \cdots ,& 0 \\
0, & 0 ,& 0 ,& \cdots ,& 0
\end{bmatrix}\end{aligned}$$ and $$\begin{aligned}
E(2;1) \ = \ \begin{bmatrix}
0 , & 1 ,& 0 ,& \cdots ,& 0 \\
0 , & 1 ,& 0 ,& \cdots ,& 0
\end{bmatrix}.\end{aligned}$$
Now, let us represent characteristic values and characteristic operators for each column of $\ell$ as follows: $$\begin{aligned}
\begin{array}{ccc}
U_{1}\ \rightarrow\ & b_{1}, & V_{1} \\
U_{2}\ \rightarrow\ & b_{2},& V_{2} \\
\vdots & \vdots & \vdots \\
U_{x}\ \rightarrow\ & b_{x},& V_{x}.
\end{array}\end{aligned}$$ Then one can establish a connection between binary matrices $E(i;0)$ and $E(i;1)$, and characteristic values $b_{i}$ and operators $V_{j}$ as follows: $$\begin{aligned}
\ell(E(i;0))_{x} \ &= \ \begin{bmatrix}
V_{i} \\
V_{i}
\end{bmatrix}\qquad \mbox{when} \quad b_{i}\ = \ 1\\
\ell(E(i;1))_{x} \ &= \ \begin{bmatrix}
V_{i} \\
V_{i}
\end{bmatrix}\qquad \mbox{when} \quad b_{i}\ = \ 0.\end{aligned}$$
**Proof of lemma \[lemma\_existence\]:** Now, let us proceed to the proof of lemma \[lemma\_existence\] for $m=1$. Without loss of generality, we can assume that $$\begin{aligned}
b_{1}\ = \ \cdots\ =\ b_{x_{0}}\ =\ 0, \qquad b_{x_{0}+1}\ =\ \cdots \ = \ b_{x}\ =\ 1\end{aligned}$$ for $x_{0}\leq x$ since permutations of columns do not affect the parities of identity generating matrices. (We will justify this later). We define the following sets of integers: $$\begin{aligned}
\textbf{b}(0)\ \equiv \ \{ 1,\cdots, x_{0} \}, \qquad \textbf{b}(1)\ \equiv \{ x_{0}+1, \cdots, x \}.\end{aligned}$$ We denote groups of Pauli operators generated by $V_{j}$ with $b_{j}=0$ and $V_{j}$ with $b_{j}=1$ as $\mathcal{V}_{0}$ and $\mathcal{V}_{1}$: $$\begin{aligned}
\mathcal{V}_{0} \ &\equiv \ \left\langle \ \{ \ V_{i} \ : \ i \ \in \ \textbf{b}(0) \ \} \ \right\rangle \\
\mathcal{V}_{1} \ &\equiv \ \left\langle \ \{ \ V_{i} \ : \ i \ \in \ \textbf{b}(1) \ \} \ \right\rangle. \end{aligned}$$
Let us show that a set of characteristic operators $\{ V_{i} \}$ for $i \in \textbf{b}(1)$ is independent : $G(\mathcal{V}_{1})=x - x_{0}$. For this purpose, we suppose that there exists some set of integers $\textbf{A}\subseteq \textbf{b}(1)$ such that $$\begin{aligned}
\prod_{i \in \textbf{A}} V_{i} \ = \ I.\end{aligned}$$ Then, the following binary matrix is an identity generating matrix: $$\begin{aligned}
B \ = \ \sum_{i \in \textbf{A}}E(i;0) \end{aligned}$$ since $$\begin{aligned}
\ell(B)_{x} \ = \ \prod_{i \in \textbf{A}} \begin{bmatrix}
V_{i}\\
V_{i}
\end{bmatrix}\ = \ \begin{bmatrix}
I\\
I
\end{bmatrix}.\end{aligned}$$ However, since $B$ is odd with $Par(B)_{i}=1$ for $i \in \textbf{A}$, this leads to a contradiction.
Next, let us analyze $\mathcal{V}_{0}$. For simplicity of discussion, we *first assume that $\{V_{i}\}$ for $i \in \textbf{b}(0)$ are independent*. We consider more general cases where $\{V_{i}\}$ for $i \in \textbf{b}(0)$ are over complete later. Then, we have $G(\mathcal{V}_{0})=x_{0}$. Now, we define the following operators for $i\in \textbf{b}(0)$: $$\begin{split}
U_{i}'\ &\equiv \ U_{i}T_{2}(U_{i}) \\
&= \ \begin{bmatrix}
V_{i} \\
V_{i}
\end{bmatrix}.
\end{split}$$ Note that $U_{i}'$ has a characteristic value $b_{i}'=1$ and a characteristic operator $V_{i}$. Notice that $$\begin{aligned}
U_{i}' \ = \ \ell(E(i;1))_{x}.\end{aligned}$$
Since $x > 2v$, there exists a set of integer $\textbf{A}$ such that $$\begin{aligned}
\prod_{i \in \textbf{A}} V_{i} \ = \ I, \qquad \textbf{A} \ \not\subseteq \ \textbf{b}(1)\quad \mbox{and} \quad \textbf{A} \ \not\subseteq \ \textbf{b}(0).\end{aligned}$$ Note that $\textbf{A}$ includes integers both from $\textbf{b}(0)$ and $\textbf{b}(1)$. Then, one notices that the following matrix $B$ is an identity generating matrix: $$\begin{aligned}
B \ = \ \sum_{i \in \textbf{A} \cap \textbf{b}(0)} E(i;1) + \sum_{i \in \textbf{A} \cap \textbf{b}(1)} E(i;0)\end{aligned}$$ since $$\begin{split}
\ell(B)_{x}\ &= \ \prod_{\{ i \in \textbf{A}\cap \textbf{b}(0) \} } U_{i}' \prod_{\{ i \in \textbf{A} \cap \textbf{b}(1) \} } U_{i} \\
&= \ \prod_{i \in \textbf{A}} \begin{bmatrix}
V_{i} \\
V_{i}
\end{bmatrix} \ = \
\begin{bmatrix}
I \\
I
\end{bmatrix}.
\end{split}$$ However, for an integer $i$ in $\textbf{A}\cap \textbf{b}(0)$, $Par(B)_{i}=1$, and $B$ is an odd matrix. This leads to a contradiction. Note that discussion above is valid under the permutations of columns.
**Proof of lemma \[lemma\_existence\], continued:** Next, let us consider the case where $\{ V_{i} \}$ for $i \in \textbf{b}(0)$ are not independent. Let us denote the number of generators for $\mathcal{V}_{0}$ as $\bar{x_{0}}=G(\mathcal{V}_{0})$ ($\bar{x_{0}}<x_{0}$). Without loss of generality, we may assume that $V_{1},\cdots,V_{\bar{x_{0}}}$ are independent since permutations do not affect parities of identity generating matrices. Here, *we change our notations slightly*: $$\begin{aligned}
\textbf{b}(0)\ \equiv \ \{ 1,\cdots, \bar{x_{0}} \}, \qquad \textbf{b}(0)'\ \equiv \ \{ \bar{x_{0}}+1,\cdots, x_{0} \}, \qquad \textbf{b}(1)\ \equiv \{ x_{0}+1, \cdots, x \}.\end{aligned}$$ and $$\begin{aligned}
\mathcal{V}_{0} \ &\equiv \ \left\langle \ \{ \ V_{i} \ : \ i \ \in \ \textbf{b}(0) \ \} \ \right\rangle \qquad G(\mathcal{V}_{0}) \ = \ \bar{x_{0}}\\
\mathcal{V}_{1} \ &\equiv \ \left\langle \ \{ \ V_{i} \ : \ i \ \in \ \textbf{b}(1) \ \} \ \right\rangle \qquad G(\mathcal{V}_{1}) \ = \ x - x_{0}. \end{aligned}$$
Since a set $\{V_{i}\}$ for $b_{i}=0$ is over complete, there are $x_{0}-\bar{x_{0}}$ sets of integers $\textbf{A}_{i}$ ($i=\bar{x_{0}}+1,\cdots,x_{0}$) such that $$\begin{aligned}
\prod_{i' \in \textbf{A}_{i}}V_{i'}\ = \ I\quad \mbox{where} \quad i' \ \leq \ i,\ \forall i' \ \in \ \textbf{A}_{i} \quad \mbox{and}\quad i \ \in \ \textbf{A}_{i}\end{aligned}$$ where the largest integer in $\textbf{A}_{i}$ is $i$. For $i = \bar{x_{0}}+1,\cdots, x_{0}$, we form the following operator: $$\begin{aligned}
U_{i}' \ \equiv \ \prod_{j \in \textbf{A}_{i}} U_{j}
\ \equiv \ \begin{bmatrix}
V_{i}' \\
V_{i}'
\end{bmatrix}\end{aligned}$$ which has a characteristic value $b_{i}'=1$ and a characteristic operator $V_{i}'$. Here, we denote a group of $\{ V_{i}'\}$ for $i \in \textbf{b}(0)'$ as $$\begin{aligned}
\mathcal{V}_{0}'\ = \ \left\langle \ \{ \ V_{i}' \ : \ i \ \in \ \textbf{b}(0)' \ \} \ \right\rangle.\end{aligned}$$ Now, we show that $\{V_{i}'\}$ for $i\in \textbf{b}(0)'$ are independent: $G(\mathcal{V}_{0}')= x_{0}-\bar{x_{0}}$. If there exists $\textbf{A} \subseteq \textbf{b}(0)'$ such that $$\begin{aligned}
\prod_{i \in \textbf{A}} V_{i}' \ = \ I,\end{aligned}$$ we have the following identity generating matrix: $$\begin{aligned}
B \ = \ \sum_{i \in \textbf{A}}\sum_{j \in \textbf{A}_{i}} E(j;0) \qquad \mbox{(mod 2)}\end{aligned}$$ since $$\begin{aligned}
\ell(B)_{x} \ = \ \prod_{i \in \textbf{A}}\begin{bmatrix}
V_{i}' \\
V_{i}'
\end{bmatrix}
\ = \ \begin{bmatrix}
I \\
I
\end{bmatrix}.\end{aligned}$$ Recall that the largest integer in $\textbf{A}_{i}$ is $i$. Let the largest integer in $\textbf{A}$ be $i_{max}$. Then, we have $Par(B)_{i_{max}}=1$, and thus, $B$ is odd. This leads to a contradiction.
So far, we have shown that $$\begin{aligned}
G(\mathcal{V}_{0}) \ = \ \bar{x_{0}},\qquad G(\mathcal{V}_{0}') \ = \ x_{0}-\bar{x_{0}},\qquad G(\mathcal{V}_{1}) \ = \ x- x_{1}.\end{aligned}$$ Since $x > 2v$, there exists a set of integers $\textbf{A}$ such that $$\begin{aligned}
\prod_{\{ i \in \textbf{A} \cap \textbf{b}(1) \} } V_{i}
\prod_{\{ i \in \textbf{A} \cap \textbf{b}(1)' \} } V_{i}'
\prod_{\{ i \in \textbf{A} \cap \textbf{b}(0) \} } V_{i} \ = \ I.\end{aligned}$$ The following matrix is the identity generating matrix: $$\begin{aligned}
B \ = \ \sum_{i \in \textbf{A} \cap \textbf{b}(0)} E(i;1) +
\sum_{i \in \textbf{A} \cap \textbf{b}(0)'} \sum_{i' \in \textbf{A}_{i}} E(i';0) +
\sum_{i \in \textbf{A} \cap \textbf{b}(1)} E(i;0) \qquad \mbox{(mod 2)}\end{aligned}$$ since $$\begin{aligned}
\ell(B)_{x} \ = \ \prod_{\{ i \in \textbf{A} : i \in \textbf{b}(0) \} }
\begin{bmatrix}
V_{i}\\
V_{i}
\end{bmatrix}
\prod_{\{ i \in \textbf{A} : i \in \textbf{b}(0)' \} }
\begin{bmatrix}
V_{i}'\\
V_{i}'
\end{bmatrix}
\prod_{\{ i \in \textbf{A} : i \in \textbf{b}(1) \} }
\begin{bmatrix}
V_{i}\\
V_{i}
\end{bmatrix} \ = \ \begin{bmatrix}
I\\
I
\end{bmatrix}.\end{aligned}$$ Since $\textbf{A} \not\subseteq \textbf{b}(0)$, $\textbf{A}$ has some element in $\textbf{b}(0)' \cup \textbf{b}(1)$. Let the largest integer in $\textbf{A}$ be $i_{max}$. Then, we have $Par(B)_{i_{max}}=1$, and thus, $B$ is odd. This leads to a contradiction. Again, permutations of columns do not affect this discussion. This completes the proof of lemma \[lemma\_existence\] for $m=1$.
Characteristic vectors
----------------------
Let us proceed to the proof for the cases where $m>1$. When $m=1$, we assigned characteristic values $0$ and $1$ to each column operator according to its symmetry. For $m>1$, we will assign a “binary vector” with $m$ components to each column operator, which we will call a *characteristic vector*. We encode “symmetries” of a column operator on these characteristic vectors.
**Characteristic vector:** We first define two maps $f_{0}$ and $f_{1}$ from $\mathcal{P}^{m'}_{col}$ to $\mathcal{P}^{m'-1}_{col}$ for $m' >0$ as follows. For a given column operator $U \in \mathcal{P}^{m'}_{col}$ which is represented as $$\begin{aligned}
U = \begin{bmatrix}
U_{1}\\
\vdots \\
U_{2^{m'}}
\end{bmatrix},\end{aligned}$$ we define $f_{0}(U), f_{1}(U) \in \mathcal{P}^{m'-1}_{col}$ as follows: $$\begin{aligned}
f_{0}(U)_j \equiv U_{j}U_{j+ 2^{m'-1}}, \qquad f_{1}(U)_j \equiv U_{j} \qquad (j=1,\cdots,2^{m'-1}).\end{aligned}$$ One may represent $f_{0}(U)$ and $f_{1}(U)$ more explicitly: $$\begin{aligned}
f_{0}(U) = \begin{bmatrix}
U_{1}U_{2^{m'-1}+1} \\
\vdots \\
U_{2^{m'-1}}U_{2^{m'}}
\end{bmatrix}, \qquad
f_{1}(U) = \begin{bmatrix}
U_{1}\\
\vdots \\
U_{2^{m'-1}}
\end{bmatrix}\end{aligned}$$ Note that $f_{0}$ and $f_{1}$ decrease the length of the column by half.
Let us denote a set of all the $m$ component binary vectors as $\textbf{B}_{vec}^{m}$. Now, for a column operator $U \in \mathcal{P}^{m}_{col}$, we assign an $m$ component binary vector $\vec{b} \in \textbf{B}^{m}_{vec}$ through the following rule.
- If $f_{0}(U)=I$, take $b_{m}=1$ and define $U^{(1)} \equiv f_{1}(U)$.
- If $f_{0}(U)\not= I$, take $b_{m}=0$ and define $U^{(1)} \equiv f_{0}(U)$.
and, iterate this procedure:
- If $f_{0}(U^{(j)})=I$, $b_{m-j}=1$ and define $U^{(j+1)}\equiv f_{1}(U^{(j)})$.
- If $f_{0}(U^{(j)})\not= I$, $b_{m-j}=0$ and define $U^{(j+1)} \equiv f_{0}(U^{(j)})$.
for $1 \leq j \leq m-1$. We define a characteristic operator $V$ of $U$ as follows: $$\begin{aligned}
V \ \equiv \ U^{(m)} \ = \ f_{b_{1}} f_{b_{2}} \cdots f_{b_{m}} (U) \ \in \ \mathcal{P}^{0}_{col}.\end{aligned}$$ Note that $V \not= I$ when $U \not=I$. It is worth presenting examples of characteristic vectors here ($m=2$): $$\begin{aligned}
\begin{bmatrix}
V \\
V\\
V\\
V
\end{bmatrix} \ \rightarrow \ (1,1), \ \
\begin{bmatrix}
V \\
I \\
V \\
I
\end{bmatrix} \ \rightarrow \ (0,1), \ \
\begin{bmatrix}
V \\
V \\
I \\
I
\end{bmatrix} \ \rightarrow \ (1,0), \ \
\begin{bmatrix}
V \\
I \\
I \\
I
\end{bmatrix} \ \rightarrow \ (0,0).\end{aligned}$$ Thus, *symmetries of column operators are encoded in characteristic vectors.*
Next, we introduce an order between binary vectors in $\textbf{B}^{m}_{vec}$. We define $$\begin{aligned}
g(\vec{b}) \ \equiv \ \sum_{j=1}^{m} b_{j}2^{j-1}\end{aligned}$$ where $\vec{b}$ is like a binary representation of an integer $g(\vec{b})$. For a given pair of $m$ component binary vectors $\vec{b}$ and $\vec{b'}$, we denote $$\begin{aligned}
\vec{b} \ < \ \vec{b'}\end{aligned}$$ if and only if $$\begin{aligned}
g(\vec{b}) \ < \ g(\vec{b'}).\end{aligned}$$ For example, for $m=3$, we have the following relations between binary vectors: $$\begin{aligned}
(0,0,0) \ <\ (1,0,0)\ <\ (0,1,0)\ <\ (1,1,0)\ <\ (0,0,1)\ <\ (1,0,1)\ <\ (0,1,1)\ <\ (1,1,1).\end{aligned}$$ Below, we shall see that a column operator with a larger characteristic vector is “more symmetric” than a column vector with a smaller characteristic vector.
**Property of characteristic vectors:** Let us briefly recall the proof of lemma \[lemma\_existence\] for $m=1$. In the proof, we constructed a column operator with $b=1$ from a column operator with $b=0$. In particular, if $U$ is a column operator with a characteristic value $b=0$ and a characteristic operator $V$: $$\begin{aligned}
U \ = \ \begin{bmatrix}
VV'\\
V'
\end{bmatrix}\end{aligned}$$ where $V'$ is some Pauli operator, we have $$\begin{aligned}
UT_{2}(U) \ = \ \begin{bmatrix}
V \\
V
\end{bmatrix}\end{aligned}$$ which is a column operator with a characteristic value $b=1$ and a characteristic operator $V$. Thus, *we can create a column operator with a larger characteristic value from a column operator with a smaller characteristic value*.
In a way similar to this, one can construct a column operator with $\vec{b'}$ from a column operator with $\vec{b}$ as long as $\vec{b'}>\vec{b}$. Let us represent a binary column $B$ as follows: $$\begin{aligned}
B \ = \ \begin{bmatrix}
B_{1,1}\\
\vdots \\
B_{1,2^{m}}
\end{bmatrix},\end{aligned}$$ and denote a set of all the binary columns as $\textbf{B}^{m}_{col}$. Here, we define a parity of $B$ as $$\begin{aligned}
Par(B) \ \equiv \ Par(B)_{1}\end{aligned}$$ by viewing a binary column $B$ as a binary matrix. For a column operator $U\in \mathcal{P}_{col}^{m}$, we define $U(B)$ as follows: $$\begin{aligned}
U(B)\ \equiv \ \prod_{j=1}^{2^{m}}T_{2}^{j-1}(U^{B_{1,j}})\ \in \ \mathcal{P}^{m}_{col}\end{aligned}$$ just like $\ell(B)$. Note that $U(B)$ is a product of translations of $U$ taken according to a binary column $B$. Then, the following lemma holds.
\[lemma\_construction\] Consider an arbitrary pair of $m$ component binary vectors $\vec{b}, \vec{b'} \in \textbf{B}^{m}_{vec}$ such that $\vec{b}<\vec{b'}$. For a given column operator $U$ which has a characteristic vector $\vec{b}$ and a characteristic operator $V$, there always exists some binary column $B \in \textbf{B}^{m}_{col}$ with an even parity $Par(B)= 0$ such that $U(B)$ has a characteristic vector $\vec{b'}$ and a characteristic operator $V$.
In other words, from a column operator $U$ with a characteristic vector $\vec{b}$, one can always create a column operator $U'$ with larger characteristic vector $\vec{b'}$ by taking a product of translations of $U$. On the other hand, it is impossible to create a column operator with a smaller characteristic vector from an operator with a larger characteristic vector. Therefore, *one can create a column operator with higher symmetries (a larger characteristic vector), but cannot create a column operator with lower symmetries (a smaller characteristic vector).*
Proof of lemma \[lemma\_construction\]
--------------------------------------
Now, we prove lemma \[lemma\_construction\] by explicitly finding a binary column $B \in \textbf{B}^{m}_{col}$ for creating $U(B)$ for every pair of $\vec{b}$ and $\vec{b'}$. In order to derive such binary matrices, we introduce a certain binary column $B(\vec{b})$, called a *characteristic column*, which can be used to change the characteristic vector of a column operator.
**Characteristic column:** Given an integer $p \in \mathbb{Z}_{2^{m}}$, one may have its binary representation by considering the inverse of $g$ denoted as $g^{-1}$: $$\begin{aligned}
\vec{p} \ \equiv \ (p_{1},\cdots,p_{m}) \ \equiv \ g^{-1}(p) \end{aligned}$$ where $p = g(\vec{p}) = \sum_{j=1}^{m-1} p_{j}2^{j-1}$. Now, we define the following sets: $$\begin{aligned}
\textbf{J}_{\vec{b}} \ = \ \{ \ \vec{a}\ \in \ \textbf{B}^{m}_{vec} \ : \ a_{j} \ \leq \ b_{j} \ \mbox{for all} \ j \ \}.\end{aligned}$$ For example, $\textbf{J}_{(1,0)} = \{(0,0),(1,0)\}$ and $\textbf{J}_{(0,0,1)} = \{(0,0,0),(0,0,1)\}$.
Based on $\textbf{J}_{\vec{b}}$, we define the *characteristic binary column* $B(\vec{b}) \in \textbf{B}^{m}_{col}$ as follows: $$\begin{aligned}
B(\vec{b})_{1,p+1} \ &= \ 1 \qquad \vec{p}\ \in \ \textbf{J}_{\vec{b}} \\
B(\vec{b})_{1,p+1} \ &= \ 0 \qquad \mbox{otherwise}.\end{aligned}$$ Here, we give some examples: $$\begin{split}
&\textbf{J}_{(0,0)} \ = \ \{ (0,0) \}, \quad \textbf{J}_{(1,0)} \ = \ \{ (0,0), (1,0) \}, \quad \textbf{J}_{(0,1)} \ = \ \{ (0,0),(0,1) \}\\ &\textbf{J}_{(1,1)} \ = \ \{ (0,0),(1,0),(0,1),(1,1) \}
\end{split}$$ and $$\begin{aligned}
B(0,0) \ = \ \begin{bmatrix}
1 \\
0 \\
0 \\
0
\end{bmatrix}, \quad
B(1,0) \ = \ \begin{bmatrix}
1 \\
1 \\
0 \\
0
\end{bmatrix}, \quad
B(0,1) \ = \ \begin{bmatrix}
1 \\
0 \\
1 \\
0
\end{bmatrix}, \quad
B(1,1) \ = \ \begin{bmatrix}
1 \\
1 \\
1 \\
1
\end{bmatrix}.\end{aligned}$$ One may see the relation between a characteristic column and a characteristic vector: $$\begin{aligned}
\begin{bmatrix}
V \\
I \\
I \\
I
\end{bmatrix} \ \rightarrow \ (0,0), \quad
\begin{bmatrix}
V \\
V \\
I \\
I
\end{bmatrix} \ \rightarrow \ (1,0), \quad
\begin{bmatrix}
V \\
I \\
V \\
I
\end{bmatrix} \ \rightarrow \ (0,1), \quad
\begin{bmatrix}
V \\
V\\
V\\
V
\end{bmatrix} \ \rightarrow \ (1,1).\end{aligned}$$ Thus, if we replace $1$ entries in $B(\vec{b})$ with $V$ and create a column operator, it has a characteristic vector $\vec{b}$.
In order to discuss changes of characteristic vectors, let us introduce the summation rule between binary vectors. We denote a summation of binary vectors $\vec{a}, \vec{b}\in \textbf{B}^{m}_{vec}$ as $\vec{a} + \vec{b}\in \textbf{B}^{m}_{vec}$ and define it as follows: $$\begin{aligned}
g(\vec{a}) + g(\vec{b}) \ = \ g(\vec{a} + \vec{b})\end{aligned}$$ when $g(\vec{a} + \vec{b}) \ \leq 2^{m}-1$. Therefore, $\vec{a} + \vec{b}$ is just like a summation of two binary “numbers” $\vec{a}$ and $\vec{b}$.
The characteristic columns defined above can be used to change a characteristic vector of a column operator, as summarized in the following lemma.
\[lemma\_property\] Let $\vec{b} < \vec{b'}$. When $U$ has a characteristic value $\vec{b}$ and a characteristic operator $V$, $$\begin{aligned}
U' \ = \ U(B(\Delta\vec{b})) \quad \mbox{where} \quad \vec{b} + \Delta\vec{b} \ = \ \vec{b'}\end{aligned}$$ has a characteristic value $\vec{b'}$ and a characteristic operator $V$.
Below, we present a proof of this lemma by finding some property of characteristic columns and a certain rule on multiplications of column operators.
**Property of characteristic columns:** There is a useful relation between a summation of vectors and characteristic columns, as summarized in the following lemma.
\[lemma\_summation\] Let $$\begin{aligned}
B(\vec{a})*B(\vec{b}) \ \equiv \ \sum_{j=1}^{2^{m}} T_{2}(B(\vec{a})^{B(\vec{b})_{1,j}})^{j-1} \qquad \mbox{(mod 2)}.\end{aligned}$$ Then, $$\begin{aligned}
B(\vec{a} + \vec{b}) \ = \ B(\vec{a})*B(\vec{b}).\end{aligned}$$
The proof involves some exercises on elementary math.
We begin by defining a summation of sets of binary vectors. For $\textbf{B}_{1},\textbf{B}_{2},\cdots,\textbf{B}_{\alpha} \ \subseteq \ \textbf{B}^{m}_{vec}$ where $\alpha$ is some positive integer, and for $\vec{b} \in \textbf{B}^{m}_{vec}$, consider decompositions: $$\begin{aligned}
\vec{b} \ = \ \vec{b_{1}} + \cdots + \vec{b_{\alpha}}, \qquad \vec{b_{i}} \ \in \ \textbf{B}_{i} \quad \mbox{for all} \ i\end{aligned}$$ and denote the number of different decompositions of $\vec{b}$ as $N(\vec{b};\textbf{B}_{1},\textbf{B}_{2},\cdots,\textbf{B}_{\alpha})$. Then, we define the following summation: $$\begin{aligned}
\textbf{B}_{1} + \textbf{B}_{2} + \cdots + \textbf{B}_{\alpha} \ \equiv \ \{ \ \vec{b} \ \in \ \textbf{B}^{m}_{vec} \ : \ N(\vec{b};\textbf{B}_{1},\textbf{B}_{2},\cdots, \textbf{B}_{\alpha}) \ = \ \mbox{odd}\ \}.\end{aligned}$$
With the summation defined above, the claim of the lemma can be written as follows. By setting $B = B(\vec{a})*B(\vec{b})$, we may notice that $$\begin{aligned}
B_{1, p+1} \ &= \ 1 \qquad \vec{p} \ \in \ \textbf{J}_{\vec{b}} + \textbf{J}_{\vec{b'}}\\
B_{1, p+1} \ &= \ 0 \qquad \mbox{otherwise}\end{aligned}$$ from a direct calculation. Therefore, we need to show that $$\begin{aligned}
\textbf{J}_{\vec{a}} + \textbf{J}_{\vec{b}} \ = \ \textbf{J}_{\vec{a} + \vec{b}}.\end{aligned}$$
The proof relies on the following sublemma.
Consider $$\begin{aligned}
{}_\alpha C_{\beta} \ \equiv \ \frac{\alpha!}{\beta!(\alpha-\beta)!}\end{aligned}$$ for ($2^{m} > \alpha \geq \beta \geq 1$). Let the binary representations of $\alpha$ and $\beta$ be $$\begin{aligned}
\vec{\alpha} \ = \ (\alpha_{1},\cdots, \alpha_{m}), \qquad \vec{\beta} \ = \ (\beta_{1},\cdots, \beta_{m})\end{aligned}$$ where $\alpha = \sum_{i=1}^{m}\alpha_{i}2^{i-1}$ and $\beta = \sum_{i=1}^{m}\beta_{i}2^{i-1}$. Then, ${}_\alpha C_{\beta}$ is odd if and only if $$\begin{aligned}
\beta_{i} \ \leq \ \alpha_{i} \qquad \mbox{for all} \ i. \end{aligned}$$
We suspect that the sublemma above has been proven somewhere else as it seems elementary. Yet, we could not find a reference, and thus, we present a proof here.
For a given integer $p \in \mathbb{Z}_{2^{m}}$, let $h(p)$ be the largest integer such that $\frac{p!}{2^{h(p)}}$ is an integer. Then, with some speculations, one may notice that $$\begin{aligned}
h(p) \ = \ \sum_{i=1}^{m} (2^{i-1}-1)p_{i}\end{aligned}$$ where $\vec{p} = (p_{1},\cdots,p_{m})$. Here, $$\begin{aligned}
{}_\alpha C_{\beta} \ = \ \frac{\alpha!}{\beta!(\alpha-\beta)!},\end{aligned}$$ and $$\begin{aligned}
h({}_\alpha C_{\beta}) \ = \ h(\alpha) - h(\beta) - h(\alpha - \beta) \ \geq \ 0.\end{aligned}$$ Then, ${}_\alpha C_{\beta}$ is odd if and only if $$\begin{aligned}
h(\beta) + h(\alpha-\beta) \ = \ h(\alpha).\end{aligned}$$ Let $\alpha - \beta \equiv \gamma$. Then, we have $$\begin{aligned}
\sum_{i=1}^{m} (2^{i-1}-1)\beta_{i} + \sum_{i=1}^{m} (2^{i-1}-1)\gamma_{i} \ = \ \sum_{i=1}^{m} (2^{i-1}-1)\alpha_{i}\end{aligned}$$ and $\beta_{i} + \gamma_{i}=\alpha_{i}$ for all $i$. This is true if and only if $$\begin{aligned}
\beta_{i} \ \leq \ \alpha_{i} \qquad \mbox{for all} \quad i.\end{aligned}$$ This completes the proof of the sublemma.
Now, let us return to the proof of the lemma. Below, we prove $\textbf{J}_{\vec{a}} + \textbf{J}_{\vec{b}} \ = \ \textbf{J}_{\vec{a} + \vec{b}}$. Let $\vec{e_{1}} \equiv (1,0,\cdots,0)$. First, we show that $$\begin{aligned}
\underbrace{\textbf{J}_{\vec{e_{1}}} +\cdots + \textbf{J}_{\vec{e_{1}}}}_{\alpha} \ = \ \textbf{J}_{\vec{\alpha}}\end{aligned}$$ where $$\begin{aligned}
\vec{\alpha} \ \equiv \ \underbrace{ \vec{e_{1}} +\cdots + \vec{e_{1}} }_{\alpha}.\end{aligned}$$ Here, notice that $$\begin{aligned}
\vec{\beta} \ \in \ \underbrace{ \textbf{J}_{\vec{e_{1}}} +\cdots + \textbf{J}_{\vec{e_{1}}} }_{\alpha}\end{aligned}$$ if and only if ${}_{\alpha} C_{\beta}$ is odd since $\textbf{J}_{ \vec{e_{1}} }$ has two elements: $(1,0,\cdots,0)$ and $(0,\cdots,0)$. Thus, $\vec{\beta} \in \textbf{J}_{\vec{\alpha}}$ from the sublemma, and we have $$\begin{aligned}
\underbrace{\textbf{J}_{\vec{e_{1}}} +\cdots + \textbf{J}_{\vec{e_{1}}}}_{\alpha} \ = \ \textbf{J}_{\vec{\alpha}}.\end{aligned}$$
Now, let us show that $$\begin{aligned}
\textbf{J}_{\vec{a}} + \textbf{J}_{\vec{b}} \ = \ \textbf{J}_{\vec{a} + \vec{b}}.\end{aligned}$$ Note that $\vec{c} \in \textbf{J}_{\vec{a} + \vec{b}}$ if and only if ${}_{a+b} C_c$ is odd. Here, we have $$\begin{aligned}
{}_{a+b} C_c \ = \ \sum_{i=0}^{c} {}_{a} C_i \cdot {}_{b} C_{c-i}\end{aligned}$$ where ${}_{x} C_{y}=0$ when $y>x$. Let us assume that ${}_{a} C_i \cdot {}_{b} C_{c-i}$ is odd for $i = \alpha_{1},\cdots,\alpha_{p}$. Notice that ${}_{a} C_i \cdot {}_{b} C_{c-i}$ is odd if and only if both ${}_{a} C_i$ and ${}_{b} C_{c-i}$ are odd. Then, ${}_{a+b} C_c$ is odd if and only if $p$ is odd. Notice that $p$ is the number of decompositions of $\vec{c}$ such that $$\begin{aligned}
\vec{c} \ = \ \vec{c_{1}} + \vec{c_{2}}\qquad (\vec{c_{1}} \ \in \ \textbf{J}_{\vec{a}} \quad \mbox{and} \quad \vec{c_{2}} \ \in \ \textbf{J}_{\vec{b}}).\end{aligned}$$ Therefore, $p$ is odd if and only if $\vec{c} \in \textbf{J}_{\vec{a}} + \textbf{J}_{\vec{b}}$. This completes the proof of the lemma.
**Multiplication of column operators:** Next, let us consider how characteristic vectors change under multiplications of column operators.
\[lemma\_multiplication\] Consider the following column operators: $$\begin{aligned}
U '' \ = \ UU'\ \not= \ I\end{aligned}$$ where $$\begin{aligned}
\begin{array}{cccc}
U & \rightarrow & V , & \vec{b}\\
U' & \rightarrow & V' , & \vec{b'}\\
U'' & \rightarrow & V'' , & \vec{b''}
\end{array}.\end{aligned}$$ Then,
- If $\vec{b} > \vec{b'}$, $\vec{b''}=\vec{b'}$ and $V'' = V'$.
- If $\vec{b'} = \vec{b}$ and $V \not= V'$, $\vec{b''}=\vec{b}$ and $V'' = VV'$.
- If $\vec{b'} = \vec{b}$ and $V = V'$, $\vec{b''} > \vec{b}$.
Below, we present the proof.
We start with the first claim. When $b_{m}'=1$, $b_{m}=1$ since $\vec{b}>\vec{b'}$. Then, $f_{0}(U)=f_{0}(U')=I$, and $f_{0}(U'')=I$. Thus, $b_{m}''=1$. When $b_{m}'=0$ and $b_{m}=0$, we have $f_{0}(U)\not=I$ and $f_{0}(U')\not=I$. Suppose $f_{0}(U'')=I$. Then, $f_{0}(U)=f_{0}(U')$, and $U^{(1)}=U'^{(1)}$. This means $\vec{b}=\vec{b'}$ which contradicts with $\vec{b}>\vec{b'}$. Thus, $f_{0}(U'')\not=I$, and $b_{m}''=0$. When $b_{m}'=0$ and $b_{m}=1$, we have $f_{0}(U)=I$ and $f_{0}(U')\not=I$, and $f_{0}(U'')\not=I$, and $b_{m}''=0$. In summary, $b_{m}'=b_{m}''$. One can repeat the same discussion and show $\vec{b}' = \vec{b}''$ and $V''=V$.
The second claim is easy to prove, so we shall skip the proof. Let us move to the third claim. Since $$\begin{aligned}
f_{b_{1}}\cdots f_{b_{m}}(U) \ = \ f_{b_{1}}\cdots f_{b_{m}}(U') \ = \ V,\end{aligned}$$ we have $f_{b_{1}}\cdots f_{b_{m}}(U'')= I$. Let the largest integer $i$ such that $f_{b_{i}}\cdots f_{b_{m}}(U'') = I$ be $i_{max}$. Then, $f_{b_{i_{max}+1}}\cdots f_{b_{m}}(U'') \not= I$ and $b_{i_{max}}''=1$. If $b_{i_{max}}=1$, $f_{b_{i_{max}+1}}\cdots f_{b_{m}}(U'') = I$ which leads to a contradiction. Thus, $b_{i_{max}}=0$.
Since $f_{b_{i_{max}+1}}\cdots f_{b_{m}}(U'') \not= I$, we have $$\begin{aligned}
b_{i} \ &= \ b_{i}'' \qquad (i_{max}+1 \ \leq \ i)\\
b_{i_{max}} \ &= \ 0, \qquad b_{i_{max}}'' \ = \ 1. \end{aligned}$$ Thus, $\vec{b''}>\vec{b}$.
**Proof of lemma \[lemma\_property\]:** Finally, let us finish the proof of lemma \[lemma\_property\]. Consider the following column operator $V(\vec{b})$ which replaces $1$ entries in $B(\vec{b})$ with $V$ and $0$ entries with $I$: $$\begin{aligned}
V(\vec{b})_{j} \ \equiv \ V^{B(\vec{b})_{1,j}}. \end{aligned}$$ Then, one may easily notice that $V(\vec{b})$ has a characteristic vector $\vec{b}$ with a characteristic operator $V$. Therefore, from lemma \[lemma\_summation\], we have $$\begin{aligned}
V(\vec{b})(B(\vec{a})) \ = \ V(\vec{a}+\vec{b}).\end{aligned}$$
For $U$ with $\vec{b}$ and $V$, we decompose $U$ with $V(\vec{b})$ as follows: $$\begin{aligned}
U \ = \ V(\vec{b})U'.\end{aligned}$$ When $U\not=V(\vec{b})$, $U'$ has a characteristic vector $\vec{b'}$ with $\vec{b'}>\vec{b}$ from lemma \[lemma\_multiplication\]. One can repeat the same decomposition and obtain: $$\begin{aligned}
U \ = \ V(\vec{b})V'(\vec{b'})V''(\vec{b''})\cdots\end{aligned}$$ where $\vec{b}<\vec{b'}<\vec{b''}$. Then, from lemma \[lemma\_summation\], we have $$\begin{aligned}
U(B(\vec{a})) \ = \ V(\vec{a}+\vec{b})V'(\vec{a}+\vec{b'})V''(\vec{a}+\vec{b''})\cdots.\end{aligned}$$ Note that $V(\vec{c})(B(\vec{a}))=I$ if $a + c >2^{m}-1$. From lemma \[lemma\_multiplication\], $U(B(\vec{a}))$ is a column operator with a characteristic vector $\vec{a}+\vec{b}$ and a characteristic operator $V$. Note that $B(\vec{a})$ is an even column when $\vec{a}\not=(0,\cdots,0)$. This completes the proof of lemma \[lemma\_property\].
The existence of an odd matrix for $m>1$
----------------------------------------
Finally, let us proceed to the proof of lemma \[lemma\_existence\], to complete the proof of theorem \[theorem\_decomposition\].
**Procedure:** Consider a logical operator $\ell$ defined inside $P(x,2^{m},1)$ with $x >2v$. Let us represent characteristic values and characteristic operators for each column of $\ell$ as follows: $$\begin{aligned}
\begin{array}{ccc}
U_{1}\ \rightarrow\ & \vec{b_{1}}, & V_{1} \\
U_{2}\ \rightarrow\ & \vec{b_{2}},& V_{2} \\
\vdots & \vdots & \vdots \\
U_{x}\ \rightarrow\ & \vec{b_{x}},& V_{x}.
\end{array}\end{aligned}$$ Without loss of generality, we may assume that $\vec{b_{i}} \leq \vec{b_{i+1}}$ for all $i$ since permutations of columns do not affect parities of binary matrices. We define a set of integers such that $\vec{b_{i}}=\vec{b}$ as $\textbf{b}(\vec{b})$: $$\begin{aligned}
\textbf{b}(\vec{b}) \ \equiv \ \{ \ i \ : \ \vec{b_{i}} \ = \ \vec{b} \ \}.\end{aligned}$$ We denote a group generated by $\{V_{i}\}$ for $i \in \textbf{b}(\vec{b})$ as $\mathcal{V}_{\vec{b}}$: $$\begin{aligned}
\mathcal{V}_{\vec{b}} \ = \ \left\langle \ \{ \ V_{i} \ : \ i \ \in \ \textbf{b}(\vec{b}) \ \} \ \right\rangle.\end{aligned}$$
Since $x > 2v$, there always exists a set of integers $\textbf{A}$ such that $$\begin{aligned}
\prod_{i \in \textbf{A}} V_{i}\ = \ I. \end{aligned}$$ We denote the largest vector in $\{\vec{b_{i}}\}_{i\in \textbf{A}}$ as $\vec{b_{\alpha}}$ and the largest integer $i$ with $\vec{b_{i}} = \vec{b_{\alpha}}$ as $i = \alpha$. Here, we define the following binary matrix $E(i')$ $$\begin{aligned}
E(i') \ \equiv \ \begin{bmatrix}
E_{1,1} ,& \cdots ,& E_{x,1} \\
\vdots & \vdots & \vdots \\
E_{1,2^{m}} ,& \cdots ,& E_{x,2^{m}}
\end{bmatrix}\end{aligned}$$ such that $$\begin{aligned}
E_{i,1} \ &= \ 1 \qquad (i \ = \ i' )\\
E_{i,j} \ &= \ 0 \qquad \mbox{otherwise}.\end{aligned}$$ For example, $$\begin{aligned}
E(1) \ = \ \begin{bmatrix}
1 ,& 0 ,& \cdots ,& 0 \\
0 ,& 0 ,& \cdots ,& 0 \\
\vdots & \vdots & \vdots & \vdots \\
0 ,& 0,& \cdots ,& 0
\end{bmatrix},\qquad E(2) \ = \ \begin{bmatrix}
0 ,& 1 ,& 0,& \cdots ,& 0 \\
0 ,& 0 ,& 0,& \cdots ,& 0 \\
\vdots & \vdots & \vdots & \vdots & \vdots \\
0 ,& 0,& 0 ,& \cdots ,& 0
\end{bmatrix}.\end{aligned}$$ Notice that $$\begin{aligned}
Par(E(i))_{j} \ = \ \delta_{i,j}. \end{aligned}$$
Now, we define the following operation between a binary column $A \in \textbf{B}^{m}_{col}$ and a binary matrix $B$: $$\begin{aligned}
B * A \ \equiv \ \sum_{j=1}^{2^{m}} T_{2}(B^{A_{1,j}})^{j-1} \qquad ( \mbox{mod 2}).\end{aligned}$$ Then, consider the following binary matrix: $$\begin{aligned}
E \ = \ \sum_{i \in \textbf{A}} E(i) * B(\Delta \vec{b_{i}}) \end{aligned}$$ where $\vec{b_{i}} + \Delta \vec{b_{i}} = \vec{b_{\alpha}}$. *Note that $E(i) * B(\Delta \vec{b_{i}}) $ is odd if and only if $\Delta \vec{b_{i}} = (0,\cdots,0)$*. This matrix can generate the following column operator: $$\begin{split}
\ell(E)_{x} \ &= \ \ell\left(\sum_{i \in \textbf{A}} E(i) * B(\Delta \vec{b_{i}}) \right)_{x}\\
&= \ \prod_{i \in \textbf{A} } U_{i} \left( B(\Delta \vec{b_{i}}) \right).
\end{split}$$ Note that $$\begin{aligned}
U_{i} \left( B(\Delta \vec{b_{i}}) \right) \end{aligned}$$ has a characteristic vector $\vec{b_{\alpha}}$ and a characteristic operator $V_{i}$ from lemma \[lemma\_summation\]. Here, we notice that $$\begin{aligned}
Par(E)_{\alpha} \ = \ 1\end{aligned}$$ and $E$ is an odd matrix since $\Delta \vec{b_{\alpha}} = (0,\cdots,0)$. Then, $\ell(E)_{x}\not=I$ since there is no odd identity generating matrix. Now, we notice that $\ell(E)_{x}$ is a column operator with a characteristic vector $\vec{b_{\alpha}'} > \vec{b_{\alpha}}$ from lemma \[lemma\_multiplication\].
We summarize the discussion so far as follows.
- From $\textbf{A}$ such that $\prod_{i \in \textbf{A}}V_{i}=I$, one can form a column operator $\ell(E)_{x}$ which has a characteristic vector $\vec{b_{\alpha}'} > \vec{b_{\alpha}}$ and a characteristic operator $V'_{\alpha}$ where $E$ is an odd matrix which satisfies $$\begin{aligned}
Par(E)_{\alpha} \ = \ 1 \quad \mbox{and} \quad
Par(E)_{j} \ = \ 0 \quad (j \ > \ \alpha).\end{aligned}$$
**Update:** Next, we “update” $U_{\alpha}$, $\vec{b_{\alpha}}$, $V_{\alpha}$ and $E(\alpha)$ to $\ell(E)_{x}$, $\vec{b_{\alpha}'}$, $V_{\alpha}'$ and $E$: $$\begin{aligned}
\begin{array}{ccc}
U_{\alpha} & \rightarrow & \ell(E)_{x} \\
\vec{b_{\alpha}} & \rightarrow & \vec{b_{\alpha}'}\\
V_{\alpha} & \rightarrow & V_{\alpha}' \\
E(\alpha) & \rightarrow & E
\end{array}\end{aligned}$$ In other words, we replace $U_{\alpha}$, $\vec{b_{\alpha}}$, $V_{\alpha}$ and $E(\alpha)$ with $E(\alpha)$, $\ell(E)_{x}$, $\vec{b_{\alpha}'}$, $V_{\alpha}'$ and $E$, and rename them as $U_{\alpha}$, $\vec{b_{\alpha}}$, $V_{\alpha}$ and $E(\alpha)$. Note that these “updated” $E(i)$ satisfy $$\begin{aligned}
Par(E(i))_{i} \ = \ 1 \quad \mbox{and} \quad Par(E(i))_{j} \ = \ 0\quad (j \ > \ i).\end{aligned}$$
Then, one may repeat the discussion above. Since $x> 2v$, there is a set $\textbf{A}'$ such that $$\begin{aligned}
\prod_{i \in \textbf{A}'} V_{i}\ = \ I.\end{aligned}$$ We denote the largest vector in $\{\vec{b_{i}}\}_{i\in \textbf{A}'}$ as $\vec{b_{\beta}}$ and the largest integer $i$ with $\vec{b_{i}} = \vec{b_{\beta}}$ as $i = \beta$. Then, consider the following binary matrix: $$\begin{aligned}
E' \ = \ \sum_{i \in \textbf{A}} E(i) * B(\Delta \vec{b_{i}}) \qquad \mbox{(mod 2)}\end{aligned}$$ where $\vec{b_{i}} + \Delta \vec{b_{i}} = \vec{b_{\beta}}$. This matrix $E'$ can generate the following column operator: $$\begin{split}
\ell(E')_{x} \ &= \ \ell\left(\sum_{i \in \textbf{A}'} E(i) * B(\Delta \vec{b_{i}}) \right)_{x} \\
&= \ \prod_{i \in \textbf{A}' } U_{i} \left( B(\Delta \vec{b_{i}}) \right).
\end{split}$$ Here, we notice that $$\begin{aligned}
Par(E')_{\beta} \ = \ 1\end{aligned}$$ since $\Delta\vec{b_{\beta}} = (0,\cdots,0)$ and $E'$ is an odd matrix. Then, $\ell(E')_{x}\not=I$. Note that $$\begin{aligned}
U_{i} \left( B(\Delta \vec{b_{i}}) \right) \end{aligned}$$ has a characteristic vector $\vec{b_{\beta}}$ and a characteristic operator $V_{i}$ from lemma \[lemma\_summation\]. Then, we notice that $\ell(E')_{x}$ is a column vector with a characteristic vector $\vec{b_{\beta}'} > \vec{b_{\beta}}$. Note that $$\begin{aligned}
Par(E')_{\beta} \ = \ 0 \quad (j>\beta).\end{aligned}$$
Then, we obtain the following observation.
- From $\textbf{A}'$ such that $\prod_{i \in \textbf{A'}'}V_{i}=I$ for “updated” $V_{i}$, one can form a column operator $\ell(E')_{x}$ which has a characteristic vector $\vec{b_{\beta}'} > \vec{b_{\beta}}$ and a characteristic operator $V'_{\beta}$. $E'$ is an odd matrix which satisfies $$\begin{aligned}
Par(E')_{\beta} \ = \ 1 \quad \mbox{and} \quad
Par(E')_{j} \ = \ 0 \quad (j \ > \ \beta).\end{aligned}$$
Here, we again “update” $U_{\beta}$, $\vec{b_{\beta}}$, $V_{\beta}$ and $E(\beta)$ to $\ell(E')_{x}$, $\vec{b_{\beta}'}$, $V_{\beta}'$ and $E'$. Then, since updated $E(i)$ always satisfy $$\begin{aligned}
Par(E(i))_{i} \ = \ 1 \quad \mbox{and} \quad Par(E(i))_{j} \ = \ 0\quad (j>i)\end{aligned}$$ one can repeat the same discussion again. In each update, characteristic vectors $\vec{b_{i}}$ increase, and at the end, one ends up with the following column operators $$\begin{aligned}
\begin{array}{cccc}
U_{1}\ \rightarrow\ & \vec{1}, & V_{1} ,& E(1) \\
U_{2}\ \rightarrow\ & \vec{1},& V_{2} ,& E(2) \\
\vdots & \vdots & \vdots & \vdots \\
U_{x}\ \rightarrow\ & \vec{1},& V_{x} ,& E(x)
\end{array}\end{aligned}$$ where $\vec{1} \equiv (1,\cdots,1)$ and $$\begin{aligned}
Par(E(i))_{i} \ = \ 1 \quad \mbox{and} \quad Par(E(i))_{j} \ = \ 0\quad (j>i)\end{aligned}$$ Then, there exists a set $\textbf{A}$ such that $$\begin{aligned}
\prod_{i \in \textbf{A}}V_{i} \ = \ I\end{aligned}$$ and, the following matrix is an identity generating matrix $$\begin{aligned}
E \ = \ \sum_{i \in \textbf{A}} E(i) \qquad \mbox{(mod 2)}.\end{aligned}$$ Let the largest integer in $\textbf{A}$ be $i_{max}$. Then, $E$ is odd since $Par(E)_{i_{max}}=1$. However, this contradicts with our original assumption that there is no odd identity generating matrix. This completes the proof of lemma \[lemma\_existence\], lemma \[lemma\_decomposition\_final\] and theorem \[theorem\_decomposition\].
Derivation of logical operators {#sec:construction}
===============================
Having showed that two-dimensional logical operators can be decomposed as a product of two-dimensional and one-dimensional centralizer operators, let us proceed to the proof of theorem \[theorem\_3dim\]. The proof owes a lot to arguments presented in [@Beni10b]. For simplicity of presentation and in order to avoid making the paper unnecessarily long, we shall skip some parts of the derivation. However, we believe that interested readers can easily construct rigorous proofs.
**Preliminaries:** We begin by providing some corollaries and lemma which are useful in the derivations of logical operators. Let us first generalize theorem \[theorem\_decomposition\] for any $n_{2}$.
\[corollary\_decomposition\] Consider a three-dimensional STS model with the system size $n_{1}= 2\cdot2^{2n_{2}v}!$, arbitrary $n_{2}$ and $n_{3}>1$. For a given logical operator $\ell$ supported inside $P(n_{1},n_{2},1)$, one can decompose $\ell$ as a product of the following centralizer operators $$\begin{aligned}
\ell \ \sim \ \ell_{a}\ell_{b}, \qquad \ell_{a},\ell_{b}\ \in \ \mathcal{C}_{P(n_{1},n_{2},1)}\end{aligned}$$ where $$\begin{aligned}
T_{1}^{\beta}(\ell_{b})\ = \ \ell_{b}, \qquad \mbox{where}\quad \beta \ \leq \ 2^{2n_{2}v}\end{aligned}$$ and $\ell_{a}$ is defined inside $P(2v, n_{2},1)$.
The proof relies on the fact that one can make a logical operator $\ell_{a}$ “quasi-periodic”.
Let us represent $n_{2}$ as $n_{2}=2^{m}\cdot n_{2}'$ where $n_{2}'$ is some odd integer. Then, for a logical operator $\ell$ defined inside $P(n_{1},n_{2},1)$, one can see that the following logical operator $$\begin{aligned}
\ell' \ = \ \prod_{j=1}^{n_{2}'}T_{2}^{(j-1)2^{m}}(\ell) \ \sim \ \ell\end{aligned}$$ is equivalent to $\ell$ since $\ell'$ is a product of an odd number of translations of $\ell$. Notice that $\ell'$ is periodic in the $\hat{2}$ direction: $$\begin{aligned}
T_{2}^{2^{m}}(\ell') \ = \ \ell'\end{aligned}$$ with the periodicity $2^{m}$. Because of this periodicity, one can form identity generating matrices in a way similar to the cases when $n_{2}=2^{m}$. Therefore, one can see that theorem \[theorem\_decomposition\] holds for any $n_{2}$.
We have seen that a two-dimensional logical operator can be decomposed as a product of a one-dimensional centralizer operator and a two-dimensional centralizer operator, as summarized in theorem \[theorem\_decomposition\]. One can further decompose a one-dimensional logical operator as a product of a one-dimensional centralizer operator and a zero-dimensional centralizer operator, as summarized in the following lemma.
\[lemma\_decomposition\_1dim\] Consider a three-dimensional STS model with the system size $n_{1}= 2\cdot2^{2n_{2}v}!$, $n_{2}=2\cdot 2^{(2v)^{2}}!$ and $n_{3}>1$ where $m$ is an arbitrary positive integer. For a given logical operator $\ell$ supported inside $P(2v,n_{2},1)$, one can decompose $\ell$ as a product of the following centralizer operators $$\begin{aligned}
\ell \ \sim \ \ell_{a}\ell_{b}, \qquad \ell_{a},\ell_{b}\ \in \ \mathcal{C}_{P(2v,n_{2},1)}\end{aligned}$$ where $$\begin{aligned}
T_{1}^{\beta}(\ell_{b})\ = \ \ell_{b}, \qquad \mbox{where}\quad \beta \ \leq \ 2^{(2v)^{2}}\end{aligned}$$ and $\ell_{a}$ is defined inside $P(2v, (2v)^{2},1)$.
We show the claim of lemma \[lemma\_decomposition\_1dim\] graphically in Fig. \[fig\_decomposition1D\]. One can prove the lemma through discussion similar to the one used in the proof of lemma \[lemma\_decomposition\]. So, we shall skip the proof.
![The claim of lemma \[lemma\_decomposition\_1dim\]. []{data-label="fig_decomposition1D"}](fig_decomposition1D.pdf){width="0.45\linewidth"}
Finally, let us extend the claim of theorem \[theorem\_2dim\] slightly.
\[corollary\_2dim\] Consider a two-dimensional STS model with even $n_{1}$ and $n_{2}$.
- Let $\ell$ be a logical operator which is periodic in the $\hat{1}$ and $\hat{2}$ directions: $$\begin{aligned}
T_{1}(\ell) \ = \ T_{2}(\ell) \ = \ \ell.\end{aligned}$$ Then, $\ell$ is a two-dimensional logical operator, and there exists a zero-dimensional logical operator $r$ which is defined inside $P(1,2v)$ and anti-commutes with $\ell$.
- Let $\ell$ be a logical operator which is defined inside $P(1,n_{2})$ and periodic in the $\hat{2}$ direction: $$\begin{aligned}
T_{2}(\ell) \ = \ \ell.\end{aligned}$$ Then, $\ell$ is a one-dimensional logical operator, and there exists another one-dimensional logical operator $r$ which is defined inside $P(n_{1},1)$, anti-commutes with $\ell$: $\{\ell,r\}=0$ and periodic in the $\hat{1}$ direction: $$\begin{aligned}
T_{1}(r) \ = \ r.\end{aligned}$$
In other words, if we find a logical operator which is periodic in both $\hat{1}$ and $\hat{2}$ directions, we readily know that it is a two-dimensional logical operator. Similarly, if we find a logical operator which is defined inside $P(1,n_{2})$ and periodic in the $\hat{2}$ direction, we readily know that it is a one-dimensional logical operator. Note that the corollary holds only for the cases where both $n_{1}$ and $n_{2}$ are even.
Since $\ell$ is periodic and system sizes are even, $\ell$ commutes with all the one-dimensional logical operators and all the two-dimensional logical operators. Therefore, $\ell$ can anti-commute only with zero-dimensional logical operators. This means that $\ell$ is a two-dimensional logical operator. The second claim can be proven in a similar way.
Now, we derive all the logical operators in a three-dimensional STS models, as described in theorem \[theorem\_3dim\]. Consider the system size analyzed in lemma \[lemma\_decomposition\_1dim\]: $(n_{1},n_{2},n_{3})= ( 2\cdot2^{2n_{2}v}!, 2\cdot 2^{(2v)^{2}}!, n_{3})$ where $n_{3}>1$ is some fixed integer. Here, we view the entire system as a two-dimensional system by considering $P(1, n_{2},1)$ as a single composite particle (Fig. \[fig\_2Dview\]). (So, the entire system is viewed as a two-dimensional lattice of one-dimensional tubes).
For a two-dimensional STS model, we already know the geometric shapes of all the logical operators, as summarized in theorem \[theorem\_2dim\]. When viewed as a two-dimensional system, “zero-dimensional” logical operators are defined inside $P(2vn_{2},n_{2},1)$. From theorem \[theorem\_decomposition\], we notice that these “zero-dimensional” logical operators can be actually defined inside $P(2v,n_{2},1)$. Then, we have the following logical operators (Fig. \[fig\_2Dview\]).
![Viewed as a two-dimensional system. []{data-label="fig_2Dview"}](fig_2Dview.pdf){width="0.50\linewidth"}
- An anti-commuting pair of logical operators in Fig. \[fig\_2Dview\](a) where $\ell$ is defined inside $P(2v,n_{2},1)$ and $r$ is periodic: $$\begin{aligned}
T_{1}(r) \ = \ T_{3}(r) \ = \ r. \end{aligned}$$
- An anti-commuting pair of logical operators in Fig. \[fig\_2Dview\](b) where $\ell$ is defined inside $P(1,n_{2},n_{3})$ and periodic: $$\begin{aligned}
T_{3}(\ell) \ = \ \ell\end{aligned}$$ and $r$ is defined inside $P(n_{1},n_{2},1)$ and periodic: $$\begin{aligned}
T_{1}(r) \ = \ r.\end{aligned}$$
Below, we analyze anti-commuting pairs of logical operators in Fig. \[fig\_2Dview\](a) and Fig. \[fig\_2Dview\](b), and derive logical operators.
**Pairs in (a):** Below, we analyze properties of logical operators described above. We start with anti-commuting pairs described in Fig. \[fig\_2Dview\]$(a)$. We stop viewing the system as a two-dimensional system for the moment. Logical operators defined inside $P(2v,n_{2},1)$ consists of periodic one-dimensional logical operators and zero-dimensional logical operators defined inside $P(2v, (2v)^{2},1)$ due to lemma \[lemma\_decomposition\_1dim\]. Let us first analyze a zero-dimensional logical operator $\ell$ defined inside $P(2v, (2v)^{2},1)$.
From theorem \[theorem\_2dim\], $\ell$ is also a logical operators for arbitrary $n_{1}$ and $n_{3}$. Consider the case when $n_{3}=1$ (Fig. \[fig\_proof\_aid1\]). Then, by viewing the system as a two-dimensional system which extends only in the $\hat{1}$ and $\hat{2}$ directions, one notices that there exists a two-dimensional logical operator $r$ which is periodic in both $\hat{1}$ and $\hat{2}$ directions: $$\begin{aligned}
T_{1}(r) \ = \ T_{2}(r) \ = \ r\end{aligned}$$ and anti-commutes with $\ell$: $\{ \ell,r\}=0$. Next, let us consider the case when $n_{3}>1$. Then, one may extend the construction of $r$ as follows (Fig. \[fig\_proof\_aid1\]): $$\begin{aligned}
r \ \rightarrow \ r' \ = \ \prod_{x=1}^{n_{3}}T_{3}^{x}(r).\end{aligned}$$ In other words, we put $r$ in a periodic way in the $\hat{3}$ direction to form $r'$. We shall call such an extension the *periodic extension*. The three-dimensional logical operator $r'$ obtained after the periodic extension of $r$ is periodic in all the directions: $$\begin{aligned}
T_{1}(r') \ = \ T_{2}(r') \ = \ T_{3}(r') \ = \ r'\end{aligned}$$ and anti-commutes with $\ell$ : $\{r',\ell \}=0$. Then, one may notice that $r'$ and $\ell$ form a pair of anti-commuting logical operators for any system size $\vec{n}$. From this discussion, we obtain the following observation (Fig. \[fig\_proof\_aid1\]).
- For zero-dimensional logical operators defined inside $P(2v, (2v)^{2},1)$, there always exists a three-dimensional logical operator $r$ which is periodic: $$\begin{aligned}
T_{1}(r) \ = \ T_{2}(r) \ = \ T_{3}(r) \ = \ r\end{aligned}$$ and anti-commutes with $\ell$: $\{\ell,r\}=0$. $\ell$ and $r$ are logical operators for any system size $\vec{n}$.
![Constructions of zero-dimensional and three-dimensional logical operators. []{data-label="fig_proof_aid1"}](fig_proof_aid1.pdf){width="0.55\linewidth"}
Next, let us consider a one-dimensional logical operator $\ell$ defined inside $P(2v,n_{2},1)$ which is periodic in the $\hat{2}$ direction: $$\begin{aligned}
T_{2}^{\beta}(\ell) \ = \ \ell\end{aligned}$$ for $\beta \leq 2^{(2v)^{2}}$. Recall that $r$ is periodic: $$\begin{aligned}
T_{1}(r) \ = \ T_{3}(r) \ = \ r\end{aligned}$$ and anti-commutes with $\ell$ : $\{r,\ell\}=0$. Then, one can decompose $r$ as a product of two centralizer operators from lemma \[lemma\_decomposition\]: $$\begin{aligned}
r \ \sim \ r_{a}r_{b}\end{aligned}$$ where $$\begin{aligned}
T_{1}(r_{a}) \ = \ T_{3}(r_{a}) \ = \ r_{a}, \qquad T_{1}(r_{b}) \ = \ T_{3}(r_{b}) \ = \ r_{b}\end{aligned}$$ and $r_{a}$ is defined inside $P(n_{1},2v,n_{3})$, and $$\begin{aligned}
T_{2}^{\beta'}(r_{b}) \ = \ r_{b}\end{aligned}$$ for $\beta' \leq 2^{2v}$. Then, we notice that $$\begin{aligned}
[r_{b},\ell] \ = \ 0\end{aligned}$$ since $T_{2}^{\beta}(\ell) = \ell$ and $T_{2}^{\beta'}(r_{b}) = r_{b}$, and $n_{2}/\beta\beta'$ is an even integer for $n_{2}=2\cdot 2^{(2v)^{2}}!$. Thus, we have $$\begin{aligned}
\{r_{a},\ell\} \ = \ 0.\end{aligned}$$
Since $r_{a}$ is periodic in the $\hat{1}$ and $\hat{3}$ directions, one can periodically extend its construction for arbitrary $n_{1}$ and $n_{3}$. Now, let us consider the system size such that $n_{1}$ and $n_{3}$ are odd. Here, note that $r_{a}$ has some equivalent logical operator $r_{a}'$ defined inside $P(n_{1},1,n_{3})$ [@Beni10b]. Then, the following operator $$\begin{aligned}
r_{a}'' \ = \ \prod_{i,j}T_{1}^{i}T_{3}^{j}(r_{a}') \ \sim \ r_{a} \end{aligned}$$ is equivalent to $r_{a}$ (Fig. \[fig\_proof\_aid2\]). Note that $r_{a}''$ is defined inside $P(n_{1},1,n_{3})$ and periodic in the $\hat{1}$ and $\hat{3}$ directions: $$\begin{aligned}
T_{1}(r_{a}'') \ = \ T_{3}(r_{a}'') \ = \ r_{a}''.\end{aligned}$$ Finally, we show that $\ell$ can be periodic in the $\hat{2}$ direction. Consider the case when $n_{1}$ and $n_{3}$ are even, and $n_{2}=1$. Then, $r_{a}''$ is also a logical operators. Now, there always exists some logical operator $\ell'$ defined inside $P(1,2v,1)$ which anti-commutes with $r_{a}''$ from corollary \[corollary\_2dim\]. One can periodically extend its construction to arbitrary $n_{2}$. We denote it $\ell''$. Then, $\ell''$ and $r_{a}''$ are logical operators for any system size. From this discussion, we obtain the following observation (Fig. \[fig\_proof\_aid2\]).
- For one-dimensional logical operators defined inside $P(2v, n_{2},1)$, there exists a two-dimensional logical operator $r$ defined inside $P(n_{1},1,n_{3})$ which is periodic: $$\begin{aligned}
T_{1}(r) \ = \ T_{3}(r) \ = \ r\end{aligned}$$ and anti-commutes with $\ell$: $\{\ell,r\}=0$. $\ell$ can be also periodic: $$\begin{aligned}
T_{2}(\ell) \ = \ \ell,\end{aligned}$$ and, $\ell$ and $r$ are logical operators regardless of the system size $\vec{n}$.
![Constructions of one-dimensional and two-dimensional logical operators. []{data-label="fig_proof_aid2"}](fig_proof_aid2.pdf){width="0.55\linewidth"}
**Pairs in (b):** Let us proceed to the analysis on pairs of logical operators in Fig. \[fig\_2Dview\](b). We consider the following anti-commuting logical operators $\ell$ and $r$.
- $\ell$ is defined inside $P(1,n_{2},n_{3})$ and periodic: $T_{3}(\ell) = \ell$.
- $r$ is defined inside $P(n_{1},n_{2},1)$ and periodic: $T_{1}(r) = r$.
Since $\ell$ is periodic in the $\hat{3}$ direction, one can decompose $\ell$ as follows: $$\begin{aligned}
\ell \ \sim \ \ell_{a} \ell_{b}, \qquad T_{3}(\ell_{a}) \ = \ \ell_{a} \quad \mbox{and} \quad T_{3}(\ell_{b}) \ = \ \ell_{b}\end{aligned}$$ where $\ell_{a}$ is defined inside $P(1,2v,n_{3})$, and $\ell_{b}$ is defined inside $P(1,n_{2},n_{3})$ and periodic: $$\begin{aligned}
T_{2}^{\beta}(\ell_{b}) \ = \ \ell_{b}\end{aligned}$$ where $\beta \leq 2^{2v}$. Thus, logical operators defined inside $P(1,n_{2},n_{3})$ consist of two-dimensional logical operators and one-dimensional logical operators defined inside $P(1,2v,n_{3})$.
Let us analyze a one-dimensional logical operator $\ell$ defined inside $P(1,2v,n_{3})$ first. We decompose $r$ defined inside $P(n_{1},n_{2},1)$ as follows: $$\begin{aligned}
r \ \sim \ r_{a} r_{b}\end{aligned}$$ where $r_{a}$ is defined inside $P(n_{1},2v,1)$, and $r_{b}$ is defined inside $P(n_{1},n_{2},1)$ and periodic: $$\begin{aligned}
T_{2}^{\beta'}(r_{b}) \ = \ r_{b}\end{aligned}$$ where $\beta'\leq 2^{2v}$. Then, we notice that $$\begin{aligned}
[\ell,r_{a}] \ = \ 0\end{aligned}$$ since there exists a translation of $r_{a}$ which does not overlap with $\ell$. Thus, we have $$\begin{aligned}
\{\ell,r_{b}\} \ = \ 0.\end{aligned}$$
Let us consider the case where $n_{1}=1$, and $n_{3}$ is even. Note that $\ell$ and $r_{b}$ are both logical operators. Then, from corollary \[corollary\_2dim\], there exists a one-dimensional logical operator $r'$ which is defined inside $P(1,n_{2},1)$, anti-commutes with $\ell$ and is periodic in the $\hat{2}$ direction: $$\begin{aligned}
T_{2}(r') \ = \ r'.\end{aligned}$$ Then, we periodically extend $r'$ in the $\hat{1}$ direction and define $r''$. Then, we notice that $r''$ is defined inside $P(n_{1},n_{2},1)$ and periodic in the $\hat{1}$ and $\hat{2}$ directions. From this discussion, we obtain the following observation.
- For a one-dimensional logical operator $\ell$ defined inside $P(1,2v,n_{3})$, there exists a two-dimensional logical operator $r$ defined inside $P(n_{1},n_{2},1)$ which is periodic: $$\begin{aligned}
T_{1}(r) \ = \ T_{2}(r) \ = \ r\end{aligned}$$ and anti-commutes with $\ell$: $\{\ell,r\}=0$. $\ell$ can be also periodic: $$\begin{aligned}
T_{3}(\ell) \ = \ \ell,\end{aligned}$$ and, $\ell$ and $r$ are logical operators regardless of the system size $\vec{n}$.
Finally, let us analyze a two-dimensional logical operator $\ell$ defined inside $P(1,n_{2},n_{3})$ with $$\begin{aligned}
T_{2}^{\beta}(\ell) \ = \ \ell. \end{aligned}$$ Since $r$ is periodic in the $\hat{1}$ direction, one can decompose it as follows: $$\begin{aligned}
r \ \sim \ r_{a} r_{b}, \qquad T_{1}(r_{a}) \ = \ r_{a} \quad \mbox{and} \quad T_{1}(r_{b}) \ = \ r_{b}\end{aligned}$$ where $r_{a}$ is defined inside $P(n_{1},2v,1)$ and $r_{b}$ are defined inside $P(n_{1},n_{2},1)$ and periodic: $$\begin{aligned}
T_{2}^{\beta'}(r_{b}) \ = \ r_{b}\end{aligned}$$ where $\beta' \leq 2^{2v}$. Then, one may notice that $$\begin{aligned}
\{ \ell, r_{b} \} \ = \ 0.\end{aligned}$$ Then, the rest is immediate, and we obtain the following observation.
- For a two-dimensional logical operator $\ell$ defined inside $P(1,n_{2},n_{3})$, there exists a one-dimensional logical operator $r$ defined inside $P(n_{1},2v,1)$ which is periodic: $$\begin{aligned}
T_{1}(r) \ = \ r\end{aligned}$$ and anti-commutes with $\ell$: $\{\ell,r\}=0$. $\ell$ can be also periodic: $$\begin{aligned}
T_{2}(\ell) \ = \ T_{3}(\ell) \ = \ \ell,\end{aligned}$$ and, $\ell$ and $r$ are logical operators regardless of the system size $\vec{n}$.
Let us recall the discussion so far. We started our analysis with the system size considered in lemma \[lemma\_decomposition\_1dim\]. Then, for each pair of anti-commuting logical operators, we found logical operators whose geometric shapes are the same as the ones described as in theorem \[theorem\_3dim\]. These logical operators are also logical operators for other system sizes because of their periodic structures and scale symmetries of the system.
It remains to sort these logical operators in a canonical form by analyzing their commutation relations. However, we shall skip this process since it is straightforward. See [@Beni10b] for a similar discussion. This completes the proof of theorem \[theorem\_3dim\].
Acknowledgments {#acknowledgments .unnumbered}
===============
I thank Eddie Farhi and Peter Shor for support at MIT. I thank Sergey Bravyi, Jeongwan Haah, Isaac Kim, Zhenghang Wang and Sam Ocko for stimulating and fruitful discussion. I thank Masahito Ueda and Spiros Michalakis for valuable comments. I gratefully acknowledge the hospitality of Caltech IQI where I enjoyed interesting discussion with group members, and particularly thank John Preskill. This work is supported by the U.S. Department of Energy under cooperative research agreement Contract Number DE-FG02-05ER41360, and by the Nakajima Foundation.
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[^1]: To the best of our knowledge, there have been no convincing experimental demonstration of error-corrections. All the demonstrations are limited only to phase-type errors, and do not correct bit-type errors. Also, it seems very difficult to reach the frequency threshold for reliable storage of logical qubits [@Dennis02]. Finally, it is extremely inefficient to keep performing error-corrections during the time one is storing a qubit if one stores a qubit for a long duration such as months or years.
[^2]: We shall discuss these proposals in detail in section \[sec:result1\].
[^3]: The definition of topological order at finite temperature will be clearly stated in section \[sec:topo\]
[^4]: Precisely speaking, the system does not return to the original state, but return to a state which is sufficiently close to the original state with a probability approaching to unity at the thermodynamic limit. Therefore, one can reliably read out the encoded bit from such a state.
[^5]: Strictly speaking, this definition is not complete. See [@Bravyi10b] for a complete definition.
[^6]: The definition used here is slightly different from the one used in [@Nussinov09] where the existence of the energy gap is not required. The difference comes apparent whether a two-dimensional Bacon’s subsystem code [@Bacon06] is classified as topologically ordered system or not. The ground state space of this subsystem code has the same reduced density matrix for any zero-dimensional regions. However, it is not likely to have a finite energy gap since it is highly frustrated, and thus, its coding properties are likely to be unstable against perturbations.
[^7]: Mathematically inclined readers may want to have more precise definitions of geometric shapes and continuous deformations. Here, we make some comments on this issue to make our discussion more rigorous. For a $D$-torus with $n_{1}\times \cdots \times n_{D}$ composite particles, we split the entire system into $n_{1}\times \cdots \times n_{D}$ hypercubic regions where each hypercube contains each composite particle, and cover the entire system completely, but without any overlap. For a given set of composite particles $R$, we define its geometric shape as a union of hypercubes which contain all the composite particles inside $R$. After assigning geometric shapes to sets of composite particles in this way, one can define a continuous map between two sets $R$ and $R'$, and introduce an equivalence relation between them in a straightforward way.
[^8]: The definition used here is similar to the one used in [@Nussinov09] except for taking the limit of $V \rightarrow 0$. This is because the definition in [@Nussinov09] is applicable to systems with gapless energy spectrum, while our definition is applicable only to systems with a finite energy gap.
[^9]: While expectation values of logical operators cannot be used as topological order parameters, geometric shapes of dressed logical operators [@Beni10b], which are operators characterizing the transformations inside the ground space of the perturbed Hamiltonians [@Hastings05], may be used. Since continuous deformations between topologically distinct logical operators are not allowed, geometric shapes of dressed logical operators must undergo some discontinuous change during the transition. Thus, some non-analyticities will be induced in ground states during quantum phase transitions between Hamiltonians with topologically distinct logical operators.
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---
abstract: 'We introduce and study the space ${\mathcal S{\mbox{Curr}}({F_N})}$ of *subset currents* on the free group $F_N$, and, more generally, on a word-hyperbolic group. A subset current on $F_N$ is a positive $F_N$-invariant locally finite Borel measure on the space $\mathfrak C_N$ of all closed subsets of $\partial F_N$ consisting of at least two points. The well-studied space ${\mbox{Curr}}(F_N)$ of geodesics currents, positive $F_N$-invariant locally finite Borel measures defined on pairs of different boundary points, is contained in the space of subset currents as a closed $\Bbb R$-linear ${\mbox{Out}}(F_N)$-invariant subspace. Much of the theory of ${\mbox{Curr}}(F_N)$ naturally extends to the ${\mathcal S{\mbox{Curr}}({F_N})}$ context, but new dynamical, geometric and algebraic features also arise there. While geodesic currents generalize conjugacy classes of nontrivial group elements, a subset current is a measure-theoretic generalization of the conjugacy class of a nontrivial finitely generated subgroup in $F_N$. If a free basis $A$ is fixed in $F_N$, subset currents may be viewed as $F_N$-invariant measures on a branching analog of the geodesic flow space for $F_N$, whose elements are infinite subtrees (rather than just geodesic lines) of the Cayley graph of $F_N$ with respect to $A$. Similarly to the case of geodesics currents, there is a continuous ${\mbox{Out}}(F_N)$-invariant co-volume form between the Outer space ${\mbox{cv}_N}$ and the space ${\mathcal S{\mbox{Curr}}({F_N})}$ of subset currents. Given a tree $T\in {\mbox{cv}_N}$ and the counting current $\eta_H\in {\mathcal S{\mbox{Curr}}({F_N})}$ corresponding to a finitely generated nontrivial subgroup $H\le F_N$, the value $\langle T, \eta_H\rangle$ of this intersection form turns out to be equal to the co-volume of $H$, that is the volume of the metric graph $T_H/H$, where $T_H\subseteq T$ is the unique minimal $H$-invariant subtree of $T$. However, unlike in the case of geodesic currents, the co-volume form ${\mbox{cv}_N}\times{\mathcal S{\mbox{Curr}}({F_N})}\to [0,\infty)$ does not extend to a continuous map ${\overline{\mbox{cv}}_N}\times{\mathcal S{\mbox{Curr}}({F_N})}\to [0,\infty)$.'
address:
- 'Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL 61801, USA http://www.math.uiuc.edu/\~kapovich/'
- 'Section de mathématiques, Université de Genève, 2-4, rue du Lièvre, c.p. 64, 1211 Genève, Switzerland http://www.unige.ch/math/folks/nagnibeda'
author:
- Ilya Kapovich
- Tatiana Nagnibeda
title: Subset currents on free groups
---
[^1]
Introduction {#sec:intro}
============
Geodesic currents were introduced in the context of hyperbolic surfaces by Bonahon in the papers [@Bo86; @Bo88] where they were used to study the Teichmüller space and mapping class groups, with various applications to 3-manifolds. In the context of free groups, geodesic currents were first investigated by Reiner Martin in his 1995 PhD thesis [@Martin], but have only become the object of systematic study in the last five years, leading to a number of interesting recent applications and developments.
A geodesic current may be thought of as a measure-theoretic analog of the notion of a conjugacy class in the group, or of a free homotopy class of a closed curve in the surface. More formally, a *geodesic current* on a free group $F_N$ is a positive Borel measure on ${\partial^2}F_N:=\{(x,y)\in {\partial}F_N\times {\partial}F_N : x\ne y\}$ which is locally finite (i.e., finite on compact subsets), $F_N$-invariant and invariant with respect to the flip map ${\partial^2}F_N\to {\partial^2}F_N$, $(x,y)\mapsto (y,x)$. Equivalently, it is a positive Borel locally finite $F_N$-invariant measure on the space of 2-element subsets of $\partial F_N$.
This paper is devoted to the study of a natural generalization of the space of geodesic currents obtained by replacing the space of 2-element subsets of $\partial F_N$ by the space $\mathfrak C_N$ of all closed subsets $S\subseteq \partial F_N$ such that $S$ consists of at least two points. The space $\mathfrak C_N$ has a natural topology given by the Hausdorff distance for the subsets of $\partial F_N$, where $\partial F_N$ is endowed with the standard visual metric provided by any choice of basis in $F_N$ (this topology is independent of the choice of the basis and coincides with the Vietoris topology). Similarly to the Cantor set ${\partial}F_N$, the space $\mathfrak C_N$ is locally compact and totally disconnected. See Subsection \[subsec:subsets\] for details about the space $\mathfrak C_N$. A *subset current* on $F_N$ is a positive locally finite $F_N$-invariant Borel measure on $\mathfrak C_N$, and we consider the space ${\mathcal S{\mbox{Curr}}({F_N})}$ of all subset currents on $F_N$ equipped with the weak-$*$ topology and with the ${\mathbb R}_{\ge 0}$-linear structure.
This definition naturally extends to the case where $F_N$ is replaced by an arbitrary word-hyperbolic group, for details see Problem \[prob:surface\] below.
The space ${\mathcal S{\mbox{Curr}}({F_N})}$ of subset currents admits a natural ${\mbox{Out}}(F_N)$-action by ${\mathbb R}_{\ge 0}$-linear homeomorphisms, and the space of geodesic currents ${\mbox{Curr}}(F_N)$ is canonically embedded in ${\mathcal S{\mbox{Curr}}({F_N})}$ as a closed ${\mbox{Out}}(F_N)$-invariant ${\mathbb R}_{\ge 0}$-linear subspace. We show below that subset currents are measure-theoretic analogs of conjugacy classes of nontrivial finitely generated subgroups in $F_N$, with geodesic currents corresponding to cyclic subgroups, and we study this connection in detail.
The space ${\mbox{Curr}}(F_N)$ of all geodesic currents on $F_N$ is a natural counterpart for the Culler-Vogtmann Outer space ${\mbox{cv}_N}$, and the present paper explores deeper levels of this interaction in a more general context of subset currents. The Outer space ${\mbox{cv}_N}$, introduced in [@CV], is a free group cousin of the Teuchmüller space and consists of $F_N$-equivariant isometry classes of free minimal discrete isometric actions of $F_N$ on ${\mathbb R}$-trees. Points of ${\mbox{cv}_N}$ may also be thought of as *marked metric graph structures* on $F_N$, see Section \[sec:background\] below for details. The space ${\mbox{cv}_N}$ comes equipped with a natural ${\mbox{Out}}(F_N)$-action that factors through to the action on the projectivized Outer space ${\mbox{CV}_N}$ (whose points are homothety classes of elements of ${\mbox{cv}_N}$). The closure ${\overline{\mbox{cv}}_N}$ of ${\mbox{cv}_N}$ with respect to the equivariant Gromov-Hausdorff convergence topology is an important object in the study of dynamics of ${\mbox{Out}}(F_N)$ and is known to consist of $F_N$-equivariant isometry classes of all *very small* minimal isometric actions of ${\mbox{Out}}(F_N)$ on ${\mathbb R}$-trees. The projectivization ${\overline{\mbox{CV}}_N}$ of ${\overline{\mbox{cv}}_N}$ is a compact finite-dimensional space analog to Thurston’s compactification of the Teichmüller space. Compared to the Teichmüller space, the geometry of the Outer space remains much less understood.
In this regard studying the interaction between the Outer space and the space of currents proved to be quite useful. This interaction is primarily given by the *geometric intersection form* or *length* pairing: in [@KL2] Kapovich and Lustig proved that there exists a unique continuous map $$\langle \, , \rangle: {\overline{\mbox{cv}}_N}\times {\mbox{Curr}}(F_N)\to {\mathbb R}_{\ge 0}$$ which is ${\mbox{Out}}(F_N)$-invariant, ${\mathbb R}_{>0}$-homogeneous with respect to the first argument, ${\mathbb R}_{\ge 0}$-linear with respect to the second argument, and has the property that for every nontrivial element $g\in F_N$ and every $T\in{\overline{\mbox{cv}}_N}$ one has $$\langle T, \eta_g\rangle =||g||_T.$$ Here $||g||_T=\inf_{x\in T} d_T(x,gx)$ is the *translation length* of $g$ with respect to $T$, and $\eta_g\in {\mbox{Curr}}(F_N)$ is the *counting current* associated to $g$ (see Subsection \[subsec:geodcurr\] for the definition). The set of scalar multiples of all counting currents is a dense subset of ${\mbox{Curr}}(F_N)$, which in particular justifies thinking about the notion of a current as generalizing that of a conjugacy class. This approach provided a number of useful recent applications to the study of the dynamics and geometry of ${\mbox{Out}}(F_N)$, such as the results of Kapovich and Lustig about various analogs of the curve complex in the free group case [@KL2]; a construction by Bestvina and Feighn, for a given finite collection of iwip elements in ${\mbox{Out}}(F_N)$, of a Gromov-hyperbolic graph with an isometric ${\mbox{Out}}(F_N)$-action, where these iwip automorphisms act as hyperbolic isometries; a result of Hamenstädt [@Ha] about lines of minima in Outer space; the work of Clay and Pettet [@CP] on realizability of an arbitrary matrix from $GL(N,{\mathbb Z})$ as the abelianization action of a hyperbolic iwip element of ${\mbox{Out}}(F_N)$, and others (see, for example, [@Ka3; @Ka4; @KL1; @KL2; @KL3; @KL4; @KL5; @CHL3; @Fra; @KN; @KN2; @CK]).
As noted above, in this paper we extend this framework in a way that allows us to study the dynamics of the action of ${\mbox{Out}}(F_N)$ on conjugacy classes of finitely generated subgroups of ${\mbox{Out}}(F_N)$ that are not necessarily cyclic. In the context of subset currents, we similarly define, given a finitely generated nontrivial subgroup $H\le F_N$, the *counting current* $\eta_H\in{\mathcal S{\mbox{Curr}}({F_N})}$. For the case where $H=\langle g\rangle\le F_N$ is infinite cyclic, we actually get $\eta_H=\eta_g$. For a finitely generated nontrivial $H\le F_N$, $\eta_H\in {\mathcal S{\mbox{Curr}}({F_N})}$ is defined (see Definition \[defn:count\]) as a sum of atomic measures on the limit set of $H$ and its conjugates. It is then shown (see Theorem \[thm:cc\]), that it can equivalently be understood in terms of the corresponding Stallings core graphs.
If a free basis $A$ is fixed in $F_N$, a convenient basis of topology is formed by cylinder sets. A subset cylinder $\mathcal SCyl_A(K)\subseteq \mathfrak C_N$ (see Definition \[defn:gcyl\] below) is determined by a finite non-degenerate subtree $K$ of the Cayley tree $X_A$ of $F_N$ with respect to $A$, and it consists of all those $S\in \mathfrak C_N$ such that the convex hull of $S$ in $X_A$ contains $K$ and such that every bi-infinite geodesic in $X_A$ with both endpoints in $S$ which intersects $K$ in a non-degenerate segment, enters and exits $K$ through vertices of degree 1 in $K$.
We develop the appropriate notion of an *occurrence* of a finite subtree $K\subseteq X_A$ in a Stallings core graph $\Delta$ (see Definition \[defn:occur\]). Namely, an occurrence of $K$ in $\Delta$ is a locally injective label-preserving morphism from $K$ into $\Delta$ which is a local homeomorphism at every point of $K$ except for the terminal vertices of leaves of $K$. Apart from being a useful tool in dealing with counting currents, the language of occurrences in core graphs produces an interesting model of non-linear (that is, not based on a segment of ${\mathbb Z}$) words, with a good notion of a subword (or factor) in such a word. Studying such models is an active subject of research in combinatorics of words (see, for example, [@ABFJ]).
As noted earlier, in the context of ${\mbox{Curr}}(F_N)$, a basic fact of the theory states that the set of all *rational currents*, that is of all ${\mathbb R}_{\ge 0}$-scalar multiples of the counting currents $\eta_g$, where $g\in F_N, g\ne 1$, is a dense subset of ${\mbox{Curr}}(F_N)$. Proofs of this fact are usually relying, in an essential way, on the commutative nature of the dynamical systems associated with ${\mbox{Curr}}(F_N)$. Thus, given a free basis $A$ of $F_N$, one can naturally view a geodesic current on $F_N$ as a positive shift-invariant Borel finite measure on the space of bi-infinite freely reduced words over $A^{\pm 1}$. Under this correspondence, the counting currents correspond exactly to the shift-invariant measures supported by periodic orbits of the shift map, and the density of the set of rational currents is then a consequence of classical results about density of periodic orbits. In the context of ${\mathcal S{\mbox{Curr}}({F_N})}$ the situation is considerably more complicated, since the symbolic dynamical systems corresponding to subset currents are no longer commutative in nature. Geometrically, a 2-element subset of ${\partial}F_N$ determines an infinite unparameterized geodesic line in the Cayley graph $X_A$ of $F_N$ with respect to a free basis $A$, so that a geodesic current may be thought of as an $F_N$-invariant measure on the space of unparameterized geodesic lines in $X_A$. Similarly, an element $S\in \mathfrak C_N$ determines an infinite subtree $Y\subseteq X_A$ without degree-one vertices, namely, the convex hull of $S$ in $X_A$. Thus, once $A$ is chosen, a subset current translates into a positive finite Borel measure on the space $\mathcal T_1(X_A)$ of infinite subtrees of $X_A$ containing $1\in F_N$ and without degree-one vertices. $F_N$-invariance of the current implies that the corresponding measure on $\mathcal T_1(X_A)$ is invariant with respect to the root change. The space $\mathcal T_1(X_A)$ may be viewed as a branching analog of the geodesic flow space for $X_A$, since its elements are infinite trees rather than lines.
Recent results of Bowen [@Bow03; @Bow09] and Elek [@Elek] about approximability of such measures on the space of rooted trees are applicable to our set-up and allow us to conclude in Theorem \[thm:dense\] that the set $\{r\eta_H| r\ge 0, H\le F_N \text{ is nontrivial and finitely generated}\}$ of *rational subset currents* is dense in ${\mathcal S{\mbox{Curr}}({F_N})}$.
The notion of a subset current is related to the study of invariant random subgroups. If $G$ is a locally compact group, an *invariant random subgroup* is a probability measure on the space $S(G)$ of all closed subgroups of $G$, such that this measure is invariant with respect to the conjugation action of $G$ on $S(G)$. The study of invariant random subgroups in various contexts goes back to the work of Stuck and Zimmer [@SZ94] and has recently become an active area of research, see for example [@AGV12; @Bow10; @Bow12; @DDMN; @D02; @Gr11; @Sa11; @Vershik; @Ve11]. If $G=F_N$ and $A$ is a free basis of $F_N$, one can view an invariant random subgroup on $F_N$ as a measure on the space of all rooted Stallings core graphs labelled by $A$, invariant with respect to root change. Note that the Stallings core graph corresponding to a nontrivial subgroup is never a tree, and, moreover, every edge in this graph is contained in some nontrivial immersed circuit. By contrast, as noted above and as we explain in greater detail in Subsection \[subsec:root\] below, a subset current on $F_N$ can be viewed as a measure on the space $\mathcal T_1(X_A)$ of infinite trees which is invariant with respect to root change. We plan to investigate deeper connections between these two notions in a future work.
In this paper we also construct (see Section \[sec:intform\]) a continuous ${\mbox{Out}}(F_N)$-invariant *co-volume form* $$\langle\, ,\, \rangle: {\mbox{cv}_N}\times{\mathcal S{\mbox{Curr}}({F_N})}\to {\mathbb R}_{\ge 0}.$$ It has properties similar to that of the geometric intersection form on geodesic currents, except that instead of translation length it computes the *co-volume*: for any $T\in{\mbox{cv}_N}$ and a nontrivial finitely generated subgroup $H\le F_N$, we have $\langle T, \eta_H\rangle= {\mbox{vol}}(H\setminus T_H)$ where $T_H$ is the convex hull of the limit set of $H$, and is also the unique minimal $H$-invariant subtree of $T$. For an infinite cyclic $H=\langle g\rangle\le F_N$ we have $||g||_T={\mbox{vol}}(H\setminus T_H)$ since in this case the quotient graph $H\setminus T_H$ is a circle. By contrast to the result of Kapovich and Lustig [@KL2] for ordinary geodesic currents, we prove (Theorem \[thm:disc\] below) that for $N\ge 3$ the co-volume form ${\mbox{cv}_N}\times{\mathcal S{\mbox{Curr}}({F_N})}\to {\mathbb R}_{\ge 0}$ does not extend to a continuous map ${\overline{\mbox{cv}}_N}\times{\mathcal S{\mbox{Curr}}({F_N})}\to {\mathbb R}_{\ge 0}$.
For a finitely generated subgroup $H\le F_N$, the *reduced rank*, ${\overline{\rm rk}}(H)$, is defined as $\max\{{\rm rk}(H)-1,0\}$, where ${\rm rk}(H)$ is the cardinality of a free basis of $H$. It turns out that reduced rank uniquely extends to a continuous ${\mbox{Out}}(F_N)$-invariant ${\mathbb R}_{\ge 0}$-linear functional ${\overline{\rm rk}}: {\mathcal S{\mbox{Curr}}({F_N})}\to {\mathbb R}_{\ge 0}$, such that for every nontrivial finitely generated $H\le F_N$ we have ${\overline{\rm rk}}(\eta_H)={\overline{\rm rk}}(H)$. As we note in Section \[sec:problems\], there are likely deeper connections arising here with the study of intersections of finitely generated subgroups of free groups.
More open problems regarding subset currents are formulated in Section \[sec:problems\]. We believe that subset currents exhibit considerably more interesting and varied geometric and dynamical behavior than geodesic currents, and can provide new and interesting information about ${\mbox{Out}}(F_N)$. Indeed, we hope this paper to serve as a starting point for investigating the space of subset currents in its different aspects, such as ${\mbox{Out}}(F_N)$-related questions, connections with invariant random subgroups, connections to the study of ergodic properties of subgroup actions on $\partial F_N$ and their Schreier graphs (see [@GKN]), etc.
We are very grateful to Lewis Bowen for illuminating and helpful conversations and to the referee for useful comments. The first author also thanks Patrick Reynolds for pointing out that for a fixed finitely generated subgroup $H\le F_N$, the co-volume function ${\overline{\mbox{cv}}_N}\to {\mathbb R}_{\ge 0}, T\mapsto ||H||_T$, is not continuous on ${\overline{\mbox{cv}}_N}$.
Outer space and the space of geodesic currents {#sec:background}
==============================================
We give here only a brief overview of basic facts related to Outer space and the space of geodesic currents. We refer the reader to [@CV; @Ka2] for more detailed background information.
Conventions regarding graphs {#subsec:conv}
----------------------------
A *graph* is a 1-complex. The set of $0$-cells of a graph $\Delta$ is denoted $V\Delta$ and its elements are called *vertices* of $\Delta$. The closed 1-cells of a graph $\Delta$ are called *topological edges* of $\Delta$. The set of all topological edges is denoted $E_{top}\Delta$. We will sometimes call the open 1-cells of $\Delta$ *open edges* of $\Delta$. The interior of every topological edge is homeomorphic to the interval $(0,1)\subseteq \mathbb R$ and thus admits exactly 2 orientations (when considered as a 1-manifold). We call a topological edge endowed with the choice of an orientation on its interior an *oriented edge* of $\Delta$. The set of all oriented edges of $\Delta$ is denoted $E\Delta$. For an oriented edge $e\in E\Delta$ changing its orientation to the opposite produces another oriented edge of $\Delta$ denoted $e^{-1}$ and called the *inverse* of $e$. Thus ${}^{-1}:E\Delta\to E\Delta$ is a fixed-point-free involution. For every oriented edge $e$ of $\Delta$ there are naturally defined (and not necessarily distinct) vertices $o(e)\in V\Delta$, called the *origin* of $e$, and $t(e)\in V\Delta$, called the *terminus* of $e$, satisfying $o(e^{-1})=t(e)$, $t(e^{-1})=o(e)$. An *orientation* on a graph $\Delta$ is a partition $E\Delta=E^+\Delta\sqcup E^-\Delta$, where for every $e\in E\Delta$ one of the edges $e, e^{-1}$ belongs to $E^+\Delta$ and the other edge belongs to $E^-\Delta$.
An *edge-path* $\gamma$ of *simplicial length* $|\gamma|=n\ge 1$ in $\Delta$ is a sequence of oriented edges $$\gamma=e_1,\dots, e_n$$ such that $t(e_i)=o(e_{i+1})$ for $i=1,\dots, n-1$. We say that $o(\gamma):=o(e_1)$ is the *origin* of $\gamma$ and that $t(\gamma)=t(e_n)$ is the *terminus* of $\gamma$. An edge-path is called *reduced* if it does not contain a back-tracking, that is a sub-path of the form $ee^{-1}$, where $e\in E\Delta$.
To every edge-path $\gamma=e_1,\dots, e_n$ in $\Delta$ there is a naturally associated continuous map $\widehat\gamma:[0,n]\to \Delta$ with $\widehat\gamma(0)=o(\gamma)$ and $\widehat\gamma(n)=t(\gamma)$. A *graph morphism* $f : \Delta \to \Delta'$ is a continuous map from a graph $\Delta$ to a graph $\Delta'$ that maps vertices of $\Delta$ to vertices of $\Delta'$ and such that each open edge of $\Delta$ is mapped homeomorphically to an open edge of $\Delta'$. Thus $f$ induces natural maps $f: E\Delta\to E\Delta'$ and $f:V\Delta\to
V\Delta'$ such that $f(e^{-1})=\left(f(e))\right)^{-1}$, $o(f(e))=f(o(e))$ and $t(f(e))=f(t(e))$ for every $e\in E\Delta$.
Markings {#subsec:mark}
--------
Let $N\ge 2$. We fix a free basis $A=\{a_1,\dots, a_N\}$ of $F_N$. We also let $R_N$ to denote the wedge of $N$ loop-edges at a vertex $x_0$. We identify $F_N$ with $\pi_1(R_N, x_0)$ by mapping each $a_i\in B$ to one of the loop edges in $R_N$. We fix this identification $F_N=\pi_1(R_N,x_0)$ for the remainder of the paper. The graph $R_N$ will be referred to as the *standard $N$-rose*. A *marking* on $F_N$ is an isomorphism $\alpha:F_N{\mathrel{\text{ \setbox0\hbox{$\rightarrow$} \rlap{\hbox to \wd0{\hss\raisebox{0.9\height}{$\sim$}\hss}}\box0
}}}\pi_1(\Gamma)$, where $\Gamma$ is a finite connected graph without degree-1 vertices.
To each marking $\alpha$ we also associate a continuous map $\widehat\alpha: R_N\to \Gamma$ such that $\widehat \alpha$ is a homotopy equivalence and such that $\widehat\alpha_\#=\alpha$. Two markings $\alpha_1:F_N{\mathrel{\text{ \setbox0\hbox{$\rightarrow$} \rlap{\hbox to \wd0{\hss\raisebox{0.9\height}{$\sim$}\hss}}\box0
}}}\pi_1(\Gamma_1)$ and $\alpha_2:F_N{\mathrel{\text{ \setbox0\hbox{$\rightarrow$} \rlap{\hbox to \wd0{\hss\raisebox{0.9\height}{$\sim$}\hss}}\box0
}}}\pi_1(\Gamma_2)$ are *equivalent* if there exists a graph isomorphism $j:\Gamma_1\to\Gamma_2$ such that for the associated continuous maps $\widehat\alpha_{1}:R_N\to\Gamma_1$ and $\widehat\alpha_{2}:R_N\to \Gamma_2$, the maps $j\circ \widehat\alpha_{1}$ and $\widehat\alpha_{2}$ are freely homotopic. We will usually only be interested in the equivalence class of a marking, and for that reason an explicit mention of a base-point in the graph $\Gamma$ in the definition of a marking will almost always be omitted.
If $\alpha:F_N{\mathrel{\text{ \setbox0\hbox{$\rightarrow$} \rlap{\hbox to \wd0{\hss\raisebox{0.9\height}{$\sim$}\hss}}\box0
}}}\pi_1(\Gamma)$ is a marking, then $\alpha$ defines an $F_N$-equivariant quasi-isometry between $F_N$ and $\widetilde\Gamma$ (endowed with the simplicial metric, giving every edge of $\widetilde\Gamma$ length $1$). This quasi-isometry induces an $F_N$-equivariant homeomorphism $\partial F_N\to\partial
\widetilde\Gamma$. When talking about markings, we will usually implicitly assume that $\partial
F_N$ is identified with $\partial\widetilde\Gamma$ via this homeomorphism and write $\partial F_N=\partial\widetilde\Gamma$.
Each marking $\alpha:F_N{\mathrel{\text{ \setbox0\hbox{$\rightarrow$} \rlap{\hbox to \wd0{\hss\raisebox{0.9\height}{$\sim$}\hss}}\box0
}}}\pi_1(\Gamma)$ provides a Hausdorff metric $d_{\alpha}$ on $\partial F_N$ as follows. For $\xi,\zeta\in \partial
F_N$, put $d_\alpha(\xi,\zeta)=\frac{1}{2^M}$ where $M$ is the length of the maximal common initial segment of the geodesic rays $[x_0,\xi)$ and $[x_0,\zeta)$ in $\widetilde \Gamma$.
A *metric graph structure* on a graph $\Delta$ is a function $\mathcal L:E\Delta\to\ (0,\infty)$ such that $\mathcal L(e)=\mathcal
L(e^{-1})$ for every $e\in E\Delta$. Equivalently, we may think of a metric graph structure on $\Delta$ as a function $\mathcal L: E_{top}\Delta\to (0,\infty)$. For an edge-path $\gamma=e_1,\dots, e_n$ in $\Delta$ the *$\mathcal L$-length* of $\gamma$ is $\mathcal L(\gamma):=\sum_{i=1}^n \mathcal L(e)$.
A *marked metric graph structure* on $F_N$ is a pair $(\alpha,
\mathcal L)$ where $\alpha:F_N{\mathrel{\text{ \setbox0\hbox{$\rightarrow$} \rlap{\hbox to \wd0{\hss\raisebox{0.9\height}{$\sim$}\hss}}\box0
}}}\pi_1(\Gamma)$ is a marking on $F_N$ and $\mathcal L$ is a metric graph structure on $\Gamma$. Two marked metric graph structures $(\alpha_1:F_N{\mathrel{\text{ \setbox0\hbox{$\rightarrow$} \rlap{\hbox to \wd0{\hss\raisebox{0.9\height}{$\sim$}\hss}}\box0
}}}\pi_1(\Gamma_1),
\mathcal L_1)$ and $(\alpha_2:F_N{\mathrel{\text{ \setbox0\hbox{$\rightarrow$} \rlap{\hbox to \wd0{\hss\raisebox{0.9\height}{$\sim$}\hss}}\box0
}}}\pi_1(\Gamma_2), \mathcal L_2)$ are *equivalent* if there exists a graph isomorphism $j:\Gamma_1\to\Gamma_2$ such that $j: (\Gamma_1, \mathcal L_1)\to
(\Gamma_2, \mathcal L_2)$ is an isometry and such that for the associated continuous maps $\widehat\alpha_{1}:R_N\to\Gamma_1$ and $\widehat\alpha_{2}:R_N\to \Gamma_2$, the maps $j\circ \widehat\alpha_{1}$ and $\widehat\alpha_{2}$ are freely homotopic. Note that if $(\alpha_1:F_N{\mathrel{\text{ \setbox0\hbox{$\rightarrow$} \rlap{\hbox to \wd0{\hss\raisebox{0.9\height}{$\sim$}\hss}}\box0
}}}\pi_1(\Gamma_1),
\mathcal L_1)$ and $(\alpha_2:F_N{\mathrel{\text{ \setbox0\hbox{$\rightarrow$} \rlap{\hbox to \wd0{\hss\raisebox{0.9\height}{$\sim$}\hss}}\box0
}}}\pi_1(\Gamma_2), \mathcal L_2)$ are equivalent marked metric graph structures on $F_N$ then the markings $\alpha_1,\alpha_2$ are equivalent.
Outer space {#subsec:outer}
-----------
Let $N\ge 2$. The *Outer space* ${\mbox{cv}_N}$ consists of all minimal free and discrete isometric actions on $F_N$ on $\mathbb R$-trees (where two such actions are considered equal if there exists an $F_N$-equivariant isometry between the corresponding trees). There are several different topologies on ${\mbox{cv}_N}$ that are known to coincide, in particular the equivariant Gromov-Hausdorff convergence topology and the so-called *length function* topology.
Every $T\in {\mbox{cv}_N}$ is uniquely determined by its *translation length function* $||.||_T:F_N\to\mathbb R$, where $||g||_T$ is the translation length of $g$ on $T$. Two trees $T_1,T_2\in {\mbox{cv}_N}$ are close if the functions $||.||_{T_1}$ and $||.||_{T_1}$ are close point-wise on a large ball in $F_N$. The closure ${\overline{\mbox{cv}}_N}$ of ${\mbox{cv}_N}$ in either of these two topologies is well-understood and known to consist precisely of all the so-called *very small* minimal isometric actions of $F_N$ on $\mathbb R$-trees, see [@BF93] and [@CL].
The automorphism group ${\mbox{Aut}}(F_N)$ has a natural continuous *right* action on ${\overline{\mbox{cv}}_N}$ (that leaves ${\mbox{cv}_N}$ invariant) defined as follows: for $T\in {\mbox{cv}_N}$ and $\phi\in {\mbox{Aut}}(F_N)$ we have $||g||_{T\phi}=||\phi(g)||_T$, where $g\in F_N$. In terms of tree actions, $T\phi$ is equal to $T$ as a metric space, but the action of $F_N$ is modified as: $g\underset{T\phi}{\cdot}
x=\phi(g)\underset{T}{\cdot} x$ where $x\in T$, $g\in F_N$. It is not hard to see that the subgroup ${\mbox{Inn}}(F_N)\le {\mbox{Aut}}(F_N)$ of inner automorphisms is contained in the kernel of the action of ${\mbox{Aut}}(F_N)$ on ${\overline{\mbox{cv}}_N}$. Hence this action quotients through to the action of ${\mbox{Out}}(F_N)$ on ${\overline{\mbox{cv}}_N}$, where ${\mbox{cv}_N}\subseteq{\overline{\mbox{cv}}_N}$ is an ${\mbox{Out}}(F_N)$-invariant dense subset. The right action of ${\mbox{Out}}(F_N)$ on ${\overline{\mbox{cv}}_N}$ can be converted into a left action as follows: for $T\in {\overline{\mbox{cv}}_N}$ and $\phi\in {\mbox{Out}}(F_N)$ put $\phi T:=T\phi^{-1}$.
The *projectivized Outer space* ${\mbox{CV}_N}=\mathbb P{\mbox{cv}_N}$ is defined as the quotient ${\mbox{cv}_N}/\sim$ where for $T_1\sim T_2$ whenever $T_2=cT_1$ in ${\mbox{cv}_N}$ for some $c>0$. One similarly defines the projectivization ${\overline{\mbox{CV}}_N}=\mathbb P{\overline{\mbox{cv}}_N}$ of ${\overline{\mbox{cv}}_N}$ as ${\overline{\mbox{cv}}_N}/\sim$ where $\sim$ is the same as above. The space ${\overline{\mbox{CV}}_N}$ is compact and contains ${\mbox{CV}_N}$ as a dense ${\mbox{Out}}(F_N)$-invariant subset. The compactification ${\overline{\mbox{CV}}_N}$ of ${\mbox{CV}_N}$ is a free group analog of the Thurston compactification of the Teichmüller space. For $T\in {\overline{\mbox{cv}}_N}$ its $\sim$-equivalence class is denoted by $[T]$, so that $[T]$ is the image of $T$ in ${\overline{\mbox{CV}}_N}$.
Every marked metric graph structure $(\alpha:F_N{\mathrel{\text{ \setbox0\hbox{$\rightarrow$} \rlap{\hbox to \wd0{\hss\raisebox{0.9\height}{$\sim$}\hss}}\box0
}}}\pi_1(\Gamma),
\mathcal L)$ defines a point in ${\mbox{cv}_N}$ as follows. Consider the universal covering tree $X=\widetilde\Gamma$ and let $d_\mathcal L$ be the metric on $X$ obtained by giving every edge of $\widetilde\Gamma$ the same length as the $\mathcal L$-length of its projection in $\Gamma$. Then $X$ is an $\mathbb R$-tree, and the action of $F_N$ on $X$ via $\alpha$ by covering transformations is a free minimal discrete isometric action on $(X,d_\mathcal L)$. Thus $(X,d_\mathcal L)$, equipped with this action of $F_N$, is a point of ${\mbox{cv}_N}$. It is well-known that for two marked metric graph structures $(\alpha_1:F_N{\mathrel{\text{ \setbox0\hbox{$\rightarrow$} \rlap{\hbox to \wd0{\hss\raisebox{0.9\height}{$\sim$}\hss}}\box0
}}}\pi_1(\Gamma_1),
\mathcal L_1)$ and $(\alpha_2:F_N{\mathrel{\text{ \setbox0\hbox{$\rightarrow$} \rlap{\hbox to \wd0{\hss\raisebox{0.9\height}{$\sim$}\hss}}\box0
}}}\pi_1(\Gamma_2),
\mathcal L_2)$ we have $(\widetilde\Gamma_1,d_{\mathcal
L_1})_{\alpha_1}=(\widetilde\Gamma_2,d_{\mathcal L_2})_{\alpha_2}$ in ${\mbox{cv}_N}$ if and only if $(\alpha_1,\mathcal L_1)$ is equivalent to $(\alpha_2,\mathcal L_2)$. Moreover, every point of ${\mbox{cv}_N}$ comes from some marked metric graph structure on $F_N$. Namely, if $T\in {\mbox{cv}_N}$, take $\Gamma=F_N\setminus
T$ and endow the edges of $\Gamma$ with the same lengths as their lifts in $T$. Since the action of $F_N$ on $T$ is free and discrete, there is a natural identification $F_N$ with $\pi_1(\Gamma)$, giving us a marking $\alpha:F_N{\mathrel{\text{ \setbox0\hbox{$\rightarrow$} \rlap{\hbox to \wd0{\hss\raisebox{0.9\height}{$\sim$}\hss}}\box0
}}}\pi_1(\Gamma)$ on $F_N$. This yields a marked metric graph structure $(\alpha:F_N{\mathrel{\text{ \setbox0\hbox{$\rightarrow$} \rlap{\hbox to \wd0{\hss\raisebox{0.9\height}{$\sim$}\hss}}\box0
}}}\pi_1(\Gamma),
\mathcal L)$ such that $T= (\widetilde\Gamma,d_{\mathcal
L})_{\alpha}$ in ${\mbox{cv}_N}$.
Geodesic currents {#subsec:geodcurr}
-----------------
Let $\partial^2 F_N:=\{ (x,y)| x,y\in \partial F_N, x\ne y\}$. The action of $F_N$ by translations on its hyperbolic boundary $\partial F_N$ defines a natural diagonal action of $F_N$ on $\partial^2 F_N$. A *geodesic current* on $F_N$ is a positive Borel measure on $\partial^2 F_N$, which is locally finite (that is finite on all compact subsets), $F_N$-invariant and is also invariant under the flip map $\partial^2 F_N\to \partial^2 F_N$, $(x,y)\mapsto (y,x)$.
The space ${\mbox{Curr}}(F_N)$ of all geodesic currents on $F_N$ has a natural $\mathbb R_{\ge 0}$-linear structure and is equipped with the weak-\* topology of point-wise convergence on continuous functions. Any choice of a marking on $F_N$ allows one to think about geodesic currents as systems of nonnegative weights satisfying certain Kirchhoff-type equations; see [@Ka2] for details. We briefly recall this construction for the case where $X_A\in {\mbox{cv}_N}$ is the Cayley tree corresponding to a free basis $A$ of $F_N$. For a non-degenerate geodesic segment $\gamma=[p,q]$ in $X_A$ the *two-sided cylinder* $Cyl_A(\gamma)\subseteq \partial^2 F_N$ consists of all $(x,y)\in \partial^2 F_N$ such that the geodesic from $x$ to $y$ in $X_A$ passes through $\gamma=[p,q]$. Given a nontrivial freely reduced word $v\in F(A)=F_N$ and a current $\mu\in {\mbox{Curr}}(F_N)$, the weight $(v;\mu)_A$ is defined as $\mu(Cyl_A(\gamma))$ where $\gamma$ is any segment in the Cayley graph $X_A$ labelled by $v$ (the fact that the measure $\mu$ is $F_N$-invariant implies that a particular choice of $\gamma$ does not matter). A current $\mu$ is uniquely determined by a family of weights $\big((v;\mu)_A\big)_{v\in F_N-\{1\}}$. The weak-\* topology on ${\mbox{Curr}}(F_N)$ corresponds to point-wise convergence of the weights for every $v\in F_N, v\ne 1$.
There is a natural left action of ${\mbox{Out}}(F_N)$ on ${\mbox{Curr}}(F_N)$ by continuous linear transformations. Specifically, let $\mu\in Curr(F_N)$, $\phi\in {\mbox{Out}}(F_N)$ and let $\Phi\in Aut(F_N)$ be a representative of $\phi$ in $Aut(F_N)$. Since $\Phi$ is a quasi-isometry of $F_N$, it extends to a homeomorphism of $\partial F_N$ and, diagonally, defines a homeomorphism of $\partial^2 F_N$. The measure $\phi\mu$ on $\partial^2 F_N$ is defined as follows. For a Borel subset $S\subseteq \partial^2 F_N$ we have $(\phi\mu)(S):=\mu(\Phi^{-1}(S))$. One then checks that $\phi\mu$ is a current and that it does not depend on the choice of a representative $\Phi$ of $\phi$.
The *space of projectivized geodesic currents* is defined as $\mathbb P{\mbox{Curr}}(F_N)={\mbox{Curr}}(F_N)-\{0\}/\sim$ where $\mu_1\sim\mu_2$ whenever there exists $c>0$ such that $\mu_2=c\mu_1$. The $\sim$-equivalence class of $\mu\in {\mbox{Curr}}(F_N)-\{0\}$ is denoted by $[\mu]$. The action of ${\mbox{Out}}(F_N)$ on ${\mbox{Curr}}(F_N)$ descends to a continuous action of ${\mbox{Out}}(F_N)$ on $\mathbb P{\mbox{Curr}}(F_N)$. The space $\mathbb PCurr(F_N)$ is compact.
For every $g\in F_N, g\ne 1$ there is an associated *counting current* $\eta_g\in Curr(F_N)$. If $A$ is a free basis of $F_N$ and the conjugacy class $[g]$ of $g$ is realized by a cyclic word $W$ (that is a cyclically reduced word in $F(A)$ written on a circle with no specified base-vertex), then for every nontrivial freely reduced word $v\in F(A)=F_N$ the weight $( v;\eta_g)_A$ is equal to the total number of occurrences of $v^{\pm 1}$ in $W$ (where an occurrence of $v$ in $W$ is a vertex on $W$ such that we can read $v$ in $W$ clockwise without going off the circle). We refer the reader to [@Ka2] for a detailed exposition on the topic. By construction, the counting current $\eta_g$ depends only on the conjugacy class $[g]$ of $g$ and it also satisfies $\eta_g=\eta_{g^{-1}}$. One can check [@Ka2] that for $\phi\in {\mbox{Out}}(F_N)$ and $g\in F_N, g\ne 1$ we have $\phi\eta_g=\eta_{\phi(g)}$. Scalar multiples $c\eta_g\in {\mbox{Curr}}(F_N)$, where $c\ge 0$, $g\in F_N, g\ne 1$ are called *rational currents*. A key fact about ${\mbox{Curr}}(F_N)$ states that the set of all rational currents is dense in ${\mbox{Curr}}(F_N)$. The set $\{[\eta_g\: g\in F_N, g\ne 1\}$ is dense in $\mathbb P{\mbox{Curr}}(F_N)$.
Intersection form {#subsec:intersection}
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In [@KL2] Kapovich and Lustig constructed a natural geometric *intersection form* that pairs trees and currents:
\[prop:int\][@KL2] Let $N\ge 2$. There exists a unique continuous map $\langle , \rangle : {\overline{\mbox{cv}}_N}\times {\mbox{Curr}}(F_N)\to \mathbb R_{\ge 0}$ with the following properties:
1. We have $\langle T, c_1\mu_1+c_2\mu_2\rangle=c_1\langle T,\mu_1\rangle+c_2\langle T,\mu_2\rangle$ for any $T\in {\overline{\mbox{cv}}_N}$, $\mu_1,\mu_2\in {\mbox{Curr}}(F_N)$, $c_1,c_2\ge 0$.
2. We have $\langle cT, \mu\rangle=c\langle T,\mu\rangle$ for any $T\in {\overline{\mbox{cv}}_N}$, $\mu\in {\mbox{Curr}}(F_N)$ and $c\ge 0$.
3. We have $\langle T\phi,\mu\rangle=\langle T, \phi\mu\rangle$ for any $T\in {\overline{\mbox{cv}}_N}$, $\mu\in {\mbox{Curr}}(F_N)$ and $\phi\in {\mbox{Out}}(F_N)$.
4. We have $\langle T, \eta_g\rangle=||g||_T$ for any $T\in {\overline{\mbox{cv}}_N}$ and $g\in F_N, g\ne 1$.
The space of subset currents {#sec:subsetcurr}
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The space $\mathfrak C_N$ {#subsec:subsets}
-------------------------
Recall that for a Hausdorff topological space $Y$, the so-called *hyper space* $\mathcal H(Y)$ consists of all non-empty closed subsets of $Y$. The space $\mathcal H(Y)$ comes equipped with the *Vietoris topology*, which has the basis consisting of all sets of the form
$$\langle U_1,...,U_n\rangle = \{B\in \mathcal H(Y) | B\subset U_1\cup ... \cup U_n ; B\cap U_i \neq \emptyset , i=1,...,n\} ,$$
where $\{U_1,...,U_n\}$ is a family of open subsets in $Y$.
If $Y$ is a compact metrizable space, then the Vietoris topology coincides with the Hausdorff topology given by the Hausdorff distance between closed subsets of $Y$, and in this case $\mathcal H(Y)$ is also compact (see e.g. [@Encycl], Chapter b-6). If $Y$ is totally disconnected then $\mathcal H(Y)$ is also totally disconnected.
\[defn:SN\] Let $N\ge 3$. We denote by $\mathfrak C_N$ the set of all closed subsets $S\subseteq \partial F_N$ such that $\#S\ge 2$. Thus $\mathfrak C_N\subseteq \mathcal H({\partial}F_N)$ and we endow $\mathfrak C_N$ with the subspace topology inherited from the Vietoris topology on $\mathcal H({\partial}F_N)$.
In view of the above remarks, the topology on $\mathfrak C_N$ coincides with the Hausdorff topology given by the Hausdorff distance $D_\alpha$ on $\mathfrak C_N$ with respect to the metric $d_\alpha$ on $\partial F_N$ corresponding to any marking $\alpha$ of $F_N$ (see \[subsec:mark\]). It is also not difficult to check directly that the topology on $\mathfrak C_N$ defined by the metric $D_\alpha$ does not depend on the choice of the marking.
The definition also straightforwardly implies that $\mathfrak C_N$ is locally compact and totally disconnected.
The condition that $\#S\ge 2$ is an important non-degeneracy condition for setting up the notion of a subset current, analogous to the assumption that $\xi\ne \zeta$ for $(\xi,\zeta)\in \partial^2 F_N$ in the definition of a geodesic current.
The space $\mathfrak C_N$ that interests us in this paper, is the complement in $\mathcal H (\partial F_N)$ of the closed subspace consisting of $1$-element subsets $\{y\}$, $y\in\partial F_N.$
It will be useful in our study of the space ${\mathcal S{\mbox{Curr}}({F_N})}$ to understand the topology on $\mathfrak C_N$ more explicitly. For this reason we describe the construction of cylindrical subsets in $\mathfrak C_N$.
\[defn:gcyl\] Let $\alpha:F_N\to \pi_1(\Gamma)$ be a marking on $F_N$, and let $X=\widetilde \Gamma$ be the universal cover of $\Gamma$.
We give each edge of $\Gamma$ and of $X$ length 1, so that $X$ can also be considered an element of ${\mbox{cv}_N}$.
Let $K\subset X$ be a non-degenerate finite simplicial subtree of $X$, that is a finite simplicial subtree with at least two distinct vertices of degree 1. Let $e_1,\dots, e_n$ be all the terminal edges of $K$, that is oriented edges whose terminal vertices are precisely all the vertices of $K$ of degree 1. (Note that $n\ge 2$ by the assumption on $K$.) For each of the edges $e_i$ denote by $Cyl_X(e_i)\subseteq \partial F_N$ the homeomorphic image of the subset in $\partial X$ consisting of all equivalence classes of geodesic rays in $X$ beginning with the edge $e_i$.
Define the [*subset cylinder*]{} $\mathcal SCyl_\alpha(K)$ to be the set $\langle Cyl_X(e_1),...,Cyl_X(e_n)\rangle \subset \mathfrak C_N$. Thus for a closed subset $S\subseteq {\partial}F_N$ with $\#S\ge 2$ we have $S\in \mathcal SCyl_\alpha(K)$ if and only if the following hold:
1. The subset $S\subseteq \partial F_N$ is closed.
2. We have $$S\subseteq \cup_{i=1}^n Cyl_X(e_i).$$
3. For each $i=1,\dots, n$ we have $$S\cap Cyl_X(e_i)\ne \emptyset.$$
The following key basic fact is a straightforward exercise in unpacking the definitions:
\[prop:topSN\] Let $\alpha$, $\Gamma$ and $X$ be as in Definition \[defn:gcyl\]. Then
1. For every non-degenerate finite subtree $K\subseteq X$ the subset $\mathcal SCyl_\alpha(K)\subset {\mathcal S{\mbox{Curr}}({F_N})}$ is compact and open.
2. The collection of all $\mathcal SCyl_\alpha(K)$, where $K$ varies over all non-degenerate finite subtrees of $X$, forms a basis for the topology on $\mathfrak C_N$ given in Definition \[defn:SN\].
\[notation:edges\] Let $X$ be a simplicial tree and let $e$ be an oriented edge of $X$. Denote by $q(e)$ the set of all oriented edges $e'$ in $X$ such that $e,e'$ is a reduced edge-path in $X$.
For any set $B$ we denote by $P_+(B)$ the set of all nonempty subsets of $B$.
The following simple lemma is key for the Kirchhoff-type formulas for subset currents (see Proposition \[prop:kirch\] below).
\[lem:disj\] Let $\Gamma, X$, $K\subset X$ be as in Definition \[defn:gcyl\] and let $e_1,\dots, e_n$ be the terminal edges of $K$, as in Definition \[defn:gcyl\]. Then for every $i=1,\dots, n$ we have $$\mathcal SCyl_\alpha(K)=\sqcup_{U\in P_+(q(e_i))} \mathcal SCyl_\alpha(K\cup U) .$$
Fix $i, 1\le i\le n$. It is obvious from the definitions that for every such nonempty $U$ we have $\mathcal SCyl_\alpha(K\cup U)\subseteq \mathcal SCyl_\alpha(K)$ and hence the union of $\mathcal SCyl_\alpha(K\cup U)$ over all such $U$ is contained in $\mathcal SCyl_\alpha(K)$.
Let $S\in \mathcal SCyl_\alpha(K)$ be arbitrary. By the definition of $\mathcal SCyl_\alpha(K)$ we have $S\cap Cyl_X(e_i)\ne\emptyset$. Let $U$ be the set of all edges $e\in EX$ such that there exists a point $\xi\in S$ that contains the geodesic ray in $X$ beginning with the edge-path $(e_i,e)$. Then $S\subseteq \mathcal SCyl_\alpha(K\cup U)$.
It follows that $$\mathcal SCyl_\alpha(K)\subseteq \cup_U \mathcal SCyl_\alpha(K\cup U)$$ with $U$ as required. We leave it as an exercise to the reader verifying that the union on the right-hand side in the above formula is a disjoint union.
Subset currents {#subsec:currents}
---------------
Observe that the left translation action of $F_N$ on $\partial F_N$ naturally extends to a left translation action by homeomorphisms on $\mathfrak
C_N$. We can now define the main notion of this paper:
A *subset current* on $F_N$ is a positive Borel measure $\mu$ on $\mathfrak C_N$ which is $F_N$-invariant and locally finite (i.e., finite on all compact subsets of $\mathfrak C_N$).
The set of all subset currents on $F_N$ is denoted ${\mathcal S{\mbox{Curr}}({F_N})}$. The space ${\mathcal S{\mbox{Curr}}({F_N})}$ is endowed with the natural weak-\* topology of convergence of integrals of continuous functions with compact support.
Define an equivalence relation $\sim$ on ${\mathcal S{\mbox{Curr}}({F_N})}-\{0\}$ as: $\mu\sim\mu'$ if $\mu=c\mu'$ for some $c>0$, where $\mu,\mu'\in {\mathcal S{\mbox{Curr}}({F_N})}-\{0\}$. For a nonzero $\mu\in{\mathcal S{\mbox{Curr}}({F_N})}$ the $\sim$-equivalence class of $\mu$ is denoted be $[\mu]$ and is called the *projective class* of $\mu$. Put ${\mathbb P{\mathcal S{\mbox{Curr}}({F_N})}}=\left({\mathcal S{\mbox{Curr}}({F_N})}-\{0\}\right)/\sim$ and endow ${\mathbb P{\mathcal S{\mbox{Curr}}({F_N})}}$ with the quotient topology.
It is not hard to check (c.f. the proof by Francaviglia [@Fra] of a similar statement for ordinary geodesic currents on $F_N$) that the weak-\* topology on ${\mathcal S{\mbox{Curr}}({F_N})}$ can be described in more concrete terms:
\[top:GC\] Let $\alpha:F_N\to \pi_1(\Gamma)$ be a marking on $F_N$, and let $X=\widetilde \Gamma$ be the universal cover of $\Gamma$.
1. Let $\mu, \mu_n\in {\mathcal S{\mbox{Curr}}({F_N})}$, where $n=1,2,\dots$. Then $\lim_{n\to\infty} \mu_n=\mu$ in ${\mathcal S{\mbox{Curr}}({F_N})}$ if and only if for every finite non-degenerate subtree $K$ of $X$ we have $$\lim_{n\to\infty} \mu_n(\mathcal SCyl_\alpha(K))= \mu(\mathcal SCyl_\alpha(K)).$$
2. For each finite non-degenerate subtree $K$ of $X$ the function $${\mathcal S{\mbox{Curr}}({F_N})}\to\mathbb R, \quad \mu\mapsto \mu(\mathcal SCyl_\alpha(K))$$ is continuous on ${\mathcal S{\mbox{Curr}}({F_N})}$.
3. Let $\mu\in {\mathcal S{\mbox{Curr}}({F_N})}$. For $\epsilon>0$ and an integer $M\ge 1$ let $U(M,\epsilon,\mu)$ be the set of all $\mu'\in {\mathcal S{\mbox{Curr}}({F_N})}$ such that for every finite non-degenerate subtree $K$ of $X$ with at most $M$ edges we have $$\left| \mu'(\mathcal SCyl_\alpha(K))- \mu(\mathcal SCyl_\alpha(K)) \right|<\epsilon.$$ Then the family $\{U(M,\epsilon, \mu): M\ge 1, 0<\epsilon<1\}$ forms a basis of open neighborhoods for $\mu$ in ${\mathcal S{\mbox{Curr}}({F_N})}$.
The following key observation follows directly from the definition of subset cylinders (see Definition \[defn:gcyl\] above) and from the fact that subset currents are $F_N$-invariant.
\[propdfn:weights\] Let $\alpha: F_N\rightarrow \pi_1(\Gamma)$ be a marking on $F_N$, let $X=\widetilde \Gamma$, and let $K$ be a finite non-degenerate subtree of $X$. Then for every element $g\in F_N$ we have $$g\mathcal SCyl_\alpha(K)=\mathcal SCyl_\alpha(gK)$$ and $$\mu(\mathcal SCyl_\alpha(K))=\mu(g\mathcal SCyl_\alpha(K)).\tag{$\ast$}$$ We denote $$(K;\mu)_\alpha:=\mu(\mathcal SCyl_\alpha(K))$$ and call it the *weight* of $K$ in $\mu$. For a given finite subtree $K$ of $X$, we denote the $F_N$-translation class of $K$ by $[K]$ (so that $[K]$ consists of all the translates of $K$ by elements of $F_N$). We put $$( [K]; \mu)_\alpha:=( K; \mu)_\alpha$$ and call it the *weight* of $[K]$ in $\mu$. In view of $(\ast)$, the weight $( [K]; \mu)_\alpha$ is well-defined and does not depend on the choice of $K$ in $[K]$. It defines a continuous function on ${\mathcal S{\mbox{Curr}}({F_N})}$.
Let $\alpha, \Gamma$ and $X$ be as in Proposition \[prop:topSN\]. Let $K$ be a finite non-degenerate subtree of $X$. Then the function $f:{\mathcal S{\mbox{Curr}}({F_N})}\to{\mathbb R}$ given by $f(\mu)=(K; \mu)_\alpha$, where $\mu\in {\mathcal S{\mbox{Curr}}({F_N})}$, is continuous.
\[not:e\] Let $\alpha, \Gamma$ and $X$ be as in Proposition \[prop:topSN\]. For every topological edge $e\in E_{top}\Gamma$ denote by $\widetilde e$ the subgraph of $X=\widetilde \Gamma$ consisting of any lift of $e$. By $F_N$-invariance of subset currents, the weight $(\widetilde e; \mu)_\alpha$ depends only on $\mu, \alpha$ and $e$ and does not depend on the choice of a lift $\widetilde e$ of $e$ to $X$. We thus denote $(e; \mu)_\alpha:=(\widetilde e; \mu)_\alpha$.
\[prop:kirch\] Let $\alpha: F_N\rightarrow \pi_1(\Gamma)$ be a marking on $F_N$, let $X=\widetilde \Gamma$, and let $K$ be a finite non-degenerate subtree of $X$ with terminal edges $e_1,\dots, e_n$, as in Definition \[defn:gcyl\]. Let $\mu\in {\mathcal S{\mbox{Curr}}({F_N})}$. Then for every $i=1,\dots, n$ we have $$(K; \mu)_\alpha=\sum_{U\in P_+(q(e_i))} (K\cup U; \mu)_\alpha ,\tag{$\bigstar$}$$ in notations of \[notation:edges\].
The statement follows directly from Lemma \[lem:disj\] and from the fact that $\mu$ is finite-additive.
Since the Borel $\sigma$-algebra on $\mathfrak C_N$ is generated by the collection of all subset cylinders, it follows, by Kolmogorov Extension Theorem, that any $F_N$-invariant system of weights on all the subset cylinders satisfying the Kirchhoff formulas actually defines a subset current:
\[prop:weights\] Let $\alpha, \Gamma$ and $X$ be as in Proposition \[prop:kirch\]. Let $\mathcal K_\Gamma$ be the set of all finite non-degenerate simplicial subtrees of $X$ and let $$\vartheta: \mathcal K_\Gamma\to [0,\infty)$$ be a function satisfying the following conditions:
1. For every $K\in \mathcal K_\Gamma$ and every $g\in F_N$ we have $\vartheta(gK)=\vartheta(K)$.
2. For every $K\in \mathcal K_\Gamma$ and every terminal edge $e$ of $K$ we have $$\vartheta(K)=\sum_{U\in P_+(q(e))} \vartheta(K\cup U).$$
Then there exists a unique $\mu\in {\mathcal S{\mbox{Curr}}({F_N})}$ such that for every $K\in \mathcal K_\Gamma$ we have $$\vartheta(K)=(K; \mu)_\alpha.$$
The space ${\mathcal S{\mbox{Curr}}({F_N})}$ is locally compact and the space ${\mathbb P{\mathcal S{\mbox{Curr}}({F_N})}}$ is compact.
It follows from the definition of ${\mathcal S{\mbox{Curr}}({F_N})}$ as the space of $F_N$-invariant locally finite positive Borel measures on $\mathfrak C_N$ that ${\mathcal S{\mbox{Curr}}({F_N})}$ is metrizable. Then to show local compactness it suffices to establish sequential local compactness of ${\mathcal S{\mbox{Curr}}({F_N})}$.
Consider a marking $\alpha:F_N{\mathrel{\text{ \setbox0\hbox{$\rightarrow$} \rlap{\hbox to \wd0{\hss\raisebox{0.9\height}{$\sim$}\hss}}\box0
}}}\pi_1(\Gamma)$. It is not hard to show, using Proposition \[prop:kirch\], Proposition \[propdfn:weights\] and a standard diagonalization argument for individual weights, that for every $C>0$ the sets $$\{\mu\in {\mathcal S{\mbox{Curr}}({F_N})}: \sum_{e\in E_{top}\Gamma} (\tilde e; \mu)_\alpha\le C \}$$ and $$\{\mu\in {\mathcal S{\mbox{Curr}}({F_N})}: \sum_{e\in E_{top}\Gamma} (\tilde e; \mu)_\alpha= C \}$$ are sequentially compact. This implies local compactness of ${\mathcal S{\mbox{Curr}}({F_N})}$ and compactness of ${\mathbb P{\mathcal S{\mbox{Curr}}({F_N})}}$.
\[rem:curr\] Recall also that elements of ${\mbox{Curr}}(F_N)$ are positive locally finite $F_N$-invariant Borel measures on the space of $2$-element subsets in $\partial F_N$ which is clearly a closed $F_N$-invariant subset of $\mathfrak C_N$. Hence the space ${\mbox{Curr}}(F_N)$ can be thought of as canonically embedded in ${\mathcal S{\mbox{Curr}}({F_N})}$, ${\mbox{Curr}}(F_N)\subseteq {\mathcal S{\mbox{Curr}}({F_N})}$, and it is not hard to see that ${\mbox{Curr}}(F_N)$ is a closed subset of ${\mathcal S{\mbox{Curr}}({F_N})}$. Moreover, once the action of ${\mbox{Out}}(F_N)$ on ${\mathcal S{\mbox{Curr}}({F_N})}$ is defined in Section \[sec:action\] below, it will be obvious that ${\mbox{Curr}}(F_N)$ is an ${\mbox{Out}}(F_N)$-invariant subset of ${\mathcal S{\mbox{Curr}}({F_N})}$. For similar reasons ${\Pr{\mbox{Curr}}({F_N})}\subseteq {\mathbb P{\mathcal S{\mbox{Curr}}({F_N})}}$ is a closed ${\mbox{Out}}(F_N)$-invariant subset.
Rational subset currents {#sec:rational}
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Counting and rational subset currents {#subsec:counting}
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Recall that for a subgroup $H\le G$ of a group $G$ the *commensurator* or *virtual normalizer* $Comm_G(H)$ of $H$ in $G$ is defined as $$Comm_G(H):=\{g\in G| [H: H\cap gHg^{-1}]<\infty, \text{ and } [gHg^{-1}: H\cap gHg^{-1}]<\infty\}.$$ It is easy to see that $Comm_G(H)$ is again a subgroup of $G$ and that $H\le Comm_G(H)$.
Suppose now that $N\ge 2$ and $H\le F_N$ is a nontrivial subgroup of $F_N$. The *limit set* $\Lambda(H)$ of $H$ in $\partial F_N$ is the set of all $\xi\in \partial F_N$ such that there exists a sequence $h_n\in H$, $n\ge 1$ satisfying $$\lim_{n\to\infty} h_n=\xi \quad\text{ in } F_N\cup \partial F_N.$$ We recall some elementary properties of limit sets. We see in particular that for every $H\le F_N, H\ne 1$ we have $\Lambda(H)\in
\mathfrak C_N$.
\[prop:basic\] Let $H\le F_N$ be a nontrivial subgroup. Then:
1. The limit set $\Lambda(H)$ is a closed $H$-invariant subset of $\partial F_N$. If $H$ is infinite cyclic, $\Lambda(H)$ consists of two distinct points; if $H$ is not cyclic, $\Lambda(H)$ is infinite.
2. For any $\xi\in \Lambda(H)$ the closure of the orbit $H\xi$ in $\partial F_N$ is equal to $\Lambda(H)$.
3. For every $g\in G$ $$\Lambda(gHg^{-1})=g\Lambda(H).$$
4. If $H\le Q\le F_N$ then $\Lambda(H)\subseteq \Lambda(Q)$.
5. Either $H$ is infinite cyclic and $\Lambda(H)$ consists of exactly two distinct points or $H$ contains a nonabelian free subgroup and $\Lambda(H)$ is uncountable.
6. Let $T\in {\mbox{cv}_N}$ and let $Conv_T(\Lambda(H))$ be the *convex hull* of $\Lambda(H)$ in $T$, that is, the union of all bi-infinite geodesics in $T$ with endpoints in $\Lambda(H)$. Then $Conv_T(\Lambda(H))=T_H$ is the unique minimal $H$-invariant subtree of $T$.
Another useful basic fact relates limit sets of finitely generated subgroups and their commensurators (see [@KS; @KM] for details).
\[prop:comm\] Let $H\le F_N$ be a nontrivial finitely generated subgroup. For a subset $Z\subseteq \partial F_N$ denote $Stab_{F_N}(Z):=\{g\in
F_N: gZ=Z\}$. Then
1. $Stab_{F_N}(\Lambda(H))=Comm_{F_N}(H)$ and $[Comm_{F_N}(H):H]<\infty$.
2. $\Lambda(H)=\Lambda (Comm_{F_N}(H))$.
3. For $H_1=Comm_{F_N}(H)$ we have $Comm_{F_N}(H_1)=H_1$.
4. Let $L\le F_N$ such that $H\le L$. Then $[L:H]<\infty$ if and only if $L\le Comm_G(H)$.
5. Suppose $H=\langle g\rangle$, where $g\in F_N, g\ne 1$. Then $H=Comm_{F_N}(H)$ if and only if $g$ is not a proper power in $F_N$, that is, if and only if $H$ is a maximal infinite cyclic subgroup of $F_N$.
\[defn:count\] Let $H\le F_N$ be a nontrivial finitely generated subgroup.
1. Suppose first that $H=Comm_{F_N}(H)$. Define the measure $\eta_H$ on $\mathfrak C_N$ as $$\eta_H:=\sum_{H_1\in [H]} \delta_{\Lambda(H_1)},$$ where $[H]$ is the conjugacy class of $H$ in $F_N$.
2. Now let $H\le F_N$ be an arbitrary nontrivial finitely generated subgroup. Put $H_0:=Comm_{F_N}(H)$ and let $m:=[H_0:H]$. Proposition \[prop:comm\] implies that $m<\infty$ and that $H_0=Comm_{F_N}(H_0)$. Then put $\eta_H:=m\eta_{H_0}$.
\[lem:count\] Let $H\le F_N$ be a nontrivial finitely generated subgroup. Then $\eta_H\in {\mathcal S{\mbox{Curr}}({F_N})}$.
It is enough to show that $\eta_H\in {\mathcal S{\mbox{Curr}}({F_N})}$ for the case where $H=Comm_{F_N}(H)$. Thus we assume that $H\le F_N$ is a nontrivial finitely generated subgroup with $H=Comm_{F_N}(H)$. Fix a free basis $A$ of $F_N$ and let $X$ be the Cayley graph of $F_N$ with respect to $A$.
Part (3) of Proposition \[prop:basic\] implies that $\eta_H$ is an $F_N$-invariant positive Borel measure on $\mathfrak C_N$. Thus it remains to check that $\eta_H(C)<\infty$ for every compact $C\subseteq \mathfrak C_N$. Every compact subset of $\mathfrak C_N$ is the union of finitely many cylinder subsets. Thus we only need to check that $\eta_H(\mathcal SCyl_{X}(K))<\infty$ for every finite non-degenerate subtree $K$ of $X$. After replacing $K$ by its $F_N$-translate, we may assume that the element $1\in F_N$ is a vertex of $K$. Also, since $\eta_H$ depends only on the conjugacy class of $H$, after replacing $H$ by a conjugate we may assume that $1\in X_H=Conv_{X}(\Lambda(H))$.
Recall from Proposition \[prop:basic\] that $Conv_X(\Lambda(H))=X_H$ is the unique minimal $H$-invariant subtree of $X$. Whenever $g\in F_N$ is such that $g\Lambda(H)=\Lambda(gHg^{-1})\in \mathcal SCyl_X(K)$, we have $1\in
Conv_X(\Lambda(H))=X_H$. Thus it suffices to show that the number of distinct translates $gX_H$ of $X_H$ that contain $1\in F_N$ is finite.
It is not hard to see, since by assumption $1\in X_H$, that for $g\in
F_N$ we have $1\in gX_H$ if and only if $g\in V(X_H)\subseteq F_N$.
Since $X_H$ is the minimal $H$-invariant subtree and $H$ is finitely generated, the quotient $H\setminus X_H$ is a finite graph. In particular $H\setminus V(X_H)$ is a finite set. Every $H$-orbit of a vertex of $X_H$ is a coset class $uH$ for some $u\in F_N$. Thus there exists a finite set $u_1,\dots, u_m\in F_N$ such that $V(X_H)=\{u_ih| h\in H, 1\le
i\le m\}$. For $g=u_ih$, where $h\in H$, $1\le i\le m$, we have $gX_H=u_ihX_H=u_iX_H$. Thus indeed there are only finitely many translates of $X_H$ that contain $1\in F_N$. Therefore $\eta_H(\mathcal SCyl_X(K))<\infty$, as required.
For a nontrivial finitely generated $H\le F_N$, we call $\eta_H\in{\mathcal S{\mbox{Curr}}({F_N})}$ given by Definition \[defn:count\] and Lemma \[lem:count\] the *counting subset current* associated to $H$.
A subset current $\mu\in{\mathcal S{\mbox{Curr}}({F_N})}$ is called *rational* if $\mu=r\eta_H$ for some $r\ge 0$ and some nontrivial finitely generated subgroup $H\le F_N$.
Definition \[defn:count\] directly implies:
Let $H\le F_N$ be a nontrivial finitely generated subgroup and let $H'=gHg^{-1}$ for some $g\in F_N$. Then $\eta_H=\eta_{H'}$.
The above statement has a partial converse:
Let $H,H'\le F_N$ be nontrivial finitely generated subgroups such that $H=Comm_{F_N}(H)$ and $H'=Comm_{F_N}(H')$. Then $\eta_H=\eta_{H'}$ if and only if $[H]=[H']$.
We have already seen that if $[H]=[H']$ then $\eta_H=\eta_{H'}$.
Suppose now that $\eta_H=\eta_{H'}$. Choose a marking $\alpha:F_N\to\pi_1(\Gamma)$ on $F_N$. Let $X=\widetilde \Gamma$.
Let $K\in\mathcal K_\Gamma$ be such that $\Lambda(H)\in \mathcal SCyl_\alpha(K)$. Since $\eta_H(\mathcal SCyl_\alpha(K))<\infty$ and $\eta_{H'}(\mathcal SCyl_\alpha(K))<\infty$, Definition \[defn:count\] implies that only finitely many distinct $F_N$-translates of $\Lambda(H)$ and $\Lambda(H')$ belong to $\mathcal SCyl_\alpha(K)$. Let these translates be $g_1\Lambda(H), \dots, g_m \Lambda(H)$ and $f_1\Lambda(H'), \dots, f_t\Lambda(H')$, with $g_1=1$. Thus we have $m+t$ distinguished points in the open set $\mathcal SCyl_\alpha(K)$. Since $\mathfrak C_N$ is metrizable and Hausdorff, we can find an open subset $V$ of $\mathcal SCyl_\alpha(K)$ such that $V$ contains exactly one of these $m+t$ points, namely the point $g_1\Lambda(H)=\Lambda(H)$. Since the cylinder subsets form a basis of open sets for $\mathfrak C_N$, there exists a finite subtree $K_1$ of $X$ such that $\Lambda(H)\in \mathcal SCyl_\alpha(K_1)\subseteq V$. Thus $\Lambda(H)\in \mathcal SCyl_\alpha(K_1)$ and no $F_N$-translate of $\Lambda(H), \Lambda(H')$, distinct from $\Lambda(H)$, belongs to $\mathcal SCyl_\alpha(K_1)$. Then $\eta_H(\mathcal SCyl_\alpha(K_1))=1$. By assumption $\eta_H=\eta_{H'}$ and hence $\eta_{H'}(\mathcal SCyl_\alpha(K_1))=1$. Thus, by definition of $\eta_{H'}$, there is a unique translate $g\Lambda(H')$, where $g\in F_N$, such that $g\Lambda(H')\in \mathcal SCyl_\alpha(K_1)$. By the choice of $K_1$, the only $F_N$-translate of $\Lambda(H), \Lambda(H')$ contained in $\mathcal SCyl_\alpha(K_1)$ is $\Lambda(H)$. Therefore $\Lambda(H)=g\Lambda(H')$ so that $\Lambda(H)=g\Lambda(H')=\Lambda(gH'g^{-1})$.
Since $H=Comm_{F_N}(H)$ and $gH'g^{-1}=Comm_{F_N}(gH'g^{-1})$, Proposition \[prop:comm\] implies that $$H=Stab_{F_N}(\Lambda(H))=Stab_{F_N}( \Lambda(gH'g^{-1}) )=gH'g^{-1},$$ and $[H]=[H']$, as required.
$\Gamma$-graphs {#subsec:graphs}
---------------
We need to introduce some terminology slightly generalizing the standard set-up of Stallings foldings for subgroups of free groups (see [@Sta; @KM]). That set-up usually involves choosing a free basis $A$ of $F_N$ and considering graphs whose edges are labelled by elements of $A^{\pm 1}$. Choosing a free basis $A$ of $F_N$ corresponds to a marking on $F_N$ that identifies $F_N$ with the fundamental group of the standard $N$-rose $R_N$. We need to relax the requirement that the graph in question be $R_N$.
\[defn:gammagraph\] Let $\Gamma$ be a finite connected graph without degree-one and degree-two vertices. A *$\Gamma$-graph* is a graph $\Delta$ together with a graph morphism $\tau:\Delta\to\Gamma$.
For a vertex $x\in V\Delta$ we say that the *type* of $x$ is the vertex $\tau(x)\in V\Gamma$. Similarly, for an oriented edge $e\in
E\Gamma$ the *type* of $e$, or the *label* of $e$ is the edge $\tau(e)$ of $\Gamma$.
Note that every covering of $\Gamma$ has a canonical $\Gamma$-graph structure. In particular, $\Gamma$ itself is a $\Gamma$-graph and so is the universal cover $\widetilde\Gamma$ of $\Gamma$. Also, every subgraph of a $\Gamma$-graph is again a $\Gamma$-graph.
Note that for a graph $\Delta$ a graph-morphism $\tau:\Delta\to\Gamma$ can be uniquely specified by assigning a label $\tau(e)\in E\Gamma$ for every $e\in E\Delta$ in such a way that $\tau(e^{-1})=(\tau(e))^{-1}$ and such that for every pair of edges $e',e'\in E\Delta$ with $o(e)=o(e')$ we have $o(\tau(e))=o(\tau(e'))\in V\Gamma$. Thus we usually will think of a $\Gamma$-graph structure on $\Delta$ as such an assignment of labels $\tau: E\Delta\to E\Gamma$. Also, by abuse of notation, we will often refer to a graph $\Delta$ as a $\Gamma$-graph, assuming that the graph morphism $\tau:\Delta\to\Gamma$ is implicitly specified.
Let $\tau_1:\Delta_1\to\Gamma$ and $\tau_2:\Delta_2\to\Gamma$ be $\Gamma$-graphs. A graph morphism $f:\Delta_1\to\Delta_2$ is called a *$\Gamma$-graph morphism*, if it respects the labels of vertices and edges, that is if $\tau_1=\tau_2\circ f$.
Let $\Delta$ be a $\Gamma$-graph. For a vertex $x\in V\Delta$ denote by $Lk_\Delta(x)$ (or just by $Lk(x)$) the [*link*]{} of $x$, namely the function $$Lk_\Delta(x): E\Gamma\to\mathbb Z_{\ge 0}$$ where for every $e\in E\Gamma$ the value $\left(Lk_\Delta(x)\right)(e)$ is the number of edges of $\Delta$ with origin $x$ and label $e$.
A $\Gamma$-graph $\Delta$ is said to be *folded* if for every vertex $x\in V\Delta$ and every $e\in E\Gamma$ we have $$\left(Lk_\Delta(x)\right)(e)\le 1.$$
If $\Delta$ is folded, we will also think of $Lk_\Delta(x)$ as a subset of $E\Gamma$ consisting of all those $e\in E\Gamma$ with $\left(Lk_\Delta(x)\right)(e)=1$, that is, of all $e\in E\Gamma$ such that there is an edge in $\Delta$ with origin $x$ and label $e$.
The following is an immediate corollary of the definitions.
Let $\Delta$ be a $\Gamma$-graph and let $\tau:\Delta\to\Gamma$ be the associated graph morphism. Then:
1. $\Delta$ is folded if and only if $\tau$ is an immersion.
2. Suppose $\Delta$ is connected. Then $\Delta$ is a covering of $\Gamma$ if and only if $Lk_\Delta(x)=Lk_\Gamma(\tau(x))$ for every $x\in V\Delta$.
3. Let $\Delta_1,\Delta_2$ be two $\Gamma$-graphs such that $\Delta_1$ is connected and $\Delta_2$ is folded. Let $x_1\in
V\Delta_1$, $x_2\in V\Delta_2$ be vertices of the same type $x\in
V\Gamma$. Then there exists at most one $\Gamma$-graph morphism $f:\Delta_1\to\Delta_2$ such that $f(x_1)=x_2$.
\[defn:core\] Referring to Stallings’ notion of [*core graphs*]{}, [@Sta], we say that a finite $\Gamma$-graph $\Delta$ is a $\Gamma$-[*core graph*]{} if $\Delta$ is *cyclically reduced*, that is it is folded and has no degree-one and degree-zero vertices.
Informally, we think of $\Gamma$- core graphs as generalizations of cyclic words, over the alphabet $\Gamma$. We need the following analog of the notion of a subword in this context:
\[defn:occur\] Let $K\subseteq \widetilde \Gamma$ be a finite non-degenerate subtree, considered together with the canonical $\Gamma$-graph structure inherited from $\widetilde \Gamma$. Let $\Delta$ be a finite $\Gamma$-core graph. An *occurrence* of $K$ in $\Delta$ is a $\Gamma$-graph morphism $f:K\to\Delta$ such that for every vertex $x$ of $K$ of degree at least $2$ in $K$ we have $Lk_K(x)=Lk_\Delta(f(x))$.
We denote the number of all occurrences of $K$ in $\Delta$ by $(K; \Delta)_\Gamma$, or just $(
K; \Delta)$.
In topological terms, a $\Gamma$-graph morphism $f:K\to\Delta$ is an occurrence of $K$ in $\Delta$ if $f$ is an immersion and if $f$ is a covering map at every point $x\in K$ (including interior points of edges) except for the degree-1 vertices of $K$. That is, for every $x\in K$, other than a degree-1 vertex of $K$, $f$ maps a small neighborhood of $x$ in $K$ homeomorphically *onto* a small neighborhood of $f(x)$ in $\Delta$.
\[prop:wd\] Let $\alpha:F_N\to\pi_1(\Gamma)$ be a marking on $F_N$ and let $X=\widetilde\Gamma$. As before (see \[prop:weights\]), let $\mathcal K_\Gamma$ denote the set of all non-degenerate finite simplicial subtrees of $X$. Let $\Delta$ be a finite $\Gamma$-core graph. Define $\vartheta_\Delta: \mathcal K_\Gamma\to \mathbb R$ as $$\vartheta_\Delta(K):=(K;\Delta)_\Gamma$$ for each $K\in \mathcal K_\Gamma$.
Then the function $\vartheta_\Delta$ satisfies the conditions of Proposition \[prop:weights\]. Thus there is a unique subset current $\mu_\Delta\in{\mathcal S{\mbox{Curr}}({F_N})}$ such that for every $K\in \mathcal
K_\Delta$ $$( K; \mu_\Delta)_\alpha=( K;\Delta)_\Gamma$$
We will see later on that, for a connected $\Delta$, the function $\vartheta_\Delta$ defines the counting current of a finitely generated subgroup of $F_N$. However, we think it is useful to have a direct argument for Proposition \[prop:wd\].
It easily follows from the definitions that for every $K\in \mathcal K_\Gamma$ and every $g\in F_N$ we have $(K;\Delta)_\Gamma=(gK;\Delta)_\Gamma$. Thus we only need to verify that the condition (2) of Proposition \[prop:weights\] holds for $\vartheta_\Delta$.
Let $K\in \mathcal K_\Gamma$ and let $e$ be a terminal edge of $K$, as in Definition \[defn:gcyl\]. Consider the set of trees $$Q(K,e)=\{ K\cup U | U\in P_+(q(e))\},$$ as in Notation \[notation:edges\]. Thus each element of $Q(K,e)$ is obtained by adding to $K$ a nonempty subset $U$ from the set of edges $q(e)$.
We now construct a function $D$ from the set $R$ of all occurrences of $K$ in $\Delta$ to the set $R'$ of all occurrences of elements of $Q(K,e)$ in $\Delta$. Put $x:=t(e)\in VK$.
Let $f:K\to\Delta$ be an occurrence of $K$. Since $\Delta$ is a core graph, the vertex $y=f(x)$ has degree bigger than 1 in $\Delta$. Note that $f(e^{-1})$ is an edge of $\Delta$ with initial vertex $y$. Put $U_0$ to be the set of the labels of all the edges in $\Delta$ with origin $y$, excluding the edge $f(e^{-1})$. Thus $U_0$ is nonempty. Let $U$ be the set of all edges in $\widetilde \Gamma$ with initial vertex $x$ and with label belonging to $U_0$. Put $K'=K\cup U$, so that $K'\in Q(K,e)$. We now extend $f$ to a morphism $f':K'\to\Delta$ by sending every edge from $U$ to the unique edge in $\Delta$ with the same label and with origin $y$. Then by construction, $f'$ is a $\Gamma$-graph morphism whose restriction to $K$ is $f$ and, moreover, $Lk_{K'}(x)=Lk_{\Delta}(y)$. Therefore $f'$ is an occurrence of $K'$ in $\Delta$. We put $D(f):=f'$.
This defines a function $D:R\to R'$. By construction, this function is injective. Moreover, $D$ is clearly onto. Indeed, if $K'=K\cup U\in
Q(K,e)$ and $f':K'\to \Delta$ is an occurrence of $K'$ in $\Delta$ then $f:=f'|_K$ is an occurrence of $K$ in $\Delta$ and $D(f)=f'$. Thus $D$ is a bijection, and hence $R$ and $R'$ have equal cardinalities. It follows that $$( K; \Delta)_\Gamma=\sum_{U\in P_+(q(e))} (K\cup U; \Delta)_\Gamma,$$ as required.
Note that if $\Delta$ is a finite $\Gamma$-core graph with connected components $\Delta_1, \dots, \Delta_k$, then each $\Delta_i$ is again a $\Gamma$-core graph and we have $$\mu_{\Delta}=\mu_{\Delta_1}+\dots+\mu_{\Delta_k}.$$
We leave it to the reader to verify the following:
\[lem:m\] Let $\Delta$ be as in Proposition \[prop:wd\] and assume that $\Delta$ is connected. Let $\widehat\Delta$ be an $m$-fold cover of $\Delta$ for some $m\ge
1$. Then $\mu_{\widehat\Delta}=m \mu_\Delta$
Counting currents and $\Gamma$-graphs {#GraphsCurrents}
-------------------------------------
We now want to relate the current constructed in Proposition \[prop:wd\] to counting currents of finitely generated subgroups of $F_N$.
Let $\alpha: F_N\to\pi_1(\Gamma)$ be a marking on $F_N$, and let $X:=\widetilde \Gamma$. Recall from Subsection \[subsec:mark\] that $\partial F_N$ and $\partial X$ are identified via the homeomorphism induced by $\alpha$. Note that if $S\in \mathfrak C_N$, the fact that $S$ has cardinality at least $2$ implies that its convex hull $Conv_X(S)$ is nonempty. Moreover, it is easy to see that $Conv_X(S)$ is an infinite subtree of $X$ without any degree-1 vertices. Let us denote by $\mathcal T(X)$ the set of all infinite subtrees of $X$ without degree-1 vertices.
The following statement is an elementary consequence of the definitions and we leave the details to the reader.
\[prop:ch\] Let $\alpha, \Gamma, X$ be as above. Then
1. For any $S\in \mathfrak C_N$ we have $\partial Conv_X(S)=S$ and for any $Y\in \mathcal T(X)$ we have $Y=Conv_X(\partial Y)$.
2. The convex hull operation yields a bijection $Conv_X: \mathfrak C_N\to \mathcal T(X)$ which is $F_N$-equivariant: for any $S\in \mathfrak C_N$ and $g\in F_N$ we have $Conv_X(gS)=gConv_X(S)$.
3. For any $S\in \mathfrak C$ we have $\partial Conv_X(S)=S$ and for any $Y\in \mathcal T(X)$ we have $Y=Conv_X(\partial Y)$.
Recall that if $T\in {\overline{\mbox{cv}}_N}$ and if $H\le F_N$ is a nontrivial subgroup, there is a unique smallest $H$-invariant subtree of $T$ denoted $T_H$, and, moreover $T_H$ has no degree-one vertices.
Propositions \[prop:ch\], \[prop:basic\] and \[prop:comm\] easily imply:
\[prop:XH\] Let $\alpha, \Gamma, X$ be as above. Let $H\le F_N$ be a nontrivial finitely generated subgroup and let $X_H$ be the minimal $H$-invariant subtree of $X$. Then:
1. $X_H\in\mathcal T(X)$ and $X_H=Conv_X(\Lambda(H))$.
2. For any $g\in F_N$, $gX_H=X_{gHg^{-1}}$.
3. $H=Comm_{F_N}(H)$ if and only if $Stab_{F_N}(X_H)=H$.
\[conv:subgroupcore\] Let $\alpha: F_N\to\pi_1(\Gamma)$ be a marking on $F_N$, and let $X:=\widetilde \Gamma$. Let $H\le F_N$ be a finitely generated subgroup and let $X_H$ be the minimal $H$-invariant subtree of $X$. Note that, being a subgraph of $\widetilde \Gamma$, the tree $X_H$ comes equipped with a canonical $\Gamma$-graph structure $X_H\to\Gamma$ (which is just the restriction to $X_H$ of the universal covering map $\widetilde \Gamma\to\Gamma$) which is $F_N$-invariant. Put $\Delta=H\setminus X_H$. Then $\Delta$ is a finite connected graph without degree-1 vertices (by minimality of $X_H$); that is, $\Delta$ is a $\Gamma$-core graph. Moreover, $\Delta$ inherits a natural $\Gamma$-graph structure from $X_H$.
It is easy to see, in view of part (2) of Proposition \[prop:XH\], that the isomorphism type of $\Delta$ as a $\Gamma$-graph depends only on the conjugacy class $[H]$ of $H$ in $F_N$. Thus we say that $\Delta$ is the $\Gamma$-[*core graph representing $[H]$*]{}.
We can also describe $\Delta$ as follows: $\Delta$ is the *core* of $\widehat \Gamma = H\setminus X$, that is, $\Delta$ is the union of all immersed (but not necessarily simple) circuits in $\widehat \Gamma$. Thus $\Delta$ and $\widehat \Gamma$ are homotopy equivalent, and $\Delta$ is obtained from $\widehat \Gamma$ by cutting-off a (possibly empty) collection of infinite tree hanging branches. Also, $\Delta$ is the smallest subgraph of $\widehat \Gamma$ whose inclusion in $\widehat \Gamma$ is a homotopy equivalence with $\widehat\Gamma$. The labelling map $\tau:\Delta\to\Gamma$ satisfies $\tau_\#(\pi_1(\Delta))=H$. In fact $\tau_\#$ is an isomorphism between $\pi_1(\Delta)$ and $H$ and sometimes, by abuse of notation, we will write $\pi_1(\Delta)=H$.
\[thm:cc\] Let $\alpha: F_N\to\pi_1(\Gamma)$ be a marking on $F_N$. Let $H\le
F_N$ be a nontrivial finitely generated subgroup such that $Comm_{F_N}(H)=H$. Let $\Delta$ be the $\Gamma$-core graph representing $[H]$. Then $\eta_H=\mu_\Delta$.
We need to show that for every finite non-degenerate subtree $K$ of $X=\widetilde \Gamma$ we have $(K; \eta_H)_\alpha=(K;\mu_\Delta)_\alpha$.
Choose a vertex $x_0$ in $K$ and let $v_0$ be the projection of $x_0$ in $\Gamma$. We may assume that $v_0$ is the base-point of $\Gamma$ and that $\alpha:F_N{\mathrel{\text{ \setbox0\hbox{$\rightarrow$} \rlap{\hbox to \wd0{\hss\raisebox{0.9\height}{$\sim$}\hss}}\box0
}}}\pi_1(\Gamma,v_0)$. Recall that $X_H=Conv_X(\Lambda
H)$ is the smallest $H$-invariant subtree of $X$. By replacing $H$ by its conjugate if necessary, we may assume that $x_0\in
X_H$. Moreover, $\Delta=H\setminus X_H$, and $(X_H,x_0)$ is canonically identified with $\widetilde
{(\Delta,y_0)}$, where $y_0$ is the image of $x_0$ under the projection $p:X_H\rightarrow\Delta$.
We need to show that the number $( K;\Delta)_\alpha$ of occurrences of $K$ in $\Delta$ is equal to the number of distinct $F_N$-translates of $X_H$ that contain $K$. We will construct a function $Q$ from the set of occurrences of $K$ in $\Delta$ to the set of $F_N$-translates of $X_H$ that contain $K$.
Let $f:K\to \Delta$ be an occurrence of $K$ in $\Delta$. Choose an edge-path $\gamma$ from $y_0$ to $f(x_0)$ in $\Delta$. The fact that both $y_0$ and $f(x_0)$ project to $v_0$ in $\Gamma$ implies that the (unique) lift $\widetilde\gamma$ of $\gamma$ to an edge-path in $X$ with origin $x_0$ has its terminal vertex of the form $gx_0$ for some (unique) $g\in F_N$. Then the definition of occurrence of $K$ in $\Delta$ and the fact that $\widetilde \Delta$ is identified with $X_H$ imply that $gK\subseteq
X_H$ and so $K\subseteq g^{-1}X_H$. We set $Q(f):=g^{-1}X_H$. We need to check that $Q(f)$ is well-defined, that is, that the translate $g^{-1}X_H$ does not depend on the particular choice of an edge-path $\gamma$ from $y_0$ to $f(x_0)$ in $\Delta$. Indeed, if $\gamma'$ is another such edge-path, then $\gamma'\gamma^{-1}\in
\pi_1(\Delta,y_0)=H$. Hence if $\gamma'$ lifts to an edge-path from $x_0$ to $g'x_0$ in $X$, we have $g'g^{-1}\in H$, so that $g^{-1}=(g')^{-1}h$ for some $h\in H$ and hence $g^{-1}X_H=(g')^{-1}X_H$. Thus the translate $g^{-1}X_H$ containing $K$ is uniquely determined by the occurrence $f:K\to \Delta$ of $K$ in $\Delta$, so that $Q(f)$ is well-defined. Hence we have constructed a map $Q$ from the set of occurrences of $K$ in $\Delta$ to the set of $F_N$-translates of $X_H$ containing $K$.
We claim that the map $Q$ is injective. Indeed, let $f_1:K\to\Delta$ be another occurrence of $K$ in $\Delta$. Let $\gamma_1$ be an edge-path in $\Delta$ from $y_0$ to $f_1(x_0)$ and let $g_1\in F_N$ be such that the lift to $X$ of $\gamma_1$ with origin $x_0$ has terminus $g_1x_0$. Suppose that $g_1^{-1}X_H=g^{-1}X_H$. Then $g_1g^{-1}X_H=X_H$ and hence $g_1g^{-1}\in Comm_{F_N}(H)$. By assumption, on $H=Comm_{F_N}(H)$, so that $g_1g^{-1}\in H$ and $g_1=hg$ for some $h\in H$.
However, since $\pi_1(\Delta,y_0)=H$, it follows that $h$ labels a closed edge-path from $y_0$ to $y_0$ in $\Delta$ and hence, up to edge-path reductions, $\gamma_1=\beta\gamma$ for some closed loop $\beta$ from $y_0$ to $y_0$ in $\Delta$. Therefore the termini of $\gamma$ and $\gamma_1$ are the same, that is $f(x_0)=f_1(x_0)\in V\Delta$. It follows, since all the graphs under consideration are folded, that $f=f_1$. Thus indeed, the map $Q$ is injective.
We claim that $Q$ is also surjective. Suppose a translate $g^{-1}X_H$ contains $K$. Then $X_H$ contains $gK$. Both $x_0$ and $gx_0$ belong to $X_H$ and hence the geodesic edge-path $[x_0,gx_0]$ is contained in $X_H$. This edge-path projects to an edge-path $\gamma$ in $\Delta=
H\setminus X_H$ from $y_0$ to the vertex $z_0$ which is the image of $gx_0$ in $\Delta$ under the projection $p: X_H\to \Delta$. Since $p$ is a covering map, $f: K\to \Delta$ defined as $f(x)=p(g(x))$, $x\in
K$, is an occurrence of $K$ in $\Delta$, with $f(x_0)=z_0$. The definition of $Q$ now gives $Q(f)=g^{-1}X_H$.
Thus $Q$ is a bijection between the set of occurrences of $K$ in $\Delta$ and the set of $F_N$-translates of $X_H$ containing $K$. It follows that $( K; \eta_H)_\alpha=(K; \mu_\Delta)_\alpha$, as required.
\[cor:cc\] Let $H\le F_N$ be any nontrivial finitely generated subgroup. Let $\alpha:F_N\to\Gamma$ be a marking and let $\Delta$ be the $\Gamma$-core graph
representing $[H]$. Then $\eta_H=\mu_\Delta$.
Put $H_1=Comm_{F_N}(H)$. Then by Proposition \[prop:comm\] $H_1=Comm_{F_N}(H_1)$, and $[H_1:H]=m<\infty$. Let $\Delta_1$ be the $\Gamma$-core graph representing $[H_1]$. Therefore by Theorem \[thm:cc\] $\eta_{H_1}=\mu_{\Delta_1}$. Moreover, $\Delta$ is an $m$-fold cover of $\Delta_1$ and hence by Lemma \[lem:m\] $\mu_\Delta=m \mu_{\Delta_1}$. Also, by definition, $\eta_H=m\eta_{H_1}$. Hence $\eta_H=m\eta_{H_1}=m\mu_{\Delta_1}=\mu_\Delta$, as required.
Rational currents are dense {#sec:dense}
===========================
\[conv:GA\] In order to avoid technical complications in the proof of the main result of this Section, Theorem \[thm:dense\] in Subsection \[subsec:result\] below, we will restrict ourselves in this section to only considering $\Gamma$-graphs for $\Gamma=R_N$, the standard $N$-rose. The universal cover $X=\widetilde R_N$ is then the Cayley graph of $F_N$ with respect to some basis. We will fix some basis $A$ in $F_N$ and we will think of $A$ as defining a marking $\alpha_A:F_N\to R_N$, that will also be fixed for the remainder of this Section.
Note that an $R_N$-structure on a graph $\Delta$ can be specified by assigning every edge $e\in E\Delta$ a label $\tau(e)\in A^{\pm 1}$ so that $\tau(e^{-1})=(\tau(e))^{-1}$.
Linear span of rational currents {#subsec:linearspan}
--------------------------------
\[prop:span\] Denote by $\mathcal SCurr_r(F_N)$ the set of all rational subset currents. Let $Span\left(\mathcal SCurr_r(F_N) \right)$ denote the $\mathbb R_{\ge 0}$-linear span of $\mathcal SCurr_r(F_N)$.
Let $N\ge 2$. Then the set $\mathcal SCurr_r(F_N)$ is a dense subset of $Span\left(\mathcal SCurr_r(F_N) \right)$.
We need to show that an arbitrary linear combination $c_1\eta_{H_1}+\dots +c_k\eta_{H_k}$ (where $k\ge 1$, $c_i\in \mathbb R_{\ge 0}$) can be approximated by currents of the form $c\eta_H$, where $c\in \mathbb R_{\ge 0}$. Arguing by induction on $k$, we see that it suffices to prove this statement for $k=2$.
Every current of the form $c_1\eta_{H_1}+c_2\eta_{H_2}$, where $c_1,c_2\in \mathbb R_{\ge 0}$ can be approximated by currents of the form $r_1\eta_{H_1}+r_2\eta_{H_2}$, where $r_2,r_2$ are positive rational numbers. Thus it suffices to approximate by rational currents every current of the form $r_1\eta_{H_1}+r_2\eta_{H_2}$ where $r_1,r_2>0$ are rational numbers. Taking $r_1,r_2$ to a common denominator, we have $r_1=p_1/q$, $r_2=p_2/q$ where $p_1,p_2,q>0$ are integers. Since dividing a rational current by $q$ again yields a rational current, it is enough to approximate by rational currents all currents of the form $p_1\eta_{H_1}+p_2\eta_{H_2}$ where $p_1,p_2$ are positive integers. However, $p_1\eta_{H_1}=\eta_{L_1}$, $p_2\eta_{H_2}=\eta_{L_2}$, where $L_i$ is a subgroup of index $p_i$ in $H_i$ for $i=1,2$. Thus we only need to show that the sum of any two counting currents can be approximated by rational currents.
Suppose now that $H_1, H_2\le F_N$ are two nontrivial finitely generated subgroups and let $\mu=\eta_{H_1}+\eta_{H_2}$. For $i=1,2$ let $\Delta_i$ be the $R_N$-core graph representing $[H_i]$, as in Convention \[conv:subgroupcore\]. Thus, by Theorem \[thm:cc\], $\eta_{H_i}=\mu_{\Delta_i}$ for $i=1,2$.
For $n=1,2,\dots $ let $\Lambda_{i,n}$ be a connected $n$-fold cover of $\Delta_i$. We now define a sequence of finite connected $R_N$-core graphs as follows. Let $n\ge 1$. First assume that each of $\Lambda_{1,n}$, $\Lambda_{2,n}$ has a vertex $v_{i,n}$ of degree $<2N$, where $i=1,2$. Then, since $A^{\pm 1}$ contains at least 4 distinct letters, there exists an $R_N$-graph $[u_1,u_2]$, which is a segment of three edges with origin denoted $u_1$ and terminus denoted $u_2$, such that identifying the origin of this segment with $v_{1,n}$ and the terminus of this segment with $v_{2,n}$ yields a folded $R_N$-graph: $$\Lambda_n:=\Lambda_{1,n}\cup \Lambda_{2,n}\cup [u_1,u_2] / \sim$$ where $u_1\sim v_{1,n}$, $u_2\sim v_{2,n}$. Note that by construction $\Lambda_n$ is connected and cyclically reduced.
Suppose now that $\Lambda_{1,n}$ has a vertex $v_{1,n}$ of degree $<2N$ but that every vertex in $\Lambda_{2,n}$ has degree $2N$.
Let $v_{2,n}$ be any vertex of $\Lambda_{2,n}$. Let $\Lambda_{2,n}'$ be obtained from $\Lambda_{2,n}$ by removing one of the edges incident to $v_{2,n}$. Note that $\Lambda_{2,n}'$ is still finite, connected and cyclically reduced.
Then, as in the previous case, there exists a simplicial segment $[u_1,u_2]$ consisting of three edges labelled by elements of $A^{\pm 1}$ such that $$\Lambda_n:=\Lambda_{1,n}\cup \Lambda_{2,n}' \cup [u_1,u_2] / \sim$$ is a folded $R_N$-graph, where $u_1\sim v_{1,n}$, $u_2\sim v_{2,n}$. Again, by construction, $\Lambda_n$ is connected and cyclically reduced. In the case where $\Lambda_{2,n}$ has a vertex of degree $<2N$ but that every vertex in $\Lambda_{1,n}$ has degree $2N$, we define $\Lambda_n$ in a similar way.
Suppose now that for $i=1,2$ every vertex of $\Lambda_{i,n}$ has degree $2N$. Then $\Lambda_{i,n}$ is a finite cover of $R_N$, so that $\Delta_1,\Delta_2$ are both finite covers of $R_N$. Let $m_i\ge 1$ be such that $\Delta_i$ is an $m_i$-fold cover of $R_N$. Choose $\Lambda_n$ to be any connected $n(m_1+m_2)$-fold cover of $R_N$.
This defines the sequence $\Lambda_{n}$, $n=1,2,\dots$ of finite, connected $R_N$-core graphs. We claim that $$\lim_{n\to\infty} \frac{1}{n} \mu_{\Lambda_n} =\mu_{\Delta_1}+\mu_{\Delta_2} \text{ in } {\mathcal S{\mbox{Curr}}({F_N})}.\tag{!}$$ We need to show that for every finite non-degenerate subtree $K$ of $X$ we have $$\lim_{n\to\infty} \frac{1}{n} (K; \Lambda_n)= (K; \Delta_1) +( K; \Delta_2).\tag{!!}$$ Let us fix such a subtree $K$. Let $n\ge 1$ be arbitrary. If $\Lambda_n$ is of the last described type, where both $\Lambda_{1,n}$ and $\Lambda_{2,n}$ are finite covers of $R_N$, then by construction $\mu_{\Lambda_n}=n(\mu_{\Delta_1}+\mu_{\Delta_2})$, and (!!) obviously holds. Suppose now that one of the other cases in the construction of $\Lambda_n$ occurs. Since the graph $\Lambda_n$ is folded, and has a vertex degree $\le 2N$, all but a bounded number (in terms of some constant depending on $K$ but independent of $n$), occurrences of $K$ in $\Lambda_n$ are disjoint from the segment $[v_{1,n}, v_{2,n}]$ and thus come from occurrences of $K$ in $\Lambda_{1,n}\sqcup \Lambda_{2,n}$. Hence $$\big| (K; \Lambda_n) - ( K; \Lambda_{1,n}) - ( K; \Lambda_{2,n}) \big| \le C_K$$ for some constant $C_K$ independent of $n$. Since $\Lambda_{i,n}$ is an $n$-fold cover of $\Delta_i$, it follows that $$\big| ( K; \Lambda_n) - n(K; \Delta_{1}) - n(K; \Delta_{2}) \big| \le C_K.$$ Dividing the above inequality by $n$ and passing to the limit as $n\to\infty$, we get (!!), as required. This implies (!). By Theorem \[thm:cc\] $\mu_{\Lambda_n}=\eta_{L_n}$ for some finitely generated nontrivial $L_n\le F_N$. Hence (!) implies that $$\lim_{n\to\infty} \frac{1}{n} \eta_{L_n} =\eta_{H_1}+\eta_{H_2} \text{ in } {\mathcal S{\mbox{Curr}}({F_N})},$$ which completes the proof.
Subset currents as measures on the space of rooted trees {#subsec:root}
--------------------------------------------------------
Recall that $\mathcal T(X)$ denotes the set of infinite subtrees of $X$ without degree-1 vertices. Denote by $\mathcal T_1(X)$ the set of all those $T\in \mathcal T(X)$ that contain the vertex $1\in F_N$. Thus all elements of $\mathcal T_1(X)$ come equipped with a base-point (or root), namely $1\in F_N$. Note also that since every element $Y\in \mathcal T_1(X)$ is folded, $Y$ does not admit any nontrivial $\Gamma$-graph automorphisms that fix the root vertex $1$. Hence we can also think of $\mathcal T_1(X)$ as the set of rooted $R_N$-graphs that are folded infinite trees without degree-1 vertices.
We equip $\mathcal T_1(X)$ with local topology, namely we say that $Y, Y'\in \mathcal T_1(X)$ are close if for some large $n\ge 1$ we have $Y\cap B_X(n)=Y'\cap B_X(n)$, where $B_X(n)$ is the ball of radius $n$ in $X$. Equivalently, $Y$ and $Y'$ are close if $\partial Y, \partial Y'\subseteq \partial X$ have small Hausdorff distance as closed subsets of $\partial X$. This makes $\mathcal T_1(X)$ a compact totally disconnected topological space.
Recall that for any finite non-degenerate subtree $K$ of $X$ we have defined a subset cylinder set $\mathcal SCyl_\alpha(K)\subseteq \mathfrak C_N$, Definition \[defn:gcyl\]. For every such $K$ containing $1\in F_N$, put $\mathcal T Cyl_\alpha(K)$ to be the set of all $Y\in \mathcal T_1(X)$ such that $\partial Y\in \mathcal SCyl_\alpha(K)$. Thus $\mathcal T Cyl_\alpha(K)$ consists of all $Y\in \mathcal T_1(X)$ such that $K\subseteq Y$ and such that whenever $\xi\in \partial Y$, then $[1,\xi)\cap K=[1,v]$ where $v$ is a vertex of degree one in $K$. The sets $\mathcal T Cyl_\alpha(K)$ are compact and open, and form a basis of open sets for the topology on $\mathcal T_1(X)$, when $K$ varies over all finite non-degenerate subtrees of $X$ containing $1\in F_N$.
The space $\mathcal T_1(X)$ has a natural (partially defined) root-change operation. For $g\in F_N$ denote by $\mathcal T_{1,g}(X)$ the set of all $Y\in \mathcal T_1(X)$ that contain the vertex $g\in F_N$. Define $r_g: \mathcal T_{1,g}(X)\to\mathcal T_1(X)$ by $r_g(Y):=g^{-1}Y$ for $Y\in \mathcal T_{1,g}(X)$. In other words, $r_g$ moves the root vertex from $1$ to $g$ in $Y$. Denote by $\mathcal M_1(X)$ the set of all finite positive Borel measures on $\mathcal T_1(X)$ that are invariant under all $r_g, g\in F_N$.
\[prop:m1\] There is a canonical $\mathbb R_{\ge 0}$-linear homeomorphism $\mathbf t: \mathcal M_1(X) \to {\mathcal S{\mbox{Curr}}({F_N})}$.
Let $\mu\in \mathcal M_1(X)$. Define a measure $\mathbf t(\mu)$ on $\mathfrak C_N$ as follows. For a finite non-degenerate subtree $K$ of $X$ containing $1$ put $$( [K]; \mathbf t(\mu))_\alpha:=\mu\left( \mathcal T Cyl_\alpha(K) \right).\tag{$\clubsuit$}$$ Invariance of $\mu$ with respect to the root-change implies that if $K, K'$ are two finite non-degenerate subtrees of $X$ that contain $1$ and such that $K'=gK$ for some $g\in F_N$, then $\mu\left( \mathcal T Cyl_\alpha(K) \right)=\mu\left( \mathcal T Cyl_\alpha(K') \right)$, so that $( [K]; \mathbf t(\mu))_\alpha$ is well-defined.
The assumption that $\mu$ is invariant with respect to $\{r_g: g\in F_N\}$ translates into the fact that $(\clubsuit)$ defines a collection of weights satisfying the requirements of Proposition \[prop:weights\] so that $\mathbf t(\mu)$ is indeed a subset current. It is then easy to check that $\mathbf t$ is an $\mathbb R_{\ge 0}$-linear homeomorphism and we leave the details to the reader.
Weak approximation by finite graphs {#subsec:approx}
-----------------------------------
Consider the set $U_{N}$ of all $R_N$-graphs with vertices of degree $\le 2N$. For every integer $R\ge 1$ denote by $U_{N,R}\subset U_N$ the set of all rooted $R$-balls in graphs from $U_{N}$. Finally, denote by $U_{N,R}^f$ the set of all $K\in U_{N,R}$ such that $K$ is a folded tree, where the root vertex is not of degree 1 and such that the distance from the root to every vertex of degree 1 of $K$ is equal to $R$.
Observe that every tree $K\in U_{N,R}^f$ admits a unique (and injective) $R_N$-graph morphism to $X$ which sends the root in $K$ to the vertex $1\in VX$. Thus we can think of trees $K\in U_{N,R}^f$ as subtrees of $X$ with root $1$.
For $R\ge 1$, $\Upsilon\in U_{N,R}$ and an $R_N$-graph $\Delta$ denote by $J(\Upsilon,R,\Delta)$ the set of all those vertices $v$ in $\Delta$ such that the $R$-ball centered at $v$ in $\Delta$ is isomorphic as a rooted $R_N$-graph to $\Upsilon$.
We say that a subset current $\mu\in {\mathcal S{\mbox{Curr}}({F_N})}$ is *normalized* if the corresponding measure $\mu' = (\mathbf t)^{-1}(\mu)\in \mathcal M_1(X)$ is of total mass $1$, $\mu'(\mathcal T_1(X))=1$. Note that if $\mu\in {\mathcal S{\mbox{Curr}}({F_N})}$ is a nonzero subset current and $\mu'=(\mathbf t)^{-1}(\mu)\in \mathcal M_1(X)$ then for $c=\mu'(\mathcal T_1(X))$ the current $\frac{1}{c}\mu\in{\mathcal S{\mbox{Curr}}({F_N})}$ is normalized.
The following definition is an adaptation, to our notations and to our context, of the definition of random weak limit of finite rooted graphs introduced by Benjamini and Schramm, [@BS].
\[defn:wa\] Let $\mu\in {\mathcal S{\mbox{Curr}}({F_N})}$ be a normalized subset current. We say that a sequence of $R_N$-graphs $\Delta_n$ *weakly approximates* $\mu\in
{\mathcal S{\mbox{Curr}}({F_N})}$ if the following conditions hold:
1. $\Delta_n\in U_{N}$ for every $n\ge 1$.
2. $\#V\Delta_n\to\infty$ as $n\to\infty$.
3. For every $R\ge 1$ and every $K\in U_{N,R}^f$ we have $$\lim_{n\to\infty} \frac{\#J(K,R,\Delta_n)}{\#V\Delta_n}=(K;\mu).$$
4. For every $R\ge 1$ and every $\Upsilon\in U_{N,R}$ such that $\Upsilon\not\in U_{N,R}^f$ we have $$\lim_{n\to\infty} \frac{\#J(\Upsilon,R,\Delta_n)}{\#V\Delta_n}=0.$$
In this case we write $$\Delta_n \xrightarrow{\rm w.a.} \mu.$$
\[lem:wa\] Let $\mu\in {\mathcal S{\mbox{Curr}}({F_N})}$ be a normalized current and let $\Delta_n$ be a sequence of finite connected $R_N$-core graphs such that $\Delta_n \xrightarrow{\rm w.a.} \mu$. Then $$\lim_{n\to\infty} \frac{1}{\# V\Delta_n} \mu_{\Delta_n}=\mu \text{ in
} {\mathcal S{\mbox{Curr}}({F_N})}.$$
Let $R\ge 1$ and let $K\in U_{N,R}^f$. Recall that we think of $K$ as a subtree of $X$ with the root of $K$ being the vertex $1\in VX$. Denote by $(K;\Delta_n)_{bad}$ the number of all occurrences $f:K\to\Delta_n$ such that $f$ is not an embedding. Then the $R$-ball in $\Delta_n$ around the $f$-image of the base-point of $K$ is not a tree. Since the set $U_{N,R}$ is finite and $\Delta_n \xrightarrow{\rm w.a.} \mu$, the definition of weak approximation then implies that $$\lim_{n\to\infty} \frac{(K;\Delta_n)_{bad}}{\#V\Delta_n}=0.$$ We also have $(K;\Delta_n)=\#J(K,R,\Delta_n)+(
K;\Delta_n)_{bad}$. Also, since $\Delta_n \xrightarrow{\rm w.a.} \mu$, we have $$\lim_{n\to\infty} \frac{\#J(K,R,\Delta_n)}{\#V\Delta_n}=(K;\mu).$$ Therefore $$\lim_{n\to\infty} \frac{(K;\Delta_n)}{\#V\Delta_n}=(K;\mu),$$ that is, $$\lim_{n\to\infty} \frac{(K;\mu_{\Delta_n})}{\#V\Delta_n}=(K;\mu).\tag{$\spadesuit$}$$
Now let $K$ be an arbitrary finite non-degenerate subtree of $X$ containing $1\in VX$. Put $R$ to be the maximum of the distances in $K$ from $1$ to other vertices of $K$. Then $K$ is precisely the $R$-ball in $K$ centered at $1$, so that $K\in U_{N,R}^f$. Therefore $(\spadesuit)$ holds for every finite subtree of $X$ containing $1$ and hence, by $F_N$-invariance of subset currents, for every finite subtree $K$ of $X$. Hence $$\lim_{n\to\infty} \frac{1}{\# V\Delta_n} \mu_{\Delta_n}=\mu \text{ in
} {\mathcal S{\mbox{Curr}}({F_N})},$$ as required.
\[lem:blowup\] Let $\mu\in {\mathcal S{\mbox{Curr}}({F_N})}$ be a normalized current and let $\Delta_n$ be a sequence of $R_N$-graphs such that $\Delta_n \xrightarrow{\rm w.a.} \mu$. Then there exists a sequence of folded $R_N$-graphs $\Delta_n'$ such that $\Delta_n' \xrightarrow{\rm w.a.} \mu$.
Let us say that a vertex $v$ of $\Delta_n$ is *folded* if the ball of radius $1$ in $\Delta_n$ around $v$ is a folded $R_N$-graph. Let $\Upsilon_N$ be the $R_N$-graph which is a simplicial circle of length $N^2$ with the label $a_1^Na_2^N\dots a_N^N$. For every non-folded vertex $v$ of $\Delta_n$ choose a copy $\Upsilon_{N,v}$ of $\Upsilon_N$. We now perform a blow-up operation on $\Delta_n$ as follows. Simultaneously, for every non-folded vertex $v$ of $\Delta_n$ we cut $\Delta_n$ open at $v$, so that $v$ gives $deg_{\Delta_n}(v)\le 2N$ new vertices, and then we attach these new vertices to $\Upsilon_{N,v}$ is a way that the resulting graph is folded. The resulting folded $R_N$-graph is denoted by $\Delta_n'$.
The definition of weak approximation implies that $\lim_{n\to\infty}
{b_n}/{\#V\Delta_n}=0$, where $b_n$ is the number of non-folded vertices of $\Delta_n$. Since the number of vertices in the graph $\Upsilon_N$ (used to blow-up non-folded vertices of $\Delta_n$) is equal to $N^2$ and does not depend on $n$, the definition of weak approximation now implies that $\Delta_n' \xrightarrow{\rm w.a.} \mu$, as required.
\[lem:loops\] Let $\mu\in {\mathcal S{\mbox{Curr}}({F_N})}$ be a normalized current and let $\Delta_n$ be a sequence of folded $R_N$-graphs such that $\Delta_n \xrightarrow{\rm w.a.} \mu$. Then there exists a sequence of $R_N$-core graphs $\Delta_n'$ such that $\Delta_n' \xrightarrow{\rm w.a.} \mu$.
For every vertex $v$ of degree $1$ in $\Delta_n$ attach a new loop-edge at $v$ with label $a\in A$ such that $a^{\pm 1}$ is different from the label of the unique edge of $\Delta_n$ starting at $v$. Denote the resulting graph by $\Delta_n'$.
Then $\Delta_n'$ is a folded and cyclically reduced $R_N$-graph. It is easy to see, from the definition of weak approximation, that $\Delta_n' \xrightarrow{\rm w.a.} \mu$.
Rational currents are dense {#subsec:result}
---------------------------
\[thm:dense\] The set $\mathcal SCurr_r(F_N)$ of all rational currents is a dense subset of ${\mathcal S{\mbox{Curr}}({F_N})}$.
Let $\mu\in {\mathcal S{\mbox{Curr}}({F_N})}$ be an arbitrary nonzero current. Then for some $c>0$ the current $\overline\mu:=\frac{1}{c}\mu$ is normalized.
A slight modification of the main result of [@Elek], together with Proposition \[prop:m1\] imply that there exists a sequence $\Delta_n$ of $R_N$-graphs such that $\Delta_n \xrightarrow{\rm
w.a.} \overline\mu$. By Lemma \[lem:blowup\] and Lemma \[lem:loops\] we may modify the sequence $\Delta_n$ to get a new sequence of $R_N$-graphs, which we again denote $\Delta_n$, such that each $\Delta_n$ is a core graph and that $\Delta_n \xrightarrow{\rm w.a.} \overline\mu$. Now Lemma \[lem:wa\] implies that $$\lim_{n\to\infty} \frac{1}{\# V\Delta_n} \mu_{\Delta_n}=\overline\mu \text{ in
} {\mathcal S{\mbox{Curr}}({F_N})}.$$
Let $\Delta_{n,1}, \dots \Delta_{n,k_n}$ be the connected components of $\Delta_n$. Then $$\mu_{\Delta_n}=\mu_{\Delta_{n,1}}+\dots+\mu_{\Delta_{n,k_n}}$$ and we see that $\overline\mu$ belongs to the closure of the $\mathbb R_{\ge
0}$-span of $\mathcal SCurr_r(F_N)$. Proposition \[prop:span\] now implies that $\overline\mu$ belongs to the closure of $\mathcal SCurr_r(F_N)$. Hence $\mu=c\overline \mu$ also belongs to the closure of $\mathcal SCurr_r(F_N)$. Since $\mu\in {\mathcal S{\mbox{Curr}}({F_N})}$ was an arbitrary nonzero current, this completes the proof.
In the proof of Theorem \[thm:dense\], instead of the result of Elek [@Elek] invoked above, alternatively one can use the results of Bowen [@Bow03; @Bow09].
The action of ${\mbox{Out}}(F_N)$ {#sec:action}
=================================
Defining the ${\mbox{Out}}(F_N)$-action {#subsec:defaction}
---------------------------------------
If $\phi\in {\mbox{Aut}}(F_N)$ is an automorphism, then $\phi$ is a quasi-isometry of $F_N$ and hence $\phi$ extends to a homeomorphism (which we will still denote by $\phi$) $\phi:\partial F_N\to\partial
F_N$. Thus for $S\in \mathfrak C_N$ we have $\phi(S)\in \mathfrak C_N$. For a subset $U\subseteq \mathfrak C_N$ we write $\phi(U):=\{\phi(S):
S\in U\}$. Thus ${\mbox{Aut}}(F_N)$ has a natural action on $\mathfrak C_N$ and on the set of subsets of $\mathfrak C_N$.
\[defn:action\] Let $\phi\in {\mbox{Aut}}(F_N)$ and let $\mu\in {\mathcal S{\mbox{Curr}}({F_N})}$.
Define a measure $\phi\mu$ on $\mathfrak C_N$ as follows:
For a Borel subset $U\subseteq \mathfrak C_N$ we put $$\phi\mu(U):=\mu\left(\phi^{-1}(U)\right).$$
\[prop:action\] Let $N\ge 2$.
1. For any $\phi\in {\mbox{Aut}}(F_N)$ and $\mu\in {\mathcal S{\mbox{Curr}}({F_N})}$ we have $\phi\mu\in {\mathcal S{\mbox{Curr}}({F_N})}$.
2. For any $\phi_1,\phi_2\in {\mbox{Aut}}(F_N)$ and $\mu\in {\mathcal S{\mbox{Curr}}({F_N})}$ we have $(\phi_1\phi_2)(\mu)=\phi_1(\phi_2\mu)$.
3. For any $\phi\in {\mbox{Aut}}(F_N)$ $\phi:{\mathcal S{\mbox{Curr}}({F_N})}\to
{\mathcal S{\mbox{Curr}}({F_N})}$ is an $\mathbb R_{\ge 0}$-linear homeomorphism.
4. If $\phi\in {\mbox{Aut}}(F_N)$ is an inner automorphism then $\phi\mu=\mu$ for every $\mu\in {\mathcal S{\mbox{Curr}}({F_N})}$.
To see (1) note that for any $g\in F_N$ and $\xi\in \partial F_N$ we have $\phi^{-1}(g\xi)=\phi^{-1}(g)\phi^{-1}(\xi)$. Hence if $U\subseteq \mathfrak C_N$ and $g\in F_N$ then, in view of $F_N$-invariance of $\mu$, we have: $$(\phi\mu)(gU)=\mu(\phi^{-1}(gU))=\mu(\phi^{-1}(g) \phi^{-1}(U))=\mu(\phi^{-1}(U))=(\phi\mu)(U).$$ Thus $\phi\mu$ is an $F_N$-invariant measure, and hence $\phi\mu\in
{\mathcal S{\mbox{Curr}}({F_N})}$ and (1) is established.
Now (2) and (3) follow directly from (1) and Definition \[defn:action\].
Part (4) follows from the fact that if $\psi\in {\mbox{Aut}}(F_N)$ is inner, that is $\psi(g)=hgh^{-1}$ for every $g\in F_N$, then $\psi(\xi) =h\xi$ for every $\xi\in \partial F_N$.
Proposition \[prop:action\] shows that Definition \[defn:action\] defines a left action of ${\mbox{Aut}}(F_N)$ on ${\mathcal S{\mbox{Curr}}({F_N})}$ by $\mathbb R_{\ge 0}$-linear homeomorphisms. Moreover, inner automorphisms of $F_N$ lie in the kernel of this action, and therefore this action factors through to the action of ${\mbox{Out}}(F_N)$ on ${\mathcal S{\mbox{Curr}}({F_N})}$.
Also, by $\mathbb R_{\ge 0}$-linearity, we have $\phi(r\mu)=r\phi(\mu)$ for $r\ge 0$, $\mu\in {\mathcal S{\mbox{Curr}}({F_N})}$ and $\phi\in {\mbox{Aut}}(F_N)$. Therefore the action of ${\mbox{Aut}}(F_N)$ on ${\mathcal S{\mbox{Curr}}({F_N})}$ also yields an action of ${\mbox{Aut}}(F_N)$ on ${\mathbb P{\mathcal S{\mbox{Curr}}({F_N})}}$ given by $\phi[\mu]:=[\phi\mu]$, where $\mu\in{\mathcal S{\mbox{Curr}}({F_N})}, \mu\ne 0$ and $\phi\in {\mbox{Aut}}(F_N)$. As before, the last action factors through to the action of ${\mbox{Out}}(F_N)$ on ${\mathbb P{\mathcal S{\mbox{Curr}}({F_N})}}$.
\[prop:r-act\] Let $H\le F_N$ be a nontrivial finitely generated subgroup and let $\phi\in {\mbox{Aut}}(F_N)$. Then $\phi(\eta_H)=\eta_{\phi(H)}$.
For $m\ge 1$, if $[H:H_1]=m$, then $[\phi(H):\phi(H_1)]=m$ and $\eta_{H_1}=m\eta_{H}$, $\eta_{\phi(H_1)}=m\eta_{\phi(H)}$. Hence, by linearity, it suffices to establish the proposition for the case $H=Comm_{F_N}(H)$. Thus assume that $H\le F_N$ is a nontrivial finitely generated subgroup with $H=Comm_{F_N}(H)$. Recall that $$\eta_{H}=\sum_{H'\sim H} \delta_{\Lambda(H')}.$$
It is easy to see that for any subgroup $G\le F_N$ we have $\phi(\Lambda(G))=\Lambda(\phi(G))$.
Let $U\subseteq \mathfrak C_N$. By Definition \[defn:action\], $\phi\eta_H(U)=\eta_H(\phi^{-1}(U))$ is equal to the number of elements of the form $\Lambda(H')$ (where $H'\sim H$) that belong to $\phi^{-1}(U)$, which, in turn, is equal to the number of elements of the form $\phi\Lambda(H')$ (where $H'\sim H$) that belong to $U$. Hence $$\begin{gathered}
\phi(\eta_H)=\sum_{H'\sim H} \delta_{\phi\Lambda(H')}=\sum_{H'\sim H} \delta_{\Lambda(\phi(H'))}=\\
=\sum_{K\sim \phi(H)} \delta_{\Lambda(K)}=\eta_{\phi(H)},\end{gathered}$$ as required.
Proposition \[prop:r-act\] implies that the action of ${\mbox{Aut}}(F_N)$ has a global nonzero fixed point, namely the current $\eta_{F_N}=\delta_{\partial F_N}$. Indeed, for any $\phi\in {\mbox{Aut}}(F_N)$ we have $$\phi\eta_{F_N}=\eta_{\phi(F_N)}=\eta_{F_N}.$$ Moreover, all scalar multiples $r\eta_{F_N}$, where $r\ge 0$, are also fixed by ${\mbox{Aut}}(F_N)$. In particular, since applying an automorphism to a subgroup of $F_N$ preserves the index of the subgroup, if $[F_N:H]<\infty$ then $\phi\eta_H=\eta_H$ for every $\phi\in {\mbox{Aut}}(F_N)$.
Local formulas {#subsec:local}
--------------
Similarly to the case of ${\mbox{Curr}}(F_N)$, we can get more explicit local formulas, with respect to a marking, for the action of ${\mbox{Out}}(F_N)$ on ${\mathcal S{\mbox{Curr}}({F_N})}$.
\[prop:local\] Let $\alpha:F_N{\mathrel{\text{ \setbox0\hbox{$\rightarrow$} \rlap{\hbox to \wd0{\hss\raisebox{0.9\height}{$\sim$}\hss}}\box0
}}}\pi_1(\Gamma)$ be a marking on $F_N$ and let $X$ denote the universal cover $\widetilde \Gamma$. As before (see \[prop:weights\]), let $\mathcal K_\Gamma$ denote the set of all non-degenerate finite simplicial subtrees of $X$. For any $\phi\in {\mbox{Out}}(F_N)$ and any $K\in\mathcal K_\Gamma$ there exist $m\ge 1$ and $K_1\dots, K_m\in \mathcal K_\Gamma$ with the following property: For every $\mu\in{\mathcal S{\mbox{Curr}}({F_N})}$ we have $$(K; \phi(\mu))_\alpha=\sum_{i=1}^m (K_i; \mu)_\alpha.$$
Since inner automorphisms are contained in the kernel of the action of ${\mbox{Aut}}(F_N)$ on ${\mathcal S{\mbox{Curr}}({F_N})}$, it suffices to prove the statement of the proposition under the assumption that $\phi\in {\mbox{Aut}}(F_N)$.
Let $\phi\in {\mbox{Aut}}(F_N)$ be arbitrary. Since $\mathcal SCyl_\alpha(K)\subseteq \mathfrak C_N$ is compact, the set $\phi^{-1}\mathcal SCyl_\alpha(K)$ is also compact. Since subset cylinders are not just compact, but also open, $\phi^{-1}\mathcal SCyl_\alpha(K)$ is covered by finitely many subset cylinders. By Lemma \[lem:disj\], after subdividing them, we may assume that $\phi^{-1}\mathcal SCyl_\alpha(K)$ is covered by finitely many pairwise disjoint subset cylinders: $$\phi^{-1}\mathcal SCyl_\alpha(K) =\sqcup_{i=1}^m \mathcal SCyl_\alpha(K_i).$$ Put $M:=\max_{i=1}^m \#EK_i$. By definition of $\phi\mu$ we have
$$\begin{gathered}
(K; \phi\mu)_\alpha =\phi\mu\left(\mathcal SCyl_\alpha(K) \right)=\mu\left(\phi^{-1}\mathcal SCyl_\alpha(K) \right)=\\
\mu\left(\sqcup_{i=1}^m \mathcal SCyl_\alpha(K_i) \right)=\sum_{i=1}^m \mu(\mathcal SCyl_\alpha(K_i) )=\sum_{i=1}^m ( K_i; \mu)_\alpha, \end{gathered}$$
as required.
As in the case of ${\mbox{Curr}}(F_N)$ [@Ka2], a more detailed argument for the proof of Proposition \[prop:local\] shows that, given $\phi\in {\mbox{Out}}(F_N)$ and $K\in \mathcal K_\Gamma$, one can algorithmically find the trees $K_1,\dots, K_m$ satisfying the conclusion of Proposition \[prop:local\].
The co-volume form {#sec:intform}
==================
Constructing co-volume form {#subsec:constr}
---------------------------
For $T\in{\mbox{cv}_N}$ and a nontrivial finitely generated subgroup $H\le F_N$ put $$||H||_T:=vol(H\setminus T_H)$$ where $T_H$ is the smallest $H$-invariant subtree of $T$. It is also the convex hull of the limit set of $H$, see Proposition \[prop:basic\].
Note that if $H=\langle g\rangle$, where $g\in F_N, g\ne 1$ then $||H||_T=||g||_T$, the translation length of $T$.
\[lem:volH\] Let $T\in{\mbox{cv}_N}$, let $H\le F_N$ be a nontrivial finitely generated subgroup and let $\phi\in {\mbox{Aut}}(F_N)$. Then $||\phi(H)||_{\phi T}=||H||_T$.
Recall, that as a metric space, $\phi T$ is equal to $T$, but the action of $F_N$ on $\phi T$ is defined as $$g\underset{\phi T\phi}{\cdot}
x=\phi^{-1}(g)\underset{T}{\cdot} x$$ where $x\in T$, $g\in F_N$. It follows that the tree $T_H\subseteq T$ is $\phi(H)$-invariant with respect to the $F_N$-action on $\phi T$, and, moreover it is the smallest $\phi(H)$-invariant subtree of $\phi T$. Also, it is easy to see that, as metric graphs, $H\setminus T_H$ is equal to $\phi(H)\setminus (\phi
T)_{\phi(H)}$. Hence $||\phi(H)||_{\phi T}=||H||_T$, as claimed.
\[defn:intform\] Let $N\ge 2$. Define a map $$\langle\, , \, \rangle: {\mbox{cv}_N}\times{\mathcal S{\mbox{Curr}}({F_N})}\to\mathbb R_{\ge 0}$$ as follows. Let $T\in{\mbox{cv}_N}$ and $\mu\in {\mathcal S{\mbox{Curr}}({F_N})}$. Let $\alpha:F_N\to\pi_1(\Gamma)$ be a marking and let $\mathcal L$ be a metric graph structure on $\Gamma$ such that $T=(\widetilde\Gamma,
d_\mathcal L)$ in ${\mbox{cv}_N}$. Put $$\langle T, \mu\rangle:=\sum_{e\in E_{top}(\Gamma)} (e;
\mu)_\alpha \mathcal L(e) \tag{$\spadesuit$}$$ where $(e;\mu)_\alpha$ is as in Notation \[not:e\]. Then the following hold:
1. The map $\langle\, , \, \rangle: {\mbox{cv}_N}\times{\mathcal S{\mbox{Curr}}({F_N})}\to\mathbb R_{\ge 0}$ is continuous, $\mathbb R_{\ge 0}$-linear with respect to the second argument and $\mathbb R_{\ge 0}$-homogeneous with respect to the first argument.
2. The map $\langle\, , \, \rangle$ is ${\mbox{Out}}(F_N)$-equivariant, that is, for any $T\in{\mbox{cv}_N}$, $\mu\in {\mathcal S{\mbox{Curr}}({F_N})}$ and $\phi\in {\mbox{Out}}(F_N)$ we have $$\langle \phi T, \phi\mu\rangle=\langle T,\mu\rangle .$$ In other words, in terms of using the right ${\mbox{Out}}(F_N)$-action on ${\mbox{cv}_N}$, for all $T\in{\mbox{cv}_N}$ and $\phi\in {\mbox{Out}}(F_N)$ we have $$\langle T\phi, \mu\rangle=\langle T,\phi\mu\rangle .$$
3. For any finitely generated nontrivial subgroup $H\le F_N$ and any $T\in {\mbox{cv}_N}$ we have $$\langle T, \eta_H\rangle=||H||_T$$ where $T_H$ is the minimal $H$-invariant subtree of $T$.
We call the map $\langle\, , \, \rangle: {\mbox{cv}_N}\times{\mathcal S{\mbox{Curr}}({F_N})}\to\mathbb R_{\ge 0}$ *the co-volume form*.
The continuity of $\langle\, , \, \rangle$ is a straightforward consequence of its definition, and the argument is essentially identical to that used in [@Ka2] to show continuity of the intersection form ${\mbox{cv}_N}\times Curr(F_N)\to \mathbb R_{\ge 0}$. We refer the reader to the proof of Proposition 5.9 in [@Ka2] for details.
The definition also directly implies that $$\langle T, c_1\mu_1+c_2\mu_2\rangle = c_1\langle T, \mu_1\rangle +c_2\langle T, \mu_2\rangle$$ for $c_1,c_2\ge 0$, $T\in {\mbox{cv}_N}$ and $\mu_1,\mu_2\in {\mathcal S{\mbox{Curr}}({F_N})}$ and that $$\langle cT, \mu\rangle= c\langle T,\mu\rangle$$ for $c\ge 0$, $T\in {\mbox{cv}_N}$, $\mu\in {\mathcal S{\mbox{Curr}}({F_N})}$. Thus (1) holds.
We now check that (3) holds. Let $H\le F_N$ be a nontrivial finitely generated subgroup. Recall that, $\Delta=H\setminus T_H$ is exactly the $\Gamma$-core graph representing the conjugacy class $[H]$ (see Convention \[conv:subgroupcore\]. By Theorem \[thm:cc\] we have $\eta_H=\mu_\Delta$. The edges of $\Delta$ in $\Delta=H\setminus T_H$ have the same lengths as the $\mathcal L$-lengths of the corresponding edges of $\Gamma$. Thus the $\mathcal L$-volume of $\Delta$ is the sum of the $\mathcal L$-lengths of all the edges of $\Delta$, so that $$\begin{gathered}
||H||_T=vol(H\setminus T_H)= vol_\mathcal L(\Delta)=\sum_{e\in E_{top}(\Gamma)} (\widetilde e; \Delta) \mathcal L(e)=\\ \sum_{e\in E_{top}(\Gamma)} (e; \mu_\Delta) \mathcal L(e)= \sum_{e\in E_{top}(\Gamma)} ( e; \eta_H) \mathcal L(e)=\langle T, \eta_H\rangle,\end{gathered}$$ and (3) is verified.
We can now show that (2) holds. Since inner automorphisms of $F_N$ act trivially on both ${\mbox{cv}_N}$ and ${\mathcal S{\mbox{Curr}}({F_N})}$, it suffices to check (2) for elements of ${\mbox{Aut}}(F_N)$ (rather than of ${\mbox{Out}}(F_N)$). Let $\phi\in {\mbox{Aut}}(F_N)$.
We need to show that for any $T\in {\mbox{cv}_N}$ and $\mu\in {\mathcal S{\mbox{Curr}}({F_N})}$. $$\langle \phi T, \phi\mu\rangle=\langle T,\mu\rangle.$$ By continuity of the intersection form, already established in (1) and by Theorem \[thm:dense\], it suffices to verify the above formula for the case where $T\in {\mbox{cv}_N}$ is arbitrary and where $\mu$ is a rational current. Thus let $\mu=c\eta_H$ where $c\ge 0$ and $H\le F_N$ is a nontrivial finitely generated subgroup. We have $$\begin{gathered}
\langle \phi T, \phi c\eta_H\rangle=c\langle \phi T, \phi\eta_H\rangle=c\langle \phi T, \eta_{\phi(H)}\rangle=\\
c||\phi(H)||_{\phi T} =c||H||_T=c\langle T, \eta_H\rangle=\langle T, c\eta_H\rangle,\end{gathered}$$ as required.
Non-existence of a continuous extension of the co-volume form to ${\overline{\mbox{cv}}_N}$ {#subsec:exist}
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It turns out that, unlike for ordinary currents, there does not exist a continuous extension of the co-volume form to ${\overline{\mbox{cv}}_N}\times{\mathcal S{\mbox{Curr}}({F_N})}$. Before proving this statement, we need to recall a few background facts regarding the dynamics of the action of iwip elements and of Dehn twists elements of ${\mbox{Out}}(F_N)$ on ${\overline{\mbox{cv}}_N}$ and ${\overline{\mbox{CV}}_N}$. Recall that an element $\phi\in {\mbox{Out}}(F_N)$ is called an *iwip* (which stands for irreducible with irreducible powers) or *fully irreducible* if there do not exist $m\ne 0$ and a proper free factor $L$ of $F_N$ such that $\phi^m([L])=[L]$, where $[L]$ is the conjugacy class of $L$.
The following North-South dynamics result for iwips is well-known and was obtained by Levitt and Lustig in [@LL]:
[@LL]\[prop:LL\] Let $N\ge 2$ and let $\phi\in {\mbox{Out}}(F_N)$ be an iwip. Then there exist unique $[T_+]=[T_+(\phi)], [T_-]=[T_-(\phi)]\in {\overline{\mbox{CV}}_N}$ and $\lambda_+=\lambda_+(\phi)>1$, $\lambda_-=\lambda_-(\phi)>1$ with the following properties:
1. We have $T_+\phi=\lambda_+ T_+$, $T_-\phi=\frac{1}{\lambda_-} T_-$, so that $[T_-]\phi=[T_-]$ and $[T_+]\phi=[T_+]$.
2. For any $[T]\in {\overline{\mbox{CV}}_N}$ such that $[T]\ne [T_-]$ we have $\lim_{n\to\infty} [T]\phi^n = [T_+]$ and for any $[T]\in {\overline{\mbox{CV}}_N}$ such that $[T]\ne [T_+]$ we have $\lim_{n\to\infty} [T]\phi^{-n} = [T_-]$.
3. For any $T\in {\mbox{cv}_N}$ we have $\lim_{n\to\infty} [T]\phi^n =[T_+]$. Moreover, given any $T\in {\mbox{cv}_N}$, we can choose a representative $T_+\in{\overline{\mbox{cv}}_N}$ of $[T_+]$ such that $$\lim_{n\to\infty} \frac{1}{\lambda_+^n} T\phi^n = T_+ \quad \text{ in } \quad {\overline{\mbox{cv}}_N}.$$
4. The action of $F_N$ on $T_+$ has dense $F_N$-orbits. Moreover, if $N\ge 3$ and $\phi\in {\mbox{Out}}(F_N)$ is an atoroidal iwip then the action of $F_N$ on $T_+$ is also free.
5. If $\theta\in {\mbox{Out}}(F_N)$ is arbitrary then for $\psi= \theta^{-1} \phi\theta$ we have $\lambda_\pm(\psi)=\lambda_{\pm}(\phi)$ and $[T_\pm (\psi)]=[T_\pm(\phi)]\theta$. Moreover, if $T\in{\mbox{cv}_N}$ and $T':=\lim_{n\to\infty} \frac{1}{\lambda_+^n} T\phi^n$ and $T'':=\lim_{n\to\infty} \frac{1}{\lambda_+^n} T\psi^n$ then $T''=T'\theta$.
The trees $[T_+(\phi)]$ and $[T_-(\phi)]$ are called the *attracting* and *repelling* trees of $\phi$ respectively.
The attracting tree $T_+(\phi)$ of an iwip $\phi$ can be understood fairly explicitly in terms of a train-track representative of $\phi$ and in fact $\lambda_+(\phi)$ is the Perron-Frobenius eigenvalue of any train-track representative of $\phi$.
Guirardel proved in [@Gui1] that for $N\ge 3$ the action of ${\mbox{Out}}(F_N)$ on $\partial {\mbox{CV}_N}={\overline{\mbox{CV}}_N}- {\mbox{CV}_N}$ is not topologically minimal and that there exists a unique proper closed ${\mbox{Out}}(F_N)$-invariant subset of $\partial {\mbox{CV}_N}$. We only need the following limited version of his result:
\[prop:Gui\][@Gui1] Let $N\ge 3$. Then there exists a unique minimal nonempty closed ${\mbox{Out}}(F_N)$-invariant subset $\mathcal M_N^{cv}\subseteq \partial {\mbox{CV}_N}$ (so that for every $[T]\in \mathcal M_N^{cv}$ the ${\mbox{Out}}(F_N)$-orbit of $[T]$ is dense in $\mathcal M_N^{cv}$). Moreover, if $T_\ast$ is the Bass-Serre tree (with all edges given length $1$) of any nontrivial free product decomposition $F_N=B\ast C$, then $[T_\ast]\in \mathcal M_N^{cv}$.
Proposition \[prop:LL\] easily implies that for any iwip $\phi\in {\mbox{Out}}(F_N)$ we have $[T_+(\phi)]\in \mathcal M_N^{cv}$.
\[prop:disc\] Let $N\ge 3$. Let $F_N=B\ast C$ be a nontrivial free product decomposition and let $T_\ast\in{\overline{\mbox{cv}}_N}$ be the corresponding Bass-Serre tree. Then there exist sequences $T_n,T_n'\in {\mbox{cv}_N}$ such that $\lim_{n\to\infty} T_n=\lim_{n\to\infty} T_n'=T_\ast$ in ${\overline{\mbox{cv}}_N}$ and that $$\lim_{n\to\infty} \langle T_n', \eta_{F_N}\rangle =1$$ but $$\lim_{n\to\infty} \langle T_n, \eta_{F_N}\rangle =0.$$
Let $\phi\in {\mbox{Out}}(F_N)$ be any iwip. Choose an arbitrary point $T\in {\mbox{cv}_N}$. Let $[T_+]=[T_+(\phi)]$ be the attracting tree of $\phi$. We may assume that $T_+(\phi)=\lim_{k\to\infty}\frac{1}{\lambda_+^k} T\phi^k$. Since both $[T_\ast]$ and $[T_+]$ belong to $\mathcal M_N^{cv}$, there exists a sequence $\theta_n\in {\mbox{Out}}(F_N)$ such that $\lim_{n\to\infty} [T_+]\theta_n= [T_\ast]$. Thus for some sequence $c_n>0$ we have $\lim_{n\to\infty} c_n T_+\theta_n= T_\ast$ in ${\overline{\mbox{cv}}_N}$. By Proposition \[prop:LL\] we know that $[T_+]\theta_n=[T_+(\psi_n)]$ where $\psi_n=\theta_n^{-1} \phi \theta_n$. Moreover, for each $n\ge 1$ we have $$T_+\theta_n=\lim_{k\to\infty} \frac{1}{\lambda_+^k} T\psi_n^k$$ Put $T_+(\psi_n):=T_+\theta_n$, and we then have $T_+\theta_n=T_+(\psi_n)= \lim_{k\to\infty} \frac{1}{\lambda_+^k} T \psi_n^k$ in ${\overline{\mbox{cv}}_N}$. For every $k\ge 1$ and $n\ge 1$ we have $$\langle \frac{1}{\lambda_+^k} T \psi_n^k, \eta_{F_N}\rangle=\frac{1}{\lambda_+^k}\langle T, \psi_n^k\eta_{F_N}\rangle=\frac{1}{\lambda_+^k}\langle T, \eta_{F_N}\rangle=\frac{{\mbox{vol}}(F_N\setminus T)}{\lambda_+^k}.$$ For each $n\ge 1$ choose $k_n\ge 1$ such that for all $k\ge k_n$ we have $c_n\langle \frac{1}{\lambda_+^k} T \psi_n^k, \eta_{F_N}\rangle=c_n \frac{{\mbox{vol}}(F_N\setminus T)}{\lambda_+^{k}}\le \frac{1}{n}$.
Since ${\overline{\mbox{cv}}_N}$ is metrizable, and since $\lim_{n\to\infty} c_n T_+(\psi_n)= T_\ast$ and $c_nT_+(\psi_n)= \lim_{k\to\infty} \frac{c_n}{\lambda_+^k} T \psi_n^k$, by a standard diagonalization argument we can find a sequence $m_n\ge k_n$ such that $\lim_{n\to\infty} \frac{c_n}{\lambda_+^{m_n}} T \psi_n^{m_n} =T_\ast$ in ${\overline{\mbox{cv}}_N}$. Put $T_n=\frac{c_n}{\lambda_+^{m_n}} T \psi_n^{m_n}$. Then by construction we have $\lim_{n\to\infty} T_n=T_\ast$ and $\langle T_n, \eta_{F_N}\rangle \le \frac{1}{n}\to 0$ as $n\to\infty$, so that $$\lim_{n\to\infty} \langle T_n, \eta_{F_N}\rangle =0.$$
To construct the sequence $T_n'$, choose a free basis $\{b_1,\dots,b_i\}$ of $B$ and a free basis $\{c_{i+1},\dots, c_N\}$ of $C$. Let $\Gamma$ be the barbell graph (with the obvious marking) consisting of a non-loop edge $e$ with $i$ loop-edges (corresponding to $b_1,\dots, b_i$) attached at $o(e)$ and with $N-i$ loop-edges (corresponding to $c_{i+1},\dots, c_N$) attached at $t(e)$. Give the edge $e$ length $1-\frac{1}{n}$ and give each of $N$ loop-edges in $\Gamma$ length $\frac{1}{Nn}$. This defines a point $T_n'\in{\overline{\mbox{cv}}_N}$ with ${\mbox{vol}}(F_N\setminus T_n')=1$. Also, by construction, $\lim_{n\to\infty} T_n'=T_\ast$ in ${\overline{\mbox{cv}}_N}$. Then $$\lim_{n\to\infty} \langle T_n', \eta_{F_N}\rangle =\lim_{n\to\infty} {\mbox{vol}}(F_N\setminus T_n')=1,$$ as required.
Proposition \[prop:disc\] immediately implies:
\[thm:disc\] Let $N\ge 3$. Then the co-volume form ${\mbox{cv}_N}\times {\mathcal S{\mbox{Curr}}({F_N})}\to R_{\ge 0}$ does not admit a continuous extension to a map ${\overline{\mbox{cv}}_N}\times {\mathcal S{\mbox{Curr}}({F_N})}\to R_{\ge 0}$.
The proof of Theorem \[thm:disc\] can be modified to cover the case $N=2$, but we only deal with the case $N\ge 3$ for simplicity.
\[rem:H\] Let $H\le F_N$ be a nontrivial finitely generated subgroup. Recall that if $T\in{\mbox{cv}_N}$ then $\langle T, \eta_H\rangle=||H||_T={\mbox{vol}}(H\setminus T_H)$. If $T\in{\overline{\mbox{cv}}_N}-{\mbox{cv}_N}$, the quotient $H\setminus T_H$ is, in general, not a nice object, and, in particular, it is not necessarily a finite metric graph. However, one can still define a reasonable notion of volume $||H||_T$ for $H\setminus T_H$. Namely, if $H$ fixes a point of $T$, put $||H||_T:=0$. Otherwise, there exists a unique minimal $H$-invariant subtree $T_H$ of $T$. In that case define $||H||_T$ as the infimum of ${\mbox{vol}}(K)$ taken over all finite subtrees $K\subseteq T_H$ such that $HK=T_H$. The proof of Proposition \[prop:disc\] exploits the fact that in general, given $H$, the function $f_H:{\overline{\mbox{cv}}_N}\to\mathbb R$, $f_H:T\mapsto ||H||_T$, is not continuous on ${\overline{\mbox{cv}}_N}$. However, one can show that $f_H$ is upper-semicontinuous.
The reduced rank functional {#sec:rrf}
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Recall that for a finitely generated free group $F$ the *rank* ${\rm rk}(F)$ is the cardinality of a free basis of $F$ and the *reduced rank* ${\overline{\rm rk}}(F)$ is defined as $${\overline{\rm rk}}(F):=\max\{{\rm rk}(F)-1, 0\}.$$ Thus is $F\ne \{1\}$ then ${\overline{\rm rk}}(F)={\rm rk}(F)-1$. If $\Delta$ is a finite connected graph and $F=\pi_1(\Delta)$ then ${\overline{\rm rk}}(F)=-\chi(\Delta)$, where $\chi(\Delta)$ is the Euler characteristic of $\Delta$. Reduced rank appears naturally in the context of the Hanna Neumann Conjecture, recently proved by Mineyev [@Min]. The conjecture (now Mineyev’s theorem) states that if $H_1,H_2\le F$ are finitely generated subgroups of a free group $F$ then $${\overline{\rm rk}}(H_1\cap H_2)\le {\overline{\rm rk}}(H_1){\overline{\rm rk}}(H_2).$$ It turns out that reduced rank extends to a $\mathbb R_{\ge 0}$-linear functional on the space of subset currents:
\[thm:rrk\] Let $N\ge 2$. Then there exists a unique continuous $\mathbb R_{\ge 0}$-linear functional $${\overline{\rm rk}}: {\mathcal S{\mbox{Curr}}({F_N})}\to\mathbb R_{\ge 0}$$ such that for every nontrivial finitely generated subgroup $H\le F_N$ we have $${\overline{\rm rk}}(\eta_H)={\overline{\rm rk}}(H).\tag{$\diamondsuit$}$$ Moreover, ${\overline{\rm rk}}$ is ${\mbox{Out}}(F_N)$-invariant, that is, for any $\phi\in{\mbox{Out}}(F_N)$ and $\mu\in {\mathcal S{\mbox{Curr}}({F_N})}$ we have ${\overline{\rm rk}}(\phi\mu)={\overline{\rm rk}}(\mu)$.
The uniqueness of ${\overline{\rm rk}}$ follows from $(\diamondsuit)$ and from the requirement that ${\overline{\rm rk}}$ be linear and continuous, since this uniquely defines ${\overline{\rm rk}}$ on the set of rational subset currents, which is dense in ${\mathcal S{\mbox{Curr}}({F_N})}$ by Theorem \[thm:dense\]. Thus it suffices to prove the existence of a functional ${\overline{\rm rk}}$ with the required properties.
Choose a free basis $A$ of $F_N$ and the corresponding marking $\alpha_A:F_N\to\pi_1(R_N)$ as in Convention \[conv:GA\]. Recall that $X=\widetilde R_N$ is the Cayley graph of $F_N$ with respect to $A$. For each $a\in A$ let $e_a$ be the topological edge in $X$ with endpoints $1,a\in F_N$. Note that every subgraph in $X$ consisting of a single edge is a translate of some $e_a$ by an element of $F_N$.
Let $\mathcal B_N$ be the set of all non-degenerate finite subtrees $K$ contained in the ball of radius $1$ in $X$ with center $1\in F_N$ such that the vertex $1\in F_N$ has degree $\ge 2$ in $K$. Note that every $K\in \mathcal B_N$ is uniquely specified by the set of its vertices of degree 1, which is a subset of $A\cup A^{-1}$ of cardinality at least $2$. Thus there are exactly $2^{2N}-2N-1$ elements in $\mathcal B_N$. Note also that if $\Delta$ is a nontrivial finite $R_N$-core graph then for every vertex $x$ of $\Delta$ there exists a unique $K\in \mathcal B_N$ such that $Lk_\Delta(x)=Lk_{K}(1)$. Define the function ${\overline{\rm rk}}:{\mathcal S{\mbox{Curr}}({F_N})}\to\mathbb R$ as follows. For $\mu\in{\mathcal S{\mbox{Curr}}({F_N})}$ $${\overline{\rm rk}}(\mu):=\sum_{a\in A} (e_a; \mu) - \sum_{K\in \mathcal
B_N} (K;\mu).$$ By construction ${\overline{\rm rk}}:{\mathcal S{\mbox{Curr}}({F_N})}\to\mathbb R$ is a continuous $\mathbb R_{\ge 0}$-linear function.
Suppose now that $H\le F_N$ is a nontrivial finitely generated subgroup. Let $\Delta$ be the finite $R_N$-core graph representing $[H]$. Then $$\begin{gathered}
{\overline{\rm rk}}(\mu_\Delta):=\sum_{a\in A} (e_a; \mu_\Delta) - \sum_{K\in \mathcal
B_N} (K;\mu_\Delta)=\\
\sum_{a\in A} (e_a; \Delta) - \sum_{K\in \mathcal
B_N} (K;\Delta).\end{gathered}$$ It is easy to see, from the definition of an occurrence (Definition \[defn:occur\]), that $\sum_{a\in A} (e_a; \Delta)=\#E_{top}\Delta$ and that $\sum_{K\in \mathcal B_N} (K;\Delta)=\#V\Delta$. Thus $${\overline{\rm rk}}(\eta_H)={\overline{\rm rk}}(\mu_\Delta)=\#E_{top}\Delta-\#V\Delta=-\chi(\Delta)={\overline{\rm rk}}(H).$$ Note that, by linearity, for any $c\ge 0$ ${\overline{\rm rk}}(c\eta_H)=c{\overline{\rm rk}}(H)\ge
0$. Thus ${\overline{\rm rk}}\ge 0$ on a dense subset of ${\mathcal S{\mbox{Curr}}({F_N})}$ and hence, by continuity, ${\overline{\rm rk}}(\mu)\ge 0$ for every $\mu\in {\mathcal S{\mbox{Curr}}({F_N})}$.
Suppose now that $\phi\in {\mbox{Aut}}(F_N)$. We claim that ${\overline{\rm rk}}(\phi\mu)={\overline{\rm rk}}(\mu)$ for every $\mu\in{\mathcal S{\mbox{Curr}}({F_N})}$. Indeed, for any finitely generated subgroup $H\le F_N$, $H$ is isomorphic to $\phi(H)$ and hence ${\overline{\rm rk}}(H)={\overline{\rm rk}}(\phi(H))$. Thus, in view of $(\diamondsuit)$, the claim holds for every rational subset current and hence, since by Theorem \[thm:dense\] rational currents are dense in ${\mathcal S{\mbox{Curr}}({F_N})}$, the claim holds for every $\mu\in {\mathcal S{\mbox{Curr}}({F_N})}$. This shows that ${\overline{\rm rk}}$ is ${\mbox{Aut}}(F_N)$-invariant, and hence ${\mbox{Out}}(F_N)$-invariant as well.
Analogs of uniform currents {#sec:uniform}
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In [@Ka2], given a free basis $A$ of $F_N$ we constructed a *uniform current* $m_A\in{\mbox{Curr}}(F_N)$ associated to $A$. Intuitively, the current $m_A$ splits equally in all directions in the Cayley graph $X$ of $F_N$ with respect to $A$.
In the setup of subset currents, given a free basis $A$ of $F_N$ one can define a natural family $m_{A,d}\in {\mathcal S{\mbox{Curr}}({F_N})}$, $d=2,\dots, 2N$ of uniform subset currents, with $m_{A,2}=m_A$ and $m_{A,2N}=\eta_{F_N}$. The current $m_{A,d}$ is supported on $d$-regular subtrees of $X$. That is, $m_{A,d}$ will have the property that $(K;m_{A,d}) >0$ if and only if $K\subseteq X$ is a finite non-degenerate subtree where every vertex has degree either $1$ or $d$ in $K$.
Before giving the explicit definition of $m_{A,d}$ we present the following computation as motivation. Since $m_{A,d}$ is a subset current, we need it to satisfy the weights condition $(\bigstar)$ from Proposition \[prop:kirch\]. Let $K$ be a non-degenerate finite subtree of $X$ where every vertex has degree $1$ or $d$ in $K$. Let $e$ be a terminal edge of $K$, as in Definition \[defn:gcyl\], and let $a\in A^{\pm 1}$ be the label of $e$. Then, in notations \[notation:edges\], $q(e)$ consists of precisely $2N-1$ distinct edges, with labels from $A^{\pm 1}-\{a^{-1}\}$. We want to choose a nonempty subset $U\subseteq q(e)$ so that in the tree $K'=K\cup U$ the terminus of $e$ has degree $d$. Thus the set $U$ needs to have cardinality $d-1$. Hence there are exactly $\binom{2N-1}{d-1}$ choices for $U\subseteq q(e)$. Since $m_{A,d}$ is supposed to be the most symmetric possible, we assign each of these choices equal weight, so we want $$(K\cup U; m_{A,d})=\frac{(K;m_{A,d})}{\binom{2N-1}{d-1}}$$ for every subset $U\subseteq q(e)$ of size $d-1$. For nonempty subsets $U\subseteq q(e)$ of size different from $d-1$ we want $(K\cup U; m_{A,d})=0$. Then, using notations \[notation:edges\], we will have $$( K; m_{A,d})=\sum_{U\in P_+(q(e))} (K\cup U; m_{A,d})$$ as required by $(\bigstar)$. Assigning every one-edge subtree of $X$ weight $1/N$ in $m_{A,d}$ and iterating the above splitting formula yields the following:
Let $2\le d\le 2N$. Then there exists a unique subset current $m_{A,d}\in {\mathcal S{\mbox{Curr}}({F_N})}$ such that
1. if $K\subseteq X$ is a finite subtree with $n\ge 1$ edges and with every vertex of degree either $1$ or $d$ in $K$, then $$( K; m_{A,d})=\frac{1}{N\left(\binom{2N-1}{d-1}\right)^{n-1}} ;$$
2. if $K\subseteq X$ is a finite subtree where some vertex has degree different from $d$ and from $1$, then $$( K; m_{A,d})=0 ;$$
3. We have $\langle X, m_{A,d}\rangle=1$.
The current $m_{A,d}\in {\mathcal S{\mbox{Curr}}({F_N})}$ is called the *uniform subset current* of grade $d$ on $F_N$ corresponding to $A$.
It is easy to check, via a direct computation, that the weights $(K; m_{A,d})$ specified above satisfy condition $(\bigstar)$ and hence, by Proposition \[prop:kirch\], there does exist a unique subset current realizing these weights. Also, for every single-edge tree $K$ we have $(K; m_{A,d})=\frac{1}{N}$, and this normalization ensures that $\langle X, m_{A,d}\rangle=1$. For $d=2N$ the above formulas give $(K; m_{A,2N})=1$ if $K\subseteq X$ is a finite $2N$-regular subtree and $(K;
m_{A,2N})=0$ if $K$ is not $2N$-regular. This shows that $m_{A,2N}=\eta_{F_N}$. Also, for $d=2$, the definition of $m_{A,2}$ yields $(K; m_{A,2})=\frac{1}{2N(2N-1)^n}$ if $K\subseteq X$ is a linear segment of length $n\ge 1$ and $(K; m_{A,2})=0$ if $K$ is not 2-regular. Thus $m_{A,2}=m_A\in{\mbox{Curr}}(F_N)$, as claimed.
In a similar way, uniform subset currents can be defined with respect to any marking $\alpha:F_N\to\pi_1(\Gamma)$, where $\Gamma$ is a $k$-regular graph (and not necessarily the standard rose).
Perhaps a more interesting notion is that of the *absolute uniform current* $m^\mathcal S_{A}\in {\mathcal S{\mbox{Curr}}({F_N})}$ associated to $A$.
If $K$ is a finite non-degenerate tree, we say that a vertex $x$ of $K$ is *interior* if $x$ has degree $\ge 2$ in $K$. Denote by $\iota(K)$ the number of interior vertices in $K$. Before giving a formal definition of $m^\mathcal S_{A}$ let us again start with some motivation. We want $m^\mathcal S_{A}$ to have the property that for a finite non-degenerate subtree $K$ of $X$ the weight $( K; m^\mathcal S_{A})$ depends only on $\iota(K)$.
We build finite subtrees of $X$ step-by-step, starting with a tree $K_0$ consisting of a single edge. Since there are exactly $N$ distinct $F_N$-translation classes of topological edges in $X$, we assign every one-edge subtree weight $\frac{1}{N}$ in $m^\mathcal S_{A}$. Note that $\iota(K_0)=0$. Arguing inductively, let $n\ge 0$ and suppose $K_n$ is already constructed and that $\iota(K_n)=n$. We choose $e$ to be any (oriented) leaf of $K$. Then $q(e)$ (see notation \[notation:edges\]) consists of precisely $2N-1$ distinct edges, with labels from $A^{\pm 1}-\{a^{-1}\}$. The set $P_+(q(e))$ of nonempty subsets of $q(e)$ has exactly $2^{2N-1}-1$ elements. We choose any nonempty subset $U$ of $q(e)$ with equal probability, and put $K_{n+1}=K\cup U$. Note that the terminus of $e$ has become an interior vertex of $K_{n+1}$, so that $\iota(K_{n+1})=n+1$. Since we are supposed to choose $U\in P_+(q(e))$ uniformly at random, we want $$(K\cup U; m_{A,d})=\frac{(K; m_{A,d})}{2^{2N-1}-1}$$ for every nonempty subset $U\subseteq q(e)$. This choice will assure that $$( K;m_{A,d})=\sum_{U\in P_+(q(e))} (K\cup U; m_{A,d})$$ as required by condition $(\bigstar)$ of Proposition \[prop:kirch\]. The above considerations lead to the following:
\[propdfn:mag\] Let $N\ge 2$, $A$ be a free basis of $F_N$ and let $X$ be the Cayley graph of $F_N$ with respect to $A$.
Then there exists a unique subset current $m_{A}^\mathcal S\in {\mathcal S{\mbox{Curr}}({F_N})}$ such that:
1. If $K\subseteq X$ is a finite non-degenerate subtree with $\iota(K)=n\ge 0$ then $$(K; m_{A}^\mathcal S)=\frac{1}{N\left( 2^{2N-1}-1 \right)^{n-1}}.$$
2. We have $\langle X, m_{A,d}\rangle=1$.
The current $m_{A}^\mathcal S\in {\mathcal S{\mbox{Curr}}({F_N})}$ is called the *absolute uniform subset current* corresponding to $A$.
It is not hard to check that the weights given in the definition of $m_A^\mathcal S$ satisfy the switch condition $(\bigstar)$ of Proposition \[prop:kirch\], which directly yields the above statement.
Note that, unlike the currents $m_{A,d}$ constructed earlier, the current $m_A^\mathcal S$ has full support, meaning that $(K;m_A^\mathcal S) >0$ for *every* finite non-degenerate subtree $K$ of $X$. Recall that, by Proposition \[prop:m1\], there is a canonical $\mathbb R_{\ge 0}$-linear homeomorphism between ${\mathcal S{\mbox{Curr}}({F_N})}$ and the space $\mathcal M_1(X)$ of finite positive Borel measures on the space $\mathcal T_1(X)$ of rooted subtrees of $X$ with root-vertex $1$, which are invariant with respect to root-change. Under this homeomorphism $m_A^\mathcal S$ corresponds to a probability measure $\mathbf m_A^\mathcal S$ on $\mathcal T_1(X)$ and $\mathbf m_A^\mathcal S$-random points of $\mathcal T_1(X)$ appear to represent an interesting class of subtrees of $X$.
Open problems {#sec:problems}
=============
Subset currents on surface groups {#subsec:surface}
---------------------------------
As noted in the introduction, the notion of a subset current makes sense for any non-elementary word-hyperbolic group $G$. Apart from $F_N$, a particularly interesting case is that of a surface group. Namely, let $\Sigma$ be a closed oriented surface of negative Euler characteristic and let $G=\pi_1(\Sigma)$. Put a hyperbolic Riemannian metric $\rho$ on $\Sigma$ so that $\widetilde \Sigma$ with the lifted metric becomes isometric to the hyperbolic plane $\mathbb H^2$. The action of $G$ on $\widetilde \Sigma=\mathbb H^2$ by covering transformations is a free isometric discrete co-compact action, so that $G$ is quasi-isometric to $\mathbb H^2$ and $\partial G$ is $G$-equivariantly homeomorphic to $\partial \mathbb H^2=\mathbb S^1$. As for a free group we can consider the space $\mathfrak(\mathbb S^1)$ of all closed subsets $S\subseteq \mathbb S^1$ such that $\#S\ge 2$. A *subset current* on $G$ is a locally finite $G$-invariant positive Borel measure on $\mathfrak(\mathbb S^1)$. The space $\mathcal S {\mbox{Curr}}(G)$ of all subset currents on $G$ again comes equipped with a natural weak-\* topology and a natural action of the mapping class group $Mod(\Sigma)$. In this context we can again define the notion of a *counting current* associated with a nontrivial finitely generated subgroup $H\le G$. If $H=Comm_G(H)$ then we put $$\eta_H:=\sum_{H_1\in [H] }\delta_{\Lambda(H_1)}$$ where $[H]$ is the conjugacy class of $H$ in $G$ and where for a subgroup $H_1\in [H]$ $\Lambda(H_1)\subseteq \mathbb S^1$ is the limit set of $H_1$ in $\mathbb S^1$. If $H\le G$ is an arbitrary nontrivial finitely generated subgroup, then, by well-known results, $H$ has a finite index $m\ge 1$ in its commensurator $H_0:=Comm_G(H)$ and we put $\eta_H:=m\ \eta_{H_0}$.
\[prob:surface\] In the above set-up, is it true that the set $$\{ c\eta_H | c\ge 0, H\le G \text{ is a nontrivial finitely generated subgroup} \}$$ is dense in $\mathcal S {\mbox{Curr}}(G)$?
Note that for every $S\in \mathfrak(\mathbb S^1)$ one can consider the convex hull $Conv(S)\subseteq \mathbb H^2$, and therefore one can geometrically view a subset current on $G$ as a $G$-invariant measure on the space of nice convex subsets of $\mathbb H^2$. If $H\le G$ is a nontrivial finitely generated subgroup of infinite index, then $H$ is a free group of finite rank $N\ge 1$ and $Conv(\Lambda H)\subseteq \mathbb H^2$ is an $H$-invariant subset which is quasi-isometric to $F_N$ (i.e. it is a quasi-tree). However, the machinery of measures on rooted graphs that we used to show that rational currents are dense in ${\mathcal S{\mbox{Curr}}({F_N})}$ is not directly applicable for tackling Problem \[prob:surface\].
Continuity of co-volume for a fixed tree $T\in{\overline{\mbox{cv}}_N}$ {#subsec:continuity}
-----------------------------------------------------------------------
In Remark \[rem:H\] we defined the notion of co-volume $||H||_T$ where $H\le F_N$ is any nontrivial finitely generated subgroup and where $T\in {\overline{\mbox{cv}}_N}$. We have seen that for a fixed $H$, the function ${\overline{\mbox{cv}}_N}\to [0,\infty)$, $T\to ||H||_T$, is not necessarily continuous on ${\overline{\mbox{cv}}_N}$ (although it is, of course, continuous if $H$ is infinite cyclic).
One can still ask if, given a fixed $T\in{\overline{\mbox{cv}}_N}$, the co-volume $||.||_T$ extends to a continuous function on ${\mathcal S{\mbox{Curr}}({F_N})}$:
\[prob:covol\] Let $T\in{\overline{\mbox{cv}}_N}$. Suppose $\mu\in {\mathcal S{\mbox{Curr}}({F_N})}$ and that $c_n\eta_{H_n}\in {\mathcal S{\mbox{Curr}}({F_N})}$ are rational currents such that $\lim_{n\to\infty} c_n\eta_{H_n}=\mu$. Does this imply that $$\lim_{n\to\infty} c_n||H_n||_{T}$$ exists and is independent of the sequence $c_n\eta_{H_n}$ approximating $\mu$? If yes, we will denote the above limit by $||\mu||_T$.
Volume equivalence {#subsec:equivalence}
------------------
Kapovich, Levitt, Schupp and Shpilrain [@KLSS] introduced and studied the notion of *translation equivalence* in free groups. Namely, two elements $g,h\in F_N$ are *translation equivalent* in $F_N$, denoted $g\equiv_t h$, if for every $T\in{\mbox{cv}_N}$ we have $||g||_T=||h||_T$. It is easy to see that $g\equiv_t h$ in $F_N$ if and only if for every $T\in{\overline{\mbox{cv}}_N}$ $||g||_T=||h||_T$. Similarly, we say (see [@Ka2]) that two currents $\mu_1,\mu_2\in{\mbox{Curr}}(F_N)$ are *translation equivalent* if for every $T\in {\mbox{cv}_N}$ $\langle T, \mu_1\rangle=\langle T,\mu_2\rangle$. Again, it clear that replacing ${\mbox{cv}_N}$ by ${\overline{\mbox{cv}}_N}$ in this definition yields the same notion. Several different sources of translation equivalence (in particular traces of $SL(2,\mathbb C)$-representations of free groups), have been exhibited in [@KLSS], and further results were obtained in [@Lee; @Lee2; @LV]. The paper [@KLSS] also defined the notion of *volume equivalence* in free groups. Two nontrivial finitely generated subgroups $H_1,H_2\le F_N$ are said to be *volume equivalent* in $F_N$, denoted $H_1\equiv_v H_2$, if for every $T\in {\mbox{cv}_N}$ we have $||H_1||_T=||H_2||_T$. Note that for nontrivial $g,h\in F_N$ we have $\langle g\rangle\equiv_v \langle h\rangle$ in $F_N$ if and only if $g\equiv_t h$ in $F_N$. It is clear that conjugate subgroups are volume equivalent and that if $H_1,H_2\le F_N$ are nontrivial finitely generated subgroups such that $Comm_{F_N}(H_1)=Comm_{F_N}(H_2)$ and such that $H_1$ and $H_2$ have the same index in $Comm_{F_N}(H_1)$, then $H_1\equiv_v H_2$ in $F_N$. Similarly, the definition of volume equivalence easily implies that if $H_1\equiv_v H_2$ in $F_N$ and $\psi: F_N\to F_M$ is an injective homomorphism then $\psi(H_1)\equiv_v \psi(H_2)$ in $F_M$. Some more interesting sources of volume equivalence were found by Lee and Ventura in [@LV], who also exhibited an example of a cyclic subgroup that is volume equivalent to a subgroup that is free of rank 2. The definition of volume equivalence naturally leads to the following question:
\[prob:ve\] Suppose $H_1\equiv_v H_2$ in $F_N$. Does this imply that for every $T\in{\overline{\mbox{cv}}_N}$ $||H_1||_T=||H_2||_T$?
By analogy with the translation equivalence case, we can also say that two subset currents $\mu_1,\mu_2\in {\mathcal S{\mbox{Curr}}({F_N})}$ are *volume equivalent* in $F_N$, denoted $\mu_1\equiv_v \mu_2$, if for every $T\in {\mbox{cv}_N}$ $\langle T, \mu_1\rangle=\langle T,\mu_2\rangle$. Thus for finitely generated subgroups $H_1,H_2\le F_N$ we have $H_1\equiv_v H_2$ if and only if $\eta_{H_1}\equiv_v \eta_{H_2}$. The notion of volume equivalence for subset currents measures the degeneracy of the co-volume form $\langle\, ,\, \rangle$ with respect to its second argument, and one can also pose an analog of Problem \[prob:ve\] for subset currents.
Generalizing the Stallings fiber product construction {#subsec:fiber}
-----------------------------------------------------
Let $H, L\le F_N$ be nontrivial finitely generated subgroups and let $\alpha:F_N{\mathrel{\text{ \setbox0\hbox{$\rightarrow$} \rlap{\hbox to \wd0{\hss\raisebox{0.9\height}{$\sim$}\hss}}\box0
}}}\pi_1(\Gamma)$ be a marking on $F_N$. Let $\Delta_H, \Delta_L$ be the finite connected $\Gamma$-core graphs representing $[H]$ and $[L]$ respectively. Then one can define (see [@Sta; @KM]) the fiber product graph $\Delta_H\times \Delta_L$. This graph is again a finite folded $\Gamma$-graph but not necessarily connected. The fundamental groups of the non-contractible connected components of $\Delta_H\times \Delta_L$ (if there are any such components) represent all the possible $F_N$-conjugacy classes of nontrivial intersections of the form $gHg^{-1}\cap L$, where $g\in F_N$. There are finitely many such non-contractible components and denote the conjugacy classes of subgroups of $F_N$ represented by them by $[U_1],\dots, [U_k]$. We thus define $$\pitchfork(\eta_H,\eta_L)=\sum_{i=1}^k \eta_{U_i}.$$ Note that ${\overline{\rm rk}}(\pitchfork(\eta_H,\eta_L))=\sum_{i=1}^k {\overline{\rm rk}}(\eta_{U_i})=\sum_{i=1}^k {\overline{\rm rk}}(U_i)$. If all connected components of $\Delta_H\times \Delta_L$ are contractible, define $\pitchfork(\eta_H,\eta_L)=0$. We can extend $\pitchfork$ by homogeneity to the set of rational subset currents as $$\pitchfork(c_1\eta_H,c_2\eta_L):=c_1c_2\pitchfork(\eta_H,\eta_L).$$ Note that $\pitchfork(c_1\eta_H,c_2\eta_L)=\pitchfork(c_2\eta_L,c_1\eta_H)$ and that $\pitchfork(c_1\eta_H,c_2\eta_L)=0$ if at least one of $H,L$ is infinite cyclic. A particularly intriguing question is the following:
\[prob:pitch\] Does the map $\pitchfork$ extends to a continuous function $$\pitchfork:{\mathcal S{\mbox{Curr}}({F_N})}\times{\mathcal S{\mbox{Curr}}({F_N})}\to{\mathcal S{\mbox{Curr}}({F_N})}?$$
If yes, then composing $\pitchfork$ with the reduced rank functional $\overline{\rm rk}$ would give us a continuous, bilinear and *symmetric* intersection functional $J:= \overline{\rm rk}\, \circ \pitchfork$ $$J : {\mathcal S{\mbox{Curr}}({F_N})}\times{\mathcal S{\mbox{Curr}}({F_N})}\to \mathbb R.$$ Moreover, in view of Mineyev’s recent proof of the Strengthened Hanna Neumann Conjecture [@Min], it would follow that for any $\mu_1,\mu_2\in {\mathcal S{\mbox{Curr}}({F_N})}$ we have $$J(\mu_1,\mu_2)\le \overline{\rm rk}(\mu_1) \overline{\rm rk}(\mu_2).$$
Random subgroup graphs and uniform currents {#subsec:random}
-------------------------------------------
Let $A$ be a free basis of $F_N$.
Let $\xi=x_1x_2\dots x_n\dots \in \partial F_N$ be a random geodesic ray over $A^{\pm 1}$, that is, $\xi$ is a random trajectory of the non-backtracking simple random walk on $F_N$ corresponding to $A$. Denote $\xi|_n=x_1\dots x_n\in F_N$. It is shown in [@Ka2] for a.e. $\xi\in \partial F_N$ we have $$\lim_{n\to\infty} \frac{\eta_{\xi|_n}}{n}=m_A \text{ in } {\mbox{Curr}}(F_N)$$ where $m_A\in {\mbox{Curr}}(F_N)$ is the *uniform current* on $F_N$ corresponding to $A$. For a random $\xi\in \partial F_N$ we may think of $\langle \xi|_n\rangle\le F_N$ as a random cyclic subgroup of $F_N$.
In Section \[sec:uniform\] we have defined a family of uniform subset currents $m_{A,d}\in {\mathcal S{\mbox{Curr}}({F_N})}$, for $d=2,\dots, 2N$, where $m_{A,2}=m_A\in {\mbox{Curr}}(F_N)$. Recall that for a finite non-degenerate subtree $K$ of the Cayley graph $X$ of $F_N$ with respect to $A$ we have $( K; m_{A,d}) >0$ if and only if $K$ is a $d$-regular tree.
Let $3\le d\le 2N-1$. Define a reasonable discrete-time random process such that at time $n$ it outputs a $d$-regular finite $R_N$-core graph $\Delta_n$ with $\lim_{n\to\infty} \#V\Delta_n=\infty$ and show that for a.e. trajectory of this process we have $$\lim_{n\to\infty}\frac{\eta_{\Delta_n}}{\#V\Delta_n}=m_{A,d} \text{ in }{\mathcal S{\mbox{Curr}}({F_N})}.$$
The same problem is also interesting for the absolute uniform current $m_A^\mathcal S\in {\mathcal S{\mbox{Curr}}({F_N})}$.
Define a reasonable discrete-time random process such that at time $n$ it outputs a finite $R_N$-core graph $\Delta_n$ with $\lim_{n\to\infty} \#V\Delta_n=\infty$ and show that for a.e. trajectory of this process we have $$\lim_{n\to\infty}\frac{\eta_{\Delta_n}}{\#V\Delta_n}=m_{A}^\mathcal S \text{ in }{\mathcal S{\mbox{Curr}}({F_N})}.$$ In particular, what happens when $\Delta_n$ is chosen using the Bassino-Nicaud-Weil [@BNW] process for generating random Stallings subgroup graphs?
Dynamics of the ${\mbox{Out}}(F_N)$-action on ${\mathbb P{\mathcal S{\mbox{Curr}}({F_N})}}$ {#subsec:dynamics}
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For ordinary currents the dynamics of the ${\mbox{Out}}(F_N)$-action on ${\Pr{\mbox{Curr}}({F_N})}$ and the interaction of this dynamics with that of the ${\mbox{Out}}(F_N)$-action on ${\overline{\mbox{cv}}_N}$ turned out to be particularly useful. For ${\mathbb P{\mathcal S{\mbox{Curr}}({F_N})}}$ the dynamics of the ${\mbox{Out}}(F_N)$-action appears to be more complicated than in the ${\Pr{\mbox{Curr}}({F_N})}$-case, particularly because, even without projectivization, ${\mbox{Out}}(F_N)$ fixes the point $\eta_{F_N}\in {\mathcal S{\mbox{Curr}}({F_N})}$.
In [@Martin] R. Martin introduced the subset $\mathcal M_N\subseteq {\Pr{\mbox{Curr}}({F_N})}$ as the closure in ${\Pr{\mbox{Curr}}({F_N})}$ of the set $$\{[\eta_g]: g\in F_N \text{ is a primitive element} \}.$$ It is easy to see that for every $N\ge 2$ $\mathcal M_N\subseteq {\Pr{\mbox{Curr}}({F_N})}$ is a closed ${\mbox{Out}}(F_N)$-invariant nonempty subset. In [@KL1] it was shown that for $N\ge 3$ $\mathcal M_N$ is the unique smallest such subset, that is, whenever $Z\subseteq {\Pr{\mbox{Curr}}({F_N})}$ is nonempty, closed and ${\mbox{Out}}(F_N)$-invariant, then $\mathcal M_N\subseteq Z$.
As noted in Remark \[rem:curr\] above, ${\Pr{\mbox{Curr}}({F_N})}\subseteq {\mathbb P{\mathcal S{\mbox{Curr}}({F_N})}}$ is a closed ${\mbox{Out}}(F_N)$-invariant subset and, similarly, ${\mbox{Curr}}(F_N)\subseteq {\mathcal S{\mbox{Curr}}({F_N})}$ is a closed ${\mbox{Out}}(F_N)$-invariant subset. The set $\{[\eta_{F_N}]\}$ is also a closed ${\mbox{Out}}(F_N)$-invariant subset of ${\mathbb P{\mathcal S{\mbox{Curr}}({F_N})}}$, so that ${\mathbb P{\mathcal S{\mbox{Curr}}({F_N})}}$ contains at least 2 minimal nonempty closed ${\mbox{Out}}(F_N)$-invariant subsets.
Let $N\ge 3$. Characterize those $[\mu]\in {\mathbb P{\mathcal S{\mbox{Curr}}({F_N})}}$ such that the closure of the ${\mbox{Out}}(F_N)$-orbit in ${\mathbb P{\mathcal S{\mbox{Curr}}({F_N})}}$ contains $\mathcal M_N$. Is it true that the closure of ${\mbox{Out}}(F_N)[\mu]$ contains $\mathcal M_N$ if and only if $[\mu]\ne [\eta_{F_N}]$?
As we note below, we do know that for every nontrivial finitely generated subgroup $H\le F_N$ of infinite index, the closure of ${\mbox{Out}}(F_N)[\eta_H]$ does contain $\mathcal M_N$.
Let $N\ge 3$ and let $\phi\in {\mbox{Out}}(F_N)$ be an atoroidal iwip (irreducible with irreducible powers) element. In this case it is known that $\phi$ has exactly two distinct fixed points in ${\Pr{\mbox{Curr}}({F_N})}$, the currents $[\mu_+]$ and $[\mu_-]$; moreover there are $\lambda_+, \lambda_- > 1$ such that $\phi\mu_+=\lambda_+\mu_+$ and $\phi^{-1}\mu_{-}=\lambda_-\mu_-$. R. Martin proved [@Martin] that the action of $\phi$ on ${\Pr{\mbox{Curr}}({F_N})}$ has North-South dynamics: If $[\mu]\in {\Pr{\mbox{Curr}}({F_N})}, [\mu]\ne [\mu_-]$ then $
\lim_{n\to\infty} \phi^n[\mu]=[\mu_+]
$ and if $[\mu]\in {\Pr{\mbox{Curr}}({F_N})}, [\mu]\ne [\mu_+]$ then $
\lim_{n\to\infty} \phi^{-n}[\mu]=[\mu_-].
$
As observed above, by Remark \[rem:curr\] we have $\mu_\pm\in {\mbox{Curr}}(F_N)\subseteq {\mathcal S{\mbox{Curr}}({F_N})}$ and $[\mu_\pm]\in {\Pr{\mbox{Curr}}({F_N})}\subseteq {\mathbb P{\mathcal S{\mbox{Curr}}({F_N})}}$.
The situation for ${\mathbb P{\mathcal S{\mbox{Curr}}({F_N})}}$ is immediately complicated by the presence of the global fixed point $[\eta_{F_N}]$. For example, for any $c_1,c_2>0$ it is not hard to see that $$\lim_{n\to\infty} \phi^n[c_1\mu_-+c_2\eta_{F_N}]=[\eta_{F_N}]
\quad \text{ and } \quad
\lim_{n\to\infty} \phi^{-n}[c_1\mu_+ + c_2\eta_{F_N}]=[\eta_{F_N}].$$
Let $N\ge 3$ and let $\phi\in{\mbox{Out}}(F_N)$ be an atoroidal iwip. Characterize those $[\mu]\in {\mathbb P{\mathcal S{\mbox{Curr}}({F_N})}}$ for which $$\lim_{n\to\infty} \phi^n[\mu]=[\mu_+].$$ Is it true that the above convergence holds for every $[\mu]\in {\mathbb P{\mathcal S{\mbox{Curr}}({F_N})}}$ which is not of the form $[\mu]=[c_1\mu_-+c_2\eta_{F_N}]$ where $c_1,c_2\ge 0$ and $|c_1|+|c_2|>0$?
Using the results of [@BFH97] we can show that if $H\le F_N$ is a nontrivial finitely generated subgroup of infinite index, then $$\lim_{n\to\infty} \phi^n[\eta_H]=[\mu_+] \text{ and } \lim_{n\to\infty} \phi^{-n}[\eta_H]=[\mu_-].$$ Since $[\mu_+]\in \mathcal M_N$, this fact does imply that the closure in ${\mathbb P{\mathcal S{\mbox{Curr}}({F_N})}}$ of the orbit ${\mbox{Out}}(F_N)[\eta_H]$ contains $\mathcal M_N$.
Subset algebraic laminations {#subsec:lamination}
----------------------------
An *algebraic lamination* on $F_N$ is a closed $F_N$-invariant subset of the space of 2-elements subsets of $({\partial}F_N)$. Algebraic laminations proved to be useful objects in the study of ${\mbox{Out}}(F_N)$ and of the Outer space. In particular, for every $T\in{\overline{\mbox{cv}}_N}$ there is a naturally defined *dual lamination* $L^2(T)$ which records some essential information about the geometry of the action of $F_N$ on $T$. For an arbitrary current $\mu\in{\mbox{Curr}}(F_N)$ its *support* is an algebraic lamination. It is proved in [@KL3] that for $\mu\in{\mbox{Curr}}(F_N)$ and $T\in{\overline{\mbox{cv}}_N}$ $\langle T,\mu\rangle=0$ if and only if ${\mbox{Supp}}(\mu)\subseteq L^2(T)$, and this fact plays a key role in understanding the interplay between the dynamics of ${\mbox{Out}}(F_N)$-actions on ${\overline{\mbox{CV}}_N}$ and ${\Pr{\mbox{Curr}}({F_N})}$.
By analogy with the set-up discussed above, we say that a *subset algebraic lamination* on $F_N$ is a closed $F_N$-invariant subset of $\mathfrak C_N$. In particular, for any $\mu\in{\mathcal S{\mbox{Curr}}({F_N})}$, the support ${\mbox{Supp}}(\mu)$ is a subset algebraic lamination. Here ${\mbox{Supp}}(\mu):=\mathfrak C_N-U$ where $U$ is the largest open subset of $\mathfrak C_N$ such that $\mu(U)=0$.
Assuming that the answer to Problem \[prob:covol\] is positive, given $T\in{\overline{\mbox{cv}}_N}$ does there exist a naturally defined subset lamination $L^2_\mathcal S(T)\subseteq \mathfrak C_N$ which captures the information about all $\mu\in {\mathcal S{\mbox{Curr}}({F_N})}$ with $||\mu||_T=0$? Or at least some naturally defined subset lamination $L^2_\mathcal S(T)\subseteq \mathfrak C_N$ which captures the information about all finitely generated $H\le F_N$ with $||H||_T=0$? If such a notion of $L^2_\mathcal S(T)$ does exist, how does $L^2_\mathcal S(T)$ look like for stable trees of various free group automorphisms?
Note that for any reasonable definition of $L^2_\mathcal S(T)$ one expects to have $L^2(T)\subseteq L^2_{\mathcal S(T)}$.
Approximation and weight realizability problem {#subsec:weight}
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We know by Theorem \[thm:dense\] that any $\mu\in {\mathcal S{\mbox{Curr}}({F_N})}$ can be approximated by rational subset currents. It would be interesting to find explicit procedures (e.g. algorithmic or probabilistic) for producing such approximations in various specific contexts.
Let $A$ be a free basis of $F_N$ and let $m_A^\mathcal S$ be the corresponding absolute uniform current. Find a natural probabilistic process producing a sequence of core graphs $\Delta_n$ such that $\lim_{n\to\infty} [\mu_{\Delta_n}] =[m_A^\mathcal S]$ in ${\mathbb P{\mathcal S{\mbox{Curr}}({F_N})}}$.
In [@Martin] Martin proves that for a marking $\alpha:F_N\to \pi_1(\Gamma)$ and a nonzero geodesic current $\mu\in Curr(F_N)$ we have $( v; \mu)_\alpha\in \mathbb Z$ for every nontrivial reduced edge-path $v$ in $\Gamma$ if and only if $\mu=\eta_{g_1}+\dots +\eta_{g_m}$ for some $m\ge 1$ and some nontrivial $g_1,\dots, g_m\in F_N$. It is natural to ask a similar question for subset currents:
Let $\alpha:F_N\to \pi_1(\Gamma)$ be a marking and let $\mu\in {\mathcal S{\mbox{Curr}}({F_N})}$ be a nonzero subset current such that for every non-degenerate finite subtree $K$ of $\widetilde\Gamma$ we have $(K; \mu)_\alpha\in \mathbb Z$. Does this imply that $\mu=\eta_{H_1}+\dots +\eta_{H_m}$ for some $m\ge 1$ and some nontrivial finitely generated subgroups $H_1,\dots, H_m\le F_N$?
In the case of ${\mbox{Curr}}(F_N)$, the proof uses Whitehead graphs for cyclic words. In the context of subset currents the corresponding objects turn out to be hyper-graphs, rather than graphs.
Recall that a *hyper-graph* $\mathbf G$ is a pair $(\mathbf V,\mathbf E)$, where $\mathbf V$ is the set of vertices and $\mathbf E$ is the set of hyper-edges. Every hyper-edge $\mathbf e$ is a nonempty subset of $\mathbf V$, whose elements are said to be *incident* to $\mathbf e$ in $\mathbf G$. We can also think of $\mathbf e$ as the characteristic function of this subset, so that $\mathbf e: \mathbf V\to\{0,1\}$ with $\mathbf e(\mathbf v)=1$ if and only if $\mathbf v$ is incident to $\mathbf e$.
Now let $\alpha:F_N{\mathrel{\text{ \setbox0\hbox{$\rightarrow$} \rlap{\hbox to \wd0{\hss\raisebox{0.9\height}{$\sim$}\hss}}\box0
}}}\pi_1(\Gamma)$ be a marking, let $X=\widetilde \Gamma$ and let $K\in \mathcal K_\Gamma$ be a finite non-degenerate subtree of $X$. Recall from Section \[sec:uniform\] that $\iota(K)$ denotes the number of interior vertices in $K$. We say that an interior vertex $x$ of $K$ is a *boundary-interior vertex* of $K$ if $x$ is an interior vertex of $K$, $x$ is adjacent to a terminal edge of $K$ and removing from $K$ all terminal edges of $K$ incident to $x$ produces a tree in which $x$ has degree $1$. That is, $x$ is an interior vertex of $K$ and the degree of $x$ in $K$ is equal to $p+1$ where $p$ is the number of terminal edges of $K$ incident to $x$. Note that every finite $K$ with $\iota(K)\ge 2$ always has at least one boundary-interior vertex. For each boundary-interior vertex $x$ of $K$ let $U_x$ be the set of topological edges of $K$ that connect $x$ to vertices of degree 1 of $K$. Let $K_x$ be obtained from $K$ by removing all the edges of $U_x$. Thus $K=K_x\cup U_x$. Note that $x$ is a vertex of degree 1 in $K_x$, so that $\iota(K_x)=m-1$.
For each $m\ge 2$ define the *level-$m$ initial hyper-graph* $\mathbf G_{\alpha,m}$ as follows. The vertex set of $\mathbf G_{\alpha,m}(\mu)$ is the set $Z_{m-1}$ of all $F_N$-translation classes $[K]$ where $K\in\mathcal K_\Gamma$ is a tree with $\iota(K)=m$. The hyper-edge set is $Z_{m}$. Every $[K']\in Z_m$ defines the incidence function $\mathbf e_{[K']}: Z_{m-1}\to \{0,1\}$ as follows. For $[K]\in Z_{m-1}$ $\mathbf e_{[K']}([K])=1$ if and only if there is a boundary-interior vertex $x$ of $K'$ such that $[K_x']=[K]$.
For $m=1$ we can also define $\mathbf G_{\alpha,1}$ in a similar way. The vertex set is $Z_0$ and the hyper-edge set is $Z_1$, but the incidence is defined slightly differently. Namely, for $[K']\in Z_1$ and an element $[K]\in Z_0$ (note that $K$ is necessarily a single topological edge of $X$) we have $\mathbf e_{[K']}([K])=1$ if and only if there is a topological edge of $K'$ which is an $F_N$-translate of $K$.
For $m\ge 1$, a *weighted level-$m$ initial hyper-graph* consists of the hyper-graph $\mathbf G_{\alpha,m}$ endowed with the *weight functions* $\theta: Z_{m-1}\to {\mathbb R}_{\ge 0}$ and $\theta: Z_{m}\to {\mathbb R}_{\ge 0}$.
Every current $\mu\in {\mathcal S{\mbox{Curr}}({F_N})}$ defines a weighted hyper-graph $\mathbf G_{\alpha,m}(\mu)$, with the weight functions $\theta([K']):=( K'; \mu)_\alpha$, $\theta([K]):=(K; \mu)_\alpha$, where $[K']\in Z_m$, $[K]\in Z_{m-1}$.
\[prob:wr\] Let $m\ge 1$ and let $(\mathbf G_{\alpha,m}, \theta)$ be a weighted initial level-$m$ hyper-graph such that the weight function $\theta$ is $\mathbb Z_{\ge 0}$–valued and satisfies the Kirchoff condition (see Proposition \[prop:kirch\]) for every $[K]\in Z_{m-1}$. Does there exist a finite $\Gamma$-core graph $\Delta$ such that $(\mathbf G_{\alpha,m}, \theta)=\mathbf G_{\alpha,m}(\mu_\Delta)$?
As shown in [@Ka2], in the ${\mbox{Curr}}(F_N)$ context the analog of the above question has positive answer. In that case initial hyper-graphs are directed graphs and the Kirchoff condition implies the existence of an Euler circuit in every connected component of the graph, when the weights are treated as multiple edges. However, it is not clear what might serve as a substitute for Euler’s theorem in the hyper-graph context, and even the proof of Theorem \[thm:dense\] does not appear to help with Problem \[prob:wr\].
[ABC]{}
M. Abert, Y. Glasner and B. Virag, *Kesten’s theorem for Invariant Random Subgroups*, preprint, 2012; arXiv:1201.3399
P. Arnoux, V. Berthé, T. Fernique, D. Jamet, *Functional stepped surfaces, flips, and generalized substitutions.* Theoret. Comput. Sci. **380** (2007), no. 3, 251–265
F. Bassino, C. Nicaud, and P. Weil, *Random generation of finitely generated subgroups of a free group.* Internat. J. Algebra Comput. **18** (2008), no. 2, 375–405
I. Benjamini, O. Schramm,
*Recurrence of distributional limits of finite planar graphs.*
Electronic Journal of Probability **6** (2001), no.23, 1–23
N. Bergeron, and D. Gaboriau, *Asymptotique des nombres de Betti, invariants $l^2$ et laminations*, Comment. Math. Helv. **79** (2004), no. 2, 362Ð395
M. Bestvina, and M. Handel, *Train tracks and automorphisms of free groups.* Ann. of Math. (2) **135** (1992), no. 1, 1–51
M. Bestvina and M. Feighn, *Outer Limits*, preprint, 1993;\
http://andromeda.rutgers.edu/\~feighn/papers/outer.pdf
M. Bestvina, M. Feighn, and M. Handel, *Laminations, trees, and irreducible automorphisms of free groups.* Geom. Funct. Anal. **7** (1997), no. 2, 215–244
M. Bestvina and M. Feighn, *The topology at infinity of ${\mbox{Out}}(F_n)$.* Invent. Math. **140** (2000), no. 3, 651–692
M. Bestvina, M. Feighn, *A hyperbolic ${\mbox{Out}}(F_n)$ complex*, Groups Geom. Dyn. **4** (2010), no. 1, 31- 58
F. Bonahon, *Bouts des variétés hyperboliques de dimension $3$.* Ann. of Math. (2) **124** (1986), no. 1, 71–158
F. Bonahon, *The geometry of Teichmüller space via geodesic currents.* Invent. Math. **92** (1988), no. 1, 139–162
F. Bonahon, *Geodesic currents on negatively curved groups.* Arboreal group theory (Berkeley, CA, 1988), 143–168, Math. Sci. Res. Inst. Publ., 19, Springer, New York, 1991
L. Bowen, *Periodicity and circle packings of the hyperbolic plane*, Geom. Dedicata **102** (2003), 213–236
L. Bowen, *Free groups in lattices.* Geom. Topol. **13** (2009), no. 5, 3021–3054
L. Bowen, *Random walks on coset spaces with applications to Furstenberg entropy*, preprint, 2010; arXiv:1008.4933.
L. Bowen, *Invariant random subgroups of the free group*, preprint, 2012; arXiv:1204.5939
M. Carette, S. Francaviglia, I. Kapovich, and A. Martino, *Spectral rigidity of automorphic orbits in free groups*, Alg. Geom Topology, to appear; arXiv:1106.0688
M. Cohen and M. Lustig, *Very small group actions on $R$-trees and Dehn twist automorphisms.* Topology **34** (1995), no. 3, 575–617
M. Clay and A. Pettet, *Twisting out fully irreducible automorphisms*, Geom. Funct. Anal. **20** (2010), no. 3, 657–689
M. Clay and A. Pettet, *Currents twisting and nonsingular matrices*, Commentarii Mathematici Helvetici **87** (2012), no. 2, 384–407
T. Coulbois, A. Hilion, and M. Lustig, *$\mathbb R$-trees and laminations for free groups I: Algebraic laminations,* J. Lond. Math. Soc. (2) **78** (2008), no. 3, 723–736
T. Coulbois, A. Hilion, and M. Lustig, *$\mathbb R$-trees and laminations for free groups II: The dual lamination of an $\mathbb
R$-tree,* J. Lond. Math. Soc. (2) **78** (2008), no. 3, 737–754
T. Coulbois, A. Hilion, and M. Lustig, *$\mathbb R$-trees and laminations for free groups III: Currents and dual $\mathbb R$-tree metrics,* J. Lond. Math. Soc. (2) **78** (2008), no. 3, 755–766
M. Culler, K. Vogtmann, *Moduli of graphs and automorphisms of free groups.* Invent. Math. **84** (1986), no. 1, 91–119
D. D’Angeli, A. Donno, M. Matter, and T. Nagnibeda, *Schreier graphs of the Basilica group*, J. Mod. Dyn. **4** (2010), no. 1, 167–205
S. G. Dani, *On conjugacy classes of closed subgroups and stabilizers of Borel actions of Lie groups*, Ergodic Theory Dynam. Systems **22** (2002), no. 6, 1697–1714
G. Elek, *On the limit of large girth graph sequences*, Combinatorica **30** (2010), no. 5, 553–563
S. Francaviglia, *Geodesic currents and length compactness for automorphisms of free groups*, Transact. Amer. Math. Soc., **361** (2009), no. 1, 161–176
R. Grigorchuk, *Some topics of dynamics of group actions on rooted trees*, The Proceedings of the Steklov Institute of Math., v. 273 (2011), 64–175
Rostislav Grigorchuk, Vadim A. Kaimanovich, and Tatiana Nagnibeda, [*Ergodic properties of boundary actions and Nielsen-Schreier theory*]{}, Adv. Math. **230** (2012), no. 3, 1340–1380
V. Guirardel, *Approximations of stable actions on $R$-trees.* Comment. Math. Helv. **73** (1998), no. 1, 89–121
V. Guirardel, *Dynamics of ${\rm Out}(F\sb n)$ on the boundary of outer space.* Ann. Sci. École Norm. Sup. (4) **33** (2000), no. 4, 433–465
U. Hamenstädt, *Word hyperbolic extensions of surface groups*, preprint, 2005; arXiv:math.GT/050524
U. Hamenstädt, *Lines of minima in Outer space*, preprint, November 2009; arXiv:0911.3620
K.P. Hart, J.-I. Nagata, J.E. Vaughan (Eds.) *Encyclopedia of General Topology*, Elsevier, 2004
A. Hatcher and K. Vogtmann, *The complex of free factors of a free group.* Quart. J. Math. Oxford Ser. (2) **49** (1998), no. 196, 459–468
V.Kaimanovich, I. Kapovich and P. Schupp, *The Subadditive Ergodic Theorem and generic stretching factors for free group automorphisms,* Israel J. Math. **157** (2007), 1–46
I. Kapovich, *The frequency space of a free group,* Internat. J. Alg. Comput. **15** (2005), no. 5-6, 939–969
I. Kapovich, *Currents on free groups,* Topological and Asymptotic Aspects of Group Theory (R. Grigorchuk, M. Mihalik, M. Sapir and Z. Sunik, Editors), AMS Contemporary Mathematics Series, vol. 394, 2006, pp. 149-176
I. Kapovich, *Clusters, currents and Whitehead’s algorithm*, Experimental Mathematics **16** (2007), no. 1, pp. 67-76
I. Kapovich, *Random length-spectrum rigidity for free groups*, Proc. AMS **140** (2012), no. 5, 1549–1560
I. Kapovich, G. Levitt, P. Schupp and V. Shpilrain, *Translation equivalence in free groups,* Transact. Amer. Math. Soc. **359** (2007), no. 4, 1527–1546
I. Kapovich and M. Lustig, *The actions of ${\mbox{Out}}(F_k)$ on the boundary of outer space and on the space of currents: minimal sets and equivariant incompatibility.* Ergodic Theory Dynam. Systems **27** (2007), no. 3, 827–847
I. Kapovich and M. Lustig, *Geometric Intersection Number and analogues of the Curve Complex for free groups*, Geometry & Topology **13** (2009), 1805–1833
I. Kapovich and M. Lustig, *Intersection form, laminations and currents on free groups*, Geom. Funct. Anal. (GAFA), 19 (2010), no. 5, 1426–1467
I. Kapovich and M. Lustig, *Domains of proper dicontinuity on the boundary of Outer space*, Illinois J. Math. **54** (2010), no. 1, pp. 89–108, special issue dedicated to Paul Schupp
I. Kapovich and M. Lustig, *Ping-pong and Outer space*, J. Topol. Anal. **2** (2010), no. 2, 173–201
I. Kapovich and A. Myasnikov, *Stallings foldings and the subgroup structure of free groups*, J. Algebra **248** (2002), no 2, 608–668
I. Kapovich and T. Nagnibeda, *The Patterson-Sullivan embedding and minimal volume entropy for Outer space*, Geom. Funct. Anal. (GAFA) **17** (2007), no. 4, 1201–1236
I. Kapovich and T. Nagnibeda, *Geometric entropy of geodesic currents on free groups*, Dynamical Numbers: Interplay Between Dynamical Systems and Number Theory, Contemporary Mathematics series, American Mathematical Society, 2010, pp. 149-176
I. Kapovich, and H. Short, *Greenberg’s theorem for quasiconvex subgroups of word hyperbolic groups.* Canad. J. Math. **48** (1996), no. 6, 1224–1244
D. Lee, *Translation equivalent elements in free groups.* J. Group Theory **9** (2006), no. 6, 809–814
D. Lee, *An algorithm that decides translation equivalence in a free group of rank two.* J. Group Theory **10** (2007), no. 4, 561–569
D. Lee, and E. Ventura, *Volume equivalence of subgroups of free groups.* J. Algebra **324** (2010), no. 2, 195-Ð217
G. Levitt and M. Lustig, *Irreducible automorphisms of $F_n$ have North-South dynamics on compactified outer space.* J. Inst. Math. Jussieu **2** (2003), no. 1, 59–72
R. Martin, *Non-Uniquely Ergodic Foliations of Thin Type, Measured Currents and Automorphisms of Free Groups*, PhD Thesis, 1995
I. Mineyev, *Submultiplicativity and the Hanna Neumann conjecture.* Ann. of Math. (2) **175** (2012), no. 1, 393–414
F. Paulin, *The Gromov topology on $R$-trees.* Topology Appl. **32** (1989), no. 3, 197–221
D. Savchuk, *Schreier Graphs of Actions of Thompsons Group $F$ on the Unit Interval and on the Cantor Set,* preprint, 2011; arXiv: 1105.4017.
John Stallings, *Topology of finite graphs*, Invent. Math. **71** (1983), no. 3, 551-Ð565
G. Stuck and R. J. Zimmer, *Stabilizers for ergodic actions of higher rank semisimple groups*, Ann. of Math. (2) **139** (1994), no. 3, 723–747
A. Vershik, [*Nonfree actions of countable groups and their characters*]{}, arXiv:1012.4604
A. Vershik, *Totally nonfree actions and infinite symmetric group*, preprint, 2011; arXiv:1109.3413
K. Vogtmann, *Automorphisms of Free Groups and Outer Space*, Geom. Dedicata **94** (2002), 1–31
[^1]: The first author was supported by the NSF grant DMS-0904200. Both authors acknowledge the support of the Swiss National Foundation for Scientific Research
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abstract: 'We construct exact solutions of the Gross-Pitaevskii equation for solitary vortices, and approximate ones for fundamental solitons, in 2D models of Bose-Einstein condensates with a spatially modulated nonlinearity of either sign and a harmonic trapping potential. The number of vortex-soliton (VS) modes is determined by the discrete energy spectrum of a related linear Schrödinger equation. The VS families in the system with the attractive and repulsive nonlinearity are mutually complementary. *Stable* VSs with vorticity $S\geq 2$ and those corresponding to higher-order radial states are reported for the first time, in the case of the attraction and repulsion, respectively.'
author:
- Lei Wu
- Lu Li
- 'Jie-Fang Zhang'
- Dumitru Mihalache
- 'Boris A. Malomed'
- 'W. M. Liu'
title: 'Exact solutions of the Gross-Pitaevskii equation for stable vortex modes'
---
The realization of Bose-Einstein condensates (BECs) in dilute quantum gases has drawn a great deal of interest to the dynamics of nonlinear excitations in matter waves, such as dark [@dark] and bright solitons [@bright], vortices [@vortex; @BECvortex], supervortices [@supervortex], etc. The work in this direction was strongly stimulated by many similarities to solitons and vortices in optics [@optics]. In the mean-field approximation, the BEC dynamics at ultra-low temperatures is accurately described by the Gross-Pitaevskii equation (GPE), the nonlinearity being determined by the $s$-wave scattering length of interatomic collisions, which can be controlled by means of the magnetic [@magnetic] or low-loss optical [@optical] Feshbach-resonance technique, making spatiotemporal management“ of the local nonlinearity possible through the use of time-dependent and/or non-uniform fields [optics]{}. In particular, a recently developed technique, which allows one to paint” arbitrary one- and two-dimensional (1D and 2D) average patterns by a rapidly moving laser beam [@painting], suggests new perspectives for the application of the nonlinearity management based on the optical Feshbach-resonance. A counterpart of the GPE, which is a basic model in nonlinear optics, is the nonlinear Schrödinger equation (NLSE) [optics]{}. In the latter case, the modulation of the local nonlinearity can be implemented too–in particular, by means of indiffusion of a dopant resonantly interacting with the light.
The search for exact solutions to the GPE/NLSE is an essential direction in the studies of nonlinear matter and photonic waves. In particular, stable soliton solutions are of great significance to experiments and potential applications, as they precisely predict conditions which should allow the creation of matter-wave and optical solitons, including challenging situations which have not yet been tackled in the experiment, such as 2D matter-wave solitons and solitary vortices [@review]. In the 1D setting with the spatially uniform nonlinearity, exact stable solutions for bright and dark solitons have been constructed in special cases, when the external potential is linear or quadratic [@homogeneous], and a specially designed inhomogeneous nonlinearity may support bound states of an arbitrary number of solitons [@beitia]. In the 2D geometry with the uniform nonlinearity, exact delocalized solutions were constructed in the presence of a periodic potential [@LDcarr2]. In the 2D geometry, 1D bright and dark solitons are unstable, with the bright ones suffering the breakup into a chain of collapsing pulses, and the dark solitons splitting into vortex-antivortex pairs [@dark; @RDS]. Another approach to finding exact solitonic states in both 1D and 2D settings (with the uniform nonlinearity), along with the supporting potentials, was proposed in the form the respective inverse problem [@Step]. As concerns vortices, while delocalized ones have been created and extensively investigated in self-repulsive BECs [@vortex], and vortex solitons (VSs) were created in photorefractive crystals equipped with photonic lattices [@photorefr], analytical solutions for localized vortices have not been reported yet, and experimental creation of stable VSs in self-attractive BEC and optical media with the fundamental cubic nonlinearity remains a challenge to the experiment.
In this work we demonstrate that exact VS solutions can be found in the 2D GPE/NLSE, with a specially designed (and experimentally realizable) profile of the radial modulation of the nonlinearity coefficient, of both attractive and repulsive types. In fact, the search for localized solutions in 2D models with diverse nonlinearity-modulation profiles has recently attracted considerable attention [@2D], although no exact solutions have been reported. We consider only nonlinearity-modulation patterns which do not change the sign, as zero-crossing would make the use of the FR problematic [@magnetic; @optical].
The scaled form of the GPE/NLSE is $i{\psi }_{t}=-\nabla ^{2}\psi +g(r)|\psi
|^{2}\psi +V(r)\psi $, where $\psi $ is the BEC macroscopic wave function, $%
\nabla ^{2}$ the 2D Laplacian, and $g(r)$ is the nonlinearity coefficient which, as well as external potential $V(r)$, is a function of radial coordinate $r$. The same equation finds a straightforward implementation as a spatial-domain model of the beam propagation in bulk media, with $t$, $%
\psi $, and $g(r)$ being the propagation distance, amplitude of the electromagnetic field, and the local Kerr coefficient.
Assuming the stationary wave function as $\psi (r,\theta ,t)=\phi (r)\exp
(iS\theta -i\mu t)$, where $\theta $ is the azimuthal angle, $S$ an integer vorticity, and $\mu $ the chemical potential (or propagation constant in optics), leads to the equation for the real stationary wave function $\phi
(r)$: $\mu \phi =-\phi ^{\prime \prime }-r^{-1}\phi ^{\prime }+g(r)\phi ^{3}+%
\left[ {S^{2}r^{-2}+}V(r)\right] \phi $. For $S\neq 0$, $\phi (r)$ should vary as $r^{|S|}$ at $r\rightarrow 0$, which is replaced by $\phi ^{\prime
}(r=0)=0$ for $S=0$. The localization requires $\phi (r=\infty )=0$.
Defining $\phi (r)\equiv \rho (r)U(R(r))$, $g(r)\equiv g_{0} r^{-2}\rho^{-6}
(r)$, with $R(r)\equiv \int_{0}^{r}s^{-1} \rho ^{-2}(s)ds $, one can find that $\rho (r)$ and $U(R)$ obey the following equations: $$\begin{aligned}
\rho ^{\prime \prime }+r^{-1}\rho ^{\prime }+\left( \mu
-V(r)-S^{2}r^{-2}\right) \rho & =Er^{-2}\rho ^{-3}, \label{evolution} \\
-d^{2}U/dR^{2}+g_{0}U^{3}& =EU, \label{final}\end{aligned}$$where $E$ and $g_{0}$ are constants. The reduction of the GPE/NLSE to Eq. (\[final\]) helps one to find exact solutions, as the latter equation is solvable in terms of elliptic functions. Then, if a solution to Eq. ([evolution]{}) is known, one can construct exact solutions to the underlying GPE/NLSE, while the nonlinearity-modulation profile admitting exact solutions is determined by $\rho (r)$. Physical solutions impose restrictions on $\rho $: from expressions for $R(r)$ and $g(r)$ it follows that $\rho $ cannot change its sign; further, it must behave as $r^{-a}$ with $a\geq 1/3$ at $r\rightarrow 0$, and diverge ($\rho \rightarrow \infty $) at $r\rightarrow \infty $, so that the nonlinearity strength is bounded and the integration in $R(r)$ converges.
{width="7cm" height="4cm"}
We begin constructing exact VS solutions for the attractive nonlinearity ($%
g_{0}<0$) when $E=0$, so that Eq. (\[evolution\]) is solvable. With the harmonic potential $V=kr^{2}$, $\rho $ can be found in terms of the Whittaker’s $\mathrm{M}$ and $\mathrm{W}$ functions [@whittaker]: $\rho
(r)=r^{-1}[c_{1}\mathrm{M}({\mu }/{4\sqrt{k}},{|S|}/{2},\sqrt{k}r^{2})+c_{2}%
\mathrm{W}({\mu }/{4\sqrt{k}},{|S|}/{2},\sqrt{k}r^{2})]$, where the restrictions on $\rho $ require $\mu <\mu _{0}=2(1+|S|)\sqrt{k}$ and $%
c_{1}c_{2}>0$. Without the trap ($k=0$), $\rho $ degenerates to $\rho
(r)=c_{3}I_{S}(\sqrt{-\mu }r)+c_{4}K_{S}(\sqrt{-\mu }r)$, with $\mu <\mu
_{0}=0$, $I_{S}$ and $K_{S}$ being the modified Bessel and Hankel functions, and the constants satisfying $c_{3}c_{4}>0$. In both cases, one has $\rho
(r)\sim r^{-|S|}$ at $r\rightarrow 0$, hence $g(r)\sim r^{6|S|-2}$ and $%
R(r)\sim r^{2|S|}$ at $r\rightarrow 0$, and $\rho (r)\rightarrow \infty $ as $r\rightarrow \infty $. Thus, the respective nonlinearity is localized and bounded, and $R(r)$ is bounded too. To meet boundary conditions $\phi
(0)=\phi (\infty )=0$, an exact solution to Eq. (\[final\]) is chosen as $$U(R)=\left( n\eta /\sqrt{-g_{0}}\right) \mathrm{cn}\left( n\eta R-K(1/\sqrt{2%
}),1/\sqrt{2}\right) , \label{ellip}$$where $n=2,4,6,\cdots $ with $n/2$ being the radial quantum number, $\eta \equiv K(\sqrt{2}/2)/R(r=\infty )$, and $K(1/%
\sqrt{2})$ is the complete elliptic integral of the first kind.
It follows from Eq. (\[ellip\]) that $U(R)\sim R$ at $R\rightarrow 0$, which implies that the amplitude of the exact VS is $\rho U\sim r^{|S|}$ at $%
r\rightarrow 0$, as it should be. Thus, for given $\mu $, $S$, and nonlinearity strength $g_{0}$ (and $k$, in the presence of the trap), one can construct an *infinite number* of exact VSs with $n/2$ bright rings surrounding the vortex core, as shown in Fig. 1. Although these VSs share the same chemical potential, their energies increase with the increase of even number $n$.
{width="6.5cm" height="3.5cm"}
Next we consider the repulsive nonlinearity, $g_{0}>0$, and include the harmonic trap to confine the system. In this case, the existence of elliptic-function solutions to Eq. (\[final\]) requires $E>0$, hence Eq. (\[evolution\]) is a nonlinear equation, which can be solved only in a numerical form. To construct the VSs in this case, one requires $\rho \sim
r^{-|S|}$ (for $S\neq 0$) at $r\rightarrow 0$, hence the nonlinear term in Eq. (\[evolution\]) may be neglected near $r=0$. Thus, $\rho$ is similar to the Neumann function, $Y_{S}(\sqrt{\mu }r)$, at $r\rightarrow 0$ for $\mu
>0$ (for $\mu <0$ it can be checked that VS solutions do not exist). On the other hand, $\rho \rightarrow \infty $ at $r\rightarrow \infty $, due to the the presence of the harmonic trap. Further, term $Er^{-2}\rho ^{-3}$ with $%
E>0$ in Eq. (\[evolution\]) guarantees the sign-definiteness of $\rho (r)$. Therefore, taking small $r_{0}$ as an initial point and using the Neumann function and its derivative at $r=r_{0}$ as initial conditions, one can numerically integrate Eq. (\[evolution\]) to obtain $R(r)$ and $g(r)$; then VSs can be constructed in the numerical form, using the exact solution to Eq. (\[final\]), $$U(R)=\sqrt{{2(E-B^{2})}/{g_{0}}}\mathrm{sn}\left( BR,\sqrt{{E}/{B^{2}}-1}%
\right) , \label{ellip_sn}$$subject to a constraint with even numbers $n$, $$\sqrt{{E}/{2}}<B\equiv n{K(\sqrt{E/B^{2}-1})}/{R(\infty )}<\sqrt{E}.
\label{B}$$From Eq. (\[B\]) it follows that $n<n_{\max }=2R(\infty )\sqrt{E}/\pi $. This implies that there is a *finite number* of the VS modes (or none, if $n_{\max }<2$), in contrast to the case of the attractive nonlinearity, where the infinite set of exact VSs was constructed. Figure 2(a) shows the number of the numerical VS solutions versus $\mu $, demonstrating that the cutoff value of the chemical potential is same as for the exact VSs with the attractive nonlinearity, and the number of VSs jumps at points $\mu =\mu
_{j}^{(S)}\equiv 2(2j+|S|-1)\sqrt{k}$, with $j=1,2,3,\cdots $, which is precisely the $j$-th energy eigenvalue of the vortex state in the corresponding linear Schrödinger equation. Thus, there are $j$ numerically exact VSs, associated with expression (\[ellip\_sn\]), where $%
n=2,4,6,\cdots ,2j$, in interval $\mu _{j}^{(S)}<\mu \leq \mu _{j+1}^{(S)}$. A similar result is true for the 1D GPE with the optical-lattice potential, where $n$ families of fundamental gap solitons exist in the $n$-th bandgap [@wubiao]. Figure 3 displays a characteristic example of the numerically found VSs, in the case when four of them exist. Similarly, it can be shown that there are infinitely many VSs, but just the first $j$ modes do not exist, when $\mu _{j}^{S}\leq \mu <\mu _{j+1}^{S}$ for the attractive nonlinearity, which demonstrates that the repulsive and attractive nonlinearities are *mutually complementary* ones, in this respect.
{width="8cm" height="2cm"}
Solutions can also be found for 2D fundamental solitons ($S=0$) supported by the following *two-tier* nonlinearity, with constant $g_{r}$: $$g(r)=g_{r}~(\mathrm{at}~0\leq r<r_{0}),~{g_{0}}{\rho ^{-6}r^{-2}}\ \left(
\mathrm{at}~r\geq r_{0}\right) . \label{gg}$$
For $r\geq r_{0}$, exact solutions can be found in the same way as above, except that now $R(r)\equiv \int_{r_{0}}^{r}ds/[s\left( \rho (s)\right)
^{2}] $. At $r<r_{0}$, if $r_{0}$ is small in comparison with the spatial scale of the external potential (for instance, $r_{0}\ll \sqrt{1/k}$, in the presence of $V=kr^{2}$), $\phi (r)$ may be approximated by a constant, $\phi
=\sqrt{[\mu -V(0)]/g_{r}}$. Because $\phi (r)$ and $\phi ^{\prime }(r)$ must be continuous at $r=r_{0}$, one then requires $\rho ^{\prime }(r_{0})=0$ and $dU(0)/dR=0$, which leads to $g_{r}=[\mu -V(0)]\left[ \rho (r_{0})U(0)\right]
^{-2}$. In this case, the solution to Eq. (\[final\]) is given by Eqs. (\[ellip\_sn\]) and (\[B\]), with $BR$ replaced by $BR+K(\sqrt{E/B^{2}-1})$, and $n=1,3,5,\cdots $ for the repulsive nonlinearity, to make $\phi
(\infty )=0$. Similar solutions can be constructed for the attractive nonlinearity. Examples of the fundamental solutions are presented in Fig. 4.
![(Color online). Fundamental solitons supported by the repulsive two-tier nonlinearity, see Eq. (\[gg\]). The constant values of $%
g_{r}$ for $n=1,3,5,7$ and 9 are respectively $0.0200,0.0201,0.0224,0.0325$, and 0.1093. Parameters are $\protect\rho (r_{0})=1$, $\protect\rho ^{\prime
}(r_{0})=0$, $r_{0}=E=1$, $k=g_{0}=0.01$, $\protect\mu =2$.](figure4.eps){width="8cm" height="2.5cm"}
We employed the linear stability analysis and direct simulations to verify the stability of the solitons. For the attractive nonlinearity, we have found that all exact VSs are subject to the azimuthal modulational instability without the external potential. As a result, the VSs break up, and eventually collapse. However, the lowest-order exact VSs with $n=2$ are *stable* in the presence of the harmonic trap, if $\mu $ is close enough to the cutoff value $\mu _{0}$, while the amplitude of the VS does not become small, see Fig. 5. We stress that, in numerous previous works, a stability region for VSs trapped in the harmonic potential was found solely for $S=1$, *all* vortices with $S\geq 2$ being conjectured unstable [@BECvortex]. The present results report the first example of *stable* vortices with $S\geq 2$ in the trapped self-attractive fields.
In the case of the repulsive nonlinearity combined with the harmonic trap, the numerically found VSs can be stable for *every* $n$ at which they exist, within some region of values of $\mu $, see Fig. 2(b). This finding is remarkable in the sense that previous works did not report stable trapped vortices in higher-order radial states. Moreover, Figs. 2(b) and 6 demonstrates that VSs with the largest number of rings may be *more stable* than its counterparts with fewer rings. An explanation to this feature is that local density peaks place themselves in troughs of the nonlinearity landscape, thus lowering the system’s energy. Similar properties are featured by the fundamental solitons. Numerical simulations show that those VSs with $n=2$ which are unstable either split into vortices with lower topological charges or exhibit a quasi-stability, periodically breaking and recovering the axial symmetry \[Fig. 6(a)\], similar to what was previously observed in the case of the attractive nonlinearity [@BECvortex; @stable], while unstable VSs with $n=4,6,\cdots$ ultimately evolve into vortices located close to zero-amplitude points.
{width="6cm" height="1.7cm"}
{width="6cm" height="1.7cm"}
In conclusion, we have constructed exact solutions of the Gross-Pitaevskii equation for solitary vortices, and approximate ones for fundamental solitons in the framework of the GPE/NLSE with 2D axisymmetric profiles of the modulation of the nonlinearity coefficient, and harmonic trapping potential. We have demonstrated that the attractive/repulsive nonlinearity supports an infinite/finite number of exact VSs. In particular, *stable* VSs with vorticity $S\geq 2$, as well as those corresponding to higher-order radial states, have been produced for the first time. The results suggest a scenario for the creation of stable vortex solitons in BEC and optics, which have not been as yet observed in experiments. The necessary BEC nonlinearity landscape can be built by means of the Feshbach-resonance technique. The corresponding nonuniform magnetic field may be created by a micro-fabricated ferromagnetic structure integrated into the matter-wave chip [@exper], or one can use the respective pattern of the laser beams. In optics, the same nonlinearity landscape may created by a nonuniform distribution of nonlinearity-enhancing dopants. The method developed in this work for finding the exact solutions, which is based on reducing the 2D equation to the solvable system of Eqs. (\[evolution\]) and (\[final\]), can be applied to other models. In particular, a challenging problem is to devise a physically relevant model admitting exact 3D solitons.
This work was supported by the NNSF of China (Grants No. 10672147, 10704049 and 10934010), the NKBRSFC (Grant No. 2006CB921400), PNSF of Shanxi (Grant No. 2007011007), and the German-Israeli Foundation through grant No. 149/2006.
[99]{} S. Burger *et al*., **83**, 5198 (1999); J. Denschlag *et al*., Science **287**, 97 (2000); B. P. Anderson *et al*., **86**, 2926 (2001).
K. E. Strecker *et al*., Nature **417**, 150 (2002); L. Khaykovich *et al*., Science **296**, 1290 (2002).
M. R. Matthews *et al*., **83**, 2498 (1999); K. W. Madison *et al*., *ibid.* **84**, 806 (2000); S. Inouye *et al*., *ibid.* **87**, 080402 (2001); J. R. Abo-Shaeer, *et al.* Science **292**, 476 (2001); A. L. Fetter, **81**, 647 (2009).
T. J. Alexander and L. Bergé, Phys. Rev. E **65**, 026611 (2002); D. Mihalache *et al*., Phys. Rev. A **73**, 043615 (2006); L. D. Carr and C. W. Clark, Phys. Rev. Lett. **97**, 010403 (2006).
H. Sakaguchi and B. A. Malomed, Europhys. Lett. **72**, 698 (2005).
Y. S. Kivshar and G. P. Agrawal, *Optical Solitons: from Fibers to Photonic Crystals* (Academic Press: San Diego, 2003); B. A. Malomed, *Soliton Management in Periodic Systems* (Springer: New York, 2006).
A. J. Moerdijk *et al*., **51**, 4852 (1995); J. L. Roberts *et al*., **81**, 5109 (1998); S. L. Cornish *et al*., *ibid.* **85**, 1795 (2000).
M. Theis *et al*., **93**, 123001 (2004); R. Ciurylo *et al*., Phys. Rev. A **71**, 030701(R) (2005); K. Enomoto *et al*., Phys. Rev. Lett. **101**, 203201 (2008); Y. N. Martinez de Escobar *et al*., *ibid*. **103**, 200402 (2009).
K. Henderson *et al*., New J. Phys. 11, 043030 (2009).
B. A. Malomed, D. Mihalache, F. Wise, and L. Torner, J. Optics B: Quant. Semicl. Opt. **7**, R53 (2005).
H. H. Chen and C. S. Liu, **37**, 693 (1976); V. N. Serkin *et al*., *ibid.* **98**, 074102 (2007); V. N. Serkin and A. Hasegawa, *ibid*. **85**, 4502 (2000); Z. X. Liang *et al*., *ibid.* **94**, 050402 (2005).
J. Belmonte-Beitia, V. M. Pérez-García, and V. Vekslerchik, **98**, 064102 (2007); J. Belmonte-Beitia *et al*., *ibid*. ** 100**, 164102 (2008).
R. M. Bradley *et al*., **77**, 033622 (2008).
G. Theocharis *et al*., **90**, 120403 (2003).
B. A. Malomed and Yu. A. Stepanyants, Chaos **20**, 013130 (2010); Y. Wang and R. Hao, Opt. Commun. **282**, 3995 (2009).
D. N. Neshev *et al*., Phys. Rev. Lett. **92**, 123903 (2004); J. W. Fleischer *et al*, *ibid*. **92**, 123904 (2004). Y. Sivan, G. Fibich, and M. I. Weinstein, Phys. Rev. Lett. **97**, 193902 (2006); H. Sakaguchi and B. A. Malomed, Phys. Rev. E 73, 026601 (2006); *ibid*. **75**, 063825 (2007); Y. Sivan *et al*., *ibid*. **78**, 046602 (2008); Y. V. Kartashov *et al*., Opt. Lett. **33**, 1747 (2008); *ibid*. **33**, 2173 (2008); *ibid*. **34**, 770 (2009); C. Hang, V. V. Konotop, and G. Huang, Phys. Rev. A **79**, 033826 (2009); D. Wang *et al*., **81**, 025604 (2010).
E. T. Whittaker and G. N. Watson, *A Course in Modern Analysis*, 4th ed. (Cambridge University Press, Cambridge, UK, 1990).
Y. Zhang and B. Wu, **102**, 093905 (2009). B. LeMesurier and P. Christiansen, Physica D 184, 226 (2003).
M. Vengalattore *et al*., J. Appl. Phys. **95**, 4404 (2004); Eur. Phys. J. D **35**, 69 (2005).
|
---
abstract: 'With the rising interest in Virtual Reality and the fast development and improvement of available devices, new features of interactions are becoming available. One of them that is becoming very popular is hand tracking, as the idea to replace controllers for interactions in virtual worlds. This experiment aims to compare different interaction types in VR using either controllers or hand tracking. Participants had to play two simple VR games with various types of tasks in those games - grabbing objects or typing numbers. While playing, they were using interactions with different visualizations of hands and controllers. The focus of this study was to investigate user experience of varying interactions (controller vs. hand tracking) for those two simple tasks. Results show that different interaction types statistically significantly influence reported emotions with Self-Assessment Manikin (SAM), where for hand tracking participants were feeling higher valence, but lower arousal and dominance. Additionally, task type of grabbing was reported to be more realistic, and participants experienced a higher presence. Surprisingly, participants rated the interaction type with controllers where both where hands and controllers were visualized as statistically most preferred. Finally, hand tracking for both tasks was rated with the System Usability Scale (SUS) scale, and hand tracking for the task typing was rated as statistically significantly more usable. These results can drive further research and, in the long term, contribute to help selecting the most matching interaction modality for a task.'
author:
-
bibliography:
- 'references.bib'
title: Influence of Hand Tracking as a way of Interaction in Virtual Reality on User Experience
---
at (current page.south) ;
Virtual Reality, Hand tracking interactions, User experience, Oculus Quest, Interaction type
INTRODUCTION & RELATED WORK
===========================
The recent popularity of Virtual Reality (VR) is resulting not only in many new applications available on the market but also in many new improvements and new features of head-mounted displays (HMD). One of the features of virtual experiences is to immerse users into other worlds [@zahorik1998presence]. Any contact with the real world such as sensors or controllers can break this feeling [@mcgloin2013video]. Therefore, new HMDs are allowing features of interacting via hand tracking to replace interactions with controllers.[^1] This new possibility can lead to very interesting use cases in virtual reality in the field of gaming, but also in others fields where virtual reality is used for serious games. Virtual reality is already being widely used in learning [@monahan2008virtual], more particularly in medicine [@mcgrath2018using] or manufacturing [@nee2013virtual]. Those fields are traditionally associated with precision, as well as hand movements are important. However, it is also important to ensure good quality and user experience to achieve good simulations and results. Based on the recent release of the build-in hand tracking technology by the company Oculus for their HMD Quest and the fact that there is still little to no scientific studies performed to test for influences on user experience, this paper aims to investigate following questions: 1) How do different interaction types (such as hand tracking or interaction via controllers) in virtual reality influence user experience?, and 2) Do different task types (such as grabbing or typing) for those interaction types have an influence on user experience? The feeling of presence and immersion are critical factors when it comes to simulations in VR, and a lot of research has been done already about it [@mania2001effects], [@brooks1999s]. An essential element in the user experience of virtual reality is to accurately represent the user’s hand in the virtual environment [@manresa2005hand], [@gratch2002creating]. This not only related to virtual reality but has already been researched in the field of gaming with other types of virtual environments [@wang2009real], putting in focus in particular algorithms on how to achieve precise hand tracking via various sensors [@penelle2014multi], [@taylor2016efficient]. There is a study reporting on hand tracking and visualization in a virtual reality simulation where a design had been developed offering to track the fingers and palm [@cameron2011hand]. The study was focusing on experiments where participants had to mimic particular hand positions and reports on performance in terms of completion time. However, little work has been done with the focus on user experience with hand tracking in virtual reality. To the best of our knowledge, there is none research done with build-in hand tracking in HMD focusing on user experience with different interaction types.
METHODS
=======
The study was conducted in a separate university’s lab room equipped with Oculus Quest head-mounted display (HMD). This device was used in this study as it is supporting interactions using hand tracking as well as interactions with controllers. For this study, two Unity game applications had been created to test two different types of tasks - grabbing and typing. The first application was a game with a goal to sort five red and five blue balls by grabbing them and putting them to the correct box, as shown in Figure \[fig:apps\]. The second application was a game where the player had to retype displayed number sequences on a virtual numerical keyboard, as shown in Figure \[fig:apps\], and repeat it for five courses. Both games had a strong focus to be played with hands, depending on the type of the game to grab or to type, and both games had the same four possible types of interactions: 1) Visualized controllers and hands, 2) Visualized controllers, but no visualization of hands, 3) Visualized hands, but no visualization of controllers, 4) Visualized hands while using hand tracking. All visualizations, for both interaction types with controllers and with hand tracking, are shown in Figure \[fig:apps\].
![Left: Preview of typing task (upper) and preview of grabbing task (below); Right (top to bottom): interaction with visualization of both controllers and hands, visualization of only controllers, visualization of only hands, and interaction with hand tracking.[]{data-label="fig:apps"}](./figures/Apps.JPG){width="40.00000%"}
At the beginning of the experiment, participants were welcomed. As part of the introduction to participants, it was explained that the assignment is to play two VR games with different tasks, using controllers and hand tracking as types of interaction. Participants also signed a consent form, and before any VR condition, they were given a questionnaire asking about demographics, their previous experience with virtual reality, and hand tracking. Also, participants were asked about their tendency to actively engage in intensive technology interaction by Affinity for Technology Interaction (ATI) Scale questionnaire [@franke2019personal]. As part of the introduction to virtual reality and interactions, participants were explained how to use controllers and hand tracking for interaction, and they were given a chance to familiarize themselves with virtual environments of both games. After they confirmed to the moderator that they feel comfortable with learned interactions, the moderator could start the experiment. Each participant was playing two games with different task types grabbing or typing four times with each of the interaction types, which means that each participant overall participated in 8 conditions. The moderator asked the participants to observe the visualization and behavior of the interaction at the beginning of each condition. After each condition, participants were asked to fill in the Self-Assessment Manikin (SAM) [@bradley1994measuring] as an emotion assessment tool that uses graphic scales. Additionally, after each condition, participants were also rating feeling of presence and realism using a standardized measurement of perceived presence with a subjective scale: the Igroup Presence Questionnaire (IPQ) [@Schubert2001]. After all conditions, the participants were asked to rate the usability of reading in virtual reality with the System Usability Scale (SUS) [@brooke1996sus] for hand tracking separately for both grabbing and typing task. Finally, at the end, participants were asked to fill in a final questionnaire asking to rank the four different interactions in their experience from best to worse. In total, there were 20 participants, eight female, and 12 male, with the average age of the participants was 30.20 years (SD = 12.74, min = 19, max = 62). When it comes to previous experience with VR technologies, 4 participants had no experience, while the others tried VR before, but nobody reported to be an expert. Further on, 9 participants reported having no experience at all with hand tracking for interactions, while others had little or some experience, and nobody was an expert with hand tracking. Data collection was conducted following ethical principles for medical research involving human subjects.
RESULTS
=======
A repeated measure Analysis of Variance (ANOVA) was performed to determine statistically significant differences to investigate the effects between conditions with different task types and interaction types. The two tasks in the application were grabbing and typing, which had to be fulfilled using different interaction types (using controllers with visualization of only controllers, controllers with visualization of only hands, controllers with visualization of hands and controllers or using hand tracking). Further on, the Friedman’s Two-Way Analysis was done to find out significant differences in preference of interaction types. Table \[tab:results\] and \[tab:results2\] provide an overview of the statistically significant results of the Friedman and the repeated measure ANOVA tests.
\[tab:results\]
\[tab:results2\]
There were statistically significant differences due to the conditions on perceived arousal, presence, and realism (see Table \[tab:results\] and Figure \[fig:task\]). For the different conditions where participants had different task types, either grabbing or typing, it can be observed that it influenced how the users have rated SAM dimension arousal. For the typing task, arousal was rater as significantly higher (M=3.83, SE=0.21) compared to the grabbing task (M=3.54, SE=0.22). Further on, participants have reported feeling significantly higher sense of presence while playing the task where they had to grab the balls (M=4.00, SE=0.15) compared to the task where they were typing (M=3.70, SE=0.17). Similarly, the feeling of realism was significantly higher for grabbing task (M=2.88, SE=0.14) compared to the typing task (M=2.69, SE=0.14). Comparing influences that were caused by different interaction types (with controllers or hand tracking), for all three dimensions of SAM questionnaire, statistically significant differences were found (see Table \[tab:results\] and Figure \[fig:inter\]). SAM dimension of arousal was rated significantly higher for interaction type using only hand tracking (M=3.38, SE=0.21) compared to the interaction with the controller where only hands were visualized (M=3.83, SE=0.21). Similarly, the dimension of valence was for hand tracking (M=2.40, SE=0.20) significantly higher compared to all other interaction types with controllers visualizes either as only controllers (M=1.65, SE=0.14), only hands (M=1.48, SE=0.12), or with visualization of both (M=1.58, SE=0.16). Interestingly, the dimension of dominance was reported to be significantly lower for hand tracking (M=3.13, SE=0.16) compared to all other conditions with visualization of both controllers and hands (M=3.85, SE=0.14), visualization of only controllers (M=3.75, SE=0.16), or visualization of only hands (M=3.70, SE=0.14).
![Mean values of SAM arousal dimension (SAM\_A), presence (PRES), and realism (REAL) over all participants for both tasks grabbing and typing. Whiskers denote 95 percent confidence intervals.[]{data-label="fig:task"}](./figures/Task.png){width="50.00000%"}
![Mean values of the SAM dimensions: arousal (SAM\_A),valence (SAM\_V), and dominance (SAM\_D) over all participants for all interaction types with controllers (visualised as both controllers and hands, only controller, only hands) and hand tracking. Whiskers denote 95 percent confidence intervals.[]{data-label="fig:inter"}](./figures/Inter.png){width="50.00000%"}
Ranking of the preferred way of interaction types, where participants had to order from best (4) interaction type to worst (1), resulted in statistically significant differences (see Table \[tab:results2\]). A side corrected pairwise comparison showed that interaction type with controllers where both where hands and controllers were visualized (M=3.30, SE=0.47) were assigned a significantly higher rank compared to all other conditions: controllers with visualizations of only controllers (M=2.35, SE=0.93), of only hands (M=2.25, SE=1.16), and using hand tracking (M=2.10, SE=1.37). A paired-samples t-test was conducted to compare average SUS scale values in hand tracking for grabbing (M=56.50, SE=18.09), and hand tracking for typing (M=75.62, SE=20.96). There was a significant difference in the scores; t(19)=3.01, p=0.007.
DISCUSSION & CONCLUSION
=======================
The type of task (grabbing vs. typing) had an impact on how participants experienced the experimental condition. During the grabbing task, participants experienced lower arousal, higher presence, and higher realism. This result can be assigned to the fact that a grabbing task is more natural, therefore resulting in a higher realism and presence experience. The higher arousal of participants during the typing task could be due to higher demand during the more unnatural task. The type of interaction (controllers visualized as both controllers and hands, the controller only, the controller only hands, and hand tracking) did not influence the experience of presence and realism. Nevertheless, there was an influence of interaction type on all SAM dimensions (valence, arousal, and dominance). It can be seen that participants felt less aroused, have a more positive experience (higher valence), and felt less dominant during the experimental condition hand tracking compared to all interaction types involving controllers. The higher valence for hand tracking can be addressed to the fact that users like the experience better. The lower arousal values could also be due to reduced demand due to the interactions imposed on the participants. The lower dominance experience could be to the fact that interacting with hand tracking leads to a lower feeling of control/precision and, therefore, also to a lower dominance rating. Finally, participants ranked the interaction type, including controllers visualized as both controllers and hands, as being more preferred than all other interaction types. The latter effect could be a combination of being used to interact with controllers and that most tasks are currently rather developed for typing selecting with a controller. Users still seem, when it comes to a visualization, preferred that not only the controller but also the hands are visualized. Finally, hand tracking for typing was rated statistically significantly more usable as hand tracking for the task grabbing on the SUS scale. This effect could be due to the fact that users are still more used to controller interaction and, therefore, also rate this experience a being more usable. This study showed that user experience using pure hand tracking interaction lead to a more positive and less arousing experience while reducing the sense of dominance/control. These results can drive further research and, in the long term, contribute to help selecting the most matching interaction modality for a task.
[^1]: https://www.oculus.com/blog/thumbs-up-hand-tracking-now-available-on-oculus-quest
|
---
abstract: 'We consider the influence of mode - mode coupling in the inflaton field on the spectrum of primordial fluctuations. To this end, we formulate a phenomenological model where the inflaton fluctuations are treated as a fluid undergoing turbulent motion. Under suitable assumptions, it is possible to estimate the size and scale of fluctuations in velocity, which upon reheating induce corresponding fluctuations in the radiation energy density. For De Sitter inflation the resulting spectrum is scale invariant on all scales of interest. The amplitude of the resulting spectrum is compatible with known observational limits. This suggests that the hypothesis of a extremely weakly coupled inflation could be relaxed without affecting the predictions of the model. Principal PACS No 98.80.Cq; additional PACS nos: 04.62.+v, 11.10.Wx, 47.27'
address: |
IAFE and Physics Department,\
University of Buenos Aires, Argentina
author:
- Esteban Calzetta and Mariana Graña
title: 'Mode - mode coupling and primordial fluctuations'
---
Introduction
============
The object of this paper is to present models of fully nonlinear fluctuations during the inflationary era conducing to a scale invariant primordial density contrast spectrum with an amplitude consistent with COBE observations.
Inflationary models were originally introduced as a solution for the so-called puzzles of standard hot Big Bang cosmology [@Peebles], namely the horizon, flatness and photon to baryon ratio problem [@guth] [@abbottpi]. A period of exponential expansion of the Universe was postulated, whereby the ratio of the Universe increased by a factor $e^{60}$ or so, followed by a period of reheating, during which the temperature of radiation was raised up to a final value of $10^9GeV$ or more (we assume units with $\hbar =c=k=1$) [@BdV]. From then on, cosmic evolution followed the lines of standard cosmology [@weinberg3'].
Although many implementations of inflation have been proposed, most attention has been devoted to the simplest scenario, which was Linde’s chaotic inflation with a single inflaton field [@linde]. In this model, inflation is powered by a scalar field $\phi ,$ the inflaton, slowly rolling down an effective potential $V\left( \phi \right) .$ We shall simplify the model further by assuming the Universe is described by a spatially flat, Friedmann - Robertson - Walker geometry. Inflation begins when the effective potential becomes the dominant form of energy density, and ends with the decay of the inflaton onto radiation. During inflation the potential energy $%
V\left( \phi \right) $ acts as an effective cosmological constant. It can be said that no satisfactory contending explanation of the cosmic puzzles is available [@TWH].
Soon after the original proposal, it was realized that inflation could perform a subtler task: to provide a framework for explaining the origin of primordial density fluctuations [@nuffield]. Quantum fluctuations of the inflaton field distort the reheating surface, inducing a primordial density contrast (see [@Peebles], we shall review this argument in greater detail below)
$$\frac{\delta \rho }\rho \sim \frac H{\dot \phi }\delta \phi
\label{pridencon}$$
$H$ being the inflationary Hubble parameter. A similar argument shows that gravitational waves are also being created, with an amplitude $h\sim H/m_p$. The validity of Eq.(\[pridencon\]) has been corroborated by several different methods [@BMPR] (for a dissenting viewpoint, see [@Grischuck]).
In order to obtain a concrete prediction from Eq.(\[pridencon\]) we must estimate the quantum fluctuations $\delta \phi $. The usual approach treats these fluctuations as a free field (for example, in the seminal paper by Starobinsky [@Starobinsky]) in its (De Sitter invariant) vacuum state or very close to it [@DSvacua]. Then a simple calculation within quantum field theory in curved spaces allows us to evaluate the $\delta \phi $ in Eq.(\[pridencon\]) as $\delta \phi \sim H$ [@BirDv]. All quantities in the right hand side of Eq.(\[pridencon\]) are evaluated as the relevant mode leaves the horizon. There are well defined ways to relate the spectrum of scalar and tensor primordial fluctuations to a given potential, which are generally described as the potential reconstruction program [@reconstruction].
As our understanding of the formalism and the complexity of observational data progresses [@COBE], it becomes clear that simple potentials such as single powers of $\phi $ are not rich enough to account for observations. Thus the potential reconstruction program generally assumes more complex functional forms, depending upon several parameters [@reconstruction]. In these more general potentials their higher derivatives may not be negligible, leading to nonlinear interactions between fluctuations. In such a case, the usual way of estimating the primordial density contrast would be invalid. The relevance of non linear fluctuations was already taken up in Ref. [@dursak]. Also several authors [@Morikawa; @Matacz; @CH95; @CG97] have considered models where fluctuation self interactions were treated perturbatively.
The main goal of this paper is to develop ways of estimating the primordial density contrast in chaotic inflation models without presupposing that couplings among fluctuations are negligible. Of course, working out the structure of quantum fluctuations on even simple states of fully nonlinear quantum field theory is a daunting task[@weinberg96]. However, a simpler alternative is available, namely, the application of hydrodynamics to describe the macroscopic behavior of quantum fluctuations. This is possible because these fluctuations, as far as it is relevant to our present concerns, may be described by a c-number energy momentum tensor subject both to the usual conservation laws and the Second Law of thermodynamics [@BirDv]. There is therefore an equivalent fluid description, consisting of a classical fluid whose energy momentum tensor and equation of state reproduce the observed ones for the quantum fluctuations. Solving the dynamics of this equivalent fluid yields answers to all relevant questions concerning the behavior of the actual quantum fluctuations. In this paper we shall not press the issue of whether the equivalent fluid is anything more than a convenient computational device.
Along with the assumption of free inflaton fluctuations, we must question whether these fluctuations are in their vacuum state. This assumption usually rests on the so called [*cosmic no hair theorem,* ]{}which states that any reasonable initial quantum state for the field will quickly relax to a De Sitter invariant state, or its closest available analogue [@CNHT]. The application of this theorem in models with a hundred or more e-foldings of inflation is justified, but these models must be rejected on the grounds that they also predict a value of the density parameter $\Omega $ unacceptably close to one (see Appendix and Ref. [@Grishchukdur]; it is also possible to develop inflationary models in open universes, see Ref. [@Cohn]). When the duration of inflation is close to minimal, it becomes likely that some large but observable scales will leave the horizon before the cosmic no hair theorem is able to operate. It is important therefore to set a realistic initial condition such as a Planckian distribution of particles with temperature $T$. The relevance of thermal effects in inflation was already pointed out (in a different context) in ref. [@Berera].
An immediate consequence of energy momentum conservation and the Second Law is that when velocities are low, the phenomenological fluid may be described within the Eckart theory of dissipative fluids [@Weinberg; @Hu] (for an analysis of the limitations of Eckart’s theory see [@Geroch]). It follows that it obeys a continuity equation and a curved space time Navier - Stokes equation. The model is then defined by giving the equation of state and the viscosity of the equivalent fluid. The advantage we gain is that these are features that can be computed locally. As far as the relevant scales are much below the curvature radius, it is possible to use for them their standard flat space time values. At high temperatures we obtain the equation of state for radiation, $p=(1/3)\rho $, and a dynamic viscosity $%
\eta \sim T^3$. Since the speed of sound, being close to light speed, is much higher than the characteristic speed of the fluid, the flow may be considered incompressible. There will be fluctuations in velocity, nevertheless, and these are the ones responsible for density fluctuations, as it is well known [@Peebles].
As we shall show below, conditions in the early stages of inflation are such that, for generic initial conditions, the flow of the equivalent fluid is highly turbulent, meaning that the corresponding Reynolds number is well over a thousand. The possible role of turbulence in the formation of cosmic structures has been studied in some detail even before inflationary models were introduced (see [@PeeblesLSS; @Raychaudhuri] for a textbook account). These early attempts were abandoned because there were no natural mechanisms for the production of primordial turbulence, and the density contrast predicted was generally too high. After inflation was proposed, the matter was taken up again by Goldman and Canuto [@GC], who studied longitudinal turbulence excited by density fluctuations in the radiation and matter dominated eras. Our work should not be understood as a continuation of this line of research, but rather as a variation on conventional models of primordial fluctuation generation (cfr. [@Peebles; @BMPR; @Starobinsky]) whereby quantum fluctuations of the inflaton field leave their imprint on the primordial density field and are subsequently washed away. In other words, the underlying physics in our model is the same as in these more familiar approaches to primordial fluctuation generation: we do not question the ultimate quantum origin of the fluctuations (another difference with the cosmic turbulence theory from the fifties), but simply borrow insights from hydrodynamics to describe the macroscopic behavior of these fluctuations, rather than rely on possibly oversimplified linearized microscopic models. The conditions of validity of our procedure are the assumptions that the energy momentum tensor of fluctuations is a c-number quantity (which ought to be true at any scale below Planck’s) and the Second Law (which, unlike the Third, has not been challenged yet to our knowledge).
A second goal of this paper is to demonstrate the application of the hydrodynamic approach by discussing some simple solutions to the non linear Navier - Stokes equation, and the resulting spectra of primordial fluctuations. Since the general solution to the Navier- Stokes equation is certainly not available, this requires the appeal to physical insights to simplify the problem.
The basic mechanism of non linear hydrodynamic evolution is the energy transport between eddies of different size through mode - mode coupling, and the viscous dissipation of small scale eddies. The fundamental issue regarding model building is to obtain a closed form energy balance equation, which allows us to follow in time the shape of the energy spectrum. This usually involves some closure hypothesis to reduce the infinite hierarchy of dynamical equations for velocity correlation functions to a manageable set [@Schilling]. Lacking anything better, we shall fall back on the time-honored hypothesis that the effect of smaller eddies at any given scale may be simulated by a scale dependent effective viscosity [@Batchelor; @UFrisch; @McComb]; for concreteness, we shall follow Heisenberg’s 1948 formulation of this idea [@Hei48].
The Heisenberg theory admits some very simple solutions with the property of self-similarity. These have been worked out by Chandrasekhar [@Chandra] and generalized to Friedmann - Robertson - Walker (FRW) backgrounds by Tomita [*et al* ]{}[@Tomita]. These solutions agree with the Kolmogorov 1941 theory in the inertial range [@McComb], failing to reproduce observations for very small eddies. Fortunately, we shall only require the solution in the opposite limit of very large eddies, where it is trustworthy[^1]. With minor adjustments, Tomita’s analysis of turbulence decay in FRW space times also provides a solution to the evolution of our equivalent fluid. Self-similarity is an appealing feature to us, as we do expect that a generic flow will be eventually brought to some sort of steady state by the inflationary expansion.
The task at hand is then to study the evolution of a typical eddy as it is blown up by the universal expansion, exchanging and dissipating energy while inside the horizon, and freezing when outside, until it reaches the reheating hypersurface and delivers its energy to radiation. By assuming that the turbulent velocity fluctuations in the eddy produce fluctuations in the energy density of radiation in the usual way, we shall be able to relate the primordial density contrast to the features of the original self similar turbulence. The resulting spectrum may be matched against the known data on the cosmic microwave background [@COBE], providing a crucial test of the inner consistency and viability of the approach. Our conclusion shall be that, insofar as the horizon remains constant during inflation, the spectrum of primordial density fluctuations produced by self similar flows is strictly scale invariant ($n=1$, see [@HZ]) at large scales. Quantitative agreement with observations may be obtained without any special fine tuning.
A few comments on the possibility of deriving present day spectra from early Universe hydrodynamics are in order; after all, one of the main criticisms against the conventional cosmic turbulence theory was that, even if turbulence were efficiently generated in the Early Universe, it would decay and become uninteresting well before recombination. This criticism does not apply to the present work, since we shall show that the results of customary inflationary models and self similar flows concerning the primordial density field immediately after reheating are identical. Further evolution of the primordial density contrast after reheating is well described by the theory of linear density fluctuations in an expanding Universe, a subject properly covered by many textbooks (cfr. [@Peebles; @BMPR; @Weinberg; @Börner]); the result, for both the conventional inflationary models and the present ones, is that these fluctuations in the primordial density contrast survive to recombination time.
The key point in our analysis is the evolution of the flow during inflation, as the different modes leave the horizon and ”freeze”, and we discuss this issue at some length. The amount of turbulence at reheating is ample enough to seed fluctuations at the level suggested by COBE data, provided the initial temperature is large enough. The lower bound in the temperature, however, is not so large that would invalidate the vacuum dominance condition which is a presupposition of Inflation. Of course, whether Inflation is likely to happen or not is a difficult question (cfr. [@me]) lying beyond the scope of this paper.
To conclude, the main objective of this paper is to show that it is possible to develop sensible models of inflation where inflaton fluctuations evolve nonlinearly and are very far from their vacuum states. The connection of the physics of primordial fluctuations to hydrodynamics opens up a wealth of new interesting phenomena, such as intermittency in the primordial spectrum [@UFrisch; @Frisch] and Burgers turbulence [@Burgers], with a strong potential impact on our understanding of the evolution of cosmic structures. Moreover, it is appealing to be able to account for a macroscopic phenomenon, such as fluctuation generation on super horizon scales, mostly on macroscopic terms (for an independent attempt in this direction, see [@zimdahl]). Most importantly, by not assuming that higher derivatives of the potential are a priori negligible, we avoid a possible conflict within the potential reconstruction program [@reconstruction].
The rest of the paper is organized as follows. In next section we provide a brief summary of hydrodynamics in flat and expanding universes, in order to set up the language for the rest of the paper. In section III we proceed to discuss the equivalent fluid description of inflaton fluctuations, and how to extract the primordial density contrast therefrom. As a simple application of the method, we consider briefly the case of free fluctuations, showing that the model leads back to the conventional results. In section IV we present Chandrasekhar’s self-similar solutions and their generalization to expanding universes, deriving the corresponding scale invariant primordial contrast; we also discuss how relevant these solutions are as actual descriptions of the inflationary period. We state our main conclusions in the final section.
Hydrodynamic flows
==================
Flows in flat space time
------------------------
The equations governing the dynamics of a fluid in local thermodynamic equilibrium are the continuity and Navier-Stokes ones, which, in the case of flat space time, read:
$$\frac{\partial \rho }{\partial t}+\left( {\bf U\cdot }\nabla \right) \rho =0
\label{contpl}$$
$$\frac{\partial {\bf U}}{\partial t}+\left( {\bf U\cdot }\nabla \right) {\bf %
U=-}\frac 1\rho \nabla p+\nu \nabla ^2{\bf U} \label{NSpl}$$
where we have assumed incompressibility, valid when typical velocities are much smaller than the sound velocity; $\nu =\eta /\rho $ is the kinematic shear viscosity. The transition from laminar to turbulent motion can be universally described by the dimensionless ”Reynolds” number:
$$R=\frac{UL}\nu$$ where $U$ is a typical velocity and $L$ a typical length scale. This number represents the order of magnitude of the ratio of the inertial to the viscous term. Low Reynolds numbers correspond to laminar motion, while high ones suggest turbulent behavior.
In general, the velocity profile displays variations in space and time. This implies that the flow must be described probabilistically. Thus, each quantity involved in (\[contpl\]-\[NSpl\]) is divided in its mean value and a fluctuation from it; for example, we write ${\bf U}=\bar U+u$, where $%
u $ stands for the fluctuating part of the velocity. In the case where motion is isotropic, the mean value $\bar U$ for the velocity must be zero, since otherwise there would be a preferred direction.
To analyze the system’s behavior, we define the two-point one-time correlation function for the velocity:
$$R_{ij}({\bf x,x}^{\prime },t)=\left\langle u_i({\bf x,}t)u_j({\bf x}^{\prime
}{\bf ,}t)\right\rangle \label{Rij}$$
In the case of homogeneous and isotropic motion, this correlation must be only a function of the time $t$ and the distance between ${\bf x}$ and ${\bf %
x}^{\prime }$, i.e. $R_{ij}({\bf x,x}^{\prime },t)=R_{ij}(r,t)$, where $%
r=\left| {\bf x-x}^{\prime }\right| .$ Observe that $R_{ii}(0,t)$ (summation over repeated indices must be understood) is twice the average energy density of the flow at time $t$. From (\[NSpl\]) we obtain the equation that this correlation must obey, namely:
$$\frac \partial {\partial t}R_{ij}(r,t)=T_{ij}(r,t)+P_{ij}(r,t)+2\nu \nabla
^2R_{ij}(r,t) \label{ecRij}$$
where $$P_{ij}(r,t)=\frac 1\rho \left( \frac \partial {\partial r_i}\left\langle p(%
{\bf x,}t{\bf )}u_j({\bf x}^{\prime },t)\right\rangle -\frac \partial {%
\partial r_j}\left\langle p({\bf x}^{\prime },t)u_i({\bf x},t)\right\rangle
\right) \label{Pij}$$ and
$$T_{ij}(r,t)=\frac \partial {\partial r_k}\left\langle u_i({\bf x},t)u_k({\bf %
x},t)u_j({\bf x}^{\prime },t)-u_i({\bf x},t)u_k({\bf x}^{\prime },t)u_j({\bf %
x}^{\prime },t)\right\rangle \label{Tij}$$
The tensor $T_{ij}$ comes form the inertia term in Navier-Stokes equation and, as it involves a product of third order in the velocity, reflects the fact that there is not a close set of equation for the correlations of successive orders but there is a hierarchy of equations instead. The problem of closing that hierarchy is known as the ”moment closure problem” [@Schilling]. Let us call $\Phi _{ij}(k,t)$ the Fourier transform of $%
R_{ij}(r,t)$. Then the energy density becomes
$$\frac 12R_{ii}(0,t)=\int E(k,t)\;dk,$$ where
$$E(k,t)=\frac 12\int \Phi _{ii}({\bf k},t)\ k^2\ d\Omega ({\bf k)} \label{ET}$$
is the energy density stored in eddies of size $k^{-1}.$ Defining $\Gamma
_{ij}$ as the Fourier transform of $T_{ij}$, we obtain from (\[ecRij\]) the equation of balance of the energy spectrum: $$-\frac \partial {\partial t}E(k,t)=T(k,t)+2\nu k^2E(k,t) \label{ecesp}$$ where $$\text{\qquad }T(k,t)=-\frac 12\int \Gamma _{ii}({\bf k},t)\ k^2\ d\Omega (%
{\bf k)} \label{ET1}$$
The inertia term $T(k,t)$ is the one that contains the mode-mode interaction, and its effect is to drain energy form the more energetic modes -typically the bigger ones- to the ones where there is major viscous dissipation -the smaller ones-.
Flows in expanding universes
----------------------------
For a curved space-time, in particular that described by a Friedmann - Robertson - Walker (FRW) background metric with zero spatial curvature ($%
ds^2=-dt^2+a^2(t)\left( dx^2+dy^2+dz^2\right) $), the generalization of the above arguments has been considered by many authors [@vonWeizsacker; @Nariai; @SMT; @OzernoiyChibisov; @NariaiyTanabe]. We follow Tomita [*et al.*]{}’s analysis [@Tomita], in which they obtain the solution for the energy spectrum in the case of homogeneous, isotropic and incompressible turbulence.
In a generic space time, we describe fluid flow from the energy density $%
\rho $, pressure $p$ and four velocity $U$. The symmetries of the FRW solution suggest using instead the commoving three velocity $u^i=U^i/U^0;$ if $U^i\ll U^0$ the flow is non relativistic, and if $\nabla {\bf u=}0,$ it is incompressible (${\bf u}=(u^1,u^2,u^3)$). Later on, we shall also use the physical three velocity $v=a(t)u.$
The corresponding continuity and Navier-Stokes equations for a Robertson-Walker background are obtained by the condition of conservation of the energy-momentum tensor [@Weinberg]. For a non relativistic incompressible fluid, with shear viscosity $\eta =\nu \left( p+\rho \right) $ (but no bulk viscosity), these reduce to:
$$\frac{\partial \rho }{\partial t}+3\frac{\dot a}a\left( p+\rho \right) =0
\label{contcur}$$
$$\frac{\partial {\bf u}}{\partial t}+\left[ \left( {\bf u\cdot }\nabla
\right) +\frac{\partial \ln \left( (p+\rho )a^5\right) }{\partial t}\right]
{\bf u=-}\frac{\nabla p}{a^2\left( p+\rho \right) }+\frac 1{a^2}\nu \nabla ^2%
{\bf u} \label{NScur}$$
where we have assumed that $p+\rho $ depends only on time. For the physical three velocity ${\bf v}$, the corresponding Navier-Stokes equation reads: $$\frac{\partial {\bf v}}{\partial t}+\left[ \frac 1a\left( {\bf v\cdot }%
\nabla \right) +\frac{\partial \ln \left( (p+\rho )a^4\right) }{\partial t}%
\right] {\bf v=-}\frac{\nabla p}{a\left( p+\rho \right) }+\frac 1{a^2}\nu
\nabla ^2{\bf v} \label{NScurv}$$ In obtaining (\[contcur\])-(\[NScurv\]) we have neglected possible perturbations to the FRW metric. The corresponding equations considering fluctuations in the metric ($g_{\mu \nu }=g_{\mu \nu }^0+h_{\mu \nu }$) have been obtained by Weinberg [@Weinberg]. The continuity equation is not corrected by gravitational perturbations, while in the Navier-Stokes equation the metric fluctuations appear explicitly only within the shear viscosity term. It can be demonstrated that these terms involving metric fluctuations are negligible for scales that are inside the horizon [@GC]. For scales bigger than the Hubble radius, since dissipation through viscosity is not effective anyway, we may still use the unperturbed Navier - Stokes equation.
The operation of Fourier transforming in the case of a Robertson-Walker cosmology is done in terms of commoving wave-numbers. In doing so, the following equation for the energy spectrum is obtained: $$-\frac \partial {\partial t}E(k,t)=T(k,t)+2\left\{ \frac{\nu k^2}{a^2}+\frac{%
\partial \ln \left( (p+\rho )a^4\right) }{\partial t}\right\} E(k,t)
\label{ecespcur}$$ where the relationship between $E(k,t)$ and $\Phi _{ij}(k,t)$ as well as between $T(k,t)$ and $\Gamma _{ij}(k,t)$ is the same as that for a flat space time, if we define $R_{ij}$ and $T_{ij}$ from correlations of physical quantities, as follows: $$\begin{aligned}
R_{ij}(r,t) &=&a^2\left\langle u_i({\bf x,}t)u_j({\bf x+r},t)\right\rangle
,\ \label{ETcur} \\
\ T_{ij}(r,t) &=&a^2\frac \partial {\partial r_k}\left( \left\langle u_i(%
{\bf x},t)u_k({\bf x},t)u_j({\bf x+r},t)\right\rangle -\left\langle u_i({\bf %
x},t)u_k({\bf x+r},t)u_j({\bf x+r},t)\right\rangle \right)\end{aligned}$$
Equivalent fluid for inflaton fluctuations
==========================================
After establishing the basic necessary notions for the description of hydrodynamic flows, our goal is to associate an equivalent fluid description to inflaton fluctuations, and to derive the spectrum of primordial density fluctuations at reheating therefrom. We shall discuss in the following sections some non trivial instances of this method.
The inflaton as a fluid
-----------------------
To describe the inflaton field from the point of view of an equivalent fluid, we need to obtain the energy density, pressure and velocity of this fluid as functionals of the state of the field. To this end, our starting point will be that in the rest frame of the fluid (quantities in this frame being labelled by a curl), the field ought to be spatially constant
$$\widetilde{\nabla }\phi =0 \label{gradcero}$$
To obtain the fluid four-velocity, we make a boost to the commoving frame. Then, the boost’s characteristic velocity will be the one we are seeking for. By the condition (\[gradcero\]) we obtain: $$u_i=-\frac{\partial _i\phi }{\dot \phi } \label{ui}$$ which is generalized to the covariant form $$u_\mu =-\frac{\partial _\mu \phi }{\sqrt{-\partial _\rho \phi \ \partial
^\rho \phi }} \label{cuadriv}$$ The energy density in the rest frame must be:
$$\widetilde{\rho }=\frac 12\left( \frac{\partial \phi }{\partial t}\right)
^2+V(\phi )$$ Using the Lorentz transformations with the four-velocity (\[cuadriv\]) we obtain the general form for the energy density: $$\rho =-\frac 12\partial _\rho \phi \ \partial ^\rho \phi +V(\phi )
\label{rho}$$ Finally, we obtain the pressure imposing an equality between the energy-momentum tensor for a perfect fluid (see for example [@Weinberg]) and that for a minimally coupled scalar field (see [@BirDv]). The resulting pressure is: $$p=-\frac 12\partial _\rho \phi \ \partial ^\rho \phi -V(\phi )
\label{presion}$$ Since the possibility of deriving a Navier - Stokes equation for the equivalent fluid rests on the conservation of $T_{\mu \nu }$, in principle only the whole inflaton field can be thus represented. However, under the approximation that the homogeneous part of the inflaton essentially contributes an effective cosmological constant, the background energy momentum tensor $T_{0\mu \nu }=\Lambda g_{\mu \nu }$ is independently conserved (even if $\Lambda $ were not constant, conservation fails only on scales too large to be cosmologically relevant), and we can associate an equivalent fluid to the inhomogeneous quantum fluctuations $\delta \phi $ alone. For this fluid, we find the physical velocity ($v^i=au^i=a^{-1}u_i$) $$v_k^i=k_{phys}^i\left( \frac{\delta \phi _k}{\dot \phi _0}\right)
\label{basoon}$$ where $\phi _0$ is the homogeneous background. We interpret this equation to mean that stochastic averages of the fluid velocity are to be identified with (symmetric) quantum expectation values of the operator in the right hand side [@BM; @CH94]. As far as the equation of state is concerned, the free energy for a massive scalar field in the high temperature limit ($T\gg
m $) [@Ramond] gives us the relationship between the pressure and the energy density, which turns out to be that for radiation, $p=\frac 13\rho $. This means that the energy density for this fluid redshifts proportional to $%
a^{-4}.$ As this result has been obtained for a flat space-time, it is valid for scales smaller than the curvature radius. When scales are bigger than the Hubble radius, which in turn takes place when the high temperature limit is no longer valid, the field’s equation of motion $\Box \phi +V^{\prime
}\left( \phi \right) =0$ turns out to be $$\frac{d^2\phi _0}{dt^2}+3H\frac{d\phi _0}{dt}+V^{\prime }(\phi _0)=0$$ for the background field $\phi _0$ (where we have neglected spatial derivatives) and $$\frac{d^2\left( \delta \phi \right) }{dt^2}+3H\frac{d\left( \delta \phi
\right) }{dt}+V^{\prime \prime }(\phi _0)\delta \phi =0 \label{ecfluct}$$ for the fluctuation $\delta \phi $ (where we have neglected spatial derivatives as well as non linear terms). The time derivative of the equation for the background field gives us an equation of motion for $\dot
\phi _0:$ $$\frac{d^2\dot \phi _0}{dt^2}+3H\frac{d\dot \phi _0}{dt}+V^{\prime \prime
}(\phi _0)\dot \phi _0=0 \label{ecfipto}$$ where we have used the constancy of $H$ during inflation. Comparing (\[ecfluct\]) and (\[ecfipto\]) we see that, as $\dot \phi _0$ and $\delta
\phi $ obey the same equation of motion, they must be related by: $\delta
\phi =\dot \phi _0\ f\left( {\bf r}\right) $, which implies that the ratio $%
\frac{\delta \phi }{\dot \phi _0}$ must be independent of time (cfr. [@GP]). For our fluid description, this means that when scales are much bigger than the Hubble radius, the velocity $u_i$ (equation (\[ui\])) must remain constant, which in turn means that the physical three velocity $%
v^i=au^i$ must redshift proportional to $a^{-1}$. As these scales are frozen out because they are outside the horizon, they cannot interact among them or be dissipated by viscosity. Thus, the Navier - Stokes equation (\[NScurv\]) reduces to $$\frac{\partial {\bf v}}{\partial t}+\frac{\partial \ln \left( (p+\rho
)a^4\right) }{\partial t}\ {\bf v=}0 \label{NSvouthor}$$ We obtain $v\propto a^{-1}$ when $\left( p+\rho \right) \propto a^{-3},$ corresponding to the equation of state of matter: $p=0$. Thus, when the scales are well outside the horizon, our fluid behaves as pressureless dust, which agrees with the well known prediction based on Virial’s theorem for the equation of state of a field undergoing oscillations [@p=0], which occurs at the final period of inflation. We must point out that the hypothesis of incompressibility is no longer valid for an equation of state of this type. Nevertheless, for scales bigger than the Hubble radius, which cannot decay through non linear interaction or dissipation by viscosity, equation (\[NSvouthor\]) is still valid, regardless of the ratio of typical velocities to the speed of sound.
Transport coefficients
----------------------
The framework to obtain transport coefficients for our fluid is linear response theory. In the limit of slowly variations in space and time of the magnitudes involved in the equation of conservation for the energy-momentum tensor, the system’s response while it is slightly displaced from equilibrium can be alternatively described by Navier-Stokes and continuity equations as well as by equilibrium expectation values of correlation functions. Matching these two descriptions, one obtains the Kubo formula for the shear viscosity [@KadMart]:
$$\eta =\frac 16\lim_{w,k\rightarrow 0}\left[ \frac 1w\int dt\int d^3{\bf r}\
e^{i\left( {\bf k\cdot r}-wt\right) }\left\langle \left[ \pi _{ij}({\bf r}%
,t),\pi ^{ij}({\bf 0,}0)\right] \right\rangle _{eq}\right] \label{eta}$$
where $\pi _{ij}$ are the traceless spatial-spatial components of the energy-momentum tensor: $$\pi _{ij}=T_{ij}-\frac 13\delta _{ij}T_{\ k}^k$$ For a minimally coupled scalar field the commutator involved in (\[eta\]) turns out to be: $$\left[ \pi _{ij}(x),\pi ^{ij}(0)\right] =\left[ \partial _i\phi (x)\partial
_j\phi (x),\partial ^i\phi (0)\partial ^j\phi (0)\right] -\frac 13\left[
\partial _i\phi (x)\partial ^i\phi (x),\partial _j\phi (0)\partial ^j\phi
(0)\right] \label{conmut}$$ These commutators can be evaluated from retarded and advanced Green functions for a massive scalar field coupled to other fields as well as to itself, which must satisfy: $$\left( \Box +\Gamma \frac \partial {\partial t}+m^2\right)
G_{R,A}(x,x^{\prime })=\delta ^4(x-x^{\prime })$$ where $\Gamma $ is the thermal width which comes from the field’s interactions. From these Green functions we obtain the Pauli-Jordan two point function $G(x,x^{\prime })=\left\langle \left[ \phi (x),\phi
(x^{\prime })\right] \right\rangle $:
$$G(x,x^{\prime })=-\frac{2i\Gamma }{\left( 2\pi \right) ^4}\int d^4p\frac{%
e^{-i\left( p_0(t-t^{\prime })-{\bf p\cdot (x-x}^{\prime })\right) }p_0}{%
\left[ (p_o-i\Gamma )^2-{\bf p}^2-m^2\right] \left[ (p_o+i\Gamma )^2-{\bf p}%
^2-m^2\right] }$$ The mean value for the anticommutator of the field $G_1(x,x^{\prime
})=\left\langle \left\{ \phi (x),\phi (x^{\prime })\right\} \right\rangle $ is obtained through the Kubo-Martin-Schwinger’s theorem [@KMS]:
$$G_1(x,x^{\prime })=-\frac{2i\Gamma }{\left( 2\pi \right) ^4}\int d^4p\frac{%
e^{-i\left( p_0(t-t^{\prime })-{\bf p\cdot (x-x}^{\prime })\right) }p_0\coth
\left( \beta p_o/2\right) }{\left[ (p_o-i\Gamma )^2-{\bf p}^2-m^2\right]
\left[ (p_o+i\Gamma )^2-{\bf p}^2-m^2\right] }$$ >From these, using Wick’s theorem [@Wick] and the c-number character of the mean value for the commutator, we obtain the commutator of the product of two fields as: $$\left\langle \left[ \phi (x)\phi (x),\phi (x^{\prime })\phi (x^{\prime
})\right] \right\rangle =\left\langle \left[ \phi (x),\phi (x^{\prime
})\right] \right\rangle \left\langle \left\{ \phi (x),\phi (x^{\prime
})\right\} \right\rangle$$ The first term involved in (\[conmut\]) turns then out to be $$\int dt\ d^3{\bf r}\ \ e^{i\left( {\bf k\cdot x-}wt\right) }\left\langle
\left[ \partial _i\phi (x)\partial _j\phi (x),\partial ^i\phi (0)\partial
^j\phi (0)\right] \right\rangle =\$$ $$\frac{-4\Gamma ^2}{\left( 2\pi \right) ^8}\int d^4p\frac{\left\{ {\bf p}^2(%
{\bf k}-{\bf p})^2+\left[ {\bf p\cdot }({\bf k}-{\bf p})\right] ^2\right\}
\left( w-p_o\right) p_0\coth \left( \beta p_o/2\right) }{\left| (p_o-i\Gamma
)^2-{\bf p}^2-m^2\right| ^2\left| (w-p_o-i\Gamma )^2-{\bf p}^2-m^2\right| ^2}%
\label{vi}$$ $\label{vi}$and a similar formula for the second term in (\[conmut\]). This gives for the shear viscosity in the high temperature limit ($T\gg
m,\Gamma $): $$\eta =const\ T\ \Gamma ^2 \label{etagama}$$ The thermal width $\Gamma $ comes from the imaginary part of the self-energy. From dimensional analysis, it must be proportional to the temperature since the only relevant scale in the high temperature limit is the temperature itself (the constant of proportionality must be much less than unity for the consistency of the high temperature limit). Assuming that the only present interaction is the one coming from a $\sigma \phi ^4$ term, $\Gamma $ must include two $\sigma $ insertions, which means that $\Gamma $ must be proportional to $\sigma ^2$ (we will estimate it as $\Gamma \sim
\sigma ^2T$). The shear dynamic viscosity becomes $$\eta \sim \sigma ^4T^3 \label{etaT}$$ A similar analysis allows us to evaluate the bulk viscosity, which turns out to be zero for a fluid with an equation of state of the type $p=\frac 13\rho
$ [@YS], in agreement with our previous assumptions.
Conversion of hydrodynamic fluctuations into primordial density contrast
------------------------------------------------------------------------
Having described the quantum field as a fluid, we will analyze the resulting spectrum of density inhomogeneities. To do so, we assume that at some time $%
t_1$ during the beginning of the inflationary phase, when all the scales relevant to cosmology were inside the horizon, the scalar field fluctuations were undergoing hydrodynamic fluctuations. Once the scales leave the Hubble radius, their energy cannot be dissipated by viscosity or by nonlinear coupling. Thus, equation (\[ecespcur\]) means that they evolve according to:
$$E(k,t>t_{out})=E(k,t=t_{out})\left[ \frac{\left( (p+\rho )a^4\right) _{out}}{%
\left( (p+\rho )a^4\right) (t)}\right] ^2 \label{Eaf}$$
where the subscript $"out"$ refers to the time when each scale leaves the horizon.
The definition of $E(k)$ (equation (\[ET\])) can be written in terms of the Fourier transform of the velocity; since the flow is statistically isotropic and homogeneous:
$$\left\langle v^i({\bf k)}v^i({\bf k}^{\prime })\right\rangle =\left( \frac{%
E(k)}{4\pi k^2}\right) \delta ^3({\bf k+k}^{\prime }) \label{viola}$$
Combining (\[Eaf\]) and (\[viola\]) we can obtain the r.m.s. value for the scalar field velocity at the time of reheating, which will be the r.m.s. value of the perturbation in the radiation’s streaming velocity. This perturbation will in turn produce fluctuations in the energy density of radiation, which will evolve in the usual way. The theory of relativistic very large wavelength fluctuations predicts $\delta \sim a^2$, where $\delta
=\delta \rho /\rho $ is the density contrast, and thus $\dot \delta \sim
H\delta $, while the continuity equation yields $\dot \delta \sim v/l$ on a scale of physical size $l$. Consistency of these two pictures leads to the relationship between the velocity at the time of reheating and the fluctuation in the energy density as
$$\left. \frac{\delta \rho }\rho \right| _{reh}=\frac{v_{reh}}{lH_{reh}}
\label{B}$$
where $H_{reh}$ is the Hubble parameter at reheating. Following this fluctuations up to the time they reenter the Hubble radius, assuming that their size is such that they are always unstable (they must be always bigger than the Jeans’s length), they grow following the law (see for example [@Peebles; @Weinberg]):
$$\delta \equiv \left. \frac{\delta \rho }\rho \right| _{ent}=\left. \frac{%
\delta \rho }\rho \right| _{reh}\left( \frac{a_{eq}}{a_{reh}}\right)
^2\left( \frac{a_{ent}}{a_{eq}}\right) \equiv H_{reh}^2\ l^2\left. \frac{%
\delta \rho }\rho \right| _{reh} \label{C}$$
where the subscript $"ent"$ means the time each scale reenters the Hubble radius (the second equation holds even if the entering time occurs before matter-radiation equality). Combining equations ((\[B\])-(\[C\])), we obtain the density contrast predicted by this theory at the time the modes reenter the Hubble radius:
$$\left\langle \delta _k\delta _{k^{\prime }}\right\rangle _{ent}=\frac{%
H_{reh}^2a_{reh}^2E(k,t=t_{reh})}{4\pi k^4}\ \delta ^3({\bf k+k}^{\prime })
\label{derhoderho}$$
This is the main result of this paper, as it relates the density contrast to a hydrodynamic variable. We shall see a nontrivial application of this formula in next section, but before, it is convenient that we pause to show explicitly how the familiar results relating to free field fluctuations are recovered in this language. Probably the most important feature of a theory where inflaton fluctuations are free is that each mode evolves independently of the other ones. Immediately after leaving the horizon they freeze, a situation that can be described phenomenologically assigning to the mode the effective equation of state of dust. This implies that $$E(k,t=t_{reh})=E(k,t=t_{out})\left( \frac{a(t=t_{out})}{a(t_{reh})}\right) ^2
\label{cuatro}$$ $$\left\langle \delta _k\delta _{k^{\prime }}\right\rangle _{ent}=\frac{%
E(k,t=t_{out}(k))}{4\pi k^2}\ \delta ^3({\bf k+k}^{\prime }) \label{cinco}$$ Let us compare this expression to the usual one in terms of quantum fluctuations. First we use Eq. (\[viola\]), neglecting any variation of $H$ or of the velocities during reheating, to get
$$\left\langle \delta _k\delta _{k^{\prime }}\right\rangle _{ent}=\ \left.
\left\langle \ v^i({\bf k)}v^i({\bf k}^{\prime })\right\rangle \right|
_{t=t_{out}(k)} \label{cello}$$
We now relate the physical velocity to field fluctuations according to Eq. (\[basoon\]); at $t=t_{out}(k)$, $k_{phys}=H$, and this reduces to $$\left\langle \delta _k\delta _{k^{\prime }}\right\rangle _{ent}=\left( \frac
H{\dot \phi }\right) ^2\left. \left\langle \delta \phi _k\delta \phi
_{k^{\prime }}\ \right\rangle \right| _{t=t_{out}(k)} \label{oboe}$$ which is the conventional result [@GP].
This shows the agreement between the fluid description and the conventional approach in this case, although of course it is only in the nonlinear case where we expect the hydrodynamic formalism to bring definite advantages.
Self similar flows and nonlinear fluctuations
=============================================
In the previous sections we set up the general formalism whereby we can associate to the evolution of quantum fluctuations during inflation an equivalent fluid description, and derive the corresponding primordial density contrast from hydrodynamic variables. Of course, to put the formalism to actual use, we must be able to solve the Navier - Stokes equations, which is in itself almost as daunting as solving the fundamental quantum field theory. However, there is in the hydrodynamic case a century of lore to draw upon [@lamb], and some well tested approximations leading to relatively simple solutions. In this section, we shall demonstrate the equivalent fluid method by investigating the spectra resulting from one of these solutions, namely self similar flows. Towards the end of the section, we shall discuss the relevance of these solutions to actual cosmology.
Self similar flows in flat and expanding universes
--------------------------------------------------
As we have seen in the previous section (Eq.(\[derhoderho\])), the key element in deriving the primordial density contrast is the energy spectrum $%
E(k)$ (Eq.(\[ET\])), which is the solution of the balance equation (Eq.(\[ecesp\])). In it, the right hand side contains the viscous dissipation as well as the inertial force $T(k,t)$. The overall effect of this term is to transfer energy from a given scale to smaller ones through mode - mode coupling; thus it is natural to model the action of the inertia term as a source of viscous dissipation, where the effective turbulent viscosity for a given mode depends on the motion of all smaller eddies [@UFrisch]. By providing closure, that is, writing this effective viscosity in terms of the spectrum itself, a closed evolution equation for $E(k)$ is obtained. Concretely, Heisenberg [@Hei48] proposed the ansatz
$$\int_0^kT(k^{\prime },t)\ dk^{\prime }=2\nu (k,t)\int_0^kE(k^{\prime },t)\
k^{\prime \ 2}\ dk^{\prime } \label{Heis1}$$
where
$$\nu (k,t)=A\int_k^\infty \sqrt{\frac{E(k^{\prime },t)}{k^{\prime 3}}}\
dk^{\prime } \label{Heis2}$$
and $A$ is a dimensionless constant. With this hypothesis (known as the Heisenberg hypothesis) as the solution to the closure problem, Chandrasekhar [@Chandra] has obtained the energy spectrum for decaying turbulence, assuming that there is a stage in the decay where the bigger eddies have sufficient amount of energy to maintain an equilibrium distribution, thus requiring that the solution for the spectrum should be self-similar. With this consideration into account he obtained an energy spectrum:
$$E(k,t)=\frac 1{A^2k_0^3t_0^2}\sqrt{\frac{t_0}t}F\left( \frac{k\sqrt{t}}{k_0%
\sqrt{t_0}}\right) \qquad$$
where $k_0$ and $t_0$ are initial conditions (namely, the wave number corresponding to the bigger eddy and its typical time of evolution). The function $F$ obeys the equation
$$\frac 14\int_0^x\left[ F(x^{\prime })-x^{\prime }\frac{dF(x^{\prime })}{%
dx^{\prime }}\right] dx^{\prime }=\left\{ \nu k_0^2t_0+\int_x^\infty \frac{%
\sqrt{F(x^{\prime })}}{x^{\prime \ 3/2}}dx^{\prime }\right\}
\int_0^xF(x^{\prime })x^{\prime \ 2}dx^{\prime } \label{ecF}$$
which predicts a Kolmogorov type behavior for an inviscid fluid ($%
R\rightarrow \infty $, $R=\frac 1{\nu k_0^2t_0}$) in the ultraviolet limit:
$$F(x)\rightarrow const\ x^{-5/3}\ (\nu =0\ ,\ x\rightarrow \infty )\label
{compkol}$$ While for nonzero viscosity:
$$F(x)\rightarrow const\ x^{-7}\ (\nu \neq 0\ ,\ x\rightarrow \infty )\label
{comp-7}$$
In the infrared limit, $F$ has the universal behavior $F(x)=4x$ $(x\ll 1)$, and thus we find a linear energy spectrum for $k\sqrt{t}\ll k_0\sqrt{t_0}.$ Chandrasekhar’s self similar solutions are easily generalized to flows in expanding Universes. The dependence on time and wave-number for the self similar energy spectrum is [@Tomita]
$$E(k,t)=v_{ti}^2\left( \frac{(p+\rho )_ia_i^4}{(p+\rho )a^4}\right) ^2\frac{%
\lambda _i^2}\lambda F(\lambda k) \label{speccur1}$$
where the subscript $i$ refers to ”initial” and $\lambda $ and $v_t$ are respectively the Taylor’s microscale and an average turbulent velocity, defined as:
$$\lambda ^2(t)\equiv 5\frac{\int E(k,t)\ dk}{\int E(k,t)\ k^2\ dk}\ \qquad
\frac 12v_t^2(t)\equiv \int E(k,t)\ dk \label{lambdayvt}$$
Their time evolution must follow the law: $$\lambda ^2(t)=\lambda _i^2+10\int_{t_i}^t\frac \eta {(p+\rho )a^2}\ dt\qquad
v_t=v_{ti}\left( \frac{(p+\rho )_ia_i^4}{(p+\rho )a^4}\right) \frac{\lambda
_i}{\lambda \left( t\right) } \label{l(t)yv(t)}$$ The viscosity for our fluid, at least at high temperature, is given by Eq. (\[etaT\]). The equation which determines de function $F(\lambda k)$ in (\[speccur1\]) turns out to be the same as in flat space time, Eq. (\[ecF\]), which means that assuming Heisenberg’s hypothesis the spectrum is linear in $k$ for scales much bigger than the Taylor’s microscale [^2]: $$E(k,t)=4v_{ti}^2\left( \frac{(p+\rho )_ia_i^4}{(p+\rho )a^4}\right)
^2\lambda _i^2k\qquad \text{for \ }\lambda k\ll 1 \label{speccur}$$
Nonlinear inflationary models
-----------------------------
We now want to place a self similar solution in the context of a inflationary scenario where, instead of regarding the inflaton fluctuations as free, we shall substitute them by an equivalent fluid, whose evolution we will assume to be self similar. We discuss at the end whether this last assumption is a reasonable one.
As before, we will assume a duration of inflation close to the minimum value ($N_{\min }\simeq 60$, where $N$ stands for the number of [*e-folds),* ]{}which can be justified by the expected quadrupole anisotropy [@Grishchukdur] as well as by the ratio of the present to the critical density (see Appendix). By this assumption, a scale whose present size equals the horizon ($\simeq 3000$ $Mpc$) leaves the Hubble radius soon after the beginning of inflation.
Unless in the free field case, here we cannot deal with each mode independently, but we must treat the whole flow subject to a phenomenological equation of state. Let us assume the self similar flow sets in at a time $t_1$ when the temperature $T>>H$ (we discuss whether this is a suitable assumption below); and that the present horizon scale leaves the horizon at or around time $t_1$. Then it is valid to use the high temperature limit for length scales close to the present horizon while they leave the Hubble radius during the inflationary phase. The fluid’s equation of state in this limit is of the $p=\frac 13\rho $ type, which means that the product $\left( p+\rho \right) a^4$ remains constant throughout the universal expansion. The factor $\left( (p+\rho )a^4\right) _{out}$ involved in (\[Eaf\]) is then independent of the particular scale being considered within this group. Thus, by (\[Eaf\]) and (\[speccur\]) we can obtain the energy spectrum for these scales while they are outside the horizon:
$$\frac{E\left( k,t>t_{out}(k)\right) }{E\left( k_0,t>t_{out}(k_0)\right) }=%
\frac{E\left( k,t=t_{out}(k)\right) }{E\left( k_0,t=t_{out}(k_0)\right) }
\label{spec}$$
where (cfr. (\[speccur\]))
$$E(k,t=t_{out})=4v_t^2(t_1)\lambda ^2(t_1)k \label{tuba}$$
$\lambda (t_1)$ being the commoving Taylor’s microscale at the time the self similar flow sets in. As at the initial time $t_1$ the only relevant scale is the temperature, we expect the initial Taylor’s microscale to be the inverse of the temperature at that time, i.e. $\lambda _{phys}(t_1)\sim
\frac 1{T(t_1)}.$ By (\[l(t)yv(t)\]) we can obtain the commoving Taylor’s microscale at later times, such that most of the scales are still inside the horizon:
$$\lambda ^2(t)=\lambda _{\ }^2(t_1)+10\int_{t_1}^t\frac \eta {\left( p+\rho
\right) a^2}dt\simeq \lambda _{\ }^2(t_1)+10\frac{\eta (t_1)}{\left( p+\rho
\right) (t_1)}\frac 1{Ha^2(t_1)} \label{lambda(t)}$$
where we have used the fact that for a viscosity dependence upon time given by (\[etaT\]) the main contribution to the integral is given by its lower limit, provided there is a difference between the times $t$ and $t_1$ of more than an e-fold. Since $$\frac{\eta (t_1)}{\lambda ^2(t_1)H\left( p+\rho \right) (t_1)}\sim \sigma ^4%
\frac{T(t_1)}H$$ this means that the commoving Taylor’s microscale freezes in its initial value, unless coupling is very strong. Let us take as reference scale $k_0$ in Eq. (\[spec\]) the inverse Taylor’s microscale (which is the last scale to leave the horizon in the high temperature regime). Then, from Eq. (\[Eaf\])
$$\frac{E\left( k_0,t_{reh}\right) }{E\left( k_0,t=t_{out}(k_0)\right) }\sim
\left( \frac{a(T_{phys}=H)}{a(t_{reh})}\right) ^2 \label{horn}$$
From Eqs. (\[spec\]) and (\[tuba\]), $$E\left( k,t_{reh}\right) \sim \left( \frac{a(T_{phys}=H)}{a(t_{reh})}\right)
^24v_t^2(t_1)\lambda ^2(t_1)k\sim \left( \frac{2v_t(t_1)}{a(t_{reh})H}%
\right) ^2k \label{fiddle}$$ and so from Eq. (\[derhoderho\]) $$\left\langle \delta _k\delta _{k^{\prime }}\right\rangle _{ent}=\frac{%
v_t^2(t_1)}\pi \frac 1{k^3}\ \delta ^3({\bf k+k}^{\prime })
\label{derhoderho2}$$ That is, a scale invariant Harrison - Zel ’dovich spectrum[@Berts] with amplitude $v_t.$
On the other hand, let us recall that the fluid velocity is related to the underlying field description by $v\sim k_{phys}\delta \phi /\dot \phi $. Now the typical wave number is $k_{phys}\sim T$, and at high temperature also $%
\delta \phi \sim T.$ From the background evolution, we have $\dot \phi \sim
V^{\prime }(\phi )/3H\sim m_p^2H/3\phi .$ The number of e-foldings $N\sim
H\phi /\dot \phi \sim \phi ^2/m_p^2,$ so $\phi \sim \sqrt{N}m_p$, $\dot \phi
\sim m_pH$, and $$v_t(t_1)\sim \frac{T^2(t_1)}{m_pH} \label{chiripa}$$ Since the ratio is necessarily less than one, the hypothesis of a non relativistic incompressible flow is seen to be consistent. The constraint of $k\ll \lambda ^{-1}$ reduces to a minimum scale above which we obtain scale invariance. Jeans’s length imposes a lower limit bigger than this (we are assuming that the scales are always unstable while they are outside the horizon, which is valid if they are bigger than the Jeans’s length).
Finally, combining the estimates for the initial Taylor’s microscale and the turbulent velocity, we obtain a Reynolds number: $$R=\frac 43\frac{\lambda _{phys}(t_1)v_t(t_1)\rho (t_1)}{\eta (t_1)}\sim
\frac{v_t(t_1)}{\sigma ^4}$$ suggesting highly turbulent motion, specially for small couplings (the self similar solutions are not dependent on high Reynolds numbers anyway). The estimate of the viscosity must be taken with a grain of salt, nevertheless, since we have ignored non perturbative effects [@YS].
Observations suggest that density fluctuations have a scale invariant spectrum with an amplitude $\delta \rho /\rho \sim 10^{-5}$ over a range of scales going from maybe as low as $100Mpc$ up to the present horizon scale at $l_{now}=3000Mpc\sim 10^{41}GeV^{-1}$[@COBE] [@Einasto]. Recall Eq. (\[app1\]) from the Appendix, relating the size of the Universe at the time $t_{exit}$ when the actual horizon’s scale left the inflationary horizon, to the Hubble parameter during inflation. At the time $t_{exit}$, the physical Taylor microscale is at most $\lambda =\left(
a(t_{exit})/a(t_1)\right) T^{-1}(t_1)=10^{-2}H^{-1}$, since a larger value would narrow too much the regime where the spectrum is scale invariant. So
$$\frac{T(t_1)}{\sqrt{m_pH}}=\left( \frac{a(t_{exit})}{a(t_1)}\right)
10^2\left( \frac H{10^{19}GeV}\right) ^{1/2}=10^{-26}\left( \frac{a(t_f)}{%
a(t_1)}\right)$$ We obtain agreement with observations (cfr.(\[derhoderho2\]) and (\[chiripa\])) provided
$$\left( \frac{a(t_f)}{a(t_1)}\right) \sim 10^{23}=e^{53}$$
Since the total run of inflation is some sixty e-foldings (no more than $70$ even in the extreme case $H=m_p$) this is not hard to obtain, provided self similarity sets in a few e-foldings after the beginning of inflation.
We have thus set an explicit model where interacting inflaton fluctuations lead to a density contrast in agreement with observations. We could rest our case at this point, but before that, we would like to discuss briefly how likely it is that this solution was actually realized in the Early Universe.
Self similar flows and our Universe
-----------------------------------
As we have seen above, a self similar turbulent flow pattern in the equivalent fluid could explain the scale invariant spectrum of primordial density fluctuations observed at scales above $100Mpc$, provided the self similar regime sets early enough. Our goal in this final section is to discuss whether this is a likely assumption regarding our own Universe.
In general, it is known that turbulent flows tend to relax towards self similarity, but it is difficult to estimate how long does it take to get there. In a rough approximation, there are two issues involved, first, on what time scale the Heisenberg closure condition becomes valid, and then, how long does it take the closed equation for the spectrum to yield self similarity.
If posed in these terms, we may observe the analogy with the problem of equilibration in the theory of dilute gases. In the latter, the fundamental description of the dynamics is the BBGKY hierarchy[@Huang]. However, after a time of the order of a collision time the hierarchy can be closed, turning into the Boltzmann equation, which in turn leads to equilibrium in times of the order of the mean free time.
In the turbulence problem, the time scale of a typical eddy is $$\tau =\frac \lambda {v_t}=\frac{m_pH}{T^3}\sim \frac 1H\left( \frac H{Tv_t}%
\right) \label{amapola}$$
We may take this as an estimate of the mean free time. The collision time can be estimated as the characteristic time for the smallest eddies in the flow (we visualize a collision as the exchange of a small eddy between two larger ones). According to Landau - Lifshitz [@Landaulif], this is $$\tau _{\min }=\tau R^{-3/4}\ll \tau \label{azucena}$$
It is simple to find parameters leading to $\tau $ of the order of an e-folding, and therefore $\tau _{\min }\ll H^{-1}$; for example, take $%
H=10^6GeV$, $T(t_1)=10^{10}GeV,$ $v_t(t_1)=10^{-5}$, with $%
T_{reh}=10^{12}GeV $. These solutions may not be available in models with such simple potentials as $\sigma \phi ^4$, but they are feasible in multiparameter models such as those invoked in the potential reconstruction program [@reconstruction]. While this does not constitute a proof, it makes it plausible that a self similar solution may be realized in the early stages of inflation.
Once the motion becomes self-similar, it may last for a rather long time. From the equation of motion Eq. (\[ecespcur\]) we expect the energy in the flow to be dissipated in a time scale of the order of $\tau _{rel}=\nu
^{-1}\lambda _{phys}^2$. With $\lambda \sim T^{-1}$ and $\nu \sim \eta
/\left( \rho +p\right) \sim \sigma ^4T^{-1}$, we obtain
$$\tau _{rel}\sim \frac 1{\sigma ^4T}\sim R\tau \label{trel}$$
This time will be generally larger than an e-folding, and might be even larger than the whole duration of inflation if $R$ is large enough.
We conclude that self similarity will be easily achieved for scales around the Taylor microscale or smaller, and will propagate to larger scales after several e-foldings. At larger scales there could be deviations from scale invariance, associated with the transient behavior of the flow. We may conjecture that as the fluid cools down typical velocities will decrease, and thereby the primordial density contrast too. Thus the model will naturally yield higher power at larger scales, reproducing the behavior of multiple inflation models [@staropol].
It bears mention that the same situation occurs in the usual treatments of the free field case. From Eq. (\[oboe\]), we may conclude that the spectrum of primordial density fluctuations has an amplitude $\delta _k\sim
(H^2/\dot \phi )\sqrt{1+n_k}$, where $n_k$ is the occupation number for that mode in the initial state of the inflaton field. Vacuum dominance requires $%
n_k\ll 1$ for $k\geq \sqrt{m_pH}$, but places no real restriction on larger scales which, in minimal models of inflation, are still observable. Thus we only obtain the conventional result (corresponding to $n_k=0$ for all modes) for specially chosen initial conditions.
Final remarks
=============
In present conventional approaches, density inhomogeneities arise from primordial fluctuations in the inflaton field, ultimately of quantum origin. Fluctuations are treated as a free field, thus forcing upon us the assumption that higher derivatives of the inflaton potential are negligible. In this paper we sought a direct estimate of the primordial density contrast generated in a nonlinear inflationary model. Instead of assuming fluctuations to behave as a free field, we consider them to be coupled, so that they can be described phenomenologically as a fluid. We showed that there are flow patterns for this fluid that reproduce observations at very large scales.
While our model takes advantage of many insights from earlier studies of the role of turbulence in structure formation (see [@GC; @Tomita] etc.), it explores a whole new aspect of the problem in the sense that it places turbulence at the origin of the primordial fluctuations, rather than being excited from them. It is therefore free from the criticisms that are usually raised against the turbulence theory of structure formation [@PeeblesLSS; @Raychaudhuri]. To the contrary, our model successfully reproduces the results of the conventional approach on very large scales, that is, a scale invariant spectrum with a density contrast of about $%
10^{-5} $. While it is not completely free of fine tuning, the fine tuning involved is the same as in the usual scenario.
In our view, the main result of our work is not that a self similar solution should be our final description of fluctuations during inflation, but rather that it is possible to make sense of the physics of fluctuations even in rather general potentials. The self similar solutions we have explored in some detail should be seen as an ideal case which will more or less approximate actual flow patterns; indeed, the same could be said of the De Sitter invariant vacuum as a description of the actual state of the field in free theories.
The connection of hydrodynamics to fluctuation generation has some interest of its own, as it provides an alternative to brute force quantum field theoretic calculations, and also yields physical insight on the macroscopic behavior of quantum fields in the Early universe. The equivalent fluid method may be used to advantage also in other regimes, such as the pre - inflationary Universe and the reheating era, where the strong back reaction phase has so far been untractable [@BdV; @Steve]. Moreover, it opens up a wealth of new phenomena, such as intermittence [@Frisch; @UFrisch] and shocks [@Burgers], which are not apparent in the customary treatments. We will continue our research in this field, which promises a most rewarding dialogue between cosmology, astrophysics, and nonlinear physics at large.
Acknowledgments
===============
We are grateful to C. Ferro Fontán and F. Minotti for pointing out refs. [@Schilling; @GCPRL] to us, and to D. Gómez for multiple conversations on turbulence theory.
This work has been partially supported by the European project CI1-CT94-0004 and by Universidad de Buenos Aires, CONICET and Fundación Antorchas.
Appendix
========
The minimal amount of inflation necessary to solve the homogeneity problem is obtained by the condition that a scale of the size of the horizon at present ($\sim 3000Mpc$) should have been inside the Hubble radius at the beginning of inflation. The Hubble radius during inflation is approximately constant. A scale whose physical size at present is $\lambda (t_0)$ was, at the end of reheating
$$\lambda (t_{reh})=\lambda (t_0)\frac{a(t_{reh})}{a(t_0)}=\lambda (t_0)\frac{%
T(t_0)}{T_{reh}}\simeq \lambda (t_0)\ 2.35\cdot 10^{-25}\left( \frac{%
10^{12}GeV}{T_{reh}}\right)$$
Write $T_{reh}=\Gamma \sqrt{m_pH}$, where $\Gamma \leq 1$ and $H$ is the Hubble parameter during inflation. Then towards the end of inflation we have
$$\lambda \left( t_f\right) =\lambda (t_0)2.35\cdot 10^{-25}\left( \frac{%
10^5GeV}H\right) ^{1/2}$$
[* *]{}This scale left the horizon at a time $t_{exit}$ with $\lambda
\left( t_{exit}\right) =(a(t_{exit})/a(t_f))\lambda \left( t_f\right)
=H^{-1} $. So
$$1=\left( \frac{a(t_{exit})}{a(t_f)}\right) \lambda (t_0)2.35\cdot
10^{-20}GeV\left( \frac H{10^5GeV}\right) ^{1/2}$$
In particular, for the present horizon scale we get
$$1=\left( \frac{a(t_{exit})}{a(t_f)}\right) 10^{21}\left( \frac H{10^5GeV}%
\right) ^{1/2} \label{app1}$$
Therefore, defining $N_{\min }=\ln \left[ a(t_f)/a(t_{exit})\right] $, which would make $t_{exit}=t_i$ for the scale of the horizon at present
$$N_{\min }=\ln \left[ 10^{21}\left( \frac H{10^5GeV}\right) ^{1/2}\right]
=64.4+\ln \sqrt{\frac H{m_p}} \label{app2}$$
[* *]{}On the other hand, inflation could not have lasted much more than this because otherwise the present density should be so fine tuned to the critical one that would contradict observations as well as most speculations on dark matter’s density. This can be seen from one of the Friedmann’s equations:
$$\Omega (t)-1=\frac k{\left( a(t)H(t)\right) ^2}$$ where $\Omega $ is the ratio of the density to the critical one and $k$ is the spatial curvature of the FRW metric. This means that the ratio (assuming instantaneous reheating):
$$\frac{\Omega (t_0)-1}{\Omega (t_i)-1}=\frac{\left( a(t_i)H(t_i)\right) ^2}{%
\left( a(t_0)H(t_0)\right) ^2}\simeq \exp 2\left[ 69.06+\ln \sqrt{\frac H{m_p%
}}-N\right] \label{app3}$$
So, if we set $\Omega (t_i)$ to lie in the interval $\left( 0,1\right) $, we conclude that if $\Omega (t_0)$ is not fine tuned to $1$, $N$ should not have been much more than $60.$
There is another argument supporting that inflation should not have spanned much more than its minimum duration based on the comparison between the expected quadrupole anisotropy and the detected one (see [@Grishchukdur]).
P.J.E.Peebles, [*Principles of Physical Cosmology*]{} (Princeton University Press, Princeton, 1993)
A. Guth, Phys. Rev. [**D 23**]{}, 347 (1981)
L. Abbott and S. Pi, [*Inflationary Cosmology* ]{}(World Scientific, Singapore, 1986).
D.Boyanovsky, H.J.de Vega and R. Holman, hep-ph 9701304 (1997)
S. Weinberg, [*The First Three Minutes*]{} (Bantam Books, New York, 1977).
A. Linde, Phys. Lett. [**129B**]{}, 177 (1981).
Y.Hu, M.S.Turner and E.Weinberg, Phys. Rev. [**D**]{} [**49**]{}, 3830 (1994)
S. Hawking, G. Gibbons and S. Siklos (eds.), [*The Very Early Universe* ]{}(Cambridge University Press, Cambridge, 1983)
R.Brandenberger, H.Feldman and V.Mukhanov, Phys. Rep[*.* ]{}[**215**]{}, 203 (1992)
L.P.Grischuck, Phys. Rev[*.* ]{}[**D**]{} [**50**]{}, 7154 (1994); J. Martin and D. Schwarz, gr-qc 9704049.
A.A.Starobinsky, in [*Field Theory, Quantum Gravity and Strings*]{}; edited by H.J$.$de Vega and N.Sánchez (Springer, Berlin, 1986)
A. Vilenkin and L. Ford, Phys. Rev. [**D 26**]{}, 1231 (1982); E. Mottola, Phys. Rev. [**D 31**]{}, 754 (1985); B. Allen, Phys. Rev. [**D 32**]{}, 3136 (1985).
N. Birrell and P. Davies, [*Quantum Field in Curved Space*]{} (Cambridge University Press, Cambridge, 1982).
E. Copeland, E. Kolb, A. Liddle and J. Lidsey, Phys. Rev. [**D 48**]{}, 2529 (1993); M. Turner, Phys. Rev. [**D 48**]{}, 3502 (1993), and Phys. Rev. [**D 48**]{}, 5539 (1993); J. Lidsey, A. Liddle, E. Kolb, E. Copeland, T. Barreiro and M. Abney, Rev. Mod. Phys. [**69**]{}, 373 (1997).
G.Hinshaw [*et al*]{}., Ap. J., [**464, L**]{}17 (1996); E.L.Wright [*et al*]{}, Ap. J. [**464**]{}, [**L** ]{}21 (1996)
R. Durrer and M. Sakellariadou, Phys. Rev. [**D 50**]{}, 6115 (1994).
M.Morikawa, Prog. Theor. Phys [**93**]{}, 685 (1995)
A.Matacz, Phys. Rev[*.* ]{}[**D 55**]{}, 1860 (1997)
E.Calzetta and B-L.Hu, Phys. Rev[*.* ]{}[**D 52**]{}, 6770 (1995)
E.Calzetta and S.Gonorazky, Phys. Rev[*.* ]{}[**D 55**]{}, 1812 (1997)
S.Weinberg, [*Quantum Field Theory*]{} (Cambridge University Press, Cambridge, 1995 and 1996).
R.Wald, Phys. Rev. [**D 28**]{}, 2118 (1983).
L.P.Grishchuck, Phys. Rev. [**D 45**]{}, 4717 (1992)
J. D. Cohn, astro-ph/9712315
A. Berera and L-Z. Fang, Phys. Rev. Lett[*.* ]{}[**74**]{}, 1912 (1995); A. Berera, Phys. Rev. Lett[*.* ]{}[**75**]{}, 3218 (1995); A. Berera, Phys. Rev. [**D 54**]{}, 2519 (1996).
S.Weinberg, [*Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity*]{} (John Wiley & Sons, 1972)
B-L.Hu, Phys. Lett. [**90 A**]{}, 375 (1982); [**97 A**]{}, 368 (1983)
R.Geroch,[* *]{}J. Math. Phys. [**36**]{}, 4226 (1995)
P.J.E.Peebles, [*The Large Scale Structure of the Universe* ]{}(Princeton University Press, Princeton, 1980)
A.K. Raychaudhuri, [*Theoretical Cosmology*]{} (Oxford University Press, Oxford, 1979)
I.Goldman and V.M.Canuto, Ap. J. [**409**]{}, 495 (1993)
O.Schilling, Ph. D. Thesis, Columbia University (1994)
G.K.Batchelor, [*The Theory of Homogeneous Turbulence* ]{} (Cambridge University Press, Cambridge, 1971)
U. Frisch, [*Turbulence* ]{} (Cambridge University Press, Cambridge, 1995)
Mc.Comb, [*The Physics of Fluid Turbulence*]{} (Oxford University Press, Oxford, 1990)
W. Heisenberg, Proc. R. Soc. Lond. [**A 195**]{}, 402 (1948).
S.Chandrasekhar, Proc. R. Soc. Lond. [**A 200**]{}, 20 (1949)
K.Tomita, H.Nariai, H. Sato, T.Matsuda and H.Takeda, Prog. Theor. Phys. [**43**]{}, 1511 (1970).
V.M.Canuto and I.Goldman, Phys. Rev. Lett[*.* ]{}[**54**]{}, 430 (1985).
E.R.Harrison, Phys. Rev. [**D**]{} [**1,**]{} 2726 (1970);Y.B.Zel’dovich, Mon. Not. Roy. Astron. Soc. [**160**]{}, 1p (1972)
G. Börner [*The Early Universe, Facts and Fiction*]{} (Springer, Berlin, 1988)
D. S. Goldwirth and T. Piran, Phys. Rev. [**D 40**]{}, 3263 (1989); Gen. Rel. Grav. 23, 7 (1991); Phys. Rep. 214, 223 (1992); D. S. Goldwirth, Phys. Rev. [**D 43**]{}, 3204 (1991); N. Deruelle and D. Goldwirth, Phys. Rev. [**D 51**]{}, 1563 (1995); E. Calzetta, Physical Review [**D 44**]{}, 3043 (1991); E. Calzetta and M. Sakellariadou, Physical Review [**D 45**]{}, 2802 (1992), Physical Review [**D 47**]{}, 3184 (1993).
U.Frisch, in [*Chaotic Behavior of Deterministic Systems -Les Houches, 1981-,* ]{}Course 13; edited by G.Ioos, R.H.Helleman and R.Stora (1983)
S. F. Shandarin and Ya. B. Zel’dovich, Rev. Mod. Phys. [**61**]{}, 185 (1989); S. Kida, J. Fluid Mech. [**93**]{}, 337 (1979).
W. Zimdahl, Phys. Rev. [**D 48**]{}, 2431 (1993).
C.F.von Weizsäcker, Ap. J. [**114**]{}, 166 (1951)
H.Nariai, Sci. Rep. Tohoku Univ. Ser. [**I 39**]{}, 213 (1956)
H.Sato, T.Matsuda and H.Takeda, Prog.Theor.Phys., [**43**]{}, 1115 (1970)
L.M.Ozernoi and G.V.Chibisov, Soviet Astron.-AJ [**14,**]{} 615 (1971)
H.Nariai and K.Tanabe, Prog. Theor. Phys. [**60,** ]{} 1583 (1978[**)**]{}
L. Landau, E, Lifshitz and L. Pitaevsky, [*Statistical Physics*]{}, Vol I (Pergamon press, Oxford, 1959).
E. Calzetta and B. L. Hu, Phys. Rev. [**D 49** ]{}, 6636 (1994).
P.Ramond, [*Field Theory, a Modern Primer*]{}, 2$^{nd}$ ed. (Addison-Wessley, 1988)
A. Guth and S-Y. Pi, Phys. Rev. Lett.[** 49**]{}, 1110 (1982).
A.A.Starobinsky, Phys. Lett. [**B 91**]{}, 99 (1980)
L.P.Kadanoff and P.C.Martin, Ann. Phys. [**24**]{}, 419 (1963)
R.J.Kubo, Phys. Soc. Jap. [**12**]{}, 570 (1957) - P.C.Martin and J.Schwinger, Phys. Rev. [**115**]{}, 1342 (1959)
R.Mills, [*Propagators for Many Particle Systems (*]{}Gordon and Breach, New York, 1969)
S.Jeon and L.G.Yaffe, Phys. Rev. [**D**]{} [**53**]{}, 5799 (1996)
H. Lamb, [*Hydrodynamics (*]{}Dover, New York, 1945).
E.Bertshinger, in [*New Insights into the Universe;* ]{} edited by[* *]{}V[*.*]{}Martínez, M.Portilla and D.Sáez. (Springer-Verlag, 1992)
J. Einasto, astro-ph/9711318.
K. [Huang,]{} [*Statistical physics*]{} (John Wiley & Sons, New York, 1987).
L. Landau and E. Lifshitz, [*Fluid Mechanics*]{} (Pergamon Press, Oxford, 1959).
P. Peter, D. Polarski and A. Starobinsky, Phys. Rev. [**D 50**]{}, 4827 (1994); J. Lesgourgues and D. Polarski, astro-ph/9710083.
B-L. Hu and S. Ramsey, Phys. Rev[*.* ]{}[**D 56**]{}, 678 (1997)
[^1]: We wish to point out that the applicability of Kolmogorov’s spectrum to large scale turbulence should not be taken for granted [@GCPRL]
[^2]: We wish to point out an ambiguity concerning the meaning of Heisenberg’s hypothesis in the case of curved spaces. For flat space time, the proportionality between the integral up to a certain wave number $k$ of the inertia and the viscous forces is given by (\[Heis1\]) and (\[Heis2\]). In the case of a FRW space time, the autosimilar solution required by Tomita [*et al.*]{} (\[speccur1\]) needs a time dependent dimensionless constant $%
A$ proportional to $\eta a^2$ for the consistency of the solution. This product does remain constant only if the dynamic shear viscosity evolves in time proportional to $a^{-2}.$ Thus unless this is the case, the solution we described looks like a natural curved space generalization of the Heisenberg - Chandrasekhar solution, but does not admit the same physical interpretation.
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author:
- 'Hicham Qasmi, Julien Barr[é]{}'
- 'Thierry Dauxois[^1]'
date: 'Received: date / Revised version: '
title: 'Links between nonlinear dynamics and statistical mechanics in a simple one-dimensional model'
---
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Introduction
============
The main goal of the paper is to study the links between the microscopic dynamics and macroscopic thermodynamical properties in a very simplified model for DNA. Both aspects are usually not studied simultaneously; in the literature, the main goal is often, either to consider dynamical aspects of coherent structures (solitons for example) in a system at zero temperature, or to derive thermodynamical properties without really considering the consequences of the existence of these coherent structures. Here, on the contrary, we will put the emphasis on the link and show that both approaches give important insights to the description of the physical properties.
Two aspects will be of particular importance and we would like to shed light on them already in the introduction. A recent method [@dauxois_peyrard], which showed how the stability of a nonlinear solution of the dynamical equations exhibits an interesting approach to describe a phase transition, will be applied in this new model: it will reveal how the small amplitude dynamics needs a careful treatment to adequately describe the thermodynamics. Second, we will describe the relationships between the maximum Lyapunov exponent which characterizes the dynamics and the thermodynamics. Using a geometric method [@pettini], we will explicitly compute the evolution of this dynamical quantity as a function of the temperature: consequences of the phase transition will therefore be explicit. From these calculations, we predict features absent in the Peyrard-Bishop model, although both models are very similar: a non monotonic behavior of the maximum Lyapunov exponent in the low temperature phase, and a jump at the critical temperature.
In Sec. \[model\] we present briefly the model. In the following Sec. \[thermodynamics\], we show how one can derive its thermodynamical properties. Then, the emphasis is put in Sec. \[domainwall\] to a special particular solution and on its use to explain the thermodynamical properties. Finally, using the powerful geometric method introduced recently to compute the largest Lyapunov exponent, we discuss in Sec. \[lyapunov\] the link between this dynamical quantity and the phase transition, a thermodynamic concept.
The Model {#model}
=========
A very simplified model has been proposed in 1989 by M. Peyrard and A. R. Bishop [@peyrardbishop] to describe DNA denaturation. This biological phenomenon leads to the breaking of the H-bonds linking both strands of DNA. This process, which appears when either the temperature increases or when the pH of the surrounding solvent is modified, was previously successfully described by Ising models [@wartellbenight]. However, the dynamical properties of the phenomenon were not captured by these static descriptions. This is why, keeping the principle of simplicity, Peyrard and Bishop described this highly complicated biomolecule by two chains of particles coupled by nonlinear springs. This step toward a more complex system, since including the minimal dynamics, was unexpectedly more successful than first understood and have lead to many studies (See Ref. [@peyrard] for a review). A large part of the results were interesting from the physical point of view: one may list in particular studies of discrete breathers modes and energy localization in systems involving nonlinear and discreteness effects [@dauxoispeyrardbishop; @dauxoispeyrardprl], existence of phase transition in one-dimensional system [@dauxois_peyrard]…Furthermore, several recent results have emphasized that this model could be successfully used to locate promotor regions [@Rasmussenbishop] for real DNA sequences. This unexpected report and similar ones explain why so simple models are still nowadays thought to be possible powerful tools to describe real DNA dynamics.
Two linear chains of particles describing phenomenologically the nucleotides describe the two different strands of DNA as schematically represented in Fig. \[dessin\]. In the simplest description, all particles of the same chain are harmonically coupled whereas the interstrand interactions are restricted to facing nucleotides; long-range interactions are neglected at this level of description. It is important to understand that because the goal is to describe denaturation dynamics, only transversal degrees of freedom are taken into account. Defined with respect to their equilibrium position, the displacement of the center of mass of the $n^{{\rm th}}$ nucleotide are called $u_n$ (resp. $v_n$) for the top (resp. bottom) chain.
Denoting $p_u$ the conjugated momentum to the spatial position $u$ and the number of nucleotides being $N$, the Hamiltonian model can be written as $$\begin{aligned}
H = \sum_{n} \Biggl[&\displaystyle \frac{p_{u,n}^2}{2 m} + \frac{K}{2} (u_n-u_{n-1})^2
+&\nonumber\\
& \displaystyle\frac{p_{v,n}^2}{2 m} + \frac{K}{2} (v_n-v_{n-1})^2
+& V\left(\frac{u_n-v_n}{\sqrt{2}}\right)
\Biggr].
\end{aligned}$$ The interstrand potential $V$ describes the effective interactions, i.e. in particular hydrogen bonds between base pairs but also the repulsion between phosphates. The canonical transformation $x_n={(u_n+v_n)}/{\sqrt{2}}$ and $y_n={(u_n-v_n)}/{\sqrt{2}}$ decouples simply both degrees of freedom since $H$ can be rewritten as $H = H_x+H_y$ where $$H_x = \sum_n \left[{ \frac{p_{x,n}^2}{2 m} + \frac{K}{2} (x_n-x_{n-1})^2 }\right]$$ and $$H_y = \sum_n \left[{ \frac{p_{y,n}^2}{2 m} + \frac{K}{2}
(y_n-y_{n-1})^2 + V\left(y_n \right) }\right].\label{defHamil}$$
The dynamics and the thermodynamics of the first part, $H_x$, which corresponds to a linear chain of harmonic oscillators, can be easily computed. We will omit this part in the remaining of the paper without loss of generality. The second part, $H_y$, on the contrary needs further developments. For the sake of simplicity, we will omit the $y$ index.
The system defined by this hamiltonian $H$ exhibits a second order phase transition as shown in the next section. The low temperature phase corresponds to states where the particles are located close to their equilibrium position, the associated DNA being in the native state: it must correspond to the bottom well of the potential $V$. On the contrary, in the high temperature phase, DNA being denaturated, the link between facing nucleotides of both strands are broken: consequently, associated particles of the model must be located in a plateau of the potential, far from their equilibrium positions since the force is vanishing. The position in this plateau will be thermodynamically chosen because of the entropy contribution to the free energy, important only at high temperature.
This simple physical description guides therefore the appropriate choice for the potential $V$. Nevertheless, the analytical calculations that we will present now are possible for only a few cases. The Morse potential $V_m(y)=D(\exp(-a_my)-1)^2$ was the first choice [@peyrardbishop; @dauxois_peyrard]. Here, we will present another possible example [@morse]: $$\label{potmorsemorceaux}
V(y)= \left\{ \begin{array}{ll}
-\displaystyle\frac{D}{\cosh^2 a y} & \mbox{if}\ y\geq 0\\
&\\
+\infty & \mbox{if }
y < 0.
\end{array}
\right.$$ As we will see, the qualitative shapes are really close but the differences of curvatures lead to several consequences. In addition, the impossibility to reach negative values for the variable $y$ gives interesting properties. One important question concerns the influence of the details of the potential $V$. We will discuss in particular the following points:
- values of the critical temperature of the phase transition for both potentials,
- consequences to the related characteristic lengths: order parameter and correlation length,
- largest Lyapunov exponent as a function of temperature.
1.5truecm 0.5truecm
Thermodynamics of the model {#thermodynamics}
===========================
The canonical partition function
--------------------------------
As it is well-known nowadays [@KrumhanslSchrieffer; @dauxoispeyrardbook], the statistical mechanics of such a one-dimensional short-range Hamiltonian can be exactly derived with the transfer integral method (See appendix \[transferinteg\]). In the framework of the continuum approximation, the solution relies on solving the following Schr[ö]{}dinger equation $$\label{equaschro}
-\frac{1}{2 \beta^2 K}\frac{{{\rm d}}^2\psi}{{{\rm d}}y^2} -\frac{D}{\cosh^2 a y } \psi = {\cal E}_k \psi,$$ if the lattice spacing between sites in the $x$-direction is set to one.
Defining the quantity $$\eta=\frac{1}{4}\left[\sqrt{1 +8\frac{{T_c}^2}{T^2}}-1\right]$$ where $$T_c=\frac{\sqrt{ K D}}{ak_B},\label{Tcexact}$$ this equation has $N_\eta =
E(\eta+1/2)$ localized states, with $E(.)$ denoting the integer part. One notes that $T_c$ corresponds to the disappearance of the last discrete state, and will be called the critical temperature.
The $(N_\eta+1)$ localized states $\psi_k$ can be expressed [@morse] in terms of hypergeometric functions as $$\begin{aligned}
\psi_k(y)& = & \frac{{\cal N}_k}{\cosh(a y)^{b_{2 k+1}}} F\bigg(-2 k-1,2
b_{2 k+1} + 2 k +2,\nonumber\\&&\hskip2.5truecm b_{2 k+1} +1; \frac{e^{- a y}}{e^{- a y}+e^{ a y}}\bigg),\end{aligned}$$ where ${\cal N}_k$ is the normalization factor of the wave function, $b_{n} = 2 \eta -n$ and finally $$\begin{aligned}
{\cal E}_k = -\frac{a^2}{2 \beta^2 K} \left({2\eta-2 k
-1}\right)^2,\end{aligned}$$ the associated eigenvalues.
The ground state which is particularly useful in the remaining of the paper can be simplified as $$\psi_0(y)= \sqrt{ 2 a \frac{2 \eta -1}{B\left({2 \eta,
\frac{1}{2}}\right)}} \frac{\sinh a y}{\cosh^{2 \eta} a
y}\label{exprpsizero}$$ by introducing the Euler function $$B(x,y)=\frac{\Gamma(x+y)}{\Gamma(x)\Gamma(y)}=\int_{0}^{1} {{\rm d}}t\, (1-t)^{x-1} t^{y-1}.$$
Discussion of the choice of the parameters set
----------------------------------------------
One of the goal of this work is to make a detailed comparison between the Morse potential and potential (\[potmorsemorceaux\]). The parameter set is obtained either by considering the effective physical interactions, or by choosing the values to get the same melting temperature. Assuming that the depth of the potential is known, both possibilities will define a relation between the width of the potentials: $a_m^{-1}$ for the Morse one, and $a^{-1}$ for potential (\[potmorsemorceaux\]).
The choice ${a_m} \approx 1.472\,a$ corresponds to the best fit. On the contrary, as the transition temperature for the Morse potential [@peyrardbishop; @dauxois_these; @dauxoispeyrardbook] is ${T_c^m}={T_c}\sqrt{8} $, the second choice $a_m=a\sqrt{8}$ would lead to the same critical temperature. Consequently, when the parameters are fitted, the critical temperature differs by a factor two!
This unexpected disagreement reveals a hidden difference between both potentials: the underlying reason is the important contribution of negative positions $y$, in the Morse case. They cannot be neglected even if the fast exponential increase was thought to play the role of the impossibility of interpenetration.
The appropriate solution is to introduce an additional parameter. The inverse width $a$ being given by the critical temperature, one could adapt the minimum of the potential to minimize the differences with the Morse potential. A fit restricted to positive $y$-values over both variables $({a_m},y_0)$ leads to an excellent agreement. This case is presented in Fig. \[fig:allure\] by the solid line.
All results below correspond to the following set of parameters $m=300$ u. a., $D=0.00094$ eV, $K=1.9$ eV.${\AA}^{-2}$ and $a=4.5$ ${\AA}^{-1}$. The value of $T_c$ will be different from the Morse case, but both potentials will be very similar as emphasized by Fig. \[fig:allure\]. This choice will allow a precise study of the negative $y$-values region and of the importance of potential curvature for the Lyapunov exponent discussed in section \[lyapunov\].
Characteristic lengths
----------------------
As usual, the thermodynamic properties of this system can be characterized by an order parameter, and its fluctuations, in the vicinity of the critical temperature $T_c$. Here, the appropriate choice is the quantity $$\ell =\left\langle \psi_0|x|\psi_0 \right\rangle
=\int_{0}^{\infty}{{\rm d}}x \, x |{\psi_0}(x)|^2
,\label{sigmacontinu}$$ which diverges for $T=T_c$. This clarifies the name critical temperature which separates, the phase with a finite order parameter (native state) from the phase with infinite order parameter, representing the denaturated state.
The associated critical exponent can even be determined by using the asymptotic expression $\int_{0}^{\infty}{{\rm d}}x\,
{x}/{\cosh^\alpha x}$ $ \stackrel{\alpha\to
0}{\sim}{1}/{\alpha^2}$. Using expression (\[exprpsizero\]) and introducing the usual reduced temperature $t=1-{T}/{T_c}$, one finally gets $${ \ell \stackrel{T\to T_c}{\sim} \frac{3}{16}\frac{1}
{a}\frac{1}{|{t}|} }.\label{asympsig}$$ In agreement with critical phenomena theory, we obtain therefore the usual critical exponent $\beta=-1$, for this second order phase transition. Fig. \[ordeparameter\] presents the evolution of $\ell$ as a function of the temperature. Formula (\[sigmacontinu\]), represented with a solid line, agrees very well with the exact result obtained with the discrete transfer operator (\[formuleA3\]). In the inset, the logarithmic plot emphasizes the critical behavior and confirms the scaling exponent $\beta=-1$.
1.5truecm 0.5truecm
Interestingly it is also possible to determine the fluctuations of the order parameter which can be computed from the following expression $${\xi_\perp^2}= \left\langle \psi_0|(x-\ell )^2|\psi_0
\right\rangle.\label{xicontinu}$$ As above, the asymptotic expression can be easily derived by taking into account that $\int_{0}^{\infty}{{\rm d}}x\,{x^2}/{\cosh^\alpha x} \stackrel{\alpha\to 0}{\sim}
{2}/{\alpha^3}$. One thus obtains in the vicinity of the critical temperature that $${
\xi_\perp\stackrel{T\to T_c}{\sim} \frac{3\sqrt{3}}{16}\frac{1}{
a}\frac{1}{|{t}|} }.\label{asympxi}$$ The critical exponent is therefore $\nu_\perp=1$. Fig. \[ordeparameter\] shows that the order parameter and its fluctuations are of the same order in this temperature regime. The inset confirms again the critical behavior and the scaling exponent $\nu=1$.
It is important to stress that all above analytical results are confirmed by the numerical but exact solution of the transfer integral operator (\[formuleA3\]), without relying on the continuum approximation: see symbols in Fig \[ordeparameter\]. As it is nowadays known [@dauxois_peyrard; @dauxoispeyrardbook], the critical exponents are valid even in the region where discreteness effects can not be neglected. To be more specific, let us introduce the parameter $$\label{defdeR}
R=\frac{D a^2}{K},$$ measuring the onsite potential with respect to the elastic coupling. For small value of this parameter, the continuum approximation is valid and results (\[asympsig\]) and (\[asympxi\]) confirmed. For the set of parameters chosen in this work, $R$ is equal to $10^{-2}$, and one get results in values slightly different from Eqs. (\[asympsig\]) and (\[asympxi\]) for $T_c$, $\ell$, and $\xi_{\perp}$. However, as shown by Fig. \[ordeparameter\] the differences are hardly distinguishable and moreover the critical exponents are fully confirmed.
Domain Wall {#domainwall}
===========
In the framework of this model, it is also interesting to discuss a new method [@dauxois_peyrard; @dauxoispeyrardbook], alternative to the usual thermodynamic one proposed in previous section, to detect the phase transition from dynamical considerations. This will illustrate again the importance of the wall located at $y=0$.
Equations of motion corresponding to Hamiltonian (\[defHamil\]) are $$m{\ddot y}_{n}=K (y_{n+1} + y_{n-1} -2 y_{n}) - \frac{ \partial
V}{\partial y_{n} }$$ or, in the continuum limit, $${\ddot y} = \frac{K}{m}\frac{ \partial ^{2 }y}{\partial x^{2 } }
-\frac{ 1}{m } \frac{ \partial V}{\partial y }. \label{EMcont}$$ The uniform profile at the minimum of $V(y)$ is a static solution of the infinite chain with free ends. Profiles verifying ${{\rm d}}^2 y /{{\rm d}}x^2 = 0$ are approximate solutions only on the plateau of the potential, since $\partial V/\partial y$ is close to zero for large $y$.
There exists also an exact, unbounded, domain-wall like solution $$y_{DW}^{\pm}(x) = \frac{1}{a}\mbox{Argsh}\, e^{\pm z} = \frac{1}{a} \ln\left [ e^{\pm z} +\sqrt{1+e^{\pm 2z}}\right] , \label{DWsol}$$ where $z=\sqrt{2R}(x-x_0)$ and $x_{0}$ is an arbitrary constant. Solution (\[DWsol\]) represents a configuration which links the stable minimum to a particular member of the metastable configurations, with a slope $\sqrt{2D/K}$. One can easily checks that this corresponds to equal contributions to the elastic and the on-site potential energy densities ($D$ per site). Consequently, the energy of the solution contains a term which is proportional to the number of sites to the right of $x_{0}$ and if lattice sites are numbered from $0$ to $N$, one has $$E_{DW}^{+}=\left( N -{ x_{0}}\right)2D + {\cal O}(N^{0
}) \quad. \label{DWen}$$ At zero temperature the profile (\[DWsol\]) is consequently not stable, and the wall spontaneously move to the right,“zipping” back the unbound portion of the double chain. This instability changes however under the influence of temperature.
At non zero temperatures, let us consider small deviations with respect to (\[DWsol\]), i.e. $$y(x,t) = y_{DW}^{+}(x-x_{0}) + \sum_{j} \alpha _{j}
f_{j}(x-x_{0})e^{-i\omega _{j}t}$$ where $|\alpha _{j}|\ll a^{-1}$. The linearized eigenfunctions $f_{j}$ satisfy the Schr[ö]{}dinger-like equation $$-\frac{{{\rm d}}^{2 }f_{j}}{ {{\rm d}}z^{2 }} +
\frac{1-2e^{2z}}{\left(1+e^{2z}\right)^2}f_{j} = \frac{ m\omega
_{j}^{2 }}{2KR }\, f_{j}. \label{lin}$$ Eq. (\[lin\]) has no bound states [@morse]. There are however scattering states: acoustic phonons oscillating on the flat portion of the potential with frequencies $$\omega_{ac}^2=\frac{2K}{m} (1-\cos q) \label{softcontinuumphonon}$$ are some of them.
In the bottom of the well of the potential, let us first forget the wall. Consequently scattering states would be optical phonons with frequencies $$\omega_{opt}^2 = \frac{2Da^2}{m}+\frac{2K}{m} (1-\cos q).\label{hardcontinuumphonon}$$
At finite temperatures, the domain wall would therefore be accompanied by a phonon cloud contributing to the free energy as $$F_{ph} = k_{B}T x_{0}\int _{0}^{\pi}\frac{ {{\rm d}}q}{\pi } \ln
\frac{\omega_{opt} }{\omega_{ac} } + ... \label{phcl}$$ where we omit terms independent of $x_{0}$. Introducing the dispersion relations, we can evaluate [@gradshteyn] the integral in (\[phcl\]) using $\int_{0}^{\pi} {{\rm d}}x \ln [1-\cos x
+ R]= {2\pi } \ln [ {(\sqrt{R}+\sqrt{R+2} )}/2].$ We obtain thus the total free energy (DW plus phonon cloud) $$F = \left( k_{B}T\ln \left[ \frac{\sqrt{R}+\sqrt{R+2} }{\sqrt{2} }
\right] -2D \right){ x_{0}} + const . \label{FDW}$$ This result describes in very simple terms why and when the phase transition occurs. At temperatures lower than $$T_{c}=\frac{ 2D}{k_{B}\ln\left[ \sqrt{R/2} + \sqrt{1+R/2} \right]
}, \label{Tcdiscrete}$$ the prefactor of $x_{0}$ in Eq. (\[FDW\]) is negative, describing the DW’s natural tendency towards high positive values of $x_{0}$: it “zips” the system back to the bound configuration. Conversely, at temperatures higher than $T_{c}$, thermal stability is achieved and the DW “opens up".
It should be noted that the value of $T_{c}$ predicted by the above DW argument coincides exactly with the result obtained for the Morse potential [@dauxois_peyrard; @dauxoispeyrardbook]. The limiting behavior in the continuum approximation, i.e. in the limit $R\ll 1$, leads for example to $T_c=2\sqrt{2KD}/(ak_B)$. This result differs nevertheless from Eq. (\[Tcexact\]).
Expression (\[Tcdiscrete\]) is consequently [*not*]{} valid for the sech$^2$-potential (\[potmorsemorceaux\]) of interest. The underlying reason is the impossibility of usual phonons to take place in the bottom of the well. The harmonic approximation of the problem leads indeed to a [*nonlinear*]{} problem because of the nonlinear condition $y>0$ introduced by the wall. Optical phonons with frequencies (\[hardcontinuumphonon\]) are therefore totally modified by the wall and cannot be computed. A way to take into account this nonlinear condition would consist in selecting only the even modes of optical phonons (\[hardcontinuumphonon\]). However, it turns out to be unexact and unsufficient. This illustrates again that the presence of the wall strongly modifies the [*dynamics*]{} of the system, which leads to important [*thermodynamic*]{} consequences.
Geometrical method to derive the largest Lyapunov exponent {#lyapunov}
==========================================================
In the theory of dynamical systems, the concept of [*Lyapunov exponent*]{} has also attracted a lot of attention [@manneville; @lichtenberg] because it defines unambiguously a sufficient condition for chaotic instability. Unfortunately, except for very few systems, it is already an extremely difficult task to derive analytically the expression of the largest one, $\lambda_1$, as a function of the energy density. As some promising results have been recently obtained to describe some properties of high-dimensional dynamical systems [@livi; @Constantoudis; @ruffo; @firpo; @VallejosAntenodo; @Kurchan], by combining tools developed in the framework of dynamical systems with concepts and methods of equilibrium statistical mechanics, the idea that both concepts could be related was proposed [@pettini].
Riemannian geometry approach {#sec:riemman}
----------------------------
The main idea is that the chaotic hypothesis is at the origin of the validity of equilibrium statistical physics, and this fact should be traced somehow in the dynamics and therefore in the largest Lyapunov exponent. The method is based on a reformulation of Hamiltonian dynamics in the language of Riemannian geometry [@pettini]: the trajectories are seen as geodesics of a suitable Riemannian manifold. The chaotic properties of the dynamics are then directly related to the curvature of the manifold and its fluctuations. Indeed, negative curvatures tend to separate initially close geodesics, and thus imply a positive Lyapunov exponent; nevertheless chaos may also be induced by positive curvatures, provided they are fluctuating, through a parametric-like instability. To approximate the curvature felt along a geodesic, the method uses a Gaussian statistical process. The mean value of this process is given by $\kappa_0$ and its variance by $\sigma_\kappa$, where $\kappa_0 $ and $\sigma_\kappa$ are the statistical average of the curvature and its fluctuations, which can be computed by standard methods of statistical mechanics.
Finally one ends up with the following expression of the largest Lyapunov exponent [@pettini] $$\label{exprlyapunov}
\lambda_1=\frac{1}{2}\left(\Lambda-\frac{4\kappa_0}{3\Lambda}\right)$$ where $$\Lambda=\left(\sigma_\kappa^2\tau+\sqrt{\left(\frac{4\kappa_0}{3}\right)^3
+\sigma_\kappa^4\tau^2}\right)^{1/3}.\label{biglambda}$$ In this definition, $\tau$, the relevant time scale associated to the stochastic process, is function of the two following timescales: $\tau_1\simeq{\pi}/{2\sqrt{\kappa_0+\sigma_\kappa}}$ is the time needed to cover the distance between two successive conjugate points along the geodesics, whereas $\tau_2\simeq{\sqrt{\kappa_0}}/{\sigma_\kappa}$ is related to the local curvature fluctuations. The general rough physical estimate ${\tau}\simeq\left({1}/{\tau_1}+{1}/{\tau_2}\right)^{-1}$ completes finally the analytical estimate of $\lambda_1$. We continue now by calculating the mean value of the curvature and its fluctuations as a function of the energy density.
Average curvature
-----------------
The curvature of the Riemannian manifold is directly given by the Laplacian of the potential. One needs therefore to compute the microcanonical average of the quantity $$\begin{aligned}
\Delta V & = & 2 K N + 2 a^2 D \sum_k g(y_k)
\label{formcurvmicro}\end{aligned}$$ and its corresponding fluctuations, where $$\begin{aligned}
g(y)& = &\frac{3}{\cosh^4 a y}-\frac{2}{\cosh^2 a y}.\end{aligned}$$ We finally obtain $$\begin{aligned}
\langle{\Delta V }\rangle_\mu =N\left(2 K + 2 a^2 D \langle
g(y_k)\rangle_{\mu}\right). \label{formmicro}\end{aligned}$$
As, in the thermodynamic limit, ensemble equivalence ensures that averages are equal in all statistical ensembles, we will compute them in the canonical one since the transfer operator method has been shown to be a powerful tool to compute thermodynamic functions, especially if the continuum approximation is valid.
Calculation of Eq. (\[formmicro\]) relies on the computation of $\langle{g(a y)}\rangle_\text{can}$, i.e. of terms such as $\langle{{1}/{\cosh^{2 \alpha}a y}}\rangle_\text{can}$. Using the transfer operator method in the continuum framework, we immediately find $$\left\langle{\frac{1}{\cosh^{2 \alpha} a
y}}\right\rangle_\text{can}=\int_{0}^{\infty}{{\rm d}}y\,
\frac{1}{\cosh^{2 \alpha}a y}\,|{\psi_0(y)}|^2.$$ With expression [@gradshteyn] $$I(\alpha)=\int_{0}^{\infty} \frac{{{\rm d}}x}{\cosh^{2\alpha}
x}=B\left({2 \alpha, \frac{1}{2}}\right),$$ and the mathematical formula $I(\alpha+1) = I(\alpha){2 \alpha}/
{(2 \alpha +1)}$, canonical averages can be simplified. Introducing the parameter $R$ defined in Eq. (\[defdeR\]), one gets $$\kappa_0 =\frac{\langle{\Delta V }\rangle_\mu }{N}= 2 K\left({1 +4R
\frac{(2\eta-1)(2\eta-3)}{(4\eta+1)(4\eta+3)}}\right).\label{expresionkappa0}$$ Above expression gives in particular the following limiting behaviors $$\lim_{T \to 0} \kappa_0=2 K+2D a^2$$ and $$\lim_{T \to T_c} \kappa_0=2K,$$ which coincide with asymptotic results for the Morse potential [@barre_dauxois].
It is however important to notice that expression (\[expresionkappa0\]) suggests that the mean curvature is not always positive, its sign being tuned by the value of $R$. As negative curvatures enhance dynamical instability, this result may have strong consequences on the largest Lyapunov exponent. A careful study [@rapporthicham], shows that for values of $R$ larger than $R_c=({31+3\sqrt{105}})/{8}$, expression (\[expresionkappa0\]) could be negative in a given interval of temperatures. This result has to be criticized since expression (\[expresionkappa0\]) has been derived in the continuum approximation, and one expects important discreteness effects for $R$ values as large as $R_c$. Numerical, but exact, resolution of the transfer integral operator shows that the curvature is actually positive for all $R$. Fig. \[fig:m\_vs\_s\] shows the curvature as a function of temperature for the parameter set chosen in this work.
Fluctuations of the curvature
-----------------------------
Contrary to the statistical averages such as the curvature $\kappa_0$, fluctuations are ensembles dependent. This is why one computes first fluctuations in the canonical ensemble, before using the Lebowitz-Percus-Verlet formula [@lebo] to get the microcanonical fluctuations.
In the canonical ensemble, using Eq. (\[formuleaciter\]), fluctuations of expression (\[formcurvmicro\]) are given by $$\begin{aligned}
\frac{\langle{\delta^2\Delta V}\rangle}{4 a^4 D^2N}\!\!\!\!\!
\!\!\!\!&\!\!=\!\!&\!\!\!\!\frac{1}{N}
\sum_{i,j}\left[{\langle{g(y_i)g(y_j)}\rangle-\langle{g(y_i)}\rangle
\langle{g(y_j)}\rangle}\right] \\
&=&\!\! {\sum_{l} \langle{g(y_N)g(y_{N-l})}\rangle-N\langle{g(y)}^2}\rangle \\
&=& \!\!\sum_{l}\sum_{q=0}^{\infty} e^{-\beta l (\varepsilon_q-\varepsilon_0)}
\left|\int{{\rm d}}y g(y)\psi_q^\star(y)\psi_0(y) \right|^2 \nonumber\\
&&\hskip 3.4truecm - N \langle g(y)^2\rangle \\
&=& \!\!\!\! \!\! \sum_{l}\sum_{q=1}^{\infty} e^{-\beta l (\varepsilon_q-\varepsilon_0)}
\left|{\int{}{}{{\rm d}}y g(y)\psi_q^\star(y)\psi_0(y) }\right|^2 \\
& \underset{ N\to \infty}{\simeq} & \!\!\!\! \sum_{q=1}^{\infty}
\frac{1}{1-e^{-\beta (\varepsilon_q-\varepsilon_0)}}
\left|{\int{}{}{{\rm d}}y\, g(y)\psi_q^\star(y)\psi_0(y)
}\right|^2.\label{formfluctfinal}\end{aligned}$$ Above expression can be used to compute the canonical fluctuations.
The microcanonical fluctuations will finally be recovered by using the Lebovitz-Percus-Verlet formula [@lebo]. For any quantity $C$ with fluctuations $\langle{\delta^2 C}\rangle$, both fluctuations are related through the formula $$\langle{\delta^2 C}\rangle_\mu = \langle{\delta^2
C}\rangle_\text{can}+\frac{\partial\langle{U}\rangle_\text{can}}{\partial\beta}^{-1}
\left(\frac{\partial\langle{C}\rangle_\text{can}}{\partial\beta}\right)^2,$$ where $\langle{U}\rangle_\text{can}$ is the averaged energy of the system.
The canonical partition function $Z$ being the product of a kinetic part $Z_T$ and a configurational one $Z_c$, the averaged energy is given by $$\langle{U}\rangle_\text{can} = -\frac{\partial\ln
Z_T}{\partial\beta}-\frac{\partial\ln Z_c}{\partial\beta}.$$ The first contribution is as usual ${N}/({2\beta})$ whereas the last one, can be simplified in the continuum approximation by using the transfer integral method (See Appendix \[transferinteg\] and in particular Eq. (\[formulaZconf\])). Denoting $\varepsilon_0$ the ground state of the associated Schr[ö]{}dinger equation, one obtains $\ln
Z_c=-N\beta \varepsilon_0$ with $$\begin{aligned}
\varepsilon_0 &=& \frac{1}{2\beta}\ln\frac{\beta K}{2\pi}
-\frac{a^2}{2 K \beta^2 }(2\eta-1)^2 .\end{aligned}$$ The averaged energy is therefore $$\langle{U}\rangle_\text{can} = N\left[\frac{1}{2\beta} +
\frac{\partial(\beta \varepsilon_0)}{\partial\beta} \right].$$ The final analytical expression for the microcanonical fluctuations is not simple. Introducing quantities $\beta_c=1/(k_BT_c)$ and $\delta=T_c/T$, one gets $$\begin{aligned}
\frac{\langle{\delta^2\kappa_0}\rangle_\mu-\langle{\delta^2\kappa_0}\rangle_\text{can}}{18432 a^4 D^2 K\beta_c}
&=& - \delta^5
\left({-16\delta^2+7\sqrt{1+8\delta^2}+5}\right)^2\nonumber
\\ && /
(1+8\delta^2)^{\frac{3}{2}} /\left({2+\sqrt{1+8\delta^2}}\right)^4 \nonumber
\\ && / ( 16 K \beta_c \delta^3
\sqrt{1+8\delta^2} -36 a^2 \delta^2 \nonumber
\\ && +40 a^2 \delta^2 \sqrt{1+8\delta^2} + 2 K \beta_c \delta\nonumber
\\ &&
\! \! \! \! \! \! \! \! \! \! \sqrt{1+8\delta^2} -3 a^2 +5 a^2 \sqrt{1+8\delta^2}) .\label{correctmicro}\end{aligned}$$ Above expressions (\[formfluctfinal\]) and (\[correctmicro\]) can be combined to compute the microcanonical fluctuations. This is what has been performed to plot the fluctuations of the curvature in Fig. \[fig:m\_vs\_s\].
Close to the critical temperature $T_c$, the continuum part of the transfer operator spectrum should be taken into account and an explicit analytical calculation is possible in principle but particularly tedious. On the contrary, one can simplify above expression in the low temperature regime as shown in next section.
Low temperature estimate
------------------------
In the low temperature regime by replacing the prefactor $(1-\exp[-\beta (\varepsilon_q-\varepsilon_0)])^{-1}$ by 1 in Eq. (\[formfluctfinal\]), one finally gets $$\begin{aligned}
\langle{\delta^2\Delta V}\rangle_\text{can} & \simeq & 4N a^4 D^2 \sum_{q=1}^{\infty} \left\langle \psi_q | g |\psi_0\right\rangle \left\langle \psi_0 | g |\psi_q\right\rangle\\[2mm]
& = & 4N a^4 D^2\left[{ \left\langle \psi_0 | g^2
|\psi_0\right\rangle- \left\langle \psi_0 | g
|\psi_0\right\rangle^2}\right].\end{aligned}$$ Combining result $$\left\langle \psi_0 |g^2|\psi_0\right\rangle=16\frac{2\eta(2\eta-1)(4\eta^2-6\eta+11)}
{(4\eta+1)(4\eta+3)(4\eta+5)(4\eta+7)}$$ with formula (\[expresionkappa0\]), one ends up with $$\langle{\delta^2 g}\rangle_\text{can} = 48\frac{(2\eta-1)
(256\eta^3-184\eta+105)}{(4\eta+1)^2(4\eta+3)^2(4\eta+5)(4\eta+7)}.$$
As the microcanonical correction (\[correctmicro\]) can be neglected in this region, the microcanonical fluctuations of the curvature are finally given in the low temperature regime by $$\begin{aligned}
\sigma_\kappa^2 &=& \frac{\langle{\delta^2\Delta V}\rangle_\text{can} }{N}\\
&\simeq&
4D^2a^4\frac{48(2\eta-1)(256\eta^3-184\eta+105)}{(4\eta+1)^2(4\eta+3)^2(4\eta+5)(4\eta+7)}
.\label{formfinalfluct}\end{aligned}$$ This expression can be used in the low temperature region to get a simpler expression for the largest Lyapunov exponent. One thus obtain that it increases quadratically with the temperature.
Largest Lyapunov exponent
-------------------------
We can now estimate the largest Lyapunov exponent $\lambda_1$ for this high-dimensional system of $N$ coupled particles in the external sech$^2$-potential (\[potmorsemorceaux\]).
As $\kappa_0$ is positive, the instability of trajectories is due to fluctuations of curvatures as reported by Pettini, Casetti and Cohen [@pettini]. Analytical expressions (\[expresionkappa0\]) for $\kappa_0$ and (\[formfinalfluct\]) for $\sigma_\kappa$ were the only missing points in the estimate (\[exprlyapunov\]) for $\lambda_1$. One can in addition notes that in the low temperature region, where $\sigma_\kappa\ll\kappa_0$ as attested by Fig. \[fig:m\_vs\_s\], one gets the asymptotic expression $$\lambda_1 \simeq\frac{\pi}{8} \frac{\sigma_\kappa^2}{{\kappa_0}^\frac{3}{2}}.\label{aprroxexpr}$$
Fig. (\[fig:lyapunov\_m\_vs\_s\]) presents the temperature evolution of the largest Lyapunov exponents. Two features should be emphasized in comparison to what has been previously reported for the Morse potential [@barre_dauxois]. One can notice a local maximum and a local minimum of the Lyapunov exponent in the low temperature region. More importantly, one has to realize that for temperatures larger than the critical one $T_c$, particles are on the plateau of the potential; the chain being equivalent to a linear chain, the largest Lyapunov exponent $\lambda_1$ if of course zero. The Lyapunov exponent should thus present a jump close to the critical temperature. It is important to check these two strong predictions by considering careful microcanonical numerical simulations with large systems. This would provide a precise test of the geometrical method to calculate Lyapunov exponents.
Conclusion
==========
We have presented a new qualitative model for DNA denaturation directly inspired by previous works [@peyrardbishop; @dauxois_these; @dauxois_peyrard]. Its complete statistical mechanics was derived, as well as all features related to the second order phase transition: not only the critical temperature, but also the critical exponents related to the order parameter and the transversal correlation length.
We have in particular emphasized the important role of the negative $y$-values for the Morse potential which were believed to be unimportant [@dauxois_these]. If critical exponents are of course not affected, the critical temperature is strongly dependent on it. Furthermore, using a geometric approach to estimate the largest Lyapunov exponent, we have computed its evolution as a function of the temperature. The results are unexpectedly qualitatively different from those obtained with the Morse potential.
This work has been partially supported by the French Minist[è]{}re de la Recherche grant ACI jeune chercheur-2001 N$^\circ$ 21-31, and the R[é]{}gion Rh[ô]{}ne-Alpes for the fellowship N$^\circ$ 01-009261-01.
Transfer integral method for the canonical partition function {#transferinteg}
=============================================================
With periodic boundary conditions, $y_0=y_N$, the configurational partition function of Hamiltonian (\[defHamil\]) can be written as $$Z_c = \int\prod_{i=0}^{N} {{\rm d}}y_i\, e^{-\beta f(y_i,y_{i-1})}
\,\delta(y_0-y_N)\label{Zc}$$ by introducing the symmetric function $$\begin{aligned}
f(y_n,y_{n-1}) &=& -\frac{D}{2}\left[\frac{1}{\cosh^2 a y_n}
+\frac{1}{\cosh^2 a y_{n-1}}\right]\nonumber\\
&&+\frac{K}{2}(y_n-y_{n-1})^2. \end{aligned}$$ Defining the transfer operator $T$ as $$\begin{aligned}
T[\phi] (y) = \int_{\mathbb{R}}{{\rm d}}x\, \phi(x) e^{-\beta f(y,x)}
,\label{formuleA3}\end{aligned}$$ its eigenvalues $\varepsilon_k$ and normalized eigenvectors $\psi_k$ are given by $T[\psi_k] =
\exp\left({-\beta\varepsilon_k}\right)\: \psi_k$.
Introducing the orthonormalization condition $$\begin{aligned}
\delta(y-y_0)=\sum_{k}\psi_k^\star(y_0)\psi_k(y)\end{aligned}$$ in Eq. (\[Zc\]), one gets [@dauxoispeyrardbook] $$\begin{aligned}
Z_c = \sum_{k} e^{-N\,\beta\,\epsilon_k}.\end{aligned}$$ If the lowest eigenvalue $\varepsilon_0$ is discrete and located in the gap below the continuum, one can simplify above expression in the limit $N\to\infty$, so that $$Z_c \simeq e^{-N\beta\varepsilon_0}.\label{formulaZconf}$$
A similar method to compute the canonical average of any function $h(y)$ leads to the following result $$\begin{aligned}
\langle{h(y)}\rangle_\text{can} = \langle{h(y_N}\rangle_\text{can}
& \underset{N\to\infty}{\simeq} & \int{}{} {{\rm d}}y \,h(y)
|\psi_0(y)|^2.\label{formuleaciter}\end{aligned}$$
Above results are valid without approximations. However, applying the continuum approximation, it is possible to go one step further since there is a mapping [@dauxois_these; @dauxoispeyrardbook] between the transfer integral operator and the following Schr[ö]{}dinger equation $$\label{equaschroappendix}
-\frac{1}{2 \beta^2 K}\frac{{{\rm d}}^2\psi}{{{\rm d}}y^2} -\frac{D}{\cosh^2 a y }\psi = {\cal E}_k \psi .$$ As the spectrum of a quantum particle in the potential (\[potmorsemorceaux\]) is known, one can derive the analytical expression of (\[formuleaciter\]) within this approximation.
[99]{} M. Peyrard, A. R. Bishop, *Physical Review Letters* [**62**]{}, 2755-2758 (1989).
R. M. Wartell, A. S. Benight, *Physics Reports* [**126**]{}, 67-107 (1985).
M. Peyrard, Nonlinearity [**17**]{}, R1-R40 (2004).
T. Dauxois , M. Peyrard, A. R. Bishop, *Physical Review E* [**47**]{}, 684-695 (1993).
T. Dauxois, M. Peyrard, *Physical Review Letters* [**70**]{}, 3935-3938 (1993).
T. Dauxois, N. Theodorakopoulos, M. Peyrard, *Journal of Statistical Physics* [**107**]{}, 869-891 (2002).
C. H. Choi, G. Kalosakas, K. O. Rasmussen, M. Hiromura, A. R. Bishop, A. Usheva, *Nucleic Acids Research* [**32**]{}, 1584-1590 (2004).
, [*Methods of Theoretical Physics*]{}, Mc Graw-Hill Book Company (1953).
M. Peyrard, T. Dauxois, [*Physique des Solitons*]{}, CNRS [É]{}ditions-EDP Sciences (2004).
, [*Physical Review B*]{} [**11**]{}, 3535-3545 (1975).
, [*Dynamique non-lin[é]{}aire et m[é]{}canique statistique d’un mod[è]{}le d’ADN*]{}, PhD thesis, Universit[é]{} de Bourgogne (1993).
H. Qasmi, Rapport de ma[î]{}trise, “Lien entre dynamique et thermodynamique autour d’un mod[è]{}le simple d’ADN”, ENS Lyon (2003).
, *Physical Review* [**153**]{}, [250-254]{} (1967).
, *Europhysics Letters* [**55**]{}, 164-170 (2001).
, [*Table of Integrals,Series, and Products, Fifth Edition*]{}, Academic Press (1994).
, [*Structures Dissipatives, Chaos et Turbulence*]{}, [Aléa Saclay]{} (1991).
, [*Regular and Chaotic Dynamics*]{}, [Springer, Berlin]{} (1992).
, *Physical Review Letters* [**74**]{}, [375-378]{} (1995). , [Physical Review E]{} [**55**]{}, [7612-7618]{} (1997).
*Physica D* [**131**]{}, [38-54]{} (1999).
, *Physical Review E* [**57**]{}, [6599-6603]{} (1998).
R. O. Vallejos, C. Anteneodo *Physical Review E* [**66**]{}, 021110 (2002). C Anteneodo, R. N. P. Maia, R. O. Vallejos, *Physical Review E* [**68**]{}, 036120 (2003).R. O. Vallejos, C. Anteneodo *Physical Review E* [**66**]{} 021110 (2002).
S. Tanase-Nicola, J. Kurchan, J. Phys. A-Math Gen. [**36**]{} 10299-10324 (2003).
, Physics Reports [**337**]{}, 237-341 (2000).
[^1]: E-mail:[email protected]
|
---
abstract: |
In this paper we will continue the study started recently in [@ABMbp] by describing and classifying all Hopf algebras $E$ that factorize through two Sweedler’s Hopf algebras. Equivalently, we shall classify all bicrossed products $H_4 \bowtie H_4$ associated to all possible matched pairs $(H_4, H_4,
\triangleright, \triangleleft)$ of Hopf algebras. There are three steps in our approach: in the first one we describe explicitly all the matched pairs $(H_4, H_4, \triangleright, \triangleleft)$ by proving that, except for the trivial one, there exists an infinite number of such matched pairs parameterized by a scalar $\lambda$ of the base field $k$. Then, for any $\lambda \in k$, we shall construct by generators and relations a $16$-dimensional, pointed, unimodular and non-semisimple quantum group ${\mathcal H}_{16, \,
\lambda}$: a Hopf algebra $E$ factorizes through $H_4$ and $H_4$ if and only if $ E \cong H_4 \ot H_4$ or $E \cong {\mathcal
H}_{16, \, \lambda}$. In the last step we classify such quantum groups by proving that there are only three isomorphism classes: $H_4 \ot H_4$, ${\mathcal H}_{16, \, 0}$ and ${\mathcal H}_{16, \,
1} \cong D(H_4)$, the Drinfel’d double of $H_4$. As a bonus to our approach the group of Hopf algebra automorphisms of these Hopf algebras are described: in particular, we prove that ${{\rm Aut}\,}_{\rm
Hopf}\big( D(H_4)\big)$ is isomorphic to a semidirect product of groups $k^* \rtimes \mathbb{Z}_2$.
address:
- 'Faculty of Mathematics and Computer Science, University of Bucharest, Str. Academiei 14, RO-010014 Bucharest 1, Romania'
- And
- 'Faculty of Engineering, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium'
author:
- Costel Gabriel Bontea
title: 'Classifying bicrossed products of two Sweedler’s Hopf algebras'
---
[^1]
Introduction {#introduction .unnumbered}
============
Let $A$ and $H$ be two given Hopf algebras. The factorization problem for Hopf algebras consists of classifying up to an isomorphism all Hopf algebras that factorize through $A$ and $H$, i.e. all Hopf algebras $E$ containing $A$ and $H$ as Hopf subalgebras such that the multiplication map $A \ot H \to E$, $a
\ot h \mapsto a h$ is bijective. The problem can be put in more general terms but we restrict ourselves to the case of Hopf algebras. For a detailed account on the subject at the level of groups the reader may consult [@ABMbp] and [@am2].
An important step in dealing with the factorization problem was made by Majid in [@majid Proposition 3.12] who generalized to Hopf algebras the construction of the bicrossed product for groups introduced by Takeuchi in [@Takeuchi]. Although in [@majid] the construction is known under the name of double cross product, we will follow [@Kassel] and call it, just like in the case of groups, the bicrossed product construction. A bicrossed product of two Hopf algebras $A$ and $H$ is a new Hopf algebra $A \bowtie H$ associated to a matched pair $(A, H, \rhd,
\lhd )$ of Hopf algebras. It is proven in [@majid] that a Hopf algebra $E$ factorizes through $A$ and $H$ if and only if $E$ is isomorphic to some bicrossed product of $A$ and $H$. Thus, the factorization problem can be stated in a computational manner: for two Hopf algebras $A$ and $H$ describe the set of all matched pairs $(A, H, \rhd, \lhd )$ and classify up to an isomorphism all bicrossed products $A \bowtie H$ of Hopf algebras. This way of approaching the problem was recently proposed in [@ABMbp] with promising results regarding new examples of quantum groups. For example, in all bicrossed products $H_4
\bowtie k[C_n]$ are described by generators and relations and are classified. They are quantum groups at roots of unity $H_{4n, \,
\omega}$ which are classified by the arithmetic of the ring $\mathbb{Z}_n$. In this paper we shall continue the study began in [@ABMbp] by classifying all Hopf algebras that factorize through two Sweedler’s Hopf algebras.
The paper is organized as follows. In [Section \[se:prel\]]{} we set the notations and recall the bicrossed product construction of two Hopf algebras. In [Section \[se:H\_4H\_4\]]{}, the main section of this paper, we classify all Hopf algebras that factorize through two Sweedler’s Hopf algebras. In order to do this, we first compute all the matched pairs $(H_4, H_4, \rhd, \lhd)$: except the trivial one, there exists an infinite number of such matched pairs parameterized by a scalar $\lambda$ of the base field $k$. Then we will describe by generators and relations all the bicrossed products $H_4 \bowtie H_4$ associated to these matched pairs. These are: $H_4 \ot H_4$ and ${\mathcal H}_{16, \, \lambda}$, where for any $\lambda \in k$, ${\mathcal H}_{16, \, \lambda}$ is the $16$-dimensional quantum group generated by $g$, $x$, $G$, $X$ subject to the relations: $$g^2 = G^2 = 1 \quad x^2 = X^2 = 0, \quad gx = -xg, \quad GX =
-XG,$$ $$gG = Gg, \quad gX = -Xg, \quad x G = - Gx, \quad xX + Xx = \lambda
\, (1 - Gg)$$ with the coalgebra structure given such that $g$ and $G$ are group-likes, $x$ is $(1, g)$-primitive and $X$ is $(1,
G)$-primitive. Finally, as the last step of our approach, using the description of the homomorphisms between two arbitrary bicrossed products given in [@ABMbp], we prove that there are only three isomorphism classes of Hopf algebras that factorize through two Sweedler’s Hopf algebras: $H_4 \ot H_4$, ${\mathcal
H}_{16, \, 0}$ and ${\mathcal H}_{16, \, 1} \cong D(H_4)$, the Drinfel’d double of $H_4$.
A different type of classification was given in [@GV Lema 2.8], using the classical theory of Hopf algebras extensions and having the cocycle bicrossproduct as a tool: any extension of $H_4$ by $H_4$ is equivalent to the trivial extension $H_4 \ot H_4
$; in particular, any cocycle bicrossproduct $H_4 \,
^{\tau}{}\#_{\sigma} H_4$ is isomorphic to the trivial tensor product $H_4 \ot H_4$. This shows that, for two given Hopf algebras, one can obtain a larger class of new types of Hopf algebras by considering the factorization problem and the bicrossed product instead of the extension problem and the cocycle bicrossedproduct. In our case, the two new types of Hopf algebras constructed in this paper, ${\mathcal H}_{16, \, 0}$ and ${\mathcal H}_{16, \, 1}$, are not extensions of $H_{4}$ by $H_{4}$ in the sense of [@AD Definition 1.2.0].
Finally, as a consequence of our results, the group of Hopf algebra automorphisms of these quantum groups are described by proving that there exists the following isomorphisms of groups: $${{\rm Aut}\,}_{\rm Hopf}( D(H_4)) \cong k^* \rtimes \mathbb{Z}_2, \quad
{{\rm Aut}\,}_{\rm Hopf}(\mathcal{H}_{16,0}) \cong (k^* \times k^*)
\rtimes \mathbb{Z}_2 \cong {{\rm Aut}\,}_{\rm Hopf}(H_4 \otimes H_4)$$
Preliminaries
=============
[\[se:prel\]]{}
Throughout this paper all algebras, coalgebras, Hopf algebras are over a commutative field $k$ and $\ot = \ot_k$. We shall use the standard notations from Hopf algebras theory: in particular, for a coalgebra $C$, we use the $\Sigma$-notation: $\Delta(c) = c_{(1)}
\ot c_{(2)}$, for any $c\in C$ (summation understood). Let $A$ and $H$ be two Hopf algebras. $A$ is a called a left $H$-module coalgebra if there exists $\rhd : H \ot A \to A$ a morphism of coalgebras such that $(A, \rhd)$ is also a left $H$-module. Similarly, $H$ is called a right $A$-module coalgebra if there exists $\lhd : H \otimes A \rightarrow H$ a morphism of coalgebras such that $(H, \lhd) $ is a right $A$-module. The actions $\rhd: H
\otimes A \to A$ and $\lhd : H \otimes A \to H$ are called *trivial* if $h \rhd a = \varepsilon_H(h) a$ and $h \lhd a =
\varepsilon_A (a) h$ respectively, for all $a \in A$ and $h \in
H$.
A *matched pair* of Hopf algebras [@majid], [@Kassel] is a quadruple $(A, H, \rhd, \lhd)$, where $A$ and $H$ are Hopf algebras, $\rhd: H \otimes A \to A$ and $\lhd: H
\otimes A \to H$ are coalgebra maps such that $(A, \rhd)$ is a left $H$-module coalgebra, $(H, \lhd)$ is a right $A$-module coalgebra and the following compatibility conditions hold: $$\begin{aligned}
h \rhd 1_{A} &{=}& \varepsilon_{H}(h)1_{A} \,\,\, 1_{H} \lhd a
= \varepsilon_{A}(a)1_{H}, {\label{eq:mp1}}\\
g \rhd (ab) &{=}& (g_{(1)} \rhd a_{(1)}) \bigl ( (g_{(2)} \lhd
a_{(2)}) \rhd b \bigl)
{\label{eq:mp2}} \\
(g h) \lhd a &{=}& \bigl( g \lhd (h_{(1)} \rhd a_{(1)}) \bigl)
(h_{(2)} \lhd a_{(2)})
{\label{eq:mp3}} \\
g_{(1)} \lhd a_{(1)} \otimes g_{(2)} \rhd a_{(2)} &{=}& g_{(2)}
\lhd a_{(2)} \otimes g_{(1)} \rhd a_{(1)} {\label{eq:mp4}}\end{aligned}$$ for all $a$, $b\in A$, $g$, $h\in H$. If $(A, H, \rhd, \lhd)$ is a matched pair of Hopf algebras then the associated *bicrossed product* $A \bowtie H$ of $A$ with $H$ is the vector space $A \ot H$ endowed with the coalgebra structure of the tensor product of coalgebras and the multiplication $${\label{eq:0010}}
(a \bowtie g) \cdot (b \bowtie h):= a (g_{(1)} \rhd b_{(1)})
\bowtie (g_{(2)} \lhd b_{(2)}) h$$ for all $a$, $b\in A$, $g$, $h \in H$, where we used $\bowtie$ for $\ot$. $A \bowtie H$ is a Hopf algebra with the antipode given by the formula: $${\label{eq:antipbic}}
S ( a \bowtie h ) := \big( 1_A \bowtie S_H (h) \big) \cdot \big(
S_A (a) \bowtie 1_H \big)$$ for all $a\in A$ and $h\in H$ [@majid2 Theorem 7.2.2], [@Kassel Theorem IX 2.3].
The basic example of a bicrossed product is the famous Drinfel’d double of a finite dimensional Hopf algebra $H$: $D(H) =
(H^*)^{\rm cop} \bowtie H$, the bicrossed product associated to a given canonical matched pair [@Kassel Theorem IX.3.5]. For others examples of bicrossed products we refer to [@ABMbp], [@Kassel], [@majid2].
We recall that a Hopf algebra $E$ *factorizes* through two Hopf algebras $A$ and $H$ if there exist injective Hopf algebra maps $i : A \to E$ and $j : H \to E$ such that the map $$A \ot H \to E, \quad a \ot h \mapsto i(a) j(h)$$ is bijective. The next fundamental result is due to Majid [@majid2 Theorem 7.2.3]: A Hopf algebra $E$ factorizes through two given Hopf algebras $A$ and $H$ if and only if there exists a matched pair of Hopf algebras $(A, H, \rhd, \lhd)$ such that $E \cong A \bowtie H$. In light of this result, the factorization problem for Hopf algebras was restated [@ABMbp] in a computational manner: Let $A$ and $H$ be two given Hopf algebras. Describe the set of all matched pairs $(A, H, \rhd,
\lhd)$ and classify up to an isomorphism all bicrossed products $A
\bowtie H$.
In order to classify the bicrossed products $A \bowtie H$ the following result, proved in [@ABMbp Theorem 2.2], is crucial. It describes the morphisms between two bicrossed products. It will also be used in determining the automorphisms group of a given bicrossed product. The result says that, given two matched pairs of Hopf algebras $(A, H, \rhd, \lhd)$ and $(A', H', \rhd',
\lhd')$, there exists a bijective correspondence between the set of all morphisms of Hopf algebras $\psi : A \bowtie H \to A'
\bowtie ' H' $ and the set of all quadruples $(u, p, r, v)$, where $u: A \to A'$, $p: A \to H'$, $r: H \rightarrow A'$, $v: H
\rightarrow H'$ are unitary coalgebra maps satisfying the following compatibility conditions: $$\begin{aligned}
u(a_{(1)}) \ot p(a_{(2)}) &{=}& u(a_{(2)}) \ot p(a_{(1)}){\label{eq:C1}}\\
r(h_{(1)}) \ot v(h_{(2)}) &{=}& r(h_{(2)}) \ot v(h_{(1)}){\label{eq:C2}}\\
u(ab) &{=}& u(a_{(1)}) \, \bigl( p (a_{(2)}) \rhd' u(b) \bigl){\label{eq:C3}}\\
p(ab) &{=}& \bigl( p (a) \lhd' u(b_{(1)}) \bigl) \, p (b_{(2)}){\label{eq:C4}}\\
r(hg) &{=}& r(h_{(1)}) \, \bigl(v(h_{(2)}) \triangleright'
r(g)\bigl){\label{eq:C5}}\\
v(hg) &{=}& \bigl(v(h) \lhd' r(g_{(1)})\bigl) \,
v(g_{(2)}){\label{eq:C6}}\\
r(h_{(1)}) \bigl(v(h_{(2)}) \triangleright' u(b) \bigl) &{=}& u
(h_{(1)} \rhd b_{(1)}) \, \Bigl( p (h_{(2)} \rhd b_{(2)}) \rhd'
r(h_{(3)} \lhd b_{(3)})
\Bigl) {\label{eq:C7}}\\
\bigl (v(h) \lhd' u(b_{(1)}) \bigl) \, p (b_{(2)}) &{=}& \Bigl( p
(h_{(1)} \rhd b_{(1)}) \lhd ' r(h_{(2)} \lhd b_{(2)}) \Bigl) \, v
(h_{(3)} \lhd b_{(3)}) {\label{eq:C8}}\end{aligned}$$ for all $a$, $b \in A$, $g$, $h \in H$. Under the above correspondence, the morphism of Hopf algebras $\psi: A \bowtie H
\to A' \bowtie ' H' $ corresponding to $(u, p, r, v)$ is given by: $${\label{eq:morfbicros}}
\psi(a \bowtie h) = u(a_{(1)}) \, \bigl( p(a_{(2)}) \rhd'
r(h_{(1)}) \bigl) \,\, \bowtie' \, \bigl( p(a_{(3)}) \lhd'
r(h_{(2)}) \bigl) \, v(h_{(3)})$$ for all $a \in A$ and $h\in H$.
The bicrossed products of two Sweedler’s Hopf algebras
======================================================
[\[se:H\_4H\_4\]]{}
In this section we are going to classify all the bicrossed products $H_4 \bowtie H_4$. Our strategy is the one of [@ABMbp]: first of all we will find all the matched pairs $(H_{4}, H_{4}, \rhd, \lhd)$, then we will describe by generators and relations all the bicrossed products associated to these matched pairs and finally, using [@ABMbp Theorem 2.2], we will decide which of these bicrossed products are isomorphic. As an offshoot of our work, the group ${{\rm Aut}\,}_{\rm Hopf} (H_{4}
\bowtie H_{4})$ of all Hopf algebra automorphisms of a given bicrossed product is computed.
Recall that Sweedler’s $4$-dimensional Hopf algebra, $H_4$, is generated by two elements, $g$ and $x$, subject to the relations: $$g^{2} = 1, \quad x^{2} = 0, \quad x g = -g x$$ The coalgebra structure and the antipode are given by: $$\Delta(g) = g \otimes g, \quad \Delta(x) = x \otimes 1 + g \otimes
x, \quad \Delta(gx) = gx \ot g + 1 \otimes gx$$ $$\varepsilon(g) = 1, \quad \varepsilon(x) = 0, \quad S(g) = g,
\quad S(x) = -gx$$ In order to avoid confusions we will denote by $\mathbb{H}_4$ a copy of $H_4$, and by $G$ and $X$ the generators of $\mathbb{H}_4$. Thus, $G^{2} = 1$, $X^{2} = 0$, $GX = - XG$, $G$ is a group-like element and $X$ is an $(1, G)$-primitive element.
Recall that, for a Hopf algebra $H$, the set of group-like elements of $H$ is denoted by $\mbox{G}(H)$, and, for $g$, $h \in
\mbox{G}(H)$, the set of $(g, h)$-primitive elements of $H$ is denoted by $\mbox{P}_{g, h} (H)$.
In order to determine the matched pairs between two Hopf algebras the following result proved [@ABMbp] is very useful.
[\[le:primitive\]]{} Let $(A, H, \rhd, \lhd)$ be a matched pair of Hopf algebras, $a$, $b \in \textnormal{G}(A)$ and $g$, $h \in \textnormal{G}(H)$. Then:
$(1)$ $g \rhd a \in \textnormal{G}(A)$ and $g \lhd a \in
\textnormal{G}(H)$;
$(2)$ If $x \in \textnormal{P}_{a, \, b}(A)$, then $g \lhd x \in
\textnormal{P}_{g\lhd a, \, g\lhd b} (H)$ and $g \rhd x \in
\textnormal{P}_{g\rhd a, \, g\rhd b} (A)$;
$(3)$ If $y \in \textnormal{P}_{g, \, h}(H)$, then $y \lhd a \in
\textnormal{P}_{g\lhd a, \, h\lhd a} (H)$ and $y \rhd a \in
\textnormal{P}_{g\rhd a, \, h\rhd a} (A)$.
In particular, if $x$ is an $(1_A, b)$-primitive element of $A$, then $g \rhd x$ is an $(1_A, g \rhd b)$-primitive element of $A$ and $g \lhd x$ is an $(g, g \lhd b)$-primitive element of $H$.
We shall use two more elementary lemmas from [@ABMbp].
[\[le:primitsw\]]{} $\textnormal{G}(H_4) = \{1, \, g \}, \quad
\textnormal{P}_{1, \, 1} (H_4) = \{0\}, \quad \textnormal{P}_{g,
\, g} (H_4) = \{0\}$ $$\textnormal{P}_{1, \, g}(H_4) = \{ \alpha - \alpha g + \beta \, x
\,\,\, | \, \alpha, \beta \in k \}, \qquad \textnormal{P}_{g, \,
1} (H_4) = \{ \alpha - \alpha g + \beta \, g x \, | \, \alpha,
\beta \in k \}$$
[\[le:morf\_H\_4H\_4\]]{}
$(1)$ $u : H_4 \rightarrow H_4$ is a unitary coalgebra map if and only if $u$ is the trivial morphism $u(h) = \varepsilon (h) 1$, for all $h\in H_4$, or there exists $\alpha$, $\beta$, $\gamma$, $\delta \in k$ such that $u(1) = 1$, $u(g) = g$, $u(x) = \alpha -
\alpha \, g + \beta \, x$, and $u(gx) = \gamma - \gamma \, g +
\delta \, gx$.
$(2)$ $u : H_4 \to H_4$ is a Hopf algebra morphism if and only if $u$ is the trivial morphism $u(h) = \varepsilon(h)1$, for all $h
\in H_4$, or there exists $\beta \in k$ such that $u(1) = 1$, $u(g) = g$, $u(x) = \beta \, x$, and $u(gx) = \beta \, gx$. In particular, ${{\rm Aut}\,}_{\rm Hopf} (H_4) \cong k^*$.
The next theorem describes the set of all matched pairs $(A =
\mathbb{H}_4, H = H_4, \rhd, \lhd)$.
[\[th:mpHH00\]]{} Let $k$ be a field of characteristic $\neq 2$. Then $(\mathbb{H}_4, H_4, \rhd , \lhd)$ is a matched pair of Hopf algebras if and only if $(\rhd$, $\lhd)$ are both the trivial actions or the pair $(\rhd$, $\lhd)$ is given by:
[l | r r r r ]{} $\rhd$ & 1 & $G$ & $X$ & $GX$\
1 & 1 & G & X & GX\
g & 1 & G & -X & -GX\
x & 0 & 0 & $\lambda - \lambda \, G$ & $\lambda - \lambda \, G$\
gx & 0 & 0 & $\lambda - \lambda \, G$ & $\lambda - \lambda\,G$\
[l | r r r r ]{} $\lhd$ & 1 & $G$ & $X$ & $GX$\
1 & 1 & 1 & 0 & 0\
g & g & g & 0 & 0\
x & x & -x & $\lambda - \lambda \, g$ & $-\lambda + \lambda \, g$\
gx & gx & -gx & $-\lambda + \lambda \, g$ & $\lambda - \lambda \, g$\
for some $\lambda \in k$.
We prove this result in three steps. The first one is [Proposition \[pr:actdr\]]{} where we describe the set of all right $\mathbb{H}_4$-module coalgebra structures $\lhd$ on $H_4$ satisfying the normalizing condition $1 \lhd h = \varepsilon(h)1$, for all $h \in \mathbb{H}_4$. There will be four such families of actions, $\lhd^{j}$, $j = 1,2,3,4$, parameterized by scalars $a$, $b$, $c$, $d\in k$.
The second step is [Proposition \[pr:actstg\]]{} where we describe the set of all left $H_4$-module coalgebra structures $\rhd$ on $\mathbb{H}_4$ satisfying the normalizing condition $h \rhd 1 = \varepsilon(h)1$, for all $h \in H_4$. There will also be four families of such actions, $\rhd^{i}$, $i = 1,2,3,4$ parameterized by scalars $s$, $t$, $u$, $v\in k$.
The final step consists of a detailed analysis of the sixteen possibilities of choice for the pair of actions $(\rhd^i,
\lhd^j)$, for all $i$, $j = 1$, $2$, $3$, $4$. This will show that the only ones that verify the axioms of a matched pair [(\[eq:mp2\])]{}-[(\[eq:mp4\])]{} are: $(\rhd^1, \lhd^1)$, i.e. the pair of trivial actions, and $(\rhd^4, \lhd^4)$, in which case the actions take the form described in the statement.
We begin with:
[\[pr:actdr\]]{} If $\lhd : H_4 \ot \mathbb{H}_4 \rightarrow H_4$ is a right $\mathbb{H}_4$-module coalgebra structure such that $1 \lhd h =
\varepsilon(h)1$, for all $h \in \mathbb{H}_4$, then $\lhd$ has one of the following forms:
[l | r r r r ]{} $\lhd^1$ & 1 & $G$ & $X$ & $GX$\
1 & 1 & 1 & 0 & 0\
$g$ & $g$ & $g$ & 0 & 0\
$x$ & $x$ & $x$ & 0 & 0\
$gx$ & $gx$ & $gx$ & 0 & 0\
[l | r r r r ]{} $\lhd^2$ & 1 & $G$ & $X$ & $GX$\
1 & 1 & 1 & 0 & 0\
$g$ & $g$ & $g$ & 0 & 0\
$x$ & $x$ & $x$ & 0 & 0\
$gx$ & $gx$ & $c-cg-gx$ & $d-dg$ & $-d+dg$\
[l | r r r r ]{} $\lhd^3$ & 1 & $G$ & $X$ & $GX$\
1 & 1 & 1 & 0 & 0\
$g$ & $g$ & $g$ & 0 & 0\
$x$ & $x$ & $a-ag-x$ & $b-bg$ & $-b+bg$\
$gx$ & $gx$ & $gx$ & 0 & 0\
[l | r r r r ]{} $\lhd^4$ & 1 & $G$ & $X$ & $GX$\
1 & 1 & 1 & 0 & 0\
$g$ & $g$ & $g$ & 0 & 0\
$x$ & $x$ & $a-ag-x$ & $b-bg$ & $-b+bg$\
$gx$ & $gx$ & $c-cg-gx$ & $d-dg$ & $-d+dg$\
where $a$, $b$, $c$, $d\in k$.
Let $\lhd : H_4 \ot \mathbb{H}_4 \rightarrow H_4$ be a right $\mathbb{H}_4$-module coalgebra structure such that $1
\lhd h = \varepsilon(h)1$, for all $h \in \mathbb{H}_4$. Thus, we have $1 \lhd G = 1$, $1 \lhd X = 0$, and $1 \lhd (GX) = 0$. We determine next the actions of $G$, $X$, and $GX$ on $g$, $x$, and $gx$. First we show that
$\lhd$ 1 $G$ $X$ $GX$
-------- ----- ----- ----- ------
$g$ $g$ $g$ 0 0
Using [Lemma \[le:primitive\]]{} we have $g \lhd G \in \{1,g\}$. We cannot have $g \lhd G = 1$, for otherwise we obtain a contradiction: $$g = g \lhd 1 = g \lhd (G^2) = (g \lhd G) \lhd G = 1 \lhd G = 1$$ Therefore $g \lhd G = g$. Next, using the fact that $X \in
\mbox{P}_{1,G}(\mathbb{H}_4)$ and $g \lhd G = g$, we deduce from [Lemma \[le:primitive\]]{} and [Lemma \[le:primitsw\]]{} that $g \lhd X = 0$. Similarly, $g \lhd (GX) = 0$. Observe also that the actions of $X$ and $GX$ on $g$ are compatible with the relations $X^2 = 0$ and $G
X =-XG$.
Next, we show that
$\lhd$ 1 $G$ $X$ $GX$
-------- ----- ----- ----- ------
$x$ $x$ $x$ 0 0
or
$\lhd$ 1 $G$ $X$ $GX$
-------- ----- -------------- ---------- -----------
$x$ $x$ $a - ag - x$ $b - bg$ $-b + bg$
for some $a$, $b\in k$.
Since $x$ is an $(1,g)$-primitive element of $H_4$, $x \lhd G \in
\mbox{P}_{1 \lhd G, g \lhd G}(H_4) = \mbox{P}_{1,g}(H_4)$. Taking into account [Lemma \[le:primitsw\]]{}, we have $x \lhd G = a - ag - bx$, for some $a$, $b \in k$. The action of $G$ is compatible with $G^2
= 1$ so $$x = x \lhd 1 = (x \lhd G) \lhd G = (a - ag + bx) \lhd G = a + ba -
(a + ba)g + b^2x$$ Thus, $b^2 = 1$ and $a (1 + b) = 0$. If $b = -1$ then there are no restrictions on $a$. Otherwise, $b = 1$ and $a = 0$. This proves that $x \lhd G = a - ag - x$, with $a \in k$, or $x \lhd G = x$.
By a straightforward computation and the results above, $x \lhd X
\in \mbox{P}_{1,g}(H_4)$. Therefore, by [Lemma \[le:primitsw\]]{}, $x \lhd
X = b - bg + cx$, for some $b$, $c \in k$ . Because $X^2=0$, we have $$0 = x \lhd 0 = (x \lhd X) \lhd X = (b - bg + cx) \lhd X = cb - cbg
+ c^2x$$ Thus, $c = 0$, and $x \lhd X = b - bg$, with $b \in k$.
If $x \lhd G = x$, then $$x \lhd (GX) = (x \lhd G) \lhd X = x \lhd X$$ and, if $x \lhd G = a - ag - x$, then $$x \lhd (GX) = (x \lhd G) \lhd X = (a - ag - x) \lhd X = -x \lhd X$$ Observe that, in both cases, $\varepsilon \big( x \lhd (GX) \big)
= 0$, and $$\Delta \big( x \lhd (GX) \big) = x_{(1)} \lhd (GX)_{(1)} \ot
x_{(2)} \lhd (GX)_{(2)}$$
It remains to see when $x \lhd (GX) = x \lhd (-XG)$. If $x \lhd G
= x$, then $$x \lhd (XG) = (x \lhd X) \lhd G = (b - bg) \lhd G = b - bg = x
\lhd X = x \lhd (GX)$$ Thus, $x \lhd (GX) = x \lhd (-XG)$ implies $x \lhd X = x \lhd (GX)
= 0$. If $x \lhd G = a - ag - x$, then $$x \lhd (XG) = (x \lhd X) \lhd G = (b - bg) \lhd G = b - bg = x
\lhd X = -x \lhd (GX)$$ In this case, the equality $x \lhd (GX) = x \lhd (-XG)$ is satisfied without further restrictions.
Finally, we show that
$\lhd$ 1 $G$ $X$ $GX$
-------- ------ ------ ----- ------
$gx$ $gx$ $gx$ 0 0
or
$\lhd$ 1 $G$ $X$ $GX$
-------- ------ --------------- ---------- -----------
$gx$ $gx$ $c - cg - gx$ $d - dg$ $-d + dg$
for some $c$, $d\in k$.
Using [Lemma \[le:primitsw\]]{} and the fact that $(gx) \lhd G \in
\mbox{P}_{g,1}(H_4)$, as one can easily see, we deduce that $(gx)
\lhd G = c - cg + dgx$. Also, $$gx = (gx) \lhd 1 = \big( (gx) \lhd G \big) \lhd G = (c - cg + dgx)
\lhd G = c + dc -(c + dc)g + d^2gx$$ therefore, $d^2 = 1$ and $c(1 + d)=0$. If $d = -1$, then nothing can be said about $c$. Otherwise, $d = 1$ and $c = 0$. It follows that $(gx) \lhd G = gx$ or $(gx) \lhd G = c - cg - gx$, with $c\in
k$.
Again, a straightforward computation shows that $(gx) \lhd X$ is $(g,1)$-primitive, hence, by [Lemma \[le:primitsw\]]{}, there exist $d$, $e
\in k$ such that $(gx) \lhd X = d - dg + egx$. Recalling that $X^2
= 0$, we have $$0 = (gx) \lhd 0 = \big( (gx) \lhd X \big) \lhd X = (d - dg + egx)
\lhd X = ed - edg + e^2gx$$ Thus, $e = 0$ and $(gx) \lhd X = d - dg$, with $d\in k$.
If $(gx) \lhd G = gx$ then $$(gx) \lhd (GX) = \big( (gx) \lhd G \big) \lhd X = (gx) \lhd X$$ and, if $(gx) \lhd G = c - cg - gx$, then $$(gx) \lhd (GX) = \big( (gx) \lhd G \big) \lhd X = (c - cg - gx)
\lhd X = -(gx) \lhd X$$ In both cases, $\varepsilon \big( (gx) \lhd (GX) \big) = 0$, and $$\Delta \big( (gx) \lhd (GX) \big) = (gx)_{(1)} \lhd (GX)_{(1)} \ot
(gx)_{2} \lhd (GX)_{2}$$
Finally, we make use of the condition $(gx) \lhd (GX) = -(gx) \lhd
(XG)$. If $(gx) \lhd G = gx$, then $$(gx) \lhd (XG) = \big( (gx) \lhd X \big) \lhd G = (d - dg) \lhd G
= d-dg = (gx) \lhd X = (gx) \lhd (GX)$$ Since $(gx) \lhd (GX) = (gx) \lhd (-XG)$, we have $(gx) \lhd
X = (gx) \lhd (GX) = 0$. If $(gx) \lhd G = c - cg - gx$, then $$(gx) \lhd (XG) = \big( (gx) \lhd X \big) \lhd G = (d - dg) \lhd G
= d - dg = (gx) \lhd X = -(gx) \lhd (GX)$$ In this case, the condition $(gx) \lhd (GX) = (gx) \lhd (-XG)$ is satisfied without further restrictions.
Analogous to [Proposition \[pr:actdr\]]{} we can prove:
[\[pr:actstg\]]{} If $\rhd : H_4 \ot \mathbb{H}_4 \to \mathbb{H}_4$ is a left $H_4$-module coalgebra structure such that $h \rhd 1 =
\varepsilon(h)1$, for all $h \in H_4$, then $\rhd$ has one of the following forms:
[l | r r r r ]{} $\rhd^1$ & 1 & $G$ & $X$ & $GX$\
1 & 1 & $G$ & $X$ & $GX$\
$g$ & 1 & $G$ & $X$ & $GX$\
$x$ & 0 & 0 & 0 & 0\
$gx$ & 0 & 0 & 0 & 0\
[l | r r r r ]{} $\rhd^2$ & 1 & $G$ & $X$ & $GX$\
1 & 1 & $G$ & $X$ & $GX$\
$g$ & 1 & $G$ & $X$ & $u-uG-GX$\
$x$ & 0 & 0 & 0 & $v-vG$\
$gx$ & 0 & 0 & 0 & $v-vG$\
[l | r r r r ]{} $\rhd^3$ & 1 & $G$ & $X$ & $GX$\
1 & 1 & $G$ & $X$ & $GX$\
$g$ & 1 & $G$ & $s-sG-X$ & $GX$\
$x$ & 0 & 0 & $t-tG$ & 0\
$gx$ & 0 & 0 & $t-tG$ & 0\
[l | r r r r ]{} $\rhd^4$ & 1 & $G$ & $X$ & $GX$\
1 & 1 & $G$ & $X$ & $GX$\
$g$ & 1 & $g$ & $s-sG-X$ & $u-uG-GX$\
$x$ & 0 & 0 & $t-tG$ & $v-vG$\
$gx$ & 0 & 0 & $t-tG$ & $v-vG$\
where $s$, $t$, $u$, $v\in k$.
One can check the validity of the statement by employing the same arguments as those used in the proof of [Proposition \[pr:actdr\]]{}. A more elegant and shorter proof can be deduced from the following elementary remark: if $\rhd: H \ot C \to C $ is a left $H$-module coalgebra on $C$ then $\lhd : C \ot H^{\rm cop}
\to C$, $c \lhd h := S(h) \rhd c$, for all $c \in C$ and $h \in H$ is a right $H^{\rm cop}$-module coalgebra on $C$, and the above correspondence is bijective if the antipode of $H$ is bijective. We apply this observation for $H = H_4$ and $C = \mathbb{H}_4$, which is just a copy of $H_4$, taking into account that the antipode of $H_4$ is bijective and $H_4^{\rm cop} \cong H_4$. In this way, the proof of [Proposition \[pr:actstg\]]{} follows from the one of [Proposition \[pr:actdr\]]{}.
We are now in a position to finish the proof of [Theorem \[th:mpHH00\]]{}.
Let $(\mathbb{H}_4, H_4, \rhd, \lhd)$ be a matched pair. Since $\lhd : H_4 \ot \mathbb{H}_4 \to H_4$ is a right $\mathbb{H}_4$-module coalgebra structure satisfying $1 \lhd h =
\varepsilon(h)1$, for all $h \in \mathbb{H}_4$, we deduce from [Proposition \[pr:actdr\]]{} that $\lhd$ is one of the $\lhd^i$’s. Similarly, $\rhd : H_4 \ot \mathbb{H}_4 \to \mathbb{H}_4$ is a left $H_4$-module coalgebra structure satisfying $h \rhd 1 =
\varepsilon(h)1$, for all $h\in H_4$, hence, $\rhd$ is one of the $\rhd^j$’s, by [Proposition \[pr:actstg\]]{}. We will prove that $(\mathbb{H}_4,
H_4,\rhd^j, \lhd^i)$ is a matched pair if and only if $(i,j)\in\{(1,1),(4,4)\}$ and, if $(i,j)=(4,4)$, then $\rhd^i$ and $\lhd^j$ are defined as we have claimed.
Firstly, $(\mathbb{H}_4, H_4,\rhd^i, \lhd^j)$ is not a matched pair, if $i = 2, 3$ or $j = 2,3$, since $$g \rhd^2 (GX) \neq (g \rhd^2 G) \left( (g \lhd^j G) \rhd^2 X
\right)$$ $$g \rhd^3 (GX) \neq (g \rhd^3 G) \left( (g \lhd^j G) \rhd^3 X
\right)$$ $$(xg) \lhd^2 G \neq \left(x \lhd^2 (g \rhd^i G) \right)(g \lhd^2 G)$$ and $$(xg) \lhd^3 G \neq \left(x \lhd^3 (g \rhd^i G) \right)(g \lhd^3 G)$$ for all $i$, $j \in \{1,2,3,4\}$, i.e., condition [(\[eq:mp2\])]{} is not satisfied. Indeed, $$g \rhd^2 (GX) = u - uG - GX \neq GX = G (g \rhd^2 X) = (g \rhd^2
G) \left( (g \lhd^j G) \rhd^2 X \right)$$ $$g \rhd^3 (GX) = GX \neq -s + sG - GX = G(s - sG - X) = G (g \rhd^3
X) = (g\rhd^3 G) \left( (g \lhd^j G) \rhd^3 X \right)$$ $$(xg) \lhd^2 G = -c + cg + gx \neq -gx = xg = (x \lhd^2 G )g =
\left(x \lhd^2 (g \rhd^i G) \right)(g \lhd^2 G)$$ and $$(xg) \lhd^3 G = -gx \neq -a + ag + gx = (a - ag -x)g = (x \lhd^3
G)g = \left(x \lhd^3 (g \rhd^i G) \right)(g \lhd^3 G)$$
Secondly, $(\mathbb{H}_4, H_4, \rhd^4, \lhd^1)$ and $(\mathbb{H}_4, H_4, \rhd^1, \lhd^4)$ are not matched pairs, since condition [(\[eq:mp4\])]{} is not verified. Indeed, $$\begin{aligned}
x_{(1)} \lhd^1 (GX)_{(1)} \ot x_{(2)} \rhd^4 (GX)_{(2)}
&= x_{(1)} \ot x_{(2)} \rhd^4(GX)\\
&= x \ot 1 \rhd^4(GX) + g \ot x \rhd^4(GX)\\
&= x \ot GX + g \ot (v - vG) \\
&= x \ot GX + vg \ot 1 - vg \ot G\end{aligned}$$ whereas $$\begin{aligned}
x_{(2)} \lhd^1 (GX)_{(2)} \ot x_{(1)} \rhd^4 (GX)_{(1)}
&= x_{(2)} \ot x_{(1)} \rhd^4 (GX)\\
&= 1 \ot x \rhd^4 (GX) + x \ot g \rhd^4 (GX)\\
&= 1 \ot (v - vG) + x \ot (u - uG - GX) \\
&= v1 \ot 1 - v1 \ot G + ux \ot 1 - ux \ot G - x \ot GX\end{aligned}$$ Similarly, $$\begin{aligned}
(gx)_{(1)} \lhd^4 X_{(1)} \ot (gx)_{(2)} \rhd^1 X_{(2)}
&= (gx) \lhd^4 X_{(1)} \ot X_{(2)}\\
&= (gx) \lhd^4 X \ot 1 + (gx) \lhd^4 G \ot X\\
&= (d - dg) \ot 1 + (c -c g - gx) \ot X \\
&= d1 \ot 1 - dg \ot 1 + c1 \ot X - cg \ot X - gx \ot X\end{aligned}$$ whereas $$\begin{aligned}
(gx)_{(2)}\lhd^4 X_{(2)} \ot (gx)_{(1)} \rhd^1 X_{(1)}
&= (gx) \lhd^4 X_{(2)} \ot X_{(1)}\\
&= (gx) \lhd^4 1 \ot X + (gx) \lhd^4 X \ot G\\
&= gx \ot X + (d - dg) \ot G\\
&= gx \ot X + d1 \ot G - dg \ot G\end{aligned}$$
We focus now our attention on when $(\mathbb{H}_4, H_4, \rhd^4,
\lhd^4)$ is a matched pair. We abandon the cumbersome notations $\rhd^4$ and $\lhd^4$, and use instead $\rhd$ and $\lhd$. We begin by verifying condition [(\[eq:mp4\])]{} from the definition of a matched pair. Evidently, the condition is verified for $(h,a) \in
\{(1,1),(1,G),(g,1),(g,G)\}$. It is also verified for $(h,a) \in
\{(1,X),(x,1),(1,GX),(gx,1),(g,X),(x,G),(g,GX),(gx,G)\}$. For example, $$\begin{aligned}
(gx)_{(1)} \lhd G_{(1)} \ot (gx)_{(2)} \rhd G_{(2)}
& = (gx) \lhd G \ot g \rhd G + 1 \lhd G \ot(gx) \rhd G\\
& = (gx) \lhd G \ot G\end{aligned}$$ and $$\begin{aligned}
(gx)_{(2)} \lhd G_{(2)} \ot (gx)_{(1)} \rhd G_{(1)}
& = g \lhd G \ot (gx) \rhd G + (gx)\lhd G \ot 1 \rhd G\\
& = (gx) \lhd G \ot G\end{aligned}$$ hence, $(gx)_{(1)} \lhd G_{(1)} \ot (gx)_{(2)} \rhd G_{(2)} =
(gx)_{(2)} \lhd G_{(2)} \ot (gx)_{(1)} \rhd G_{(1)}$.
For $(h,a)=(x,X)$ we have $$\begin{aligned}
x_{(1)} \lhd X_{(1)} \ot x_{(2)} \rhd X_{(2)} & = x \lhd X \ot 1
\rhd 1 + x \lhd G \ot 1 \rhd X + g \lhd X \ot x \rhd 1 +\\
& \hspace{5mm} g \lhd G \ot x \rhd X\\
& = (b - bg) \ot 1 + (a - ag - x) \ot X + g \ot (t - tG)\\
& = b1 \ot 1 + (t - b)g \ot 1 - tg \ot G + a 1\ot X - ag \ot X - x
\ot X\end{aligned}$$ $$\begin{aligned}
x_{(2)} \lhd X_{(2)} \ot x_{(1)} \rhd X_{(1)} & = 1 \lhd 1 \ot x
\rhd X + 1 \lhd X \ot x \rhd G + x \lhd 1 \ot g \rhd X +\\
& \hspace{5mm} x \lhd X \ot g \rhd G\\
& = 1 \ot (t - tG) + x \ot (s - sG - X) + (b - bg) \ot G\\
& = t1 \ot 1 + (b - t)1 \ot G - bg \ot G + sx \ot 1 - sx \ot G - x
\ot X\end{aligned}$$ Since $x_{(1)} \lhd X_{(1)} \ot x_{(2)} \rhd X_{(2)} = x_{(2)}
\lhd X_{(2)} \ot x_{(1)} \rhd X_{(1)}$, we must have $t = b$ and $a = s = 0$.
Let $(h,a) = (gx,GX)$. Then $$\begin{aligned}
(gx)_{(1)} \lhd & (GX)_{(1)} \ot (gx)_{(2)} \rhd (GX)_{(2)}\\
& = (gx) \lhd (GX) \ot g \rhd G + (gx) \lhd 1 \ot g \rhd (GX) +
1 \lhd (GX) \ot (gx) \rhd G +\\
& \hspace{5mm} 1 \lhd 1 \ot (gx) \rhd (GX)\\
& = (-d + dg) \ot G + gx \ot (u - uG - GX) + 1 \ot (v - vG)\\
&= v1 \ot 1 - (v+d)1 \ot G + dg \ot G + ugx \ot 1 - ugx \ot G - gx
\ot GX\end{aligned}$$ $$\begin{aligned}
(gx)_{(2)}\lhd &(GX)_{(2)} \ot (gx)_{(1)} \rhd (GX)_{(1)}\\
& = g \lhd G \ot (gx) \rhd (GX) + g \lhd (GX) \ot (gx) \rhd 1 +
(gx) \lhd G \ot 1 \rhd (GX) +\\
&\hspace{5mm} (gx) \lhd (GX) \ot 1 \rhd 1\\
&=g \ot (v - vG) + (c - cg - gx) \ot GX + (-d + dg) \ot 1\\
&=-d1 \ot 1 + (v + d)g \ot 1 - vg \ot G + c1 \ot GX - cg \ot GX -
gx \ot GX.\end{aligned}$$ Since $$(gx)_{(1)} \lhd (GX)_{(1)} \ot (gx)_{(2)} \rhd (GX)_{(2)} =
(gx)_{(2)} \lhd (GX)_{(2)} \ot (gx)_{(1)} \rhd (GX)_{(1)}$$ we must have $v = -d$ and $c = u = 0$.
Now, if $(h,a) = (x,GX)$, then $$\begin{aligned}
x_{(1)} \lhd (GX)_{(1)}& \ot x_{(2)} \rhd (GX)_{(2)}\\
& = x \lhd (GX) \ot 1 \rhd G + x \lhd 1 \ot 1 \rhd (GX) +
g \lhd (GX) \ot x \rhd G +\\
&\hspace{5mm} g \lhd 1 \ot x \rhd (GX)\\
&=(-b + bg) \ot G + x \ot GX + g \ot (v - vG)\\
&=-b1 \ot G + (b - v)g \ot G + vg \ot 1 + x \ot GX\end{aligned}$$ and $$\begin{aligned}
x_{(2)} \lhd (GX)_{(2)}& \ot x_{(1)} \rhd (GX)_{(1)}\\
& = 1 \lhd G \ot x \rhd (GX) + 1 \lhd (GX) \ot x \rhd 1 +
x \lhd G \ot g \rhd (GX) +\\
&\hspace{5mm} x \lhd (GX) \ot g \rhd 1\\
&=1 \ot (v - vG) + (-x) \ot (-GX) + (-b + bg) \ot 1\\
&=(v - b)1 \ot 1 - v1 \ot G + bg \ot 1 + x \ot GX.\end{aligned}$$ Since $$x_{(1)} \lhd (GX)_{(1)} \ot x_{(2)} \rhd (GX)_{(2)} = x_{(2)} \lhd
(GX)_{(2)} \ot x_{(1)} \rhd (GX)_{(1)}$$ we must have $v = b$.
Finally, if $(h,a) = (gx,X)$, then $$\begin{aligned}
(gx)_{(1)} \lhd X_{(1)}& \ot (gx)_{(2)} \rhd X_{(2)}\\
& = (gx) \lhd X \ot g \rhd 1 + (gx) \lhd G \ot g \rhd X +
1 \lhd X \ot (gx) \rhd 1 +\\
&\hspace{5mm} 1 \lhd G \ot (gx) \rhd X\\
&=(d - dg) \ot 1 + (-gx) \ot (-X) + 1 \ot (t - tG)\\
&=(d + t)1 \ot 1 - dg \ot 1 - t1 \ot G + gx \ot X\end{aligned}$$ $$\begin{aligned}
(gx)_{(2)} \lhd X_{(2)}& \ot (gx)_{(1)} \rhd X_{(1)}\\
& = g \lhd 1 \ot (gx) \rhd X + g \lhd X \ot (gx) \rhd G +
(gx) \lhd 1 \ot 1 \rhd X +\\
& \hspace{5mm} (gx) \lhd X \ot 1 \rhd G\\
&= g \ot (t - tG) + gx \ot X + (d - dg) \ot G\\
&= tg \ot 1 - (t + d)g \ot G + d1 \ot G + gx \ot X\end{aligned}$$ Since $t = b$ and $d = -v = -b$, we have $$(gx)_{(1)} \lhd X_{(1)} \ot (gx)_{(2)} \rhd X_{(2)} = (gx)_{(2)}
\lhd X_{(2)} \ot (gx)_{(1)} \rhd X_{(1)}$$
Thus, condition [(\[eq:mp4\])]{} is satisfied when $a = c = s = u =0$, $t = v = b$ and $d = -b$. It is now easy to see that, in these circumstances, conditions [(\[eq:mp2\])]{} and [(\[eq:mp3\])]{} are compatible with the relations $G^2 = g^2 = 1$, $X^2 = x^2 = 0$, $gx = -xg$ and $GX = -XG$, i.e. that $h \rhd (G^2) = h \rhd 1$, $h
\rhd (X^2) = h \rhd 0$, $h \rhd (GX) = h \rhd (-XG)$, for all $h
\in H_4$, and $(g^2) \lhd a = 1 \lhd a$, $(x^2) \lhd a = 0 \lhd
a$, $(gx) \lhd a = (-xg) \lhd a$, for all $a \in \mathbb{H}_4$. We will do the verifications for $h = gx$ and $a = GX$ and leave the rest to the reader. We have $$\begin{aligned}
(gx) \rhd (G^2) & = \left( (gx) \rhd G \right) \left( (g \lhd G)
\rhd G \right) + (1 \rhd G) \left( \left( (gx) \lhd G \right)
\rhd G \right)\\
& = G \left( -(gx) \rhd G \right)\\
& = 0 = (gx) \rhd 1\end{aligned}$$ $$\begin{aligned}
(gx) \rhd (X^2) & = ((gx) \rhd X) \left( (g \lhd 1) \rhd X \right)
+ (1 \rhd X) \left( ((gx) \lhd 1) \rhd X \right) + \\
&\hspace{5mm} ((gx) \rhd G) \left( (g \lhd X) \rhd X \right) +
(1 \rhd G) \left( ((gx) \lhd X) \rhd X \right)\\
& = (b - bG)(g \rhd X) + X((gx) \rhd X) + G \left( (-b + bg) \rhd
X \right)\\
& = (b - bG)(-X) + X(b - bG) + G(-bX - bX)\\
& = 0 = (gx) \rhd 0\end{aligned}$$ and $$\begin{aligned}
(gx) \rhd (XG) & = ((gx) \rhd X) \left( (g \lhd 1) \rhd G \right)
+ (1 \rhd X) \left( ((gx) \lhd 1) \rhd G \right) +\\
&\hspace{5mm} ((gx) \rhd G) \left( (g \lhd X) \rhd G \right) +
(1 \rhd G) \left( ((gx) \lhd X) \rhd G \right)\\
&= (b - bG)(g \rhd G) + X((gx) \rhd G) +
G \left( (-b + bg) \rhd G \right)\\
&= (b - bG)G + G(-bG + bG)\\
&= -b + bG\\
&= (gx) \rhd (-GX)\end{aligned}$$ Also $$\begin{aligned}
(g^2) \lhd (GX) &= \left( g \lhd (g \rhd (GX)) \right) (g \lhd G)
+ \left( g \lhd (g \rhd 1) \right) (g \lhd (GX))\\
& = (g \lhd (-GX))g\\
& = 0 = 1 \lhd (GX)\end{aligned}$$ $$\begin{aligned}
(x^2) \lhd (GX) & = \left( x \lhd (x \rhd (GX)) \right) (1 \lhd G)
+ \left( x \lhd (x \rhd 1) \right) (1 \lhd (GX)) +\\
&\hspace{5mm} \left( x \lhd (g \rhd (GX)) \right) (x \lhd G) +
\left( x \lhd (g \rhd 1) \right) (x \lhd (GX))\\
&= x \lhd (b - bG) + \left( x \lhd (-GX) \right) (-x) + (x \lhd 1)(-b + bg)\\
&= bx + bx + (b - bg)(-x) + x(-b + bg)\\
&= 0 = 0 \lhd (GX)\end{aligned}$$ and $$\begin{aligned}
(xg) \lhd (GX) & = \left( x \lhd (g \rhd (GX)) \right) (g \lhd G)
+ \left( x \lhd (g \rhd 1) \right) (g \lhd (GX))\\
& = \left( x \lhd (-GX) \right)g\\
& = (b - bg)g\\
& = -b + bg\\
& = (-gx) \lhd (GX)\end{aligned}$$
We are able to describe and classify all Hopf algebras that factorize through two Sweedler’s Hopf algebras.
[\[th:clH\_4H\_4\]]{} Let $k$ be a field of characteristic $\neq 2$. Then:
$(1)$ A Hopf algebra $E$ factorizes through $\mathbb{H}_4$ and $H_4$ if and only if $E \cong \mathbb{H}_4 \ot H_4$ or $E \cong
{\mathcal H}_{16, \, \lambda}$, for some $\lambda \in k$, where ${\mathcal H}_{16, \, \lambda}$ is the Hopf algebra generated by $g$, $x$, $G$, $X$ subject to the relations: $$g^2 = G^2 = 1 \quad x^2 = X^2 = 0, \quad gx = -xg, \quad GX =
-XG,$$ $$gG = Gg, \quad gX = -Xg, \quad x G = - Gx, \quad xX + Xx = \lambda
\, (1 - Gg)$$ with the coalgebra structure given by $$\Delta (g) = g \ot g, \quad \Delta (x) = x\ot 1 + g \ot x, \quad
\Delta (G) = G \ot G, \quad \Delta (X) = X\ot 1 + G \ot X,$$ $$\varepsilon (g) = \varepsilon (G) = 1, \qquad \varepsilon (x) =
\varepsilon (X) = 0$$
$(2)$ ${\mathcal H}_{16, \, \lambda}$ is a pointed, unimodular and non-semisimple $16$-dimensional Hopf algebra.
$(3)$ Up to an isomorphism of Hopf algebras, there are only three Hopf algebras that factorize through $\mathbb{H}_4$ and $H_4$, namely $$\mathbb{H}_4 \ot H_4, \qquad {\mathcal H}_{16, \, 0} \qquad {\rm
and} \qquad {\mathcal H}_{16, \, 1} \cong D(H_4)$$ where $D(H_4)$ is the Drinfel’d double of $H_4$.
$(1)$ The Hopf algebra ${\mathcal H}_{16, \, \lambda}$ is the explicit description of the bicrossed product $\mathbb{H}_4
\bowtie H_4$ associated to the non-trivial matched pair given in [Theorem \[th:mpHH00\]]{}. In $\mathbb{H}_4 \bowtie H_4$ we make the canonical identifications: $G = G \bowtie 1$, $X = X \bowtie 1$, $
g = 1 \bowtie g$, $x = 1 \bowtie x$. The defining relations of ${\mathcal H}_{16, \, \lambda}$ follow easily. For instance: $$\begin{aligned}
x \, X &=& (1 \bowtie x) (X \bowtie 1) = x_{(1)} \rhd X_{(1)} \,
\bowtie \, x_{(2)} \lhd X_{(1)} \\
&=& (\lambda - \lambda G) \bowtie 1 - X \bowtie x + G \bowtie (\lambda - \lambda \, g)\\
&=& \lambda 1 \bowtie 1 - X \bowtie x - \lambda \, G \bowtie g =
\lambda \, 1 - Xx - \lambda \, Gg\end{aligned}$$
$(2)$ ${\mathcal H}_{16, \, \lambda}$ is pointed because, as a coalgebra, it is the tensor product of two pointed coalgebras [@Montgomery Lemma 5.1.10]. Since $H_{4}$ is non-semisimple, so is ${\mathcal H}_{16, \, \lambda}$ [@Montgomery Corollary 3.2.3]. ${\mathcal H}_{16, \, \lambda}$ is unimodular since $(X+GX)(x - gx)$ is simultaneously a non-zero left and right integral. For instance, $$\begin{aligned}
x(X + GX)(x - gx) &= \big( -Xx + \lambda(1 - Gg) + GXx - \lambda
(
G - g) \big)(x -gx) \\
& = \lambda (x - gx - Ggx + Gx - Gx + Ggx + gx -x) = 0\end{aligned}$$ and $$\begin{aligned}
(X + GX)(x - gx) X &= (X + GX) \big( -Xx + \lambda(1 - Gg) - Xgx -
\lambda (g - G)\big) \\
&= \lambda (X + GX + GXg + Xg - Xg - GXg - GX - X) = 0\end{aligned}$$
$(3)$ ${\mathcal H}_{16, \, \lambda} \cong {\mathcal H}_{16, \,
1}$, for $\lambda \in k^{*}$, since the defining relations for ${\mathcal H}_{16, \, 1}$ can be obtained from that of ${\mathcal
H}_{16, \, \lambda}$ by replacing $X$ with $\lambda^{-1} X$.
We prove next that $\mathbb{H}_4 \ot H_4$, ${\mathcal H}_{16, \,
0}$ and ${\mathcal H}_{16, \, 1}$ are non-isomorphic Hopf algebras. It will be useful to regard ${\mathcal H}_{16, \,
\lambda}$ as the bicrossed product $\mathbb{H}_4 \bowtie_{\lambda}
H_4 $ associated to the non-trivial matched pair $(\rhd_{\lambda},
\lhd_{\lambda})$ given in [Theorem \[th:mpHH00\]]{}, for then we can use our description of morphisms between two bicrossed products.
From [@ABMbp Theorem 2.2] we know that if $\psi: \mathbb{H}_4
\bowtie H_4 \to \mathbb{H}_4 \bowtie^{\prime} H_4$ is such a morphism then $$\psi(a \bowtie h) = u \left( a_{(1)} \right) \left( p \left(
a_{(2)} \right) \rhd' r \left( h_{(1)} \right) \right) \bowtie'
\left( p \left (a_{(3)} \right) \lhd' r \left( h_{(2)} \right)
\right) v \left( h_{(3)} \right)$$ for all $a \in \mathbb{H}_4$ and $h \in H_4$, where $u :
\mathbb{H}_4 \to \mathbb{H}_4$, $p : \mathbb{H}_4 \to H_4$, $r :
H_4 \to \mathbb{H}_4$, and $v : H_4 \to H_4$ are unitary coalgebra maps satisfying conditions [(\[eq:C1\])]{}-[(\[eq:C8\])]{}. From [Lemma \[le:morf\_H\_4H\_4\]]{} we know the description of unitary coalgebra maps between two Sweedler’s Hopf algebras. We will show that if one of the pairs $(u,p)$, $(r,v)$ and $(u,r)$ consists of maps both trivial or both non-trivial then $\psi$ is not an isomorphism.
In order to do this, we first describe the pairs $(u,p)$ of unitary coalgebra maps, $u : \mathbb{H}_4 \to \mathbb{H}_4$ and $p
: \mathbb{H}_4 \to H_4$, that satisfy ${(\ref{eq:C1})}$, ${(\ref{eq:C3})}$ and ${(\ref{eq:C4})}$. What we will find is that if $u$ and $p$ are both non-trivial, then $u(G) = G$, $u(X) = u(GX) = 0$, $p(G) = g$ and $p(X) = p(GX) = 0$, and, if one of them is non-trivial and the other trivial, then the former is a Hopf algebra morphism.
Suppose $u$ and $p$ are both non-trivial, and let $a_u$, $a_p$, $b_u$, $b_p$, $c_u$, $c_p$, $d_u$, $d_p \in k$ such that $u(X) =
a_u - a_uG + b_uX$, $u(GX) = c_u - c_uG + d_uGX$, $p(X) = a_p -
a_pg + b_px$, and $p(GX) = c_p - c_pg + d_pgx$. Using condition [(\[eq:C1\])]{}, we have $$u(X) \ot p(1) + u(G) \ot p(X) = u(1) \ot p(X) + u(X) \ot p(G)$$ from which we deduce that $b_p = b_u = 0$ and $a_p = a_u$. Again, from condition [(\[eq:C1\])]{}, we have $$u(GX) \ot p(G) + u(1) \ot p(GX) = u(G) \ot p(GX) + u(GX) \ot p(1)$$ from which we obtain $d_p = d_u =0$ and $c_p = c_u$. From condition [(\[eq:C3\])]{} we have: $$u(GX) = u(G) \left( p(G) \rhd' u(X) \right) = G \left( g \rhd'
(a_u - a_uG) \right) = G(a_u - a_uG) = -a_u + a_uG$$ Since $u(GX) = c_u - c_uG$, we must have $c_u = -a_u$. Also $u(GX)
= -u(XG)$, since $GX = -XG$. But $$\begin{aligned}
u(XG) & = u(X) \left( p(1) \rhd' u(G) \right) + u(G) \left( p(X)
\rhd' u(G) \right)\\
& = (a_u - a_uG)G + G \left( (a_u - a_ug) \rhd' G \right)\\
& = -a_u + a_uG + G(a_uG - a_uG)\\
& = -a_u + a_uG\\
& = u(GX)\end{aligned}$$ so $a_u=0$. Therefore, $u(X) = u(GX) = p(X) = p(GX) = 0$ and this concludes our first claim.
If $u$ is trivial and $p$ is non-trivial then the conditions [(\[eq:C1\])]{} and [(\[eq:C3\])]{} are trivially fulfilled and condition [(\[eq:C4\])]{} becomes $p(ab) = p(a)p(b)$, for all $a$, $b \in
\mathbb{H}_4$. Thus, $p$ is a Hopf algebra morphism in this case. Similarly, $u$ is a Hopf algebra morphism if $u$ is non-trivial and $p$ is trivial.
Observe that the above arguments also apply for pairs $(r,v)$ of unitary coalgebra maps, $r : H_4 \to \mathbb{H}_4$ and $v : H_4
\to H_4$, that satisfy conditions ${(\ref{eq:C2})}$, ${(\ref{eq:C5})}$ and ${(\ref{eq:C6})}$.
We are now able to prove our assertion that $\psi$ is not an isomorphism if one of the pairs $(u, p)$, $(r, v)$, and $(u, r)$ consists of maps that are both trivial or both nontrivial. We have $$\psi (X \bowtie 1) = u(X) \bowtie' 1 + u(G) \bowtie' p(X)$$ and $$\psi (1 \bowtie x) = r(x) \bowtie' 1 + r(g) \bowtie' v(x)$$ hence, $\psi (X \bowtie 1) = 0$ if $u$ and $p$ are both trivial or both non-trivial, and $\psi (1 \bowtie x) = 0$ if $r$ and $v$ are both trivial or both non-trivial. Since $X \bowtie 1$ and $1
\bowtie x$ are non-zero elements, $\psi$ is not injective in these cases. Suppose next that $u$ and $r$ are both trivial. If $p$ or $v$ is trivial then $\psi$ is not an isomorphism as we have already seen. If $p$ and $v$ are both non-trivial, then, considering the previous remarks, they are Hopf algebra morphisms. Using [Lemma \[le:morf\_H\_4H\_4\]]{} we deduce that $p(X)=a x$ and $v(x)=b
x$, for some $a$, $b$ in $k$. Since $$\psi( X \bowtie x) = 1 \bowtie' p(X) v(x) = ab 1 \bowtie' x^2 = 0$$ and $X \bowtie x \neq 0$, it follows that $\psi$ is not injective, hence not an isomorphism. Finally, suppose that $u$ and $r$ are both non-trivial. Then $\psi$ is not an isomorphism if $p$ or $v$ is non-trivial, so consider $p$ and $v$ trivial. Then $u$ and $r$ are Hopf algebra morphisms, hence there exist $a$, $b \in k$ such that $u(X)=aX$ and $r(x)=bX$. Since $$\psi( X \bowtie x) = u(X)r(x) \bowtie' 1 = ab X^2 \bowtie' 1 = 0$$ we deduce that $\psi$ is not injective in this case either.
We have reduced so far the number of possibilities for $(u,p,r,v)$ when $\psi$ is an isomorphism. More precisely, if $\psi$ is an isomorphism then $u$ and $v$ are both trivial and $p$ and $r$ are both non-trivial, or $u$ and $v$ are both non-trivial and $p$ and $r$ are both trivial.
In what follows we will consider the Hopf algebras $\mathbb{H}_4
\ot H_4$, $\mathcal{H}_{16,0}$ and $\mathcal{H}_{16,1}$ and the morphisms between them associated to the two remaining possibilities for $(u,p,r,v)$.
Let $\psi: \mathbb{H}_4 \ot H_4 \to \mathcal{H}_{16,0}$ be a morphism associated to $(u,p,r,v)$. Suppose $u$ and $v$ are trivial and $p$ and $r$ are non-trivial. Thus, $p$ and $r$ are Hopf algebra morphisms. Since the actions associated to $\mathbb{H}_4\ot H_4$ are the trivial ones, condition [(\[eq:C7\])]{} takes the following form: $$\varepsilon(b)r(h) = p(b) \rhd_0 r(h)$$ Consider $a \in k$ such that $r(x) = aX$. Then $$aX = \varepsilon(G)r(x) = p(G) \rhd_0 r(x) = ag \rhd_0 X = -aX$$ hence $a = 0$. From $$\psi (1 \ot x) = r(x) \bowtie_0 1 = 0$$ it follows that $\psi$ is not an isomorphism. Suppose now that $u$ and $v$ are nontrivial and $p$ and $r$ are trivial. Then $u$ and $v$ are Hopf algebra morphisms and condition [(\[eq:C7\])]{} takes the form: $$v(h) \rhd_0 u(b) = \varepsilon(h)u(b)$$ Let $a\in k$ such that $u(X) = a X$. From $$aX = \varepsilon(g)u(X) = v(g) \rhd_0 u(X) = ag \rhd_0 X = -aX$$ we deduce that $a = 0$ and, consequently, that $u(X)=0$. Since $$\psi (X \ot 1) = u(X) \bowtie_0 1 = 0$$ it follows that $\psi$ cannot be an isomorphism. Thus $\mathbb{H}_4 \ot H_4 \ncong \mathcal{H}_{16,0}$ as all morphisms from $\mathbb{H}_4 \ot H_4$ to $\mathcal{H}_{16,0}$ are not injective.
Observe that, if we replace in the previous paragraph $\mathcal{H}_{16,0}$ with $\mathcal{H}_{16,1}$ and $\rhd_0$ with $\rhd_1$, then everything else remains the same, since $g \rhd_1 X
= -X$. We deduce, therefore, that $\mathbb{H}_4 \ot H_4 \ncong
\mathcal{H}_{16,1}$ also.
Consider now $\psi : \mathcal{H}_{16,0} \to \mathcal{H}_{16,1}$ a morphism associated to $(u,p,r,v)$. If $u$ and $v$ are trivial and $p$ and $r$ are non-trivial, then $p$ and $r$ are Hopf algebra morphisms and condition [(\[eq:C7\])]{} becomes $$\varepsilon(b)r(h) = p \left( h_{(1)} \rhd_0 b_{(1)} \right)
\rhd_1 r \left( h_{(2)} \lhd_0 b_{(2)} \right)$$ Let $a$, $b \in k$ such that $p(X) = ax$ and $r(x) = bX$. Since $$0 = \varepsilon(X)r(x) = p \left( x_{(1)} \rhd_0 X_{(1)} \right)
\rhd_1 r \left( x_{(2)} \lhd_0 X_{(2)} \right) = -p(X) \rhd_1 r(x)
= -ab + abG$$ we deduce that $ab = 0$, hence $p(X) = 0$ or $r(x) = 0$. But then $\psi (X \bowtie_0 1) = 1 \bowtie_1 p(X) = 0$ or $\psi (1
\bowtie_0 x) = r(x) \bowtie_1 1 = 0$, which shows that $\psi$ is not injective. If $u$ and $v$ are non-trivial and $p$ and $r$ are trivial, then $u$ and $v$ are Hopf algebra morphisms and condition [(\[eq:C7\])]{} has the following form: $$v(h) \rhd_1 u(b) = u(h \rhd_0 b)$$ Let $a$, $b \in k$ such that $u(X) = aX$ and $v(x) = bx$. From $$0 = u(x \rhd_0 X) = v(x) \rhd_1 u(X) = abx \rhd_1X = ab - abG$$ we deduce that $ab = 0$. Thus $u(x) = 0$ or $v(x) = 0$. Since $\psi(X \bowtie_0 1) = u(X) \bowtie_1 0$ and $\psi(1 \bowtie_0 x)
= 1 \bowtie_1 v(x)$ it follows that $\psi$ is not injective. We conclude that $\mathcal{H}_{16,0} \ncong \mathcal{H}_{16,1}$, as all morphisms from $\mathcal{H}_{16,0}$ to $\mathcal{H}_{16,1}$ are not injective.
Finally, we shall prove that ${\mathcal H}_{16, \, 1} \cong
D(H_4)$ and this will finish the proof. Observe first that $\left(
H_4^* \right)^{\rm cop} \simeq \mathbb{H}_4$, so $D(H_4)$ factorizes through $\mathbb{H}_4$ and $H_4$. Indeed, if $\{ 1^*,
g^*, x^*, (gx)^* \}$ is the dual basis of $\{1,g,x,gx\}$ then $$(1^{\ast} - g^{\ast}) \ast (1^{\ast} - g^{\ast}) = 1^{\ast} +
g^{\ast} = 1_{H_4^{\ast}}$$ $$(x^{\ast} + (gx)^{\ast}) \ast (x^{\ast} + (gx)^{\ast}) = 0$$ $$(1^{\ast} - g^{\ast}) \ast (x^{\ast} + (gx)^{\ast}) = -x^{\ast} +
(gx)^{\ast} = -(x^{\ast} + (gx)^{\ast}) \ast (1^{\ast} - g^{\ast})$$ where $\ast$ is the convolution product in $H_4^*$, hence, as an algebra, $H_4^* = k \langle G,X| G^2 = 1, X^2 = 0, GX = -XG
\rangle$, where $G = 1^* - g^*$, and $X = x^* +(gx)^*$. Recall now that, if $H$ is a Hopf algebra with basis $\{e_1,...,e_n\}$ and $\{e_1^*,...,e_n^*\}$ is the dual basis, then $$\Delta_{H^*}(e_i) = \sum_{j,l = 1}^n e_i^*(e_je_l) e_j^* \ot e_l^*$$ Using this fact we obtain that
$\Delta_{H_4^*}(G) = G \ot G$ and $\Delta_{H_4^*}(X) =
1 \ot X + X \ot G$
Therefore, $\left( H_4^* \right)^{\rm cop} \cong \mathbb{H}_4$, so $D(H_4)$ factorizes indeed through $\mathbb{H}_4$ and $H_4$. In order to see which of the three algebras from [Theorem \[th:clH\_4H\_4\]]{} $D(H_4)$ is, recall the Drinfel’d double as a matched pair. If $H$ is a finite dimensional Hopf algebra, then $\left( \left( H^*
\right)^{\rm cop}, H, \rhd, \lhd \right)$ is a matched pair, where
$\rhd : H \ot \left( H^* \right)^{\rm cop} \to \left( H^*
\right)^{\rm cop}$, $h \rhd h^* = h^* \left(
S^{-1}(h_{(2)})?h_{(1)} \right)$
$\lhd : H \ot \left( H^* \right)^{\rm cop} \to H$, $h \lhd
h^* = h^* \left( S^{-1}(h_{(3)})h_{(1)} \right) h_{(2)}$
for all $h \in H$, $h^* \in \left( H^* \right)^{\rm cop}$, and $D(H) \simeq \left( H^* \right)^{\rm cop} \bowtie H$. In our case, we have $$\begin{aligned}
x \lhd X & = X \left( S^{-1}(1)x \right) + X \left( S^{-1}(1)g
\right) x + X \left( S^{-1}(x)g \right) g\\
& = X(x) + X(g)x + X(-x)g\\
& =1 - g\end{aligned}$$ which shows that $D(H_4) \simeq \mathbb{H}_4 \bowtie_1 H_4$.
$\mathcal{H}_{16,0}$ is not the dual of $D(H_4)$ for otherwise $D(H_4)^*$ would be unimodular and so would $H_{4}$ ([@Radford_MQHA], Corollary 4).
The classification of pointed Hopf algebras of dimension 16 over an algebraically closed field of characteristic zero was done in [@Caenepeel; @Dascalescu]. Since the coradical of $\mathcal{H}_{16, \, \lambda}$ is $k[\mbox{G}(\mathbb{H}_{4})] \ot
k[\mbox{G}(H_{4})]$, we have $\mbox{G} (\mathcal{H}_{16, \,
\lambda}) = \{ 1, g, G,$ $gG \} \simeq \mathbb{Z}_{2} \times
\mathbb{Z}_{2}$. With the notations of [@Caenepeel; @Dascalescu Theorem 5.2], we have $\mathbb{H}_{4} \ot H_{4}
\simeq H_{(3)}$, $\mathcal{H}_{16,0} \simeq H_{(4)}$ and $\mathcal{H}_{16,1} \simeq H_{(5)}$.
As a consequence of our approach, we can describe the group of Hopf algebra automorphisms of the Hopf algebras obtained in [Theorem \[th:clH\_4H\_4\]]{}. We begin with the Drinfel’d double, $D(H_4)$.
[\[pr:autDH\]]{} Let $k$ be a field of characteristic $\neq 2$. Then there exists an isomorphism of groups $${{\rm Aut}\,}_{\rm Hopf}(D(H_4)) \cong k^* \rtimes_f \mathbb{Z}_2$$ where $k^* \rtimes_f \, \mathbb{Z}_2$ is the semidirect product of the group $k^*$ of the units of $k$ with $\mathbb{Z}_2$ associated to the action as automorphisms $f : \mathbb{Z}_2 \to {{\rm Aut}\,}(k^*)$, $f (1 + 2\mathbb{Z}) (\alpha) = \alpha^{-1}$, for all $\alpha \in
k^*$.
Let $\psi : \mathcal{H}_{16,1} \to \mathcal{H}_{16,1}$ be an endomorphism of the Drinfel’d double, $D(H_4) =
\mathcal{H}_{16,1}$, associated to $(u, p, r, v)$, where $u :
\mathbb{H}_4 \to \mathbb{H}_4$, $p : \mathbb{H}_4 \to H_4$, $r :
H_4 \to \mathbb{H}_4$, and $v : H_4 \to H_4$ are unitary coalgebra maps satisfying conditions [(\[eq:C1\])]{}-[(\[eq:C8\])]{}. We know, from the investigation in the proof of [Theorem \[th:clH\_4H\_4\]]{} that if $\psi$ is an isomorphism then $u$ and $v$ are both trivial and $p$ and $r$ are both non-trivial, or $u$ and $v$ are both non-trivial and $p$ and $r$ are both trivial.
We continue with the investigation further in order to describe all the automorphisms of $D(H_4)$. Suppose first that $u$ and $v$ are trivial and $p$ and $r$ are non-trivial. Then conditions [(\[eq:C4\])]{} and [(\[eq:C5\])]{} imply that $p$ and $r$ are Hopf algebra morphisms, conditions [(\[eq:C7\])]{} and [(\[eq:C8\])]{} become $$\varepsilon(b)r(h) = p(h_{(1)} \rhd_1 b_{(1)}) \rhd_1 r(h_{(2)}
\lhd_1 b_{(2)})$$ $$\varepsilon(h)p(b) = p(h_{(1)} \rhd_1 b_{(1)}) \lhd_1 r(h_{(2)}
\lhd_1 b_{(2)})$$ for all $b \in \mathbb{H}_4$ and $h \in H_4$, and the rest of the conditions are trivially fulfilled. Since $p$ and $r$ are Hopf algebra morphisms, it follows from [Lemma \[le:morf\_H\_4H\_4\]]{} that there exist $\alpha$, $\beta \in k$ such that $p(X) = \alpha \, x$, $p(GX) = \alpha \, gx$, $r(x) = \beta \, X$ and $r(gx) = \beta \,
GX$. By a routine check one can verify that conditions [(\[eq:C7\])]{} and [(\[eq:C8\])]{} are satisfied if and only if $\alpha \cdot \beta =
1$. For example, [(\[eq:C7\])]{} is true for $b = X$ and $h = x$ if and only if $$\begin{aligned}
\varepsilon(X)r(x) & = -p(X) \rhd_1 r(x) + p(G) \rhd_1 r(1-g)\\
& = 1 - \alpha \beta + (\alpha \beta - 1)G\end{aligned}$$ if and only if $\alpha \beta = 1$, since $\varepsilon(X)r(x) = 0$. Thus, $p$ and $r$ are actually Hopf algebra isomorphisms inverse to one another. It is also a routine check to verify that $\psi^2
= {{\rm Id}\,}_{D(H_4)}$, for example by looking at the elements of a basis for $D(H_4)$. Therefore, $\psi$ is an isomorphism. We will denote the isomorphism obtained in this way and associated to $\alpha \in
k^*$ by $\psi_{\alpha}$.
Suppose now that $u$ and $v$ are non-trivial and $p$ and $r$ are trivial. Then conditions [(\[eq:C3\])]{} and [(\[eq:C6\])]{} imply that $u$ and $v$ are Hopf algebras morphisms, conditions [(\[eq:C7\])]{} and [(\[eq:C8\])]{} become $$v(h) \rhd_1 u(b) = u(h \rhd_1 b)$$ $$v(h) \lhd_1 u(b) = v(h \lhd_1 b)$$ for all $b \in \mathbb{H}_4$ and $h \in H_4$, while the rest of the conditions are trivially fulfilled. By [Lemma \[le:morf\_H\_4H\_4\]]{}, $u(X) = \alpha \, X$ and $v(x) = \beta \, x$ for some $\alpha$, $\beta \in k$. A straightforward verification shows that [(\[eq:C7\])]{} and [(\[eq:C8\])]{} are satisfied if and only if $\alpha
\cdot \beta = 1$. For example, condition [(\[eq:C7\])]{} for $b = X$ and $h = x$ is equivalent to $$\alpha \beta (1 - G) = 1 - G$$ which is true if and only if $\alpha \beta = 1$. If we denote by $\varphi_{\alpha}$ the morphism $\psi$ associated to $\alpha \in
k^*$ then we may remark that $\varphi_{\alpha}
\varphi_{\alpha^{-1}} = \varphi_{\alpha^{-1}} \varphi_{\alpha} =
{{\rm Id}\,}_{D(H_4)}$, hence $\varphi_{\alpha}$ is an isomorphism.
Thus, the set of Hopf automorphisms of $D(H_4)$ is ${{\rm Aut}\,}_{\rm
Hopf}(D(H_4)) = \{ \psi_{\alpha} \vert \alpha \in k^* \} \cup \{
\varphi_{\alpha} \vert \alpha \in k^* \}$, a disjoint union of two sets indexed by $k^*$. The elements of ${{\rm Aut}\,}_{\rm Hopf}(D(H_4))$ multiply, as it can easily be seen, according to the following rules $$\psi_{\alpha} \psi_{\beta} = \varphi_{\alpha^{-1} \beta}, \qquad
\varphi_{\alpha} \varphi_{\beta} = \varphi_{\alpha \beta}, \qquad
\psi_{\alpha} \varphi_{\beta} = \psi_{\alpha \beta}, \qquad
\varphi_{\alpha} \psi_{\beta} = \psi_{\alpha^{-1} \beta}$$ for all $\alpha$, $\beta \in k^*$. In order to see that ${{\rm Aut}\,}_{\rm
Hopf}(D(H_4))$ is isomorphic to the semidirect product of $k^*$ and $\mathbb{Z}_2$, $k^* \rtimes_f \mathbb{Z}_2$, associated to $f
: \mathbb{Z}_2 \to {{\rm Aut}\,}(k^*)$, $f (1 + 2\mathbb{Z}) (\alpha) =
\alpha^{-1}$, for all $\alpha \in k^*$, we define $$\Gamma : k^* \rtimes_f \mathbb{Z}_2 \to {{\rm Aut}\,}_{\rm Hopf}(D(H_4)),
\qquad \Gamma(\alpha, \hat{0}) = \varphi_{\alpha}, \qquad
\Gamma(\alpha, \hat{1}) = \psi_{\alpha^{-1}}$$ for all $\alpha \in k^*$. Obviously, $\Gamma$ is bijective and, moreover, it is a morphism of groups. Indeed, recalling that $k^*
\rtimes_f \mathbb{Z}_2$ is $k^* \times \mathbb{Z}_2$ with the multiplication $(\alpha, \hat{m}) \cdot (\beta, \hat{n}) = (\alpha
f(\hat{m})(\beta), \hat{m} + \hat{n})$, for all $\alpha$, $\beta
\in k^*$ and $\hat{m}$, $\hat{n} \in \mathbb{Z}_2$, we have $$\Gamma \left( (\alpha, \hat{0}) \cdot (\beta, \hat{0}) \right) =
\Gamma (\alpha \beta, \hat{0}) = \varphi_{\alpha \beta} =
\varphi_{\alpha} \varphi_{\beta} = \Gamma (\alpha, \hat{0}) \Gamma
(\beta, \hat{0})$$ $$\Gamma \left( (\alpha, \hat{0}) \cdot (\beta, \hat{1}) \right) =
\Gamma (\alpha \beta, \hat{1}) = \psi_{\alpha^{-1} \beta^{-1}} =
\varphi_{\alpha} \psi_{\beta^{-1}} = \Gamma (\alpha, \hat{0})
\Gamma (\beta, \hat{1})$$ $$\Gamma \left( (\alpha, \hat{1}) \cdot (\beta, \hat{0}) \right) =
\Gamma (\alpha \beta^{-1}, \hat{1}) = \psi_{\alpha^{-1} \beta} =
\psi_{\alpha^{-1}} \varphi_{\beta} = \Gamma (\alpha, \hat{1})
\Gamma (\beta, \hat{0})$$ $$\Gamma \left( (\alpha, \hat{1}) \cdot (\beta, \hat{1}) \right) =
\Gamma (\alpha \beta^{-1}, \hat{0}) = \varphi_{\alpha \beta^{-1}}
= \psi_{\alpha^{-1}} \psi_{\beta^{-1}} = \Gamma (\alpha, \hat{1})
\Gamma (\beta, \hat{1})$$ for all $\alpha$, $\beta \in k^*$. Therefore, ${{\rm Aut}\,}_{\rm
Hopf}(D(H_4)) \cong k^* \rtimes_f \mathbb{Z}_2$, which concludes the proof.
[\[pr:autH\_0\]]{} Let $k$ be a field of characteristic $\neq 2$. Then there exist isomorphism of groups $${{\rm Aut}\,}_{\rm Hopf}(\mathcal{H}_{16,0}) \cong (k^* \times k^*)
\rtimes_g \mathbb{Z}_2 \cong {{\rm Aut}\,}_{\rm Hopf}(\mathbb{H}_4 \otimes
H_4)$$ where $(k^* \times k^*) \rtimes_g \, \mathbb{Z}_2$ is the semidirect product of the group $k^* \times k^*$ with $\mathbb{Z}_2$ associated to the action as automorphisms $g :
\mathbb{Z}_2 \to {{\rm Aut}\,}(k^* \times k^*)$, $g (1 + 2\mathbb{Z})
(\alpha, \beta) = (\beta, \alpha)$, for all $(\alpha, \beta) \in
k^* \times k^*$.
The proof of this result is similar to the one of [Proposition \[pr:autDH\]]{}. We start with $\mathcal{H}_{16,0}$. Let $\psi : \mathcal{H}_{16,0}
\to \mathcal{H}_{16,0}$ be an endomorphism of $\mathcal{H}_{16,0}$, associated to $(u, p, r, v)$, where $u :
\mathbb{H}_4 \to \mathbb{H}_4$, $p : \mathbb{H}_4 \to H_4$, $r :
H_4 \to \mathbb{H}_4$, and $v : H_4 \to H_4$ are unitary coalgebra maps satisfying conditions [(\[eq:C1\])]{}-[(\[eq:C8\])]{}. We look for necessary and sufficient conditions for $\psi$ to be an isomorphism. One necessary condition, as we have seen in the proof of [Theorem \[th:clH\_4H\_4\]]{}, is that $u$ and $v$ be both trivial and $p$ and $r$ be both non-trivial, or $u$ and $v$ be both non-trivial and $p$ and $r$ be both trivial.
If $u$ and $v$ are trivial and $p$ and $r$ are non-trivial then conditions [(\[eq:C4\])]{} and [(\[eq:C5\])]{} imply that $p$ and $r$ are Hopf algebra morphisms, conditions [(\[eq:C7\])]{} and [(\[eq:C8\])]{} become $$\varepsilon(b)r(h) = p(h_{(1)} \rhd_0 b_{(1)}) \rhd_0 r(h_{(2)}
\lhd_0 b_{(2)})$$ $$\varepsilon(h)p(b) = p(h_{(1)} \rhd_0 b_{(1)}) \lhd_0 r(h_{(2)}
\lhd_0 b_{(2)})$$ for all $b \in \mathbb{H}_4$ and $h \in H_4$, and the rest of the conditions are trivially fulfilled. Since $p$ and $r$ are Hopf algebra morphisms there exist $\alpha$, $\beta \in k$ such that $p(X) = \alpha \, x$, $p(GX) = \alpha \, gx$, $r(x) = \beta \, X$ and $r(gx) = \beta \, GX$. A routine check shows that conditions [(\[eq:C7\])]{} and [(\[eq:C8\])]{} are satisfied without further restrictions on $\alpha$ and $\beta$. By looking at the matrix of $\psi$ with respect to the basis $\{ef \mid e = 1, G, X, GX,\,\, f
= 1, g, x, gx\}$, we find that $\psi$ is an isomorphism if and only if $\alpha \cdot \beta \neq 0$. We denote by $\psi_{\alpha,
\beta}$ the isomorphism obtained in this way and associated to $(\alpha, \beta) \in k^* \times k^*$.
If $u$ and $v$ are non-trivial and $p$ and $r$ are trivial then conditions [(\[eq:C3\])]{} and [(\[eq:C6\])]{} imply that $u$ and $v$ are Hopf algebras morphisms, conditions [(\[eq:C7\])]{} and [(\[eq:C8\])]{} become $$v(h) \rhd_0 u(b) = u(h \rhd_0 b)$$ $$v(h) \lhd_0 u(b) = v(h \lhd_0 b)$$ for all $b \in \mathbb{H}_4$ and $h \in H_4$, while the rest of the conditions are trivially fulfilled. By [Lemma \[le:morf\_H\_4H\_4\]]{}, $u(X) = \alpha \, X$ and $v(x) = \beta \, x$ for some $\alpha$, $\beta \in k$. Again, [(\[eq:C7\])]{} and [(\[eq:C8\])]{} are satisfied without further restrictions on $\alpha$ and $\beta$, and $\psi$ is an isomorphism if and only if $\alpha \cdot \beta \neq 0$. We denote by $\varphi_{\alpha, \beta}$ the isomorphism associated to $(\alpha, \beta) \in k^* \times k^*$ in this way.
Thus, the set of Hopf automorphisms of $\mathcal{H}_{16,0}$ is ${{\rm Aut}\,}_{\rm Hopf}(\mathcal{H}_{16,0}) = \{ \psi_{\alpha, \beta}
\vert (\alpha, \beta) \in k^* \times k^*\} \cup \{
\varphi_{\alpha, \beta} \vert (\alpha, \beta) \in k^* \times k^*
\}$, where the two sets are disjoint. The elements of ${{\rm Aut}\,}_{\rm
Hopf}(\mathcal{H}_{16,0})$ multiply, as it can easily be seen, according to the following rules $$\psi_{\alpha, \beta} \psi_{\gamma, \delta} = \varphi_{\beta
\gamma, \alpha \delta}, \quad \varphi_{\alpha, \beta}
\varphi_{\gamma, \delta} = \varphi_{\alpha \gamma, \beta \delta},
\quad \psi_{\alpha, \beta} \varphi_{\gamma, \delta} = \psi_{\alpha
\gamma, \beta \delta}, \quad \varphi_{\alpha, \beta} \psi_{\gamma,
\delta} = \psi_{\beta \gamma, \alpha \delta}$$ for all $\alpha$, $\beta$, $\gamma$, $\delta \in k^*$. An isomorphism with the semidirect product of $k^* \times k^*$ with $\mathbb{Z}_2$ associated to $g : \mathbb{Z}_2 \to {{\rm Aut}\,}(k^* \times
k^*)$, $f (1 + 2\mathbb{Z}) (\alpha, \beta) = (\beta, \alpha)$, for all $(\alpha, \beta) \in k^* \times k^*$, can be noticed, since, in $(k^* \times k^*) \rtimes_g \mathbb{Z}_2$, elements multiply in the following way $$\big( (\beta, \alpha), \hat{1} \big) \cdot \big( (\delta, \gamma),
\hat{1} \big) = \big((\beta \gamma, \alpha \delta), \hat{0} \big)
\qquad \big( (\alpha, \beta), \hat{0} \big) \cdot \big( (\gamma,
\delta), \hat{0} \big) = \big((\alpha \gamma, \beta \delta),
\hat{0} \big)$$ $$\big( (\beta, \alpha), \hat{1} \big) \cdot \big( (\gamma, \delta),
\hat{0} \big) = \big( (\beta \delta, \alpha \gamma), \hat{1} \big)
\qquad \big( (\alpha, \beta), \hat{0} \big) \cdot \big( (\delta,
\gamma), \hat{1} \big) = \big( (\alpha \delta, \beta \gamma),
\hat{1} \big)$$ for all $\alpha$, $\beta$, $\gamma$, $\delta \in k^*$. Thus, $
\Gamma : (k^* \times k^*) \rtimes_g \mathbb{Z}_2 \to {{\rm Aut}\,}_{\rm
Hopf}(\mathcal{H}_{16,0})$, $\Gamma \big( (\beta, \alpha), \hat{1}
\big) = \psi_{\alpha, \beta}$, $\Gamma \big( (\alpha, \beta),
\hat{0} \big) = \varphi_{\alpha, \beta}$, for all $\alpha$, $\beta
\in k^*$, is an isomorphism of groups.
The proof that ${{\rm Aut}\,}_{\rm Hopf}(\mathbb{H}_4 \otimes H_4) \cong
(k^* \times k^*) \rtimes_g \mathbb{Z}_2$ is just the same as for $\mathcal{H}_{16,0}$ so we omit it.
Acknowledgement
===============
The author wishes to thank Professor Gigel Militaru for suggesting the problem and for the help given in improving this paper.
[99]{}
Agore, A.L., Bontea, C.G., Militaru G. - Classifying bicrossed products of Hopf algebras, arXiv:1205.6110.
Agore, A.L., Militaru, G. - Schreier type theorems for bicrossed product, [*Cent. Eur. J. Math.*]{}, [**10**]{}(2012), 722-739.
Andruskiewitsch, N. and Devoto, J. - Extensions of Hopf algebras, [*Algebra i Analiz*]{} [**7**]{} (1995), 22–61.
Caenepeel, S., Dascalescu, S., Raianu, S. - Classifying pointed Hopf algebras of dimension 16, Communications in Algebra, 28(2), 541-568, 2000.
Garcia, G. A. and Vay, C. - Hopf algebras of dimension 16, [*Algebr. Represent. Theory*]{}, [**13**]{} (2010), 383–405.
Kassel, C. - Quantum groups, Graduate Texts in Mathematics 155. Springer-Verlag, New York, 1995.
Larson, R.G., Radford, D.E. - Semisimple Hopf Algebras, J. Algebra 171 (1995), no. 1, 5–-35.
Majid, S. - Physics for algebraists: non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction, [*J. Algebra*]{}, [**130**]{} (1990), 17–64.
Majid, S. - Foundations of quantum groups theory, Cambridge University Press, 1995.
Montgomery, S. - Hopf Algebras and Their Actions on Rings, CBMS Reg. Conf. Series 82, Providence, R.I., 1993.
Radford, D.E., - Minimal Quaitriangular Hopf Algebras, J. Algebra 157 (1993), 285-315.
Takeuchi, M. - Matched pairs of groups and bismash products of Hopf algebras, [*Comm. Algebra*]{} [**9**]{}(1981), 841–882.
[^1]: This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, grant no. 88/05.10.2011.
|
---
abstract: 'Statistical spoken dialogue systems have the attractive property of being able to be optimised from data via interactions with real users. However in the reinforcement learning paradigm the dialogue manager (agent) often requires significant time to explore the state-action space to learn to behave in a desirable manner. This is a critical issue when the system is trained on-line with real users where learning costs are expensive. Reward shaping is one promising technique for addressing these concerns. Here we examine three recurrent neural network (RNN) approaches for providing reward shaping information in addition to the primary (task-orientated) environmental feedback. These RNNs are trained on returns from dialogues generated by a simulated user and attempt to diffuse the overall evaluation of the dialogue back down to the turn level to guide the agent towards good behaviour faster. In both simulated and real user scenarios these RNNs are shown to increase policy learning speed. Importantly, they do not require prior knowledge of the user’s goal.'
author:
- |
Pei-Hao Su, David Vandyke, Milica Ga[š]{}ić,\
[**Nikola Mrk[š]{}ić, Tsung-Hsien Wen Steve Young**]{}\
Department of Engineering, University of Cambridge, Cambridge, UK\
[{phs26, djv27, mg436, nm480, thw28, sjy}@cam.ac.uk]{}
bibliography:
- 'sd2015.bib'
title: 'Reward Shaping with Recurrent Neural Networks for Speeding up On-Line Policy Learning in Spoken Dialogue Systems'
---
Introduction {#sec:intro}
============
Spoken dialogue systems (SDS) offer a natural way for people to interact with computers. With the ability to learn from data (interactions) statistical SDS can theoretically be created faster and with less man-hours than a comparable handcrafted rule based system. They have also been shown to perform better [@POMDP-review]. Central to this is the use of partially observable Markov decision processes (POMDP) to model dialogue, which inherently manage the uncertainty created by errors in speech recognition and semantic decoding [@POMDP_williams].
The dialogue manager is a core component of an SDS and largely determines the quality of interaction. Its behaviour is controlled by a *policy* which maps belief states to system actions (or distributions over sets of actions) and this policy is trained in a reinforcement learning framework [@RL] where rewards are received from the environment, the most informative of which occurs only at the dialogues conclusion, indicating task success or failure.
It is the sparseness of this environmental reward function which, by not providing any information at intermediate turns, requires exploration to traverse deeply many sub-optimal paths. This is a significant concern when training SDS on-line with real users where one wishes to minimise client exposure to sub-optimal system behaviour. In an effort to counter this problem, *reward shaping* [@RS_Andrew] introduces domain knowledge to provide earlier informative feedback to the agent (additional to the environmental feedback) for the purpose of biasing exploration for discovering optimal behaviour quicker. Reward shaping is briefly reviewed in Section \[sub:rs\].
In the context of SDS, have motivated the use of reward shaping via analogy to the ‘social signals’ naturally produced and interpreted throughout a human-human dialogue. This non-statistical reward shaping model used heuristic features for speeding up policy learning.
As an alternative, one may consider attempting to handcraft a finer grained environmental reward function. For example, proposed diffusing expert ratings of dialogues to the state transition level to produce a richer reward function. Policy convergence may occur faster in this altered POMDP and dialogues generated by a task based simulated user may also alleviate the need for expert ratings. However, unlike reward shaping, modifying the environmental reward function also modifies the resulting optimal policy.
We recently proposed convolutional and recurrent neural network (RNN) approaches for determining dialogue success. This was used to provide a reinforcement signal for learning on-line from real users without requiring any prior knowledge of the user’s task [@Su_2015]. Here we extend the RNN approach by introducing new training constraints in order to combine the merits of the above three works: (1) diffusing dialogue level ratings down to the turn level to (2) add reward shaping information for faster policy learning, whilst (3) not requiring prior task knowledge which is simply unavailable on-line.
In Section \[sec:models\] we briefly describe potential based reward shaping before introducing the RNNs we explore for producing reward shaping signals (basic RNN, long short-term memory (LSTM) and gated recurrent unit (GRU)). The features the RNNs use along with the training constraint and loss are also described. The experimental evaluation is then presented in Section \[sec:exp\]. Firstly, the estimation accuracy of the RNNs is assessed. The benefit of using the RNN for reward shaping in both simulated and real user scenarios is then also demonstrated. Finally, conclusions are presented in Section \[sec:conclusions\].
RNNs for Reward Shaping {#sec:models}
=======================
Reward Shaping {#sub:rs}
--------------
Reward shaping provides the system with an extra reward signal $F$ in addition to environmental reward $R$, making the system learn from the composite signal $R + F$. The shaping reward $F$ often encodes expert knowledge that complements the sparse signal $R$. Since the reward function defines the system’s objective, changing it may result in a different task. When the task is modelled as a fully observable Markov decision process (MDP), defined formal requirements on the shaping reward as a difference of any potential function $\phi$ on consecutive states $s$ and $s'$ which preserves the optimality of policies. Based on this property, further extended it to POMDP by proof and empirical experiments: $$%\vspace{-3mm}
F(b_t, a, b_{t+1}) = \gamma \phi(b_{t+1}) - \phi(b_{t})
\label{eqn:reward_shaping}$$ where $\gamma$ is the discount factor, $b_t$ the belief state at turn $t$, and $a$ the action leading $b_t$ to $b_{t+1}$.
However determining an appropriate potential function for an SDS is non-trivial. Rather than hand-crafting the function with heuristic knowledge, we propose using an RNN to predict proper values as in the following.
Recurrent Neural Network Models {#sec:rnns}
-------------------------------
![[RNN with three types of hidden units: basic, LSTM and GRU. The feature vectors ${\bf f}_t$ extracted at turns $t=1,\dots,T$ are labelled ${\bf f}_t$.]{}](fig/RNNmodels.png){width="75mm"}
\[fig:models\]
The RNN model is a subclass of neural network defined by the presence of feedback connections. The ability to succinctly retain history information makes it suitable for modelling sequential data. It has been successfully used for language modelling [@mikolov2011extensions] and spoken language understanding [@RNNonSLU].
However, observed that basic RNNs suffer from vanishing/exploding gradient problems that limit their capability of modelling long context dependencies. To address this, long short-term memory [@hochreiter1997long] and gated recurrent unit [@chung2014empirical] RNNs have been proposed. In this paper, all three types of RNN (basic/LSTM/GRU) are compared.
Reward Shaping with RNN Prediction {#sub:RNNRS}
----------------------------------
The role of the RNN is to solve the regression problem of predicting the scalar return of each completed dialogue. At every turn $t$, input feature $f_t$ are extracted from the belief/action pair and used to update the hidden layer $h_t$. From dialogues generated by a simulated user [@userSim] supervised training pairs are created which consist of the turn level sequence of these feature vectors $f_t$ along with the scalar dialogue return as scored by an objective measure of task completion. Whilst the RNN models are trained on dialogue level supervised targets, we hypothesise that their subsequent turn level predictions can guide policy exploration via acting as informative reward shaping potentials.
To encourage good turn level predictions, all three RNN variants are trained to predict the dialogue return not with the final output of the network, but with the constraint that their scalar outputs from each turn $t$ should sum to predict the return for the whole dialogue. This is shown in Figure \[fig:models\]. A mean-square-error (MSE) loss is used (see Appendix \[appendix\]). The trained RNNs are then used directly as the reward shaping potential function $\phi$, using the RNN scalar output at each turn.
The feature inputs $f_t$ for all RNNs consisted of the following sections: the real-valued belief state vector formed by concatenating the distributions over user discourse act, method and goal variables [@BUDS], one-hot encodings of the user and summary system actions, and the normalised turn number. This feature vector was extracted at every turn (system + user exchange).
Experiments {#sec:exp}
===========
Experimental Setup
------------------
In all experiments the Cambridge restaurant domain was used, which consists of approximately 150 venues each having 6 attributes (slots) of which 3 can be used by the system to constrain the search and the remaining 3 are informable properties once a database entity has been found.
The shared core components of the SDS in all experiments were a domain independent ASR, a confusion network (CNet) semantic input decoder [@CNET], the BUDS [@BUDS] belief state tracker that factorises the dialogue state using a dynamic Bayesian network and a template based natural language generator. All policies were trained by GP-SARSA [@GPRL] and the summary action space contains 20 actions. Per turn reward was set to -1 and final reward 20 for task success else 0.
With this ontology, the size of the full feature vector was 147. The turn number was expressed as a percentage of the maximum number of allowed turns, here 30. The one-hot user dialogue act encoding was formed by taking only the most likely user act estimated by the CNet decoder.
Neural Network Training {#sub:nnt}
-----------------------
![[RMSE of return prediction by using RNN/LSTM/GRU, trained on 18K and 1K dialogues and tested on sets *testA* and *testB* (see text).]{}](fig/RNNRS.png){width="80mm"}
\[fig:NN\_training\]
Here results of training the 3 RNNs on the simulated user dialogues are presented. Two training sets were used consisting of 18K and 1K dialogues to verify the model robustness. In all cases a separate validation set consisting of 1K dialogues was used for controlling overfitting. Training and validation sets were approximately balanced regarding objective success/failure labels and collected at a 15% semantic error rate (SER). Prediction results are shown in Figure \[fig:NN\_training\] on two test sets; *testA:* 1K dialogues, balanced regarding objective labels, at 15% SER and *testB:* containing 12K dialogues collected at SERs of $0,15,30$ and $45$ as the data occurred (*i.e.* with no balancing regarding labels).
Root-MSE (RMSE) results of predicting the dialogue return are depicted in Figure \[fig:NN\_training\]. The models with LSTM and GRU units achieved a slight improvement in most cases over the basic RNN. Notice that the model with GRU even reached comparable results when trained with 1K training data compared to 18K. The results from the 1K training set indicate that the model can be developed from limited data. This enables datasets to be created by human annotation, avoiding the need for a simulated user. The results on set *testB* also show that the models can perform well in situations with varying error rates as would be encountered in real operating environments. Note that the dataset could also be created from human’s annotation which avoids the need for a simulated user. We next examine the RNN-based reward shaping for policy training with a simulated user.
Policy Learning with Simulated User {#sub:sim}
-----------------------------------
Since the aim of reward shaping is to enhance policy learning speed, we focus on the first 1000 training dialogues. Figure \[fig:NN\_training\] shows that the GRU RNN attained slightly better performance than the other two RNN models, albeit with no statistical significance. Thus for clearer presentation of the policy training results we plot only the GRU results, using the model trained on 18K dialogues.
To show the effectiveness of using RNN with GRU for predicting reward shaping potentials, we compare it with the hand-crafted (HDC) method for reward shaping proposed by that requires prior knowledge of the user’s task, and a baseline policy using only the environmental reward. Figure \[fig:gp\_sim\] shows the learning curve of the reward for the three systems. After every 50 training iterations each system was tested with 1000 dialogues and averaged over 10 policies. The simulated user’s SER was set to 15%.
We see that reward shaping indeed provides the agent with more information, increasing the learning speed. Furthermore, our proposed RNN method further outperforms the hand-crafted system, whilst also being able to be applied on-line.
![[Policy training via simulated user with (GRU/HDC) and without (baseline) reward shaping. Standard errors are also shown.]{}](fig/RNNRS_simulation.png){width="80mm"}
\[fig:gp\_sim\]
Policy Learning with Human Users {#sub:real}
--------------------------------
Based on the above results, the same GRU model was selected for training a policy on-line with humans. Two systems were trained with users recruited via Amazon Mechanical Turk: a baseline was trained with only the environmental reward, and another system was trained with an additional shaping reward predicted by the proposed GRU. Learning began from a random policy in all cases.
Figure \[fig:gp\_mturk\] shows the on-line learning curve of the reward when training the systems with 400 dialogues. The moving average was calculated using a window of 100 dialogues and each result was averaged over three policies in order to reduce noise. It can be seen that by adding the RNN based shaping reward, the policy learnt quicker in the important initial phase of policy learning.
![[Learning curves of reward with standard errors during on-line policy optimisation for the baseline (black) and proposed (green) systems.]{}](fig/RNNRS_realuser.png){width="70mm"}
\[fig:gp\_mturk\]
Conclusions {#sec:conclusions}
===========
This paper has shown that RNN models can be trained to predict the dialogue return with a constraint such that subsequent turn level predictions act as good reward shaping signals that are effective for accelerating policy learning on-line with real users. As in many other applications, we found that gated RNNs such as LSTM and GRU perform a little better than basic RNNs.
In the work described here, the RNNs were trained using a simulated user and this simulator could have been used to bootstrap a policy for use with real users. However our supposition is that RNNs could be trained for reward prediction which are substantially domain independent and hence have wider applications via domain adaptation and extension [@MGdistribution2015; @polTransRS]. Testing this supposition will be the subject of future work.
Acknowledgements
================
Pei-Hao Su is supported by Cambridge Trust and the Ministry of Education, Taiwan. David Vandyke and Tsung-Hsien Wen are supported by Toshiba Research Europe Ltd, Cambridge Research Lab.
Training Constraint/Loss Function {#appendix}
=================================
For all RNN models the following MSE loss function is used on a per-dialogue basis: $$\mbox{MSE}=\left({R}-\sum_{t=1}^{T}r_{t}\right)^2$$ where the current dialogue has $T$ turns, ${R}$ is the return and training target, and $r_t$ is the scalar prediction output by the RNN model at each turn.
|
---
abstract: 'We rigorously justify the mean-field limit of an $N$-particle system subject to Brownian motions and interacting through the Newtonian potential in ${\mathbb{R}}^3$. Our result leads to a derivation of the Vlasov-Poisson-Fokker-Planck (VPFP) equations from the regularized microscopic $N$-particle system. More precisely, we show that the maximal distance between the exact microscopic trajectories and the mean-field trajectories is bounded by $N^{-\frac{1}{3}+\varepsilon}$ ($\frac{1}{63}\leq\varepsilon<\frac{1}{36}$) with a blob size of $N^{-\delta}$ ($\frac{1}{3}\leq\delta<\frac{19}{54}-\frac{2\varepsilon}{3}$) up to a probability of $1-N^{-\alpha}$ for any $\alpha>0$. Moreover, we prove the convergence rate between the empirical measure associated to the regularized particle system and the solution of the VPFP equations. The technical novelty of this paper is that our estimates rely on the randomness coming from the initial data and from the Brownian motions.'
author:
- 'Hui Huang[^1], Jian-Guo Liu[^2], Peter Pickl[^3] [^4]'
bibliography:
- 'meanfield.bib'
title: |
On the mean-field limit for the\
Vlasov-Poisson-Fokker-Planck system
---
[ Coupling method, propagation of chaos, concentration inequality, Wasserstein metric.]{}
Introduction
============
Systems of interacting particles are quite common in physics and biosciences, and they are usually formulated according to first principles (such as Newton’s second law). For instance, particles can represent galaxies in cosmological models [@aarseth2003gravitational], molecules in a fluid [@jabin2004identification], or ions and electrons in plasmas [@vlasov1968vibrational]. Such particle systems are also relevant as models for the collective behavior of certain animals like birds, fish, insects, and even micro-organisms (such as cells or bacteria) [@bolley2011stochastic; @carrillo2010particle; @motsch2011new]. In this paper, we are interested in the classical Newtonian dynamics of $N$ indistinguishable particles interacting through pair interaction forces and subject to Brownian noise. Denote by $x_i\in {\mathbb{R}}^3$ and $v_i\in{\mathbb{R}}^3$ the position and velocity of particle $i$. The evolution of the system is given by the following stochastic differential equations (SDEs), $$\label{IBM}
dx_i=v_idt,\quad dv_i=\frac{1}{N-1}\sum\limits_{j\neq i}^Nk(x_i-x_j)dt+\sqrt{2\sigma}dB_i,\quad i=1,\cdots,N,$$ where $k(x)$ models the pairwise interaction between the individuals, and $\{B_i\}_{i=1}^N$ are independent realizations of Brownian motions which count for extrinsic random perturbations such as random collisions against the background. In the presence of friction, model is known as the interacting Ornstein-Uhlenbeck model in the probability or statistical mechanics community. In particular, we refer readers to [@olla1991scaling; @tremoulet2002hydrodynamic] by Olla, Varadhan and Tremoulet for the scaling limit of the Ornstein-Uhlenbeck system. In this manuscript, we take the interaction kernel to be the Coulombian kernel $$\label{couke}
k(x) = a \frac{x}{|x|^{3} },$$ for some real number $a$. The case $a>0$ corresponds, for example, to the electrostatic (repulsive) interaction of charged particles in a plasma, while the case $a<0$ describes the attraction between massive particles subject to gravitation. We refer readers to [@jeans1915theory; @vlasov1968vibrational] for the original modelings.
Since the number $N$ of particles is large, it is extremely complicated to investigate the microscopic particle system directly. Fortunately, it can be studied through macroscopic descriptions of the system based on the probability density for the particles on phase space. These macroscopic descriptions are usually expressed as continuous partial differential equations (PDEs). The analysis of the scaling limit of the interacting particle system to the macroscopic continuum model is usually called the *mean-field limit*. For the second order particle system , it is expected to be approximated by the following Vlasov-Poisson-Fokker-Planck (VPFP) equations $$\label{vlasovoriginal}
\left\{
\begin{aligned}
&\partial_t f(x,v,t)+v\cdot\nabla_x f(x,v,t)+k\ast\rho(x,t)\cdot\nabla_v f(x,v,t)=\sigma\Delta_v f(x,v,t),\\
&f(x,v,0)=f_0(x,v),
\end{aligned}
\right.$$ where $f(x,v,t):~(x,v,t) \in{\mathbb{R}}^3\times{\mathbb{R}}^3\times[0,\infty)\rightarrow {\mathbb{R}}^+$ is the probability density function in the phase space $(x,v)$ at time $t$, and $$\label{chargeden}
\rho(x,t)=\int_{{\mathbb{R}}^3} f(x,v,t)dv,$$ is the charge density introduced by $f(x,v,t)$. We denote by $E(x,t):=k\ast\rho(x,t)$ the Coulombian or gravitational force field.
The intent of this research is to show the mean-field limit of the particle system towards the Vlasov-Poisson-Fokker-Planck equations . In particular, we quantify how close these descriptions are for a given $N$. Where $\sigma=0$ (there is no randomness coming from the noise), mean-field limit results for interacting particle systems with globally Lipschitz forces have been obtained by Braun and Hepp [@braun1977vlasov] and Dobrushin [@dobrushin1979vlasov]. Bolley, Cañizo and Carrilo [@bolley2011stochastic] presented an extension of the classical theory to the particle system with only locally Lipschitz interacting force. Such case concerning kernels $k\in W_{loc}^{1,\infty}$ are also used in the context of neuroscience [@bossy2015clarification; @touboul2014propagation]. The last few years have seen great progress in mean-field limits for singular forces by treating them with an $N$-dependent cut-off. In particular, Hauray and Jabin [@jabin2015particles] discussed mildly singular force kernels satisfying $|k(x)|\leq \frac{C}{|x|^\alpha}$ with $\alpha<d-1$ in dimensions $d\geq3$. For $1<\alpha<d-1$, they performed the mean-field limit for typical initial data, where they chose the cut-off to be $N^{-\frac{1}{2d}}$. For $\alpha< 1$, they prove molecular chaos without cut-off. Unfortunately, their method fails precisely at the Coulomb threshold when $\alpha= d-1$. More recently, Boers and Pickl [@boers2016mean] proposed a novel method for deriving mean-field equations with interaction forces scaling like $\frac{1}{|x|^{3\lambda-1}}$ $(5/6<\lambda<1)$, and they were able to obtain a cut-off as small as $ N^{-\frac{1}{d}}$. Furthermore, Lazarovici and Pickl [@lazarovici2015mean] extended the method in [@boers2016mean] to include the Coulomb singularity and they obtained a microscopic derivation of the Vlasov-Poisson equations with a cut-off of $N^{-\delta}$ $(0<\delta<\frac{1}{d})$. More recently, the cut-off parameter was reduced to as small as $N^{-\frac{7}{18}}$ in [@grass2019microscopic] by using the second order nature of the dynamics. Where $\sigma>0$, the random particle method for approximating the VPFP system with the Coulombian kernel was studied in [@havlak1996numerical], where the initial data was chose on a mesh and the cut-off parameter can be $N^{-\delta}$ $(0<\delta<\frac{1}{d})$. Most recently, Carrilo $et.al.$ [@carrillo2018propagation] also investigated the singular VPFP system but with the i.i.d. initial data, and obtained the propagation of chaos through a cut-off of $N^{-\delta}$ $(0<\delta<\frac{1}{d})$, which was a generalization of [@lazarovici2015mean]. We also note that Jabin and Wang [@jabin2016mean] rigorously justified the mean-field limit and propagation of chaos for the Vlasov systems with $L^\infty$ forces and vanishing viscosity ($\sigma_N\rightarrow 0$ as $N\rightarrow \infty$) by using a relative entropy method. Lastly, for a general overview of this topic we refer readers to [@carrillo2010particle; @jabin2014review; @jabin2017mean; @spohn2004dynamics].
When the interacting kernel $k$ is singular, it poses problems for both theory and numerical simulations. An easy remedy is to regularize the force with an $N$-dependent cut-off parameter and get $k^N$. The delicate question is how to choose this cut-off. On the one hand, the larger the cut-off is, the smoother $k^N$ will be and the easier it will be to show the convergence. However, the regularized system is not a good approximation of the actual system. On the other hand, the smaller the cut-off is, the closer $k^N$ is to the real $k$, thus the less information will be lost through the cut-off. Consequently, the necessary balance between accuracy (small cut-off) and regularity (large cut-off) is crucial. The analyses we reviewed above tried to justify that. In this manuscript, we set $\sigma>0$. Compared with the recent work [@carrillo2018propagation], the main technical innovation of this paper is that we fully use the randomness coming from the initial conditions [*and*]{} the Brownian motions to significantly improve the cut-off. Note that in [@carrillo2018propagation] the size of cut-off can be very close to but larger than $N^{-\frac{1}{d}}$. However we manage to reduce the cut-off size to be smaller than $N^{-\frac{1}{d}}$ (see Remark \[remark\]), which is a sort of average minimal distance between $N$ particles in dimension $d$. This manuscript significantly improves the ideas presented in [@garcia2017]. There the potential is split up into a more singular and less singular part. The less singular part is controlled in the usual manner while the mixing coming from the Brownian motion is used to estimate the more singular part. The technical innovation in the present paper is that the possible number of particles subject to the singular part of the interaction can be bounded due to the fact that the support of the singular part is small using a Law of Large Numbers argument. Again using the Law of Large Numbers based on the randomness coming from the Brownian motion, we show that the leading order of the singular part of the interaction can be replaced by its expectation value. This step is a key point of the present manuscript. The replacement by the expectation value, i.e. the integration of the force against the probability density, gives the regularization of the singular part and gives a significant improvement of our estimates. This is carried out in Lemma \[tildaXficedtime\], the proof of which can be found in section \[prolem\]. [@garcia2017] and the present paper are, to our knowledge, so far the only results where the mixing from the Brownian motion has been used in the derivation of a mean-field limit for an interacting many-body system.
As a companion of , some also consider the first order stochastic system $$\label{IBM1}
dx_i=\frac{1}{N-1}\sum\limits_{j\neq i}^Nk(x_i-x_j)dt+\sqrt{2\sigma}dB_i,\quad i=1,\cdots,N.$$ As before, one can expect that as the number of the particles $N$ goes to infinity we can get the continuous description of the dynamics as the following nonlinear PDE $$\partial_t f(x,t)+\nabla\cdot (f(k\ast f))=\sigma\Delta_xf\,,$$ where $f(x,t)$ is now the spatial density.
The particle system has many important applications. One of the best known classical applications is in fluid dynamics with the Biot-Savart kernel $$k(x)=\frac{1}{2\pi}(\frac{-x_2}{|x|^2},\frac{x_1}{|x|^2})\,.$$ It can be treated by the well-known vortex method introduced by Chorin in 1973 [@chorin1973numerical]. The convergence of the vortex method for two and three dimensional inviscid ($\sigma=0$) incompressible fluid flows was first proved by Hald $et\,al.$ [@hald; @HOH], Beale and Majda [@BM; @BM1]. When the effect of viscosity is involved ($\sigma>0$), the vortex method is replaced by the so called random vortex method by adding a Brownian motion to every vortex. The convergence analysis of the random vortex method for the Navier-Stokes equation was given by [@GJ; @LD; @MCPM; @OH] in the 1980s. For more recent results we refer to [@duerinckx2016mean; @fournier2014propagation; @jabin2018quantitative; @serfaty2018mean]. Another well-known application of the system is to choose the interaction to be the Poisson kernel $$k(x)=-C_d\frac{x}{|x|^d},\quad d\geq 2\,,$$ where $C_d>0$ and $k$ is set to be attractive. Now the system coincides with the particle models to approximate the classical Keller-Segel (KS) equation for chemotaxis [@keller1970initiation; @PCS]. We mainly refer to [@garcia2017; @fetecau2018propagation; @HH2; @HH1; @huilearning; @liu2016propagation; @YZ] for the mean-field limit of the KS system. Concerning the size of the cut-off, more specifically, [@liu2016propagation] chose the cut-off to be $(\ln N)^{-\frac{1}{d}}$, which was significantly improved in [@HH2], where the cut-off size can be as small as $N^{-\frac{1}{d(d+1)}}\log(N)$. In [@garcia2017; @HH1], the cut-off size was almost optimal, coming fairly close to $N^{-\frac{1}{d}}$. Many techniques used in this manuscript are adapted from these papers. For the Poisson-Nernst-Planck equation ($k$ is set to be repulsive), [@liu2016propagation] proved the mean-field limit without a cut-off.
The rest of the introduction will be split into three parts: We start with introducing the microscopic random particle system in Section \[intro1\]. Then we present some results on the existence of the macroscopic mean-field VPFP equations in Section \[intro2\]. Lastly, our main theorem will be stated in Section \[intro3\], where we prove the closeness of the approximation of solutions to VPFP equations by the microscopic system.
Microscopic random particle system {#intro1}
----------------------------------
We are interested in the time evolution of a system of $N$-interacting Newtonian particles with noise in the $N\to\infty$ limit. The motion of the system studied in this paper is described by trajectories on phase space, i.e. a time dependent $\Phi_t:\mathbb{R}\to\mathbb{R}^{6N}$. We use the notation $$\label{Phit}
\Phi_t:=\left(X_t,V_t\right):=\left(x^t_1,\ldots, x^t_N,v^t_1,\ldots v^t_N\right),$$ where $x^t_j$ stands for the position of the $j^{\text{th}}$ particle at time $t$ and $v^t_j$ stands for the velocity of the $j^{\text{th}}$ particle at time $t$. The system is a Newtonian system with a noise term coupled to the velocity, whose evolution is governed by a system of SDEs of the type $$\begin{aligned}
\label{eq:motion}
\left\{
\begin{aligned}
&d x_i^t=v_i^tdt,\quad i=1,\cdots,N\,,\\
&d v_i^t=\frac{1}{N-1}\sum_{j\neq i}^Nk(x_i^t-x_j^t)dt + \sqrt{2\sigma}dB_i^t\;,
\end{aligned}
\right.
\end{aligned}$$ where $k$ is the Coulomb kernel modeling interaction between particles and $B_i^t$ are independent realizations of Brownian motions. We regularize the kernel $k$ by a blob function $0\leq \psi(x)\in C^2({\mathbb{R}}^3)$, $\mbox{supp }\psi(x)\subseteq B(0,1)$ and $\int_{{\mathbb{R}}^3}\psi(x)dx=1$. Let $\psi_\delta^N=N^{3\delta}\psi(N^{\delta}x)$, then the Coulomb kernel with regularization has the form $$\label{eq:force}
k^N(x) = k \ast \psi_\delta^N.$$ Thus one has the regularized microscopic $N$-particle system for $i=1,2\cdots,N$ $$\begin{aligned}
\label{eq:regpar}
\left\{
\begin{aligned}
& d x_i^t=v_i^t dt,\\
&d v_i^t=\frac{1}{N-1}\sum_{i\neq j}^Nk^N(x_i^t-x_j^t)dt + \sqrt{2\sigma}dB_i^t\;.
\end{aligned}
\right.
\end{aligned}$$ Here the initial condition $\Phi_0$ of the system is independently, identically distributed (i.i.d.) with the common probability density given by $f_0$. And the corresponding regularized VPFP equations are $$\label{vlasov}
\left\{
\begin{aligned}
&\partial_t f^N(x,v,t)+v\cdot\nabla_x f^N(x,v,t)+k^N\ast\rho^N(x,t)\cdot\nabla_v f^N(x,v,t)=\sigma\Delta_v f^N(x,v,t),\\
&\rho^N(x,t)=\int_{{\mathbb{R}}^3}f^N(x,v,t)dv,\\
&f^N(x,v,0)=f_0(x,v).
\end{aligned}
\right.$$
Existence of classical solutions to the Vlasov-Poisson-Fokker-Planck system {#intro2}
---------------------------------------------------------------------------
The existence of weak and classical solutions to VPFP equations and related systems has been very well studied. Degond [@degond] first showed the existence of a global-in-time smooth solution for the Vlasov-Fokker-Planck equations in one and two space dimensions in the electrostatic case. Later on, Bouchut [@BO; @BO1] extended the result to three dimensions when the electric field was coupled through a Poisson equation, and the results were given in both the electrostatic and gravitational case. Also, Victory and O’Dwyer [@V-O-B] showed existence of classical solutions for VPFP equations when the spacial dimension is less than or equal to two, and local existence for all other dimensions. Then Bouchut in [@BO] proved the global existence of classical solutions for the VPFP system in dimension $d=3$. His proof relied on the techniques introduced by Lions and Perthame [@lions1991propagation] concerning the existence to the Vlasov-Poisson system in three dimensions. The long time behavior of the VPFP system was studied by Ono and Strauss [@ono2000regular], Carpio [@carpio1998long] and Carrillo $et\,al.$ [@soler1997asymptotic].
The existence results in [@V-O-B] and [@BO] are most appropriate for this work. We summarize them in the following theorem, which is also used in [@havlak1996numerical Theorem 2.1].
\[existence\] (Classical solutions of the VPFP equations) Let the initial data $0\leq f_0(x,v)$ satisfies the following properties:
a) $f_0\in W^{1,1}\cap W^{1,\infty}({\mathbb{R}}^6)$;
b) there exists a $m_0>6$, such that $$(1+|v|^2)^{\frac{m_0}{2}}f_0\in W^{1,\infty}({\mathbb{R}}^6)\,.$$
Then for any $T>0$, the VPFP equations admits a unique classical solution on $[0,T]$.
The proof of the above theorem given in [@V-O-B] and [@BO] indicates that the map $$t\rightarrow E(\cdot,t):=F\ast\rho(\cdot,t)\,,$$ is a continuous map from $[0,T]$ to $W^{1,\infty}({\mathbb{R}}^3)$. This implies that initial smooth data remains smooth for all time intervals $[0,T]$. So if we assume the initial data satisfies the following for any $k\geq 1$
a) $f_0\in W^{k,1}\cap W^{k,\infty}({\mathbb{R}}^6)$;
b) there exists a $m_0>6$, such that $$(1+|v|^2)^{\frac{m_0}{2}}f_0\in W^{k,1}\cap W^{k,\infty}({\mathbb{R}}^6)\,.$$
Then the unique classical solution $f$ maintains the regularity on $[0,T]$ for any $k\geq 1$: $$\max\limits_{0\leq t\leq T}{\lVert(1+|v|^2)^{\frac{m_0}{2}}f_t\rVert}_{W^{k,1}\cap W^{k,\infty}({\mathbb{R}}^6)}<\infty\,.$$
The present paper also needs the uniform-in-time $L^\infty$ bound of the charge density $\rho$: $$\label{Linfrho}
\max\limits_{0\leq t\leq T}{\lVert\rho(\cdot,t)\rVert}_{W^{1,\infty}({\mathbb{R}}^3)}<\infty\,,$$ which was obtained in [@pulvirenti2000infty] by means of the stochastic characteristic method under the assumption the $f_0$ is compactly supported in velocity. We also note that [@carrillo2018propagation] provided a proof of the local-in-time $L^\infty$ bound for $\rho$ by employing Feynman-Kac’s formula and assuming the initial data has polynomial decay.
In this paper, we assume that the initial data $f_0$ satisfies the following assumption:
\[assum\] The initial data $0\leq f_0(x,v)$ satisfies
1. $f_0\in W^{1,1}\cap W^{1,\infty}({\mathbb{R}}^6)$;
2. there exists a $m_0>6$, such that $$(1+|v|^2)^{\frac{m_0}{2}}f_0\in W^{1,1}\cap W^{1,\infty}({\mathbb{R}}^6);$$
3. $f_0(x,v)=0$ when $|v|>Q_v$.
The above assumption makes sure that we have the regularity needed for this article: for any $T>0$, $$\begin{aligned}
\label{regularity}
\max\limits_{t\in[0,T]}{\lVert\rho(\cdot,t)\rVert}_{W^{1,\infty}({\mathbb{R}}^3)}&+\max\limits_{0\leq t\leq T}{\lVert(1+|v|^2)^{\frac{m_0}{2}}f_t\rVert}_{W^{1,1}\cap W^{1,\infty}({\mathbb{R}}^6)}\leq C_{f_0},\end{aligned}$$ where $C_{f_0}$ depends only on $\|f_0\|_{W^{1,1}\cap W^{1,\infty}({\mathbb{R}}^6)}$, ${\lVert(1+|v|^2)^{\frac{m_0}{2}}f_0\rVert}_{W^{1,1}\cap W^{1,\infty}({\mathbb{R}}^6)}$ and $Q_v$. Note that the charge density $\rho$ satisfies $$\partial_t\rho(x,t)+\int_{{\mathbb{R}}^3}v\cdot \nabla_xf(x,v,t)dv=0\,.$$ Thus we have $$\begin{aligned}
\label{partialtrho}
\max\limits_{t\in[0,T]}{\lVert\partial_t\rho(\cdot,t)\rVert}_{L^{\infty}({\mathbb{R}}^3)}&\leq \max\limits_{t\in[0,T]}{\left\lVert\int_{{\mathbb{R}}^3}|v|| \nabla_xf(x,v,t)|dv\right\rVert}_{L^{\infty}({\mathbb{R}}^3)}\notag \\
&\leq \max\limits_{t\in[0,T]}{\lVert(1+|v|^2)^{\frac{m_0}{2}}f_t\rVert}_{W^{1,\infty}({\mathbb{R}}^6)}\int_{{\mathbb{R}}^3}\frac{|v|}{(1+|v|^2)^{\frac{m_0}{2}}}dv\leq C_{f_0}\,.\end{aligned}$$ We also note that equivalently one can estimate a bound for $f^N$ and $\rho^N$ uniformly in $N$.
The assumption that $f_0$ is compactly supported in the velocity variable is not required for the existence of the VPFP system. However it is used to get the $L^\infty$ bound of the charge density $\rho$ (see in [@pulvirenti2000infty]) and also in the proof of Lemma \[lmQ\] (see in ).
All our estimates below are also possible in the presence of sufficiently smooth external fields. Due to the fluctuation-dissipation principle it is more natural to add an external, velocity-dependent friction force to the system.
Statement of the main results {#intro3}
-----------------------------
Our objective is to derive the macroscopic mean-field PDE from the regularized microscopic particle system . We will do this by using probabilistic methods as in [@garcia2017; @HH1; @HH2; @lazarovici2015mean]. More precisely, we shall prove the convergence rate between the solution of VPFP equations and the empirical measure associated to the regularized particle system $\Phi_t$ satisfying . We assume that the initial condition $\Phi_0$ of the system is independently, identically distributed (i.i.d.) with the common probability density given by $f_0$.
Given the solution $f^N$ to the mean-field equation , we first construct an auxiliary trajectory $\Psi_t$ from . Then we prove the closeness between $\Phi_t$ and $\Psi_t$. For the auxiliary trajectory $$\label{Psi}
\Psi_t:=\left(\overline X_t,\overline V_t\right)=\left(\overline x^t_1,\ldots,\overline x^t_N,\overline v^t_1,\ldots \overline v^t_N\right),$$ we shall consider again a Newtonian system with noise, however, this time not subject to the pair interaction but under the influence of the external mean field $k^N\ast \rho^N(x,t)$ $$\begin{aligned}
\label{eq:mean}
\left\{
\begin{aligned}
&d\overline x_i^t=\overline v_i^t dt,\quad i=1,\cdots,N\,,\\
&d\overline v_i^t=\int_{{\mathbb{R}}^3} k^N(\overline x_i^t-x)\rho^N(x,t)dxdt+\sqrt{2\sigma}dB_i^t\,.
\end{aligned}
\right.\end{aligned}$$ Here we let $\Psi_t$ have the same initial condition as $\Phi_t$ (i.i.d. with the common density $f_0$). Since the particles are just $N$ identical copies of evolution, the independence is conserved. Therefore the $\Psi_t$ are distributed i.i.d. according to the common probability density $f^N$. We remark that the VPFP equation is Kolmogorov’s forward equation for any solution of , and in particular their probability density $f^N$ solves . This i.i.d. property will play a crucial role below, where we shall use the concentration inequality (see in Lemma \[central\]) on some functions depending on $\Psi_t$.
Our main result states that the $N$-particle trajectory $\Phi_t$ starting from $\Phi_0$ (i.i.d. with the common density $f_0$) remains close to the mean-field trajectory $\Psi_t$ with the same initial configuration $\Phi_0=\Psi_0$ during any finite time $[0,T]$. More precisely, we prove that the measure of the set where the maximal distance $\max\limits_{t\in[0,T]}{\lVert\Phi_t-\Psi_t\rVert}_\infty$ on $[0,T]$ exceeds $N^{-\lambda_2}$ decreases exponentially as the number $N$ of particles grows to infinity. Here the distance ${\lVert\Phi_t-\Psi_t\rVert}_\infty$ is measured by $$\label{distance}
{\lVert\Phi_t-\Psi_t\rVert}_\infty:=\sqrt{\log(N)}\|X_t-\overline X_t\|_\infty+ \|V_t-\overline V_t\|_\infty.$$
\[mainthm\] For any $T>0$, assume that trajectories $\Phi_t=(X_t,V_t)$, $\Psi_t=(\overline X_t,\overline V_t)$ satisfy and respectively with the initial data $\Phi_0=\Psi_0$, which is i.i.d. sharing the common density $f_0$ that satisfies Assumption \[assum\]. Then for any $\alpha>0$ and $0<\lambda_2<\frac{1}{3}$, there exists some $0<\lambda_1<\frac{\lambda_2}{3}$ and a $N_0\in \mathbb{N}$ which both depend only on $\alpha$, $T$ and $C_{f_0}$, such that for $N\geq N_0$, the following estimate holds with the cut-off index $\delta\in\left[\frac{1}{3},\min\left\{\frac{\lambda_1+3\lambda_2+1}{6},\frac{1-\lambda_2}{2}\right\}\right)$ $$\mathbb{P}\left(\max\limits_{t\in[0,T]}{\lVert\Phi_t-\Psi_t\rVert}_\infty \leq N^{-\lambda_2} \right)\geq 1-N^{-\alpha},$$ where ${\lVert\Phi_t-\Psi_t\rVert}_\infty$ is defined in .
\[remark\] In particular, for any $\frac{1}{63}\leq\varepsilon<\frac{1}{36}$, choosing $\lambda_2=\frac{1}{3}-\varepsilon$ and $\lambda_1=\frac{1}{9}-\varepsilon$, we have a convergence rate $N^{-\frac{1}{3}+\varepsilon}$ with a cut-off size of $N^{-\delta}$ $(\frac{1}{3}\leq\delta<\frac{19}{54}-\frac{2\varepsilon}{3})$. In other words, the cut-off parameter $\delta$ can be chosen very close to $\frac{19}{54}$ and in particular larger than $\frac{1}{3}$, which is a significant improvement over previous results in the literature.
**Strategy of the proof.** The strategy is to obtain a Gronwall-type inequality for $\max\limits_{t\in[0,T]}{\lVert\Phi_t-\Psi_t\rVert}_\infty $. Notice that $$\frac{d (V_t-\overline V_t)}{d t}=K^N(X_t)-\overline K^N(\overline X_t),$$ where $K^N(X_t)$ and $\overline K^N(\overline X_t)$ are defined as $$(K^N(X_t))_i:=\frac{1}{N-1}\sum_{j\neq i}k^N(x_i^t-x_j^t);\quad (\overline K^N(\overline X_t))_i:=\int_{{\mathbb{R}}^3} k^N(\overline x_i^t-x)\rho^N(x,t)dx.$$
One can compute $$\begin{aligned}
\label{strategy}
&\frac{d {\lVert\Phi_t-\Psi_t\rVert}_\infty}{dt} \leq\sqrt{\log(N)}{\left\|}V_t-\overline V_t{\right\|}_\infty+{\left\|}K^N(X_t)-\overline K^N(\overline X_t){\right\|}_\infty \notag \\
\leq &\sqrt{\log(N)}{\left\|}V_t-\overline V_t{\right\|}_\infty+{\left\|}K^N(X_t)- K^N(\overline X_t){\right\|}_\infty +{\left\|}K^N(\overline X_t)-\overline K^N(\overline X_t){\right\|}_\infty \,. \end{aligned}$$ If the force $k^N$ is Lipschitz continuous with a Lipschitz constant independent of $N$, the desired convergence follows easily [@braun1977vlasov; @dobrushin1979vlasov]. However the force considered here becomes singular as $N\rightarrow \infty$, hence it does not satisfy a uniform Lipschitz bound.
The first term in is already a sufficient bound in view of Gronwall’s Lemma.
$\bullet$ By the Law of Large Numbers, carried out in detail for our purpose here in Proposition \[propconsis\], we show for any $T>0$ $$\label{strategy1}
\max\limits_{t\in[0,T]}{\left\|}K^N(\overline X_t)-\overline K^N(\overline X_t){\right\|}_\infty \preceq CN^{2\delta-1}\log(N)\,,$$ where for convenience we abused the notation $a\preceq b$ to denote $a\leq b$ except for an event with probability approaching zero.
This direct error estimate can be seen as a consistency of the two evolutions in high probability.
$\bullet$ In Proposition \[propstab\], we show that the propagation of errors, coming from the second term in , is stable. This stability is important to be able to close the Gronwall argument. We show that for any $T>0$ $$\label{strategy2}
\| K^N(X_t)-K^N(\overline X_t) \|_\infty \preceq C\log(N){\left\|}X_t-\overline X_t{\right\|}_\infty+C\log^2(N)N^{-\lambda_3}\,, \mbox{ for all }t\in[0,T]\,,$$ holds under the condition that $$\label{stabcond}
\max\limits_{t\in[0,T]}{\lVert\Phi_t-\Psi_t\rVert}_\infty\preceq N^{-\lambda_2}\,.$$ Here it is crucial to ensure the constant $\lambda_3$ satisfies $2\delta-1\leq -\lambda_3 <-\lambda_2<0$. The function of this additional condition will be clear later (see Remark \[ATexplain\]).
To get this improvement of the cutoff parameter compared to previous results in the literature, we make use of the mixing caused by the Brownian motion. Therefore we split potential $K^N:=K_1^N+K_2^N$, where $K_2^N$ is chosen to have a wider cut-off of order $N^{-\lambda_2}>N^{-\delta}$. The less singular part $K_2^N$ is controlled in the usual manner [@boers2016mean; @HH1; @HH2; @lazarovici2015mean] (see in estimate ). $$\| K_2^N(X_t)-K_2^N(\overline X_t) \|_\infty \preceq C\log(N){\left\|}X_t-\overline X_t{\right\|}_\infty.$$
Thus we are left with the force $K_1^N$. We shall first estimate the number of particles that will be present in the support of $K_1^N$. Since the latter is small, this number will always be very small compared to $N$.
Under the condition we can not track the particles of the Newtonian time evolution with an accuracy larger than $N^{-\lambda_2}$. Thus – without using the Brownian motion – we have to assume the worst case scenario, which is all particles giving the maximal possible solution to the force, i.e. sitting close to the edge of the cutoff region, and all forces summing up, i.e. all particles sitting on top of each other.
But the Brownian motion in our system will lead to mixing. For a short time interval the effect of mixing will be much larger than the effect of the correlations coming from the pair interaction, and we can make use of the independence of the Brownian motions. This mixing, which happens on a larger spacial scale than the range of the potential, causes the particles to be distributed roughly equally over the support of the interaction resulting in a cancellation of the leading order of $K_1^N$.
It follows for the more singular part $K_1^N$ that $$\| K_1^N(X_t)-K_1^N(\overline X_t) \|_\infty\preceq C\log^2(N)N^{-\lambda_3}\,,$$
This is mainly carried out in Section \[prolem\] (the proof of Lemma \[tildaXficedtime\]).
$\bullet$ Combining consistency and stability , we conclude that for any $0< T_1\leq T$ $$\frac{d {\lVert\Phi_t-\Psi_t\rVert}_\infty}{dt} \preceq C\log(N){\left\|}X_t-\overline X_t{\right\|}_\infty+C\log^2(N)N^{-\lambda_3}\,, \mbox{ for all }t\in(0,T_1]\,,$$ holds provided that $$\max\limits_{t\in[0,T_1]}{\lVert\Phi_t-\Psi_t\rVert}_\infty\preceq N^{-\lambda_2}\,,$$ where $-\lambda_3<-\lambda_2<0$. This implies a generalized Gronwall’s inequality (see Lemma \[lmprior\]), which leads to $$\max\limits_{t\in[0,T]}{\lVert\Phi_t-\Psi_t\rVert}_\infty\preceq N^{-\lambda_2}\,.$$ Hence it completes our proof.
To quantify the convergence of probability measures, we give a brief introduction on the topology of the $p$-Wasserstein space. In the context of kinetic equations, it was first introduced by Dobrushin [@dobrushin1979vlasov]. Consider the following probability space with finite $p$-th moment: $$\mathcal{P}_p(\mathbb{R}^{d})=\big\{\mu|~\mu \mbox{ is a probability measure on } \mathbb{R}^{d} \mbox{ and } \int_{\mathbb{R}^{d}}|x|^p d\mu(x)<+\infty\big\}.$$ We denote the Monge-Kantorovich-Wasserstein distance in $\mathcal{P}_p(\mathbb{R}^{d})$ as follows $$W_p^p(\mu,\nu)=\inf_{\pi\in\Lambda(\mu,~\nu)}\Big\{\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|x-y|^pd\pi(x,y)\Big\}=\inf_{X\sim \mu,Y\sim \nu}\Big\{{\mathbb{E}}[|X-Y|^p]\Big\},$$ where $\Lambda(\mu,~\nu)$ is the set of joint probability measures on ${\mathbb{R}^{d}}\times{\mathbb{R}^{d}}$ with marginals $\mu$ and $\nu$ respectively and $(X,Y)$ are all possible couples of random variables with $\mu$ and $\nu$ as respective laws. For notational simplicity, the notation for a probability measure and its probability density is often abused. So if $\mu,\nu$ have densities $\rho_1,\rho_2$ respectively, we also denote the distance as $W_p^p(\rho_1,\rho_2)$. For further details, we refer the reader to the book of Villani [@villani2008optimal].
Following the same argument as [@lazarovici2015mean Corollary 4.3], Theorem \[mainthm\] implies molecular chaos in the following sense:
For any $T>0$, let $F_0^N:=\otimes^N f_0$ and $F_t^N$ be the $N$-particle distribution evolving with the microscopic flow starting from $F_0^N$. Then the $k$-particle marginal $$^{(k)}F_t^N(z_1,\cdots,z_k):=\int F_t^N(Z)dz_{k+1}\cdots dz_{N}$$ converges weakly to $\otimes^kf_t$ as $N\rightarrow\infty$ for all $k\in N$, where $f_t$ is the unique solution of the VPFP equations with $f_t|_{t=0}= f_0$. More precisely, under the assumptions of Theorem \[mainthm\], for any $\alpha>0$, there exists some constants $C>0$ and $N_0>0$ depending only on $\alpha$, $T$ and $C_{f_0}$, such that for $N\geq N_0$, the following estimate holds $$W_1\left(^{(k)}F_t^N,\otimes^kf_t\right)\leq k\exp\left(TC\sqrt{\log (N)}\right)N^{-\lambda_2},\quad\forall\,~ 0\leq t\leq T,$$ where $\lambda_2$ is used in Theorem \[mainthm\].
Another result from Theorem \[mainthm\] is the derivation of the macroscopic mean-field VPFP equations from the microscopic random particle system . We define the empirical measure associated to the microscopic $N$-particle systems and respectively as $$\label{empirical}
\mu_\Phi(t):=\frac{1}{N}\sum_{i=1}^{N}\delta(x-x_i^t)\delta(v-v_i^t),\quad \mu_\Psi(t):=\frac{1}{N}\sum_{i=1}^{N}\delta(x-\overline x_i^t)\delta(v-\overline v_i^t).$$
The following theorem shows that under additional moment control assumptions on $f_0$, the empirical measure $\mu_\Phi(t)$ converges to the solution of VPFP equations in $W_p$ distance with high probability.
\[cor\] Under the same assumption as in Theorem \[mainthm\], let $f_t$ be the unique solution to the VPFP equations with the initial data satisfying Assumption \[assum\] and $\mu_\Phi(t)$ be the empirical measure defined in with $\Phi_t$ being the particle flow solving . Let $p\in[1,\infty)$ and assume that there exists $m>2p$ such that $\iint_{{\mathbb{R}}^6}|x|^mf_0(x,v)dxdv<+\infty$. Then for any $T>0$ and $\kappa<\min\{\frac{1}{6},\frac{1}{2p},\delta\}$, there exists a constant $C_1$ depending only on $T$ and $C_{f_0}$ and constants $C_2$, $C_3$ depending only on $m$, $p$, $\kappa$, such that for all $N\geq e^{\left(\frac{C_1}{1-3\lambda_2}\right)^2}$ it holds that $$\begin{aligned}
{\mathbb{P}}\bigg(\max\limits_{t\in[0,T]}W_p(f_t,\mu_\Phi(t))&\leq N^{-\kappa+1-3\lambda_2}+N^{-\lambda_2}\bigg)\notag\\
&\geq 1-C_2\left(e^{-C_3N^{1-\max\{6,2p\}\kappa}}+N^{1-\frac{m}{2p}}\right).\end{aligned}$$ where $\delta$ and $\lambda_2$ are used in Theorem \[mainthm\].
This theorem provides a derivation of the VPFP equations from an interacting $N$-particle system, bridging the gap between the microscopic descriptions in terms of agent based models and macroscopic or hydrodynamic descriptions for the particle probability density.
Preliminaries
=============
In this section we collect the technical lemmas that are used in the proofs of the main theorems. Throughout this manuscript, generic constants will be denoted generically by $C$ (independent of $N$), even if they are different from line to line. We use ${\lVert\cdot\rVert}_p$ for the $L^p$ ($1\leq p\leq \infty$) norm of a function. Moreover if $v=(v_1,\cdots,v_N)$ is a vector, then ${\lVertv\rVert}_\infty:=\max\limits_{i=1,\cdots,N}|v_i|$.
Local Lipschitz bound
---------------------
First let us recall some estimates of the regularized kernel $k^N$ defined in :
\[lmkenerl\](Regularity of $k^N$)
(i) $k^N(0)=0$, $k^N(x)=k(x)$, for any $|x|\geq N^{-\delta}$ and $|k^N(x)|\leq |k(x)|$, $x\in\mathbb{R}^3$;
(ii) $|\partial^\beta k^N(x)|\leq CN^{(2+|\beta|)\delta},\mbox{for
any } x\in\mathbb{R}^3$;
(iii) $\|k^N\|_{2}\leq CN^{\frac{\delta}{2}}.$
The estimate $(i)$ has been proved in [@RY Lemma 2.1] and $(ii)$ follows from [@BM Lemma 5.1]. As for $(iii)$, it is a direct result of Young’s inequality.
Next we define a cut-off function $\ell^N$, which will provide the local Lipschitz bound for $k^N$.
\[defLN\] Let $$\label{lN}
\ell^N(x)=\left\{
\begin{aligned}
&\frac{6^3}{|x|^{3}}, && \text{ if } |x|\geq 6N^{-\delta},\\
&N^{3\delta}, && \text{ else },
\end{aligned}
\right.$$ and $L^N: {\mathbb{R}}^{3N}\rightarrow {\mathbb{R}}^N$ be defined by $(L^N(X_t))_i:=\frac{1}{N-1}\sum\limits_{i\neq j}\ell^N(x_i^t-x_j^t)$. Furthermore, we define $\overline L^N(\overline X_t)$ by $( \overline
L^N(\overline X_t))_i:=\int_{{\mathbb{R}}^3} \ell^N(\overline
x_i^t-x)\rho^N(x,t)dx$.
We summarize our first observation of $k^N$ and $\ell^N$ in the following lemma:
\[lmmid\] There is a constant $C>0$ independent of $N$ such that for all $x,y\in\mathbb{R}^3$ with $|x-y|\leq N^{-\lambda_2}\gg N^{-\delta}$ $(\lambda_2<\delta)$ the following holds: $$\begin{aligned}
\frac{\left|\nabla k^N(x)\right|}{\ell^N(y)}\leq CN^{3(\delta-\lambda_2)},
\end{aligned}$$ where $k^N$ is the regularization of the Coulomb kernel and $\ell^N$ satisfies Definition \[defLN\].
Let us first consider the case $|y|<2 N^ {-\lambda_2}$. It follows from the bound from Lemma \[lmkenerl\] and the decrease of $\ell^ N$ that $$\begin{aligned}
\label{case1}
\frac{\left|\nabla k^N(x)\right|}{\ell^N(y)}\leq \frac{N^{3\delta}}{\ell^
N(2 N^ {-\lambda_2})}=CN^{3(\delta-\lambda_2)},
\end{aligned}$$ where we used $2N^{-\lambda_2}>6N^ {-\delta}$, thus $\ell^ N( 2N^
{-\lambda_2})=27N^{3\lambda_2}$.
Next we consider the case $|y|\geq 2 N^ {-\lambda_2}$. It follows that $|x|\geq N^ {-\lambda_2}$ and thus by Lemma \[lmkenerl\] (i)
$$\begin{aligned}
\label{case2}
\frac{\left|\nabla k^N(x)\right|}{\ell^N(y)}\leq \frac{C|x|^{-3}}{|y|^{-3}}\leq C \frac{(|y|-N^
{-\lambda_2})^{-3} }{|y|^{-3}}\leq C,
\end{aligned}$$
where in the last step we used $|x|\geq (|y|-N^
{-\lambda_2})\geq\frac{|y|}{2}$ for $|y|\geq 2N^ {-\lambda_2}$. Collecting and finishes the proof.
Recall the notations $$\label{FN}
(K^N(X_t))_i:=\frac{1}{N-1}\sum_{j\neq i}k^N(x_i^t-x_j^t),\quad (K^N(\overline X_t))_i:=\frac{1}{N-1}\sum_{j\neq i}k^N(\overline x_i^t-\overline x_j^t),$$ and we have the local Lipschitz continuity of $K^N$:
\[lmlip\] If $\|X_t-\overline X_t\|_\infty\leq 2N^{-\delta}$, then it holds that $$\label{lmlipeq}
{\left\|}K^N(X_t)-K^N(\overline X_t){\right\|}_\infty\leq C\|L^N(\overline X_t)\|_\infty\|X_t-\overline X_t\|_\infty,$$ for some $C>0$ independent of $N$.
For any $\xi\in{\mathbb{R}}^3$ with $|\xi|<4N^{-\delta}$, we claim that $$\label{lemcla}
|k^N(x+\xi)-k^N(x)|\leq C\ell^N(x)|\xi|,$$ where $\ell^N(x)$ is defined in . Indeed, for $|x|<6N^{-\delta}$, estimate holds due to the fact that ${\lVert\nabla k^N\rVert}_\infty\leq N^{3\delta}$. For $|x|\geq 6N^{-\delta}$, there exists $s\in[0,1]$ such that $$|k^N(x+\xi)-k^N(x)|\leq |\nabla k^N(x+s\xi)||\xi|,$$ where $$|\nabla k^N(x+s\xi)|\leq C|x+s\xi|^{-3}.$$ The right hand side of the above expression takes its largest value when $s=1$ and $$|x+s\xi|^{-3}\leq |x(1-\frac{|\xi|}{|x|})|^{-3}.$$ Since $|\xi|<4N^{-\delta}$ and $|x|\geq 6N^{-\delta}$, it follows that $\frac{|\xi|}{|x|}<\frac{2}{3}$. Therefore, we get $$|k^N(x+\xi)-k^N(x)|\leq C\left(\frac{3}{|x|}\right)^2|\xi|\leq C\frac{|\xi|}{|x|^3}.$$
Applying claim one has $$\begin{aligned}
|(K^N(X_t))_i-(K^N(\overline X_t))_i|&\leq \frac{1}{N-1}\sum\limits_{j\neq i}^{N}| k^N(x_i^t-x_j^t)-k^N(\overline x_i^t-\overline x_j^t)|\notag\\
&\leq \frac{1}{N-1}\sum\limits_{j\neq i}^{N}C\ell^N(\overline x_i^t-\overline x_j^t)|x_i^t-x_j^t-\overline x_i^t+\overline x_j^t|\notag \\
&\leq C( L^N(\overline X_t))_i\|X_t-\overline X_t\|_\infty\leq C\|L^N(\overline X_t)\|_\infty\|X_t-\overline X_t\|_\infty,
\end{aligned}$$ which leads to .
The following observations of $k^N$ and $\ell^N$ turn out to be very helpful in the sequel:
\[converse\] Let $\ell^N(x)$ be defined in Definition \[defLN\] and $\rho \in W^{1,1}\cap W^{1,\infty} ({\mathbb{R}}^3)$. Then there exists a constant $C>0$ independent of $N$ such that $$\label{con1}
{\lVert\ell^N\ast \rho\rVert}_\infty\leq C\log(N)({\lVert\rho\rVert}_1+{\lVert\rho\rVert}_\infty),\quad{\lVert(\ell^N)^2\ast \rho\rVert}_\infty\leq CN^{(3\delta)}({\lVert\rho\rVert}_1+{\lVert\rho\rVert}_\infty);$$ and $$\label{con2}
{\lVert k^N\ast \rho\rVert}_\infty\leq C({\lVert\rho\rVert}_1+{\lVert\rho\rVert}_\infty),\quad {\lVert\nabla k^N\ast \rho\rVert}_\infty\leq C({\lVert\nabla\rho\rVert}_1+{\lVert\nabla\rho\rVert}_\infty).$$
We only prove one of the estimates above, since all the estimates can be obtained through the same procedure. One can estimate $$\begin{aligned}
&{\lVert\ell^N\ast\rho\rVert}_\infty={\left\lVert\int_{{\mathbb{R}}^3}\ell^N(x-y)\rho(y)dy\right\rVert}_\infty\notag \\
\leq &{\left\lVert\int_{|x-y|<6N^{-\delta}}\ell^N(x-y)\rho(y)dy\right\rVert}_\infty+{\left\lVert\int_{6N^{-\delta}
\leq|x-y|\leq 1}\ell^N(x-y)\rho(y)dy\right\rVert}_\infty\notag\\
&+{\left\lVert\int_{1
\leq|x-y|}\ell^N(x-y)\rho(y)dy\right\rVert}_\infty.
\end{aligned}$$ We estimate the first term $$\begin{aligned}
\label{ter1}
{\left\lVert\int_{|x-y|<6N^{-\delta}}\ell^N(x-y)\rho(y)dy\right\rVert}_\infty\leq {\lVert\rho\rVert}_\infty{\lVert\ell^N\rVert}_\infty|B(6N^{-\delta})|\leq \frac{4\pi}{3}(6N^{-\delta})^3N^{3\delta}{\lVert\rho\rVert}_\infty\leq C{\lVert\rho\rVert}_\infty,\end{aligned}$$ where $B(r)$ denotes the ball with radius $r$ in ${\mathbb{R}}^3$. The second term is bounded by $$\begin{aligned}
\label{ter2}
{\left\lVert\int_{6N^{-\delta}\leq|x-y|\leq 1}\ell^N(x-y)\rho(y)dy\right\rVert}_\infty\leq {\lVert\rho\rVert}_\infty\int_{6N^{-\delta}\leq|y|\leq 1}\frac{C}{|y|^3}dy\leq C\log(N){\lVert\rho\rVert}_\infty.\end{aligned}$$ It is easy to compute the last term $$\label{ter3}
{\left\lVert\int_{1
\leq|x-y|}\ell^N(x-y)\rho(y)dy\right\rVert}_\infty\leq C{\lVert\rho\rVert}_1.$$ Collecting estimates , and , one has $${\lVert\ell^N\ast\rho\rVert}_\infty\leq C{\lVert\rho\rVert}_{\infty}+C\log(N){\lVert\rho\rVert}_\infty+C{\lVert\rho\rVert}_1\leq C\log(N)({\lVert\rho\rVert}_\infty+{\lVert\rho\rVert}_1).$$
Law of Large Numbers
--------------------
Also, we need the following concentration inequality to provide us the probability bounds of random variables:
\[central\] Let $Z_1,\cdots,Z_N$ be $i.i.d.$ random variables with $\mathbb{E}[Z_i]=0,$ $\mathbb{E}[Z_i^2]\leq g(N)$ and $|Z_i|\leq C\sqrt{Ng(N)}$. Then for any $\alpha>0$, the sample mean $\bar{Z}=\frac{1}{N}\sum_{i=1}^{N}Z_i$ satisfies $${\mathbb{P}}\left(|\bar{Z}|\geq\frac{C_\alpha \sqrt{g(N)}\log(N)}{\sqrt{N}}\right)\leq N^{-\alpha},$$ where $C_\alpha$ depends only on $C$ and $\alpha$.
The proof can be seen in [@GJ Lemma 1], which is a direct result of Taylor’s expansion and Markov’s inequality.
Recall the notation $$\label{barFN}
(\overline K^N(\overline X_t))_i:=\int_{{\mathbb{R}}^3} k^N(\overline x_i^t-x)\rho^N(x,t)dx.$$ We can introduce the following version of the Law of Large Numbers:
\[lmlarge\] At any fixed time $t\in[0,T]$, suppose that $\overline X_t$ satisfies the mean-field dynamics , $K^N$ and $\overline K^N$ are defined in and respectively, $L^N$ and $\overline L^N$ are introduced in Definition \[defLN\]. For any $\alpha>0$ and $\frac{1}{3}\leq\delta<1$, there exist a constant $C_{1,\alpha}>0$ depending only on $\alpha$, $T$ and $C_{f_0}$ such that $$\label{largef}
{\mathbb{P}}\left({\left\|}K^N(\overline X_t)-\overline K^N(\overline X_t){\right\|}_\infty\geq C_{1,\alpha} N^{2\delta-1}\log (N)\right)\leq N^{-\alpha},$$ and $$\label{largel}
{\mathbb{P}}\left({\left\|}L^N(\overline X_{t})-\overline{L}^N(\overline X_{t}){\right\|}_\infty\geq C_{1,\alpha} N^{3\delta-1}\log (N)\right)\leq N^{-\alpha}.$$
We can prove this lemma by using Lemma \[central\]. Due to the exchangeability of the particles, we are ready to bound $$(K^N(\overline X_t))_1-(\overline K^N(\overline X_t))_1=\frac{1}{N-1}\sum_{j=2}^Nk^N(\overline x_1^t-\overline x_j^t)-\int_{{\mathbb{R}}^3} k^N(\overline x_1^t-x)\rho^N(x,t)dx=\frac{1}{N-1}\sum_{j=2}^{N}Z_j,$$ where $$Z_j:=k^N(\overline x_1^t-\overline x_j^t)-\int_{{\mathbb{R}}^3} k^N(\overline x_1^t-x)\rho^N(x,t)dx.$$ Since $\overline x_1^t$ and $\overline x_j^t$ are independent when $j\neq 1$ and $k^N(0)=0$, let us consider $\overline x_1^t$ as given and denote $\mathbb{E'}[\cdot]=\mathbb{E}[\cdot|\overline x_1^t]$. It is easy to show that $\mathbb{E}'[Z_j]=0$ since $$\begin{aligned}
\mathbb{E}'\left[k^N(\overline x_1^t-\overline x_j^t)\right]&=\iint_{{\mathbb{R}}^6} k^N(\overline x_1^t-x)f^N(x,v,t)dxdv\notag \\
&=\int_{{\mathbb{R}}^3} k^N(\overline x_1^t-x)\rho^N(x,t)dx.
\end{aligned}$$
To use Lemma \[central\], we need a bound for the variance $$\mathbb{E}'\big[|Z_j|^2\big]=\mathbb{E}'\left[\left|k^N(\overline x_1^t-\overline x_j^t)-\int_{{\mathbb{R}}^3} k^N(\overline x_1^t-x)\rho^N(x,t)dx\right|^2\right].$$ Since it follows from Lemma \[converse\] that $$\int_{{\mathbb{R}}^3} k^N(\overline x_1^t-x)\rho^N(x,t)dx\leq C({\lVert\rho^N\rVert}_1+{\lVert\rho^N\rVert}_\infty),$$ it suffices to bound $$\mathbb{E'}\big[k^N(\overline x_1^t-\overline x_j^t)\big]=\int_{{\mathbb{R}}^3} k^N(\overline x_1^t-x)\rho^N(x,t)dx\leq C({\lVert\rho^N\rVert}_1+{\lVert\rho^N\rVert}_\infty)\leq C(T,C_{f_0}),$$ and $$\mathbb{E'}\big[k^N(\overline x_1^t-\overline x_j^t)^2\big]=\int_{{\mathbb{R}}^3} k^N(\overline x_1^t-x)^2\rho^N(x,t)dx\leq {\lVert\rho^N\rVert}_\infty{\lVertk^N\rVert}_2^2\leq C(T,C_{f_0})N^{\delta},$$ where we have used ${\lVertk^N\rVert}_2\leq CN^{\frac{\delta}{2}}$ in Lemma \[lmkenerl\] $(iii)$. Hence one has $$\label{Esqure}
\mathbb{E}'\big[|Z_j|^2\big]\leq CN^{\delta}.$$
So the hypotheses of Lemma \[central\] are satisfied with $g(N)=CN^{4\delta-1}$. In addition, it follows from $(ii)$ in Lemma \[lmkenerl\] that $|Z_j|\leq CN^{2\delta}\leq C\sqrt{Ng(N)}$. Hence, using Lemma \[central\], we have the probability bound $${\mathbb{P}}\left(\left|(K^N(\overline X_t))_1-(\overline K^N(\overline X_t))_1\right|\geq C(\alpha,T,C_{f_0}) N^{2\delta-1}\log (N)\right)\leq N^{-\alpha}.$$ Similarly, the same bound also holds for all other indexes $i=2,\cdots,N$, which leads to $$\label{residual1'}
{\mathbb{P}}\left({\left\|}K^N(\overline X_t)-\overline K^N(\overline X_t){\right\|}_\infty\geq C(\alpha,T,C_{f_0}) N^{2\delta-1}\log (N)\right)\leq N^{1-\alpha}.$$ Let $C_{1,\alpha}$ be the constant $C(\alpha,T,C_{f_0}) $ in , then we conclude .
To prove , we follow the same procedure as above $$(L^N(\overline X_t))_1-(\overline L^N(\overline X_t))_1=\frac{1}{N-1}\sum_{j=2}^N\ell^N(\overline x_1^t-\overline x_j^t)-\int_{{\mathbb{R}}^3} \ell^N(\overline x_1^t-x)\rho^N(x,t)dx=\frac{1}{N-1}\sum_{j=2}^{N}Z_j,$$ where $$Z_j=\ell^N(\overline x_1^t-\overline x_j^t)-\int_{{\mathbb{R}}^3} \ell^N(\overline x_1^t-x)\rho^N(x,t)dx.$$ It is easy to show that $\mathbb{E}'[Z_j]=0$. To use Lemma \[central\], we need a bound for the variance. One computes that $$\mathbb{E'}\big[\ell^N(\overline x_1^t-\overline x_j^t)\big]=\int_{{\mathbb{R}}^3} \ell^N(\overline x_1^t-x)\rho^N(x,t)dx\leq C\log(N)({\lVert\rho\rVert}_1+{\lVert\rho\rVert}_\infty)\leq C(T,C_{f_0})\log(N),$$ and $$\mathbb{E'}\big[\ell^N(\overline x_1^t-\overline x_j^t)^2\big]=\int_{{\mathbb{R}}^3} \ell^N(\overline x_1^t-x)^2\rho^N(x,t)dx\leq CN^{3\delta}({\lVert\rho\rVert}_1+{\lVert\rho\rVert}_\infty)\leq C(T,C_{f_0})N^{3\delta},$$ where we have used the estimates of $\ell^N$ in Lemma \[converse\]. Hence one has $$\mathbb{E}'\big[|Z_j|^2\big]\leq CN^{3\delta}.$$
So the hypotheses of Lemma \[central\] are satisfied with $g(N)=CN^{6\delta-1}$. In addition, it follows from Definition \[defLN\] that $|Z_j|\leq CN^{3\delta}\leq C\sqrt{Ng(N)}$. Hence, we have the probability bound $${\mathbb{P}}\left(\left|(L^N(\overline X_t))_1-(\overline L^N(\overline X_t))_1\right|\geq C(\alpha,T,C_{f_0}) N^{3\delta-1}\log (N)\right)\leq N^{-\alpha},$$ by Lemma \[central\], which leads to $$\label{residual1''}
{\mathbb{P}}\left({\left\|}L^N(\overline X_t)-\overline L^N(\overline X_t){\right\|}_\infty\geq C(\alpha,T,C_{f_0}) N^{3\delta-1}\log (N)\right)\leq N^{1-\alpha}.$$ Thus, follows from .
Proof of Theorem \[mainthm\]
============================
We do the proof by following the idea in [@HH2; @HH1], which is that consistency and stability imply convergence. This at least in principle corresponds to the Lax’s equivalence theorem of proving the convergence of a numerical algorithm, which is that stability and consistency of an algorithm imply its convergence.
Consistency
-----------
In order to obtain the consistency error for the entire time interval, we divide $[0,T]$ into $M+1$ subintervals with length $\Delta \tau=N^{-\frac{\gamma}{3}}$ for some $\gamma>4$ and $\tau_k=n\Delta \tau$, $k=0,\cdots,M+1$. The choice of $\gamma$ will be clear from the discussion below. Here the choice of $\Delta \tau$ is only for the purpose of proving consistency and it can be sufficiently small. Note that it is different from $\Delta t$ in the proof of stability in the next subsection.
First, we establish the following lemma on the traveling distance of $\overline X_t$ in a short time interval $[\tau_k,\tau_{k+1}]$:
\[lmQ\] Assume that $(\overline X_t,\overline V_t)$ satisfies the mean-field dynamics . For $\gamma>4$ it holds $$\label{lmQeq1}
{\mathbb{P}}\left(\max\limits_k\max\limits_{t\in[\tau_k,\tau_{k+1}]}{\left\|}\overline X_t-\overline X_{\tau_k}{\right\|}_\infty\geq C_BN^{-\frac{\gamma-1}{3}}\right)\leq C_BN^{\frac{\gamma-1}{3}}\exp(-C_BN^{\frac{2}{3}}),$$ where $C_B$ depends only on $T$ and $C_{f_0}$.
Notice that for $t\in [\tau_k,\tau_{k+1}]$ $$\begin{aligned}
\label{Qdiff}
\overline X_t-\overline X_{\tau_k}&=\int_{\tau_k}^{t}\overline V_sds=\int_{\tau_k}^{t}\int_{\tau_k}^{s}\overline K^N(\overline X_\tau)d\tau ds+\sqrt{2\sigma}\int_{\tau_k}^{t}(B(s)-B(\tau_k))ds+\int_{\tau_k}^{t}\overline V_{\tau_k}ds,\notag \\
&=:I_1^k(t)+I_2^k(t)+I_3^k(t),
\end{aligned}$$ where $$\overline V_{\tau_k}=V_0+\int_{0}^{\tau_k}\overline K^N(\overline X_s)ds+\sqrt{2\sigma}B(\tau_k).$$
The estimate of $I_1^k(t)$ follows from Lemma \[converse\] $$\int_{\tau_k}^{t}\int_{\tau_k}^{s}\overline K^N(\overline X_\tau)d\tau ds\leq (\Delta t)^2\|\overline K^N\|_\infty\leq CN^{-\frac{2\gamma}{3}}.$$ So we have $$\label{I1}
\max\limits_k\max\limits_{t\in[\tau_k,\tau_{k+1}]}{\lVertI_1^k(t)\rVert}_\infty\leq CN^{-\frac{2\gamma}{3}}.$$
To estimate $I_2^k(t)$, recall a basic property of Brownian motion [@freedman1983brownian Chap. 1.2]: $$\label{Bproperty}
{\mathbb{P}}\left(\max\limits_{t\leq s\leq t+\Delta t}\|B(s)-B(t)\|_\infty\geq b\right)\leq C_1(\sqrt{\Delta t}/b)\exp(-C_2b^2/\Delta t),$$ which leads to $$\label{57}
{\mathbb{P}}\left(\max\limits_{t\in[\tau_k,\tau_{k+1}]}\|B(t)-B(\tau_k)\|_\infty\geq N^{-\frac{1}{3}}\right)\leq C_1N^{-\frac{\gamma-2}{6}}\exp(-C_2N^{\frac{\gamma-2}{3}}),$$ where we choose $b=N^{-\frac{1}{3}}$.
Since $\max\limits_{t\in[\tau_k,\tau_{k+1}]}{\lVertI_2^k(t)\rVert}_\infty\leq \Delta t \sqrt{2\sigma} \max\limits_{t\in[\tau_k,\tau_{k+1}]}\|B(t)-B(\tau_k)\|_\infty$, it follows from that $${\mathbb{P}}\left(\max\limits_{t\in[\tau_k,\tau_{k+1}]}{\lVertI_2^k(t)\rVert}_\infty\geq C N^{-\frac{\gamma+1}{3}}\right)\leq C_1N^{-\frac{\gamma-2}{6}}\exp(-C_2N^{\frac{\gamma-2}{3}}),$$ which leads to $$\label{I2}
{\mathbb{P}}\left(\max\limits_k\max\limits_{t\in[\tau_k,\tau_{k+1}]}{\lVertI_2^k(t)\rVert}_\infty\geq C N^{-\frac{\gamma+1}{3}}\right)\leq C_1N^{\frac{\gamma+2}{6}}\exp(-C_2N^{\frac{\gamma-2}{3}}),$$ where we used the fact that $n\leq\frac{T}{\Delta t}=TN^{\frac{\gamma}{3}}$.
Lastly, we prove the estimate of $I_3^k(t)$. It is obvious that $$\int_{0}^{\tau_k}\overline K^N(\overline X_s)ds\leq n \Delta t {\lVert\overline K^N\rVert}_\infty\leq CT,$$ and it follows from that $${\mathbb{P}}(\|B(\tau_k)\|_\infty\geq N^{\frac{1}{3}})\leq C_1N^{-\frac{1}{3}}\sqrt{T}\exp(-C_2N^{\frac{2}{3}}/T).$$ Moreover, it follows from the assumption in Theorem \[existence\] $b)$ the distribution $f_0^v(v)$ of $V_0$ has a compact support: $$f_0^v(v)=\int_{{\mathbb{R}}^3}f_0(x,v)dx=0,\mbox{ when }|v|> Q_v .$$ Then one has $$\label{compactv}
{\mathbb{P}}(\|V_0\|_\infty\geq N^{\frac{1}{3}})=\int_{|v|\geq N^{\frac{1}{3}}}f_0^v(v)dv=0, \mbox{ when }N> Q_v ^3.$$ It follows from that $$\begin{aligned}
\max\limits_{t\in[\tau_k,\tau_{k+1}]}\|I_3^k(t)\|_\infty&=\int_{\tau_k}^{t}\|\overline V_{\tau_k}\|_\infty ds\leq N^{-\frac{\gamma}{3}}\left(\|V_0\|_\infty+\sqrt{2\sigma}\|B(\tau_k)\|_\infty+\int_{0}^{\tau_k}\overline K^N(\overline X_s)ds\right)\notag\\
&\leq N^{-\frac{\gamma}{3}}(\|V_0\|_\infty+\sqrt{2\sigma}\|B(\tau_k)\|_\infty)+CN^{-\frac{\gamma}{3}},
\end{aligned}$$ then it yields $$\begin{aligned}
&{\mathbb{P}}\left(\max\limits_{t\in[\tau_k,\tau_{k+1}]}{\left\|}I_3^k(t){\right\|}_\infty \geq3N^{-\frac{\gamma-1}{3}}\right)\notag \\
\leq &{\mathbb{P}}\left(N^{-\frac{\gamma}{3}}\|V_0\|_\infty\geq N^{-\frac{\gamma-1}{3}}\right)+{\mathbb{P}}\left(\sqrt{2\sigma}N^{-\frac{\gamma}{3}}\|B(\tau_k)\|_\infty\geq N^{-\frac{\gamma-1}{3}}\right)+{\mathbb{P}}\left(CN^{-\frac{\gamma}{3}}\geq N^{-\frac{\gamma-1}{3}}\right)\notag \\
\leq &0+CN^{-\frac{1}{3}}\exp(-CN^{\frac{2}{3}}) +0\leq CN^{-\frac{1}{3}}\exp(-CN^{\frac{2}{3}}) ,
\end{aligned}$$ which leads to
$$\begin{aligned}
\label{I3}
{\mathbb{P}}\left(\max\limits_k\max\limits_{t\in[\tau_k,\tau_{k+1}]}{\left\|}I_3^k(t){\right\|}_\infty\geq3N^{-\frac{\gamma-1}{3}}\right)\leq CN^{\frac{\gamma-1}{3}}\exp(-CN^{\frac{2}{3}}).
\end{aligned}$$
Then it follows from , and that $$\begin{aligned}
&{\mathbb{P}}\left(\max\limits_k\max\limits_{t\in[\tau_k,\tau_{k+1}]}{\left\|}\overline X_t-\overline X_{\tau_k}{\right\|}_\infty\geq CN^{-\frac{\gamma-1}{3}}\right)\notag\\
\leq &C_1N^{\frac{\gamma+2}{6}}\exp(-C_2N^{\frac{\gamma-2}{3}})+CN^{\frac{\gamma-1}{3}}\exp(-CN^{\frac{2}{3}}) \leq CN^{\frac{\gamma-1}{3}}\exp(-CN^{\frac{2}{3}}) ,
\end{aligned}$$ for $\gamma>4$, which completes the proof of .
Now we can prove the consistency error for the entire time interval $[0,T]$.
\[propconsis\](Consistency) For any $T>0$, let $(\overline X_t,\overline V_t)$ satisfy the mean-field dynamics with initial density $f_0(x,v)$, $K^N$ and $\overline K^N$ be defined in and respectively. For any $\alpha>0$ and $\frac{1}{3}\leq\delta<1$, there exist a constant $C_{2,\alpha}>0$ depending only on $\alpha$, $T$ and $C_{f_0}$ such that $$\label{consistency}
{\mathbb{P}}\left(\max\limits_{t\in[0,T]}{\left\|}K^N(\overline X_t)-\overline K^N(\overline X_t){\right\|}_\infty\geq C_{2,\alpha} N^{2\delta-1}\log (N)\right)\leq N^{-\alpha},$$ and $$\label{consistency1}
{\mathbb{P}}\left(\max\limits_{t\in[0,T]}{\left\|}L^N(\overline X_t)-\overline L^N(\overline X_t){\right\|}_\infty\geq C_{2,\alpha} N^{3\delta-1}\log (N)\right)\leq N^{-\alpha}.$$
Denote the events: $$\label{eventH}
\mathcal{\overline H}:=\left\{\max\limits_k\max\limits_{t\in[\tau_k,\tau_{k+1}]}{\left\|}\overline X_t-\overline X_{\tau_k}{\right\|}_\infty\leq C_BN^{-\frac{\gamma-1}{3}}\right\},$$ and $${\mathcal{C}}_{\tau_k}:=\left\{{\left\|}K^N(\overline X_{\tau_k})-\overline K^N(\overline X_{\tau_k}){\right\|}_\infty\leq C_{1,\alpha} N^{2\delta-1}\log (N)\right\},$$ where $C_B$ and $C_{1,\alpha} $ are used in Lemma \[lmlarge\] and Lemma \[lmQ\] respectively. According to Lemma \[lmlarge\] and Lemma \[lmQ\], one has $$\label{CHevent}
{\mathbb{P}}({\mathcal{C}}_{\tau_k}^c)\leq N^{-\alpha},\quad {\mathbb{P}}(\mathcal{\overline H}^c)\leq C_BN^{\frac{\gamma-1}{3}}\exp(-C_BN^{\frac{2}{3}}),$$ for any $\alpha>0$ and $\gamma>4$.
Furthermore, we denote $$\label{Btn}
{\mathcal{B}}_{\tau_k}:=\left\{{\left\|}L^N(\overline X_{\tau_k})-\overline{L}^N(\overline X_{\tau_k}){\right\|}_\infty\leq C_{1,\alpha} N^{3\delta-1}\log (N)\right\},$$ then one has $$\label{Bevent}
{\mathbb{P}}({\mathcal{B}}_{\tau_k}^c)\leq N^{-\alpha},$$ by Lemma \[lmlarge\]. Also, under the event ${\mathcal{B}}_{\tau_k}$, it holds that $$\label{Btnresult}
\|L^N(\overline X_{\tau_k})\|_\infty\leq \|\overline L^N(\overline X_{\tau_k})\|_\infty+C_{1,\alpha}N^{3\delta-1}\log (N)\leq C(\alpha, T, C_{f_0})N^{3\delta-1}\log (N),$$ where we have used $\|\overline L^N(\overline X_{\tau_k})\|_\infty\leq C\log (N)$ from Lemma \[converse\].
For all $t\in[\tau_k,\tau_{k+1}]$, under the event ${\mathcal{B}}_{\tau_k}\cap{\mathcal{C}}_{\tau_k}\cap \mathcal{\overline H}$, we obtain $$\begin{aligned}
&{\left\|}K^N(\overline X_t)-\overline K^N(\overline X_t){\right\|}_\infty\notag\\
\leq &{\left\|}K^N(\overline X_t)-K^N(\overline X_{\tau_k}){\right\|}_\infty+{\left\|}K^N(\overline X_{\tau_k})-\overline K^N(\overline X_{\tau_k}){\right\|}_\infty+{\left\|}\overline K^N(\overline X_{\tau_k})-\overline K^N(\overline X_t){\right\|}_\infty\notag\\
\leq& C\|L^N(\overline X_{\tau_k})\|_\infty\|\overline X_t-\overline X_{\tau_k}\|_\infty+C_{1,\alpha} N^{2\delta-1}\log (N)+C{\left\|}\overline X_{t}-\overline X_{\tau_k}{\right\|}_\infty+CN^{-\frac{\gamma}{3}}\notag\\
\leq&C(\alpha, T, C_{f_0}) N^{3\delta-1}\log (N)N^{-\frac{\gamma-1}{3}}+C_{1,\alpha} N^{2\delta-1}\log (N)\notag\\
\leq&C(\alpha, T, C_{f_0}) N^{2\delta-1}\log (N),\end{aligned}$$ due to the fact that $3\delta+1<4<\gamma$. In the second inequality we have used the local Lipschitz bound of $K^N$ $${\left\|}K^N(X_t)-K^N(\overline X_{\tau_k}){\right\|}_\infty\leq C\|L^N(\overline X_{\tau_k})\|_\infty\|X_t-\overline X_{\tau_k}\|_\infty,$$ under the event $\mathcal{\overline H}$ (see in Lemma \[lmlip\]). To bound the third term ${\left\|}\overline K^N(\overline X_{\tau_k})-\overline K^N(\overline X_t){\right\|}_\infty$, we used the uniform control of $\max\limits_{\tau_k\leq t \leq \tau_{k+1}}{\lVert\partial_t\rho^N\rVert}_{L^\infty({\mathbb{R}}^3)}$ in . Indeed, $$\begin{aligned}
&\| k^N\ast \rho_{t}(X_{t})- k^N\ast \rho_{\tau_k}(X_{\tau_k})\|_\infty\notag\\
\leq& \| k^N\ast \rho_{t}(X_{t})- k^N\ast \rho_{t}(X_{\tau_k})\|_\infty+\| k^N\ast \rho_{t}(X_{\tau_k})- k^N\ast \rho_{\tau_k}(X_{\tau_k})\|_\infty\notag\\
\leq &C{\left\|}\overline X_{t}-\overline X_{\tau_k}{\right\|}_\infty+C\Delta \tau\leq C{\left\|}\overline X_{t}-\overline X_{\tau_k}{\right\|}_\infty+CN^{-\frac{\gamma}{3}}.
\end{aligned}$$ In the third inequality we have used and . This yields that $$\max\limits_{t\in[0,T]}{\left\|}K^N(\overline X_t)-\overline K^N(\overline X_t){\right\|}_\infty\leq C(\alpha, T, C_{f_0}) N^{2\delta-1}\log (N),$$ holds under the event $\bigcap\limits_{k=0}^{M}{\mathcal{B}}_{\tau_k}\cap{\mathcal{C}}_{\tau_k}\cap\mathcal{\overline H}$. Therefore it follows from and that $$\begin{aligned}
\label{68}
&{\mathbb{P}}\left(\max\limits_{t\in[0,T]}{\left\|}K^N(\overline X_t)-\overline K^N(\overline X_t){\right\|}_\infty\geq C(\alpha, T, C_{f_0})N^{2\delta-1}\log (N)\right)\notag \\
\leq &\sum\limits_{k=0}^MP({\mathcal{B}}_{\tau_k}^c)+\sum\limits_{k=0}^MP({\mathcal{C}}_{\tau_k}^c)+P(\mathcal{\overline H}^c)\notag \\
\leq &TN^{-\frac{3\alpha-\gamma}{3}}+TN^{-\frac{3\alpha-\gamma}{3}}+C_BN^{\frac{\gamma-1}{3}}\exp(-C_BN^{\frac{2}{3}})
\leq N^{-\alpha'}.
\end{aligned}$$ Denote $C_{2,\alpha'}$ to be the constant $C(\alpha, T, C_{f_0})$ in . Since $\alpha>0$ is arbitrary and so is $\alpha'$, holds true. The proof of can be done similarly.
Stability
---------
In this subsection we obtain the stability result.
\[defA\] Let ${\mathcal{A}}_T$ be the event given by $$\label{eventA}
{\mathcal{A}}_T:=\left\{\max\limits_{t\in[0,T]} \sqrt{\log(N)}{\left\|}X_t-\overline X_t{\right\|}_\infty+{\left\|}V_t-\overline V_t{\right\|}_\infty\leq N^{-\lambda_2}\right\}.$$
\[propstab\] (Stability) For any $T>0$, assume that the trajectories $\Phi_t=(X_t,V_t)$, $\Psi_t=(\overline X_t,\overline V_t)$ satisfy and respectively with the initial data $\Phi_0=\Psi_0$ which is i.i.d. sharing the common density $f_0$ satisfying Assumption \[assum\]. Let $K^N$ be introduced in . For any $0<\lambda_2<\frac{1}{3}$, $0<\lambda_1<\frac{\lambda_2}{3}$ and $\frac{1}{3}\leq \delta <1$, we denote the event: $$\begin{aligned}
\mathcal{S}_T(\Lambda):=\bigg\{&\| K^N(X_t)-K^N(\overline X_t) \|_\infty \leq \Lambda\log(N){\left\|}X_t-\overline X_t{\right\|}_\infty\notag\\
&+\Lambda\log^2(N)(N^{6\delta-1-\lambda_1-4\lambda_2}+N^{3\lambda_1-2\lambda_2}+N^{2\delta-1}),~\forall~t\in[0,T]\bigg\}.
\end{aligned}$$ Then for any $\alpha>0$, there exists some $C_{3,\alpha}>0$ and a $N_0\in\mathbb{N}$ depending only on $\alpha$, $T$ and $C_{f_0}$ such that $$\begin{aligned}
&\mathbb{P}\left({\mathcal{A}}_T \cap \mathcal{S}_T^c(C_{3,\alpha})\right)\leq N^{-\alpha},
\end{aligned}$$ for all $N\geq N_0$.
This proposition is one of the crucial statements in our paper. Proving propagation of chaos for systems like the one we consider under the assumptions of Lipschitz-continuous forces is standard, as explained in the introduction. The forces we consider are more singular. However our techniques allow us to show that the Lipschitz condition encoded in the definition of $\mathcal{S}$ holds typically, i.e. with probability close to one. In this sense, Proposition \[propstab\] is only helpful if we find an argument that ${\mathcal{A}}_T$ holds. But as long as we have good estimates on the difference of the forces and thus the growth of $\max\limits_{t\in[0,T]} \sqrt{\log(N)}{\left\|}X_t-\overline X_t{\right\|}_\infty+{\left\|}V_t-\overline V_t{\right\|}_\infty$, we are in fact able to control ${\mathcal{A}}_T$. This control is done by a generalization of Gronwalls Lemma, which will be introduced in our next step (Lemma \[lmprior\]).
Let $\alpha>0$. First, we write $\mathcal{S}_T(\Lambda)$ as the intersection of non-overlapping sets $\{\mathcal{S}_n(\Lambda)\}_{n=0}^{M'}$, where $$\begin{aligned}
\mathcal{S}_n(\Lambda):=\bigg\{&\| K^N(X_t)-K^N(\overline X_t) \|_\infty \leq \Lambda\log(N){\left\|}X_t-\overline X_t{\right\|}_\infty\notag\\
&+\Lambda\log^2(N)(N^{6\delta-1-\lambda_1-4\lambda_2}+N^{3\lambda_1-2\lambda_2}+N^{2\delta-1}),~\forall~t\in[t_n,t_{n+1}]\,0\leq n\leq M'\bigg\},
\end{aligned}$$ with $\Delta t:=t_{n+1}-t_n=N^{-\lambda_1}$, then $\mathcal{S}_T(\Lambda)=\bigcap\limits_{n=0}^{M'}\mathcal{S}_n(\Lambda)$. Note that here the choice of $\Delta t$ is for the purpose of proving stability and it is different from $\Delta \tau$ in the proof of consistency.
To prove this proposition, we split the interaction force $k^N$ into $k^N=k_1^N+k_2^N$, where $k_2^N$ is the result of choosing a wider cut-off of order $N^{-\lambda_2}>N^{-\delta}$ in the force kernel $k$ and $$\label{k1}
k_1^N:=k^N-k_2^N,\quad k_2^N=k\ast\psi_{\lambda_2}^N,$$ which means that for $k_2^N$ and $\ell_2^N$ we choose $\delta=\lambda_2$ in and respectively.
Following the approach in [@garcia2017], we introduce the following auxiliary trajectory $$\label{tilde}
\left\{
\begin{aligned}
&d\widetilde x_i^t= \widetilde v_i^tdt,\\
&d \widetilde v_i^t=\int_{{\mathbb{R}}^3} k^N(\widetilde x_i^t-x)\rho^N(x,t)dxdt + \sqrt{2\sigma}dB_i^t\;.
\end{aligned}
\right.$$ We consider the above auxiliary trajectory with two different initial phases. For any $1\leq n\leq M'$ and $t\in[t_n,t_{n+1}]$, we consider the auxiliary trajectory starting from the initial phase $$\label{tilde1}
(\widetilde x_i^{t_{n-1}}, \widetilde v_i^{t_{n-1}})=(x_i^{t_{n-1}}, v_i^{t_{n-1}}),$$ where $(x_i^{t_{n-1}}, v_i^{t_{n-1}})$ satisfies at time $t_{n-1}$. However when $n=1$, i.e. $t\in[0,t_1]$, the initial phase of the auxiliary trajectory is chosen to be $(\widetilde x_i^{0}, \widetilde v_i^{0})=(x_i^{0}, v_i^{0})$, which has the distribution $f_0$. Moreover in the latter case the distribution of $(\widetilde x_i^{t}, \widetilde v_i^{t})$ is exactly $f_t^N$, which solves the regularized VPFP equations with the initial data $f_0$.
For later reference let us estimate the difference $\|\overline X_t -\widetilde X_t\|_\infty$ and $\|\overline V_t -\widetilde V_t\|_\infty$. Using the equations of these trajectories, we have for $t\in[t_n,t_{n+1}],$ $$\frac{d}{dt}\|\overline X_t -\widetilde X_t\|_\infty=\|\overline V_t -\widetilde V_t\|_\infty,$$ and $$\begin{aligned}
\frac{d}{dt}\|\overline V_t -\widetilde V_t\|_\infty=&\|\overline K^N(\overline X_t) -\overline K^N(\widetilde X_t)\|_\infty\notag
\\\leq& \max_{1\leq j\leq N}|k^N \ast \rho^N(\cdot,t) (\overline x_j)-k^N\ast \rho^N(\cdot,t) (\widetilde x_j)|\notag
\\\leq& \max_{1\leq j\leq N}|\overline x_j-\widetilde x_j|\|\nabla k^N\ast\rho^N(\cdot,t)\|_\infty\notag
\\\leq&C({\lVert\nabla \rho^N\rVert}_1+{\lVert\nabla \rho^N\rVert}_\infty)\|\overline X_t -\widetilde X_t\|_\infty \notag
\\\leq& C \|\overline X_t -\widetilde X_t\|_\infty\;,
\end{aligned}$$ where $C$ depends only on $T$ and $C_{f_0}$. Summarizing, we get $$\frac{d}{dt}\left(\|\overline X_t -\widetilde X_t\|_\infty+\|\overline V_t -\widetilde V_t\|_\infty\right)\leq C\left(\|\overline X_t -\widetilde X_t\|_\infty+\|\overline V_t -\widetilde V_t\|_\infty\right)$$ Using Gronwall’s inequality it follows that $$\begin{aligned}
\label{overlineminustilde}
\max_{t_n \leq t \leq t_{n+1}}\left(\|\overline X_t -\widetilde X_t\|_\infty+\|\overline V_t -\widetilde V_t\|_\infty\right)&\leq \exp(C\Delta t)(\|\overline X_{t_n} - X_{t_n}\|_\infty+\|\overline V_{t_n} - V_{t_n}\|_\infty)\notag\\
&\leq \exp(CN^{-\lambda_1})N^{-\lambda_2}\leq CN^{-\lambda_2},
\end{aligned}$$ under the event $\mathcal{A}_T$ defined in .
Then for any $t\in[t_n,t_{n+1}]$, one splits the error $$\begin{aligned}
&\| K^N(X_t)-K^N(\overline X_t) \|_\infty \notag \\
\leq &{\lVertK_2^N(X_t)-K_2^N(\overline X_t) \rVert}_\infty+{\lVertK_1^N(X_t)-K_1^N(\widetilde X_t) \rVert}_\infty+{\lVertK_1^N(\widetilde X_t)-K_1^N(\overline X_t) \rVert}_\infty \notag \\
=:&\mathcal{I}_1+\mathcal{I}_2+\mathcal{I}_3.
\end{aligned}$$
First, let us compute $\mathcal{I}_1$: $$\begin{aligned}
\| K_2^N(X_t)-K_2^N(\overline X_t) \|_\infty \leq C\|L_2^N(\overline X_{t})\|_\infty{\left\|}X_t-\overline X_t{\right\|}_\infty,
\end{aligned}$$ where we have used the local Lipschitz bound of $K_2^N$ under the event ${\mathcal{A}}_T$ (see in Lemma \[lmlip\]). Furthermore, we denote $$\label{eventB2}
{\mathcal{B}}_2:=\left\{\max\limits_{t\in[0,T]}{\left\|}L_2^N(\overline X_{t})-\overline L_2^N(\overline X_{t}){\right\|}_\infty\leq C_{2,\alpha}N^{3\lambda_2-1}\log(N)\right\}.$$ Since Proposition \[propconsis\] also holds for the case $\lambda_2<\frac{1}{3}$, one has $$\label{estB2}
{\mathbb{P}}({\mathcal{B}}_2^c)\leq N^{-\alpha}.$$
Under the event ${\mathcal{B}}_2$, it holds that $$\label{useb2}
\|L_2^N(\overline X_{t})\|_\infty\leq \|\overline L_2^N(\overline X_{t})\|_\infty+C_{2,\alpha}N^{3\lambda_2-1}\log(N)\leq C\log(N),$$ since $\lambda_2<\frac{1}{3}$, where $\|\overline L_2^N(\overline X_{t})\|_\infty\leq C\log(N)$ follows from Lemma \[converse\]. Hence, one has $$\label{I1'}
\mathcal{I}_1\leq\| K_2^N(X_t)-K_2^N(\overline X_t) \|_\infty \leq C\log(N){\left\|}X_t-\overline X_t{\right\|}_\infty,\quad \forall~t\in[t_n,t_{n+1}],$$ under event ${\mathcal{A}}_T\cap{\mathcal{B}}_2$.
To estimate $\mathcal{I}_2$, notice that by triangle inequality and one has $$\begin{aligned}
{\lVertX_t-\widetilde{X}_t\rVert}_\infty\leq&\int_{t_n}^{t}{\lVertV_s-\widetilde V_s\rVert}_\infty ds\leq
\int_{t_n}^{t}{\lVertV_s-\overline V_s\rVert}_\infty +{\lVert\overline V_s- \widetilde V_s\rVert}_\infty ds
\\\leq& \Delta t \max\limits_{s\in[t_n,t]}\left( {\lVertV_s-\overline V_s\rVert}_\infty+{\lVert\overline V_s- \widetilde V_s\rVert}_\infty \right)
\\\leq& C N^{-\lambda_1-\lambda_2},
\end{aligned}$$ under the event ${\mathcal{A}}_T$, which leads to $$\begin{aligned}
&{\lVertK_1^N(X_t)-K_1^N(\widetilde X_t) \rVert}_\infty\leq ({\lVert\nabla K_1^N(X_t)\rVert}_\infty+{\lVert\nabla K_1^N(\widetilde X_t)\rVert}_\infty){\lVertX_t-\widetilde{X}_t\rVert}_\infty \notag \\
\leq& CN^{3(\delta-\lambda_2)}{\lVertL^N(\overline X_t) \rVert}_\infty{\lVertX_t-\widetilde{X}_t\rVert}_\infty
\leq CN^{3\delta-\lambda_1-4\lambda_2} {\lVertL^N(\overline X_t) \rVert}_\infty.
\end{aligned}$$ Here the bound $\frac{{\lVert\nabla K_1^N(X_t)\rVert}_\infty}{{\lVert L^N(\overline X_t)\rVert}_\infty}\leq CN^{3(\delta-\lambda_2)}$ uses Lemma \[lmmid\] since $$\left\| X_t-\overline{X}_t\right\|_\infty\leq N^{-\lambda_2}\gg N^{-\delta}.$$ And a similar estimate leads to $\frac{{\lVert\nabla K_1^N(\widetilde X_t)\rVert}_\infty}{{\lVert L^N(\overline X_t)\rVert}_\infty}\leq CN^{3(\delta-\lambda_2)}$.
We denote the event $$\label{eventB3}
{\mathcal{B}}_3:=\left\{\max\limits_{t\in[0,T]}{\left\|}L^N(\overline X_{t})-\overline L^N(\overline X_{t}){\right\|}_\infty\leq C_{2,\alpha}N^{3\delta-1}\log(N)\right\}.$$ It has been proved in Proposition \[propconsis\] that $$\label{estB3}
{\mathbb{P}}({\mathcal{B}}_3^c)\leq N^{-\alpha}.$$ Then under the event ${\mathcal{B}}_3$ it follows that $$\|L^N(\overline X_{t})\|_\infty\leq \|\overline L^N(\overline X_{t})\|_\infty+C_{2,\alpha}N^{3\delta-1}\log(N) \leq CN^{3\delta-1}\log(N),$$ since $\|\overline L^N(\overline X_{t})\|_\infty\leq C\log(N)$ and $\frac{1}{3}\leq \delta<1$. Thus, we have $$\label{I2'}
\mathcal{I}_2={\lVertK_1^N(X_t)-K_1^N(\widetilde X_t) \rVert}_\infty\leq CN^{6\delta-1-\lambda_1-4\lambda_2}\log(N),\quad \forall~t\in[t_n,t_{n+1}],$$ under the event ${\mathcal{A}}_T\cap{\mathcal{B}}_3$.
The estimate of $\mathcal{I}_3$ is a result of Lemma \[tildaX\]. Indeed, we denote the event $$\begin{aligned}
\label{eventG}
\mathcal{G}_n:=\bigg\{&\max\limits_{t\in[t_n,t_{n+1}]}\left\| K_1^{N}(\widetilde X_t)- K_1^{N}(\overline X_t)\right\|_\infty\leq C_{4,\alpha}N^{2\delta-1} \log(N)\notag \\
&+C_{4,\alpha}\log^2(N)N^{3\lambda_1-\lambda_2}\|k_1^N\|_1\bigg\},
\end{aligned}$$ so by Lemma \[tildaX\] one has that for any $0\leq n \leq M'$ $$\label{estG}\mathbb{P}\left(\mathcal{A}_T\cap\mathcal{G}_n^c\right)\leq N^{-\alpha}.$$ Furthermore, it holds that $$\begin{aligned}
\label{I3'}
\mathcal{I}_3=&{\lVert K_1^{N}(\widetilde X_t)- K_1^{N}(\overline X_t)\rVert}_\infty
\leq C_{4,\alpha}N^{2\delta-1}\log(N)+C_{4,\alpha}\log^2(N)N^{3\lambda_1-2\lambda_2}\notag \\
\leq&C(\alpha,T,C_{f_0})\log^2(N)(N^{3\lambda_1-2\lambda_2}+N^{2\delta-1}), \quad \forall~t\in[t_n,t_{n+1}],
\end{aligned}$$ under the event $\mathcal{G}_n$, where we have used the fact that ${\lVertk_1^N\rVert}_1\leq CN^{-\lambda_2}$. Indeed, it is easy to compute that $$\label{k1est}
{\lVertk_1^N\rVert}_1={\lVertk^N-k_2^N\rVert}_1\leq C\int_{0\leq |x|\leq N^{-\lambda_2}}\frac{1}{|x|^2}dx\leq CN^{-\lambda_2}.$$
Collecting , and yields that $$\begin{aligned}
&\| K^N(X_t)-K^N(\overline X_t) \|_\infty \notag\\
\leq &C\log(N){\left\|}X_t-\overline X_t{\right\|}_\infty+CN^{6\delta-1-\lambda_1-4\lambda_2}\log(N)+C\log^2(N)(N^{3\lambda_1-2\lambda_2}+N^{2\delta-1})\notag\\
\leq&C\log(N){\left\|}X_t-\overline X_t{\right\|}_\infty+C\log^2(N)(N^{6\delta-1-\lambda_1-4\lambda_2}+N^{3\lambda_1-2\lambda_2}+N^{2\delta-1}),~\forall t\in[t_n,t_{n+1}],
\end{aligned}$$ under the event ${\mathcal{B}}_2\cap{\mathcal{B}}_3\cap{\mathcal{A}}_T\cap \mathcal{G}_n$, where $C$ depends on $\alpha$, $T$ and $C_{f_0}$. To distinguish it from other constants we will denote this $C$ by $C_{3,\alpha}$. This implies ${\mathcal{B}}_2\cap{\mathcal{B}}_3\cap{\mathcal{A}}_T\cap \mathcal{G}_n\subseteq \mathcal S_n(C_{3,\alpha})$, which yields that $${\mathcal{B}}_2\cap{\mathcal{B}}_3\cap{\mathcal{A}}_T\cap(\bigcap_{n=0}^{M'}\mathcal{G}_{n})\subseteq \left( \mathcal \bigcap_{n=0}^{M'}S_n(C_{3,\alpha})\right)=S_T(C_{3,\alpha}).$$ It follows that $$\mathbb{P}\left({\mathcal{A}}_T\cap S_T^c(C_{3,\alpha})\right)\leq \mathbb{P}\left({\mathcal{B}}_2^c\right)+\mathbb{P}\left({\mathcal{B}}_3^c\right)+\sum_{n=0}^{M'}\mathbb{P}\left({\mathcal{A}}_T\cap\mathcal{G}_{n}^c\right)\leq (M'+3) N^{-\alpha}\leq 2TN^{\lambda_1-\alpha}\leq N^{-\alpha'},$$ where we used the estimates in , and. Here $\alpha$ is arbitrary and so is $\alpha'$.
\[tildaX\] Assume that the event ${\mathcal{A}}_T$ holds. Consider two trajectories $(\widetilde X_t,\widetilde V_t)$, $(\overline X_t,\overline V_t)$ on $t\in[t_n,t_{n+1}]$ satisfying - and respectively. When $1\leq n\leq M'$, the two different initial phases are chosen to be $(X_{t_{n-1}},V_{t_{n-1}})$ and $(\overline X_{t_{n-1}},\overline X_{t_{n-1}})$ at time $t=t_{n-1}$, and when $n=0$ the two different initial phases are chosen to be $(X_{0},V_{0})$ and $(\overline X_{0},\overline V_{0})$ at time $t=0$. Then for any $\alpha>0$, there exists a $C_{4,\alpha}>0$ depending only on $\alpha$, $T$ and $C_{f_0}$ such that for $N$ sufficiently large it holds that $$\begin{aligned}
\label{tildaXeq}
{\mathbb{P}}\bigg(&\max\limits_{t\in[t_n,t_{n+1}]} \left\| K_1^{N}(\widetilde X_t)- K_1^{N}(\overline X_t)\right\|_\infty\geq C_{4,\alpha} N^{2\delta-1}\log(N) \notag \\
&+C_{4,\alpha} \log^2(N) N^{3\lambda_1-\lambda_2}\|k_1^N\|_1\bigg)\leq N^{-\alpha},
\end{aligned}$$ where we require $t_{n+1}-t_n=N^{-\lambda_1}$ with $0<\lambda_1<\frac{\lambda_2}{3}$ and $0<\lambda_2<\frac{1}{3}$. Here $$(K_1^N(\widetilde X_t))_i=\frac{1}{N-1}\sum\limits_{j\neq i}^Nk_1^N(\widetilde x_i^t-\widetilde x_j^t)\,,\quad (K_1^N(\overline X_t))_i=\frac{1}{N-1}\sum\limits_{j\neq i}^Nk_1^N(\overline x_i^t-\overline x_j^t),\quad t\in[t_n,t_{n+1}]\,,$$ where $k_1^N$ is defined in .
Lemma \[tildaX\] is used in the proof of Proposition \[propstab\]. It follows from the following estimate of the term in at any fixed time $t\in[t_n,t_{n+1}]$, a statement which will later be generalized to hold for the maximum of $\max t\in[t_n,t_{n+1}]$.
\[tildaXficedtime\] Under the same assumptions as in Lemma \[tildaX\], for any $\alpha>0$, there exists $C_{5,\alpha}>0$ depending only on $\alpha$, $T$ and $C_{f_0}$ such that for $N$ sufficiently large it holds that for any fixed time $t\in[t_n,t_{n+1}]$ $$\begin{aligned}
\label{tildaXfixedtimeeq}
{\mathbb{P}}\bigg(&\left\| K_1^{N}(\widetilde X_t)- K_1^{N}(\overline X_t)\right\|_\infty\geq C_{5,\alpha} N^{2\delta-1}\log(N) \notag \\
&+C_{5,\alpha} \log^2(N) N^{3\lambda_1-\lambda_2}\|k_1^N\|_1\bigg)\leq N^{-\alpha}.
\end{aligned}$$
The proof of Lemma \[tildaXficedtime\] is carried out in Section \[prolem\]. The novel technique in the proof used the fact that $k_1^N$ has a support with the radius $N^{-\lambda_2}$ (small). This means that in order to contribute to the interaction, $\widetilde x_j^t$ (or $\overline x_j^t$) has to get close enough (less than $N^{-\lambda_2}$) to $\widetilde x_i^t$ (or $\overline x_i^t$). Due to the effect of Brownian motion we get mixing of the positions of the particles over the whole support of $k_1^N$. Using a Law of Large Numbers argument one can show that the leading order of the interaction can in good approximation be replaced by the respective expectation value. Due to symmetry of $k_1^N$ this expectation value is zero. Significant fluctuations of the interaction $k_1^N$ have very small probability.
We follow the similar procedure as in Proposition \[propconsis\]. We divide $[t_n,t_{n+1}]$ into $M+1$ subintervals with length $\Delta \tau=N^{-\frac{\gamma}{3}}$ for some $\gamma>4$ and $\tau_k=k\Delta \tau$, $k=0,\cdots,M+1$. Recall the event $\mathcal{\overline H}$ as in and denote the event $$\label{eventH1}
\mathcal{\widetilde H}:=\left\{\max\limits_k\max\limits_{t\in[\tau_k,\tau_{k+1}]}{\left\|}\widetilde X_t-\widetilde X_{\tau_k}{\right\|}_\infty\leq C_BN^{-\frac{\gamma-1}{3}}\right\}.$$ It follows from Lemma \[lmQ\] that $$\label{CHevent1}
{\mathbb{P}}(\mathcal{\overline H}^c),{\mathbb{P}}(\mathcal{\widetilde H}^c)\leq C_BN^{\frac{\gamma-1}{3}}\exp(-C_BN^{\frac{2}{3}}),$$ for any $\gamma>4$. Furthermore we denote the event $$\begin{aligned}
\mathcal{G}_{\tau_k}:=\bigg\{&\left\| K_1^{N}(\widetilde X_{\tau_k})- K_1^{N}(\overline X_{\tau_k})\right\|_\infty\leq C_{5,\alpha}N^{2\delta-1} \log(N)\notag \\
&+C_{5,\alpha}\log^2(N)N^{3\lambda_1-\lambda_2}\|k_1^N\|_1\bigg\}\end{aligned}$$ in , then it follow from Lemma \[tildaXficedtime\] that $$\label{Cg}
{\mathbb{P}}(\mathcal{G}_{\tau_k}^c)\leq N^{-\alpha}.$$
For all $t\in[\tau_k,\tau_{k+1}]$, under the event $\mathcal{G}_{\tau_k}\cap \mathcal{\overline H}\cap \mathcal{\widetilde H}$, we obtain $$\begin{aligned}
&\left\| K_1^{N}(\widetilde X_t)- K_1^{N}(\overline X_t)\right\|_\infty\\
\leq& \left\| K_1^{N}(\widetilde X_t)- K_1^{N}(\widetilde X_{\tau_k})\right\|_\infty+\left\| K_1^{N}(\widetilde X_{\tau_k})- K_1^{N}(\overline X_{\tau_k})\right\|_\infty+\left\| K_1^{N}(\overline X_{\tau_k})- K_1^{N}(\overline X_t)\right\|_\infty\\
\leq& {\lVert\nabla K_1^{N}\rVert}_\infty\left({\lVert\widetilde X_t-\widetilde X_{\tau_k}\rVert}_\infty+{\lVert\overline X_t-\overline X_{\tau_k}\rVert}_\infty\right)+\left\| K_1^{N}(\widetilde X_{\tau_k})- K_1^{N}(\overline X_{\tau_k})\right\|_\infty\\
\leq& CN^{3\delta-\frac{\gamma-1}{3}}+C_{5,\alpha}N^{2\delta-1} \log(N)+C_{5,\alpha}\log^2(N)N^{3\lambda_1-\lambda_2}\|k_1^N\|_1\\
\leq &C(\alpha,T,C_{f_0})N^{2\delta-1} \log(N)+C(\alpha,T,C_{f_0})\log^2(N)N^{3\lambda_1-\lambda_2}\|k_1^N\|_1\end{aligned}$$ when $\gamma>4$ is sufficiently large. This yields that under the event $\bigcap_{k=0}^M\mathcal{G}_{\tau_k}\cap\mathcal{\overline H}\cap \mathcal{\widetilde H}$ it holds that $$\begin{aligned}
&\max\limits_{t\in[t_n,t_{n+1}]} \left\| K_1^{N}(\widetilde X_t)- K_1^{N}(\overline X_t)\right\|_\infty\\
\leq &C(\alpha,T,C_{f_0})N^{2\delta-1}\log(N)+C(\alpha,T,C_{f_0})\log^2(N) N^{3\lambda_1-\lambda_2}\|k_1^N\|_1.\end{aligned}$$ Therefore it follows from and that $$\begin{aligned}
\label{gnesti}
{\mathbb{P}}\bigg(&\max\limits_{t\in[t_n,t_{n+1}]} \left\| K_1^{N}(\widetilde X_t)- K_1^{N}(\overline X_t)\right\|_\infty\geq C(\alpha,T,C_{f_0})N^{2\delta-1}\log(N) \notag \\
&+C(\alpha,T,C_{f_0}) \log^2(N) N^{3\lambda_1-\lambda_2}\|k_1^N\|_1\bigg)\leq \sum_{k=0}^{M}{\mathbb{P}}(\mathcal{G}_{\tau_k}^c)+{\mathbb{P}}(\mathcal{\overline H}^c)+{\mathbb{P}}(\mathcal{\widetilde H}^c)\notag\\
\leq& TN^{-\frac{3\alpha-\gamma}{3}}+2C_BN^{\frac{\gamma-1}{3}}\exp(-C_BN^{\frac{2}{3}})\leq N^{-\alpha'}.\end{aligned}$$ Denote $C_{4,\alpha'}$ to be the constant $C(\alpha, T, C_{f_0})$ in . Since $\alpha>0$ is arbitrary and so is $\alpha'$, holds true. This completes the proof of Lemma \[tildaX\].
Convergence and the proof of Theorem \[mainthm\]
------------------------------------------------
In this section, we achieve the convergence by using the consistency from Proposition \[propconsis\] and the stability from Proposition \[propstab\]. To do this, we first prove the following Gronwall-type inequality.
\[lmprior\] For any $T>0$, let $e(t)$ be a non-negative continuous function on $[0,T]$ with the initial data $e(0)=0$ and $\lambda_2,~\lambda_3$ be two universal constants satisfying $0<\lambda_2<\lambda_3$. Assume that for any $0< T_1\leq T$ the function $e(t)$ satisfies the following differential inequality that holds with $C>0$ independent of $N>0$ $$\label{priorineq}
\frac{de(t)}{dt}\leq C\sqrt{\log(N)}e(t)+C\log^2(N)N^{-\lambda_3},\quad 0<t\leq T_1,$$ provided that $$\label{priorcon}
\max\limits_{t\in[0,T_1]}e(t)\leq N^{-\lambda_2},$$ holds. Then $e(t)$ is uniformly bounded on $[0,T]$. Furthermore there is a $N_0\in\mathbb{N}$ depending only on $C$ and $T$ such that for all $N\geq N_0$ $$\label{priorres}
\max\limits_{t\in[0,T]}e(t)\leq N^{-\lambda_2}.$$
\[ATexplain\] The lemma is in fact a generalization of Gronwall’s Lemma. In Gronwall’s Lemma it is assumed that holds and is the consequence. Here we have a weaker condition: namely we assume that holds under the additional assumption . But as long as holds we can control the growth of $e(t)$ via Gronwall’s inequality to make sure that remains valid on an even larger interval.
We prove the lemma by contradiction: we assume that there is a $t\in [0,T]$ with $e(t)\geq N^{-\lambda_2}$ and show that for $N\geq N_0$ with some $N_0\in\mathbb{N}$ specified below, we get a contradiction.
It follows that the infimum over all times $t$ where $e(t)$ is larger than or equal to $N^{-\lambda_2}$ exists and we define $$T_*=\inf_{}\{0\leq t \leq T:e(t)\geq N^{-\lambda_2}\}.$$ We get by continuity of $e(t)$ together with $e(0)=0$ that $T^*>0$, $$\label{contra}e(T_*)=N^{-\lambda_2}\text{ and }\max_{0\leq t \leq T_*}e(t)=N^{-\lambda_2}\;.$$
Since implies , we get for $T_1=T_*$ that
$$\frac{de(t)}{dt}\leq C\sqrt{\log(N)}e(t)+C\log^2(N)N^{-\lambda_3},\quad 0<t\leq T_*.$$ Gronwall’s Lemma gives that $$e(t)\leq e^{C\sqrt{\log(N)}t}\log^2(N)N^{-\lambda_3},$$ in particular $$e(T_*)\leq e^{C\sqrt{\log(N)}T_*}\log^2(N)N^{-\lambda_3}.$$
Since $ e^{C\sqrt{\log(N)}T_*}$ and $\log^2(N)$ are asymptotically bounded by any positive power of $N$, we can find a $N_0\in\mathbb{N}$ depending only on $C$ and $T_*$ such that for any $N\geq N_0$ $$e^{C\sqrt{\log(N)}T_*}\log^2(N)< N^{\lambda_3-\lambda_2},\quad \mbox{ for }0<\lambda_2<\lambda_3,$$ and hence $$e(T_*)< N^{-\lambda_2}\text{ for any }N\geq N_0\:.$$
Thus we get a contradiction to for all $N\geq N_0$ and the lemma is proven.
We now return to the proof of Theorem \[mainthm\]. Denote the event $${\mathcal{C}}_{T}:=\left\{\max\limits_{t\in[0,T]}{\left\|}K^N(\overline X_t)-\overline K^N(\overline X_t){\right\|}_\infty\leq C_{2,\alpha} N^{2\delta-1}\log (N)\right\},$$ and consider the quantity $e(t)$ defined as $$e(t):={\lVert\Phi_t-\Psi_t\rVert}_\infty =\sqrt{\log(N)}{\left\|}X_t-\overline X_t{\right\|}_\infty+{\left\|}V_t-\overline V_t{\right\|}_\infty.$$ Recall that $${\mathcal{A}}_{T}=\left\{\max\limits_{t\in[0,T]} e(t)\leq N^{-\lambda_2}\right\}.$$
To prove the theorem we will show that under the assumptions $\mathcal{C}_{T}$ and $\mathcal{A}_{T}^c\cup\mathcal{S}_{T}(C_{3,\alpha})$ it follows that $$\label{statement}\max\limits_{t\in[0,T]}e(t)\leq N^{-\lambda_2}.$$ Let us explain why proving under the assumptions $\mathcal{C}_{T}$ and $\mathcal{A}_{T}^c\cup\mathcal{S}_{T}(C_{3,\alpha})$ proves the theorem: Since $\mathcal{C}_{T}$ is the consistency in Proposition \[propconsis\], i.e. $\mathbb{P}\left(\mathcal{C}_{T}^c\right)\leq N^{-\alpha}$, and by Proposition \[propstab\] one has $\mathbb{P}\left(\mathcal{A}_{T}\cap\mathcal{S}_{T}^c(C_{3,\alpha})\right)\leq N^{-\alpha}$, this implies that $${\mathbb{P}}\left(\max\limits_{t\in[0,T]}{\lVert\Phi_t-\Psi_t\rVert}_\infty \geq N^{-\lambda_2}\right)\leq {\mathbb{P}}(\mathcal{C}_{T}^c) +{\mathbb{P}}({\mathcal{A}}_{T}\cap\mathcal{S}_{T}^c(C_{3,\alpha}) )\leq 2N^{-\alpha}\;.$$ It follows that for any $\alpha>0$, there exists some $N_0\in \mathbb{N}$ such that $${\mathbb{P}}\left(\max\limits_{t\in[0,T]}{\lVert\Phi_t-\Psi_t\rVert}_\infty \leq N^{-\lambda_2}\right)\geq 1- N^{-\alpha}$$ for all $N\geq N_0$, which proves Theorem \[mainthm\].
To prove the statement we use Lemma \[lmprior\]. We will show that for any $0<T_1\leq T$, under the additional assumption ${\mathcal{A}}_{T_1}$, the following differential inequality holds $$\label{gwin}
\frac{d e(t)}{dt}
\leq C \sqrt{\log(N)} e(t) +C \log^2(N)N^{-\lambda_3},\mbox{ for all } t\in(0,T_1]\,,$$ for some $\lambda_3>\lambda_2$. Then Lemma \[lmprior\] states that in fact holds which, as explained above, proves Theorem \[mainthm\]. Note that since $e(0)=0$, according to the general Gronwall’s inequality in Lemma \[lmprior\], the assumption ${\mathcal{A}}_{T_1}$ can be removed.
Since ${\mathcal{A}}_{T}\subseteq {\mathcal{A}}_{T_1}$, we have to prove under the assumption that ${\mathcal{A}}_{T}\cap\mathcal{C}_{T}\cap(\mathcal{A}_{T}^c\cup\mathcal{S}_{T}(C_{3,\alpha}))={\mathcal{A}}_{T}\cap\mathcal{C}_{T}\cap\mathcal{S}_{T}(C_{3,\alpha})$ holds. Let us recall the assumptions $\mathcal{C}_{T}$, $\mathcal{S}_{T}(C_{3,\alpha})$ and ${\mathcal{A}}_{T}$ for easier reference. They hold if $$\begin{aligned}
(i)\;\;&\max\limits_{t\in[0,T]}{\left\|}K^N(\overline X_t)-\overline K^N(\overline X_t){\right\|}_\infty\leq C_{2,\alpha} N^{2\delta-1}\log (N)\label{assum1},\\
(ii)\;\;&\| K^N(X_t)-K^N(\overline X_t) \|_\infty \leq C_{3,\alpha}\log(N){\left\|}X_t-\overline X_t{\right\|}_\infty\notag\\
&+C_{3,\alpha}\log^2(N)(N^{6\delta-1-\lambda_1-4\lambda_2}+N^{3\lambda_1-2\lambda_2}+N^{2\delta-1}),~\forall~t\in[0,T]\label{assum2} ,\\
(iii)\;\;&\max\limits_{0\leq t\leq T}e(t)\leq N^{-\lambda_2}.\end{aligned}$$ Notice that for any $0<T_1\leq T$ $${\mathcal{A}}_{T}\subseteq {\mathcal{A}}_{T_1},\quad {\mathcal{C}}_{T}\subseteq {\mathcal{C}}_{T_1}, \quad \mathcal{S}_{T}(C_{3,\alpha})\subseteq \mathcal{S}_{T_1}(C_{3,\alpha}).$$ Using the fact that $\frac{d\|x\|_{\infty}}{dt}\leq \|\frac{dx}{dt}\|_{\infty}$, one has for all $t\in(0,T_1]$ $$\begin{aligned}
\frac{d e(t)}{dt}&\leq \sqrt{\log(N)}{\left\|}V_t-\overline V_t{\right\|}_\infty+{\left\|}K^N(X_t)-\overline K^N(\overline X_t){\right\|}_\infty \notag \\
&\leq \sqrt{\log(N)}{\left\|}V_t-\overline V_t{\right\|}_\infty+{\left\|}K^N(X_t)- K^N(\overline X_t){\right\|}_\infty +{\left\|}K^N(\overline X_t)-\overline K^N(\overline X_t){\right\|}_\infty.\end{aligned}$$ It follows that $$\begin{aligned}
\label{et}
\frac{d e(t)}{dt}
&\leq \sqrt{\log(N)}{\left\|}V_t-\overline V_t{\right\|}_\infty \notag \\
&~~+C_{3,\alpha}\log(N){\left\|}X_t-\overline X_t{\right\|}_\infty+C_{3,\alpha}\log^2(N)(N^{6\delta-1-\lambda_1-4\lambda_2}+N^{3\lambda_1-2\lambda_2}+N^{2\delta-1})\notag \\
&~~+C_{2,\alpha} N^{2\delta-1}\log (N)\notag\\
&\leq C(\alpha,T,C_{f_0})\sqrt{\log(N)} e(t) \notag \\
&\quad+C(\alpha,T,C_{f_0})\log^2(N)(N^{6\delta-1-\lambda_1-4\lambda_2}+N^{3\lambda_1-2\lambda_2}+N^{2\delta-1})\notag\\
&\leq C(\alpha,T,C_{f_0})\sqrt{\log(N)} e(t) +C(\alpha,T,C_{f_0})\log^2(N) N^{-\lambda_3},\end{aligned}$$ where in the first inequality we used assumptions and and in the second inequality we used the fact that $$\sqrt{\log(N)}{\left\|}V_t-\overline V_t{\right\|}_\infty+\log(N){\left\|}X_t-\overline X_t{\right\|}_\infty=\sqrt{\log(N)} e(t).$$ Here we denote $$-\lambda_3:=\max\left\{6\delta-1-\lambda_1-4\lambda_2,3\lambda_1-2\lambda_2,2\delta-1\right\}.$$
Notice that for $$0<\lambda_2<1/3;~0<\lambda_1<\frac{\lambda_2}{3};~\frac{1}{3}\leq\delta<\min\left\{\frac{\lambda_1+3\lambda_2+1}{6},\frac{1-\lambda_2}{2}\right\},$$ one has $-\lambda_3<-\lambda_2$. In other words, we obtain that for $\lambda_2<\lambda_3$ $$\begin{aligned}
\frac{d e(t)}{dt}\leq C(\alpha,T,C_{f_0})\sqrt{\log(N)} e(t) +C(\alpha,T,C_{f_0})\log^2(N) N^{-\lambda_3},\mbox{ for all } t\in(0,T_1]\,,\end{aligned}$$ which verifies and the theorem is proven.
Proof of Theorem \[cor\]
========================
In order to prove the error estimate between $f_t$ and $\mu_\Phi(t)$, let us split the error into three parts $$\begin{aligned}
W_p(f_t,\mu_\Phi(t))&\leq W_p(f_t,f_t^N)+W_p(f_t^N,\mu_\Psi(t))+W_p(\mu_\Psi(t),\mu_\Phi(t)).\end{aligned}$$ The Theorem \[cor\] is proven once we obtain the respective error estimates of those three parts.
$\bullet$*The first term $W_p(f_t,f_t^N)$*. The convergence of this term is a deterministic result: solutions of the regularized VPFP equations approximate solutions of the original VPFP equations as the width of the cut-off goes to zero. It follows from [@carrillo2018propagation Lemma 3.2] that $$\label{term1}
\max\limits_{t\in[0,T]}W_p(f_t,f_t^N)\leq N^{-\delta}e^{C_1\sqrt{\log(N)}},$$ where $p\in[1,\infty)$, $N>3$ and $C_1$ depends only on $T$ and $C_{f_0}$. The proof is inspired by the method of Leoper [@loeper2006uniqueness]. Note that here we can’t follow the method in [@lazarovici2015mean] directly since the support of $f^N$ and $f$ are not compact in our present case.
$\bullet$*The second term $W_p(f_t^N,\mu_\Psi(t))$*. This term concerns the sampling of the mean-field dynamics by discrete particle trajectories. The convergence rate has been proved in [@lazarovici2015mean Corollary 9.4] by using the concentration estimate of Fournier and Guillin [@fournier2015rate]. We summarize the result as follows: let $p\in[1,\infty)$, $\kappa<\min\{\delta,\frac{1}{6},\frac{1}{2p}\}$ and $N>3$. Assume that there exists $m>2p$ such that $$\iint_{{\mathbb{R}}^6}(|x|^m+|v|^m)f_0(x,v)dxdv<+\infty.$$ Then there exist constants $C_2$ and $C_3$ such that it holds $$\begin{aligned}
\label{term2}
{\mathbb{P}}\bigg(\max\limits_{t\in[0,T]}W_p(f_t^N,\mu_\Psi(t))&\leq \sqrt{\log (N)}N^{-\kappa}e^{C_2\sqrt{\log(N)}}\bigg)\notag \\
&\geq 1-C_3\left(e^{-C_4N^{1-\max\{6,2p\}\kappa}}+N^{1-\frac{m}{2p}}\right).
\end{aligned}$$
$\bullet$*The third term $W_p(\mu_\Psi(t),\mu_\Phi(t))$*. The convergence of this term is a direct result of Theorem \[mainthm\]. Indeed, it follows from [@lazarovici2015mean Lemma 5.2] that for all $p\in[0,\infty]$ $$\max\limits_{t\in[0,T]}W_p(\mu_\Psi(t),\mu_\Phi(t))\leq \max\limits_{t\in[0,T]}{\lVert\Psi(t)-\Phi(t)\rVert}_\infty.$$ Then we choose $\alpha=\frac{m}{2p}-1$ in Theorem \[mainthm\] so that $$\label{term3}
\mathbb{P}\left(\max\limits_{t\in[0,T]}W_p(\mu_\Psi(t),\mu_\Phi(t)) \leq N^{-\lambda_2} \right)\geq 1-N^{1-\frac{m}{2p}}.$$
$\bullet$*Convergence of $W_p(f_t,\mu_\Phi(t))$*. Collecting estimates , and and choosing $\kappa <\min\{\delta,\frac{1}{6},\frac{1}{2p}\}$, it follows that $$\begin{aligned}
{\mathbb{P}}\bigg(\max\limits_{t\in[0,T]}W_p(f_t,\mu_\Phi(t))&\leq (1+\sqrt{\log (N)})N^{-\kappa}e^{C_5\sqrt{\log(N)}}+N^{-\lambda_2}\bigg)\notag\\
&\geq 1-C_6\left(e^{-C_7N^{1-\max\{6,2p\}\kappa}}+N^{1-\frac{m}{2p}}\right),
\end{aligned}$$ where $C_5$ depends only on $T$ and $C_{f_0}$, and $C_6$, $C_7$ depend only on $m$, $p$, $\kappa$. We can simplify this result by demanding $N\geq e^{\left(\frac{2C_5}{1-3\lambda_2}\right)^2}$, which yields $N^{1-3\lambda_2}\geq (1+\sqrt{\log (N)})e^{C_5\sqrt{\log(N)}}$. Hence we conclude that $$\begin{aligned}
{\mathbb{P}}\bigg(\max\limits_{t\in[0,T]}W_p(f_t,\mu_\Phi(t))&\leq N^{-\kappa+1-3\lambda_2}+N^{-\lambda_2}\bigg)\notag\\
&\geq 1-C_6\left(e^{-C_7N^{1-\max\{6,2p\}\kappa}}+N^{1-\frac{m}{2p}}\right).
\end{aligned}$$
The proof of Lemma \[tildaXficedtime\] {#prolem}
======================================
In this section, we present the proof of Lemma \[tildaXficedtime\], which provides the distance between $K_1^{N}(\widetilde X_t)$ and $K_1^{N}(\overline X_t)$ ($t\in[t_n,t_{n+1}]$), where $(\widetilde X_t,\widetilde V_t)$, $(\overline X_t,\overline V_t)$ satisfying - and respectively with two different initial phases $(X_{t_{n-1}},V_{t_{n-1}})$ and $(\overline X_{t_{n-1}},\overline X_{t_{n-1}})$ at time $t=t_{n-1}$ when $1\leq n\leq M'$, or $(X_{0},V_{0})$ and $(\overline X_{0},\overline V_{0})$ at time $t=0$ when $n=0$. To do this, we introduce the following stochastic process: For time $0\leq s \leq t$ and $a:=(a_x,a_v)\in {\mathbb{R}}^{6N}$, let $Z^{a,N}_{t,s}:=(Z^{a,N}_{x,t,s},Z^{a,N}_{v,t,s})$ be the process starting at time $s$ at the position $(a_x,a_v)$ and evolving from time $s$ up to time $t$ according to the mean-field force $\overline K^N$: $$\begin{aligned}
\label{lastprocess}
\begin{cases}
dZ^{a,i,N}_{x,t,s}&=Z^{a,i,N}_{v,t,s} dt,\quad t>s,\\
dZ^{a,i,N}_{v,t,s}&=\int_{{\mathbb{R}}^3} k^N(Z^{a,i,N}_{x,t,s}-x)\rho^N(x,t)dx+\sqrt{2\sigma}dB_i^t,\quad i=1,\cdots,N,
\end{cases}\end{aligned}$$ and $$(Z^{a,i,N}_{x,s,s},Z^{a,i,N}_{v,s,s})=(a_x^i,a_v^i),\quad \mbox{at }t=s.$$ Note that here $(Z^{a,i,N}_{x,t,s},Z^{a,i,N}_{v,t,s})$, $i=1,\cdots,N$ are independent. Furthermore $(Z^{a,N}_{x,t,s},Z^{a,N}_{v,t,s})$ has the strong Feller property (see [@Girsanov59] Definition (A)), implying in particular that it has a transition probability density $u^{a,N}_{t,s}$ which is given by the product $u^{a,N}_{t,s}:=\prod_{i=1}^N u^{a,i,N}_{t,s}$. Hence each term $u^{a,i,N}_{t,s}$ is the transition probability density of $(Z^{a,i,N}_{x,t,s},Z^{a,i,N}_{v,t,s})$ and is also the solution to the linearized equation for $t>$: $$\partial _t u^{a,i,N}_{t,s}+v\cdot\nabla_x u^{a,i,N}_{t,s}+k^N\ast \rho^N\cdot \nabla_v u^{a,i,N}_{t,s}=\Delta_vu^{a,i,N}_{t,s},\quad u^{a,i,N}_{s,s}=\delta_{a_i},$$ where $\rho^N=\int_{{\mathbb{R}}^3}f^N(t,x,v)dv$, and $f^N$ solves the regularized VPFP equations with initial condition $f_0$.
Consider now the process $Z^{a,N}_{t,s}$ and $Z^{b,N}_{t,s}$ for two different starting points $a,b\in {\mathbb{R}}^{6N}$. It is intuitively clear that the probability density $u^{a,i,N}_{t,s}$ and $u^{b,i,N}_{t,s}$ are just a shift of each other. The next lemma gives an estimate for the distance between any two densities in terms of the distance between the starting points $a$ and $b$ and the elapsed time $t-s$. The proof is carried out in Appendix A.
\[transition\] \[lemma\]There exists a positive constant $C$ depending only on $C_{f_0}$ and $T$ such that for each $N \in \mathbb{N}$, any starting points $a,b \in
\mathbb{R}^{6 N}$ and any time $0 < t \leqslant T$, the following estimates for the transition probability densities $u_{t, s}^{a,i,N}$ resp. $u_{t,
s}^{b,i,N}$ of the processes $Z_{t, s}^{a,i,N}$ resp. $Z_{t, s}^{b,i,N}$ given by (\[lastprocess\]) hold for $t-s<\min\{1,T-s\}$:
(i) $ \| u_{t, s}^{a,i,N} \|_{\infty,1}
\leqslant C \left((t - s)^{-\frac{9}{2}}+1\right)$,
(ii) $\| u_{t, s}^{a,i,N} -
u_{t, s}^{b,i,N} \|_{\infty,1} \leqslant C | a - b|\left((t - s)^{- 6}+1\right).$
The norm $\|\cdot\|_{p,q}$ denotes the $p$-norm in the $x$ and $q$-norm in the $v$-variable, i.e. for any $f:\mathbb{R}^3\times\mathbb{R}^3\to\mathbb{R} $ $$\|f\|_{p,q}:=\left(\int_{{\mathbb{R}}^3} \left(\int_{{\mathbb{R}}^3} |f(x,v)|^q dv\right)^{p/q}dx\right)^{1/p}.$$
To this end one assumes $\Delta t=t_{n+1}-t_n=N^{-\lambda_1}$. Next we define for $t\in[t_n,t_{n+1}]$ the random sets $$\label{Mtn}
M_{t_n}^t:=\left\{2\leq j\leq N: \left| x_1^{t_n}- x_j^{t_n}+(t-t_n)(v^{t_n}_1- v^{t_n}_j)\right|\leq N^{-\lambda_2}+ \log (N) \Delta t^{\frac{3}{2}} \right\}$$ and $$\label{-Mtn}
\overline M_{t_n}^t:=\left\{2\leq j\leq N: \left| \overline x_1^{t_n}- \overline x_j^{t_n}+(t-t_n)(\overline v^{t_n}_1- \overline v^{t_n}_j)\right|\leq 3N^{-\lambda_2}+ \log (N) \Delta t^{\frac{3}{2}} \right\} .$$ Here $M_{t_n}^t$ is at time $t_n$ the set of indices of those particles $x_j^{t_n}$ which are in the ball of radius $N^{-\lambda_2}+ \log (N) \Delta t^{\frac{3}{2}} $ around $x_1^{t_n}+(t-t_n)(v^{t_n}_1- v^{t_n}_j)$, and $\overline M_{t_n}^t$ is an intermediate set introduced to help to control $M_{t_n}^t$. Note that under the event $\mathcal{A}_T$, we have $M_{t_n}^t\subseteq \overline M_{t_n}^t$.
We also define random sets for $t\in[t_n,t_{n+1}]$ $$\label{Stn}
\mathcal{S}_{t_n}^t=\left\{\mbox{card }(M_{t_n}^t)< 2C_\ast N\left(3N^{-\lambda_2}+ \log (N) \Delta t^{\frac{3}{2}} \right)^2 \right\},$$ and $$\label{-Stn}
\overline{\mathcal{S}}_{t_n}^t=\left\{\mbox{card }(\overline M_{t_n}^t) < 2C_\ast N\left(3N^{-\lambda_2}+ \log (N) \Delta t^{\frac{3}{2}} \right)^2 \right\}\;,$$ where $C_\ast$ will be defined later. Here $\mathcal{S}_{t_n}^t$ indicates the event where the number of particles inside the set $M_{t_n}^t$ is smaller than $2C_\ast N\left(3N^{-\lambda_2}+ \log (N) \Delta t^{\frac{3}{2}} \right)^2 $, and the event $\overline{\mathcal{S}}_{t_n}^t$ is introduced to help estimate ${\mathbb{P}}(\mathcal{S}_{t_n}^t)$.
Our next lemma provides the probability estimate of the event where particle $\widetilde x_{j}^t$ (or $\overline x_{j}^t$) is close to $\widetilde x_{1}^t$ (or $\overline x_{1}^t$) (distance smaller than $N^{-\lambda_2}$) during a short time interval $t-t_n$, which contributes to the interaction of $k_1^N$ defined in , since the support of $k_1^N$ has radius $N^{-\lambda_2}$.
\[unlikely\] Let $(\widetilde x_{j}^t,\widetilde v_{j}^t)$ satisfy and on $t\in[t_n,t_{n+1}]$ and the random set $M_{t_n}^t$ satisfy , then for any $\alpha>0$, there exists some constant $N_0>0$ depending only on $\alpha$, $T$ and $C_{f_0}$ such that for all $N\geq N_0$ it holds $$\begin{aligned}
{\mathbb{P}}\left(\min_{t\in[t_n,t_{n+1}]}\max_{j\in( M^ t_{t_n})^c}\left\{\left|\widetilde x_{1}^t-\widetilde x_{j}^t\right|\right\}<N^{-\lambda_2}\;\;\right)&\leq N^ {-\alpha},
\\{\mathbb{P}}\left(\min_{t\in[t_n,t_{n+1}]}\max_{j\in(\overline M^ t_{t_n})^c}\left\{\left|\overline x_{1}^t-\overline x_{j}^t\right|\right\}<N^{-\lambda_2}\;\;\right)&\leq N^ {-\alpha}.\end{aligned}$$ This means that for some particle index $j$ outside $M_{t_n}^t$, $\widetilde x_{j}^t$ for some $t\in[t_n,t_{n+1}]$ such that $\left|\widetilde x_{1}^t-\widetilde x_{j}^t\right|<N^{-\lambda_2}$ (i.e. $\widetilde x_{j}^t$ contributes to the interaction of $k_1^N$) with probability less than $N^ {-\alpha}$. Here ${\mathbb{P}}$ is understood to be taken on the initial condition $\widetilde x_{j}^{t_n}$.
Let $(1,j)$ be fixed and $a_1^t:=(a^t_{1,x},a^t_{1,v}),b_j^t:=(b^t_{j,x},b^t_{j,v})\in\mathbb{R}^6$ satisfy the stochastic differential equations $$da^t_{1,x}=a^ t_{1,v} dt,~da^ t_{1,v}= \sqrt{2\sigma}dB_1^ t;\quad db^t_{j,x}=b^t_{j,v} dt,~db^t_{j,v}=\sqrt{2\sigma}dB_j^ t,\quad t_n<t\leq t_{n+1},$$ with the initial data $a_1^{t_n}=0$ and $b_j^{t_n}=0$. Here $B_j^t$ is the same as in . It follows from the evolution equation that $$d \left(\widetilde x_1^t - a^t_{1,x}\right)= \left(\widetilde v^{t}_1 - a^t_{1,v}\right)dt \text{ and }d \left(\widetilde v^{t}_1 - a^t_{1,v}\right)= \overline k^N(\widetilde x^ t_1) dt,\quad t_n<t\leq t_{n+1},$$ and $$d \left(\widetilde x^{t}_j - b^t_{j,x}\right)= \left(\widetilde v^{t}_j - b^t_{j,v}\right)dt \text{ and }d \left(\widetilde v^{t}_j - b^t_{j,v}\right)= \overline k^N(\widetilde x^ t_j) dt,\quad t_n<t\leq t_{n+1},$$ where $$\overline k^N:=k^N\ast\rho^N,$$ which is bounded by ${\lVert \overline k^N\rVert}_\infty\leq C({\lVert\rho^N\rVert}_1+{\lVert\rho^N\rVert}_\infty)$ according to Lemma \[converse\]. Integrating twice we get for any $s\geq t_n$ $$\left(\widetilde v^{s}_j - b^s_{j,v}\right)= \widetilde v^{t_n}_j +\int_{t_n}^s \overline k^N(\widetilde x^\tau_j) d\tau$$ and $$\left(\widetilde x^{t}_j - b^t_{j,x}\right)=\widetilde x_j^{t_n}+\int_{t_n}^t \left(\widetilde v^{t_n}_j +\int_{t_n}^s \overline k^N(\widetilde x^\tau_j) d\tau\right) ds\;.$$ And by the same argument one has $$\begin{aligned}
\widetilde x^{t}_1-\widetilde x^{t}_j-(a^t_{1,x}-b^t_{j,x})=\widetilde x_1^{t_n}-\widetilde x_j^{t_n}+
\int_{t_n}^t \left(\widetilde v^{t_n}_1-\widetilde v^{t_n}_j+\int_{t_n}^s \overline k^N(\widetilde x^\tau_1) d\tau -\int_{t_n}^s \overline k^N(\widetilde x^\tau_j) d\tau\right) ds\;.\end{aligned}$$ Since $\overline k^N(\widetilde x^\tau_j)$ is bounded by ${\lVert \overline k^N\rVert}_\infty\leq C({\lVert\rho^N\rVert}_1+{\lVert\rho^N\rVert}_\infty)$ according to Lemma \[converse\]., it follows that there is a constant $0<C<\infty$ depending only on ${\lVert\overline k^N\rVert}_\infty$ such that $$\begin{aligned}
\label{esta}
\left|\widetilde x^{t}_1-\widetilde x^{t}_j\right|\geq \left|\widetilde x_1^{t_n}-\widetilde x_j^{t_n}+(t-t_n)(\widetilde v^{t_n}_1-\widetilde v^{t_n}_j)\right|-\left|a^t_{1,x}-b_{j,x}^t\right|-C\Delta t^2, \mbox{ for all } t\in[t_n,t_{n+1}].\end{aligned}$$
For $j\in (M_{t_n}^t)^c$ for some $t\in[t_n,t_{n+1}]$, i.e. $$\label{161}
\left|\widetilde x_1^{t_n}-\widetilde x_j^{t_n}+(t-t_n)(\widetilde v^{t_n}_1-\widetilde v^{t_n}_j)\right|\geq N^{-\lambda_2}+\log (N) \Delta t^{\frac{3}{2}}$$ together with $\min\limits_{t\in[t_n,t_{n+1}]}\max\limits_{j\in( M^ t_{t_n})^c}\left\{\left|\widetilde x_{1}^t-\widetilde x_{j}^t\right|\right\}<N^{-\lambda_2}$, and imply $$\max_{t\in[t_n,t_{n+1}]}\min_{j\in( M^ t_{t_n})^c}\left\{\left|a^t_{1,x}-b_{j,x}^ t\right|\right\}>(\ln N-C) \Delta t^{\frac{3}{2}}.$$ Hence $$\begin{aligned}
\label{min1}\nonumber
&{\mathbb{P}}\left(\min_{t\in[t_n,t_{n+1}]}\max_{j\in( M^ t_{t_n})^c}\left\{\left|\widetilde x_{1}^t-\widetilde x_{j}^t\right|\right\}<N^{-\lambda_2}\;\;\right) \notag\\
\leq&{\mathbb{P}}\left(\max_{t\in[t_n,t_{n+1}]}\min_{j\in( M^ t_{t_n})^c}\left\{\left|a^t_{1,x}-b_{j,x}^ t\right|\right\}>(\ln N-C) \Delta t^{\frac{3}{2}}\right) \notag
\\\leq&{\mathbb{P}}\left(\max_{t\in[t_n,t_{n+1}]}\min_{j\in( M^ t_{t_n})^c}\left\{\left|a^t_{1,v}-b_{j,v}^ t\right|\right\}>(\ln N-C) \Delta t^{\frac{1}{2}}\right),\end{aligned}$$ where we used $a_x^ t=\int_{t_n}^ta_v^sds$ and $b_x^ t=\int_{t_n}^tb_v^sds$ in the second inequality. In the same way we can argue that $$\begin{aligned}
\label{min2}
&{\mathbb{P}}\left(\min_{t\in[t_n,t_{n+1}]}\max_{j\in(\overline M^ t_{t_n})^c}\left\{\left|\overline x_{1}^t-\overline x_{j}^t\right|\right\}<N^{-\lambda_2}\;\;\right)\notag\\
\leq&{\mathbb{P}}\left(\max_{t\in[t_n,t_{n+1}]}\min_{j\in(\overline M^ t_{t_n})^c}\left\{\left|a^t_{1,v}-b_{j,v}^ t\right|\right\}>(\ln N-C) \Delta t^{\frac{1}{2}}\right)\;.\end{aligned}$$
Due to independence the difference $c_{j,v}^t=(c_{j,1}^t,c_{j,2}^t,c_{j,3}^t)=a_{1,v}^t-b_{j,v}^t$ is itself a Wiener process [@wiener1923] since $$dc_{j,v}^t=d (a_{1,v}^t-b_{j,v}^t)= d(B_1^t-B_j^t).$$ Splitting up this Wiener process into its three spacial components we get $$\begin{aligned}
\label{estc}
&{\mathbb{P}}\left(\max_{t\in[t_n,t_{n+1}]}\min_{j\in( M^ t_{t_n})^c}\left\{\left|a^t_{1,v}-b_{j,v}^ t\right|\right\}>(\ln N-C) \Delta t^{\frac{1}{2}}\right) \notag\\
\leq& 3{\mathbb{P}}\left(\max_{t\in[t_n,t_{n+1}]}\min_{j\in( M^ t_{t_n})^c}\left\{\left|c^ t_{j,1}\right|\right\}>(\ln N-C) \Delta t^{\frac{1}{2}}\right)\notag\\
\leq &6{\mathbb{P}}\left(\max_{t\in[t_n,t_{n+1}]} \min_{j\in( M^ t_{t_n})^c}\left\{c^ t_{j,1}\right\} >(\ln N-C) \Delta t^{\frac{1}{2}}\right)\notag\\
=&12 {\mathbb{P}}\left( \min_{j\in( M^ t_{t_n})^c} \left\{c^ {t_{n+1}}_{j,1}\right\}>(\ln N-C) \Delta t^{\frac{1}{2}}\right)\;.\end{aligned}$$ where in the last equality we used the reflection principle based on the Markov property [@Levy40].
Recall that the time evolution of $a_{1,v}^ t$ and $b_{j,v}^t$ are standard Brownian motions, i.e. the density is a Gaussian with standard deviation $\sigma_t=\sigma(t-t_n)^{\frac{1}{2}}$. Due to the independence of $a^t_{1,v}$ and $b^t_{j,v}$, $c_{j,1}^t$ is also normal distributed with the standard deviation of order $(t-t_n)^{\frac{1}{2}}$. Hence for $N$ sufficiently large, following from , it holds that $${\mathbb{P}}\left(\max_{t\in[t_n,t_{n+1}]}\min_{j\in( M^ t_{t_n})^c}\left\{\left|a^t_{1,v}-b_{j,v}^ t\right|\right\}>(\ln N-C) \Delta t^{\frac{1}{2}}\right)\leq N^{-\alpha}\,$$ and $${\mathbb{P}}\left(\max_{t\in[t_n,t_{n+1}]}\min_{j\in(\overline M^ t_{t_n})^c}\left\{\left|a^t_{1,v}-b_{j,v}^ t\right|\right\}>(\ln N-C) \Delta t^{\frac{1}{2}}\right)\leq N^{-\alpha}\,$$ With and the lemma follows.
Now we have all the estimates needed for the proof of Lemma \[tildaXficedtime\].
We show that under the event $\mathcal{A}_T$ defined in , for any $\alpha>0$ there exists a $C_\alpha$ depending only on $\alpha$, $T$ and $C_{f_0}$ such that at any fixed time $t\in[t_n,t_{n+1}]$
$$\begin{aligned}
\label{goal}
{\mathbb{P}}\bigg(&\left| \frac{1}{N-1}\sum_{j\neq 1}^N\left(k_1^N(\widetilde x_1^t-\widetilde x_j^t)- k_1^N(\overline x_1^t-\overline x_j^t)\right)\right|\notag\\
&\geq C_{\alpha}N^{2\delta-1}\log(N) +C_\alpha\log^2(N)N^{3\lambda_1-\lambda_2}\|k_1^N\|_1\bigg)\leq N^{-\alpha}.
\end{aligned}$$
This is done under the event $\mathcal{A}_T$ in three steps:
- We prove that for any $t\in [t_n,t_{n+1}]$ the number of particles inside $M_{t_n}^t$ is larger than $$\label{defM}
M_\ast:= 2C_\ast N\left(3N^{-\lambda_2}+ \log (N) \Delta t^{3/2}\right)^2$$ with probability less than $N^{-\alpha}$. Note that $M_\ast$ is used as a bound in the definition of and . For any $t\in [t_n,t_{n+1}]$ we prove that $$\label{a}
{\mathbb{P}}\left(\mbox{card }(M_{t_n}^t)> M_\ast\right)={\mathbb{P}}((\mathcal{S}_{t_n}^t)^c)\leq{\mathbb{P}}((\overline{\mathcal{S}}_{t_n}^t)^c)\leq N^{-\alpha}.$$
- We prove that at any fixed time $t\in [t_n,t_{n+1}]$, particles outside $M_{t_n}^t$ contribute to the interaction of $k_1^N$ with probability less than $N^{-\alpha}$, namely $$\begin{aligned}
\label{b}
{\mathbb{P}}\bigg(&\left| \frac{1}{N-1}\sum_{j\in (M_{t_n}^t)^c}\left(k_1^N(\widetilde x_1^t-\widetilde x_j^t)- k_1^N(\overline x_1^t-\overline x_j^t)\right)\right|> 0\bigg)\leq N^{-\alpha}.
\end{aligned}$$
- According to step $(2)$ above, at any fixed time $t\in [t_n,t_{n+1}]$, particles outside $M_{t_n}^t$ do not contribute to the interaction of $k_1^N$ with high probability, so we only consider particles that are inside $M_{t_n}^t$. And we know already from step $(1)$ above that the number of particles inside $M_{t_n}^t$ is larger than $M_\ast$, with low probability. To prove , we only need to prove $$\begin{aligned}
\label{c}
{\mathbb{P}}\bigg( \mathcal{X}(M_{t_n}^t)\cap\left\{\mbox{ card }(M_{t_n}^t)\leq M_\ast\right\}\bigg)\leq N^{-\alpha}\,
\end{aligned}$$ at any fixed time $t\in [t_n,t_{n+1}]$, where the event $\mathcal{X}(M_{t_n}^t)$ is defined by $$\begin{aligned}
\label{eventX}
\mathcal{X}(M_{t_n}^t):=\bigg\{&\bigg|\frac{1}{N-1}\sum_{j\in M_{t_n}^t}\left(k_1^N(\widetilde x_1^t-\widetilde x_j^t)- k_1^N(\overline x_1^t-\overline x_j^t)\right)\bigg|\notag \\
&\geq C_{\alpha}N^{2\delta-1}\log(N) +C_\alpha\log^2(N)N^{3\lambda_1-\lambda_2} \|k_1^N\|_1\bigg\}.
\end{aligned}$$
\[figure\] {width="99.00000%"}
$\bullet \textit{Step 1:}$ To prove the first part of , note that on the event $\mathcal A_T$ defined in and assuming that $t\in[t_n,t_{n+1}]$
$$\left| x_1^{t_n}- x_j^{t_n}+(t-t_n)(v_1^{t_n}- v_j^{t_n})\right|\leq N^{-\lambda_2}+ \log (N) \Delta t^{\frac{3}{2}}\,$$ implies $$\left| \overline x_1^{t_n}- \overline x_j^{t_n}+(t-t_n)(\overline v_1^{t_n}- \overline v_j^{t_n})\right|\leq 3N^{-\lambda_2}+ \log (N) \Delta t^{\frac{3}{2}}\;.$$
Hence $M_{t_n}^t\subseteq \overline M_{t_n}^t$ and thus for any $R>0$, $\mbox{card }(\overline M_{t_n}^t)<R$ implies that $\mbox{card } (M_{t_n }^t)\leq \mbox{card } (\overline M_{t_n }^t)<R$, consequently $\mathcal{S}_{t_n}^t\supseteq\overline{\mathcal{S}}_{t_n}^t$, i.e. $(\mathcal{S}^ t_{t_n})^c\subseteq(\overline{\mathcal{S}}^t_{t_n})^c$ .
The second part of is trivial. For the third part we use the independence of the $\overline x$-particles. Note that the law of $(\overline x^j_{t_n}, \overline v^j_{t_n})$ has a density $f^N(x,v,t_n)$. For any $j\in\left\{2,\ldots,N\right\}$ the probability to find $j\in \overline M_{t_n}^t$ for any $t\in[t_n,t_{n+1}]$ is given by $$\begin{aligned}
{\mathbb{P}}\left(j\in \overline M_{t_n}^t\right)&=\int_{{\mathbb{R}}^3} \int_{B_{R}(\Xi^t)} f^N(x,v,t_n)dx dv,\end{aligned}$$ where the center $\Xi^t$ of the ball is given by $\Xi^t=\overline x_1^{t_n}+(t-t_n)(\overline v_1^{t_n}- v)$, and the radius of the ball is given by $R=3N^{-\lambda_2}+ \log (N) \Delta t^{3/2} $.
Define $$\label{defgN}
g^N(x,v,s):=f^N(x-vs,v,t_n)$$ which then satisfies the following transport equation $$\begin{cases}
\partial_sg^N(x,v,s)+v\cdot \nabla_x g^N(x,v,s)=0,~ 0<s\leq \Delta t,\\
g^N(x,v,0)=f^N(x,v,t_n).
\end{cases}$$
Then one has $$\int_{B_{R}(\Xi^t)} f^N(x,v,t_n)dx=\int_{B_{R}( \Xi_0^t)} g^N(x,v,t-t_n)dx,$$ where the center $\Xi_0^t$ of the ball is given by $\Xi_0^t=\overline x_1^{t_n}+(t-t_n)\overline v_1^{t_n}$, in particular the integration area is independent of $v$. It follows that the probability of finding $j\in\overline M_{t_n}^t$ for any $t\in[t_n,t_{n+1}]$ is equivalent to $$\label{gfunc}
{\mathbb{P}}\left(j\in \overline M_{t_n}^t\right)=\int_{{\mathbb{R}}^3} \int_{B_{R}( \Xi_0^t)} g^N(x,v,t-t_n)dx dv.$$
Next, we compute for $0<s\leq \Delta t$ $$\begin{aligned}
\bar \rho^N (x,s)&:=\int_{{\mathbb{R}}^3}g^N(x,v,s)dv=\int_{|v|\leq r(s)}g^N(x,v,s)dv+\int_{|v|> r(s)}g^N(x,v,s)dv \notag\\
&\leq C_1{\lVertg^N(\cdot,\cdot,s)\rVert}_\infty r(s)^3+\frac{1}{r(s)^6} \int_{|v|> r(s)}|v|^6g^N(x,v,s)dv\notag\\
&= 2C_1^{\frac{2}{3}}{\lVertg^N(\cdot,\cdot,s)\rVert}_\infty^{\frac{2}{3}} \left(\int_{|v|> r(s)}|v|^6g^N(x,v,s)dv\right)^{\frac{1}{3}},
\end{aligned}$$ where we have chosen $$r(s)=\left(\frac{\int_{|v|> r(s)}|v|^6g^N(x,v,s)dv}{C_1{\lVertg^N(\cdot,\cdot,s)\rVert}_\infty}\right)^{\frac{1}{9}}.$$ It follows that $$\begin{aligned}
\int_{{\mathbb{R}}^3}|\bar \rho ^N(x,s)|^3dx& \leq 8C_1^2{\lVertg^N(\cdot,\cdot,s)\rVert}_\infty^2\iint_{{\mathbb{R}}^6}|v|^6g^N(x,v,s)dxdv\notag\\
&=8C_1^2{\lVertf^N(x-vs,v,t_n)\rVert}_\infty^2\iint_{{\mathbb{R}}^6}|v|^6f^N(x-vs,v,t_n)dxdv\notag\\
&\leq C\left(\|f^N(\cdot,\cdot,t_n)\|_{L^{\infty}({\mathbb{R}}^6)},~
\||v|^{6}f^N(\cdot,\cdot,s)\|_{L^1({\mathbb{R}}^6)}\right),
\end{aligned}$$ which leads to $$\max\limits_{s\in[0,\Delta t]}{\lVert\bar \rho^N(\cdot,s)\rVert}_{3}\leq C_2,$$ because of , where $C_2$ depends only on $T$, and $C_{f_0}$. It follows from that $$\begin{aligned}
{\mathbb{P}}\left(j\in \overline M_{t_n}^t\right)&=\int_{B_{R}(\Xi_0^t)} \bar \rho^N(x,t-t_n)dx\leq {\lVert\bar \rho^N\rVert}_{3} |B_{R}(\Xi_0^t)| ^{\frac{2}{3}}\notag\\
&\leq C_2(\frac{4}{3}\pi) ^{\frac{2}{3}}\left(3N^{-\lambda_2}+ \log (N) \Delta t^{\frac{3}{2}}\right)^2\notag \\
&=C_\ast\left(3N^{-\lambda_2}+ \log (N) \Delta t^{3/2}\right)^2 =:p\;,\end{aligned}$$ where we define $C_\ast:=C_2(\frac{4}{3}\pi) ^{\frac{2}{3}}$, which depends only on $T$ and $C_{f_0}$.
The probability of finding $k$ particles inside the set $\overline M_{t_n}^t$ is thus bounded from above by the binomial probability mass function with parameter $p$ at position $k$, i.e. for any natural number $0\leq A\leq N$ and any $t\in [t_n,t_{n+1}]$ $${\mathbb{P}}\left(\mbox{card }(\overline M_{t_n}^t)\geq A\right)\leq\sum_{j=A}^ N \begin{pmatrix}
N \\
j
\end{pmatrix}
p^j (1-p)^{N-j}.$$ Binomially distributed random variables have mean $Np$ and standard deviation $\sqrt{Np(1-p)}<\sqrt{Np}$, and the probability to find more than $Np+ a \sqrt{Np}$ particles in the set $\overline M_{t_n}^t$ is exponentially small in $a$, i.e. there is a sufficiently large $N$ for any $\alpha>0$ and any $t\in [t_n,t_{n+1}]$ such that $${\mathbb{P}}\left(\mbox{card }(\overline M_{t_n}^t)\geq Np+a\sqrt{Np}\right)\leq a^{-\alpha}\;.$$ This is because of the central limit theory and so the binomial distribution can be seen as a normal distribution when $N$ is sufficiently large. Since $p\geq C N^{-3\lambda_2}$, we get that $\sqrt{Np}>C N^ {\frac{1}{2}(1-3\lambda_2)}$ $(\lambda_2<1/3)$. Hence the probability of finding more than $2Np=Np+\sqrt{Np}\sqrt{Np}$ (i.e. $a=\sqrt{Np}>C N^ {\frac{1}{2}(1-3\lambda_2)}$) particles is the set $\overline M_{t_n}^t$ is smaller than any polynomial in $N$, i.e. there is a $C_\alpha$ for any $\alpha>0$ and any $t\in [t_n,t_{n+1}]$ such that $${\mathbb{P}}((\overline{\mathcal{S}}_{t_n}^t)^c)={\mathbb{P}}\left(\mbox{card }(\overline M_{t_n}^t)\geq 2Np\right)\leq N^{-\alpha}.$$
$\bullet \textit{Step 2:}$ For it is sufficient to show that for any $\alpha>0$ there is a sufficiently large $N$ such that for some $j\in (M_{t_n}^t)^c$ $${\mathbb{P}}\bigg(\max\limits_{t\in[t_n,t_{n+1}]}\left| \frac{1}{N-1} \left(k_1^N(\widetilde x_1^t-\widetilde x_j^t)- k_1^N(\overline x_1^t-\overline x_j^t)\right)\right|> 0\bigg)\leq N^{-\alpha}.$$ The total probability we have to control in is at maximum the $N$-fold value of this. The key to prove that is Lemma \[unlikely\]. To have an interaction $k_1^N(\widetilde x_1^t-\widetilde x_j^t)\neq 0$ for all $t\in[t_n,t_{n+1}]$ the distance between particle $1$ and particle $j$ has to be reduced to a value smaller than $N^ {-\lambda_2}$. Due to the Brownian motion, this is possible, but suppressed. Due to the fast decay of the Gaussian it is very unlikely that $k_1^N(\widetilde x_1^t-\widetilde x_j^t)\neq 0$. The probability is smaller than any polynomial in $N$ (see Lemma \[unlikely\]).The same holds true for $k_1^N(\overline x_1^t-\overline x_j^t)$.
In more detail: due to the cut-off $N^{-\lambda_2}$ we introduced for $k_1^N$
$$\begin{aligned}
&{\mathbb{P}}\bigg( \max\limits_{t\in[t_n,t_{n+1}]}\left| \frac{1}{N-1}\sum_{j\in (M_{t_n}^t)^c}\left(k_1^N(\widetilde x_1^t-\widetilde x_j^t)- k_1^N(\overline x_1^t-\overline x_j^t)\right)\right|> 0\bigg)
\\\leq&
{\mathbb{P}}\bigg(\max\limits_{t\in[t_n,t_{n+1}]}\left| \frac{1}{N-1}\sum_{j\in (M_{t_n}^t)^c}k_1^N(\widetilde x_1^t-\widetilde x_j^t) \right|> 0\bigg)
\notag\\
&+
{\mathbb{P}}\bigg(\max\limits_{t\in[t_n,t_{n+1}]}\left| \frac{1}{N-1}\sum_{j\in (M_{t_n}^t)^c} k_1^N(\overline x_1^t-\overline x_j^t)\right|> 0\bigg)
\\\leq& N{\mathbb{P}}\left(\min_{t\in[t_n,t_{n+1}]}\max_{j\in( M^ t_{t_n})^c}\left\{\left|\widetilde x_{1}^t-\widetilde x_{j}^t\right|\right\}<N^{-\lambda_2}\;\;\right)\\
&+N{\mathbb{P}}\left(\min_{t\in[t_n,t_{n+1}]}\max_{j\in(\overline M^ t_{t_n})^c}\left\{\left|\widetilde x_{1}^t-\widetilde x_{j}^t\right|\right\}<N^{-\lambda_2}\;\;\right),\end{aligned}$$
where we used the fact that $(\overline M^ t_{t_n})^c\subseteq ( M^ t_{t_n})^c$ in the last inequality. With Lemma \[unlikely\] we get the bound for .
$\bullet \textit{Step 3:}$ To get we prove that for any natural number $$\begin{aligned}
0\leq M&\leq M_\ast=2C_\ast N\left(3N^{-\lambda_2}+ \log (N) \Delta t^{3/2}\right)^2 \end{aligned}$$ one has $$\begin{aligned}
{\mathbb{P}}\big(\mathcal{X}(M_{t_n}^t)\cap\{\mbox{ card }(M_{t_n}^t)=M\}\big)\leq N^{-\alpha},\end{aligned}$$ where the event $\mathcal{X}(M_{t_n}^t)$ is defined in . This can be recast without relabeling $j$ as $$\begin{aligned}
\label{toshow}
{\mathbb{P}}\bigg(&\left|\frac{1}{N-1}\sum_{j=1}^{M}\left(k_1^N(\widetilde x_1^t-\widetilde x_j^t)- k_1^N(\overline x_1^t-\overline x_j^t)\right)\right|\notag \\
&\geq C_{\alpha}N^{2\delta-1}\log(N)
+C_\alpha\log^2(N)N^{3\lambda_1-\lambda_2} \|k_1^N\|_1\bigg)\leq N^{-\alpha}.\end{aligned}$$
\[central2\] Let $Z_1,\cdots,Z_M$ be independent random variables with $\mathbb{E}[|Z_i|]\leq CM^{-2}$ and $|Z_i|\leq C$ for any $i\in\{1,\cdots,M\}$. Then for any $\alpha>0$, it holds that $${\mathbb{P}}\left(\sum_{i=1}^{M}|Z_i|\geq C_\alpha \ln(M)\right)\leq M^{-\alpha},$$ where $C_\alpha$ depends only on $C$ and $\alpha$.
We first split the random variables $Z_i=Z_i^a+Z_i^b$ such that $Z_i^a$ and $Z_i^b$ are sequences of independent random variables with $$\mathbb{P}(|Z_i^a|>0)= M^{-1}\: \text{and } |Z_i^b|\leq CM^{-1}.$$ This can be achieved by defining $$Z_i^a(\omega)=\begin{cases} Z_i(\omega) &\mbox{if } Z_i(\omega)>\gamma,\\
0 & \mbox{else }. \end{cases}$$ and $Z_i^b=Z_i-Z_i^a$. Here we choose $\gamma$ such that $\mathbb{P}(|Z_i^a|>0)= M^{-1}$. Applying Markov’s inequality, one computes $$\begin{aligned}
M^{-1}=\mathbb{P}(|Z_i^a|>0)={\mathbb{P}}(Z_i>\gamma)\leq{\mathbb{P}}\left(|Z_i|>\frac{\gamma}{\mathbb{E}[|Z_i|]}\mathbb{E}[|Z_i|]\right)\leq \frac{\mathbb{E}[|Z_i|]}{\gamma}\leq C\frac{M^{-2}}{\gamma}.
\end{aligned}$$ This implies that $\gamma\leq CM^{-1}$.
For the sum of $Z_i^b$ we get the trivial bound $$\sum_{j=1}^M |Z_i^b|\leq CM^{-1} M=C\;.$$ Thus the lemma follows if we can show that $$\label{Za}
{\mathbb{P}}\left(\sum_{i=1}^M|Z_i^a|\geq C_\alpha \ln(M)\right)\leq N^{-\alpha},$$ where $C_\alpha$ has been changed.
Let $$X_i(\omega)=\begin{cases} 0 &\mbox{if } Z_i^a(\omega)=0, \\
1 & \mbox{else }. \end{cases}$$ Since $|Z_i|\leq C$, one has $$\sum_{i=1}^M|Z_i^a|= \sum_{i=1}^MX_i|Z_i|\leq C\sum_{i=1}^MX_i.$$ Then it follows that $$\sum_{i=1}^M|Z_i^a|\geq C_\alpha \ln (M)\Longrightarrow\sum_{j=1}^M X_i\geq \frac{C_\alpha}{C}\ln (M).$$ Noticing that $X_i$ are i.i.d. Bernoulli random variables with ${\mathbb{P}}(X_i=1)=\mathbb{P}(|Z_i^a|>0)= M^{-1}$, we get $$\begin{aligned}
&{\mathbb{P}}\left(\sum_{i=1}^M|Z_i^a|\geq C_\alpha \ln (M)\right)\leq {\mathbb{P}}\left(\sum_{j=1}^M X_i\geq \frac{C_\alpha}{C}\ln (M)\right)
\\
= &\sum_{j=a}^M \frac{M!}{j!(M-j)!} M^{-j}(1-M^{-1})^{M-j}
\leq \sum_{j=a}^M \frac{M^j}{j!} M^{-j}
\leq \frac{2}{a!}\,,
\end{aligned}$$ where $a=\frac{C_\alpha}{C}\ln(M)$. Notice the decay property of the factorial $$\begin{aligned}
(\ln(M))!\geq \left(\frac{\ln(M)}{2}\right)^{\frac{\ln(M)}{2}}=\exp\left(\frac{\ln(M)}{2}\ln\left(\frac{\ln(M)}{2}\right)\right)=M^{\frac{1}{2}\ln\left(\frac{\ln(M)}{2}\right)}.
\end{aligned}$$ Thus one chooses $M$ large enough and concludes , which proves the lemma.
Using the lemma above, now we proceed to prove . Define $$Z_j:=N^{-2\delta }\left(k_1^N(\widetilde x_1^t-\widetilde x_j^t)- k_1^N(\overline x_1^t-\overline x_j^t)\right).$$ It follows that $|Z_j|$ is bounded and $$\begin{aligned}
\mathbb{E}(|Z_j|)\leq CN^{-2\delta }{\lVertk_1^N\rVert}_1\begin{cases}
\|u_{t,t_{n-1}}^{a,N}\|_{\infty,1},~&\mbox{ for }1\leq n\leq M'\,\\
\|f_t^N\|_{\infty,1},~&\mbox{ for } n=1\,
\end{cases}\leq C N^{\frac{9}{2} \lambda_1-2\delta-\lambda_2}\,,\end{aligned}$$ where we use the fact $\|u_{t,t_{n-1}}^{a,N}\|_{\infty,1}\leq CN^{\frac{9}{2}\lambda_1}$ $(\Delta t\leq t-t_{n-1}\leq 2\Delta t)$ from $(i)$ in Lemma \[transition\], $\|f_t^N\|_{\infty,1}\leq C_{f_0}$ and ${\lVertk_1^N\rVert}_1\leq N^{-\lambda_2}$ from . Using Lemma \[central2\] with $M=N^{\delta+\frac{\lambda_2}{2}-\frac{9}{4}\lambda_1}$ one obtains $$\begin{aligned}
{\mathbb{P}}\left(\frac{1}{N-1}\sum_{j=1}^{N^{\delta+\frac{\lambda_2}{2}-\frac{9}{4}\lambda_1}}\left|\left(k_1^N(\widetilde x_1^t-\widetilde x_j^t)- k_1^N(\overline x_1^t-\overline x_j^t)\right)\right|\geq C_\alpha N^{2\delta-1}\ln(N)\right)\leq N^{-\alpha},\end{aligned}$$ which leads to for $M=N^{\delta+\frac{\lambda_2}{2}-\frac{9}{4}\lambda_1}$. It is obvious that $$\sum_{i=1}^{M}|Z_i|\leq \sum_{i=1}^{N^{\delta+\frac{\lambda_2}{2}-\frac{9}{4}\lambda_1}}|Z_i|,$$ for any $M\leq N^{\delta+\frac{\lambda_2}{2}-\frac{9}{4}\lambda_1}$. Thus one concludes holds for the case $M\leq N^{\delta+\frac{\lambda_2}{2}-\frac{9}{4}\lambda_1}$.
For the remaining $M$ we note that $$2C_\ast N\left(3N^{-\lambda_2}+ \log (N)N^{-\frac{3}{2}\lambda_1}\right)^2\leq 4C_\ast N \log^2 (N)N^{-3\lambda_1},$$ due to the fact that $0<\lambda_1<\frac{2}{3}\lambda_2$. Thus we are left to prove for the case $$\label{remainingM}N^{\delta+\frac{\lambda_2}{2}-\frac{9}{4}\lambda_1}<M\leq 4C_\ast N \log^2 (N)N^{-3\lambda_1}\;.$$ This can be done by Lemma \[central\], which we repeat below for easier reference:
#### Lemma \[central\]
Let $Z_1,\cdots,Z_{M}$ be $i.i.d.$ random variables with $\mathbb{E}[Z_i]=0,$ $\mathbb{E}[Z_i^2]\leq g(M)$ and $|Z_i|\leq C\sqrt{M g(M)}$. Then for any $\alpha>0$, the sample mean $\bar{Z}=\frac{1}{M}\sum_{i=1}^{M}Z_i$ satisfies $${\mathbb{P}}\left(|\bar{Z}|\geq\frac{C_\alpha \sqrt{g(M)}\log(M)}{\sqrt{M}}\right)\leq M^{-\alpha},$$ where $C_\alpha$ depends only on $C$ and $\alpha$.
For any fixed $t\in[t_n,t_{n+1}]$ we choose $Z_j^t:=\frac{M}{N-1} k_1^N(\widetilde x_1^t-\widetilde x_j^t)-\frac{M}{N-1} {\mathbb{E}}[k_1^N(\widetilde x_1^t-\widetilde x_j^t)]$ and $g(M):=CM N^{4\delta-2}$, where $N^{\delta+\frac{\lambda_2}{2}-\frac{9}{4}\lambda_1}<M\leq 4C_\ast N \log^2 (N)N^{-3\lambda_1}$. Then following the same argument as in , the condition $$\begin{aligned}
\mathbb{E}[(Z_j^t)^2]\leq&C\frac{M^2}{(N-1)^2} N^{\delta}\begin{cases}
\|u_{t,t_{n-1}}^{a,N}\|_{\infty,1},~&\mbox{ for }1\leq n\leq M'\,\\
\|f_t^N\|_{\infty,1},~&\mbox{ for } n=1\,
\end{cases}\leq C M^2 N^{\delta-2}N^{\frac{9}{2}\lambda_1}\leq g(M),$$ is satisfied. We can also deduce that $$\begin{aligned}
|Z_j^t|\leq C\frac{M}{N-1}N^ {2\delta}\leq\sqrt{M (CM N^{4\delta-2})}=\sqrt{M g(M)}.$$ Applying Lemma \[central\] we obtain at any fixed time $t\in[t_n,t_{n+1}]$ $$\label{part1}
{\mathbb{P}}\left(\left|\frac{1}{N-1}\sum_{j=1}^{M}\left( k_1^N(\widetilde x_1^t-\widetilde x_j^t)-\mathbb{E}[k_1^N(\widetilde x_1^t-\widetilde x_j^t)]\right)\right|\geq C_\alpha N^ {2\delta-1}\log (N)\right)\leq N^{-\alpha},$$ and similarly $$\label{part2}
{\mathbb{P}}\left(\left|\frac{1}{N-1}\sum_{j=1}^{M}\left( k_1^N(\overline x_1^t-\overline x_j^t)-\mathbb{E}[k_1^N(\overline x_1^t-\overline x_j^t)]\right)\right|\geq C_\alpha N^ {2\delta-1}\log (N)\right)\leq N^{-\alpha}\;.$$
It is left to control the difference $$\left|\frac{1}{N-1}\sum_{j=1}^{M}\left({\mathbb{E}}[k_1^N(\widetilde x_1^t-\widetilde x_j^t)]-{\mathbb{E}}[ k_1^N(\overline x_1^t-\overline x_j^t)]\right)\right|,$$ where $M$ satisfies . This can be done by using Lemma \[transition\]. For any $t\in[t_n,t_{n+1}]$, when $1\leq n\leq M'$ we write $a=(\widetilde X_{t_{n-1}},\widetilde V_{t_{n-1}})=(X_{t_{n-1}},V_{t_{n-1}})$ and $b=(\overline X_{t_{n-1}},\overline V_{t_{n-1}})$. Then it follows that $$\begin{aligned}
\label{transresult}
&\left|\frac{1}{N-1}\sum_{j=1}^{M}\left({\mathbb{E}}[k_1^N(\widetilde x_1^t-\widetilde x_j^t)]-{\mathbb{E}}[ k_1^N(\overline x_1^t-\overline x_j^t)]\right)\right|\notag\\
=&\frac{1}{N-1}\bigg|\sum\limits_{j=1}^{M}\int k_1^N(x_1-x_j)\big(u^{a,1,N}_{t,t_{n-1}}(x_1,v_1)u^{a,j,N}_{t,t_{n-1}}(x_j,v_j)\notag \\
& \qquad\qquad-u^{b,1,N}_{t,t_{n-1}}(x_1,v_1)u^{b,j,N}_{t,t_{n-1}}(x_j,v_j)\big)dx_1dv_1dx_jdv_j\bigg| \notag\\
\leq& \frac{1}{N-1}\sum\limits_{j=1}^{M}\left|\int k_1^N(x_1-x_j)u^{a,1,N}_{t,t_{n-1}}(x_1,v_1)\left(u^{a,j,N}_{t,t_{n-1}}(x_j,v_j)-u^{b,j,N}_{t,t_{n-1}}(x_j,v_j)\right)dx_1dv_1dx_jdv_j\right|\notag \\
&+\frac{1}{N-1}\sum\limits_{j=1}^{M}\left|\int k_1^N(x_1-x_j)u^{b,j,N}_{t,t_{n-1}}(x_1,v_1)\left(u^{a,1,N}_{t,t_{n-1}}(x_j,v_j)-u^{b,1,N}_{t,t_{n-1}}(x_j,v_j)\right)dx_1dv_1dx_jdv_j\right|\notag \\
\leq& \frac{1}{N-1}\sum\limits_{j=1}^{M} \left({\lVertu^{a,j,N}_{t,t_{n-1}}-u^{b,j,N}_{t,t_{n-1}}\rVert}_{\infty,1}{\lVertk_1^N\ast\rho_{t,t_{n-1}}^{a,1,N}\rVert}_1+{\lVertu^{a,1,N}_{t,t_{n-1}}-u^{b,1,N}_{t,t_{n-1}}\rVert}_{\infty,1}{\lVertk_1^N\ast\rho_{t,t_{n-1}}^{b,j,N}\rVert}_1\right)\notag \\
\leq& \frac{1}{N-1}\sum\limits_{j=1}^{M} C(t-s)^{-6}|a_i-b_i|({\lVertk_1^N\rVert}_1{\lVert\rho_{t,t_{n-1}}^{a,1,N}\rVert}_1+{\lVertk_1^N\rVert}_1{\lVert\rho_{t,t_{n-1}}^{b,j,N}\rVert}_1)\notag \\
\leq& C\log^2(N)N^{3\lambda_1}|a_i-b_i|{\lVertk_1^N\rVert}_1\leq C\log^2(N)N^{3\lambda_1-\lambda_2}{\lVertk_1^N\rVert}_1,
\end{aligned}$$ where $\rho_{t,t_{n-1}}^{a,1,N}(x_1)=\int_{{\mathbb{R}}^3} u_{t,t_{n-1}}^{a,1,N}(x_1,v_1)dv_1$. Here we have used the fact that when $1\leq n\leq M'$ $${\lVertu^{a,j,N}_{t,t_{n-1}}-u^{b,j,N}_{t,t_{n-1}}\rVert}_{\infty,1}\leq C|a_i-b_i|((t-t_{n-1})^{-6}+1)\leq C|a_i-b_i|N^{6\lambda_1}$$ by Lemma \[transition\] since $N^{-\lambda_1}\leq t-t_{n-1}\leq 2N^{-\lambda_1}$. When $n=1$, since $a=(\widetilde X_{0},\widetilde V_{0})=(X_{0},V_{0})=(\overline X_{0},\overline V_{0})=b$, one has $$\left|\frac{1}{N-1}\sum_{j=1}^{M}\left({\mathbb{E}}[k_1^N(\widetilde x_1^t-\widetilde x_j^t)]-{\mathbb{E}}[ k_1^N(\overline x_1^t-\overline x_j^t)]\right)\right|=0\,.$$ Collecting , and we get for $M$ satisfying , which finishes the proof of for any $M$. Hence we conclude .
$\bullet \textit{Step 4:}$ Now we prove . To see this, we split the summation $\sum_{j\neq 1}^N$ into two parts: the part where $j\in M _{t_n}^t$ and the part where $j\in (M _{t_n}^t )^c$ $$\begin{aligned}
{\mathbb{P}}\bigg(&\left| \frac{1}{N-1}\sum_{j\neq 1}^N\left(k_1^N(\widetilde x_1^t-\widetilde x_j^t)- k_1^N(\overline x_1^t-\overline x_j^t)\right)\right|\geq 2C_{\alpha}N^{2\delta-1}\log(N) \notag \\
&+2C_\alpha\log^2(N)N^{3\lambda_1-\lambda_2} \|k_1^N\|_1\bigg) \leq{\mathbb{P}}\left(\mathcal{X}(M_{t_n}^t)\right)+{\mathbb{P}}\left(\mathcal{X}((M_{t_n}^t)^c)\right)\,,\end{aligned}$$ where $\mathcal{X}(M_{t_n}^t)$ is defined in and $$\begin{aligned}
\label{eventX1}
\mathcal{X}((M_{t_n}^t)^c):=\bigg\{&\big|\frac{1}{N-1}\sum_{j\in (M_{t_n}^t)^c}\left(k_1^N(\widetilde x_1^t-\widetilde x_j^t)- k_1^N(\overline x_1^t-\overline x_j^t)\right)\big|\notag \\
&\geq C_{\alpha}N^{2\delta-1}\log(N) +C_\alpha\log^2(N)N^{3\lambda_1-\lambda_2} \|k_1^N\|_1\bigg\}.\end{aligned}$$ For the part in the event $\mathcal{X}((M_{t_n}^t)^c)$ where $j\in (M _{t_n}^t )^c$, it follows from that $$\begin{aligned}
{\mathbb{P}}\left(\mathcal{X}((M_{t_n}^t)^c)\right)\leq N^{-\alpha}.\end{aligned}$$ Thus we have $$\begin{aligned}
\label{186}
{\mathbb{P}}\bigg(&\left| \frac{1}{N-1}\sum_{j\neq 1}^N\left(k_1^N(\widetilde x_1^t-\widetilde x_j^t)- k_1^N(\overline x_1^t-\overline x_j^t)\right)\right|\geq 2C_{\alpha}N^{2\delta-1}\log(N) \notag \\
&+2C_\alpha\log^2(N)N^{3\lambda_1-\lambda_2} \|k_1^N\|_1\bigg)\leq {\mathbb{P}}\left(\mathcal{X}(M_{t_n}^t)\right) +N^{-\alpha}.\end{aligned}$$
Next we split the summation $\sum_{j\in M _{t_n}^t }$ in the event $\mathcal{X}(M_{t_n}^t)$ into two cases: the case where $\mbox{ card }(M_{t_n}^t)\leq M_\ast$ and the case where $\mbox{ card }(M_{t_n}^t)> M_\ast$. Here $M_\ast$ is defined in . $$\begin{aligned}
\label{187}
{\mathbb{P}}\left(\mathcal{X}(M_{t_n}^t)\right)& \leq {\mathbb{P}}\big(\mathcal{X}(M_{t_n}^t)\cap \left\{\mbox{ card }(M_{t_n}^t)\leq M_\ast\right\}\big) +{\mathbb{P}}\big(\mathcal{X}(M_{t_n}^t)\cap \left\{\mbox{ card }(M_{t_n}^t)> M_\ast\right\}\big) \notag \\
&\leq{\mathbb{P}}\big(\mathcal{X}(M_{t_n}^t)\cap \left\{\mbox{ card }(M_{t_n}^t)> M_\ast\right\}\big)+N^{-\alpha},\end{aligned}$$ where in the last inequality we used .
According to , for any $t\in[t_n,t_{n+1}]$ one has $${\mathbb{P}}\left(\mbox{card }(M_{t_n}^t)> M_\ast\right)\leq N^{-\alpha},$$ which leads to $$\begin{aligned}
{\mathbb{P}}\big(\mathcal{X}(M_{t_n}^t)\cap \left\{\mbox{ card }(M_{t_n}^t)> M_\ast\right\}\big)\leq {\mathbb{P}}\left(\mbox{card }(M_{t_n}^t)> M_\ast\right)\leq N^{-\alpha}.\end{aligned}$$ Therefore it follows from that $$\begin{aligned}
{\mathbb{P}}\left(\mathcal{X}(M_{t_n}^t)\right)\leq 2N^{-\alpha}.\end{aligned}$$ Together with , it implies $$\begin{aligned}
\label{goal1}
{\mathbb{P}}\bigg(&\left| \frac{1}{N-1}\sum_{j\neq 1}^N\left(k_1^N(\widetilde x_1^t-\widetilde x_j^t)- k_1^N(\overline x_1^t-\overline x_j^t)\right)\right|\geq 2C_{\alpha}N^{2\delta-1}\log(N) \notag \\
&+2C_\alpha\log^2(N)N^{3\lambda_1-\lambda_2} \|k_1^N\|_1\bigg)\leq 3N^{-\alpha}.\end{aligned}$$
Finally, since the particles are exchangeable, the same result holds for changing $(\widetilde x_1^t,\overline x_1^t)$ in into $(\widetilde x_i^t,\overline x_i^t)$, $i=2,\cdots, N$, which completes the proof of Lemma \[tildaXficedtime\].
[**Acknowledgments:**]{} H.H. is partially supported by NSFC (Grant No. 11771237). The research of J.-G. L. is partially supported by KI-Net NSF RNMS (Grant No. 1107444) and NSF DMS (Grant No. 1812573).
Appendix {#appendix .unnumbered}
========
Proof of Lemma \[transition\]
=============================
First, let us consider the fundamental solution $G(x,v, t)$ of the equation $$\partial_t G + v\cdot \nabla_xG=\Delta_v G,\;\;G\mid_{t=0}=\delta(x)\delta(v),$$ which can be calculated explicitly as $$\label{gexpl}
G(x,v, t)=C \frac1{t^6}\exp\left(-\frac{|v|^2}{4t}-\frac{3|x- tv/2|^2}{t^3}\right),$$ where $C$ is a normalization constant. The following lemma states some estimates of the fundamental solution.
Let $G(x,v, t)$ be defined in and $p\in[1,\infty]$. There exists a $C_p$ such that for any $j\in\mathbb{N}_0$ the following holds $$\begin{aligned}
\label{heat1}
\left\||x|^ j\nabla_v G \right\|_{p,1}\leq C_p t^{\frac{-10p+3jp+9}{2p}}\;,\quad \left\||x|^ j\nabla_x G \right\|_{p,1}\leq C_p t^{\frac{-12p+3jp+9}{2p}}\; ,
\end{aligned}$$ and $$\begin{aligned}
\label{heat3}
\left\|G\left(\cdot-\frac{1}{2}\left(a_i-b_i\right)\right)-G\left(\cdot-\frac{1}{2}\left(b_i-a_i\right)\right)\right\|_{p,1}\leq C_p|a_i-b_i|\left(t^{\frac{-12p+9}{2p}}+t^{\frac{-10p+9}{2p}}\right)\;,
\end{aligned}$$ as well as $$\begin{aligned}
\label{heat2}
\left\||\cdot|\left(G\left(\cdot-\frac{1}{2}\left(a_i-b_i\right)\right)-G\left(\cdot-\frac{1}{2}\left(b_i-a_i\right)\right)\right)\right\|_{p,1}\leq C_p|a_i-b_i|\left(t^{\frac{-7p+9}{2p}}+t^{\frac{-9p+9}{2p}}\right)\;.
\end{aligned}$$ The norm $\|\cdot\|_{p,q}$ denotes the $p$-norm in the $x$ and $q$-norm in the $v$-variable, i.e. for any $f:\mathbb{R}^3\times\mathbb{R}^3\to\mathbb{R} $ $$\|f\|_{p,q}:=\left(\int_{{\mathbb{R}}^3} \left(\int_{{\mathbb{R}}^3} |f(x,v)|^q dv\right)^{p/q}dx\right)^{1/p}.$$
It is easy to compute that $$G=C \frac1{t^6} \exp\left(-\frac{3|x|^2}{4t^3}\right) \exp\left(-\frac{|v - \frac{3x}{2t}|^2}{t}\right)$$ and $$\nabla_vG=C\frac1{t^6} \exp\left(-\frac{3|x|^2}{4t^3}\right) \exp\left(-\frac{|v - \frac{3x}{2t}|^2}{t}\right)\left(-\frac{2v}{t}+\frac{3x}{t^2}\right).$$
Now we can do the calculation of $\int_{{\mathbb{R}}^3}|G|dv$ and $\int_{{\mathbb{R}}^3}|\nabla_vG|dv$: $$\begin{aligned}
\label{G}
\int_{{\mathbb{R}}^3}|G|dv&=C\frac1{t^6} \exp\left(-\frac{3|x|^2}{4t^3}\right) \int_{{\mathbb{R}}^3}\exp\left(-\frac{|v - \frac{3x}{2t}|^2}{t}\right)dv\notag\\
&\leq C\frac1{t^{9/2}} \exp\left(-\frac{3|x|^2}{4t^3}\right),
\end{aligned}$$ and $$\begin{aligned}
\label{vG}
\int_{{\mathbb{R}}^3}|\nabla_vG|dv&=C\frac1{t^6} \exp\left(-\frac{3|x|^2}{4t^3}\right) \int_{{\mathbb{R}}^3}\exp\left(-\frac{|v - \frac{3x}{2t}|^2}{t}\right)\left(-\frac{2v}{t}+\frac{3x}{t^2}\right)dv\notag\\
&\leq C\frac1{t^5} \exp\left(-\frac{3|x|^2}{4t^3}\right) \int_{{\mathbb{R}}^3}u\exp\left(-u^2\right)du\leq C\frac1{t^5} \exp\left(-\frac{3|x|^2}{4t^3}\right) ,
\end{aligned}$$ respectively. As a direct result from and , one has $$\left\||\cdot|^jG \right\|_{\infty,1}\leq C\frac1{t^{(9-3j)/2}}\hspace{1cm}\left\||\cdot|^ j\nabla_v G \right\|_{\infty,1}\leq C\frac1{t^{5-3j/2}} .$$ For $1\leq p<\infty$ $$\begin{aligned}
\left\||\cdot|^jG \right\|_{p,1}&\leq Ct^{-{9/2}}\left(\int_{{\mathbb{R}}^3}|x|^{pj}\exp\left(-\frac{3px^2}{4t^3}\right)dx\right)^{\frac{1}{p}}\notag \\
&\leq C_pt^{-\frac{9}{2}+\frac{9+3pj}{2p}}\left(\int_{{\mathbb{R}}^3}|y|^ j\exp\left(-y^2\right)dy\right)^{\frac{1}{p}}\leq C_p t^{\frac{-9p+3jp+9}{2p}},\label{Gnorm}
\end{aligned}$$ and $$\begin{aligned}
\left\||\cdot|^j\nabla_v G \right\|_{p,1}&\leq Ct^{-5}\left(\int_{{\mathbb{R}}^3}|x|^ j\exp\left(-\frac{3px^2}{4t^3}\right)dx\right)^{\frac{1}{p}}\notag \\
&\leq C_pt^{-5+\frac{9+3jp}{2p}}\left(\int_{{\mathbb{R}}^3}|y|^ j\exp\left(-y^2\right)dy\right)^{\frac{1}{p}}\leq C_p t^{\frac{-10p+3jp+9}{2p}}.
\end{aligned}$$ We also have $$\begin{aligned}
\nabla_xG= \frac1{t^6} \exp\left(-\frac{3|x|^2}{4t^3}\right) \exp\left(-\frac{|v - \frac{3x}{2t}|^2}{t}\right)
\left(-\frac{6x}{t^2}+\frac{3v}{t}\right),
\end{aligned}$$ which leads to $$\begin{aligned}
\label{xG}
\int_{{\mathbb{R}}^3}|\nabla_xG|dv&=C\frac1{t^6} \exp\left(-\frac{3|x|^2}{4t^3}\right) \int_{{\mathbb{R}}^3}\exp\left(-\frac{|v - \frac{3x}{2t}|^2}{t}\right)\left(-\frac{6x}{t^3}+\frac{3v}{t^2}\right)dv\notag\\
&\leq C\frac1{t^8} \exp\left(-\frac{3|x|^2}{4t^3}\right) \int_{{\mathbb{R}}^3}\exp\left(-\frac{|v - \frac{3x}{2t}|^2}{t}\right)\left(-\frac{2x}{t}+v\right)dv\notag \\
&\leq Ct^{-8+\frac{3}{2}} \exp\left(-\frac{3|x|^2}{4t^3}\right) \int_{{\mathbb{R}}^3}(\sqrt{t}u-\frac{x}{2t})\exp\left(-u^2\right)du \notag\\
&\leq C\frac1{t^6} \exp\left(-\frac{3|x|^2}{4t^3}\right)+C\frac1{t^6} \frac{x}{t^{\frac{3}{2}}}\exp\left(-\frac{3|x|^2}{4t^3}\right).
\end{aligned}$$ It follows from the above that $$\left\||\cdot|^j\nabla_x G \right\|_{\infty,1}\leq C\frac1{t^{6-3j/2}} .$$ For $1\leq p<\infty$ $$\begin{aligned}
&{\lVert|\cdot|^j\nabla_x G\rVert}_{p,1}\notag\\
\leq& C\frac1{t^6}\left(\left(\int_{{\mathbb{R}}^3}|x|^j\exp\left(-\frac{3px^2}{4t^3}\right)dx\right)^{\frac{1}{p}} +\left(\int_{{\mathbb{R}}^3}|x|^ j\exp\left(-\frac{3p|x|^2}{4t^3}\right) \left (\frac{x}{t^{\frac{3}{2}}}\right)^pdx\right)^{\frac{1}{p}}\right)\notag \\
\leq &C_pt^{\frac{-12p+3jp+9}{2p}}\left(\left(\int_{{\mathbb{R}}^3}|y|^ j\exp\left(-y^2\right)dy\right)^{\frac{1}{p}}+\left(\int_{{\mathbb{R}}^3}|y|^j\exp\left(-py^2\right)|y|^pdy\right)^{\frac{1}{p}}\right)\notag\\
\leq &C_pt^{\frac{-12p+3jp+9}{2p}},
\end{aligned}$$ which concludes the proof of .
As a direct result of we can prove . Indeed, $$\begin{aligned}
&\left|G\left(\cdot-\frac{1}{2}\left(a_i-b_i\right)\right)-G\left(\cdot-\frac{1}{2}\left(b_i-a_i\right)\right)\right|\notag\\
\leq& |a_i-b_i|\int_0^1\left|\nabla G\left(\cdot-\frac{1}{2}(b_i-a_i)+s(b_i-a_i)\right)\right|ds \notag\\
\leq & |a_i-b_i|\int_0^1\left|\nabla_v G\left(\cdot-\frac{1}{2}(b_i-a_i)+s(b_i-a_i)\right)\right|ds \notag\\
&+|a_i-b_i|\int_0^1\left|\nabla_x G\left(\cdot-\frac{1}{2}(b_i-a_i)+s(b_i-a_i)\right)\right|ds,
\end{aligned}$$ which leads to $$\begin{aligned}
&\left\|\left(G\left(\cdot-\frac{1}{2}\left(a_i-b_i\right)\right)-G\left(\cdot-\frac{1}{2}\left(b_i-a_i\right)\right)\right)\right\|_{p,1}\notag \\
\leq &C|a_i-b_i|\left({\lVert\nabla_v G\rVert}_{p,1}+{\lVert\nabla_x G\rVert}_{p,1}\right)
\leq C_p|a_i-b_i|\left(t^{\frac{-12p+9}{2p}}+t^{\frac{-10p+9}{2p}}\right).
\end{aligned}$$
Next we prove : $$\begin{aligned}
&|\cdot| \left(G\left(\cdot-\frac{1}{2}\left(a_i-b_i\right)\right)-G\left(\cdot-\frac{1}{2}\left(b_i-a_i\right)\right)\right)\notag\\
\leq& \left(|\cdot-\frac{1}{2}\left(a_i-b_i\right)| G\left(\cdot-\frac{1}{2}\left(a_i-b_i\right)\right)-|\cdot-\frac{1}{2}\left(b_i-a_i\right)| G\left(\cdot-\frac{1}{2}\left(b_i-a_i\right)\right)\right)
\notag\\&+\frac{1}{2}|a_i-b_i|\left(G\left(\cdot-\frac{1}{2}\left(a_i-b_i\right)\right)+G\left(\cdot-\frac{1}{2}\left(b_i-a_i\right)\right)\right).
\end{aligned}$$ In view of , the $(p,1)$-norm of the terms in the last line have the right bound. With the other term we proceed as above, using the function $H=|\cdot|G$: $$\begin{aligned}
&
\left(|\cdot-\frac{1}{2}\left(a_i-b_i\right)| G\left(\cdot-\frac{1}{2}\left(a_i-b_i\right)\right)-|\cdot-\frac{1}{2}\left(b_i-a_i\right)| G\left(\cdot-\frac{1}{2}\left(b_i-a_i\right)\right)\right)
\notag\\\leq& |a_i-b_i|\int_0^1\left|\nabla H\left(\cdot-\frac{1}{2}(b_i-a_i)+s(b_i-a_i)\right)\right|ds \notag\\
\leq & |a_i-b_i|\int_0^1\left|\nabla_v H\left(\cdot-\frac{1}{2}(b_i-a_i)+s(b_i-a_i)\right)\right|ds \notag\\
&+|a_i-b_i|\int_0^1\left|\nabla_x H\left(\cdot-\frac{1}{2}(b_i-a_i)+s(b_i-a_i)\right)\right|ds.
\end{aligned}$$ It follows from our estimates in that $$\begin{aligned}
&\left\||\cdot-\frac{1}{2}\left(a_i-b_i\right)| G\left(\cdot-\frac{1}{2}\left(a_i-b_i\right)\right)-|\cdot-\frac{1}{2}\left(b_i-a_i\right)| G\left(\cdot-\frac{1}{2}\left(b_i-a_i\right)\right)\right\|_{p,1}
\\\leq & C|a_i-b_i|\left(\left\||\cdot|\nabla_v G\right\|_{p,1}+\left\||\cdot|\nabla_x G\right\|_{p,1}+\left\| G\right\|_{p,1}\right)\notag\\
\leq&C_p|a_i-b_i|\left(t^{\frac{-7p+9}{2p}}+t^{\frac{-9p+9}{2p}}\right),
\end{aligned}$$ which leads to .
The proof of the estimates follows the ideas of [@garcia2017 Lemma 2]. However, the evolution equation for the present system is more difficult to handle, and in particular, the spacial overlap is suppressed for short periods of time since we have a noise term in the momentum variable only. Both estimates can be proved in the same way. We just give the proof for the more difficult part $(ii)$, which can be easily adapted for part $(i)$. Without loss of generality we set $s=0$ and $t<1$. What we need to show then is $$\| u_{t, s}^{a, i,N} -
u_{t, s}^{b,i, N} \|_{\infty,1} \leqslant C | a_i - b_i|\left((t - s)^{- 6}+1\right)$$ holds for all $i=1,\cdots,N.$
Note that the force $\overline k_t^N(x):=k^N\ast \rho^N_t$ we consider is globally Lipschitz and $L^\infty$ because of , thus there exists a $C>0$ independent of $N$ such that $$\label{lipschf}
\max_{0\leq t\leq T;x,y\in\mathbb{R}^3}\frac{|\overline k_t(x)-\overline k_t(y)|}{|x-y|}\leq C\;.$$
Let $c_t$ be the trajectory on phase space following the Newtonian equations of motion with respect to the force $\overline k_t^N$, starting with $\frac{1}{2}(a_i+b_i)$ at time $0$, i.e. $$c_t=(x^c_t,v^c_t),\hspace{1cm}\frac{d}{dt}x^c_t=v^c_t,\hspace{1cm} \frac{d}{dt}v^c_t=\overline k_t^N(x^c_t),\hspace{1cm} c_0=\frac{1}{2}(a_i+b_i)\;.$$
We use the trajectory $c$ to change the frame of inertia that we use to look at $u_{t, s}^{d,i,N}$ for $d\in\{a,b\}$, i.e. we define for any $t>0$ the density $w_{t, 0}^{a,i,N}$ on phase space by $$w_{t, 0}^{a,i,N}((x,v)):=u_{t, 0}^{a,i,N}((x,v)+c_t)\,.$$
From the evolution equation of $u_{t, s}^{d,i,N}$ for $d\in\{a,b\}$ and $c_t$ one gets directly $$\label{defw}
\frac{\partial}{\partial t} w_{t, 0}^{d,i,N}\left(x,v\right):=\Delta_v w_{t, 0}^{d,i,N}\left(x,v\right)-\nabla_x w_{t, 0}^{d,i,N}\cdot v-\nabla_v w_{t, 0}^{d,i,N}\cdot \left(\overline k_t^N\left(x+x^c_t\right)-\overline k_t^N\left(x^c_t\right)\right),$$ with $w_{0, 0}^{a,i,N}=\delta\left(\cdot-\left(\frac{1}{2}\left(a_i-b_i\right)\right)\right)$ and $w_{0, 0}^{b,i,N}=\delta\left(\cdot-\left(\frac{1}{2}\left(b_i-a_i\right)\right)\right)$.
Since $w$ is built from $u$ by translation we have for any $1\leq p \leq\infty$ $$\label{diffu}
\| u_{t, 0}^{a,i,N} -
u_{t, 0}^{b,i,N} \|_{p,1}=\| w_{t, 0}^{a,i,N} -
w_{t, 0}^{b,i,N} \|_{p,1}\;.$$
Before proceeding we would like to explain the advantage of looking at $w$ instead of $u$ first on a heuristic level. The difficulties arise when dealing with short periods of time. There the $u^{d}$, $d\in\{a,b\}$ are roughly given by a Gaussian around the center at $\frac{1}{2}(a+b)$, respectively the $w^ {d}$ are roughly given by a Gaussian around the center at $0$. Here the force term of $w$ – which is zero at $x=0$ – suppresses the last term of . Thus $w $ will be very close to the heat-kernel $G_t$ of our time evolution.
Using and the properties of the heat kernel we get $$\begin{aligned}
\nonumber
w_{t, 0}^{a,i,N}=& G_t\ast \delta\left(\cdot-\left(\frac{1}{2}\left(a_i-b_i\right)\right)\right) - \int_{0}^t G_{t-s}\ast \left(\nabla_v w_{s, 0}^{a,i,N}\cdot \left(\overline k_s\left(\cdot+x^c_s\right)-\overline k_s\left(x^c_s\right)\right)\right) ds
\\=& G_t\left(\cdot-\frac{1}{2}\left(a_i-b_i\right)\right) - \int_{0}^t \nabla_vG_{t-s}\ast\left( w_{s, 0}^{a,i,N} \left(\overline k_s\left(\cdot+x^c_s\right)-\overline k_s\left(x^c_s\right)\right)\right) ds ,\label{wformula}
\end{aligned}$$ and $$\begin{aligned}
w_{t, 0}^{b,i,N}= G_t\left(\cdot-\frac{1}{2}\left(b_i-a_i\right)\right) - \int_{0}^t \nabla_vG_{t-s}\ast\left( w_{s, 0}^{b,i,N} \left(\overline k_s\left(\cdot+x^c_s\right)-\overline k_s\left(x^c_s\right)\right)\right) ds,
\end{aligned}$$ thus $$\begin{aligned}
\label{diffw}
w_{t, 0}^{a,i,N}-w_{t, 0}^{b,i,N}=& \left(G_t\left(\cdot-\frac{1}{2}\left(a_i-b_i\right)\right)-G_t\left(\cdot-\frac{1}{2}\left(b_i-a_i\right)\right)\right) \\&- \int_{0}^t \nabla_vG_{t-s}\ast\left( \left( w_{s, 0}^{a,i,N}-w_{s, 0}^{b,i,N}\right) \left(\overline k_s\left(\cdot+x^c_s\right)-\overline k_s\left(x^c_s\right)\right)\right) ds.\nonumber
\end{aligned}$$
Defining $\eta_{t, 0}^{ N}:\mathbb{R}^6\to\mathbb{R}^+_0$ by $\eta_{t, 0}^{ N}(x,v):=|(x,v)|\left|w_{t, 0}^{a,i,N}-w_{t, 0}^{b,i,N}\right|$ and using , we can find a constant $C$ such that $$\begin{aligned}
\label{etaform}
\eta_{t, 0}^{ N}\leq & |\cdot| \left|G_t\left(\cdot-\frac{1}{2}\left(a_i-b_i\right)\right)-G_t\left(\cdot-\frac{1}{2}\left(b_i-a_i\right)\right)\right| \\&+ C \left|\int_{0}^t \nabla_v G_{t-s}\ast\eta_{s, 0}^{ N} ds\right|.\nonumber
\end{aligned}$$ Using the properties of the heat kernel , and Young’s inequality in , we get $$\begin{aligned}
\left\|\eta_{t, 0}^{ N}\right\|_{1,1}\leq & C|a_i-b_i|+ C \int_{0}^t (t-s)^{-\frac{1}{2}}\left\|\eta_{s, 0}^{ N} \right\|_{1,1}ds.
\end{aligned}$$ Applying a generalized Gronwall’s inequality with weak singularities [@henry2006geometric Lemma 7.1.1] leads to $$\label{l1}
\|\eta_{t, 0}^{N}\|_{1,1}\leq C |a_i-b_i| \hspace{1cm}\text{ uniform in }t\in[0,T]\;.$$ Further gives for any $1\leq p\leq\infty$ and $t\in[0,T]$ $$\begin{aligned}
\label{use}
\left\|\eta_{t, 0}^{N}\right\|_{p,1}\leq & \left\||\cdot|G_t\left(\cdot-\frac{1}{2}\left(a_i-b_i\right)\right)-G_t\left(\cdot-\frac{1}{2}\left(b_i-a_i\right)\right)\right\|_{p,1} \\&+ C\int_{0}^{t/2} \left\|\nabla_v G_{t-s}\ast\eta_{s, 0}^{ N}\right\|_{p,1}ds
+ C \int_{t/2}^t \left\|\nabla_v G_{t-s}\ast\eta_{s, 0}^{ N}\right\|_{p,1} ds.\nonumber
\end{aligned}$$ Using Young’s inequality we get for $1+p^{-1}=\frac{9}{10}+q^{-1}$ and $t\in[0,T]$, $$\begin{aligned}
\left\|\eta_{t, 0}^{ N}\right\|_{p,1}\leq & C|a_i-b_i|t^{\frac{-9p+9}{2p}} + C \int_{0}^{t/2} \left\|\nabla_v G_{t-s}\right\|_{p,1} \left\|\eta_{s, 0}^{ N}\right\|_{1,1} ds
\\&+ C \int_{t/2}^{t} \left\|\nabla_v G_{t-s}\right\|_{10/9,1} \left\|\eta_{s, 0}^{N}\right\|_{q,1} ds.
\end{aligned}$$ Due to , one has $\left\|\nabla_v G_{t-s}\right\|_{10/9,1}\leq C (t-s)^{-19/20}$. This and give $$\begin{aligned}
\label{iteration}
\left\|\eta_{t, 0}^{ N}\right\|_{p,1}\leq C|a_i-b_i|t^{\frac{-9p+9}{2p}} + C|a_i-b_i| \int_{0}^{t/2} \left\|\nabla_v G_{t-s}\right\|_{p,1} ds
+ C \max_{t/2\leq s \leq t}\left\|\eta_{s, 0}^{N}\right\|_{q,1}.
\end{aligned}$$
We use this formula starting at $p_1=1$ and setting $p_{k+1} = \frac{10p_k}{10-p_k}$. Therefore, starting with our estimate for $ \left\|\eta_{t, 0}^{a,i,N}\right\|_{1,1}$ (see ) we can then iteratively estimate the $L^p$ norms of $\eta_{t, 0}^{ N}$ for higher exponents, i.e. $$\begin{aligned}
\label{iteration1}
\left\|\eta_{t, 0}^{ N}\right\|_{p_{k+1},1}\leq C|a_i-b_i|t^{\frac{-9p_{k+1}+9}{2p_{k+1}}} + C|a_i-b_i| \int_{0}^{t/2} \left\|\nabla_v G_{t-s}\right\|_{p_{k+1},1} ds
+ C \max_{t/2\leq s \leq t}\left\|\eta_{s, 0}^{N}\right\|_{p_{k},1}.
\end{aligned}$$ The exponent $p_{k+1} = \infty$ is attained after $k = 10$ steps. It follows that $$\begin{aligned}
\label{etaest}
\left\|\eta_{t, 0}^{ N}\right\|_{\infty,1}\leq C|a_i-b_i|(t^{\frac{-9}{2}}+1)\;.
\end{aligned}$$
Having good control of $\|\eta_{t, 0}^{ N}\|_{\infty,1}$ we can now estimate $w_{t, 0}^{a,i,N}-w_{t, 0}^{b,i,N}$ using : $$\begin{aligned}
\left\|w_{t, 0}^{a,i,N}-w_{t, 0}^{b,i,N}\right\|_{\infty,1}\leq& \left\|G_t\left(\cdot-\frac{1}{2}\left(a_i-b_i\right)\right)-G_t\left(\cdot-\frac{1}{2}\left(b_i-a_i\right)\right)\right\|_{\infty,1} \\&+ \int_{0}^t \left\| \nabla_vG_{t-s}\ast\left( \left( w_{s, 0}^{a,i,N}-w_{s, 0}^{b,i,N}\right) \left(\overline k_s\left(\cdot+x^c_s\right)-\overline k_s\left(x^c_s\right)\right)\right)\right\|_{\infty,1} ds\nonumber
\\\leq& C|a_i-b_i|t^{-6} +C\int_{0}^{t/2} \left\| \nabla_vG_{t-s}\right\|_{\infty,1} \left\|\eta_{s, 0}^{N}\right\|_{1,1} ds\nonumber
\\&+\int_{t/2}^t \left\| \nabla_vG_{t-s}\right\|_{1,1} \left\|\eta_{s, 0}^{N}\right\|_{\infty,1} ds\nonumber
\\\leq& C|a_i-b_i|t^{-6}+C|a_i-b_i|\int_{0}^{t/2}(t-s)^{-5}ds+C|a_i-b_i|\int_{t/2}^t(t-s)^{-\frac{1}{2}} (s^{-\frac{9}{2}}+1)ds \notag\\
\leq & C|a_i-b_i|\left(t^{-6}+t^{-4}+t^{\frac{1}{2}}\right)\leq C|a_i-b_i|(t^{-6}+1).
\end{aligned}$$ With statement $(ii)$ of the lemma follows.
[^1]: Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada. Email: [email protected]
[^2]: Departments of Physics and Mathematics, Duke University, Durham, NC, USA. Email: [email protected]
[^3]: Mathematisches Institut, Universität München, München, Germany. Email: [email protected]
[^4]: Duke Kunshan University, Kunshan, Jiangsu, China. Email: [email protected]
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---
abstract: 'Recent work of An, Drummond-Cole, and Knudsen [@ADK], as well as the author [@R], has shown that the homology groups of configuration spaces of graphs can be equipped with the structure of a finitely generated graded module over a polynomial ring. In this work we study this module structure in certain families of graphs using the language of $\operatorname{FI}$-algebras recently explored by Nagel and Römer [@NR]. As an application we prove that the syzygies of the modules in these families exhibit a range of stable behaviors.'
address: 'University of Michigan Department of Mathematics, 530 Church St., Ann Arbor, MI 48109'
author:
- Eric Ramos
title: 'An application of the theory of $\operatorname{FI}$-algebras to graph configuration spaces'
---
Introduction
============
There has been a recent raising interest in studying *configuration spaces of graphs* due to their connections with robotics and topological motion planning [@Fa; @G]. In this paper, a *graph* is a 1-dimensional, compact simplicial complex. Its *$n$-stranded configuration space* is the topological space $$U{\mathcal{F}}_n(G) := \{(x_1,\ldots,x_n) \in G^n \mid x_i \neq x_j\}/{\mathfrak{S}}_n.$$
In this paper, we explore the homology groups of these configuration spaces from a relatively new perspective, following the work of An, Drummond-Cole, and Knudsen [@ADK], as well as the author [@R].
For each $q \geq 0$, and each graph $G$, the *total $q$-th homology group* is the graded abelian group $$\mathcal{H}_q(G) := \bigoplus_{n \geq 0} H_q(U{\mathcal{F}}_n(G))$$ Writing $A_G$ for the integral polynomial ring with variables indexed by the edges of $G$, the following theorem was proven in the case where $G$ is a tree by the author [@R], and in the general case by An, Drummond-Cole, and Knudsen.
The graded abelian group $\mathcal{H}_q(G)$ can be equipped with the structure of a finitely generated, graded module over $A_G$.
It therefore becomes natural to ask whether one can use techniques from commutative algebra to deduce facts about these homology groups. In [@R], the author showed that the total homology groups could be decomposed into sums of graded shifts of square-free monomial ideals, in the cases wherein $G$ is a tree. Other than that work, however, very little has been done in this direction. In this work, we study the modules $\mathcal{H}_q(G)$ for specific families of graphs $G$, using recent work of White and the author [@RW], and Lütgehetmann [@L], as well as Nagel and Römer [@NR].
In [@RW], White and the author introduced the notion of a *finitely generated $\operatorname{FI}$-graph* (see Definition \[figraphdef\]). For the purposes of this introduction, we restrict to a specific natural class of examples of this structure. Fix two graphs, $G, H$, as well as a choice of vertex $v_G,v_H$ in each. Then for each $n \geq 0$ we may define $$G_n := G \bigvee^n H.$$ The sequence of graphs $\{G_n\}$ is notable for us, as it carries an action by the category $\operatorname{FI}$ of finite sets and injections. Indeed, if we set $[n] := \{1,\ldots, n\}$ and let $f:[n] \hookrightarrow [r]$ be an injection, then we obtain an injective homomorphism of graphs (see Definition \[graphdef\]) $$G(f):G_n \rightarrow G_r$$ which fixes the vertices of $G$, while permuting the vertices of $H$ according to $f$. This structure was first exploited in the study of configuration spaces by Lütgehetmann in [@L]. In particular, the following is proven.
Let $G_n$ be as above. Then for each $q,m \geq 0$, the functor $$[n] \mapsto H_q(U{\mathcal{F}}_m(G_n))$$ from $\operatorname{FI}$ to abelian groups is finitely generated in the sense of [@CEF].
Therefore, if we fix the number of points being configured, but allow our graph to grow in a certain way, one obtains a finiteness result. On the other hand, we have also seen that if we fix the graph, and allow the number of points being configured to increase, we obtain a similar finiteness result. The main theorem of this work is that these two types of finiteness are compatible with one another in the appropriate sense.
\[mainthmab\] Let $G_n$ be as above. Then for each $q \geq 0$, the functor $$[n] \mapsto \mathcal{H}_q(G_n)$$ from $\operatorname{FI}$ to graded abelian groups is finitely generated as a graded module over the functor $$[n] \mapsto A_{G_n}$$ from $\operatorname{FI}$ to ${{\mathbb{Z}}}$-algebras, in the sense of [@NR] (see also Definition \[fgdef\]).
As previously stated, the most general theorem involves studying a larger class of families of graphs (see Theorem \[mainthm\]). That being said, it is unclear whether the scope this theorem extends to all finitely generated $\operatorname{FI}$-graphs (see Definition \[figraphdef\]). This difficulty is fundamentally related to the fact that the $\operatorname{FI}$-algebra $$[n] \mapsto A_{G_n}$$ is not necessarily Noetherian in general. It is an interesting question to ask whether the conclusion of Theorem \[mainthmab\] will hold for all $\operatorname{FI}$-graphs, even in the case where $G_n = K_n$ is the complete graph on $n$ vertices.
Coming back to the goal of studying the commutative algebra of $\mathcal{H}_q(G)$, we note that Theorem \[mainthmab\] has concrete consequences related to the Betti numbers of these modules. If $M$ is a finitely generated graded module over the polynomial ring $k[x_1,\ldots,x_n]$, where $k$ is a field, then it admits a minimal free resolution $$0 \rightarrow F^{(n)} \rightarrow \ldots \rightarrow F^{(0)} \rightarrow M \rightarrow 0$$ Tensoring with the residue field $k$, we define the *$j$-th Betti number in homological degree $p$* to be $$\beta_{p,j}(M) := \dim_k(\operatorname{Tor}_p(M,k)_j).$$ Knowledge of the Betti numbers allows one to access the number of generators in each grade of the $p$-th syzygies. Our second main result is that the Betti numbers of the modules $\mathcal{H}_q(G_n)$ stabilize in a strong sense.
\[bettithm\] Let $G_n$ be as above, and let $k$ be a field. Then for each $p,j \geq 0$, the function $$n \mapsto \beta_{p,j}(\mathcal{H}_q(G_n;k))$$ agrees with a polynomial for $n \gg 0$. Moreover, there exists a finite list of integers $$j_0(\mathcal{H}_q(G_n;k),p) < \ldots < j_t(\mathcal{H}_q(G_n;k),p)$$ such that for all $n \gg 0$ $$\beta_{n,p,j}(\mathcal{H}_q(G_n;k)) \neq 0 \iff j \in \{j_0(\mathcal{H}_q(G_n;k),p), \ldots, j_t(\mathcal{H}_q(G_n;k),p)\}.$$
It has been known since at least [@KP Theorem 3.16] that the $A_G$-module $\mathcal{H}_1(G)$ is generated in degree $\leq 2$, for all graphs $G$. This result has also appeared in An, Drummond-Cole, and Knudsen [@ADK Proposition 5.6]. To the knowledge of the author, the present work is the first to provide some evidence of such uniform boundedness for the higher syzygies of $\mathcal{H}_q$, for $q \geq 2$.
Graph configuration spaces
==========================
\[graphdef\] A **graph** is a 1-dimensional compact simplicial complex. We call the 0-simplicies of $G$ its **vertices**, $V(G)$, while the 1-simplices are its **edges**, $E(G)$. If $v$ is a vertex of $G$, then the **degree** of $v$, denoted $\mu(v)$, is the number of edges adjacent to $v$. A vertex is said to be **essential** if it has $\mu(v) \neq 2$. A **homomorphism of graphs** is a map between vertices $$\phi:V(G) \rightarrow V(G')$$ which preserves adjacency.
If $G$ is a graph, the **(unordered) $n$-strand configuration space of $G$**, denoted $U{\mathcal{F}}_n(G)$, is given by $$U{\mathcal{F}}_n(G) := \{(x_1,\ldots,x_n) \in G^n \mid x_i \neq x_j\}/{\mathfrak{S}}_n$$ where ${\mathfrak{S}}_n$ acts by permuting coordinates in the natural way.
Our main concern will be developing a better understanding of the homology groups $H_q(U{\mathcal{F}}_n(G))$. While much of the current literature on the subject has been concerned with their relation to $\pi_1(U{\mathcal{F}}_n(G))$ (see [@FS; @KKP], for instance), in this work we will approach their study from a more recent algebraic perspective.
Writing $E(G)$ for the set of edges of $G$, We will use $A_G$ to denote the polynomial ring $$A_G := {{\mathbb{Z}}}[x_e \mid e \in E(G)]$$
In [@R], the author showed that the total $q$-homology group, $\mathcal{H}_q(G) = \bigoplus_n H_q(U{\mathcal{F}}_n(G))$ could be endowed with the structure of a finitely generated, graded module over $A_G$ whenever $G$ was a tree. Following this, An, Drummond-Cole, and Knudsen showed that this could be extended to all graphs [@ADK]. Our interest in this paper will be to understand the syzygies of the module $\mathcal{H}_q$ for a certain family of graphs, which we call edge-linear (see Definition \[edgelin\]). In particular, we prove facts about the Betti numbers, in the commutative algebraic sense, by applying the techniques of $\operatorname{FI}$-algebras recently developed by Nagel and Römer [@NR], building on the work of Sam [@Sa; @Sa2], Snowden [@Sn], and others.
Let $G$ be a graph. A **half-edge** of $G$ is a pair $h = (v,e)$ where $v$ is a vertex and $e$ is an edge adjacent to $v$. We will use $v(h)$ and $e(h)$ to denote the vertex and edge associated to $h$, respectively. The set of half edges with $v(h) = v$ will be written $H(v)$.
For a vertex $v \in V(G)$, we set $Sw(v)$ to be the abelian group freely generated by the symbols $$Sw(v) = <\emptyset, v, h \in H(v)>.$$ We consider $Sw(v)$ to be a bigraded abelian group by setting $$|v| = (0,1), |\emptyset| = (0,0), |h| = (1,1).$$ The **Swiatkowski complex** associated to $G$ is the differential bigraded $A_G$-module $$Sw(G) = A_G \otimes \bigotimes_v Sw(v),$$ with $$|e| = (0,1)$$ and differential defined by $$\partial(h) = e(h) - v(h), \partial(e) = \partial(v) = \partial(\emptyset) = 0.$$
Expressed more classically, we may think of the Swiatkowski complex in the following way. For each $q \geq 0$, write $Sw_q(G)$ to be the subgroup of $Sw(G)$ generated by pure tensors of grade $(q,x)$ for some $x \geq 0$. Then $Sw_q(G)$ can be thought of as a graded module over $A_G$. The Swiatkowski complex is the complex of graded $A_G$-modules $$Sw_{\star}(G) : \ldots \rightarrow Sw_q(G) \stackrel{\partial}\rightarrow Sw_{q-1}(G) \rightarrow \ldots \rightarrow Sw_0(G) \rightarrow 0$$ where $\partial$ is as above. We will use these two perspectives interchangeably in what follows.
Because it will be relevant later, we note that the assignment $G \mapsto Sw(G)$ is functoral. In particular, given an injective homomorphism of graphs $\phi:G \rightarrow G'$ one obtains a morphism $Sw(G) \rightarrow Sw(G)$. The reader can check that this morphism preserves all the relevent structure of the complex including the bigrading, as well as the differential.
We will follow the notation of An, Drummond-Cole, and Knudsen regarding shifts in the grading of bigraded modules. If $M$ is a differential bigraded module over some algebra $A$, then we we will write $M\{1\}$ to denote the bigraded module with $$M\{1\}_{(a,b)} = M_{(a,b-1)}.$$ Similarly, we will write $M[1]$ to denote the module with bigrading $$M[1]_{(a,b)} =M_{(a-1,b)}.$$ The differentials of these modules follow the usual sign conventions for degree shifting.
The Swiatkowski complex was orginally expressed in the above form by An, Drummond-Cole, and Knudsen in [@ADK], though they credit work of Swiatkowski [@Sw] for the origin of the idea. The following theorem is extremely relevant for the remainder of this paper.
\[swiahomology\] There is an isomorphism of functors from the category of graphs and injective homomorphisms to bigraded abelian groups, $$\mathcal{H}_\star(G) \cong H_\star(Sw(G))$$ In particular, for any graph $G$ the total $q$-homology group $\mathcal{H}_q(G)$ carries the structure of a finitely generated graded module over the polynomial ring $A_G$.
In the case where $G$ is a tree, the structure of $\mathcal{H}_q(G)$ as a $A_G$-module was explored by the author [@R]. It is shown in that work that $\mathcal{H}_q(G)$ decomposes as a direct sum of graded shifts of square-free monomial ideals. In the work [@ADK], An, Drummond-Cole and Knudsen also explore $\mathcal{H}_q(G)$ in the case where $G$ has maximum valency 3.
For the purpose of doing computations later in this work, we will need to present a simplification of the Swiatkowski complex.
Let $G$ be a graph with no isolated vertices. For each vertex $v$ of $G$ we may enumerate the half-edges containing $v$ by $\{h_i\}$. Define a bigraded subgroup of $S(v)$ by, $$\widetilde{S}(v) := <\emptyset, h_{i,j}>_{i,j},$$ where $h_{i,j} = h_i - h_j$. Then the **reduced Swiatkowski complex** is the submodule of $Sw(G)$ defined by $$\widetilde{Sw}(G) := A_G \otimes \bigotimes_{v} \widetilde{S}(v)$$
One observes that for any pair of half edges $h_i,h_j$ containing some fixed vertex $v$, one has $$\partial(h_{i,j}) = e(h_i) - e(h_j)$$ It follows that $\widetilde{Sw}(G)$ is indeed a differential bigraded submodule of the Swiatkowski complex.
If $G$ does not contain any isolated vertices, then the inclusion $$\widetilde{Sw}(G) \hookrightarrow Sw(G)$$ is a quasi-isomorphism.
$\operatorname{FI}$-algebras and edge-linear $\operatorname{FI}$-graphs
=======================================================================
For each integer $n \geq 0$, we will write $[n]$ to denote the finite set $[n] := \{1,\ldots,n\}$. The category $\operatorname{FI}$ is that whose objects are the sets $[n]$ and whose morphisms are set injections.
If $k$ is a commutative Noetherian ring, then an **$\operatorname{FI}$-module over $k$** is a functor $M$ from $\operatorname{FI}$ to the category of $k$-modules. We will often write $M_n$ to denote $M([n])$.
The representation theory of $\operatorname{FI}$ has recently risen to popularity due in large part to the seminal works of authors such as Church, Ellenberg, Farb, and Nagpal [@CEF; @CEFN], Djament and Vespa [@D; @DV], Sam and Snowden [@SS; @SS2], and many others. In this work we will be primarily concerned with two constructions stemming from $\operatorname{FI}$: $\operatorname{FI}$-algebras and $\operatorname{FI}$-graphs.
\[fgdef\] If $k$ is a commutative Noetherian ring, an $\operatorname{FI}$-algebra is a functor $\mathbf{A}$ from $\operatorname{FI}$ to the category of $k$-algebras. Given an $\operatorname{FI}$-algebra $\mathbf{A}$, an **$\mathbf{A}$-module** is an $\operatorname{FI}$-module $M$ satisfying the following:
1. $M_n$ is an $\mathbf{A}_n$-module for each $n$;
2. if $f:[n] \hookrightarrow [m]$ is an injection of sets then the following diagram commutes for all $a \in A_n$ $$\begin{CD}
M_n @> M(f)>> M_m\\
@V a VV @V A(f)(a) VV\\
M_n @> M(f) >> M_m
\end{CD}$$
An $\mathbf{A}$-module $M$ is said to be **finitely generated** so long as there is a finite collection of elements $\{v\} \subseteq \bigoplus M_n$ which no proper sub $\mathbf{A}$-module contains. We say that a finitely generated $\mathbf{A}$-module $M$ is **Noetherian** if every sub $\mathbf{A}$-module of $M$ is also finitely generated. We say that $\mathbf{A}$ is itself **Noetherian** if all finitely generated modules over $\mathbf{A}$ are Noetherian.
The simplest example of an $\operatorname{FI}$-algebra is that whose points are defined by $\mathbf{A}_n = k$, and whose transition maps are all the identity. In this case $\mathbf{A}$-modules are no different than $\operatorname{FI}$-modules over $k$.
The second example of an $\operatorname{FI}$-algebra, and the example which will be the fundamental object of study in this paper, is that given on points by $\mathbf{A}_n = k[x_1,\ldots,x_n]$, and whose transition maps act on indices in the obvious way.
Note that for any fixed $d \geq 0$, and any $\operatorname{FI}$-algebra $\mathbf{A}$, we may always define a new algebra given by the tensor power $$(\mathbf{A}^{\otimes d})_n := (\mathbf{A}_n)^{\otimes d}$$
It is famously known that all finitely generated $\operatorname{FI}$-modules over Noetherian rings are themselves Noetherian (see [@CEFN] for one such proof). However, finitely generated modules over a general $\operatorname{FI}$-algebra do not share this property. This is true even if we assume that $\mathbf{A}$ is itself finitely generated. Despite this, one can prove the following.
\[noeth\] If $k$ a commutative Noetherian ring, the $\operatorname{FI}$-algebra $\mathbf{A}$ given by $\mathbf{A}_n = k[x_1,\ldots,x_n]$ is Noetherian. More generally, for any fixed integer $d \geq 0$, $\mathbf{A}^{\otimes d}$ is Noetherian.
It is still open whether the Noetherian property of $\operatorname{FI}$-algebras is preserved by tensor products. The above special case was proven by Nagel and Römer in the provided source.
Theorem \[noeth\] has a variety of interesting consequences in the case where we keep track of the natural grading on $\mathbf{A}_n$.
We say that an $\operatorname{FI}$-algebra $\mathbf{A}$ over a field $k$ is **standard graded** if every algebra $\mathbf{A}_n$ is a graded $k$-algebra generated in degree 1 with $\mathbf{A}_{n,0} = k$, and every induced map $A(f)$ is a morphism of graded $k$-algebras. We say that an $\mathbf{A}$-module $M$ is **graded** if each $M_n$ is a graded module over $\mathbf{A}_n$, and each induced map $M(f)$ preserves this grading.
If $M$ is a finitely generated graded $\mathbf{A}$-module, then for each $p,n \geq 0, j \in {{\mathbb{Z}}}$ we define $\beta_{n,p,j}(M)$ to be the Betti number $$\beta_{n,p,j}(M) := \dim_k (\operatorname{Tor}_p^{\mathbf{A}_n}(M_n,k)_j).$$
\[betti\] Let $\mathbf{A}$ be a finitely generated standard graded $\operatorname{FI}$-algebra over a field $k$. Then for any integers $p,j \geq 0$, and any finitely generated graded $\mathbf{A}$-module $M$, the function $$n \mapsto \beta_{n,p,j}(M)$$ agrees with a polynomial for $n \gg 0$. Moreover, there exists a finite list of integers $$j_0(M,p) < \ldots < j_t(M,p)$$ such that for all $n \gg 0$ $$\beta_{n,p,j}(M_n) \neq 0 \iff j \in \{j_0(M,p), \ldots, j_t(M,p)\}.$$
The first part of Corollary \[betti\] does not explicitly appear in [@NR], although it is a clear consequence of the proof of [@NR Theorem 7.7].
One way to think about Corollary \[betti\] is that if $M$ is a finitely generated graded $\mathbf{A}$-module, then in each fixed degree $p$ the $p$-th syzygies eventually stabilize. One should note, however, that it is not the case that this stabilization is uniform across all $p$. Namely, one should not expect the regularity of $M_n$ to be eventually independent of $n$.
**For the remainder of the paper, we will reserve $\mathbf{A}$ to mean the $\operatorname{FI}$-algebra defined by $\mathbf{A}_n = {{\mathbb{Z}}}[x_1,\ldots,x_n]$.**
The second $\operatorname{FI}$-structure which will be important to us are $\operatorname{FI}$-graphs.
\[figraphdef\] An $\operatorname{FI}$-graph is a functor $G_{\bullet}$ from $\operatorname{FI}$ to the category of graphs with graph homomorphisms. As with $\operatorname{FI}$-algebras, we will often write $G_n$ to denote $G_{\bullet}([n])$ and $G(f)$ to denote $G_{\bullet}(f)$. We say that $G_{\bullet}$ is finitely generated if for $n \gg 0$, the vertex set $V(G_{n+1})$ is equal to $\cup_f G(f)(V(G_n))$, where the union is over injections $f:[n] \hookrightarrow [n+1]$.
We say that $G_{\bullet}$ is torsion-free if the induced maps $G(f)$ are injective for all $f$.
\[exlin\] The most typical example of an $\operatorname{FI}$-graph is the complete graph $K_n$. This is the graph whose vertices are labeled by $[n]$, and whose edges are labeled by pairs $\{i,j\}$ with $i \neq j$.
For the purposes of this paper a more relevant example is given as follows. Let $(G,v_G)$ and $(H,v_H)$ be any pair of graphs with a choice of vertex. Then we define $$G_n := G \bigvee^n H.$$ For any injection, the induced map on $G_{\bullet}$ is defined by fixing the copy of $G$, and permuting the copies of $H$. For example, if $G$ is a single vertex, and $H$ is an edge, then $G_n$ is the star graph, which has a unique essential vertex of degree $n$. This example was first discussed in the context of $\operatorname{FI}$-graphs by Lütgehetmann in [@L], although he does not use the language of $\operatorname{FI}$-graphs.
$\operatorname{FI}$-graphs were first introduced by the author and Graham White in [@RW]. Configuration spaces of $\operatorname{FI}$-graphs are shown to exhibit a variety of stable behaviors in that work, extending results of Lütgehetmann from [@L]. In this work we combine these stabilization results with the aforementioned $A_G$-action.
\[edgelin\] Let $G_{\bullet}$ be a finitely generated $\operatorname{FI}$-graph. Then it is proven in [@RW] that the function $$n \mapsto |E(G_n)|$$ agrees with a polynomial for all $n \gg 0$. We say that $G_{\bullet}$ is **edge-linear** if it is finitely generated, and $|E(G_n)|$ agrees with a linear polynomial for all $n \gg 0$.
The second example given in Example \[exlin\] is edge-linear. Moreover, for any fixed $m \geq 1$ the $\operatorname{FI}$-graph defined by $G_n = K_{n,m}$, the complete bipartite graph on $(n+m)$-vertices, is also edge-linear.
Both of these examples can be thought of as being construct via the following gluing procedure. Let $G$ and $G'$ be two graphs, each sharing a common subgraph $H$. Then for any $n \geq 0$, we obtain a new graph $$G \bigsqcup_{H}^n G'$$ by gluing $n$ copies of $G'$ to $G$ along $H$. This family of graphs may be given an $\operatorname{FI}$-graph structure, where the symmetric group acts by permuting the copies of $G'$, while fixing $G$. Graphs of this kind were originally studied by Lütgehetmann in [@L]. The Classification Theorem \[class\] essentially implies that all edge-linear $\operatorname{FI}$-graphs are isomorphic to something of the form $G \bigsqcup_H^{\bullet}G'$, for some $G,G',H$.
Edge-linear $\operatorname{FI}$-graphs will be useful in this work due to their connection to the $\operatorname{FI}$-algebra $\mathbf{A}$. We explore this connection as a corollary to the following classification result.
\[class\] Let $G_{\bullet}$ be an edge-linear $\operatorname{FI}$-graph. Then there exists a pair of graphs $G,G'$, with a common subgraph $H$, such that for all $n \gg 0$ $$G_n \cong G \bigsqcup_H^n G'$$
We begin by observe that an edge-linear $\operatorname{FI}$-graph must also have linear growth in its number of vertices. Indeed, this follows from the graph-theoretic inequality $$|V(G)| \leq 2|E(G)|/\delta(G)$$ where $\delta(G)$ is the minimum degree of a vertex of $G$. Appealing to the classification theorem for $\operatorname{FI}$-sets [@RSW Theorem A], we conclude that $V(G_n)$ is isomorphic to a disjoint union of some number of copies of the ${\mathfrak{S}}_n$-action on $[n]$, along with some number of invariant vertices.
The only ${\mathfrak{S}}_n$-equivariant relation on $[n]$ is the identity relation $\{(i,i) \mid i \in [n]\}$, and the anti-identity relation $\{(i,j) \mid i \neq j\}$. In the latter case, the number of edges does not grow linearly in $n$. Therefore, there are no edges between two vertices in the same ${\mathfrak{S}}_n$-orbits. If we write two distinct ${\mathfrak{S}}_n$-orbits of vertices as $${\mathcal{O}}_1 = \{(i,1) \mid i \in [n]\}, {\mathcal{O}}_2 = \{(i,2) \mid i \in [n]\}$$ then the same argument before shows that there cannot be edges between vertices of the form $(i,1)$ and $(j,2)$ if $i \neq j$. Finally, if there is an edge between $(i,1)$ and $(i,2)$ for some $i$, then there must be an edge between $(i,1)$ and $(i,2)$ for all $i$. This then leads into the construction of the desired graphs $G,G',$ and $H$: let $\widetilde{G}$ be the graph whose vertices are labeled by ${\mathfrak{S}}_n$-orbits of vertices of $G_n$, and whose edges indicate the existence of an edge in $G_n$ between two elements of the orbits. We let $G$ be the induced subgraph of $\widetilde{G}$ whose vertices correspond to singleton orbits, let $G'$ be any subgraph of $\widetilde{G}$ containing all the non-singleton orbits and all edges adjacent to these orbits, and let $H$ be the intersection in $\widetilde{G}$ of $G$ and $G'$.
It was revealed in the above proof that the choices of $G'$ and $H$ are not necessarily unique, though $G$ is unique.
The asymptotic nature of Theorem \[class\] is necessary. For instance, one may define an $\operatorname{FI}$-graph which agrees with the bipartite graph $K_{n,n}$ for $n \leq 10$, and collapses to a single edge for all larger $n$. This observation inspires the following definition.
Let $G_{\bullet}$ be an edge-linear $\operatorname{FI}$-graph, and let $m$ be sufficiently large so that $G_n \cong G \bigsqcup_H^n G'$ for all $n \geq m$, and for some $G,G',H$. The **tail** of $G_{\bullet}$, $G_{\bullet}^{\gg}$, is the edge-linear $\operatorname{FI}$-graph, $$G_n^{\gg} := \begin{cases} \emptyset &\text{ if $n < m$}\\ G_n &\text{ otherwise.}\end{cases}$$
We also define an $\operatorname{FI}$-algebra $\mathbf{A}_{G_{{\bullet}}}$ by setting $$\mathbf{A}_{G_{{\bullet}},n} := A_{G \bigsqcup_H^n G'}$$ for all $n \geq 0$, where $G,G'$, and $H$ are as above.
While the tail of an $\operatorname{FI}$-graph depends on the choice of $m$, this choice will not matter in what follows.
\[algfg\] Let $G_{\bullet}$ be an edge-linear $\operatorname{FI}$-graph. Then there is some $d \geq 0$ such that $\mathbf{A}_{G_{{\bullet}}}$ is a quotient of $$\mathbf{A}^{\otimes d}.$$ In particular, $\mathbf{A}_{G_{{\bullet}}}$ is Noetherian.
Appealing to Theorem \[class\], it follows that $E(G_n)$ has $a+b$ ${\mathfrak{S}}_n$-orbits: $a$ orbits isomorphic to the action of ${\mathfrak{S}}_n$ on $[n]$, and $b$ singleton orbits. We conclude that $\mathbf{A}_{G_{{\bullet}}}$ is isomorphic to a quotient of $\mathbf{A}^{\otimes (a+b)}$. Namely, the quotient which identifies the variables of the first $a$ tensor factors with the elements in the non-singleton orbits of $E(G_n)$, and identities the variables of the final $b$ tensor factors with the $b$ singleton orbits.
The second part of the proposition follows from the fact that any quotient of a Noetherian $\operatorname{FI}$-algebra must be Noetherian, and Theorem \[noeth\].
The goal of the next section is to show that if $G_{\bullet}$ is an edge-linear $\operatorname{FI}$-graph, then the assignments $$n \mapsto \mathcal{H}_q(G_{\bullet})$$ defines a finitely generated module over $\mathbf{A}_{G_{{\bullet}}}$. To do this we will make critical use of the fact that $\mathbf{A}_{G_{{\bullet}}}$ is Noetherian.
The proofs of the main theorems
===============================
For the remainder of this section we fix an edge-linear $\operatorname{FI}$-graph $G_{\bullet}$. **In fact, we substitute $G_{\bullet}$ by its tail $G^\gg_{\bullet}$ so that $G_{\bullet}$ is torsion free**. We will also assume that $G_n$ does not contain any isolated vertices for all $n$. In view of Corollary \[betti\] and Proposition \[algfg\], it will suffice to show that the $\operatorname{FI}$-module $$\mathcal{H}_q(G_{\bullet}) := \bigoplus_{n \geq 0} H_q(U{\mathcal{F}}_n(G_{\bullet}))$$ is finitely generated and graded over the $\operatorname{FI}$-algebra $\mathbf{A}_{G_{\bullet}}$. We accomplish this by proving that the collection of reduced Swiatkowski complexes $\{\widetilde{Sw}(G_n)\}_{n \geq 0}$ can be extended to a complex of graded $\mathbf{A}_{G_{\bullet}}$-modules, $\widetilde{Sw}(G_{\bullet})$. We will then show that the terms in the complex $\widetilde{Sw}(G_{\bullet})$ are finitely generated $\mathbf{A}_{G_{\bullet}}$-modules. The Noetherian property (Proposition \[algfg\]) will imply that $\mathcal{H}_q(G_{\bullet})$ is finitely generated, as desired.
For any injection of sets $f:[n] \hookrightarrow [r]$, and any $q \geq 0$ the following diagram commutes $$\begin{CD}
Sw_q(G_r) @ > \partial >> Sw_{q-1}(G_r)\\
@A Sw(G(f)) AA @A Sw(G(f)) AA\\
Sw_q(G_n) @> \partial >> Sw_{q-1}(G_n)
\end{CD}$$ In particular, the collection of Swiatkowski complexes $\{Sw(G_n)\}_{n \geq 0}$ can be extended to a complex of graded $\mathbf{A}_{G_{\bullet}}$-modules.
This follows from the definition of $\partial$, as well as the definition of $Sw(G(f))$. Indeed, we have for any half edge $h = (v,e)$ $$\partial(Sw(G(f))(h)) = \partial(G(f)(e),G(f)(v)) = G(f)(e) - G(f)(v) = Sw(G(f))(\partial(h)).$$
One observes that, for any injection $f:[n] \hookrightarrow [r]$, $Sw(G(f))$ restricts to a map $\widetilde{Sw}(G_n) \rightarrow \widetilde{Sw}(G_r)$. The above lemma therefore shows that the reduced Swiatokowski complex can also be extended to a complex of graded $\mathbf{A}_{G_{\bullet}}$-modules.
For each $q\geq 0$, the $\mathbf{A}_{G_{\bullet}}$-module $\widetilde{Sw}_q(G_{\bullet})$ is finitely generated.
One should observe that this lemma is false if we insist on working with the usual Swiatkowski complex instead of the reduced one.
We begin by defining an $\operatorname{FI}$-set, in the sense of [@RSW], $X_{q,d}(G_{\bullet})$ For each $n$, the elements of $X_{q,d}(G_{\bullet})$ are tuples $$(m,\widetilde{h}_1,\ldots,\widetilde{h}_q)$$ where $m$ is a monomial in $A_{G_n}$ of degree $d$, $\widetilde{h}_i$ is a pair of distinct half-edges containing a common vertex of $G_n$, and $\widetilde{h}_i, \widetilde{h}_j$ are associated to different vertices whenever $i \neq j$. It isn’t hard to see that $X_{q,d}(G_{\bullet})$ is a $\operatorname{FI}$-subset of a product of finitely generated $\operatorname{FI}$-sets. This implies that it is itself finitely generated (see [@RSW Proposition 4.2]).
Note that we may associated to each element of $X_{q,d}(G_n)$ a generating element of $\widetilde{Sw}_q(G_n)$. In fact, we have that $$\widetilde{Sw}(G_{\bullet}) \cong \bigoplus_{d \geq 0} {{\mathbb{Z}}}X_{q,d}(G_{\bullet})$$ where ${{\mathbb{Z}}}X_{q,d}(G_{\bullet})$ is the $\operatorname{FI}$-module ${{\mathbb{Z}}}$-linearization of $X_{q,d}(G_{\bullet})$. This shows that $\widetilde{Sw}(G_{\bullet})$ is finitely generated, as desired.
As previously stated, these two lemmas and the Noetherian property immediately lead to the following consequence.
\[mainthm\] Let $G_{\bullet}$ be an edge-linear $\operatorname{FI}$-graph. For each $q\geq 0$, the $\mathbf{A}_{G_{\bullet}}$-module $\mathcal{H}_q(G_{\bullet})$ is finitely generated.
The two proceeding lemma do not use edge-linearity of the $\operatorname{FI}$-graph $G_{\bullet}$. Indeed, the only thing preventing one from concluding an analog of Theorem \[mainthm\] for all $\operatorname{FI}$-graphs is the failing of Noetherianity outside of the linear case. It is unclear whether one should expect such a finite generation result in these cases.
Base change from ${{\mathbb{Z}}}$ to any field will preserve finite generation of $\operatorname{FI}$-modules, and so we also obtain the following as a corollary to the above
Let $G_{\bullet}$ be an edge-linear $\operatorname{FI}$-graph, and let $k$ be a field. Then for each $p,j \geq 0$, the function $$n \mapsto \beta_{p,j}(\mathcal{H}_q(G_n;k))$$ agrees with a polynomial for $n \gg 0$. Moreover, there exists a finite list of integers $$j_0(\mathcal{H}_q(G_n;k),p) < \ldots < j_t(\mathcal{H}_q(G_n;k),p)$$ such that for all $n \gg 0$ $$\beta_{n,p,j}(\mathcal{H}_q(G_n;k)) \neq 0 \iff j \in \{j_0(\mathcal{H}_q(G_n;k),p), \ldots, j_t(\mathcal{H}_q(G_n;k),p)\}.$$
An inductive method for computation
===================================
In this section we outline an inductive method with which one can compute the $\mathbf{A}_{G_{\bullet}}$-module $\mathcal{H}_q(G_{\bullet})$. This method extends a similar method originally described by An, Drummond-Cole, and Knudsen [@ADK Proposition 5.13]. We then conclude the section by performing some example computations
As with the previous section, we fix a edge-linear $\operatorname{FI}$-graph $G_{\bullet}$, which we make torsion free by working with its tail.
The theory of finitely generated $\operatorname{FI}$-graphs tells us that for $n \gg 0$ the ${\mathfrak{S}}_n$-orbits of vertices of $G_n$ eventually stabilize (see, [@RSW Proposition 3.6]), in the sense that the transition maps of $G_{\bullet}$ induce isomorphisms $$V(G_n)/{\mathfrak{S}}_n \cong V(G_{n+1})/{\mathfrak{S}}_{n+1}.$$ Let $m \gg 0$ be in this stable range, and let $v$ be a vertex of $G_m$ which is fixed by ${\mathfrak{S}}_m$. We will often say that $v$ is an **invariant vertex of $G_{\bullet}$**. Then all images of $v$ under all transition maps of $G_{\bullet}$ are also invariant under the action of the relevant symmetric group. By abuse of notation, we say that $v$ is a vertex of $G_n$ for all $n \geq m$.
The **blow up of $G_{\bullet}$ at $v$** is an $\operatorname{FI}$-graph $Bl_v(G_{\bullet})$, defined as follows. For $n \geq m$, $Bl_v(G_n)$ is the graph with vertex set $$V(Bl_v(G_n)) := (V(G_n) - \{v\}) \cup (\{v\} \times H(v)),$$ where $H(v)$ is the set of half-edges containing $v$. The edges of $Bl_v(G_n)$ are the edges of $G_n$, altered in the obvious way to account for the new vertices. We also set $Bl_v(G_n) = \emptyset$ for $n < m$. See [@ADK Figure 9] for a pictorial representation of the blow up.
Let $f:[n] \hookrightarrow [r]$ be an injection of sets. Then the transition map induced by $f$ is defined as follows. On vertices not of the form $v \times h$, the transition map agrees with $G(f)$. Otherwise, we have $$Bl_v(G_{\bullet})(f)(v \times h) = v \times G(f)(h).$$ This is well defined, as $v$ was an invariant vertex.
We will often talk of invariant vertices of an $\operatorname{FI}$-graph, without making explicit reference to what degree they originate from. Because much of what we care above is asymptotic, this shouldn’t cause confusion.
Following the techniques of [@ADK], our goal will be to relate $\mathcal{H}_q(G_{\bullet})$ to $\mathcal{H}_q(Bl_v(G_{\bullet}))$. We first observe that $\mathcal{H}_q(Bl_v(G_{\bullet}))$ is a $\mathbf{A}_{G_{\bullet}}$-module. Indeed, the construction of the blowup does not erase, or meaningfully alter edges.
Let $v$ be an invariant vector of $G_{\bullet}$. Then there is an inclusion of complexes $$\widetilde{Sw}(Bl_v(G_{\bullet})) \hookrightarrow \widetilde{Sw}(G_{\bullet}).$$
one can realize $\widetilde{Sw}(Bl_v(G_n))$ as the subcomplex of $\widetilde{Sw}(G_n))$ generated by pure tensors whose $v$-th term is $\emptyset$. The fact that this extends to the desired inclusion follows from the fact that $v$ was chosen to be an invariant vector.
The above proof immediately points to how one can describe the cokernel of $\widetilde{Sw}(Bl_v(G_{\bullet})) \hookrightarrow \widetilde{Sw}(G_{\bullet})$. To accomplish this we will need some notation. If $v$ is an invariant vector of $G_{\bullet}$, then we define an $\operatorname{FI}$-module $\widetilde{H}(v)$ to be that for which $$\widetilde{H}(v)_n = {{\mathbb{Z}}}<h_i-h_j \mid h_i,h_j \in H(v)_n>,$$ where $H(v)_n$ is the set of half-edges containing $v$, thought of as a vertex of $G_n$. We now extend the above lemma as follows. Note that this result generalizes a lemma of An, Drummond-Cole, and Knudsen [@ADK].
Let $v$ be an invariant vector of $G_{\bullet}$. Then there is an exact sequence of complexes $$0 \rightarrow \widetilde{Sw}(Bl_v(G_{\bullet})) \rightarrow \widetilde{Sw}(G_{\bullet}) \rightarrow \widetilde{H}(v) \otimes \widetilde{Sw}(Bl_v(G_{\bullet}))\{1\}\rightarrow 0$$
It therefore follows that computing $\mathcal{H}_q(G_{\bullet})$ can often times be reduced to computing $\mathcal{H}_q(Bl_v(G_{\bullet}))$. This inductive approach becomes even more appealing when one considers how simple its base case is. Indeed, the classification theorem implies that after sufficiently many blow-ups one is left with some fixed number of disjoint edges, invariant under the action of the symmetric group, paired with $n$ disjoint copies of some graph, which the symmetric group permutes.
Take $G_{\bullet}$ to be the star graph $$G_n = K_{n,1}$$ This graph has a unique invariant vertex, and $Bl_v(G_n)$ is a disjoint union of $n$ edges. It is fact that $U{\mathcal{F}}_n(G_n)$ only has homology in degrees 0 and 1 (see, for instance, [@Sw]). We therefore have, $$0 \rightarrow \mathcal{H}_1(G_{\bullet}) \rightarrow S^{({\bullet}-1,1)} \otimes \mathcal{H}_0(Bl_v(G_{\bullet}))\{1\} \rightarrow \mathcal{H}_0(Bl_v(G_{\bullet})) \rightarrow \mathcal{H}_0(G_{\bullet}) \rightarrow 0,$$ where $S^{({\bullet}-1,1)}$ is the $\operatorname{FI}$-module for which $S^{(n-1,1)}$ is the standard representation of ${\mathfrak{S}}_n$. Examining this exact sequence from right to left, we note that $\mathcal{H}_0(G_n) = {{\mathbb{Z}}}$ for all $n \geq 0$, while $\mathcal{H}_0(Bl_v(G_{\bullet})) \cong \mathbf{A}$. The map $$\mathcal{H}_0(Bl_v(G_{\bullet})) \rightarrow \mathcal{H}_0(G_{\bullet}) \rightarrow 0$$ identifies all of the disjoint edges of $Bl_v(G_{\bullet})$. In particular, the kernel of this map is the module ${\mathfrak{m}}_{{\bullet}}$, defined on points by $${\mathfrak{m}}_n = (x_i - x_j \mid i \neq j)$$
On the other hand, the map $$S^{({\bullet}-1,1)} \otimes \mathbf{A}\{1\} \rightarrow {\mathfrak{m}}_{\bullet}\rightarrow 0$$ is given by $$(e_i - e_j) \otimes 1 \mapsto x_i-x_j.$$ The kernel of this map is the submodule of $S^{({\bullet}-1,1)} \otimes \mathbf{A}\{1\}$ generated in degree 3 by $$(e_1-e_2) \otimes x_3 + (e_3-e_1) \otimes x_2 + (e_2-e_3) \otimes x_1$$
In particular, we conclude that all of the homology classes in $H_1(U{\mathcal{F}}_n(G_n))$ are determined by $H_1(U{\mathcal{F}}_2(G_3))$. This recovers [@ADK Lemma 5.5]. A straight forward computation also verifies that for all $d \gg 0$, the ${{\mathbb{Z}}}[x_1,\ldots,x_d]$-module $$\mathcal{H}_1(G_d)$$ is generated in grade 2, with relations generated in grades $\leq 3$, verifying the second part of Theorem \[bettithm\] for the first two Betti numbers.
[aaaa]{} B. H. An, G. C. Drummond-Cole and B. Knudsen, *Subdivisional spaces and graph braid groups*, [[](http://arxiv.org/abs/1708.02351)]{}. T. Church, J.S. Ellenberg and B. Farb, *$\operatorname{FI}$-modules and stability for representations of symmetric groups*, Duke Math. J. 164, no. 9 (2015), 1833-1910. T. Church, J.S. Ellenberg, B. Farb, and R. Nagpal, *$\operatorname{FI}$-modules over Noetherian rings*, Geom. Topol. 18 (2014) 2951-2984. T. Church and B. Farb, *Representation theory and homological stability*, Advances in Mathematics (2013), 250-314. A. Djament, *Des propriétés de finitude des foncteurs polynomiaux*, [[](http://arxiv.org/abs/1308.4698)]{}. A. Djament, and C. Vespa, *Foncteurs faiblement polynomiaux*, [[](http://arxiv.org/abs/1308.4106)]{}. M. Farber, *Topological complexity of motion planning* Discrete and Computational Geometry 29 (2003), 211-221. D. Farley and L. Sabalka, *Discrete Morse theory and graph braid groups*, Algebr. Geom. Topol. 5 (2005), 1075-1109 (electronic). <http://www.users.miamioh.edu/farleyds/FS1.pdf>. R. Ghrist, *Configuration spaces and braid groups on graphs in robotics*, Knots, braids, and mapping class groups - papers dedicated to Joan S. Birman (New York, 1998), AMS/IP Stud. Adv. Math., 24, Amer. Math. Soc., Providence, RI (2001), 29–40. <https://www.math.upenn.edu/~ghrist/preprints/birman.pdf>. D. Lütgehetmann, *Representation Stability for Configuration Spaces of Graphs*, [[](http://arxiv.org/abs/1701.03490)]{}. K. H. Ko and H. W. Park, *Characteristics of graph braid groups*, Discrete Comput Geom (2012) 48: 915. [[](http://arxiv.org/abs/1101.2648)]{}. J. H. Kim, K. H. Ko, and H. W. Park, *Graph braid groups and right-angled Artin groups*, Trans. Amer. Math. Soc. 364 (2012), 309-360. [[](http://arxiv.org/abs/0805.0082)]{}. U. Nagel and T. Römer, *FI- and OI-modules with varying coefficients*, [[](http://arxiv.org/abs/1710.09247)]{}. E. Ramos, *Stability phenomena in the homology of tree braid groups*, to appear, Algebraic and Geometric Topology, [[](http://arxiv.org/abs/1609.05611)]{}. E. Ramos and G. White, *Families of nested graphs with compatible symmetric-group actions*, [[](http://arxiv.org/abs/1711.07456)]{}. E. Ramos, D Speyer, and G. White, *FI-sets with relations*, [[](http://arxiv.org/abs/1804.04238)]{}. S. Sam, *Syzygies of bounded rank symmetric tensors are generated in bounded degree*, Math. Ann., to appear. [[](http://arxiv.org/abs/1608.01722)]{}. S. Sam, Steven V Sam, *Ideals of bounded rank symmetric tensors are generated in bounded degree*, Invent. Math. 207 (2017), no. 1, 1–21, [[](http://arxiv.org/abs/1510.04904)]{}. A. Snowden, *Syzygies of Segre embeddings and $\Delta$-modules*, Duke Math. J. 162 (2013), no. 2, 225-277, [[](http://arxiv.org/abs/1006.5248)]{}. J. Swiatkowski, *Estimates for homological dimension of configuration spaces of graphs* Colloquium Mathematicum 89.1 (2001): 69-79. S. Sam and A. Snowden, *$GL$-equivariant modules over polynomial rings in infinitely many variables*, Trans. Amer. Math. Soc. 368 (2016), 1097-1158. S. Sam, and A. Snowden, *Gröbner methods for representations of combinatorial categories*, J. Amer. Math. Soc. 30 (2017), 159-203. [[](http://arxiv.org/abs/1409.1670)]{}.
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author:
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Qichuan Geng$^1$Hong ZhangXiaojuan Qi$^2$Ruigang Yang$^{3}$Zhong Zhou$^1$Gao Huang$^{4}$\
$^1$Beihang University $^2$University of Oxford $^3$University of Kentucky$^4$ Tsinghua University\
[{zhaokefirst,zz}@buaa.edu.com]{}\
bibliography:
- 'citation.bib'
title: Gated Path Selection Network for Semantic Segmentation
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abstract: 'We study the eta invariants of compact flat spin manifolds of dimension $n$ with holonomy group $\mathbb{Z}_p$, where $p$ is an odd prime. We find explicit expressions for the twisted and relative eta invariants and show that the reduced eta invariant is always an integer, except in a single case, when $p=n=3$. We use the expressions obtained to show that any such manifold is trivial in the appropriate reduced equivariant spin bordism group.'
title: The eta invariant and equivariant bordism of flat manifolds with cyclic holonomy group of odd prime order
---
[Peter B. Gilkey$^{(*)}$, Roberto J. Miatello$^{(**)}$ and Ricardo A. Podestá$^{(**)}$]{}
[PG: Mathematics Department, University of Oregon, Eugene, OR 97403, USA]{}
[[email protected]]{}
[RM, RP: FaMAF-CIEM, Universidad Nacional de Córdoba, 5000 Córdoba, Argentina]{}
[[email protected], [email protected]]{}
Introduction {#intro}
============
If $M$ is a Riemannian manifold having finite holonomy group $F$, we shall say that $M$ is an *$F$-manifold*. As it is well known, any such manifold is flat, by the Ambrose-Singer theorem [@AS]. Let $p$ be an odd prime. Throughout this paper, $M$ will be a compact flat Riemannian $n$-manifold with cyclic holonomy group of order $p$, $F \simeq {\mathbb Z}_p$, that is, in the above terminology, a flat ${\mathbb Z}_p$-manifold. Such a manifold is of the form $M_{\Gamma}={\Gamma}\backslash {\mathbb R}^n$, with ${\Gamma}$ a Bieberbach group such that ${\Lambda}\backslash {\Gamma}\simeq {\mathbb Z}_p$, where ${\Lambda}$ denotes the translation lattice of ${\Gamma}$. Flat ${\mathbb Z}_p$-manifolds have been fully classified by Charlap [@Ch65], who used Reiner’s classification [@Re] of integral representations of the group ${\mathbb Z}_p$. A convenient description of these manifolds will be given in §\[S2-ZPM\] and §\[zpabc\].
It turns out that any ${\mathbb Z}_p$-manifold $M=M_\Gamma$ is spin, that is, it admits spin structures defined on its tangent bundle. Actually, we show that such $M$ admits exactly $2^{\beta_1}$ spin structures, where $\beta_1 = \beta_1(M)$ is the first Betti number of $M$. In particular, we shall see that there is a unique spin structure such that the corresponding homomorphism $\varepsilon : \Gamma \rightarrow \textrm{Spin} (n)$ is trivial on the translation lattice $\Lambda$ of $\Gamma$, called the spin structure of *trivial type*, or *trivial* spin structure, for short.
One of the main goals of this paper is to obtain a rather explicit expression for the reduced eta invariant of an arbitrary spin ${\mathbb Z}_p$-manifold $M$, associated to the spinorial Dirac operator twisted by characters. We will make use of results in [@MiPo06], where the spectra of twisted Dirac operators on flat bundles over arbitrary compact flat spin manifolds is determined. Also, we refer to [@MR09] for a survey on the spectral geometry of flat manifolds and for a more complete bibliography than we can present here.
In a general setting, let $M$ be an arbitrary Riemannian $n$-manifold, let $D$ be a self adjoint partial differential operator of Dirac type, and we let $Spec_D(M)=\{\lambda_n\}$ be the set of eigenvalues of $D$, counted with multiplicities. Results of Seeley [@Se-67] show that the *eta series* $$\label{etaseries}
\eta_D(s) := \sum_{\lambda_i\ne
0}\operatorname{sign}(\lambda_i)|\lambda_i|^{-s}$$ is holomorphic for $\operatorname{Re}(s)>n$. Furthermore, $\eta_D(s)$ has a meromorphic extension to $\mathbb{C}$ called the *eta function* (that we still denote by $\eta_D(s)$) with isolated simple poles on the real axis. In their study of the index theorem for manifolds with boundary, Atiyah, Patodi, and Singer showed that $0$ is a regular value of the eta function ([@APS76-I; @APS76-II; @APS76-III], see also [@Gi81]) and defined the *eta invariant* $$\label{eta invariant APS}
\eta_D := \eta_D(0),$$ as a measure of the spectral asymmetry of $D$, and the invariant $$\label{etainvariant}
\bar \eta_D := {\tfrac}12 \big( \eta_D + \dim \ker D \big) \,,$$ which is also referred to as the eta invariant by some authors.
We now return to ${\mathbb Z}_p$-manifolds. Let ${\varepsilon}_h$ be a spin structure on a ${\mathbb Z}_p$-manifold $M$. We consider the *spin Dirac operator* $D_{\ell}$ twisted by a character $\rho_\ell$ of ${\mathbb Z}_p$, for $0\le \ell \le p-1$ (with $\ell=0$ corresponding to the untwisted case), acting on smooth sections of the twisted spinor bundle of $(M,{\varepsilon}_h)$ (see §\[S2-STDO\]). Denote the associated eta series by $\eta_{\ell,h}(s)$ and the corresponding eta invariants by $\eta_{\ell,h}$ and $\bar \eta_{\ell,h}$, respectively.
In §\[specasymm\] we study the spectrum $Spec_{D_{\ell}}(M,{\varepsilon}_h)$ and determine its contribution to formula . We show that non trivial eta series can only occur for the so called *exceptional* ${\mathbb Z}_p$-manifolds, in the terminology of Charlap [@Ch88]. Using the information on the spectrum in §\[specasymm\], in Theorem \[thm.eta series\] we obtain explicit expressions for the eta function $\eta_{\ell,h}(s)$ in terms of Hurwitz zeta functions $\zeta(s,{\frac}jp)$, with $1\le j \le p-1$ (see ). From these expressions for $\eta_{\ell,h}(s)$ we get formulas for the eta invariants $\eta_{\ell,h}$, by evaluation at $s=0$ (Theorem \[thm.etainvs\]).
It is known that the dimension of the kernel of $D_{\ell}$ on compact flat manifolds is non zero only for the spin structure of trivial type [@MiPo06]. In Proposition \[harmonics\] we give an expression for $\dim \ker D_{\ell}$, which, together with our previous computations, and in light of , yield an expression for $\bar \eta_{\ell,h}$ and for the difference $\bar\eta_{\ell,h} - \bar\eta_{0,h}$ of $M$ (see Remark \[remark\] and Corollary \[eta untwisted\]).
The integrality of $\bar \eta_{\ell,h}$, except in a single case, is one of the main results in this paper.
\[twisted etas mod Z\] Let $p$ be an odd prime and let $\ell \in {\mathbb N}_0$, with $0\le \ell \le p-1$. Consider a ${\mathbb Z}_p$-manifold $M$ of dimension $n$ equipped with a spin structure ${\varepsilon}$. Then $$\bar \eta_{\ell} \equiv 0 \mod {\mathbb Z}$$ unless $p=n=3$, and in this case $\bar \eta_{\ell} \equiv {\frac}23 \mod {\mathbb Z}$. Furthermore, in all cases, $\bar \eta_{\ell} - \bar \eta_{0} \equiv 0 \mod {\mathbb Z}$.
The theorem says that the eta invariants $\bar \eta_{\ell}$, $0\le \ell \le p-1$, are integers except in the case of the so called *tricosm*, the only 3-dimensional ${\mathbb Z}_3$-manifold (see Example \[tricosm\]).
We point out that for certain ${\mathbb Z}_p$-manifolds with $p\equiv 3$ mod 4, there is a close connection between the spectral invariants and invariants coming from number theory. More precisely, when $\beta_1(M) = 1$ and $(n-1)/(p-1)$ is odd, the eta invariants $\eta_{\ell}$ are given in terms of sums involving Legendre symbols $({\frac}jp)$ with $0\le j \le p-1$ (see Remark \[rem. hp\]). Moreover, in the untwisted case, $\eta_{0}$ is a simple multiple of the class number $h_{-p}$ of the imaginary quadratic field ${\mathbb Q}(\sqrt {-p})$. Namely, these manifolds have only 2 spin structures and we have $$\eta_{0,1} = -4 \, p^{{\frac}{a-1}2} \, \tfrac{h_{-p}}{\omega_{-p}}, \qquad \eta_{0,2} = \big( \big( {\tfrac}{2}p \big) -1 \big) \, \eta_{0,1},$$ where $\omega_{-p}$ is the number of $p^\mathrm{{th}}$-roots of unity in ${\mathbb Q}(\sqrt {-p})$.
As a main application, in Section \[sect-5\] we use the computations in Section \[sect-4\] to study the equivariant spin bordism group of ${\mathbb Z}_p$-manifolds. We recall that two compact closed spin manifolds $M_1$ and $M_2$ are said to be *bordant* if there exists a compact spin manifold $N$ so that the boundary of $N$ is $M_1 \cup -M_2$ with the inherited spin structure, where $-M_2$ denotes $M_2$ with the opposite orientation. If $M_1$ and $M_2$ have equivariant ${\mathbb Z}_p$-structures (see Section \[sect-5\]), we say $N$ is an *equivariant bordism* between $M_1$ and $M_2$ if the ${\mathbb Z}_p$-structure extends over $N$. In this situation $M_1$ and $M_2$ are said to be *equivariant Spin-bordant*. Bordism is an important topological concept first investigated by Thom. Theorem \[thm-1.2\] below states that any ${\mathbb Z}_p$-manifold with the canonical ${\mathbb Z}_p$-structure is equivariant bordant to the same manifold with the trivial ${\mathbb Z}_p$-structure; i.e., vanishes in the reduced equivariant bordism group. We postpone until Section 5 a more precise description.
As it is known, the eta invariant is an analytic spectral invariant, that gives rise to topological invariants which completely detect the equivariant $\mathbb{Z}_p$ spin bordism groups. The integrality results of Theorem \[twisted etas mod Z\] then yield the following geometric and topological result, one of the main motivations for this investigation.
\[thm-1.2\] Let $(M, {\varepsilon},\sigma_p)$ and $(M, {\varepsilon},\sigma_0)$ denote a ${\mathbb Z}_p$-manifold $M$ which is equipped with a spin structure ${\varepsilon}$ together with the canonical and the trivial equivariant $\mathbb{Z}_p$-structures $\sigma_p$ and $\sigma_0$ respectively. Then $$[(M, {\varepsilon},\sigma_p)] - [(M, {\varepsilon},\sigma_0)]=0$$ in the reduced equivariant spin bordism group $\RMSpin_n(B\mathbb{Z}_p)$.
It is worth putting these groups into a bit of a historical context. The equivariant spin bordism groups are important in algebraic topology as they are closely related to Brown-Peterson homology. In [@BBDG] the eta invariant was used to compute $BP_*(BG)$ where $G$ was a spherical space form group; this computation yields the additive structure of $\RMSpin_*(B\mathbb{Z}_p)$ for $p$ an odd prime. But in addition to their topological importance, these groups have also appeared in a geometric setting, for instance, in connection with spin manifolds with finite fundamental group admitting a metric of positive scalar curvature (see Remark \[peters-remark\]).
A brief outline of the paper is as follows. In Section 2 we start by giving a somewhat detailed description of the structure of ${\mathbb Z}_p$-manifolds and of their spin structures. Sections 3 through 5 are devoted to the proofs of the main results. In Sections 3 we study the spectrum and the eta series, in Section 4 we give the results concerning eta invariants and in Section 5 we settle the result on spin bordism. In these proofs we use a number of auxiliary formulas, stated and proved in Section 6. Namely, we need formulas for trigonometric products (§\[sect-a-STP\]), for twisted character Gauss sums (§\[Sect-A-TCGS\]) and sums involving Legendre symbols (§\[Sect-a-SILS\]). We have presented this material at the end to avoid interrupting the flow of our discussion of the main results.
${\mathbb Z}_p$-manifolds {#sect-2}
=========================
Compact flat manifolds {#S2-CFM}
----------------------
Any compact flat $n$-manifold is isometric to a quotient of the form $$M_{\Gamma}= {\Gamma}\backslash{\mathbb R}^n$$ where ${\Gamma}$ is a Bieberbach group, that is, a discrete, cocompact, torsion-free subgroup ${\Gamma}$ of $\text{I}({\mathbb R}^n)$, the isometry group of ${\mathbb R}^n$. Thus, one has that any element ${\gamma}\in \text{I}({\mathbb R}^n) \simeq {\text{O}(n)}\rtimes {\mathbb R}^n$ decomposes uniquely as ${\gamma}= B L_b$, where $B \in {\text{O}(n)}$, $L_b$ denotes translation by $b\in {\mathbb R}^n$, and furthermore, multiplication is given by $$\label{semidirectproduct}
BL_b \cdot CL_c = BCL_{C^{-1}b+c} \,.$$
The pure translations in ${\Gamma}$ form a normal, maximal abelian subgroup of finite index, $L_{\Lambda}$, ${\Lambda}$ a lattice in ${\mathbb R}^n$ that is $B$-stable for each $BL_b \in {\Gamma}$. The restriction to ${\Gamma}$ of the canonical projection $\text{I}({\mathbb R}^n) \rightarrow {\text{O}(n)}$ given by $BL_b\mapsto B$ is a homomorphism with kernel $L_{{\Lambda}}$ and its image $F$ is a finite subgroup of ${\text{O}(n)}$. Thus, we have an exact sequence of groups $$\label{exseq}
0 \rightarrow \Lambda \rightarrow {\Gamma}\rightarrow F \rightarrow 1.$$ The group $F \simeq {\Lambda}\backslash {\Gamma}$ is called the [*holonomy group*]{} of ${\Gamma}$. The action by conjugation $BL_{\lambda}B^{-1}=L_{B{\lambda}}$ of ${\Lambda}\backslash {\Gamma}$ on ${\Lambda}$ defines a representation $F \rightarrow \mathrm{GL}_n({\mathbb Z})$ called the [*integral holonomy representation*]{}, or, for short, the [*holonomy representation*]{}. In general, the integral holonomy representation is far from determining a flat manifold uniquely.
We note that in any compact flat $n$-manifold, we have that $$\label{nB}
n_B := \dim \ker (B - {\text{\sl Id}}_n) = \dim \, ({\mathbb R}^n)^B \ge 1$$ for every $BL_b \in {\Gamma}$ (see for instance [@MR09]) and that $M_{\Gamma}$ is orientable if and only if $F\subset {\text{SO}(n)}$.
${\mathbb Z}_p$-manifolds {#S2-ZPM}
-------------------------
A *${\mathbb Z}_p$-manifold* is a compact flat manifold $M_{\Gamma}= {\Gamma}\backslash {\mathbb R}^n$ with holonomy group $F\simeq {\mathbb Z}_p$. Hence ${\Gamma}= \langle BL_b, {\Lambda}\rangle$ is torsion-free, with $B\in {\text{O}(n)}$ of order $p$ and $b\in {\mathbb R}^n \smallsetminus \Lambda$.
By , $M_{\Gamma}$ can be thought to be constructed from a ${\mathbb Z}_p$-action on $\Lambda$. Thus, as a ${\mathbb Z}_p$-module, ${\Lambda}$ is of the form given by Reiner in [@Re], i.e. ${\Lambda}$ is isomorphic to $$\label{Reiner}
{\Lambda}(a,b,c,\mathfrak a) := \mathfrak{a} \oplus (a-1)\, \mathcal{O} \oplus b \,{\mathbb Z}[{\mathbb Z}_p] \oplus c \, {\text{\sl Id}},$$ where $a,b,c$ are non negative integers satisfying $a+b>0$ and $$\label{dimension}
n=a(p-1)+bp+c,$$ $\xi$ is a primitive $p^{\mathrm{th}}$-root of unity, $\mathcal{O} = {\mathbb Z}[\xi]$ is the full ring of algebraic integers in the cyclotomic field ${\mathbb Q}(\xi)$ and $\mathfrak{a}$ is an ideal in $\mathcal{O}$. Also, ${\mathbb Z}[{\mathbb Z}_p]$ denotes the group ring over ${\mathbb Z}$, and ${\text{\sl Id}}\simeq
{\mathbb Z}$ stands for the trivial ${\mathbb Z}_p$-module.
The ${\mathbb Z}_p$-actions on the modules $\mathcal{O}$, $\mathfrak{a}$ and ${\mathbb Z}[{\mathbb Z}_p]$ are given by multiplication by $\xi$. In the bases $1,\xi,\ldots,\xi^{p-2}$ of $\mathcal{O}$ and $1,\xi,\ldots,\xi^{p-1}$ of ${\mathbb Z}[{\mathbb Z}_p]$, the actions of the generator are given, in matrix form, respectively by [$$\label{blocks}
C_p= \left( \begin{smallmatrix}
0 & & & & -1 \\
1 & 0 & & & -1 \\
& 1 & & & -1 \\
& & \ddots & & \vdots \\
& & & 0 & -1 \\
& & & 1 & -1
\end{smallmatrix} \right) \in \mathrm{GL}_{p-1}({\mathbb Z}),
\qquad
J_p= \left( \begin{smallmatrix}
0 & & & & 1 \\
1 & 0 & & & 0 \\
& 1 & & & 0 \\
& & \ddots & & \vdots \\
& & & 0 & 0 \\
& & & 1 & 0
\end{smallmatrix} \right) \in \mathrm{GL}_{p}({\mathbb Z}).$$]{} If $\mathfrak{a} = \mathcal{O}\alpha$ is principal, we may use the ${\mathbb Z}$-basis $\alpha,\xi\alpha,\ldots,\xi^{p-2}\alpha$ of $\mathfrak{a}$ and the action of the generator is again described by the matrix $C_p$. For a general ideal this action is given by a more complicated integral matrix that we shall denote by $C_{p, \mathfrak{a}}$. We note that $C_{p, \mathfrak{a}}^p = {\text{\sl Id}}$, the action has no fixed points, and the eigenvalues of $C_{p, \mathfrak{a}}$ are again all primitive $p^{\mathrm{th}}$-roots of 1. In particular, $C_{p, \mathfrak{a}}$ is conjugate to $C_p$ in $\mathrm{GL}(n, {\mathbb Q})$.
Note that $J_p \in \mathrm{SO}(p)$ but $C_p \in \mathrm{SL}_{p-1}({\mathbb Z}) \smallsetminus \mathrm{O}(p-1),$ and furthermore, $n_{J_p}=1$, $n_{C_p}=n_{C_{p,a}}=0$. Since $\det \,C_p = \det \, J_p =1$ we have $F\subset \mathrm{SO}(n)$, and hence $M_{\Gamma}$ is orientable.
Using , Charlap was able to give a full classification of flat ${\mathbb Z}_p$-manifolds up to affine equivalence classes in [@Ch65] (see also [@Ch88]). He distinguished between two cases, that he called exceptional and non-exceptional manifolds.
A ${\mathbb Z}_p$-manifold $M$ is called *exceptional* if the lattice of translation is, as a ${\mathbb Z}_p$-module, isomorphic to $\Lambda(a,0,1,\mathfrak a)$ for some ideal $\mathfrak a$ in $\mathcal{O}={\mathbb Z}[\xi]$; that is, if $(b,c)=(0,1)$. Otherwise, $M$ is called *non-exceptional*.
The following proposition collects several standard facts on the structure of ${\mathbb Z}_p$-manifolds. We include a sketch of the proof to make the paper more self-contained.
\[propZp\] Let $M_{\Gamma}= {\Gamma}\backslash {\mathbb R}^n$ be a ${\mathbb Z}_p$-manifold with $\Gamma = \langle {\gamma},\, \Lambda \rangle$, where ${\gamma}= BL_b$. (i) $(BL_b)^p= L_{b_p}$ where $b_p = \sum_{j=0}^{p-1} B^j b \in L_{\Lambda}\smallsetminus (\sum_{j=0}^{p-1} B^j ) \Lambda$.
\(ii) As a ${\mathbb Z}_p$-module, $\Lambda\simeq {\Lambda}(a,b,c,\mathfrak a)$ as in with $c\ge 1$ and $a,b,c$ uniquely determined by the isomorphism class of ${\Gamma}$.
\(iii) $\Gamma$ is conjugate in $\mathrm{I}({\mathbb R}^n)$ to a Bieberbach group $\tilde\Gamma = \langle \tilde{\gamma},\, \Lambda \rangle$ where $\tilde{\gamma}=BL_{\tilde b}$ for which one further has that $B\tilde b = \tilde b$ and $\tilde b
\in \frac 1p \Lambda \smallsetminus {\Lambda}$.
\(iv) One has that $$H_1(M_{\Gamma},{\mathbb Z}) \simeq {\mathbb Z}_p^a \oplus {\mathbb Z}^{b+c}, \qquad H^1(M_{\Gamma},{\mathbb Z}) \simeq {\mathbb Z}^{b+c},$$ and hence $n_B = b+c = \beta_1$, where $\beta_1$ is the first Betti number of $M_{\Gamma}$.
\(v) We have that $n_B = 1 \: \Leftrightarrow \: (b,c)=(0,1)$. In this case, there is a ${\mathbb Z}$-basis $f_1, \dots, f_n$ of $\Lambda$ such that $\Lambda_0 = {\mathbb Z}f_n$ and $\langle f_j, f_n \rangle=0$ for any $1\le j \le n-1$. Furthermore, the element ${\gamma}= BL_b$ as above can be chosen so that $b = \frac 1p f_n$.
\(i) By repeatedly applying , we get $(BL_b)^p= L_{\sum_{j=1}^{p} B^{-j} b} \in {\Gamma}$ and hence $b_p = \sum_{j=0}^{p-1} B^j b \in {\Lambda}$. Furthermore, $b_p \not \in (\sum_{j=0}^{p-1} B^j) \Lambda$. In fact, if $b_p = \sum_{j=0}^{p-1} B^j \lambda$ with $\lambda \in \Lambda$, then we would have $(BL_p L_{-\lambda})^p = {\text{\sl Id}}$, which contradicts the torsion freeness of ${\Gamma}$.
\(ii) By Charlap’s classification [@Ch65] (see also [@Ch88]), the translation lattice $\Lambda$ must be one of the Reiner’s ${\mathbb Z}_p$-module described in . Also, the torsion-free condition on ${\Gamma}$ implies that $c\ge 1$. By (iv), ${\Gamma}$ determines $a$ and $b+c$. Since $n = (a+b)(p-1)+ (b+c)$, the number $a+b$ is also determined and hence so are $b$ and $c$.
\(iii) If $BL_b \in \Gamma$ as in the statement, we have that $b = b_+ + b'$ where one has $Bb_+ = b_+$ and $b'
\in
\ker (B-{\text{\sl Id}})^\perp$. Furthermore, $b_p'=0$ since $b_p'= (\sum_{j=0}^{p-1} B^j) b'$ lies in $\ker
(B-{\text{\sl Id}})^\perp \cap \ker (B-{\text{\sl Id}})$. Thus $(BL_b)^p = L_{p b_+}$ is a translation in $\Gamma$, hence $pb_+ \in \Lambda$ and $b_+ \ne 0$.
If $v \in {\mathbb R}^n$, then $L_v BL_b L_{-v} = BL_{b+(B^{-1} -{\text{\sl Id}})v}$. Now, we have $\text{Im}(B^{-1} -{\text{\sl Id}})$ = $\ker (B^{-1}-{\text{\sl Id}})^\perp$ = $\ker (B-{\text{\sl Id}})^\perp$, so, one can choose $v$ so $(B^{-1} -{\text{\sl Id}})v = -b'$. In this way, conjugation of $\Gamma$ by $L_v$ changes $\Gamma$ into a Bieberbach group generated by $\tilde {\gamma}= BL_{b_+}$ and $\Lambda$, where $\tilde {\gamma}$ satisfies $Bb_+ = b_+$, $pb_+ \in \Lambda$ and $b_+ \notin {\Lambda}$, as desired.
\(iv) These groups are given in [@Ch88], pp. 153, Exercise 7.1 (iv). For completeness, we give a sketch of the proof by explicit calculations. We note that the result for $H_1(M_{\Gamma},{\mathbb Z})$ implies the one for $H^1(M_{\Gamma},{\mathbb Z})$, by the universal coefficient theorem, and furthermore, the formula for $H^1(M_{\Gamma},{\mathbb Z})$ in turn implies that $\beta_1 = b+c =n_B$.
Thus, it suffices to compute $H_1(M_{\Gamma},{\mathbb Z}) \simeq {\Gamma}/[{\Gamma}, {\Gamma}]$. Since $[L_{\lambda},L_{{\lambda}'}]={\text{\sl Id}}$ and $[{\gamma},L_{\lambda}] = B L_b L_{\lambda}L_{-b} B^{-1} L_{-{\lambda}} = L_{(B - {\text{\sl Id}}){\lambda}}$, we have that $$\label{commutator}
[{\Gamma}, {\Gamma}] = \langle [{\gamma},L_{\lambda}] : {\lambda}\in {\Lambda}\rangle = L_{(B-{\text{\sl Id}}){\Lambda}} \,.$$
In order to compute $(B-{\text{\sl Id}}){\Lambda}$ in our case we use a basis of ${\Lambda}$ such that the action of $B$ is represented by matrices as in . We shall denote by ${\Lambda}_ \mathcal{O}$ the sum of all submodules of ${\Lambda}$ of type $\mathcal{O}$ or $\mathfrak{a}$, by ${\Lambda}_R$ the sum of those of type ${\mathbb Z}[{\mathbb Z}_p]$ and by $\Lambda_0$ the sum of the trivial submodules.
We first note that if $f_1, \dots, f_{p-1}$ is a ${\mathbb Z}$-basis of a module $N$ of type $\mathcal{O}$ or $\mathfrak{a}$, then a basis of $(B-{\text{\sl Id}})N$ is given by $f_2-f_1,\ldots,f_{p-1}-f_{p-2}, -\sum_{j=1}^{p-1} f_j - f_{p-1}$, or else we can use the basis $f_2-f_1,f_3-f_2,\ldots,f_{p-1}-f_{p-2}, p f_{p-1}$. This implies that $N/(B-{\text{\sl Id}})N \simeq {\mathbb Z}_p$, hence $${\Lambda}_ \mathcal{O}/(B-{\text{\sl Id}}){\Lambda}_ \mathcal{O} \simeq {\mathbb Z}_p^a.$$
Similarly, if $f'_1, \dots, f'_{p}$ is a ${\mathbb Z}$-basis of a module $N'$ of type ${\mathbb Z}[{\mathbb Z}_p]$, then a basis of $(B-{\text{\sl Id}})N'$ is given by $f'_2-f'_1, \ldots , f'_{p-1}-f'_{p-2}, f'_p - f'_{p-1}, f_1' - f_{p}'$. This implies that $N'/(B-{\text{\sl Id}})N'\simeq {\mathbb Z}$. Finally, for a summand of trivial type, $N'' \simeq {\mathbb Z}$, we clearly have $(B-{\text{\sl Id}})N''=0$. Thus $${\Lambda}_R \oplus {\Lambda}_0 \, / \, (B-{\text{\sl Id}}){\Lambda}_R \simeq {\mathbb Z}^{b+c}.$$
Now $(BL_b)^p= L_{pb_+}$ and $pb_+ \in {\Lambda}$ is fixed by $B$, hence one has that $pb_+
\in
{\Lambda}^B = {\Lambda}_R^B \oplus {\Lambda}_0$, since $({\Lambda}_ \mathcal{O})^B=0$ (a module of type $N$ has no $B$-fixed vectors). For a module of type $N'$ we have that $(N')^B ={\mathbb Z}(\sum_{i=1}^p f'_j) \simeq {\mathbb Z}$, by using a basis $f'_j,\, i\le j \le p,$ as above. Hence ${\Lambda}_R^B \simeq {\mathbb Z}^b$.
Putting all this information together, by , it is not hard to check that $${\Gamma}/[{\Gamma}, {\Gamma}] \simeq \langle BL_b, L_ {{\Lambda}_R^B \oplus {\Lambda}_0} \rangle \, / \, (B-{\text{\sl Id}}){\Lambda}_R\,\simeq {\mathbb Z}_p^a \oplus {\mathbb Z}^{b+c},$$ and hence the assertions in (iv) are proved.
\(v) Since $n_B=b+c$ and $b\ge 0$, $c\ge 1$, it is clear that $n_B=1$ if and only if $(b,c)=(0,1)$.
Now, by (ii), we may assume that $b_+ = b$, after conjugation of $\Gamma$ by $L_v$ in $\mathrm{I}({\mathbb R}^n)$ if necessary. By the description of the lattice in (ii), and since $(b,c) = (0,1)$, there is a ${\mathbb Z}$-basis $f_1, \dots, f_n$ of $\Lambda$ such that $\Lambda_0 = {\mathbb Z}f_n$ and $pb = af_n$ with $a \in {\mathbb Z}$, and where $(p,a)=1$, since $b \notin \Lambda$. Now, if $s,t \in{\mathbb Z}$ are such that $sa+tp=1$, then $spb = sa f_n = f_n - tp f_n$, so $sb = \frac {f_n}p - t f_n$. Hence, since $tf_n \in \Lambda$, we may change the generator $\gamma$, of ${\Gamma}$ mod ${\Lambda}$, by $\tilde {\gamma}:= (BL_b)^s L_{t f_n}= B^s L_{{\frac}1p f_n}$, which has the asserted properties. Finally, we note that since $B$ has no fixed points on $({\Lambda}_0)^\perp$ then $\langle f_j, f_n \rangle=0$ for any $1\le j \le n-1$.
This completes the proof.
\[cfm dim3\] The compact flat manifolds are classified, up to affine equivalence, only in low dimensions $n\le 6$ ([@HW] $n\le 3$, [@BBNWZ] $n=4$ and [@CZ] $n=5,6$). In dimension 3 there are 10 compact flat manifolds, half of them having cyclic holonomy groups $F\simeq {\mathbb Z}_2,{\mathbb Z}_3,{\mathbb Z}_4, {\mathbb Z}_6$ (see [@Wo]). These manifolds were described in [@CR], where they were called *platycosms*. Out of these, there is only one ${\mathbb Z}_3$-manifold, the *tricosm*. That is, there is only one 3-dimensional ${\mathbb Z}_p$-manifold with $p$ an odd prime. It is denoted by $\mathcal{G}_3$ in [@Wo] and by $c3$ in [@CR].
The models $\zp$ {#zpabc}
----------------
We shall now give some explicit models of ${\mathbb Z}_p$-manifolds. In particular, we will show that for any triple, $a,b,c\in {\mathbb N}_0$, $c\ge 1$ and any ideal $\mathfrak{a}$ in $\mathcal {O}$, one can construct a compact flat manifold with translation lattice ${\Lambda}$ such that, as a ${\mathbb Z}_p$-module it satisfies . For $a,b,c \in {\mathbb N}_0$, $c\ge 1$, and any ideal $\mathfrak{a}$, let $C_p, J_p$, and $C_{p, \mathfrak{a}}$ be as in the previous subsection and define matrices $C, C' \in {\text{GL}}_n({\mathbb Z})$ of the form $$\label{matrices C}
\begin{split}
& C = \text{diag}(\underbrace{C_p,\ldots,C_p}_{a},\underbrace{J_p,\ldots,J_p}_{b},\underbrace{1,\ldots,1}_{c})\,,
\\
& C' = \text{diag}(C_{p, \mathfrak{a}}, \underbrace{C_p,\ldots,C_p}_{a-1}, \underbrace{J_p,\ldots,J_p}_{b},
\underbrace{1,\ldots,1}_{c})\,,
\end{split}$$ where $\mathfrak{a}$ is not principal.
Note that, actually, $C$ is just a particular case of $C'$ when $\mathfrak{a}= \mathcal{O}$ (or when $\mathfrak{a}$ is principal) and $C'$ depends on $a,b,c$ and $\mathfrak{a}$, though this is not apparent in the notation. Although $C, C' \not \in \mathrm{O}(n)$, they can be conjugated into $\text{I}({\mathbb R}^n)$. Indeed, the eigenvalues of $C_p$ and $C_{p, \mathfrak{a}}$ are exactly the primitive $p^{\mathrm{th}}$-roots of unity and the eigenvalues of $J_p$ are all $p^{\mathrm{th}}$-roots of unity. Thus, if $\sim$ means conjugation in ${\text{GL}}_n({\mathbb R})$, then $C_p \sim B_p$, $C_{p, \mathfrak{a}} \sim B_p$ and $J_p \sim (\begin{smallmatrix} B_p & \\ & 1 \end{smallmatrix})$, where $$\label{Bp-matrix}
B_p := \text{diag}\big( B({\tfrac}{2\pi}{p}), B({\tfrac}{2\cdot 2\pi}{p}), \ldots,
B({\tfrac}{2q\pi}{p})\big),$$ with $q=[{\frac}{p-1}2]$ and $B(t) = \left( \begin{smallmatrix} \cos t & -\sin t
\\ \sin t & \cos t \end{smallmatrix}\right)$, $t \in {\mathbb R}$. That is, there exists a matrix $X_{\mathfrak a} \in {\text{GL}}_{n-c}({\mathbb R}) \subset {\text{GL}}_n({\mathbb R})$ such that $X_{\mathfrak a} C' X^{-1}_{\mathfrak a} = B \in \text{SO}(n-c) \subset \text{SO}(n)$, where $$\label{diag Bp-matrix}
B = \text{diag}(\underbrace{B_p,\ldots,B_p}_{a+b},\underbrace{1,\ldots,1}_{b+c}).$$
We now define a lattice in ${\mathbb R}^n$ by $${\Lambda}_{p,a}^{b,c}(\mathfrak{a}) := X_{\mathfrak a} {\mathbb Z}^{n-c} \stackrel{\perp}{\oplus} {\mathbb Z}^{c} = X_{\mathfrak{a}} L_{{\mathbb Z}^n} X_{\mathfrak{a}}^{-1}.$$ Thus, as a ${\mathbb Z}_p$-module, ${\Lambda}_{p,a}^{b,c}(\mathfrak{a})$ decomposes as in , with $c{\text{\sl Id}}$ orthonormal to its complement $\mathfrak{a} \oplus (a-1)\mathcal{O}\oplus b{\mathbb Z}[{\mathbb Z}_p]$. With these ingredients, we define an $n$-dimensional Bieberbach group: $$\label{Gpa}
{\Gamma}_{p,a}^{b,c} (\mathfrak a) := \langle BL_{{\tfrac}{e_n}{p}}, {\Lambda}_{p,a}^{b,c}(\mathfrak{a}) \rangle,$$ where $e_n$ is the canonical vector, and the corresponding flat Riemannian $n$-manifold $$\label{mzp}
M_{p,a}^{b,c} (\mathfrak a):= {\Gamma}_{p,a}^{b,c} (\mathfrak a) \backslash {\mathbb R}^n.$$
\[essential\] As we shall see, in the study of eta invariants of ${\mathbb Z}_p$-manifolds it will essentially suffice to look at exceptional ${\mathbb Z}_p$-manifolds, i.e. those with $\beta_1 =1$. Proposition \[propZp\] (vi) says that any exceptional ${\mathbb Z}_p$-manifold $M$ is diffeomorphic to some $M_{p,a}^{0,1}(\mathfrak a)$ as in , i.e. having $b = {\tfrac}1p e_n$.
As mentioned in the Introduction, the integrality result in Theorem \[twisted etas mod Z\] will be proved to hold for every ${\mathbb Z}_p$-manifold except for a single one, the so called tricosm. We now give a description of this manifold.
\[tricosm\] In our previous description, the tricosm $c3$ (see Remark \[cfm dim3\]) corresponds to $M_{3,1} = M_{3,1}^{0,1}(\mathcal{O})$, with $\mathcal{O} = {\mathbb Z}[{\frac}{2\pi i}{3}]$. So, as a ${\mathbb Z}_3$-module, we have $\Lambda = {\mathbb Z}[e^{{\frac}{2\pi i}{3}}] \oplus {\mathbb Z}$ and the integral representation of $F\simeq {\mathbb Z}_3$ is given by the matrix $\tilde C_3 = \mathrm{diag}(C_3,1) =
\left( \begin{smallmatrix} 0 & -1 & \\ 1 & -1 & \\ & & 1 \end{smallmatrix} \right)$. Then $M_{3,1} = \langle BL_{{\frac}{e_3}{3}}, L_{f_1}, L_{f_2}, L_{e_3} \rangle \backslash {\mathbb R}^3$ where $f_1,f_2,e_3$ is a ${\mathbb Z}$-basis of $\Lambda_{3,1} = X{\mathbb Z}^2 \oplus {\mathbb Z}$, $B = \left( \begin{smallmatrix} -1/2 & -\sqrt 3/2 & \\ \sqrt 3/2 & -1/2 & \\ & & 1 \end{smallmatrix} \right) \in \mathrm{SO}(3)$ and $X\in \mathrm{GL}_3({\mathbb R})$ is such that $X \tilde C_3 X^{-1} = B$.
Spin group and spin representations {#S2-SGSR}
-----------------------------------
Let $Cl(n)$ denote the Clifford algebra of ${\mathbb R}^n$ with respect to the standard inner product. Inside the group of units of $Cl(n)$ there is the spin group, $\text{Spin}(n)$, which is a compact, simply connected Lie group if $n\geq 3$. There is a canonical epimorphism $$\label{picovering}
\pi:\text{Spin}(n)\rightarrow \text{SO}(n)$$ given by $\pi(v)(x) = vxv^{-1}$, with kernel $\{\pm 1\}$. A maximal torus of $\text{Spin}(n)$ has the form $T=\left\{x(t_1,\ldots,t_m) \,:\, t_j \in {\mathbb R}, \, 1\le j\le m \right\}$ where $m = [\frac n2]$, $$\label{xelements}
x(t_1,\dots,t_m) := \prod_{j=1}^m (\cos t_j + \sin t_j \, e_{2j-1}e_{2j})
\in \text{Spin}(n)$$ and $e_1,\ldots,e_n$ is the canonical basis of ${\mathbb R}^n$. For convenience, if $a\in {\mathbb N}$, we shall use the notation $$\label{notacionxj}
x_{a}(t_1,t_2,\ldots,t_q) := x(\underbrace{t_1,t_2,\ldots,t_q}_{1}, \ldots,
\underbrace{t_1,t_2,\ldots,t_q}_{a}) \,.$$
Set $x_0(t_1,\ldots,t_m):= \mathrm{diag}(B(t_1),\ldots, B(t_m),1)$ if $n=2m+1$ and omit the 1 if $n=2m$. A maximal torus in $\text{SO}(n)$ is $T_0=\left\{x_0(t_1,\dots,t_m) \;:\; t_i \in {\mathbb R}\right\}$. The restriction map $\pi:{T}\rightarrow T_0$ is a 2-fold covering and $$\label{2 fold}
\pi(x(t_1,\dots,t_m)) = x_0(2t_1,\ldots,2t_m) \,.$$
Let $(L_n,{\text{S}}_n)$ denote the [*spin representation*]{} of ${\text{Spin}(n)}$, which is the restriction to ${\text{Spin}(n)}$ of any irreducible representation of the complex Clifford algebra $Cl(n)\otimes {\mathbb C}$. It has complex dimension $2^m$ with $m=[\tfrac n2]$. If $n$ is odd, $(L_n,{\text{S}}_n)$ is irreducible while, if $n$ is even, ${\text{S}}_n = \text{S}_n^+
\oplus \text{S}_n^-$ where each ${\text{S}}_n^\pm$ is irreducible of dimension $2^{m-1}$. The representations $L_n^\pm := {L_n}_{|S_n^{\pm}}$ are called the [*half-spin representations*]{}. It is known that the values of the characters of $L_n$ and $L^\pm_n$ on the torus $T$ are given by (see [@MiPo06], Lemma 6.1) $$\label{spinpmchars}
\begin{split}
& \chi_{_{L_n}} (x(t_1,\dots,t_m)) = 2^{m} \prod_{j=1}^m \cos t_j, \\
& \chi_{_{L^\pm_n}} (x(t_1,\dots,t_m))= 2^{m-1}\Big( \prod_{j=1}^m \cos t_j
\pm i^m \prod_{j=1}^m \sin t_j \Big).
\end{split}$$
Spin structures {#S2-SpSt}
---------------
It is a well known fact that spin structures on a compact flat spin manifold $M_{\Gamma}$ are in a 1–1 correspondence with group homomorphisms $$\label{spindiagram}
{\varepsilon}:{\Gamma}{\rightarrow}{\text{Spin}(n)}\qquad \text{ such that } \qquad \pi ({\varepsilon}(\gamma)) = B,$$ for any ${\gamma}=BL_b \in {\Gamma}$ (see [@Fr97], [@LM] or [@Pf]), where $\pi$ is as in (\[picovering\]).
Any morphism ${\varepsilon}$ as in is determined by the generators of ${\Gamma}$. Let $M_{\Gamma}$ be a ${\mathbb Z}_p$-manifold with ${\Gamma}= \langle {\gamma}, L_{\Lambda}\rangle$ and let $f_1,\ldots,f_n$ be a ${\mathbb Z}$-basis of $\Lambda$. Since $r({\Lambda})={\text{\sl Id}}$, we have ${\varepsilon}({\Lambda}) \in \{\pm 1\}$, and hence ${\varepsilon}$ is determined by ${\varepsilon}({\gamma})$ and $$\label{delta_j}
\delta_j := {\varepsilon}(L_{f_j}) \in \{\pm 1\}, \qquad j=1,\ldots,n.$$
Since every $F$-manifold with $|F|$ odd is spin ([@Va], Corollary 1.3), the ${\mathbb Z}_p$-manifolds considered are spin. The determination of such structures for ${\mathbb Z}_p$-manifolds were previously considered in some particular cases (see [@MiPo06] and [@MiPo09] for the exceptional manifolds $M_{p,a}^{0,1}(\mathfrak a)$, and in [@SS] in the case of $M_{p,1}^{0,1}$ and $p$ any odd integer, not necessarily prime). In fact, it is known that the exceptional manifolds $M_{p,a}^{0,1}(\mathfrak a)$ admit only two spin structures, ${\varepsilon}_1, {\varepsilon}_2$, one of which, ${\varepsilon}_1$, is of trivial type ([@MiPo09]). They are given, for $h=1,2$, by $$\label{eq.spinstructuresX}
\begin{split}
& {\varepsilon}_h(L_{f_1}) = \cdots = {\varepsilon}_h(L_{f_{n-1}}) = 1, \qquad {\varepsilon}_h(L_{f_n}) = (-1)^{h+1},
{\medskip}\\
& {\varepsilon}_h({\gamma}) = (-1)^{a[{\frac}{q+1}{2}] + h + 1} \; x_a\big({\tfrac}{\pi}p,{\tfrac}{2\pi}p,\dots, {\tfrac}{q\pi}p \big),
\end{split}$$ in the notation of .
Although in the sequel we will not need the explicit description of the spin structures of an arbitrary ${\mathbb Z}_p$-manifold, we will now give it for completeness. To this end, we will use the following abuse of notation $$\label{notation}
{\varepsilon}_{|{\Lambda}} = ({\varepsilon}(L_{f_1}),\ldots,{\varepsilon}(L_{f_n})).$$
\[spinstructs\] Let $p$ be an odd prime and let $M$ be a ${\mathbb Z}_p$-manifold with lattice of translations ${\Lambda}\simeq {\Lambda}(a,b,c,\mathfrak a)$. Then $M$ admits exactly $2^{b+c}=2^{\beta_1}$ spin structures, only one of which is of trivial type.
If, in particular, $M=\zp$ then the spin structures are explicitly given by $$\label{eq.spinstructures}
\begin{split}
& {{\varepsilon}}_{|{\Lambda}} = \big( \underbrace{1,\ldots,1}_{a(p-1)}, \underbrace{\delta_1, \ldots,\delta_1}_p,
\ldots, \underbrace{\delta_b,\ldots,\delta_b}_p, \delta_{b+1},\ldots,\delta_{b+c-1},
(-1)^{h+1} \big) {\medskip}\\
& {\varepsilon}({\gamma}) = (-1)^{(a+b)[{\frac}{q+1}{2}] + h + 1}
\; x_{a+b}\big({\tfrac}{\pi}p, {\tfrac}{2\pi}p,\dots, {\tfrac}{q\pi}p \big),
\end{split}$$ in the notations of , and .
Let $M={\Gamma}\backslash {\mathbb R}^n$ be a ${\mathbb Z}_p$-manifold. By the results in [@MiPo06] (see also [@MiPo09], [@MiPo04]), a group homomorphism ${\varepsilon}: {\Gamma}\rightarrow {\text{Spin}(n)}$ as in determines a spin structure on $M$ if and only if it satisfies the following conditions:
- ${\varepsilon}(L_{B{\lambda}})={\varepsilon}(L_{\lambda}) \quad \text{ for any } {\lambda}\in {\Lambda}$,
- ${\varepsilon}({\gamma})^p = {\varepsilon}({\gamma}^p) = {\varepsilon}(L_{pb_+})$,
where ${\gamma}=BL_{b}$ with $b=b_+ + b'$ and $b' \perp b_+$. Furthermore, the orthogonal projection of $b_+$ on ${\Lambda}_0 = c{\text{\sl Id}}$ is not $0$.
We fix a ${\mathbb Z}$-basis $f_1,\ldots,f_{n-1},f_n=e_n$ of ${\Lambda}$.
*Case 1, $M \simeq \zp$* First, suppose that $M = \zp$ and assume a group homomorphism ${\varepsilon}:{\Gamma}_{p,a}^{b,c}(\mathfrak a) \rightarrow {\text{Spin}(n)}$ as in is given.
By , and we have $$\label{vep(g)}
{\varepsilon}({\gamma}) = \pm (-1)^{(a+b)[{\frac}{q+1}{2}]} \; x_{a+b}\big({\tfrac}{\pi}p, {\tfrac}{2\pi}p,\dots, {\tfrac}{q\pi}p \big)$$ where $q={\frac}{p-1}{2}$.
We note that in this case, since $b_+ = {\frac}{e_n}{p}$, condition will only give a condition for ${\varepsilon}$ on ${\mathbb Z}e_n$. To determine the action of ${\varepsilon}$ on $$({\mathbb Z}e_n)^\perp = {\Lambda}_\mathcal{O} \oplus {\Lambda}_R \oplus {\Lambda}_0' = \mathcal{O}^{\oplus a} \oplus R^{\oplus b} \oplus {\text{\sl Id}}^{\oplus (c-1)},$$ we will use condition $\textsf{C1}$ together with the integral matrix $C'$ given in . Let $\tilde {\Gamma}:= \langle C'L_{{\frac}{e_n}{p}}, \tilde {\Lambda}\rangle \subset \mathrm{Aff}({\mathbb R}^n)$ where $\tilde {\Lambda}= {\Lambda}(a,b,c,\mathfrak a)$ is as in . By the description in §\[zpabc\], we have $X_{\mathfrak a} \tilde {\Gamma}X_{\mathfrak a}^{-1} = {\Gamma}_{p,a}^{b,c}(\mathfrak a)$. Now, define $\tilde {\varepsilon}: \tilde {\Gamma}{\rightarrow}{\text{Spin}(n)}$ by $\tilde{\varepsilon}= {\varepsilon}\circ I_{X_{\mathfrak a}}$ where $I_{X_{\mathfrak a}}$ is conjugation by $X_{\mathfrak a}$. Since $${\varepsilon}(L_{(B-{\text{\sl Id}}){\Lambda}_{p,a}^{b,c}(\mathfrak a)}) = {\varepsilon}( X_{\mathfrak a} L_{(C'-{\text{\sl Id}}) {\Lambda}} X_{\mathfrak a}^{-1}) = \tilde{\varepsilon}(L_{(C'-{\text{\sl Id}}){\Lambda}})$$ we have that ${\varepsilon}(L_{(B-{\text{\sl Id}}){\Lambda}_{p,a}^{b,c}(\mathfrak a)})=1$ if and only if $\tilde{\varepsilon}(L_{(C'-{\text{\sl Id}}){\Lambda}}) = 1$.
*Step 1.* Here we will show that ${\varepsilon}_{|{\Lambda}_\mathcal{O}}\equiv 1$. For any summand of type $\mathcal{O}={\mathbb Z}[\xi]$ in , there is a ${\mathbb Z}$-basis of the form $\{e,\xi\,e,\ldots,\xi^{p-2}\,e\}$. Hence by condition we must have $1 = \tilde{\varepsilon}(\xi\,e - e)=\cdots = \tilde{\varepsilon}(\xi^{p-2}\, e - \xi^{p-3}\,e) = \tilde{\varepsilon}(\xi^{p-1}\, e - \xi^{p-2}\,e)$. Thus, we have $\tilde{\varepsilon}(e) = \tilde{\varepsilon}( \xi\,e)= \cdots = \tilde{\varepsilon}(\xi^{p-2}\,e) = \tilde{\varepsilon}(e) \, \tilde{\varepsilon}(\xi\,e) \cdots \tilde{\varepsilon}(\xi^{p-2}\,e)$, where in the last equality we have used that $\xi^{p-1}=-\sum_{i=0}^{p-2} \xi^i$. This implies $\tilde{\varepsilon}(e)^{p-2} = 1$, and hence $\tilde{\varepsilon}(e)=1$ since $p$ is odd. Therefore, $\tilde{\varepsilon}(\xi^j\, e)=1$ for every $0\le j \le p-2$ and thus $\tilde{\varepsilon}(\lambda )=1$ for any $\lambda \in {\Lambda}_\mathcal{O}$.
Now, given a summand of type $\mathfrak{a}$ in ${\Lambda}$, there exist $e_1$, $e_2 \in \mathfrak{a}$ such that $\mathfrak{a} = \mathcal{O}e_1 + \mathcal{O}e_2$. By the same argument as in the case of $\mathcal{O}$, we conclude that $\tilde{\varepsilon}_{|\mathcal{O}e_1 } = \tilde{\varepsilon}_{|\mathcal{O}e_2 }
=1$. Hence $\tilde{\varepsilon}_{|\mathfrak{a}}=1$.
In this way, for any $\lambda \in {\Lambda}_\mathcal{O}$ we have $${\varepsilon}(L_{{\lambda}}) = {\varepsilon}(L_{X_{\mathfrak a} {\lambda}}) = {\varepsilon}(X_{\mathfrak a} L_{{\lambda}} X_{\mathfrak a}^{-1}) = \tilde{\varepsilon}(L_{{\lambda}}) =1.$$ In particular, $\delta_j = {\varepsilon}(L_{f_j}) = 1$ for $1\le j \le a(p-1)$.
*Step 2.* We now study the action of ${\varepsilon}$ on the block ${\Lambda}_R \oplus {\Lambda}_0$. For each summand of type $R={\mathbb Z}[{\mathbb Z}_p]$ in , there is a ${\mathbb Z}$-basis $\{e,\xi\,e,\ldots,\xi^{p-1}\,e\}$. Thus, by condition we have $\tilde{\varepsilon}(e) = \tilde{\varepsilon}( \xi \,e) = \cdots = \tilde{\varepsilon}(\xi^{p-1}\,e) \in \{\pm 1 \}$ since there are no extra restrictions. Hence $\tilde {\varepsilon}_{|R} = \delta$, where $\delta\in \{\pm 1\}$ and, proceeding as before, we have $${\varepsilon}(L_{f_{a(p-1)+jp+1}}) = \cdots = {\varepsilon}(L_{f_{a(p-1)+jp+p}}) = \delta_j, \qquad 0 \le j \le b-1.$$ Also, for trivial summands, it is clear that ${\varepsilon}(L_{f_i}) = \delta_i$ for $n-c+1 \le i \le n$. For $i\ne n$ there are no restrictions.
*Step 3.* Since ${\gamma}^p = L_{e_n}$, conditions and are linked together and give a restriction which determines both the value of ${\varepsilon}(L_{e_n})$ and the sign in . Indeed, since ${\gamma}^p=L_{e_n}$ we have $$\delta_n = {\varepsilon}({\gamma})^p = \sigma \, x_{a+b}({\tfrac}{\pi}p,{\tfrac}{2\pi}p,\ldots,{\tfrac}{q\pi}p)^p
= \sigma \, x_{a+b}(\pi,2\pi,\ldots,q\pi) = \sigma (-1)^{t+1}$$ with $\sigma = \pm (-1)^{(a+b)[{\frac}{q+1}{2}]}$ and where we have used that $x(\theta)^k = x(k\theta)$ for any $\theta\in {\mathbb R}$, $k\in{\mathbb Z}$ and the commutativity in ${\mathbb C}l(n)$ of the elements $e_{2i-1}e_{2i}$ and $e_{2j-1}e_{2j}$ for $i \not = j$.
Putting $\pm = (-1)^{h+1}$ with $h=1,2$, we get the expressions in .
*Case 2, $M$ arbitrary*. The proof is entirely analogous, where we now start with the $b+c-1$ characters $\delta_j$ corresponding to ${\Lambda}_R$ and ${\Lambda}_0'=(c-1){\text{\sl Id}}$, and where we have $(BL_b)^p =L_{pb_+}$ with $pb_+ \in {\Lambda}_R^B \oplus {\Lambda}_0$ (in place of $pb_+ =f_n$).
Then the equation ${\varepsilon}({\gamma})^p = {\varepsilon}({\gamma}^p) = {\varepsilon}(L_{p b_+})$ imposes a condition linking the $\delta$’s and we again have $2^{b+c}$ spin structures as in the case of the model before.
It is known that if a manifold $M$ is spin, the inequivalent spin structures are classified by $H^1(M,{\mathbb Z}_2)$ ([@LM], [@Fr97]). For $M$ a ${\mathbb Z}_p$-manifold, by Proposition \[propZp\] (iv) and the universal coefficients theorem (or also directly), one can prove that $H^1(M,{\mathbb Z}_2) \simeq H_1(M,{\mathbb Z}_2) \simeq {\mathbb Z}_2^{b+c}$. Hence, the number of spin structures of a ${\mathbb Z}_p$-manifold is $2^{b+c}=2^{\beta_1}$. In Proposition \[spinstructs\] we give a direct proof of this fact together with an explicit description of these structures in the case of the models $M=\zp$.
Twisted eta series {#sect-3}
==================
Spectrum of twisted Dirac operators {#S2-STDO}
-----------------------------------
Let $(M_{\Gamma},{\varepsilon})$ be a compact flat spin $n$-manifold with lattice of translations $\Lambda$. Let $\rho : \Gamma \rightarrow U(V)$ be a unitary representation such that $\rho_{|\Lambda} =1$. Consider the *twisted Dirac operator* $$D_\rho = \sum_{i=1}^n L_n(e_i) \, {\frac}{\partial}{\partial x_i},$$ where $\{e_1,\ldots,e_n\}$ is an orthonormal basis of ${\mathbb R}^n$ and $L_n$ is the spin representation, acting on smooth sections of the twisted spinor bundle $$\mathcal{S}_\rho({M_\Gamma},{\varepsilon}) = {\Gamma}\backslash({\mathbb R}^n\times ({\text{S}}\otimes V)){\rightarrow}{\Gamma}\backslash {\mathbb R}^n$$ of ${M_\Gamma}$ (see [@MiPo06] for details).
Let ${\Lambda}^\ast_{{\varepsilon}} = \{ u \in {\Lambda}^\ast : {\varepsilon}(L_{\lambda}) = e^{2\pi i
{\lambda}\cdot u} \text{ for any } {\lambda}\in {\Lambda}\}$, where ${\Lambda}^*$ is the dual lattice. The nonzero eigenvalues of $D_\rho$ are of the form $\pm 2\pi \mu$ with $\mu = ||v||$ for some $v \in {\Lambda}_{\varepsilon}^*$. In [@MiPo06], Theorem 2.5, it is shown that the multiplicities $d_{\rho,\mu}^\pm$ of $\pm 2\pi
\mu$ for $({M_\Gamma},{\varepsilon})$ are given, for $n$ odd, by $$\label{eq.multipodd}
d_{\rho,{\mu}}^\pm({\Gamma},{\varepsilon}) = {\tfrac}{1}{|F|} \sum_{{\gamma}=BL_b \in {\Lambda}\backslash
{\Gamma}} \chi_\rho({\gamma}) \sum _{u \in ({\Lambda}_{{\varepsilon},\mu}^\ast)^B}
e^{-2\pi i u\cdot b} \;\chi_{_{L_{n-1}^{\pm \sigma(u,x_{\gamma})}}}(x_{\gamma}).$$ Here, $\chi_\rho$ and $\chi_{L_{n-1}^\pm}$ are the characters of $\rho$ and of the half spin representations, respectively, and for ${\gamma}= BL_b \in {\Gamma}$ we have ${\Lambda}_{{\varepsilon},\mu}^\ast = \{ v \in {\Lambda}_{\varepsilon}^* \,:\, ||v||=\mu \}$ and $$\label{lattice emuB}
({\Lambda}_{{\varepsilon},\mu}^\ast)^B = \{ v \in {\Lambda}_{{\varepsilon},\mu}^* \,:\, Bv=v \}.$$ Furthermore, $x_{\gamma}\in T$ is a fixed element in the maximal torus of ${\text{Spin}(n)}$, conjugate in ${\text{Spin}(n)}$ to ${\varepsilon}({\gamma})$, and $\sigma(u, x_{\gamma})$ is a sign depending on $u$ and on the conjugacy class of $x_{\gamma}$ in $\text{Spin}(n-1)$ (see Definition 2.3 in [@MiPo06]).
Relative to the multiplicity of the 0 eigenvalue, i.e. the dimension of the space of harmonic spinors, it is shown in [@MiPo06] that $$\label{eq.harmonicspinors}
d_{\rho,0}(\Gamma,{\varepsilon}) = \left\{ \begin{array}{ll} {\tfrac}{1}{|F|} \,
\sum\limits_{{\gamma}\in {\Lambda}\backslash {\Gamma}} \, \chi_{_\rho}({\gamma}) \;
{\chi}_{_{L_{n}}}({\varepsilon}({\gamma})) & \qquad \text{if } {\varepsilon}_{|{\Lambda}}= 1, {\smallskip}\\ 0 &
\qquad \text{if } {\varepsilon}_{|{\Lambda}} \ne 1.
\end{array} \right.$$
Spectral asymmetry {#specasymm}
------------------
Consider an arbitrary ${\mathbb Z}_p$-manifold $M$ of dimension $n$ as in , with $p$ an odd prime, equipped with a spin structure ${\varepsilon}$. The formula for the multiplicity of the eigenvalues involves the character $\chi_{\rho}$ of a representation $\rho : {\mathbb Z}_p \rightarrow U(V)$. Thus, we will consider for each $0\le \ell \le p-1$, the Dirac operator $D_\ell$ twisted by the characters $$\label{eq.characters}
\rho_\ell : {\mathbb Z}_p \rightarrow \mathbb{S}^1 \subset {\mathbb C}^*, \qquad k \mapsto
e^{{\frac}{2\pi ik \ell }{p}}.$$
\[symmetric spectrum\] By Corollary 2.6 in [@MiPo06], valid for arbitrary compact flat manifolds, if $n$ is even, or else, if $n$ is odd and $n_B\ge 2$ for every $BL_b\in {\Gamma}$, then the spectrum of $D_\ell$ is symmetric. That is, one has that $d_{\ell,\mu}^+({\Gamma},{\varepsilon}) = d_{\ell,\mu}^- ({\Gamma},{\varepsilon})$. Hence $\eta_{\ell,{\varepsilon}}(s) \equiv 0$ and, in particular, $\eta_{\ell,{\varepsilon}} = 0$. In the case of ${\mathbb Z}_p$-manifolds, since $n_B= b+c$ and $c\ge 1$, we see that for the non-exceptional ones, i.e. those with $(b,c) \ne (0,1)$, the eta invariant is just given by $\bar \eta_{\ell,{\varepsilon}} = {\tfrac}12 \dim \ker D_\ell$. This computation will be done in the next section (see ).
By the previous remark and Remark \[essential\], in the computations of $\eta_\ell(s)$ and $\eta_\ell$, we will need only consider *exceptional* ${\mathbb Z}_p$-manifolds of the form $M_{p,a}^{0,1}(\mathfrak a)$.
By , we can write $$\label{eq.etaseriesdif}
\eta_{\ell,h}(s):= \eta_{\ell}({\Gamma},{\varepsilon}_h)(s) = \sum_{\pm 2\pi \mu \in
\mathcal{A}_{\ell,h}} {\frac}{ d_{\ell,\mu,h}^+ - d_{\ell,\mu,h}^-}{(2\pi \mu)^s}$$ for $\mathrm{Re}(s)>n$, where $d_{\ell,\mu,h}^\pm$ stand for $d_{\rho_\ell,\mu}^\pm({\Gamma},{\varepsilon}_h)$ as given in (\[eq.multipodd\]) and $\mathcal{A}_{\ell,h}$ denotes the *asymmetric spectrum*, that is $$\mathcal{A}_{\ell,h} = \{ \pm 2\pi \mu \in Spec_{D_{\ell,h}}(M) : d_{\ell,\mu, h}^+\ne d_{\ell,\mu,h}^-\}.$$
To this end, we will first compute the differences, $d_{\ell,\mu,h}^+ - d_{\ell,\mu,h}^-$ in Proposition \[prop.Deltas\], and then the series in , in Theorem \[thm.eta series\].
\[lem.Deltafinal\] Let $p$ be an odd prime and $\ell \in {\mathbb N}$ with $0\le \ell \le p-1$. Let $M$ be an exceptional ${\mathbb Z}_p$-manifold with a spin structure ${\varepsilon}_h$, $h=1,2$. Then $$\begin{aligned}
&&d_{\ell,\mu,h}^+ - d_{\ell,\mu,h}^- \\
&=& (-1)^{({\frac}{p^2-1}{8})a+1} \, i^{m+1} \, 2 \, p^{{\frac}{a}{2}-1} \, \sum_{k=1}^{p-1} (-1)^{k(h+1)} \, \big({\tfrac}kp \big)^a
\, e^{{\frac}{2\pi i k\ell}{p}} \sin({\tfrac}{2\pi \mu k}p),\end{aligned}$$ where $\big( {\frac}{\cdot}p \big)$ is the Legendre symbol and $d_{\ell,\mu,h}^\pm$ denotes the multiplicity of the eigenvalue $\pm 2\pi \mu$ of $D_\ell$.
Given an exceptional ${\mathbb Z}_p$-manifold $M_{\Gamma}$, by Proposition \[propZp\] (iii), we may assume that ${\Gamma}=\langle {\gamma}, {\Lambda}\rangle$ with ${\gamma}= BL_b \in {\Gamma}$, $B^p={\text{\sl Id}}$ and $b={\frac}{e_n}{p}$.
We begin by computing the expression in . Note that the holonomy group is $F \simeq \{{\text{\sl Id}},B,B^2,\ldots,B^{p-1}\}$. Since ${\varepsilon}_h({\gamma}^k) = {\varepsilon}_h({\gamma})^k \in T$, we can take $x_{{\gamma}^k} = {\varepsilon}_h({\gamma}^k)$ and hence $\sigma(e_n,x_{{\gamma}^k})=1$ for every $1\le k \le n-1$, by the definition of $\sigma$ (see [@MiPo06]). Hence, according to (\[eq.multipodd\]), if $b_k$ is defined by the relation ${\gamma}^k = B^k L_{b_k}$, we obtain $$\label{multipoddzp}
d_{\ell,\mu,h}^\pm = {\tfrac}1p \, \sum_{k=0}^{p-1} \rho_\ell(k) \sum _{u \in ({\Lambda}_{{\varepsilon}_h,\mu}^\ast)^{B^k}} e^{-2\pi i u \cdot b_k} \; \chi_{{L_{n-1}^{\pm \sigma}}}({\varepsilon}_h({{\gamma}}^k))$$ where $\sigma = \sigma(u,x_{{\gamma}^k})$.
Now, since ${\Lambda}= \big( \mathfrak a \oplus (a-1)\mathcal{O} \big) \stackrel{\perp}{\oplus} {\mathbb Z}e_n$ and $({\mathbb R}^n)^{B^k} = {\mathbb R}e_n$, $1\le k \le n-1$, we have that $({\Lambda}_{{\varepsilon}_h}^\ast)^{B^k} = {\mathbb Z}e_n$ if $h=1$ and $({\Lambda}_{{\varepsilon}_h}^\ast)^{B^k} = ({\mathbb Z}+{\tfrac}12) e_n$ if $h=2$. Hence, $$\label{eq.latis}
({\Lambda}_{{\varepsilon}_h,\mu}^\ast)^{B^k} = \{\pm \mu e_n\}$$ with $\mu \in {\mathbb N}$ for ${\varepsilon}_1$ and $\mu \in {\mathbb N}_0+{\tfrac}12$ for ${\varepsilon}_2$.
In this way, using and the fact that $b_k = {\frac}{k}{p}e_n$, we see that reduces to $$\label{eq.multips}
d_{\ell,\mu,h}^\pm = \tfrac 1p \,\Big(2^{m-1}|{\Lambda}_{{\varepsilon}_h,\mu}^*| + \sum_{k=1}^{p-1} e^{{\frac}{2\pi
i k\ell}{p}} S_{\mu,h}^\pm(k)
\Big)$$ where we have put $$\label{eq.skmus}
S_{\mu,h}^\pm(k) := e^{{\frac}{-2 \pi i \mu k}p} \chi_{L_{n-1}^\pm}({\varepsilon}_h({\gamma}^k))
+ e^{{\frac}{2 \pi i \mu k}p} \chi_{L_{n-1}^\mp}({\varepsilon}_h({\gamma}^k)).$$ Here we have used that $\sigma(-u,\gamma)=-\sigma(u,\gamma)$ and that $\sigma(e_n,{\gamma}^k)=1$.
Now, using that $x(\theta)^k=x(k\theta)$ for $\theta\in {\mathbb R}, k\in {\mathbb Z}$, we have that $$\label{eq.spinstructure gk}
{\varepsilon}_h({\gamma}^k)= (-1)^{s_{h,k}} \; x_{a}\big({\tfrac}{k\pi}p, {\tfrac}{2k\pi}p,\dots,
{\tfrac}{qk\pi}p \big)$$ for $1\le k\le p$, where $$\label{eq.signs}
s_{h,k} := k([{\tfrac}{q+1}{2}]a + h + 1).$$ Thus, by and using , we obtain $$\chi_{_{L^\pm_{n-1}}}({\varepsilon}_h({\gamma}^k)) = (-1)^{s_{h,k}} \, 2^{m-1} \Big\{
\Big(\prod_{j=1}^q \cos({\tfrac}{jk\pi}p)\Big)^{a} \pm i^m \Big(\prod_{j=1}^q
\sin({\tfrac}{jk\pi}p)\Big)^{a} \Big\}.$$ Substituting this expression into we see that $$\begin{gathered}
\label{eq.skhmus}
S_{\mu,h}^\pm(k) = (-1)^{s_{h,k}} \,2^{m} \\ \times \Big\{
\cos(\tfrac{2k\pi \mu}p) \Big(\prod_{j=1}^q \cos(\tfrac{jk\pi}p)
\Big)^{a} \mp i^{m+1} \sin(\tfrac{2k\pi \mu}p) \Big(\prod_{j=1}^q
\sin({\tfrac}{jk\pi}p) \Big)^{a} \Big\}.\end{gathered}$$ Hence, by and , we obtain $$\begin{aligned}
d_{\ell,\mu,h}^+ - d_{\ell,\mu,h}^- & = & \tfrac 1p \, \sum_{k=1}^{p-1} e^{{\frac}{2\pi i
k\ell}{p}} \big( S_{\mu,h}^+(k) - S_{\mu,h}^-(k) \big)
\\
&=& -\tfrac{(2i)^{m+1}}{p} \, \sum_{k=1}^{p-1} (-1)^{s_{h,k}} \, e^{{\frac}{2\pi
i k\ell}{p}} \, \sin({\tfrac}{2k\pi \mu}p)\, \Big(\prod_{j=1}^q
\sin \big({\tfrac}{jk\pi}p\big)\Big)^{a}.\end{aligned}$$ Now, by Lemma \[lem.prodtrigs\](i), , and also using that $(-1)^{\frac}{p^2-1}8 =
(-1)^{[{\frac}{q+1}{2}]}$ and $aq=m$, we arrive at the desired expression.
Our next goal is to find explicit expressions for $d^+_{\ell,\mu,h} - d^-_{\ell,\mu,h}$. First, we fix some notations. Set $p=2q+1$ and $n=2m+1$. Then, since $b=0$ and $c=1$, we get $n=a(p-1)+1 = 2aq+1$. Thus, $m=aq$ and we have $$\label{conditions2}
a \text{ even} \: \Rightarrow \: m=2r, \quad a \text{ odd} \: \Rightarrow
\: \left\{ \begin{array}{ll}
p = 4t+1 \quad \Leftrightarrow \quad m=2r, {\smallskip}\\
p = 4t+3 \quad \Leftrightarrow \quad m=2r+1.
\end{array} \right.$$
The proof of Lemma \[lem.Deltafinal\] shows that $\mu \in {\mathbb N}$ if $h=1$ and $\mu \in {\mathbb N}_0 + {\tfrac}12$ if $h=2$.
\[prop.Deltas\] Let $p=2q+1$ be a prime and let $\ell \in {\mathbb N}_0$ be chosen with $0\le \ell \le p-1$. Consider $D_\ell$ acting on an exceptional ${\mathbb Z}_p$-manifold of dimension $n = a(p-1)+1$ equipped with a spin structure ${\varepsilon}_h$, $h=1,2$. Put $r=[{\frac}n4]$.
\(i) If $a$ is even, then $d_{0,\mu,1}^+ - d_{0,\mu,1}^- = d_{0,\mu,2}^+ - d_{0,\mu,2}^- =0$ and if $\ell\ne0$ we have $$d_{\ell,\mu,h}^+ - d_{\ell,\mu,h}^- = \left\{ \begin{array}{ll} \pm (-1)^{r} \, p^{{\frac}a2}
& \qquad \textrm{if } p \,|\, h(\ell \mp \mu), {\medskip}\\ 0 & \qquad \textrm{otherwise.} \end{array}\right.$$
\(ii) If $a$ is odd, then $$\begin{aligned}
d_{\ell,\mu,h}^+ - d_{\ell,\mu,h}^- = (-1)^{q+r} \, \Big( \big({\tfrac}{2(\ell - \mu) }{p} \big) - \big( {\tfrac}{2(\ell + \mu)}{p} \big) \Big) \, p^{{\frac}{a-1}{2}}.\end{aligned}$$ In particular, for $\ell=0$ we have $$d_{0,\mu,h}^+ - d_{0,\mu,h}^-
= \left\{ \begin{array}{ll}
0 & \quad \text{if } p\equiv 1 \,(4), {\smallskip}\\
(-1)^{r} \, 2 \, \big({\tfrac}{2\mu}{p} \big) \, p^{{\frac}{a-1}{2}} & \quad
\text{if } p\equiv 3 \,(4),
\end{array} \right.$$ where $\big( {\frac}{\cdot}p \big)$ denotes the Legendre symbol and $\mu\in {\tfrac}12{\mathbb N}_0$.
We define the integer $$\label{eq:cmu}
c_\mu = c(\mu,h) := \mu-{\tfrac}{\delta_{h,2}}{2} \in {\mathbb N}_0,$$ where $\delta_{_{h,2}}$ is the Kronecker delta function. Note that the expression in Lemma \[lem.Deltafinal\] can be written as $$\label{eq.Delta a}
d_{\ell,\mu,h}^+ - d_{\ell,\mu,h}^- \left\{ \begin{array}{ll}
-i^{m+1} \,2\,p^{{\frac}{a}{2}-1} F_h^{\chi_{_0}}(\ell,c_\mu) & \qquad \text{if
$a$ even,} {\medskip}\\
- i^{m+1} \,2\, p^{{\frac}{a}{2}-1} \,(-1)^{({\frac}{p^2-1}{8})} \,
F_h^{\chi_p}(\ell,c_\mu) & \qquad \text{if $a$ odd,}
\end{array} \right.$$ in the notations of Definition \[defi.sums\].
Assertion (i) follows directly from Proposition \[prop.Qh\] and from the previous expression. Relative to assertion (ii), we can apply Proposition \[prop.Fh\] and the fact that $\big( {\frac}2p \big) =
(-1)^{{\frac}{p^2-1}{8}}$ to get $$\begin{aligned}
d_{\ell,\mu,1}^+ - d_{\ell,\mu,1}^- & = i^m \, \delta(p) \, p^{{\frac}{a-1}{2}} \, \big( {\tfrac}2p \big) \, \Big( \big( {\tfrac}{\ell-\mu}{p} \big) - \big({\tfrac}{\ell+\mu}{p}\big) \Big), \\ d_{\ell,\mu,2}^+ - d_{\ell,\mu,2}^- & = i^m \, \delta(p) \, p^{{\frac}{a-1}{2}} \, \Big(
\big( {\tfrac}{2(\ell-c_\mu)-1}{p} \big) - \big({\tfrac}{2(\ell+c_\mu)+1}{p}\big) \Big),\end{aligned}$$ where $\delta(p)=1$ if $p\equiv 1 \,(4)$ and $\delta(p)=i$ if $p\equiv 3 \,(4)$. Note that by $i^m \delta(p) = (-1)^{q+r}$. The result follows from and the multiplicativity of $({\tfrac}.p)$. In particular, for $\ell=0$ we get the remaining assertion.
Eta series {#S3-ES}
----------
We are now in a position to explicitly compute the twisted eta function $\eta_{\ell,h}(s)$ of a general spin ${\mathbb Z}_p$-manifold $(M,{\varepsilon}_h)$. We shall see that the expressions will be given in terms of Hurwitz zeta functions $$\label{hurwitz}
\zeta(s,\alpha) = \sum_{n\ge 0} {\tfrac}{1}{(n+\alpha)^s}, \qquad
\mathrm{Re}(s)>1, \quad \alpha \in (0,1].$$
\[thm.eta series\] Let $p$ be an odd prime, $\ell\in {\mathbb N}_0$ with $0 \le \ell\le p-1$. Let $(M,{\varepsilon}_h)$, be an exceptional spin ${\mathbb Z}_p$-manifold of dimension $n=a(p-1)+1$. Put $r=[{\tfrac}n4]$ and $t=[{\frac}{p}{4}]$. Then, the eta series is given as follows:
*(i)* Let $a$ be even. Then $\eta_{0,h}(s) =0$, $h=1,2,$ and for $1\le \ell \le p-1$ we have $$\begin{aligned}
& \eta_{\ell,1}
(s) = {\tfrac}{(-1)^r}{(2\pi p)^s} \, p^{{\frac}a2} \, \big( \zeta(s,{\tfrac}\ell p) -
\zeta(s,{\tfrac}{p-\ell}{p}) \big), \\ \\
& \eta_{\ell,2}
(s) = \left\{ \begin{array}{ll} {\tfrac}{(-1)^r}{(2\pi p)^{s}} \,\ p^{{\frac}a2} \,
\Big( \zeta(s,{\tfrac}12 + {\tfrac}\ell p) - \zeta(s,{\tfrac}12 - {\tfrac}\ell p) \Big) &
\quad 1 \le \ell \le
q, {\medskip}\\
{\tfrac}{(-1)^{r}}{(2\pi p)^{s}} \, p^{{\frac}a2} \, \Big( \zeta(s,{\tfrac}12 -
{\tfrac}{p-\ell}{p}) - \zeta(s,{\tfrac}12 + {\tfrac}{p-\ell}{p}) \Big) & \quad q< \ell <p.
\end{array} \right.\end{aligned}$$
*(ii)* Let $a$ be odd. Then, for $0\le \ell\le p-1$ we have $$\begin{aligned}
\eta_{\ell,1}(s) & = {\tfrac}{(-1)^{t+r}}{(2\pi p)^s} \, p^{{\frac}{a-1}{2}} \, \sum_{j=1}^{p-1}
\Big( ({\tfrac}{\ell -j}{p}) - ({\tfrac}{\ell +
j}{p}) \Big) \, \zeta(s,{\tfrac}jp),
{\smallskip}\\
\eta_{\ell,2}(s) & = {\tfrac}{(-1)^{q+r}}{(\pi p)^s} \, p^{{\frac}{a-1}{2}} \, \sum_{j=0}^{p-1}
\left( ({\tfrac}{2\ell-(2j+1)}{p}) -
({\tfrac}{2\ell+(2j+1)}{p}) \right) \, \zeta(s,{\tfrac}{2j+1}{2p}).\end{aligned}$$ In particular, $\eta_{0,h}(s)=0$ for $p\equiv 1 \, (4)$.
By and , we have to compute the series $$\label{etaseriesdif2}
\eta_{\ell,h}(s) = {\tfrac}{1}{\pi^s} \, \sum_{c=1}^{\infty}
{\frac}{d_{\ell,\mu,h}^+ - d_{\ell,\mu,h}^-}{(2c-\delta_{_{h,2}})^s}.$$
We first prove (i). Let $a$ be even. By Proposition \[prop.Deltas\] we have that $d_{0,\mu,h}^+ - d_{0,\mu,h}^- = 0$ and hence $\eta_{0,h}(s) = 0$, $h=1,2$. Also, for $1 \le \ell \le p-1$ we have $ d_{\ell,\mu,h}^+ - d_{\ell,\mu,h}^- = \pm (-1)^r \, p^{{\tfrac}a2}$ if $p \,|\, h(\ell \mp \mu)$ where $\mu = c_\mu + {\frac}{\delta_{_{h,2}}}{2}$ with $c_\mu \in {\mathbb N}_0$; and $ d_{\ell,\mu,h}^+ - d_{\ell,\mu,h}^- = 0$ otherwise. Let $c=c_\mu
\ge 1$.
\(a) Take $h=1$ and $p\,|\, \ell \mp c$. Then $c = \mp (pk-\ell)$ for some $k\in{\mathbb Z}$ and $$\begin{aligned}
c=\ell - pk \ge 1 \: \Leftrightarrow \: k\le 0, \qquad c=pk - \ell \ge
1 \: \Leftrightarrow \: k\ge 1.\end{aligned}$$ Thus, by we get $$\begin{aligned}
\eta_{\ell,1}(s) &=&
{\tfrac}{(-1)^r}{(2\pi)^s}\, p^{{\frac}a2} \, \Big( \sum_{k\le 0} {\tfrac}{1}{{(\ell -
pk)}^s} - \sum_{k\ge 1} {\tfrac}{1}{{(pk-\ell)}^s} \Big)\\& =&
{\tfrac}{(-1)^r}{(2\pi p)^s} \, p^{{\frac}a2} \, \big( \zeta(s,{\tfrac}\ell p) -
\zeta(s, {\tfrac}{p-\ell} p) \big).\end{aligned}$$
\(b) Take $h=2$ and $p\,|\, 2(\ell \mp c) \mp 1$. Then $2(\ell \mp c) \mp
1 = pk$, with $k$ odd. Thus, we have $2c+1=\pm
(2\ell-pk)$, $k$ odd, and $$\begin{aligned}
& 2\ell-pk \ge 1 \quad \Leftrightarrow \quad \left\{ \begin{array}{ll}
k\le -1 & \quad \text{if } 1\le \ell \le q, {\smallskip}\\ k\le 1 &
\quad \text{if } q<\ell <p, \end{array}\right. {\medskip}\\ & pk-2\ell \ge 1
\quad \Leftrightarrow \quad
\left\{ \begin{array}{ll} k\ge 1 & \quad \:\:\:\, \text{if }
1\le \ell \le q, {\smallskip}\\ k\ge 3 & \quad \:\:\:\, \text{if } q<\ell <p.
\end{array}\right.\end{aligned}$$
Assume $1\le \ell \le q$. We have $$\begin{aligned}
\eta_{\ell,2}(s) &=& {\tfrac}{(-1)^r}{({2\pi p})^s}\, p^{{\frac}a2} \, \Big(
\sum_{\begin{smallmatrix}
k\le -1 \\ k \text{ odd} \end{smallmatrix}} {\frac}{1}{{(\frac \ell p - \frac
k2)}^s} - \sum_{\begin{smallmatrix} k\ge 1 \\ k \text{ odd}
\end{smallmatrix}}
{\frac}{1}{{( \frac k2- \frac \ell p)}^s} \Big) \\
&=& {\tfrac}{(-1)^r}{(2\pi p)^s} \, p^{{\frac}a2} \, \Big( \sum_{n=0}^{\infty}
{\frac}{1}{{(n+ {\tfrac}12 +{\tfrac}\ell p)}^s} -
\sum_{n=0}^\infty {\frac}{1}{{(n+ {\tfrac}12- {\tfrac}\ell p)}^s} \Big) \\
&=& {\tfrac}{(-1)^r}{(2\pi p)^s} \, p^{{\frac}a2} \, \big(
\zeta(s,{\tfrac}12 + {\tfrac}\ell p) - \zeta(s,{\tfrac}12 - {\tfrac}\ell{p}) \big),\end{aligned}$$ since $0<{\frac}12 \pm {\frac}\ell{p} <1$ for $1\le \ell \le q$.
The case $q<\ell<p$ is a bit more involved. We have $$\begin{aligned}
\label{etaell2}
\eta_{\ell,2}(s) &=& {\tfrac}{(-1)^r}{(2 \pi p)^s}\, p^{{\frac}a2} \, \Big(
\sum_{\begin{smallmatrix} k\le 1 \\ k \text{ odd}
\end{smallmatrix}} {\frac}{1}{{( {\tfrac}\ell p - {\tfrac}k 2)}^s} -
\sum_{\begin{smallmatrix}
k\ge 3 \\ k \text{ odd} \end{smallmatrix}}
{\frac}{1}{{({\tfrac}k 2- {\tfrac}\ell p)}^s} \Big).\end{aligned}$$ Now, the first sum in this expression equals $$\begin{aligned}
{\frac}{1}{{({\tfrac}\ell p - {\tfrac}12 )}^s} +
\sum_{\begin{smallmatrix} k\ge 1 \\ k \text{ odd} \end{smallmatrix}}
{\frac}{1}{{({\tfrac}k2 + {\tfrac}{\ell}p)}^s} & = & {\frac}{1}{{({\tfrac}{\ell}{p}-{\tfrac}12)}^s} +
\sum_{n = 0}^{\infty} {\frac}{1}{(n+ {\tfrac}12 + {\frac}{\ell}{p})^s}
\\ & = &
\sum_{n = 0}^{\infty} {\frac}{1}{(n + {\tfrac}12 +
{\frac}{\ell-p}{p})^s} =
\zeta(s,{\tfrac}12 - {\tfrac}{p-\ell}{p})\end{aligned}$$ since $0<{\frac}{p-\ell}{p} < {\frac}12$ for $q<\ell<p$. Similarly, the second sum equals $$\begin{aligned}
\sum_{n\ge 1} {\frac}{1}{{(n+{\tfrac}12 -{\tfrac}{\ell}{p})}^s} =
\sum_{n \ge 0} {\frac}{1}{{(n+ {\tfrac}12+ {\tfrac}{p-\ell}{p})}^s} =
\zeta (s,{\tfrac}12 +{\tfrac}{p-\ell}{p}),\end{aligned}$$ By substituting in we obtain the second expression for $\eta_{\ell,2}(s)$ in (i).
We now check (ii). Let $a$ be odd and $0\le \ell \le p-1$. By using and Proposition \[prop.Deltas\] (ii), and writing $c=pt+j$ with $t\ge 0$, $0 \le j \le p-1$, we get $$\begin{aligned}
\eta_{\ell,1}(s) & = & {\tfrac}{(-1)^{q+r}}{(2\pi)^s} ({\tfrac}2p) \, p^{{\frac}{a-1}2}
\sum_{c=1}^{\infty} {\frac}{({\frac}{\ell-c}{p})-({\frac}{\ell+c}{p})}{c^s}
\\ &=&
{\tfrac}{(-1)^{t+r}}{(2\pi p)^s} p^{{\frac}{a-1}2} \sum_{j=1}^{p-1} \big(
({\tfrac}{\ell-j}{p})-({\tfrac}{\ell+j}{p}) \big) \sum_{t=0}^\infty
{\tfrac}{1}{(t+ {\tfrac}jp)^s}\end{aligned}$$ where we have used that $(-1)^q ({\tfrac}2p) = (-1)^{t}$. This gives the expression of $\eta_{\ell,1}
(s)$. Similarly $$\begin{aligned}
\eta_{\ell,2}
(s) & = & {\tfrac}{(-1)^{q+r}}{2\pi^s} p^{{\frac}{a-1}2} \sum_{c=0}^{\infty}
{\frac}{({\frac}{2(\ell-c)-1}{p})-({\frac}{2(\ell+c)+1}{p})}{(c+{\tfrac}12)^s} \\
&=& {\tfrac}{(-1)^{q+r}}{(2\pi p)^s} p^{{\frac}{a-1}2} \sum_{j=0}^{p-1} \Big(
\big( {\tfrac}{2\ell-(2j+1)}{p} \big) - \big( {\tfrac}{2\ell+(2j+ 1)}{p} \big)
\Big) \sum_{t=0}^\infty {\tfrac}{1}{\big(t+{\tfrac}{2j+1}{2p}\big)^s}.\end{aligned}$$ Now, using that $\sum_{t=0}^\infty \big(t+{\frac}{2j+1}{2p}\big)^{-s} =
\zeta(s,{\tfrac}{2j+1}{2p})$ in the previous equations, we obtain the expression in the statement. The remaining assertion is clear and the theorem is thus proved.
In the particular case when $\ell=0$, $b+c=1$ (i.e. $\beta_1=1$), $a$ is odd and $p\equiv 3\,(4)$ (see Theorem 3.3 and Corollary 3.4), the untwisted eta series $\eta_{0,h} (s)$ were computed in [@MiPo09]. Some easy calculations show that the expressions given there coincide with the corresponding ones in Theorem \[thm.eta series\].
Twisted eta invariants {#sect-4}
======================
Twisted and relative eta invariants {#S3-TERLI}
-----------------------------------
Here we compute the twisted eta invariants $\eta_\ell$ and $\bar \eta_\ell$, for any $0\le \ell \le p-1$, the dimension of the kernel of $D_\ell$ and the twisted relative eta invariants, i.e. the differences $\bar \eta_\ell - \bar \eta_0$.
We will need the following notations. For $h=1,2$ we set $$\label{aux sums2}
S_h^{\pm}(\ell,p) := \sum_{j=1}^{p+\big[{\frac}{h\ell}p \big]\,p-h\ell-1} \big( {\tfrac}jp \big)
\quad \pm \quad \sum_{j=1}^{h\ell-\big [{\frac}{h\ell}p\big ]\,p-1} \big( {\tfrac}jp \big).$$
where $\big(\frac{\cdot}{p}\big)$ stands for the Legendre symbol modulo $p$. Note that $$\label{s0p=0}
S_1^\pm(0,p) = S_1^\pm(0,p) = 0$$ since $\sum_{j=1}^{p-1} \big( {\tfrac}jp \big) =0$.
\[thm.etainvs\] Let $p=2q+1$ be an odd prime and let $\ell \in {\mathbb N}$ be such that $0\le \ell \le p-1$. Let $M$ be an exceptional ${\mathbb Z}_p$-manifold of dimension $n=a(p-1)+1$. Put $r=[{\tfrac}n4]$ and $t=[{\tfrac}p4]$. The twisted eta invariants of $(M,{\varepsilon}_h)$ are given as follows.
\(i) If $a$ is even then $\eta_{0,h}(0) = 0$ and for $\ell \ne 0$ we have $$\eta_{\ell,1} = (-1)^{r} p^{{\frac}{a}{2}-1} (p -2\ell), \qquad
\eta_{\ell,2} = (-1)^{r} p^{{\frac}{a}{2}-1}2([{\tfrac}{2\ell}p]\,p -\ell).$$ (ii) If $a$ is odd then $$\begin{aligned}
\eta_{\ell,1} & = \left\{ \begin{array}{ll} (-1)^{t+r+1} p^{{\frac}{a-1}{2}}
\, S_1^-(\ell,p) & \qquad \quad p\equiv 1\,(4), {\medskip}\\ (-1)^{t+r} p^{{\frac}{a-1}{2}} \big( S_1^+(\ell,p) + {\tfrac}2p
\sum\limits_{j=1}^{p-1} \big({\tfrac}{j}{p}\big)j \big) & \qquad \quad p\equiv 3\,(4),
\end{array} \right.
{\medskip}\\
\eta_{\ell,2} & = \left\{
\begin{array}{ll}
(-1)^{q+r+1} p^{{\frac}{a-1}{2}} \big( S_2^-(\ell,p) - \big( {\tfrac}{2}{p} \big)
S_1^-(\ell,p) \big) & \quad \: p\equiv 1\,(4), {\medskip}\\
(-1)^{q+r} p^{{\frac}{a-1}{2}} \big\{ S_2^+(\ell,p) + \big( {\tfrac}{2}{p} \big)
\, S_1^+(\ell,p) \, + \\ \hfill + \, \big(1- ({\tfrac}2p)
\big) {\tfrac}2p \sum\limits_{j=1}^{p-1} \big( {\tfrac}{j}{p} \big) j \big\} & \quad
\: p\equiv 3\,(4).
\end{array} \right.\end{aligned}$$ In particular, if $a$ is odd, we have that $\eta_{0,1} = 0$ for $p\equiv 1\,(4)$ and that $\eta_{0,2} = 0$ for both $p\equiv 1\, (4)$ or $p\equiv 7 \,(8)$.
We need only evaluate the expressions in Theorem \[thm.eta series\] at $s=0$, using that $\zeta(0,\alpha) = {\tfrac}{1}{2} - \alpha$.
\(i) If $a$ is even, by Theorem \[thm.eta series\] (i) we have $$\eta_{\ell,1}(0) = (-1)^r p^{{\frac}a2} \big[({\tfrac}12 - {\tfrac}{\ell}{p}) - ({\tfrac}12
- {\tfrac}{p-\ell}{p})\big] = (-1)^r p^{{\frac}a2} (1-{\tfrac}{2\ell}{p})$$ Proceeding similarly, we have $$\eta_{\ell,2}(0) = \left\{ \begin{array}{ll}
(-1)^{r} \, p^{{\frac}{a}{2} -1}\, (-2\ell) & \quad 1\le \ell \le q, {\medskip}\\
(-1)^r p^{{\frac}{a}{2}-1}\, 2(p-\ell) & \quad q< \ell <p,
\end{array} \right.$$ from where the expression in the statement follows.
\(ii) Assume now that $a$ is odd. By Theorem \[thm.eta series\] (ii), we have $$\begin{aligned}
\eta_{\ell,1}(0) & = (-1)^{t+r} \, p^{{\frac}{a-1}{2}} \, \sum_{j=1}^{p-1} \Big( ({\tfrac}{\ell
-j}{p}) - ({\tfrac}{\ell + j}{p}) \Big) \,
({\tfrac}12 - {\tfrac}jp),
{\medskip}\\
\eta_{\ell,2}
(0) & = (-1)^{q+r} \, p^{{\frac}{a-1}{2}} \, \sum_{j=0}^{p-1} \left(
({\tfrac}{2\ell-(2j+1)}{p}) - ({\tfrac}{2\ell+(2j+1)}{p}) \right) \, ({\tfrac}{p-1}{2p} -
{\tfrac}{j}{p}).\end{aligned}$$ Now, by applying Lemma \[lem.legendres\], in the notations of , we have $$\begin{aligned}
\eta_{\ell,1}(0) & = (-1)^{t+r+1} \, p^{{\frac}{a-3}{2}} \, S_1(\ell,p), {\medskip}\\
\eta_{\ell,2}(0) & = (-1)^{q+r+1} \, p^{{\frac}{a-3}{2}} \, S_2(\ell,p).\end{aligned}$$ Finally, using Proposition \[prop.dif legendres\] we get the desired expressions.
The remaining assertions follow from and thus the theorem is now proved.
We will now show the integrality of the eta invariants $\eta_\ell$ (except for the 3-dimensional ${\mathbb Z}_3$-manifold $M_{3,1}$) and study their parity. To this end, we first recall the Dirichlet class number formula for a negative discriminant $D$ in the particular case $D=-p$, with $p \equiv 3 \,(4)$ a positive odd prime. It is given by $$\label{eq.dirichlet}
h_{-p} = -{\tfrac}{\omega_{-p}}{2p}\, \sum _{j=0}^{p-1} \big( {\tfrac}{j}{p} \big) j
\in {\mathbb Z},$$ where $h_{-p}$ is the class number of the imaginary quadratic field ${\mathbb Q}(\sqrt{-p})$ and $\omega_{-p}$ is the number of $p^{\mathrm{th}}$-roots of unity in that field. Hence, $\omega_{-p}=6$ if $p=3$ and $\omega_{-p}=2$ if $p\ge 5$.
\[parity\] Let $p$ be an odd prime and $\ell \in {\mathbb N}$ with $0\le \ell \le p-1$. Let $(M,{\varepsilon}_h)$ be an exceptional spin ${\mathbb Z}_p$-manifold, $h=1,2$.
\(i) If $(p,a) \ne (3,1)$ then $\eta_{\ell,h} \in {\mathbb Z}$. Furthermore, $\eta_{0,h}$ is even and, if $\ell \ne 0$, then $\eta_{\ell,1}$ is odd and $\eta_{\ell,2}$ is even.
\(ii) If $(p,a) = (3,1)$ then $$\eta_{\ell,1} = \left\{ \begin{array}{rl} -2/3 & \quad \ell=0, {\smallskip}\\ 1/3 & \quad \ell=1,2,
\end{array}\right. \quad \text{ and } \quad \eta_{\ell,2} = 4/3 \quad \ell=0,1,2.$$
\(i) Let $(p,a) \ne (3,1)$. If $a$ is even it is clear from the expressions in (i) of Theorem \[thm.etainvs\] that $\eta_{\ell,h} \in {\mathbb Z}$ and $\eta_{0,h} \in 2{\mathbb Z}$. For $\ell \ne 0$, we also have that $\eta_{\ell,1}$ is odd and $\eta_{\ell,2}$ is even.
We now let $a$ be odd. We will first show that the values at 0 are integers. It is clear that the sums $S_1^\pm(\ell,p), S_2^\pm(\ell,p) \in {\mathbb Z}$. In the case $p\equiv 3\,(4)$ there is another term to consider. By , it follows that $${\tfrac}{1 }{p}\, \sum _{j=0}^{p-1} \big( {\tfrac}{j}{p} \big) j = -{\tfrac}{2h_{-p} }{\omega_{-p}} = \left \{ \begin{array}{ll}-h_{-p}& \quad p\ge 5, {\smallskip}\\ -2/3 &\quad p=3, \end{array} \right.$$ since $h_{-3}=1$. In this way, ${\tfrac}{1}{p}\, \sum _{j=0}^{p-1} \big({\tfrac}{j}{p} \big) j \in {\mathbb Z}$ for $p\ge 5$, while for $p=3$ we have that $p^{{\frac}{a-1}{2}} \, {\tfrac}{1}{p}\, \sum _{j=0}^{p-1} \big( {\tfrac}{j}{p} \big) j = 3^{{\frac}{a-1}{2}} \, {\tfrac}{(-2)}{3} \in {\mathbb Z}$ for $a>1$. In any case, we see that $\eta_{\ell,h} \in {\mathbb Z}$ for $(p,a)\ne(3,1)$.
We now consider the parity of the sums $S_h^\pm(\ell,p)$, $h=1,2$. If $\ell=0$, all the sums are zero. If $\ell\ne 0$, $S_1^\pm(\ell,p) \equiv (p-\ell-1) + (\ell - 1) \equiv p \mod 2$, hence it is odd. Similarly we verify that $S_2^\pm(\ell,p)$ is odd, in this case.
Now, making use of these parity considerations and looking at the expressions in (i) of Theorem \[thm.etainvs\] for $a$ odd, we see that again $\eta_{0,h} \in 2{\mathbb Z}$ and $\eta_{\ell,1}$ is odd and $\eta_{\ell,2}$ is even, for $\ell\ne 0$.
\(ii) Suppose $(p,a)=(3,1)$. We need evaluate the expressions in Theorem \[thm.etainvs\] (ii) for $p\equiv 3\,(4)$. We have that $q=1$ and $r=t=0$. Using that $({\tfrac}13)=1$, $({\tfrac}23)=-1$ and $\sum_{j=1}^{2} ({\tfrac}j3)j = -1$ we have $$\label{sl12}
\eta_{\ell,1} = S_1^+(\ell,3) - {\tfrac}23, \qquad \eta_{\ell,2} = {\tfrac}43 + S_1^+(\ell,3) - S_2^+(\ell,3).
$$ Now, using we have $S_1^+(0,3) = S_2^+(0,3) = 0$, by , and it is easy to check that $S_1^+(\ell,3) = S_2^+(\ell,3) = 1$ for $\ell=1,2$. Substituting these values in the previous equations the result follows.
\[rem. hp\] By using the formula , the expressions for the eta invariants $\eta_{\ell,h}$ in Theorem can be put in terms of class numbers $h_{-p}$ when $a$ is odd and $p\equiv 3\,(4)$. In particular, by , for an exceptional manifold in the untwisted case $\ell = 0$, we get the expressions $$\eta_{0,1} = -4 \, p^{{\frac}{a-1}2} \, \tfrac{h_{-p}}{\omega_{-p}}, \qquad \eta_{0,2} = \big\{ \big( {\tfrac}{2}p \big) -1 \big\} \, \eta_{0,1},$$ where we have used that $r=[{\frac}n4]$ and $t=[{\frac}p4]$. Thus, for $p=3$ we have $\eta_{0,1} = -2 \cdot 3^{{\frac}{a-3}2}$ and $\eta_{0,2} = 4 \cdot 3^{{\frac}{a-3}2}$. For $p\ge 7$ we may conclude that $$\eta_{0,1} = -2 \, p^{{\frac}{a-1}2} \, h_{-p} \qquad \text{and} \qquad \eta_{0,2} = \left\{ \begin{array}{ll} 0 & \qquad p\equiv 7 \; (8), \\ 4 \, p^{{\frac}{a-1}2}\, h_{-p} & \qquad p\equiv 3 \; (8). \end{array} \right.$$ These expressions coincide with the ones obtained in [@MiPo09], Theorem 4.1.
It is known that the dimension of the kernel of the Dirac operator $D_{\ell}$ coincides with the number of independent harmonic spinors, which in turn equals the multiplicity of the eigenvalue 0. That is, $$\dim \ker D_{\ell} = d_{\ell,0}.$$ We now compute this invariant for an arbitrary ${\mathbb Z}_p$-manifold.
\[harmonics\] Let $p=2q+1$ be an odd prime. Let $M$ be a ${\mathbb Z}_p$-manifold with a spin structure ${\varepsilon}_h$. Then, $d_{\ell,0,h}=0$ for any nontrivial spin structure ${\varepsilon}_h$, $h\ne 1$, while for the trivial spin structure ${\varepsilon}_1$ we have $$\label{eq.dimker}
d_{\ell,0,1} = {\tfrac}{2^{{\frac}{b+c-1}{2}}}{p} \Big( 2^{(a+b)q} +
(-1)^{({\frac}{p^2-1}{8}){(a+b)}} \, \big(p \delta_{\ell,0} - 1\big) \Big)$$ where $0\le \ell \le p-1$ and $\delta_{\ell,0}$ is the Kronecker delta function.
In particular, if $b+c>1$ then $d_{\ell,0,1}$ is even for any $0\le \ell \le p-1$ while if $b+c=1$ then $d_{0,0,1}$ is even and $d_{\ell,0,1}$ is odd for $\ell \ne 0$.
By we have $d_{\ell,0,h}=0$ for $h \ne 1$ and $$\begin{aligned}
d_{\ell,0,1} = {\tfrac}1p \, \sum_{k=0}^{p-1} \, e^{{\frac}{2\pi i k\ell}{p}}
\; {\chi}_{_{L_{n}}}({\varepsilon}_1({\gamma}^k)).\end{aligned}$$ Using and the fact that $x(\theta)^k=x(k\theta)$, $\theta\in {\mathbb R}$, $k\in {\mathbb Z}$, we have that ${\varepsilon}_1({\gamma}^k) = (-1)^{k[{\frac}{q+1}{2}](a+b)} \; x_{a+b}\big({\tfrac}{k\pi}p,
{\tfrac}{2k\pi}p,\dots, {\tfrac}{qk\pi}p \big)$. Now, applying , we get $$d_{\ell,0,1} = {\tfrac}{2^m}{p} \, \sum_{k=0}^{p-1} \, (-1)^{k[\frac{q+1}{2}](a+b)} \, \Big(\prod_{j=1}^q \cos
\big({\tfrac}{jk\pi}p\big)\Big)^{a+b} \, e^{{\frac}{2\pi i k\ell}{p}} \,.$$
By (ii) in Lemma \[lem.prodtrigs\], for $k>0$ we have $$\Big(\prod_{j=1}^q \cos \big({\tfrac}{jk\pi}p\big)\Big)^{a+b} =
{\frac}{(-1)^{(k-1)({\frac}{p^2-1}{8})(a+b)}}{2^{(a+b)q}} \,.$$
Thus, we get $$d_{\ell,0,1} = {\tfrac}{2^m}{p} \, \Big( 1 + {\tfrac}{(-1)^{(\frac{p^2-1}{8})(a+b)}}{2^{q(a+b)}} \, \sum_{k=1}^{p-1}
e^{{\frac}{2\pi i k\ell}{p}} \Big) \,.$$ Expression now follows from the fact that $\sum_{k=1}^{p-1} e^{{\frac}{2\pi ik\ell}{p}}$ equals $p-1$ for $\ell=0$ and $-1$ for $1\le \ell \le p-1$. Since $2m+1 = n = a(p-1) + bp + c$ and $p=2q+1$ we have that $b+c$ is odd and $m = (a+b)q +
({\tfrac}{b+c-1}{2})$.
The remaining assertions are now clear from and the proposition readily follows.
\[remark\] By , for a ${\mathbb Z}_p$-manifold we have $$\label{eta res + har}
\bar \eta_{\ell,h} = {\tfrac}12 (\eta_{\ell,h} + d_{\ell,0,h}) \,. $$ Using Theorem \[thm.etainvs\] and Proposition \[harmonics\] one could easily write down explicit expressions for the twisted eta invariants $\bar \eta_{\ell,h}$ and the relative eta invariants $\bar \eta_{\ell,h} - \bar \eta_{0}$ for $1\le \ell \le p-1$. These formulas are too complicated to write them down, because of the sums $S_h^\pm(\ell,p)$ appearing in the expression for $\eta_{\ell,h}$. However, in the untwisted case they get explicit and closed expressions (see Corollary \[eta untwisted\]). On the other hand, we are mainly interested in their values modulo ${\mathbb Z}$ (see Theorem ).
\[eta untwisted\] Let $p$ be an odd prime. In the untwisted case, i.e. $\ell=0$, the eta invariants of an arbitrary ${\mathbb Z}_p$-manifold $(M,{\varepsilon}_h)$ have the following expressions.
\(i) If $M$ is non-exceptional, i.e. $(b,c) \ne (0,1)$, then $$\bar \eta_{0,1} = {\tfrac}1p \, 2^{{\frac}{b+c-3}{2}} \big( 2^{(a+b)({\frac}{p-1}{2})} + (-1)^{({\frac}{p^2-1}{8}){(a+b)}} \, (p - 1) \big)$$ and $\bar \eta_{0,h} = 0$ for $h\ne 1$.
\(ii) If $M$ is exceptional, i.e. $(b,c) = (0,1)$, then $$\bar \eta_{0,1} = \left\{ \begin{array}{ll}
{\tfrac}{1}{2p} \big( 2^{{\frac}{n-1}{2}} + (-1)^{({\frac}{p+1}{8})(n-1)} (p-1) \big) -2 \, p^{{\frac}{a-1}{2}}
\, {\frac}{h_{-p}}{\omega_{-p}} & a \text{ odd, } p \equiv 3\,(4), \\ {\smallskip}{\tfrac}{1}{2p} \big( 2^{{\frac}{n-1}{2}} + (-1)^{({\frac}{p+1}{8})(n-1)} (p-1) \big)
& \quad \text{otherwise,} \end{array} \right.$$ and $$\bar \eta_{0,2} = \left\{ \begin{array}{ll}
\big( 1 - ({\tfrac}2p) \big) \, 2 \ p^{{\frac}{a-1}{2}} \, {\frac}{h_{-p}}{\omega_{-p}} & \qquad \qquad a \text{ odd, } p \equiv 3\,(4), \\ {\smallskip}0 & \qquad \qquad \text{otherwise.} \end{array} \right.$$
The result follows directly by substituting the expressions obtained in Theorem \[thm.etainvs\], Proposition \[harmonics\] and Remark \[rem. hp\] in , and considering the different cases involved.
We are now in a position to prove Theorem \[twisted etas mod Z\], one of the main results in the paper.
Proof of Theorem 1.1 {#proof-of-theorem-1.1 .unnumbered}
--------------------
We need study the integrality (or not) of $\bar \eta_{\ell,h}$ in , by looking at the parity of the numbers $\eta_{\ell,h}$ and $d_{\ell,0,h}$.
In the non-exceptional case, i.e. $(b,c)\ne (0,1)$, by using Corollary \[parity\] and Proposition \[harmonics\] we have that $\eta_{\ell,h}=0$, and hence $\bar \eta_{\ell,h} = {\tfrac}12 d_{\ell,0,h} \in {\mathbb Z}$ for any $0 \le \ell \le p-1$. In particular, $\bar \eta_{\ell,h}=0$ for $\ell\ne 0$.
In the exceptional case, i.e. $(b,c)=(0,1)$, we have the following results. If $(p,a)\ne (3,1)$ then $$\begin{aligned}
& \bar \eta_{0,1} = {\tfrac}12 (\mathrm{even} + \mathrm{even}) \in {\mathbb Z}, & \bar \eta_{0,2} = {\tfrac}12 (\mathrm{even} + 0) \in {\mathbb Z}, \\
& \bar \eta_{\ell,1} = {\tfrac}12 (\mathrm{odd} + \mathrm{odd}) \in {\mathbb Z}, & \bar \eta_{\ell,2} = {\tfrac}12 (\mathrm{even }+ 0) \in {\mathbb Z},\end{aligned}$$ for $\ell \ne 0$. Thus, we have $\bar \eta_{\ell,h} \equiv 0$ mod ${\mathbb Z}$ in this case.
If $(p,a)=(3,1)$ then $$\begin{aligned}
& \bar \eta_{0,1} = {\tfrac}12 ({\tfrac}23 +0) = -{\tfrac}13, & \bar \eta_{0,2} = {\tfrac}12 ({\tfrac}43 + 0) = {\tfrac}23, \\
& \bar \eta_{\ell,1} = {\tfrac}12 (-{\tfrac}13 + 1) = {\tfrac}23, & \bar \eta_{\ell,2} = {\tfrac}12 ({\tfrac}43 + 0) = {\tfrac}23,\end{aligned}$$ and now we have $\bar \eta_{\ell,h}\equiv \frac 23$ mod ${\mathbb Z}$.
The remaining assertion is now clear and the result follows. $\square$
Equivariant spin bordism {#sect-5}
========================
This section is devoted to the proof of Theorem \[thm-1.2\]. We first review the basic definitions and notions. Let $p$ be an odd prime. Let $M$ be a compact oriented smooth manifold of dimension $n$ without boundary. An *equivariant* [*$\mathbb{Z}_p$-structure*]{} $\sigma$ on $M$ is a principal $\mathbb{Z}_p$-bundle $$\mathbb{Z}_p \rightarrow P \rightarrow M.$$ This structure can also be regarded as being either a representation of the fundamental group of each connected component of $M$ to $\mathbb{Z}_p$, or as being the homotopy class of a smooth map from $M$ to the classifying space $B\mathbb{Z}_p$. These are equivalent formulations and this explains the utility of the concept. The [*trivial $\mathbb{Z}_p$-structure*]{} $\sigma_0$ is defined by taking the product principal bundle $P=M\times\mathbb{Z}_p$ or, equivalently, by taking the trivial representation of the fundamental group, or else, equivalently, by taking the constant map from $M$ to $B\mathbb{Z}_p$.
Let $(M_i,\pbgs_i)$ be compact oriented spin manifolds of dimension $n$. Let $M_1-M_2$ be the disjoint union of $M_1$ and $M_2$ where we give $M_2$ the opposite orientation. One says that $M_1$ and $M_2$ are [*Spin-bordant*]{} if there exists a compact spin manifold $N$ with boundary, so that the boundary of $N$ is $M_1-M_2$ and so that the spin structure on $N$ restricts to induce the given spin structures on the manifolds $M_i$. Spin-bordism induces an equivalence relation; let $[(M,\pbgs)]$ denote the associated equivalence class and let $\MSpin_n$ be the collection of equivalence classes. Disjoint union and Cartesian product gives $\MSpin_*$ the structure of a graded unital ring. We refer to [@ABP66; @ABP67; @ABP69; @Th54] for further details concerning these and related structures.
Additionally suppose $\sigma_i$ are equivariant $\mathbb{Z}_p$-structures on the manifolds $M_i$. One says that $(M_1,\sigma_1,\pbgs_1)$ is [*$\mathbb{Z}_p$-equivariant Spin-bordant*]{} to $(M_2,\sigma_2, \pbgs_2)$ if in addition the bounding manifold $N$ admits an equivariant $\mathbb{Z}_p$-structure which restricts to given structures on the manifolds $M_i$. Again, this is an equivalence relation and we let $\MSpin_n(B\mathbb{Z}_p)$ denote the associated equivariant spin bordism groups.
We wish to focus on the $\mathbb{Z}_p$-structure. Forgetting the $\mathbb{Z}_p$-structure defines the [*forgetful map*]{} from $\MSpin_n(B\mathbb{Z}_p)$ to $\MSpin_n$ which splits by the inclusion which associates to every spin manifold the trivial $\mathbb{Z}_p$-structure $\sigma_0$. The *reduced equivariant bordism* groups $\RMSpin_n(B\mathbb{Z}_p)$ are the kernel of the forgetful map, that is $[(M,\sigma,\pbgs)]$ belongs to $\RMSpin_n(B\mathbb{Z}_p)$ if and only if $[(M,\pbgs)]=0$ in $\MSpin_n$. These groups play much the same role in studying equivariant bordism as the reduced homology groups play in the study of homology – one has a natural isomorphism $$\MSpin_n(B\mathbb{Z}_p) = \RMSpin_n(B\mathbb{Z}_p)\oplus \MSpin_n.$$ Cartesian product makes $\RMSpin_*(B\mathbb{Z}_p)$ into an $\MSpin_*$-module. We refer to Bahri *et al*. [@BBDG; @BBG; @BG87; @BG87a] for details concerning the additive structure of these and other related groups. The natural projection $\pi$ from $\MSpin_n(B\mathbb{Z}_p)$ to $\RMSpin_n(B\mathbb{Z}_p)$ is the object of study in Theorem \[thm-1.2\] and is defined by $$\pi(M,\varepsilon,\sigma)=[(M,\varepsilon,\sigma)]-[(M,\varepsilon,\sigma_0)]\in
\RMSpin_n(B\mathbb{Z}_p)\,.$$ The following result follows from Lemma 3.4.2, Lemma 3.4.3, and Theorem 3.44 of [@Gi-88]:
\[thm-?\] Let $p$ be an odd prime.
\(i) If $n$ is even, then $\RMSpin_n(B\mathbb{Z}_p)=0$.
\(ii) If $n$ is odd, then $\RMSpin_n(B\mathbb{Z}_p)$ is a finite group and all the torsion in $\RMSpin_n(B\mathbb{Z}_p)$ is $p$-torsion. Furthermore, $\RMSpin_n(B\mathbb{Z}_p)$ is generated as a $\MSpin_*$-module by the diagonal lens spaces $\mathbb{S}^{2k-1}/\mathbb{Z}_p$ for $2k-1\le n$.
The characteristic numbers of $\MSpin_*$ are the Pontrjagin numbers, the Stiefel-Whitney numbers, and connective $K$-theory numbers. By contrast, the characteristic numbers of $\RMSpin_*$ are given by the twisted eta invariant defined previously and these lie in $\mathbb{Q}/\mathbb{Z}$ and are torsion invariants. Let $D$ be the Dirac operator and let $\tau$ be a representation of the spin group. We let $\eta_\ell^\tau$ be the eta invariant of the Dirac operator with coefficients in the bundle defined by the representation $\tau$ and twisted by the character $\ell$. The following result follows from Lemma 3.4.2, Lemma 3.4.3, and Theorem 3.44 of [@Gi-88] – see also the discussion in Lemma 4.7.3 and Lemma 4.7.4 of [@Gi-95]. It motivated our investigation of the eta invariant for flat $\mathbb{Z}_p$-manifolds in the first instance:
\[thm-2.3\] Let $p$ be an odd prime. Let $M$ be an oriented manifold of dimension $n$. Let $\varepsilon$ be a spin structure on $M$ and let $\sigma$ be an equivariant $\mathbb{Z}_p$-structure on $M$. Let $\mathcal{M}:=(M,\varepsilon,\sigma)$.
\(i) Let $1\le \ell\le p-1$ and let $\tau$ be a representation of the spin group. Then:
\(a) $(\bar\eta_\ell^\tau - \bar\eta_0^\tau)(\mathcal{M})$ takes values in $\mathbb{Z}[\frac1p]/\mathbb{Z}$.
\(b) If $\pi(\mathcal{M})=0$ in $\RMSpin_n(B\mathbb{Z}_p)$, then $(\bar \eta_\ell^\tau - \bar \eta_0^\tau)(\mathcal{M})=0$ in $\mathbb{R}/\mathbb{Z}$.
\(ii) If the twisted relative eta invariants $(\bar \eta_\ell^\tau - \bar \eta_0^\tau)(\mathcal{M})$ vanish for all $\tau$ and $\ell$, then $\pi(\mathcal{M})$ vanishes in $\RMSpin_n(B\mathbb{Z}_p)$.
We can now prove one of the two main results in the paper.
Proof of Theorem \[thm-1.2\] {#proof-of-theorem-thm-1.2 .unnumbered}
----------------------------
Let $(M,{\varepsilon})$ be a spin ${\mathbb Z}_p$-manifold. The canonical equivariant ${\mathbb Z}_p$-structure $\sigma_p$ is defined by the cover $${\mathbb Z}_p \rightarrow T_\Lambda \rightarrow M,$$ where $T_\Lambda$ is the associated torus. The trivial equivariant ${\mathbb Z}_p$-structure $\sigma_0$ is defined by the cover ${\mathbb Z}_p \rightarrow M \times {\mathbb Z}_p \rightarrow M$. The associated principal Spin bundle is flat and defined by an equivariant $\mathbb{Z}_{2p}$-structure on $M$ which may or may not reduce to a $\mathbb{Z}_p$-structure. Let $\tau$ be a representation of $\mathrm{Spin}(n)$ and let $0\le\ell\le p-1$. Since the spin structure is flat and arises from a representation of $\mathbb{Z}_{2p}$, the bundle defined by the representation $\tau$ and twisted by the character $\ell$ is flat and is defined by some representation $\nu$ of $\mathbb{Z}_{2p}$. We may decompose $$\mathbb{Z}_{2p} = \mathbb{Z}_2 \oplus \mathbb{Z}_p.$$ Let $\vartheta$ be the non-trivial character of $\mathbb{Z}_2$. We may then decompose the representation $\nu = \nu_1 \oplus \nu_2 \vartheta$ where $\nu_i \in \operatorname{Rep}(\mathbb{Z}_p)$. Expand the representations $\nu_1$ and $\nu_2$ in terms of the characters $\rho_i$ in the form: $$\nu_1=\sum_{0\le i\le p-1}n_i\rho_i\quad\text{and}\quad
\nu_2=\sum_{0\le i\le p-1}\tilde n_i\rho_i\,.$$ Here $n_i=n_i(\tau,\ell)$ and $\bar n_i=\bar n_i(\tau,\ell)$ are non-negative integers. Let $\tilde{{\varepsilon}}$ be the associated spin structure on $M$ arising from the $\mathbb{Z}_2$ twisting $\vartheta$. Let $\mathcal{M} = (M, {\varepsilon}, \sigma)$ and $\tilde{\mathcal{M}} = (M, \tilde {\varepsilon}, \sigma)$. The above discussion then yields: $$\begin{aligned}
&&(\bar \eta_\ell^\tau - \bar \eta_0^\tau)(\mathcal{M}) = \sum_{i=1}^{p-1} \big\{ n_i(\bar \eta_i - \bar \eta_0)(\mathcal{M}) + \tilde n_i(\bar \eta_i - \bar \eta_0)(\tilde{\mathcal{M}}) \big\}.\end{aligned}$$ Theorem \[twisted etas mod Z\] shows that $(\bar \eta_\ell^\tau - \bar \eta_0^\tau)(\mathcal{M})$ vanishes in $\mathbb{Q}/\mathbb{Z}$. Theorem \[thm-1.2\] now follows by Theorem \[thm-2.3\] (ii) since $\tau$ and $\ell$ were arbitrary.
\[peters-remark\] Here we show how the reduced equivariant spin bordism groups come up in some questions in geometry. Let $M$ be a connected spin manifold with finite fundamental group $\pi$ which admits a metric of positive scalar curvature. The formula of Lichnerowicz [@L63] shows that the kernel of the Dirac operator is necessarily trivial. From this it follows that the index of the spin operator vanishes and hence the generalized $\hat A$-genus vanishes – the generalized $\hat A$-genus is a topological invariant which can be computed purely combinatorially; it takes values either in $\mathbb{Z}$ or in $\mathbb{Z}_2$ depending upon the underlying dimension of the manifold. Stolz [@S92] used the absolute spin bordism groups $\MSpin_*$ to show that the generalized $\hat A$-genus was the only obstruction to $M$ admitting a metric of positive scalar curvature if the fundamental group $\pi$ was trivial. If $\pi=\mathbb{Z}_p$ or, more generally, if $\pi$ is a spherical space form group, then one can define an equivariant $\hat A$-genus and establish similar topological necessary and sufficient conditions for $M$ to admit a metric of positive scalar curvature [@BGS97]. We refer to [@GLP99] for further details about this area; the reduced equivariant spin bordism groups $\RMSpin_n(B\mathbb{Z}_p)$ and the associated eta invariants play a central role in the discussion.
Appendix: additional computations {#appendix}
=================================
Here we gather all the extra computations that were needed to obtain the results in Section \[sect-3\]. We compute some trigonometric products of special values used to determine the asymmetric contribution of the eigenvalues to the eta series, some twisted Gauss sums appearing in the eta series and several sums involving Legendre symbols appearing in the computation of the eta-invariants.
We recall here that the Legendre symbol $\big({\frac}{\cdot}{p} \big)$ is $p$-periodic and satisfies $$\label{legendre2p}
({\tfrac}2p) = (-1)^{{\frac}{p^2-1}{8}}, \qquad ({\tfrac}{-1}{p}) = (-1)^{{\frac}{p-1}{2}}.$$
Some trigonometric products {#sect-a-STP}
---------------------------
\[lem.prodtrigs\] Let $p=2q+1$ be an odd prime and let $k
\in {\mathbb N}$. Then we have $$\begin{aligned}
\label{prod sines}
(i) \qquad \prod_{j=1}^q \sin({\tfrac}{jk\pi}p) & = \left\{
\begin{array}{ll}
(-1)^{(k-1)({\frac}{p^2-1}{8})}\left( {\frac}kp\right ) \, 2^{-q} \, \sqrt p & \quad
\text{ if }\; (k,p)=1, {\medskip}\\
0 & \quad \text{ if } \; (k,p)>1,
\end{array} \right.
\\ \\
(ii) \qquad \prod_{j=1}^q \cos({\tfrac}{jk\pi}p) & = \left\{
\begin{array}{ll} (-1)^{(k-1)({\frac}{p^2-1}{8})} 2^{-q} & \qquad
\qquad \:\:\, \text{ if }\; (k,p)=1, {\medskip}\\ (-1)^{{\frac}kp [{\frac}{q+1}2]} & \qquad \qquad \:\:\; \text{ if } \; (k,p)>1.
\end{array} \right.\end{aligned}$$
Formula (i) in Lemma \[lem.prodtrigs\] is proved in [@MiPo09], Lemma 3.2. We check the second expression in the case $(k,p)=1$. By (i) in the Lemma, we have $$\begin{aligned}
\prod_{j=1}^q \cos({\tfrac}{jk \pi}{p}) & = &
{\frac}{\prod\limits_{j=1}^q \sin({\frac}{2jk \pi}{p})}{2^q \prod\limits_{j=1}^q
\sin({\frac}{j k \pi}{p})} =
{\frac}{(-1)^{(2k-1)({\frac}{p^2-1}{8})} \left( {\frac}{2k}{p} \right
)}{(-1)^{(k-1)({\frac}{p^2-1}{8})}\left({\frac}kp \right)2^q}.\end{aligned}$$ Canceling terms and using the assertion in the proposition follows.
Twisted character Gauss sums {#Sect-A-TCGS}
----------------------------
Here we will compute the values of certain twisted character Gauss sums.
We recall the character Gauss sum associated to the quadratic Dirichlet character given by the Legendre symbol $({\frac}{\cdot}{p})$ modulo $p$ for $l\in {\mathbb N}$, $$\label{eq:Gausssum}
G(l,p) = \sum_{k=0}^{p-1} \left({\tfrac}kp \right) e^{{\tfrac}{2\pi i l k}{p}}$$ and the special values $$\label{eq.G(1,p)}
G(1,p) = \delta(p) \sqrt{p}, \qquad G(l,p) = G(1,p)\left({\tfrac}lp \right)$$ where we have put $$\label{deltap} \delta(p) := \left\{ \begin{array}{ll} 1 &
\qquad p\equiv 1 \;(4), {\smallskip}\\ i &
\qquad p\equiv 3 \;(4).
\end{array} \right.$$
\[defi.sums\] Let $p=2q+1$ be an odd prime, $l,c\in {\mathbb N}$ and $h=1,2$. For $\chi$ a character mod $p$ define the sums $$\begin{aligned}
\label{eq.Gh} & G_h^\chi(l) := \sum_{k=1}^{p-1} (-1)^{k(h+1)} \, \chi(k)
\:
e^{{\tfrac}{\pi i k \,(2l + \delta_{_{h,2}})}{p}}, {\smallskip}\\
\label{eq.Fh} & F_{h}^\chi(l,c) := \sum_{k=1}^{p-1} (-1)^{k(h+1)} \, \chi(k)
\; e^{{\tfrac}{2 \pi i l k}{p}} \, \sin \big({\tfrac}{ \pi k\,(2c +
\delta_{_{h,2}})}{p}\big).\end{aligned}$$
We are interested in these sums only for $\chi = \chi_0$, the trivial character $\mod p$, and for $\chi =
({\frac}{\cdot}{p})$, the quadratic character mod $p$ given by the Legendre symbol. We will denote these characters by $\chi_{_0}$ and $\chi_p$, respectively.
Thus, for example, $G_1^{\chi_p}(l) = G(l,p)$ is the standard character Gauss sum in and $G_2^{\chi_p}(l,p)$ corresponds to the shifted alternating Gauss sum $$\label{H(l,p)}
G_2^{\chi_p}(l) = \sum_{k=0}^{p-1} (-1)^k \left({\tfrac}kp \right)
e^{{\tfrac}{(2l+1)\pi ik}{p}} = {\tfrac}{p}{G(1,p)} \left({\tfrac}{q-l}p \right).$$ (See [@MiPo09], Theorem 5.1. *Note:* there $G_2^{\chi_p}(l,p)$ was denoted by $\tilde{H}(l,p)$. We note that a factor $p$ is missing in that expression, although not in their proof.)
We will also make use of the identity $$\label{leg q-l}
\big( {\tfrac}{l-q}{p}\big) = \big( {\tfrac}{2}{p}\big)\big( {\tfrac}{2l-2q}{p}\big) =
\big( {\tfrac}{2}{p}\big)\big( {\tfrac}{2l+1}{p}\big).$$
### **Computation of the sums $G_h^\chi(l)$**
We now find the values of $G_h^\chi(l)$ for $\chi = \chi_{_0}, \chi_p$. These sums are modifications of sums of $p^{\mathrm{th}}$-roots of unity and of Gauss sums.
\[prop.Rh\] Let $p$ be an odd prime and $l\in {\mathbb N}$. Then, $$\begin{aligned}
G_1^{\chi_{_0}}(l) & = \left\{ \begin{array}{ll} p-1 & \quad p \,|\, l,
{\medskip}\\ -1 & \quad p \nmid l, \end{array}\right. \quad
\text{and} \quad G_2^{\chi_{_0}}(l) = \left\{ \begin{array}{ll} p-1 & \quad
p \,|\, 2l + 1, {\medskip}\\ -1 & \quad p \nmid 2l + 1.
\end{array} \right.\end{aligned}$$ In particular, $G_h^{\chi_{_0}}(l) \in {\mathbb Z}$, $h=1,2$.
Since $G_h^{\chi_{_0}}(l)$ is $p$-periodic we may assume that $0\le l \le
p-1$.
By we have $G_1^{\chi_{_0}}(l) = \sum_{k=1}^{p-1} e^{{\frac}{2l \pi
i k}{p}}$. Clearly, $G_1^{\chi_{_0}}(0) = p-1$ and if $1\le l \le p-1$, then $1 + G_1^{\chi_{0}}(l) = \sum_{k=0}^{p-1} e^{{\frac}{2l
\pi i k}{p}} = 0$, and hence $G_1^{\chi_{_0}}(l) = -1$.
Now, $G_2^{\chi_{_0}}(l) = \sum_{k=1}^{p-1} (-1)^{k}e^{{\frac}{(2l + 1) \pi i
k}{p}}$, by . If $p\,|\,2l+1$ then $2l+1 = p\alpha$ with $\alpha$ odd. Thus, $G_2^{\chi_{_0}}(l) =
\sum_{k=1}^{p-1} (-1)^k(-1)^k = p-1$. If $p\nmid
2l+1$ then, denoting by $\omega_l = e^{{\frac}{(2l+1)\pi i k}{p}}$ and using geometric summation, we have $$1 + G_2^{\chi_{_0}}(l) = \sum_{k=0}^{p-1} (-1)^k \, \omega_l^k =
\frac{\omega_l^p+1}{\omega_l+1} = 0,$$ since $\omega_l^p = -1$ and $\omega_l\ne 1$. Thus, $G_2^{\chi_{_0}}(l) = -1$ in this case.
\[prop.Gh\] Let $p$ be an odd prime and $l\in {\mathbb N}$. Then, $$G_1^{\chi_p}(l) = \delta(p) \left({\tfrac}l p \right) \sqrt{p} \qquad \text{and}
\qquad G_2^{\chi_p}(l) = \delta(p) \left({\tfrac}2p \right)
\left({\tfrac}{2l+1}{p} \right) \sqrt{p},$$ where $\delta(p)$ is as defined in . In particular, $G_1^{\chi_p}(l)=0$ if $p\,|\,l$ and $G_2^{\chi_p}(l)=0$ if $p\,|\,2l+1$.
If $h=1$, we have $G_1^{\chi_p}(l) = G(l,p) = \delta(p) \big({\tfrac}l p \big)
\sqrt{p}$ by and . If $h=2$, by and , we have $$G_2^{\chi_p}(l) = {\tfrac}{p}{G(1,p)} \left({\tfrac}{q-l}p \right) =
{\tfrac}{1}{\delta(p)} \big({\tfrac}{-2}{p} \big) ({\tfrac}{2l+1}{p}) \sqrt p = \delta(p)
\big({\tfrac}{2}{p} \big) ({\tfrac}{2l+1}{p}) \sqrt
p$$ since ${\frac}{1}{\delta(p)} \big({\tfrac}{-1}{p} \big) = \delta(p)$, and the result follows.
### **Computation of the sums $F_h^\chi(l,c)$**
We now find the values of $F_h^{\chi}(l,c)$ for $\chi = \chi_{_0}, \chi_p$.
\[prop.Qh\] Let $p$ be an odd prime, $l\in {\mathbb N}_0$ and $c \in {\mathbb N}$. If $p\,|\,l$ then $F_h^{\chi_{_0}}(l,c)=0$. If $p \nmid l $, then $$F_1^{\chi_{_0}}(l,c) = \left\{ \begin{array}{cl} \pm {\frac}{ip}{2}, & \:
\text{if } p\,|\,l \mp c, {\medskip}\\
0 & \: \text{otherwise}, \end{array} \right. \quad
F_2^{\chi_{_0}}(l,c) = \left\{ \begin{array}{cl} \pm {\frac}{ip}{2} & \: \text{if
} p\,|\,2(l \mp c) \mp 1, {\medskip}\\
0 & \: \text{otherwise}.
\end{array}\right.$$
By , we have $$\begin{aligned}
&&F_1^{\chi_{_0}}(l,c) = \sum_{k=1}^{p-1} e^{{\tfrac}{2 \pi i k l}{p}} \, \sin
\big({\tfrac}{2c \pi k}{p}\big), \\
&&F_2^{\chi_{_0}}(l,c) = \sum_{k=1}^{p-1} (-1)^k \, e^{{\tfrac}{2 \pi i k
l}{p}} \, \sin \big({\tfrac}{(2c+1) \pi k}{p}\big)\,.\end{aligned}$$
If $p\,|\,l$, then $e^{{\frac}{2 \pi i k l}{p}}=1$ and hence we have $F_1^{\chi_{_0}}(l,c) = \mathrm{Im} \, G_1^{\chi_{_0}}(c) = 0$ and $F_2^{\chi_{_0}}(l,c) = \mathrm{Im} \, G_2^{\chi_{_0}}(c) = 0$.
Now, if $p\nmid l$, then using trigonometric identities \[prop.Rh\] we have that the real and imaginary parts of $F_1^{\chi_{_0}}(l,c)$ are respectively given by $$\begin{aligned}
\sum_{k=1}^{p-1} \cos \big( {\tfrac}{2 \pi k l}{p} \big) \, \sin \big({\tfrac}{2c \pi
k}{p} \big) &=& {\tfrac}12 \sum_{k=1}^{p-1} \sin \big({\tfrac}{2\pi
k (l + c)}{p}\big) - \sin \big({\tfrac}{2 \pi k(l-c)}{p}\big) \\ &=& {\tfrac}12 \big\{
\mathrm{Im} \, G_1^{\chi_{_0}}(l+c) - \mathrm{Im} \,
G_1^{\chi_{_0}}(l-c) \big\},\end{aligned}$$ $$\begin{aligned}
\sum_{k=1}^{p-1} \sin \big( {\tfrac}{2 \pi k l}{p} \big) \, \sin \big( {\tfrac}{2c
\pi k}{p} \big) & = & {\tfrac}12 \sum_{k=1}^{p-1} \cos
\big({\tfrac}{2\pi k (l-c)}{p}\big) - \cos \big({\tfrac}{2 \pi k (l+c)}{p}\big) \\
&=& {\tfrac}12 \big\{ \mathrm{Re} \, G_1^{\chi_{_0}}(l-c) - \mathrm{Re} \,
G_1^{\chi_{_0}}(l+c) \big\}.\end{aligned}$$ Thus, by Proposition \[prop.Rh\] we get $$\mathrm{Re} \, F_1^{\chi_{_0}}(l,c) =0, \qquad \mathrm{Im} \,
F_1^{\chi_{_0}}(l,c) = \left\{ \begin{array}{cc} \pm {\frac}{p}2 & \quad
p\,|\,l \mp c, {\smallskip}\\ 0 & \quad \text{otherwise},
\end{array}\right.$$ Similarly, we have $$\begin{aligned}
\mathrm{Re} \, F_2^{\chi_{_0}}(l,c) & = & \sum_{k=1}^{p-1} (-1)^k \, \cos
\big( {\tfrac}{2 \pi k l}{p} \big) \, \sin
\big( {\tfrac}{(2c+1) \pi k}{p} \big)
\\ &=& {\tfrac}12 \sum_{k=1}^{p-1} (-1)^k \, \big\{ \sin \big({\tfrac}{(2(l+c)+1)\pi
k}{p}\big) - \sin \big({\tfrac}{(2(l-c)-1) \pi k)}{p}\big) \big\} \\
&=& {\tfrac}12 \big\{ \mathrm{Im} \, G_2^{\chi_{_0}}(l+c) - \mathrm{Im} \,
G_2^{\chi_{_0}}(l-c-1) \big\},\end{aligned}$$ $$\begin{aligned}
\mathrm{Im} \, F_2^{\chi_{_0}}(l,c) & = & \sum_{k=1}^{p-1} (-1)^k \, \sin
\big( {\tfrac}{2 \pi k l}{p} \big) \,
\sin \big( {\tfrac}{(2c+1) \pi k}{p} \big) \\
&=& {\tfrac}12 \sum_{k=1}^{p-1} (-1)^k \big\{ \cos \big({\tfrac}{(2(l-c)-1)\pi k
}{p}\big) - \cos \big({\tfrac}{(2(l+c)+1) \pi k}{p}\big) \big\} \\
&=& {\tfrac}12 \big\{ \mathrm{Re} \, G_2^{\chi_{_0}}(l-c-1) - \mathrm{Re} \,
G_2^{\chi_{_0}}(l+c) \big\}\end{aligned}$$ and hence by Proposition \[prop.Rh\] $$\mathrm{Re} \, F_2^{\chi_{_0}}(l,c) = 0, \qquad \mathrm{Im} \,
F_2^{\chi_{_0}}(l,c)
= \left\{ \begin{array}{cc} \pm {\frac}{p}2 & \: p\,|\,2(l \mp c) \mp 1, {\smallskip}\\ 0 & \: \text{otherwise}.
\end{array}\right.$$
We thus get the expressions in the statement.
\[prop.Fh\] Let $p$ be an odd prime and $l,c \in {\mathbb N}$. Thus, we have $$F_h^{\chi_p}(l,c) = \left\{ \begin{array}{ll}
\, i \, \delta(p) \Big( \big( {\tfrac}{l-c}{p} \big) - \big( {\tfrac}{l+c}{p} \big)
\Big) \,{\frac}{\sqrt p}{2} & \qquad h=1, {\medskip}\\
i\, \delta(p) \, \Big({\frac}{2}{p}\Big) \Big( \big( {\tfrac}{2(l-c)-1}{p} \big) -
\big( {\tfrac}{2(l+c)+1}{p} \big)
\Big)\,{\frac}{\sqrt p}{2} & \qquad h=2,
\end{array} \right.$$ where $\delta(p)$ is defined in .
In particular, if $p\,|\,l$ then $$\begin{aligned}
F_1^{\chi_p}(l,c) & = \left\{ \begin{array}{ll} 0 & \qquad \qquad p \equiv
1 \,(4), {\smallskip}\\
\big( {\frac}{ c}{p} \, \big) \sqrt p & \qquad \qquad p \equiv 3\,(4),
\end{array} \right. {\medskip}\\
F_2^{\chi_p}(l,c) & = \left\{ \begin{array}{ll} 0 & \quad \, p \equiv
1 \,(4), {\smallskip}\\
\big( {\frac}2p \big) \big( {\frac}{2c+1}{p} \big) \, \sqrt p & \quad \, p \equiv
3\,(4).
\end{array} \right.\end{aligned}$$
By , we have $$\begin{aligned}
F_1^{\chi_p}(l,c) & = & \sum_{k=1}^{p-1} \big({\tfrac}kp\big) \, e^{{\tfrac}{2 \pi i
k l}{p}} \, \sin \big({\tfrac}{2c \pi k}{p}\big),
\\ F_2^{\chi_p}(l,c) & = & \sum_{k=1}^{p-1} (-1)^k \, \big({\tfrac}kp \big) \,
e^{{\tfrac}{2 \pi i k l}{p}} \, \sin \big({\tfrac}{(2c+1) \pi k}{p}\big).\end{aligned}$$
If $h=1$, using trigonometric identities, the real and imaginary parts of $F_1^{\chi_p}(l,c)$ are respectively given by $$\begin{aligned}
\sum_{k=1}^{p-1} \big({\tfrac}kp\big) \, \cos \big( {\tfrac}{2 \pi k l}{p} \big) \,
\sin \big({\tfrac}{2c \pi k}{p} \big) &=& {\tfrac}12 \sum_{k=1}^{p-1}
\big({\tfrac}kp\big) \, \big\{ \sin \big({\tfrac}{2\pi k (l+c)}{p}\big) - \sin
\big({\tfrac}{2 \pi k (l-c)}{p}\big) \big\}
\\ &=& {\tfrac}12 \big\{ \mathrm{Im} \, G_1^{\chi_p}(l+c) - \mathrm{Im} \,
G_1^{\chi_p}(l-c) \big\},\end{aligned}$$ $$\begin{aligned}
\sum_{k=1}^{p-1} \big( {\tfrac}{k}{p} \big) \, \sin \big( {\tfrac}{2 \pi k l}{p} \big)
\, \sin \big( {\tfrac}{2c \pi k}{p} \big) &=& {\tfrac}12
\sum_{k=1}^{p-1} \big({\tfrac}kp\big) \, \big\{ \cos \big( {\tfrac}{2\pi k (l-c)}{p}
\big) - \cos \big( {\tfrac}{2 \pi k (l+c)}{p} \big) \big\}
\\ &=& {\tfrac}12 \big\{ \mathrm{Re} \, G_1^{\chi_p}(l-c) - \mathrm{Re} \,
G_1^{\chi_p}(l+c) \big\}\end{aligned}$$ Thus, by Proposition \[prop.Gh\] we get $$\begin{split}
\mathrm{Re} \, F_1^{\chi_p}(l,c) & = \left\{ \begin{array}{ll} 0 & \qquad p
\equiv 1 \,(4), {\smallskip}\\ \, \Big( \big( {\frac}{l+c}{p} \big) -
\big( {\frac}{l-c}{p} \big) \Big) {\frac}{\sqrt p}{2} & \qquad p \equiv 3\,(4),
\end{array}\right. {\medskip}\\
\mathrm{Im} \, F_1^{\chi_p}(l,c) & = \left\{ \begin{array}{ll} \, \Big(
\big( {\frac}{l-c}{p} \big) - \big( {\frac}{l+c}{p} \big)
\Big) {\frac}{\sqrt p}{2}& \qquad p \equiv 1 \,(4), {\smallskip}\\ 0 & \qquad p \equiv 3
\,(4), \end{array} \right.
\end{split}$$ and hence $$\label{enero}
F_1^{\chi_p}(l,c) = i\delta(p) \Big( \big( {\tfrac}{l+c}{p} \big) - \big(
{\tfrac}{l-c}{p} \big) \Big) {\tfrac}{\sqrt p}{2}.$$
Similarly, for $h=2$, we have $$\begin{aligned}
\mathrm{Re} \, F_2^{\chi_p}(l,c) & = & \sum_{k=1}^{p-1} (-1)^k \, \big(
{\tfrac}{k}{p} \big) \, \cos \big( {\tfrac}{2 \pi k l}{p} \big) \, \sin
\big( {\tfrac}{(2c+1) \pi k}{p} \big) \\
&=& {\tfrac}12 \sum_{k=1}^{p-1} (-1)^k \, \big( {\tfrac}{k}{p} \big) \, \big\{\sin
\big({\tfrac}{(2(l+c)+1)\pi
k}{p}\big) - \sin \big({\tfrac}{(2(l-c-1)+1) \pi k)}{p}\big) \big\} \\
&=& {\tfrac}12 \big\{ \mathrm{Im} \, G_2^{\chi_p}(l+c) - \mathrm{Im} \,
G_2^{\chi_p}(l-c-1) \big\}\end{aligned}$$ $$\begin{aligned}
\mathrm{Im} \, F_2^{\chi_p}(l,c) & = & \sum_{k=1}^{p-1} (-1)^k \, \big(
{\tfrac}{k}{p} \big) \, \sin \big( {\tfrac}{2 \pi k l}{p} \big) \, \sin
\big( {\tfrac}{(2c+1) \pi k}{p} \big) \\ &=& {\tfrac}12 \sum_{k=1}^{p-1} (-1)^k \,
\big( {\tfrac}{k}{p} \big) \big\{ \cos \big({\tfrac}{(2(l-c-1)+1)\pi k
}{p}\big) - \cos \big({\tfrac}{(2(l+c)+1) \pi k}{p}\big) \big\} \\
&=& {\tfrac}12 \big\{ \mathrm{Re} \, G_2^{\chi_p}(l-c-1) - \mathrm{Re} \,
G_2^{\chi_p}(l+c) \big\}\end{aligned}$$ Again, by Proposition \[prop.Gh\] we get $$\begin{split}
\mathrm{Re} \, F_2^{\chi_p}(l,c) & = \left\{ \begin{array}{ll} 0 & \quad p
\equiv 1 \,(4), {\smallskip}\\ \big({\frac}2p \big) \Big(
\big({\frac}{2(l+c)+1}{p} \big) - \big({\frac}{2(l-c)-1}{p}\big) \Big) {\frac}{\sqrt p }{2}
& \quad p \equiv 3 \,(4), \end{array} \right. \\
\mathrm{Im} \, F_2^{\chi_p}(l,c) & = \left\{ \begin{array}{ll} \big({\frac}2p
\big) \Big( \big({\frac}{2(l-c)-1}{p}\big) -
\big({\frac}{2(l+c)+1}{p}\big) \Big) {\frac}{\sqrt p}{2} & \quad p \equiv 1 \,(4), {\smallskip}\\ 0 & \quad p \equiv 3 \,(4),
\end{array} \right.
\end{split}$$ and hence $$\label{febrero}
F_2^{\chi_p}(l,c) = i\delta(p) \big({\tfrac}2p \big) \Big(
\big({\tfrac}{2(l-c)-1}{p} \big) - \big({\tfrac}{2(l+c)+1}{p}\big) \Big)
{\tfrac}{\sqrt p }{2}.$$
By and we get the first formula in the statement. The remaining assertion is easy to check, and the proposition follows.
Sums involving Legendre symbols {#Sect-a-SILS}
-------------------------------
Here we compute some sums involving Legendre symbols that were used in the body of the paper. We will use the fact that $\sum_{j=1}^{p-1}\big({\frac}jp\big) =0$ for any prime $p$.
\[lem.legendres\] Let $p$ be an odd prime and $\ell \in {\mathbb N}$ with $0\le \ell \le p-1$. Then, $$\begin{aligned}
\label{klj}
& \sum_{j=1}^{p-1} \big( {\tfrac}{k \ell \pm j}{p} \big ) = - \big( {\tfrac}{k
\ell}{p} \big), \qquad k\in {\mathbb Z},\\ \label{2l-2j+1} &
\sum_{j=0}^{p-1} \big( {\tfrac}{2 \ell \pm (2j+1)}{p} \big) = 0.\end{aligned}$$
First, note that $$\sum_{j=1}^{p-1} \big( {\tfrac}{\ell+j}{p} \big) = \sum_{\begin{smallmatrix}
j=1 \\ j\ne \ell \end{smallmatrix}}^{p-1} \big( {\tfrac}{j}{p} \big)
= - \big( {\tfrac}{\ell}{p} \big),$$ and hence also, $$\sum_{j=1}^{p-1} \big( {\tfrac}{\ell-j}{p} \big) = \sum_{j=1}^{p-1} \big(
{\tfrac}{-(j-\ell)}{p} \big) =
\big( {\tfrac}{-1}{p} \big) \sum_{j=1}^{p-1} \big( {\tfrac}{j+(p-\ell)}{p} \big) = -
\big( {\tfrac}{-1}{p} \big) \big( {\tfrac}{p-\ell}{p} \big) = - \big(
{\tfrac}{\ell}{p} \big).$$
Since $p$ is prime, for any $k\in {\mathbb N}$ coprime with $p$, the sets $\{1,2,\ldots,p-1\}$ and $\{k,2k,\ldots,(p-1)k\}$ coincide modulo $p$ and thus, by $p$-periodicity of the Legendre symbol, we have $$\sum_{j=1}^{p-1} \big( {\tfrac}{k\ell \pm j}{p} \big) = \sum_{j=1}^{p-1} \big(
{\tfrac}{k\ell \pm kj}{p} \big) =
\big( {\tfrac}{k}{p} \big) \sum_{j=1}^{p-1} \big( {\tfrac}{\ell \pm j}{p} \big) = -
\big( {\tfrac}{k}{p} \big) \big( {\tfrac}{\ell}{p} \big).$$
On the other hand, if $p\,|\,k$, both sides of vanish, and the first equation in the statement is proved.
For the second equation, splitting the sum $\sum_{j=1}^{2p-1} \big(
{\tfrac}{2 \ell \pm j}{p} \big)$ into sums over even and odd indices, we get $$\begin{aligned}
\sum_{j=0}^{p-1} \big( {\tfrac}{2 \ell \pm (2j+1)}{p} \big) &=& \sum_{j=1}^{2p-1}
\big( {\tfrac}{2 \ell \pm j}{p} \big) -
\sum_{j=1}^{p-1} \big( {\tfrac}{2 \ell \pm 2j}{p} \big) = \sum_{j=0}^{p-1} \big(
{\tfrac}{2 \ell \pm j}{p} \big) - ({\tfrac}2p) \sum_{j=0}^{p-1}
\big( {\tfrac}{\ell \pm j}{p} \big),\end{aligned}$$ and by we get .
\[prop.legendres\] Let $p=2q+1$ be a prime and $\ell \in
{\mathbb N}$ with $0\le \ell \le p-1$. Then, $$\begin{aligned}
(i) & \qquad \qquad \begin{array}{lcl}
\sum\limits_{j=1}^{p-1} \big( {\tfrac}{\ell + j}{p} \big) j & = &
p \sum\limits_{j=1}^{\ell - 1} \big( {\tfrac}{j}{p} \big) +
\sum\limits_{j=1}^{p-1} \big({\tfrac}{j}{p} \big) j,
{\medskip}\\
\sum\limits_{j=1}^{p-1} \big( {\tfrac}{\ell - j}{p} \big) j & = & \big(
{\tfrac}{-1}{p}\big) \Big( p \sum\limits_{j=1}^{p - \ell - 1}
\big( {\tfrac}{j}{p} \big) + \sum\limits_{j=1}^{p-1} \big( {\tfrac}{j}{p} \big) j
\Big),
\end{array}
{\medskip}\\
(ii) & \qquad \qquad \begin{array}{lcl}
\sum\limits_{j=1}^{p-1} \big( {\tfrac}{2\ell + j}{p} \big) j & = & p
\sum\limits_{j=1}^{2 \ell - \big[{\frac}{2\ell}p \big]\,p - 1} \big( {\tfrac}{j}{p}
\big) +
\sum\limits_{j=1}^{p-1} \big({\tfrac}{j}{p} \big) j
{\medskip}\\
\sum\limits_{j=1}^{p-1} \big( {\tfrac}{2 \ell - j}{p} \big) j & = & \big(
{\tfrac}{-1}{p}\big) \Big( p \sum\limits_{j=1}^{p + \big[{\frac}{2\ell}p \big ]\,p -
2\ell - 1} \big( {\tfrac}{j}{p} \big) + \sum\limits_{j=1}^{p-1} \big( {\tfrac}{j}{p}
\big) j \Big).
\end{array}\end{aligned}$$
If $\ell =0$, then by the class number formula there is nothing to prove. So, we assume that $\ell \ne 0$. We want to compute the sums $\sum_{j=1}^{p-1} \big( {\frac}{h\ell \pm j}{p} \big)
j$ for $h=1,2$. Let us first consider the case $h=1$. By Lemma \[lem.legendres\] we have $$\label{ljs} \sum_{j=1}^{p-1} \big( {\tfrac}{\ell+j}{p} \big) j =
\sum_{j=1}^{p-1} \big( {\tfrac}{\ell+j}{p} \big) (\ell+j) - \ell
\sum_{j=1}^{p-1} \big( {\tfrac}{\ell + j}{p} \big) = \sum_{j=1}^{p-1} \big(
{\tfrac}{\ell+j}{p} \big) (\ell+j) + \ell \big( {\tfrac}{\ell}{p} \big).$$
Now, since $1\le j \le p-1$, $0\le j \le p-1$, we have $2\le j+\ell \le
2p-2$ and hence $j+\ell$ can be uniquely written as $$\label{j+l}
j+\ell=pq_j + r_j, \qquad 0 \le r_j \le p-1, \: q_j = \left\{
\begin{array}{ll}
0 & \: \text{if } j<p-\ell, {\smallskip}\\
1 & \: \text{if } j\ge p-\ell.
\end{array}\right.$$ Then, from , and using , we have that $$\begin{aligned}
\sum_{j=1}^{p-1} \big( {\tfrac}{\ell+j}{p} \big) j &=& \sum_{j=1}^{p-\ell-1}
\big( {\tfrac}{r_j}{p} \big) r_j + \sum_{j=p-\ell}^{p-1} \big(
{\tfrac}{r_j}{p} \big) (p+r_j) + \ell \big( {\tfrac}{\ell}{p} \big) \\ &=&
\sum_{j=1}^{p-1} \big( {\tfrac}{r_j}{p} \big) r_j + p
\sum_{j=p-\ell}^{p-1} \big( {\tfrac}{r_j}{p} \big) + \ell\big( {\tfrac}{\ell}{p}
\big).\end{aligned}$$ Thus, using that $$(r_0,r_1,\ldots,r_{p-\ell-1},r_{p-\ell},r_{p-\ell+1},\ldots,r_{p-1}) =
(\ell,\ell+1,\ldots,p-1,0,1,\ldots,\ell-1).$$ we get $$\label{skoll}
\sum_{j=1}^{p-1} \big( {\tfrac}{\ell+j}{p} \big) j = \sum_{\begin{smallmatrix}
j=1 \\ j\ne \ell \end{smallmatrix}}^{p-1}
\big( {\tfrac}{j}{p} \big) j + p \sum_{j=1}^{\ell-1} \big( {\tfrac}{j}{p} \big) +
\ell \big( {\tfrac}{\ell}{p} \big) = p \sum_{j=1}^{\ell-1}
\big( {\tfrac}{j}{p} \big) + \sum_{j=1}^{p-1} \big( {\tfrac}{j}{p} \big) j.$$ By the previous expression we also have $$\label{campinas-}
\sum_{j=1}^{p-1} \big( {\tfrac}{\ell-j}{p} \big) j = \big( {\tfrac}{-1}{p} \big)
\sum_{j=1}^{p-1} \big( {\tfrac}{j+(p-\ell)}{p} \big) j = \big(
{\tfrac}{-1}{p} \big) \Big( p \sum_{j=1}^{p-\ell-1} \big( {\tfrac}{j}{p} \big) +
\sum_{j=1}^{p-1} \big( {\tfrac}{j}{p} \big) j \Big).$$
Now, consider $h=2$. If $1 \le \ell \le q$ then $2 \le 2\ell \le p-1$ and we can use directly with $2\ell$ in place of $\ell$. In the other case, if $q+1 \le \ell \le p-1$ then $1\le 2\ell - p
\le p-2$ and, by , we have $$\sum_{j=1}^{p-1} \big( {\tfrac}{2\ell+j}{p} \big) j = \sum_{j=1}^{p-1} \big(
{\tfrac}{2 \ell - p + j}{p} \big) j
= p \sum_{j=1}^{2\ell-p-1} \big( {\tfrac}{j}{p} \big) + \sum_{j=1}^{p-1} \big(
{\tfrac}{j}{p} \big) j.$$ In the remaining case, proceeding as before and using , one gets the desired result in the statement, and thus the proposition follows.
\[lem.legendres3\] Let $p$ be an odd prime and $\ell \in {\mathbb N}$ with $0\le \ell \le p-1$. Then, $$\label{legendre odd}
\sum_{j=0}^{p-1} \big( {\tfrac}{2\ell \pm (2j+1)}{p} \big) j = \sum_{j=1}^{p-1}
\big( {\tfrac}{2\ell \pm j}{p} \big) j - ({\tfrac}2p) \sum_{j=1}^{p-1}
\big( {\tfrac}{\ell \pm j}{p} \big) j.$$
We first note that $$\begin{aligned}
2\sum_{j=0}^{p-1} \big( {\tfrac}{2\ell \mp (2j+1)}{p} \big) j &=&
\sum_{j=0}^{p-1} \big( {\tfrac}{2\ell \mp (2j+1)}{p} \big) (2j+1) \\&=&
\sum_{j=1}^{2p-1} \big( {\tfrac}{2\ell \mp j}{p} \big) j - \sum_{j=1}^{p-1} \big(
{\tfrac}{2\ell \mp 2j}{p} \big) 2j \\ &=& \sum_{j=1}^{p-1}
\big( {\tfrac}{2\ell \mp j}{p} \big) j + \sum_{j=p}^{2p-1} \big( {\tfrac}{2\ell \mp
j}{p} \big) j - 2 ({\tfrac}2p) \sum_{j=1}^{p-1} \big( {\tfrac}{\ell \mp
j}{p} \big) j,\end{aligned}$$ where in the first equality we have used . The second sum in the r.h.s. of the above expression equals $$\begin{aligned}
\sum_{h=0}^{p-1} \big( {\tfrac}{2\ell \mp (p+h)}{p} \big) (p+h) = p
\sum_{h=0}^{p-1} \big( {\tfrac}{2\ell \mp h}{p} \big) + \sum_{h=0}^{p-1} \big(
{\tfrac}{2\ell \mp h}{p} \big) h = \sum_{j=1}^{p-1} \big(
{\tfrac}{2\ell \mp j}{p} \big) j.\end{aligned}$$ Substituting this expression in the first one we get the desired result.
We want to compute the sums $$\label{S1yS2}
\begin{split}
& S_1(\ell,p) := \sum_{j=1}^{p-1} \Big( \big( {\tfrac}{\ell - j}{p} \big) - \big(
{\tfrac}{\ell + j}{p} \big) \Big) j, \\ & S_2(\ell,p) :=
\sum_{j=0}^{p-1} \Big( \big( {\tfrac}{2\ell - (2j+1)}{p} \big) - \big( {\tfrac}{2\ell
+ (2j+1)}{p} \big) \Big) j,
\end{split}$$ for $0\le \ell \le p-1$. We are now in a position to prove the results that were used in Section 3.
\[prop.dif legendres\] Let $p$ be an odd prime and $\ell \in {\mathbb N}$ with $0\le \ell \le p-1$. Then, in the notations in we have $$S_1(\ell,p) = \left\{ \begin{array}{ll} p \, S_1^-(\ell,p) & \qquad p \equiv
1 \,(4), {\smallskip}\\
-p \, S_1^+(\ell,p) - 2\sum\limits_{j=1}^{p-1} ({\tfrac}{j}{p})j & \qquad
p\equiv 3 \,(4), \end{array}\right.$$ and $$S_2(\ell,p) = \left\{ \begin{array}{lc} p \Big( S_2^-(\ell,p) - ({\tfrac}2p)
S_1^-(\ell,p) \Big) & p\equiv 1\,(4), {\smallskip}\\
-p \Big( S_2^+(\ell,p) - ({\tfrac}2p) S_1^+(\ell,p) \Big) + 2 \big( ({\tfrac}2p)-1
\big) \sum\limits_{j=1}^{p-1} \big( {\tfrac}jp \big)j & p\equiv
3\,(4), \end{array} \right.$$ where $S_h(\ell,p)$ and $S_h^{\pm}(\ell,p)$ are defined in and respectively.
By Lemma \[prop.legendres\] (i), we have $$\begin{aligned}
S_1(\ell,p) & = & \big( {\tfrac}{-1}{p}\big) \Big( p \sum_{j=1}^{p - \ell - 1}
\big( {\tfrac}{j}{p} \big) + \sum_{j=1}^{p-1} \big( {\tfrac}{j}{p}
\big) j \Big) - \Big( p \sum_{j=1}^{\ell - 1} \big( {\tfrac}{j}{p} \big) +
\sum_{j=1}^{p-1} \big({\tfrac}{j}{p} \big) j \Big).\end{aligned}$$ By using and we get the first expression in the statement.
On the other hand, by Lemma \[lem.legendres3\] we have that [$$\begin{aligned}
S_2(\ell,p) & = & \Big( \sum_{j=1}^{p-1} \big( {\tfrac}{2\ell - j}{p} \big) j -
({\tfrac}2p) \sum_{j=1}^{p-1} \big( {\tfrac}{\ell - j}{p} \big) j
\Big) - \Big( \sum_{j=1}^{p-1} \big( {\tfrac}{2\ell + j}{p} \big) j - ({\tfrac}2p)
\sum_{j=1}^{p-1} \big( {\tfrac}{\ell + j}{p} \big) j \Big) \\
&=& \sum_{j=1}^{p-1} \Big( \big( {\tfrac}{2\ell - j}{p} \big) - \big( {\tfrac}{2\ell
+ j}{p} \big) \Big) j - ({\tfrac}2p) \sum_{j=1}^{p-1} \Big(
\big( {\tfrac}{\ell - j}{p} \big) - \big( {\tfrac}{\ell + j}{p} \big) \Big) j.\end{aligned}$$]{} By using Lemma \[prop.legendres\] (ii) we see that $S_2(\ell,p)$ equals $$\begin{aligned}
&&({\tfrac}{-1}{p}) \Big( \sum_{j=1}^{p+ \big[{\frac}{2\ell}p \big]\,p-2\ell-1} \big(
{\tfrac}{j}{p} \big) + \sum_{j=1}^{p-1} \big( {\tfrac}{j}{p} \big) j \Big)\\&& -
\Big( \sum_{j=1}^{2\ell-\big [{\frac}{2\ell}p \big ]\,p-1} \big( {\tfrac}{j}{p}
\big) + \sum_{j=1}^{p-1} \big( {\tfrac}{j}{p} \big) j \Big) -
({\tfrac}2p) S_1(\ell,p),\end{aligned}$$ and now applying and using we get the desired result, and hence the proposition follows.
Acknowledgements {#acknowledgements .unnumbered}
================
The research of P. Gilkey was supported by Project MTM2006-01432 (Spain) and by the University of Córdoba (Argentine). R. Podestá wishes to thank the hospitality at the Universidad Autónoma de Madrid (Spain).
[AAA]{}
W. Ambrose, I. M. Singer, a theorem on holonomy, Trans. Am. Math. Soc., [**75**]{} (428–443) 1953.
D. W. Anderson, E. H. Brown Jr., F. P. Peterson, *Spin cobordism*, Bull. Am. Math. Soc. [**72**]{} (256–260), 1966.
D. W. Anderson, E. H. Brown Jr., F. P. Peterson, *The structure of the Spin cobordism ring*, Ann. Math. [**86**]{} (271–298), 1967.
D. W. Anderson, E. H. Brown Jr., F. P. Peterson, *Pin cobordism and related topics*, Comment. Math. Helv. [**44**]{} (462–468), 1969.
T. Apostol, [*Introduction to analytic number theory*]{}, Springer Verlag UTM, New York, 1998.
M. F. Atiyah, F. K. Patodi, I. M. Singer, *Spectral asymmetry and Riemannian geometry I*, Math. Proc. Camb. Philos. Soc. [**77**]{} (43–69), 1975.
M. F. Atiyah, F. K. Patodi, I. M. Singer, *Spectral asymmetry and Riemannian geometry II*, Math. Proc. Camb. Philos. Soc. [**78**]{} (405–432), 1975.
M. F. Atiyah, F. K. Patodi, I. M. Singer, *Spectral asymmetry and Riemannian geometry III*, Math. Proc. Camb. Philos. Soc. [**79**]{} (71–99), (1976).
A. Bahri, M. Bendersky, D. Davis, P. Gilkey, *The complex bordism of groups with periodic cohomology*, Trans. Am. Math. Soc. [**316**]{} (673–687), 1989.
A. Bahri, M. Bendersky, P. Gilkey, *The relationship between complex bordism and K-theory for groups with periodic cohomology*, Contemp. Math. [**96**]{} (19–31), 1989.
Brown, H., Bülow, R., Neubüser, J., Wondratschok, H., Zassenhaus, H., *Crystallographic groups of four-dimensional space*, Wiley, New York, 1978.
A. Bahri, P. B. Gilkey, *Pin${}\sp c$ cobordism, and equivariant Spin${}\sp c$ cobordism for cyclic 2-groups*, Proc. of the Am. Math. Soc. [**99**]{} (380–382), 1987.
A. Bahri, P. B. Gilkey, *The eta invariant, Pin${}\sp
c$ bordism, and equivariant Spin${}\sp c$ bordism for cyclic 2-groups*, Pacific Journal [**128**]{} (1–24), 1987.
B. Berndt, R. Evans, K. Williams, *Gauss and Jacobi sums*, Canadian Math. Soc. Series, Vol. **21**, Wiley-Interscience, 1998.
B. Botvinnik, P. Gilkey, and S. Stolz, [*T*he Gromov Lawson Rosenberg conjecture for groups with periodic cohomology]{}, Journal Differential Geometry **46** (374–405), 1997.
Conway, J. H., Rossetti, J. P., *Describing the platycosms*, Math. Res. Lett. **13** (475–494), 2006.
L. Charlap, [*Compact flat Riemannian manifolds I*]{}, Ann. of Math. [**81**]{} (15–30), 1965.
L. Charlap, [*Bieberbach groups and flat manifolds*]{}, Springer Verlag, Universitext, 1988.
Cid, C., Schulz, T., *Computation of Five and Six dimensional Bieberbach groups*, Experiment Math. [**10**]{} (109–115) 2001.
T. Friedrich, [*Dirac operator in Riemannian geometry*]{}, Amer. Math. Soc. GSM [**25**]{}, 1997.
P. B. Gilkey, [*The Residue of the Global $\eta$ Function at the Origin*]{}, Adv. in Math. [**40**]{} (290–307), 1981.
P. B. Gilkey, [*The geometry of spherical space form groups*]{}, Series in Pure Mathematics **7**. Singapore, World Scientific 1988.
P. B. Gilkey, [*Invariance theory, the heat equation and the Atiyah-Singer index theorem*]{} $2^{\operatorname{nd}}$ ed., Studies in Advanced Mathematics, Boca Raton, FL, CRC Press, 1995.
P. Gilkey, J. Leahy, J.H. Park, [*Spectral Geometry, Riemannian Submersions, and the Gromov-Lawson Conjecture*]{}, Boca Raton, FL, CRC Press, 1999.
Hantzsche, W., Wendt, H., *Dreidimensionale euklidische Raumformen*, Math. Ann. [**10**]{} (593–611) 1935.
H. B. Lawson, M. L. Michelsohn, [*Spin geometry*]{}, Princeton University Press, New Jersey, 1989.
A. Lichnerowicz, [*Spineurs harmoniques*]{}, C. R. Acad.Sci. Paris [**257**]{} (7–9), 1963.
R. J. Miatello, R. A. Podestá, [*Spin structures and spectra of ${\mathbb Z}_2^k$-manifolds*]{}, Math. Zeitschrift **247** (319–335), 2004.
R. J. Miatello, R. A. Podestá, [*The spectrum of twisted Dirac operators on compact flat manifolds*]{}, Trans. Amer. Math. Society **358** (4569–4603), 2006.
R. J. Miatello, R. A. Podestá, *Eta invariants and class numbers*, Pure and Applied Math. Quart. **5** (1–26), 2009.
R. J. Miatello, J. P. Rossetti, *Spectral properties of flat manifolds*, Contemporary Math, to appear.
F. Pfäffle, [*The Dirac spectrum of Bieberbach manifolds*]{}, J. Geom. Phys. [**35**]{} (367–385), 2000.
I. Reiner, [*Integral representations of cyclic groups of prime order*]{}, Proc. Amer. Math. Soc. **8** (142–145), 1957.
M. Sadowski, A. Szczepanski, [*Flat manifolds, harmonic spinors and eta invariants*]{}, Adv. in Geometry **6** (287–300), 2006.
S. Stolz, [*Simply connected manifolds of positive scalar curvature*]{}, Ann. of Math. [**136**]{} (1992), 511–540.
R. T. Seeley, *Complex powers of an elliptic operator*, Proc. Sympos. Pure Math. [**10**]{} (288–307), 1967.
R. Thom, *Quelques propriétés globales des variétés différentiables*, Comment. Math. Helv. [**28**]{} (17–86), 1954.
A. Vasquez, [*Flat Riemannian manifolds*]{}, J. Diff. Geometry [**4**]{} (367–382), 1970.
J. Wolf, [*Spaces of constant curvature*]{}, Mc Graw-Hill, New York, 1967.
|
---
abstract: |
The cosmic star formation histories are evaluated for different minimum masses of the initial halo structures, with allowance for realistic gas outflows. With a minimum halo mass of $10^{7}$–$10^{8}\
\mathrm{M}_{\odot}$ and a moderate outflow efficiency, we reproduce both the current baryon fraction and the early chemical enrichment of the IGM. The intensity of the formation rate of “normal” stars is also well constrained by the observations: it has to be dominated by star formation in elliptical galaxies, except perhaps at very low redshift. The fraction of baryons in stars is predicted as are also the type Ia and II supernova event rates. Comparison with SN observations in the redshift range $z=0-2$ allows us to set strong constraints on the time delay of type Ia supernovae (a total delay of $\sim$4 Gyr is required to fit the data), the lower end of the mass range of the progenitors (2 - 8 $\mathrm{M}_{\odot}$) and the fraction of white dwarfs that reproduce the type Ia supernova (about 1 per cent). The intensity in the initial starburst of zero metallicity stars below 270 $\mathrm{M_{\odot}}$ must be limited in order to avoid premature overenrichment of the IGM. Only about 10 - 20 % of the metals present in the IGM at $z = 0$ have been produced by population III stars at very high $z$. The remaining 80 - 90 % are ejected later by galaxies forming normal stars, with a maximum outflow efficiency occurring at a redshift of about 5. We conclude that $10^{-3}$ of the mass in baryons must lie in first massive stars in order to produce enough ionizing photons to allow early reionization of the IGM by $z\sim 15$.
author:
- Frédéric Daigne
- 'Keith A. Olive'
- Joe Silk
- Felix Stoehr
- Elisabeth Vangioni
bibliography:
- 'dossv4.bib'
title: 'Hierarchical Growth and Cosmic Star Formation: Enrichment, Outflows and Supernova Rates.'
---
Introduction
============
Chemical evolution is a key to understanding the hybrid role of massive star formation in the early universe with regard to the epoch of reionization, the heavy element abundances of the oldest stars and the high $z$ intergalactic medium, and the mass outflows associated with galaxy formation. In addition, predicted supernova rates provide us with an independent probe of the early epoch of star formation. Combining abundance and supernova rate predictions allows us to develop an improved understanding of both the cosmic star formation history and of the enrichment of the IGM, as well as to elucidate the nature of Population III (see e.g. the recent review of @ciardi:05).
The prevalent view is that the first stars encompassed the mass range 100 to 1000 M$_\odot.$ Recent support for this possibility stems from the need to reionize the Universe at high redshift [@cen:03a; @haiman:03; @wyithe:03; @bromm:04a] as indicated by the WMAP first-year data [@kogut:03]. The possibility for early reionization by the first galaxies was considered by @ciardi:03 and further support for this hypothesis is based on chemical abundance patterns at low metallicity [@wasserburg:00; @oh:01; @qian:01; @qian:05].
However, the robustness of the conclusion that very massive stars (VMS) were necessarily present among the first stars has been questioned [@venkatesan:03b; @tumlinson:04; @tumlinsona:05]. A “normal" initial mass function (IMF) may be capable of producing ionization consistent with WMAP [@venkatesan:03a; @wyithe:03], and within these constraints, it was argued [@venkatesan:04] that a broad set of chemical abundances may be better fit using the yields of @umeda:03 for stars with masses in the 1- 50 M$_\odot$ range than with the yields from pair-instabilty supernovae (PISN) [@heger:02]. Indeed, these results were confirmed in @daigne:04 [hereafter DOSVA] where it was argued that a top-heavy IMF without VMS supplied a better fit to low metallicity abundance data while still accounting for the early re-ionization of the Universe when using a detailed model of cosmic chemical evolution. Furthermore, when VMS were assumed to be the predominant source for Pop III, it was found that there could not be enough of these stars to have accounted for reionization without producing anomalous chemical abundance ratios in both old halo stars and in the IGM. Even partial suppression of the yields of pair instability SNe via rotation and fall-back should not vitiate this conclusion in view of the anomalous abundance ratios and the large discrepancy found by DOSVA.
In this paper, we improve on an earlier model for primordial star formation and chemical evolution (DOVSA) by using a more realistic model of structure formation to study the role of baryonic outflows and the enrichment of the IGM. Our principal result is that only if Population III predominantly consisted of stars in the “normal” mass range, but with a top-heavy IMF, that is to say dominated by stars in the mass range 40 to 100 M$_\odot,$ can one simultaneously account for reionization and chemical enrichment, both of galaxies and of the IGM. However, this mass range is not capable of sufficient Si production and therefore, more massive stars may be necessary to account for the inferred abundance of Si in the IGM. Such a bimodal approach to star formation allows us to account both for the cosmic star formation history and global chemical evolution of the universe. The predicted supernova rates agree with those observed. We also infer a baryonic outflow rate that allows us to reconcile the observed baryon fraction in massive galaxies with that observed in clusters, the CMB and the Lyman alpha forest, as well as that predicted by primordial nucleosynthesis.
The outline of this paper is as follows. In § \[sec:method\] we present the details of the chemical evolution model, especially the treatment of the structure formation and of the galactic winds. We also outline the key parameters used in the models which affect our conclusions. In § \[sec:model0\], we discuss the normal mode of star formation and we show that the minimum masses of the halos of star forming structures as well as the efficiency of the galactic winds are well constrained by the observations. This allows us to select a few realistic models. In § \[sec:SNae\], we compute the expected SNII and SNIa rates in these models and deduce the time delay inferred for SNIa to fit the data. § \[sec:popIII\] is devoted to the addition of an initial starburst of massive stars in our scenario. We summarize our results in § \[sec:conclusions\].
Hierarchical formation scenario and associated galactic winds {#sec:method}
=============================================================
A cosmic evolutionary model
---------------------------
The chemical evolution model used in this paper has been described in DOVSA. It is a generalized version of standard models designed to follow one specific structure such as the Milky Way (for a review, see @tinsley:80). We describe baryons in the Universe by two large reservoirs. The first is associated with collapsed structures (hereafter the “structures”) and is divided in two sub-reservoirs: the gas (hereafter the “interstellar medium” or ISM) and the stars and their remnants (hereafter the “stars”). The second reservoir corresponds to the medium in between the collapsed structures (hereafter the “intergalactic medium” or IGM). The evolution of the baryonic mass of these reservoirs, i.e. $M_\mathrm{IGM}(t)$ of the IGM, $M_\mathrm{ISM}(t)$ of the ISM and $M_{*}(t)$ of the stars is governed by a set of differential equations (see section 2 in DOVSA): $$\frac{dM_\mathrm{IGM}}{dt}=-\frac{dM_\mathrm{struct}}{dt}=-a_\mathrm{b}(t)+o(t),$$
$$\frac{dM_\mathrm{*}}{dt}=\Psi(t)-e(t)\ \mathrm{and}\ \frac{dM_\mathrm{ISM}}{dt}=\frac{dM_\mathrm{struct}}{dt}-\frac{dM_\mathrm{*}}{dt}\ .$$
In addition, we have $M_\mathrm{ISM}(t)+M_\mathrm{*}(t)=M_\mathrm{struct}(t)$, corresponding to the total baryonic mass of the structures, and $M_\mathrm{IGM}(t)+M_\mathrm{struct}(t)=\mathrm{constant}$, which is the total baryonic mass of the Universe. As can be seen, these equations are controlled by four rates which represent four fundamental processes (see sketch in Figure \[fig:schemacode\]): the formation of structures through the accretion of baryons from the IGM, $a_\mathrm{b}(t)$; the formation of stars through the transfer of baryons from the ISM, $\Psi(t)$; the ejection of enriched gas by stars, $e(t)$ and the outflow of baryons from the structures into the IGM, $o(t)$.
We track the chemical composition of the ISM and the IGM separately as a function of time (or redshift). The differential equations governing the evolution of the mass fraction $X_{i}^\mathrm{ISM}$ ($X_{i}^\mathrm{IGM}$) of element $i$ in the ISM (IGM) are given by equations (6) and (7) in section 2 in DOVSA. In addition, we also compute the ionizing UV fluxes from the stars (equation (12) in section 3 in DOVSA) and the rate of explosive events (type Ia and gravitational collapse supernovae).\
The age $t$ of the Universe which appears in the equations is related to the redshift by $$\frac{dt}{dz}=\frac{9.78 h^{-1}\ \mathrm{Gyr}}{(1+z)\sqrt{\Omega_\mathrm{\Lambda}+\Omega_\mathrm{m}\left(1+z\right)^{3}}}\ ,$$ assuming the cosmological parameters of the so-called “concordance model”, with a density of matter $\Omega_\mathrm{m}=0.27$ and a density of “dark energy” $\Omega_\mathrm{\Lambda}=0.73$ (Spergel et al. 2003), and taking $H_{0}=71\ \mathrm{km/s/Mpc}$ for the Hubble constant ($h=0.71$). This allows us to trace all the quantities we describe as a function of redshift. The input stellar data (lifetimes, mass and type of the remnant, metal yields, UV flux) are taken to be dependent both of the mass and the metallicity of the star (see DOVSA for more details).\
Building upon our original model, we have included the following improvements: (i) the baryon accretion rate, $a_\mathrm{b}(t)$, by the structures is now computed in the framework of the hierarchical scenario of structure formation, [@press:74; @sheth:99; @jenkins:01; @wyithe:03b]. This is described in section \[sec:ab\]. (ii) the baryon outflow rate, $o(t)$, from the structures now includes a redshift-dependent efficiency, which accounts for the increasing escape velocity of the structures as the galaxy assembly is in progress. This is described in section \[sec:o\]. Note that in DOVSA, the outflow contained two sources: $o(t)=o_\mathrm{w}(t)+o_\mathrm{SN}(t)$ where $o_\mathrm{w}(t)$ is a global outflow powered by stellar explosions (galactic winds) and $o_\mathrm{SN}(t)$ corresponds to stellar supernova ejecta that are flushed directly out of the structures. Since it was shown in DOVSA that the second term has a very small effect on the results, we have neglected it in this work (in DOVSA’s notation, this is equivalent to setting $\alpha=0$).
Hierarchical formation scenario {#sec:ab}
-------------------------------
As seen in DOVSA, the term $a_\mathrm{b}(t)$ which stands for the process of structure formation can have a strong impact on the high redshift evolution, intensity of the star formation process, ionizing flux, and the enrichment of the IGM, as it governs the size of the reservoir of baryons available for star formation. In DOVSA, although this term was estimated in an oversimplified way, we assumed only that structure formation efficiency decays exponentially. This first attempt allowed us to point out the key role of this term in the generalized chemical evolution model we have implemented. Here, we improve the model with a more realistic description of structure formation.
We adopt the framework of the hierarchical scenario where small structures are formed first. At redshift $z$, the comoving density of dark matter halos in the mass range $[M,M+dM]$ is $f_\mathrm{PS}(M,z)dM$, with $$\int_{0}^{\infty} dM\ M f_\mathrm{PS}(M,z) = \rho_\mathrm{DM}\ ,$$ where $\rho_\mathrm{DM}$ is the comoving dark matter density. The distribution function of halos $f_\mathrm{PS}(M,z)$ is computed using the method described in @jenkins:01 using a code provided by A. Jenkins. It follows the standard theory [@press:74], including the modification of @sheth:99 and assumes a primordial power spectrum with a power-law index $n=1$ and the fitting formula to the exact transfer function for non-baryonic cold dark matter given by @bond:84. We adopt a rms amplitude $\sigma_{8}=0.9$ for mass density fluctuations in a sphere of radius $8\ h^{-1}\ \mathrm{Mpc}$.\
We assume that the baryon distribution traces the dark matter distribution without any bias so that the density of baryons is just proportional to the density of dark matter by a factor $\Omega_{b}/\left(\Omega_{m}-\Omega_{b}\right)$. We take a baryonic density $\Omega_\mathrm{b}=0.044$ [@spergel:03]. We parametrize the fact that stars can form only in structures which are suitably dense by defining the minimum mass $M_\mathrm{min}$ of a dark matter halo of the collapsed structures where star formation occurs. This mass could be related in principle with the critical temperature at which the cooling processes become efficient enough to allow star formation. In fact, this critical temperature, and hence the minimum mass $M_\mathrm{min}$, should evolve with redshift $z$ as the cooling processes of the hot gas in structures depend strongly on the chemical composition and ionizing state of the gas. It is however beyond the scope of this study to include such a detailed analysis so we prefer to keep $M_\mathrm{min}$ constant and to consider it as a free parameter of the model. The fraction of baryons at redshift $z$ which are in such structures is then given by $$f_\mathrm{b,struct}(z) = \frac{\int_{M_\mathrm{min}}^{\infty} dM\ M f_\mathrm{PS}(M,z)}{\int_{0}^{\infty} dM\ M f_\mathrm{PS}(M,z)}\ .
\label{eq:fb}$$ Therefore, the mass flux $a_\mathrm{b}$ can be estimated by $$\begin{aligned}
a_\mathrm{b}(t) & = & \Omega_\mathrm{b}\left(\frac{3H_{0}^{2}}{8\pi G}\right)\ \left(\frac{dt}{dz}\right)^{-1}\ \left|\frac{d f_\mathrm{b,struct}}{dz}\right|\nonumber\\
& = & 1.2 h^{3}\ \mathrm{M_{\odot}/yr/Mpc^{3}}\ \left(\frac{\Omega_\mathrm{b}}{0.044}\right)\left(1+z\right)\sqrt{\Omega_\mathrm{\Lambda}+\Omega_\mathrm{m}\left(1+z\right)^{3}}\left|\frac{df_\mathrm{b,struct}}{dz}\right|\ .\end{aligned}$$ This is the new expression of $a_\mathrm{b}(t)$ used in the model. Even if it represents a significative improvement compared to the treatment of the same term in DOVSA, one should keep in mind that this derivation is only partially consistent, as the actual fraction of baryons in collapsed structures with dark matter halos of mass $M\ge M_\mathrm{min}$ will differ from the value given by equation \[eq:fb\] due to the outflow of baryons from the star-forming structures towards the IGM. However, since $o(t)$ is always small compared to $a_\mathrm{b}(t)$, this is a small correction for most of the evolutionary history.\
Finally, this formalism allows us to indirectly fix the initial redshift $z_\mathrm{init}$ where the first stars form in the simulation. This redshift is a priori smaller than the redshift $z_\mathrm{PS}(M_\mathrm{min})$ at which the first dark matter halos of mass $M_\mathrm{min}$ appear, as the star formation process can be delayed due to the time necessary to cool the collapsing gas. Instead of fixing the value of $z_\mathrm{init}$, we prefer to fix the size of the gas reservoir when the star formation starts, i.e. the initial fraction $f_\mathrm{b,struct}(z_\mathrm{init})$ of baryons in collapsed structures which form stars. We usually adopt $f_\mathrm{b,struct}(z_\mathrm{init})=1\ \%$ but we also consider $0.1\ \%$ and $5\ \%$ for some specific cases. Supernova feedback generically gives a star formation efficiency of 1 or 2 percent per dynamical time [@silk:03] and an inefficiency of this order is inferred in essentially all current epoch star-forming galaxies [@kenni:03].
The corresponding initial redshift $z_\mathrm{init}$ is computed from equation \[eq:fb\] for a given minimum mass $M_\mathrm{min}$ (see table \[tab:model0\]). Note that this redshift is usually smaller than that found in hydrodynamical simulations of collapsing gas for the redshift of formation of the first zero-metallicity stars (population III stars): $z\sim 20-30$ [@bromm:04b; @yoshida:04; @bromm:04a]. However, the first stars formed in these simulations are found to form in collapsed structures with dark matter halos of mass $M\sim 10^{6}\ \mathrm{M_{\odot}}$ which represents only a very small fraction ($\la 10^{-3}$) of the baryons in the Universe at these redshifts. Therefore, this very primordial star formation activity is completely negligible in comparison with the global evolution of the Universe.
Outflows {#sec:o}
--------
We compute the outflow by $$o(t) = \frac{2\epsilon}{v_\mathrm{esc}^{2}(z)}\int_{\max{\left(8\ \mathrm{M_{\odot}},m_\mathrm{d}(t)\right)}}^{m_\mathrm{up}} dm\ \Phi(m)\Psi\left(t-\tau(m)\right) E_\mathrm{kin}(m)\ ,$$ where $\Phi(m)$ is the initial mass function (IMF) of stars, $\tau(m)$ is the lifetime of a star of mass $m$ and $m_\mathrm{d}$ is the mass of stars that die at age $t$, $E_\mathrm{kin}(m)$ is the kinetic energy released by the explosion of a star of mass $m$, $\epsilon$ is the fraction of the kinetic energy of supernovae that is available to power the outflow and $v_\mathrm{esc}^{2}(z)$ is the mean square of the escape velocity of structures at redshift $z$ [see e.g. @scully:97]. The escape velocity is obtained by mass-averaging the gravitational potential over the mass distribution at redshift $z$: $$v_\mathrm{esc}^{2}(z) = \frac{\int_{M_\mathrm{min}}^{\infty} dM\ f_\mathrm{PS}(M,z) M \left(2 G M / R\right)}{\int_{M_\mathrm{min}}^{\infty} dM\ f_\mathrm{PS}(M,z) M}\ ,$$ where $R$ is the radius of a dark matter halo of mass $M$. The resulting velocity is plotted in Figure \[fig:velocity\] for different minimum masses $M_\mathrm{min}$. Notice the strong evolution with $z$: outflows at high redshift are much more efficient and a large fraction (if not all) of the gas in the structures is ejected and mixed in the intergalactic medium, thus contributing to the early enrichment of the IGM.
The total efficiency of this outflow process will depend both on $\epsilon$ and $v_\mathrm{esc}^{2}(z)$ and will therefore vary with redshift. An approximation of the efficiency can be obtained in the “instantaneous recycling limit” where the lifetime of massive stars is neglected by comparing the instantaneous outflow rate with the star formation rate. In this case we have $$r_\mathrm{outflow} = \frac{o(t)}{\Psi(t)} \simeq \frac{2\epsilon\ \bar{e}_\mathrm{kin}}{v_\mathrm{esc}^{2}(z)} ,$$ where $\bar{e}_\mathrm{kin}$ is the kinetic energy per stellar mass, averaged over the initial mass function: $$\bar{e}_\mathrm{kin} = \int_{m_\mathrm{inf}}^{m_\mathrm{up}} dm\ \Phi(m) E_\mathrm{kin}(m)\ .$$ For a power-law IMF of slope $-(1+x)$, a minimal mass $m_\mathrm{inf}$ and a constant supernova kinetic energy $E_\mathrm{kin}(m)\simeq \mathrm{constant}$, this kinetic energy per stellar mass can be approximated by $$\bar{e}_\mathrm{kin} \simeq \frac{x-1}{x}\frac{E_\mathrm{kin}}{m_\mathrm{inf}}\times\left\lbrace\begin{array}{ll}
\left(8\ \mathrm{M_{\odot}} / m_\mathrm{inf}\right)^{-x}\ & \mathrm{if}\ m_\mathrm{inf} < 8\ \mathrm{M}_{\odot}\ \mathrm{(normal\ mode)}\\
1 & \mathrm{if}\ m_\mathrm{inf} \ge 8\ \mathrm{M}_{\odot}\ \mathrm{(massive\ mode)}
\end{array}\right.\ .
\label{ebar}$$ The derivation of eq. \[ebar\] assumes a properly normalized IMF (see eq. \[norm\] below). Typical values for $v_\mathrm{esc}^{2}(z)$ at different redshifts can be obtained from Figure \[fig:velocity\]. These result in the following efficiencies for $x=1.3$ and $M_\mathrm{min}=10^{7}\ \mathrm{M_{\odot}}$: $$\mathit{r}_\mathrm{outflow} \simeq 0.052\
\left(\frac{\epsilon}{0.01}\right)
\left(\frac{E_\mathrm{kin}}{10^{51}\ \mathrm{erg}}\right)
\left(\frac{m_\mathrm{inf}}{0.1\ \mathrm{M_\odot}}\right)^{0.3}
\left(\frac{v_\mathrm{esc}^{2}(z)}{3\times 10^{48}\ \mathrm{erg/M_{\odot}}}\right)^{-1}$$ for a normal mode of stellar formation at $z\sim 1$, $$\mathit{r}_\mathrm{outflow} \simeq 1.9\
\left(\frac{\epsilon}{0.01}\right)
\left(\frac{E_\mathrm{kin}}{10^{51}\ \mathrm{erg}}\right)
\left(\frac{m_\mathrm{inf}}{0.1\ \mathrm{M_\odot}}\right)^{0.3}
\left(\frac{v_\mathrm{esc}^{2}(z)}{8\times 10^{46}\ \mathrm{erg/M_{\odot}}}\right)^{-1}$$ for a normal mode of stellar formation at $z\sim 8$, $$\mathit{r}_\mathrm{outflow} \simeq 5.8\
\left(\frac{\epsilon}{0.01}\right)
\left(\frac{E_\mathrm{kin}}{10^{51}\ \mathrm{erg}}\right)
\left(\frac{m_\mathrm{inf}}{40\ \mathrm{M_\odot}}\right)^{-1}
\left(\frac{v_\mathrm{esc}^{2}(z)}{2\times 10^{46}\ \mathrm{erg/M_{\odot}}}\right)^{-1}$$ for a massive mode of stellar formation at $z\sim 16$ ($m_\mathrm{inf}=40\ \mathrm{M_{\odot}}$) and $$\mathit{r}_\mathrm{outflow} \simeq 8.2\
\left(\frac{\epsilon}{0.01}\right)
\left(\frac{E_\mathrm{kin}}{5\times 10^{51}\ \mathrm{erg}}\right)
\left(\frac{m_\mathrm{inf}}{140\ \mathrm{M_\odot}}\right)^{-1}
\left(\frac{v_\mathrm{esc}^{2}(z)}{2\times 10^{46}\ \mathrm{erg/M_{\odot}}}\right)^{-1}$$ for a very massive mode of stellar formation at $z\sim 16$ ($m_\mathrm{inf}=140\ \mathrm{M_{\odot}}$). These estimates are confirmed by results described below. One should note that the outflow is a self-regulating process and though it may be large compared with the SFR at a given moment, it will turn off as the SFR decreases. Thus we see that the outflow can be ultra-efficient during the early population III phase and can still be quite efficient at high redshift even in the normal mode of star formation. In contrast, outflows are very inefficient at low redshift.
Model parameters
----------------
Before we describe the models in more detail, it will be useful to outline the adjustable parameters that we will consider. While this list of parameters is not exclusive, they are the parameters which most affect the results and shape our conclusions.
1. The minimum mass $M_\mathrm{min}$ of dark matter halos of star-forming structures. As described above, the distribution of star-forming structures spans a mass range with some minimal halo mass. Structures with masses smaller than $M_\mathrm{min}$ are considered as part of the IGM and do not participate in the star formation process. As we will see, when the initial baryon fraction is fixed, there is a tight relationship between $M_\mathrm{min}$ and the redshift at which star formation begins. There are few direct constraints on $M_\mathrm{min}$, and we will consider values from $10^6$ – $10^{11}$ M$_\odot$.
2. The initial fraction $f_\mathrm{b,struct}(z_\mathrm{init})$ of baryons within the structures when star formation begins. We will for the most part assume that star formation begins when the baryon fraction in a structure is 1%. For a given value of $M_\mathrm{min}$, eq. (\[eq:fb\]) can be used to determine the redshift at which star formation begins. Thus, we can alternatively specify, $z_\mathrm{init}$, and compute the initial baryon fraction.
3. The efficiency of the outflow $\epsilon$. This parameter too, is largely undetermined. Its value can be fixed within the model as it sensitively controls the final baryon fraction in structures, $f_\mathrm{b,struct}(z=0)$. For models with an exponentially decreasing star formation rate (as we consider below), increasing $\epsilon$ leads to a decrease in the final value of $f_\mathrm{b,struct}(z=0)$ which we constrain to lie in the range of $61\pm 18\ \%$ [see e.g. @fukugita:04] resulting in values of $\epsilon < 0.02$.
4. The star formation rate $\Psi(t)$. In principle, there are many possible analytical forms for the star formation. For example, it is common to assume that the SFR is proportional to the gas mass fraction, $\sigma$, (or some power of $\sigma$). As we describe below, we cannot obtain satisfactory fits to both the cosmic SFR and the observed type II supernova rates when $\Psi \propto \sigma$ for the normal mode ((0.1 – 100) M$_\odot$) of star formation. In contrast, we find very good fits with an exponentially decreasing rate of the form $\Psi = \nu_1 e^{-t/\tau}$. Similarly, we assume an exponential form for the massive population III mode as well, although here we assume a metallicity-dependent cutoff so that $\Psi = \nu_2 e^{-Z/Z_\mathrm{crit}}$. The timescale $\tau$ is determined largely by the observed cosmic star formation rate (at low redshift), while the intensity $\nu_1$ is constrained by the observed metal abundances in the ISM and supernova rates. $\nu_2$ is taken to be as large as possible in order to achieve a high ionizing flux at high redshift, while avoiding the overproduction of metals which would appear as a prompt initial enrichment of the ISM and the IGM. Finally, we assume a critical metallicity of $10^{-4}$ [@bromm:03; @yoshida:04], though our results do not change significantly if $Z_\mathrm{crit}$ were a factor of 10 higher.
5. The initial mass function of stars $\Phi(m)$. As for the SFR, we must specify an IMF for the normal and massive modes. In each case, we take $\Phi(m) \propto m^{-(1+x)}$ for $m_\mathrm{inf}\le m \le m_\mathrm{sup}$, with $$\int_{m_\mathrm{inf}}^{m_\mathrm{sup}} dm\ m \Phi(m)=1\ .
\label{norm}$$ The normal mode is assumed to contain stars between 0.1 and 100 M$_\odot$. The slope of the IMF is constrained by both the type II supernova rate and the metallicity of the ISM.
6. Specific parameters for type Ia supernovae. Modelling type Ia supernovae requires several additional parameters which include the mass range $[m_\mathrm{SN\ Ia,min};m_\mathrm{SN\ Ia,max}]$ of the progenitors, the fraction $f_\mathrm{SN\ Ia}$ of these progenitors which will produce a type Ia supernova, and time delay $\Delta t_\mathrm{SN\ Ia}$ between the formation of the white dwarf and the explosion (so that for a progenitor of mass $m$, the total delay between the star formation and the type Ia supernova is $\tau(m)+\Delta t_\mathrm{SN\ Ia}$).
The normal mode of star formation {#sec:model0}
=================================
In all of the models considered below, there is a normal mode of star formation comprising of stars with masses between 0.1 and 100 M$_\odot$. Once the normal mode is adjusted, a massive population III mode is added at high redshift for $Z < Z_\mathrm{c}$. Because the massive mode is cut off rather early (its main effect is to supply a prompt initial enrichment of the ISM and IGM as well as provide an ionizing source at high redshift), most observable constraints are tied to the normal mode. Thus, we begin with a more detailed description of the normal mode and its observational consequences.
We listed the main parameters of the model in the previous section. The model consists of a superposition of a normal mode of star formation and an early massive mode at high redshift, when the global metallicity in the star-forming structures is still very low (population III stars). We will describe this mode in section \[sec:popIII\]. We focus here on the normal mode of star formation. We fix the mass range of star formation to $m_\mathrm{inf}=0.1\ \mathrm{M_{\odot}}$ and $m_\mathrm{sup}=100\ \mathrm{M_\odot}$ so that the only parameter needed to define the IMF is its slope, namely $x_{1}$, which is usually estimated to be in the range $1.30$ [@salpeter:55] to $1.7$ [@scalo:86].
As noted above, we employ an exponentially decreasing SFR (which is representative of elliptical galaxies) as parametrizations such as a Schmidt-law yield significantly poorer fits to the observational data. Hence we take, $$\Psi(t) = \nu_{1} \exp{(-(t-t_\mathrm{init})/\tau_{1})}\ ,$$ where $t_\mathrm{init}$ is the initial age of the Universe at $z_\mathrm{init}$ where the star formation starts in the model, $\tau_{1}$ is a timescale of the order of $2-3\ \mathrm{Gyr}$ and $\nu_{1}=f_{1} m_\mathrm{struct}(t) / \tau_{1}$ with $f_{1}$ being a fraction governing the efficiency of the star formation.\
Since many of the observational constraints used to fix the parameters pertain to relatively low redshift, we performed a detailed scan of the parameter space including only the normal mode. The parameter grid was chosen to be: $M_\mathrm{min}=10^{6}$, $10^{7}$ , $10^{8}$, $10^{9}$, and $10^{11}\ \mathrm{M_{\odot}}$; $\epsilon=0$, $10^{-3}$, $2\times 10^{-3}$, $3\times 10^{-3}$, [**$5\times 10^{-3}$**]{}, $7\times 10^{-3}$, $10^{-2}$, $2\times 10^{-2}$ and $3\times 10^{-2}$; $\nu_{1}=0.1$, 0.2, 0.3, 0.4, 0.5, 0.6 and $0.7\ \mathrm{Gyr^{-1}}$; $\tau_{1}=2.2$, 2.4, 2.6, 2.8, 3.0, 3.2 Gyr; $x_1=1.3$, 1.35 and 1.4. In all cases, we have assumed that the onset of star formation begins when the baryon fraction in structures is 1%. We have tested that this initial fraction has a very weak impact on the results concerning the normal mode of star formation. It is on the other hand of great importance for the population III stars and its effect will be studied in section \[sec:popIII\]. One should also note that there are additional parameters associated with the rate of type Ia supernovae. These are directly constrained by observations and will be fixed in section \[sec:SNae\].\
To determine the parameters which best fit the observations, we performed $\chi^2$ analysis over the parameter grid. Included in the $\chi^2$ analysis are six sets of observational data: (1) The observed cosmic star formation rate up to $z\sim 5$ [@hopkins:04]. The data were binned and averaged in redshift leading to somewhat larger observational uncertainties at a given redshift than typically reported for a single measurement; (2) The observed rate of type II supernovae up to $z\sim 0.7$ [@dahlen:04]; (3) The present fraction of baryons in structures, $f_\mathrm{b,struct}(z=0)\simeq 61\pm 18\ \%$ [@fukugita:04]; (4) The present fraction of baryons in stars, $f_\mathrm{b,*}(z=0)\simeq 6\pm 6\ \%$ [@fukugita:04]; (5,6) The evolution of the metal content in the ISM and IGM from $z\sim 5$ to $z=0$. Specifically, for the ISM we took $\log_{10}{Z/Z_\mathrm{\odot}}=-0.3\pm 0.2$ at $z=0$ and $-0.9\pm 0.2$ at $z=4$. For the IGM we took, $\log_{10}{Z/Z_\mathrm{\odot}}=-2.5\pm 0.2$ at $z=0$ and $-3.1\pm 0.2$ at $z=4$ [@prochaska:03; @ledoux:03; @songaila:01; @schaye:03; @aguirre:04]. These constraints are chosen so that the metallicity in the ISM (IGM) is always larger (smaller) than in the most (least) metallic DLA. The dominant effect of including the massive mode will fall on the metallicity of the IGM and will be discussed in section 5 below.
The results of the $\chi^2$ minimization are listed in Table \[tab:model0\] separately for each value of $M_\mathrm{min}$. Having fixed the onset of star formation to correspond to an initial baryon fraction of 1%, each value of $M_\mathrm{min}$ leads to a distinct redshift for initial star formation. This value of $z_\mathrm{init}$ is also given in the table. In each case, we found a best fit IMF slope of $x_1 = 1.3$. It should be remembered that when we add in the massive mode, these ‘best’ fits have to be modified in order to take into account the metals produced by population III stars. This will affect only the efficiency of the outflow, $\epsilon$, as the other parameters control the later phases of evolution. Of the sampled values of $M_\mathrm{min}$, $M_\mathrm{min} = 10^7$ M$_\odot$ was found to give the best fit. However if the observed SFR at high redshift ($z>3$) has been underestimated, the best fit model may be at lower values of $M_\mathrm{min}$. For example, we find that $M_\mathrm{min}=10^{6}\ \mathrm{M_{\odot}}$ becomes a better fit if the high redshift SFR data are increased by a factor $\sim 3$. Also shown in the table are the output values for the baryon fraction in structures and in stars. Note that none of the models accurately reproduce the stellar baryon fraction at $z = 0$. In each case, we over-produce stars at a level of about 1$\sigma$.
[ccccccc]{}\
$M_{\mathrm{min}}$ & $z_{\mathrm{init}}$ & $\epsilon$ & $\nu_{1}$ & $\tau_{1}$ & $f_{\mathrm{b,struct}}$ & $f_{\mathrm{b,*}}$\
($M_{\odot}$) & & & (Gyr$^{-1}$) & (Gyr) &\
$10^{6}$ & 18.2 & $2\times 10^{-3}$ & 0.2 & 2.8 & 63 % & 14 %\
$\mathbf{10^{7}}$ & **16.0** & $\mathbf{3\times 10^{-3}}$ & **0.2** & **2.8** & **61 %** & **13 %**\
$10^{8}$ & 13.7 & $5\times 10^{-3}$ & 0.2 & 2.8 & 58 % & 11 %\
$10^{9}$ & 11.3 & $ 10^{-2}$ & 0.2 & 3.0 & 54 % & 11 %\
$10^{11}$ & 6.57 & $1.5\times 10^{-2}$ & 0.5 & 2.2 & 44 % & 11 %\
The results of our best fit models are plotted in Figure \[fig:model0SFRZ\] for each value $M_\mathrm{min}$ as indicated in the upper panel. Because we have fixed the initial baryon fraction in structures, each value of $M_\mathrm{min}$ corresponds to a different initial redshift. As one can see, each of the models gives a satisfactory fit to the global SFR, save perhaps the case with $M_\mathrm{min} = 10^{11}$ M$_\odot$. The data shown (taken from @hopkins:04 has already been corrected for extinction. If we compare the cosmic SFR calculated by @croton:05a (see also @springel:03), we find good agreement at high redshift, particularly with the $M_\mathrm{min} = 10^{7}$ M$_\odot$ model. However, at $z<5$ their SFR is somewhat lower than ours by a factor of about 2. Moreover, in our model, about 60 per cent of all stars formed by z = 3 compared with 50 per cent at the same redshift in their model. The predicted type II supernova rates are also an excellent fit to the existing data at low redshift. The predicted rates are substantially higher at higher redshift and lower $M_\mathrm{min}$.
As noted above, while the baryon fraction in structures is fit quite well, the baryon fraction in stars at z = 0 (or equivalently $\Omega_b^*$) is somewhat high. This output is clearly linked to the SFR which is in fact well fit and could be even higher than shown here. For example, we show in Figure \[fig:model0HighSFR\] resulting predictions when the astration rate $\nu_1$ has been doubled for our best fit model with $M_\mathrm{min}=10^{7}\ \mathrm{M_{\odot}}$. As one can see, the baryon fraction in stars now rises to nearly 30% and clearly disagrees with observational determinations of this quantity. Finally, coming back to Figure \[fig:model0SFRZ\], we also show the evolution of the metallicity in both the ISM and IGM (dashed curves). While the ISM metallicity rises very quickly initially, evolution is more gradual in the IGM where it is controlled by the outflow. Furthermore, we see that the evolution of the metallicity in the IGM is dependent on $M_\mathrm{min}$. Since models with large $M_\mathrm{min}$ begin rather late (e.g. the model with $M_\mathrm{min} = 10^{11}\ \mathrm{M_{\odot}}$ begins at $z \approx 6.5$) these models have difficulty in producing a suitably large metallicity in the IGM at redshift 2 - 3. Note that we have only included the normal mode here, and contributions from population III stars will only serve to increase the metallicity in both the ISM and IGM.
Supernova rates {#sec:SNae}
===============
Among the important constraints imposed on these models of cosmic chemical evolution are the rates of type Ia and II supernovae. These are intimately linked to the choice of an IMF and SFR and represent an independent probe of the star forming universe at high redshift. While type II rates directly trace the star formation history, there is a model dependent time delay between the formation (and lifetime) of the progenitor star and the type Ia explosion. Since existing data is available only for relatively low redshifts, we discuss the predicted SN rates in the context of Model 0 (the normal mode of star formation). Indeed, at these redshifts the massive mode is not operational.
Type II supernovae
------------------
The predicted rates for type II supernovae were discussed briefly in the previous section, where we showed the results as a function of redshift for several choices of $M_\mathrm{min}$ in Figure \[fig:model0SFRZ\]. In Figure \[fig:model0SN\], we focus on the low redshift range and concentrate on the best fit model with $M_\mathrm{min}=10^{7}\ \mathrm{M_{\odot}}$. The data shown are taken from the Great Observatories Origins Deep Survey (GOODS) [@dahlen:04 and references therein]. Of the 42 supernovae observed, 25 are associated with type Ia and 17 with core collapse supernovae (type II as well as type Ib,c).
The GOODS data for core collapse supernovae have been placed in two bins at $z = 0.3 \pm 0.2$ and $z = 0.7 \pm 0.2$. Their results (which have been corrected for the effects of extinction) show SN rates which are significantly higher than the local rate (at $z = 0$) which is taken from @cappellaro:99. Since the timescale between star formation and the core collapse explosion is very short (particularly for very massive stars), there will be no contribution of Population III stars to type II rates at $z \le 1$.
As remarked above, the SN type II rate is directly related to the overall SFR and therefore to the assumed astration rate $\nu_1$ and our best fit model does an excellent job at describing the existing data. Note that, had we chosen a Schmidt law ($\Psi \propto \sigma_{\mathrm gas})$, it would not be possible to simultaneously fit both the local ($z = 0$) data as well as the GOODS data. In general, models with a Schmidt law can not account for a large enhancement in the SN rate. Therefore once the peak of the SFR has been fixed (at $z \approx 1$), the local SN rate is too large. The slope of the type II rate (with respect to redshift) is also dependent on the IMF slope, $x_1$. In Figure \[fig:SNII\], we show this dependence displaying the predicted type II rates for $x_1 = 1.1, 1.3$ and 1.5. Of course the choice of $\nu_1$ and $x_1$ also has a strong effect on the metallicity and detailed chemical history as discussed in more detail in the next section when Population III stars are included in the analysis.
It is important to note that while the data at $z \le 1$ is well described by the model, the same model predicts much higher SN rates at higher redshift. Indeed, the SN type II rate peaks at a redshift $z \approx 3$ at a rate which is nearly 8 times the observed rate at $z \approx 0.7$. It will be interesting to see whether future data will be able to probe the SN rate at higher redshift. These elevated rates may be detectable in spite of the expected increase in dust extinction due to the early production of metals. An alternative to the direct detection of SN at high redshift is the indirect detection of SN through neutrinos. The predicted neutrino background is very sensitive to the assumed chemical evolution model [@ando:04; @iocco:05; @dosv:05].
Type Ia supernovae
------------------
As in the case for type II supernovae, the data (also taken from GOODS [@dahlen:04]) is binned into four redshift bins at $z = 0.4$, 0.8, 1.2, and 1.6, each with a spread of $\pm 0.2$. These are shown in the lower panel of Figure \[fig:model0SN\] along with data from @reiss:00, @hardin:00, @pain:02, @madgwick:03, @tonry:03, @strolger:04, and @blanc:04. Once again, the local ($z = 0$) data is taken from @cappellaro:99. Unlike the type II data, there appears to be a definite peak in the rate of type Ia supernovae around $z \sim 1$.
As the progenitors of type Ia supernovae are stars with intermediate masses between 2 and 8 M$_\odot$, the rate for SN type Ia is controlled entirely by parameters within the normal mode of star formation. There are however two additional parameters that must be adjusted to fit the observed data, namely the fraction of intermediate mass stars that explode as SN type Ia, and the time delay between the death of the progenitor and the SN explosion. In most previous models of Galactic chemical evolution, this delay was taken to be of order 1 Gyr. Recently, many studies have been devoted to the progenitors of SNIa supernovae, including the analysis of time delays and the mass ranges (@han:04, @yungelson:04, @garnavich:05 and @belczynski:05). The GOODS data are now consistent with significantly larger delays (of order 4 Gyr, including the lifetime of the star) and exclude delays less than 2 Gyr at the 95 % confidence level [@strolger:04; @dahlen:04]. In the context of the normal mode (Model 0), we find that these observations are well reproduced if $\sim 1\ \%$ of intermediate mass stars lead to type Ia supernovae and if the typical delay between the formation of the white dwarf and the explosion is $\sim 3-3.5\ \mathrm{Gyr}$. The resulting comparison between our theoretical prediction and the data is shown in the lower panel of Figure \[fig:model0SN\] where the effect of the time delay is quite apparent and is chiefly responsible for the peak in the SN rate at $z \sim 1$. While this comparison is made using our best fit model with $M_\mathrm{min} = 10^{7}\ \mathrm{M_{\odot}}$, the resulting rate for type Ia supernovae would be quite similar for other choices of $M_\mathrm{min}$. For an alternative approach see @scann:05 [@mannucci:05].
To test the sensitivity of the result to the time delay, we show in Figure \[fig:SNIa\] the predicted SNIa rates using a time delay of 2.0 and 4.0 Gyr, compared with our best fit value of 3.2 Gyr. Also shown in Figure \[fig:SNIa\] (upper panel) is the sensitivity to the adopted intermediate mass range. Our best fit model assumes a range of 2 - 8 M$_\odot$. Also plotted are models where the lower end of the mass range is 1.5 and 3 M$_\odot$.
Finally, in the upper panel of Figure \[fig:model0SN\], we compare our predictions of type Ia to type II supernovae. We see that these are quite consistent with the local rate [@cappellaro:99]. The ratio drops off at redshifts $z \ga 1$ due to the time delay in producing type Ia events. Furthermore, when SN Ia explosions do occur, structures are larger and mass outflows are suppressed (see discussion in the next section). Thus, as a consequence of the SN Ia time delay, we predict that while $\sim 50\ \%$ of iron in structures is produced by type Ia supernovae, this fraction is only $\sim 10\ \%$ in the IGM.
Population III stars {#sec:popIII}
====================
Necessity of an initial starburst of massive stars
--------------------------------------------------
As shown in @daigne:04, the normal mode of star formation, labelled Model 0 there as well as here, is not capable of reionizing the Universe at high redshift. Indeed, at a redshift $z = 17$, only 1.6 ionizing photons per baryon are available compared to the requisite value of approximately 20 assuming a clumpiness factor, $C_\mathrm{H\;\scriptscriptstyle{II}} = 10$ [@ricotti:04b]. It was also shown in @daigne:04 that Model 0 alone is not capable of explaining the observed abundance patterns in the extremely iron-poor stars, CS 22949-037 [@depagne:02; @israelian:04], HE 0107-5240 [@christlieb:04; @bessell:04], HE 1327-2326 (@frebel:05) and G 77-61 (@plez:05).
The necessity of a massive mode of star formation, active at high redshift has become an integral part of our emerging picture of the growth of galactic structures. However, there is an active debate as to the exact nature of the massive mode. As in @daigne:04, we consider three possibilities for the massive mode. 1) stars with masses in the range 40 – 100 M$_\odot$, Model 1; these stars terminate as type II supernovae. 2) stars with masses in the range 140 – 260 M$_\odot$, Model 2a; these stars terminate as pair-instability supernovae (PISN). 3) stars with masses in the range 270 – 500 M$_\odot$, Model 2b; these stars terminate as black holes through total collapse and do not contribute any metal enrichment.
In all cases, we assume a bimodal birthrate function of the form $$B(m,t,Z) = \Phi_1(m) \Psi_1(t) + \Phi_2(m) \Psi_2(Z)$$ where the two IMFs are normalized independently as in eq. (\[norm\]). The SFR, $\Psi_2$, is now expressed as a function of metallicity and cuts off once a critical metallicity is reached, as described below. Furthermore, we assume the same slope of the IMF, $x_1 = x_2 = 1.3$ so that $\Phi_1$ and $\Phi_2$ differ only in their mass range. In principle, we can choose to start the normal mode of star formation either simultaneously with the massive mode, or sequentially, that is, when the $Z > Z_\mathrm{crit}$. In the latter case, one can argue that the initial injection of metals by Pop III stars is responsible for the formation of the first extremely metal poor Pop II (or Pop II.5) stars [@mackey:03; @salvaterra:04; @johnson:05]. Since critical metallicity is achieved very rapidly (within 3 Myr), our results for these two choices are almost indistinguishable.
As the cooling process of the gas depends strongly on its chemical composition, it is believed that the evolution of the mass range of the IMF is mainly governed by the global metallicity [@fang:04]. As noted above, we assume a transition from population III to the normal formation mode at a critical metallicity $Z_\mathrm{crit}$ [@bromm:03; @yoshida:04] by defining the SFR of the massive mode by $$\Psi(t)_2 = \nu_{2} \exp{\left(- Z / Z_\mathrm{crit}\right)}\ ,$$ with $\nu_{2} = f_{2} m_\mathrm{struct}(t)$. We adopt $Z_\mathrm{crit}/Z_\mathrm{\odot}=10^{-4}$ .
Because the massive stars adopted in Models 1 and 2a are efficient in producing heavy elements, the duration of the massive phase in these models is relatively brief. In contrast, no heavy elements are produced in Model 2b, and therefore, the massive mode continues to affect the evolution of structures until the metallicity of the [*normal*]{} mode reaches the critical value, which is also relatively fast, due to the short lifetimes of massive stars.
Most of the constraints discussed above must be applied at relatively low redshift ($z \la 5$). As a result they fix the parameters of Model 0. For the most part these are unchanged, with one important exception. The metallicity of the IGM now receives two distinct contributions. The first from the massive mode, which will appear as a prompt initial enrichment, and the second from the outflows of the normal mode which contribute at lower redshifts. In principle, these contributions can be distinguished with precise abundance data as a function of redshift. If the metallicity in the IGM originates from the massive mode, we expect IGM abundances which are constant with respect to $z$. On the other hand, if there is a significant contribution from the outflows due to the normal mode, we should see abundances which vary with redshift. Unfortunately, the current data does not allow us to distinguish between these two possibilities. Indeed, one would expect that the IGM contains contributions from both modes, but the relative contribution in fact becomes another parameter of our model.
In what follows, we will consider two possibilities for the role of the massive mode in the enrichment of the IGM. In one case, we will assume that nearly all ($\sim 90 \%$) of the IGM metallicity originates from massive stars. Increasing the contribution of the massive mode allows us to maximize its ionizing efficiency. In section 3, we discussed the role of the normal mode. There, we had effectively set $\nu_2 = 0$, and the efficiency of outflow, $\epsilon$, controlled both the baryon fraction in structures and the metallicity of the IGM. When $\nu_2$ is maximized, we will clearly be forced to reduce the value of $\epsilon$ to avoid the overproduction of metals in the IGM. Despite the low value of $\epsilon$, outflow is more efficient at early times and the enrichment of the IGM is also more efficient due to the high level of SNe (large $\nu_2$). Consequently, in this case, most of the IGM metals comes from pop III stars and we expect that $Z_\mathrm{IGM}$ will be constant with $z$. Note that although the value of $Z_\mathrm{crit}$ is very small, ISM metallicities and even IGM metallicities much greater than $Z_\mathrm{crit}$ can be produced by the massive mode. This is because of the finite lifetimes of the massive stars. Although these lifetimes are short, they are nevertheless of order a few Myr and hence metals continue to be injected into the ISM (and IGM) after $Z > Z_\mathrm{crit}$ is reached and the formation of new massive stars is quenched.
We will also consider an intermediate case where the IGM receives equal contributions from the massive and normal modes at $z = 2.5$. In this case, $\nu_2$ is again chosen to maximize the efficiency of reionization, and $\epsilon$ will be closer to the value found in section 3.
In Table 2, we display the four parameters for Models 1, 2a, and 2b, for both sets of assumptions concerning the contribution of Pop III to the IGM. Models 1, 2a, and 2b refer to the case where the IGM receives equal contributions from the normal and massive modes (at $z \approx 2.5$), whereas Models, 1e and 2ae, refer to the extreme case where 90% of the metallicity at $z \simeq 2.5$ is derived from the massive mode. Note that there is no Model 2be, as no metals are ejected into either the ISM or IGM from massive mode stars in this case, and all of the low redshift metallicity is due to the normal mode. As noted above, the outflow efficiencies for Model 1 are within a factor of 2 of those for Model 0 (cf. Table 1). In contrast, the efficiencies for Model 1e are significantly lower. However as one can see in Table 2, the low outflow efficiency is compensated for by a large SFR ($\nu_2$) greatly increasing the models capacity for reionization. The final baryon fraction in structures and stars is very similar in these models to that found in Model 0. In general, $f_{\mathrm{b,struct}}$ is larger than the values in Table 1 by 1 – 2 % for Models 1, 2a, and 2b, and by 2 – 4 % for Models 1e and 2ae. $f_{\mathrm{b,*}}$ is unchanged from Model 0. As we will see, there is also a close correlation between the intensity of the SFR associated with the massive mode, reionization and the metal enrichment provided by these Pop III stars.
[cccc]{}\
Model & $M_{\mathrm{min}}$ & $\epsilon$ & $\nu_{2}$\
& ($M_{\odot}$) & & (Gyr$^{-1}$)\
1 & $10^{6}$ & $1.5 \times 10^{-3}$ & 30\
1e & $10^{6}$ & $5 \times 10^{-5}$ & 260\
1 & $10^{7}$ & $2 \times 10^{-3}$ & 60\
1e & $10^{7}$ & $6 \times 10^{-5}$ & 340\
1 & $10^{8}$ & $2.5\times 10^{-3}$ & 80\
1e & $10^{8}$ & $1.8\times 10^{-4}$ & 290\
1& $10^{9}$ & $ 5 \times 10^{-3}$ & 100\
1 & $10^{11}$ & $7\times 10^{-3}$ & 200\
2a & $10^{7}$ & $1 \times 10^{-3}$ & 9\
2ae & $10^{7}$ & $8 \times 10^{-5}$ & 40\
2b & $10^{7}$ & $3 \times 10^{-3}$ & 100\
Once the ratio of massive/normal mode contribution to the IGM is adopted, we apply the observational constraints to fix the values of $\epsilon$ and $\nu_2$. In addition to the constraint based on the overall metallicity of the IGM, we must require that the massive mode produces a sufficient number of ionizing photons. We will also look at the evolutionary history of several individual element abundances including C, O, Si, and Fe. Finally, we compare the results to the observed abundances of several extremely iron-poor stars CS 22949-037 [@depagne:02; @israelian:04], HE 0107-5240 [@christlieb:04; @bessell:04], HE1327-2326 [@frebel:05], and G 77-61 [@plez:05]. They represent the mean abundance in the Universe at this epoch.
Model 1
-------
We begin by discussing the results of the bimodal model which combines the normal mode (Model 0) with the massive mode denoted as Model 1 which includes stars with masses in the range 40 –100 M $_\odot$. In this model (as well as in Model 2a described below), star formation begins at a very high rate and falls precipitously as metals are injected into the ISM. In inhomogeneous models of structure formation, we would expect this narrow burst to broaden [@scannapieco:03]. As in the case of Model 0, the onset of star formation is determined by $M_\mathrm{min}$, and the initial value for the baryon fraction in structures (fixed to be 1 %). Results for the SFR, baryon fraction and metallicity are shown in Figure \[fig:model1SFRstarsZ\] for the intermediate case where Model 1 stars contribute 50% to the IGM metallicity at z = 2 to 3 and 20% at z = 0. With exception of the metallicity, shown in the third panel, the effect of the Population III stars is minor as the SFR and baryon fraction is very similar here to the case of Model 0 shown in Figure \[fig:model0SFRZ\]. Note that at high redshift, the fraction of the baryons in massive stars is about $10^{-3}$ as seen in the insert to the middle panel of Figure \[fig:model1SFRstarsZ\]. As one can see in the lower panel, the metallicity in the ISM reaches values far in excess of $Z_\mathrm{crit}$ due to the finite lifetime of the massive stars relative to the speed at which the metallicity is attained. Notice also that once the metallicity from Pop III stars is produced, the ISM metallicity later decreases as a result of the accretion of metal-free gas as the structures grow.
The rate of accretion as well as the rate of outflow is shown in Figure \[fig:model1Outflow\]. We see here that outflows from Pop III stars are very efficient at high redshift but their duration is very short due to the rapid increase in metallicity in the ISM. It is important to remember that the mass flux of outflows is very sensitive to $\epsilon$. As a consequence of the luminosity density obtained by the Sloan Digital Sky Survey (SDSS, @blanton:03) and by the 2dF galaxy redshift Survey [@croton:05b] together with the evaluation of the mass rate triggered by galactic winds [@veilleux:05] we estimate that the actual mass flux of outflows is of the order of 0.01 to 0.1 M$_\odot$/yr/Mpc$^3$. At $z = 2$, the results for Model 1 show outflows at the lower end of this range.
Among the chief motivating factors in developing a model of cosmic chemical evolution is the early reionization of the Universe. Several studies suggest that an early burst of star formation as in Model 1, is sufficient [@venkatesan:03a; @wyithe:03; @venkatesan:03b; @tumlinson:04; @daigne:04]. In Figure \[fig:model1Reionization\], we show the number of ionizing photons per baryon produced for each of our choices of $M_\mathrm{min}$. The procedure for computing this stellar ionizing flux is explained in DOVSA. It is important to remember that only a fraction $f_\mathrm{esc}$ of these UV photons will escape the structures and therefore be available to ionize the IGM. The effective value of $f_\mathrm{esc}$ is poorly known but could vary from about 1 to 30 %. The minimum number of photons required for complete reionization (see DOVSA) is also plotted for three possible clumpiness factors. The ionizing potential clearly increases with $M_\mathrm{min}$ and decreasing redshift. The ratio of this minimum number (dashed line) to that for the stellar ionizing photons (solid line) gives the minimum fraction $f_\mathrm{esc}$ necessary to fully reionize the IGM. It appears that all models are able to fully reionize the IGM for a clumpiness factor $C_\mathrm{H\ II} \sim 10$ and $f_\mathrm{esc} \sim 25 \% \to 1 \%$ for $M_\mathrm{min}=10^{6}\to 10^{11}\ \mathrm{M_\odot}$. When the clumpiness factor increases, only models with high $M_\mathrm{min}$ are still able to fully reionize the IGM with $f_\mathrm{esc} < 30 \%$. Typically, if $C_\mathrm{H\ II}\sim 30$ (100), we need $M_\mathrm{min} \ga 10^{7}\ \mathrm{M_{\odot}}$ ($10^{9}\ \mathrm{M_{\odot}}$). There is therefore a tendency to favor a high $M_\mathrm{min}$ and low reionization redshift if the clumpiness factor is high.
In Figure \[fig:ZcRinit\], we show the dependence of these results on the choice of $Z_\mathrm{c}$, the critical metallicity which shuts down massive star formation, and the initial baryon fraction in structures which allows star formation. In all of the models discussed up to now, we have taken $Z_\mathrm{c} = 10^{-4} Z_\odot$ and $f_\mathrm{b,struct}(z_\mathrm{init}) = 1\%$. We see in Figure \[fig:ZcRinit\] that results are almost independent of $Z_\mathrm{c}$. Increasing $Z_\mathrm{c}$ leads to a modest change in the number of ionizing photons per baryon. The reason is that the first generation of stars formed are in fact capable of producing a metallicity much larger than either $10^{-4}$ or $5\times 10^{-3} Z_\odot$, and massive star formation is rapidly terminated once these stars pollute the ISM. On the other hand, the results for the number of ionizing photons per baryon are quite sensitive to the choice of $f_\mathrm{b,struct}(z_\mathrm{init})$ as is the initial redshift for star formation. Our standard choice for $f_\mathrm{b,struct}(z_\mathrm{init})$ of 1% corresponds to our conservative estimate of the minimum baryon fraction where sufficient dissipation occurs to allow star formation. Most galaxy formation simulations adopt a value of 10% (cf. [@abadi:03]).
Decreasing $f_\mathrm{b,struct}(z_\mathrm{init})$ by a factor of 10 to 0.1% allows star formation to begin earlier (in this case at a redshift $z \sim 21$ for $M_\mathrm{min}=10^{7}\ \mathrm{M_\odot}$), when the IGM is still quite dense, and reionization does not occur, even if $C_\mathrm{H\ II} = 10$. If $f_\mathrm{b,struct}(z_\mathrm{init})$ is increased to 5%, star formation occurs later ($z \sim 11$ for $M_\mathrm{min}=10^{7}\ \mathrm{M_{\odot}}$), when the IGM is less dense, and reionization occurs quite easily, even if $C_\mathrm{H\ II} = 100$ (only a fraction $f_\mathrm{esc} \sim 13 \%$ is needed in this case).
Next, we show results for the evolution of C, O, Si, and Fe as a function of redshift in Model 1. These are found in Figures \[fig:model1OC\] and \[fig:model1FeSi\] (the yields vary with metallicity and come from @woosley:95). Overall, one sees that the chemical abundances are well reproduced. In the cases of O and C, we see quite clearly the effects of the hierarchical growth of structure. After the initial burst of star formation by massive stars, the ISM abundances within structures is diluted as metal-free baryons are accreted. As the normal mode of star formation begins to eject metals, these abundances subsequently begin to rise again. Unfortunately, observations at sufficiently high redshift do not exist at present to test this predicted feature. Because of the rapid ejection of metals in Model 1, we are able to achieve the high abundances of C and O seen in the extremely iron-poor stars indicated in Figure \[fig:model1OC\], in contrast to what is possible in Model 0, especially for C (see DOVSA).
Model 1 predictions in the IGM agree with the estimate of the oxygen abundance in the Lyman alpha forest (@telfer:02, @simcoe:04, @aracil:04, @bergeron:05). Note that only OVI is directly observed. Therefore, the estimate of the total O abundance is limited by the uncertainties of the ionization state of this element. We can reasonably consider that \[O/H\] can be as high as -2.5 (P. Petitjean, private communication). OVI was recently used to ascertain the distribution of metals in the IGM [@tumlinsonb:05]. Similarly, the IGM abundances of C agree well with the determination by @aguirre:04 and are (as expected) in excess of the CIV determinations of @songaila:01 [@songaila:05]. In contrast the models currently fit the SiIV abundances and underproduce the total Si abundance based on the adopted yields of type II supernovae in Model 1. Indeed, we see from Fig. \[fig:model1FeSi\] that the Si abundance derived from SiIV observations [@aguirre:04] is clearly underproduced by stars in the mass range defined by Model 1. As we will see below in section 5.5, this translates in to a predicted value for \[Si/C\] which is low compared with the determination of @aguirre:04. Furthermore, we note that as $M_\mathrm{min}$ is increased, it becomes more difficult to reproduce the IGM abundances.
To test the importance of the normal mode of star formation on the evolution of the chemical abundances, we also consider a set of models in which the efficiency of outflow is reduced, thereby increasing the relative contribution of Population III stars to the overall metallicity in the IGM. To compensate for the reduction in outflow, the SFR of the massive mode is increased (cf. Model 1e in Table 2) which enhances the ability of the population to reionize the IGM. In Figure \[fig:extremModel1SFRFracBZ\], we show the SFR, baryon fraction, and metallicity for Model 1e compared with the analogous result for Model 1, restricting attention to $M_\mathrm{min} = 10^7$ M$_\odot$. With the exception of $\epsilon$, other Model 0 parameters are left unchanged. The difference in the SFR between Models 1 and 1e is predominantly in the initial burst of massive star formation. The massive SFR is typically 5 times larger initially, while at lower redshifts the SFR, which is determined by $\nu_1$ is identical. Similarly, the baryon fraction in stars is the same for both Models 1 and 1e, but 1e exhibits a strong peak (close to $f_b^* = 1$), which later relaxes due to hierarchical growth. Unfortunately it is not possible to observationally test for these differences. In contrast, however, the overall metallicity does present us with a real test (requiring suitably accurate data) between Models 1 and 1e. In Model 1, the IGM metallicity is clearly changing with redshift. For example, when $M_\mathrm{min} = 10^7$ M$_\odot$, the metallicity increases by a factor of about 20 from a redshift of 16 to 0 and by a factor of about 2 from a redshift of 5 to 0. In Model 1e, the evolution of $Z$ is very nearly constant in the IGM, with its final value fixed at very high redshift.
One of the key benefits of Model 1e is its ionizing potential relative to Model 1. In Figure \[fig:extremModel1FluxOutflow\], we compare the number of ionizing photons per baryon in Models 1 and 1e. As one can see, whereas Model 1 (with $M_\mathrm{min} = 10^7$ M$_\odot$) provided the minimum number of ionizing photons when $C_\mathrm{H\ II}=30$, Model 1e produces a factor of 5 times more ionizing photons per baryon at high redshift.
As explained earlier, to achieve a high ionization efficiency, we increased the SFR parameter $\nu_2$. However, in order to avoid overproduction of IGM metals formed by Pop III stars, we are required to adjust $\epsilon$ downwards. Consequently, while outflow is very efficient at very high redshift, it is suppressed at later times due to hierarchical growth and is unable to eject metals coming from subsequent structures. The mass flux of outflows in Models 1 and 1e are compared in Figure \[fig:extremModel1FluxOutflow\]. Recall that observations indicate a flux of 0.01 – 0.1 M$_\odot$/yr/Mpc$^3$. We see that the predicted outflow in Model 1 is ten times higher than that in Model 1e. Model 1 seems to be in better agreement with the data than Model 1e, which is probably insufficient to fit the high outflows observed in galaxies at $z = 2$ (see also @bertone:05 [@rupke:05]).
Despite the advantage in ionization potential, it is also difficult to reconcile the chemical history of Model 1e with IGM observations. In Figures \[fig:extremModel1OC\] and \[fig:extremModel1FeSi\], we compare the evolution of C, O, Si, and Fe in Models 1 and 1e. The early burst of star formation in Model 1e produces a prompt initial enrichment of the IGM in C and O, which show very little evolution at lower redshifts. In contrast, the evolution of Si and Fe is particularly poor, as this models fails to reproduce the observed IGM abundances of these elements at low redshift.
Model 2a
--------
Next we consider a massive mode made of stars with masses in the range 140 – 260 M$_\odot$ which explode as pair-instability supernovae. In Figure \[fig:model2SFRZ\], we compare the star formation rate, ionization potential and metallicity in Model 2a with that of Model 1. In Model 2a, the initial rapid burst of star formation leads to large metallicities at high redshift. In the ISM, these are diluted by hierarchical growth. In the IGM, where there is no dilution, the metallicity remains relatively flat. Our results confirm that PISN models are less efficient at reionization than a more standard IMF as in Model 1. Indeed, even for a clumpiness factor, $C_\mathrm{H\ II} \sim 10$, Model 2a would require a photon escape fraction in excess of 50% placing this model in a difficult position to explain the early reionization of the IGM.
We have also considered a more extreme version of Model 2a (2ae) in which approximately 90% of the IGM oxygen abundance at $z = 2.5$ originates in Pop III. As with the relation between Models 1 and 1e, Model 2ae has a star formation rate characterized by $\nu_2$ which is about 5 times larger than Model 2a. This enhancement is seen in the early SFR is shown in Figure \[fig:extremModel2aSFRFracBZ\] where the evolution of the baryon density and overall metallicity are also compared with Model 2a.
In the more extreme case, (Model 2ae), the number of ionizing photons per baryon (shown in Figure \[fig:extremModel2aFluxOutflow\]) is increased to about 100 at $z \sim 16$, making this version of the PISN model acceptable based on reionization. However, as was the case in Model 1e, the mass flux of outflows at low redshift is very small in Model 2ae.
The evolution of C, O, Si, and Fe are shown in Figures \[fig:extremModel2aOC\] and \[fig:extremModel2aFeSi\]. This model is acceptable for all nuclei (see however section 5.5 for a comparison of abundance ratios with the those observed in iron-poor stars), and predicts little evolution in the IGM.
Model 2b
--------
Finally, we turn to the case of Model 2b, where the massive mode is made of stars with masses in the range 270 – 500 M$_\odot$ which collapse directly to black holes without any ejection of heavy elements into either the ISM or the IGM. The star formation rate, ionization potential and metallicity in Model 2b are also shown in Figure \[fig:model2SFRZ\]. As one can see, the SFR and the number of ionizing photons per baryon are quite high relative to Models 1 and 2a. Because these stars do not produce any metals, the chemical evolution of this model is identical to that of Model 0. This is mostly problematic in regard to the very iron-poor stars whose abundances could not be explained in this case.
Abundance Ratios
----------------
Before concluding, we examine and compare a number of abundance ratios in Models 1 and 2a. In particular, we consider the evolution of O/Fe, C/Fe, and Si/C. These ratios are of particular interest when trying to model the very iron-poor stars which exhibit anomalously high ratios of O/Fe, C/Fe and to some extent Si/C. The latter is of potential interest at lower redshifts where some derivations of the IGM abundance of Si/C are relatively high coming from SiIV and CIV observations.
As shown in Figure \[fig:model1OFeCFeSiC\], the extremely iron-poor stars observed over the last several years show values of \[O/Fe\] between 2-4, which is significantly higher than typical Population II values which range between 0.5 and 1. A similar pattern is seen for \[C/Fe\] which is also very large in these stars. Model 1 reproduces these patterns quite well because high C,O/Fe ratios are expected in massive type II supernova explosions. The Si/C is also suppressed, and the fit to HE 0107-5240 is also quite good whereas the model somewhat underpredicts the Si/C ratio for CS 22949-037. This result reinforces the notion that these stars are indeed very old and reveal the composition of the gas at the end of Population III. In contrast, Model 2a and PISN are not capable of achieving the element ratios observed in these stars.
At low redshift, one can attempt to use the IGM abundances of Si/C to try to differentiate between models. From Figure \[fig:model1OFeCFeSiC\], we see that Model 1 predicts a \[Si/C\] ratio of order -0.25 whereas in Model 2 it is closer to 1 at a redshift between 2 and 4. This is due to the large Si yields found in PISN models [@heger:02]. There have been some recent observations of ionized Si and C [@songaila:01; @schaye:03; @aguirre:04] from which a model-dependent ratio was inferred [@hm:96]. Indeed the result is highly dependent on the assumption of the UV background model. The most extreme case is that of a UV background powered solely quasars. In this case, \[Si/C\] = 1.48, far exceeding the model predictions found here. When a quasar plus galaxy model is considered, the ratio drops to 0.77, which is close to the result of Model 2a. When the UV flux is softened at high redshift, the ratio drops to 0.48. This ratio may present a serious challenge to Model 1 which clearly underproduces Si. We note however, that the resulting Si abundance is sensitive to several parameter choices. For example, lowering $m_{inf}$ to 20 M$_\odot$ results in more Si (but aversely affects the C,O/Fe ratios as shown in DOVSA. Increasing the slope of the IMF and more importantly decreasing the massive mode astration factor, $\nu_2$, both lead to enhanced Si production. The latter effect can be seen in Fig. \[fig:extremModel1FeSi\], where we show the effect of increasing $\nu_2$ by a factor of 5 leads to decrease in Si production by a factor of about 30, with little change in C. While it is tempting to argue for a model such as 2a on the basis of \[Si/C\] [@qian:05], given the large uncertainty inherent in the UV model, it is not clear which model (early star formation and chemical evolution or the UV background model) is being tested by these observations. The total IGM element abundances inferred from ionized values remain uncertain. Nevertheless, Model 1 currently predicts a value of \[Si/C\] which is too small and may signal the need for the presence of some component of more massive stars such as in Model 2a.
Conclusions {#sec:conclusions}
===========
We have calculated the cosmic star formation history corresponding to different minimum masses for the initial halo structures, spanning $10^{6}$ to $10^{11}\ \mathrm{M_{\odot}}$. We include realistic gas outflows from the structures, powered by the kinetic energy of supernovae and which take into account the increase of the escape velocity due to the growth of structure. We then deduce the baryon content and the chemical composition of the structures and of the intergalactic medium (IGM). We show that with a minimum halo mass of $10^{7}$–$10^{8}\
\mathrm{M}_{\odot}$ and a moderate outflow efficiency , we are able to reproduce both the fraction of baryons in the structures at the present time and the early chemical enrichment of the IGM. The intensity of the formation rate of “normal” stars is also well constrained by the observations: it has to be dominated by star formation in elliptical galaxies, except perhaps at very low redshift. The fraction of baryons in stars is also predicted as are also the type Ia and II supernova event rates. The comparison with SN observations in the redshift range $z=0-2$ allows us to set strong constraints on the time delay of type Ia supernovae (a total delay of $\sim$4 Gyr is required to fit the data), on the lower end of the mass range of the progenitors (2 - 8 $\mathrm{M}_{\odot}$) and on the fraction of white dwarfs that reproduce the type Ia supernova (about 1 per cent). The type II supernova rate is also well fitted in the same range of redshifts in our models and it is directly correlated to the cosmic SFR. We incorporate an improved treatment of structure formation compared to our previous work [@daigne:04] that leads to new insights into the initial mass function (IMF) of the population III stars at high redshift. We compare three possible mass ranges: 40-100 $\mathrm{M}_{\odot}$ (normal supernovae), 140-260 $\mathrm{M}_\odot$ (pair-instability supernovae) and 270-500 $\mathrm{M_{\odot}}$.
We have demonstrated that the fraction in the initial starburst of zero metallicity stars below 270 $\mathrm{M_{\odot}}$ must be limited in order to avoid premature overenrichment of the IGM. Specifically, we predict that about 10 - 20 % of the metals present in the IGM at $z = 0$ have been produced by population III stars at very high $z$. The remaining 80 - 90 % are ejected later by galaxies forming normal stars, with a maximum efficiency of the outflow occurring at a redshift of about 5. In full agreement with @daigne:04, because of the chemical constraints, including both for the IGM and very metal-poor halo stars, $10^{-3}$ of the baryons must lie in the first massive stars in order to produce enough ionizing photons to allow early reionization of the IGM by $z\sim 15$. The case of the very massive mass range (270-500 $\mathrm{M}_{\odot}$) is highly efficient regarding the ionizing flux but cannot reproduce alone the global chemical evolution. A massive component of stars with masses in the 40 – 100 $\mathrm{M}_{\odot}$ account for the observed ISM and IGM abundances with the possible exception of IGM Si. While more massive stars in the range 140 – 260 $\mathrm{M}_{\odot}$ produce ample amounts of Si, they can not produce the C,O/Fe ratios observed in extremely iron-poor stars, nor are they able to efficiently reionize the IGM at high redshift. If future observations bear out a high Si/C ratio in the IGM, we would be led to consider a hybrid model of massive stars.
In summary, we have demonstrated the sensitivity of chemical evolution to different constraints, including the minihalo mass range, the role of outflows and the redshift of structure formation, correlated with different ranges of massive stars. We have evaluated the rates as a function of epoch of both SNII and SNIa. We conclude that only 10 to 20 % of the metals in the IGM are produced by Population III. Addition of a massive star component to Population III with masses in the conventional stellar mass range provides an effective means of simultaneously accounting for both early reionization and the chemical evolution both of the oldest extreme metal-poor stars and of the IGM.
The authors gratefully thank Yannick Mellier and Patrick Petitjean for frequent discussions and Adrian Jenkins for giving us the code computing the dark matter mass functions. The work of K.A.O., F.D. and E.V. was supported by the Project “INSU - CNRS/USA”, and the work of K.A.O. was also supported partly by DOE grant DE–FG02–94ER–40823.
|
---
author:
- 'M.-C. ARNAUD [^1] [^2] [^3]'
title: Lyapunov exponents of minimizing measures for globally positive diffeomorphisms in all dimensions
---
[*Key words:* ]{} discrete weak KAM theory, symplectic twist maps, Lyapunov exponents, Aubry-Mather theory, minimizing measures, Green bundles.\
[*2010 Mathematics Subject Classification:*]{} 37C40, 37D25, 37H15, 583E30, 58E35, 49M99, 49J52
Introduction {#introduction .unnumbered}
============
At the end of the 19th century, motivated by the restricted 3-body problem, H. Poincare introduced the study of the area preserving diffeomorphisms near an elliptic fixed point.\
Then, in the ’30s, Birkhoff began the study of the exact symplectic twist maps : after a symplectic change of coordinates (action-angle), these maps represent what happens near an elliptic fixed point of a generic area preserving diffeomorphism (see [@Bir1]).\
In the ’80s, S. Aubry & P. Le Daeron and J. Mather proved the existence of invariant minimizing measures for these twist maps (see [@ALD] and [@Mat1]). As proved by P. Le Calvez, these minimizing measures are in general hyperbolic (see [@LC1]). For such minimizing measures, I proved in [@Arn1] that there is a link between the fact that they are hyperbolic and the regularity in some sense of their support and I proved in [@Arn4] that there is a link between the size of the Lyapunov exponents and the mean angle of the Oseledet’s splitting when the minimizing measure is hyperbolic. A fundamental tool to obtain such results is the pair of [*Green bundles*]{}, that are two bundles in lines that are defined along the support of the minimizing measures.
A natural question is then: what happens in higher dimension?\
Let us explain what is a twist map in this setting (see for example [@Gol1] or [@Arn3]).
The $2n$-dimensional annulus is ${\mathbb {A}}_n={\mathbb {T}}^n\times {\mathbb {R}}^n$ endowed with its usual symplectic form $\omega$. More precisely, if $q=(q_1, \dots, q_n) \in{\mathbb {T}}^n$ and $p=(p_1, \dots , p_n)\in{\mathbb {R}}^n$ then $\displaystyle{\omega=dq\wedge dp=\sum_{i=1}^ndq_i\wedge dp_i}$. .\
Let us recall that a diffeomorphism $f$ of ${\mathbb {A}}_n$ is [*symplectic*]{} if it preserves the symplectic form: $f^*\omega=\omega$.\
We denote by $\pi:{\mathbb {A}}_n\rightarrow {\mathbb {T}}^n$ the projection $(q, p)\mapsto q$.\
At every $x=(q,p)\in {\mathbb {A}}_n$, we define the vertical subspace $V(x)=\ker D\pi(x)\subset T_x {\mathbb {A}}_n$ as being the tangent subspace at $x$ to the fiber $\{ q\}\times {\mathbb {R}}^n$.
A [*globally positive*]{} diffeomorphism of ${\mathbb {A}}_n$ is a symplectic $C^1$-diffeomorphism $f:{\mathbb {A}}_n\rightarrow {\mathbb {A}}_n$ that is homotopic to ${\rm Id}_{{\mathbb {A}}_n}$ and that has a lift $F: {\mathbb {R}}^n\times {\mathbb {R}}^n\rightarrow {\mathbb {R}}^n\times {\mathbb {R}}^n$ that admits a $C^2$ generating function $S: {\mathbb {R}}^n\times {\mathbb {R}}^n\rightarrow {\mathbb {R}}$ such that:
1. there exists $\alpha>0$ such that: $\frac{\partial^2 S}{\partial q\partial Q}(q, Q)(v, v)\leq -\alpha\| v\|^2$;
2. $F$ is implicitly given by: $$F(q,p)=(Q, P)\Longleftrightarrow \left\{\begin{matrix}p=-\frac{\partial S}{\partial q}(q, Q)\\ P=\frac{\partial S}{\partial Q}(q, Q)
\end{matrix}\right.$$ where $ \|.\|$ is the usual Euclidean norm in ${\mathbb {R}}^n$.
When we use a symplectic change of basis near a completely elliptic periodic point of a generic symplectic diffeomorphism in any dimension, we obtain a Birkhoff normal form defined on a subset on ${\mathbb {A}}_n$ by $(q, p)\mapsto (q+b.p+o(\| p\|), p+o(\|p\|)$ where the torsion $b$ is a symmetric non-degenerate matrix. When $b$ is positive definite, this normal form is a a globally positive diffeomorphims on some bounded subannulus ${\mathbb {T}}^n\times [a, b]^n$ (see for example [@Mos2] or [@Arn6]).
If $f$, $F$ satisfy the above hypotheses, the restriction to any fiber $\{q\}\times {\mathbb {R}}^n$ of $\pi\circ F$ and $\pi\circ F^{-1}$ are diffeomorphisms. Moreover, for every $k\geq 2$, $q_0, q_k\in{\mathbb {R}}^n$, the function $\displaystyle{\hat{\mathcal {F}}:({\mathbb {R}}^n)^{k-1}\rightarrow {\mathbb {R}}}$ defined by $\displaystyle{\hat{\mathcal {F}}(q_1, \dots, q_{k-1})={\mathcal {F}}(q_0, \dots, q_k)=\sum_{j=1}^kS(q_{j-1}, q_j)}$ has a minimum, and at every critical point for $\hat{\mathcal {F}}$, the following sequence is a piece of orbit for $F$: $$(q_0, -\frac{\partial S}{\partial q}(q_0, q_1)), (q_1, \frac{\partial S}{\partial Q}(q_0, q_1)), (q_2, \frac{\partial S}{\partial Q}(q_1, q_2)), \dots , (q_k, \frac{\partial S}{\partial Q}(q_{k-1}, q_k)).$$
In the 2-dimensional case ($n=1$), J. Mather and Aubry & Le Daeron proved in [@ALD] and [@Mat1] the existence of orbits $(q_i,p_i)_{i\in{\mathbb {Z}}}$ for $F$ that are [*globally minimizing*]{}. This means that for every $\ell\in{\mathbb {Z}}$ and every $k\geq 2$, $(q_{\ell+1}, \dots, q_{\ell+k-1})$ is minimizing the function $\hat{\mathcal {F}}$ defined by: $$\hat{\mathcal {F}}(q_{\ell+1}, \dots, q_{\ell+k-1})=\sum_{i=\ell+1}^{k} S(q_{i-1}, q_i).$$ Then each of these orbits $(q_i, p_i)_{i\in{\mathbb {Z}}}$ is supported in the graph of a Lipschitz map defined on a closed subset of ${\mathbb {T}}$, and there exists a bi-Lipschitz orientation preserving homeomorphisms $h:{\mathbb {T}}\rightarrow{\mathbb {T}}$ such that $(q_i)_{i\in{\mathbb {Z}}}=(h^i(q_0))_{i\in{\mathbb {Z}}}$. Hence each of these orbits has a [*rotation number*]{}. Moreover, for each rotation number $\rho$, there exists a minimizing orbit that has this rotation number and there even exist a [*minimizing measure*]{}, i.e. an invariant measure the support of whose is filled by globally minimizing orbits, such that all the orbits contained in the support have the same rotation number $\rho$. These supports are sometimes called [*Aubry-Mather sets*]{}.\
For the globally positive diffeomorphisms in higher dimension, a discrete weak KAM and an Aubry-Mather theories were developped by E. Garibaldi & P. Thieullen in [@GarThi]. They prove that there exist some globally minimizing orbits and measures (the support of whose is compact and a Lipschitz graph) in ${\mathbb {A}}_n$ for all $n\geq 1$.\
Two Lagrangian subbundles of $T{\mathbb {A}}_n$ can be defined along the support of the minimizing measures of any globally positive diffeomorphism. They are called [*Green bundles*]{}, denoted by $G_-$ and $G_+$[^4] and their existence is proved in [@BiaMac] and [@Arn4]. We will prove that for any ergodic minimizing measure, the almost eveywhere dimension of the intersection of the two Green bundles gives the number of zero Lyapunov exponents of this measure:
\[nbexp\] Let $\mu$ be an ergodic minimizing measure of a globally positive diffeomorphism of ${\mathbb {A}}_n$. Let $p$ be the almost everywhere dimension of the intersection $G_-\cap G_+$ of the two Green bundles. Then $\mu$ has exactly $2p$ zero Lyapunov exponents, $n-p$ positive Lyapunov exponents and $n-p$ negative Lyapunov exponents.
Then we will explain that there is a link between the angle between the two Green bundles and the size of the positive Lyapunov exponents. To do that, let us introduce some notations.
We associate an almost complex structure $J$ and then a Riemannian metric $(.,.)_x$ defined by: $(v,u)_x=\omega (x)(v, Ju)$ to the symplectic form $\omega$ of ${\mathbb {A}}_n$; from now on, we work with this fixed Riemannian metric of ${\mathbb {A}}_n$.\
We choose on $ G_+(x)$ an orthonormal basis and complete it in a symplectic basis whose last vectors are in $V(x)$.\
In these coordinates, $G_+$ is the graph of the zero-matrix and $G_-$ is the graphs of a negative semi-definite symmetric matrix that is denoted by $-\Delta S$.\
In these coordinates, along the support of a minimizing measure, the image $Df.V$ of the vertical (resp. $Df^{-1}V$) is transverse to the vertical and then the graph of a symmetric matrix $S_1$ (resp. $S_{-1}$).\
For a positive semi-definite symmetric matrix $S$ that is not the zero matrix, we decide to denote by $q_+(S)$ its smallest positive eigenvalue.
\[sizeexp\] Let $\mu$ be an ergodic minimizing measure of a globally positive diffeomorphism of ${\mathbb {A}}_n$ that has at least one non-zero Lyapunov exponent. We denote the smallest positive Lyapunov exponent of $\mu$ by $\lambda (\mu)$ and an upper bound for $\| S_1-S_{-1}\|$ above ${\rm supp} \mu$ by $C$. Then we have: $$\lambda (\mu)\geq\frac{1}{2}\int \log\left( 1+\frac{1}{C}q_+(\Delta S (x))\right) d\mu (x).$$
In fact, Garibaldi and Thieullen prove the existence of measures that have a stronger property than being minimizing: they are strongly minimizing[^5]. They prove that the supports of these strongly minimizing measures are Lipschitz graphs $M$ above a compact subset of ${\mathbb {T}}^n$. In general, these graphs are not contained in a smooth graph. But we can define at every point $m\in M$ its [*limit contingent cone*]{} $\widetilde C_mM$ that is an extension of the notion of tangent space to a manifold [^6].\
Let us recall that we defined in [@Arn5] an order $\leq$ between the Lagrangian subspaces of $T_x{\mathbb {A}}_n$ that are transverse to the vertical. If ${\mathcal {L}}_-$, ${\mathcal {L}}_+$ are two such subspaces such that ${\mathcal {L}}_-\leq {\mathcal {L}}_+$, we say that a vector $v\in T_x{\mathbb {A}}_n$ is between ${\mathcal {L}}_-$ and ${\mathcal {L}}_+$ if there exists a third Lagrangian subspace ${\mathcal {L}}$ such that $v\in{\mathcal {L}}$ and ${\mathcal {L}}_-\leq {\mathcal {L}}\leq {\mathcal {L}}_+$.\
We will prove that the limit contingent cone to the support of every strongly minimizing measure is between some [*modified Green bundles*]{} $\widetilde G_-$ and $\widetilde G_+$[^7].
\[Tcone\] Let $\mu$ be a strongly minimizing measure of a globally positive diffeomorphism of ${\mathbb {A}}_n$ et let ${\rm supp}\mu$ be its support. Then $$\forall x\in {\rm supp} \mu, \widetilde G_-(x)\leq \widetilde C_x({\rm supp}\mu)\leq \widetilde G_+(x).$$
Hence, the more irregular ${\rm supp}\mu$ is, i.e. the bigger the limit contingent cone is, the more distant $\widetilde G_-$ and $\widetilde G_+$ (and thus $G_-$ and $G_+$ too) are from each other and the larger the positive Lyapunov exponents are.
We define too a notion of $C^1$-isotropic graph (see subsection \[sscone\]) that generalized the notion of $C^1$ isotropic manifold (for the symplectic form). Then we deduce from theorem \[Tcone\]:
\[Cisotropic\] Let $\mu$ be an ergodic strongly minimizing measure of a globally positive diffeomorphism of ${\mathbb {A}}_n$ all exponents of whose are zero. Then ${\rm supp}\mu$ is $C^1$-isotropic almost everywhere.
Some related results {#some-related-results .unnumbered}
--------------------
Theorem \[nbexp\] is an extension of a result that we proved for the autonomous Tonelli Hamiltonians in [@Arn4]. The ideas of the proof are more or less the same ones as in [@Arn4], but some adaptions are needed because we cannot use any continuous dependence in time.\
A particular case of theorem \[sizeexp\] was proved in [@Arn4]: the weak hyperbolic case, where the two Green bundles are almost everywhere transverse. Here we fill the gap by using the reduced Green bundles.\
The inequality given in theorem \[Tcone\] is completely new, even if an analogue to corollary \[Cisotropic\] was given in [@Arn2] for Tonelli Hamiltonians.\
1\) A discrete weak KAM theory is given in [@Gom1] too by D. Gomes, but the condition used by the author there is the convexity of a Lagrangian function that is not the generating function, and this condition is different from the one we use. But Garibaldi & Thieullen results can be used.
2\) There exists too an Aubry-Mather theory for time-one maps of time-dependent Tonelli Hamiltonians (see for example [@Ber1]). Even when the manifold $M$ is ${\mathbb {T}}^n$, the time-one map is not necessarily a globally positive diffeomorphism of ${\mathbb {A}}_n$. Moreover, except for the 2-dimensional annulus (see [@Mos1]), it is unknown if a globally positive diffeomorphism is always the time-one map of a time-dependent Tonelli Hamiltonian (see theorem 41.1 in [@Gol1] for some partial results). In this article, we won’t speak about these time-one maps and will focus on the globally positive diffeomorphisms.
Structure of the article {#structure-of-the-article .unnumbered}
------------------------
In section \[sGreen\], after explaining the construction of the classical Green bundles and the restricted Green bundles, we will prove theorem \[nbexp\].\
We will then explain in section \[sAngles\] that the mean angle between the two Green bundles gives the size of the smallest positive Lyapunov exponent.\
Section \[sKAM\] is devoted to some reminders in discrete weak KAM theory and to the proofs of theorem \[Tcone\] and corollary \[Cisotropic\].\
There are two parts in the appendix. The first one is used in subsection \[fred\] and the second one is used in subsection \[sscone\].
Green bundles {#sGreen}
=============
Classical Green bundles
-----------------------
We recall some classical results that are in [@Arn4]. For the definition of the order between Lagrangian subspaces that are transverse to the vertical, see subsection \[sscomp\] of the appendix. Let $f:{\mathbb {A}}_n\rightarrow {\mathbb {A}}_n$ be a globally positive diffeomorphism.
If $k\in{\mathbb {Z}}$ and $x\in{\mathbb {A}}_n$, we denote by $G_k(x)$ the Lagrangian subspace $G_k(x)=(Df^k).V(f^{-k}x)$.
Let $x\in{\mathbb {A}}_n$ be a point the orbit of whose is minimizing. Then the sequence $(G_k(x))_{k\geq 1}$ is a strictly decreasing sequence of Lagrangian subspaces of $T_x({\mathbb {A}}_n)$ that are transverse to $V(x)$ and $(G_{-k}(x))_{k\geq 1}$ is an increasing sequence of Lagrangian subspaces of $T_x({\mathbb {A}}_n)$ that are transverse to $V(x)$. The two Green bundles are $x$ are the Lagrangian subspaces $$G_-(x)=\lim_{k\rightarrow +\infty} G_{-k}(x)\quad{\rm and}\quad G_+(x)=\lim_{k\rightarrow +\infty}G_k(x).$$
It is proved in [@Arn4] that the two Green bundles are transverse to the vertical and verify: $$\forall k\geq 1, G_{-k}<G_{-(k+1)}<G_-\leq G_+<G_{k+1}<G_k.$$ In general these two bundles are not continuous, but they depend in a measurable way to $x$. Moreover, they are semicontinuous is some sense. Let us recall some properties that are proved in [@Arn4].
Assume that the orbit of $x$ is minimizing. Then
1. $G_-$ and $G_+$ are invariant by the linearized dynamics, i.e. $Df.G_\pm=G_\pm\circ f$;
2. for every compact $K$ such that the orbit of every point of $K$ is minimizing, the two Green bundles restricted to $K$ are uniformly far from the vertical;
3. (dynamical criterion) if the orbit of $x$ is minimizing and relatively compact in ${\mathbb {A}}_n$, if $\displaystyle{\liminf_{k\rightarrow+\infty} \| D(\pi\circ f^k)(x)v\|\leq +\infty}$ then $v\in G_-(x)$,\
if $\displaystyle{\liminf_{k\rightarrow+\infty} \| D(\pi\circ f^{-k})(x)v\|\leq +\infty}$ then $v\in G_+(x)$.
An easy consequence of the dynamical criterion and the fact that the Green bundles are Lagrangian is that when there is a splitting of $T_x(T^*M)$ into the sum of a stable, a center and a unstable bundle $T_x(T^*M)=E^s(x)\oplus E^c(x)\oplus E^u(x)$, for example an Oseledets splitting, then we have $$E^s\subset G_-\subset E^s\oplus E^c\quad{\rm and} \quad E^u\subset G_+\subset E^u\oplus E^c.$$ Let us give the argument of the proof. Because of the dynamical criterion, we have $E^s\subset G_-$. Because the dynamical system is symplectic, the symplectic orthogonal subspace to $E^s$ is $(E^s)^\bot=E^s\oplus E^c$ (see e.g. [@BocVia1]). Because $G_-$ is Lagrangian, we have $G_-^\bot=G_-$. We obtain then $G_-^\bot=G_-\subset E^{s\bot}=E^s\oplus E^c$.\
Let us note the following straightforward consequence: for a minimizing measure, the whole information concerning the positive (resp. negative) Lyapunov exponents is contained in the restricted linearized dynamics $Df_{|G_+}$ (resp. $Df_{|G_-}$).
From $E^s\subset G_-\subset E^s\oplus E^c$ and $E^u\subset G_+\subset E^u\oplus E^c$, we deduce that $G_-\cap G_+\subset E^c$. Hence $G_-\cap G_+$ is an isotropic subspace (for $\omega$) of the symplectic space $E^c$. We deduce that $\dim (E^c)\geq 2\dim (G_-\cap G_+)$. When $E^s\oplus E^c\oplus E^u$ designates the Oseledet splitting of some minimizing measure, what is proved in [@Arn2] is that this inequality is an equality for the Tonelli Hamiltonian flows and we will prove here the same result for the globally positive diffeomorphisms of ${\mathbb {A}}_n$.
Reduced Green bundles {#fred}
---------------------
The reduced Green bundles were introduced in [@Arn2] for the Tonelli Hamiltonian flows. We will give a similar construction.
We assume that $\mu$ is a minimizing ergodic measure and that $p\in [0, n]$ is so that at $\mu$-almost every point $x$, the intersection of the Green bundles $G_+(x)$ and $G_-(x)$ is $p$-dimensional. We deduce from the above comments that for $\mu$ almost every $x\in {\mathbb {A}}_n$: $G_+(x)\cap G_-(x)\subset E^c(x)$ and $E^s(x)\oplus E^u(x)= \left( E^c(x)\right)^\bot\subset G_+(x)^\bot+ G_-(x)^\bot=G_-(x)+ G_+(x)$.
We introduce the two notations: $E(x)=G_-(x)+G_+(x)$ and $R(x)= G_-(x)\cap G_+(x)$. We denote the reduced space: $F(x)=E(x)/R(x)$ by $F(x)$ and we denote the canonical projection $p~: E\rightarrow F$ by $p$. As $G_-$ and $G_+$ are invariant by the linearized dynamics $Df$, we may define a reduced cocycle $M ~: F\rightarrow F$. But $M $ is not continuous, because $G_-$ and $G_+$ don’t vary continuously.\
Moreover, we introduce the notation: ${\mathcal {V}}(x)=V(x)\cap E(x)$ is the trace of the linearized vertical on $E(x)$ and $v(x)=p({\mathcal {V}}(x))$ is the projection of ${\mathcal {V}}(x)$ on $F(x)$. We introduce a notation for the images of the reduced vertical $v(x)$ by $M^k$: $g_k(x)=M^kv(f^{-k}x)$.\
Of course, we define an order on the set of the Lagrangian subspaces of $F(x)$ that are transverse to $v(x)$ exactly as this was done in the non-reduced case.
The subspace $E(x)$ of $T_x{\mathbb {A}}_n$ is co-isotropic with $E(x)^{\bot}=R(x)$. Hence $F(x)$ is nothing else than the symplectic space that is obtained by symplectic reduction of $E(x)$. We denote its symplectic form by $\Omega$. Then we have: $\forall (v, w)\in E(x)^2, \Omega (p(v), p(w))=\omega (v,w)$. Moreover, $M$ is a symplectic cocycle.\
We can notice, too, that $\dim E(x)=\dim (G_-(x)+G_+(x))=\dim G_-(x)+\dim G_+(x)-\dim (G_-(x)\cap G_+(x))=2n-p$ and deduce that $\dim F(x)=\dim E(x)-\dim (G_-(x)\cap G_+(x))=2(n-p)$.
If $L$ is any Lagrangian subspace of $T_x{\mathbb {A}}_n$, we denote $(L\cap E(x))+R(x)$ by $\tilde{L}$ and $p(\tilde{L})$ by $l$.
\[bernard1\] If $L\subset T_x{\mathbb {A}}_n$ is Lagrangian, then $\tilde{L}$ is also Lagrangian and $l=p(\tilde L)=p(L\cap E(x))$ is a Lagrangian subspace of $F(x)$. Moreover, $p^{-1}(l)=\tilde{L}$ . In particular, $v(x)$ is a Lagrangian subspace of $F(x)$ and $p^{-1}(v(x))={\mathcal {V}}(x)+R(x)$.
The proof is given in [@Arn2].
\[L2\] The subspace $v(x)$ is a Lagrangian subspace of $F(x)$. Moreover, for every $k\not=0$, $g_k(f^kx)=M^kv(x)$ is transverse to $v(f^k(x))$
The first result is contained in lemma \[bernard1\].\
Let us consider $k\not=0$ and let us assume that $ M^kv(x) \cap v(f^kx)\not=\{ 0\}$. We may assume that $k>0$ (or we replace $x$ by $f^k(x)$ and $k$ by $-k$).
Then there exists $v\in {\mathcal {V}}(x)\backslash \{ 0\}$ such that $Df^k(x)v\in {\mathcal {V}}(f^k x)+(G_-(f^kx)\cap G_+(f^kx))$. Let us write $Df^k(x)v=w+g$ with $w\in {\mathcal {V}}(f^kx)$ and $g\in R(f^kx)$. We know that the orbit has no conjugate vectors (because the measure is minimizing); hence $g\not=0$.
Moreover, we know that $Df^k V(x)$ is strictly above $G_-(f^kx)$, i.e. that: $$\forall h\in G_-(f^kx), \forall h'\in V(f^kx), h+h'\in (Df^k V(x))\backslash \{ 0\}\Rightarrow \omega (h, h+h')> 0.$$ We deduce that: $\omega (g, w+g)>0$.
This contradicts: $Df^kv\in E(f^kx)=\left(G_+(f^kx)\cap G_-(f^kx)\right)^\bot\subset ({\mathbb {R}}g)^\bot$.
\[bernard2\] Let $L_1$, $L_2$ be two Lagrangian subspaces of $T_x{\mathbb {A}}_n$ transverse to $V(x)$ such that at least one of them is contained in $E(x)$. Then, if $L_1<L_2$ (resp. $L_1\leq L_2$), we have: $l_1$ and $l_2$ are transverse to $v(x)$ and $l_1 < l_2$ (resp. $l_1\leq l_2$). We deduce that $p(G_-)<p(G_+)$.
The proof is given in [@Arn2].
\[linegred\] If $\mu$ is a minimizing measure, for every $x\in{\rm supp}\mu$, for all $0<k<m$, we have: $$g_{-k}(x)<g_{-m}(x)<p(G_-)<p(G_+)<g_m(x)<g_k(x).$$
We cannot use the proof given in [@Arn2] that use in a crucial way the continuous dependence on time. Let us prove by iteration on $k\geq 1$ that $p(G_+)<g_{k+1}<g_k$, i.e. that $g_{k+1}\in{\mathcal {P}}(p(G_+), g_{k})$ with the notations of the appendix.\
Because $G_+<G_1$ and because of lemma \[bernard2\], we have $p(G_+)<g_1$ i.e. $g_1\in{\mathcal {P}}(p(G_+),v)$. Taking the image by $M$, we deduce: $g_2\in {\mathcal {P}}(p(G_+), g_1)$. We deduce from proposition \[proappbis\] (see the appendix) that $p(G_+)<g_2<g_1$. The result for $g_k$ with $k\geq 1$ is just an iteration, and the result for $k\leq -1$ is very similar.
We have: $\displaystyle{\lim_{k\rightarrow +\infty}g_k=p(G_+)\quad{\rm and} \quad \lim_{k\rightarrow +\infty}g_{-k}=p(G_-)}$
From lemma \[linegred\], we deduce that the $(g_k)_{k\geq 1}$ converges to $g_+\geq p(G_+)$ and that $(g_{-k})_{k\geq 1}$ converges to $g_-\leq p(G_-)$.\
Let us assume for example that $g_+\not=p(G_+)$. Then $W=p^{-1}(g_+)$ is transverse to $V$ and invariant by $Df$.\
Moreover, for every $w\in W$ and $v\in G_+$, we have: $\omega(w, v)= \Omega(p(w), p(v))$. We deduce that $G_+\leq W$. We choose a Lagrangian subspace $L$ of $T_x({\mathbb {A}}_n)$ such that $W<L$ and $G_1<L$.\
Because $G_1<L$, we have $L\in {\mathcal {P}}(G_1, V)$ and then for every $k\geq 1$: $Df^k(L\circ f^{-k} )\in {\mathcal {P}}(G_{k+1}, G_k)$, hence, by proposition \[proappbis\], $G_{k+1}<Df^kL<G_k$. Note that this implies that $G_+<Df^kL$.\
Because $G_+\leq W<L$, we have $W\in \overline{ {\mathcal {P}}(G_+, L)}$ by proposition \[proappter\] and then $W=Df^kW\circ f^{-k}\in \overline{{\mathcal {P}}(G_+, Df^kL\circ f^{-k})}$, and then $G_+\leq W\leq Df^kL\circ f^{-k}$ by proposition \[proappbis\].\
We have finally proved $$\forall k\geq 1, G_+\leq W\leq Df^kL\circ f^{-k}<G_k.$$ Taking the limit, we obtain: $W=G_+$.
The two Lagrangian subbundles $g_-=p(G_-)$ and $g_+=p(G_+)$ are the two [*reduced Green bundles*]{}.
Weak hyperbolicity of the reduced cocycle {#sshyp}
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With the notations of subsection \[fred\], we will now explain why the reduced cocycle is weakly hyperbolic and why $\mu$ has exactly $2p$ zero Lyapunov exponents. The proof is very similar to the one given in [@Arn2] for the Tonelli Hamiltonian flows, we just translate it to the discrete case.
We choose at every point $x\in{\rm supp}\mu$ some (linear) symplectic coordinates $(Q,P)$ of $F(x)$ such that $v(x)$ has for equation: $Q=0$ and $g_+(x)$ has for equation $P=0$. We will be more precise on this choice later. Then the matrix of $M^k(x)=M(f^{k-1}(x))\dots M(x)$ in these coordinates is a symplectic matrix: $M^k(x)=\begin{pmatrix}
a_k(x)&b_k(x)\\
0&d_k(x)\\
\end{pmatrix}$. As $M^k(x)v(x)=g_k(f^kx)$ is a Lagrangian subspace of $E(f^kx)$ that is transverse to the vertical, then $\det b_k(x)\not=0$ and there exists a symmetric matrix $s_k^+(f^kx)$ whose graph is $g_k(f^kx)$, i.e: $d_k(x)=s_k^+(f^k(x))b_k(x)$. Moreover, the family $(s^+_k(x))_{k>0}$ being decreasing and tending to zero (because by hypothesis the horizontal is $g_+$), the symmetric matrix $s^+_k(f^kx)$ is positive definite. Moreover, the matrix $M^k(x)$ being symplectic, we have: $$\left(M^k(x)\right)^{-1}=\begin{pmatrix} {}^td_k(x)&-{}^tb_k(x)\\
0&{}^ta_k(x)\\
\end{pmatrix}$$ and by definition of $g_{-k}(x)$, if it is the graph of the matrix $s^-_k(x)$ (that is negative definite), then: ${}^ta_k(x)=-s_k^-(x){}^tb_k(x)$ and finally: $$M^k(x)=\begin{pmatrix}
-b_k(x)s_k^-(x)& b_k(x)\\
0& s_k^+(f^kx)b_k(x)\\
\end{pmatrix}$$ Let us be now more precise in the way we choose our coordinates; as explained at the end of the introduction, we may associate an almost complex structure $J$ and then a Riemannian metric $(.,.)_x$ defined by: $(v,u)_x=\omega (x)(v, Ju)$ with the symplectic form $\omega$ of ${\mathbb {A}}_n$; from now on, we work with this fixed Riemannian metric of ${\mathbb {A}}_n$. We choose on $ G_+(x)=p^{-1}(g_+(x))$ an orthonormal basis whose last vectors are in $R(x)$ and complete it in a symplectic basis whose last vectors are in $V(x)$. We denote the associated coordinates of $T_x{\mathbb {A}}_n$ by $(q_1, \dots, q_n, p_1, \dots, p_n)$. These (linear) coordinates don’t depend in a continuous way on the point $x$ (because $G_+$ doesn’t), but in a bounded way. Then $G_-(x)=p^{-1}(g_-(x))$ is the graph of a symmetric matrix whose kernel is $R(x)$ and then on $G_-(x),$ we have: $p_{n-p+1}=\dots =p_n=0$. An element of $E(x)$ has coordinates such that $p_{n-p+1}=\dots =p_n=0$, and an element of $F(x)=E(x)/R(x)$ may be identified with an element with coordinates $(q_1, \dots , q_{n-p}, 0, \dots , 0, p_1, \dots , p_{n-p}, 0, \dots , 0)$. We then use on $F(x)$ the norm $\displaystyle{\sum_{i=1}^{n-p}(q_i^2+p_i^2)}$, which is the norm for the Riemannian metric of the considered element of $F(x)$. Then this norm depends in a measurable way on $x$.
\[LJ\] For every $\varepsilon >0$, there exists a measurable subset $J_\varepsilon$ of ${\rm supp}\mu$ such that:
1. $\mu (J_\varepsilon)\geq 1-\varepsilon$;
2. on $J_\varepsilon$, $(s_k^+)$ and $(s_k^-)$ converge uniformly ;
3. there exists two constants $\beta=\beta(\varepsilon)>\alpha=\alpha(\varepsilon)>0$ such that: $\forall x\in J_\varepsilon,\beta{\bf 1}\geq -s_-(x)\geq \alpha {\bf 1}$ where $g_-$ is the graph of $s_-$.
This is a consequence of Egorov theorem and of the fact that $\mu$-almost everywhere on ${\rm supp}\mu$, $g_+$ and $g_-$ are transverse and then $-s_-$ is positive definite. We deduce:
\[LCVU\] Let $J_\varepsilon$ be as in the previous lemma. On the set $\{ (k,x)\in{\mathbb {N}}\times J_\varepsilon, f^k(x)\in J_\varepsilon\}$, the sequence of conorms $(m(b_k(x)))$ converge uniformly to $+\infty$, where $m(b_k)=\| b_k^{-1}\| ^{-1}$.
Let $k, x$ be as in the lemma.\
The matrix $M_k(x)=\begin{pmatrix}
-b_k(x)s_k^-(x)& b_k(x)\\
0& s_k^+(f^kx)b_k(x)\\
\end{pmatrix}
$ being symplectic, we have:\
$-s_k^-(x){}^tb_k(x)s_k^+(f^kx)b_k(x)={\bf 1}$ and thus $-b_k(x)s_k^-(x){}^tb_k(x)s_k^+(f^kx)={\bf 1}$ and:\
$b_k(x)s_k^-(x){}^tb_k(x)=-\left(s_k^+(f^kx)\right)^{-1}$.\
We know that on $J_\varepsilon$, $(s_k^+)$ converges uniformly to zero. Hence, for every $\delta>0$, there exists $N=N(\delta) $ such that: $k\geq N\Rightarrow \| s_k^+(f^kx)\|\leq \delta$. Moreover, we know that $\| s_k^-(x)\|\leq \beta$. Hence, if we choose $\delta'=\frac{\delta^2}{\beta}$, for every $k\geq N=N(\delta')$ and $x\in J_\varepsilon$ such that $f^kx\in J_\varepsilon$, we obtain: $$\forall v\in{\mathbb {R}}^p,\beta \| {}^tb_k(x)v\|^2= {}^tv b_k(x)(\beta{\bf 1}){}^tb_k(x)v\geq - {}^tv b_k(x)s_k^-(x){}^tb_k(x)v={}^tv\left(s_k^+(f^kx)\right)^{-1}v$$ and we have: ${}^tv\left(s_k^+(f^kx)\right)^{-1}v\geq \frac{\beta}{\delta^2}\| v\|^2$ because $s_k^+(f^kx)$ is a positive definite matrix that is less than $\frac{\delta^2}{\beta}{\bf 1}$. We finally obtain: $\| {}^tb_k(x)v\|\geq \frac{1}{\delta}\| v\|$ and then the result that we wanted. From now we fix a small constant $\varepsilon>0$, associate a set $J_\varepsilon$ with $\varepsilon$ via lemma \[LJ\] and two constants $0<\alpha<\beta$; then there exists $N\geq 0$ such that $$\forall x\in J_\varepsilon, \forall k\geq N, f^k(x)\in J_\varepsilon\Rightarrow m(b_k(x))\geq \frac{2}{\alpha}.$$
Let $J_\varepsilon$ be as in lemma \[LJ\]. For $\mu$-almost point $x$ in $J_\varepsilon$, there exists a sequence of integers $(j_k)=(j_k(x))$ tending to $+\infty$ such that: $$\forall k\in {\mathbb {N}}, m(b_{j_k}(x)s^-_{j_k}(x))\geq \left( 2^\frac{1-\varepsilon}{2N}\right)^{j_k}.$$
As $\mu$ is ergodic for $f$, we deduce from Birkhoff ergodic theorem that for almost every point $x\in J_\varepsilon$, we have: $$\lim_{\ell\rightarrow +\infty}\frac{1}{\ell}\sharp \{ 0 \leq k\leq \ell-1; f^k(x)\in J_\varepsilon\}=\mu (J_\varepsilon)\geq 1-\varepsilon.$$ We introduce the notation: $N(\ell)=\sharp \{ 0 \leq k\leq \ell-1; f^k(x)\in J_\varepsilon\}$.\
For such an $x$ and every $\ell\in{\mathbb {N}}$, we find a number $n(\ell)$ of integers: $$0=k_1\leq k_1+N\leq k_2\leq k_2+N\leq k_3\leq k_3+N\leq \dots \leq k_{n(\ell)}\leq \ell$$ such that $f^{k_i}(x)\in J_\varepsilon$ and $n(\ell)\geq [\frac{N(\ell)}{N}]\geq \frac{N(\ell)}{N}-1$. In particular, we have: $\frac{n(\ell)}{\ell}\geq\frac{1}{N}(\frac{N(\ell)}{\ell}-\frac{N}{\ell})$, the right term converging to $\frac{\mu (J_\varepsilon)}{N}\geq \frac{1-\varepsilon}{N}$ when $\ell$ tends to $+\infty$. Hence, for $\ell$ large enough, we find: $n(\ell)\geq 1+ \ell \frac{1-\varepsilon}{2N}$.\
As $f^{k_i}(x)\in J_\varepsilon$ and $k_{i+1}-k_i\geq N$, we have: $m(b_{k_{i+1}-k_i}(f^{k_i}(x)))\geq \frac{2}{\alpha}$. Moreover, we have: $m(s_{k_{i+1}-k_i}^-(f^{k_i}x))\geq \alpha$; hence: $$m(b_{k_{i+1}-k_i}(f^{k_i}x)s_{k_{i+1}-k_i}^-(f^{k_i}x))\geq 2.$$ But the matrix $-b_{k_{n(\ell)}}(x)s^-_{k(n(\ell))}(x)$ is the product of $n(\ell)-1$ such matrix. Hence: $$m(b_{k_{n(\ell)}}(x)s^-_{k(n(\ell))}(x))\geq 2^{n(\ell)-1}\geq 2^{\ell\frac{1-\varepsilon}{2N}}\geq \left( 2^\frac{1-\varepsilon}{2N}\right)^{k_{n(\ell)}}.$$
Let us now come back to the whole tangent space $T_x{\mathbb {A}}_n$ with a slight change in the coordinates that we use. We defined the symplectic coordinates $(q_1, \dots , q_n, p_1, \dots , q_n)$ and now we use the non symplectic ones:\
$(Q_1, \dots, Q_n,P_1, \dots , P_n)=(q_{n-p+1}, \dots, q_n, q_1, \dots, q_{n-p}, p_1, \dots , p_n)$. Then:
1. $(Q_1, \dots , Q_p)$ are coordinates in $R(x)$;
2. $(Q_1, \dots , Q_n)$ are coordinates in $G_+(x)$;
3. $(Q_1, \dots , Q_n, P_{1}, \dots , P_{n-p})$ are coordinates of $E(x)=G_+(x)+G_-(x)$.
We write then the matrix of $Df^k(x)$ in these coordinates $(Q_1, \dots , Q_n, P_1, \dots , P_n)$ (which are not symplectic): $$\begin{pmatrix}
A^1_k(x)&A^2_k(x)&A^3_k(x)&A^4_k(x)\\
0&b_k(x)s_k^-(x)&b_k(x)&A^5_k(x)\\
0&0& s_k^+(f^kx)b_k(x)&A^6_k(x)\\
0&0&0&A^7_k(x)\\
\end{pmatrix}$$ where the blocks correspond to the decomposition $T_x{\mathbb {A}}_n=E_1(x)\oplus E_2(x)\oplus E_3(x)\oplus E_4(x)$ with $\dim E_1(x)=\dim E_4(x)=p$ and $\dim E_2(x)=\dim E_3(x)=n-p$.\
We have noticed that $E_1(x)=E(x)\subset E^c(x)$ and that $G_+(x)=E_1(x)\oplus E_2(x)$.\
If $x\in J_\varepsilon$, we have found a sequence $(j_k)$ of integers tending to $+\infty$ so that: $$\forall k\in {\mathbb {N}}, m(b_{j_k}(x)s^-_{j_k}(x))\geq \left( 2^\frac{1-\varepsilon}{2N}\right)^{j_k}.$$ We deduce: $$\forall v\in E_2(x)\backslash \{ 0\}, \frac{1}{j_k}\log\left( \| b_{j_k}(x)s^-_{j_k}(x)v\|\right)\geq \frac{1-\varepsilon}{2N}\log 2 +\frac{\|v\|}{j_k};$$ and because $E_1(x)\subset E^c(x)$: $$\forall v\in G_+(x)\backslash E_1(x), \liminf_{k\rightarrow +\infty}\frac{1}{k}\log \| Df^k(x)v\|\geq \frac{1-\varepsilon}{2N}\log 2.$$ Hence there are at least $n-p$ Lyapunov exponents bigger than $ \frac{1-\varepsilon}{2N}\log 2$ and then bigger than $0$ for the linearized dynamics. Because this dynamics is symplectic, we deduce that it has at least $n-p$ negative Lyapunov exponents (see [@BocVia1] ). As we noticed that the linearized flow has at least $2p$ zero Lyapunov exponents, we deduce that $\mu$ has exactly $n-p$ positive Lyapunov exponents, exactly $n-p$ negative Lyapunov exponents and exactly $2p$ zero Lyapunov exponents.\
This finishes the proof of theorem \[nbexp\].
Let us notice that we proved too that for $\mu$ almost every $x\in {\rm supp} \mu$, we have: $E^u(x)\subset G_+(x)$, and then $G_+(x)=E^u(x)\oplus R(x) $.
Size of the Lyapunov exponents and angle between the two Green bundles {#sAngles}
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The idea to prove theorem \[sizeexp\] is to use the reduced Green bundles that we introduced just before and to adapt the proof that we gave in [@Arn4] in the case of weak hyperbolicity. We use the same notations as in section \[sGreen\].\
The Lagrangian bundles $g_-$ and $g_+$ being transverse to the vertical at every point of ${\rm supp}\mu$, there exist two symmetric matrices ${\mathbb {S}}$ and ${\mathbb {U}}$ such that $g_-$ (resp. $g_+$) is the graph of ${\mathbb {S}}$ (resp. ${\mathbb {U}}$) in the coordinates$(q_1, \dots , q_{n-p}, p_1, \dots , p_{n-p})$ that we defined at the beginning of subsection \[sshyp\]. We denote by $(e_1, \dots , e_{2(n-p)})$ the associated symplectic basis. As $g_-$ and $g_+$ are transverse $\mu$-almost everywhere, we know that there exists $\varepsilon>0$ such that $A_\varepsilon=\{ x\in {\rm supp}\mu; {\mathbb {U}}-{\mathbb {S}}\geq \varepsilon {\bf 1}\}$ has positive $\mu$-measure. We use the notation $x_k=f^k(x)$.We may then assume that $x_0\in A_\varepsilon$ and that $\{ k\geq 0; {\mathbb {U}}(x_k)-{\mathbb {S}}(x_k)>\varepsilon{\bf 1}\}$ is infinite. Let us notice that in this case, $g_-$ and $g_+$ are transverse along the whole orbit of $x_0$ (but ${\mathbb {U}}-{\mathbb {S}}$ can be very small at some points of this orbit). Let us note too that in fact ${\mathbb {U}}=0$.\
Hence, for every $k\in{\mathbb {N}}$, there exists a unique positive definite matrix $S_0(x_k)$ such that: $S_0(x_k)^2={\mathbb {U}}(x_k)-{\mathbb {S}}(x_k)$. Let us recall that a matrix $M=\begin{pmatrix} a & b\\ c&d\\ \end{pmatrix}$ of dimension $2(n-p)$ is symplectic if and only if its entries satisfy the following equalities: $${}^tac={}^tca;\quad {}^tbd={}^tdb;\quad {}^tda-{}^tbc={\bf 1}.$$ We define along the orbit of $x_0$ the following change of basis: $P=\begin{pmatrix} S_0^{-1}& S_0^{-1}\\ {\mathbb {S}}S_0^{-1}& {\mathbb {U}}S_0^{-1}\\ \end{pmatrix}.$ Then it defines a symplectic change of coordinates, whose inverse is: $$Q=P^{-1}=\begin{pmatrix} 0&{\bf 1}\\ -{\bf 1}& 0\\ \end{pmatrix} {}^tP\begin{pmatrix} 0&-{\bf 1}\\ {\bf 1}& 0\\ \end{pmatrix}=\begin{pmatrix} S_0^{-1}{\mathbb {U}}& -S_0^{-1}\\ -S_0^{-1}{\mathbb {S}}& S_0^{-1}\\ \end{pmatrix}.$$ We use this symplectic change of coordinates along the whole orbit of $x_0$. More precisely, if we denote the matrix of $M^k$ in the usual canonical basis $e=(e_i)$ by $M_k$, then the matrix of $M^k$ in the basis $Pe=(Pe_i)$ is denoted by $\tilde M_k$; we have then: $\tilde M_k(x_h)= P^{-1}(x_{h+k})M_k(x_h)P(x_h)$. Note that the image of the horizontal (resp. vertical) Lagrangian plane by $P$ is $g_-$ (resp. $g_+$). As the bundles $g_-$ and $g_+$ are invariant by $M$, we deduce that $\tilde M_k=\begin{pmatrix} \tilde a_k&0 \\
0 &\tilde d_k\\ \end{pmatrix}$; we have ${}^t\tilde a_k \tilde d_k={\bf 1}$ because this matrix is symplectic.\
Moreover, we know that: $M_k(x_h)=\begin{pmatrix} -b_k(x_h)s_{-k}(x_h)& b_k(x_h)\\
c_k(x_h)& s_k(x_{k+h})b_k(x_h)\\ \end{pmatrix}$ where $g_k(x_h)=M^k.v(x_{h-k})$ is the graph of $s_k(x_h)$.\
Writing that $\tilde M_k(x_h)=\begin{pmatrix} \tilde a_k(x_h)&0 \\
0 &\tilde d_k(x_h)\\ \end{pmatrix}= P^{-1}(x_{h+k})M_k(x_h)P(x_h)$, we obtain firstly: $$\begin{matrix} S_0(x_{h+k})^{-1}&{}^tb_k(x_h)S_0(x_h)^{-1}=\hfill\\ &S_0(x_{h+k})^{-1}({\mathbb {S}}(x_{h+k})-s_{k}(x_{h+k}))b_k(x_h)(s_{-k}(x_h)-{\mathbb {S}}(x_h))S_0(x_h)^{-1};
\end{matrix}$$ $$\begin{matrix}-S_0(x_{h+k})^{-1}&{}^tb_k(x_h)S_0(x_h)^{-1}\hfill\\
&=S_0(x_{h+k})^{-1}({\mathbb {U}}(x_{h+k})-s_{k}(x_{h+k}))b_k(x_h)({\mathbb {U}}(x_h)-s_{-k}(x_h))S_0(x_h)^{-1}.
\end{matrix}$$ We deduce that: $\tilde a_k(x_h)=S_0(x_{h+k})b_k(x_h)({\mathbb {S}}(x_h)-s_{-k}(x_h))S_0(x_h)^{-1}$ and:\
$\tilde d_k(x_h)=S_0(x_{h+k})b_k(x_h)({\mathbb {U}}(x_h)-s_{-k}(x_h))S_0(x_h)^{-1}$.
Because of the changes of basis that we used, $(\tilde a_k(x_h))_k$ represents the linearized dynamics $(M^k_{|g_-(x_h)})_k$ restricted to $g_-$ and $(\tilde d_k(x_h))_k$ the linearized dynamics restricted to $g_+$. Hence we need to study $(\tilde d_k(x_h))$ to obtain some information about the positive Lyapunov exponents of $\mu$. Let us compute:\
${}^t\tilde d_k (x_h)=\tilde a_k(x_h)^{-1}= S_0(x_h)({\mathbb {S}}(x_h)-s_{-k}(x_h))^{-1}b_k(x_h)^{-1}S_0(x_{h+k})^{-1}$; we deduce:\
$${}^t\tilde d_k(x_h)\tilde d_k(x_h)=S_0(x_h)({\mathbb {S}}(x_h)-s_{-k}(x_h))^{-1}({\mathbb {U}}(x_h)-s_{-k}(x_h))S_0(x_h)^{-1}$$ $$=S_0(x_h)({\mathbb {S}}(x_h)-s_{-k}(x_h))^{-1}({\mathbb {U}}(x_h)-{\mathbb {S}}(x_h)+{\mathbb {S}}(x_h)-s_{-k}(x_h))S_0(x_h)^{-1}$$ $$={\bf 1}+ S_0(x_h)({\mathbb {S}}(x_h)-s_{-k}(x_h))^{-1} S_0(x_h)$$ $$={\bf 1}+({\mathbb {U}}(x_h)-{\mathbb {S}}(x_h))^\frac{1}{2}({\mathbb {S}}(x_h)-s_{-k}(x_h))^{-1} ({\mathbb {U}}(x_h)-{\mathbb {S}}(x_h))^\frac{1}{2}.$$ Let us denote the conorm of $a$ (for the usual Euclidean norm of ${\mathbb {R}}^{n-p}$) by: $m(a)=\| a^{-1}\|^{-1}$. Then we have: $$m(\tilde d_k(x_h))^2=m({}^t\tilde d_k(x_h)\tilde d_k(x_h));$$ Let us recall that on ${\rm supp}\mu$, $G_+$ is uniformly far from the vertical. This implies that $S_1-S_{-1}$ is uniformly bounded on the (compact) support of $\mu$ (see the notations before theorem \[sizeexp\] for the definition of $S_1$ and $S_{-1}$). Then their restriction to $q_{n-p+1}=\dots =q_n=0$ is uniformly bounded too (by the same constant); let $C$ designate $\sup \| s_1-s_{-1}\|$ above the support of $\mu$. We have then: $m(\tilde d_k(x_h))^2\geq 1+\frac{1}{C}m(({\mathbb {U}}-{\mathbb {S}})(x_h))$; indeed, we know that: $s_1-s_{-1}\geq {\mathbb {S}}-s_{-k} >0$.
The entry $\tilde d_k$ being multiplicative, we deduce that: $$m(\tilde d_k(x_0))^2\geq \prod_{n=0}^{k-1} (1+\frac{1}{C}m({\mathbb {U}}(x_h)-{\mathbb {S}}(x_h)))$$ and: $$\frac{1}{k}\log m(\tilde d_k(x_0))\geq \frac{1}{2k}\sum_{n=0}^{k-1}\log (1+\frac{1}{C}m({\mathbb {U}}(x_h)-{\mathbb {S}}(x_h))).$$ When $k$ tends to $+\infty$, we deduce from Birkhoff’s ergodic theorem that: $$(*)\quad \liminf_{k\rightarrow \infty}\frac{1}{k}\log m(\tilde d_k(x_0))\geq\frac{1}{2}\int \log\left( 1+\frac{1}{C}m({\mathbb {U}}(x)-{\mathbb {S}}(x))\right) d\mu (x).$$ Let us recall that $(\tilde d_k(x_0))$ represents the dynamics along $g_+$, but the change of basis that we have done is not necessarily bounded. To obtain a true information about the Lyapunov positive exponents of $(M^k)$ , we need to have a result for the matrix $D_k$ of $(M^k_{|g_+(x_0)})$ in the basis $(e_1, \dots , e_{n-p})$ of $g_+$ whose matrix in the usual coordinates is: $\begin{pmatrix} {\bf 1}\\ {\mathbb {U}}\\ \end{pmatrix}=\begin{pmatrix} {\bf 1}\\ {\bf 0} \\ \end{pmatrix}$. Since $(\tilde d_k)$ is the matrix of $M^k$ in the basis whose matrix is $\begin{pmatrix} S_0^{-1}\\ {\mathbb {U}}S_0^{-1} \\ \end{pmatrix}=\begin{pmatrix} S_0^{-1}\\ {\bf 0} \\ \end{pmatrix}$, we deduce that: $D_k(x_0)=S_0(x_k)\tilde d_k(x_0)S_0(x_0)^{-1}$ and:\
$m(D_k(x_0))\geq m(S_0(x_k)) m(\tilde d_k(x_0)) m(S_0(x_0)^{-1})=\left( m({\mathbb {U}}(x_k)-{\mathbb {S}}(x_k))\right)^\frac{1}{2} m(\tilde d_k(x_0)) m(S(x_0)^{-1})$.\
We have $(*)$ and we know that: $\displaystyle{\liminf_{k\rightarrow \infty} m({\mathbb {U}}(x_k)-{\mathbb {S}}(x_k))\geq \varepsilon}$. We deduce: $$\lambda (\mu)\geq \liminf_{k\rightarrow \infty}\frac{1}{k}\log m(D_k(x_0))\geq\frac{1}{2}\int \log\left( 1+\frac{1}{C}m({\mathbb {U}}(x)-{\mathbb {S}}(x))\right) d\mu (x).$$
Because $\Delta S$ is a symmetric positive semi-definite matrix, we have: $q_+(\Delta S)=q_+(\Delta S_{|(\ker \Delta S)^\bot})$. If we look at the definition of the coordinates $(q_i, p_i)$, we note that: $\Delta S_{|(\ker \Delta S)^\bot}=-{\mathbb {S}}={\mathbb {U}}-{\mathbb {S}}$. Hence we have proved theorem \[sizeexp\].
Shape of the support of the minimizing measures and Lyapunov exponents {#sKAM}
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Some reminders about discrete weak KAM theory {#ssweakkam}
---------------------------------------------
The general reference for what is contained in this section is the article of Garibaldi & Thieullen [@GarThi] and the results that they obtain are very similar to the ones obtained by A. Fathi in the setting of the time-continuous weak K.A.M. theory (see [@Fat1]). The dynamics that we study here are contained in the ones that they study and that are called “ferromagnetic”. In [@GarThi], a big part of the article deals with a Lagrangian function $L:{\mathbb {R}}^n\times {\mathbb {R}}^n\rightarrow {\mathbb {R}}$ that is defined by $L(x, v)=S(x, x+v)$ (let us recall that $S$ is a generating function for $F$) and the action ${\mathcal {F}}$ is denoted by ${\mathcal {L}}$ by them. They prove the existence of a unique $\overline {\mathcal {L}}\in {\mathbb {R}}$ such that there exists two ${\mathbb {Z}}^n$-periodic continuous functions $u_-, u_+:{\mathbb {R}}^n\rightarrow {\mathbb {R}}$ such that: $$\forall x\in{\mathbb {R}}^n, u_-(y)=\inf_{x\in{\mathbb {R}}^n} u_-(x)+S(x,y)-\overline{\mathcal {L}}\quad{\rm and}\quad u_+(x)=\sup_{y\in{\mathbb {R}}^n} u_+(y)-S(x,y)+\overline{\mathcal {L}}$$ and that the infimum (resp. supremum) is attained at some point.
[*(Garibaldi-Thieullen)*]{}\
With the above notations and assumptions: $$\bar {\mathcal {L}}=\inf _\mu \int_{{\mathbb {R}}^n\times{\mathbb {R}}^n} S(x,y)d\tilde\mu(x,y);$$ where the infimum is taken on the set of the Borel probability measures that are invariant by $f$ and $\tilde \mu$ is any lift of $\mu$ to a fundamental domain of ${\mathbb {R}}^n\times {\mathbb {R}}^n$ for the projection $(x,y)\mapsto (x, -\frac{\partial S}{\partial x}(x,y))$ onto ${\mathbb {T}}^n\times {\mathbb {R}}^n$. Moreover the infimum is attained for some invariant $\mu$.
Then such a measure $\mu$ where the infimum is attained is a minimizing measure, but all the minimizing measures are not like that. We define:
A configuration $(x_k)_{k\in{\mathbb {Z}}}$ of points of ${\mathbb {R}}^n$ is [*strongly minimizing*]{} if for any pairs $m<\ell$ et $m'<\ell'$ and any configuration $(y_k)_{k\in{\mathbb {Z}}}$ satisfying $y_{m'}-x_m\in{\mathbb {Z}}^n$ and $y_{\ell'}-x_\ell\in{\mathbb {Z}}^n$, we have: $$\bar {\mathcal {F}}(x_m, x_{m+1}, \dots, x_\ell)\leq \bar {\mathcal {F}}(y_{m'}, \dots, y_{\ell'}).$$ The corresponding orbit for $f$ is then [*strongly minimizing*]{}.
It is not hard to see that if $\mu$ is a Borel probability measure invariant by $f$ then it satisfies the equality in the proposition above if and only if its support is filled by strongly minimizing orbits.
The union of the supports of all the measures $\tilde\mu$ where $\mu$ is strongly minimizing is called the [*Mather set*]{} and is denoted by ${\mathcal {M}}(S)$.
We introduce the notations $\bar S(x,y)=S(x,y)-\bar{\mathcal {L}}$ and $$\bar{\mathcal {F}}(x_1, \dots x_m)={\mathcal {F}}(x_1, \dots, x_m)-(m-1)\bar{\mathcal {L}}=\sum_{i=1}^{m-1}\bar S(x_i, x_{i+1}).$$
From now on, we will call $\bar{\mathcal {F}}$ the action and we will consider minimizing orbits for this action (in fact minimizing orbits are the same for the two actions).
Let $u:{\mathbb {R}}^n\rightarrow {\mathbb {R}}$ be a ${\mathbb {Z}}^n$-periodic and continuous function. Then
1. $u$ is a [*subaction*]{} with respect to $S$ if: $$\forall x, y\in{\mathbb {R}}^n, u(y)-u(x)\leq \bar S(x,y);$$
2. $u$ is [*backward calibrated*]{} if it is a subaction and $$\forall y\in{\mathbb {R}}^n, u(y)=\inf_{x\in{\mathbb {R}}^n}(u(x)+\bar S(x, y));$$
3. $u$ is [*forward calibrated*]{} if it is a subaction and $$\forall x\in{\mathbb {R}}^n, u(x)=\sup_{y\in{\mathbb {R}}^n}(u(y)-\bar S(x, y)).$$
Let $K\geq 0$ be a constant. A function $u:{\mathbb {R}}^n\rightarrow {\mathbb {R}}$ is $K$-[*semiconcave*]{} if for every $x_0\in{\mathbb {R}}^N$, there exists $p_{x_0}\in{\mathbb {R}}^n$ (that is non-necessarily unique) such that: $$\forall x\in {\mathbb {R}}^n, \| x-x_0\|\leq 1\Rightarrow u(x)\leq u(x_0)+p_{x_0}(x-x_0)+K\| x-x_0\|^2.$$ Then $p_{x_0}$ is a [*superdifferential*]{} for $u$ at $x_0$. The function $u$ is $K$-[*semiconvex*]{} if $-u$ is semiconcave.
Let us recall some well-known properties of semiconcave functions (see for example [@CanSin]); we assume that $u$ is $K$-semiconcave.
1. if $x_0$ is a local minimizer for $u$, then $u$ is differentiable at $x_0$;
2. a infimum of $K$-semiconcave functions is $K$-semiconcave;
3. every semiconcave function is Lipschitz.
A consequence of these properties is that any backward calibrated subaction is semiconcave and any forward calibrated subaction is semiconvex.
If $u:{\mathbb {R}}^n\times {\mathbb {R}}^n\rightarrow {\mathbb {R}}$ is a subaction, then: $${\mathcal {N}}(u)=\{ (x,y)\in{\mathbb {R}}^n\times{\mathbb {R}}^n; u(y)=u(x)+\bar S(x,y)=u(x)+S(x,y)-\bar{\mathcal {L}}\}.$$
Note that for every $(x, y)\in {\mathcal {N}}(u)$, then $u$ is differentiable at $x$ and $du(x)=-\frac{\partial S}{\partial x}(x,y)$. Indeed, the map $ (z\mapsto u(z)+\bar S(z, y))$ is semiconcave and $x$ is a minimizer. Hence $u$ is differentiable a $x$ and $du(x)+\frac{\partial S}{\partial x}(x,y)=0$.
Let us give a result that is very similar to a one given in [@Ber1] in the time-continuous case.
If $u_-:{\mathbb {R}}^n\rightarrow {\mathbb {R}}$ is a backward calibrated subaction, then for every $y\in {\mathbb {R}}^n$, we denote by $\Sigma (y)$ the set of the $x\in {\mathbb {R}}^n$ where: $$u_-(y)=u_-(x)+\bar S(x,y).$$
\[Pbackward\] Let $u_-:{\mathbb {R}}^n\rightarrow {\mathbb {R}}$ be a backward calibrated subaction. Then, if $y\in{\mathbb {R}}^n$ and $x\in \Sigma (y)$, $\frac{\partial S}{\partial y}(x,y)$ is a superdifferential for $u_-$ at $y$.\
Moreover, $u_-$ is differentiable at $y$ if and only if $\Sigma(y)=\{ x\}$ has exactly one element. Then in this case $du_-(x)=-\frac{\partial S}{\partial x}(x, y)$ and $du_-(y)=\frac{\partial S}{\partial y}(x,y)$.
There is of course a similar statement for the forward calibrated subactions.
Assume that $x\in\Sigma (y)$. Then if $z\in {\mathbb {R}}^n$ satisfies $\| z-x\| \leq 1$, we have: $$\begin{matrix}u_-(z)&\leq u_-(x)+S(x,z) \leq u_-(x)+S(x, y)+(S(x,z)-S(x,y))\hfill\\
&\leq
u_-(y)+S(x,z)-S(x,y)\leq u_-(y)+\frac{\partial S}{\partial y}(x,y)(z-y)+K\| z-y\|^2.\end{matrix}$$ Hence $\frac{\partial S}{\partial y}(x,y)$ is a superderivative for $u_-$ at $y$.\
Assume that $\Sigma (y)$ has at least two elements $x_1$ and $x_2$. Then $\frac{\partial S}{\partial y}(x_1,y)$ and $\frac{\partial S}{\partial y}(x_2,y)$ are two superderivatives for $u_-$ at $y$. Because of the twist condition and because $x_1\not=x_2$, we have $\frac{\partial S}{\partial y}(x_1,y)\not= \frac{\partial S}{\partial y}(x_2,y)$. The function $u_-$ has then two superderivatives at $y$ and then is not differentiable at $y$.\
Assume now hat $\Sigma (y)=\{ x\}$ has exactly one element. Let $y'$ be close to $y$. Then every element $x'$ of $\Sigma (y')$ is close to $x$ and we have: $$\begin{matrix} u_-(y')&=u(x')+S(x', y')=u(x')+S(x', y)+(S(x', y')-S(x', y))\hfill\\&\geq u_-(y)+\frac{\partial S}{\partial y}(x', y)(y'-y)+o(\| y'-y\|)\hfill\\
&\geq u_-(y)+\frac{\partial S}{\partial y}(x, y)(y'-y)+o(\| y'-y\|).\hfill\end{matrix}.$$ This proves that $u$ is differentiable at $y$ and that $du(y)= \frac{\partial S}{\partial y}(x,y)$. The fact that $du(x) =-\frac{\partial S}{\partial x}(x,y)$ is a consequence of the remarks that we made previously.
We deduce from proposition \[Pbackward\] that if a backward calibrated subaction $u_-:{\mathbb {R}}^n\rightarrow {\mathbb {R}}$ is differentiable at $x_0$, if we use the notation $\Sigma (x_i)=\{ x_{i+1}\}$, then $u$ is differentiable at every $x_i$ and $du(x_i)=-\frac{\partial S}{\partial x}(x_i,x_{i-1})=\frac{\partial S}{\partial y}(x_{i+1}, x_{i})$, i.e. $(x_i, du(x_i))_{i\in{\mathbb {N}}}=(x_i, \frac{\partial S}{\partial y}(x_{i+1},x_i))_{i\in{\mathbb {N}}}$ is a backward orbit for $F$. Moreover, the configuration $(x_i)_{i\geq 0}$ is strongly minimizing.
[(Garibaldi-Thieullen)]{} For any subaction $u$, we have: $\emptyset\not={\mathcal {M}}(S)\subset {\mathcal {N}}(u)$.
Moreover they prove:
[(Garibaldi-Thieullen)]{} For any backward calibrated subaction $u_-$, there exists a forward calibrated subaction $u_+$ such that:
1. $u_-\leq u_+$;
2. $u_{-|{\mathcal {M}}(S)}=u_{+|{\mathcal {M}}(S)}.$
Such a pair $(u_-, u_+)$ will be called a pair of [*conjugate calibrated subactions*]{} and we introduce the notation
If $(u_-, u_+)$ is a pair of conjugate calibrated subactions, we denote by ${\mathcal {I}}(u_-, u_+)$ the set: $${\mathcal {I}}(u_-, u_+)=\{ x\in{\mathbb {R}}^n; u_-(x)=u_+(x)\}.$$
Note that ${\mathcal {M}}(S)\subset {\mathcal {I}}(u_-, u_+)$. Note too that $u_-$ and $u_+$ are differentiable above ${\mathcal {I}}(u_-, u_+)$ with the same derivative. We use the following notation $$\widetilde {\mathcal {I}}(u_-, u_+)=\{ (x, du_-(x)); x\in {\mathcal {I}}(u_-, u_+)\}.$$
Mañé potential and images of the vertical fiber
-----------------------------------------------
In the discrete case, an action potential can be defined that is an analogue of the one given by R. Mañé in [@Man1]:
Let $m\geq 1$ be an integer. The [*action potential*]{} ${\mathcal {A}}_m: {\mathbb {R}}^n\times{\mathbb {R}}^n\rightarrow {\mathbb {R}}$ is defined by: $$\forall x, y\in{\mathbb {R}}^n, {\mathcal {A}}_m(x, y)=\inf\{\sum_{i=1}^m\bar S(x_{i-1}, x_i); x_0=x, x_m=y\}.$$
Let us give a result that is very similar to a statement given by P. Bernard in [@Ber1]. We use a notation:
If $m\geq 1$ is an integer and $x, y\in{\mathbb {R}}^n$, then $\Sigma_m(x, y)\subset ({\mathbb {R}}^n)^{m+1}$ is the set of the $(x_0, x_1, \dots, x_m)$ such that $x_0=x$, $x_m=y$ and $${\mathcal {A}}_m(x, y)=\sum_{i=1}^m\bar S(x_{i-1}, x_i).$$
\[Ppotential\] Let $m\geq 1$ be an integer. Then ${\mathcal {A}}_m$ is semiconcave. Let $x, y\in{\mathbb {R}}^n$ be two points. Then $\Sigma(x,y)\not=\emptyset$ and if $(x_0, \dots, x_m)\in \Sigma_m(x,y)$, it is the projection of a unique orbit for $F$ that is: $$(x_0, -\frac{\partial S}{\partial x}(x_0, x_1)), (x_1, -\frac{\partial S}{\partial x}(x_1, x_2)= \frac{\partial S}{\partial y}(x_0, x_1)), \dots, (x_m, \frac{\partial S}{\partial y}(x_{m-1}, x_m));$$ and $( \frac{\partial S}{\partial x}(x_0, x_1), \frac{\partial S}{\partial y}(x_{m-1}, x_m))$ is a superdifferential for ${\mathcal {A}}_m$ at $(x,y)$. Moreover,the following assertions are equivalent:
1. ${\mathcal {A}}_m$ is differentiable with respect to $x$ at $(x, y)$;
2. ${\mathcal {A}}_m$ is differentiable with respect to $y$ at $(x, y)$;
3. $\Sigma(y)=\{ (x_0, \dots , x_m)\}$ has exactly one element.
The function ${\mathcal {A}}_m$ is the infimum of a uniformly semiconcave, bounded from below and coercive familiy. Hence it is semiconcave and the infimum is attained.\
If $(x_0, \dots, x_m)\in \Sigma_m(x, y)$, we have an infimum and then the partial derivatives vanish and: $$\frac{\partial S}{\partial y}(x_0, x_1)+\frac{\partial S}{\partial x}(x_1, x_2)=0, \dots , \frac{\partial S}{\partial y}(x_{m-2}, x_{m-1})+\frac{\partial S}{\partial x}(x_{m-1}, x_m)=0.$$ This implies that $(x_0, \dots, x_m)$ is the projection of a unique orbit, that is: $$(x_0, -\frac{\partial S}{\partial x}(x_0, x_1)), (x_1, -\frac{\partial S}{\partial x}(x_1, x_2)= \frac{\partial S}{\partial y}(x_0, x_1)), \dots, (x_m, \frac{\partial S}{\partial y}(x_{m-1}, x_m)).$$ Moreover, $$\begin{matrix}{\mathcal {A}}_m(x',y')&\leq \bar S(x', x_1)+\dots +\bar S(x_{m-1}, y')\hfill\\
&\leq{\mathcal {A}}_m(x, y)+(\bar S(x', x_1)-\bar S(x,x_1))+(\bar S(x_{m-1}, y')-\bar S(x_{m-1}, x))\hfill\\
&\leq {\mathcal {A}}_m(x, y)+\frac{\partial S}{\partial x}(x, x_1)(x'-x)+\frac{\partial S}{\partial y}(x_{m-1}, y)(y'-y)\\
& \hfill +K(\| x-x'\|^2+\| y-y'\|^2)
\end{matrix}$$ hence $( \frac{\partial S}{\partial x}(x_0, x_1), \frac{\partial S}{\partial y}(x_{m-1}, x_m))$ is a superdifferential for ${\mathcal {A}}_m$ at $(x,y)$.\
Let us now assume that $\Sigma_m(x,y)$ contains at least two distinct elements $(x_0, \dots, x_m)$ and $(y_0, \dots , y_m)$. We know that they are the projections of two distinct orbits, one joining $(x, -\frac{\partial S}{\partial x}(x_0, x_1))$ to $(y, \frac{\partial S}{\partial y}(x_{m-1}, x_m))$ and the other one joining $(x, -\frac{\partial S}{\partial x}(y_0, y_1))$ to $(y, \frac{\partial S}{\partial y}(y_{m-1}, y_m))$. Because the orbits are distinct, the points are not the same and then $\frac{\partial S}{\partial x}(x_0, x_1)\not= \frac{\partial S}{\partial x}(y_0, y_1)$ and $\frac{\partial S}{\partial y}(x_{m-1}, x_m)\not= \frac{\partial S}{\partial y}(y_{m-1}, y_m)$. Hence ${\mathcal {A}}_m$ has two distinct superderivatives with respect to $x$ and two distinct superderivatives with respect to $y$ at $(x,y)$.\
Let us assume that $\Sigma_m(x,y)$ contains exactly one element $(x_0, \dots, x_m)$. Let $(x', y')$ be close to $(x,y)$. Then every element $(x'_0, \dots, x'_m)$ of $\Sigma_m(x',y')$ is close to $(x_0, \dots, x_m)$ and we have: $$\begin{matrix} {\mathcal {A}}_m(x', y')&=\displaystyle{\sum_{i=1}^m}\bar S (x'_{i-1}, x'_i) =\bar S(x, x'_1)+\displaystyle{\sum_{i=2}^{m-1}}\bar S (x'_{i-1}, x'_i)+\hfill\\ & +\bar S(x'_{m-1}, y)+ \bar S(x', x'_1)-\bar S(x, x'_1)+\bar S(x'_{m-1}, y')-\bar S(x'_{m-1}, y)\hfill\\
&\geq {\mathcal {A}}_m(x, y)+\bar S(x', x'_1)-\bar S(x , x'_1)+\bar S(x'_{m-1}, y')-\bar S(x'_{m-1}, y)\hfill\\
&\geq {\mathcal {A}}_m(x, y)+\frac{\partial\bar S}{\partial x}(x, x'_1)(x'-x)+\frac{\partial\bar S}{\partial y}(x'_{m-1}, y)(y'-y)+o(\| x-x'\|)+o(\| y'-y\|)\hfill\\
&\geq {\mathcal {A}}_m(x, y)+\frac{\partial\bar S}{\partial x}(x, x_1)(x'-x)+\frac{\partial\bar S}{\partial y}(x_{m-1}, y)(y'-y)+o(\| x-x'\|)+o(\| y'-y\|).\hfill\end{matrix}.$$ This proves that ${\mathcal {A}}_m$ is differentiable at $(x,y)$.
At every $x\in{\mathbb {R}}^n$ we denote by ${\mathcal {V}}(x)$ the fiber $\{ x\}\times {\mathbb {R}}^n $ of ${\mathbb {R}}^n\times {\mathbb {R}}^n$.
\[Pvertical\] Let $(x,y)$ be a point of differentiability of ${\mathcal {A}}_m$. Then $(y, \frac{\partial {\mathcal {A}}_m}{\partial y}(x,y))\in F^m({\mathcal {V}}(x))$ and $(x, -\frac{\partial {\mathcal {A}}_m}{\partial x}(x,y))\in F^{-m}({\mathcal {V}}(y))$.
We use proposition \[Ppotential\]. As $(x,y)$ is a point of differentiability of ${\mathcal {A}}_m$, $\Sigma_m(x,y)=\{ (x_0, \dots , x_m)\}$ has only one element and this is the projection of the $F$-orbit $$(x_0, -\frac{\partial S}{\partial x}(x_0, x_1)), (x_1, -\frac{\partial S}{\partial x}(x_1, x_2)= \frac{\partial S}{\partial y}(x_0, x_1)), \dots, (x_m, \frac{\partial S}{\partial y}(x_{m-1}, x_m)).$$ Moerover, we have $\frac{\partial {\mathcal {A}}_m}{\partial x}(x,y)=\frac{\partial S}{\partial x}(x, x_1)$ and $\frac{\partial {\mathcal {A}}_m}{\partial y}(x,y)=\frac{\partial S}{\partial y}(x_{m-1}, y)$. We deduce that $F^m(x, -\frac{\partial {\mathcal {A}}_m}{\partial x}(x,y))=(y, \frac{\partial {\mathcal {A}}_m}{\partial y}(x,y))$ and then proposition \[Pvertical\].
\[C1\] We assume that a piece of orbit $(x_i,y_i)_{i\in [0, m+1]}$ for $F$ is minimizing. Then ${\mathcal {A}}_m$ is as regular as $F$ is in a neighbourhood of $(x_0,x_m)$ and in a neighborhood of $(x_1, x_{m+1})$.
We prove the first assertion.\
Let us prove that $DF^{m}(V(x_0))$ is transverse to $V(x_m)$. We use the results that are contained in section 2.3. of [@Arn4] (especially proposition 6). Let us use the notation: $$\bar {\mathcal {F}}(y_0, \dots , y_{m+1})=\sum_{i=0}^m\bar S(y_i, y_{i+1})$$ and $(x_0, \dots , x_{m+1})$ is a minimizer of $\bar{\mathcal {F}}$ among the $(y_0, \dots , y_{m+1})$ such that $y_0=x_0$ and $y_{m+1}=x_{m+1}$. We denote by ${\mathcal {H}}={\mathcal {H}}(x_0, \dots, x_{m+1})$ the Hessian of ${\mathcal {F}}$ with fixed ends at $(x_0, \dots, x_{m+1})$. Then it is positive semidefinite. The kernel of ${\mathcal {H}}$ is the set of projections $(\delta x_i)_{1\leq i\leq m}$ of infinitesimal orbits $(\delta x_i, \delta y_i)_{1\leq i\leq m}$ along the orbit $(x_i,y_i)_{1\leq i\leq m}$ such that their extension $(\delta x_i, \delta y_i)_{i\in{\mathbb {Z}}}$ satisfies $\delta x_0=0$ and $\delta x_{m+1}=0$.\
Let us assume that $DF^{m}(V(x_0))$ is not transverse to $V(x_m)$. Then there exists an infinitesimal orbit $(\delta x_i, \delta y_i)_{0\leq i\leq m}$ that is not the $(0, 0)$ orbit and that satisfies $\delta x_0=\delta x_m=0$. Then $( 0, \delta x_1, \dots , \delta x_{m-1}, 0,0)$ is in the isotropic cone for ${\mathcal {H}}(x_{0}, \dots, x_{m+1})$ and because ${\mathcal {H}}(x_0, \dots, x_{m+1})$ is positive semi-definite, $( 0, \delta x_1, \dots , \delta x_{m-1}, 0,0)$ is in the kernel of $( 0, \delta x_1, \dots , \delta x_{m-1}, 0,0)$. This implies that it is an infinitesimal orbit and then the $0$-orbit.\
We have then proved that $DF^{m}(V(x_0))$ is transverse to $V(x_m)$ and this implies that $DF^{-m}(V(x_m))$ is transverse to $V(x_0)$. Hence $F^m({\mathcal {V}}(x_0))$ (resp. $F^{-m}({\mathcal {V}}(x_m))$) is a manifold that is a graph as smooth as $F$ is in a neighborhood of $(x_m, y_m)$ (resp. $(x_0, y_0)$).\
If $(x'_0, x_m')$ is closed to $(x_0, x_m)$, we have noticed that every element of $\Sigma(x'_0, x_m')$ is closed to the unique element of $\Sigma (x_0, x_m)$. Hence it corresponds to an orbit $(x'_i, y'_i)_{0\leq i\leq m}$ that is closed to $(x_i, y_i)_{0\leq i\leq m}$. Moreover, $F^m({\mathcal {V}}(x'_0))$ (resp. $F^{-m}({\mathcal {V}}(x'_m))$) is a manifold that is a graph as smooth as $F$ is in a neighborhood of $(x'_m, y'_m)$ (resp. $(x'_0, y'_0)$) because it is close to $F^m({\mathcal {V}}(x_0))$ (resp. $F^{-m}({\mathcal {V}}(x_m))$). Hence there is only one choice for $(x_0', y_0')$ above $x_0'$ close to $x_0$ on $F^{-m}({\mathcal {V}}(x'_m))$ and it smoothly depends on $(x'_0, x'_m)$ and we have the same result for the choice of $y'_m$. This means that $\Sigma_m(x'_0, x'_m)$ has only one element, hence ${\mathcal {A}}_m$ is differentiable at $(x_0', x_m')$. Morever, $\frac{\partial {\mathcal {A}}_m}{\partial x}(x'_0, x'_m)=-y'_0$ and $\frac{\partial {\mathcal {A}}_m}{\partial y}(x'0, x'_m)=y'_m$ smoothly depend on $(x'_0, x'_m)$.
Comparison between Ma\~ né’s potential and subactions
-----------------------------------------------------
A consequence of the definition of a subaction is that if $u:{\mathbb {R}}^n\rightarrow {\mathbb {R}}$ is a subaction, then: $\forall x, y\in{\mathbb {R}}^n, u(y)-u(x)\leq {\mathcal {A}}_m(x, y)$.
Let $u_-:{\mathbb {R}}^n\rightarrow {\mathbb {R}}$ be a backward calibrated subaction and let $u_+:{\mathbb {R}}^n\rightarrow {\mathbb {R}}$ be a forward calibrated subaction. Let $x_0$ be a point of differentiability for $u_-$ (resp. $u_+$). Then the backward (resp. forward) orbit of $(x_0, du_-(x_0))$ (resp. $(x_0, du_+(x_0))$) is on the graph of $du_-$ (resp. $du_+$) and is denoted by $(x_i, du_-(x_i))_{i\in{\mathbb {N}}}$ (resp. $(x_i, du_+(x_i))_{i\in{\mathbb {N}}}$). Then $(x_i)$ is strongly minimizing, ${\mathcal {A}}_m$ is differentiable at every $(x_m, x_0)$ (resp. $(x_0, x_m)$) with $m\geq 1$ and for every $x\in{\mathbb {R}}^n$ $$u_-(x)-u_-(x_0)-du_-(x_0)(x-x_0)\leq {\mathcal {A}}_m(x_m, x)-{\mathcal {A}}_m(x_m, x_0)-\frac{\partial {\mathcal {A}}_m}{\partial y}(x_m, x_0)(x-x_0)$$ (resp. $$u_+(x)-u_+(x_0)-du_+(x_0)(x-x_0)\geq {\mathcal {A}}_m(x_0, x_m)- {\mathcal {A}}_m(x, x_m)+\frac{\partial {\mathcal {A}}_m}{\partial x}(x_0, x_m)(x-x_0)).$$
We prove the result for $u_-$. We assume that $x_0$ is a point of differentiability for $u_-$.\
We deduce from the remark after proposition \[Pbackward\] that the backward orbit of $(x_0, du_-(x_0))$ is on the graph of $du_-$ and we denote it by $(x_i, du_-(x_i))_{i\in{\mathbb {N}}}$ . We deduce from the same remark that $(x_i)$ is strongly minimizing. We deduce from proposition \[Pbackward\] that $\Sigma (x_i)$ has only one element. Hence $\Sigma (x_m, x_0)$ has only one element and then ${\mathcal {A}}_m$ is differentiable at $(x_m, x_0)$.\
As $u_-$ is a subaction, we have $$\forall x\in{\mathbb {R}}^n, u_-(x)-u_-(x_0)=u_-(x)-u_-(x_m)+u_-(x_m)-u_-(x_0)\leq {\mathcal {A}}_m(x_m, x)-{\mathcal {A}}_m(x_m, x_0)$$ because $u_-(x_m)-u_-(x_0)={\mathcal {A}}_m(x_m, x_0)$. As the two functions vanish for $x=x_0$ and are differentiable with respect to $x$, we deduce $u_-'(x_0)=\frac{\partial {\mathcal {A}}_m}{\partial y}(x_m, x_0)$ and then the wanted inequality.
\[Palg\] We assume that $(u_-, u_+)$ is a pair of conjugate calibrated subaction. Let $x\in{\mathcal {I}}(u_-, u_+)$ be a point, $(y_n)$ be a sequence of points of ${\mathbb {R}}^n$ converging to $x$, and $(\lambda_n)$ be a sequence of positive real numbers so that the two limits (written in charts) $\displaystyle{\lim_{n\rightarrow \infty} \frac{y_n-x}{\lambda_n}=X}$ and $\displaystyle{Y=\lim_{n\rightarrow \infty} \frac{du_-(y_n)-du_-(x)}{\lambda_n} }$ (resp. $\displaystyle{ \lim_{n\rightarrow \infty} \frac{du_+(y_n)-du_+(x)}{\lambda_n} }$) exist. Then we have: $\forall k\in{\mathbb {R}}^n,$ $$Y.k \leq \frac{1}{2} \bigg( \frac{\partial^2 {\mathcal {A}}_m}{\partial y^2}(x_{-m}, x)(k,k)+ \frac{\partial^2 {\mathcal {A}}_m}{\partial y^2}(x_{-m}, x)(X,X)+ \frac{\partial^2 {\mathcal {A}}_m}{\partial x^2}(x, x_m) (X-k,X-k)\bigg)$$ where $(x_i, du_-(x_i))_{i\in{\mathbb {Z}}}$ is the orbit of $(x, du_-(x))$ (resp: $\forall k\in{\mathbb {R}}^n,$ $$\frac{1}{2}\bigg( -\frac{\partial^2 {\mathcal {A}}_m}{\partial x^2}(x, x_m)(k,k)-\frac{\partial^2 {\mathcal {A}}_m}{\partial x^2}(x, x_m)(X,X)- \frac{\partial^2 {\mathcal {A}}_m}{\partial y^2}(x_{-m}, x)(k-X,k-X) {\bigg)}\leq Y.k )$$
The proof is an adapted version of the proof of proposition 18 in [@Arn2]. We just prove the first inequality.\
Let $x\in{\mathcal {I}}(u_-, u_+)$ and let $z$ be a point of differentiability of $u_-$. We denote the negative orbit of $(z, du_-(z))$ by $(z_{-i}, du_-(z_{-i})_{i\in{\mathbb {N}}}$.Then we have:
1. $u_-({z}+h)-u_-(z)-du_-({z})h\leq {\mathcal {A}}_m(z_{-m}, z+h)-{\mathcal {A}}_m(z_{-m}, z)-\frac{\partial {\mathcal {A}}_m}{\partial y}(z_{-m}, z)h$;
2. $u_-({z})-u_-(x)-du_-(x)({z}-x)\leq {\mathcal {A}}_m(x_{-m} ,z)-{\mathcal {A}}_m(x_{-m}, x)-\frac{\partial {\mathcal {A}}_m}{\partial y}(x_{-m}, x)(z-x))$;
3. $
{\mathcal {A}}_m(x, x_m)- {\mathcal {A}}_m(z+h, x_m)+\frac{\partial {\mathcal {A}}_m}{\partial x}(x, x_m)(z+h-x)\leq u_+({z}+h)-u_+(x)-du_+(x)({z}+h-x)$.
Hence, by adding these three inequalities and using that $u_-(x)=u_+(x)$, $du_-(x)=du_+(x)$ and $u_+\leq u_-$:\
$(du_-(x) -du_-({z}))h$\
1truecm$ \leq {\mathcal {A}}_m(z_{-m}, z+h)-{\mathcal {A}}_m(z_{-m}, z)-\frac{\partial {\mathcal {A}}_m}{\partial y}(z_{-m}, z)h + {\mathcal {A}}_m(x_{-m} ,z)-{\mathcal {A}}_m(x_{-m}, x)$\
2truecm$-\frac{\partial {\mathcal {A}}_m}{\partial y}(x_{-m}, x)(z-x)) - {\mathcal {A}}_m(x, x_m)+ {\mathcal {A}}_m(z+h, x_m)-\frac{\partial {\mathcal {A}}_m}{\partial x}(x, x_m)(z+h-x)$.\
We now consider a sequence $(y_k)$ of points of differentiability of $u_-$ that converges to $x$ such that $\forall k, y_k \not=x$, a vector $K$ with fixed norm $\| K\|=\mu>0$ and the sequence $(h_k)=(\lambda_k K)$ where $(\lambda_k)$ is a sequence of positive numbers tending to zero. we denote by $(z_{-i}^k, du_-(z_{-i}^k))$ the backward orbit of $(y_k, du_-(y_k))$ for $F$. We have proved that:
$(du_-(x) -du_-({y_k}))h$\
1truecm$ \leq {\mathcal {A}}_m(z^k_{-m}, y_k+h_k)-{\mathcal {A}}_m(z^k_{-m}, y_k)-\frac{\partial {\mathcal {A}}_m}{\partial y}(z^k_{-m},y_k)h_k + {\mathcal {A}}_m(x_{-m} ,y_k)-{\mathcal {A}}_m(x_{-m}, x)$\
1truecm$-\frac{\partial {\mathcal {A}}_m}{\partial y}(x_{-m}, x)(y_k-x)) - {\mathcal {A}}_m(x, x_m)+ {\mathcal {A}}_m(y_k+h_k, x_m)-\frac{\partial {\mathcal {A}}_m}{\partial x}(x, x_m)(y_k+h_k-x)$.\
We assume that $\displaystyle{\lim_{k\rightarrow +\infty} \frac{y_k-x}{\lambda_k}=X}$ and $\displaystyle{\lim_{k\rightarrow +\infty} \frac{du_-(y_k)-du_-(x)}{\lambda_k}=Y}$. We have proved in corollary \[C1\] that ${\mathcal {A}}_m$ is as regular as $F$ is in a neighborhood of $(x_{-m}, x)$, $(z^k_{-m}, y_k)$ and $(x, x_m)$. Moreover, we have the following lemma that is lemma 18 in [@Arn2]:
\[Aubrylipschitz\]There exists a constant $K>0$ such that, for every $q\in{\mathcal {I}}(u_-, u_+)$ and every ${q'}\in M$ where $u_-$ (resp. $u_+$) is differentiable, then $\| du_-(q)-du_-({q'})\|\leq K\| q-{q'}\|$ (resp. $\| du_+(q)-du_+({q'})\|\leq K\| q-{q'}\|$ ). In particular, $du_-$ and $du_+$ are continuous at every point of ${\mathcal {I}}(u_-, u_+)$.
This lemma implies that $(y_k, du_-(y_k))$ is closed to $(x, du_-(x))$ and then that $z_{-m}^k$ is close to $x_{-m}$. Hence we obtain:
$$\begin{matrix} (du_-(x) -&du_-({y_k}))h_k \leq \frac{1}{2} ( \frac{\partial^2{\mathcal {A}}_m}{\partial y^2} (z_{-m}^k, y_k)(h_k, h_k) + \frac{\partial^2{\mathcal {A}}_m}{\partial y^2} (x_{-m}, x)(y_k-x, y_k-x)\hfill\\
&+\frac{\partial^2{\mathcal {A}}_m}{\partial x^2}(x, x_m) (y_k+h_k-x, y_k+h_k-x) +o(\| h_k\|^2+\| y_k+h_k-x\|^2) ) .\end{matrix}$$ We multiply by $\frac{1}{\lambda_n^2}$ and take the limit and obtain $$-Y.K\leq \frac{1}{2}( \frac{\partial^2{\mathcal {A}}_m}{\partial y^2}(x_{-m}, x )(K, K)+\frac{\partial^2{\mathcal {A}}_m}{\partial y^2}(x_{-m}, x)(X, X)+\frac{\partial^2{\mathcal {A}}_m}{\partial x^2}(x, x_m)(X+K, X+K)).$$ Changing $K$ into $-K$, we obtain the wanted inequality.
Links between the tangent cone to the support of a strongly minimizing measure and the Green bundles {#sscone}
----------------------------------------------------------------------------------------------------
The notion of contingent cone was introduced by G. Bouligand in [@Bou].
Let $A\subset {\mathbb {R}}^n\times {\mathbb {R}}^n$ be a subset of ${\mathbb {R}}^n\times {\mathbb {R}}^n$ and let $a\in A$ be a point of $A$. Then the [*contingent cone* ]{} to $A$ at $a$ is defined as being the set of all the limit points of the sequences $t_k(a_k-a)$ where $(t_k)$ is a sequence of real numbers and $(a_k)$ is a sequence of elements of $A$ that converges to $a$. This cone is denoted by $C_aA$ and it is a subset of $T_a({\mathbb {R}}^n\times {\mathbb {R}}^n)$.
We introduce an extension to this definition that is
Let $A\subset {\mathbb {R}}^n\times {\mathbb {R}}^n$ be a subset of ${\mathbb {R}}^n\times {\mathbb {R}}^n$ and let $a\in A$ be a point of $A$. Then the [*limit contingent cone*]{} to $A$ at $a$ is the set of the limit points of sequences $v_k\in C_{a_k}A$ where $(a_k)$ is any sequence of points of $A$ that converges to $a$. It is denoted by $\widetilde C_aA$.
In general , these tangent cones are not Lagrangian subspaces. Because we need to compare them to Lagrangian subspaces, we give a definition:
Let ${\mathcal {L}}_-\leq {\mathcal {L}}_+$ be two Lagrangian subspaces of $T_x( {\mathbb {R}}^n\times {\mathbb {R}}^n) $ that are transverse to the vertical. If $v\in T_x ({\mathbb {R}}^n\times {\mathbb {R}}^n )$ is a vector, we say that $v$ is between ${\mathcal {L}}_-$ and ${\mathcal {L}}_+$ and write ${\mathcal {L}}_-\leq v\leq {\mathcal {L}}_+$ if there exists a third Lagrangian subspace in $T_x( {\mathbb {R}}^n\times {\mathbb {R}}^n) $ such that:
1. $v\in{\mathcal {L}}$;
2. ${\mathcal {L}}_-\leq {\mathcal {L}}\leq {\mathcal {L}}_+$.
A part $B$ of $T_x( {\mathbb {R}}^n\times {\mathbb {R}}^n) $ is between ${\mathcal {L}}_-$ and ${\mathcal {L}}_+$ if $\forall v\in B, {\mathcal {L}}_-\leq v\leq {\mathcal {L}}_+$. Then we write ${\mathcal {L}}_-\leq B\leq {\mathcal {L}}_+$.
We introduce the two modified Green bundles. We use the constant $c_0=\frac{\sqrt{13}}{3}-\frac{5}{6}$.
We denote by $S_\pm(x):{\mathbb {R}}_n\rightarrow {\mathbb {R}}^n$ the linear operator such that $G_\pm(x)$ is the graph of $S_\pm(x)$: $G_\pm(x)=\{ (v, S_\pm(x)v); v\in{\mathbb {R}}^n\}$. Then the [*modified Green bundles*]{} $G_\pm$ are defined by: $$\widetilde G_-(x)=\{ (v, (S_-(x)-c_0(S_+(x)-S_-(x)))v); v\in {\mathbb {R}}^n\}$$[and]{}$$\widetilde G_+(x)=\{ (v, (S_+(x)+c_0(S_+(x)-S_-(x)))v); v\in {\mathbb {R}}^n\}.$$
Let $(u_-, u_+)$ be a pair of conjugate calibrated subactions. Then $$\forall x\in {\mathcal {I}}(u_-, u_+), \widetilde G_-(x, du_-(x))\leq \widetilde C_{(x, du_-(x))}\widetilde {\mathcal {I}}(u_-, u_+)\leq \widetilde G_+(x, du_-(x)).$$
A consequence of proposition \[Palg\] and proposition \[Pbilin\] is that: $$\forall x\in {\mathcal {I}}(u_-, u_+), \widetilde G_-(x, du_-(x))\leq C_{(x, du_-(x))}\widetilde {\mathcal {I}}(u_-, u_+)\leq \widetilde G_+(x, du_-(x)).$$ Then the conclusion of the proposition comes from the definition of the limit contingent cone and the semicontinuity property of the Green bundles (see for example [@Arn5]) and then of the modified Green bundles.
As ${\mathcal {M}}(S)\subset \widetilde {\mathcal {I}}(u_-, u_+)$, we deduce the following corollary and then theorem \[Tcone\].
We have: $\forall x\in {\mathcal {M}}(S), \widetilde G_-(x)\leq \widetilde C_{x}{\mathcal {M}}(S)\leq \widetilde G_+(x, du_-(x)).$
A subset $A$ of ${\mathbb {R}}^n\times {\mathbb {R}}^n$ is $C^1$-isotropic at some point $a\in A$ if $\widetilde C_aA$ is contained in some Lagrangian subspace.
For example, a $C^1$ submanifold is $C^1$-isotropic if it isotropic.\
Corollary \[Cisotropic\] that is given in the introduction is just a consequence of theorem \[Tcone\] and theorem \[nbexp\].
Appendix
========
Comparison of Lagrangian subspaces {#sscomp}
----------------------------------
Let us assume that $(E, \omega)$ is a symplectic $2n$-dimensional space. Let $L_1$, $L_2$ be two transverse Lagrangian subspaces of $E$. Then the set the Lagrangian subspaces of $E$ that are transverse to $L_1$ and $L_2$ is open in the Grassmann space ${\mathcal {L}}$ of the Lagrangian subspaces of $E$. Moreover it has exactly $n+1$ connected component. Let us be more precise.
If $L\in{\mathcal {L}}$ is transverse to $L_2$, then it is the graph of a linear map $\ell:L_1\rightarrow L_2$. We then define a quadratic form $q(L_1, L_2, ;L)$ on $L_1$ by: $$\forall v\in L_1, q(L_1, L_2; L)(v)=\omega (v, \ell(v)).$$ Then $L$ is transverse to both $L_1$ and $L_2$ if and only if $q(L_1, L_2; L)$ is non-degenerate and the connected components of the set the Lagrangian subspaces of $E$ that are transverse to $L_1$ and $L_2$ correspond to the signature of this quadratic form.\
We will denote by ${\mathcal {P}}(L_1, L_2)$ the set of the $L\in{\mathcal {L}}$ that correspond to a positive definite quadratic form.
\[propapp\] Let $L_1, L_2\in{\mathcal {L}}$ be two transverse Lagrangian subspaces of $E$. Then
1. if $M:E\rightarrow E$ is a symplectic isomorphism, we have: $M({\mathcal {P}}(L_1, L_2))={\mathcal {P}}(M(L_1), M(L_2))$;
2. if $L\in{\mathcal {P}}(L_1, L_2)$, then ${\mathcal {P}}(L_1, L)\cup {\mathcal {P}}(L, L_2)\subset {\mathcal {P}}(L_1, L_2)$.
The proof of the first assertion is elementary.\
For the second one, let us begin by proving that ${\mathcal {P}}(L_1, L)\subset {\mathcal {P}}(L_1, L_2)$. Let $W\in {\mathcal {P}}(L_1, L)$. For $w\in W\backslash\{ 0\}$, we write $w=\ell_1+\ell$ with $\ell_1\in L_1$, $\ell\in L$. Then we have: $\omega(\ell_1, \ell)>0$. As $\ell\in L\backslash \{ 0\} $ and $L\in {\mathcal {P}}(L_1, L_2)$, we can write $\ell =\ell_1'+\ell_2'$ with $\ell_i'\in L_i$ and we have $\omega (\ell'_1, \ell'_2)>0$.\
Finally we have proved that $w=(\ell_1+\ell_1')+\ell_2'$ with $\ell_1+\ell_1'\in L_1$ and $\ell_2'\in L_2$ and $\omega(\ell_1+\ell_1', \ell_2')=\omega(\ell_1, \ell_2')+\omega(\ell_1', \ell_2')= \omega(\ell_1, \ell_1'+\ell_2')+\omega(\ell_1', \ell_2')>0$.\
the proof of the second inclusion is very similar.
In the particular case where $E=T_x{\mathbb {A}}_n={\mathbb {R}}^n\times{\mathbb {R}}^n$, we define an order relation on the set ${\mathcal {H}}$ of Lagrangian subspaces that are transverse to $V(x)$ in the following way.
If $L_1, L_2\in{\mathcal {H}}$,
1. we say that $L_1$ is stricly under $L_2$ and write $L_1<L_2$ if $L_2\in {\mathcal {P}}(L_1, V(x))$;
2. we say that $L_1$ is under $L_2$ and write $L_1\leq L_2$ if $L_2$ is in the closure of ${\mathcal {P}}(L_1, V(x))$.
Note that $L_1\leq L_2$ if and only if $q(L_1, V(x); L_2)$ is positive semi-definite. A consequence of proposition \[propapp\] is that $<$ and $\leq$ are transitive.\
We can then define what is a decreasing or increasing sequence of elements of ${\mathcal {H}}$.
\[proappbis\] If $L_1, L_2, L_3\in{\mathcal {H}}$, if $L_1<L_2$ and $L_3\in{\mathcal {P}}(L_1, L_2)$, then $L_1<L_3$ and $L_3<L_2$.
Let us prove the first inequality. We assume that $L_1<L_2$, i.e. $L_2\in {\mathcal {P}}(L_1, V(x))$. We know by proposition \[propapp\] that ${\mathcal {P}}(L_1, L_2)\cup {\mathcal {P}}(L_2, V(x))\subset {\mathcal {P}}(L_1, V(x))$. We deduce that $L_3\in {\mathcal {P}}(L_1, V(x))$ i.e. $L_1<L_3$.\
We explain how to prove the second inequality. We choose a basis $(e_1,\dots,e_n)$ of $L_3$ and complete it with $f_1, \dots, f_n\in V(x)$ in such a way that the basis is symplectic.\
Then there exist two symmetric matrices $S_1$ and $S_2$ such that $L_i$ is the graph of the linear map $\phi_i: L_3\rightarrow V(x)$ with matrix $S_i$ in the bases $(e_1, \dots, e_n)$, $(f_1, \dots, f_n)$. Because $L_1<L_3$, we know that $S_1$ is negative definite. We want to prove that $S_2$ is positive definite.\
Let us write that $L_3\in{\mathcal {P}}(L_1, L_2)$. This means that for all $v\in {\mathbb {R}}^n\backslash\{ 0\}$, if $(v, 0)=(v_1, S_1v_1)+(v_2, S_2v_2)$, then ${}^tv_1S_2v_2-{}^tv_2S_1v_1>0$. This can be reformulated in the following way. $$\forall w\in{\mathbb {R}}^n, -{}^twS_2S_1^{-1}S_2w+{}^twS_2 w>0.$$ Let $s$ be the positive definite matrix such that $s^2=-S_1$. If $s_2=s^{-1}S_2s^{-1}$, we obtain $$\forall u\in {\mathbb {R}}^n, {}^tus_2^2u+{}^tus_2u>0.$$ If $\lambda_1, \dots, \lambda_n$ are the eigenvalues of $s_2$, we deduce that $\lambda_i^2+\lambda_i>0$ i.e. $\lambda_i<-1$ or $\lambda_i>0$. Moreover, we know that $L_1<L_2$, hence $0<-S_1+S_2$, i.e. $0<{\bf 1}_n+s_2$ and $\lambda_i>-1$. We deduce that $\lambda_i>0$ and $S_2$ is positive definite.
\[proappter\] If $L_1, L_2, L_3\in{\mathcal {H}}$, if $L_1<L_3<L_2$ then $L_3\in{\mathcal {P}}(L_1, L_2)$.
As in the proof of proposition \[proappbis\], we choose a basis $(e_1,\dots,e_n)$ of $L_3$ and complete it with $f_1, \dots, f_n\in V(x)$ in such a way that the basis is symplectic. Then there exist two symmetric matrices $S_1$ and $S_2$ such that $L_i$ is the graph of the linear map $\phi_i: L_3\rightarrow V(x)$ with matrix $S_i$ in the bases $(e_1, \dots, e_n)$, $(f_1, \dots, f_n)$. We know that $S_1$ is negative definite and $S_2$ is positive definite.\
We want to prove that $L_3\in{\mathcal {P}}(L_1, L_2)$. This means that for all $v\in {\mathbb {R}}^n\backslash\{ 0\}$, if $(v, 0)=(v_1, S_1v_1)+(v_2, S_2v_2)$, then ${}^tv_1S_2v_2-{}^tv_2S_1v_1>0$. As $S_2$ is positive definite and $S_1$ is negative definite, the conclusion is straightforward.
A result in bilinear algebra
----------------------------
\[Pbilin\] Let $Q_-$, $Q_+$ be two quadratic forms on ${\mathbb {R}}^n$ such that $Q_-\leq Q_+$ and let $(X, Y)\in {\mathbb {R}}^n\times {\mathbb {R}}^n$ be such that: $$\forall K\in{\mathbb {R}}^n, Y.K \leq \frac{1}{2} \bigg( Q_+(K,K)+ Q_+(X,X)- Q_-(X-K,X-K)\bigg)$$ and $$\forall K\in{\mathbb {R}}^n, \frac{1}{2}\bigg( Q_-(K,K)+Q_-(X,X)- Q_+(K-X,K-X) {\bigg)}\leq Y.K .$$ Then there exists a quadratic form $\sigma$ such that:
1. $Q_--(\frac{\sqrt{13}}{3}-\frac{5}{6})(Q_+-Q_-)\leq \sigma \leq Q_++(\frac{\sqrt{13}}{3}-\frac{5}{6})(Q_+-Q_-)$;
2. $Y={}^t\sigma(X, .)$.
1\) Note that $\frac{\sqrt{13}}{3}-\frac{5}{6}<\frac{1}{2}$, hence we obtain the same inequalities by replacing $\frac{\sqrt{13}}{3}-\frac{5}{6}$ by $\frac{1}{2}$.\
2) We gave in [@Arn2] an example in dimension $n=2$ that proves that in general, we cannot improve the first point into $Q_-\leq \sigma\leq Q_+$.
$\Delta Q= Q_+-Q_-$; $\Delta Y_+=Y-{}^tQ_+(X, .)$ and $\Delta Y_-=Y-{}^tQ_-(X, .)$.\
We use the constant: $c_0=\frac{\sqrt{13}}{3}-\frac{5}{6}$.
Note that $\Delta Y_--\Delta Y_+={}^t\Delta Q(X, .)$.
Using the above notations, we rewrite the two inequalities: $$\forall K\in{\mathbb {R}}^n, \Delta Y_+.K\leq \frac{1}{2}\Delta Q(X-K, X-K)\quad{\rm and}\quad \Delta Y_-.K\geq -\frac{1}{2}\Delta Q(X-K, X-K).$$ We deduce that $\Delta Y_+, \Delta Y_-\in {\mathrm{Im}}^t\Delta Q=(\ker \Delta Q)^\bot$. We then use the restriction of $\Delta Q$ to ${\mathrm{Im}}^t\Delta Q={\mathbb {R}}^d$, hence $\Delta Q$ is positive definite and we want to prove that there exists a quadratic form $\sigma$ on ${\mathbb {R}}^d$ such that $-(1+c_0)\Delta Q \leq \sigma\leq c_0 \Delta Q$ and $\Delta Y_+ ={}^t \sigma (X, )$.
As $\Delta Q$ is positive definite, there exists a symmetric automorphism $L: {\mathbb {R}}^d\rightarrow{\mathbb {R}}^d$ such that $\Delta Q(L(X))=\| X\|^2$ ($\| .\|$ is the usual Euclidean norm). We introduce the notations $x=L^{-1}X$, $y_+={}^tL\Delta Y_+$ and $y_-={}^tL\Delta Y_-$. Note that $y_--y_+=x$. The inequalities are rewritten as: $$\forall k\in {\mathbb {R}}^d, y_+.k\leq\frac{1}{2}\| k-x\|^2\quad{\rm and}\quad y_-.k\geq -\| x-k\|^2.$$ We now want to find $\eta=\sigma\circ L$ such that $y_+={}^t\eta (x,.)$ and $-(1+c_0)\|.\|^2 \leq \eta\leq c_0 \|.\|^2.$ Using an orthogonal change of basis, we can assume that $x=(\mu, 0, \dots, 0)$ and we can multiply all the inequalities by $\mu^2$ and assume that $\mu=1$. We use the notations $y_+=(y_i)_{1\leq i\leq d}$, and $k=(k_i)_{1\leq i\leq d}$. We have $x=(1, 0, \dots, 0)$. Then the inequalities become: $$\sum_{i=1}^d y_i.k_i\leq \frac{1}{2}(k_1-1)^2+\frac{1}{2}\sum_{i=2}^dk_i^2\quad{\rm and}\quad k_1+\sum_{i=1}^d y_i.k_i\geq -\frac{1}{2}(k_1-1)^2-\frac{1}{2}\sum_{i=2}^dk_i^2.$$ They can be rewritten as follows $$(k_1-1-y_1)^2 +1+\sum_{i=2}^d(k_i-y_1)^2\geq (y_1+1)^2+\sum_{i=2}^dy_i^2$$ and $$\sum_{i=1}^d(k_i+y_i)^2+1\geq \sum_{i=1}^d y_i^2.$$ As $(k_i)_{1\leq i\leq d}$ can be any element of ${\mathbb {R}}^d$, this is equivalent to: $$(y_1+1)^2+\sum_{i=2}^dy_i^2\leq 1\quad{\rm and}\quad 1\geq \sum_{i=1}^d y_i^2.$$ Then we choose the quadratic form $\eta$. Its matrix in the canonical basis is $$S=\begin{pmatrix}y_1&y_2&y_3&\dots &y_{d-1}&y_d\\
y_2&-\frac{1}{2}&0&\dots &0&0\\
.&.&.&\dots&.&.\\
y_d&0&0&\dots&0&-\frac{1}{2}
\end{pmatrix}$$ i.e. the only entries that may be non-zero are on the first line, on the first column and on the diagonal. If ${\bf 1}$ is the identity matrix, we have to prove that $c_0{\bf 1}-S$ and $(1+c_0){\bf 1}+S$ are positive semidefinite. We have $$c_0{\bf 1}-S= \begin{pmatrix}c_0-y_1&-y_2&-y_3&\dots &-y_{d-1}&-y_d\\
-y_2&c_0+\frac{1}{2}&0&\dots &0&0\\
.&.&.&\dots&.&.\\
-y_d&0&0&\dots&0&c_0+\frac{1}{2}
\end{pmatrix}$$ The restriction of $c_0\| .\|^2-\eta$ to $\{0\}\times {\mathbb {R}}^{d-1}$ is positive definite. Hence to prove that this quadratic form is positive, we only have to prove that the determinant of $c_0{\bf 1}-S$ is non-negative. We then compute it. Note that when $d=1$, we have: $\delta(1)=c_0-y_1$. Moreover, if $d\geq 2$, we have $$\delta(d)=\det(c_0{\bf 1}-S)=(c_0+\frac{1}{2})\delta(d-1) + (-1)^{d} y_d\det \begin{pmatrix}
-y_2&c_0+\frac{1}{2}&0&\dots &0\\
.&.&.&\dots&c_0+\frac{1}{2}\\
-y_d&0&0&\dots&0
\end{pmatrix}$$ and thus $$\delta (d)=(c_0+\frac{1}{2})\delta(d-1) + (-1)^{d}(-1)^{d-1} y_d^2(c_0+\frac{1}{2})^{d-2}=(c_0+\frac{1}{2})\delta(d-1) - y_d^2(c_0+\frac{1}{2})^{d-2} .$$ We finally deduce: $$\delta (d)=(c_0+\frac{1}{2})^{d-1}\left((c_0+\frac{1}{2})(c_0-y_1)-\sum_{i=2}^dy_i^2\right).$$ We have proved that $\displaystyle{(y_1+1)^2+\sum_{i=2}^dy_i^2\leq 1}$, hence we have: $$\begin{matrix}\delta (d)&\geq (c_0+\frac{1}{2})^{d-1}\left((c_0+\frac{1}{2})(c_0-y_1)+(1+y_1)^2-1\right)\\
&\geq (c_0+\frac{1}{2})^{d-1}\left((y_1+ \frac{3}{4}-\frac{c_0}{2})^2+\frac{3}{4}c_0^2+\frac{5}{4}c_0-\frac{9}{16}\right).
\end{matrix}$$ As $\frac{3}{4}c_0^2+\frac{5}{4}c_0-\frac{9}{16}=0$, we conclude that $c_0\|.\|^2-\eta$ is positive semidefinite.\
Let us now prove that $(1+c_0){\bf 1}+S$ is positive semidefinite. We compute $$(1+c_0){\bf 1}+S=\begin{pmatrix}1+c_0+y_1&y_2&y_3&\dots &y_{d-1}&y_d\\
y_2& \frac{1}{2}+c_0&0&\dots &0&0\\
.&.&.&\dots&.&.\\
y_d&0&0&\dots&0& \frac{1}{2}+c_0
\end{pmatrix}$$ Then the restriction of $\eta+(1+c_0)\|.\|^2$ to $\{ 0\}\times {\mathbb {R}}^{d-1}$ is positive definite and we just have to prove that $\det ((1+c_0){\bf 1}+S)$ is non negative. Using the computations that we did for $\delta (d)$ (we replace $y_i$ by $-y_i$ and $y_1$ by $-(1+y_1)$), we obtain: $$\det((1+c_0){\bf 1}+S)=(c_0+\frac{1}{2})^{d-1}\left((c_0+\frac{1}{2})(c_0+1+y_1)-\sum_{i=2}^dy_i^2\right).$$ We have proved that $\displaystyle{1\geq \sum_{i=1}^d y_i^2}$ hence we deduce $$\begin{matrix}\det((1+c_0){\bf 1}+S)&\geq (c_0+\frac{1}{2})^{d-1}\left((c_0+\frac{1}{2})(c_0+1+y_1)+y_1^2-1\right)\hfill\\
&\geq(c_0+\frac{1}{2})^{d-1}\left( (y_1+\frac{c_0}{2}+\frac{1}{4})^2+\frac{3}{4}c_0^2+\frac{5}{4}c_0-\frac{9}{16}\right)\hfill\\
&\geq(c_0+\frac{1}{2})^{d-1}(y_1+\frac{c_0}{2}+\frac{1}{4})^2.\hfill
\end{matrix}$$ Then the quadratic form $(1+c_0)^2+\eta$ is positive semidefinite.
[cc]{}
M.-C. Arnaud, Type des points fixes des difféomorphismes symplectiques de ${\mathbb {T}}^n\times {\mathbb {R}}^n$. (French) \[The type of fixed points of the symplectic diffeomorphisms of ${\mathbb {T}}^n\times {\mathbb {R}}^n$\] [*Mém. Soc. Math. France*]{} (N.S.) No. 48 (1992), 63 pp.
M.-C. Arnaud, [*Fibrés de Green et régularité des graphes $C^0$-Lagrangiens invariants par un flot de Tonelli*]{}, Ann. Henri Poincaré [**9**]{} (2008), no. 5, 881–926.
M.-C. Arnaud, [*The link between the shape of the Aubry-Mather sets and their Lyapunov exponents*]{}, Annals of Mathematics, [**174-3**]{} (2011), p 1571-1601
M.-C. Arnaud, [*Green bundles, Lyapunov exponents and regularity along the supports of the minimizing measures*]{}. Ann. Inst. H. Poincaré Anal. Non Linéaire 29 (2012), no. 6, 989–1007
M.-C. Arnaud, [*Lower and upper bounds for the Lyapunov exponents of twisting dynamics: a relationship between the exponents and the angle of Oseledets’ splitting*]{}. Ergodic Theory Dynam. Systems 33 (2013), no. 3, 693-712.
M.-C. Arnaud, [*Lyapunov exponents for conservative twisting dynamics: a survey*]{}, preprint 2014
S. Aubry & P. Y. Le Daeron. [*The discrete Frenkel-Kontorova model and its extensions. I. Exact results for the ground-states.*]{} Phys. D 8 (1983), no. 3, 381–422.
P. Bernard. *The dynamics of pseudographs in convex Hamiltonian systems.* J. Amer. Math. Soc. 21 (2008), no. 3, 615–669.
M. L Bialy & R. S. MacKay, * Symplectic twist maps without conjugate points*. Israel J. Math. [**141**]{}, 235–247 (2004).
G. D. Birkhoff, Surface transformations and their dynamical application, [*Acta Math.*]{} [**43**]{} (1920) 1-119.
J. Bochi & M. Viana, *Lyapunov exponents: how frequently are dynamical systems hyperbolic?* Modern dynamical systems and applications, 271–297, Cambridge Univ. Press, Cambridge, 2004.
G. Bouligand. Introduction à la géométrie infinitésimale directe (1932) [*Librairie Vuibert*]{}s, Paris.
P. Cannarsa & C. Sinestrari, Semiconcave functions, Hamilton-Jacobi equations, and optimal control. [*Progress in Nonlinear Differential Equations and their Applications*]{}, 58. Birkhäuser Boston, Inc., Boston, MA, 2004. xiv+304 pp
A. Fathi, [*Weak KAM theorems in Lagrangian dynamics*]{}, book in preparation.
E. Garibaldi & P. Thieullen, [*Minimizing orbits in the discrete Aubry-Mather model.*]{} Nonlinearity 24 (2011), no. 2, 563-611
C. Golé, Symplectic twist maps. Global variational techniques. Advanced Series in Nonlinear Dynamics, 18. [*World Scientific Publishing Co., Inc.*]{}, River Edge, NJ, 2001. xviii+305 pp
D.A. Gomes, [*Viscosity solution methods and the discrete Aubry-Mather problem.*]{} Discrete Contin. Dyn. Syst. 13 (2005), no. 1, 103-116
P. Le Calvez, [*Les ensembles d’Aubry-Mather d’un difféomorphisme conservatif de l’anneau déviant la verticale sont en général hyperboliques*]{}. (French) \[The Aubry-Mather sets of a conservative diffeomorphism of the annulus twisting the vertical are hyperbolic in general\] C. R. Acad. Sci. Paris Sér. I Math. [**306**]{} , no. 1, 51–54 (1988).
R. Man' e, [*Lagrangian flows : the dynamics of globally minimizing orbits*]{}, Int. Pitman Res. Notes Math. Ser., [**362**]{}, 120-131 (1996).
J. N. Mather. [*Existence of quasiperiodic orbits for twist homeomorphisms of the annulus.*]{} Topology 21 (1982), no. 4, 457–467.
J. Moser, [*Proof of a generalized form of a fixed point theorem due to G. D. Birkhoff.*]{} Geometry and topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976), pp. 464-494. Lecture Notes in Math., Vol. 597, Springer, Berlin, 1977.
J. Moser, [*Monotone twist mappings and the calculus of variations.*]{} Ergodic Theory Dynam. Systems [**6-3**]{} (1986), 401–413
[^1]: ANR-12-BLAN-WKBHJ
[^2]: Avignon Université , Laboratoire de Mathématiques d’Avignon (EA 2151), F-84 018 Avignon, France. e-mail: [email protected]
[^3]: membre de l’Institut universitaire de France
[^4]: Their definition is recalled in section \[sGreen\]
[^5]: see subsection \[ssweakkam\] for the definition
[^6]: see section \[sscone\] for the exact definition
[^7]: see section \[sscone\] for the precise definition
|
---
abstract: 'In Einstein-Maxwell-Chern-Simons theory the extremal Reissner-Nordström solution is no longer the single extremal solution with vanishing angular momentum, when the Chern-Simons coupling constant reaches a critical value. Instead a whole sequence of rotating extremal $J=0$ solutions arises, labeled by the node number of the magnetic U(1) potential. Associated with the same near horizon solution, the mass of these radially excited extremal solutions converges to the mass of the extremal Reissner-Nordström solution. On the other hand, not all near horizon solutions are also realized as global solutions.'
author:
- |
[**Jose Luis Blázquez-Salcedo$^1$**]{}, [**Jutta Kunz$^2$**]{},\
[**Francisco Navarro-Lérida$^3$**]{}, [**Eugen Radu$^2$**]{}\
$^1$ Dept. de Física Teórica II, Ciencias Físicas\
Universidad Complutense de Madrid, E-28040 Madrid, Spain\
$^2$ Institut für Physik, Universität Oldenburg\
Postfach 2503, D-26111 Oldenburg, Germany\
$^3$ Dept. de Física Atómica, Molecular y Nuclear, Ciencias Físicas\
Universidad Complutense de Madrid, E-28040 Madrid, Spain
title: Sequences of extremal radially excited rotating black holes
---
[**Introduction.–**]{} Higher-dimensional black hole spacetimes have received much interest in recent years, associated with various developments in gravity and high energy physics. In particular, the first successful statistical counting of black hole entropy in string theory was performed for an extremal static Reissner-Nordström (RN) black hole in five spacetime dimensions [@Strominger:1996sh].
However, in odd dimensions the Einstein-Maxwell (EM) action may be supplemented by a Chern-Simons (CS) term. In 5 dimensions, for a certain value of the CS coefficient $\lambda=\lambda_{\rm SG}$, the theory corresponds to the bosonic sector of $D=5$ supergravity, where rotating black hole solutions are known analytically [@Breckenridge:1996sn; @Breckenridge:1996is; @Cvetic:2004hs; @Chong:2005hr]. A particular interesting subset of these black holes, the BMPV [@Breckenridge:1996is] solutions, corresponds to extremal cohomogeneity-1 solutions, where both angular momenta have equal magnitude, $|J_1|=|J_2|=|J|$. These black holes have a non-rotating horizon, although their angular momentum is nonzero. It is stored in the Maxwell field, with a negative fraction of the total angular momentum stored behind the horizon [@Gauntlett:1998fz; @Herdeiro:2000ap; @Townsend:2002yf]. As conjectured in [@Gauntlett:1998fz], supersymmetry is associated with the borderline between stability and instability, since for $\lambda>\lambda_{\rm SG}$ a rotational instability arises, where counterrotating black holes appear [@Kunz:2005ei]. Moreover, when the CS coefficient is increased beyond the critical value of $2\lambda_{\rm SG}$, EMCS black holes - with the horizon topology of a sphere - are no longer uniquely characterized by their global charges [@Kunz:2005ei].
Focussing on extremal solutions with equal magnitude angular momenta, we here reanalyze 5-dimensional EMCS black holes in the vicinity and beyond the critical value of the CS coupling constant $\lambda_{\rm SG}$. We obtain these cohomogeneity-1 solutions numerically, solving the field equations with appropriate boundary conditions.
These extremal black holes are associated with analytical near horizon solutions, obtained in the entropy function formalism. Surprisingly, however, certain sets of near horizon solutions are associated with more than one global solution, whereas other sets of near horizon solutions do not possess global counterparts. In particular, we find whole sequences of radially excited extremal solutions, all with the same area and angular momenta for a given charge.
[**The model.–**]{} We consider the EMCS action with Lagrangian [@Gauntlett:1998fz] $$\begin{aligned}
{\cal L}= \frac{1}{16\pi G_5} [\sqrt{-g}(R - F^{2}
)
- \frac{2\lambda}{3\sqrt{3}}\varepsilon^{\mu\nu\alpha\beta\gamma}A_{\mu}F_{\nu\alpha}F_{\beta\gamma}] ,
\label{Lag}\end{aligned}$$ with curvature scalar $R$, Newton’s constant $G_5$, gauge potential $A_\mu $, field strength tensor $ F_{\mu \nu} = \partial_\mu A_\nu -\partial_\nu A_\mu $, and CS coupling constant ${ \lambda}$ (with $\lambda_{\rm SG}=1$).
To obtain stationary cohomogeneity-1 solutions we employ for the metric the parametrization [@Kunz:2006eh] $$\begin{aligned}
\label{metric}
&&ds^2 = -f(r) dt^2 + \frac{m(r)}{f(r)}(dr^2 + r^2 d\theta^2) \nonumber \\
&& + \frac{l(r)}{f(r)}r^2 \sin^2\theta \left( d \varphi -\frac{\omega(r)}{r}dt
\right)^2 \nonumber \\
\nonumber
&&+\frac{l(r)}{f(r)}r^2 \cos^2\theta \left( d \psi -\frac{\omega(r)}{r}dt \right)^2
\nonumber \\
&&+\frac{m(r)-l(r)}{f(r)}r^2 \sin^2\theta \cos^2\theta(d \varphi -d \psi)^2,\end{aligned}$$ and for the gauge potential $$A_\mu dx^\mu = a_0(r) dt + a_k(r) (\sin^2 \theta d\varphi+\cos^2 \theta d\psi).$$ To obtain asymptotically flat solutions, the metric functions should satisfy the following set of boundary conditions at infinity: $f|_{r=\infty}=m|_{r=\infty}=l|_{r=\infty}=1$, $\omega|_{r=\infty}=0$. For the gauge potential we choose a gauge such that $a_0|_{r=\infty}=a_{\varphi}|_{r=\infty}=0$. In isotropic coordinates the horizon is located at $r_{\rm H}=0$. An expansion at the horizon yields $f(r) = f_4 r^4 + f_{\alpha} r^{\alpha} + \dots$, $m(r) = m_2 r^2 + m_{\beta} r^{\beta} + \dots$, $l(r) = l_2 r^2 + l_{\gamma} r^{\gamma} + \dots$, $\omega(r) = \Omega_H r + \omega_2 r^2 + \dots$, $a_0(r) = a_{0,0} + a_{0,\lambda} r^{\lambda} + \dots$, $a_k(r) = a_{k,0} + a_{k,\mu} r^{\mu} + \dots$. Interestingly, the coefficients $\alpha$, $\beta$, $\gamma$, $\lambda$, $\mu$ and $\nu$ can be non-integer.
The global charges of these solutions can be read from the asymptotic expansion [@Kunz:2005ei] $$\begin{aligned}
f=1-\frac{\ 8 G_5 M}{3 \pi r^{2}} + \dots \ ,~
\omega=\frac{4 G_5J}{\pi r^{3}} + \dots \ , \nonumber \\
a_0=-\frac{G_5Q}{\pi r^{2}} + \dots \ ,~
a_{\varphi}=\frac{G_5 { \mu}_{\rm mag}}{\pi r^{2}} + \dots \ ,
$$ together with their magnetic moment ${ \mu}_{\rm mag}$. These extremal solutions satisfy the Smarr formula [@Gauntlett:1998fz] $
M = 3 \Omega_H J + \Phi_{\rm H} Q ,
$ and the first law $
dM = 2 \Omega_H dJ+\Phi_{\rm H} dQ ,
$ where $A_{\rm H}=2\pi^2 \sqrt{ {l_2} } {m_2}/{f_4}^{3/2} $ is the horizon area, $\Omega_H$ is the horizon angular velocity, and $\Phi_{\rm H}=-(a_{0,0}+\Omega_H a_{k,0})$ is the horizon electrostatic potential.
[**The near horizon solutions.–**]{} A partial analytical understanding of the properties of the solutions can be achieved by studying their near horizon expression in conjunction with the attractor mechanism [@Astefanesei:2006dd]. The advantage of the latter is that we can compute the physical charges and obtain semi-analytic expressions for the entropy as a function of electric charge and angular momentum. To apply the entropy function for the near horizon geometry of the extremal EMCS solutions, one uses the ansatz [@Kunduri:2007qy] $$\begin{aligned}
\label{metric_ansatz}
&&ds^2 = v_1(\frac{d\rho^2}{\rho^2}-\rho^2dt^2) + v_2
[
\sigma_1^2+\sigma_2^2+v_3 (\sigma_3 -\alpha \rho dt)^2
], \nonumber \\ && A_\mu dx^\mu =-e \rho dt +p \sigma_3,
$$ (where $\sigma_1^2+\sigma_2^2=d\bar \theta^2+\sin^2\bar \theta d\psi^2$), $\sigma_3=d\phi+\cos \bar \theta d \psi$, with $\bar \theta=2 \theta$, $\phi_1-\phi_2=\phi$, $\phi_1+\phi_2=\psi$), and constants $v_a$, $e$, $p$ and $\alpha$.
In the entropy function formalism, the entropy can be found from the extremum of the entropy function $S=2\pi (\alpha J+e \hat q-f)$ in which $ f =\int d \theta d \varphi_1 d\varphi_2 \sqrt{-g} {\cal L}$ and $J=\partial f/\partial \alpha$, $\hat q=\partial f/\partial e$. However, the analysis is somehow intricate due to the presence of the CS term. For example, for $\lambda \neq 0$, the constant $\hat q$ cannot be identified with the electric charge and the extremization equations $\partial F/\partial v_a= \partial F/\partial e=\partial F/\partial \alpha=0$ should be used together with the Maxwell-Chern-Simons equations [@Suryanarayana:2007rk]. The near horizon solution is found in terms of $p,v_1$ (this holds also for $S,J$ and $Q$). However, for a generic nonzero $\lambda \neq \lambda_{SC}$, it is not possible to write an explicit expression of $S=A_H/4G_5$ as a function of $Q,J$. Instead, a straightforward numerical study of the algebraic relations reveals a rather complicated picture, with several branches of solutions. For example, two different near horizon solutions may exist with the same global charges $J,Q$ (see Fig. \[fig4\]).
[**The results.–**]{} The global solutions are found by solving numerically the EMCS equations subject to the boundary conditions described above. In the numerical calculations [@numerics], we introduce a compactified radial coordinate $\bar{r}= r/(1+r)$ and employ units such that $16 \pi G_5=1$.
We start by exhibiting in Fig. \[fig1\] the scaled angular momentum $j=J/M^{3/2}$ versus the scaled charge $q=Q/M$. The results there and also in Figures \[fig3\], \[fig4\] and \[fig5\] are for a CS coupling constant $\lambda=5$; however, a similar picture has been found for other values of $\lambda>2$. Fig. \[fig1\] exhibits the domain of existence of EMCS black holes, since all non-extremal black hole solutions reside within the outer boundary, formed by extremal black holes.
For spinning solutions, the CS term breaks the symmetry $Q\to -Q$. The extremal solutions with negative charge form the left outer boundary. This contains the extremal static solution where $J$ vanishes. The extremal solutions with positive charge, on the other hand, represent a much more interesting set of solutions.
First of all, these solutions extend much further in $q$. At $q_{\rm max}$ the solutions are singular and possess zero area. Second, the right outer boundary does not contain the static solution. Instead, two rotating $J=0$ solutions are encountered at a value $q_1$ of the scaled charge that is still considerably larger than the static value. Thus for the same global charges, there are two distinct (but symmetric) solutions at $q_1$. Interestingly, however, the branches of extremal black hole solutions also extend inside the domain of existence, where they form an intriguing pattern of branches. In particular, a whole sequence of rotating $J=0$ solutions arises. Found at $q_2,q_3,\dots$ , all these solutions come in pairs with the same global charges.
Investigating this sequence of extremal $J=0$ solutions for fixed charge $Q$ in more detail we realize, that these solutions constitute a set of radially excited extremal solutions, that can be labeled by the node number $n$ of the magnetic gauge potential $a_k(r)$, as seen in Fig. \[fig2\], or equivalently, of the metric function $\omega(r)$. The first node always refers to spatial infinity. We have constructed solutions with up to 30 nodes; thus it is likely that they form an infinite sequence.
With increasing node number $n$, the mass $M_n$ converges monotonically from below to the mass of the extremal static black hole, $M_{\rm RN}$ (see Fig. \[fig3\]). Surprisingly, however, the horizon area has the same value for all solutions of the sequence and the same holds for the magnitude of the horizon angular momentum. For comparison, we exhibit in Fig. \[fig4\] the area versus the angular momentum as obtained within the near horizon formalism for the same charge $Q$ (Figs. \[fig2\]-\[fig6\] are for a given value $Q=8 \sqrt{3} \pi^2$; for completeness, the charge $-Q$ is also shown).
The set of near horizon solutions exhibits only three $J=0$ solutions, the extremal static one and two (symmetric) rotating solutions with lower area. The latter have precisely the values found for all solutions of the sequence. Thus these two near horizon solutions are not only associated with a single global solution each, but with whole sequences of global solutions.
Let us next address the full set of extremal solutions for fixed charge $Q$ inside the domain of existence. As expected, sequences of radially excited solutions [@non-extr] exist also for nonzero $J$. As seen in Fig. \[fig3\] two (symmetric) $n=1$ branches of solutions extend from the $n=1$ $J=0$ solutions up to a local minimum resp. maximum of the angular momentum. There cusps $C_{1,2}$ are encountered formed with higher mass $n=2$ branches. These higher mass branches then pass the $n=2$ $J=0$ solutions and end at bifurcations $B_{1,2}$ with the opposite $n=1$ branches.
Considering this same set of solutions in Fig. \[fig4\] we see, that the cusps $C_{1,2}$ encountered by the global solutions do not correspond to the cusps $C_{\rm nh}$ of the solutions obtained in the near horizon fomalism, since the branches of global solutions end before the branches of near horizon solutions end. Thus there are sets of near horizon solutions which do not possess counterparts in the sets of global solutions [@Galtsov].
The bifurcations $B_{1,2}$ of the $n=2$ branches with the opposite $n=1$ branches are clearly visible in Fig. \[fig5\], where we exhibit the horizon angular velocity $\Omega_{\rm H}$ of the extremal solutions versus the angular momentum $J$. On the $n=2$ branches bifurcations $B_{2,3}$ with $n=3$ branches arise. Passing the $n=3$ $J=0$ solutions, these branches end in cusps $C_{3,4}$, formed with branches of the $n=4$ solutions. The $n=4$ branches pass the $n=4$ $J=0$ solutions and end in bifurcation points $B_{3,4}$ on the $n=3$ branches. This pattern then repeats again and again for the higher node branches.
In this way a whole sequence of branches is generated. Since the cusps $C_{n,n+1}$ occur at decreasing values of $|J|$, the number of extremal global solutions for fixed $|J|$ decreases with increasing $|J|$, whenever a cusp is passed. We conclude that a given near horizon solution can correspond to [*i) more than one global solution*]{} (possibly even an infinite set), [*ii) precisely one global solution*]{}, or [*iii) no global solution at all*]{}.
Interestingly, the presence of the bifurcation points $B_{n,n+1}$ indicates, that at each of these points there are two distinct solutions with the same global charges. They possess, however, different values of the area and thus correspond to different near horizon solutions.
Let us finally address the $\lambda$ dependence of the observed pattern of extremal solutions. This is shown in Fig. \[fig6\] where we exhibit the mass $M$ versus the CS coupling $\lambda$ of the rotating $J=0$ solutions with one to seven nodes. We note, that we do not find rotating $J=0$ solutions below $\lambda=2$, and the mass $M_n$ of the solutions with $n$ nodes approaches the mass $M_{\rm RN}$ of the static extremal solution, as $\lambda$ is decreased towards $\lambda=\lambda_{SG}$.
[**Further remarks.–**]{} The results in this work show that the intuition based on known exact solutions cannot be safely applied in the general case. Working in EMCS theory, we have shown that beyond a critical value of the CS coupling $\lambda$, for fixed charge $Q$, a sequence of branches of extremal radially excited black holes arises. To our knowledge, this is the first example of black holes with Abelian fields which form excited states reminiscent of radial excitations of atoms [@note2].
Also, these black holes clearly illustrate that the relation between the global solutions and the near horizon solutions may be rather intricate. Since there are near horizon solutions that do not correspond to global solutions, while other near horizon solutions correspond to one or more global solutions, possibly even infinitely many.
As we decrease the CS coupling below the critical value, this intriguing pattern of global solutions disappears. However, in the limit $\lambda \to 0$ when EM theory is obtained, we again encounter two branches of extremal solutions. The first branch is connected to the static RN black hole and has horizon area $A_H=\pi 3^{3/4}J^2 Q^{-3/2}/\sqrt{2}+3^{1/4}\sqrt{2}Q^{3/2}/48\pi$. The second branch originates at the Myers-Perry solution, and has the rather unusual property to possess an entropy independent of the electric charge, $A_H=J/2$. Both branches are associated with near horizon solutions, that are only partly realized as global configurations.
Finally, we conjecture that extremal black holes with similar properties may also exist in other theories, in particular in a $D=4$ EM-dilaton theory.\
[**Acknowledgement**]{} We gratefully acknowledge support by the Spanish Ministerio de Ciencia e Innovacion, research project FIS2011-28013, and by the DFG, in particular, the DFG Research Training Group 1620 ”Models of Gravity”. J.L.B is supported by the Spanish Universidad Complutense de Madrid.
[000]{} A. Strominger and C. Vafa, Phys. Lett. B [**379**]{}, 99 (1996) \[arXiv:hep-th/9601029\]. J. C. Breckenridge, D. A. Lowe, R. C. Myers, A. W. Peet, A. Strominger and C. Vafa, Phys. Lett. B [**381**]{}, 423 (1996) \[arXiv:hep-th/9603078\]. J. C. Breckenridge, R. C. Myers, A. W. Peet and C. Vafa, Phys. Lett. B [**391**]{}, 93 (1997) \[arXiv:hep-th/9602065\]. M. Cvetic, H. Lu and C. N. Pope, Phys. Lett. B [**598**]{}, 273 (2004) \[arXiv:hep-th/0406196\]. Z. W. Chong, M. Cvetic, H. Lu and C. N. Pope, Phys. Rev. Lett. [**95**]{}, 161301 (2005) \[arXiv:hep-th/0506029\]. J. P. Gauntlett, R. C. Myers and P. K. Townsend, Class. Quant. Grav. [**16**]{}, 1 (1999) \[arXiv:hep-th/9810204\]. C. A. R. Herdeiro, Nucl. Phys. B [**582**]{}, 363 (2000) \[hep-th/0003063\]. P. K. Townsend, Annales Henri Poincare [**4**]{}, S183 (2003) \[hep-th/0211008\]. J. Kunz and F. Navarro-Lerida, Phys. Rev. Lett. [**96**]{}, 081101 (2006) \[hep-th/0510250\]. J. Kunz, F. Navarro-Lerida and J. Viebahn, Phys. Lett. B [**639**]{}, 362 (2006) \[hep-th/0605075\]. D. Astefanesei, K. Goldstein, R. P. Jena, A. Sen and S. P. Trivedi, JHEP [**0610**]{} (2006) 058 \[hep-th/0606244\]. H. K. Kunduri and J. Lucietti, JHEP [**0712**]{} (2007) 015 \[arXiv:0708.3695 \[hep-th\]\]. N. V. Suryanarayana and M. C. Wapler, Class. Quant. Grav. [**24**]{} (2007) 5047 \[arXiv:0704.0955 \[hep-th\]\]. We employ a collocation method for boundary-value ordinary differential equations, equipped with an adaptive mesh selection procedure [@COLSYS]. Typical mesh sizes include $10^3-10^4$ points, the solutions having a relative accuracy of $10^{-8}$.
U. Ascher, J. Christiansen, R. D. Russell, Mathematics of Computation [**33**]{} (1979) 659; ACM Transactions [**7**]{} (1981) 209. A family of radially excited solutions exists also in the non-extremal case for large enough values of $\lambda$, though with a finite maximal value of $n$. This feature was also found for extremal black holes in D=4 Gauss-Bonnet gravity [@Chen:2008hk]. C. -M. Chen, D. V. Gal’tsov and D. G. Orlov, Phys. Rev. D [**78**]{} (2008) 104013 \[arXiv:0809.1720 \[hep-th\]\]. This resembles also the case of black holes with non-Abelian gauge fields [@Volkov:1998cc]. M. S. Volkov and D. V. Gal’tsov, Phys. Rept. [**319**]{} (1999) 1 \[hep-th/9810070\].
|
---
abstract: 'Present and planned dark matter detection experiments search for WIMP-induced nuclear recoils in poorly known background conditions. In this environment, the maximum gap statistical method provides a way of setting more sensitive cross section upper limits by incorporating known signal information. We give a recipe for the numerical calculation of upper limits for planned directional dark matter detection experiments, that will measure both recoil energy and angle, based on the gaps between events in two-dimensional phase space.'
author:
- Shawn Henderson
- Jocelyn Monroe
- Peter Fisher
bibliography:
- 'patchPaper.bib'
date: 'July 22, 2008'
title: The Maximum Patch Method for Directional Dark Matter Detection
---
\[sec:introduction\]Introduction
================================
Dark matter comprises approximately 25% of the mass of the universe [@r:spergel2003]. The generic dark matter candidate is a weakly interacting massive particle (WIMP). If WIMPs are supersymmetric particles, the predicted mass is in the range of 10 to 10$^4$ GeV/c$^2$, and the expected cross section lies in the range of $10^{-42}$ to $10^{-48}$ cm$^2$ [@r:ellis2005]. Many experiments seek to detect dark matter particles via their elastic scattering interactions with detector nuclei [@r:gaitskell2004]. Recent measurements [@r:xenon2007; @r:cdms2005] limit the cross section to be less than approximately $5\times10^{-44}$ cm$^2$. Given the small size of the expected WIMP cross section, discrimination against backgrounds is of paramount importance in direct detection experiments. For the same reason, there is much to gain by optimizing the statistical methods used to interpret experimental data as upper limits on the WIMP interaction cross section [@r:yellin].
In this paper, we develop a new statistic, the maximum patch, for setting limits on the WIMP-nucleus interaction cross section. This method is motivated by directional dark matter experiments, which seek to measure both the nuclear recoil energy, and the recoil angle of the struck nucleon in WIMP-nucleon interactions. In section \[sec:overview\] we introduce the theoretical distributions used for setting limits, and in sections \[sec:1dstatistics\] and \[sec:2dstatistics\] we discuss limit setting techniques within the context of one- and two-dimensional WIMP detection experiments. In section \[sec:comparison\] we compare results in the cases of (i) observing a signal with no background, (ii) observing some signal and some background, and (iii) observing only background.
\[sec:overview\]Setting Dark Matter Cross Section Limits
========================================================
Direct detection experiments typically measure the energy deposited by the recoil nucleus [@r:gaitskell2004], infer the true nuclear recoil energy, and set upper limits on the WIMP-nucleus interaction cross section by comparing the theoretical distribution with this one-dimensional data set. The theoretical event rate distribution is given by [@r:lewin] $$\frac{dN}{dE} \ = \ \frac{R_0}{E_0 r} \frac{1}{2 \pi v_0^2} \int_{v_{threshold}}^{v_{max}} \frac{1}{v} f(v,v_E) d^3v
\label{eq:1Ddistribution}$$ where $E$ is the nuclear recoil energy, $E_0=\frac{1}{2} m_D v_0^2$ is the dark matter particle’s kinetic energy, $r = 4 m_D m_T / (m_D + m_T)^2$ with dark matter particle mass $m_D$ and target nucleus mass $m_T$, $v_{threshold}$ and $v_{max}$ are the minimum observable and escape velocities of the dark matter (taken to be the dark matter velocity that produces a maximum recoil energy equal to the experimental lower limit recoil energy threshold and $\infty$ here respectively, for simplicity), $v_E$ = 244 km/s is the Earth’s velocity relative to the dark matter halo, and $f(v,v_E)$ is the dark matter velocity distribution function, assumed here to be a Maxwell-Boltzmann distribution with RMS velocity $v_{0}$ = 230 km/s. The normalization factor $R_0$ is the event rate per unit mass with $(v_{threshold},v_{max})$ = (0,$\infty$), $$R_0 \ = \ \frac{2}{\sqrt{\pi}} \frac{N_0}{A} \frac{\rho_D}{m_D} v_0 \sigma_0
\label{eq:r0}$$ where $N_0$ is Avogadro’s number, $A$ is the atomic mass of the target, $\rho_D$ is the dark matter density, taken here to be 0.3 $(GeV/c^2)/cm^3$, and $\sigma_0$ is the zero momentum-transfer dark matter-nucleus interaction cross section. We use the fact that the differential interaction rate scales simply with $\sigma_0$ in the following discussion of limit setting techniques.
A new thrust in the field of WIMP searches has been to develop detector sensitivity in a second dimension, the nuclear recoil angle [@r:Burgos:2007gv; @r:Miuchi:2007ga; @r:Dujmic:2007bd]. The WIMP-nucleus interaction signal is expected to be highly anisotropic in recoil direction because of the earth’s motion with respect to the WIMP halo [@r:Spergel:1987kx]. In contrast, the backgrounds of most WIMP experiments are relatively isotropic in recoil angle in the detector coordinate system [@r:mei2005], and therefore this experimental approach can provide increased discrimination against backgrounds. It has recently been suggested that WIMP direct detection searches relying on a recoil energy signature alone may be insufficient for distinguishing WIMP events from nuclear recoils if the WIMP-nucleus cross section is smaller than the coherent scattering cross section for solar neutrinos [@r:jmonroe]. This makes directional detection particularly attractive, since solar neutrino-induced recoils point back to the sun, unlike recoils from WIMP interactions. Directional detection could also potentially probe the velocity distribution of our galaxy’s dark matter halo [@r:Host:2007fq]. The theoretical distribution as a function of nuclear recoil energy $E$ and recoil angle $\psi$ (where $\psi$ is the angle in the detector lab frame between the nuclear recoil track observed in the detector and the direction the dark matter “wind” is blowing, which is normally taken to be the vector pointing from the constellation Cygnus to Earth) is given by [@r:Spergel:1987kx; @r:lewin] $$\frac{d^2N(v_E,\infty)}{dE d(cos\psi)} \ = \ \frac{1}{2}\frac{R_0}{E_0 r}exp\biggl(-\frac{(v_E cos\psi - v_{min})^2}{v_0^2}\biggr)
\label{eq:2Ddistribution}$$ where $v_{min} \ = \ (E / E_0 r)^{1/2} v_0$ is the smallest dark matter particle velocity which can produce a nuclear recoil with energy $E$.
Given a theoretical distribution, one can compare with an observation to set a limit on the WIMP-nucleus interaction cross section. The usual method to obtain an upper limit at some confidence level is to vary the theoretical parameters until the appropriate cumulative probability distribution function (CDF) takes on the confidence level desired ($0.9$ for a $90$% confidence level upper limit) when evaluated at the observed statistic (e.g. the number of observed events). In the following two sections, we discuss upper limit calculations using the one- and two-dimensional theoretical distributions respectively, together with several statistics of interest.
\[sec:1dstatistics\]Dark Matter Statistics in One Dimension
===========================================================
Here we compare the traditional Poisson method with the maximum gap, a statistic often used in dark matter experiments for obtaining an upper limit with one-dimensional data [@r:xenon2007; @r:cdms2005]. While the traditional Poisson method is based solely on the number of counts observed [@r:pdg], the maximum gap procedure incorporates what is known about the shape of the expected signal into the limit determination [@r:yellin]. For the discussion that follows, consider a series of nuclear recoil energy measurements $\{E_{1},...,E_{N}\}$ where $N$ is the total number of measurements. Assume the data points are distributed with a known theoretical function $dN(\vec{\lambda})/dE$, where the $\vec{\lambda}$ are the parameters of the theoretical model $dN/dE$ given in equation \[eq:1Ddistribution\]. Assuming standard values for the dark matter halo parameters, there is only one free parameter which we then vary to set upper limits, the zero-momentum transfer dark matter-nucleus interaction cross section $\sigma_{0}$ in equation \[eq:r0\].
\[subsec:poisson\]Poisson Method
--------------------------------
A straightforward cross section upper limit on a given data set can be obtained by employing the Poisson method. To set a limit, we are interested in the probability, given a value of the cross section $\sigma_{0}$ in a theoretical distribution $dN/dE$, that the total number of events observed in our data is equal to a certain value or less. If we are conservative and assume no knowledge of the background distribution and therefore that all observed events are signal, then an upper limit at some desired confidence level may be set by adjusting $\sigma_{0}$ in $dN/dE$ until the total number of events $\mu$ expected, given by integrating $dN/dE$ over the whole experimental range, is such that it satisfies equation \[eq:PoissonCDF\]. $$\alpha=e^{-\mu}\displaystyle\sum_{m=0}^N \frac{\mu^{m}}{m!}
\label{eq:PoissonCDF}$$ Here, $1-\alpha$ is the confidence level of the upper limit set in this way, and $N$ is the number of observed data events. In order to incorporate knowledge of expected backgrounds into equation \[eq:PoissonCDF\], we must use a modified form of this relation that assumes the overall normalization of the background, which is often poorly understood in dark matter direct detection experiments (for instance, see equation 32.35 in [@r:pdg]). The most conservative approach is to assume no knowledge of the backgrounds, and so all events are signal candidates. In this case, any observed events considerably degrade the upper limit obtained with equation \[eq:PoissonCDF\]. This is particularly true for scenarios in which a background fills a small subset of the full experimental acceptance. For nuclear recoil signals, as the energy detection threshold is lowered, the sensitivity to backgrounds increases. Background events observed near detection threshold are counted with the same significance as events in higher recoil energy sub-intervals of the experimental acceptance. Direct dark matter detection experiments gain sensitivity to WIMP events by lowering their energy thresholds, since the WIMP-nucleon interaction rate is expected to peak at low nuclear recoil energies. In this scenario, the Poisson method can lead to overly conservative upper limits on the WIMP-nucleon interaction cross section, in the presence of backgrounds.
\[subsec:maxgap\]Maximum Gap Method
-----------------------------------
For a given $\sigma_{0}$, the “gap” for a pair of data points is defined to be [@r:yellin] $$x_{i}=\int^{E_{i+1}}_{E_{i}} \frac{dN}{dE}(\sigma_{0}) dE
\label{eq:gapDefinition}$$ where $x_{i}$ is the value obtained by integrating $dN/dE$ between the observed energy values $[E_{i},E_{i+1}]$ for $i=0,..,N$ (see figure 1 in [@r:yellin]) and $E_{0}$ and $E_{N+1}$ are the lower and upper recoil energy experimental thresholds. A set of $N$ recoil energy measurements yields an $(N-1)$-dimensional vector $\vec{x}$ of gaps. The “maximum gap” for a set of $N$ recoil energy measurements is defined to be the largest member of the set of all gaps that can be computed from the data. This quantity depends on an integral over the hypothesized theoretical distribution, and through that integral, on the WIMP interaction cross section $\sigma_{0}$. The larger the maximum gap given a $\sigma_0$, the larger the discrepancy between the number of points observed in data and the number of points expected. Therefore, the maximum gap allows a powerful statistical test between the measured data and the normalization of the theoretical signal distribution.
To set a limit, we are interested in the probability, given a value of the cross section $\sigma_{0}$ in a theoretical distribution $dN/dE$, that the maximum gap for a given set of data is equal to a certain value or less. This is described by the CDF of the maximum gap, and can be analytically calculated [@r:yellin], $$C_{0}(x,\mu)=\displaystyle\sum_{k=0}^m \frac{(kx-\mu)^{k}e^{-kx}}{k!}\left(1+\frac{k}{\mu-kx}\right)
\label{eq:maximumGapCDF}$$ where $\mu$ is the total number of events expected in the experimental range ($dN/dE$ integrated from lower to upper limit energy threshold), and $m$ is the greatest integer $\leq$ $\mu/x$. An upper limit at a given confidence level is obtained from equation \[eq:maximumGapCDF\] by adjusting the input cross section $\sigma_{0}$ until the function $C_{0}$ above, evaluated at the maximum gap of the data, yields $0.90$. The interpretation is that in an ensemble of experiments, on each of which the upper limit setting technique is employed, $90$% will obtain an upper limit greater than $\sigma_0$.
\[subsec:limittechniques\]Discussion of Limit Setting Techniques
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The maximum gap statistic possesses a number of nice qualities, particularly in the presence of background, which motivate the generalization of the method to two dimensions for directional dark matter detection experiments. We summarize the main results here; see [@r:yellin] for a rigorous discussion.
First, the maximum gap is unchanged under a one-one transformation of the variable in which the events are distributed. This can be seen by making a transformation from recoil energy in the above discussion to a variable $\rho$ such that $\rho$ is equal to the total number of events expected between the point $E$ and the lower energy threshold. In $\rho$, equation \[eq:1Ddistribution\] is a uniformly distributed, unit density function. This calculation is treated in more detail in Appendix I. This means that the maximum gap does not depend on the form of the theoretical distribution. Second, the method can be used for an arbitrarily large number of observed data points, and requires no binning of the data points. Most importantly, this statistic provides a conservative upper limit on the true WIMP cross section that can considerably out-perform the Poisson upper limit setting technique.
In WIMP detection experiments there are often low energy backgrounds from processes which are difficult to model accurately with simulations, such as MeV-scale neutron interactions or $^{238}$U and $^{232}$Th decay progeny in the detector materials. It is unlikely that the measured maximum gap will be found in an interval of an experiment’s acceptance that contains both signal WIMP events and a large number of background events. The presence of sizable background would significantly shorten the gap sizes expected from signal alone. We thus expect the maximum gap to occur in the data in an interval where the background does not dominate. In this way, the maximum gap method automatically selects recoil energy intervals that are characteristic of the expected WIMP signal alone.
Another striking advantage of the maximum gap method is that, for a fixed gap size, it is independent of the total number of events observed. In the Poisson case, each additional point observed inflates the upper limit an experimenter sets on their data. On the other hand, if an experimenter observes a large gap in their data, the limit set on that data is unchanged if the number of points observed outside of the maximum gap is one or one million.
\[sec:2dstatistics\]A New Dark Matter Statistic for Two Dimensions
==================================================================
For directional dark matter detection experiments, it is desirable to preserve the benefits of the maximum gap method for setting upper limits. Towards this end, we need to generalize the method to two dimensions. We consider a series of measurements $\{(E_{1},\cos(\psi)_{1}),...,(E_{N},\cos(\psi)_{N})\}$, where $N$ is the total number of measurements of the energy of nuclear recoils $E$, and $\psi$ is the nuclear recoil angle in a dark matter detection experiment measured from the vector pointing from the constellation Cygnus to the Earth in the detector lab frame. In general, the two-dimensional rate will be given by a function $d^{2}N(\vec{\lambda})/d(\cos(\psi))dE$, where the $\vec{\lambda}$ are the theoretical parameters of the model. As in one dimension, the model for the two-dimensional differential WIMP-nucleon interaction rate, equation \[eq:2Ddistribution\], under standard dark matter halo assumptions, depends only on $\sigma_{0}$, the true WIMP interaction cross section. In the following we describe an algorithm for obtaining general, multi-dimensional CDFs, focusing on the two-dimensional case. We then apply the usual prescription for setting upper limits, varying $\sigma_0$, and find that our two-dimensional limit setting technique has the correct $90$% coverage.
\[sec:mc\_cdf\]Monte Carlo Generated Cumulative Distribution Functions
----------------------------------------------------------------------
To calculate the CDFs for an arbitrary statistic of interest (SI), we simplify matters by asking an easier question than “what is the probability that the SI is less than or equal to a certain value” and instead ask “what is the probability that, given an observation of $N$ events, the SI is less than or equal to a certain value.” The advantage of the latter question is that it can be addressed with Monte Carlo methods, and leads naturally to the resolution of the first question.
We start by generalizing the one-dimensional case. The theoretical distribution in equation \[eq:1Ddistribution\], assuming a value for $\sigma_0$, gives a concrete form for $dN/dE$, the expectation for how the observed events are distributed in nuclear recoil energy. Then, we draw $N$ events from this distribution. We compute the value of the SI on this fake data, whether it be the maximum gap of the distribution, or some other SI. We repeat this procedure many times, until we have an SI frequency distribution, given $N$ observed events. We do this in turn for $N=1,2,...,N_{max}$ stopping for $N_{max}$ so large that the Poisson probability (equation \[eq:PoissonProbability\]) for observing $N_{max}$ events is negligible. This results in an array $\vec{h}=\{h_{0},...,h_{N_{max}}\}$ of histograms, where $h_{0}$ is the frequency plot of the SI given the observation of zero events, $h_{1}$ is the frequency plot of the SI given the observation of one event, etc.
In each of the $h_{i}$, $i=1,...,N_{max}$, we have $N_{toys}$ entries, where $N_{toys}$ is the number of toy experiments, for each number of observed events, that we performed. By normalizing each of the $h_{i}$ by $N_{toys}$, we obtain the probability distribution of the SI, for each of the $i$ events observed subclasses considered. The resulting normalized vector of histograms can be properly interpreted as the probability distribution functions (PDFs) of the SI $\hat{h}=\vec{h}/N_{toys}$.
For setting an upper limit, we need the CDFs of the SI. To construct these, we take each histogram member $\hat{h}_{i}$ of $\hat{h}$ in turn and create a new histogram, $\hat{h}_{c,i}$, assigning to each bin $b'_{mi}$ of $\hat{h}_{c,i}$ the value given in equation \[eq:binToBinSum\], where $b_{ki}$ is the $k^{th}$ bin of $\hat{h}_{i}$, and $k$ and $m$ index the bins of the PDF and CDF histograms, respectively. $$b'_{mi}=\displaystyle\sum_{k=1}^m b_{ki}
\label{eq:binToBinSum}$$ For convenience, $500$ bin CDF and PDF histograms were generated for the maximum gap studies in this paper, and $300$ bin CDF and PDF histograms were generated for the maximum patch studies.
In this way we numerically turn each of the PDFs in $\hat{h}$ into a CDF in $\hat{h}_{c}$ for the SI. Now we have $\hat{h}_{c}$, a vector of CDFs, each corresponding to a number of observed events. The probability of observing a certain number of events is determined by the Poisson probability of observing $N$ events given $\mu$ total events, $$P(\mu,N)=\frac{\mu^{N}}{N!}e^{-\mu}.
\label{eq:PoissonProbability}$$ In order to construct the full CDF of the SI for the input theoretical distribution $dN(\sigma_{0})/dE$, we add all of the histograms in $\hat{h}_{c}$ together, weighting each by the probability in equation \[eq:PoissonProbability\] for seeing that number of events. This yields the following equation for the Monte Carlo generated CDF for the SI, $$C_{SI}(x_{SI},\mu)=\displaystyle\sum_{k=0}^{N_{max}} \hat{h}_{c,k}(x_{SI}) \left ( \frac{\mu^{k}}{k!}e^{-\mu} \right )
\label{eq:MCMaximumGapCDF}$$ Here $x_{SI}$ is the value of the SI at which we want to know the value of the CDF, and $\mu$ is the total number of events expected in the experimental range. The notation $\hat{h}_{c,k}(x_{SI})$ means to evaluate the value of the $k^{th}$ histogram in the $\hat{h}_{c}$ vector of SI CDFs at $x_{SI}$. This can be done in a number of different ways; we can take this as the height of the bin in $\hat{h}_{c,k}$ that $x_{SI}$ falls into (thereby obtaining a discrete CDF), or, perform a bin-to-bin interpolation, thus turning the $\hat{h}_{c,k}$’s into smooth functions of $x_{SI}$. In the results presented in this paper, the $\hat{h}_{c,k}$ histograms are interpolated using splines.
We have now in equation \[eq:MCMaximumGapCDF\] manufactured the analogue to equation \[eq:maximumGapCDF\] for the maximum gap statistic for an arbitrary SI. Note however that for the maximum gap statistic this discussion is unnecessarily complicated because the maximum gap is unchanged under a one-one transformation of the theoretical distribution $dN/dE$ in $E$, as shown in Appendix I. This property allows one to transform any distribution into a unit density, uniformly distributed function. Thus whenever we change $\sigma_{0}$ for the maximum gap statistic, we need not draw events from a different distribution, we may instead always use a unit density, uniform distribution.
We validate the Monte Carlo CDF generating scheme by comparing the frequency distribution of upper limits resulting from the Monte Carlo version of the maximum gap method with the result obtained using the analytic CDF in equation \[eq:maximumGapCDF\]. We find that the results agree within numerical errors.
\[sec:maxpatchmethod\]Maximum Patch Method
------------------------------------------
Analogously to the one-dimensional gap, we may construct a “patch” as a subset of an experiment’s total acceptance, which is determined by $(E_{0},\cos(\psi)_{0})$, the energy and angle lower limit experimental threshold of measurement, and $(E_{N+1},\cos(\psi)_{N+1})$, the energy and angle upper limit experimental threshold of measurement. We define a patch for a set of $N$ data points to be $$y_{ijk}=\int^{\cos(\psi)_{j}}_{\cos(\psi)_{k}} \int^{E_{i+1}}_{E_{i}} \frac{d^{2}N}{d(\cos(\psi))dE}(\sigma_{0}) dE d(\cos(\psi))
\label{eq:patchDefinition}$$ where $i$ ranges from $0$ to $N$ and $j$ and $k$ range, independently, from $1$ to $N$. We require that $\cos(\psi)_{j}>\cos(\psi)_{k}$ and that $E_{i}<E_{j}<E_{i+1}$ and $E_{i}<E_{k}<E_{i+1}$. We also include the additional patch candidate not picked up by this prescription for every $i$; namely that one which has as its borders in $\cos(\psi)$ the upper and lower angular limits $[\cos(\psi)_{0},\cos(\psi)_{N+1}]$. Equation \[eq:patchDefinition\] describes rectangular patches, whose limits are $[E_{i},E_{i+1}]$ in $E$ and $[\cos(\psi)_{k},\cos(\psi)_{j}]$ (plus $[\cos(\psi)_{0},\cos(\psi)_{N+1}]$) in $\cos(\psi)$. Some of the rectangles described by equation \[eq:patchDefinition\] may have points inside their boundaries; these are disqualified from being maximum patches, for the same reason that gaps with points in them are disqualified in the maximum gap method. In principle any two-dimensional shape can be used to define a patch; we have chosen to use rectangles for ease of computation. It is possible that sensitivity could be gained in this method by considering patch geometries other than rectangles.
To set an upper limit we are interested in the maximum value that $y_{ijk}$ takes in equation \[eq:patchDefinition\] for all acceptable values of $j$, $k$ and $i$. The situation is illustrated in figure \[fig:method2D\], where the spikes are Monte Carlo generated fake data points in $E$ and $\cos(\psi)$ (4 in total) and the smooth curve represents $d^{2}N(\sigma_{0})/d(\cos(\psi))dE$ for a given cross section value. The maximum patch candidate in this example has boundaries that extend from the lower to the upper $\cos(\psi)$ threshold, and from $5-15$ $keV$. Note that this particular maximum patch candidate is also a maximum gap candidate in $E$-space.
![\[fig:method2D\]An illustration of a maximum patch candidate. The spikes are hypothetical measured data points in an experiment, and the smooth curve is, for some assumed cross section, the expected distribution of signal between the two lowest energy data points. The volume under the curve is one of the $y_{ijk}$’s of equation \[eq:patchDefinition\] for this dataset, and thus a maximum patch candidate. Note that if we project the data points and theoretical distribution onto the energy interval, this is also a maximum gap candidate in the nuclear recoil energy dimension.](maximumPatchMethodPlot.eps){width="9cm"}
Our algorithm for computing the maximum patch on a set of observed two-dimensional data points is described in detail in Appendix II.
![\[fig:maximumPatchAlgorithmExample\]An example illustrating the maximum patch algorithm for 1 assumed data point in the experimental acceptance. If the data point is not on one of the four boundaries, there are four patches that contribute. Our algorithm is detailed in Appendix II; the order in which our algorithm computes the above maximum patch candidates is upper left, upper right, lower left and lower right.](maximumPatchAlgorithmExample.eps){width="9cm"}
An example of the patch-finding algorithm for $1$ observed event is shown in figure \[fig:maximumPatchAlgorithmExample\]. Once the maximum patch is found, one calculates the PDFs and sums them, appropriately weighted, to produce the CDFs as in section \[sec:mc\_cdf\]. Figure \[fig:maximumPatchCDFs\] shows the resulting CDF for several expected numbers of events $\mu$.
We note that we are able to use an analogue to the simplified CDF generation scheme mentioned at the end of \[sec:mc\_cdf\] for the maximum gap method, with one important change. Unlike the maximum gap statistic, the maximum patch is not unchanged under a one-one transformation of the theoretical distribution, $d^{2}N/d(\cos(\psi))dE$, in $E$ and $\cos(\psi)$. Therefore, to set limits on data distributed according to $d^{2}N/d(\cos(\psi))dE$ with CDFs generated from a unit density, uniformly distributed function, one must use the transformation given in Appendix C of [@r:yellin2].
![\[fig:maximumPatchCDFs\]Cumulative probability distribution functions (CDFs) for various total expected numbers of events $\mu$ in the experimental acceptance (equation \[eq:MCMaximumGapCDF\] for the maximum patch SI, equation \[eq:patchDefinition\]). A horizontal line is drawn at the $90$% confidence level. So, for instance, for $\mu=5$, $90$% of the time, the maximum patch of a toy signal experiment will be less than $\approx4$.](maximumPatchCDFs.eps){width="9cm"}
\[subsec:recipe\]The Recipe
---------------------------
The recipe for the experimenter wishing to use the maximum patch method to set an upper limit on the dark matter cross section in her experiment is as follows, assuming a measurement of a vector of $N$ two-dimensional data points $\vec{D}=\{(E_{1},\cos(\psi)_{1}),...,(E_{N},\cos(\psi)_{N})\}$. We take as an explicit example a direction sensitive dark matter direct detection experiment in this recipe, but this method can be used for any two-dimensional dataset for which the distribution of the signal is known.
1. Given a predicted two-dimensional rate $d^{2}N(\sigma_{0})/d(\cos(\psi))dE$, the experimenter calculates the maximum patch of their data for some starting value $\sigma_1$ for the WIMP-nucleon interaction cross section by applying the recipe for calculating the maximum patch of a set of two-dimensional data points outlined in Appendix II.
2. The experimenter then must evaluate equation \[eq:MCMaximumGapCDF\] for the case where the statistic of interest is the maximum patch. The maximum patch CDF for $\mu$ total expected events can either be calculated by following the Monte Carlo procedure outlined in section \[sec:mc\_cdf\] or by referencing the tables provided at the end of this paper in Appendix III.
3. Evaluating equation \[eq:MCMaximumGapCDF\] at the maximum patch of the data yields the confidence level at which $\sigma_1$ is an upper limit on the WIMP-nucleon interaction cross section. If this is less (more) than the confidence level desired, the cross section guess is increased (decreased) to a new guess $\sigma_2$. The experimenter again calculates the maximum patch of the data for this assumed cross section and then, via equation \[eq:MCMaximumGapCDF\], calculates the CL at which $\sigma_2$ is an upper limit on the WIMP-nucleon interaction cross section. This procedure is iterated for as many guesses $\sigma_N$ as required to set the desired confidence level upper limit on the WIMP-nucleon cross section for the data.
\[subsec:validation\]Validation of the Maximum Patch Method
-----------------------------------------------------------
We validate the maximum patch prescription described above by checking that the coverage of our upper limit setting technique is correct. To do this, we generate many ensembles of toy Monte Carlo experiments, each with different WIMP-nucleon interaction cross sections, and with signal events distributed according to equation \[eq:2Ddistribution\]. We set an upper limit on each toy experiment using the maximum patch technique. Note that to test coverage at the $90$% confidence level, we must choose cross sections that yield an expected number of events $\mu>2.3026$. Below this, no cross section upper limit can be set with a confidence level as high as $90$% (see figure \[fig:maximumPatchCDFs\]). For each input cross section $\sigma_0$, and the associated total number of events $\mu$, we perform 10,000 toy experiments, where the observed number of events is Poisson-distributed about $\mu$, and $(E,\cos(\psi))$ for the events are distributed according to equation \[eq:2Ddistribution\]. For each ensemble of 10,000 experiments with different input $\sigma_0$’s, we record the percentage of the time our upper limit is higher than $\sigma_0$. For an upper limit requested at $90$% CL, the percentage should be $90$% within statistical errors, which is the definition of correct coverage. The results of this study, for $\sigma_0$’s such that $\mu=1,...,30$ are shown in figure \[fig:upperLimitCoveragesPlot\]. The stepping behaviour observed in the Poisson limit coverage is expected, due to the discrete nature of the statistic. Figure \[fig:upperLimitCoveragesPlot\] also shows the coverage computed in this way for the maximum gap method (with the CDFs computed via our Monte Carlo technique) and the Poisson method. All methods are observed to have the correct coverage, within statistical and numerical errors.
![\[fig:upperLimitCoveragesPlot\] For toy Monte Carlo experiments with pure signal events, the coverage as a function of signal events for the maximum gap, maximum patch and Poisson methods. This demonstrates that our upper limit setting methods have the correct coverage, within statistical errors for $\mu>2.23026$, as expected. Each point in this plot corresponds to 10,000 MC toy experiments. The errors shown are statistical.](upperLimitCoveragesPlot.eps){width="9cm"}
\[sec:comparison\]Comparison of Methods
=======================================
Having built the maximum patch method, we compare it to various other methods for setting upper limits, in several circumstances of interest. Our goal is to highlight the impact of directionality for dark matter direct detection experiments. Unless otherwise stated, for the various comparisons in this paper we arbitrarily assume a WIMP mass of $60$ GeV, and we use the Xenon10 experiment’s acceptance and target properties [@r:xenon2007] to construct limits (ignoring subtleties like quenching factors, form factors and efficiencies).
First, in the absence of background and with a sizable signal, we would like to verify that the maximum patch method has not only the same coverage as the Poisson method (see subsection \[subsec:validation\]) but also that its performance is comparable as a method for setting upper limits. Towards this end, we employ the ensembles of 10,000 toy Monte Carlo experiments from subsection \[subsec:validation\], recording the median upper limit obtained by the maximum patch, maximum gap and Poisson methods as a function of $\mu$ for each 10,000 event sub-sample generated with a different input $\sigma_0$. This comparison is shown in figure \[fig:upperLimitFrequencyMedians\].
![\[fig:upperLimitFrequencyMedians\]The median upper limit cross section obtained by our implementation of the Poisson, maximum gap and maximum patch procedures divided by the true input cross section $\sigma_{0}$ as a function of input cross section (and thereby, as a function of the total number of expected events in the experimental interval). Each point in this plot corresponds to 10,000 MC toy experiments.](upperLimitFrequencyMediansPlot.eps){width="9cm"}
Figure \[fig:upperLimitFrequencyDist\] shows the frequency distribution of upper limits set by the maximum gap, maximum patch and Poisson methods on the 10,000 event ensemble with an input $\sigma_0$ such that a total of $\mu=7$ signal events are expected. These three distributions are used to generate the $\mu=7$ point in both figure \[fig:upperLimitFrequencyMedians\] and figure \[fig:upperLimitCoveragesPlot\]. The computed coverages in figure \[fig:upperLimitFrequencyDist\] for the maximum gap, maximum patch and Poisson methods are $(90\pm1)$%, $(90\pm1)$% and $(92\pm1)$%, respectively, which is correct, within statistical errors.
![\[fig:upperLimitFrequencyDist\]The frequency distribution of upper limits $\sigma_{UL}$ obtained by applying the Poisson, maximum gap and maximum patch procedures to 10,000 toy Monte Carlo experiments with an input cross section $\sigma_{0}$ shown as a black line on the graph. $\sigma_{0}$ was chosen to yield $7$ total expected events in the experimental interval. The percent of the time that our procedure for each method sets a correct upper limit (a limit above the true input cross section $\sigma_0$) is $(90\pm1)$%, $(90\pm1)$% and $(92\pm1)$% for the maximum gap, maximum patch and Poisson methods, respectively.](upperLimitFrequencyDistPlot.eps){width="9cm"}
We note that in the case of pure signal, the Poisson method outperforms the maximum gap and maximum patch methods by a factor of $\approx1.2$. This is expected; the maximum gap procedure has already been demonstrated to give looser upper limits than the Poisson method in the case of pure signal (see [@r:yellin], figure 3(a)). This could be resolved, as in [@r:yellin] by considering gaps containing greater than zero events, which is termed the optimal gap method. The extension of the optimal gap approach to two dimensions is not considered here. We also note that little sensitivity is gained by using the maximum patch method versus the maximum gap method in the case of pure signal.
Realistically, WIMP direct detection experiments have backgrounds, and so a more interesting comparison is how the maximum gap, maximum patch and Poisson methods do in the presence of sizable backgrounds. For this test, we populate the lower half of our nuclear recoil energy range, and the lower half of our nuclear recoil angular range, with a background drawn according to a flat distribution. We caution that all of our results including background events are highly dependent on this particular background distribution choice. Figure \[fig:upperLimitFrequencyDist2\] shows the frequency distribution of upper limits obtained by applying the maximum gap method, the maximum patch method, and the Poisson method to 10,000 toy experiments generated in this way with a total number of expected background events of $7$, and a total expected number of WIMP signal events of $5$. The coverage is $(100\pm1)$%, $(95\pm1)$% and $(100\pm1)$% for the maximum gap, maximum patch and Poisson methods respectively, which is not at the confidence level requested due to the presence of the large background. The median $90$% confidence level upper limit cross sections, from the maximum gap, maximum patch and Poisson techniques are compared in figure \[fig:upperLimitFrequencyMedians2\], as a function of the total number of expected input background events $\mu=1,2,...,30$ (with $5$ expected signal events in each toy experiment). The total number of background events in a given Monte Carlo experiment, like the total number of signal events, is drawn randomly from a Poisson distribution with the mean given by the total number of expected events.
Figure \[fig:upperLimitFrequencyMedians2\] shows that in the presence of a sizable WIMP signal, the maximum patch procedure provides stronger upper limits than the Poisson or maximum gap procedures as the amount of background contamination increases. The Poisson method does so poorly because it uses only the total number of events to set upper limits, assuming that they are all signal, yielding an increasingly inflated upper limit as the number of background events injected into the toy experiments increases. The maximum patch method outperforms the maximum gap procedure because it includes an extra dimension in which signal and background are differently distributed. The observation that the maximum gap and maximum patch limits seem to asymptotically flatten is due to the overly simplistic background chosen for these studies; eventually the maximum patch or gap is always outside of the lower $E$ interval or $E-\cos(\psi)$ quadrant, and thus characteristic only of the WIMP-signal input which, within statistical fluctuations, is identical outside of the lower $E$ interval or $E-\cos(\psi)$ quadrant as the backgrounds are confined there.
![\[fig:upperLimitFrequencyDist2\]The frequency distribution of upper limits $\sigma_{UL}$ obtained by applying the Poisson, maximum gap and maximum patch procedures to 10,000 toy Monte Carlo experiments with an input cross section $\sigma_{0}$ shown as a black line on the graph. $\sigma_{0}$ was chosen to yield $5$ total expected signal events in the experimental interval. Background events were included in these experiments with a uniform distribution in the lower half of the recoil energy and recoil angle experimental acceptance. For this plot, the total number of expected background events was set at $7$. The percent of the time that our procedure for each method sets a correct upper limit (a limit above the true input cross section $\sigma_0$) is $(100\pm1)$%, $(95\pm1)$% and $(100\pm1)$% for the maximum gap, maximum patch and Poisson methods, respectively.](upperLimitFrequencyDistPlot2.eps){width="9cm"}
![\[fig:upperLimitFrequencyMedians2\]The median upper limit cross section obtained by our implementation of the Poisson, maximum gap and maximum patch procedures divided by the true input cross section $\sigma_{0}$ as a function of background events injected into our toy experiments, uniformly distributed in the lower half recoil energy and recoil angle interval. The point at number of background events equals $7$ comes from dividing the medians of the frequency distributions in figure \[fig:upperLimitFrequencyDist2\] by the true input cross section for the Poisson, maximum gap and maximum patch limit setting procedures. Each point in this plot corresponds to 10,000 MC toy experiments and has, in addition to background, a signal component generated with a cross section corresponding to $5$ total expected signal events.](upperLimitFrequencyMediansPlot2.eps){width="9cm"}
The most probable situation in current dark matter experiments is that the true cross section lies well below the detectable range. To study this scenario, we perform 10,000 toy experiments as above, with $\sigma_0=1\times10^{-46}$ cm$^2$ (or $\approx0$ total expected signal events), and the same distributions of background events as in figures \[fig:upperLimitFrequencyDist2\] and \[fig:upperLimitFrequencyMedians2\] (flat in the $E-\cos(\psi)$ plane, and confined to the lower $E-\cos(\psi)$ quadrant). Figure \[fig:upperLimitFrequencyDist3\] shows the frequency distribution of upper limits obtained by applying the maximum gap method, the maximum patch method, and the Poisson method to 10,000 toy experiments with an expected number of background events of $7$ and negligible signal. The coverage is $(100\pm1)$%, $(100\pm1)$% and $(100\pm1)$% for the maximum gap, maximum patch and Poisson methods, respectively, which is not at the confidence level requested due to the presence of a large background and no signal. The median $90$% confidence level upper limit cross sections, from the maximum gap, maximum patch and Poisson techniques are compared in figure \[fig:upperLimitFrequencyMedians3\] as a function of input background events in the case of negligible signal.
In the presence of a negligible WIMP signal and increasing backgrounds, the maximum patch procedure provides by far the most restrictive upper limit as the amount of background contamination increases. From figure \[fig:upperLimitFrequencyMedians3\] it is clear that the Poisson method limit is not competitive as the number of backgrounds increases, and the maximum patch method outperforms the maximum gap method by at least a factor of $2$ for more than $1$ expected background events for this particular background distribution.
![\[fig:upperLimitFrequencyDist3\]The frequency distribution of upper limits $\sigma_{UL}$ obtained by applying the Poisson, maximum gap and maximum patch procedures to 10,000 toy Monte Carlo experiments with an input cross section $\sigma_{0}=1\times10^{-46}$ shown as a black line on the graph, chosen to give $\approx0$ events for the Monte Carlo toy exposures. Background events were included in these experiments with a uniform distribution in the lower half of the recoil energy and recoil angle experimental acceptance. For this plot, the total number of expected background events was set at $7$. The percent of the time that our procedure for each method sets a correct upper limit (a limit above the true input cross section $\sigma_0$) is $(100\pm1)$%, $(100\pm1)$% and $(100\pm1)$% for the maximum gap, maximum patch and Poisson methods, respectively.](upperLimitFrequencyDistPlot3.eps){width="9cm"}
![\[fig:upperLimitFrequencyMedians3\]The median upper limit cross section obtained by our implementation of the Poisson, maximum gap and maximum patch procedures divided by the true input cross section $\sigma_{0}$ as a function of background events injected into our toy experiments, uniformly distributed in the lower half recoil energy and recoil angle interval for a negligible input signal cross section. The point at number of background events equals $7$ comes from dividing the medians of the frequency distributions in figure \[fig:upperLimitFrequencyDist3\] by the true input cross section for the Poisson, maximum gap and maximum patch limit setting procedures. Each point in this plot corresponds to 10,000 MC toy experiments.](upperLimitFrequencyMediansPlot3.eps){width="9cm"}
In typical dark matter detection experiments, upper limits on the WIMP-nucleon cross section are reported as a function of WIMP mass. In order to compare the performance of the maximum gap, maximum patch and Poisson methods of setting upper limits as a function of WIMP mass, we generated 10,000 Monte Carlo toy datasets at a variety of different WIMP masses. Adopting the likely scenario for a given dark matter experiment, we hypothesize a WIMP-nucleon interaction cross section of $\sigma_0=1\times10^{-46}$ for these toy experiments and a background with an average number of events equal to $10$ (again, flat in the $E-\cos(\psi)$ plane, and confined to the lower $E-\cos(\psi)$ quadrant). The result of this study is the three limit curves in figure \[fig:upperLimitFrequencyMediansPlotVaryingMdark\]. The solid bands represent the RMS widths of the upper limit frequency plots (much like figures \[fig:upperLimitFrequencyDist3\], \[fig:upperLimitFrequencyDist2\] and \[fig:upperLimitFrequencyDist\]) obtained on the 10,000 experiments at each given mass point. The shape of the maximum gap limits at low WIMP masses is an artifact of the background choice. For low WIMP masses, the maximum patch method is by far the most sensitive to the cross section (between $\approx10-100$ GeV) with a very small RMS. At higher WIMP masses, the difference between the maximum gap and maximum patch limit techniques diminishes, but both consistently outperform the Poisson limit setting method. We note that the studies in this paper are all based on one assumed background shape. Different background distributions will lead to different results.
![\[fig:upperLimitFrequencyMediansPlotVaryingMdark\]The median upper limit cross section obtained by our implementation of the Poisson, maximum gap and maximum patch procedures divided by the true input cross section $\sigma_{0}$ as a function of assumed WIMP mass, with $10$ background events on average that are uniformly distributed in the lower half recoil energy and recoil angle interval for a negligible input signal cross section. Each point in this plot corresponds to 10,000 MC toy experiments. The bands correspond to the RMS widths of the upper limit frequency distributions obtained using the Poisson, maximum gap and maximum patch methods, respectively, at each mass point sampled.](upperLimitFrequencyMediansPlotVaryingMdark.eps){width="9cm"}
\[sec:conclusions\]Conclusions
==============================
In this paper, we have developed a new method for setting upper limits in two dimensions. The motivation for our maximum patch method is directional dark matter detection, but it is generally applicable to any two-dimensional data sets for which the distribution of the signal is known. The approach is an extension of the one-dimensional maximum gap method [@r:yellin], a statistic often used to set limits on the WIMP-nucleon interaction cross section in direct detection dark matter experiments. To directly detect dark matter requires unprecedented control and understanding of backgrounds. The great advantage of the maximum gap and patch methods is that they require no knowlege of the background distribution to set conservative upper limits on the WIMP nucleon scattering cross section. The scattering kinematics of a dark matter signal in one and two-dimensional direct detection experiments are relatively simple. This information is included in a straightforward way in the maximum gap and patch methods. In particular, the maximum patch method incorporates information about the large expected angular anisotropy of dark matter scattering into the limit setting procedure. We demonstrate that for simplistic background assumptions, the maximum patch method and two-dimensional dark matter detection yield a large gain in sensitivity, especially at low WIMP masses, over one-dimensional dark matter detection.
\[sec:app1\]Appendix I {#secapp1appendix-i .unnumbered}
======================
Our goal is to prove, in detail, the assertion in [@r:yellin] that the maximum gap is unchanged under a one-one transformation of the variable in which the events are distributed, and thus that the maximum gap is independent of the particular way in which the events are distributed for a given $\sigma_{0}$ in one dimension (by $dN(\sigma_{0})/dE$). The total number of events in the experimental window is given by $\mu$, where $$\mu=\int^{+\infty}_{-\infty} dE \frac{dN}{dE}.
\label{eq:appendix1}$$ Here, if $E$ is outside of the experimental thresholds, $dN/dE=0$, and $dN/dE$ is strictly positive.
We wish to change variables from $E$ in equation \[eq:appendix1\] to $\rho$, where $\rho$ [@r:yellin] is $$\rho(E)=\int^{E}_{-\infty} dE'\frac{dN}{dE'}
\label{eq:appendix2}$$ By the chain rule, $$\frac{dN}{d\rho}=\frac{dN}{dE}\frac{dE}{d\rho}
\label{eq:appendix3}$$ and $dE/d\rho$=$(d\rho/dE)^{-1}$. $d\rho/dE$ from equation \[eq:appendix2\] is then given by the fundamental theorem of calculus to be $$\frac{d\rho}{dE}=\frac{d}{dE}\left(\int^{E}_{-\infty} dE'\frac{dN}{dE'}\right)=\frac{dN}{dE}(E)
\label{eq:appendix4}$$ Thus, equation \[eq:appendix3\] implies that $$\frac{dN}{d\rho}=\frac{dN}{dE}\left(\frac{d\rho}{dE}\right)^{-1}=\frac{\left(\frac{dN}{dE}\right)}{\left(\frac{dN}{dE}\right)}=1
\label{eq:appendix5}$$ and that in the new variable $\rho$, the differential rate is a unit density function, of length $\mu$.
\[sec:app2\]Appendix II {#secapp2appendix-ii .unnumbered}
=======================
Our method for calculating the maximum patch is as follows. A similar scheme is put forth in [@r:yellin2]. Call the set of measured two-dimensional data points on which we wish to determine the maximum patch, $\vec{D}=\{(E_{1},\cos(\psi)_{1}),...,(E_{N},\cos(\psi)_{N})\}$, where as above, $(E_{0},\cos(\psi)_{0})$ and $(E_{N+1},\cos(\psi)_{N+1})$ we define to be the upper and lower recoil energy and recoil angle thresholds of our experiment.
1. Order $\vec{D}$ in $E$. Call the new, $E$-ordered vector $\vec{D}'$.
2. Loop through the intervals $[E_{i},E_{j}]$ in $\vec{D}'$ such that $i$ runs from $0$ to $N+1$ and $j$ runs from $i+1$ to $N+1$.
3. Inside each $E$ interval in this loop, if there are no points with an $E_{k}$ such that $E_{i}<E_{k}<E_{j}$, then compute equation \[eq:patchDefinition\] with $E$ limits $[E_{i},E_{j}]$ and $\cos(\psi)$ limits $[\cos(\psi)_{0},\cos(\psi)_{N+1}]$.
4. If there are $N_{p}$ points with an $E_{k}$ such that $E_{i}<E_{k}<E_{j}$, make a list of them, $\vec{P}=\{(E_{0},\cos(\psi)_{0}),...,(E_{k},\cos(\psi)_{k}),...,$ $(E_{N+1},\cos(\psi)_{N+1})\}$ of length $N_{p}+2$, ordered in $E$, adding the points $(E_{0},\cos(\psi)_{0})$ and $(E_{N+1},\cos(\psi)_{N+1})$ (the threshold points - that’s where the $+2$ comes from in the total number of points in $\vec{P}$) to the front, and back of the list, respectively. Order $\vec{P}$ by $E$, calling the new $E$-ordered vector $\vec{P'}$. Then loop through the intervals $[\cos(\psi)_{P',m},\cos(\psi)_{P',n}]$, in $\vec{P'}$ such that $m$ runs from $0$ to $N_{p}+2$ and $n$ runs from $m+1$ to $N_{p}+2$, where the subscript $(P',m)$ denotes the $m^{th}$ member of the ordered vector $\vec{P'}$. Each iteration of this loop will provide a maximum patch candidate with $E$ limits $[E_{i},E_{j}]$ and $\cos(\psi)$ limits $[\cos(\psi)_{P',m},\cos(\psi)_{P',n}]$.
5. For each of these candidates, check to see if there is a point $D'_{d}$ in $\vec{D'}$ such that $E_{i}<E_{d}<E_{j}$ and $\cos(\psi)_{P',m}<\cos(\psi)_{d}<\cos(\psi)_{P',n}$. If so, then there is a point inside our patch candidate, which means it is not a maximum patch candidate; throw it out.
6. If a maximum patch candidate has passed all of the above criterion, stick it in a vector of some arbitrary length $\vec{y}$. This vector exhausts all possible rectangles in the two-dimensional $E$-$\cos(\psi)$ plane.
7. To find the maximum patch of the data, loop through all of the elements $y_{i}$ of $\vec{y}$, and store the largest $y_{i}$ value; this is the maximum patch.
\[sec:app3\]Appendix III {#secapp3appendix-iii .unnumbered}
========================
Tables I and II below record the $\hat{h}_{c,k}(x_{SI})$’s of the maximum patch statistic for $k=1,...,100$ ($\hat{h}_{c,0}(x_{SI})$ is $0$ for all maximum patch values except for the patch equal to the total number of expected events). The maximum patch CDF is computed by interpolating these points into smooth curves and using them to evaluate equation \[eq:MCMaximumGapCDF\]. Note that the smoothed curves obtained from the tables are functions of $(y/\mu)$, the maximum patch value divided by the total number of expected events. The work in this paper was done using $200$ CDFs in the sum of equation \[eq:MCMaximumGapCDF\]. In Tables I and II, a deviation from a cumulative probability of $100$% for $(y/\mu)=1$ is observed at the $0.1$% level for $\mu=67$ and greater. The reader is advised to use the tables below only for $\mu=50$ or less, where they have been verified to give correct coverage at the $1$% level.
The reader is cautioned that these tables were generated using a unit density, uniformly distributed theoretical function as input, and thus cannot be directly applied, as in the maximum gap case, to data to obtain an upper limit. The data must be transformed such that it is uniformly distributed assuming a given model as its true distribution. The necessary transformation can be found in Appendix C of [@r:yellin2]. This subtlety can be avoided by generating a set of CDFs for every parameter change in the theoretical model considered for the data, but in most applications this will be prohibitively computationally intensive.
This work was supported by the NSF Graduate Research Fellowship program, the MIT Pappalardo Fellowship program, and the MIT Kavli Institute. We wish to thank Steve Yellin for helpful discussions.
(y/$\mu$) n=1 n=2 n=3 n=4 n=5 n=6 n=7 n=8 n=9 n=10 n=11 n=12 n=13 n=14 n=15 n=16 n=17 n=18
------------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- --
$\leq$ 0.17 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
0.20 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.002 0.004 0.006 0.013 0.022 0.038
0.24 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.003 0.008 0.017 0.030 0.056 0.093 0.141 0.192 0.251
0.28 0.000 0.000 0.000 0.000 0.000 0.000 0.003 0.011 0.024 0.047 0.081 0.131 0.188 0.263 0.339 0.420 0.483 0.556
0.32 0.000 0.000 0.000 0.000 0.002 0.012 0.034 0.071 0.119 0.189 0.271 0.359 0.447 0.534 0.606 0.679 0.735 0.786
0.36 0.000 0.000 0.000 0.005 0.024 0.063 0.121 0.199 0.292 0.409 0.509 0.596 0.676 0.748 0.796 0.836 0.878 0.907
0.40 0.000 0.000 0.007 0.034 0.091 0.170 0.283 0.395 0.505 0.627 0.706 0.768 0.828 0.869 0.902 0.930 0.956 0.969
0.44 0.000 0.003 0.030 0.102 0.204 0.318 0.464 0.583 0.679 0.775 0.842 0.884 0.920 0.948 0.965 0.976 0.986 0.992
0.48 0.000 0.015 0.083 0.204 0.345 0.481 0.619 0.730 0.814 0.875 0.924 0.950 0.969 0.980 0.987 0.993 0.997 0.997
0.52 0.003 0.056 0.171 0.324 0.491 0.631 0.749 0.841 0.895 0.936 0.965 0.979 0.986 0.991 0.996 0.997 0.998 0.999
0.56 0.017 0.115 0.276 0.448 0.628 0.755 0.843 0.911 0.943 0.969 0.983 0.990 0.993 0.997 0.998 0.999 1.000 1.000
0.60 0.042 0.190 0.400 0.578 0.732 0.838 0.911 0.951 0.973 0.987 0.994 0.996 0.997 0.999 0.999 1.000 1.000 1.000
0.64 0.080 0.286 0.518 0.691 0.817 0.902 0.954 0.975 0.987 0.993 0.997 0.998 0.999 1.000 1.000 1.000 1.000 1.000
0.68 0.134 0.376 0.625 0.788 0.891 0.946 0.975 0.986 0.993 0.997 0.998 0.999 1.000 1.000 1.000 1.000 1.000 1.000
0.72 0.194 0.491 0.726 0.865 0.944 0.975 0.990 0.994 0.998 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.76 0.276 0.608 0.820 0.916 0.975 0.991 0.997 0.998 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.80 0.368 0.705 0.877 0.953 0.989 0.997 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.84 0.471 0.801 0.935 0.980 0.996 0.998 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.88 0.581 0.883 0.969 0.995 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.92 0.717 0.946 0.991 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.96 0.859 0.985 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
1.00 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
(y/$\mu$) n=19 n=20 n=21 n=22 n=23 n=24 n=25 n=26 n=27 n=28 n=29 n=30 n=31 n=32 n=33 n=34 n=35 n=36
$\leq$0.10 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
0.12 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.002 0.003 0.003 0.007
0.14 0.000 0.000 0.000 0.000 0.000 0.001 0.002 0.002 0.003 0.006 0.009 0.014 0.021 0.033 0.046 0.059 0.074 0.093
0.16 0.001 0.002 0.003 0.005 0.009 0.015 0.025 0.036 0.052 0.069 0.093 0.117 0.143 0.168 0.199 0.232 0.273 0.315
0.18 0.010 0.014 0.025 0.036 0.055 0.084 0.119 0.155 0.193 0.227 0.271 0.313 0.352 0.396 0.437 0.482 0.521 0.560
0.20 0.057 0.081 0.117 0.154 0.200 0.249 0.293 0.344 0.385 0.433 0.492 0.533 0.571 0.611 0.645 0.682 0.717 0.751
0.22 0.164 0.209 0.269 0.321 0.370 0.423 0.476 0.528 0.576 0.625 0.676 0.715 0.744 0.777 0.800 0.824 0.854 0.874
0.24 0.313 0.368 0.432 0.493 0.547 0.596 0.651 0.696 0.731 0.767 0.804 0.835 0.855 0.879 0.893 0.908 0.928 0.939
0.26 0.486 0.548 0.609 0.655 0.703 0.748 0.786 0.820 0.841 0.870 0.895 0.913 0.929 0.940 0.950 0.958 0.966 0.972
0.28 0.627 0.686 0.736 0.781 0.814 0.847 0.874 0.893 0.912 0.930 0.946 0.954 0.962 0.968 0.975 0.980 0.985 0.987
0.30 0.745 0.789 0.826 0.858 0.881 0.904 0.926 0.941 0.956 0.966 0.975 0.980 0.985 0.987 0.991 0.993 0.995 0.996
0.32 0.826 0.866 0.895 0.917 0.932 0.946 0.958 0.966 0.975 0.982 0.989 0.992 0.994 0.995 0.996 0.997 0.998 0.999
0.34 0.887 0.915 0.931 0.947 0.956 0.965 0.975 0.980 0.987 0.990 0.994 0.996 0.997 0.998 0.999 0.999 1.000 1.000
0.36 0.931 0.950 0.963 0.972 0.977 0.981 0.988 0.990 0.993 0.995 0.996 0.998 0.999 1.000 1.000 1.000 1.000 1.000
0.38 0.960 0.975 0.982 0.987 0.990 0.991 0.995 0.996 0.998 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.40 0.979 0.986 0.991 0.994 0.995 0.997 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
$\geq$0.42 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
(y/$\mu$) n=37 n=38 n=39 n=40 n=41 n=42 n=43 n=44 n=45 n=46 n=47 n=48 n=49 n=50 n=51 n=52 n=53 n=54
$\leq$0.08 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
0.10 0.000 0.000 0.000 0.000 0.001 0.001 0.001 0.001 0.002 0.002 0.004 0.005 0.008 0.011 0.013 0.017 0.022 0.034
0.12 0.010 0.013 0.021 0.025 0.034 0.045 0.057 0.067 0.084 0.098 0.118 0.142 0.167 0.190 0.214 0.240 0.267 0.301
0.14 0.114 0.144 0.172 0.198 0.228 0.261 0.297 0.329 0.358 0.390 0.422 0.458 0.497 0.527 0.557 0.586 0.614 0.640
0.16 0.353 0.396 0.434 0.469 0.504 0.540 0.582 0.617 0.648 0.674 0.700 0.724 0.751 0.775 0.795 0.813 0.831 0.847
0.18 0.603 0.641 0.673 0.706 0.731 0.763 0.786 0.806 0.828 0.847 0.864 0.876 0.893 0.901 0.914 0.925 0.934 0.943
0.20 0.783 0.812 0.834 0.857 0.876 0.893 0.909 0.920 0.932 0.940 0.948 0.955 0.960 0.965 0.970 0.973 0.977 0.982
0.22 0.897 0.913 0.924 0.937 0.947 0.956 0.964 0.969 0.975 0.979 0.982 0.984 0.986 0.988 0.990 0.991 0.993 0.993
0.24 0.951 0.959 0.965 0.973 0.979 0.982 0.984 0.986 0.989 0.991 0.993 0.994 0.996 0.997 0.997 0.999 0.999 0.999
0.26 0.978 0.982 0.987 0.990 0.991 0.993 0.994 0.994 0.995 0.997 0.998 0.998 0.998 0.999 0.999 0.999 1.000 1.000
0.28 0.990 0.992 0.995 0.995 0.996 0.996 0.997 0.997 0.998 0.999 0.999 0.999 0.999 0.999 0.999 0.999 1.000 1.000
$\geq$0.30 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
(y/$\mu$) n=55 n=56 n=57 n=58 n=59 n=60 n=61 n=62 n=63 n=64 n=65 n=66 n=67 n=68 n=69 n=70 n=71 n=72
------------ ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- --
$\leq$0.07 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
0.08 0.001 0.001 0.001 0.001 0.001 0.001 0.002 0.003 0.003 0.004 0.005 0.006 0.008 0.010 0.012 0.014 0.017 0.020
0.09 0.006 0.008 0.011 0.015 0.018 0.023 0.029 0.033 0.041 0.045 0.053 0.061 0.073 0.084 0.097 0.109 0.125 0.137
0.10 0.041 0.050 0.067 0.079 0.092 0.110 0.127 0.145 0.162 0.184 0.205 0.221 0.244 0.266 0.290 0.316 0.348 0.367
0.11 0.156 0.178 0.205 0.228 0.249 0.276 0.299 0.324 0.354 0.381 0.409 0.436 0.460 0.484 0.512 0.541 0.567 0.592
0.12 0.327 0.351 0.383 0.412 0.436 0.470 0.500 0.526 0.552 0.577 0.600 0.626 0.655 0.674 0.699 0.722 0.738 0.758
0.13 0.504 0.534 0.570 0.594 0.616 0.640 0.665 0.690 0.713 0.733 0.754 0.770 0.791 0.810 0.828 0.841 0.853 0.865
0.14 0.669 0.696 0.722 0.746 0.764 0.780 0.800 0.816 0.836 0.852 0.864 0.873 0.884 0.895 0.911 0.917 0.924 0.932
0.15 0.781 0.800 0.822 0.837 0.854 0.864 0.879 0.890 0.904 0.914 0.923 0.929 0.936 0.943 0.953 0.955 0.959 0.962
0.16 0.859 0.874 0.890 0.899 0.911 0.917 0.926 0.934 0.943 0.947 0.956 0.959 0.963 0.968 0.975 0.976 0.979 0.982
0.17 0.915 0.927 0.936 0.943 0.950 0.952 0.958 0.963 0.968 0.971 0.976 0.978 0.980 0.983 0.987 0.988 0.989 0.991
0.18 0.950 0.957 0.963 0.968 0.971 0.973 0.978 0.981 0.986 0.987 0.988 0.990 0.991 0.993 0.995 0.995 0.995 0.996
0.19 0.974 0.978 0.981 0.984 0.986 0.987 0.990 0.991 0.994 0.995 0.995 0.996 0.996 0.997 0.998 0.998 0.998 0.999
0.20 0.985 0.987 0.989 0.991 0.993 0.994 0.995 0.996 0.998 0.998 0.998 0.998 0.999 0.999 0.999 0.999 1.000 1.000
0.21 0.991 0.993 0.993 0.995 0.997 0.997 0.998 0.998 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.22 0.995 0.996 0.996 0.997 0.998 0.999 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.23 0.998 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
$\geq$0.24 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
(y/$\mu$) n=73 n=74 n=75 n=76 n=77 n=78 n=79 n=80 n=81 n=82 n=83 n=84 n=85 n=86 n=87 n=88 n=89 n=90
$\leq$0.05 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
0.06 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.001 0.001
0.07 0.000 0.001 0.001 0.003 0.004 0.005 0.005 0.007 0.007 0.010 0.012 0.015 0.018 0.021 0.023 0.027 0.031 0.039
0.08 0.024 0.029 0.035 0.042 0.050 0.058 0.068 0.078 0.087 0.101 0.114 0.128 0.140 0.156 0.172 0.186 0.205 0.222
0.09 0.154 0.173 0.190 0.206 0.226 0.246 0.271 0.295 0.314 0.342 0.363 0.386 0.409 0.429 0.449 0.472 0.498 0.518
0.10 0.389 0.414 0.437 0.458 0.478 0.502 0.528 0.552 0.570 0.593 0.613 0.632 0.653 0.668 0.686 0.703 0.721 0.737
0.11 0.613 0.634 0.653 0.672 0.690 0.706 0.728 0.743 0.760 0.775 0.789 0.804 0.814 0.824 0.836 0.847 0.860 0.869
0.12 0.774 0.790 0.803 0.818 0.831 0.843 0.852 0.861 0.873 0.882 0.893 0.902 0.909 0.916 0.922 0.927 0.933 0.940
0.13 0.876 0.890 0.896 0.904 0.912 0.920 0.925 0.930 0.938 0.944 0.950 0.956 0.960 0.963 0.966 0.969 0.971 0.975
0.14 0.938 0.946 0.950 0.956 0.960 0.964 0.967 0.971 0.975 0.977 0.979 0.981 0.984 0.986 0.987 0.988 0.989 0.989
0.15 0.965 0.971 0.974 0.977 0.979 0.981 0.982 0.985 0.989 0.991 0.992 0.993 0.994 0.995 0.996 0.996 0.997 0.997
0.16 0.984 0.986 0.987 0.989 0.991 0.992 0.992 0.993 0.994 0.996 0.996 0.996 0.996 0.997 0.998 0.998 0.998 0.999
0.17 0.994 0.995 0.995 0.996 0.997 0.997 0.997 0.997 0.998 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999
0.18 0.998 0.998 0.998 0.998 0.998 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999 1.000 1.000 1.000
0.19 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
$\geq$0.20 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
(y/$\mu$) n=91 n=92 n=93 n=94 n=95 n=96 n=97 n=98 n=99 n=100
$\leq$0.05 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
0.06 0.001 0.002 0.002 0.003 0.003 0.004 0.006 0.007 0.010 0.011
0.07 0.045 0.054 0.060 0.068 0.075 0.083 0.095 0.105 0.116 0.125
0.08 0.238 0.258 0.275 0.298 0.317 0.338 0.358 0.379 0.399 0.421
0.09 0.536 0.556 0.574 0.596 0.611 0.630 0.647 0.666 0.680 0.691
0.10 0.751 0.765 0.778 0.793 0.801 0.811 0.824 0.837 0.848 0.858
0.11 0.877 0.887 0.893 0.903 0.908 0.913 0.919 0.926 0.929 0.936
0.12 0.944 0.949 0.953 0.958 0.959 0.963 0.966 0.968 0.970 0.974
0.13 0.977 0.979 0.981 0.983 0.985 0.986 0.988 0.988 0.989 0.991
0.14 0.990 0.990 0.991 0.993 0.993 0.994 0.995 0.995 0.995 0.996
0.15 0.998 0.998 0.998 0.998 0.999 0.999 0.999 0.999 0.999 1.000
0.16 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.17 0.999 0.999 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
$\geq$0.18 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
|
---
abstract: |
The following question is studied and answered:
Is it possible to stably approximate $f^\prime$ if one knows:
1\) $f_\delta\in L^\infty(\R)$ such that $\|f-f_\delta\|<\delta$,
and
2\) $f\in C^\infty(\R)$, $\|f\|+\|f^\prime\|\leq c$?
Here $\|f\|:=\sup_{x\in\R} |f(x)|$ and $c>0$ is a given constant. By a stable approximation one means $\|L_\delta f_\delta-f^\prime\|\leq \eta(\delta)\to 0$ as $\delta\to 0$. By $L_\delta f_\delta$ one denotes an estimate of $f^\prime$. The basic result of this paper is the inequality for $\|L_\delta f_\delta-f^\prime\|$, a proof of the impossibility to approximate stably $f^\prime$ given the above data 1) and 2), and a derivation of the inequality $\eta(\delta)\leq c\delta^{\frac a{1+a}}$ if 2) is replaced by $\|f\|_{1+a}\leq m_{1+a}$, $0<a\leq 1$. An explicit formula for the estimate $L_\delta f_\delta$ is given.
author:
- |
A.G. Ramm\
Mathematics Department, Kansas State University,\
Manhattan, KS 66506-2602, USA\
[*[email protected]*]{}
title: ' Inequalities for the derivatives [^1] [^2] '
---
Introduction
============
The classical problem of theoretical and computational mathematics is the problem of estimation of the derivative $f^\prime$ of a function from various data.
Inequalities between the derivatives are known (Landau-Hadamard, Kolmogorov [@HLP]-[@L], [@R1]), for example:
$$m_k\leq c_{nk}m_0^{\frac{n-k}{n}} m_n^{\frac{k}{n}},
\tag{1.1}$$
where $$m_k:=\|f^{(k)}\|:=\sup_{x\in I} |f^{(k)}(x)|, \,\, I=\R,$$ and $c_{nk}$ are some constants. In particular, if $I=\R$, then $$m_1\leq \sqrt{2m_0m_2},
\tag{1.2}$$ if $I=(0,\infty)$, then $$m_1\leq 2\sqrt{m_0m_2},
\tag{1.3}$$ if $I=(0,h)$, $h\geq 2\sqrt{\frac{m_0}{m_2}}$, then (1.3) holds, if $I=(0,h)$, $h < 2\sqrt{\frac{m_0}{m_2}}$, then $$\
m_1\leq \frac{2}{h} m_0+\frac{h}{2}m_2.
\tag{1.4}$$ These inequalities can be found in [@HLP]-[@L].
In pratice the following problem is of great interest. Suppose that $f(x)\in C^\infty(\R)$ is unknown, but one knows $m_j$, $j=0,1,2$, and one knows $f_\delta\in L^\infty(\R)$ such that $$\|f_\delta-f\|\leq\delta.
\tag{1.5}$$ Can one estimate $f^\prime(x)$ stably? In other words, can one find an operator $L_\delta$ such that $$\|L_\delta f_\delta-f^\prime\|\leq \eta(\delta)\to 0
\qquad\hbox{as}\quad \delta\to 0.
\tag{1.6}$$ The operator $L_\delta$ can be linear or nonlinear, in general.
This problem was investigated in [@R2], where it was proved that the operator $$L_\delta f_\delta:=
\frac{f_\delta(x+h(\delta))-f_\delta(x-h(\delta))}{2h(\delta)},
\qquad h(\delta):=\sqrt{\frac{2\delta}{m_2}}
\tag{1.7}$$ yields the estimate: $$\|L_\delta f_\delta-f^\prime\|
\leq \varepsilon(\delta):=\sqrt{2m_2\delta},
\tag{1.8}$$ under the assumptions $m_2<\infty$ and (1.5).
Inequality (1.8) is quite convenient practically. The original result of [@R2] was the first of its kind and generated many papers in which the choice of the discretization parameter was used for a stable solution of various ill-posed problems, in particular stable differentiation of random functions and applications in electrical engineering (see [@MR]-[@R6] and references therein).
In [@R1 pp.82-84] one can find a proof of the following interesting fact: [*among all linear and nonlinear operators $T$, the operator $L_\delta$, defined in (1.7), gives the best possible estimate of $f^\prime$ on the class of all $f\in{\mathcal K}(\delta,m_2)$.*]{} Here $$\mathcal K(\delta,m_j):=\{f:f\in C^j(\R),
\quad m_j<\infty, \quad \|f-f_\delta\|\leq\delta\}.
\tag{1.9}$$ In other words, the following inequality holds [@R1 p.82]: $$\mathop{\inf}_{T} \sup_{f\in\mathcal K(\delta,m_2)}
\|T f_\delta-f^\prime\|\geq\varepsilon(\delta):=\sqrt{2m_2\delta},
\tag{1.10}$$ where $T$ runs through the set of all linear and nonlinear operators $T:L^\infty(\R)\to L^\infty(\R)$.
In this paper we investigate and answer the following questions:
[Given $f_\delta\in L^\infty(\R)$ such that (1.5) holds, and a number $m_j$, $\|f^{(j)}\|\leq m_j$, $f\in C^\infty(\R)$, $j=0,1$, can one estimate stably $f^\prime$?]{}
In other words, does there exist an operator $T$ such that $$\sup_{f\in\mathcal K(\delta,m_j)}
\|Tf_\delta-f^\prime\|\leq\eta(\delta)\to 0
\qquad\hbox{as}\quad \delta\to 0,
\tag{1.11}$$ where $j=0$ or $j=1$?
It is similar to Question 1 but now it is assumed that $j=1+a>1$: $$\|f^{(1+a)}\|:=m_{1+a}<\infty, \quad 0<a\leq 1,
\tag{1.12}$$ where $\|f^{(1+a)}\|:=\|f^{\prime^{(a)}}\|$, and $$\|g^{(a)}\|:=\sup_{x,y\in\R}
\frac{|g(x)-g(y)|}{|x-y|^a} +\|g\|,
\qquad 0<a\leq 1.
\tag{1.13}$$
The basic results of this paper are summarized in Theorem 1.
There does not exist an operator $T$ such that inequality (1.11) holds for $j=0$ or for $j=1$. There exists such an operator if $j>1$.
In the proof of Theorem 1 an explicit formula is given for $T$ and an explicit inequality (2.8) is given for the error estimate.
In section 2 proofs are given. In the course of these proofs we derive inequalities for the quantity $$\gamma_j:=\gamma_j(\delta):=
\gamma_j(\delta,m_j):=\inf_T\sup_{f\in K(\delta,m_j)}
\|Tf_\delta-f^\prime\|
\tag{1.14}$$ In [@R7] the theory presented in this paper is developed further and numerical examples of its applications are given.
Proof of Theorem 1
==================
Let $f_\delta(x)=0$, and consider $f_1(x):=-\frac{M}{2}x(x-2h)$, $0\leq x\leq 2h$, and $f_1(x)$ is extended to the whole real axis in such a way that $\|f^{(j)}_1\|=\sup_{0\leq x\leq 2h}\|f^{(j)}_1\|$, $j=0,1,2,$ are preserved. It is known that such an extension is possible. Let $f_2(x)=-f_1(x)$. Denote $(Tf_\delta)(0):=(T0)(0):=b$.
Since $$\|Tf_\delta-f^\prime_1\|
\geq \left| (Tf_\delta)(0)-f^\prime_1(0)\right|
=|b-Mh|, \notag$$ and $$\|Tf_\delta-f^\prime_2\|\geq|b+Mh|,
\notag$$ one has $$\gamma_j(\delta)\geq \inf_{b\in\R}
\max \left\{|b-Mh|,|b+Mh|\right\}=Mh
\tag{2.1}$$
Inequality (1.5) with $f_\delta(x)=0$ implies $$\sup_x |f_s(x)|=\frac{Mh^2}{2}\leq \delta,\qquad s=1,2.
\tag{2.2}$$
Let us take $\frac{Mh^2}{2}=\delta$, then $$h=\sqrt{\frac{2\delta}{M}},\qquad Mh=\sqrt{2\delta M}.
\tag{2.3}$$ If $j=0$, then (2.2) implies $m_0=\delta$. Since $M$ can be chosen arbitrary for any $\delta>0$ and $m_0=\delta$, inequality (2.1) with $j=0$ proves that estimate (1.11) is false on the class ${\mathcal K}(\delta,m_0)$, and in fact $\gamma_0(\delta)\to\infty$ as $M\to\infty$.
This estimate is also false on the class ${\mathcal K}(\delta, m_1)$. Indeed, for $f_1(x)$ and $f_2(x)$ one has $$m_1=\|f^\prime_1\|= \|f^\prime_2\|= \sup_{0\leq x\leq 2h} |M(x-h)|
=Mh=\sqrt{2\delta M}.
\tag{2.4}$$ If $m_1\leq c<\infty$, then one can find $M$ such that $m_1=\sqrt{2\delta M}=c$, thus $Mh=c$, and by (2.1) one gets $$\gamma_1(\delta)\geq c>0,\qquad \delta\to 0,
\tag{2.5}$$ so that (1.11) is false.
Let us assume now that (1.12) holds. Take $Tf_\delta:=L_{\delta,h}f_\delta$, where $L_{\delta,h}f_\delta$ is defined as in (1.7) but $h$ replaces $h(\delta)$. One has, using the Lagrange formula, $$\begin{align}
\|L_{\delta,h}f_\delta-f^\prime\|
=&\|L_{\delta,h}(f_\delta-f)\|
+\left\|L_{\delta,h}f-f^\prime \right\|
\notag \\
\leq & \frac{\delta}{h}
+\left\|\frac{f(x+h)-f(x-h)-2hf^\prime(x)}{2h} \right\|
\notag \\
\leq &\frac{\delta}{h}
+\left\|\frac{[f^\prime(y)
-f^\prime(x)] h+[f^\prime(z)-f^\prime(x)]h}{2h} \right\|
\notag \\
\leq & \frac{\delta}{h} + m_{1+a}h^a:=\varepsilon_a(\delta,h).
\tag{2.6} \end{align}
\notag$$ where $y$ and $z$ are the intermediate points in the Lagrange formula.
Minimizing the right-hand side of (2.6) with respect to $h\in(0,\infty)$ yields $$h_a(\delta)=\left(\frac{\delta}{am_{1+a}}\right)^{\frac{1}{1+a}},
\qquad \varepsilon_a(\delta)=c_a\delta^{\frac{a}{1+a}},
\qquad 0<a\leq 1,
\tag{2.7}$$ where $c_a:=\left( am_{1+a} \right)^{\frac{1}{1+a}}
+\frac{m_{1+a}}{\left(am_{1+a}\right)^\frac{a}{1+a}}$.
From (2.6) and (2.7) the following inequality follows: $$\sup_{f\in {\mathcal K}(\delta,m_{1+a})}
\|L_\delta f_\delta-f^\prime\|
\leq c_a \delta^{\frac{a}{1+a}},
\qquad 0<a\leq 1.
\tag{2.8}$$
Theorem 1 is proved.
[10]{}
G. Hardy, J. Littlewood, G. Polya, [**Inequalities**]{}, Cambridge Univ. Press, London, 1951.
A. Kolmogorov, On inequalities between derivatives, Uchen. Zapiski Moscow Univ. Math., 30, (1939), 3-16.
E. Landau, Einige Ungleichungen für zweimal differentierbare Functionen, Proc. Lond. Math. Soc., 2, N13, (1913), 43-49.
T.Miller, A. G. Ramm, Estimates of the derivatives of random functions II. J. Math. Anal. Appl., 110, (1985), 429-435.
A.G. Ramm, [**Random fields estimation theory**]{}, Longman Scientific and Wiley, New York, 1990.
, On numerical differentiation. Izvestija vuzov, Mathem., 11, (1968), 131-135.
, Stable solutions of some ill-posed problems, Math. Meth. in appl. Sci., 3, (1981), 336-363.
, Estimates of the derivatives of random functions. J. Math. Anal. Appl., 102, (1984), 244-250.
, On simultaneous approximation of a function and its derivative by interpolation polynomials. Bull. Lond. Math. Soc., 9, (1977), 283-288.
, Simplified optimal differentiators. Radiotech.i Electron., 17, (1972), 1325-1328.
A.G.Ramm, A.B.Smirnova, On stable numerical differentiation. (to appear)
[^1]: key words: stable numerical differentiation, inequalities for the derivatives
[^2]: Math subject classification: 65D25, 65M10
|
---
author:
- Yada Nandukumar
- Pinaki Pal
title: 'Oscillatory instability and routes to chaos in Rayleigh-Bénard convection: effect of external magnetic field'
---
Introduction
============
The study of thermal convection of electrically conducting fluids in presence of magnetic field fascinated researchers for many years due to its relevance in astrophysical and geophysical problems [@chandra:book_1961; @proctor:others]. Convection in presence of magnetic field is considered to play a significant role in the formation of sunspots, solar granulation, magnetic field generation of stars and planets. Besides these, magnetoconvection is also important in the engineering applications like crystal growth [@hurle:book], and nuclear heat exchanger [@gailitis:2002]. These natural as well as industrial problems involve complex geometry and scientists therefore often consider simplified model of magnetoconvection to understand the basic physics.
Rayleigh-Bénard convection (RBC) [@chandra:book_1961; @bodenschatz:ahlers] is a simplified model of convection and it is studied for many years to understand properties of convection both in presence and absence of magnetic field. The linear theory for Rayleigh-Bénard magnetoconvection has been extensively developed by Chandrasekhar [@chandra:book_1961]. It is known that the primary instability is greatly effected by the presence of vertical magnetic field [@chandra:book_1961]. On the other hand, horizontal magnetic field does not modify the primary instability and remains same as in the absence of magnetic field [@chandra:book_1961; @busse:1983; @clever:1989; @busse:1989; @pesch:2006]. However, secondary as well as other higher order instabilities are significantly affected by the presence of horizontal magnetic field.
There has been extensive theoretical as well as experimental studies on the effect of external magnetic field on the convective flow of electrically conducting fluids to understand the nonlinear aspects of the problem. Early experiments [@lehnert:nakagawa] considered inhomogeneous as well as homogeneous external magnetic fields and found the stabilizing effects of it on the convective flow. Later a series of theoretical works [@busse:1983; @clever:1989; @busse:1989; @pesch:2006] considered both horizontal as well as vertical external magnetic fields and observed that magnetic field plays an important role in inhibiting the oscillatory convection.
The effect of horizontal magnetic field on the convective flow of liquid metals has been investigated in several experimental studies [@fauve:1981; @fauve:JPL_1984; @fauve:1984; @burr:2002; @yanagisawa:2013]. Fauve et al. [@fauve:1981; @fauve:JPL_1984] found that horizontal magnetic field have a stabilizing effect on the convective flow and stronger magnetic field can orient the flow along the magnetic field direction. The phenomena of flow reversal and pattern dynamics in presence of horizontal magnetic field have been experimentally investigated in liquid metals [@yanagisawa:2013]. An interesting two parameter experimental study on the effect of horizontal magnetic field on the routes to chaos in the convective flow of mercury has been carried out by Libchaber et al. [@fauve:1983]. They found that the routes to chaos depend on the magnitude of the external magnetic field as the value of the Rayleigh number is increased. Recently, intermittency routes to chaos has also been reported in a model of magnetoconvection [@macek:2014].
In this article, we consider RBC of low Prandtl-number fluids in presence of uniform horizontal magnetic field. We perform three dimensional direct numerical simulations of the governing equations under Boussinesq approximation with free-slip boundary conditions and investigate the effect of uniform horizontal magnetic field on the onset of oscillatory instability. Magnetic field is found to push the oscillatory instability onset towards higher Rayleigh number ($\mathrm{Ra}$). The supercritical Rayleigh number for the onset of oscillation shows a scaling with $\mathrm{Q}$ as $\mathrm{Q}^{\alpha}$, $\alpha = 1.8$ for low Prandtl numbers ($\mathrm{Pr}$). We then focus our investigation on the effect of magnetic field on the route to chaos in liquid metals ($\mathrm{Pr}\sim 10^{-2}$). DNS shows a period doubling cascade to chaos for low values of $\mathrm{Q}$ and a quasiperiodic route to chaos for higher values of $\mathrm{Q}$. These results show similarity with the experimental results [@fauve:1983]. To understand detailed bifurcation structure associated with different routes to chaos, we construct a low dimensional model from the DNS data and perform bifurcation analysis.
Hydromagnetic system
====================
We consider a thin horizontal layer of electrically conducting fluid of thickness $d$, kinematic viscosity $\nu$, thermal diffusivity $\kappa$, magnetic diffusivity $\lambda$ and coefficient of volume expansion $\alpha$ kept between two horizontal thermally conducting plates in presence of uniform horizontal external magnetic field $\vec{\bf{B_0}} = (0, B_0, 0)$. The system is heated from below and a uniform temperature gradient $\beta$ between the two plates is maintained. The dimensionless magnetohydrodynamic equations under Boussinesq approximation are given by: $$\begin{aligned}
{\partial_t{\bf v}} + ({\bf v}{\cdot}\nabla){\bf v} &=& -\nabla p + \nabla^2 {\bf v} + \mathrm{Ra}\theta\hat{\bf e}_3\nonumber\\
&+& Q\left[\partial_y{\bf b} + \mathrm{Pm}({\bf b}{\cdot}\nabla){\bf b}\right],\label{eq:velocity1}\\
\mathrm{Pr}[{\partial_t \theta}+({\bf v}{\cdot}\nabla)\theta]&=&\left[{\nabla}^2 \theta + v_3\right],\label{eq:theta1}\\
\mathrm{Pm}[{\partial_t{\bf b}} + ({\bf v}{\cdot}\nabla){\bf b} &-& ({\bf b}{\cdot}\nabla){\bf v}] = {\nabla}^2 {\bf b} + \partial_y{\bf v},\label{eq:magnetic1}\\
\nabla{\cdot}{\bf v} &=& \nabla{\cdot}{\bf b}=0,\label{eq:continuity1}
%{\nabla}^2 {\bf b}&=&-\partial_y{\bf v},\label{eq:magnetic field}\end{aligned}$$ where ${\bf v}(x,y,z,t) \equiv (v_1,v_2,v_3)$ is the velocity field, ${\bf b}(x,y,z,t)=b(b_1,b_2,b_3)$ the induced magnetic field due to convection, $\theta(x,y,z,t)$ the deviation in the temperature field from the steady conduction profile, $p$ the pressure, $g$ the acceleration due to gravity and $\hat{\bf e}_3$ is the vertically directed unit vector. For nondimensionalization, the units $d$, $d^2/\nu$, ${\nu\beta{d}}/{\kappa}$ and ${B_0\nu}/{\lambda}$ have been used for length, time, temperature and induced magnetic field. Four dimensionless parameters namely $\mathrm{Ra} = ({\alpha \beta g d^4})/({\nu\kappa})$, the Rayleigh number, $\mathrm{Pr} = {\nu}/{\kappa}$, the Prandtl number, $\mathrm{Pm} = {\nu}/{\lambda}$, the magnetic Prandtl number and $Q = ({B_0^2d^2})/({\rho_0\nu\lambda})$, the Chandrasekhar number appear in the process, where $\rho_0$ is the reference density of the fluid. The boundaries are assumed to be stress-free, maintained at fixed temperatures and perfectly conducting, which imply, $$\begin{aligned}
v_3 &=& \partial_{z}v_1 = \partial_{z}v_2 = \theta = 0 \quad \mbox{and} \nonumber\\
b_3 &=& \partial_{z}b_1 = \partial_{z}b_2 = 0 \quad \mbox{at}\quad z = 0, 1.\label{eq:bcs} \end{aligned}$$ Also, we assumed periodic boundary conditions along horizontal directions. In this paper, we investigate convective flow of low magnetic Prandtl number fluids e.g. liquid metals for which $\mathrm{Pm} \sim 10^{-6}$. Therefore, for simplicity we consider the limit $\mathrm{Pm}\rightarrow 0$ [@meneguzzi:1987] and in this limit, the equations (\[eq:velocity1\]) and (\[eq:magnetic1\]) reduce to: $$\begin{aligned}
{\partial_t{\bf v}} + ({\bf v}{\cdot}\nabla){\bf v} &=& -\nabla p + \nabla^2 {\bf v} + \mathrm{Ra}\theta\hat{\bf e}_3 + Q\partial_y{\bf b},\label{eq:velocity}\\
{\nabla}^2 {\bf b}&=&-\partial_y{\bf v}.\label{eq:magnetic}\end{aligned}$$ As a result, the induced magnetic field becomes slaved to velocity field (see equation (\[eq:magnetic\])). Now the equations (\[eq:theta1\]), (\[eq:continuity1\]), (\[eq:velocity\]) and (\[eq:magnetic\]) together with the boundary conditions (\[eq:bcs\]) form the mathematical model of the physical system under consideration. The critical Rayleigh number and wave number for the onset of convection in this case are $\mathrm{Ra_c} = {27\pi^4}/{4}$ and $k_c = {\pi}/{\sqrt{2}}$. We define another parameter called reduced Rayleigh number by $r = {\mathrm{Ra}}/{\mathrm{Ra_c}}$ for the subsequent sections.
Direct Numerical Simulations {#sec:DNS}
============================
We perform direct numerical simulations of the governing equations (\[eq:theta1\]), (\[eq:continuity1\]), (\[eq:velocity\]) and (\[eq:magnetic\]) with boundary conditions (\[eq:bcs\]) using the psuedo-spectral code [@verma:Arxiv_2011]. In the simulation code, vertical velocity ($v_3$), vertical vorticity ($\omega_3$) and temperature ($\theta$) fields are expanded with the set of orthogonal basis functions either with respect to $\{e^{i(lk_cx+mk_cy)}\sin{(n\pi z)}: l, m, n = 0,1, 2, \dots\}$ or with respect to $\{e^{i(lk_cx+mk_cy)}\cos{(n\pi z)}: l, m, n = 0, 1, 2, \dots\}$ whichever is compatible with the [*free-slip*]{} boundary conditions, where $k_c$ is the wave number. Therefore, the expanded fields take the following form: $$\begin{aligned}
v_3 (x,y,z,t) &=& \sum_{l,m,n} W_{lmn}(t)e^{ik_c(lx+my)}\sin{(n\pi z)},\\
\omega_3 (x,y,z,t) &=& \sum_{l,m,n} Z_{lmn}(t)e^{ik_c(lx+my)}\cos{(n\pi z)},\\
\theta (x,y,z,t)&=& \sum_{l,m,n} T_{lmn}(t)e^{ik_c(lx+my)}\sin{(n\pi z)}.\end{aligned}$$
The horizontal components of velocity ($v_1$ and $v_2$) are determined from the equation of continuity (\[eq:continuity1\]) and the induced magnetic field components are then derived from the equation (\[eq:magnetic\]). For time advancement, fourth order Runge-Kutta (RK4) scheme is used. The grid resolution for the simulation is taken to be $32^3$ with time step $\Delta t = 0.001$. To test the convergence, some simulations are repeated with $64^3$ grid and found no change in the results.
![Time series as well as PSD of the mode $W_{101}$ computed from the DNS for $\mathrm{Pr}=0.025$ and $\mathrm{Q}=12$.[]{data-label="fig:period2_DNS"}](Q12_timeseries_DNS.eps){height="6.5cm" width="8.5cm"}
We first perform DNS of the governing equations to investigate the effect of magnetic field on the onset of oscillatory instability. Fig. \[fig:Pr\_r\_o(Q0,Q2.5,Q5)\] shows the variation of the reduced Rayleigh number threshold $r_0(\mathrm{Q},\mathrm{Pr})$ for the onset of oscillatory instability as a function of Prandtl number for three different values of the Chandrasekhar number $\mathrm{Q}$ as obtained from the DNS. The figure clearly shows that the imposed horizontal magnetic field inhibits the onset of oscillatory instability [@busse:1972]. This result is consistent with the earlier theoretical as well as experimental results [@busse:1989; @fauve:1981; @fauve:JPL_1984]. We compute the relative distance from the threshold of oscillatory instability ${(\mathrm{Ra}_\mathrm{Q}^{o}-\mathrm{Ra}_0^{o})}/{\mathrm{Pr}}$ as a function of $\mathrm{Q}$ from the DNS data for $\mathrm{Pr} = 0.025$ and $0.1$, where $\mathrm{Ra}_\mathrm{Q}^{o}$ and $\mathrm{Ra}_0^{o}$ are the threshold values of oscillatory solutions in presence $(\mathrm{Q}\neq 0)$ and absence of external magnetic field $(\mathrm{Q}=0)$ respectively. Here we note that for $\mathrm{Pr} = 0.1$, the oscillatory solution bifurcates from steady $2D$ rolls [@busse:1972], while for $\mathrm{Pr} = 0.025$, it bifurcates from steady cross rolls solution [@mishra:2010]. We find that ${(\mathrm{Ra}_\mathrm{Q}^{o}-\mathrm{Ra}_0^{o})}/{\mathrm{Pr}}$ scales with $\mathrm{Q}$ as $\mathrm{Q}^{\alpha}$, where $\alpha = 1.8$. Fig. \[fig:Power\_law(DNS)\] represents this scaling as obtained from the DNS for two Prandtl numbers. Similar scaling law was observed in the earlier numerical and experimental investigations [@busse:1983; @fauve:JPL_1984], where they found $\alpha = 1.2$. The difference in the exponent of the scaling from the earlier investigations may be attributed to the difference in velocity boundary conditions.
Now we focus to understand the effect of external magnetic field on routes to chaos of the oscillatory solutions in mercury ($\mathrm{Pr} = 0.025$) as the value of reduced Rayleigh number is increased. In absence of the magnetic field, mercury shows period doubling route to chaos [@fauve:1983]. We find that external magnetic field of smaller magnitude does not change the route to chaos. The time series of $W_{101}$ as well as power spectral density (PSD) plotted in Fig. \[fig:period2\_DNS\] for four values of $r$ clearly show the period doubling route to chaos for $\mathrm{Q} = 12$ in mercury. On the other hand, DNS for $\mathrm{Q}=150$ show quasiperiodic route to chaos. Now to understand the bifurcation structure associated with $\mathrm{Q}$ dependent routes to chaos in mercury, we construct a low dimensional model from the DNS data.
Model construction
==================
From the DNS data we identify $12$ vertical velocity: $W_{101}$, $W_{011}$, $W_{111}$, $W_{202}$, $W_{022}$, $W_{031}$, $W_{301}$, $W_{103}$, $W_{013}$, $W_{112}$, $W_{211}$, $W_{121}$ , $15$ vertical vorticity: $Z_{100}$, $Z_{010}$, $Z_{110}$, $Z_{111}$, $Z_{112}$, $Z_{310}$, $Z_{130}$, $Z_{120}$, $Z_{210}$, $Z_{102}$, $Z_{012}$, $Z_{201}$, $Z_{021}$, $Z_{121}$, $Z_{211}$ and $13$ temperature: $T_{101}$, $T_{011}$, $T_{112}$, $T_{111}$, $T_{202}$, $T_{022}$, $T_{103}$, $T_{013}$, $T_{301}$, $T_{031}$, $T_{121}$, $T_{211}$, $T_{002}$ most energetic real modes. Now projecting the hydromagnetic equations on these modes, we get a set of $40$ coupled nonlinear ordinary differential equations, which is the low dimensional model for the present investigation.
Analysis of the model and DNS results
=====================================
We integrate the low dimensional model using the $ode45$ solver of MATLAB. First we observe that horizontal magnetic field inhibit the oscillatory instability as observed in the DNS. Then we compute the quantity ${(\mathrm{Ra}_\mathrm{Q}^{o}-\mathrm{Ra}_0^{o})}/{\mathrm{Pr}}$ as a function of $\mathrm{Q}$ for $\mathrm{Pr} = 0.1$ and $0.025$. We find that ${(\mathrm{Ra}_\mathrm{Q}^{o}-\mathrm{Ra}_0^{o})}/{\mathrm{Pr}}$ also scales with $\mathrm{Q}$ as $\mathrm{Q}^{\alpha}$, where $\alpha = 1.3$. The exponent found in this case is much closer to the experimental value $\alpha = 1.2$ [@fauve:1983].
Now we perform the bifurcation analysis of the model, using a continuation software [@dhooge:matcont_2003]. Fig. \[fig:bif\_Q20\] shows the bifurcation diagram constructed from the model for $\mathrm{Pr = 0.025}$ and $\mathrm{Q} = 20$. In the bifurcation diagram, the extremum values of $W_{101}$ for different solutions have been plotted as a function of $r$ in the range $0.99\le r \le 1.70$. The trivial stable and unstable conduction state have been shown with solid and dashed cyan curve in the bifurcation diagram. The conduction solution or the zero solution is stable for $r < 1$ which becomes unstable via pitchfork bifurcation at $r = 1$. A stable two dimensional ($2D$) rolls solution ($W_{101}\neq 0$ and $W_{011} = 0$ or vice versa) appears. It is shown with solid blue curve in the figure. The stable $2D$ rolls observed in our simulation is due to the choice of the small aspect ratio ($\frac{2\pi}{k_c}: \frac{2\pi}{k_c}$) of the periodicity intervals as also observed in [@meneguzzi:1987]. It is interesting to note that Busse and Bolton [@busse_bolton:1984] had shown that convective rolls with critical wave number is always unstable for $\mathrm{Pr} < 0.543$, under the action of long-wavelength instabilities which are suppressed in our simulation. We also do not observe the secondary instabilities reported in [@busse_bolton:1984], because of this. As the value of $r$ is increased further, the stable $2D$ rolls become unstable via pitchfork bifurcation and stable stationary cross rolls (CR) solution ($W_{101}\neq 0$ and $W_{011}\neq 0$) is generated. The unstable $2D$ rolls are shown with dashed blue curve and stable CR solutions are shown with solid black curve. The appearance of CR from 2D rolls via pitchfork bifurcation has already been reported by Mishra et al. [@mishra:2010] for $\mathrm{Pr} = 0.02$ in absence of magnetic field and they nicely explained the bifurcation structure associated with the stationary squares (SQ) reported in [@thual:1992]. For $\mathrm{Pr} = 0.025$, we also observe similar bifurcation structure near the onset of convection. For low values of $\mathrm{Q}$, the bifurcation scenario is modified but appearance of CR from stationary $2D$ rolls is observed in our simulation.
The CR solutions become unstable via supercritical Hopf bifurcation (HB) at $r = 1.0398$ and stable limit cycles are generated (solid red curves in Fig. \[fig:bif\_Q20\]). Physically these limit cycles represent oscillatory cross rolls solutions (OCR). The unstable CR solutions continue to exist (dashed black curve) in the regime of stable OCR solutions. It is apparent from the bifurcation diagram that the size of the limit cycles of the OCR solutions initially grows and then decrease with the increase of the value of $r$ and eventually the unstable CR solution becomes stable at $r = 1.4434$, via inverse HB. Again stable CR is observed in the regime $1.4434 \le r \le 1.5925$ (solid black curve). This CR branch again undergoes a pitchfork bifurcation at $r = 1.5925$ and another stable CR branch comes out (solid pink curve). Note that the black CR solution exists as unstable solution. The pink CR solution then undergoes a supercritical HB and stable limit cycle (solid green curve) is generated. This limit cycles undergoes a period doubling cascade (brown dots) as the value of $r$ is increase and eventually chaos is observed. A zoomed view of this period doubling cascade has been shown in the inset.
We now explore the ($\mathrm{Q}, r$) parameter space for $\mathrm{Pr} = 0.025$ and compute a phase diagram (Fig. \[fig:phase\_diagram\]) from the model. The diagram delimits different solution regimes of the model with different colours. To describe the phase diagram, first we note that the cyan island exist for $0\leq \mathrm{Q} \leq 112.6$. The regime $0\leq \mathrm{Q} \leq 112.6$ can also be subdivided into two parts namely $0\leq \mathrm{Q} \leq 60$ and $60 < \mathrm{Q} \leq 112.6$.
For $0 \leq \mathrm{Q} \leq 60$, as the value of $r$ is increased from $r = 1$, a bifurcation structure similar to $\mathrm{Q} = 20$ is observed. This means for $\mathrm{Q} \leq 60$, as the value of $r$ is raised slowly from $r = 1$, stable 2D rolls regime appears via supercritical pitchfork bifurcation of the conduction solution at $r = 1$ (green regime in Fig. \[fig:phase\_diagram\]). Note that 2D rolls regime becomes very thin as $\mathrm{Q} \rightarrow 0$. As the value of $r$ is increased further, CR regime (white region in Fig. \[fig:phase\_diagram\]) is stabilized after pitchfork bifurcation of the 2D rolls solutions. Then a pair of supercritical HBs, one forward and the other reverse, occur to the CR solutions. These are associated with the lower and upper boundaries of the cyan island. In between these two HBs, oscillatory cross rolls solutions are observed. After the reverse HB at the upper boundary of the cyan region, the stationary CR regime (white region in Fig. \[fig:phase\_diagram\]) again becomes stabilized. For a further increase in the value of $r$, supercritical HB occurs at the lower boundary of the red region in the Fig. \[fig:phase\_diagram\] and time dependent regime starts. Near the lower boundary of the red region, periodic oscillatory cross rolls solutions are observed. These periodic solutions become chaotic via period doubling route for $\mathrm{Q} \leq 60$ for higher $r$. Note that similar period doubling route to chaos is also observed in DNS for low values of $\mathrm{Q}$.
For $\mathrm{Q} > 60$, the bifurcation sequence till the value of $r$ reaches the lower boundary of the red region is similar as $\mathrm{Q} \leq 60$. As the value of $r$ is increased further, quasiperiodic transition to chaos occurs. A similar quasiperiodic route to chaos is also observed in DNS for moderate values of $\mathrm{Q}$. The bifurcation sequence observed in DNS is also similar to model. Now we emphasize that DNS also shows qualitatively model like flow regimes and bifurcation structure in ($\mathrm{Q} - r$) parameter space for $\mathrm{Pr} = 0.025$. To show this we compute the bifurcation boundaries for $\mathrm{Q} = 20, 40, 60$ and $80$ and plotted in the Fig. \[fig:phase\_diagram\]. The blue circles, black square, blue triangle and filled black circles respectively represent branch points, and three HB points obtained from DNS in the Fig. \[fig:phase\_diagram\]. Note that for $\mathrm{Q} = 100$, we observe only one Hopf for higher value of the Rayleigh number in DNS and the stable limit cycle solution generated out of it becomes chaotic via quasiperiodic route as the value of the Rayleigh number increased further. This implies that the oscillatory cross rolls regime in DNS corresponding to the cyan colored portion in the Fig. \[fig:phase\_diagram\] terminates for a value of $\mathrm{Q}$ between $80$ and $100$ in DNS.
The bifurcation scenario for $\mathrm{Q} > 112.6$ becomes qualitatively different in the model. The cyan regime in the Fig. \[fig:phase\_diagram\] does not exist any more. Two different CR regimes intercepted by cyan regime in Fig. \[fig:phase\_diagram\] is now connected and only CR is observed in a large range of $r$. The stationary CR solutions becomes unstable via supercritical HB at the lower boundary of the red region and periodic oscillatory solutions are born. These periodic solutions become chaotic via quasiperiodic route as the value of $r$ is increased further. In the model, we observe similar bifurcation scenario till $\mathrm{Q} = 250$ for $\mathrm{Pr} = 0.025$. We have not considered $\mathrm{Q} > 250$ in the model, as beyond this, the model results start deviating significantly from the DNS.
To understand the details of the bifurcation structure for the higher values of $\mathrm{Q}$, we construct a bifurcation diagram (Fig. \[fig:bif\_Q250\]) from the model for $\mathrm{Q} = 250$ with $\mathrm{Pr} = 0.025$. We observe that the bifurcation structure is drastically modified for higher values of $\mathrm{Q}$. The trivial conduction state is stable for $r < 1$ and becomes unstable at $r = 1$ via a pitchfork bifurcation. The stable and unstable conduction solutions are shown with solid and dashed cyan curves in Fig. \[fig:bif\_Q250\]. Stable 2D rolls (solid blue line in Fig. \[fig:bif\_Q250\]) solution is originated.
The 2D rolls solution again undergoes a pitchfork bifurcation at $r = 1.368$ and stable CR solution branch is born (solid black curve in Fig. \[fig:bif\_Q250\]). This CR solutions remains stable for a large range of $r$ and looses stability via HB at $r = 6.652$. Stable limit cycles are generated (solid green curves in Fig. \[fig:bif\_Q250\]). As $r$ is increased further, the limit cycles become unstable and two frequency quasiperidic solutions are observed (light brown dots in Fig. \[fig:bif\_Q250\]). The qusiperiodic solution becomes chaotic for higher values of $r$. We observe similar bifurcation sequence for $\mathrm{Q} = 100$, $150$, $200$ $250$, $300$ and $350$ in DNS.
We now explore high $\mathrm{Q}$ regime in DNS. DNS for $\mathrm{Q} = 400$ shows quasiperiodic route to chaos but with qualitatively different bifurcations scenario. In this case, the regime of stable $2D$ rolls is greatly enhanced and it does not bifurcate to CR. The $2D$ rolls become unstable via HB and periodic wavy rolls are generated and eventually becomes chaotic via quasiperiodic route for value of $r$. The time series of the mode $W_{101}$ have been shown in Fig. \[fig:time\_series\_Q400\] for three different values of $r$ with $\mathrm{Q} = 400$. The figure clearly shows a quasiperiodic route to chaos. We then consider values of $\mathrm{Q}$ upto $3900$ and found quasiperiodic routes to chaos only in DNS. It is interesting to note that we do not observe subharmonic route to chaos like [@fauve:1984] for high values of $\mathrm{Q}$ in our simulation. The Nusselt number plot for DNS with high $\mathrm{Q}$ have been shown in Fig. \[fig:nusselt\]. From the figure, a sharp change in the slope of Nusselt number is observed for $\mathrm{Q}\neq 0$. The heat flux corresponding to the stationary solutions follow a common upper boundary with higher slope and it deviates from the upper boundary as flow becomes time dependent for all nonzero $\mathrm{Q}$.
Conclusions
===========
In this article, we have presented the results of our investigation on the effect of external horizontal magnetic field on oscillatory instability and routes to chaos in RBC of low Prandtl-number fluids with free-slip boundary conditions. We perform direct numerical simulations as well as low dimensional modeling for the investigation. We find that oscillatory instability is inhibited due to the horizontal magnetic field and the supercritical Rayleigh number shows a scaling with $\mathrm{Q}$ similar to experimental result. DNS results also show that the routes to chaos depends on the magnitude of external magnetic field. Then we carry out detailed investigation with $\mathrm{Pr} = 0.025$, the Prandtl number of mercury by DNS and low dimensional modeling. A large range of $\mathrm{Q}$ ($0 < \mathrm{Q} \leq 3900$) has been considered for the investigation. We find that the route to chaos is period doubling for lower values of $\mathrm{Q}$ and quasiperiodic routes are observed for higher values of $\mathrm{Q}$. These results show similarity with the experimental results [@fauve:1983].\
[99]{} S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (Cambridge University Press, Cambridge, 1961). M.R.E. Proctor and N.O. Weiss, [Rep. Prog. Phys.]{} [**45**]{}, 1317 (1982); F. Cattaneo, T. Chiuch and D. W. Hughes, [Mon. Not. R. Astr. Soc.]{} [**247**]{}, 6 (1990); K. Zhang and G. Schubert, Annu. [Rev. Fluid Mech.]{} [**32**]{}, 409 (2000); R.F. Stein and A. Nordlund, Astrophys. J. [**642**]{}, 1246 (2006).
D.T.J. Hurle and R.W. Series, edited by D.T.J. Hurle (North Holland, Amsterdam, 1994). A. Gailitis et.al., Rev. Mod. Phys. [**74**]{}, 973 (2002). E. Bodenschatz, W. Pesch, and G. Ahlers, Annu. Rev. Fluid Mech. [**32**]{}, 709 (2000); G. Ahlers, S. Grossmann, and D. Lohse, Rev. Mod. Phys. [**81**]{}, 503 (2009). J. Mech.Theor. Appl. [**2**]{} 495 (1983). R. M. Clever and F. H. Busse, J. Fluid Mech. [**201**]{}, 507 (1989). , Phy. Rev. A [**40**]{}, 1954 (1989). F. H. Busse and W. Pesch, Geophys. Astrophys. Fluid Dyn. [**100**]{}, 139 (2006). L. Lehnert and N. C. Little, Tellus [**9**]{}, 97 (1957); Y. Nakagawa, Proc. R. Soc. London Ser. A [**240**]{}, 108 (1957). F. H. Busse and R. M. Clever, Phys. Fluids [**25**]{}, 931 (1982); E. Knobloch and N.O. Weiss, Physica D [**9**]{}, 379 (1983); P. Sulem , Prog. Astro. Aeronaut. [**100**]{}, 125 (1985); O.M. Podvigina, Phys. Rev. E [**81**]{}, 056322 (2010); P. Pal, and K. Kumar, Eur. Phys. J. B [**85**]{}, 201 (2012); A. Basak, R. Raveendran, and K. Kumar, Phys. Rev. E [**90**]{}, 033002 (2014). S. Fauve, C. Laroche and A. Libchaber, [J. Phys. Lett.]{}, [**42**]{}, L455 (1981). S. Fauve, C. Laroche and A. Libchaber, [J. Phys. Lett.]{}, [**45**]{}, 101(1984). S. Fauve , [Phys. Rev. Lett.]{} [**52**]{}, 1774 (1984). U. Burr and U. Müller, J. Fluid Mech. [**453**]{}, 345 (2002). T. Yanagisawa et al., [Phys. Rev. E]{}, [**88**]{}, 063020 (2013). A. Libchaber, S. Fauve, and C. Laroche, [Physica D 7]{}, 73 (1983). W. M. Macek, and M. Strumik, [Phys. Rev. Lett.]{} [**112**]{}, 074502 (2014). M. Meneguzzi , [J. Fluid Mech.]{} [**182**]{}, 169 (1987). M. K. Verma , [Pramana]{} [**81**]{} 617 (2013). F. H. Busse, [J. Fluid Mech.]{} [**52**]{}, 97 (1972). , [Europhys. Lett.]{} [**89**]{}, 44003 (2010). A. Dhooge, W. Govaerts, and Y. A. Kuznetsov, ACM Trans. Math. Softw. [**29**]{}, 141 (2003). F. H. Busse and E. W. Bolton, [J. Fluid Mech.]{} [**146**]{}, 115 (1984); E. W. Bolton and F. H. Busse, [J. Fluid Mech.]{} [**150**]{}, 487 (1985). , [J. Fluid Mech.]{} [**240**]{}, 229 (1992).
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abstract: 'Electron collisions, described by stochastic differential equations (SDEs), were simulated using a second-order weak convergence algorithm. Using stochastic analysis, we constructed an SDE for energetic electrons in Lorentz plasma to incorporate the collisions with heavy static ions as well as background thermal electrons. Instead of focusing on the errors of each trajectory, we initiated from the weak convergence condition to directly improve the numerical accuracy of the distribution and its moments. A second-order weak convergence algorithm was developed, estimated the order of the algorithm in a collision simulation case, and compared it with the results obtained using the Euler-Maruyama method and the Cadjan-Ivanov method. For a fixed error level, the second order method largely reduce the computation time compared to the Euler-Maruyama method and the Cadjan-Ivnaov method. Finally, the backward runaway phenomenon was numerically studied and the probability of this runaway was calculated using the new method.'
author:
- Wentao Wu
- Jian Liu
- Hong Qin
bibliography:
- 'stochastic.bib'
title: 'Stochastic simulation of collisions using high-order weak convergence algorithms'
---
[^1]
Introduction
============
Monte Carlo method is a approach that calculates the distribution or expectation through random sampling and simulations. It is commonly used to simulate collision in plasma by first solving the corresponding Newton equation with random forces and then statistically reconstructing the distribution function. Random force was originally formulated by Paul Langevin[@Langevin1908] and the equation was named after him. It was proposed as an intuitive approach to explain Brownian motion after diffusion theory was developed by Einstein[@Einstein1906]. A rigorous mathematical framework for this remained undiscovered until Wiener and Ito formulated the integral theories of the Wiener process and the stochastic differential equation (SDE) theory[@Ito1974; @Klebaner2005; @Oeksendal2003; @Karatzas1991; @higham2001algorithmic]. According to their works, the effect of the random force was interpreted as an integral in a Wiener process rather than a simple random variable plugged into an ordinary differential equation. Thus, the Langevin equation was reformed as an SDE and its connection to diffusion theory was revealed by the famous Feynman-Kac formula, which boosted the research on SDEs. In most applications[@Albright2002; @Albright2003; @Cadjan1999; @Cadjan1997; @Castejon2003; @Fernandez2012; @Manheimer1997; @Rosin2014; @shkarofsky1992numerical], the SDE is solved with numerical algorithms. Various numerical algorithms have been proposed to solve this equation. Most studies simply adopted the Euler-Maruyama method[@Jones1996; @Manheimer1997; @Albright2002; @Albright2003], and some used the Milstein method[@mil1975approximate].
Specific to the research of collision in plasma physics, SDE have mainly been used to study the collision effects. By applying the traditional Langevin approach, Jones and Manheimer[@Jones1996; @Manheimer1997] separately developed Coulomb collision models in particle-in-cell (PIC) simulations. Cadjan and Ivanov[@Cadjan1999; @Cadjan1997] first expressed the Lorentz-collision operator in a modern SDE form. Subsequently, Albright developed the quiet direct simulation Monte Carlo (QDSMC)[@Albright2002; @Albright2003] technique utilizing the Ito stochastic integral. These methods have been applied to the study of wave-particle interactions[@Castejon2003] and runaway electrons[@Fisch1987; @Karney1986]. Additionally, Cadjan and Ivanov proposed a numerical method[@Cadjan1999; @Cadjan1997], which is referred to as the Cadjan-Ivanov method in this study.
However, the numerical errors in traditional procedures for solving the Langevin equations have not been carefully discussed. Traditional improvement of numerical schemes mainly focused on finding a correct and accurate path. As a result, the measurement of the accuracy of a numerical solution was judged by its deviation from theoretical paths. This criterion can be strictly explained using the mathematical concept of strong convergence. While, for Monte Carlo simulation of collisions, it is more important to pursue an accurate distribution function or its moments instead of the path itself. In other words, although the strong convergence condition implies the correctness of the distribution function, it need not be necessarily met as the condition is too strong. Consequently, the viability of a numerical algorithm should be evaluated based on the deviation of the numerical distribution from the theoretical distribution, which is referred to as the weak convergence condition.
The order of both the Euler-Maruyama method and the Cadjan-Ivanov method was 0.5 under strong convergence. The path of a numerical solution generated by these traditional methods converges to an exact solution with a numerical error on the order of $\sqrt{\mathrm{\Delta }}$. From the viewpoint of weak convergence, the distribution of numerical solutions of the two methods is convergent to the distribution of exact solutions on the order of one. Occasionally, even the strong convergence order has been improved to 1.0, as with the Milstein method[@Kloeden1992; @Milstein2004], while the weak convergence order remained at this value. More precisely, the accuracy of the distribution function does not improve even if the path of a particle is calculated with a more accurate algorithm. Therefore, to improve the accuracy of an Monte Carlo simulation, increasing the convergence rate from a weak convergence aspect is more efficient and economic.
We developed a second order Monte Carlo simulation algorithm using modern SDE framework under weak convergence for extended Lorentz collision operator, which not only includes the collision effects of energetic electron with background ions yielded the standard Lorentz operator, but also the collision with background electrons. The SDE of the energetic electrons are derived from the Boltzmann equation for such electrons with extended Lorentz collision terms using the Dynkin formula[@Klebaner2005; @Karatzas1991]. By further assuming the diffusion coefficient matrix to be symmetric, an analytical SDE form can be derived using Cadjan and Ivanov’s decomposition method[@Cadjan1997]. By applying the Ito-Taylor expansion, the solution of the Ito SDE can be expanded into the sum of a series of Ito multiple integrals[@Kloeden1992]. Under weak convergence conditions, these integrals can be further simplified by replacing the distribution of the increment of the Wiener process-which is a normal distribution-with a three-point distribution[@kloeden1992higher; @Kloeden1992]. By dropping the remainder terms, a second-order algorithm under the weak convergence condition can be derived.
To test the performance of the second-order method, the numerical order of the algorithm was estimated in a backward runaway[@Karney1986] simulation and compared with those estimated using the Euler-Maruyama and the Cadjan-Ivanov methods. To estimate the error of each method and time steps, a good enough solution was treat as the theoretical value. The good enough solution was required to meet two criterions. First, the solution should be close to that of nearest larger time steps using the same method, which is tested using a Welch’s t-test[@welch1947]. Second, the solution should be close to that of different methods using the same time step, which is tested usign a ANOVA F-test[@fisher1936use]. Numerical orders of algorithms were estimated using ordinary least squares (OLS) regression between the logarithms of errors and the logarithms of time steps. The mean parallel velocity and energy was evaluated in samples as the first and the second order moment function of solutions for investigation. For both of the two physical quantity, the second-order method is near 2.0 higher than the order of the Euler-Maruyama and the Cadjan-Ivanov methods which are both near 1.0. Meanwhile, to achieve the same error level of $\exp(-9)$, the second order use the least time nearly half of the Cadjan-Ivanov method and 1/20 of the Euler-Maruyama method.
To demonstrate the usage of the second-order method with extended Lorentz operator, a practical case of backward runaway phenomenon is simulated. Although this is a runaway phenomenon[@Dreicer1959; @Wesson2011], the electrons initially travel in the direction identical to a DC electric field. The backward runaway probability is the probability that such runaway occurs at a given initial electron velocity. Using the second-order algorithm, the runaway phenomenon was simulated and visualized at different times and the runaway probability was calculated. In addition, when the initial velocity ran over the $v_{||}-v_\perp$ velocity space, the contour of the backward runaway probability was plotted.
The remainder of this paper is organized as follows. In Section II, we derive the stochastic differential equation governing the electron collision effect in plasmas. In Section III, a second-order algorithm under a weak convergence condition is presented and compared with the Euler-Maruyama and the Cadjan-Ivanov method. Finally, in Section IV, the backward runaway probability is calculated using the second order method.
Collision Model of Electrons
============================
The evolution of the electron distribution is governed by a Boltzmann equation with a collision operator. We used $f$ to denote the distribution function of electrons. The collision operator on the right-hand side of the equation encapsulates the collision effects as a partial derivative of the distribution function. Eq. presents the dominant Boltzmann equation of the electron distribution function, including the collision effects: $$\frac{\partial f}{\partial t}+\bm{v}\cdot \frac{\partial f}{\partial \bm{x}}+\frac{q}{m}(\bm{E}+\bm{v}\times \bm{B})\cdot \frac{\partial f}{\partial \bm{v}}={\left(\frac{\partial f}{\partial t}\right)}_{\mathrm{coll}}.
\label{eqn:Boltzmann}$$
Among the varieties of collision operators, a suitable choice for high-energy electrons in plasmas is the Lorentz collision operator[@helander2005collisional], which is expressed as follows: $${\left(\frac{\partial f}{\partial t}\right)}_L=\frac{Z_i\mathrm{\Gamma }}{2v^3}\frac{\partial }{\partial \mu }\left(1-{\mu }^2\right)\frac{\partial f}{\partial \mu }\,
\label{eqn:lorentz}$$ where $\mu =v_{||}/v$ represents the velocity parallel to electric field, $Z_i$ is the charge carried by ions, and $\mathrm{\Gamma }$ is a constant defined by $\mathrm{\Gamma }=n_eq^4\mathrm{ln}\mathrm{\Lambda }/4\pi {\epsilon }^2_0m^2_e$, indicating the collision intensity. This operator reflects the collision effects of electrons with ions under two assumptions. The first assumption states that the ions are uniformly distributed in the background, whereas the second states that the mass of ions is much greater than that of electrons.
To incorporate the effects of collisions between the energetic and background electrons and ions under the velocity limit condition $v\gg v_T$ for the extended Lorentz operator, a velocity friction term is added[@Karney1986; @shkarofsky1992numerical], which is expressed as follows: $${\left(\frac{\partial f}{\partial t}\right)}_{friction}=\mathrm{\Gamma }\frac{\bm{v}}{v^3}\cdot \frac{\partial f}{\partial \bm{v}}.
\label{eqn:friction}$$ In addition, the original pitch-angle-scattering effect caused by the ion-electron collision $$\left(\frac{\partial f}{\partial t}\right)_{scatter}=\frac{\mathrm{\Gamma Z_i}}{2v^3}\frac{\partial }{\partial \mu }\left(1-{\mu }^2\right)\frac{\partial f}{\partial \mu }
\label{eqn:scatter_ei}$$ was enhanced by the electron-electron collision $$\left(\frac{\partial f}{\partial t}\right)_{scatter}=\frac{\mathrm{\Gamma }}{2v^3}\frac{\partial }{\partial \mu }\left(1-{\mu }^2\right)\frac{\partial f}{\partial \mu }.
\label{eqn:scatter_ee}$$ The extended Lorentz collision operator including both effects for energetic electrons can be expressed as follows: $${\left(\frac{\partial f}{\partial t}\right)}_{\mathrm{extended}}=\mathrm{\Gamma }\frac{\bm{v}}{v^3}\cdot \frac{\partial f}{\partial \bm{v}}+\frac{\mathrm{\Gamma }\left(1+Z_i\right)}{2v^3}\frac{\partial }{\partial \mu }\left(1-{\mu }^2\right)\frac{\partial f}{\partial \mu }.
\label{eqn:colltotal}$$
To solve the Boltzmann equation Eq. with the extended Lorentz collision operator Eq. , we determined an SDE using the Dynkin formula[@Oeksendal2003; @Klebaner2005] on the collisional Boltzmann equation. We initiated from the Boltzmann equation for energetic electrons in Cartesian coordinates, i.e., $$\frac{\partial f}{\partial t}+\left(\frac{q\bm{E}}{m}+\frac{q}{m}\bm{v}\times \bm{B}-\mathrm{\Gamma }\frac{\bm{v}}{v^3}\right)\cdot \frac{\partial f}{\partial \bm{v}}+\frac{{(Z}_i\mathrm{+1)}\mathrm{\Gamma }}{2}\frac{\partial }{\partial \bm{v}}\cdot \left(\frac{v^2I-\bm{vv}}{v^3}\cdot \frac{\partial f}{\partial \bm{v}}\right)=0.
\label{eqn:cartcoor}$$ The corresponding Ito SDE has the following general form: $$d\bm{v}\left(t\right)=\bm{\mu }\left(\bm{v},t\right)dt+\sum^m_{j=1}{{\bm{\sigma }}_j\left(\bm{v},t\right)dW^j}.
\label{eqn:generalSDE}$$ Required the distribution of the solution of Eq. exactly solves the original Boltzmann equation Eq. , the coefficient of Eq. should satisfy the following relations: $$\begin{aligned}
\bm{\sigma }{\bm{\sigma }}^T&=\left(Z_i+1\right)\mathrm{\Gamma }\cdot \frac{v^2\bm{I}-\bm{vv}}{v^3}, \label{eqn:condsigma} \\
\bm{\mu }&=\frac{q\bm{E}}{m}+\frac{q}{m}\bm{v}\times \bm{B}-(2+Z_i)\mathit{\Gamma}\frac{\bm{v}}{v^3}. \label{eqn:condmu}\end{aligned}$$ According to Dynkin formula, the distribution of solution of such an Ito SDE solves the Boltzmann equation. It can be directly shown that the right-hand side of Eq. comprises the entries in a positive definite matrix. Furthermore, the $\bm{\sigma} $ matrix is the square root of this matrix. Therefore, $\bm{\sigma}$ can be numerically solved via Cholesky decomposition. By further demanding for $\bm{\sigma} $ to be symmetric, $\bm{\sigma}$ can be solved in a closed form using Cadjan and Ivanov’s decomposition method[@Cadjan1999; @Cadjan1997] as follows: $$\bm{\sigma }=\sqrt{\frac{\left(Z_i+1\right)\mathrm{\Gamma }}{v}}\left(\bm{I}-\frac{\bm{vv}}{v^2}\right).
\label{eqn:sigmacloseform}$$ Plugging the Eq. and Eq. into Eq. , the extended Ito SDE can be rewritten in vector form as follows: $$d\bm{v}=\left(\frac{q}{m}\bm{E}\bm{+}\frac{q}{m}\bm{v}\times \bm{B}-(2+Z_i)\mathrm{\Gamma }\frac{\bm{v}}{v^3}\right)dt-\sqrt{\frac{{(Z}_i+1)\mathrm{\Gamma }}{v^5}}\bm{v}\times \bm{v}\times d\bm{W}.
\label{eqn:totalsde}$$
The left-hand side of this SDE is the infinitesimal increment of the velocity process, whereas the right-hand side is the sum of the two terms. The first term is the total force experienced by a particle. It contains a Lorentz force generated by an external field and the friction arising from collisions. The second term is the effect of pitch-angle scattering. This term corresponds to the random force in the Langevin equation; however, it is written here in a handy notation that allows it be understood as an Ito integral in the Wiener process.
Higher-order weak convergence algorithms
========================================
Traditional measurement of an algorithm for solving an SDE in Monte Carlo simulation for collisions is focus on the path accuracy, which can be expressed by the definition of the strong convergence condition. To express it formally, let $X_T$ be the theoretical solution and $X^{\Delta }$ be the numerical solution. If there exist constants $C$ and ${\Delta }_0$ independent of the time step $\Delta $, for any $\Delta \in (0,{\Delta }_0)$, $$E\left[\left|X_T-X^{\Delta }(T)\right|\right]\le C{\Delta }^{\gamma },
\label{eqn:strong_conv}$$ we can say that $X^{\Delta }$ is strongly convergent[@kloeden1992higher; @Kloeden1992] to $X_T$ with order $\gamma $.
While, for Monte Carlo simulation of collisions, it is the distribution function or its moments rather than the sample path itself matters. Therefore, the measurement of a numerical algorithm should be established based on the difference between the distributions, which is the weak convergence condition. Formally, this condition can be expressed as follows: $$\left|E\left[g(X_T)\right]-E\left[g(X^{\Delta}(T))\right]\right|\le C{\Delta }^{\beta },
\label{eqn:weak_conv}$$ where $g\in C^{2\left(\beta +1\right)}\left({\mathcal{R}}^d,\mathcal{R}\right)$ is any continuous function. If a numerical solution, $X^{\Delta }$, satisfies Eq. , it is weakly convergent[@Kloeden1992] to $X$ at time $T$ with order $\beta $. Some simple choices of function $g$ help make this definition more intuitive. If random process $X(t)$ is taken as the velocity process and $g(X)=X$, weak convergence gives the expectation convergence condition, which implies the correct mean velocity. If $g(X)=(X-\bar{X})^2$, this then yields the variance convergence condition, which implies the correct temperature. For all continuous $g$, if Eq. hold, which guarantees all order of the moments.
The strong convergence condition sufficient guarantees the week convergence condition[@Kloeden1992]. However, to meet the week convergence condition, the strong condition is not necessarily to be satisfied. What is worse, the improvement of strong convergence order does not sufficiently results in the improvement of week convergence order. The Milstein method[@Kloeden1992], for example, have the first order strong convergence higher than the Euler-Maruyama which has the 1/2 order strong convergence. But they have the same the first order of weak convergence. Therefore, this kind of accuracy increment of sample path with additional computation cost is in vain for the distribution accuracy improvement. As a consequence, instead of improving the accuracy of electron trajectories, we directly seek more accurate distribution functions. Thus, a higher-order weak convergence algorithm is employed.
A second-order weak convergence numerical method is constructed as follows. To find the weak solution of the SDE with the general form $$dX^k={\mu }^k\left(t,X\right)dt+\sum^m_{j=1}{{\sigma }^{k,j}\left(t,X\right)dW_j},$$ a function of its solution can be expanded into multiple Ito integrals[@Klebaner2005; @Kloeden1992] as follows: $$f\left(X_{\tau }\right)=\sum_{\alpha\in\mathcal{A}} I_\alpha{\left[f_\alpha\left(\rho ,X_{\rho}\right)\right]}_{\rho ,\tau }+\sum_{\alpha\in\mathcal{B}(\mathcal{A})} I_\alpha{\left[f_\alpha\left(\cdot ,X_{\cdot }\right)\right]}_{\rho ,\tau },
\label{eqn:Ito_multiples}$$ where $\mathcal{A}$ is a hierarchical set, $\mathcal{B}(\mathcal{A})$ are the corresponding remainder sets[@Kloeden1992], and $\rho$ and $\tau$ are two stopping times that indicating the start and end time points. This is a stochastic analog of the deterministic Taylor expansion. The first term on the right-hand side of Eq. is the expansion series to a certain order, and the second term is the remainder term. By dropping the remainder term, an approximation of function $f$ is obtained.
Because the weak convergence condition is used, the simplification can go further. Each Ito integral can be estimated using the following equation[@Kloeden1992]: $$I_{\alpha }{\left[f_{\alpha }(t_0,X_{t_0})\right]}_{t_0,t}\approx f_{\alpha }\left(t_0,X_{t_0}\right){\hat{I}}_{\alpha ,t_0,t}.$$ Further substitution of multiple Ito integrals must satisfy the convergence condition. As for second-order weak convergence, the condition can be simplified to $$%\left|E\left[\hat{I}\right]\right|+\left|E\left[{\hat{I}}^2\right]-\mathrm{\Delta }\right|+\left|E\left[{\hat{I}}^3\right]\right|+\left|E\left[{\hat{I}}^4\right]-3{\mathrm{\Delta }}^2\right|+\left|E\left[{\hat{I}}^5\right]\right|\le K{\mathrm{\Delta }}^3
\left|E[\hat{I}]\right|+\left|E[{\hat{I}}^2]-\mathrm{\Delta }\right|+\left|E[{\hat{I}}^3]\right|+\left|E[{\hat{I}}^4]-3{\mathrm{\Delta }}^2\right|+\left|E[{\hat{I}}^5]\right|\le K{\mathrm{\Delta }}^3.$$ We introduce a simple three-point distribution of random variables[@Kloeden1992] $\mathrm{\Delta }{\hat{W}}^j$ as $\hat{I}$, which is defined as follows: $$P\left(\mathrm{\Delta }{\hat{W}}^j=\pm \sqrt{3\mathrm{\Delta }}\right)=\frac{1}{6},\qquad P\left(\mathrm{\Delta }{\hat{W}}^j=0\right)=\frac{2}{3}. \label{eqn:dwhat}.$$ It can be verified that in both the three-point and classic Gaussian $N(0;\mathrm{\Delta })$ distributions, the random variables meet the moment conditions. However, it is much easier and cheaper to generate and compute $\Delta\hat{W}$. This simplifies the calculation while maintaining the order of weak convergence.
An explicit second-order weak convergence algorithm is then constructed as follows[@Kloeden1992]. We define two supporting vector variables as follows: $$\begin{aligned}
{\overline{R}}^j_{\pm } &=X_n+\mu \left(X_n\right)\mathrm{\Delta }\pm {\sigma }^j(X_n)\sqrt{\mathrm{\Delta }},\\
{\overline{U}}^j_{\pm } &=X_n\pm {\sigma }^j\left(X_n\right)\sqrt{\mathrm{\Delta }}.\end{aligned}$$ Using these variables, numerical multiple integrals of the first and second order are constructed as follows: $$\mathrm{\Upsilon}_{c1}=\frac{1}{4}\sum_{j=1}^m\left[\sigma^j(\overline{R}^j_+)+\sigma^j(\overline{R}^j_-)+2\sigma^j(Y_n)+\sum_{ \substack{ r=1\\ r\neq j}}^m
\left(\sigma^j(\overline{U}^r_+)+\sigma^j(U^r_-)-2\sigma^j(Y_n)\right)\right]\mathrm{\Delta}{\hat{W}^j},$$ $$\mathrm{\Upsilon}_{c2}=\frac{{\mathrm{\Delta}}^{-\frac{1}{2}}}{4}\sum_{j=1}^m\left[\left(\sigma^j(\overline{R}^j_+)-\sigma^j(\overline{R}^j_-)\right)\left(\left(\mathrm{\Delta}{\hat{W}}^j\right)^2-\mathrm{\Delta }\right)+\sum^m_{\substack{r=1\\ r\neq j}}
\left(\sigma^j(\overline{U}^r_+)-\sigma^j(\overline{U}^r_-)\right)(\mathrm{\Delta }\hat{W}^j\mathrm{\Delta }\hat{W}^r+V_{r, j})\right],$$ where $$V=\begin{pmatrix}
-\mathrm{\Delta } & \cdots & {\xi }_{j_1,{\ j}_m} \\
\vdots & \ddots & \vdots \\
{-\xi }_{j_1,{\ j}_m} & \cdots & -\mathrm{\Delta } \end{pmatrix}$$ is a stochastic matrix with entries ${\xi }_{ij}$ set to be random variables taking the value $\Delta$ or $-\Delta$ with the same probability.
Finally, the second-order algorithm can be expressed as follows: $$\begin{aligned}
\overline{\mathrm{\Upsilon }}&=Y_n+\mu \left(Y_n\right)\mathrm{\Delta }+\sum^m_{j=1}{b^j\mathrm{\Delta }{\hat{W}}^j}, \label{eqn:finalalg1} \\
Y_{n+1}&=Y_n+\frac{1}{2}\left(a\left(\overline{\mathrm{\Upsilon }}\right)+a\left(Y_n\right)\right)\mathrm{\Delta }+{\mathrm{\Upsilon }}_{c1}+{\mathrm{\Upsilon }}_{c2}. \label{eqn:finalalg2}\end{aligned}$$
The first two terms on the right-hand side of Eq. correspond to the deterministic Euler method using a predictor-corrector scheme. The remaining two terms are the modifications for diffusion. ${\mathrm{\Upsilon }}_{c1}$ yields the first-order integral term of the Ito expansion, and ${\mathrm{\Upsilon }}_{c2}$ improves the accuracy of the final results up to the second order.
Backward runaway probability
============================
Phenomenon and theory
---------------------
The runaway effect, in a uniform electric field in space and without a magnetic field, results from the competition between collisional friction and external electric forces. A collision comprises two sources[@Karney1986]: the effect of energetic electrons colliding with heavy and static ions and the effect of energetic electrons colliding with background thermal electrons. Both kinds of collision will affect pitch-angle scattering. But only the collision with background electrons yields a frictional force on energetic electrons. If the electric force is greater than the collisional friction, the electron will be further accelerated. Higher velocity reduces the friction, hence the acceleration continues. This is called the runaway phenomenon. On the contrary, if the collisional friction is greater than the electric force, the electron will be slowed down. Lower velocity increases the friction until the velocity is reduced to the thermal velocity at the end. This is call being stopped.
The backward runaway mechanism is similar to that for standard forward runaways but with its initial velocity direction opposite to that of the electric force. Because the initial velocity moves in opposition to the electric force, the electron is pulled to slow down by the electric force at the beginning. When the speed is reduced to near the thermal velocity of electrons $v_T$, the high-velocity approximation is violated and the collision operator is no longer valid[@Fisch1987]. The collision frequency is assumed to be constant[@Fisch1987] there and the electrons are subsequently thermalized to be normally distributed in velocity space as background electrons. But if an electron gains a sufficient perpendicular velocity from pitch-angle scattering[@Fernandez2012; @Fisch1987], it can avoid this stopping phenomenon. Moreover, if the electric force exerted on the unstopped electron overcomes the dynamic friction, the electron can be indefinitely accelerated as forward runaway. This is called the backward runaway. Because the collision force of the scattering effect is a random effect, there is an uncertainty whether an electron will be stopped or classified as runaway. This uncertainty is measured by the backward runaway probability, which can be estimated by the fraction of electrons that are not stopped out of the total number of electrons.
According to the dynamics of backward runaway, the collision operator for energetic electrons in Eq. , which includes collision effects from both background ions and electrons, is used. By introducing the characteristic variables, the dominate equation Eq. \[eqn:totalsde\] is simplified into a dimensionless form. We introduce the Dreicer velocity[@Dreicer1959] $v_D$ and the characteristic collision time $\tau =v^3_D/\mathrm{\Gamma }$. The renormalized variables are $\tilde{v}=v/v_D$, $\tilde{t}=t/\tau $, $\tilde{E}=qE/F_D$, and $\tilde{W}=W(\tau \tilde{t})/\sqrt{\tau }\ $, which is a standard Wiener process as $W(t)$. The critical force is defined as $F_D=\mathrm{\Gamma }m_e/v^3_d$. Then, the dimensionless SDE can be written as follows: $$d\tilde{v}=\left(\tilde{E}-\left(2+Z_i\right)\frac{\tilde{v}}{{\tilde{v}}^3}\right)d\tilde{t}+{\sqrt{1+Z_i}\tilde{v}}^{-5/2}\tilde{v}\times \tilde{v}\times d\tilde{W}.
\label{eqn:dimless}$$
When the velocity of an energetic electron is less than the Dreicer velocity, it is labeled as a stopped electron. As a result, Eq. is no longer valid and the velocity of the stopped electron is simulated by sampling from the thermal distribution of background electrons. Furthermore, we did not consider the secondary electron emission problem, i.e., once an energetic electron is stopped, there is no way for it to become excited again.
At the end of simulation at time $T$, the runaway probability $P_r$ can be defined as follows: $$P_r=E\left[I_r(X_T)\right],
\label{eqn:def_runaway_prob}$$ where $I_r(X)$ is an indicator function defined as follows: $$I_r(X)=
\begin{cases}
1 & X\ge v_D, \\
0 & otherwise.
\end{cases}
\label{eqn:indicator_func}$$
Estimation of the order of algorithms
-------------------------------------
The weak order of the algorithm is defined according to Eq. . We estimate the numerical order of the second method, the Euler-Maruyama and the Cadjan-Ivanov method in the backward runaway simulation case. The numerical orders of the three methods are compared. The numerical order is calculated according to the following procedure. First, given the time step and numerically solve the SDE Eq. . By averaging the function $g(X^\Delta_T)$ to estimate the expectation $E[g(X^\Delta_T)]$. Second, change the step size and calculate the expectation $E[g(X^\Delta_T)]$ with the same way. Third, perform the ordinary least square (OLS) on the logarithms of the error of expectation $E[g(X^\Delta_T)]$ and the logarithms of the time steps $\Delta_t$ to obtain the slope, which is the order estimated.
The third in the procedure above need to know the theoretical solution to calculate errors in practice, but the theoretical solution of equation eq. is unknown. To overcome this difficulty, we find out a good enough solution to replace the theoretical solution. The solution is so accurate that first, it cannot be distinguished from that of nearest larger time steps and second, it cannot be distinguished from the solution of different method at the same time step.
To test the first requirement, since the variances of solution differ in different time steps, the Welch’s unequal variance t-test is used to detect the difference of two solutions. If the null hypothesis of the test cannot be rejected, we cannot tell the difference and thus it satisfies the first requirement. In addition, Shapiro test is performed to check the normality of solution distribution in order to meet the presumption of Welch’s t-test.
To test the second requirement, the analysis of variance (ANOVA) technique is used. The ANOVA F-test is utilized to detect the means of different method is equal or not. If the solution is normally distributed and the F-test fails to reject the null hypothesis, the second requirement is satisfied.
Given an SDE, the estimation of $E[g(X^\Delta_T)]$ and corresponding variance is calculated according to the following algorithm.
calculate the mean and variance of $E[g(X^\Delta_T)]$ of M batches
In the simulation, we set $N = 100,000$ samples in each batch and $M = 30$ total batches. The initial time step is $\Delta\tilde{t}=1.0$. The simulation scans from $K=0$ where $\Delta\tilde{t}=2^{0}$ to $K=6$ where $\Delta\tilde{t}=2^{-6}$ in total 7 different steps. All simulations end at $\tilde{t}=1.0$. In addition, $Z_i=1$ and the electric field is $\tilde{E}=\{1,0,0\}$, and the initial velocity is $\tilde{v} = \{3,0,0\}$.
The samples of solution at different time step using the Euler-Maruyama method and the weak order method are plotted in the Fig. \[fig:weak\_property\]. Noticed that the second order method only guarantees the accuracy of the distribution function and the path is simplified using a three point distribution, Eq. , the solution from the second order method is first spitted into grids but keep the moment correct. With collision accumulation, the distribution obtained from different method converges.
![The samples of step size being $2^{-7}$ obtained from the second order method at time step 1, 10, 50 are plotted in the left column (a), (b) and (c). And the results of the Euler-Maruyama method at corresponding time step 1, 10, 50 are plotted in the right column (d), (e) and (f).\[fig:weak\_property\]](01_weak_strong_compare){width="0.8\linewidth"}
To demonstrate the numerical order of the algorithm, we choose two typical functions with clear physical meanings. Noticed average of the $y$ and $z$ components of the solution are both 0 due to the system’s symmetry, our first function is chosen to be the $x$ velocity component, i.e., $g_1(X)=X_x$, which is a first-order moment function. The second is the energy function $g_2(X)=X_x^2+X_y^2+X_z^2$, which is a second-order moment function.
The expectation of $g_1$ and $g_2$ with respect to time step is plotted in Fig.(\[fig:moments\_values\]). All three numerical methods converge to the same value with decreasing time steps as $K$ increasing. The first moments, $E[g_1(X^\Delta)]$ are plotted in Panel (a) using the three algorithms and the second moments, $E[g_2(X^\Delta)]$, are plotted in Panel (b).
![The first-order moment functions $E[g_1(X^\Delta)]$s using the Cadjan-Ivanov method (blue), the Euler-Maruyama method (orange), and the second-order method (green) are plotted in (a). The second-order moment functions $E[g_2(X^\Delta)]$ using the three methods are plotted in (b).\[fig:moments\_values\]](02_Eg_1vsK-Eg_2vsK){width="0.8\linewidth"}
As discussed above, we next choose a good enough solution as the theoretical solution that satisfied the two requirements. First, solutions with the same method in two different time steps is investigated. The solution of $K=7$ is selected as potential target and is compared with the solution of $K=6$. To quantitatively demonstrate that the two solutions under $K=5$ and $K=6$ are sufficiently close, we perform Welch’s unequal variance t-tests for the functions of the first- and second-order moments. The normality of different method in $K=5$ and $K=6$ are tested using Shapiro test, and the statistics are listed in table. \[tab:shapiro\_test\] Under Welch’s unequal variance t-test on $E[g_1(X^\Delta)]$ and $E[g_2(X^\Delta)]$, We calculate the statistics ($t$), degrees of freedom ($\nu$), and corresponding p-values in table. \[tab:ttest\_K\].
From table. \[tab:ttest\_K\], when the time step is reduced from $K=6$ to $K=7$, the $p$-value of second order method on two moments function are all above 50%. We cannot reject the null hypothesis that the two solution are the same, so the solution of second order method with $K=7$ passes the first test.
---------------------------- ---------------- -------------- -------------- -------------- -------------- -- -- -- -- -- --
Expectation Methods
W-statistic p-value W-statistic p-value
\*[$E[g_1(X_T^\Delta)]$]{} Euler-Maruyama 0.9288509488 0.0457747914 0.9448584914 0.1229689941
Cadjan-Inanov 0.9723019600 0.6038923264 0.9418374896 0.1019422486
Second-Order 0.9591118097 0.2939443886 0.9595355392 0.3014061153
\*[$E[g_2(X_T^\Delta)]$]{} Euler-Maruyama 0.9620262384 0.3486383259 0.9357385635 0.0698712692
Cadjan-Inanov 0.9853706956 0.9433261156 0.9483121037 0.1523194462
Second-Order 0.9637868404 0.3855869174 0.9749118686 0.6801327467
---------------------------- ---------------- -------------- -------------- -------------- -------------- -- -- -- -- -- --
---------------------------- ---------------- ---- -------------- -------------- ------- -------------- -------------- ---------------- --------------- -------------- --
Expection Methods t $\nu$ p-value
M Mean Std M Mean Std (two-sided)
\*[$E[g_1(X_T^\Delta)]$]{} Euler-Maruyama 30 1.5181471154 0.0010572291 30 1.5167099144 0.0009388375 -5.4738626822 57.2007133978 0.0000010189
Cadjan-Inanov 30 1.5161602266 0.0011350644 30 1.5155411902 0.0008255942 -2.3751147326 52.9744270096 0.0211956264
Second-Order 30 1.5155835766 0.0011201136 30 1.5154497894 0.0011551838 -0.4477535215 57.9449611347 0.6560006370
\*[$E[g_2(X_T^\Delta)]$]{} Euler-Maruyama 30 3.6075405807 0.0011471987 30 3.6011830706 0.0007982998 -24.4960634146 51.7508692636 0.0000000000
Cadjan-Inanov 30 3.5952511325 0.0005412185 30 3.5945317579 0.0007720412 -4.1554688610 52.4304111506 0.0001202918
Second-Order 30 3.5944935081 0.0008078345 30 3.5943761437 0.0008143028 -0.5510099036 57.9963114240 0.5837434234
---------------------------- ---------------- ---- -------------- -------------- ------- -------------- -------------- ---------------- --------------- -------------- --
Expectation Methods Set Sum Squares Degree of Freedom Mean Squares F-statistic p-value
-------------------------- -------------------- ------------ -------------- ------------------- -------------- ---------------- --------------
\*[$E[g_1(X^\Delta)]$]{} Euler-Maruyama and Treatments 0.0000296219 2 0.0000148109 14.8238525409 0.0000028839
Cadjan-Ivanov and Error 0.0000869241 87 0.0000009991
Second Order Total 0.0001165460 89
\*[$E[g_1(X^\Delta)]$]{} Cadjan-Ivanov and Treatments 0.0000001253 1 0.0000001253 0.1201698569 0.7301048951
Second-Order Error 0.0000604817 58 0.0000010428
Total 0.0000606070 59
\*[$E[g_2(X^\Delta)]$]{} Euler-Maruyama and Treatments 0.0009059843 2 0.0004529922 700.0487590997 0.0000000000
Cadjan-Ivanov and Error 0.0000562965 87 0.0000006471
Second-Order Total 0.0009622808 89
\*[$E[g_2(X^\Delta)]$]{} Cadjan-Ivanov and Treatments 0.0000003632 1 0.0000003632 0.5666709067 0.4546294465
Second-Order Error 0.0000371781 58 0.0000006410
Total 0.0000375413 59
Expectation Method slope mean slope std. slope t-statistics p-value $R^2$
-------------------------- ---------------- ------------ ------------ -------------------- --------- -------
\*[$E[g_1(X^\Delta)]$]{} Euler-Maruyama -0.9833 0.011 -93.483 0.000 0.999
Cadjan-Ivanov -1.0038 0.018 -56.783 0.000 0.998
Second-order -1.7603 0.087 -20.265 0.000 0.990
\*[$E[g_2(X^\Delta)]$]{} Euler-Maruyama -0.9091 0.020 -45.462 0.000 0.997
Cadjan-Ivanov -1.1313 0.042 -27.137 0.000 0.993
Second-order -2.0024 0.027 -75.199 0.000 0.999
For the second test, given the fixed time step to be $K=7$, the equality of mean of solutions from the different methods are tested using AVOVA technique. Since the normality has been verified using shapiro test before in table. \[tab:shapiro\_test\], we directly conduct the AVOVA. We perform the AVOVA on two groups of methods, one consists of all three methods and the other group only consists of the Cadjan-Ivanov and the second order method. The analysis results are listed in table. \[tab:ttest\_methods\].
According to table. \[tab:ttest\_methods\], in the group of all three methods, the means are significantly different regardless of $E[g_1(X_T^\Delta)]$ or $E[g_2(X_T^\Delta)]$, the main error is bought by the Euler-Maruyama method. But in the group containing only the Cadjan-Inanov and the second order method, we cannot reject the null hypothesis of the means are the same. Therefore, we select the solution from the second order method under $K=7$ as the accurate solution and used to compute error of other methods and other time steps.
The error is calculated and the numerical order of algorithm is obtained from OLS regression as slope according to the logarithms of errors and logarithms of time steps. The result is plotted in Fig. \[fig:logerr\] and statistics are shown in table. \[tab:slopes\]
![Logarithms of errors of $E[g_1(X^\Delta)]$ (a) and $E[g_2(X^\Delta)]$ (b) at different logarithms of time steps $K$ using the Euler-Maruyama method (blue)，the Cadjan-Ivanov (orange) and the second-order methods (green).\[fig:logerr\]](03_loglog_err){width="0.8\linewidth"}
The errors of the first order moments $E[g_1(X_T^\Delta)]$ and second order moments $E[g_2(X_T^\Delta)]$ are plotted in subplots Fig. \[fig:logerr\]a and Fig. \[fig:logerr\]b, respectively. We can observe that the errors decrease when the time step is decreased for all algorithms. The order of the algorithm is estimated by the slope in table. \[tab:slopes\]. The Euler-Maruyama and the Cadjan-Ivanov methods have slope of 0.98 and 1.00 respectively for $E[g_1(X_T^\Delta)]$. And the second order has a slope of 1.76. For $E[g_2(X_T^\Delta)]$, the Euler-Maruyama and the Cadjan-Ivanov have the slope of 0.92 and 1.21. The second order method has a slope of 2.00. As expected, the second order method has a slope near 2.0 higher than these of the Euler-Maruyama and the Cadjan-Ivanov method near 1.0 for both functions. Even the other method may have a smaller error at large time steps, with the time step decreasing, the second order can reduce the error much faster. After $K>3$, the second order method perform better than all other methods.
In the aspect of time consumption, to reduce the error of $E[g_1(X_T^\Delta)]$ to the level of $\exp(-9)$, it costs the Euler-Maruyama method 26.374s and the Cadjan-Ivanov method 3.411s. But the the second order method only uses 1.510s. To reduce the error of $E[g_2(X_T^\Delta)]$ to the same level, it costs the Euler-Maruyama method 95.94s and the Cadjan-Ivanov method 6.796s. The the second order method uses 5.780s. The higher requirement of accuracy, the less time the second order method uses compared to the Euler-Maruyama method and the Cadjan-Ivanov method.
Runaway probability contour
---------------------------
The fixed ion background is set to contain protons that have $Z_i=1$. The electric field is uniformly distributed in the space and taken to be $\tilde{E}=(1,0,0)$ and no magnetic field is set. The initial velocities of the electrons are $\tilde{v}_{||}=10$ and $\tilde{v}_{\bot}=0$. Considering the time step to be $\mathrm{\Delta }\tilde{t}=0.01$, the simulation ends at $\tilde{t}=30$.
We solve the SDE Eq. to obtain 30 batches of solutions, and each batch contains 10,000 samples. One of the typical batches on different time is plotted in Fig. \[fig:runaway\_sim\]. The velocities are plotted in velocity space at three different time points: $\tilde{t}=6,\ \tilde{t}=14$, and $\tilde{t}=20$. The horizontal and vertical axes denote the parallel velocity $\tilde{v}_{||}$ and the vertical velocity $\tilde{v}_{\bot }$, respectively. Red samples are those whose velocity is less than $v_D$. They are thermally distributed as background electrons. On the contrary, blue points are samples that are considered to be eventual runaways.
The transient runaway probability time is calculated as the fraction of unstopped electron at a given time. Using time step size on $K=5$, we calculate the transient runaway probability of electron starting from different initial velocity.
From Fig. \[fig:trans\_runaway\_prob\], we verified that the t end to be 30 is a valid choice.
The runaway probability in Eq. in expectation form can be approximated by the average of function $I_r(X_T)$. Noticed that the indicator function take the value of one only when the sample is runaway, this expression is equivalent to the fraction of runaway sample to the total sample number. By calculating the ratio of runaway, the runaway probability is obtained.
Next, we calculated the backward runaway probability for electrons initiated under different conditions. We use $N=3,000$ samples and simulate one batch for each initial velocity. The time step is set to 0.01, and the time ends at $\tilde{t}=20$ and $Z_i=1$. The parallel velocity $v_{||}$ varies from $-8.0$ to $3.0$, whereas the perpendicular velocity ranges from $0.0$ to $5.0$. The contour is plotted in Fig. \[fig:runaway\_prob\].
It is evident from the figure that for $v_{||}$, the backward runaway probability increases as $v_{\bot }$ grows and that for a fixed $v_{\bot }$, this probability increases as $v_{||}$ moves away from 0. Electrons with $v_{||}$ greater than $v_D$ runaway almost certainly, allowing the forward runaway case to be recovered.
![The contour plot of backward runaway probability in velocity space obtained from a Monte Carlo simulation of 3,000 samples. The time step is 0.002, and the time ends at $\tilde{t}=20$. Nuclear charge is set to $Z_i=1$.\[fig:runaway\_prob\]](06_runaway_prob){width="0.8\linewidth"}
Discussion
==========
Starting with the Boltzmann equation of energetic electrons colliding with static heavy ions and background thermal electrons, we developed an SDE for the energetic electrons under both collision effects. Instead of aiming to improve the accuracy of the path of the sample, we rather aimed to improve the accuracy of the distribution. A second-order algorithm was developed to solve the SDE under the weak convergence condition, improving the accuracy with which the moments and distributions were calculated. The numerical weak order was estimated in the backward-runaway case and compared with that estimated using the widely applied the Euler-Maruyama and the Cadjan-Ivanov methods. The runaway probability was defined and shown to converge with order same as that of a weak-order algorithm. A backward runaway case was simulated, and the contours of runaway probability were calculated.
In future research, structure-preserving algorithms, such as symplectic algorithms[@Milstein2002a] or volume-preserving algorithms[@Qin2013], could be developed to solve the SDE. In addition, a system containing an electron source can be solved using the Feynman-Kac formula, after which the collision problem with the source term can thus be solved. The energetic electron simulation program may also inspire many applications in geophysics and space physics. The advantage of adopting the weak convergence condition is that it improves the calculation accuracy of the physical quantities associated with distributed computation and specific moment calculations.
acknowledgments
[^1]: Corresponding author, [email protected]
|
---
abstract: 'We introduce some new logics of imperfect information by adding atomic formulas corresponding to *inclusion* and *exclusion* dependencies to the language of first order logic. The properties of these logics and their relationships with other logics of imperfect information are then studied. Furthermore, a game theoretic semantics for these logics is developed. As a corollary of these results, we characterize the expressive power of independence logic, thus answering an open problem posed in (Grädel and Väänänen, 2010).'
address: |
Faculteit der Natuurwetenschappen, Wiskunde en Informatica\
Institute for Logic, Language and Computation\
Universiteit van Amsterdam\
P.O. Box 94242, 1090 GE AMSTERDAM, The Netherlands\
Phone: +31 020 525 8260\
Fax (ILLC): +31 20 525 5206
author:
- Pietro Galliani
bibliography:
- 'biblio.bib'
title: 'Inclusion and Exclusion Dependencies in Team Semantics\'
---
dependence ,independence ,imperfect information ,team semantics ,game semantics ,model theory 03B60 ,03C80 ,03C85
Introduction
============
The notions of dependence and independence are among the most fundamental ones considered in logic, in mathematics, and in many of their applications. For example, one of the main aspects in which modern predicate logic can be thought of as superior to medieval term logic is that the former allows for quantifier alternation, and hence can express certain complex patterns of dependence and independence between variables that the latter cannot easily represent. A fairly standard example of this can be seen in the formal representations of the notions of *continuity* and *uniform continuity*: in the language of first order logic, the former property can be expressed as $\forall x (\forall \epsilon > 0)(\exists \delta > 0) \forall x' ( |x - x'| < \delta \rightarrow |f(x) - f(x')| < \epsilon)$, while the latter can be expressed as $(\forall \epsilon > 0)(\exists \delta > 0) \forall x \forall x' (|x - x'| < \delta \rightarrow |f(x) - f(x')| < \epsilon)$. The difference between these two expressions should be clear: in the first one, the value of the variable $\delta$ is a function of the values of the variables $x$ and $\epsilon$, while in the second one it is a function of the value of the variable $\epsilon$ alone. This very notion of functional dependence also occurs, at first sight rather independently, as one of the fundamental concepts of Database Theory, and in that context it proved itself to be highly useful both for the specification and study of *normal forms* and for that of *constraints* over databases.[^1]\
*Logics of imperfect information* are a family of logical formalisms whose development arose from the observation that not all possible patterns of dependence and independence between variables may be represented in first order logic. Among these logics, *dependence logic* [@vaananen07] is perhaps the one most suited for the analysis of the notion of dependence itself, since it isolates it by means of *dependence atoms* which correspond, in a very exact sense, to functional dependencies of the exact kind studied in Database Theory. The properties of this logic, and of a number of variants and generalizations thereof, have been the object of much research in recent years, and we cannot hope to give here an exhaustive summary of the known results. We will content ourselves, therefore, to recall (in Subsection \[subsect:deplog\]) the ones that will be of particular interest for the rest of this work.\
*Independence logic* [@gradel10] is a recent variant of dependence logic. In this new logic, the fundamental concept that is being added to the first order language is not *functional dependence*, as for the case of dependence logic proper, but *informational independence*: as we will see, this is achieved by considering *independence atoms* ${y ~\bot_{x}~ z}$, whose informal meaning corresponds to the statement “for any fixed value of $x$, the sets of the possible values for $y$ and $z$ are independent”. Just as dependence logic allows us to reason about the properties of functional dependence, independence logic does the same for this notion. Much is not known at the moment about independence logic; in particular, one open problem mentioned in [@gradel10] concerns the *expressive power* of this formalism over open formulas. As we will see, a formula in a logic of imperfect information defines, for any suitable model $M$, the family of its *trumps*, that is, the family of all sets of assignments (*teams*, in the usual terminology of dependence logic) which satisfy the formula. This differs from the case of first order logic, in which formulas satisfy or do not satisfy single assignments, and the intuitive reason for this should be understandable: asking whether a statement such as “the values of the variables $x$ and $y$ are independent” holds with respect of a single variable assignment is meaningless, since such an assertion can be only interpreted with respect to a family of *possible assignments*. A natural question is then *which* families of sets of possible variable assignments may be represented in terms of independence logic formulas.[^2] An upper bound for the answer is in [@gradel10] already: all classes of sets of assignments which are definable in independence logic correspond to second order relations which are expressible in existential second order logic. In this work, we will show that this is also a *lower bound*: a class of sets of assignments is definable in independence logic if and only if it is expressible in existential second order logic. This result, which we will prove as Corollary \[coro:ILform\], implies that independence logic is not merely a formalism obtained by adding an arbitrary, although reasonable-looking, new kind of atomic formula to the first order language. It – and any other formalism equivalent to it – is instead a natural upper bound for a general family of logics of imperfect information: in particular, if over finite models an arbitrary logic of imperfect information characterizes only teams which are in NP then, by Fagin’s theorem [@fagin74], this logic is (again, over finite models) equivalent to some fragment of independence logic.\
The way in which we reach this result is also perhaps of some interest. Even though functional dependence and informational independence are certainly very important notions, they are by no means the only ones of their kind that are of some relevance. In the field of database theory, a great variety of other constraints over relations[^3] has indeed been studied. Two of the simplest such constraints are *inclusion dependencies* and *exclusion dependencies*, whose definitions and basic properties we will recall in Subsection \[subsect:incexc\]; then, in Subsections \[subsect:inclog\] and \[subsect:exclog\], we will develop and study the corresponding logics.[^4] As we will see, “exclusion logic” is equivalent, in a strong sense, to dependence logic, while “inclusion logic” is properly contained in independence logic but incomparable with dependence logic. Then, in Subsection \[subsect:ielogic\], we will consider *inclusion/exclusion logic*, that is, the logic obtained by adding atoms for inclusion *and* exclusion dependencies to the language of first order logic, and prove that it is equivalent to independence logic.\
Section \[sect:gamesem\] develops a *game theoretic semantics* for inclusion/exclusion logic. A game-theoretic semantics assigns truth values to expressions according to the properties of certain *semantic games* (often, but not always, in terms of the existence of *winning strategies* for these games). Historically, the first semantics for logics of imperfect information were of this kind; and even though, for many purposes, team semantics is a more useful and clearer formalism, we will see that studying the relationship between game semantics and team semantics allows us to better understand certain properties of the semantic rules for disjunction and existential quantification. Then, in Section \[sect:defin\], we examine the classes of teams definable by inclusion/exclusion logic formulas (or equivalently, by independence logic formulas), and we prove that these are precisely the ones corresponding to second order relations definable in existential second order logic.\
Finally, in the last section we show that, as a consequence of this, some of the most general forms of dependency studied in database theory are expressible in independence logic. This, in the opinion of the author, suggests that logics of imperfect information (and, in particular, independence logic) may constitute an useful theoretical framework for the study of such dependencies and their properties.
Dependence and independence logic
=================================
In this section, we will recall a small number of known results about dependence and independence logic. Some of the basic definitions of these logics will be left unsaid, as they will be later recovered in a slightly more general setting in Subsection \[subsect:laxstrict\]. This section and that subsection, taken together, can be seen as a very quick crash course on the field of logics of imperfect information; the reader who is already familiar with such logics can probably skim through most of it, paying however some attention to the discussion of independence logic of Subsection \[subsect:indlog\], the alternative semantic rules of Definition \[def:altsem\] and the subsequent discussion.
Dependence logic {#subsect:deplog}
----------------
Dependence logic [@vaananen07] is, together with IF logic ([@hintikka96], [@tulenheimo09]), one of the most widely studied logics of imperfect information. In brief, it can be described as the extension of first order logic obtained by adding *dependence atoms* $=\!\!(t_1 \ldots t_n)$ to its language, with the informal meaning of “The value of the term $t_n$ is functionally determined by the values of the terms $t_1 \ldots t_{n-1}$”.\
This allows us to express patterns of dependence and independence between variables which are not expressible in first order logic: for example, in the formula $\forall x \exists y \forall z \exists w (=\!\!(z, w) \wedge \phi(x,y,z,w))$ the choice of the value for the variable $w$ depends only on the value of the variable $w$, and not from the values of the variables $x$ and $y$ - or, in other words, this expression is equivalent to the branching quantifier ([@henkin61]) sentence $$\left(\begin{array}{c c}
\forall x & \exists y \\
\forall z & \exists w
\end{array}\right)
\phi(x, y, z, w)$$ and the corresponding Skolem normal form is $\exists f \exists g \forall x \forall z \phi(x, f(x), z, g(z))$.
The idea of allowing more general patterns of dependence and independence between quantifiers than the ones permitted in first order logic was, historically, the main reason for the development of logics of imperfect information: in particular, [@hintikka96] argues that the restriction on these patterns forced by first order logic has little justification, and that hence logics of imperfect information are a more adequate formalism for reasoning about the foundations of mathematics.
No such claim will be made or discussed in this work. But in any case, the idea of allowing more general patterns of dependence and independence between quantifiers seems a very natural one. In IF logic, the notion of dependence is, however, inherently connected with the notion of quantification: for example, the above expression would be written in it as $\forall x \exists y \forall z (\exists z / x, y) \phi(x, y, z, w)$, where $(\exists z / x, y)$ is to be read as “there exists a $z$, independent from $x$ and $y$, such that …”. Dependence logic and its variants, instead, prefer to separate the notion of dependency from the notion of quantification: in this second group of logics of imperfect information, dependence patterns between quantifiers are exactly as first order logic and our linguistic intuitions would suggest, but dependence atoms may be used to specify that the value of a certain variable (or, in general, of a certain term) must be a function of certain other values. This corresponds precisely to the notion of *functional dependence* which is one of the central tools of Database Theory; and indeed, as we will recall later in this work, the satisfaction conditions for these atoms are in a very precise relationship with the formal definition of functional dependence.
This, at least in the opinion of the author, makes dependence logic an eminently suitable formalism for the study of the notion of functional dependence and of its properties; and as we will see, one of the main themes of the present work will consist in the development and study of formalisms which have a similar sort of relationship with other notions of dependency.\
We will later recall the full definition of the *team semantics* of dependence logic, an adaptation of Hodges’ compositional semantics for IF-logic ([@hodges97]) and one of the three equivalent semantics for dependence logic described in [@vaananen07].[^5] It is worth noting already here, though, that the key difference between Hodges semantics and the usual Tarskian semantics is that in the former semantics the satisfaction relation $\models$ associates to every first order model[^6] $M$ and formula $\phi$ a set of *teams*, that is, a set of sets of assignments, instead of just a set of assignments as in the latter one.\
As discussed in [@hodges07], the fundamental intuition behind Hodges’ semantics is that a team is a representation of an *information state* of some agent: given a model $M$, a team $X$ and a suitable formula $\phi$, the expression $$M \models_X \phi$$ asserts that, from the information that the “true” assignment $s$ belongs to the team $X$, it is possible to infer that $\phi$ holds - or, in game-theoretic terms, that the Verifier has a strategy $\tau$ which is winning for all plays of the game $G(\phi)$ which start from any assignment $s \in X$.
The satisfaction conditions for the dependence atom is then given by the following semantic rule **TS-dep**:
\[def:dep\] Let $M$ be a first order model, let $X$ be a team over it, let $n \in \mathbb N$, and let $t_1 \ldots t_n$ be terms over the signature of $M$ and with variables in $\dom(X)$. Then
TS-dep:
: $M \models_X =\!\!(t_1 \ldots t_n)$ if and only if, for all $s, s' \in X$ such that $t_i \langle s \rangle = t_i\langle s'\rangle \mbox{ for } i = 1 \ldots n-1$, $t_n\langle s \rangle = t_n\langle s'\rangle$.
This rule corresponds closely to the definition of *functional dependency* commonly used in Database Theory ([@codd72]): more precisely, if $X(t_1 \ldots t_n)$ is the relation $\{(t_1\langle s\rangle, \ldots, t_n\langle s \rangle) : s \in X\}$ then $$M \models_X =\!\!(t_1 \ldots t_n) \Leftrightarrow X(t_1 \ldots t_n) \models \{t_1 \ldots t_{n-1}\} \rightarrow t_n$$ where the right hand expression states that, in the relation $X(t_1 \ldots t_n)$, the value of the last term $t_n$ is a function of the values of $t_1 \ldots t_{n-1}$.
Another formulation of the truth condition of a dependence atom\
$=\!\!(t_1 \ldots t_n)$, easily seen to be equivalent to this one, is the following: a team $X$ satisfies such an atom if and only if a rational agent $\alpha$, whose beliefs about the identity of the “true” assignment $s$ are described by $X$, would be capable of inferring the value of $t_n$ from the values of $t_1 \ldots t_{n-1}$.[^7] A special case of dependence atom, useful to consider in order to clarify our intuitions, is constituted by *constancy atoms* $=\!\!(t)$: applying the above definitions, we can observe that $M \models_X =\!\!(t)$ if and only if the value $t\langle s\rangle$ is the same for all assignments $s \in X$ - or, using the agent metaphor, if and only if an agent $\alpha$ as above *knows* the value of $t$.[^8]\
The following known results will be of some use for the rest of this work:
\[theo:DLloc\] Let $M$ be a first order model and let $\phi$ be a dependence logic formula over the signature of $M$ with free variables in ${\vec{v}}$. Then, for all teams $X$ with domain ${\vec{w}} \supseteq {\vec{v}}$, if $X'$ is the restriction of $X$ to ${\vec{v}}$ then $$M \models_X \phi \Leftrightarrow M \models_{X'} \phi.$$
As an aside, it is worth pointing out that the above property does not hold for most variants of $IF$-logic: for example, if $\dom(M) = \{0,1\}$ and $X = \{(x:0, y:0), (x:1, y:1)\}$ it is easy to see that $M \models_X (\exists z / y) z = y$, even though for the restriction $X'$ of $X$ to $\free((\exists z/y) z=y) = \{y\}$ we have that $M \not \models_{X'} (\exists z/y) z=y$.[^9]
\[theo:DLdc\] Let $M$ be a model, let $\phi$ be a dependence logic formula over the signature of $M$, and let $X$ be a team over $M$ with domain ${\vec{v}} \supseteq \free(\phi)$ such that $M \models_X \phi$. Then, for all $X' \subseteq X$, $$M \models_{X'} \phi.$$
\[theo:DLsent\] For every dependence logic sentence $\phi$, there exists a $\Sigma_1^1$ sentence $\Phi$ such that $$M \models_{\{\emptyset\}} \phi \Leftrightarrow M \models \Phi.$$
Conversely, for every $\Sigma_1^1$ sentence $\Phi$ there exists a dependence logic sentence $\phi$ such that the above holds.
\[theo:DLform\] For every dependence logic formula $\phi$ and every tuple of variables ${\vec{x}} \supseteq \free(\phi)$ there exists a $\Sigma_1^1$ sentence $\Phi(R)$, where $R$ is a $|{\vec{x}}|$-ary relation which occurs only negatively in $\Phi$, such that $$M \models_X \phi \Leftrightarrow M \models \Phi(\rel(X))$$ for all teams $X$ with domain ${\vec{x}}$.[^10]
Conversely, for all such $\Sigma_1^1$ sentences there exists a dependence logic formula $\phi$ such that the above holds with respect to all *nonempty* teams $X$.
Independence logic {#subsect:indlog}
------------------
Independence logic [@gradel10] is a recently developed logic which substitutes the dependence atoms of dependence logic with *independence atoms* ${{\vec{t}}_2 ~\bot_{{\vec{t}}_1}~ {\vec{t}}_3}$, where ${\vec{t}}_1 \ldots {\vec{t}}_3$ are tuples of terms (not necessarily of the same length).
The intuitive meaning of such an atom is that the values of the tuples ${\vec{t}}_2$ and ${\vec{t}}_3$ are informationally independent for any fixed value of ${\vec{t}}_1$; or, in other words, that all information about the value of ${\vec{t}}_3$ that can be possibly inferred from the values of ${\vec{t}}_1$ and ${\vec{t}}_2$ can be already inferred from the value of ${\vec{t}}_1$ alone.\
More formally, the definition of the team semantics for the independence atom is as follows:
\[def:indep\] Let $M$ be a first order model, let $X$ be a team over it and let ${\vec{t}}_1, {\vec{t}}_2$ and ${\vec{t}}_3$ be three finite tuples of terms (not necessarily of the same length) over the signature of $M$ and with variables in $\dom(X)$. Then
TS-indep:
: $M \models_X {{\vec{t}}_2 ~\bot_{{\vec{t}}_1}~ {\vec{t}}_3}$ if and only if for all $s, s' \in X$ with ${\vec{t}}_1\langle s\rangle = {\vec{t}}_1\langle s\rangle$ there exists a $s'' \in X$ such that ${\vec{t}}_1\langle s''\rangle {\vec{t}}_2\langle s''\rangle = {\vec{t}}_1 \langle s\rangle {\vec{t}}_2\langle s\rangle$ and ${\vec{t}}_1 \langle s''\rangle {\vec{t}}_3\langle s''\rangle = {\vec{t}}_1\langle s'\rangle {\vec{t}}_3\langle s'\rangle$.
We refer to [@gradel10] for a discussion of this interesting class of atomic formulas and of the resulting logic. Here we only mention a few results, found in that paper, which will be useful for the rest of this work:[^11]
\[theo:DL2IL\] Dependence atoms are expressible in terms of independence atoms: more precisely, for all suitable models $M$, teams $X$ and terms $t_1 \ldots t_n$ $$M \models_X =\!\!(t_1 \ldots t_n) \Leftrightarrow M \models_X {t_n ~\bot_{t_1 \ldots t_{n-1}}~ t_n}.$$
\[theo:ILsent\] Independence logic is equivalent to $\Sigma_1^1$ (and therefore, by Theorem \[theo:DLsent\], to dependence logic) over sentences: in other words, for every sentence $\phi$ of independence logic there exists a sentence $\Phi$ of existential second order logic such that $$M \models_{\{\emptyset\}} \phi \Leftrightarrow M \models \Phi.$$ and for every such $\Phi$ there exists a $\phi$ such that the above holds.
There is no analogue of Theorem \[theo:DLdc\] for independence logic, however, as the classes of teams corresponding to independence atoms are not necessarily downwards closed: for example, according Definition \[def:indep\] the formula ${x ~\bot_{\emptyset}~ y}$ holds in the team $$\{(x:0, y:0), (x:0, y:1), (x:1,y:0), (x:1, y:1)\}$$ but not in its subteam $\{(x:0,y:0), (x:1,y:1)\}$.\
The problem of of finding a characterization similar to that of Theorem \[theo:DLform\] for the classes of teams definable by formulas of independence logic was left open by Grädel and Väänänen, who concluded their paper by stating that ([@gradel10])
> *The main open question raised by the above discussion is the following, formulated for finite structures:*\
> **Open Problem:** *Characterize the NP properties of teams that correspond to formulas of independence logic.*
In this paper, an answer to this question will be given, as a corollary of an analogous result for a new logic of imperfect information.
Team semantics
==============
In this section, we will introduce some of the main concepts that we will need for the rest of this work and then we will test them on a relatively simple case. Subsection \[subsect:laxstrict\] contains the basic definitions of team semantics, following for the most part the treatment of [@vaananen07]; and furthermore, in this subsection we introduce two variant rules for disjunction and existential quantification which, as we will later see, will be of significant relevance. Then, in Subsection \[subsect:constancy\], we will begin our investigations by examining *constancy logic*, that is, the fragment of dependence logic obtained by adding constancy atoms to the language of first order logic. The main result of that subsection will be a proof that constancy logic is expressively equivalent to first order logic over sentences, and, hence, that it is strictly less expressive than the full dependence logic. This particular consequence is a special case of the far-reaching *hierarchy theorem* of [@durand11], which fully characterizes the expressive powers of certain fragments of dependence logic.
First order (team) logic, in two flavors {#subsect:laxstrict}
----------------------------------------
In this subsection, we will present and briefly discuss the team semantics for first order logic, laying the groundwork for reasoning about its extensions while avoiding, as far as we are able to do so, all forms of semantical ambiguity.\
As we will see, some special care is required here, since certain rules which are equivalent with respect to dependence logic proper will not be so with respect to these new logics. As it often is the case for logics of imperfect information, the game theoretic approach to semantics (which we will discuss in Section \[sect:gamesem\]) will be of support and clarification for our intuitions concerning the intended interpretations of operators.\
But let us begin by recalling some basic definitions from [@vaananen07]:
Let $M$ be a first order model, and let ${\vec{v}}$ be a tuple of variables.[^12] Then a *team* $X$ for $M$ with *domain* ${\vec{v}}$ is simply a set of assignments with domain ${\vec{v}}$ over $M$.
\[defin:relteams\] Let $M$ be a first order model, $X$ be a team for $M$ with domain ${\vec{v}}$, and let ${\vec{t}} = t_1 \ldots t_k$ be a tuple of terms with variables in ${\vec{v}}$. Then we write $X({\vec{t}})$ for the relation $$X({\vec{t}}) = \{(t_1\langle s\rangle \ldots t_k\langle s \rangle) : s \in X\}.$$ Furthermore, if ${\vec{w}}$ is contained in ${\vec{v}}$ we will write $\rel_{{\vec{w}}}(X)$ for $X({\vec{w}})$; and, finally, if $\dom(X) = {\vec{v}}$ we will write $\rel(X)$ for $\rel_{{\vec{v}}}(X)$.
Let $X$ be any team in any model, and let $V$ be a set of variables contained in $\dom(X)$. Then $$X_{\upharpoonright V} = \{s_{\upharpoonright V} : s \in X\}$$ where $s_{\upharpoonright V}$ is the restriction of $s$ to $V$, that is, the only assignment $s'$ with domain $V$ such that $s'(v) = s(v)$ for all $v \in V$.
The team semantics for the first order fragment of dependence logic is then defined as follows:
\[def:team\_fol\] Let $M$ be a first order model, let $\phi$ be a first order formula in negation normal form[^13] and let $X$ be a team over $M$ with domain ${\vec{v}} \supseteq \free(\phi)$. Then
TS-atom:
: If $\phi$ is a first order literal, $M \models_X \phi$ if and only if, for all assignments $s \in X$, $M \models_s \phi$ in the usual first order sense;
TS-$\vee_L$:
: If $\phi$ is $\psi \vee \theta$, $M \models_X \phi$ if and only if there exist two teams $Y$ and $Z$ such that $X = Y \cup Z$, $M \models_Y \psi$ and $M \models_Z \theta$;
TS-$\wedge$:
: If $\phi$ is $\psi \wedge \theta$, $M \models_X \phi$ if and only if $M \models_X \psi$ and $M \models_X \theta$;
TS-$\exists_S$:
: If $\phi$ is $\exists x \psi$, $M \models_X \phi$ if and only if there exists a function $F: X \rightarrow \dom(M)$ such that $M \models_{X[F/x]}\psi$, where $$X[F/x] = \{s[F(s)/x] : s \in X\};\footnote{Sometimes, we will write $X[F_1 F_2 \ldots F_n / x_1 \ldots x_n]$, or even $X[{\vec{F}}/{\vec{x}}]$, as a shorthand for $X[F_1/x_1][F_2/x_2]\ldots[F_n/x_n]$.}$$
TS-$\forall$:
: If $\phi$ is $\forall x \psi$, $M \models_X \phi$ if and only if $M \models_{X[M/x]} \psi$, where $$X[M/x] = \{s[m/x] : s \in X\}.\footnote{Sometimes, we will write $X[M/x_1 x_2 \ldots x_n]$, or even $X[M/{\vec{x}}]$, as a shorthand for $X[M/x_1][M/x_2]\ldots[M/x_n]$.}$$
Over singleton teams, this semantics coincides with the usual one for first order logic:
\[propo:FO\_Team2Tarski\] Let $M$ be a first order model, let $\phi$ be a first order formula in negation normal form over the signature of $M$, and let $s$ be an assignment with $\dom(s) \supseteq \free(\phi)$. Then $M \models_{\{s\}} \phi$ if and only if $M \models_{s} \phi$ with respect to the usual Tarski semantics for first order logic.
Furthermore, as the following proposition illustrates, the team semantics of first order logic is compatible with the intuition, discussed before, that teams represent states of knowledge:
\[propo:FOflat\] Let $M$ be a first order model, let $\phi$ be a first order formula in negation normal form over the signature of $M$, and let $X$ be a team with $\dom(X) \supseteq \free(\phi)$. Then $M \models_X \phi$ if and only if, for all assignments $s \in X$, $M \models_{\{s\}} \phi$.[^14]
On the other hand, these two proposition also show that, for first order logic, all the above machinery is quite unnecessary. We have no need of carrying around such complex objects as teams, since we can consider any assignment in a team individually!\
Things, however, change if we add dependence atoms $=\!\!(t_1 \ldots t_n)$ to our language, with the semantics of rule **TS-dep** (Definition \[def:dep\] here). In the resulting formalism, which is precisely *dependence logic* as defined in [@vaananen07], not all satisfaction conditions over teams can be reduced to satisfaction conditions over assignments: for example, a “constancy atom” $=\!\!(x)$ holds in a team $X$ if and only if $s(x) = s'(x)$ for all $s, s' \in X$, and verifying this condition clearly requires to check *pairs* of assignments at least![^15]\
When studying variants of dependence logic, similarly, it is necessary to keep in mind that semantic rules which are equivalent with respect to dependence logic proper may not be equivalent with respect to these new formalisms. In particular, two alternative definitions of disjunction and existential quantification exist which are of special interest for this work’s purposes:[^16]
\[def:altsem\] Let $M$, $X$, $\phi$, $\psi$ and $\theta$ be as usual. Then
TS-$\vee_S$:
: If $\phi$ is $\psi \vee \theta$, $M \models_X \phi$ if and only if there exist two teams $Y$ and $Z$ such that $X = Y \cup Z$, $Y \cap Z = \emptyset$, $M \models_Y \psi$ and $M \models_Z \theta$;
TS-$\exists_L$:
: If $\phi$ is $\exists x \psi$, $M \models_X \phi$ if and only if there exists a function $H: X \rightarrow \part(\dom(M))\backslash \emptyset$ such that $M \models_{X[H/x]}\psi$, where $$X[H/x] = \{s[m/x] : s \in X, m \in H(s)\}.$$
The subscripts of $\cdot_S$ and $\cdot_L$ of these rules and of the corresponding ones of Definition \[def:team\_fol\] allow us to discriminate between the *lax* operators $\vee_L$ and $\exists_L$ and the *strict* ones $\vee_S$ and $\exists_S$. This distinction will be formally justified in Section \[sect:gamesem\], and in particular by Theorems \[theo:game\_team\_lax\] and \[theo:game\_team\_strict\]; but even at a glance, this grouping of the rules is justified by the fact that **TS-$\vee_S$** and **TS-$\exists_S$** appear to be stronger conditions than **TS-$\vee_L$** and **TS-$\exists_L$**. We can then define two alternative semantics for first order logic (and for its extensions, of course) as follows:
\[def:team\_fol\_lax\] The relation $M \models_X^L \phi$, where $M$ ranges over all first order models, $X$ ranges over all teams and $\phi$ ranges over all formulas with free variables in $\dom(X)$, is defined as the relation $M \models_X \phi$ of Definition \[def:team\_fol\] (with additional rules for further atomic formulas as required), but substituting Rule **TS-$\exists_S$** with Rule **TS-$\exists_L$**.
\[def:team\_fol\_strict\] The relation $M \models_X^S \phi$, where $M$ ranges over all first order models, $X$ ranges over all teams and $\phi$ ranges over all formulas with free variables in $\dom(X)$, is defined as the relation $M \models_X \phi$ of Definition \[def:team\_fol\] (with additional rules for further atomic formulas as required), but substituting Rule **TS-$\vee_L$** with Rule **TS-$\vee_S$**.
For the cases of first order and dependence logic, the lax and strict semantics are equivalent:
\[propo:laxeqstrict\] Let $\phi$ be any formula of dependence logic. Then $$M \models_X^S \phi \Leftrightarrow M \models^L_X \phi$$ for all suitable models $M$ and teams $\phi$.
This is easily verified by structural induction over $\phi$, using the downwards closure property (Theorem \[theo:DLdc\]) to take care of disjunctions and existentials (and, moreover, applying the Axiom of Choice for the case of existentials). We verify the case corresponding to existential quantifications, as an example: the one corresponding to disjunctions is similar but simpler, and the the others are trivial.\
Suppose that $M \models_X^S \exists x \phi$: then, by rule **TS-$\exists_S$**, there exists a function $F: X \rightarrow \dom(M)$ such that $M \models_{X[F/x]}^S \phi$. Now define the function $H: X \rightarrow \part(\dom(M)) \backslash \{\emptyset\}$ so that, for all $s \in X$, $H(s) = \{F(s)\}$: then $X[H/x] = X[F/x]$, and therefore by induction hypothesis $M \models_{X[H/x]}^L \phi$, and hence by rule **TS-$\exists_L$** $M \models_X^L \exists x \phi$. Conversely, suppose that $M \models_X^L \exists x \phi$: then, by rule **TS-$\exists_L$**, there exists a function $H: X \rightarrow \part(\dom(M)) \backslash \{\emptyset\}$ such that $M \models_{X[H/x]}^L \phi$. Then, by the Axiom of Choice, there exists a *choice function* $F: X \rightarrow \dom(X)$ such that, for all $s \in X$, $F(s) \in H(s)$; therefore, $X[F/x] \subseteq X[H/x]$ and, by downwards closure, $M \models_{X[F/x]}^L \phi$. But then by induction hypothesis $M \models_{X[F/x]}^S \phi$ and, by rule **TS-$\exists_L$**, $M \models_X^S \phi$.
As we will argue in Section \[subsect:inclog\], for the logics that we will study for which a difference exists between lax and strict semantics the former will be the most natural choice; therefore, from this point until the end of this work the symbol $\models$ written without superscripts will stand for the relation $\models^L$.
Constancy logic {#subsect:constancy}
---------------
In this section, we will present and examine a simple fragment of dependence logic. This fragment, which we will call *constancy logic*, consists of all the formulas of dependence logic in which only dependence atoms of the form $=\!\!(t)$ occur; or, equivalently, it can be defined as the extension of (team) first order logic obtained by adding *constancy atoms* to it, with the semantics given by the following definition:
\[def:const\] Let $M$ be a first order model, let $X$ be a team over it, and let $t$ be a term over the signature of $M$ and with variables in $\dom(X)$. Then
TS-const:
: $M \models_X =\!\!(t)$ if and only if, for all $s, s' \in X$, $t\langle s\rangle = t\langle s'\rangle$.
Clearly, constancy logic is contained in dependence logic. Furthermore, over open formulas it is more expressive than first order logic proper, since, as already mentioned, the constancy atom $=(x)$ is a counterexample to Proposition \[propo:FOflat\].\
The question then arises whether constancy logic is properly contained in dependence logic, or if it coincides with it. This will be answered through the following results:
\[propo:const\_out\] Let $\phi$ be a constancy logic formula, let $z$ be a variable not occurring in $\phi$, and let $\phi'$ be obtained from $\phi$ by substituting one instance of $=\!\!(t)$ with the expression $z = t$.
Then $$M \models_X \phi \Leftrightarrow M \models_X \exists z (=\!\!(z) \wedge \phi').$$
The proof is by induction on $\phi$.
1. If the expression $=\!\!(t)$ does not occur in $\phi$, then $\phi' = \phi$ and we trivially have that $\phi \equiv \exists z (=\!\!(z) \wedge \phi)$, as required.
2. If $\phi$ is $=\!\!(t)$ itself then $\phi'$ is $z = t$, and $$\begin{aligned}
& M \models_X \exists z (=\!\!(z) \wedge z = t) \Leftrightarrow
\exists m \in \dom(M)\mbox{ s.t. } M \models_{X[m/z]} z = t \Leftrightarrow\\
&\Leftrightarrow \exists m \in \dom(M) \mbox{ s.t. } t\langle s\rangle = m \mbox{ for all } s \in X \Leftrightarrow M \models_X =\!\!(t)
\end{aligned}$$ as required, where we used $X[m/z]$ as a shorthand for $\{s(m/z) : s \in X\}$.
3. If $\phi$ is $\psi_1 \vee \psi_2$, let us assume without loss of generality that the instance of $=\!\!(t)$ that we are considering is in $\psi_1$. Then $\psi'_2 = \psi_2$, and since $z$ does not occur in $\psi_2$ $$\begin{aligned}
&M \models_X \exists z(=\!\!(z) \wedge (\psi'_1 \vee \psi_2)) \Leftrightarrow \exists m \mbox{ s.t. } M \models_{X[m/z]} \psi'_1 \vee \psi_2 \Leftrightarrow\\
&\Leftrightarrow \exists m, X_1, X_2 \mbox{ s.t. } X_1 \cup X_2 = X, M \models_{X_1[m/z]} \psi'_1 \mbox{ and } M \models_{X_2[m/z]} \psi_2 \Leftrightarrow\\
&\Leftrightarrow \exists m, X_1, X_2 \mbox{ s.t. } X_1 \cup X_2 = X, M \models_{X_1[m/z]} \psi'_1 \mbox{ and } M \models_{X_2} \psi_2 \Leftrightarrow\\
&\Leftrightarrow X_1, X_2 \mbox{ s.t. } X_1 \cup X_2 = X, M \models_{X_1} \exists z (=\!\!(z) \wedge \psi'_1) \mbox{ and } M \models_{X_2} \psi_2 \Leftrightarrow\\
&\Leftrightarrow X_1, X_2 \mbox{ s.t. } X_1 \cup X_2 = X, M \models_{X_1} \psi_1 \mbox{ and } M \models_{X_2} \psi_2 \Leftrightarrow\\
&\Leftrightarrow M \models_{X} \psi_1 \vee \psi_2
\end{aligned}$$ as required.
4. If $\phi$ is $\psi_1 \wedge \psi_2$, let us assume again that the instance of $=\!\!(t)$ that we are considering is in $\psi_1$. Then $\psi_2' = \psi_2$, and $$\begin{aligned}
&M \models_X \exists z(=\!\!(z) \wedge \psi'_1 \wedge \psi_2) \Leftrightarrow\\
&\Leftrightarrow \exists m \mbox{ s.t. } M \models_{X[m/z]} \psi'_1 \mbox{ and } M \models_{X[m/z]} \psi_2 \Leftrightarrow\\
&\Leftrightarrow M \models_X \exists z(=\!\!(z) \wedge \psi'_1) \mbox{ and } M \models_X \psi_2 \Leftrightarrow\\
&\Leftrightarrow M \models_X \psi_1 \mbox{ and } M \models_X \psi_2 \Leftrightarrow\\
&\Leftrightarrow M \models_X \psi_1 \wedge \psi_2.
\end{aligned}$$
5. If $\phi$ is $\exists x \psi$, $$\begin{aligned}
&M \models_X \exists z (=\!\!(z) \wedge \exists x \psi') \Leftrightarrow \\
&\Leftrightarrow \exists m \mbox{ s.t. } M \models_{X[m/z]} \exists x \psi' \Leftrightarrow\\
&\Leftrightarrow \exists m, \exists H: X[m/z] \rightarrow \part(\dom(M)) \backslash \{\emptyset\} \mbox{ s.t. } M \models_{X[m/z][H/x]} \psi' \Leftrightarrow\\
&\Leftrightarrow \exists H': X \rightarrow \part(\dom(M)) \backslash \{\emptyset\}, \exists m \mbox{ s.t. } M \models_{X[H'/x][m/z]} \psi' \Leftrightarrow\\
&\Leftrightarrow \exists H': X \rightarrow \part(\dom(M)) \backslash \{\emptyset\} \mbox{ s.t. } M \models_{X[H'/x]} \exists z (=\!\!(z) \wedge \psi') \Leftrightarrow\\
&\Leftrightarrow \exists H': X \rightarrow \part(\dom(M)) \backslash \{\emptyset\}, \mbox{ s.t. } M \models_{X[H'/x]} \psi \Leftrightarrow\\
&\Leftrightarrow M \models_X \exists x \psi.
\end{aligned}$$
6. If $\phi$ is $\forall x \psi$, $$\begin{aligned}
&M \models_X \exists z(=\!\!(z) \wedge \forall x \psi') \Leftrightarrow\\
&\Leftrightarrow \exists m \mbox{ s.t. } M \models_{X[m/z]} \forall x \psi' \Leftrightarrow \\
&\Leftrightarrow \exists m \mbox{ s.t. } M \models_{X[m/z][M/x]} \psi' \Leftrightarrow \\
&\Leftrightarrow \exists m \mbox{ s.t. } M \models_{X[M/x][m/z]} \psi' \Leftrightarrow \\
&\Leftrightarrow M \models_{X[M/x]} \exists z (=\!\!(z) \wedge \psi') \Leftrightarrow \\
&\Leftrightarrow M \models_{X[M/x]} \psi \Leftrightarrow \\
&\Leftrightarrow M \models_{X} \forall x \psi.
\end{aligned}$$
As a corollary of this result, we get the following normal form theorem for constancy logic:[^17]
Let $\phi$ be a constancy logic formula. Then $\phi$ is logically equivalent to a constancy logic formula of the form $$\exists z_1 \ldots z_n \left( \bigwedge_{i=1}^n =\!\!(z_i) \wedge \psi(z_1 \ldots z_n)\right)$$ for some tuple of variables ${\vec{z}} = z_1 \ldots z_n$ and some first order formula $\psi$.
Repeatedly apply Proposition \[propo:const\_out\] to “push out” all constancy atoms from $\phi$, thus obtaining a formula, equivalent to it, of the form $$\exists z_1 (=\!\!(z_1) \wedge \exists z_2 (=\!\!(z_2) \wedge \ldots \wedge \exists z_n(=\!\!(z_n) \wedge \psi(z_1 \ldots z_n)))$$ for some first order formula $\psi(z_1 \ldots z_n)$. It is then easy to see, from the semantics of our logic, that this is equivalent to $$\exists z_1 \ldots z_n(=\!\!(z_1) \wedge \ldots \wedge =\!\!(z_n) \wedge \psi(z_1 \ldots z_n))$$ as required.
The following result shows that, over sentences, constancy logic is precisely as expressive as first order logic:
\[coro:const\_remove\] Let $\phi = \exists {\vec{z}} \left(\bigwedge_i =\!\!(z_i) \wedge \psi({\vec{z}})\right)$ be a constancy logic sentence in normal form.
Then $\phi$ is logically equivalent to $\exists {\vec{z}} \psi({\vec{z}})$.
By the rules of our semantics, $M \models_{\{\emptyset\}} \psi$ if and only if there exists a family $A_1 \ldots A_n$ of nonempty sets of elements in $\dom(M)$ such that, for $$X = \{(z_1:= m_1 \ldots z_n:= m_n) : (m_1 \ldots m_n) \in A_1 \times \ldots \times A_n\},$$ it holds that $M \models_X \psi$. But $\psi$ is first-order, and therefore, by Proposition \[propo:FOflat\], this is the case if and only if for all $m_1 \in A_1, \ldots, m_n \in A_n$ it holds that $M \models_{\{(z_1:m_1, \ldots z_n:m_n)\}} \psi$.
But then $M \models_{\{\emptyset\}} \phi$ is and only if there exist $m_1 \ldots m_n$ such that this holds;[^18] and therefore, by Proposition \[propo:FO\_Team2Tarski\], $M \models_{\{\emptyset\}} \phi$ if and only if $M \models_\emptyset \exists z_1 \ldots z_n \psi(z_1 \ldots z_n)$ according to Tarski’s semantics, or equivalently, if and only if $M \models_{\{\emptyset\}} \exists z_1 \ldots z_n \psi(z_1 \ldots z_n)$ according to team semantics.
Since, by Theorem \[theo:DLsent\], dependence logic is strictly stronger than first order logic over sentences, this implies that constancy logic is strictly weaker than dependence logic over sentences (and, since sentences are a particular kind of formulas, over formulas too).\
The relation between first order logic and constancy logic, in conclusion, appears somewhat similar to that between dependence logic and independence logic - that is, in both cases we have a pair of logics which are reciprocally translatable on the level of sentences, but such that one of them is strictly weaker than the other on the level of formulas. This discrepancy between translatability on the level of sentences and translatability on the level of formulas is, in the opinion of the author, one of the most intriguing aspects of logics of imperfect information, and it deserves further investigation.
Inclusion and exclusion in logic
================================
This section is the central part of the present work. We will begin it by recalling two forms of non-functional dependency which have been studied in Database Theory, and some of their known properties. Then we will briefly discuss their relevance in the framework of logics of imperfect information, and then, in Subsection \[subsect:inclog\], we will examine the properties of the logic obtained by adding atoms corresponding to the first sort of non-functional dependency to the basic language of team semantics. Afterward, in Subsection \[subsect:equilog\] we will see that nothing is lost if we only consider a simpler variant of this kind of dependency: in either case, we obtain the same logical formalism, which - as we will see - is strictly more expressive than first order logic, strictly weaker than independence logic, but incomparable with dependence logic. In Subsection \[subsect:exclog\], we will then study the other notion of non-functional dependency that we are considering, and see that the corresponding logic is instead equivalent, in a very strong sense, to dependence logic; and finally, in Subsection \[subsect:ielogic\] we will examine the logic obtained by adding *both* forms of non-functional dependency to our language, and see that it is equivalent to independence logic.
Inclusion and exclusion dependencies {#subsect:incexc}
------------------------------------
Functional dependencies are the forms of dependency which attracted the most interest from database theorists, but they certainly are not the only ones ever considered in that field.\
Therefore, studying the effect of substituting the dependence atoms with ones corresponding to other forms of dependency, and examining the relationship between the resulting logics, may be - in the author’s opinion, at least - a very promising, and hitherto not sufficiently explored, direction of research in the field of logics of imperfect information.[^19] First of all, as previously mentioned, teams correspond to states of knowledge. But often, relations obtained from a database correspond precisely to information states of this kind;[^20] and therefore, some of the dependencies studied in database theory may correspond to constraints over the agent’s beliefs which often occur in practice, as is certainly the case for functional dependencies.[^21]
Moreover, and perhaps more pragmatically, database researchers have already performed a vast amount of research about the properties of many of these non-functional dependencies, and it does not seem unreasonable to hope that this might allow us to derive, with little additional effort of our own, some useful results about the corresponding logics.\
The present paper will, for the most part, focus on *inclusion* ([@fagin81], [@casanova82]) and *exclusion* ([@casanova83]) dependencies and on the properties of the corresponding logics of imperfect information. Let us start by recalling and briefly discussing these dependencies:
Let $R$ be a relation, and let ${\vec{x}}$, ${\vec{y}}$ be tuples of attributes of $R$ of the same length. Then $R \models {\vec{x}} \subseteq {\vec{y}}$ if and only if $R({\vec{x}}) \subseteq R({\vec{y}})$, where $$R({\vec{z}}) = \{r({\vec{z}}) : r \mbox{ is a tuple in } R\}.$$
In other words, an inclusion dependency ${\vec{x}} \subseteq {\vec{y}}$ states that all values taken by the attributes ${\vec{x}}$ are also taken by the attributes ${\vec{y}}$. It is easy to think up practical examples of inclusion dependencies: one might for instance think of the database consisting of the relations (Person, Date\_of\_Birth), (Father, Children$_F$) and (Mother, Children$_M$).[^22] Then, in order to express the statement that every father, every mother and every child in our knowledge base are people and have a date of birth, we may impose the restrictions $$\left\{
\begin{array}{l}
\mbox{Father} \subseteq \mbox{Person}, ~\mbox{Mother} \subseteq \mbox{Person},\\
\mbox{Children}_F \subseteq \mbox{Person}, ~\mbox{Children}_M \subseteq \mbox{Person}
\end{array}
\right\}.$$ Furthermore, inclusion dependencies can be used to represent the assertion that every child has a father and a mother, or, in other words, that the attributes Children$_F$ and Children$_M$ take the same values: $$\{\mbox{Children}_F \subseteq \mbox{Children}_M,~ \mbox{Children}_M \subseteq \mbox{Children}_F\}.$$ Note, however, that inclusion dependencies do not allow us to express all “natural” dependencies of our example. For instance, we need to use functional dependencies in order to assert that everyone has exactly one birth date, one father and one mother:[^23] $$\{\mbox{Person}\rightarrow \mbox{Date\_of\_Birth},~ \mbox{Children}_F \rightarrow \mbox{Father},~ \mbox{Children}_M \rightarrow \mbox{Mother}\}.$$
In [@casanova82], a sound and complete axiom system for the implication problem of inclusion dependencies was developed. This system consists of the three following rules:
I1:
: For all ${\vec{x}}$, $\vdash {\vec{x}} \subseteq {\vec{x}}$;
I2:
: If $|{\vec{x}}| = |{\vec{y}}| = n$ then, for all $m \in \mathbb N$ and all $\pi: 1 \ldots m \rightarrow 1 \ldots n$, $${\vec{x}} \subseteq {\vec{y}} \vdash x_{\pi(1)} \ldots x_{\pi(m)} \subseteq y_{\pi(1)} \ldots y_{\pi(m)};$$
I3:
: For all tuples of attributes of the same length ${\vec{x}}$, ${\vec{y}}$, and ${\vec{z}}$, $${\vec{x}} \subseteq {\vec{y}}, {\vec{y}} \subseteq {\vec{z}} \vdash {\vec{x}} \subseteq {\vec{z}}.$$
Let $\Gamma$ be a set of inclusion dependencies and let ${\vec{x}}$, ${\vec{y}}$ be tuples of relations of the same length. Then $$\Gamma \vdash {\vec{x}} \subseteq {\vec{y}}$$ can be derived from the axioms **I1**, **I2** and **I3** if and only if all relations which respect all dependencies of $\Gamma$ also respect ${\vec{x}} \subseteq {\vec{y}}$.
However, the combined implication problem for inclusion and functional dependencies is undecidable ([@mitchell83], [@chandra85]).\
Whereas inclusion dependencies state that all values of a given tuple of attributes also occur as values of another tuple of attributes, *exclusion* dependencies state that two tuples of attributes have no values in common:
Let $R$ be a relation, and let ${\vec{x}}$, ${\vec{y}}$ be tuples of attributes of $R$ of the same length. Then $R \models {\vec{x}} ~|~ {\vec{y}}$ if and only if $R({\vec{x}}) \cap R({\vec{y}}) = \emptyset$, where $$R({\vec{z}}) = \{r({\vec{z}}) : r \mbox{ is a tuple in } R\}.$$
Exclusion dependencies can be thought of as a way of partitioning the elements of our domain into *data types*, and of specifying which type corresponds to each attribute. For instance, in the example $$\mbox{(Person, Date\_of\_birth)}\times \mbox{(Father, Children$_F$)}\times \mbox{(Mother, Children$_M$)}$$ considered above we have two types, corresponding respectively to *people* (for the attributes Person, Father, Mother, Children$_F$ and Children$_M$) and *dates* (for the attribute Date\_of\_birth). The requirement that no date of birth should be accepted as a name of person, nor vice versa, can then be expressed by the set of exclusion dependencies $$\{ A ~|~ \mbox{Date\_of\_birth} : A = \mbox{Person}, \mbox{Father}, \mbox{Mother}, \mbox{Children}_M, \mbox{Children}_F\}.$$
Other uses of exclusion dependencies are less common, but they still exist: for example, the statement that no one is both a father and a mother might be expressed as $\mbox{Father} ~|~ \mbox{Mother}$.
In [@casanova83], the axiom system for inclusion dependencies was extended to deal with both inclusion and exclusion dependencies as follows:
1. *Axioms for inclusion dependencies:*
I1:
: For all ${\vec{x}}$, $\vdash {\vec{x}} \subseteq {\vec{x}}$;
I2:
: If $|{\vec{x}}| = |{\vec{y}}| = n$ then, for all $m \in \mathbb N$ and all $\pi: 1 \ldots m \rightarrow 1 \ldots n$, $${\vec{x}} \subseteq {\vec{y}} \vdash x_{\pi(1)} \ldots x_{\pi(m)} \subseteq y_{\pi(1)} \ldots y_{\pi(m)};$$
I3:
: For all tuples of attributes of the same length ${\vec{x}}$, ${\vec{y}}$ and ${\vec{z}}$, $${\vec{x}} \subseteq {\vec{y}}, {\vec{y}} \subseteq {\vec{z}} \vdash {\vec{x}} \subseteq {\vec{z}};$$
2. *Axioms for exclusion dependencies:*
E1:
: For all ${\vec{x}}$ and ${\vec{y}}$ of the same length, ${\vec{x}} ~|~ {\vec{y}} \vdash {\vec{y}} ~|~ {\vec{x}}$;
E2:
: If $|{\vec{x}}| = |{\vec{y}}| = n$ then, for all $m \in \mathbb N$ and all $\pi: 1 \ldots m \rightarrow 1 \ldots n$, $$x_{\pi(1)} \ldots x_{\pi(m)} ~|~ y_{\pi(1)} \ldots y_{\pi(m)} \vdash {\vec{x}} ~|~ {\vec{y}};$$
E3:
: For all ${\vec{x}}$, ${\vec{y}}$ and ${\vec{z}}$ such that $|{\vec{y}}| = |{\vec{z}}|$, ${\vec{x}} ~|~ {\vec{x}} \vdash {\vec{y}} ~|~ {\vec{z}}$;
3. *Axioms for inclusion/exclusion interaction:*
IE1:
: For all ${\vec{x}}$, ${\vec{y}}$ and ${\vec{z}}$ such that $|{\vec{y}}| = |{\vec{z}}|$, ${\vec{x}} ~|~ {\vec{x}} \vdash {\vec{y}} \subseteq {\vec{z}}$;
IE2:
: For all ${\vec{x}}, {\vec{y}}, {\vec{z}}, {\vec{w}}$ of the same length, ${\vec{x}} ~|~ {\vec{y}}, {\vec{z}} \subseteq {\vec{x}}, {\vec{w}} \subseteq {\vec{y}} \vdash {\vec{z}} ~|~ {\vec{w}}$.
The above system is sound and complete for the implication problem for inclusion and exclusion dependencies.
It is not difficult to transfer the definitions of inclusion and exclusion dependencies to team semantics, thus obtaining *inclusion atoms* and *exclusion atoms*:
\[def:inc\_exc\] Let $M$ be a first order model, let ${\vec{t}}_1$ and ${\vec{t}}_2$ be two finite tuples of terms of the same length over the signature of $M$, and let $X$ be a team whose domain contains all variables occurring in ${\vec{t}}_1$ and ${\vec{t}}_2$. Then
TS-inc:
: $M \models_X {\vec{t}}_1 \subseteq {\vec{t}}_2$ if and only if for every $s \in X$ there exists a $s' \in X$ such that ${\vec{t}}_1\langle s\rangle = {\vec{t}}_2\langle s'\rangle$;
TS-exc:
: $M \models_X {\vec{t}}_1 ~|~ {\vec{t}}_2$ if and only if for all $s, s' \in X$, ${\vec{t}}_1\langle s\rangle \not= {\vec{t}}_2\langle s'\rangle$.
Returning for a moment to the agent metaphor, the interpretation of these conditions is as follows.\
A team $X$ satisfies ${\vec{t}}_1 \subseteq {\vec{t}}_2$ if and only if all possible values that the agent believes possible for ${\vec{t}}_1$ are also believed by him or her as possible for ${\vec{t}}_2$ - or, by contraposition, that the agent cannot exclude any value for ${\vec{t}}_2$ which he cannot also exclude as a possible value for ${\vec{t}}_1$. In other words, from this point of view an inclusion atom is a way of specify a state of *ignorance* of the agent: for example, if the agent is a chess player who is participating to a tournament, we may want to represent the assertion that the agent *does not know* whether he will play against a given opponent using the black pieces or the white ones. In other words, if he believes that he *might* play against a given opponent when using the white pieces, he should also consider it possible that he played against him or her using the black ones, and vice versa; or, in our formalism, that his belief set satisfies the conditions $$\begin{aligned}
&\mbox{Opponent\_as\_White} \subseteq \mbox{Opponent\_as\_Black},\\
&\mbox{Opponent\_as\_Black} \subseteq \mbox{Opponent\_as\_White}.\end{aligned}$$
This very example can be used to introduce a new dependency atom ${\vec{t}}_1 \bowtie {\vec{t}}_2$, which might perhaps be called an *equiextension atom*, with the following rule:
\[def:equi\] Let $M$ be a first order model, let ${\vec{t}}_1$ and ${\vec{t}}_2$ be two finite tuples of terms of the same length over the signature of $M$, and let $X$ be a team whose domain contains all variables occurring in ${\vec{t}}_1$ and ${\vec{t}}_2$. Then
TS-equ:
: $M \models_X {\vec{t}}_1 \bowtie {\vec{t}}_2$ if and only if $X({\vec{t}}_1) = X({\vec{t}}_2)$.
It is easy to see that this atom is different, and strictly weaker, from the first order formula $${\vec{t}}_1 = {\vec{t}}_2 := \bigwedge_i (({\vec{t}}_1)_i = ({\vec{t}}_2)_i).$$ Indeed, the former only requires that the sets of all possible values for ${\vec{t}}_1$ and for ${\vec{t}}_2$ are the same, while the latter requires that ${\vec{t}}_1$ and ${\vec{t}}_2$ coincide in all possible states of things: and hence, for example, the team $X = \{(x:0,y:1),(x:1, y:0)\}$ satisfies $x \bowtie y$ but not $x = y$.
As we will see later, it is possible to recover inclusion atoms from equiextension atoms and the connectives of our logics.\
On the other hand, an exclusion atom specifies a state of *knowledge*. More precisely, a team $X$ satisfies ${\vec{t}}_1 ~|~ {\vec{t}}_2$ if and only if the agent can confidently exclude all values that he believes possible for ${\vec{t}}_1$ from the list of the possible values for ${\vec{t}}_2$. For example, let us suppose that our agent is also aware that a boxing match will be had at the same time of the chess tournament, and that he knows that no one of the participants to the match will have the time to play in the tournament too - he has seen the lists of the participants to the two events, and they are disjoint. Then, in particular, our agent knows that no potential winner of the boxing match is also a potential winner of the chess tournament, even know he is not aware of who these winners will be. In our framework, this can be represented by stating our agent’s beliefs respect the exclusion atom $$\mbox{Winner\_Boxing} ~|~ \mbox{Winner\_Chess}.$$
This is a different, and stronger, condition than the first order expression $\mbox{Winner\_Boxing} \not = \mbox{Winner\_Chess}$: indeed, the latter merely requires that, in any possible state of things, the winners of the boxing match and of the chess tournament are different, while the former requires that *no possible* winner of the boxing match is a potential winner for the chess tournament. So, for example, only the first condition excludes the scenario in which our agent does not know whether T. Dovramadjiev, a Bulgarian chessboxing champion, will play in the chess tournament or in the boxing match, represented by the team of the form $$X = \begin{array}{c | l l l}
& \mbox{Winner\_Boxing} & \mbox{Winner\_Chess} & \ldots\\
\hline
s_0 & \mbox{T. Dovramadjiev} & \mbox{V. Anand} & \ldots\\
s_1 & \mbox{T. Woolgar} & \mbox{T. Dovramadijev} & \ldots\\
\ldots & \ldots & \ldots
\end{array}$$
Inclusion logic {#subsect:inclog}
---------------
In this section, we will begin to examine the properties of *inclusion logic* - that is, the logic obtained adding to (team) first order logic the *inclusion atoms* ${\vec{t}}_1 \subseteq {\vec{t}}_2$ with the semantics of Definition \[def:inc\_exc\].\
A first, easy observation is that this logic does not respect the downwards closure property. For example, consider the two assignments $s_0 = (x:0, y:1)$ and $s_1 = (x:1,y:0)$: then, for $X = \{s_0, s_1\}$ and $Y = \{s_0\}$, it is easy to see by rule **TS-inc** that $M \models_X x \subseteq y$ but $M \not \models_Y x \subseteq y$.\
Hence, the proof of Proposition \[propo:laxeqstrict\] cannot be adapted to the case of inclusion logic. The question then arises whether inclusion logic with strict semantics and inclusion logic with lax semantics are different; and, as the next two propositions will show, this is indeed the case.
\[propo:dislaxneqstr\] There exist a model $M$, a team $X$ and two formulas $\psi$ and $\theta$ of inclusion logic such that $M \models_X^L \psi \vee \theta$ but $M \not \models_X^S \psi \vee \theta$.
Let $\dom(M) = \{0,1, 2, 3, 4\}$, let $X$ be the team $$X = \begin{array}{c | c c c}
~ & x & y & z\\
\hline
s_0 & 0 & 1 & 2\\
s_1 & 1 & 0 & 3\\
s_2 & 4 & 3 & 0
\end{array}$$ and let $\psi = x \subseteq y$, $\theta = y \subseteq z$.
- $M \models_X^L \psi \vee \theta$:\
Let $Y = \{s_0, s_1\}$ and $Z = \{s_1, s_2\}$. Then $Y \cup Z = X$, $Y(x) = \{0,1\} = Y(y)$ and $Z(y) = \{0,3\} = Z(z)$.\
Hence, $M \models_Y^L x \subseteq y$ and $M \models_Z^L y \subseteq z$, and therefore $M \models_X^L x \subseteq y \vee y \subseteq z$ as required.
- $M \not \models_X^S \psi \vee \theta$:\
Suppose that $X = Y \cup Z$, $Y \cap Z = \emptyset$, $M \models_X^S x \subseteq y$ and $M \models_Z^S y \subseteq z$.\
Now, $s_2$ cannot belong in $Y$, since $s_2(x) = 4$ and $s_i(y) \not = 4$ for all assignments $s_i$; therefore, we necessarily have that $s_2 \in Z$. But since $M \models_Z^S y \subseteq z$, this implies that there exists an assignment $s_i \in Y$ such that $s_i(z) = s_2(y) = 3$. The only such assignment in $X$ is $s_1$, and therefore $s_1 \in Y$.\
Analogously, $s_0$ cannot belong in $Z$: indeed, $s_0(y) = 1 \not = s_i(z)$ for all $i \in 0 \ldots 2$. Therefore, $s_0 \in Y$; and since $M \models_Y^S x \subseteq y$, there exists an $s_i \in Y$ with $s_i(y) = s_0(x) = 0$. But the only such assignment in $X$ is $s_1$, and therefore $s_1 \in Y$.\
In conclusion, $Y = \{s_0, s_1\}$, $Z = \{s_1, s_2\}$ and $Y \cap Z = \{s_1\} \not = \emptyset$, which contradicts our hypothesis.
\[propo:exlaxneqstr\] There exist a model $M$, a team $X$ and a formula $\phi$ of inclusion logic such that $M \models_X^L \exists x \phi$ but $M \not \models_X^S \exists x \phi$.
Let $\dom(M) = \{0,1\}$, let $X$ be the team $$X = \begin{array}{c | c c }
~ & y & z\\
\hline
s_0 & 0 & 1
\end{array}$$ and let $\phi$ be $y \subseteq x \wedge z \subseteq x$.
- $M \models_X^L \exists x \phi$:\
Let $H: X \rightarrow \part(\dom(M))$ be such that $H(s_0) = \{0,1\}$.
Then $$X[H/x] = \begin{array}{c | c c c}
~ & y & z & x\\
\hline
s'_0 & 0 & 1 & 0\\
s'_1 & 0 & 1 & 1\\
\end{array}$$ and hence $X[H/x](y), X[H/x](z) \subseteq X[H/x](x)$, as required.
- $M \not \models_X^S \exists x \psi$:\
Let $F$ be any function from $X$ to $\dom(M)$. Then $$X[F/x] = \begin{array}{c | c c c}
~ & y & z & x\\
\hline
s''_0 & 0 & 1 & F(s_0)\\
\end{array}$$ But $F(s_0) \not = 0$ or $F(s_0) \not = 1$; and in the first case $M \not \models_{X[F/x]}^S y \subseteq x$, while in the second one $M \not \models_{X[F/x]}^S z \subseteq x$.
Therefore, when studying the properties inclusion logic it is necessary to specify whether we are are using the strict or the lax semantics for disjunction and existential quantification. However, only one of these choices preserves *locality* in the sense of Theorem \[theo:DLloc\], as the two following results show:
\[propo:strict\_nonlocal\] The strict semantics does not respect locality in inclusion logic (or in any extension thereof). In other words, there exists a a model $M$, a team $X$ and two formulas $\psi$ and $\theta$ such that $M \models_X^S \psi \vee \theta$, but for $X' = X_{\upharpoonright \free(\phi \vee \psi)}$ it holds that $M \not \models_{X'}^S \psi \vee \theta$ instead; and analogously, there exists a model $M$, a team $X$ and a formula $\xi$ such that $M \models_X^S \exists x \xi$, but for $X' = X_{\upharpoonright \free(\exists x \xi)}$ we have that $M \not \models_{X'}^S \exists \xi$ instead.
1. Let $\dom(M) = \{0 \ldots 4\}$, let $\psi$ and $\theta$ be $x \subseteq y$ and $y \subseteq z$ respectively, and let $$X = \begin{array}{c | c c c c}
~ & x & y & z & u\\
\hline
s_0 & 0 & 1 & 2 & 0\\
s_1 & 1 & 0 & 3 & 0\\
s_2 & 1 & 0 & 3 & 1\\
s_3 & 4 & 3 & 0 & 0
\end{array}$$ Then $M \models_X^S \psi \vee \theta$: indeed, for $Y = \{s_0, s_1\}$ and $Z = \{s_2, s_3\}$ we have that $X = Y \cup Z$, $Y \cap Z = \emptyset$, $M \models_Y \psi$ and $M \models_Z \theta$, as required. However, the restriction $X'$ of $X$ to $\free(\psi \vee \theta) = \{x,y,z\}$ is the team considered in the proof of Proposition \[propo:dislaxneqstr\], and - as was shown in that proof - $M \not \models_X^S \psi \vee \theta$.
2. Let $\dom(M) = \{0,1\}$, let $\xi$ be $y \subseteq x \wedge z \subseteq x$, and let $$X = \begin{array}{c | c c c}
~ & y & z & u\\
\hline
s_0 & 0 & 1 & 0\\
s_1 & 0 & 1 & 1
\end{array}$$ Then $M \models_X^S \exists x \xi$: indeed, for $F: X \rightarrow \dom(M)$ defined as $$\begin{aligned}
&F(s_0) = 0;\\
&F(s_1) = 1;\\
\end{aligned}$$ we have that $$X[F/x] = \begin{array}{c | c c c c}
~ & y & z & u & x\\
\hline
s'_0 & 0 & 1 & 0 & 0\\
s'_1 & 0 & 1 & 1 & 1
\end{array}$$ and it is easy to check that this team satisfies $\xi$. However, the restriction $X'$ of $X$ to $\free(\exists x \xi) = \{y, z\}$ is the team considered in the proof of Proposition \[propo:exlaxneqstr\], and - again, as shown in that proof - $M \not \models_X^S \exists x \psi$.
\[theo:strict\_local\] Let $M$ be a first order model, let $\phi$ be any inclusion logic formula, and let $V$ be a set of variables with $\free(\phi) \subseteq V$. Then, for all suitable teams $X$, $$M \models_X^L \phi \Leftrightarrow M \models_{X_{\upharpoonright V}}^L \phi$$
The proof is by structural induction on $\phi$.
In Section \[subsect:ielogic\], Theorem \[theo:ielocal\], we will prove the same result for an extension of inclusion logic; so we refer to that theorem for the details of the proof.
Since, as we saw, inclusion logic is not downwards closed, by Theorem \[theo:DLdc\] it is not contained in dependence logic. It is then natural to ask whether dependence logic is contained in inclusion logic, or if dependence and inclusion logic are two incomparable extensions of first order logic.\
This is answered by the following result, and by its corollary:
Let $\phi$ be any inclusion logic formula, let $M$ be a first order model and let $(X_i)_{i \in I}$ be a family of teams with the same domain such that $M \models_{X_i} \phi$ for all $i \in I$. Then, for $X = \bigcup_{i \in I} X_i$, we have that $M \models_X \phi.$
By structural induction on $\phi$.
1. If $\phi$ is a first order literal, this is obvious.
2. Suppose that $M \models_{X_i} {\vec{t}}_1 \subseteq {\vec{t}}_2$ for all $i \in I$. Then $M \models_X {\vec{t}}_1 \subseteq {\vec{t}}_2$. Indeed, let $s \in X$: then $s \in X_i$ for some $i \in I$, and hence there exists another $s' \in X_i$ with $s'({\vec{t}}_2) = s({\vec{t}}_1)$. Since $X_i \subseteq X$ we then have that $s' \in X$, as required.
3. Suppose that $M \models_{X_i} \psi \vee \theta$ for all $i \in I$. Then each $X_i$ can be split into two subteams $Y_i$ and $Z_i$ with $M \models_{Y_i} \psi$ and $M \models_{Z_i} \theta$. Now, let $Y = \bigcup_{i \in I} Y_i$ and $Z = \bigcup_{i \in I} Z_i$: by induction hypothesis, $M \models_Y \psi$ and $M \models_Z \theta$. Furthermore, $Y \cup Z = \bigcup_{i \in I} Y_i ~\cup~ \bigcup_{i \in I} Z_i = \bigcup_{i \in I} (Y_i \cup Z_i) = X$, and hence $M \models_X \psi \vee \theta$, as required.
4. Suppose that $M \models_{X_i} \psi \wedge \theta$ for all $i \in I$. Then for all such $i$, $M \models_{X_i} \psi$ and $M \models_{X_i} \theta$; but then, by induction hypothesis, $M \models_X \psi$ and $M \models_X \theta$, and therefore $M \models_X \psi \wedge \theta$.
5. Suppose that $M \models_{X_i} \exists x \psi$ for all $i \in I$, that is, that for all such $i$ there exists a function $H_i: X_i \rightarrow \part(\dom(M)) \backslash \{\emptyset\}$ such that $M \models_{X_i[H_i/x]} \psi$. Then define the function $H : X \rightarrow \part(\dom(M)) \backslash \{\emptyset\}$ so that, for all $s \in X$, $H(s) = \bigcup \{H_i(s): s \in X_i\}$. Now, $X[H/x] = \bigcup_{i \in I} (X_i[H_i/x])$, and hence by induction hypothesis $M \models_{X[H/x]} \psi$, and therefore $M \models_X \exists x \psi$.
6. Suppose that $M \models_{X_i} \forall x \psi$ for all $i \in I$, that is, that $M \models_{X_i[M/x]} \psi$ for all such $i$. Then, since $\bigcup_{i \in I} (X_i[M/x]) = \left(\bigcup_{i \in I} X_i\right)[M/x] = X[M/x]$, by induction hypothesis $M \models_{X[M/x]} \psi$ and therefore $M \models_X \forall x \psi$, as required.
There exist constancy logic formulas which are not equivalent to any inclusion logic formula.
This follows at once from the fact that the constancy atom $=\!\!(x)$ is not closed under unions.\
Indeed, let $M$ be any model with two elements $0$ and $1$ in its domain, and consider the two teams $X_0 = \{(x:0)\}$ and $X_1 = \{(x:1)\}$: then $M \models_{X_0} =\!\!(x)$ and $M \models_{X_1} =\!\!(x)$, but $M \not \models_{X_0 \cup X_1} =\!\!(x)$.
Therefore, not only inclusion logic does not contain dependence logic, it does not even contain constancy logic!\
Now, by Theorem \[theo:DL2IL\] we know that dependence logic is properly contained in independence logic. As the following result shows, inclusion logic is also (properly, because dependence atoms are expressible in independence logic) contained in independence logic:
\[theo:Inc2IL\] Inclusion atoms are expressible in terms of independence logic formulas. More precisely, an inclusion atom ${\vec{t}}_1 \subseteq {\vec{t}}_2$ is equivalent to the independence logic formula $$\phi := \forall v_1 v_2 {\vec{z}} (({\vec{z}} \not = {\vec{t}}_1 \wedge {\vec{z}} \not = {\vec{t}}_2) \vee (v_1 \not = v_2 \wedge {\vec{z}} \not = {\vec{t}}_2) \vee ((v_1 = v_2 \vee {\vec{z}} = {\vec{t}}_2) \wedge {{\vec{z}} ~\bot~ v_1 v_2})).$$ where $v_1$, $v_2$ and ${\vec{z}}$ do not occur in ${\vec{t}}_1$ or ${\vec{t}}_2$ and where, as in [@gradel10], ${{\vec{z}} ~\bot~ v_1 v_2}$ is a shorthand for ${{\vec{z}} ~\bot_{\emptyset}~ v_1v_2}$.
Suppose that $M \models_X {\vec{t}}_1 \subseteq {\vec{t}}_2$. Then split the team $X' = X[M/v_1 v_2 {\vec{z}}]$ into three teams $Y$, $Z$ and $W$ as follows:
- $Y = \{ s \in X' : s({\vec{z}}) \not = {\vec{t}}_1\langle s\rangle \mbox{ and } s({\vec{z}}) \not = {\vec{t}}_2\langle s\rangle\}$;
- $Z = \{ s \in X' : s(v_1) \not = s(v_2) \mbox{ and } s({\vec{z}}) \not = {\vec{t}}_2\langle s\rangle\}$;
- $W = X' \backslash (Y \cup Z) = \{s \in X' : s({\vec{z}}) = {\vec{t}}_2\langle s\rangle \mbox{ or } (s({\vec{z}}) = {\vec{t}}_1\langle s\rangle \mbox{ and } s(v_1) = s(v_2))\}$.
Clearly, $X' = Y \cup Z \cup W$, $M \models_Y z \not = t_1 \wedge z \not = t_2$ and $M \models_Z v_1 \not = v_2 \wedge z \not = t_2$; hence, if we can prove that $$M \models_W ((v_1 = v_2 \vee {\vec{z}} = {\vec{t}}_2)) \wedge {{\vec{z}} ~\bot~ v_1 v_2}$$ we can conclude that $M \models_X \phi$, as required.
Now, suppose that $s \in W$ and $s(v_1) \not = s(v_2)$: then necessarily $s({\vec{z}}) = {\vec{t}}_2$, since otherwise we would have that $s \in Z$ instead. Hence, the first conjunct $v_1 = v_2 \vee {\vec{z}} = {\vec{t}}_2$ is satisfied by $W$.
Now, consider two assignments $s$ and $s'$ in $W$: in order to conclude this direction of the proof, we need to show that there exists a $s'' \in W$ such that $s''({\vec{z}}) = s({\vec{z}})$ and $s''(v_1 v_2) = s'(v_1 v_2)$. There are two distinct cases to examine:
1. If $s({\vec{z}}) = {\vec{t}}_2 \langle s \rangle$, consider the assignment $$s'' = s[s'(v_1)/v_1][s'(v_2)/v_2]:$$ by construction, $s'' \in X'$. Furthermore, since $s''({\vec{z}}) = {\vec{t}}_2\langle s\rangle = {\vec{t}}_2\langle s''\rangle$, $s''$ is neither in $Y$ nor in $Z$. Hence, it is in $W$, as required.
2. If $s({\vec{z}}) \not = {\vec{t}}_2\langle s\rangle$ and $s \in W$, then necessarily $s({\vec{z}}) = {\vec{t}}_1\langle s \rangle$ and $s(v_1) = s(v_2)$.
Since $s \in W \subseteq X[M/v_1 v_2 {\vec{z}}]$, there exists an assignment $o \in X$ such that $${\vec{t}}_1\langle o\rangle = {\vec{t}}_1 \langle s\rangle = s({\vec{z}});$$ and since $M \models_X {\vec{t}}_1 \subseteq {\vec{t}}_2$, there also exist an assignment $o' \in X$ such that $${\vec{t}}_2\langle o'\rangle = {\vec{t}}_1\langle o\rangle = s({\vec{z}}).$$
Now consider the assignment $s'' = o'[s'(v_1)/v_1][s'(v_2)/v_2][s({\vec{z}})/{\vec{z}}]$: by construction, $s'' \in X'$, and since $$s''({\vec{z}}) = s({\vec{z}}) = {\vec{t}}_2 \langle o'\rangle = {\vec{t}}_2\langle s''\rangle$$ we have that $s'' \in W$, that $s''({\vec{z}}) = s({\vec{z}})$ and that $s''(v_1 v_2) = s'(v_1 v_2)$, as required.
Conversely, suppose that $M \models_X \phi$, let $0$ and $1$ be two distinct elements of the domain of $M$, and let $s \in X$.
By the definition of $\phi$, the fact that $M \models_X \phi$ implies that the team $X[M/v_1v_2{\vec{z}}]$ can be split into three teams $Y$, $Z$ and $W$ such that $$\begin{aligned}
& M \models_Y {\vec{z}} \not = {\vec{t}}_1 \wedge {\vec{z}} \not = {\vec{t}}_2;\\
& M \models_Z v_1 \not = v_2 \wedge {\vec{z}} \not = {\vec{t}}_2;\\
& M \models_W (v_1 = v_2 \vee {\vec{z}} = {\vec{t}}_2) \wedge {{\vec{z}} ~\bot~ v_1 v_2}.\end{aligned}$$ Then consider the assignments $$h = s[0/v_1][0/v_2][{\vec{t}}_1\langle s\rangle/{\vec{z}}]$$ and $$h' = s[0/v_1][1/v_2][{\vec{t}}_2\langle s\rangle/{\vec{z}}]$$ Clearly, $h$ and $h'$ are in $X[M/v_1v_2{\vec{z}}]$. However, neither of them is in $Y$, since $h({\vec{z}}) = {\vec{t}}_1\langle h\rangle$ and $h'({\vec{z}}) = {\vec{t}}_2\langle h'\rangle$, nor in $Z$, since $h(v_1) = h(v_2)$ and, again, since $h'({\vec{z}}) = {\vec{t}}_2\langle h'\rangle$. Hence, both of them are in $W$.
But we know that $M \models_W {{\vec{z}} ~\bot~ v_1 v_2}$, and thus there exists an assignment $h'' \in W$ with $$h''({\vec{z}}) = h({\vec{z}}) = {\vec{t}}_1 \langle s \rangle$$ and $$h''(v_1 v_2) = h'(v_1 v_2) = 01.$$
Now, since $h''(v_1) \not = h''(v_2)$, since $h'' \in W$ and since $$M \models_W v_1 = v_2 \vee {\vec{z}} = {\vec{t}}_2,$$ it must be the case that $h''({\vec{z}}) = {\vec{t}}_2\langle h''\rangle$.
Finally, this $h''$ corresponds to some $s'' \in X$; and for this $s''$, $${\vec{t}}_2\langle s''\rangle = {\vec{t}}_2\langle h''\rangle = h''({\vec{z}}) = h({\vec{z}}) = {\vec{t}}_1\langle s\rangle.$$ This concludes the proof.
The relations between first order (team) logic, constancy logic, dependence logic, inclusion logic and independence logic discovered so far are then represented by Figure \[fig:logrel1\].
$\\$
However, things change if we take in consideration the the expressive power of these logics with respect to their sentences only. Then, as we saw, first order logic and constancy logic have the same expressive power, in the sense that every constancy logic formula is equivalent to some first order formula and vice versa, and so do dependence and independence logic. What about inclusion logic sentences?
At the moment, relatively little is known by the author about this. In essence, all that we know is the following result:
Let $\psi({\vec{x}}, {\vec{y}})$ be any first order formula, where ${\vec{x}}$ and ${\vec{y}}$ are tuples of disjoint variables of the same arity. Furthermore, let $\psi'({\vec{x}}, {\vec{y}})$ be the result of writing $\lnot \psi({\vec{x}}, {\vec{y}})$ in negation normal form. Then, for all suitable models $M$ and all suitable pairs ${\vec{a}}$, ${\vec{b}}$ of constant terms of the model, $$M \models_{\{\emptyset\}} \exists {\vec{z}} ({\vec{a}} \subseteq {\vec{z}} \wedge {\vec{z}} \not = {\vec{b}} \wedge \forall {\vec{w}}(\psi'({\vec{z}}, {\vec{w}}) \vee {\vec{w}} \subseteq {\vec{z}}))$$ if and only if $M \models \lnot [\traclo_{{\vec{x}}, {\vec{y}}} ~\psi]({\vec{a}}, {\vec{b}})$, that is, if and only if the pair of tuples of elements corresponding to $({\vec{a}}, {\vec{b}})$ is not in the transitive closure of $\{({\vec{m}}_1, {\vec{m}}_2) : M \models \psi({\vec{m}}_1, {\vec{m}}_2)\}$.
Suppose that $M \models_{\{\emptyset\}} \exists {\vec{z}} ({\vec{a}} \subseteq {\vec{z}} \wedge {\vec{z}} \not = {\vec{b}} \wedge \forall {\vec{w}}(\psi'({\vec{z}}, {\vec{w}}) \vee {\vec{w}} \subseteq {\vec{z}}))$. Then, by definition, there exists a tuple of functions ${\vec{H}} = H_1 \ldots H_n$ such that
1. $M \models_{\{\emptyset\}[{\vec{H}}/{\vec{z}}]} {\vec{a}} \subseteq {\vec{z}}$, that is, ${\vec{a}} \in {\vec{H}}(\{\emptyset\})$;
2. $M \models_{\{\emptyset\}[{\vec{H}}/{\vec{z}}]} {\vec{z}} \not = {\vec{b}}$, and therefore ${\vec{b}} \not \in {\vec{H}}(\{\emptyset\})$;
3. $M \models_{\{\emptyset\}[{\vec{H}}/{\vec{z}}][{\vec{M}}/{\vec{w}}]} \psi'({\vec{z}}, {\vec{w}}) \vee {\vec{w}} \subseteq {\vec{z}}$.
Now, the third condition implies that whenever $M \models \psi({\vec{m}}_1, {\vec{m}}_2)$ and ${\vec{m}}_1$ is in ${\vec{H}}(\{\emptyset\})$, ${\vec{m}}_2$ is in ${\vec{H}}(\{\emptyset\})$ too. Indeed, let $Y = \{\emptyset\}[{\vec{H}}/{\vec{z}}][{\vec{M}}/{\vec{w}}]$: then, by the semantics of our logic, we know that $Y = Y_1 \cup Y_2$ for two subteams $Y_1$ and $Y_2$ such that $M \models_{Y_1} \psi'({\vec{z}}, {\vec{w}})$ and $M \models_{Y_2} {\vec{w}} \subseteq {\vec{z}}$. But $\psi'$ is logically equivalent to the negation of $\psi$, and therefore we know that, for all $s \in Y_1$, $M \not \models \psi(s({\vec{z}}), s({\vec{w}}))$ in the usual Tarskian semantics.\
Suppose now that ${\vec{m}}_1 \in {\vec{H}}(\{\emptyset\})$ and that $M \models \psi({\vec{m}}_1, {\vec{m}}_2)$. Then $s = ({\vec{z}}:={\vec{m}}_1, {\vec{w}}:={\vec{m}}_2)$ is in $Y$; but it cannot be in $Y_1$, as we saw, and hence it must belong to $Y_2$. But $M \models_{Y_2} {\vec{w}} \subseteq {\vec{z}}$, and therefore there exists another assignment $s' \in Y_2$ such that $s'({\vec{z}}) = s({\vec{w}}) = {\vec{m}}_2$. But we necessarily have that $s'({\vec{z}}) \in {\vec{H}}(\{\emptyset\})$, and therefore ${\vec{m}}_2 \in {\vec{H}}(\{\emptyset\})$, as required.
So, ${\vec{H}}(\{\emptyset\})$ is an set of tuples of elements of our models which contains the interpretation of ${\vec{a}}$ but not that of ${\vec{b}}$ and such that $${\vec{m}}_1 \in H(\{\emptyset\}), M \models \psi({\vec{m}}_1), {\vec{M}}_2 \Rightarrow {\vec{m}}_2 \in H(\{\emptyset\}).$$ This implies that $M \models \lnot [\traclo_{{\vec{x}}, {\vec{y}}} ~\psi]({\vec{a}}, {\vec{b}})$, as required.\
Conversely, suppose that $M \models \lnot [\traclo_{{\vec{x}}, {\vec{y}}} ~\psi]({\vec{a}}, {\vec{b}})$: then there exists a set $A$ of tuples of elements of the domain of $M$ which contains the interpretation of ${\vec{a}}$ but not that of ${\vec{b}}$, and such that it is closed by transitive closure for $\psi({\vec{x}}, {\vec{y}})$. Then, by choosing the functions ${\vec{H}}$ so that ${\vec{h}}(\{\emptyset\}) = A$, it is easy to verify that $M$ satisfies our inclusion logic sentence.
As a corollary, we have that inclusion logic is strictly more expressive than first order logic over sentences: for example, for all finite linear orders $M = (\dom(M), <, S, 0, e)$, where $S$ is the successor function, $0$ is the first element of the linear order and $e$ is the last one, we have that $$M \models \exists z (0 \subseteq z \wedge z \not = e \wedge \forall w ( w \not = S(S(z)) \vee w \subseteq z))$$ if and only if $|M|$ is odd. It is not difficult to see, for example through the Ehrenfeucht-Fraïssé method ([@hodges97b]), that this property is not expressible in first order logic.
Equiextension logic {#subsect:equilog}
-------------------
Let us now consider *equiextension logic*, that is, the logic obtained by adding to first order logic (with the lax team semantics) equiextension atoms ${\vec{t}}_1 \bowtie {\vec{t}}_2$ with the semantics of Definition \[def:equi\].
It is easy to see that equiextension logic is contained in inclusion logic:
Let ${\vec{t}}_1$ and ${\vec{t}}_2$ be any two tuples of terms of the same length. Then, for all suitable models $M$ and teams $X$, $$M \models_X {\vec{t}}_1 \bowtie {\vec{t}}_2 \Leftrightarrow M \models_X {\vec{t}}_1 \subseteq {\vec{t}}_2 \wedge {\vec{t}}_2 \subseteq {\vec{t}}_1.$$
Obvious.
Translating in the other direction, however, requires a little more care:
Let ${\vec{t}}_1$ and ${\vec{t}}_2$ be any two tuples of terms of the same length. Then, for all suitable models $M$ and teams $X$, $M \models_X {\vec{t}}_1 \subseteq {\vec{t}}_2$ if and only if $$M \models_X \forall u_1 u_2 \exists {\vec{z}} ({\vec{t}}_2 \bowtie {\vec{z}} \wedge (u_1 \not = u_2 \vee {\vec{z}} = {\vec{t}}_1))$$ where $u_1, u_2$ and ${\vec{z}}$ do not occur in ${\vec{t}}_1$ and ${\vec{t}}_2$.
Suppose that $M \models_X {\vec{t}}_1 \subseteq {\vec{t}}_2$. Then let $X' = X[M/u_1 u_2]$, and pick the tuple of functions ${\vec{H}}$ used to choose ${\vec{z}}$ so that $${\vec{H}}(s) = \left\{\begin{array}{l l}
\{{\vec{t}}_1 \langle s\rangle\}, & \mbox{ if } s({\vec{u}}_1) = s({\vec{u}}_2);\\
\{{\vec{t}}_2\langle s\rangle\}, & \mbox{ otherwise}
\end{array}\right.$$ for all $s \in X'$.[^24]\
Then, for $Y = X'[{\vec{H}}/{\vec{z}}]$, by definition we have that $M \models_{Y} u_1 \not = u_2 \vee {\vec{z}} = {\vec{t}}_1$, and it only remains to verify that $M \models_{Y} {\vec{t}}_2 \bowtie {\vec{z}}$, that is, that $Y({\vec{t}}_2) = Y({\vec{z}})$.
- $Y({\vec{t}}_2) \subseteq Y({\vec{z}})$:\
Let $h \in Y$. Then there exists an assignment $s \in X$ with ${\vec{t}}_2\langle s\rangle = {\vec{t}}_2\langle h\rangle$. Now let $0$ and $1$ be two distinct elements of $M$, and consider the assignment $h' = s[0/u_1][1/u_2][{\vec{H}}/{\vec{z}}]$. By construction, $h' \in Y$; and furthermore, by the definition of ${\vec{H}}$ we have that $h'({\vec{z}}) = {\vec{t}}_2\langle s \rangle = {\vec{t}}_2\langle h\rangle$, as required.
- $Y({\vec{z}}) \subseteq Y({\vec{t}}_2)$:\
Let $h \in Y$. Then, by construction, $h({\vec{z}})$ is ${\vec{t}}_1\langle h\rangle$ or ${\vec{t}}_2 \langle h\rangle$. But since $X({\vec{t}}_1) \subseteq X({\vec{t}}_2)$, in either case there exists an assignment $s \in X$ such ${\vec{t}}_2\langle s\rangle = h({\vec{z}})$. Now consider $h' = s[0/u_1][1/u_2][{\vec{H}}/{\vec{z}}]$: again, $h' \in Y$ and $h'({\vec{z}}) = {\vec{t}}_2\langle h'\rangle = {\vec{t}}_2\langle s\rangle = h({\vec{z}})$, as required.
Conversely, suppose that $M \models_X \forall u_1 u_2 \exists {\vec{z}} ({\vec{t}}_2 \bowtie {\vec{z}} \wedge (u_1 \not = u_2 \vee {\vec{z}} = {\vec{t}}_1))$, and that therefore there exists a tuple of functions ${\vec{H}}$ such that, for $Y = X[M/u_1 u_2][{\vec{H}}/{\vec{z}}]$, $M \models_Y {\vec{t}}_2 \bowtie {\vec{z}} \wedge (u_1 \not = u_2 \vee {\vec{z}} = {\vec{t}}_1)$. Then consider any assignment $s \in X$, and let $h = s[0/u_1][0/u_2][{\vec{H}}/{\vec{z}}]$. Now, $h \in Y$ and $h({\vec{z}}) = {\vec{t}}_1\langle s\rangle$; but since $M \models_Y {\vec{t}}_2 \bowtie {\vec{z}}$, this implies that there exists an assignment $h' \in Y$ such that ${\vec{t}}_2 \langle h' \rangle = h({\vec{z}}) = {\vec{t}}_1\langle s\rangle$. Finally, $h'$ derives from some assignment $s' \in X$, and for this assignment we have that ${\vec{t}}_2\langle s\rangle = {\vec{t}}_2\langle h'\rangle = {\vec{t}}_1\langle s\rangle$ as required.
As a consequence, inclusion logic is precisely as expressive as equiextension logic:
Any formula of inclusion logic is equivalent to some formula of equiextension logic, and vice versa.
Exclusion logic {#subsect:exclog}
---------------
With the name of *exclusion logic* we refer to (lax, team) first order logic supplemented with the *exclusion atoms* ${\vec{t}}_1 ~|~ {\vec{t}}_2$, with the satisfaction condition given in Definition \[def:inc\_exc\].\
As the following results show exclusion logic is, in a very strong sense, equivalent to dependence logic:
\[theo:excl\_to\_dep\] For all tuples of terms ${\vec{t}}_1$ and ${\vec{t}}_2$, of the same length, there exists a dependence logic formula $\phi$ such that $$M \models_X \phi \Leftrightarrow M \models_X {\vec{t}}_1 ~|~ {\vec{t}}_2$$ for all suitable models $M$ and teams $X$.
This follows immediately from Theorem \[theo:DLform\], since the satisfaction condition for the exclusion atom is downwards monotone and expressible in $\Sigma_1^1$.\
For the sake of completeness, let us write a direct translation of exclusion atoms into dependence logic anyway.
Let ${\vec{t}}_1$ and ${\vec{t}}_2$ be as in our hypothesis, let ${\vec{z}}$ be a tuple of new variables, of the same length of ${\vec{t}}_1$ and ${\vec{t}}_2$, and let $u_1, u_2$ be two further unused variables. Then $M \models_X {\vec{t}}_1 ~|~ {\vec{t}}_2$ if and only if $$M \models_X \forall {\vec{z}} \exists u_1 u_2 (=\!\!({\vec{z}}, u_1) \wedge =\!\!({\vec{z}}, u_2) \wedge ((u_1 = u_2 \wedge {\vec{z}} \not = {\vec{t}}_1) \vee (u_1 \not = u_2 \wedge {\vec{z}} \not = {\vec{t}}_2))).$$
Indeed, suppose that $M \models_X {\vec{t}}_1 ~|~ {\vec{t}}_2$, let $X' = X[M/{\vec{z}}]$, and let $0, 1$ be two distinct elements in $\dom(M)$.
Then define the functions $H_1$ and $H_2$ as follows:
- For all $s' \in X'$, $H_1(s') = \{0\}$;
- For all $s'' \in X'[H_1/u_1]$, $H_2(s'') = \left\{\begin{array}{l l}
\{0\} &\mbox{ if } s''({\vec{z}}) \not \in X({\vec{t}}_1);\\
\{1\} & \mbox{ if } s''({\vec{z}}) \in X({\vec{t}}_1).
\end{array}\right.$
Then, for $Y = X'[H_1 H_2/u_1 u_2]$, we have that $M \models_Y =\!\!({\vec{z}}, u_1)$ and that $M \models_Y =\!\!({\vec{z}}, u_2)$, since the value of $u_1$ is constant in $Y$ and the value of $u_2$ in $Y$ is functionally determined by the value of ${\vec{z}}$.
Now split $Y$ into the two subteams $Y_1$ and $Y_2$ defined as $$\begin{aligned}
&Y_1 = \{s \in Y : s(u_2) = 0\};\\
&Y_2 = \{s \in Y : s(u_2) = 1\}.\end{aligned}$$
Clearly, $M \models_{Y_1} u_1 = u_2$ and $M \models_{Y_2} u_1 \not = u_2$; hence, we only need to verify that $M \models_{Y_1} {\vec{z}} \not = {\vec{t}}_1$ and that $M \models_{Y_2} {\vec{z}} \not = {\vec{t}}_2$.\
For the first case, let $h$ be any assignment in $Y_1$: then, by definition, $h({\vec{z}}) \not = {\vec{t}}_1\langle s\rangle$ for all $s \in X$. But then $h({\vec{z}}) \not = {\vec{t}}_1\langle h'\rangle$ for all $h' \in Y_1$, and since this is true for all $h \in Y_1$ we have that $M \models_{Y_1} {\vec{z}} \not = {\vec{t}}_1$, as required.
For the second case, let $h$ be in $Y_2$ instead: then, again by definition, $h({\vec{z}}) = {\vec{t}}_1\langle s\rangle$ for some $s \in X$. But $M \models_X {\vec{t}}_1 ~|~ {\vec{t}}_2$, and hence $h({\vec{z}}) \not = {\vec{t}}_2\langle s'\rangle$ for all $s' \in X$; and as in the previous case, this implies that $h({\vec{z}}) \not = {\vec{t}}_2(h')$ for all $h' \in Y_2$ and, since this argument can be made for all $h \in Y_2$, $M \models_{Y_2} {\vec{z}} \not = {\vec{t}}_2$.\
Conversely, suppose that $$M \models_X \forall {\vec{z}} \exists u_1 u_2 (=\!\!({\vec{z}}, u_1) \wedge =\!\!({\vec{z}}, u_2) \wedge ((u_1 = u_2 \wedge {\vec{z}} \not = {\vec{t}}_1) \vee (u_1 \not = u_2 \wedge {\vec{z}} \not = {\vec{t}}_2))).$$ Then there exist two functions $H_1$ and $H_2$ such that, for $Y = X[M/{\vec{z}}][H_1 H_2 / u_1 u_2]$, $$M \models_Y =\!\!({\vec{z}}, u_1) \wedge =\!\!({\vec{z}}, u_2) \wedge ((u_1 = u_2 \wedge {\vec{z}} \not = {\vec{t}}_1) \vee (u_1 \not = u_2 \wedge {\vec{z}} \not = {\vec{t}}_2)).$$
Now, let $s_1$ and $s_2$ be any two assignments in $X$: in order to conclude the proof, I only need to show that ${\vec{t}}_1 \langle s_1\rangle \not = {\vec{t}}_2\langle s_2\rangle$. Suppose instead that ${\vec{t}}_1 \langle s_1 \rangle = {\vec{t}}_2 \langle s_2\rangle = {\vec{m}}$ for some tuple of elements ${\vec{m}}$, and consider two assignments $h_1, h_2$ such that $$h_1 \in \{s_1[{\vec{m}}/{\vec{z}}]\}[H_1 H_2 / u_1 u_2];\footnote{This team and the next one are actually singletons, because $H_1$ and $H_2$ must satisfy the dependency conditions.}$$ and $$h_2 \in \{s_2[{\vec{m}}/{\vec{z}}]\}[H_1 H_2 / u_1 u_2].$$ Then $h_1, h_2 \in Y$; and furthermore, since $h_1({\vec{z}}) = h_2({\vec{z}})$ and $M \models =\!\!({\vec{z}}, u_1) \wedge =\!\!({\vec{z}}, u_2)$, it must hold that $h_1({\vec{u}}_1) = h_2({\vec{u}}_1)$ and $h_1({\vec{u}}_2) = h_2({\vec{u}}_2)$.
Moreover, $M \models_Y (u_1 = u_2 \wedge {\vec{z}} \not = {\vec{t}}_1) \vee (u_1 \not = u_2 \wedge {\vec{z}} \not = {\vec{t}}_2)$, and therefore $Y$ can be split into two subteams $Y_1$ and $Y_2$ such that $$M \models_{Y_1} (u_1 = u_2 \wedge {\vec{z}} \not = {\vec{t}}_1)$$ and $$M \models_{Y_2} (u_1 \not = u_2 \wedge {\vec{z}} \not = {\vec{t}}_2).$$
Now, as we saw, the assignments $h_1$ and $h_2$ coincide over $u_1$ and $u_2$, and hence either $\{h_1, h_2\} \subseteq Y_1$ or $\{h_1, h_2\} \subseteq Y_2$. But neither case is possible, because $$h_1({\vec{z}}) = {\vec{m}} = {\vec{t}}_1\langle s_1\rangle = {\vec{t}}_1\langle h_1\rangle$$ and therefore $h_1$ cannot be in $Y_1$, and because $$h_2({\vec{z}}) = {\vec{m}} = {\vec{t}}_2\langle s_2\rangle = {\vec{t}}_2\langle h_2\rangle$$ and therefore $h_2$ cannot be in $Y_2$.\
So we reached a contradiction, and this concludes the proof.
\[theo:dep\_to\_excl\] Let $t_1 \ldots t_n$ be terms, and let $z$ be a variable not occurring in any of them. Then the dependence atom $=\!\!(t_1 \ldots t_n)$ is equivalent to the exclusion logic expression $$\phi = \forall z ( z = t_n \vee (t_1 \ldots t_{n-1} z ~|~ t_1 \ldots t_{n-1} t_n)),$$ for all suitable models $M$ and teams $X$.
Suppose that $M \models_X =\!\!(t_1 \ldots t_n)$, and consider the team $X[M/z]$. Now, let $Y =\{s \in X[M/z] : s(z) = t_n\langle s\rangle\}$ and let $Z = X[M/z] \backslash Y$.
Clearly, $Y \cup Z = X[M/x]$ and $M \models_Y z = t_n$; hence, if we show that $Z \models t_1 \ldots t_{n-1} z ~|~ t_1 \ldots t_{n-1} t_n$ we can conclude that $M \models_X \phi$, as required.
Now, consider any two $s, s' \in Z$, and suppose that $t_i\langle s\rangle = t_i\langle s'\rangle$ for all $i = 1 \ldots n-1$. But then $s(z) \not = t_n\langle s'\rangle$: indeed, since $M \models_X =\!\!(t_1 \ldots t_n)$, by the locality of dependence logic and by the downwards closure property we have that $M \models_Z =\!\!(t_1 \ldots t_n)$ and hence that $t_n\langle s\rangle = t_n\langle s'\rangle$.
Therefore, if we had that $s(z) = t_n\langle s'\rangle$, it would follow that $s(z) = t_n\langle s'\rangle = t_n\langle s\rangle$ and $s$ would be in $Y$ instead.
So $s(z) \not = t_n\langle s'\rangle$, and since this holds for all $s$ and $s'$ in $Z$ which coincide over $t_1 \ldots t_{n-1}$ we have that $$M \models_Z t_1 \ldots t_{n-1} z ~|~ t_1 \ldots t_{n-1}t_n,$$ as required.\
Conversely, suppose that $M \models_X \phi$, and let $s, s' \in X$ assign the same values to $t_1 \ldots t_{n-1}$. Now, by the definition of $\phi$, $X[M/z]$ can be split into two subteams $Y$ and $Z$ such that $M \models_Y z = t_n$ and\
$M \models_Z (t_1 \ldots t_{n-1} z ~|~ t_1 \ldots t_{n-1} t_n)$.
Now, suppose that $t_n\langle s\rangle = m$ and $t_n \langle s'\rangle = m'$, and that $m \not = m'$: then $s[m'/z]$ and $s'[m/z]$ are in $s[M/z]$ but not in $Y$, and hence they are both in $Z$. But then, since ${\vec{t}}_i\langle s\rangle = {\vec{t}}_i\langle s'\rangle$ for all $i = 1 \ldots n-1$, $$t_n\langle s' \rangle = m' = s[m'/z](z) \not = t_n\langle s'[m/z]\rangle = t_n\langle s'\rangle
$$ which is a contradiction. Therefore, $m = m'$, as required.
\[coro:DLeqEL\] Dependence logic is precisely as expressive as exclusion logic, both with respect to definability of sets of teams and with respect to sentences.
Inclusion/exclusion logic {#subsect:ielogic}
-------------------------
Now that we have some information about inclusion logic and about exclusion logic, let us study *inclusion/exclusion logic* (I/E logic for short), that is, the formalism obtained by adding both inclusion and exclusion atoms to the language of first-order logic.\
By the results of the previous sections, we already know that inclusion atoms are expressible in independence logic and that exclusion atoms are expressible in dependence logic; furthermore, by Theorem \[theo:DL2IL\], dependence atoms are expressible in independence logic.
Then it follows at once that I/E logic is contained in independence logic:
For every inclusion/exclusion logic formula $\phi$ there exists an independence logic formula $\phi^*$ such that $$M \models_X \phi \Leftrightarrow M \models_X \phi^*$$ for all suitable models $M$ and teams $X$.
Now, is I/E logic properly contained in independence logic?\
As the following result illustrates, this is not the case:
Let ${{\vec{t}}_2 ~\bot_{{\vec{t}}_1}~ {\vec{t}}_3}$ be an independence atom, and let $\phi$ be the formula $$\begin{aligned}
& \forall {\vec{p}} {\vec{q}} {\vec{r}} ~ \exists u_1 u_2 u_3 u_4
\left( \bigwedge_{i=1}^4 =\!\!({\vec{p}} {\vec{q}} {\vec{r}}, u_i) \wedge
((u_1 \not = u_2 \wedge ({\vec{p}} {\vec{q}} ~|~ {\vec{t}}_1 {\vec{t}}_2))
\vee\right.\\
&~ ~\left. \vee (u_1 = u_2 \wedge u_3 \not= u_4 \wedge ({\vec{p}} {\vec{r}} ~|~ {\vec{t}}_1 {\vec{t}}_3)) \vee (u_1 = u_2 \wedge u_3 = u_4 \wedge ({\vec{p}} {\vec{q}} {\vec{r}} \subseteq {\vec{t}}_1 {\vec{t}}_2 {\vec{t}}_3))) \right)\end{aligned}$$ where the dependence atoms are used as shorthands for the corresponding exclusion logic expressions, which exist because of Theorem \[theo:dep\_to\_excl\], and where all the quantified variables are new.
Then, for all suitable models $M$ and teams $X$, $$M \models_X {{\vec{t}}_2 ~\bot_{{\vec{t}}_1}~ {\vec{t}}_3} \Leftrightarrow M \models_X \phi.$$
Suppose that $M \models_X {{\vec{t}}_2 ~\bot_{{\vec{t}}_1}~ {\vec{t}}_3}$, and consider the team $X' = X[M/{\vec{p}} {\vec{q}} {\vec{r}}]$.
Now, let $0$ and $1$ be two distinct elements of the domain of $M$, and let the functions $F_1 \ldots F_4$ be defined as follows:
- For all $s \in X'$, $F_1(s) = 0$;
- For all $s \in X'[F_1/u_1]$, $$F_2(s) = \left\{\begin{array}{l l}
0 & \mbox{ if there exists a } s' \in X \mbox{ such that } {\vec{t}}_1 \langle s'\rangle {\vec{t}}_2 \langle s'\rangle = s({\vec{p}}) s({\vec{q}});\\
1 & \mbox{ otherwise;}
\end{array}
\right.$$
- For all $s \in X'[F_1/u_1][F_2/u_2]$, $F_3(s) = 0$;
- For all $s \in X'[F_1/u_1][F_2/u_2][F_3/u_3]$, $$F_4(s) = \left\{\begin{array}{l l}
0 & \mbox{ if there exists a } s' \in X \mbox{ such that } {\vec{t}}_1 \langle s'\rangle {\vec{t}}_3 \langle s'\rangle = s({\vec{p}}) s({\vec{r}});\\
1 & \mbox{ otherwise.}
\end{array}
\right.$$
Now, let $Y = X'[F_1/u_1][F_2/u_2][F_3/u_3][F_4/u_4]$: by the definitions of $F_1 \ldots F_4$, it holds that all dependencies are respected. Let then $Y$ be split into $Y_1$, $Y_2$ and $Y_3$ according to:
- $Y_1 = \{s \in Y : s(u_1) \not = s(u_2)\}$;
- $Y_2 = \{s \in Y : s(u_3) \not = s(u_4)\} \backslash Y_1$;
- $Y_3 = Y \backslash (Y_1 \cup Y_2)$.
Now, let $s$ be any assignment of $Y_1$: then, since $s(u_1) \not = s(u_2)$, by the definitions of $F_1$ and $F_2$ we have that $$\forall s' \in Y, s({\vec{p}}) s({\vec{q}}) \not = {\vec{t}}_1\langle s'\rangle {\vec{t}}_2\langle s'\rangle$$ and, in particular, that the same holds for all the $s' \in Y_1$. Hence, $$M \models_{Y_1} u_1 \not = u_2 \wedge ({\vec{p}} {\vec{q}} ~|~ {\vec{t}}_1 {\vec{t}}_2),$$ as required.
Analogously, let $s$ be any assignment of $Y_2$: then $s(u_1) = s(u_2)$, since otherwise $s$ would be in $Y_1$, $s(u_3) \not = s(u_4)$ and $$\forall s' \in Y, s({\vec{p}}) s({\vec{r}}) \not = {\vec{t}}_1\langle s'\rangle {\vec{t}}_3\langle s'\rangle$$ and therefore $$M \models_{Y_2} u_1 = u_2 \wedge u_3 \not= u_4 \wedge ({\vec{p}} {\vec{r}} ~|~ {\vec{t}}_1 {\vec{t}}_3).$$ Finally, suppose that $s \in Y_3$: then, by definition, $s(u_1) = s(u_2)$ and $s(u_3) = s(u_4)$. Therefore, there exist two assignments $s'$ and $s''$ in $X$ such that $${\vec{t}}_1\langle s'\rangle {\vec{t}}_2\langle s'\rangle = s({\vec{p}}) s({\vec{q}})$$ and $${\vec{t}}_1\langle s''\rangle {\vec{t}}_3\langle s''\rangle = s({\vec{p}}) s({\vec{r}})$$ But by hypothesis we know that $M \models_X {{\vec{t}}_2 ~\bot_{{\vec{t}}_1}~ {\vec{t}}_3}$, and $s'$ and $s''$ coincide over ${\vec{t}}_1$, and therefore there exists a new assignment $h \in X$ such that $${\vec{t}}_1\langle h\rangle {\vec{t}}_2\langle h\rangle {\vec{t}}_3\langle h\rangle = s({\vec{p}}) s({\vec{q}}) s({\vec{r}}).$$
Now, let $o$ be the assignment of $Y$ given by $$o = h[{\vec{t}}_1\langle h\rangle {\vec{t}}_2\langle h\rangle {\vec{t}}_3\langle h\rangle / {\vec{p}} {\vec{q}} {\vec{r}}][F_1 \ldots F_4 /u_1 \ldots u_4]:$$ by the definitions of $F_1 \ldots F_4$ and by the construction of $o$, we then get that $$o(u_1) = o(u_2) = o(u_3) = o(u_4) = 0$$ and therefore that $o \in Y_3$.
But by construction, $${\vec{t}}_1\langle o\rangle {\vec{t}}_2\langle o\rangle {\vec{t}}_3\langle o\rangle = {\vec{t}}_1\langle h\rangle {\vec{t}}_2\langle h\rangle {\vec{t}}_3\langle h\rangle = s({\vec{p}}) s({\vec{q}}) s({\vec{r}}),$$ and hence $$M \models_{Y_3} {\vec{p}} {\vec{q}} {\vec{r}} \subseteq {\vec{t}}_1 {\vec{t}}_2 {\vec{t}}_3$$ as required.\
Conversely, suppose that $M \models_X \phi$, and let $s, s' \in X$ be such that ${\vec{t}}_1 \langle s\rangle = {\vec{t}}_1 \langle s'\rangle$. Now, consider the two assignments $h, h' \in X' = X[M/{\vec{p}} {\vec{q}} {\vec{r}}]$ given by $$h = s[{\vec{t}}_1\langle s\rangle/{\vec{p}}][{\vec{t}}_2\langle s\rangle/{\vec{q}}][{\vec{t}}_3\langle s'\rangle/{\vec{r}}]$$ and $$h' = s'[{\vec{t}}_1\langle s\rangle/{\vec{p}}][{\vec{t}}_2\langle s\rangle/{\vec{q}}][{\vec{t}}_3\langle s'\rangle/{\vec{r}}].$$
Now, since $M \models_X \phi$, there exist functions $F_1 \ldots F_4$, depending only on ${\vec{p}}$, ${\vec{q}}$ and ${\vec{r}}$, such that\
$Y = X'[F_1/u_1][F_2/u_2][F_3/u_3][F_4/u_4]$ can be split into three subteams $Y_1$, $Y_2$ and $Y_3$ and $$\begin{aligned}
&M \models_{Y_1} (u_1 \not = u_2 \wedge ({\vec{p}} {\vec{q}} ~|~ {\vec{t}}_1 {\vec{t}}_2));\\
&M \models_{Y_2} (u_1 = u_2 \wedge u_3 \not= u_4 \wedge ({\vec{p}} {\vec{r}} ~|~ {\vec{t}}_1 {\vec{t}}_3));\\
&M \models_{Y_3} (u_1 = u_2 \wedge u_3 = u_4 \wedge ({\vec{p}} {\vec{q}} {\vec{r}} \subseteq {\vec{t}}_1 {\vec{t}}_2 {\vec{t}}_3)).\end{aligned}$$ Now, let $$o = h[F_1/u_1][F_2/u_2][F_3/u_3][F_4/u_4]$$ and $$o' = h'[F_1/u_1][F_2/u_2][F_3/u_3][F_4/u_4]:$$ since the $F_i$ depend only on ${\vec{p}} {\vec{q}} {\vec{r}}$ and the values of these variables are the same for $h$ and for $h'$, we have that $o$ and $o'$ have the same values for $u_1 \ldots u_4$, and therefore that they belong to the same $Y_i$.
But they cannot be in $Y_1$ nor in $Y_2$, since $$o({\vec{p}})o({\vec{q}}) = o'({\vec{p}})o'({\vec{q}}) = {\vec{t}}_1\langle s\rangle {\vec{t}}_2\langle s\rangle = {\vec{t}}_1\langle o\rangle {\vec{t}}_2\langle o\rangle$$ and $$o({\vec{p}})o({\vec{r}}) = o'({\vec{p}})o'({\vec{r}}) = {\vec{t}}_1\langle s'\rangle {\vec{t}}_3\langle s'\rangle = {\vec{t}}_1\langle o'\rangle {\vec{t}}_3\langle o'\rangle;$$ therefore, $o$ and $o'$ are in $Y_3$, and there exists an assignment $o'' \in Y_3$ with $${\vec{t}}_1\langle o''\rangle {\vec{t}}_2\langle o''\rangle {\vec{t}}_3\langle o''\rangle = o({\vec{p}}) o({\vec{q}}) o({\vec{r}}) = {\vec{t}}_1\langle s\rangle{\vec{t}}_2\langle s\rangle {\vec{t}}_3\langle s'\rangle$$ and, finally, there exists a $s'' \in X$ such that ${\vec{t}}_1\langle s''\rangle {\vec{t}}_2\langle s''\rangle {\vec{t}}_3\langle s''\rangle = {\vec{t}}_1\langle s\rangle{\vec{t}}_2\langle s\rangle {\vec{t}}_3\langle s'\rangle$, as required.
Independence logic and inclusion/exclusion logic are therefore equivalent:
\[coro:IL\_eq\_IE\] Any independence logic formula is equivalent to some inclusion/exclusion logic formula, and any inclusion/exclusion logic formula is equivalent to some independence logic formula.
Figure \[fig:logrel2\] summarizes the translatability[^25] relations between the logics of imperfect information which have been considered in this work.
Let us finish this section verifying that I/E logic (and, as a consequence, also inclusion logic, equiextension logic and independence logic) with the lax semantics is local:
\[theo:ielocal\] Let $M$ be a first order model, let $\phi$ be any I/E logic formula and let $V$ be a set of variables such that $\free(\phi) \subseteq V$. Then, for all suitable teams $X$, $$M \models_X \phi \Leftrightarrow M \models_{X_{\upharpoonright V}} \phi$$
The proof is by structural induction on $\phi$.
1. If $\phi$ is a first order literal, an inclusion atom or an exclusion atom then the statement follows trivially from the corresponding semantic rule;
2. Let $\phi$ be of the form $\psi \vee \theta$, and suppose that $M \models_X \psi \vee \theta$. Then, by definition, $X = Y \cup Z$ for two subteams $Y$ and $Z$ such that $M \models_Y \psi$ and $M \models_Z \theta$. Then, by induction hypothesis, $M \models_{Y_{\upharpoonright V}} \psi$ and $M \models_{Z_{\upharpoonright V}} \theta$. But $X_{\upharpoonright V} = Y_{\upharpoonright V} \cup Z_{\upharpoonright V}$: indeed, $s \in X$ if and only if $s \in Y$ or $s \in Z$, and hence $s_{\upharpoonright V} \in X_{\upharpoonright V}$ if and only if it is in $Y_{\upharpoonright V}$ or in $Z_{\upharpoonright V}$. Hence, $M \models_{X_{\upharpoonright V}} \psi \vee \theta$, as required.
Conversely, suppose that $M \models_{X_{\upharpoonright V}} \psi \vee \theta$, that is, that $X_{\upharpoonright V} = Y' \cup Z'$ for two subteams $Y'$ and $Z'$ such that $M \models_{Y'} \psi$ and $M \models_{Z'} \theta$. Then define $Y = \{s \in X : s_{\upharpoonright V} \in Y'\}$ and $Z = \{s \in X : s_{\upharpoonright V} \in Z'\}$. Now, $X = Y \cup Z$: indeed, if $s \in X$ then $s_{\upharpoonright V}$ is in $X_{\upharpoonright V}$, and hence it is in $Y'$ or in $Z'$, and on the other hand if $s$ is in $Y$ or in $Z$ then it is in $X$ by definition. Furthermore, $Y_{\upharpoonright V} = Y'$ and $Z_{\upharpoonright V} = Z'$,[^26] and hence by induction hypothesis $M \models_Y \psi$ and $M \models_Z \theta$, and finally $M \models_X \psi \vee \theta$.
3. Let $\phi$ be of the form $\psi \wedge \theta$. Then $M \models_X \psi \wedge \theta$ if and only if $M \models_X \psi$ and $M \models_X \theta$, that is, by induction hypothesis, if and only if $M \models_{X_{\upharpoonright V}} \psi$ and $M \models_{X_{\upharpoonright V}} \theta$. But this is the case if and only if $M \models_{X_{\upharpoonright V}} \psi \wedge \theta$, as required.
4. Let $\phi$ be of the form $\exists x \psi$, and suppose that $M \models_X \exists x \psi$. Then there exists a function $H: X \rightarrow \part(\dom(M)) \backslash \{\emptyset\}$ such that $M \models_{X[H/x]} \psi$. Then, by induction hypothesis, $M \models_{(X[H/x])_{\upharpoonright V \cup \{x\}}} \psi$.\
Now consider the function $H': X_{\upharpoonright V} \rightarrow \part(\dom(M)) \backslash \emptyset$ which assigns to every $s' \in X_{\upharpoonright V}$ the set $$H'(s') = \bigcup \{H(s) : s \in X, s' = s_{\upharpoonright V}\}.$$ Then $H'$ assigns a nonempty set to every $s' \in X_{\upharpoonright V}$, as required; and furthermore, $X_{\upharpoonright V}[H'/x]$ is precisely $(X[H/x])_{\upharpoonright V \cup \{x\}}$.[^27] Therefore, $M \models_{X_{\upharpoonright V}} \exists x \psi$, as required.
Conversely, suppose that $M \models_{X_{\upharpoonright V}} \exists x \psi$, that is, that $M \models_{X_{\upharpoonright V}[H'/x]} \psi$ for some $H'$. Then define the function $H: X \rightarrow \part(\dom(M)) \backslash \{x\}$ so that $H(s) = H'(s_{\upharpoonright V})$ for all $s \in X$; now, $X_{\upharpoonright V}[H'/x] = (X[H/x])_{\upharpoonright V \cup \{x\}}$,[^28] and hence by induction hypothesis $M \models_X \exists x \psi$.
5. For all suitable teams $X$, $X[M/x]_{\upharpoonright V \cup \{x\}} = X_{\upharpoonright V}[M/x]$; and hence, $M \models_{X_{\upharpoonright V}} \forall x \psi \Leftrightarrow M \models_{X[M/x]_{\upharpoonright V \cup \{x\}}} \psi \Leftrightarrow M \models_{X[M/x]} \psi \Leftrightarrow M \models_X \forall x \psi$, as required.
Game theoretic semantics {#sect:gamesem}
========================
By this point, we have developed a team semantics for inclusion/exclusion logic and we have examined the relations between it and other logics of imperfect information. In this section, an equivalent game theoretic semantics for inclusion/exclusion logic will be developed; once this is done, the semantics for inclusion logic and for exclusion logic will simply be the restrictions of this semantics to the corresponding sublanguages. The connection between game semantics and team semantics, moreover, will allow us to revisit and further justify the distinction between lax and strict connectives introduced in Section \[subsect:laxstrict\]. However, we will not discuss here the history or the motivations of game theoretic semantics, nor its connections to other game-theoretical approaches to formal semantics. The interested reader is referred to [@hintikka83] and [@hintikkasandu97] for a more philosophically oriented discussion of game theoretic semantics; in the rest of this section, we will content ourselves to present such a semantics for the case of I/E logic and prove its equivalence to team semantics.
Let $\phi$ be an I/E logic formula, let $M$ be a first order model over a signature containing that of $\phi$ and let $X$ be a team over $M$ whose domain contains all free variables of $\phi$. Then the game $G^M_X(\phi)$ is defined as follows:
- There are two players, called $I$ and $II$;[^29]
- The positions of the game are expressions of the form $(\psi, s)$, where $\psi$ is an instance of a subformula of $\phi$ and $s$ is an assignment whose domain contains all free variables of $\psi$;
- The initial positions are all those of the form $(\phi, s)$ for $s \in X$;
- The terminal positions are those of the form $(\alpha, s)$, where $\alpha$ is a first order literal, an inclusion atom, or an exclusion atom;
- If $p = (\psi, s)$ is not a terminal position, the set $S(p)$ of its *successors* is defined according to the following rules:
1. If $\psi$ is of the form $\theta_1 \vee \theta_2$ or $\theta_1 \wedge \theta_2$ then $S(p) = \{(\theta_1, s), (\theta_2, s)\}$;
2. If $\psi$ is of the form $\exists x \theta$ or $\forall x \theta$ then $S(p) = \{(\theta, s[m/x]) : m \in \dom(M)\}$;
- If $p = (\psi, s)$ is not a terminal position, the *active player* $T(p) \in \{I, II\}$ is defined according to the following rules:
1. If $\psi$ is of the form $\theta_1 \vee \theta_2$ or $\exists x \theta$ then $T(p) = II$;
2. If $\psi$ is of the form $\theta_1 \wedge \theta_2$ or $\forall x \theta$ then $T(p) = I$;
- A terminal position $p = (\alpha, s)$ is *winning* for Player $II$ if and only if
- $\alpha$ is a first order literal and $M \models_s \alpha$ in the usual first order sense, or
- $\alpha$ is an inclusion atom ${\vec{t}}_1 \subseteq {\vec{t}}_2$ and $s$ is any assignment, or
- $\alpha$ is an exclusion atom ${\vec{t}}_1 ~|~ {\vec{t}}_2$ and $s$ is any assignment.
If a terminal position is not winning for Player $II$, it is winning for Player $I$.
The definitions of *play*, *complete play* and *winning play* are straightforward:
Let $G^M_X(\phi)$ be a semantic game as above. Then a *play* for $G^M_X(\phi)$ is a finite sequence of positions $p_1 \ldots p_n$ such that
- $p_1$ is an initial position;
- For all $i = 2 \ldots n$, $p_i \in S(p_{i-1})$.
Such a play is said to be *complete* if, furthermore, $p_n$ is a terminal position; and it is *winning* for Player $II$ \[$I$\] if and only if $p_n$ is a winning position for $II$ \[$I$\].
However, it will be useful to consider *non-deterministic strategies* rather than deterministic ones only:
Let $G^M_X(\phi)$ be a semantic game as above. Then a *strategy* for Player $II$ $[I]$ in $G^M_X(\phi)$ is a function $\tau$ sending each position $p = (\psi, s)$ with $T(p) = II$ $[I]$ into some $\tau(p) \in \mathcal P(S(p)) \backslash \emptyset$.\
Such a strategy is said to be *deterministic* if, for all such $p$, $\tau(p)$ is a singleton.
A play $p_1 \ldots p_n$ is said to *follow* a strategy $\tau$ for $II$ \[$I$\] if and only if, for all $i \in 1 \ldots n-1$, $$T(p_i) = II~[I] \Rightarrow p_{i+1} \in \tau(p_i).$$
A strategy $\tau$ for $II$ $[I]$ is *winning* for Player $II$ \[$I$\] if and only if all complete plays ${\vec{p}}$ which follow $\tau$ are winning for $II$ \[$I$\].\
The set of all plays of $G^M_X(\phi)$ in which Player $\rho \in \{I, II\}$ follows strategy $\tau$ will be written as $P(G^M_X(\phi), \rho:\tau)$.
So far, inclusion and exclusion atoms play little role in our semantics, as they always correspond to winning positions for Player $II$. Similarly to dependence atoms in [@vaananen07], however, inclusion and exclusion atoms restrict the *set of strategies* available to Player $II$. This is modeled by the following definition of *uniform strategy*:
Let $G^M_X(\phi)$ be a semantic game as above. Then a strategy $\tau$ for Player $II$ is said to be *uniform* if and only if, for all complete plays $p_1 \ldots p_n = {\vec{p}} \in P(G^M_X(\phi), II:\tau)$,
1. If $p_n$ is of the form $p_n = ({\vec{t}}_1 \subseteq {\vec{t}}_2, s)$ then there exists a play $q_1 \ldots q_{n'} \in P(G^M_X(\phi), II:\tau)$ such that $q_{n'} = ({\vec{t}}_1 \subseteq {\vec{t}}_2, s')$ for the same instance of the inclusion atom and such that ${\vec{t}}_2 \langle s'\rangle = {\vec{t}}_1\langle s\rangle$;
2. If $p_n$ is of the form $p_n = ({\vec{t}}_1 ~|~ {\vec{t}}_2, s)$ then for all plays $q_1 \ldots q_{n'} \in P(G^M_X(\phi), II:\tau)$ such that $q_{n'} = ({\vec{t}}_1 ~|~ {\vec{t}}_2, s')$ for the same instance of the exclusion atom it holds that ${\vec{t}}_1\langle s\rangle \not = {\vec{t}}_2 \langle s'\rangle$.
This notion of uniformity also makes it clear why in inclusion logic there is a difference between working with non-deterministic and with deterministic strategies: whereas the uniformity condition for dependence atoms restrict the information available to Player $II$ thorough the game, the one for inclusion atoms requires that the set of *possible plays*, given a strategy for Player $II$, is *closed* with respect to certain monotonically increasing operators. This phenomenon does not occur for the uniformity conditions of exclusion atoms, whose form is more similar to the conditions of the dependence atom in [@vaananen07]. The next definition and the lemmas following it will be of some use in order to prove the main result of this section:
Let $G^M_X(\phi)$ be a game as in our previous definitions and let $\tau$ be a strategy for Player $II$ in it. Furthermore, let $\psi$ be an instance of a subformula of $\phi$ and let $$Y = \{s : \mbox{ there is a play in } P(G^M_X, II: \tau) \mbox{ passing through } (\psi, s)\}.$$ Furthermore, let $\tau'$ be the restriction of $\tau$ to $G^M_Y(\psi)$, in the sense that $\tau'(\theta, s) = \tau(\theta, s)$ for all $\theta$ contained in $\psi$ and for all assignments $s$. Then we say that $(Y, \psi, \tau')$ is a $M$-*successor* of $(X, \phi, \tau)$, and we write $$(Y, \psi, \tau') \leq_M (X, \phi, \tau).$$
From a game-theoretical perspective, the notion of $M$-successor can be seen as a generalization of the notion of the concepts of *subgame* and *substrategy* to multiple initial positions and to games of imperfect information.
\[lemma:MSucch\_Uniq\] Let $G^M_X(\phi)$ be a semantic game for I/E logic, and let $\psi$ be an instance of a subformula in $\phi$. Then there exists precisely one team $Y$ and precisely one strategy $\tau'$ for $G^M_Y(\psi)$ such that $(Y, \psi, \tau') \leq_M (X, \phi, \tau)$.
Obvious from definition.
\[lemma:MSucc\_Play\] Let $G^M_X(\phi)$ be a semantic game as usual, and let $\tau$ be a strategy for Player $II$ in it. Furthermore, let $\psi$ be an instance of a subformula of $\phi$ and let $Y$, $\tau'$ be such that $(Y, \psi, \tau') \leq_M (X, \phi, \tau)$. Then
1. For any play $p_1 \ldots p_n = {\vec{p}} \in P(G^M_X(\phi), II:\tau)$ passing through the subformula $\psi$ there exist a $k \in 1 \ldots n$ such that $p_k \ldots p_n$ is a play in $P(G^M_Y(\psi), II:\tau')$;
2. For any play $q_1 \ldots q_m = {\vec{q}} \in P(G^M_Y(\psi), II:\tau')$ there exists a $k \in 1 \ldots n$ and positions $p_1 \ldots p_{k}$ of the game $G^M_X(\phi)$ such that $p_1 \ldots p_k q_1 \ldots q_m$ is a play in $(G^M_X(\psi), II:\tau)$.
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1. Consider any play $p_1 \ldots p_n$ as in our hypothesis, and let $k \in 1 \ldots n$ be such that $p_k = (\psi, s)$ for some assignment $s$. Then, by definition of $M$-successors, $s \in Y$ and $p_k$ is a possible initial position of $G^M_Y(\psi)$; furthermore, again by the definition of $M$-successor, we have that, for all $i = k \ldots n-1$, $\tau'(p_i) = \tau(p_i) \ni p_{i+1}$.\
Hence, $p_k \ldots p_n$ is a play in $P(G^M_Y(\psi), II:\tau')$, as required.
2. Consider any play $q_1 \ldots q_m$ as in our hypothesis, and hence let $q_1 = (\psi, s)$ for some $s \in Y$. Then, by definition, there exists a play $p_1 \ldots p_n$ in $P(G^M_X(\psi), II:\tau)$ such that $p_{k+1} = q_1 = (\psi, s)$ for some $k \in 0 \ldots n-1$. But $\tau'$ behaves like $\tau$, and hence $\tau(q_i) = \tau'(q_i) \ni q_{i+1}$ for all $i = 1 \ldots m-1$. Thus, $p_1 \ldots p_{k}q_1 \ldots q_m$ is a play in $(G^M_X(\psi), II:\tau)$, as required.
\[lemma:MSucc\_Sub\] Let $G^M_X(\phi)$ be a semantic game as usual, and let $\tau$ be a strategy for Player $II$ in it. Furthermore, let $\psi$ be an instance of a subformula of $\phi$ and let $Y$, $\tau'$ be such that $(Y, \psi, \tau') \leq_M (X, \phi, \tau)$.\
Then
1. If $\tau$ is winning for $II$ in $G^M_X(\phi)$ then $\tau'$ is winning for $II$ in $G^M_Y(\psi)$;
2. If $\tau$ is uniform in $G^M_X(\phi)$ then $\tau'$ is uniform in $G^M_Y(\psi)$;
3. If $\tau$ is deterministic in $G^M_X(\phi)$ then $\tau'$ is deterministic in $G^M_Y(\psi)$.
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1. Suppose that $\tau$ is winning[^30], and consider any play $q_1 \ldots q_m = {\vec{q}} \in P(G^M_Y(\psi), II: \tau')$. Then, by Lemma \[lemma:MSucc\_Play\], there exists a play $p_1 \ldots p_n \in P(G^M_X(\phi), II:\tau)$ such that $p_k \ldots p_n = q_1 \ldots q_m$ for some $k \in 1 \ldots m$. But $\tau$ is a winning strategy for $II$ in $G^M_X(\phi)$ and therefore $p_n$ is a winning position, as required.
2. Suppose that $\tau$ is uniform, and consider any play $q_1 \ldots q_m = {\vec{q}} \in P(G^M_Y(\psi), II: \tau')$.\
Then, again, there exists a play $p_1 \ldots p_n = {\vec{p}} \in P(G^M_X(\psi), II:\tau)$ such that $p_k \ldots p_n = q_1 \ldots q_m$ for some $k$.\
Now suppose that $p_n = q_m = ({\vec{t}}_1 \subseteq {\vec{t}}_2, s)$: then, since $\tau$ is a uniform strategy, there exists another play $p'_1 \ldots p'_{n'}$ in $(G^M_X(\phi), II:\tau)$ such that $p'_{n'} = ({\vec{t}}_1 \subseteq {\vec{t}}_2, s')$ for the same instance of the inclusion atom and for a $s'$ such that $t_2 \langle s'\rangle = t_1 \langle s\rangle$.\
Since $p_n$ and $p'_{n'}$ correspond the same dependency atom of ${\vec{p}}$, it must be the case that the play $p'_1 \ldots p'_{n'}$ passes through $\psi$; and therefore, by Lemma \[lemma:MSucc\_Play\], there exists some $j \in 1 \ldots n'$ such that $p'_j \ldots p'_{n'}$ is a play in $P(G^M_Y(\psi), \tau')$, thus satisfying the uniformity condition for $\tau'$.\
Now suppose that $p_n = q_m = ({\vec{t}}_1 ~|~ {\vec{t}}_2, s)$ instead, and consider any other play $q'_1 \ldots q'_{m'} \in P(G^M_Y(\psi), \tau')$ such that $q'_m = ({\vec{t}}_1 ~|~ {\vec{t}}_2, s')$ for the same instance of the exclusion atom. Then there exist positions $p'_1 \ldots p'_{k'}$ such that $p'_1 \ldots p'_{k'}q'_1 \ldots q'_{m'}$ is a play in $P(G^M_X(\phi), II:\tau)$. But $\tau$ is uniform, and therefore $s({\vec{t}}_1) \not = s'({\vec{t}}_2)$, as required.
3. This follows trivially by the definition of $M$-successor.
\[lemma:MSucc\_Sup\] Let $G^M_X(\phi)$ be a semantic game for I/E logic and let $\tau$ be a strategy for $II$ in it. Furthermore, let $\psi_1 \ldots \psi_t$ be an enumeration of all immediate subformulas of $\phi$, and let $Y_1 \ldots Y_t$, $\tau_1 \ldots \tau_t$ be such that $(Y_i, \psi_i, \tau_i) \leq_M (X, \phi, \tau)$ for all $i \in 1 \ldots t$. Then
1. If all $\tau_i$ are winning in $G^M_{Y_i}(\psi_i)$ then $\tau$ is winning in $G^M_X(\phi)$;
2. If all $\tau_i$ are uniform in $G^M_{Y_i}(\psi_i)$ then $\tau$ is uniform in $G^M_X(\phi)$;
3. If all $\tau_i$ are deterministic in $G^M_{Y_i}(\psi_i)$ and $T(\phi) = I$[^31] then $\tau$ is deterministic;
4. If all $\tau_i$ are deterministic in $G^M_{Y_i}(\psi_i)$, $T(\phi) = II$ and $|\tau(\phi, s)| = 1$ for all $s \in Y$ then $\tau$ is deterministic.
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1. Suppose that all $\tau_i$ are winning for the respective games, and consider any play $p_1 \ldots p_n = {\vec{p}} \in P(G^M_X(\phi), II:\tau)$. Then $p_2$ is of the form $(\psi_i, s)$ for some $i \in 1 \ldots t$ and some $s \in Y_i$; and therefore, $p_2 \ldots p_n \in P(G^M_{Y_i}(\psi), II:\tau_i)$. But $\tau_i$ is winning, and hence $p_n$ is a winning position for Player $II$, as required.
2. Suppose that all $\tau_i$ are uniform, and consider any play $p_1 \ldots p_n = {\vec{p}} \in P(G^M_X(\phi), II: \tau)$: then, once again, $p_2 \ldots p_n \in P(G^M_{Y_i}(\psi_i), II:\tau_i)$ for some $i$.\
Suppose now that $p_n$ is $({\vec{t}}_1 \subseteq {\vec{t}}_2, s)$: since $\tau_i$ is uniform, there exists another play $q_1 \ldots q_m = {\vec{q}} \in P(G^M_{Y_i}(\psi_i), II:\tau_i)$ such that $q_m = ({\vec{t}}_1 \subseteq {\vec{t}}_2, s')$ for the same instance of the inclusion atom and $${\vec{t}}_1 \langle s\rangle = {\vec{t}}_2 \langle s'\rangle.$$ Finally, ${\vec{q}}$ is contained in a play of $(G^M_X(\phi), II:\tau)$ and hence the uniformity condition is respected for $\tau$.\
Suppose instead that $p_n$ is $({\vec{t}}_1 ~|~ {\vec{t}}_2, s)$, and consider any other play $p'_1 \ldots p'_{n'}$ of $P(G^M_X(\phi), II:\tau)$ such that $p'_{n'}$ is $({\vec{t}}_1 ~|~ {\vec{t}}_2, s')$ for the same instance of ${\vec{t}}_1 ~|~ {\vec{t}}_2$. Now, since the same exclusion atom is reached, it must be the case that $p'_2 \ldots p'_{n'}$ is in $P(G^M_Y(\psi_i), II:\tau_i)$ too, for the same $i$; but then, since $\tau_i$ is uniform, ${\vec{t}}_1 \langle s\rangle \not = {\vec{t}}_2 \langle s'\rangle$, as required.
3. Let $p$ be any position in $G^M_X(\phi)$ such that $T(p) = II$. Then $p$ corresponds to a subformula of some $\psi_i$, and hence $|\tau(p)| = |\tau_i(p)| = 1$.
4. Let $p$ be any position in $G^M_X(\phi)$ such that $T(p) = I$. If $p$ is $(\phi, s)$ for some $s \in Y$, then $|\tau(p)| = 1$ by hypothesis; and otherwise, $p$ corresponds to a subformula of some $\psi_i$, and as in the previous case $|\tau(p)| = |\tau_i(p)| = 1$.
Finally, the connection between semantic games and team semantics is given by the following theorem:
\[theo:game\_team\_lax\] Let $M$ be a first order model, let $\phi$ be an inclusion logic formula over the signature of $M$ and let $X$ be a team over $M$ whose domain contains all free variables of $\phi$. Then Player $II$ has a uniform winning strategy in $G^M_X(\phi)$ if and only if $M \models_X \phi$ (with respect to the lax semantics).
The proof is by structural induction on $\phi$.
1. If $\phi$ is a first order literal then the only strategy available to $II$ in $G^M_X(\phi)$ is the empty one. This strategy is always uniform, and the plays which follow it are of the form ${\vec{p}} = p_1 = (\phi, s)$, where $s$ ranges over $X$. Such a play is *winning* for $II$ if and only if $M \models_s \phi$ in the usual first-order sense; and hence, the strategy is winning for $II$ if and only if $M \models_s \phi$ for all $s \in X$, that is, if and only if $M \models_X \phi$.
2. If $\phi$ is an inclusion atom ${\vec{t}}_1 \subseteq {\vec{t}}_2$ then, again, the only strategy available to Player $II$ is the empty one and the plays which follow it are those of the form ${\vec{p}} = p_1 = ({\vec{t}}_1 \subseteq {\vec{t}}_2, s)$ for some $s \in X$.
By the definition of the winning positions of $G^M_X(\phi)$, this strategy is winning; hence, it only remains to check whether it is uniform.
Now, in order for the strategy to be uniform it must be the case that for all plays ${\vec{p}} = p_1 = ({\vec{t}}_1 \subseteq {\vec{t}}_2, s)$ where $s \in X$ there exists a play ${\vec{q}} = q_1 = ({\vec{t}}_1 \subseteq {\vec{t}}_2, s')$, again for $s' \in X$, such that ${\vec{t}}_2 \langle s' \rangle = {\vec{t}}_1 \langle s \rangle$. But this can be the case if and only if $\forall s \in X \exists s' \in X \mbox{ s.t. } {\vec{t}}_1 \langle s\rangle = {\vec{t}}_2\langle s'\rangle$, that is, if and only if $M \models_X {\vec{t}}_1 \subseteq {\vec{t}}_2$.
3. If $\phi$ is an exclusion atom ${\vec{t}}_1 ~|~ {\vec{t}}_2$, the only strategy for $II$ in $G^M_X(\phi)$ is, once again, the empty one. This strategy is always winning, and it is uniform if and only if for all plays ${\vec{p}} = p_1 = ({\vec{t}}_1 ~|~ {\vec{t}}_2, s)$ and ${\vec{q}} = q_1 = ({\vec{t}}_1 ~|~ {\vec{t}}_2, s')$ (for $s, s' \in X$) it holds that ${\vec{t}}_1 \langle s \rangle \not = {\vec{t}}_2 \langle s'\rangle$.
But this is the case if and only if $M \models_X {\vec{t}}_1 ~|~ {\vec{t}}_2$, as required.
4. If $\phi$ is a disjunction $\psi \vee \theta$, suppose that $\tau$ is a uniform winning strategy for $II$ in $G^M_X(\psi \vee \theta)$. Then define the teams $Y, Z \subseteq X$ as follows: $$\begin{aligned}
&Y = \{s \in X : (\psi, s) \in \tau(\psi \vee \theta, s)\};\\
&Z = \{s \in X : (\theta, s) \in \tau(\psi \vee \theta, s)\}.
\end{aligned}$$ Then $Y \cup Z = X$: indeed, for all $s \in X$ it must be the case that $\emptyset \not = \tau(\psi \vee \theta, s) \subsetneq \{(\psi, s), (\theta, s)\}$. Furthermore, $Y \cap Z = \emptyset$.\
Now consider the following two strategies for $II$ in $G^M_Y(\psi)$ and $G^M_Z(\theta)$ respectively:
- $\tau_1(p) = \tau(p)$ for all positions $p$ of $G^M_Y(\psi)$;
- $\tau_2(p) = \tau(p)$ for all positions $p$ of $G^M_Z(\theta)$.
Since all positions of $G^M_Y(\psi)$ and of $G^M_Z(\theta)$ are also positions of $G^M_X(\psi \vee \theta)$, $\tau_1$ and $\tau_2$ are well-defined.\
Furthermore, $(Y, \psi, \tau_1) \leq_M (X, \phi, \tau)$ and $(Z, \psi, \tau_2) \leq_M (X, \phi, \tau)$; therefore, by Lemma \[lemma:MSucc\_Sub\], $\tau_1$ and $\tau_2$ are uniform and winning for $G^M_Y(\psi)$ and $G^M_Z(\theta)$. By induction hypothesis, this implies that $M \models_Y \psi$ and $M \models_Z \theta$, and by the definition of the semantics for disjunction, this implies that $M \models_X \psi \vee \theta$.
Conversely, suppose that $M \models_X \psi \vee \theta$: then, by definition, there exist teams $Y$ and $Z$ such that $X = Y \cup Z$, $M \models_Y \psi$ and $M \models_Z \theta$. Then, by induction hypothesis, there exist uniform winning strategies $\tau_1 $ and $\tau_2$ for $II$ in $G^M_Y(\psi)$ and $G^M_Z(\theta)$ respectively. Then define the strategy $\tau$ for $II$ in $G^M_X(\psi \vee \theta)$ as follows:
- $\tau(\psi \vee \theta, s) = \left\{
\begin{array}{l l}
\{(\psi, s)\} & \mbox{ if } s \in Y \backslash Z;\\
\{(\theta, s)\} & \mbox{ if } s \in Z \backslash Y;\\
\{(\psi, s), (\theta, s)\} & \mbox{ if } s \in Y \cap Z;
\end{array}
\right.$
- If $p$ is $(\chi, s)$ for some $s$ and some formula $\chi$ contained in $\psi$, then $\tau(p) = \tau_1(p)$;
- If $p$ is $(\chi, s)$ for some $s$ and some $\chi$ contained in $\theta$, then $\tau(p) = \tau_2(p)$.
Then, by construction, we have that $(Y, \psi, \tau_1), (Z, \theta, \tau_2) \leq_M (X, \psi \vee \theta, \tau)$; furthermore, $\psi$ and $\theta$ are all the immediate subformulas of $\psi \vee \theta$, and $\tau_1$ and $\tau_2$ are winning and uniform by hypothesis. Therefore, by Lemma \[lemma:MSucc\_Sup\], $\tau$ is a uniform winning strategy for $G^M_X(\psi \vee \theta)$, as required.
5. If $\phi$ is $\psi \wedge \theta$, suppose again that $\tau$ is a uniform winning strategy for $II$ in $G^M_X(\psi \wedge \theta)$. Then consider the two strategies for $II$ in $G^M_X(\psi)$ and $G^M_Z(\theta)$, respectively, defined as
- $\tau_1(p) = \tau(p)$ for all positions $p$ of $G^M_X(\psi)$;
- $\tau_2(p) = \tau(p)$ for all positions $p$ of $G^M_X(\theta)$.
Then $(X, \psi, \tau_1), (X, \theta, \tau_2) \leq_M (X, \psi \wedge \theta, \tau)$, and therefore by Lemma \[lemma:MSucc\_Sub\] $\psi$ and $\theta$ are uniform winning strategies. Hence, by induction hypothesis, $M \models_X \psi$ and $M \models_X \theta$, and therefore $M \models_X \psi \wedge \theta$.
Conversely, suppose that $M \models_X \psi \wedge \theta$. Then $M \models_X \psi$ and $M \models_X \theta$, and therefore $II$ has uniform winning strategies $\tau_1$ and $\tau_2$ for $G^M_X(\psi)$ and $G^M_X(\theta)$ respectively. Now define the strategy $\tau$ for $II$ in $G^M_X(\psi \wedge \theta)$ as follows: $$\mbox{for all } s \in X, \tau(\chi, s) = \left\{\begin{array}{l l}
\tau_1(\chi, s) & \mbox{ if } \chi \mbox{ is contained in } \psi;\\
\tau_2(\chi, s) & \mbox{ if } \chi \mbox{ is contained in } \theta.
\end{array}
\right.$$ Then $(X, \psi, \tau_1), (X, \theta, \tau_2) \leq_M (X, \psi \wedge \theta, \tau)$ and $\psi, \theta$ are all immediate subformulas of $\psi \wedge \theta$; hence, by Lemma \[lemma:MSucc\_Sup\], $\tau$ is a uniform winning strategy for $II$ in $G^M_X(\phi)$, as required.
6. If $\phi$ is $\exists x \psi$, suppose that $\tau$ is a uniform winning strategy for $II$ in $G^M_X(\exists x \psi)$. Then define the function $H : X \rightarrow \mathcal P(\dom(M)) \backslash \emptyset$ as $H(s) = \{m \in M: (\psi, s[m/x]) \in \tau(\exists x \psi, s)\}$ and consider the following strategy $\tau'$ for $II$ in $G^M_{X[H/x]}(\psi)$: $$\tau'(p) = \tau(p) \mbox{ for all suitable } p.$$ $\tau'$ is well-defined, because any position of $G^M_{X[H/x]}(\psi)$ is also a possible position of $G^M_X(\exists x \psi)$. Furthermore, $(X[H/x], \psi, \tau') \leq_M (X, \exists x \psi, \tau)$, and therefore $\tau'$ is a uniform winning strategy for $II$ in $G^M_{X[H/x]}(\psi)$. By induction hypothesis, this implies that $M \models_{X[H/x]} \psi$, and hence that $M \models_X \exists x \psi$.\
Conversely, suppose that $M \models_X \exists x \psi$; then, there exists a function $H$ such that $M \models_{X[H/x]} \psi$. By induction hypothesis, this means that there exists a winning strategy $\tau'$ for $II$ in $G^M_{X[H/x]}(\psi)$. Now consider the following strategy $\tau$ for $II$ in $G^M_{X}(\exists x \psi)$: $$\begin{aligned}
&\tau(\exists x \psi, s) = \{(\psi, s[m/x]) : m \in H(s)\};\\
&\tau(\theta, s) = \tau'(\tau, s) \mbox{ for all } \tau \mbox{ contained in } \psi \mbox{ and all } s.
\end{aligned}$$ Then $(X[H/x], \psi, \tau') \leq_M (X, \exists x \psi, \tau)$, and $\psi$ is the only direct subformula of $\exists x \psi$; hence, $\tau$ is uniform and winning, as required.
7. If $\phi$ is $\forall x \psi$, suppose that $\tau$ is a uniform winning strategy for $II$ in $G^M_X(\forall x \psi)$. Then consider the strategy $\tau'$ for $II$ in $G^M_{X[M/x]}(\psi)$ given by $$\begin{aligned}
&\tau'(\theta, s) = \tau(\theta, s) \mbox{ for all } \theta \mbox{ contained in } \psi \mbox{ and all } s.
\end{aligned}$$ Then $(X[M/x], \psi, \tau') \leq_M (X, \forall x \psi, \tau)$, and hence $\tau'$ is uniform and winning. By induction hypothesis, this means that $M \models_{X[M/x]} \psi$, and hence that $M \models_X \forall x \psi$.\
Conversely, suppose that $M \models_X \forall x \psi$. Then $M \models_{X[M/x]} \psi$, and hence there exists a uniform winning strategy $\tau'$ for $II$ in $G^M_{X[M/x]}(\psi)$. Then consider the strategy $\tau$ for $II$ in $G^M_X(\psi)$ given by $$\begin{aligned}
&\tau(\theta, s) = \tau'(\theta, s) \mbox{ for all } \theta \mbox{ contained in } \psi \mbox{ and all } s.
\end{aligned}$$ This strategy is well-defined, since the first move of $G^M_X(\forall x \psi)$ is Player $I$’s; furthermore, $$(X[M/x], \psi, \tau') \leq_M (X, \forall x \psi, \tau)$$ and therefore $\tau$ is uniform and winning, as required.
Hence, we have a game theoretic semantics which is equivalent to the lax team semantics for inclusion/exclusion logic; and of course, the game theoretic semantics for inclusion and exclusion logic are simply the restrictions of this semantics to the corresponding languages. As was argued previously, the strict team semantics for disjunction and existential quantification is somewhat less natural when it comes to inclusion logic or I/E logic. However, there exists a link between strict team semantics and *deterministic* strategies:
\[theo:game\_team\_strict\] Let $M$ be a first order model, let $\phi$ be an inclusion logic formula over the signature of $M$ and let $X$ be a team over $M$ whose domain contains all free variables of $\phi$. Then Player $II$ has a uniform, *deterministic* winning strategy in $G^M_X(\phi)$ if and only if $M \models_X \phi$ (with respect to the *strict* semantics).
The proof is by structural induction over $\phi$, and it runs exactly as for the lax case. The only differences occur in the cases of disjunction and existential quantification, in which the determinism of the strategies poses a restriction on the choices available to Player $II$ and for which the proof runs as follows:
- If $\phi$ is a disjunction $\psi \vee \theta$, suppose that $\tau$ is a uniform, deterministic winning strategy for $II$ in $G^M_X(\psi \vee \theta)$. Then define the teams $Y, Z \subseteq X$ as follows: $$\begin{aligned}
&Y = \{s \in X : \tau(\psi \vee \theta, s) = \{(\psi, s)\}\};\\
&Z = \{s \in X : \tau(\psi \vee \theta, s) = \{(\theta, s)\}\}.
\end{aligned}$$ Then $Y \cup Z = X$: indeed, for all $s \in X$ it must be the case that $\emptyset \not = \tau(\psi \vee \theta, s) \subseteq \{(\psi, s), (\theta, s)\}$, and hence $s$ is in $Y$ or in $Z$ (or in both). Furthermore, $Y \cap Z = \emptyset$.\
Now consider the following two strategies for $II$ in $G^M_Y(\psi)$ and $G^M_Z(\theta)$ respectively:
- $\tau_1(p) = \tau(p)$ for all positions $p$ of $G^M_Y(\psi)$;
- $\tau_2(p) = \tau(p)$ for all positions $p$ of $G^M_Z(\theta)$.
Since all positions of $G^M_Y(\psi)$ and of $G^M_Z(\theta)$ are also positions of\
$G^M_X(\psi \vee \theta)$, $\tau_1$ and $\tau_2$ are well-defined. Furthermore, they are deterministic, since $\tau$ is so, and $(Y, \psi, \tau_1), (Z, \theta, \tau_2) \leq_M (X, \phi, \tau)$; therefore, $\tau_1$ and $\tau_2$ are uniform and winning for $G^M_Y(\psi)$ and $G^M_Z(\theta)$. By induction hypothesis, this implies that $M \models_Y \psi$ and $M \models_Z \theta$; and by the definition of the (strict) semantics for disjunction, this implies that $M \models_X \psi \vee \theta$.\
Conversely, suppose that $M \models_X \psi \vee \theta$, according to the strict semantics: then, by definition, there exist teams $Y$ and $Z$ such that $X = Y \cup Z$, $Y \cap Z = \emptyset$, $M \models_Y \psi$ and $M \models_Z \theta$. Then, by induction hypothesis, there exist uniform, deterministic winning strategies $\tau_1$ and $\tau_2$ for $II$ in $G^M_Y(\psi)$ and $G^M_Z(\theta)$ respectively. Then define the strategy $\tau$ for $II$ in $G^M_X(\psi \vee \theta)$ as follows:
- $\tau(\psi \vee \theta, s) = \left\{
\begin{array}{l l}
\{(\psi, s)\} & \mbox{ if } s \in Y;\\
\{(\theta, s)\} & \mbox{ if } s \in Z.\\
\end{array}
\right.$
- If $p$ is $(\chi, s)$ and $\chi$ is contained in $\psi$ then $\tau(p) = \tau_1(p)$;
- If $p$ is $(\chi, s)$ and $\chi$ is contained in $\theta$ then $\tau(p) = \tau_2(p)$.
Then, by construction, we have that $$(Y, \psi, \tau_1), (Z, \theta, \tau_2) \leq_M (X, \psi \vee \theta, \tau);$$ and furthermore, $\psi$ and $\theta$ are all the immediate subformulas of $\psi \vee \theta$, and $\tau_1$ and $\tau_2$ are winning and uniform by hypothesis. Therefore $\tau$ is a uniform, deterministic winning strategy for $G^M_X(\psi \vee \theta)$, as required.
- If $\phi$ is $\exists x \psi$, suppose that $\tau$ is a uniform, deterministic winning strategy for $II$ in $G^M_X(\exists x \psi)$. Then define the function $F : X \rightarrow \dom(M)$ so that, for every $s \in X$, $F(s)$ is the unique element $m$ of the model such that $\tau(\exists x \psi, s) = \{(\psi, s[m/x])\}$ and consider the following strategy $\tau'$ for $II$ in $G^M_{X[F/x]}(\psi)$: $$\tau'(p) = \tau(p) \mbox{ for all suitable } p.$$ $\tau'$ is well-defined, because any position of $G^M_{X[F/x]}(\psi)$ is also a possible position of $G^M_X(\exists x \psi)$. Furthermore, $(Y[F/x], \psi, \tau') \leq_M (Y, \exists x \psi, \tau)$, and therefore $\tau'$ is a uniform, deterministic winning strategy for $II$ in $G^M_{X[F/x]}(\psi)$. By induction hypothesis, this implies that $M \models_{X[F/x]} \psi$, and hence that $M \models_X \exists x \psi$ (with respect to the strict semantics).\
Conversely, suppose that $M \models_X \exists x \psi$ according to the strict semantics; then, there exists a $F$ such that $M \models_{X[F/x]} \psi$. By induction hypothesis, this means that there exists a uniform, deterministic winning strategy $\tau'$ for $II$ in $G^M_{X[F/x]}(\psi)$. Now consider the following strategy $\tau$ for $II$ in $G^M_{X}(\exists x \psi)$: $$\begin{aligned}
&\tau(\exists x \psi, s) = \{(\psi, s[F(s)/x])\};\\
&\tau(\theta, s) = \tau'(\tau, s) \mbox{ for all } \tau \mbox{ contained in } \psi.
\end{aligned}$$ Then $(X[F/x], \psi, \tau') \leq_M (X, \exists x \psi, \tau)$, and $\psi$ is the only direct subformula of $\exists x \psi$; hence, $\tau$ is uniform, deterministic and winning, as required.
In [@forster06], Thomas Forster considers the distinction between deterministic and nondeterministic strategies for the case of the logic of branching quantifiers and points out that, in the absence of the Axiom of Choice, different truth conditions are obtained for these two cases. In the same paper, he then suggests that
> Perhaps advocates of branching quantifier logics and their descendents will tell us which semantics **\[that is, the deterministic or nondeterministic one\]** they have in mind.
Dependence logic, inclusion logic, inclusion/exclusion logic and independence logic can certainly be seen as descendents of branching quantifier logic, and the present work strongly suggests that the semantics that we “have in mind” is the nondeterministic one. As we just saw, the deterministic/nondeterministic distinction in game theoretic semantics corresponds precisely to the strict/lax distinction in team semantics; and indeed, as seen in Subsection \[subsect:laxstrict\], for dependence logic proper (which is expressively equivalent to branching quantifier logic), the lax and strict semantics are equivalent modulo the Axiom of Choice (Proposition \[propo:laxeqstrict\]).
But for inclusion logic and its extensions, we have that lax and strict (and, hence, nondeterministic and deterministic) semantics are not equivalent, even in the presence of the Axiom of Choice (Propositions \[propo:dislaxneqstr\] and \[propo:exlaxneqstr\]), and that only the lax one satisfies Locality in the sense of Theorem \[theo:DLloc\] (see Proposition \[propo:strict\_nonlocal\] and Theorems \[theo:strict\_local\], \[theo:ielocal\] for the proof).
Furthermore, as stated before, Fredrik Engström showed in [@engstrom10] that the lax semantics for existential quantification arises naturally from his treatment of generalized quantifiers in dependence logic.
All of this, in the opinion of the author at least, makes a convincing case for the adoption of the nondeterministic semantics (or, in terms of team semantics, of the lax one) as the natural semantics for the study of logics of imperfect information, thus suggesting an answer to Thomas Forster’s question.
Definability in I/E logic (and in independence logic) {#sect:defin}
=====================================================
In [@kontinenv09], Kontinen and Väänänen characterized the expressive power of dependence logic formulas (Theorem \[theo:DLform\] here), and, in [@kontinennu09], Kontinen and Nurmi used a similar technique to prove that a class of teams is definable in team logic ([@vaananen07b]) if and only if it is expressible in full second order logic.\
In this section, I will attempt to find an analogous result for I/E logic (and hence, through Corollary \[coro:IL\_eq\_IE\], for independence logic). One direction of the intended result is straightforward:
\[theo:IE2Sigma11\] Let $\phi({\vec{v}})$ be a formula of I/E logic with free variables in ${\vec{v}}$. Then there exists an existential second order logic formula $\Phi(A)$, where $A$ is a second order variable with arity $|{\vec{v}}|$, such that $$M \models_X \phi({\vec{v}}) \Leftrightarrow M \models \Phi(\rel_{{\vec{v}}}(X))$$ for all suitable models $M$ and teams $X$.
The proof is an unproblematic induction over the formula $\phi$, and follows closely the proof of the analogous results for dependence logic ([@vaananen07]) or independence logic ([@gradel10]).
The other direction, instead, requires some care:[^32]
\[theo:Sigma112IL\] Let $\Phi(A)$ be a formula in $\Sigma_1^1$ such that $\free(\Phi) = \{A\}$, and let ${\vec{v}}$ be a tuple of distinct variables with $|{\vec{v}}| = \arity(A)$. Then there exists an I/E logic formula $\phi({\vec{v}})$ such that $$M \models_X \phi({\vec{v}}) \Leftrightarrow M \models \Phi(\rel_{{\vec{v}}}(X))$$ for all suitable models $M$ and nonempty teams $X$.
It is easy to see that any $\Phi(A)$ as in our hypothesis is equivalent to the formula $$\Phi^*(A) = \exists B ( \forall {\vec{x}} (A {\vec{x}} \leftrightarrow B {\vec{x}}) \wedge \Phi(B)),$$ in which the variable $A$ occurs only in the conjunct $\forall {\vec{x}}(A {\vec{x}} \leftrightarrow B {\vec{x}})$. Then, as in [@kontinenv09], it is possible to write $\Phi^*(A)$ in the form $$\exists {\vec{f}}~ \forall {\vec{x}} {\vec{y}} ( (A {\vec{x}} \leftrightarrow f_1({\vec{x}}) = f_2({\vec{x}})) \wedge \psi({\vec{x}}, {\vec{y}}, {\vec{f}})),$$ where ${\vec{f}} = f_1 f_2 \ldots f_n$, $\psi({\vec{f}}, x, y)$ is a quantifier-free formula in which $A$ does not appear, and each $f_i$ occurs only as $f({\vec{w}}_i)$ for some fixed tuple of variables ${\vec{w}}_i \subseteq {\vec{x}} {\vec{y}}$.
Now define the formula $\phi({\vec{v}})$ as $$\forall {\vec{x}} {\vec{y}} ~\exists {\vec{z}} \left(
\bigwedge_i =\!\!({\vec{w}}_i, z_i) \wedge ( (({\vec{v}} \subseteq {\vec{x}} \wedge z_1 = z_2) \vee ({\vec{v}} ~|~ {\vec{x}} \wedge z_1 \not = z_2)) \wedge \psi'({\vec{x}}, {\vec{y}}, {\vec{z}}))\right),$$ where $\psi'({\vec{x}}, {\vec{y}}, {\vec{z}})$ is obtained from $\psi({\vec{x}}, {\vec{y}}, {\vec{f}})$ by substituting each $f_i({\vec{w}}_i)$ with $z_i$, and the dependence atoms are used as shorthands for the corresponding expressions of I/E logic.
Now we have that $M \models_X \phi({\vec{v}}) \Leftrightarrow M \models \Phi^*(\rel_{{\vec{v}}}(X))$:
Indeed, suppose that $M \models_X \phi({\vec{v}})$. Then, by construction, for each $i = 1 \ldots n$ there exists a function $F_i$, depending only on ${\vec{w}}_i$, such that for $Y = X[M/{\vec{x}} {\vec{y}}][{\vec{F}} / {\vec{z}}]$ $$M \models_{Y} (({\vec{v}} \subseteq {\vec{x}} \wedge z_1 = z_2) \vee ({\vec{v}} ~|~ {\vec{x}} \wedge z_1 \not = z_2)) \wedge \psi'({\vec{x}}, {\vec{y}}, {\vec{z}}).$$
Therefore, we can split $Y$ into two subteams $Y_1$ and $Y_2$ such that $M \models_{Y_1} {\vec{v}} \subseteq {\vec{x}} \wedge z_1 = z_2$ and $M \models_{Y_2} {\vec{v}} ~|~ {\vec{x}} \wedge z_1 \not = z_2$.
Now, for each $i$ define the function $f_i$ so that, for every tuple ${\vec{m}}$ of the required arity, $f_i({\vec{m}})$ corresponds to $F_i(s)$ for an arbitrary $s \in X[M/{\vec{x}} {\vec{y}}]$ with $s({\vec{w}}_i) = {\vec{m}}$, and let $o$ be any assignment with domain ${\vec{x}} {\vec{y}}$.
Thus, if we can prove that $M \models_o ((\rel_{{\vec{v}}}(X)) {\vec{x}} \leftrightarrow f_1({\vec{x}}) = f_2 ({\vec{x}})) \wedge \psi({\vec{x}}, {\vec{y}}, {\vec{f}})$ then the left-to-right direction of our proof is done.
First of all, suppose that $M \models_o (\rel_{{\vec{v}}}(X)) {\vec{x}}$, that is, that $o({\vec{x}}) = {\vec{m}} = s({\vec{v}})$ for some $s \in X$.
Then choose an arbitrary tuple of elements ${\vec{r}}$ and consider the assignment $h = s[{\vec{m}}/{\vec{x}}][{\vec{r}}/{\vec{y}}][{\vec{F}}/{\vec{z}}] \in Y$. This $h$ cannot belong to $Y_2$, since $h({\vec{v}}) = s({\vec{v}}) = {\vec{m}} = h({\vec{x}})$, and therefore it is in $Y_1$ and $h(z_1) = h(z_2)$.
By the definition of the $f_i$, this implies that $f_1 ({\vec{m}}) = f_2 ({\vec{m}})$, as required.\
Analogously, suppose that $M, \not \models_o (\rel_{{\vec{v}}}(X)){\vec{x}}$, that is, that $o({\vec{x}}) = {\vec{m}} \not = s({\vec{v}})$ for all $s \in X$. Then pick an arbitrary such $s \in X$ and an arbitrary tuple of elements ${\vec{r}}$, and consider the assignment $$h = s[{\vec{m}}/{\vec{x}}][{\vec{r}}/{\vec{y}}][{\vec{F}}/{\vec{z}}] \in Y.$$
If $h$ were in $Y_1$, there would exist an assignment $h' \in Y_1$ such that $h'({\vec{v}}) = h({\vec{x}}) = {\vec{m}}$; but this is impossible, and therefore $h \in Y_2$. Hence $h(z_1) \not = h(z_2)$, and therefore $f_1({\vec{m}}) \not = f_2({\vec{m}})$.
Putting everything together, we just proved that $$M \models_o R{\vec{x}} \Leftrightarrow f_1({\vec{x}}) = f_2 ({\vec{x}})$$ for all assignments $o$ with domain ${\vec{x}} {\vec{y}}$, and we still need to verify that $M \models_o \psi({\vec{x}}, {\vec{y}}, f)$ for all such $o$.
But this is immediate: indeed, let $s$ be an arbitrary assignment of $X$, and construct the assignment $$h = s[o({\vec{x}} {\vec{y}})/{\vec{x}} {\vec{y}}][{\vec{F}}/{\vec{z}}] \in X[M/{\vec{x}} {\vec{y}}][{\vec{F}}/{\vec{z}}].$$
Then, since $M \models_{X[M/{\vec{x}} {\vec{y}}][{\vec{F}}/{\vec{z}}]} \psi'({\vec{x}}, {\vec{y}}, {\vec{z}})$ and $\psi'({\vec{x}}, {\vec{y}}, {\vec{z}})$ is first order, $M \models_{\{h\}} \psi'({\vec{x}}, {\vec{y}}, {\vec{z}})$; but $\psi'({\vec{x}}, {\vec{y}}, {\vec{f}}({\vec{x}} {\vec{y}}))$ is equivalent to $\psi({\vec{x}}, {\vec{y}}, {\vec{f}})$ and $h(z_i) = f(h({\vec{w}}_i)) = f(o({\vec{w}}_i))$, and therefore $$M \models_o \psi({\vec{x}}, {\vec{y}}, {\vec{f}})$$ as required. $\\$
Conversely, suppose that $M \models_s (\rel_{{\vec{v}}}(X)){\vec{x}} \leftrightarrow (f_1({\vec{x}}) = f_2 ({\vec{x}})) \wedge \psi({\vec{x}}, {\vec{y}}, {\vec{f}})$ for all assignments $s$ with domain ${\vec{x}} {\vec{y}}$ and for some fixed choice of the tuple of functions ${\vec{f}}$.
Then let ${\vec{F}}$ be such that, for all assignments $h$ and for all $i$, $$F_i(h) = f_i(h({\vec{w}}_i))$$ and consider $Y = X[M/{\vec{x}} {\vec{y}}][F/{\vec{z}}]$.
Clearly, $Y$ satisfies the dependency conditions; furthermore, it satisfies $\psi'({\vec{x}}, {\vec{y}}, {\vec{z}})$, because for every assignment $h \in Y$ and every $i \in 1 \ldots n$ we have that $h(z_i) = F_i(h) = f_i(h({\vec{w}}_i))$.
Finally, we can split $Y$ into two subteams $Y_1$ and $Y_2$ as follows: $$\begin{aligned}
& Y_1 = \{o \in Y : o({\vec{z}}_1) = o({\vec{z}}_2)\};\\
& Y_2 = \{o \in Y : o({\vec{z}}_1) \not = o({\vec{z}}_2)\}.\end{aligned}$$
It is then trivially true that $M \models_{Y_1} z_1 = z_2$ and $M \models_{Y_2} z_1 \not= z_2$, and all that is left to do is proving that $M \models_{Y_1} {\vec{v}} \subseteq {\vec{x}}$ and $M \models_{Y_2} {\vec{v}} ~|~ {\vec{x}}$.
As for the former, let $o \in Y_1$: then, since $o(z_1) = o(z_2)$, $f_1(o({\vec{x}})) = f_2(o({\vec{x}}))$.
This implies that $o({\vec{x}}) \in \rel_{{\vec{v}}}(X)$, and hence that there exists an assignment $s' \in X$ with $s'({\vec{v}}) = o({\vec{x}})$.
Now consider the assignment $$o' = s'[o({\vec{x}} {\vec{y}})/{\vec{x}} {\vec{y}}][{\vec{F}}/{\vec{z}}]:$$ since in $Y$ the values of ${\vec{z}}$ depend only on the values of ${\vec{x}} {\vec{y}}$ and since $o(z_1) = o(z_2)$, we have that $o'(z_1) = o'(z_2)$ and hence $o' \in Y_1$ too. But $o'({\vec{v}}) = s'({\vec{v}}) = o({\vec{x}})$, and since $o$ was an arbitrary assignment of $Y_1$, this implies that $M \models_{Y_1} {\vec{v}} \subseteq {\vec{x}}$.
Finally, suppose that $o \in Y_2$. Then, since $o(z_1) \not = o(z_2)$, we have that $f_1(o({\vec{x}})) \not= f_2(o({\vec{x}}))$; and therefore, $o({\vec{x}}) \not \in \rel_{{\vec{v}}}(X)$, that is, for all assignments $s \in X$ it holds that $s({\vec{v}}) \not = o({\vec{x}})$. Then the same holds for all $o' \in Y_2$.
This concludes the proof.
Since by Corollary \[coro:IL\_eq\_IE\] we already know independence logic and I/E logic have the same expressive power, this has the following corollary:
\[coro:ILform\] Let $\Phi(A)$ be an existential second order formula with $\free(\Phi) = A$, and let ${\vec{v}}$ be any set of variables such that $|{\vec{v}}| = \arity(A)$. Then there exists an independence logic formula $\phi({\vec{v}})$ such that $$M \models_X \phi({\vec{v}}) \Leftrightarrow M \models \Phi(\rel_{{\vec{v}}}(X))$$ for all suitable models $M$ and teams $X$.
Finally, by Fagin’s Theorem ([@fagin74]) this gives an answer to Grädel and Väänänen’s question:
All NP properties of teams are expressible in independence logic.
This result has far-reaching consequences. First of all, it implies that independence logic (or, equivalently, I/E logic) is the most expressive logic of imperfect information which only deals with existential second order properties. Extensions of independence logic can of course be defined; but unless they are capable of expressing some property which is not existential second order (as, for example, is the case for the intuitionistic dependence logic of [@yang10], or for the $BID$ logic of [@abramsky08]), they will be expressively equivalent to independence logic proper. As (Jouko Väänänen, private communication) pointed out, this means that independence logic is *maximal* among the logics of imperfect information which always generate existential second order properties of teams. In particular, *any* dependency condition which is expressible as an existential second order property over teams can be expressed in independence logic: and as we will see in the next section, this entails that such a logic is capable of expressing a great amount of the notions of dependency considered by database theorists.
Equality generating dependencies, tuple generating dependencies and independence logic {#sect:eqtupgen}
======================================================================================
In Database Theory, two of the most general notions of dependence are *tuple generating* and *equality generating* dependencies.
In brief, a *tuple generating dependency* over a database relation $R$ is a sentence of the form $$\Delta(A) = \forall x_1 \ldots x_n (\phi(x_1 \ldots x_n) \rightarrow \exists z_1 \ldots z_k \psi(x_1 \ldots x_n, z_1 \ldots z_k))$$ where $A$ is a second order variable with arity equal to the number of attributes of $R$.[^33] and $\phi$ and $\psi$ are conjunctions of atoms of the form $A {\vec{t}}$ or ${\vec{t}}_1 = {\vec{t}}_2$ for some terms ${\vec{t}}$, ${\vec{t}}_1$ and ${\vec{t}}_2$ in the empty vocabulary and with free variables in $x_1 \ldots x_n$.
An *equality generating dependency* is defined much in the same way, except that $\psi$ is a single equality atom instead.\
Then, given a domain of predication $M$, a relation $R$ is said to satisfy a (tuple-generating or equality-generating) dependency $\Delta$ if and only if $M \models \Delta(R)$ in the usual first order sense.
As an example of the expressive power of tuple-generating and equality-generating dependencies, let us observe that dependency atoms correspond to equality generating dependencies and that independence atoms correspond to tuple generating dependencies: indeed, for example, $M \models_X =\!\!(x, y)$ if and only if $$M \models \forall x y_1 y_2 {\vec{z}}_1 {\vec{z}}_2 ((\rel(X))xy_1{\vec{z}}_1 \wedge (\rel(X))x y_2 {\vec{z}}_2 \rightarrow y_1 = y_2)$$ where $|{\vec{z}}_1| = |{\vec{z}}_2| = |\dom(X) \backslash \{x,y\}|$, and $M \models_X {y ~\bot_{x}~ z}$ if and only if $$\begin{aligned}
M \models & \forall x y_1 y_2 z_1 z_2 {\vec{w}}_1 {\vec{w}}_2 ( ( (\rel(X)) x y_1 z_1 {\vec{w}}_1 \wedge (\rel(X)) x y_2 z_2 {\vec{w}}_2) \rightarrow\\
& \rightarrow \exists {\vec{w}}_3 (\rel(X)) x y_1 z_2 {\vec{w}}_3).\end{aligned}$$
From the main result of the previous section, it is easy to see that I/E logic (and, as a consequence, independence logic) is capable to express all tuple and equality generating dependencies:
Let $\Delta(A)$ be a tuple generating or equivalent generating dependency, and let ${\vec{v}}$ be a tuple of distinct variables with $|{\vec{v}}| = \arity(A)$. Then there exists an I/E logic (or independence logic) formula $\phi({\vec{v}})$ such that $$M \models_X \phi({\vec{v}}) \Leftrightarrow M \models \Delta(\rel_{{\vec{v}}}(X))$$ for all suitable models $M$ and all teams $X$ with ${\vec{v}} \subseteq \dom(X)$.
$\Delta(A)$ is definable by a first order formula, and hence by Theorem \[theo:Sigma112IL\] it is expressible in I/E logic (and therefore by independence logic too, by Corollary \[coro:IL\_eq\_IE\]).
Hence, many of the properties which are discussed in the context of Database Theory can be expressed through independence logic. The vast expressive power of this formalism comes with a very high computational cost, of course; but it is the hope of the author that the result of this work may provide a justification to the study of this logic (and, more in general, of logics of imperfect information) as a general theoretic framework for reasoning about knowledge bases.
Acknowledgements
================
The author wishes to thank Jouko Väänänen for many valuable insights and suggestions. Furthermore, he thanks Erich Grädel for having suggested a better notation for inclusion and exclusion dependencies, and Allen Mann for having mentioned Thomas Forster’s paper. Finally, the author thankfully acknowledges the support of the EUROCORES LogICCC LINT programme.
[^1]: We will not discuss these issues in any detail in this work; for a handy reference, we suggest [@date03] or any other database theory textbook.
[^2]: The analogous question for dependence logic was answered in [@kontinenv09], and we will report that answer as Theorem \[theo:DLform\] of the present work.
[^3]: Such constraints are usually called *dependencies*, for historical reasons; but they need not correspond to anything resembling the informal idea of dependency.
[^4]: Subsection \[subsect:equilog\] briefly considers the case of *equiextension dependencies* and shows that, for our purposes, they are equivalent to inclusion dependencies.
[^5]: The readers interested in a more thorough explanation of the team semantics and of the two game theoretic semantics for dependence logic are referred to [@vaananen07] itself.
[^6]: In all this paper, I will assume that first order models have at least two elements in their domain.
[^7]: Decomposing the notion further, this is equivalent to stating that if the values of $t_1 \ldots t_{n-1}$ for the true assignment $s \in X$ were announced to the agent then he or she would also learn the value of $t_{n}$. The properties of this sort of announcement operators for dependence logic are discussed in [@galliani10].
[^8]: The existence of a relation between these notions and the ones studied in the field of epistemic modal logic is clear, but to the knowledge of the author the matter has not yet been explored in full detail. See [@vanbenthem06] for some intriguing reflections about this topic.
[^9]: This is a typical example of *signalling* ([@hintikka96], [@janssen06]), one of the most peculiar and, perhaps, problematic aspects of $IF$-logic.
[^10]: Here $\rel(X)$ is the relation corresponding to the team $X$, as in Definition \[defin:relteams\].
[^11]: Another interesting result about independence logic, pointed out by Fredrik Engström in [@engstrom10], is that the semantic rule for independence atoms corresponds to that of *embedded multivalued dependencies*, in the same sense in which the one for dependence atoms corresponds to *functional* ones.
[^12]: Or, equivalently, a *set* of variables; but having a fixed ordering of the variables as part of the definition of team will simplify the definition of the correspondence between teams and relations. With an abuse of notation, we will identify this tuple of variables with the underlying set whenever it is expedient to do so.
[^13]: Since the negation is not a semantic operation in dependence logic ([@burgess03], [@kontinenv10]), it is useful to assume that all formulas are in negation normal form. It is of course possible to adapt these definitions to formulas not in negation normal form, but in order to do so for the cases of dependence or independence logic it would be necessary to define two distinct relationships $\models^+$ and $\models^-$, as in [@vaananen07]. Since, for the purposes of this work, this would offer no significant advantage and would complicate the definitions, it was chosen to avoid the issue by requiring all formulas to be in negation normal form instead.
[^14]: In other words, first order formulas are *flat* in the sense of [@vaananen07].
[^15]: That is, all constancy atoms - and, more in general, all dependence atoms - are $2$*-coherent* but not $1$*-coherent* in the sense of [@kontinen_ja10].
[^16]: The rule **TS**-$\exists_L$ is also discussed in [@engstrom10], in which it is shown that it arises naturally from treating the existential quantifier as a *generalized quantifier* ([@mostowski57], [@lindstrom66]) for dependence logic.
[^17]: This normal form theorem is very similar to the one of dependence logic proper found in [@vaananen07]. See also [@durand11] for a similar, but not identical result, developed independently, which Arnaud Durand and Juha Kontinen use in that paper in order to characterize the expressive powers of subclasses of dependence logic in terms of the maximum allowed width of their dependence atoms.
[^18]: Indeed, if this is the case we can just choose $A_1 = \{m_1\}, \ldots, A_n = \{m_n\}$, and conversely if $A_1 \ldots A_n$ exist with the required properties we can simply select arbitrary elements of them for $m_1 \ldots m_n$.
[^19]: Apart from the present paper, [@engstrom10], which introduces *multivalued dependence atoms*, is also a step in this direction. The resulting “multivalued dependence logic” is easily seen to be equivalent to independence logic.
[^20]: As a somewhat naive example, let us consider the problem of finding a spy, knowing that yesterday he took a plane from London’s Heathrow airport and that he had at most 100 EUR available to buy his plane ticket. We might then decide to obtain, from the airport systems, the list of the destinations of all the planes which left Heathrow yesterday and whose ticket the spy could have afforded; and this list - that is, the list of all the places that the spy might have reached - would be a state of information of the kind which we are discussing.
[^21]: For example, our system should be able to represent the assertion that the flight code always determines the destination of the flight.
[^22]: Equivalently, one may consider the Cartesian product of these relations, as per the universal relation model ([@fagin84]).
[^23]: The simplest way to verify that these conditions are not expressible in terms of inclusion dependencies is probably to observe that inclusion dependencies are *closed under unions*: if the relations $R$ and $S$ respect ${\vec{x}} \subseteq {\vec{y}}$, so does $R \cup S$. Since functional dependencies as the above ones are clearly *not* closed under unions, they cannot be represented by inclusions.
[^24]: As an aside, it can be observed that, since ${\vec{H}}$ always selects singletons, this whole argument can be adapted to the case of strict semantics without any difficulties. Therefore, strict equiextension logic is equivalent to strict inclusion logic and, by Proposition \[propo:strict\_nonlocal\], does not satisfy locality either.
[^25]: To be more accurate, Figure \[fig:logrel2\] represents the translatability relations between the logics which we considered, *with respect to all formulas*. Considering sentences only would lead to a different graph.
[^26]: By definition, $Y_{\upharpoonright V} \subseteq Y'$ and $Z_{\upharpoonright V} \subseteq Z'$. On the other hand, if $s' \in Y'$ then $s' \in X_{\upharpoonright V}$, and hence $s'$ is of the form $s_{\upharpoonright V}$ for some $s \in X$, and therefore this $s$ is in $Y$ too, and finally $s' = s_{\upharpoonright V} \in Y_{\upharpoonright V}$. The same argument shows that $Z' \subseteq Z_{\upharpoonright V}$.
[^27]: Indeed, suppose that $s' \in X[H/x]$: then there exists a $s \in X$ such that $s' = s[m/x]$ for some $m \in H(s)$. Then $s_{\upharpoonright V} \in X_{\upharpoonright V}$, and moreover $m \in H'(s_{\upharpoonright V})$ by the definition of $H'$, and hence $s'_{\upharpoonright V \cup \{x\}} = s_{\upharpoonright V}[m/x] \in X_{\upharpoonright V}[H'/x]$.
Conversely, suppose that $h' \in X_{\upharpoonright V}[H'/x]$: then there exists a $h \in X_{\upharpoonright V}$ such that $h' = h[m/x]$ for some $m \in H'(h)$. But then there exists a $s \in X$ such that $h = s_{\upharpoonright V}$ and such that $m \in H(s)$; and therefore, $s[m/x] \in X[H/x]$, and finally $h' = h[m/x] = (s[m/x])_{\upharpoonright V \cup \{x\}} \in (X[H/x])_{\upharpoonright V \cup \{x\}}$.
[^28]: In brief, for all $s \in X$ and all $m \in \dom(M)$ we have that $m \in H'(s_{\upharpoonright V})$ if and only if $m \in H(s)$, by definition. Hence, for all such $s$ and $m$, $s_{\upharpoonright V}[m/x] \in X_{\upharpoonright V}[H'/x]$ if and only if $s[m/x] \in X[H/x]$.
[^29]: These players can also be named Falsifier and Verifier, or Abelard and Eloise.
[^30]: Here and in the rest of the work, when we write “winning” without specifying the player we mean “winning for Player $II$”.
[^31]: With a slight abuse of notation, we say that $T(\psi) = \alpha$ if $T(\psi, s) = \alpha$ for all suitable assignments $s$. In other words, $T(\psi) = I$ if $\psi$ is of the form $\psi_1 \wedge \psi_2$ or of the form $\forall v \psi_1$, and $T(\psi) = II$ if $\psi$ is of the form $\psi_1 \vee \psi_2$ or $\exists v \psi_1$.
[^32]: The details of this proof are similar to the ones of [@kontinenv09] and [@kontinennu09].
[^33]: In other words, if we consider $R$ as a relation in first order logic then $\arity(A) = \arity(R)$.
|
---
abstract: 'We study the algebra $\cI^{QM}$ of iterated integrals of quasimodular forms for $\SL_2(\bZ)$, which is the smallest extension of the algebra $QM_{\ast}$ of quasimodular forms, which is closed under integration. We prove that $\cI^{QM}$ is a polynomial algebra in infinitely many variables, given by Lyndon words on certain monomials in Eisenstein series. We also prove an analogous result for the $M_{\ast}$-subalgebra $\cI^{M}$ of $\cI^{QM}$ of iterated integrals of modular forms.'
address: 'Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111, Bonn, Germany'
author:
- Nils Matthes
title: On the algebraic structure of iterated integrals of quasimodular forms
---
Introduction
============
Quasimodular forms are generalizations of modular forms, which have first been introduced in [@KanekoZagier], in a context motivated by mathematical physics. The $\bC$-algebra $QM_{\ast}$ of quasimodular forms for the full modular group $\SL_2(\bZ)$ can be defined, in a slightly ad hoc fashion, as the polynomial ring $\bC[E_2,E_4,E_6]$, where $E_{2k}$ denotes the normalized Eisenstein series of weight $2k$: $$E_{2k}(\tau)=1-\frac{4k}{B_{2k}}\sum_{n=1}^{\infty}n^{2k-1}\frac{q^n}{1-q^n}, \quad q=e^{2\pi i\tau},$$ where $B_{2k}$ are the Bernoulli numbers. In particular, $QM_{\ast}$ contains the algebra of modular forms $M_{\ast} \cong \bC[E_4,E_6]$.
The derivative of a quasimodular form (of weight $k$) is again a quasimodular form (of weight $k+2$); this was essentially already known to Ramanujan (cf. [@123], Proposition 15). On the other hand, the integral of a quasimodular form is in general not quasimodular. For example, a primitive of $E_2$ would have to be of weight zero, but every quasimodular form of weight zero is constant.
The goal of this paper is to study the smallest algebra extension of $QM_{\ast}$, which is closed under integration. For this, the idea is to iteratively adjoin primitives to $QM_{\ast}$, which eventually leads to adjoining all (indefinite) *iterated integrals* $$\label{eqn:iterEich}
I(f_1,\ldots,f_n;\tau)=(2\pi i)^n\idotsint\limits_{\tau \leq \tau_1 \leq \ldots \leq \tau_n\leq i\infty}f_1(\tau_1)\ldots f_n(\tau_n)\dd\tau_1\ldots\dd\tau_n,$$ where $f_1,\ldots,f_n$ are quasimodular forms (a precise definition will be given in Section \[ssec:2.2\]). The integrals have first been studied by Manin [@Man] and later by Brown [@Brown:MMV] and Hain [@Hain:HodgeDeRham], in the case where all the $f_i$ are modular forms.[^1] In all of these treatments, the focus lies rather on arithmetic aspects of these iterated integrals, for example their special values at cusps of the upper half-plane. By contrast, we study them solely as holomorphic functions of $\tau$. It is also worth noting that even in the modular case, the iterated integrals we study in the present paper are slightly more general than the ones introduced in [@Brown:MMV; @Hain:HodgeDeRham; @Man]. For example, if $f(\tau)$ is a modular form of weight $k$, then the integral $\int_{\tau}^{i\infty}f(\tau_1)\tau_1^n\dd\tau_1$ is an iterated integral of modular forms in the sense of the present paper *for every* $n \geq 0$, while [@Brown:MMV; @Hain:HodgeDeRham; @Man] also require $n \leq k-2$.
Now let $\cI^{QM}$ be the $QM_{\ast}$-algebra generated by all the integrals , which is the smallest algebra extension of $QM_{\ast}$, closed under integration. It turns out that $\cI^{QM}$ is not finitely generated, but still has a manageable structure, which is captured by the notion of shuffle algebra (which is just the graded dual of the tensor algebra with a certain commutative multiplication, the so-called shuffle product) [@Reu]. More precisely, let $V=\bC \cdot E_2\oplus M_{\ast}$ be the $\bC$-vector space spanned by all modular forms and the Eisenstein series $E_2$, and let $\bC\langle V\rangle$ be the shuffle algebra on $V$. Our main result is the following.
The $QM_{\ast}$-linear morphism $$\begin{aligned}
\varphi^{QM}: QM_{\ast} \otimes_{\bC} \bC\langle V\rangle &\rightarrow \cI^{QM}\notag\\
[f_1|\ldots|f_n] &\mapsto I(f_1,\ldots,f_n;\tau)\end{aligned}$$ is an isomorphism of $QM_{\ast}$-algebras.
A similar result holds for the $M_{\ast}$-subalgebra $\cI^M$ of $\cI^{QM}$ of iterated integrals of modular forms (cf. Theorem \[thm:mainmod\]).[^2] The surjectivity of $\varphi^{QM}$ can be reduced to the fact that every quasimodular form can be written uniquely as a polynomial in $n$-th derivatives of modular forms and the Eisenstein series $E_2$ (cf. [@123], Proposition 20). The proof of injectivity is more elaborate and amounts to showing that iterated integrals of modular forms and the Eisenstein series $E_2$ are linearly independent over $QM_{\ast}$. It extends a result of [@LMS] which dealt with iterated integrals of Eisenstein series. In both cases, the key is to use a general result on linear independence of iterated integrals [@DDMS]. It would be interesting to prove similar results for quasimodular forms for congruence subgroups.
The Milnor–Moore theorem [@MM] states that if $k$ has characteristic zero, then $k\langle V\rangle$ is isomorphic to a polynomial algebra (usually in infinitely many variables). Fixing a (totally ordered) basis $\cB$ of $V$, Radford [@Radford] has given explicit generators of $k\langle V\rangle$ in terms of Lyndon words on $\cB$ (cf. Section \[sec:4\]). Using this, we get the following theorem.
Let $\cB$ be a basis of $\bC \cdot E_2 \oplus M_{\ast}$. We have a natural isomorphism $$\label{eqn:canisom}
\cI^{QM} \cong QM_{\ast}[Lyn(\cB^*)],$$ where the right hand side is the polynomial $QM_{\ast}$-algebra on the set $Lyn(\cB^*)$ of Lyndon words of $\cB$.
Again, a similar result holds for $\cI^M$. Since $QM_{\ast}$ has an explicit basis given by monomials in the Eisenstein series $E_2$, $E_4$ and $E_6$, the isomorphism can be made completely explicit, and may be viewed as an analog of the isomorphism $QM_{\ast} \cong \bC[E_2,E_4,E_6]$ [@KanekoZagier].
Finally, we note that classically, integrals of modular forms play an important role in Eichler–Shimura theory, where they give rise to group-cocycles (say for $\SL_2(\bZ)$ or more generally for some congruence subgroup thereof) with values in homogeneous polynomials. This has been generalized by Manin [@Man], and later by Brown [@Brown:MMV] and Hain [@Hain:HodgeDeRham], who attach certain non-abelian cocycles to iterated integrals of modular forms. Although it is not the main focus of this article, in the appendix we show how one can attach cocycles to quasimodular forms (for $\SL_2(\bZ)$), partly since we found no mention of this in the literature. On the other hand, we leave the definition and study of cocycles attached to iterated integrals of quasimodular forms for future investigation.
The plan of the paper is as follows. In Section \[sec:2\], we collect the necessary background on quasimodular forms and their iterated integrals. In Section \[sec:3\], we prove a linear independence result for iterated integrals of quasimodular forms. This result is then put to use in Section \[sec:4\], where the main results are proved. In the appendix, we discuss the above-mentioned generalization of the classical Eichler–Shimura theory to quasimodular forms for $\SL_2(\bZ)$.
[**Acknowledgments:**]{} Very many thanks to Pierre Lochak for bringing the article [@DDMS] to the author’s attention. Also, many thanks to Francis Brown, Erik Panzer and the referees for corrections as well as very helpful suggestions and to Don Zagier for inspiring discussions on the appendix. The results of this paper were found while the author was a PhD student at Universität Hamburg under the supervision of Ulf Kühn.
Preliminaries {#sec:2}
=============
Throughout the paper, all modular and quasimodular forms will be for $\SL_2(\bZ)$. We fix some notation. Let $\fH=\{z \in \bC \, \vert \, \im(z)>0 \}$ be the upper half-plane with canonical coordinate $\tau$. For every $k \in \bZ$, we have a group action of $\SL_2(\bZ)$ on the set of all functions $f: \fH \rightarrow \bC$ (not necessarily holomorphic), defined by $(\gamma,f) \mapsto f\vert_k\gamma$, where $$(f\vert_k\gamma)(\tau):=(c\tau+d)^{-k}f\left(\frac{a\tau+b}{c\tau+d} \right).$$ For fixed $\tau \in \fH$, we also define a map $X: \SL_2(\bZ) \rightarrow \bC$ by $X(\gamma)=\frac{1}{2\pi i}\frac{c}{c\tau+d}$. Note that $X$ has infinite, and thus Zariski dense, image.
Recap of modular forms
----------------------
Denote by $M_k$ the space of modular forms of weight $k \in \bZ$. By definition, these are the holomorphic functions $f: \fH \rightarrow \bC$, which satisfy $f\vert_k\gamma=f$ for all $\gamma \in \SL_2(\bZ)$, and which are “holomorphic at the cusp”. The latter condition means that in the Fourier expansion $f(\tau)=\sum_{n \in \bZ}a_nq^n$ (which exists since for $\gamma=\left(\begin{smallmatrix}1&1\\0&1\end{smallmatrix} \right) \in \SL_2(\bZ)$, the condition $f\vert_k\gamma=f$ is just $f(\tau+1)=f(\tau)$ for all $\tau$), all $a_n=0$ for $n<0$. Examples of modular forms include the Eisenstein series $$E_{2k}(\tau)=1-\frac{4k}{B_{2k}}\sum_{n=1}^{\infty}n^{2k-1}\frac{q^n}{1-q^n}=1-\frac{4k}{B_{2k}}\sum_{n=1}^{\infty}\left( \sum_{d\vert n}d^{2k-1} \right)q^n,$$ which is a modular form of weight $2k$, for $k \geq 2$ (the $B_{2k}$ are Bernoulli numbers). The $\bC$-vector space of all modular forms $M_{\ast}$ is a graded (for the weight) $\bC$-algebra $M_{\ast}=\bigoplus_{k \in \bZ}M_k$, which is well-known to be isomorphic to the polynomial algebra $\bC[E_4,E_6]$. Proofs of all these facts and much more on modular forms can be found for example in [@123].
Quasimodular forms
------------------
Quasimodular forms are a generalization of modular forms, which have first been introduced in [@KanekoZagier] (see also [@BlochOk], §3 and [@123], §5.3). The definition we give here is due to W. Nahm[^3] and is also used for example in [@MartinRoyer].
\[dfn:qmod\] Let $k,p \in \bZ$ with $p \geq 0$. A *quasimodular form* of weight $k$ and depth $\leq p$ is a function $f:\fH \rightarrow \bC$ with the following property: there exist holomorphic functions $f_r: \fH \rightarrow \bC$, for $0 \leq r \leq p$, which have Fourier expansions $\sum_{n=0}^{\infty}a_nq^n$, such that $$\label{eqn:qmod}
(f\vert_k\gamma)(\tau)=\sum_{r=0}^pf_r(\tau)X(\gamma)^r, \quad \mbox{for all }\gamma \in \SL_2(\bZ).$$ We denote by $QM^{\leq p}_k$ the $\bC$-vector space of quasimodular forms of weight $k$ and depth $\leq p$ and set $$QM_k:=\bigcup_{p\geq 0}QM^{\leq p}_k, \quad QM_{\ast}:=\bigoplus_{k\in\bZ}QM_k.$$
\[rmk:unique\]
1. It is clear from the definition that, if $f_1 \in QM_{k_1}^{\leq p_1}$, $f_2 \in QM_{k_2}^{\leq p_2}$, then $f_1f_2 \in QM_{k_1+k_2}^{\leq p_1+p_2}$. In other words, $QM_{\ast}$ is a graded (for the weight) and filtered (for the depth) $\bC$-algebra.
2. Using that $X$ is Zariski dense, it is easy to see that the functions $f_r(\tau)$ are uniquely determined by $f(\tau)$. Also, applying with $\gamma=\left(\begin{smallmatrix}1&0\\0&1\end{smallmatrix}\right)$, we see that $f_0(\tau)=f(\tau)$. In particular, every quasimodular form is holomorphic on $\fH$ and at the cusp.
Every modular form is a quasimodular form of depth zero, more precisely, $M_k=QM_k^{\leq 0}$. An example of a quasimodular form, which is not modular is the Eisenstein series of weight two $
E_2(\tau)=1-24\sum_{n=1}^{\infty}n\frac{q^n}{1-q^n},
$ which transforms as $$\label{eqn:E2trans}
(E_2\vert_2\gamma)(\tau)=E_2(\tau)+12X(\gamma)=E_2(\tau)-\frac{6i}{\pi}\frac{c}{c\tau+d},$$ for all $\gamma \in \SL_2(\bZ)$. In particular, $E_2 \in QM_2^{\leq 1} \setminus M_2$.
The following proposition recalls basic properties of $QM_{\ast}$ that will be of use later.
\[prop:qmod\]
1. The $\bC$-algebra $QM_{\ast}$ is closed under the differential operator $D:=\frac{1}{2\pi i}\frac{d}{d\tau}=q\frac{d}{dq}$. More precisely, for $f$ quasimodular of weight $k$ and depth $\leq p$, we have $$\label{eqn:diff}
(D(f)\vert_{k+2}\gamma)(\tau)=\sum_{r=0}^{p+1}(D(f_r)(\tau)+(k-r+1)f_{r-1}(\tau))X(\gamma)^r.$$ In particular, $D(QM_k^{\leq p}) \subset QM_{k+2}^{\leq p+1}$ for all $k,p \in \bZ$.
2. We have $$QM_k=\begin{cases}\{0\}, & \mbox{if $k<0$} \\ \bC \cdot E_2, & \mbox{if $k=2$}\\ D(QM_{k-2}) \oplus M_k &\mbox{else.}\end{cases}$$ In particular, $QM_{\ast}=\bC \cdot E_2 \oplus D(QM_{\ast}) \oplus M_{\ast}$, and $$QM_{\ast}\cong\bC[E_2,E_4,E_6]$$ as graded $\bC$-algebras.
For (i), simply apply $D$ to both sides of . The first equality in (ii) follows from [@123], Proposition 20.(iii), and the isomorphism $QM_{\ast} \cong \bC[E_2,E_4,E_6]$ is essentially a consequence of this, but can also be proved independently (cf. [@BlochOk], Proposition 3.5.(ii)).
\[rmk:qmodfun\] Relaxing the condition in the definition of quasimodular forms that every $f_r$ be a holomorphic function, one can define the notion of *weakly quasimodular form* of weight $k$ and depth $\leq p$ as a meromorphic function $f: \fH \rightarrow \bC$ satisfying , but where the functions $f_r(\tau)$ are only required to be meromorphic on $\fH$ and have Fourier series of the form $\sum_{n=-M}^{\infty}a_nq^n$ ($f_r$ is “meromorphic at the cusp”). As in the case of quasimodular forms, one shows easily that the functions $f_r(\tau)$ are uniquely determined by $f(\tau)$ (cf. Remark \[rmk:unique\]). Moreover, Proposition \[prop:qmod\].(i) generalizes straightforwardly to weakly quasimodular forms.
We end this subsection with a short lemma, for which we couldn’t find a suitable reference. Denote by $\Delta=\frac{1}{1728}(E_4^3-E_6^2)$ Ramanujan’s cusp form of weight $12$.
\[lem:E2diff\] Let $g \in QM_{\ast} \setminus \{0\}$ and $\alpha \in \bC$ such that $$\label{eqn:E2diff}
D(g)=(\alpha E_2) \cdot g.$$ Then $\alpha$ is a non-negative integer, and $g=\beta \Delta^{\alpha}$ for some $\beta \in \bC \setminus \{0\}$.
Let $g=\sum_{n=0}^{\infty}a_nq^n$, so that $D(g)=\sum_{n=0}^{\infty}na_nq^n$. Comparing coefficients on both sides of yields that $\alpha$ equals the smallest integer $m \geq 0$ such that $a_m \neq 0$. On the other hand, $
\frac{D(\Delta)}{\Delta}=E_2
$ (cf. [@123], proof of Proposition 7), and from the chain rule $\frac{D(\Delta^{\alpha})}{\Delta^{\alpha}}=\alpha E_2$, which gives the result.
Iterated integrals on the upper half-plane {#ssec:2.2}
------------------------------------------
Iterated integrals of modular forms have been considered first by Manin (for cusp forms) [@Man], and later by Brown (in general) [@Brown:MMV]. They are generalizations of the classical Eichler integrals [@Eichler; @Lang] $$\label{eqn:EichlerInt}
\int_{\tau}^{i\infty}f(z)z^m\dd z, \quad m=0,\ldots,k-2$$ where $f$ is a cusp form of weight $k$. Extending to a general modular form poses the problem of logarithmic divergences, which arise from the constant term in the Fourier series of $f$. A procedure for regularizing such integrals is described in [@Brown:MMV], and we borrow it to define iterated integrals of quasimodular forms. Since it is perhaps not so well-known, we give some details, for the convenience of the reader.
Let $W \subset \cO(\fH)$ be the $\bC$-subalgebra of holomorphic functions $f: \fH \rightarrow \bC$, which have an everywhere convergent Fourier series $f(\tau)=\sum_{n=0}^{\infty}a_nq^n$ with $q=e^{2\pi i\tau}$. Note that $QM_{\ast}\subset W$. For $f(\tau) \in W$, let $f^{\infty}=a_0$, and $f^0(\tau)=f(\tau)-f^{\infty}=\sum_{n=1}^{\infty}a_nq^n$. Let $\bC\langle W\rangle$ (sometimes denoted by $T^c(W)$) be the shuffle algebra [@Reu], i.e. the graded dual of the tensor algebra $T(W)=\bigoplus_{k \geq 0} W^{\otimes n}$ on $W$, where the grading is by the length of tensors. Elements of $(W^{\otimes n})^{\vee}$ will be written using bar notation $[f_1|f_2|\ldots|f_n]$, and a general element of $\bC\langle W\rangle$ is a $\bC$-linear combination of those. The product on $\bC\langle W\rangle$ is the shuffle product $\shuffle$, which is defined on the basic elements by $$\label{eqn:shuffleproduct}
[f_1|\ldots|f_r] \shuffle [f_{r+1}|\ldots|f_{r+s}]=\sum_{\sigma \in \Sigma_{r,s}}[f_{\sigma(1)}|\ldots|f_{\sigma(r+s)}],$$ where $\Sigma_{r,s}$ denotes the set of all the permutations on the set $\{1,\ldots,r+s\}$ such that $\sigma^{-1}(1)<\ldots<\sigma^{-1}(r)$ and $\sigma^{-1}(r+1)<\ldots<\sigma^{-1}(r+s)$.
Define a $\bC$-linear map $R: \bC\langle W\rangle \rightarrow \bC\langle W\rangle$ by the formula $$R[f_1|\ldots|f_n]=\sum_{i=0}^n(-1)^{n-i}[f_1|\ldots|f_i] \shuffle [f_n^{\infty}|\ldots|f_{i+1}^{\infty}].$$ Following [@Brown:MMV], Section 4, we make the following definition.
\[dfn:maindef\] For $f_1,\ldots,f_n \in W$, define their regularized iterated integral $$\label{eqn:maindef}
I(f_1,\ldots,f_n;\tau):=(2\pi i)^n\sum_{i=0}^n(-1)^{n-i}\int_{\tau}^{i\infty}R[f_1|\ldots|f_i]\int_0^{\tau}[f_n^{\infty}|\ldots|f_{i+1}^{\infty}],$$ where $\displaystyle \int\limits_a^b[f_1|\ldots|f_n]:=\int\limits_{0 \leq t_1 \leq \ldots\leq t_n\leq 1}(\gamma_a^b)^*(f_1(\tau_1)\dd\tau_1)\ldots (\gamma_a^b)^*(f_n(\tau_n)\dd\tau_n)$ denotes the usual iterated integral along the straight line path $\gamma_a^b$ from $a$ to $b$.
Using the change of variables $\tau \mapsto q=e^{2\pi i\tau}$, it is easy to see that $I(f_1,\ldots,f_n;\tau) \in W[\log(q)]$, where $\log(q):=2\pi i\tau$. By the same token, if all of the $f_i$ have rational Fourier coefficients, then $I(f_1,\ldots,f_n;\tau)$ will also have rational coefficients, as a series in $q$ and $\log(q)$.
The functions $I(f_1,\ldots,f_n;\tau)$ satisfy the following properties.
1. The product of any two of them is given by the shuffle product: $$\label{eqn:shuffle}
I(f_1,\ldots,f_r;\tau)I(f_{r+1},\ldots,f_{r+s};\tau)=\sum_{\sigma \in \Sigma_{r,s}}I(f_{\sigma(1)},\ldots,f_{\sigma(r+s)};\tau).$$
2. They satisfy the differential equation $$\label{eqn:diffeq}
\frac{1}{2\pi i}\frac{d}{d\tau}\Bigr\vert_{\tau=\tau_0}I(f_1,\ldots,f_n;\tau)=-f_1(\tau_0)I(f_2,\ldots,f_n;\tau_0).$$
3. We have the integration by parts formulas $$\begin{aligned}
I(f_1,\ldots,f_i,D(g),f_{i+1},\ldots,f_n;\tau)&=I(f_1,\ldots,f_i,gf_{i+1},\ldots,f_n;\tau)\notag\\
&-I(f_1,\ldots,f_ig,f_{i+1},\ldots,f_n;\tau), \label{eqn:intparts}\end{aligned}$$ as well as $$I(D(g),f_2,\ldots,f_n;\tau)=I(gf_2,f_3,\ldots,f_n;\tau)-g(\tau)I(f_2,\ldots,f_n;\tau),$$ and $$I(f_1,\ldots,f_{n-1},D(g);\tau)=g(i\infty)I(f_1,\ldots,f_{n-1};\tau)-I(f_1,\ldots,f_{n-1}g;\tau).$$
Using the definition , all of these follow from the analogous properties for usual iterated integrals (cf. e.g. [@Hai]).
A criterion for linear independence of iterated integrals
---------------------------------------------------------
Let $\operatorname{Frac}(W)$ be the field of fractions of the $\bC$-algebra $W$ introduced in the last subsection. By the quotient rule, it is easy to see that $\operatorname{Frac}(W)$ is closed under $D=\frac{1}{2\pi i}\frac{d}{d\tau}$.
The following theorem is a special case of the main result of [@DDMS].
\[thm:DDMS\] Let $\cF=(f_i)_{i \in I}$ be a family of elements of $W$, and let $\cC \subset \operatorname{Frac}(W)$ be a subfield, which is closed under $D$ and contains $\cF$. The following are equivalent:
1. The family of iterated integrals $(I(f_1,\ldots,f_n;\tau) \, \vert \, f_i \in I, \, n\geq 0)$ is linearly independent over $\cC$.
2. The family $\cF$ is linearly independent over $\bC$, and we have $$\label{eqn:trivint}
D (\cC) \cap \spn_{\bC}(\cF)=\{0\}.$$
This is the special case of Theorem 2.1 in [@DDMS], with the notation of *loc.cit.*, $k=\bC$, $(\cA,\dd)=(\operatorname{Frac}(\cO(\fH)),D)$, $X=\{A_{f_i} \, \vert \, f_i \in \cF \}$, $M=-\sum_{i \in I}f_iA_{f_i}$ and $S=\sum_{n \geq 0}\sum_{f_{i_1},\ldots,f_{i_n} \in S}I(f_1,\ldots,f_n;\tau)\cdot A_{f_1}\ldots A_{f_n}$. Note that it follows from that $$D (S)=M\cdot S,$$ as required in Theorem 2.1 of [@DDMS].
Variants of Theorem \[thm:DDMS\] have been known before (cf. [@Brown:thesis], Lemma 3.6).
Linear independence of iterated integrals of quasimodular forms {#sec:3}
===============================================================
In this section, we apply Theorem \[thm:DDMS\] to deduce linear independence of a large family of iterated integrals of quasimodular forms. More precisely, our main result is the following theorem.
\[thm:linind\] Let $\cB$ be a $\bC$-linearly independent family of elements of $\bC \cdot E_2 \oplus M_{\ast}$. Then the family of iterated integrals $$(I(f_1,\ldots,f_n;\tau) \, \vert \, f_i \in \cB)$$ is linearly independent over $\operatorname{Frac}(QM_{\ast}) \cong \bC(E_2,E_4,E_6)$.
Two auxiliary lemmas
--------------------
For the proof of Theorem \[thm:linind\], we need two lemmas.
\[lem:1\] Let $f,g \in \bC[E_2,E_4,E_6]$ such that $g \neq 0$ and such that $f$ and $g$ are coprime. Assume that $D\left(\frac fg\right) \in \bC[E_2,E_4,E_6]$. Then $g=\beta \Delta^{\alpha}$ for some $\alpha \in \bZ_{\geq 0}$ and some $\beta \in \bC \setminus \{0\}$, where $\Delta:=\frac{1}{1728}(E_4^3-E_6^2)$ is Ramanujan’s cusp form of weight 12.
By the quotient rule, we have $$D\left(\frac fg\right)=\frac{D(f)g-fD(g)}{g^2}=\frac{D(f)-f\frac{D(g)}{g}}{g}.$$ The left hand side is contained in $\bC[E_2,E_4,E_6]$ by assumption, and since also $D(f)$ and $g$ are in $\bC[E_2,E_4,E_6]$, we have $f\frac{D(g)}{g} \in \bC[E_2,E_4,E_6]$. But then, as $f$ and $g$ have no common factor, $g$ must divide $D(g)$, i.e. there exists $h \in \bC[E_2,E_4,E_6]$ such that $$\label{eqn:product}
D(g)=gh.$$ Since the operator $D: QM_{\ast} \rightarrow QM_{\ast}$ is homogeneous of weight $2$ (cf. Proposition \[prop:qmod\].(i)), we have $h \in QM_2$, i.e. $h=\alpha E_2$ with $\alpha \in \bC$. In other words, $g$ solves the differential equation $D(g)=(\alpha E_2)\cdot g$. But by Lemma \[lem:E2diff\], $\alpha$ must be a non-negative integer and $g=\beta\Delta^{\alpha}$ for some $\beta\in \bC\setminus \{0\}$.
\[lem:2\] Let $f$ be a weakly quasimodular form, such that its derivative $D(f)$ is a quasimodular form. Then $f$ is a quasimodular form.
It is no loss of generality to assume that $f$ is of weight $k \in \bZ$ and depth $\leq p$, where $p \geq 0$. By the definition of weakly quasimodular forms (cf. also Remark \[rmk:unique\]), there exist uniquely determined meromorphic functions $f_r(\tau)$, for $0\leq r\leq p$, such that $$(f\vert_k\gamma)(\tau)=\sum_{r=0}^pf_r(\tau)X(\gamma)^r,$$ for all $\gamma \in \SL_2(\bZ)$. Therefore, we only need to show that every $f_r(\tau)$ is holomorphic, including at the cusp.
To this end, by Proposition \[prop:qmod\].(i), we know that $$\label{eqn:auxalmhol}
(D(f)\vert_{k+2}\gamma)(\tau)=\sum_{r=0}^{p+1}(D(f_r)(\tau)+(k-r+1)f_{r-1}(\tau))X(\gamma)^r,$$ and since $D(f)$ is a quasimodular form by assumption, every coefficient of is holomorphic, including at the cusp.
The constant term, with respect to $X(\gamma)$, in equals $D(f_0)(\tau)$, which is holomorphic by assumption. But a meromorphic function whose derivative is holomorphic everywhere is itself holomorphic everywhere. An easy induction argument, using that the coefficients of are holomorphic, now shows that in fact every $f_r(\tau)$ is holomorphic.
Proof of Theorem \[thm:linind\]
-------------------------------
We will use the criterion of Theorem \[thm:DDMS\] in the case where $\cC=\operatorname{Frac}(QM_{\ast})$ and $\cF=\cB$. Since $\cB$ is linearly independent over $\bC$ by assumption, it is enough to prove that if $h \in \operatorname{Frac}(QM_{\ast})$ then $$D(h)=\sum_{f \in \cB}\alpha_f f, \, \alpha_f \in \bC \quad \Rightarrow \quad \alpha_f=0, \, \mbox{ for all } f \in \cB.$$ Also, since $\cB$ spans a subspace of $\bC \cdot E_2 \oplus M_{\ast}$, it clearly suffices to prove that $D(h) \in \bC \cdot E_2 \oplus M_{\ast}$ implies that $D(h)=0$, or equivalently that $h$ is constant. Thus, the following proposition completes the proof of Theorem \[thm:linind\].
\[prop:technical\] Let $h \in \operatorname{Frac}(QM_{\ast}) \cong \bC(E_2,E_4,E_6)$, such that $D(h) \in \bC \cdot E_2\oplus M_{\ast}$. Then $h$ is constant.
Write $h=\frac fg$ with $f,g \in \bC[E_2,E_4,E_6]$, $g\neq 0$ and such that $f$ and $g$ are coprime. Writing $f$ as a $\bC$-linear combination of its homogeneous components, it is enough to show the proposition for $f$ homogeneous of weight $k_f$.
First, we know from Lemma \[lem:1\] that $g=\beta\Delta^{\alpha}$ for some $\alpha \in \bZ_{\geq 0}$ and $\beta \in \bC \setminus \{0\}$, where $\Delta$ is Ramanujan’s cusp form of weight $12$. In particular, $g$ is a cusp form of weight $k_g=12\alpha$.
Since $f$ is quasimodular of weight $k_f$ and depth $\leq p$, there exist holomorphic (including at the cusp) functions $f_r(\tau)$, for $0\leq r\leq p$, such that $$(f\vert_{k_f}\gamma)(\tau)=\sum_{r=0}^pf_r(\tau)X(\gamma)^r,$$ for all $\gamma \in \SL_2(\bZ)$. Setting $h_r(\tau):=\frac{f_r}{g}(\tau)$, we also have, for $k:=k_f-k_g$ $$\label{eqn:5}
(h\vert_k\gamma)(\tau)=\sum_{r=0}^ph_r(\tau)X(\gamma)^r.$$ Moreover, the functions $h_r(\tau)$ are meromorphic, thus, $h$ is a weakly quasimodular form (of weight $k$ and depth $\leq p$). By assumption, $D(h)$ is a quasimodular form (necessarily of weight $k+2$ and depth $\leq p+1$), and using Lemma \[lem:2\], this implies that $h \in QM^{\leq p}_k$, therefore every $h_r(\tau)$ is holomorphic, including at the cusp.
Summarizing, we have seen that $h \in \operatorname{Frac}(QM_{\ast})$ such that $D(h) \in QM_{\ast}$ implies that $h \in QM_{\ast}$. But we even have $D(h) \in \bC \cdot E_2\oplus M_{\ast}$ by assumption, and therefore Proposition \[prop:qmod\].(ii) now implies that $h$ is constant, as was to be shown.
Iterated integrals of quasimodular forms and shuffle algebras {#sec:4}
=============================================================
We describe the $QM_{\ast}$-algebra of iterated integrals of quasimodular forms, which is the smallest algebra, which contains $QM_{\ast}$ and is closed under integration. Using the results of the last section, we show that it is canonically isomorphic to an explicit shuffle algebra. A similar result holds for the $M_{\ast}$-subalgebra of iterated integrals of modular forms.
The algebra of iterated integrals of quasimodular forms
-------------------------------------------------------
Define $\cI^{QM}$ to be the $QM_{\ast}$-module generated by all iterated integrals of quasimodular forms: $$\cI^{QM}=\spn_{QM_{\ast}}\{ I(f_1,\ldots,f_n;\tau) \, \vert \, f_i \in QM_{\ast} \}.$$ We also denote by $\cI^{QM}_n$ the $QM_{\ast}$-linear submodule, which is spanned by all of the $I(f_1,\ldots,f_r;\tau)$ with $r \leq n$.
The subspaces $\cI^{QM}_n$ define an ascending filtration $\cI^{QM}_{\bullet}$ on $\cI^{QM}$, called the length filtration (in analogy with the length filtration on iterated integrals [@Hai]). It follows from that $\cI^{QM}$ is a filtered $QM_{\ast}$-algebra. However, the length is not a grading, as shown by the next result.
\[prop:length\] Let $f_1,\ldots,f_n$ be quasimodular forms. Then $$I(f_1,\ldots,f_{i-1},D(f_i),f_{i+1},\ldots,f_n;\tau) \in \cI^{QM}_{n-1}.$$
This is an immediate consequence of the integration by parts formula .
$\cI^{QM}$ as a shuffle algebra
-------------------------------
We let $V$ be the $\bC$-vector space $\bC \cdot E_2 \oplus M_{\ast}$, and denote by $\bC\langle V\rangle$ the shuffle algebra on $V$ (cf. Section \[ssec:2.2\]). Recall that this is the graded dual of the tensor algebra $T(V)$, whose grading is given by the length of tensors. Elements of $\bC\langle V\rangle$ are $\bC$-linear combination of the basic elements $[f_1|\ldots|f_n]$, and the product on $\bC\langle V\rangle$ is the shuffle product .
The following theorem is the main result of this paper.
\[thm:main\] The $QM_{\ast}$-linear map $$\begin{aligned}
\label{eqn:varphi}
\varphi^{QM}: QM_{\ast} \otimes_{\bC} \bC\langle V\rangle &\rightarrow \cI^{QM}\\
[f_1|\ldots|f_n] &\mapsto I(f_1,\ldots,f_n;\tau) \notag\end{aligned}$$ is an isomorphism of $QM_{\ast}$-algebras.
Let $\cB$ be a basis of $V$, so that the family $([f_1|\ldots|f_n] \, \vert \, f_i \in \cB)$ is a basis of $\bC\langle V\rangle$. The injectivity of $\varphi^{QM}$ follows from the $\operatorname{Frac}(QM_{\ast})$-linear independence of the family $$\label{eqn:family}
\cF=(I(f_1,\ldots,f_n;\tau) \, \vert \, f_i \in \cB),$$ which is a consequence of Theorem \[thm:linind\].
In order to obtain the surjectivity, we need to prove that the family generates $\cI^{QM}$. To this end, we prove inductively that for every $n\geq 0$, we have $\cI^{QM}_n \subset \spn_{QM_{\ast}}\cF$. The case $n=0$ is trivial. Now let $n\geq 1$ and assume that for every $r \leq n-1$, we have $\cI^{QM}_r \subset \spn_{QM_{\ast}}\cF$. Given quasimodular forms $f_1,\ldots,f_n$, we can write $f_i=g_i+D(h_i)$, where $g_i \in\bC \cdot E_2 \oplus M_{\ast}$ and $h_i \in D(QM_{\ast})$ by Proposition \[prop:qmod\].(ii). Then by linearity $$\begin{aligned}
I(f_1,\ldots,f_n;\tau)&=I(g_1,\ldots,g_n;\tau)\notag\\&+\sum_{i=1}^nI(g_1,\ldots,g_{i-1},D(h_i),g_{i+1},\ldots,g_n)+\ldots, \label{eqn:sum}\end{aligned}$$ where the $\ldots$ above signifies iterated integrals, which have at least two $D(h_i)$ as integrands. The first term on the right is contained in $\spn_{QM_{\ast}}\cF$, since $g_i \in \bC \cdot E_2 \oplus M_{\ast}$ for every $i$ and $\cB$ is a basis. On the other hand, all other terms in the sum are iterated integrals, which contain at least one $D(h_i)$. By Proposition \[prop:length\], it thus follows that $I(f_1,\ldots,f_n;\tau) \equiv I(g_1,\ldots,g_n;\tau) \mod \cI^{QM}_{n-1}$, and we conclude using the induction hypothesis. Finally, it is clear that $\varphi^{QM}$ is a homomorphism of algebras, since both sides of are endowed with the shuffle product.
The algebra of iterated integrals of modular forms
--------------------------------------------------
In this section, we study the subalgebra $\cI^M$ of $\cI^{QM}$, generated by iterated integrals of modular forms.
\[dfn:algitermod\] Define $\cI^M$ to be the $M_{\ast}$-module generated by all iterated integrals of modular forms: $$\cI^{M}=\spn_{M_{\ast}}\{ I(f_1,\ldots,f_n;\tau) \, \vert \, f_i \in M_{\ast} \}.$$
As in the case of $\cI^{QM}$, the length of iterated integrals defines the length filtration $\cI^{M}_{\bullet}$ on $\cI^{M}$, and $\cI^M$ is a filtered $M_{\ast}$-subalgebra of $\cI^{QM}$. We let $\bC\langle M_{\ast}\rangle$ be the shuffle algebra on the $\bC$-vector space $M_{\ast}$.
\[thm:mainmod\] The $M_{\ast}$-linear map $$\begin{aligned}
\label{eqn:varphi2}
\varphi^M: M_{\ast} \otimes_{\bC} \bC\langle M_{\ast}\rangle &\rightarrow \cI^{M}\\
[f_1|\ldots|f_n] &\mapsto I(f_1,\ldots,f_n;\tau) \notag\end{aligned}$$ is an isomorphism of $M_{\ast}$-algebras.
The morphism $\varphi^M$ is surjective by definition. It is also injective, since for a basis $\cB_M$ of $M_{\ast}$, the iterated integrals $I(f_1,\ldots,f_n;\tau)$ with $f_i \in \cB_M$ are linearly independent over $M_{\ast}$ by Theorem \[thm:linind\], as $M_{\ast} \subset \operatorname{Frac}(QM_{\ast})$.
A polynomial basis for $\cI^{QM}$
---------------------------------
Recall from Proposition \[prop:qmod\].(ii) that $QM_{\ast}$ is isomorphic to the polynomial algebra $\bC[E_2,E_4,E_6]$. A similar, but slightly more involved statement holds for the $QM_{\ast}$-algebra $\cI^{QM}$ of iterated integrals of quasimodular forms. Namely, $\cI^{QM}$ is a polynomial algebra over $QM_{\ast}$ in infinitely many variables, which are given by certain Lyndon words.
In the following, if $(S,<)$ is a totally ordered set, we will endow the free monoid $S^*$ on $S$ with the lexicographical order induced by $<$. Also, the *length* of $w$ is simply the number of letters of $w$.
A *Lyndon word* on $S^*$ is a non-trivial word, $w \in S^* \setminus \{1\}$, such that for all factorizations $w=uv$ with $u,v \neq 1$, we have $w<v$. We denote by $Lyn(S^*)$ the set of all Lyndon words on $S^*$.
Let $S=\{a,b\}$ with total order $a<b$. Then the Lyndon words on $S^*$ of length at most four are $$a,b,ab,aab,abb,aaab,aabb,abbb.$$
Now for a field $k$ and any set $S$, define $k\langle S\rangle$ to be the shuffle algebra on the free $k$-vector space generated by $S$. If $k$ is of characteristic zero, then by the Milnor–Moore theorem [@MM], $k\langle S\rangle$ is isomorphic to a polynomial algebra (in possibly infinitely many variables). The following refinement is due to Radford.
\[thm:rad\] If $k$ has characteristic zero, then $k\langle S\rangle$ is freely generated, as a $k$-algebra, by the set of Lyndon words $Lyn(S^*)$. Equivalently, $k\langle S\rangle \cong k[Lyn(S^*)]$, the polynomial algebra on $Lyn(S^*)$.
Returning to quasimodular forms, consider again the $\bC$-vector space $V=\bC \cdot E_2 \oplus M_{\ast}$, and let $\cB=\cup_{k\geq 0}\cB_k$ be the homogeneous basis of $V$, given by $\cB_k=\{E_4^aE_6^b \, \vert \, 4a+6b=k \}$ for $k \neq 2$, and $\cB_2=\{E_2\}$. The basis $\cB$ can be ordered for the lexicographical order as follows: if $E_4^aE_6^b,E_4^{a'}E_6^{b'} \in \cB_k$, then $$E_4^aE_6^b < E_4^{a'}E_6^{b'} :\Leftrightarrow a<a', \mbox{ or } a=a', \mbox{ and } b< b',$$ and if $f \in \cB_k$, $g \in \cB_{k'}$ with $k<k'$, then $f<g$.
Now, since for $f_1,\ldots,f_n \in \cB$, the iterated integrals $I(f_1,\ldots,f_n;\tau)$ are linearly independent over $QM_{\ast}$ (by Theorem \[thm:linind\]), we can canonically identify the set of all $I(f_1,\ldots,f_n;\tau)$ with the free monoid $\cB^*$, and order $\cB^*$ for the lexicographical ordering induced from the order on $\cB$ above. The next result is a formal consequence of Theorems \[thm:main\], \[thm:mainmod\] and \[thm:rad\].
\[thm:polynomial\] The elements of $Lyn(\cB^*)$ are algebraically independent over $QM_{\ast}$ and we have a natural isomorphism of $QM_{\ast}$-algebras $$QM_{\ast}[Lyn(\cB^*)] \cong \cI^{QM},$$ which is filtered for the length, where the left hand side is the polynomial $QM_{\ast}$-algebra on $Lyn(\cB^*)$. Explicitly, the isomorphism maps an element $w=f_1\ldots f_n \in Lyn(\cB^*)$ to the iterated integral $I(f_1,\ldots,f_n;\tau)$. Similarly, we have a natural isomorphism of $M_{\ast}$-algebras $$M_{\ast}[Lyn(\cB^*_M)] \cong \cI^M,$$ where $\cB_M=\cB \setminus \{E_2\}$.
The following table gives all elements of $Lyn(\cB^*)$ involving iterated integrals of length at most two of quasimodular forms of total weight at most $12$. For ease of notation, we have dropped the $\tau$ from $I(f_1,\ldots,f_n;\tau)$.
0 1 2
------ --- ------------------------ ------------------------------------------------------------- -- -- -- --
$0$ — $I(1)$ —
$2$ — $I(E_2)$ —
$4$ — $I(E_4)$ $I(1,E_4)$
$6$ — $I(E_6)$ $I(1,E_6)$, $I(E_2,E_4)$
$8$ — $I(E_4^2)$ $I(1,E_4^2)$, $I(E_2,E_6)$
$10$ — $I(E_4E_6)$ $I(1,E_4E_6)$, $I(E_2,E_4^2)$, $I(E_4,E_6)$
$12$ — $I(E_4^3)$, $I(E_6^2)$ $I(1,E_4^3)$, $I(1,E_6^2)$, $I(E_2,E_4E_6)$, $I(E_4,E_4^2)$
Also, the list of all elements of $Lyn(\cB^*)$ consisting of iterated integrals of length at most three of quasimodular forms of total weight $12$ is given by $$\begin{aligned}
\{&I(E_4^3),\, I(E_6^2),\, I(1,E_4^3),\, I(1,E_6^2),\, I(E_2,E_4E_6),\, I(E_4,E_4^2), \notag\\
&I(1,1,E_4^3),\, I(1,1,E_6^2),\, I(1,E_2,E_4E_6),\, I(1,E_4,E_4^2),\, I(1,E_6,E_6),\notag\\
&I(1,E_4^2,E_4),\, I(1,E_4E_6,E_2),\, I(E_2,E_2,E_4^2),\, I(E_2,E_4,E_6),\, I(E_2,E_6,E_4) \}.\end{aligned}$$
Eichler–Shimura for quasimodular forms
======================================
In this appendix, we show how one can attach one-cocycles to quasimodular forms. This extends the classical Eichler–Shimura theory of the cocycles attached to modular forms, and is probably well-known to the experts, but the author does not know of a suitable reference for the precise statements.
Throughout this appendix, we will freely use some elementary concepts from the cohomology of groups, for which we refer to [@Weibel:Hom], Ch. 6.
Cocycles attached to modular forms
----------------------------------
We begin by briefly recalling how modular forms give rise to cocycles for $\SL_2(\bZ)$. A standard reference is [@Lang], Ch. VI.
For $d \geq 0$, let $\bQ[X,Y]_d$ be the $\bQ$-vector space of homogeneous polynomials in $X$ and $Y$ of degree $d$. It is a right $\SL_2(\bZ)$-module by defining $$P(X,Y)\vert_\gamma=P(aX+bY,cX+dY), \quad \mbox{for }\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix} \in \SL_2(\bZ), \quad P \in \bQ[X,Y]_d.$$ With this action, given a modular form $f$ of weight $k \geq 2$, it is straightforward to verify that the holomorphic differential one-form $$\underline{f}(\tau):=(2\pi i)^{k-1}f(\tau)(X-\tau Y)^{k-2} \dd\tau \in \Omega^1(\fH) \otimes_{\bQ} \bQ[X,Y]_{k-2}$$ is $\SL_2(\bZ)$-invariant, where $\SL_2(\bZ)$ acts on $\fH$ in the usual way via fractional linear transformations. Fixing a base point $\tau_0$ of $\fH$ (possibly $i\infty$), it follows from the $\SL_2(\bZ)$-invariance that the function $$r_{f,\tau_0}: \SL_2(\bZ) \rightarrow \bC[X,Y]_{k-2}, \quad \gamma \mapsto \int_{\tau}^{\tau_0}\underline{f}(\tau)-\left(\int_{\gamma.\tau}^{\tau_0}\underline{f}(\tau)\right)\bigg\vert_\gamma,$$ (regularized as in Section \[ssec:2.2\], if $\tau_0=i\infty$) is a one-cocycle, i.e. it satisfies $r_{f,\tau_0}(\gamma_1 \gamma_2)=r_{f,\tau_0}(\gamma_1)\vert_{\gamma_2}+r_{f,\tau_0}(\gamma_2)$ for all $\gamma_1,\gamma_2 \in \SL_2(\bZ)$. Its cohomology class does not depend on $\tau_0$, and we denote this class simply by $[r_f]$.
The same construction can also be applied to the complex conjugate $\overline{\underline{f}(\tau)}:=(-2\pi i)^{k-1}\overline{f(\tau)}(X-\overline{\tau}Y)^{k-2}\dd\overline{\tau}$ of the one-form $\underline{f}(\tau)$, and we denote by $[r_{\overline{f}}]$ the resulting cohomology class.
\[thm:ES\] For every $k \geq 2$, the morphism $$\begin{aligned}
M_k \oplus \overline{S}_k &\rightarrow H^1(\SL_2(\bZ),\bQ[X,Y]_{k-2}) \otimes_{\bQ}\bC,\\
(f,\overline{g}) &\mapsto [r_f]+[r_{\overline{g}}],\end{aligned}$$ is an isomorphism of $\bC$-vector spaces. Here, $\overline{S}_k$ denotes the complex conjugate of the $\bC$-vector space of cusp forms of weight $k$.
Cocycles for the braid group
----------------------------
The fact that $r_f$ is a cocycle hinges on the modularity of $f$. In order to incorporate quasimodular forms into the picture, we need to consider instead of $\SL_2(\bZ)$ the braid group $B_3=\langle \sigma_1,\sigma_2 : \sigma_1\sigma_2\sigma_1=\sigma_2\sigma_1\sigma_2 \rangle$ on three strands. It is a central extension $$\label{eqn:ses}
1 \longrightarrow \bZ \longrightarrow B_3 \longrightarrow \SL_2(\bZ) \longrightarrow 1,$$ and also the fundamental group of the quotient of $\bC^{\times} \times \fH$ by the $\SL_2(\bZ)$-action $$\gamma.(z,\tau)=((c\tau+d)z,\gamma.\tau), \quad \mbox{for }\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix} \in \SL_2(\bZ),$$ where $\SL_2(\bZ)$ acts on $\fH$ as before. We refer to [@Hain:Modulispaces], §8, for more details and further equivalent descriptions of $B_3$.
Next, we compute the cohomology groups $H^1(B_3,\bQ[X,Y]_d)$, where $B_3$ acts on $\bQ[X,Y]_d$ via the projection $B_3\rightarrow \SL_2(\bZ)$.
\[prop:ESM\] We have canonical isomorphisms $$\begin{aligned}
H^1(B_3,\bQ[X,Y]_d) &\cong \begin{cases}H^1(\SL_2(\bZ),\bQ[X,Y]_d), & \mbox{for }d \geq 1,\\
\bQ, & \mbox{for }d=0.
\end{cases}\end{aligned}$$
The Hochschild–Serre spectral sequence ([@Weibel:Hom], Ch. 6.8.3) associated to the extension yields an exact sequence $$\label{eqn:ses2}
0 \rightarrow H^1(\SL_2(\bZ),\bQ[X,Y]_d) \rightarrow H^1(B_3,\bQ[X,Y]_d) \rightarrow H^1(\bZ,\bQ[X,Y]_d)^{\SL_2(\bZ)} \rightarrow 0,$$ where we have used that $H^2(\SL_2(\bZ),\bQ[X,Y]_d)=\{0\}$, as $\SL_2(\bZ)$ has virtual cohomological dimension equal to one. The proposition now follows easily from this.
Quasimodular forms and braid group cocycles
-------------------------------------------
In light of Theorem \[thm:ES\], Proposition \[prop:ESM\] suggests to attach a one-cocycle $B_3 \rightarrow \bC$ to the Eisenstein series $E_2$. Indeed, this can be done as follows.
First, the modular transformation property of $E_2$ implies that the differential one-form $$\label{eqn:diffform}
2\pi iE_2(\tau)\dd \tau-12\frac{\dd z}{z} \in \Omega^1(\bC^{\times} \times \fH)$$ is $\SL_2(\bZ)$-invariant, i.e. it descends to the quotient $\SL_2(\bZ) \backslash (\bC^{\times} \times \fH)$. Denote by $$\underline{E_2}(\xi,\tau):=\varphi^*\left(2\pi iE_2(\tau)\dd \tau-12\frac{\dd z}{z}\right)=2\pi iE_2(\tau)\dd\tau-12\dd \xi \in \Omega^1(\bC \times \fH),$$ the pull-back of along the universal covering map $\varphi: \bC\times \fH \rightarrow \SL_2(\bZ)\backslash (\bC^{\times}\times \fH)$. Clearly, $\underline{E_2}(\xi,\tau)$ is $B_3$-invariant and it follows that for any base point $(\xi_0,\tau_0)$ (for example $(\xi_0,\tau_0)=(0,i\infty)$), the function $$\begin{aligned}
r_{E_2,(\xi_0,\tau_0)}: B_3 &\rightarrow \bC \notag\\
\gamma &\mapsto \int_{(\xi,\tau)}^{(\xi_0,\tau_0)}\underline{E_2}(\xi,\tau)-\left(\int_{\gamma.(\xi,\tau)}^{(\xi_0,\tau_0)}\underline{E_2}(\xi,\tau)\right)\bigg\vert_\gamma \label{eqn:E2cocycle}\end{aligned}$$ is a well-defined cocycle (again, regularization is needed if $\tau_0=i\infty$).
The integral $I(E_2;\tau)$ introduced in Section \[ssec:2.2\] is actually equal to $\int_{\tau}^{i\infty} \underline{E_2}(\xi,\tau)$, where we embed $\fH$ into $\bC \times \fH$ by $\tau \mapsto (0,\tau)$. However, that embedding is not $B_3$-equivariant, and indeed the integral $I(E_2;\tau)$ does not give rise to a cocycle for $B_3$; for this, one really needs to lift the form $2\pi iE_2(\tau)\dd\tau$ to the form $\underline{E_2}(\xi,\tau)$.
Now since the cocycle $r_{E_2,(\xi_0,\tau_0)}$ is non-zero, its cohomology class (which is again independent of the choice of base point $(\xi_0,\tau_0)$) is non-trivial. The Eichler–Shimura theorem (Theorem \[thm:ES\]) together with Proposition \[prop:ESM\] then implies the next result.
\[cor:ESM\] For every $k \geq 2$, the morphism $$\begin{aligned}
V_k \oplus \overline{S}_k &\rightarrow H^1(B_3,\bQ[X,Y]_{k-2}) \otimes_{\bQ}\bC,\\
(f,\overline{g}) &\mapsto [r_f]+[r_{\overline{g}}],\end{aligned}$$ where $V:=M_{\ast} \oplus \bC\cdot E_2$, is an isomorphism of $\bC$-vector spaces.
One can also attach a cocycle $r_{f,\tau_0}$ to a general quasimodular form $f \in QM_k$ of weight $k$ as follows. By Proposition \[prop:qmod\].(ii), we know that $f$ can be written uniquely as a $\bC$-linear combination of derivatives of modular forms and of derivatives of $E_2$. Thus, we can write $$f=\sum \lambda_g \cdot D^{p_g}(g), \quad \lambda_g \in \bC, \, p_g \geq 0,$$ where $g$ is either a modular form of weight $k-2p_g$ or $g=E_2$. Therefore, we may define $r_{f,\tau_0}: B_3 \rightarrow \bC[X,Y]_{\leq k-2}:=\bigoplus_{0\leq d\leq k-2}\bC[X,Y]_d$ by $$r_{f,\tau_0}:=\sum \lambda_g \cdot r_{g,\tau_0}.$$ Using this definition, one sees in particular that the cocycles of quasimodular forms can be expressed in terms of the cocycles attached to modular forms and to $E_2$. This is of course in line with Corollary \[cor:ESM\].
In [@Brown:MMV; @Hain:HodgeDeRham; @Man], certain non-abelian $\SL_2(\bZ)$-cocycles given in terms of iterated integrals of modular forms are studied. It would be natural to try and extend this theory to non-abelian $B_3$-cocycles attached to iterated integrals of quasimodular forms (perhaps along the lines suggested in [@Hain:HodgeDeRham], §14), but this is beyond the scope of the present paper.
[^1]: More precisely, Manin only defined iterated integrals of cusp forms, and the extension to all modular forms is due to Brown.
[^2]: After this paper has been submitted for publication, the author learned that, in the case of iterated integrals of modular forms, a very similar result has also been proved by Brown (cf. [@Brown:EquivIterIntEis], Proposition 4.4), using a slightly different method.
[^3]: Cf. [@123], Section 5.3.
|
---
abstract: 'In this paper we compute the signature for a family of knots $W(k,n)$, the weaving knots of type $(k,n)$. By work of E. S. Lee the signature calculation implies a vanishing theorem for the Khovanov homology of weaving knots. Specializing to knots $W(3,n)$, we develop recursion relations that enable us to compute the Jones polynomial of $W(3,n)$. Using additional results of Lee, we compute the ranks of the Khovanov Homology of these knots. At the end we provide evidence for our conjecture that, asymptotically, the ranks of Khovanov Homology of $W(3,n)$ are [*normally distributed*]{}.'
author:
- |
Rama Mishra [^1]\
Department of Mathematics\
IISER Pune\
Pune, India
- |
Ross Staffeldt [^2]\
Department of Mathematical Sciences\
New Mexico State University\
Las Cruces, NM 88003 USA
bibliography:
- 'knotinvariantslist.bib'
title: 'The Jones Polynomial and Khovanov Homology of Weaving Knots $W(3,n)$'
---
Introduction {#Introduction}
============
Weaving knots originally attracted interest, because it was conjectured that their complements would have the largest hyperbolic volume for a fixed crossing number. Here is the weaving knot $W(3,4)$.
$$\xygraph{ !{0;/r1.5pc/:}
!{\hcap}[u]
!{\hcap[3]}[u]
!{\hcap[5]}[llllllll]
!{\xcaph[-8]@(0)}[dl]
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Enumerating strands $1, \ldots, p$ from the outside inward, our example is the closure of the braid $(\sigma_1 \sigma_2^{-1})^4$ on three strands. Thus, $\sigma_1$ is a righthand twist involving strands 1 and 2, and $\sigma_2$ is a righthand twist involving strands 2 and 3, and so on.
In words, the weaving knot $W(p,q)$ is obtained from the torus knot $T(p,q)$ by making the standard diagram of the torus knot alternating. Symbolically, $T(p,q)$ is the closure of the braid $(\sigma_1 \sigma_2 \cdots \sigma_{p-1})^q$, and $W(p,q)$ is the closure of the braid $(\sigma_1 \sigma_2^{-1} \cdots \sigma_{p-1}^{\pm 1})^q$. Obviously, the parity of $p$ is important. If the greatest common divisor $\gcd(p,q) > 1$, then $T(p,q)$ and $W(p,q)$ are both links with $\gcd(p,q)$ components. In general we are interested only in the cases when $W(p,q)$ is an actual knot.
In [@Weaving_vol] the main result is the following theorem.
\[Theorem 1.1, [@Weaving_vol]\] If $p \geq 3$ and $q \geq 7$, then $$v_{{\rm oct}}(p-2)\,q\,\biggl(1 - \frac{(2\pi)^2}{q^2}\biggr)^{3/2} \leq {\rm vol}(W(p,q))
< \bigl(v_{{\rm oct}}(p-3) + 4\,v_{{\rm tet}}) q.$$
Champanerkar, Kofman, and Purcell call these bounds asymptotically sharp because their ratio approaches 1, as $p$ and $q$ tend to infinity. Since the crossing number of $W(p,q)$ is known to be $(p{-}1)q$, the volume bounds in the theorem imply $$\lim_{p,q \to \infty}\frac{{\rm vol}(W(p,q))}{c(W(p,q))} = v_{{\rm oct}} \approx 3.66$$ Their study raised the question of examining the asymptotic behavior of other invariants of weaving knots. In this paper we start a study of the asymptotic behavior of Khovanov homology of weaving knots.
Briefly, since weaving knots are alternating knots by definition, we may specialize certain properties of the Khovanov homology of alternating knots to get started. This is accomplished in section \[Weaving\], where we explain how to calculate the signature of weaving knots. The second main ingredient in our analysis is the fact that for alternating knots knowing the Jones polynomial is equivalent to knowing the Khovanov homology. How this works explicitly in our examples is explained in section \[Jones-to-Khovanov\]. In section \[Hecke\] we prepare to follow the development of the Jones polynomial in [@Jones_poly86], starting from representations of braid groups into Hecke algebras. For weaving knots $W(3,n)$, which are naturally represented as the closures of braids on three strands, we develop recursive formulas for their representations in the Hecke algebras. These formulas are used in computer calculations of the Jones polynomials we need. Section \[JonesPoly\] builds on the recursion formulas to develop information about the Jones polynomials $V_{W(3,n)}(t)$. After we explain how to obtain the two-variable Poincaré polynomial for Khovanov homology in section \[Jones-to-Khovanov\], we present the results of calculations in a few relatively small examples. Our observation is that the distributions of dimensions in Khovanov homology resemble normal distributions. We explore this further in section \[Data\], where we present tables displaying summaries of calculations for weaving knots $W(3,n)$ for selected values of $n$ satisfying $\gcd(3, n) = 1$ and ranging up to $n=326$. The standard deviation $\sigma$ of the normal distribution we attach to the Khovanov homology of a weaving knot is a significant parameter. The geometric significance of this number is an open question.
Generalities on Weaving knots {#Weaving}
=============================
We have already mentioned that weaving knots are alternating by definition. Various facts about alternating knots facilitate our calculations of the Khovanov homology of weaving knots $W(3,n)$. For example, we appeal first to the following theorem of Lee.
\[locateKHLee\] For any alternating knot $L$ the Khovanov invariants ${\mathcal H}^{i,j}(L)$ are supported in two lines $$j = 2i -\sigma(L) \pm 1. \qed$$
We will see that this result also has several practical implications. For example, to obtain a vanishing result for a particular alternating knot, it suffices to compute the signature. Indeed, it turns out that there is a combinatorial formula for the signature of oriented non-split alternating links. To state the formula requires the following terminology.
\[crossings\] For a link diagram $D$ let $c(D)$ be the number of crossings of $D$, let $x(D)$ be number of negative crossings, and let $y(D)$ be the number of positive crossings. For an oriented link diagram, let $o(D)$ be the number of components of $D(\emptyset)$, the diagram obtained by $A$-smoothing every crossing.
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\[fig:crossings\_smoothings\]
In words, $A$-regions in a neighborhood of a crossing are the regions swept out as the upper strand sweeps counter-clockwise toward the lower strand. An $A$-smoothing removes the crossing to connect these regions. With these definitions, we may cite the following proposition.
\[basic\_signature\] For an oriented non-split alternating link $L$ and a reduced alternating diagram $D$ of $L$, $\sigma(L) = o(D) - y(D) -1$.
We now use this result to compute the signatures of weaving knots. For a knot or link $W(m, n)$ drawn in the usual way, the number of crossings $c(D) = (m{-}1)n$. In particular, for $W(2k{+}1,n)$, $c\bigl(W(2k{+}1,n)\bigr)= 2kn$; for $W(2k, n)$, $c\bigl( W(2k, n) \bigr) = (2k{-}1)n$. Evaluating the other quantities in definition \[crossings\], we calculate the signatures of weaving knots.
\[weavingsignature\] For a weaving knot $W(2k{+}1,n)$, $\sigma\bigl( W(2k{+}1,n) \bigr) = 0$, and for $W(2k, n)$, $\sigma\bigl( W(2k, n) \bigr) = -n{+}1$.
Consider first the example $W(3,n)$, illustrated by figures \[fig:w34\] and \[fig:w34a\] for $W(3,4)$ drawn below. After $A$-smoothing the diagram, the outer ring of crossings produces a circle bounding the rest of the smoothed diagram. On the inner ring of crossings the $A$-smoothings produce $n$ circles in a cyclic arrangement. Therefore $o\bigl( W(3,n) \bigr) = 1 + n$. The outer ring of crossings consists of positive crossings and the inner ring of crossings consists of negative crossings, so $x(D) = y(D) = n$. Applying the formula of theorem \[basic\_signature\], we obtain the result $\sigma\bigl( W(3,n) \bigr)=0$.
For the general case $W(2k{+}1, n)$, we have the following considerations. The crossings are organized into $2k$ rings. Reading from the outside toward the center, we have first a ring of positive crossings, then a ring of negative crossings, and so on, alternating positive and negative. Thus $y(D) = kn$. Considering the $A$-smoothing of the diagram of $W(2k{+}1,n)$, as in the special case, a bounding circle appears from the smoothing of the outer ring. A chain of $n$ disjoint smaller circles appears inside the second ring. No circles appear in the third ring, nor in any odd-numbered ring thereafter. On the other hand, chains of $n$ disjoint smaller circles appear in each even-numbered ring. Since there are $k$ even-numbered rings, we have $o(D) = 1 + kn$. Applying the formula of proposition \[basic\_signature\] $$\sigma\bigl( W(2k{+}1, n)\bigr) = o(D) - y(D) -1 = (1+kn) - kn -1 = 0.$$ These figures illustrate the main points of the $W(3,n)$-cases, and, as explained above, the main points of the $W(2k{+}1, n)$-cases.
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For the case $W(2k, n)$, we show $W(4,5)$ below in figures \[fig:w45\] and \[fig:w45a\] as an example. Our standard diagram may be organized into $2k{-}1$ rings of crossings. In each ring there are $n$ crossings, so the total number of crossings is $c(D) = (2k{-}1)n$. In our standard representation, there is an outer ring of $n$ positive crossings, next a ring of $n$ negative crossings, alternating until we end with an innermost ring of $n$ positive crossings. There are thus $k$ rings of $n$ positive crossings and $k{-}1$ rings of $n$ negative crossings. Therefore, $y(D) = kn$ and $x(D) = (k{-}1)n$. Considering the $A$-smoothing of the diagram, a bounding circle appears from the smoothing of the outer ring. As before, a chain of $n$ disjoint smaller circles appears inside the second ring and in each successive even-numbered ring. As previously noted, there are $k{-}1$ of these rings. No circles appear in odd-numbered rings, until we reach the last ring, where an inner bounding circle appears. Thus, $o(D) = 1 + (k{-}1)n + 1 = (k{-}1)n + 2$. Consequently, $$\sigma\bigl( W(2k, n)\bigr) = o(D) - y(D) -1 = \bigl((k{-}1)n + 2\bigr) - kn - 1
= -n{+}1. \qedhere$$
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\[locateKH\] For a weaving knot $W(2k{+}1,n)$ the non-vanishing Khovanov homology ${\mathcal H}^{i,j}\bigl( W(2k{+}1, n) \bigr)$ lies on the lines $$j = 2i \pm 1.$$ For a weaving knot $W(2k, n)$ the non-vanishing Khovanov homology ${\mathcal H}^{i,j}\bigl( W(2k, n) \bigr)$ lies on the lines $$j = 2i + n -1 \pm 1$$
Substitute the calculations made in lemma \[weavingsignature\] into the formula of theorem \[locateKHLee\].
Recursion in the Hecke algebra {#Hecke}
==============================
We review briefly the definition of the Hecke algebra $H_{N+1}$ on generators $T_1$ through $T_N$, and we define the representation of the braid group $B_3$ on three strands in $H_3$. Theorem \[heckerecursion\] sets up recursion relations for the coefficients in the expansion of the image in $H_3$ of the braid $(\sigma_1 \sigma_2^{-1})^{n}$, whose closure is the weaving knot $W(3,n)$. The recursion relations are essential for automating the calculation of the Jones polynomial for the knots $W(3,n)$. Proposition \[C121\] uses these relations developed in theorem \[heckerecursion\] to prove a vanishing result for one of the coefficients. Being able to ignore one of the coefficients speeds up the computations slightly.
\[Heckealgebras\] Working over the ground field $K$ containing an element $q \neq 0$, the Hecke algebra $H_{N+1}$ is the associative algebra with $1$ on generators $T_1$, …, $T_N$ satisfying these relations. $$\begin{aligned}
T_iT_j &= T_jT_i, \quad \text{whenever ${\lverti-j\rvert} \geq 2$,} \label{commutativity}
\\
T_iT_{i+1}T_i &= T_{i+1}T_iT_{i+1}, \quad \text{for $1 \leq i \leq N{-}1$,} \label{interchange}
\intertext{and, finally,}
T_i^2 &= (q{-}1)T_i + q,\quad \text{for all $i$.} \label{inverse}\end{aligned}$$ It is well-known [@Jones_poly86] that $(N{+}1)!$ is the dimension of $H_{N+1}$ over $K$.
Recasting the relation $T_i^2 = (q{-}1)T_i + q$ in the form $q^{-1} \bigl(T_i - (q-1)\bigr)\cdot T_i = 1$ shows that $T_i$ is invertible in $H_{N+1}$ with $T_i^{-1} = q^{-1}\bigl( T_i - (q-1) \bigr)$. Consequently, the specification $\rho(\sigma_i) = T_i$, combined with relations and , defines a homomorphism $\rho \colon B_{N+1} {\rightarrow}H_{N+1} $ from $B_{N+1}$, the group of braids on $N{+}1$ strands, into the multiplicative monoid of $H_{N+1}$.
For work in $H_3$, choose the ordered basis $\{1, T_1, T_2, T_1T_2, T_2T_1, T_1T_2T_1\}$. The word in the Hecke algebra corresponding to the knot $W(3,n)$ is formally $$\label{eq:basicw3n}
\rho\bigl( (T_1T_2^{-1})^n\bigr)
= q^{-n}\bigl(C_{n,0}+ C_{n,1}\cdot T_1 + C_{n,2} \cdot T_2 + C_{n,12} \cdot T_1T_2
+ C_{n,21} \cdot T_2T_1 + C_{n,121} \cdot T_1T_2T_1 \bigr),$$ where the coefficients $C_{n,*} = C_{n,*}(q)$ of the monomials in $T_1$ and $T_2$ are polynomials in $q$. For $n = 1$, $$\rho ( \sigma_1 \sigma_2^{-1}) = T_1T_2^{-1} = q^{-1}\cdot\bigl( T_1 ( -(q{-}1) + T_2) \bigr)
= q^{-1}\bigl( -(q{-}1)\cdot T_1 + T_1T_2 \bigr),$$ so we have initial values $$\label{initialCs}
C_{1,0}(q) = 0, \; C_{1,1}(q) = -(q{-}1), \; C_{1,2}(q) = 0, \; C_{1,12}(q) = 1,
\; C_{1,21}(q) = 0, \; \text{and} \; C_{1,121}(q) = 0.$$
\[heckerecursion\] These polynomials satisfy the following recursion relations. $$\begin{aligned}
C_{n,0}(q) &= q^2\cdot C_{n-1,21}(q) - q(q{-}1)\cdot C_{n-1,1}(q) \label{cn0}
\\
C_{n,1}(q) &= - (q{-}1)^2\cdot C_{n-1,1}(q) - (q{-}1)\cdot C_{n-1,0}(q) + q^2\cdot C_{n-1,121}(q) \label{cn1}
\\
C_{n,2}(q) &= q\cdot C_{n-1,1}(q) \label{cn2}
\\
C_{n,12}(q) &= (q{-}1)\cdot C_{n-1,1}(q) + C_{n-1,0}(q) \label{cn12}
\\
\begin{split}
C_{n, 21}(q) &= -(q{-}1)\cdot C_{(n-1),2}(q) + q\cdot C_{n-1,12}(q)\\
& \hspace{3em} - (q{-}1)^2\cdot C_{n-1,21}(q) + q(q{-}1)\cdot C_{n-1, 121}(q) \label{cn21}
\end{split}
\\
C_{n,121}(q) &= C_{n-1,2}(q) + (q{-}1)\cdot C_{n-1,21}(q) \label{cn121}\end{aligned}$$
We have $$\begin{gathered}
\label{exp0}
\rho( T_1T_2^{-1} )^n = \rho( T_1T_2^{-1} )^{n-1} \cdot \rho (T_1T_2^{-1})
\\
= q^{-n}\bigl(C_{n-1,0}+ C_{n-1,1}\cdot T_1 + C_{n-1,2} \cdot T_2 + C_{n-1,12} \cdot T_1T_2
+ C_{n-1,21} \cdot T_2T_1 + C_{n-1,121}\cdot T_1T_2T_1 \bigr)
\\ \cdot \bigl( -(q{-}1)\cdot T_1 + T_1T_2 \bigr)
\\
= q^{-n}\biggl( -(q{-}1)C_{n-1,0}\cdot T_1 -(q{-}1)C_{n-1,1}\cdot T_1^2 -(q{-}1)C_{n-1,2} \cdot T_2T_1
\\
-(q{-}1)C_{n-1,12} \cdot T_1T_2T_1 -(q{-}1)C_{n-1,21} \cdot T_2T_1^2 -(q{-}1)C_{n-1,121}\cdot T_1T_2T_1^2
\\
+ C_{n-1,0} \cdot T_1T_2+ C_{n-1,1}\cdot T_1^2T_2 + C_{n-1,2} \cdot T_2T_1T_2
\\
+ C_{n-1,12} \cdot T_1T_2T_1T_2 + C_{n-1,21} \cdot T_2T_1^2T_2 + C_{n-1,121}\cdot T_1T_2T_1^2T_2 \biggr)
\\
= q^{-n}\biggl( \Bigl( -(q{-}1)C_{n-1,0}\cdot T_1 -(q{-}1)C_{n-1,2} \cdot T_2T_1 -(q{-}1)C_{n-1,12} \cdot T_1T_2T_1 + C_{n-1,0} \cdot T_1T_2\Bigr)
\\
+ \Bigl\{-(q{-}1)C_{n-1,1}\cdot T_1^2 -(q{-}1)C_{n-1,21} \cdot T_2T_1^2 -(q{-}1)C_{n-1,121}\cdot T_1T_2T_1^2 + C_{n-1,1}\cdot T_1^2T_2
\\
+C_{n-1,2} \cdot T_2T_1T_2 + C_{n-1,12} \cdot T_1T_2T_1T_2 + C_{n-1,21} \cdot T_2T_1^2T_2 + C_{n-1,121}\cdot T_1T_2T_1^2T_2 \Bigr\} \biggr)
\end{gathered}$$ after collecting powers of $q$ and expanding. In the last grouping, the first four terms inside the parentheses $( \ )$ involve only elements of the preferred basis; the second eight terms in the pair of braces $\{\ \}$ all require further expansion, as follows. $$\begin{aligned}
\begin{split} -(q{-}1)C_{n-1,1}\cdot T_1^2 &= -(q{-}1)C_{n-1,1}\cdot ((q-1)T_1 + q)
\\ &= -(q-1)^2C_{n-1,1}\cdot T_1 - q(q-1)C_{n-1,1} \end{split} \label{expa}
\\
\begin{split} -(q{-}1)C_{n-1,21} \cdot T_2T_1^2 &= -(q{-}1)C_{n-1,21} \cdot T_2((q-1)T_1 + q)
\\ &= -(q-1)^2C_{n-1,21}\cdot T_2T_1 - q(q-1)C_{n-1,21} \cdot T_2 \end{split} \label{expb}
\\
\begin{split} -(q{-}1)C_{n-1,121}\cdot T_1T_2T_1^2 &= -(q{-}1)C_{n-1,121} \cdot T_1T_2((q-1)T_1 + q)
\\ & = -(q-1)^2C_{n-1,121}\cdot T_1T_2T_1 - q(q-1)C_{n-1,121} \cdot T_1T_2 \end{split} \label{expc}
\\
\begin{split} C_{n-1,1}\cdot T_1^2T_2 &= C_{n-1,1}\cdot( (q{-}1) T_1 + q) T_2
\\ &= (q{-}1)C_{n-1,1}\cdot T_1T_2 + qC_{n-1,1} \cdot T_2
\end{split} \label{expd}
\\
C_{n-1,2} \cdot T_2T_1T_2 &= C_{n-1,2} \cdot T_1T_2T_1 \label{expe}
\\
\begin{split}
C_{n-1,12} \cdot T_1T_2T_1T_2 &= C_{n-1,12}\cdot T_1^2T_2T_1 = C_{n-1,12}((q{-}1)T_1 + q)T_2T_1
\\ &= (q{-}1)C_{n-1,12}\cdot T_1T_2T_1 + qC_{n-1,12}\cdot T_2T_1
\end{split} \label{expf}
\\
\begin{split}
C_{n-1,21} \cdot T_2T_1^2T_2 &= C_{n-1,21}\cdot T_2 ((q{-}1)T_1 + q) T_2 = (q{-}1)C_{n-1,21} \cdot T_2T_1T_2 + qC_{n-1,21} \cdot T_2^2
\\ &= (q{-}1)C_{n-1,21} \cdot T_1T_2T_1 + qC_{n-1,21}\cdot ((q{-}1)T_2 + q)
\\ &= (q{-}1)C_{n-1,21} \cdot T_1T_2T_1 + q(q{-}1)C_{n-1,21}\cdot T_2 + q^2C_{n-1,21})
\end{split} \label{expg}
\\
\begin{split}
C_{n-1,121}\cdot T_1T_2T_1^2T_2 &= C_{n-1,121} \cdot T_1T_2((q{-}1)T_1 + q)T_2
\\ &= (q{-}1)C_{n-1,121} \cdot T_1T_2T_1T_2 + qC_{n-1,121} \cdot T_1T_2^2
\\ &= (q{-}1)C_{n-1,121} \cdot T_1^2T_2T_1 + qC_{n-1,121}\cdot T_1((q{-}1)T_2+ q)
\\ &= (q{-}1)C_{n-1,121} \cdot ((q{-}1)T_1+ q)T_2T_1 + qC_{n-1,121}\cdot T_1((q{-}1)T_2+ q)
\\ &= (q{-}1)^2C_{n-1,121}\cdot T_1T_2T_1 + q (q{-}1)C_{n-1,121}\cdot T_2T_1
\\ & \hspace{3em} + q(q{-}1)C_{n-1,121}\cdot T_1T_2+ q^2C_{n-1,121} \cdot T_1
\end{split} \label{exph}\end{aligned}$$ Collecting the constant terms from and , we get $$\begin{aligned}
C_{n,0} &= - q(q-1)C_{n-1,1} + q^2C_{n-1,21}.
\intertext{Collecting coefficients of $T_1$ from \eqref{exp0}, \eqref{expa}, \eqref{exph}, we get}
C_{n,1} &= -(q{-}1)C_{n-1,0} -(q-1)^2C_{n-1,1} + q^2C_{n-1,121}.
\intertext{Collecting coefficients of $T_2$ from \eqref{expb}, \eqref{expd}, and \eqref{expg}, we get}
C_{n,2} &= - q(q-1)C_{n-1,21} + qC_{n-1,1} + q(q{-}1)C_{n-1,21} = qC_{n-1,1}.
\intertext{Collecting coefficients of $T_1T_2$ from \eqref{exp0}, \eqref{expc}, \eqref{expd}, and \eqref{exph}, we get}
C_{n,12} &= C_{n-1,0} - q(q-1)C_{n-1,121} + (q{-}1)C_{n-1,1} + q(q{-}1)C_{n-1,121} = C_{n-1,0} + (q{-}1)C_{n-1,1}
\intertext{Collecting coefficients of $T_2T_1$ from \eqref{exp0}, \eqref{expb}, \eqref{expf}, and \eqref{exph}, we get}
C_{n,21} &= -(q{-}1)C_{n-1,2} -(q-1)^2C_{n-1,21} + qC_{n-1,12} + q (q{-}1)C_{n-1,121}.
\intertext{Collecting coefficients of $T_1T_2T_1$ from \eqref{exp0}, \eqref{expc}, \eqref{expe}, \eqref{expf}, \eqref{expg}, and \eqref{exph},
we get}
C_{n,121} &= -(q{-}1)C_{n-1,12} -(q-1)^2C_{n-1,121} + C_{n-1,2} \\ &+ (q{-}1)C_{n-1,12} + (q{-}1)C_{n-1,21} + (q{-}1)^2C_{n-1,121}
\\
&= C_{n-1,2} + (q{-}1)C_{n-1,21}\end{aligned}$$ Up to simple rearrangements and expansion of notation, these are formulas through .
\[secondCs\] Applying the recursion formulas just proved to the table of initial polynomials, or by computing $\rho\bigl( (\sigma_1 \sigma_2^{-1})^2 \bigr)$ directly from the definitions, we find $$\begin{aligned}
C_{2,0}(q) &= q^2 \cdot C_{1,21}(q) - q(q{-}1)\cdot C_{1,1}(q) = q(q{-}1)^2,
\\
C_{2,1}(q) &= -(q{-}1)^2\cdot C_{1,1}(q)-(q{-}1)\cdot C_{1,0}(q) = (q{-}1)^3,
\\
C_{2,2}(q) &= q \cdot C_{1,1}(q) = -q(q{-}1),
\\
C_{2,12}(q) &= (q{-}1)\cdot C_{1,1}(q) + C_{1,0}(q) = -(q{-}1)^2,
\\
C_{2,21}(q) &= -(q{-}1)\cdot C_{1,2}(q) + q \cdot C_{1,12}(q) - (q{-}1)^2\cdot C_{1,21}(q) = q,
\\
C_{2,121}(q) &=0.\end{aligned}$$
As a first application, we have the following vanishing result.
\[C121\] For all $n$, $C_{n,121}(q) = 0$.
For $n \geq 1$, we claim $C_{n+1,121}(q) = 0$. Make the inductive assumption that $C_{k,121}(q) = 0$ for $1 \leq k \leq n$. Apply , , and the inductive hypothesis to write $$\begin{gathered}
C_{n+1,121}(q) = C_{n,2}(q) + (q{-}1)\cdot C_{n,21}(q)
\\
\shoveleft = C_{n,2}(q)
\\
+ (q{-}1)\Bigl( -(q{-}1)\cdot C_{n-1,2}(q) + q\cdot C_{n-1,12}(q) - (q{-}1)^2\cdot C_{n-1,21}(q)
+ q(q{-}1)\cdot C_{n-1, 121} \Bigr)
\\
= C_{n,2}(q) +
(q{-}1)\bigl( -(q{-}1)\cdot C_{n-1,2}(q) + q\cdot C_{n-1,12}(q) -
(q{-}1)^2\cdot C_{n-1,21}(q) \bigr).\end{gathered}$$ Using to replace the first term $C_{n,2}(q)$ and to replace the third term factor $C_{n-1,12}(q)$ on the right, $$\begin{aligned}
C_{n+1, 121}(q) &= q\cdot C_{n-1,1}(q) - (q{-}1)^2C_{n-1,2}(q) + q(q{-}1)\bigl( (q{-}1)C_{n-2,1}(q)+C_{n-2,0}(q) \bigr)
\\
&- (q{-}1)^3C_{n,21}(q)
\\
&= q\cdot C_{n-1,1}(q) - (q{-}1)^2C_{n-1,2}(q) + (q{-}1)^2\Bigl( q C_{n-2,1}(q) \Bigr) + q(q{-}1)C_{n-2,0}(q)
\\ &- (q{-}1)^3C_{n,21}(q)
\\
&= q\cdot C_{n-1,1}(q) - (q{-}1)^2C_{n-1,2}(q) + (q{-}1)^2C_{n-1,2}(q)+ q(q{-}1)C_{n-2,0}(q)
\\ &- (q{-}1)^3C_{n,21}(q),\end{aligned}$$ and using in reverse to rewrite the term $q C_{n-2,1}(q) $. Making the obvious cancellation, $$\begin{aligned}
C_{n+1, 121} &= q\cdot C_{n-1,1}(q)+ q(q{-}1) \cdot C_{n-2,0}(q) - (q{-}1)^3 \cdot C_{n,21}(q)
\\
&=q\bigl( C_{n-1,1}+ (q{-}1)C_{n-2,0}\bigr) - (q{-}1)^3 \cdot C_{n-1, 21}
\\
&= q\Bigl(\bigl(-(q{-}1)^2 \cdot C_{n-2,1} - (q{-}1) \cdot C_{n-2,0}\bigr) + (q{-}1) \cdot C_{n-2,0}\biggr) - (q{-}1)^3 \cdot C_{n-1, 21},\end{aligned}$$ since $$\begin{aligned}
C_{n-1,1}(q) &= - (q{-}1)^2\cdot C_{n-2,1}(q) - (q{-}1)\cdot C_{n-2,0}(q) + q^2\cdot C_{n-2,121}(q)
\\
& = - (q{-}1)^2\cdot C_{n-2,1}(q) - (q{-}1)\cdot C_{n-2,0}(q)\end{aligned}$$ by and the inductive hypothesis. Therefore, $$\begin{aligned}
C_{n+1,121}(q) &= -q(q{-}1)^2\cdot C_{n-2,1}(q) - (q{-}1)^3\cdot C_{n-1, 21}(q)
\\
&= -(q{-}1)^2\cdot C_{n-1,2}(q) - (q{-}1)^3\cdot C_{n-1,21}(q),
\intertext{using \eqref{cn2} in the form $ C_{n-1,2}(q) = q\cdot C_{n-2,1}(q) $,}
&= -(q{-}1)^2\bigl( C_{n-1, 2}(q) - (q{-}1)\cdot C_{n-1,21} (q)\bigr)
\\
&= -(q{-}1)^2\cdot C_{n, 121}(q) = 0,\end{aligned}$$ using and the inductive hypothesis.
Obtaining the Jones Polynomial {#JonesPoly}
==============================
Following the construction given in [@Jones_poly86 p.288] we work over the function field $K = {{\mathbf C}}(q,z)$, and we put $w=1{-}q{+}z$. Let $H_{N+1}$ be the Hecke algebra over $K$ corresponding to $q$ with $N$ generators as in definition \[Heckealgebras\]. The starting point is the following theorem.
\[traces\] For $N \geq 1$ there is a family of trace functions $\operatorname{Tr}\colon H_{N+1} {\rightarrow}K$ compatible with the inclusions $H_N {\rightarrow}H_{N+1}$ satisfying
1. $\operatorname{Tr}(1) = 1$,
2. $\operatorname{Tr}$ is $K$-linear and $ \operatorname{Tr}(ab) = \operatorname{Tr}(ba)$,
3. If $a, b \in H_N$, then $\operatorname{Tr}(aT_Nb) = z\operatorname{Tr}(ab)$.
Property 3 enables the calculation of $\operatorname{Tr}$ on basis elements of $H_{N+1}$ through use of the defining relations and induction. For $H_3$, note that $$\operatorname{Tr}(T_1) = \operatorname{Tr}(T_2) = z, \quad \operatorname{Tr}(T_1T_2) = \operatorname{Tr}(T_2T_1) = z^2, \quad \operatorname{Tr}(T_1T_2T_1) = z \operatorname{Tr}(T_1^2) = z \bigl((q{-}1)z + q\bigr).$$ The next step toward the Jones polynomial of the knot that is the closure of the braid $\alpha \in B_{N+1}$ is given by the formula $$V_{\alpha}(q,z) = \Bigl(\frac{1}{z}\Bigr)^{(N + e(\alpha))/2}
\cdot \Bigl( \frac{q}{w} \Bigr)^{(N-e(\alpha))/2}\cdot \operatorname{Tr}\bigl(\rho(\alpha)\bigr),$$ where $e(\alpha)$ is the exponent sum of the word $\alpha$. The expression defines an element in the quadratic extension $K(\sqrt{q/zw})$. For the weaving knot $W(3,n)$, viewed as the closure of $(\sigma_1\sigma_2^{-1})^n$, we have the exponent sum $e=0$, and $N=2$, and $$\rho\bigl( (\sigma_1\sigma_2^{-1})^n \bigr)= (T_1T_2^{-1})^n
= q^{-n}\bigl( C_{n,0}(q) + C_{n,1}(q)\cdot T_1 + C_{n,2}(q) \cdot T_2 + C_{n,12}(q) \cdot T_1T_2 + C_{n,21}(q) \cdot T_2T_1\bigr) ,$$ thanks to proposition \[C121\], which says the expression for $(T_1T_2^{-1})^n$ requires only the use of the basis elements $1$, $T_1$, $T_2$, $T_1T_2$ and $T_2T_1$. Then we have $$\begin{gathered}
V_{(\sigma_1\sigma_2^{-1})^n}(q,z) \\ = \Bigl(\frac{1}{z}\Bigr)\cdot \Bigl( \frac{q}{w} \Bigr)\cdot q^{-n}
\operatorname{Tr}\bigl(C_{n,0}(q) + C_{n,1}(q)\cdot T_1 + C_{n,2}(q) \cdot T_2 + C_{n,12}(q) \cdot T_1T_2
+ C_{n,21}(q) \cdot T_2T_1 \bigr)
\\
= \Bigl(\frac{q}{zw}\Bigr)\cdot q^{-n} \cdot \bigl( C_{n,0}(q) + C_{n,1}(q) \cdot z + C_{n,2}(q) \cdot z + C_{n,12}(q) \cdot z^2
+ C_{n,21} (q) \cdot z^2 \bigr),\end{gathered}$$ using the facts that $\operatorname{Tr}T_1 = \operatorname{Tr}T_2 = z$ and $\operatorname{Tr}T_1T_2 = \operatorname{Tr}T_2T_1 = z^2$. Consequently, the sums $ C_{n,1} + C_{n,2} $ and $C_{n,12} + C_{n,21}$ are essential for understanding the two-variable Jones polynomial of $W(3,n)$, the closure of $\alpha=(\sigma_1\sigma_2^{-1})^n $. Making the substitutions $$q = t, \quad z = \frac{t^2}{1+t}, \quad w = \frac{1}{1+t}$$ leads to the one-variable Jones polynomial $$\begin{gathered}
V_{W(3,n)}(t) = \frac{t(1{+}t)^2}{t^2} \cdot t^{-n} \cdot
\Bigl( C_{n,0}(t) + (C_{n,1}(t)+C_{n,2}(t))\cdot \frac{t^2}{1{+}t} + (C_{n,12}(t) + C_{n,21}(t)) \cdot \frac{t^4}{(1{+}t)^2}\Bigr)
\\
= t^{-n-1}\cdot\bigl( (1{+}t)^2\cdot C_{n,0}(t) + (1{+}t)\cdot( C_{n,1}(t) + C_{n,2}(t) )\cdot t^2 + (C_{n,12}(t) + C_{n,21}(t))\cdot t^4 \bigr).\end{gathered}$$
For $W(3,1)$, which is the unknot, we have $$\begin{aligned}
V_{W(3,1)}(t)
&= t^{-2}\cdot\bigl( (1{+}t)^2\cdot C_{1,0}(t) + (1{+}t)\cdot( C_{1,1}(t) + C_{1,2}(t) )\cdot t^2 + (C_{1,12}(t) + C_{1,21}(t))\cdot t^4 \bigr)
\\
&= t^{-2}\cdot\bigl( (1{+}t)^2\cdot 0 + (1{+}t)\cdot(-(t-1) + 0 )\cdot t^2 + (1 + 0 )\cdot t^4 \bigr)
\\
&= t^{-2}\cdot ( (1{-}t^2) t^2 + t^4 ) = 1.
\end{aligned}$$
\[jonesfig8knot\] For $W(3,2)$, which is the figure-8 knot, we have $$\begin{aligned}
V_{W(3,2)}(t)
&=t^{-3}\cdot \bigl( (1{+}t)^2\cdot C_{2,0}(t) + (1{+}t)\cdot( C_{2,1}(t) + C_{2,2}(t) )\cdot t^2
+ (C_{2,12}(t) + C_{2,21}(t))\cdot t^4 \bigr)
\\
&=t^{-3}\cdot \bigl( (1{+}t)^2\cdot t(t{-}1)^2 +(1{+}t)\cdot( (t{-}1)^3 - t(t{-}1) ) \cdot t^2
+( -(t{-}1)^2+ t ) \cdot t^4 \bigr)
\\
&= t^{-3}\cdot \bigl( t^5 - t^4 + t^3 -t^2 + t \bigr) = t^2 - t + 1 -t^{-1} + t^{-2}
\end{aligned}$$
Now we take a closer look at the formal expression $$\begin{gathered}
V_{W(3,n)}(t) =
\\
= t^{-n-1}\cdot\bigl( (1{+}t)^2\cdot C_{n,0}(t) + (1{+}t)\cdot( C_{n,1}(t) + C_{n,2}(t) )\cdot t^2 + (C_{n,12}(t) + C_{n,21}(t))\cdot t^4 \bigr)\end{gathered}$$ for the Jones polynomial of the weaving knot $W(3,n)$.
\[degrees1\] We have a uniform bound on the degrees of the polynomials $C_{n,*}$. Namely, $$\deg(C_{n,*}) \leq 2n-1,$$ and the sharper bounds $$\deg(C_{n,2}) \leq 2n{-}2, \quad \deg(C_{n,12}) \leq 2n{-}2, \quad \text{and} \quad \deg(C_{n,21}) \leq 2n{-}3.$$
These are easy arguments by induction, using either the formulas or the formulas in example \[secondCs\] to start the inductions. Use the recursion formulas through along with the fact that $C_{n, 121}= 0$, proved in proposition \[C121\], for the inductive step. We have $$\begin{aligned}
\begin{split}
\deg(C_{n,0}) &\leq \max\{ \deg(C_{n-1,21})+ 2, \; \deg(C_{n-1,1}) + 2\}
\\
&\leq \max\{(2n{-}5)+2, (2n{-}3)+2 \} = 2n{-}1;
\end{split}
\\
\begin{split}
\deg(C_{n,1}) &\leq \max\{ \deg(C_{n-1,1}) + 2, \; \deg(C_{n-1,0}) + 1\}
\\
&\leq \max\{(2n{-}3)+2, (2n{-}3)+1\} = 2n-1;
\end{split}
\\
\deg(C_{n,2}) &= \deg(C_{n-1,1}) + 1 \leq (2n{-}3) + 1 = 2n-2;
\\
\begin{split}
\deg(C_{n,12}) &\leq \max\{ \deg(C_{n-1,1}) + 1, \deg(C_{n-1,0}) \}
\\
&\leq \max \{ (2n{-}3)+1, 2n-3 \} = 2n-2;
\end{split}
\\
\begin{split}
\deg(C_{n,21}) &\leq \max\{ \deg(C_{n-1,2}) + 1, \deg( C_{n-1,12}) +1, \deg(C_{n-1,21}) + 2 \}
\\
&\leq \max\{ (2n{-}4) + 1, (2n{-}4) + 1, (2n{-}5)+ 2\} = 2n{-}3.
\end{split}
\end{aligned}$$
Accordingly, set $$\begin{aligned}
C_{n,0}(q) &= \sum_{i=0}^{2n-1} c_{n,0,i}q^i, &
C_{n,1}(q) &= \sum_{i=0}^{2n-1} c_{n,1,i}q^i, &
C_{n,2}(q) &= \sum_{i=0}^{2n-2} c_{n,2,i}q^i,
\\
C_{n,12}(q) &= \sum_{i=0}^{2n-2} c_{n,12,i}q^i,
& &\text{and} &
C_{n,21}(q) &= \sum_{i=0}^{2n-3} c_{n,21,i}q^i.\end{aligned}$$
\[Cn0coefficients\] In the polynomial $C_{n,0}(q)$, the constant term $c_{n,0,0} = 0$ for all $n \geq 1$, and the degree one coefficient $c_{n,0,1}=(-1)^{n-2}$ for $n \geq 2$.
The first polynomial $C_{1,0}(q) = 0$, and setting $q=0$ in the recurrence relation immediately yields $$c_{n,0,0} = C_{n,0}(0) = 0.$$ Differentiate the recursion relation with respect to $q$, obtaining $$C_{n,0}'(q) = \bigl( 2q\cdot C_{n-1,21}(q)+ q^2\cdot C_{n-1,21}'(q) \bigr)
- \bigl( (2q{-}1)\cdot C_{n-1,1}(q)+ q(q{-}1) \cdot C_{n-1,1}'(q)\bigr) .$$ Substituting $q=0$ yields immediately $c_{n,0,1} = C_{n,0}'(0) = C_{n-1,1}(0) = c_{n-1,1,0}$. We have $$C_{n,1}(q) = - (q{-}1)^2\cdot C_{n-1,1}(q) - (q{-}1)\cdot C_{n-1,0}(q),$$ simplifying relation using proposition \[C121\] to set $C_{n-1,121}(q)=0$. Now we prove $c_{n,1,0} = (-1)^{n-1}$ for all $n \geq 1$. We have $C_{1,1}(q) = -(q-1)$, so $c_{1,1,0} = 1$ as claimed. Substituting $q=0$ and using $c_{n,0,0}=0$ we get $$c_{n,1,0} = C_{n,1}(0) = -(-1)^2\cdot C_{n-1,1}(0) - (-1) \cdot C_{n-1,0}(0) = -c_{n-1,1,0} + 0 = -(-1)^{n-2}=(-1)^{n-1}$$ Thus $c_{n,0,1} = c_{n-1,1,0} = (-1)^{n-2}$, as claimed.
Thus, we improve the expression for $C_{n,0}(q)$ slightly, obtaining $C_{n,0}(q) = \sum_{i=1}^{2n-1} c_{n,0,i}q^i$. Now we examine $$\begin{aligned}
V_{W(3,n)}(t)
&= t^{-n-1}\cdot \bigl( (1{+}t)^2\cdot C_{n,0}(t) + (t^2{+}t^3)\cdot( C_{n,1}(t) + C_{n,2}(t) ) + t^4 \cdot (C_{n,12}(t) + C_{n,21}(t)) \bigr)
\\
\begin{split}
&= t^{-n-1}\cdot \Biggl( (1{+}t)^2\cdot \biggl(\sum_{i=1}^{2n-1} c_{n,0,i}t^i\biggr)
\\
& \hspace{0.15\linewidth} + (t^2{+}t^3)\cdot \Bigl( \sum_{i=0}^{2n-1} c_{n,1,i}t^i + \sum_{i=0}^{2n-2} c_{n,2,i}t^i \Bigr)
\\
& \hspace{0.30\linewidth} + t^4 \cdot \biggl(\sum_{i=0}^{2n-2} c_{n,12,i}t^i + \sum_{i=0}^{2n-3} c_{n,21,i}t^i\biggr) \Biggr).
\end{split}\end{aligned}$$ In the expression for $V_{W(3,n)}(t)$ the highest degree term is apparently $$t^{-n-1}\cdot\bigl( c_{n,1,2n-1} \cdot t^{2n+2} + c_{n,12,2n-2} \cdot t^{2n+2} \bigr),$$ but we claim this term is actually zero.
Using $\deg(C_{n-1,1}) \leq 2n-3$, $\deg(C_{n-1,0}) \leq 2n-3$, the fact that $C_{n, 121}=0$, and simplifying the recursion formulas and , respectively, to $$C_{n,1}(q) = - (q{-}1)^2\cdot C_{n-1,1}(q) - (q{-}1)\cdot C_{n-1,0}(q)
\; \text{and} \;
C_{n,12}(q) = (q{-}1)\cdot C_{n-1,1}(q) + C_{n-1,0}(q),$$ we compute $$c_{n,1,2n-1} + c_{n,12,2n-2} = -c_{n-1,1,2n-3} + c_{n-1,1,2n-3} = 0. \qedhere$$
So the degree of the highest term in $V_{W(3,n)}$ is no more than the degree of $t^{-n-1}\cdot t^{2n+1}$ which is $n$. Lemma \[Cn0coefficients\] shows that the term of lowest degree is $\pm t^{-n-1} \cdot t = \pm t^{-n}$, so the span of the Jones polynomial $V_{W(3,n)}$ is no more than $2n$. Of course, Kauffman [@States Theorem 2.10] has proved that the span of the polynomial is, in fact, precisely $2n$.
From the Jones Polynomial to Khovanov homology {#Jones-to-Khovanov}
==============================================
In this section we amplify Theorem \[locateKH\], at least the first part of it.
For a weaving knot $W(2k{+}1,n)$ the non-vanishing Khovanov homology ${\mathcal H}^{i,j}\bigl( W(2k{+}1, n) \bigr)$ lies on the lines $$j = 2i \pm 1.$$ For a weaving knot $W(2k, n)$ the non-vanishing Khovanov homology ${\mathcal H}^{i,j}\bigl( W(2k, n) \bigr)$ lies on the lines $$j = 2i + n -1 \pm 1$$
We have the following definition of the bi-graded Euler characteristic associated to Khovanov homology. $$Kh(L)(t,Q) \stackrel{{\rm def}}{=} \sum t^iQ^j \dim {\mathcal H}^{i,j}(L)$$
\[Lee1\] For an oriented link $L$, the graded Euler characteristic $$\sum_{i,j \in {{\mathbf Z}}} (-1)^iQ^j \dim {\mathcal H}^{i,j}(L)$$ of the Khovanov invariant ${\mathcal H}(L)$ is equal to $(Q^{-1}{+}Q)$ times the Jones polynomial $V_L(Q^2)$ of $L$.
In terms of the associated polynomial $Kh(L)$, $$\label{JonesfromKhovanov}
Kh(L)(-1, Q) = (Q^{-1}+Q)V_L(Q^2).$$
\[Lee4\] For an alternating knot $L$, its Khovanov invariants ${\mathcal H}^{i,j}(L)$ of degree difference $(1,4)$ are paired except in the $0$th cohomology group.
This fact may be expressed in terms of the polynomial $Kh(L)$, as follows. There is another polynomial $Kh'(L)$ in one variable and an equality $$\label{polynomialperiodicity}
Kh(L)(t, Q) = Q^{-\sigma(L)}\bigl\{ (Q^{-1}{+}Q) + (Q^{-1}+tQ^2\cdot Q)\cdot Kh'(L)(tQ^2) \bigr\}$$ When we combine theorems \[Lee1\] and \[Lee4\], we find that the bi-graded Euler characteristic and the Jones polynomial of an alternating link determine one another. Obviously, the equality shows that one knows $V_L$ if one knows $Kh(t,Q)$.
To obtain $Kh(t,Q)$ from $V_L(Q^2)$ requires a certain amount of manipulation. Implementing these manipulations in [*Maple*]{} and [*Mathematica*]{} is an important step in our experiments. Setting $t{=}-1$ in and combining with equation , one has $$\begin{aligned}
(Q^{-1} + Q) \cdot V_L(Q^2) &= Q^{-\sigma(L)}\bigl\{ (Q^{-1}{+}Q) + (Q^{-1}- Q^3)\cdot Kh'(L)(-Q^2) \bigr\}.
\intertext{Consequently,}
V_L(Q^2) &= Q^{-\sigma(L)}\bigl\{ 1 + \frac{(Q^{-1}- Q^3)}{(Q^{-1}{+}Q)}\cdot Kh'(L)(-Q^2) \bigr\}
\\
&= Q^{-\sigma(L)}\bigl\{ 1 + (1 - Q^2)\cdot Kh'(L)(-Q^2)\bigr\}.
\intertext{Furthermore,}
Q^{\sigma(L)} \cdot V_L(Q^2) -1 &= (1 - Q^2)\cdot Kh'(L)(-Q^2),
\intertext{or}
Kh'(L)(-Q^2) &= (1 - Q^2)^{-1}\cdot \bigl(Q^{\sigma(L)} \cdot V_L(Q^2) -1\bigr) .\end{aligned}$$ Replacing $Q^2$ in the last equation by $-tQ^2$ is the last step to obtain $Kh'(L)$ from the Jones polynomial. Within a computer algebra system, one must first replace $Q^2$ by $-X$ and then replace $X$ by $tQ^2$. Once one has $Kh'(L)(tQ^2)$, one obtains $Kh(t,Q)$ directly from equation .
We have computed $V_{W(3,2)}(t) = t^{-2} - t^{-1} + 1 - t + t^2$ in example \[jonesfig8knot\], so $$\begin{aligned}
Kh'\bigl(W(3,2)\bigr)(-Q^2) &= (1- Q^2)^{-1} \cdot \bigl( Q^0 \cdot ( Q^{-4} - Q^{-2} - Q^2 + Q^4) \bigr)
\\
&= (1-Q^2)^{-1} \cdot \bigl( (1 - Q^2) \cdot ( Q^{-4} - Q^2) \bigr)
\\
&= Q^{-4} - Q^2.
\end{aligned}$$ It follows that $Kh'\bigl( W(3,2) \bigr) (tQ^2) = t^{-2}Q^{-4} + tQ^2$, and $$\begin{gathered}
Kh\bigl(W(3,2)\bigr) (t, Q) = (Q+Q^{-1})+ (Q^{-1} + tQ^3)(t^{-2}Q^{-4} + tQ^2)
\\
= t^{-2}Q^{-5} + t^{-1}Q^{-1} + Q^{-1} + Q + tQ + t^2Q^5.\end{gathered}$$
Khovanov homology examples {#Khovanov}
==========================
Once one has the Khovanov polynomial one can make a plot of the Khovanov homology in an $(i,j)$-plane as in this example. The Betti number $\dim KH^{i,j}\bigl( W(3,11) \bigr)$ is plotted at the point with coordinates $(i,j)$.
![Khovanov homology of $W(3,11)$[]{data-label="fig:w311"}](KHW3_11.eps){width="4in"}
Clearly, as $n$ gets larger, it is going to be harder to make sense of such plots. Notice that the $(1,4)$-periodicity of the Khovanov homology for these knots makes the information on one of the lines $j-2i = \pm 1$ redundant.
Taking advantage of this feature, we simplify by recording the Betti numbers along the line $j- 2i = 1$. In order to study the asymptotic behavior of Khovanov homology we have to normalize the data. This is done by computing the total rank of the Khovanov homology along the line and dividing each Betti number by the total rank. We obtain normalized Betti numbers that sum to one.
This raises the possibility of approximating the distribution of normalized Betti numbers by a probability distribution. For our baseline experiments we choose to use the normal $N(\mu, \sigma^2)$ probability density function $$f_{\mu, \sigma^2}(x) = \frac{1}{\sigma \sqrt{2\pi}} \exp \Bigl( - \frac{(x-\mu)^2}{2\sigma^2} \Bigr)$$ Fit a quadratic function $q_n(x)= -(\alpha \, x^2 - \beta\, x + \delta)$ to the logarithms of the normalized Khovanov dimensions along the line $j=2i+1$ and exponentiate the quadratic function. Since the total of the normalized dimensions is 1, we normalize the exponential, obtaining $$\rho_n(x) = A_n e^{q_n(x)} \quad \text{satisfying} \quad \int_{-\infty}^{\infty} \rho_n(x) \; dx = 1.$$ To obtain a formula for $A_n$, complete the square $$q_n(x) = -\alpha \cdot \bigl( x - (\beta/2\alpha) \bigr)^2 +\bigl( (\beta^2/4\alpha) - \delta\bigr).$$ Then consider $$\begin{aligned}
1 &= A_n \int_{-\infty}^{\infty} \exp q_n(x) \; dx
\\
&= A_n \cdot
\int_{-\infty}^{\infty} \exp \bigl((\beta^2/4\alpha) - \delta \bigr)
\cdot \exp \bigl( -\alpha \cdot \bigl( x -(\beta/2\alpha) \bigr)^2\bigr) \; dx
\\
&= A_n \cdot \bigl((\beta^2/(4\alpha) - \delta \bigr) \cdot
\int_{-\infty}^{\infty} \exp \bigl( -\alpha \cdot \bigl( x -(\beta/2\alpha) \bigr)^2\bigr) \; dx
\\
&= A_n \cdot \bigl((\beta^2/4\alpha) - \delta\bigr) \cdot \sqrt{\pi/\alpha}\end{aligned}$$ Thus, the expression for $A_n$ is $$A_n = \exp -\bigl((\beta^2/4\alpha) - \delta\bigr)\cdot \sqrt{\alpha/\pi}.$$
Equating the expressions $$\rho_n(x) = \frac{1}{\sigma_n \sqrt{2\pi}} \exp \Bigl( - \frac{(x-\mu_n)^2}{2\sigma_n^2} \Bigr)
\quad \text{and} \quad
\rho_n(x) = A_n \exp ( q_n(x))$$ $\mu_n = \beta/2\alpha$ and the efficient way to the parameter $\sigma_n$ is by solving the equation $$\frac{1}{\sigma_n \sqrt{2\pi}} = \rho_n( \frac{\beta}{2\alpha} ) = A_n \exp( q_n(\beta/2\alpha))
= \exp -\bigl( \frac{\beta^2}{4\alpha} - \delta\bigr)\cdot \sqrt{\frac{\alpha}{\pi}}
\exp\bigl( (\beta^2/4\alpha) - \delta \bigr)$$ obtaining $\sigma_n = 1/ \sqrt{2\alpha} $.
Working this out for $W(3,10)$, and carrying only 3 decimal places, the raw dimensions are
-------- ----- ----- ----- ----- ----- ----- ----- ----- ----- -----
$i$ -9 -8 -7 -6 -5 -4 -3 -2 -1 0
$\dim$ 1 9 36 94 196 346 529 721 879 970
$i$ 1 2 3 4 5 6 7 8 9 10
$\dim$ 971 879 721 529 346 196 94 36 9 1
-------- ----- ----- ----- ----- ----- ----- ----- ----- ----- -----
and, to three significant digits, the logarithms of the normalized dimensions are
----- ------- ------- ------- ------- ------- ------- ------- ------- ------- -------
$i$ -9 -8 -7 -6 -5 -4 -3 -2 -1 0
-17.9 -15.7 -14.3 -13.3 -12.6 -12.0 -11.6 -11.3 -11.1 -11.0
$i$ 1 2 3 4 5 6 7 8 9 10
-11.0 -11.1 -11.3 -11.6 -12.0 -12.6 -13.3 -14.3 -15.7 -17.9
----- ------- ------- ------- ------- ------- ------- ------- ------- ------- -------
Fitting a quadratic to this information, we get $$q_{10}(x) = -10.7 + 0.0720\, x - 0.0720\, x^2,\quad \alpha = \beta = 0.0720, \quad \delta = 10.7.$$ To three significant digits $\mu_{10} = 0.500$ and $\sigma_{10} = 2.64$.
By the symmetry of Khovanov homology, the mean $\mu_n$ approaches $1/2$ rapidly, so this parameter is of little interest. On the other hand, relating the parameter $\sigma_n$ to some geometric quantity, say, some hyperbolic invariant of the complement of the link, is a very interesting problem.
For $W(3,10)$, the density function is $$\rho_{10}(x) = 11686.8431618280538\,\sqrt {{\pi }^{-1}}
{{\rm e}^{- 10.7018780565714309+ 0.0716848579220777243\,x- 0.0716848579220778631
\,{x}^{2}}}$$ When placed into standard form, $\mu_{10} =0.5000054030 $ and $\sigma_{10} = 2.640882970 $. Here is the comparison plot.
![normalized homology of $W(3,10)$ compared with density function[]{data-label="fig:w310comp"}](khw3_10_11apr17.eps){height="2.0in"}
For the knot $W(3,11)$ the expression for the density function is $$\rho_{11}(x) =29676.8676257830375\,\sqrt {{\pi }^{-1}}{{\rm e}^{-
11.6724860231789886+ 0.0661625395821569817\,x- 0.0661623073574252735
\,{x}^{2}}}$$ When placed into standard form, $\mu_{11} =0.5000017550 $ and $\sigma_{11} = 2.749031276$. Here is the comparison plot.
![normalized homology of $W(3,11)$ compared with density function[]{data-label="fig:w311comp"}](khw3_11_11apr17.eps){height="2.0in"}
For $W(3,22)$, the density function is $$\rho_{22}(x) =833596689.149608016\,\sqrt {{\pi }^{-1}}{{\rm e}^{-
22.2219365040983057+ 0.0353061029354434300\,x- 0.0353061029347388616
\,{x}^{2}}}$$ When placed into standard form, $\mu_{22} =0.500000000 $ and $\sigma_{22} = 3.763224354$. Here is the comparison plot.
![normalized homology of $W(3,22)$ compared with density function[]{data-label="fig:w322comp"}](khw3_22_11apr17.eps){height="2.0in"}
For $W(3,23)$, the density function is $$\begin{gathered}
\rho_{23}(x) = 2113964949.23002362\,\sqrt {{\pi }^{-1}} \\
\cdot {{\rm e}^{-
23.1731352596503442+ 0.0338545815354610105\,x- 0.0338545815348441914
\,{x}^{2}}}\end{gathered}$$ When placed into standard form, $\mu_{23} =0.5000000000 $ and $\sigma_{23} =3.843052143 $. Here is the comparison plot.
![normalized homology of $W(3,23)$ compared with density function[]{data-label="fig:w323comp"}](khw3_23_11apr17.eps){height="2.0in"}
worksheets and, later, [*Mathematica*]{} notebooks will be available at URL prepared by the second-named author.
Data Tables {#Data}
===========
This section contains tables of data generated using [*Maple*]{} to implement some of the results of earlier sections. The first table collects data for weaving knots $W(3,n)$ for $n \equiv 1 \mod 3$. The first column lists the dimension; the second column lists the total dimension of the Khovanov homology lying along the line $j = 2i{+}1$; and the third column lists the dimension of the vector space ${\mathcal H}^{0,1}\bigl( W(3,n)\bigr)$. In section \[Khovanov\] we approximate the distribution of normalized Khovanov dimenstions by a standard normal distribution, and we have displayed graphics comparing the actual distribution with the approximation.
To quantify those visual impressions, we compute two total deviations. Let $$\text{Total dimension} = \sum_{i=-2n}^{2n+1} \dim {\mathcal H}^{i, 2i+1}\bigl( W(3,n) \bigr).$$ For the $L^2$-comparison, we compute $$\Biggl( \sum_{i = -2n}^{2n+1} \biggl( \rho_n(i) - \frac{\dim {\mathcal H}^{i, 2i+1}\bigl( W(3,n) \bigr)}{\text{Total dimension}} \biggr)^2 \Biggr)^{1/2}$$ For the $L^1$-comparison, we compute $$\sum_{i = -2n}^{2n+1} {\Bigl\lvert \rho_n(i) - \frac{\dim {\mathcal H}^{i, 2i+1}\bigl( W(3,n) \bigr)}{\text{Total dimension} } \Bigr\rvert}$$ The $L^2$ comparisons appear to tend to 0, whereas the $L^1$ comparisons appear to be growing slowly.
----- -------------------------- ---------------------------- ---------- ------------------ ------------------
$n$ Total dimension $\dim {\mathcal H}^{0,1}$ $\sigma$ $L^2$-comparison $L^1$ comparison
10 7563 970 2.64088 0.040509 0.134828
13 135721 15418 2.95616 0.0411329 0.150599
16 2435423 250828 3.24564 0.040792 0.155995
19 43701901 4146351 3.51395 0.040145 0.161336
22 784198803 69337015 3.76322 0.039413 0.165763
25 14071876561 1169613435 3.99810 0.038678 0.167576
28 252509579303 19864129051 4.22032 .037971 0.167790
31 4531100550901 339205938364 4.43167 0.0373026 0.170736
34 81307300336923 5818326037345 4.63358 0.0366758 0.172391
37 1459000305513721 100173472277125 4.82718 0.036089 0.173119
40 26180698198910063 1730135731194046 5.01342 0.035541 0.173178
43 469793567274867421 29963026081609060 5.19305 0.035027 0.173811
46 8430103512748703523 520131503664409798 5.36671 0.034547 0.175059
49 $1.51272\cdot 10^{20}$ $ 9.04765\cdot10^{18}$ 5.53502 0.0340935 0.175779
52 $ 2.71447\cdot 10^{21}$ $ 1.57670\cdot 10^{20} $ 5.69838 0.033667 0.176100
55 $ 4.87091\cdot 10^{22} $ $ 2.75210\cdot 10^{21} $ 5.85721 0.033265 0.176098
58 $ 8.74050\cdot 10^{23} $ $ 4.81071\cdot 10^{22} $ 6.01187 0.032885 0.175898
61 $ 1.56842\cdot 10^{25} $ $ 8.42017\cdot 10^{23}$ 6.16267 0.032524 0.176778
64 $ 2.81441\cdot 10^{26} $ $ 1.47552\cdot 10^{25} $ 6.30989 0.032182 0.177369
67 $5.05026\cdot 10^{27}$ $ 2.58843\cdot 10^{26} $ 6.45376 0.031857 0.177716
70 $ 9.06233\cdot 10^{28} $ $4.54520\cdot 10^{27}$ 6.59451 0.031547 0.177859
73 $1.62617\cdot 10^{30}$ $7.98842\cdot 10^{28}$ 6.73233 0.031251 0.177831
76 $ 2.91804\cdot 10^{31}$ $1.40517 \cdot 10^{30}$ 6.86740 0.030968 0.177657
79 $5.23621\cdot 10^{32}$ $ 2.47359 \cdot 10^{31}$ 6.99986 0.030697 0.177995
82 $9.39600\cdot 10^{33} $ $4.35747 \cdot 10^{32}$ 7.12988 0.030437 0.178445
85 $1.68604\cdot 10^{35} $ $7.68116 \cdot 10^{33}$ 7.25757 0.030188 0.178746
88 $ 3.02548\cdot 10^{36} $ $1.35483 \cdot 10^{35}$ 7.38305 0.029948 0.178918
91 $ 5.42901\cdot 10^{37} $ $ 2.39106 \cdot 10^{36} $ 7.50645 0.029718 0.178976
94 $ 9.74196\cdot 10^{38}$ $ 4.22211 \cdot 10^{37} $ 7.62786 0.029496 0.178935
97 $ 1.74812\cdot 10^{40} $ $7.45910 \cdot 10^{38}$ 7.74736 0.029282 0.178807
100 $3.13688\cdot 10^{41} $ $1.31840 \cdot 10^{40} $ 7.86506 0.029075 0.178890
121 $1.87923\cdot 10^{50} $ $ 7.18477\cdot 10^{48} $ 8.64424 0.027805 0.179577
142 $1.12580\cdot 10^{59} $ $ 3.97500 \cdot 10^{57} $ 9.35886 0.026769 0.180247
163 $6.74436\cdot 10^{67} $ $2.22337 \cdot 10^{66}$ 10.0227 0.025900 0.180596
184 $4.04037\cdot 10^{76}$ $ 1.25398 \cdot 10^{75} $ 10.6453 0.025156 0.180629
205 $ 2.42049\cdot 10^{85}$ $7.11854 \cdot 10^{83} $ 11.2334 0.024508 0.180907
247 $8.68689\cdot 10^{102}$ $2.32816 \cdot 10^{101} $ 12.3258 0.023423 0.181027
289 $3.11764\cdot 10^{120} $ $7.72623 \cdot 10^{118} $ 13.3289 0.022542 0.181268
----- -------------------------- ---------------------------- ---------- ------------------ ------------------
: Data for $W(3,n)$ with $n \equiv 1 \mod 3$
----- --------------------------- --------------------------- ---------- ------------------ ------------------
$n$ Total dimension $\dim {\mathcal H}^{0,1}$ $\sigma$ $L^2$-comparison $L^1$ comparison
11 19801 2431 2.74903 0.040906 0.141925
14 355323 38983 3.05533 0.041079 0.153170
17 6376021 637993 3.33710 0.040595 0.156595
20 114413063 10591254 3.59850 0.039905 0.163190
23 2053059121 177671734 3.84305 0.039166 0.166596
26 36840651123 3004390818 4.07348 0.038438 0.167789
29 661078661101 51124396786 4.29190 0.037744 0.168941
32 11862575248703 874400336044 4.49997 0.037089 0.171411
35 212865275815561 15018149469823 4.69899 0.036476 0.172723
38 3819712389431403 258853011125599 4.89004 0.035903 0.173203
41 68541957733949701 4474997964407374 5.07400 0.035366 0.173083
44 1229935526821663223 77563025486587315 5.25158 0.034864 0.174290
47 22070297525055988321 1347390412214087833 5.42341 0.034392 0.175346
50 $ 3.96035\cdot 10^{20} $ $ 2.34525 5.59000 0.033949 0.175926
\cdot 10^{19} $
53 $ 7.10657\cdot 10^{21} $ $ 4.08927 5.75181 0.033531 0.176131
\cdot 10^{20} $
56 $ 1.27522\cdot 10^{23} $ $ 7.14133 5.90921 0.033136 0.176037
\cdot 10^{21} $
59 $ 2.28829\cdot 10^{24} $ $ 1.24888 6.06255 0.032763 0.176227
\cdot 10^{23} $
62 $ 4.10617\cdot 10^{25} $ $ 2.18679 6.21213 0.032408 0.177005
\cdot 10^{24} $
65 $ 7.36823\cdot 10^{26} $ $ 3.83347 6.35821 0.032072 0.177510
\cdot 10^{25} $
68 $ 1.32218\cdot 10^{28} $ $ 6.72713 6.50102 0.031752 0.177785
\cdot 10^{26} $
71 $ 2.37255\cdot 10^{29} $ $ 1.18163 6.64077 0.031446 0.177867
\cdot 10^{28} $
74 $ 4.25736\cdot 10^{30} $ $ 2.07736 6.77765 0.031155 0.177787
\cdot 10^{29} $
77 $ 7.63953\cdot 10^{31} $ $ 3.65504 6.91183 0.030876 0.177602
\cdot 10^{30} $
80 $ 1.37086\cdot 10^{33} $ $ 6.43571 7.04347 0.030609 0.178163
\cdot 10^{31} $
83 $ 2.45990\cdot 10^{34} $ $ 1.13397 7.17269 0.030353 0.178561
\cdot 10^{33} $
86 $ 4.41412\cdot 10^{35} $ $ 1.99933 7.29963 0.030107 0.178817
\cdot 10^{34} $
89 $ 7.92082\cdot 10^{36} $ $ 3.52717 7.42441 0.029871 0.178949
\cdot 10^{35} $
92 $ 1.42133\cdot 10^{38} $ $ 6.22605 7.54714 0.029643 0.178972
\cdot 10^{36} $
95 $ 2.55048\cdot 10^{39} $ $ 1.09958 7.66790 0.029424 0.178901
\cdot 10^{38} $
98 $ 4.57665\cdot 10^{40} $ $ 1.94290 7.78679 0.029212 0.178747
\cdot 10^{39} $
119 $ 2.74175\cdot 10^{49} $ $ 1.05696 8.57308 0.027914 0.179650
\cdot 10^{48} $
140 $ 1.64251\cdot 10^{58} $ $ 5.84051 9.29316 0.026859 0.180257
\cdot 10^{56} $
161 $ 9.83989\cdot 10^{66} $ $ 3.26385 9.96138 0.025977 0.180552
\cdot 10^{65} $
182 $ 5.89483\cdot 10^{75} $ $ 1.83951 10.5875 0.025223 0.180539
\cdot 10^{74} $
203 $ 3.53144\cdot 10^{84} $ $ 1.04367 11.1787 0.024566 0.180926
\cdot 10^{83} $
245 $ 1.26740\cdot 10^{102} $ $ 3.41053 12.2759 0.023469 0.181064
\cdot 10^{100} $
287 $ 4.54858\cdot 10^{119} $ $ 1.13115 13.2829 0.022580 0.181221
\cdot 10^{118} $
329 $ 1.63244\cdot 10^{137} $ $ 3.79224 14.2187 0.021838 0.181399
\cdot 10^{135} $
----- --------------------------- --------------------------- ---------- ------------------ ------------------
: Data for $W(3,n)$ with $n \equiv 2 \mod 3$
[^1]: We thank the office of the Dean of the College of Arts and Sciences, New Mexico State University, for arranging a visiting appointment.
[^2]: We thank the office of the Vice-President for Research, New Mexico State University, for arranging a visit to IISER-Pune to discuss implementation of an MOU between IISER-Pune and NMSU.
|
---
abstract: 'The use of dynamical symmetries or spectrum generating algebras for the solution of the nuclear many-body problem is reviewed. General notions of symmetry and dynamical symmetry in quantum mechanics are introduced and illustrated with simple examples such as the SO(4) symmetry of the hydrogen atom and the isospin symmetry in nuclei. Two nuclear models, the shell model and the interacting boson model, are reviewed with particular emphasis on their use of group-theoretical techniques.'
author:
- 'P. Van Isacker'
title: Symmetries in Nuclei
---
Introduction {#s_intro}
============
In the [*Oxford Dictionary of Current English*]{} symmetry is defined as the ‘right correspondence of parts; quality of harmony or balance (in size, design etc.) between parts’. The word is derived from Greek where it has the meaning ‘with proportion’ or ‘with order’. In modern theories of physics it has acquired a more precise meaning but the general idea of seeking to order physical phenomena still remains. Confronted with the bewildering complexity exhibited by the multitude of physical systems, physicists attempt to extract some simple regularities from observations, and the fact that they can do so is largely due to the presence of symmetries in the laws of physics. Although one can never hope to explain all observational complexities entirely on the basis of symmetry arguments alone, these are nevertheless instrumental in establishing correlations between and (hidden) regularities in the data.
The mathematical theory of symmetry is called group theory and its origin dates back to the nineteenth century. Of course, the notion of symmetry is present implicitly in many mathematical studies that predate the birth of group theory and goes back even to the ancient Greeks, in particular Euclid. It was, however, Évariste Galois who perceived the importance of the group of permutations to answer the question whether the roots of a polynomial equation can be algebraically represented or not. (A readable summary of the solution of this problem is given in the first chapter of Gilmore’s book [@Gilmore08].) In the process of solving that long-standing mathematical problem he invented group theory as well as Galois theory which studies the relation between polynomials and groups. The mathematical theory of groups developed further throughout the nineteenth century and made another leap forward in 1873 when Sophus Lie proposed the concept of a Lie group and its associated Lie algebra.
For a long time it was assumed that group theory was a branch of mathematics without any application in the physical sciences. This state of affairs changed with the advent of quantum mechanics, and it became clear that group theory provides a powerful tool to understand the structure of quantum systems from a unified perspective. After the introduction of symmetry transformations in abstract spaces (associated, for example, with isospin, flavor, color, etc.) the role of group theory became even central.
The purpose of these lecture notes is to introduce, explain and illustrate the concepts of symmetry and dynamical symmetry. In Sect. \[s\_qm\] a brief reminder is given of the central role of symmetry in quantum mechanics and of its relation with invariance and degeneracy. There exist two standard examples to illustrate the idea that symmetry implies degeneracy and [*vice versa*]{}, namely the hydrogen atom and the harmonic oscillator. In Sect. \[s\_hydrogen\] the first of them is analyzed in detail. Section \[s\_dsym\] describes the process of symmetry breaking and, in particular, dynamical symmetry breaking in the sense as it is used in these lecture notes. This mechanism is illustrated in Sect. \[s\_isospin\] with a detailed example, namely isospin and its breaking in nuclei. Sections \[s\_shell\] and \[s\_ibm\] then present the nuclear shell model and the interacting boson model, respectively, with a special emphasis on the symmetry techniques that have been used in the context of these models. Finally, in Sect. \[s\_conc\], a summary of these lecture notes is given.
Symmetry in quantum mechanics {#s_qm}
=============================
The starting point of any discussion of symmetry is that the laws of physics should be invariant with respect to certain transformations of the reference frame, such as a translation or rotation, or a different choice of the origin of the time coordinate. This observation leads to three fundamental conservation laws: conservation of linear momentum, angular momentum and energy. In some cases an additional space-inversion symmetry applies, yielding another conserved quantity, namely parity. In a relativistic framework the above transformations on space and time cannot be considered separately but become intertwined. The laws of nature are then invariant under the Lorentz transformations which operate in four-dimensional space–time.
These transformations and their associated invariances can be called ‘geometric’ in the sense that they are defined in space–time. In quantum mechanics, an important extension of these concepts is obtained by also considering transformations that act in abstract spaces associated with intrinsic variables such as spin, isospin (in atomic nuclei), flavor and color (of quarks) etc. It is precisely these ‘intrinsic’ invariances which have lead to the preponderance of symmetry applications in the quantum physics.
To be more explicit, consider a transformation acting on a physical system, that is, an operation that transforms the coordinates $\bar r_i$ and the momenta $\bar p_i$ of the particles that constitute the system. Such transformations are of a geometric nature. For a discussion of symmetry in quantum-mechanical systems this definition is too restrictive and the appropriate generalization is to consider, instead of the geometric transformations themselves, the corresponding transformations in the Hilbert space of quantum-mechanical states of the system. The action of the geometric transformation on spin variables ([*i.e.*]{}, components of the spin vector) is assumed to be identical to its action on the components of the angular momentum vector $\bar\ell=\bar r\wedge\bar p$. Furthermore, it can be shown [@Blaizot97] that a correspondence exists between the geometric transformations in physical space and the transformations induced by it in the Hilbert space of quantum-mechanical states. This correspondence, however, is not necessarily one-to-one; that is only the case if the system is ‘bosonic’ (consists of any number of integer-spin bosons and/or an even number of half-integer-spin fermions). If the system is ‘fermionic’ (contains an odd number of fermions), the correspondence is two-to-one and the groups, formed by the geometric transformations and by the corresponding transformations in the Hilbert space of quantum-mechanical states, are not isomorphic but rather homomorphic.
No distinction is made in the following between geometric and quantum-mechanical transformations; all elements $g_i$ will be taken as operators acting on the Hilbert space of quantum-mechanical states.
Symmetry {#ss_sym}
--------
A time-independent Hamiltonian $H$ which commutes with the generators $g_k$ that form a Lie algebra G, $$\forall g_k\in{\rm G}:
[H,g_k]=0,
\label{e_sym1}$$ is said to have a symmetry G or, alternatively, to be invariant under G. The determination of operators $g_k$ that leave invariant the Hamiltonian of a given physical system is central to any quantum-mechanical description. The reasons for this are profound and can be understood from the correspondence between geometrical and quantum-mechanical transformations. It can be shown [@Blaizot97] that the transformations $g_k$ with the symmetry property (\[e\_sym1\]) are induced by geometrical transformations that leave unchanged the corresponding classical Hamiltonian. In this way the classical notion of a conserved quantity is transcribed in quantum mechanics in the form of the symmetry property (\[e\_sym1\]) of the time-independent Hamiltonian.
Degeneracy and state labeling {#ss_deg}
-----------------------------
A well-known consequence of a symmetry is the occurrence of degeneracies in the eigenspectrum of $H$. Given an eigenstate $|\gamma\rangle$ of $H$ with energy $E$, the condition (\[e\_sym1\]) implies that the states $g_k|\gamma\rangle$ all have the same energy, $$H g_k|\gamma\rangle=
g_kH|\gamma\rangle=
Eg_k|\gamma\rangle.
\label{e_deg0}$$ An arbitrary eigenstate of $H$ shall be written as $|\Gamma\gamma\rangle$, where the first quantum number $\Gamma$ is different for states with different energies and the second quantum number $\gamma$ is needed to label degenerate eigenstates. The eigenvalues of a Hamiltonian that satisfies (\[e\_sym1\]) depend on $\Gamma$ only, $$H|\Gamma\gamma\rangle=
E(\Gamma)|\Gamma\gamma\rangle,
\label{e_deg1}$$ and, furthermore, the transformations $g_k$ do not admix states with different $\Gamma$, $$g_k|\Gamma\gamma\rangle=
\sum_{\gamma'}a^\Gamma_{\gamma'\gamma}(k)
|\Gamma\gamma'\rangle.
\label{e_deg2}$$
This simple discussion of the consequences of a Hamiltonian symmetry illustrates the relevance of group theory in quantum mechanics. Symmetry implies degeneracy and eigenstates that are degenerate in energy provide a Hilbert space in which irreducible representations of the symmetry group are constructed. Consequently, the irreducible representations of a given group directly determine the degeneracy structure of a Hamiltonian with the symmetry associated to that group.
Eigenstates of $H$ can be denoted as $|\Gamma\gamma\rangle$ where the symbol $\Gamma$ labels the irreducible representations of ${\rm G}$. Note that the same irreducible representation might occur more than once in the eigenspectrum of $H$ and, therefore, an additional multiplicity label $\eta$ should be introduced to define a complete labeling of eigenstates as $|\eta\Gamma\gamma\rangle$. This label shall be omitted in the subsequent discussion.
A sufficient condition for a Hamiltonian to have the symmetry property (\[e\_sym1\]) is that it is a Casimir operator which by definition commutes with all generators of the algebra. The eigenequation (\[e\_deg1\]) then becomes $$C_m[{\rm G}]
|\Gamma\gamma\rangle=
E_m(\Gamma)
|\Gamma\gamma\rangle.
\label{e_deg3}$$ In fact, all results remain valid if the Hamiltonian is an analytic function of Casimir operators of various orders. The energy eigenvalues $E_m(\Gamma)$ are functions of the labels that specify the irreducible representation $\Gamma$, and are known for all classical Lie algebras [@Wybourne74].
These concepts can be illustrated with the example of the hydrogen atom which is discussed in detail in the next section.
The hydrogen atom {#s_hydrogen}
=================
The Hamiltonian for a particle of charge $-e$ and mass $m_{\rm e}$ in a Coulomb potential $e/r$ is given by $$H_{\rm H}={\frac{p^2}{2m_{\rm e}}}-{\frac{e^2}{r}}=
{-\frac{\hbar^2}{2m_{\rm e}}}\nabla^2-{\frac{e^2}{r}}.
\label{e_hydham}$$ This is taken here as a model Hamiltonian for the hydrogen atom. The Hamiltonian is independent of the spin of the electron which leads to a two-fold degeneracy of all states corresponding to spin-up and spin-down. Electron spin is ignored in the following and the symmetry properties of the spatial part only of the electron wave function are studied.
The solutions of the associated Schrödinger equation, $H_{\rm H}\tilde\phi(\bar r)=E\tilde\phi(\bar r)$, are well known from standard quantum mechanics. The energies of the stationary states are $$E(n)=-{\frac{m_{\rm e}e^4}{2\hbar^2n^2}}\equiv
-{\frac{R_{\rm H}}{n^2}},
\label{e_hyener}$$ where $R_{\rm H}$ is the Rydberg constant and $n$ the so-called principal quantum number. The electron wave functions are $$\tilde\phi_{n\ell m_\ell}(r,\theta,\varphi)=
\tilde R_{n\ell}(r)
Y_{\ell m_\ell}(\theta,\varphi),
\label{e_hywave}$$ with $\tilde R_{n\ell}(r)$ and $Y_{\ell m_\ell}(\theta,\varphi)$ known functions[^1]. The $Y_{\ell m_\ell}(\theta,\varphi)$ are spherical harmonics which occur for any central potential with spherical symmetry. The $\tilde R_{n\ell}(r)$ are radial wave functions whose exact form is not of concern here. The solution of the differential equation $H_{\rm H}\tilde\phi(\bar r)=E\tilde\phi(\bar r)$ also leads to the conditions $$n=1,2,\dots,
\qquad
\ell=0,1,\dots,n-1,
\qquad
m_\ell=-\ell,-\ell+1,\dots,+\ell.$$ The energy spectrum of the hydrogen atom is shown in Fig. \[f\_hydspec\].
![The energy spectrum of the hydrogen atom.[]{data-label="f_hydspec"}](f_hydspec.eps){width="11.5cm"}
The energy eigenvalues $E(n)$ only depend on $n$ and not on $\ell$ or $m_\ell$. A given level with energy $E(n)$ is thus $n^2$-fold degenerate since $$\sum_{\ell=0}^{n-1}(2\ell+1)=n^2.$$ The nature of this degeneracy will be explained using symmetry arguments and, in addition, it will be shown that the entire spectrum can be determined with algebraic methods without recourse to boundary conditions of differential equations.
The Hamiltonian of the hydrogen atom is rotationally \[or SO(3)\] invariant. This is obvious on intuitive grounds since the properties of the hydrogen atom do not change under rotation. Formally, it follows from the following commutation property: $$[H_{\rm H},L_\mu]=0,$$ where $L_\mu$ are the components of the angular momentum operator[^2], $\bar L=(\bar r\wedge\bar p)=-i\hbar(\bar r\wedge\bar\nabla)$. It is of interest to look more closely at the origin of the vanishing commutator between $H_{\rm H}$ and $L_\mu$. The Hamiltonian of the hydrogen atom consists of two parts, kinetic and potential, and [*both*]{} commute with $L_\mu$ since $$%{-\frac{\hbar^2}{2m_{\rm e}}}
[\nabla^2,L_\mu]=0,
\qquad
%-e^2
[r^{-1},L_\mu]=0,$$ where use is made of commutation relations like $$[\bar\nabla,r^k]=k\,r^{k-2}\bar r,
\quad
[\nabla^2,\bar r]=2\bar\nabla.$$ Since the components $L_\mu$ form an SO(3) algebra, $$[L_\mu,L_\nu]=i\hbar\sum_{\rho=1}^3\epsilon_{\mu\nu\rho}L_\rho,$$ and since $L_\mu$ commutes with $H_{\rm H}$, one concludes that the Hamiltonian of the hydrogen atom has an SO(3) symmetry. This explains part of the observed degeneracy, namely, levels with a given $\ell$ are $(2\ell+1)$-fold degenerate.
To understand the origin of the [*complete*]{} degeneracy of the hydrogen spectrum, it is instructive to consider first the Kepler problem of the motion of a single planet around the sun which is the classical analogue of the hydrogen atom. Besides angular momentum, there is another conserved quantity because there is no precession of the planetary orbit, that is, the major axis of its elliptic trajectory is fixed. In contrast to the conservation of angular momentum which is valid for [*all*]{} central potentials, the absence of precession is a specific property of the Newtonian $1/r$ potential. The associated conserved quantity is known from classical mechanics, $$\bar R_{\rm cl}=
{\frac{\bar p\wedge\bar L}{m_{\rm e}}}-
e^2{\frac{\bar r}{r}}.$$ This vector is known as the Runge–Lenz (or also Lenz–Pauli) vector and its three components are conserved for a $1/r$ potential, that is, not only its direction (along the major axis of the orbit) but also its magnitude is conserved (see Fig. \[f\_kepler\]).
![The angular momentum vector $\bar L$ and the Runge–Lenz vector $\bar R$ in the classical Kepler problem of a planet orbiting the sun.[]{data-label="f_kepler"}](f_kepler.eps){width="6.5cm"}
The latter property follows from the relation $$\bar R_{\rm cl}^2=e^4+{\frac{2E}{m_{\rm e}}}\bar L^2,
\label{e_rule}$$ which shows that $\bar R_{\rm cl}^2$ can be expressed in terms of the energy and the angular momentum, both of which are conserved.
The construction of the quantum-mechanical equivalent of the Runge–Lenz vector is done in the usual way and yields $$\bar R'=
-{\frac{\hbar^2}{2m_{\rm e}}}
[\bar\nabla\wedge(\bar r\wedge\bar\nabla)-
(\bar r\wedge\bar\nabla)\wedge\bar\nabla]-
e^2{\frac{\bar r}{r}}.$$ The relation (\[e\_rule\]) between the energy and the moduli of the angular momentum and Runge–Lenz vectors converts to $$\bar R'^2=e^4+{\frac{2H_{\rm H}}{m_{\rm e}}}\left(\bar L^2+\hbar^2\right).
\label{e_hydrel1}$$ From the classical analysis one expects $R'_\mu$ to commute with $H_{\rm H}$, $$[H_{\rm H},R'_\mu]=0,$$ which is indeed confirmed through explicit calculation. Unlike in the case of the angular momentum, however, it is only the entire Hamiltonian which commutes with the Runge–Lenz vector, and [*not*]{} the kinetic and potential parts separately since $${-\frac{\hbar^2}{2m_{\rm e}}}[\nabla^2,R'_\mu]=
e^2[r^{-1},R'_\mu]=
{\frac{\hbar^2e^2}{m_{\rm e}}}
\left[
{\frac{1}{r}}\nabla_\mu-
{\frac{r_\mu}{r^3}}(1+\bar r\cdot\bar\nabla)\right]\neq0.$$ Just as in the classical Kepler problem with its exceptional precessionless orbits, one finds that the commutator with the Runge–Lenz vector vanishes for a $1/r$ potential but not in general.
It is now established that both vectors $\bar L$ and $\bar R'$ commute with the Hamiltonian of the hydrogen atom and hence are constants of motion, but the symmetry of the system still needs to be determined. This can be done from the commutation relations among $L_\mu$ and $R'_\mu$ which read $$[L_\mu,R'_\nu]=i\hbar\sum_{\rho=1}^3\epsilon_{\mu\nu\rho}R'_\rho,
\qquad
[R'_\mu,R'_\nu]=i\hbar\sum_{\rho=1}^3\epsilon_{\mu\nu\rho}
{\frac{-2H_{\rm H}}{m_{\rm e}}}L_\rho,$$ together with the SO(3) relations among $L_\mu$. Since the commutation relations among $R'_\mu$ do [*not*]{} give back $L_\rho$, one cannot claim that $L_\mu$ and $R'_\mu$ form a Lie algebra. In the space of eigenvectors corresponding to a single, negative eigenvalue, the following alternative operators can be introduced: $$R_\mu=\sqrt{\frac{m_{\rm e}}{-2H_{\rm H}}}R'_\mu.$$ In general, the square-root of an operator has problematic properties but not in this case since it acts in a space of constant eigenvalue. Note also that one may rely here on the fact that neither $L_\mu$ nor $R_\mu$ or $R'_\mu$ can connect to states with a different energy eigenvalue, since they all commute with $H_{\rm H}$. The commutation relations among $L_\mu$ and $R_\mu$ now close, $$[L_\mu,R_\nu]=i\hbar\sum_{\rho=1}^3\epsilon_{\mu\nu\rho}R_\rho,
\qquad
[R_\mu,R_\nu]=i\hbar\sum_{\rho=1}^3\epsilon_{\mu\nu\rho}L_\rho,$$ and the algebra consisting of $L_\mu$ and $R_\mu$ can be identified with SO(4), associated with the group of rotations in four dimensions. The relation (\[e\_hydrel1\]) between the Hamiltonian and the conserved quantities $\bar L^2$ and $\bar R^2$ can be rewritten as $$H_{\rm H}=
-{\frac{\hbar^2R_{\rm H}}{\bar L^2+\bar R^2+\hbar^2}}.
\label{e_hydrel2}$$ The operator occurring at the right-hand side of this identity, $\bar L^2+\bar R^2$, can be identified with $C_2[{\rm SO}(4)]$, the quadratic Casimir operator of SO(4). The hydrogen atom provides thus a simple example in which a Hamiltonian can be written in terms of the Casimir operator of its symmetry algebra.
In general, if the symmetry group of a Hamiltonian is determined, its degeneracy structure follows automatically from the irreducible representations which can be looked up in monographs on group theory. In the case of SO(4) the analysis can be worked out with simple methods by converting to the operators $$P_\mu={\frac 1 2}\left(L_\mu+R_\mu\right),
\qquad
Q_\mu={\frac 1 2}\left(L_\mu-R_\mu\right),$$ in terms of which the commutation relations become $$[P_\mu,P_\nu]=i\hbar\sum_{\rho=1}^3\epsilon_{\mu\nu\rho}P_\rho,
\quad
[Q_\mu,Q_\nu]=i\hbar\sum_{\rho=1}^3\epsilon_{\mu\nu\rho}Q_\rho,
\quad
[P_\mu,Q_\nu]=0.$$ The components $P_\mu$ commute with $Q_\nu$ and, furthermore, each set separately forms an SO(3) algebra. This, in fact, proves the isomorphism ${\rm SO}(4)\simeq{\rm SO}(3)\otimes{\rm SO}(3)$. Instead of relying on SO(4) representation theory, one can therefore use well-known results from SO(3). Since the operators $P^2$, $P_z$, $Q^2$ and $Q_z$ commute with each other, and since they all commute with $H_{\rm H}$, they form a (complete) set of commuting operators. The eigenstates of $H_{\rm H}$ can then be labeled with the eigenvalues of the operators in this set and, in particular, with $p(p+1)\hbar^2$ and $q(q+1)\hbar^2$, the eigenvalues of the operators $\bar P^2$ and $\bar Q^2$. The allowed values of the labels $p$ and $q$ are those of angular momentum, integer or half-integer, and for each value of $p$ ($q$) there are $2p+1$ ($2q+1$) allowed substates. Furthermore, eigenstates of $H_{\rm H}$ necessarily have $p=q$ because the angular momentum and the Runge–Lenz vectors are orthogonal, $\bar L\cdot\bar R=0$, which implies $$\left(\bar L+\bar R\right)^2=
\left(\bar L-\bar R\right)^2
\Rightarrow
\bar P^2=\bar Q^2
\Rightarrow
p(p+1)=q(q+1).$$ The allowed energy eigenvalues are now immediately obtained from (\[e\_hydrel2\]) since the operator $\bar L^2+\bar R^2=2\bar P^2+2\bar Q^2$ has the eigenvalue $4p(p+1)\hbar^2$, $$E(p)=-{\frac{\hbar^2R_{\rm H}}{4p(p+1)\hbar^2+\hbar^2}}=
-{\frac{R_{\rm H}}{(2p+1)^2}},
\qquad
p=0,{\textstyle{\frac 1 2}},1,\dots.$$ This coincides with the result (\[e\_hyener\]) obtained from the standard quantum-mechanical derivation.
The hydrogen atom provides a beautiful application of symmetry. The degeneracies observed in the energy spectrum are higher than what is obtained from just rotational invariance. This requires the existence of a larger symmetry which is indeed found to be the case. Another illustration of this principle is provided by the spectrum of the harmonic oscillator in which case the underlying symmetry turns out to be U(3) [@Moshinsky69].
A final comment concerns the method followed here to determine the eigenspectrum of the hydrogen atom. The standard way to do so is to solve the time-independent Schrödinger equation and to find the allowed values of the various quantum numbers from boundary conditions on the eigenfunctions. The procedure followed here is entirely different and exclusively based on the knowledge of a set of constants of motion which commute with the Hamiltonian, together with their mutual commutation relations. A crucial feature is that the Hamiltonian can be expressed in terms of the Casimir operator of the symmetry algebra. Although elegant and compact, the method itself does not provide an expression for the wave functions of stationary states. This ‘algebraic’ solution method of the problem of the hydrogen atom was proposed by Pauli in 1926 [@Pauli26].
Dynamical symmetry breaking {#s_dsym}
===========================
The concept of a dynamical symmetry for which (at least) two algebras ${\rm G}_1$ and ${\rm G}_2$ with ${\rm G}_1\supset{\rm G}_2$ are needed can now be introduced. The eigenstates of a Hamiltonian $H$ with symmetry ${\rm G}_1$ are labeled as $|\Gamma_1\gamma_1\rangle$. But, since ${\rm G}_1\supset{\rm G}_2$, a Hamiltonian with ${\rm G}_1$ symmetry necessarily must also have a symmetry ${\rm G}_2$ and, consequently, its eigenstates can also be labeled as $|\Gamma_2\gamma_2\rangle$. Combination of the two properties leads to the eigenequation $$H|\Gamma_1\eta_{12}\Gamma_2\gamma_2\rangle=
E(\Gamma_1)|\Gamma_1\eta_{12}\Gamma_2\gamma_2\rangle,
\label{e_dsym1}$$ where the role of $\gamma_1$ is played by $\eta_{12}\Gamma_2\gamma_2$. The irreducible representation $\Gamma_2$ may occur more than once in $\Gamma_1$, and hence an additional quantum number $\eta_{12}$ is needed to uniquely label the states. Because of ${\rm G}_1$ symmetry, eigenvalues of $H$ depend on $\Gamma_1$ only.
In many examples in physics (several are discussed below), the condition of ${\rm G}_1$ symmetry is too strong and a [*possible*]{} breaking of the ${\rm G}_1$ symmetry can be imposed via the Hamiltonian $$H'=
\kappa_1C_{m_1}[{\rm G}_1]+
\kappa_2C_{m_2}[{\rm G}_2],$$ which consists of a combination of Casimir operators of ${\rm G}_1$ [*and*]{} ${\rm G}_2$. The symmetry properties of the Hamiltonian $H'$ are now as follows. Since $[H',g_k]=0$ for all $g_k$ in ${\rm G}_2$, $H'$ is invariant under ${\rm G}_2$. The Hamiltonian $H'$, since it contains $C_{m_2}[{\rm G}_2]$, does not commute, in general, with all elements of ${\rm G}_1$ and for this reason the ${\rm G}_1$ symmetry is broken. Nevertheless, because $H'$ is a combination of Casimir operators of ${\rm G}_1$ and ${\rm G}_2$, its eigenvalues can be obtained in closed form, $$%\left(
%\kappa_1C_{m_1}[{\rm G}_1]+
%\kappa_2C_{m_2}[{\rm G}_2]\right)
H'
|\Gamma_1\eta_{12}\Gamma_2\gamma_2\rangle=
\left[
\kappa_1E_{m_1}(\Gamma_1)+
\kappa_2E_{m_2}(\Gamma_2)\right]
|\Gamma_1\eta_{12}\Gamma_2\gamma_2\rangle.$$ The conclusion is thus that, although $H'$ is not invariant under ${\rm G}_1$, its eigenstates are the same as those of $H$ in (\[e\_dsym1\]). The Hamiltonian $H'$ is said to have ${\rm G}_1$ as a dynamical symmetry. The essential feature is that, although the eigenvalues of $H'$ depend on $\Gamma_1$ [*and*]{} $\Gamma_2$ (and hence ${\rm G}_1$ is not a symmetry), the eigenstates do not change during the breaking of the ${\rm G}_1$ symmetry. As the generators of ${\rm G}_2$ are a subset of those of ${\rm G}_1$, the dynamical symmetry breaking splits but does not admix the eigenstates. A convenient way of summarizing the symmetry character of $H'$ and the ensuing classification of its eigenstates is as follows: $$\begin{array}{ccc}
{\rm G}_1&\supset&{\rm G}_2\\
\downarrow&&\downarrow\\
\Gamma_1&&\eta_{12}\Gamma_2
\end{array}.$$ This equation indicates the larger algebra ${\rm G}_1$ (sometimes referred to as the dynamical algebra or spectrum generating algebra) and the symmetry algebra ${\rm G}_2$, together with their associated labels with possible multiplicities.
Many concrete examples exist in physics of the abstract idea of dynamical symmetry. Perhaps the best known in nuclear physics concerns isospin symmetry and its breaking by the Coulomb interaction which is discussed in the next section.
Isospin symmetry {#s_isospin}
================
The starting point in the discussion of isospin symmetry is the observation that the masses of the neutron and proton are very similar, $m_{\rm n}c^2=939.55$ MeV and $m_{\rm p}c^2=938.26$ MeV, and that both have a spin of ${\frac 1 2}$. Furthermore, experiment shows that, if one neglects the contribution of the electromagnetic interaction, the forces between two neutrons are about the same as those between two protons. More precisely, the strong nuclear force between two nucleons with anti-parallel spins is found to be (approximately) independent of whether they are neutrons or protons. This indicates the existence of a symmetry of the strong interaction, and isospin is the appropriate formalism to explore the consequences of that symmetry in nuclei. The equality of the masses and the spins of the nucleons is not sufficient for isospin symmetry to be valid and the charge independence of the nuclear force is equally important. This point was emphasized by Wigner [@Wigner37] who defined isospin for complex nuclei as we know it today and who also coined the name of ‘isotopic spin’.
Because of the near-equality of the masses and of the interactions between nucleons, the Hamiltonian of the nucleus is (approximately) invariant with respect to transformations between neutron and proton states. For one nucleon, these can be defined by introducing the abstract space spanned by the two vectors $$|{\rm n}\rangle=
\left[\begin{array}{c}
1\\0
\end{array}\right],
\qquad
|{\rm p}\rangle=
\left[\begin{array}{c}
0\\1
\end{array}\right].$$ The most general transformation among these states (which conserves their normalization) is a unitary $2\times2$ matrix. A matrix close to the identity can be represented as $$\left[\begin{array}{ccc}
1+\epsilon_{11}&&\epsilon_{12}\\
\epsilon_{21}&&1+\epsilon_{22}
\end{array}\right],$$ where the $\epsilon_{ij}$ are infinitesimal complex numbers. Unitarity imposes the relations $$\epsilon_{11}+\epsilon_{11}^*=
\epsilon_{22}+\epsilon_{22}^*=
\epsilon_{12}+\epsilon_{21}^*=0.$$ An additional condition is found by requiring the determinant of the unitary matrix to be equal to $+1$, $$\epsilon_{11}+\epsilon_{22}=0,$$ which removes the freedom to make a simultaneous and identical change of phase for the neutron and the proton. The infinitesimal, physical transformations between a neutron and a proton can therefore be parametrized as $$\left[\begin{array}{ccc}
1-{\frac 1 2}i\epsilon_z&&-{\frac 1 2}i(\epsilon_x-i\epsilon_y)\\
-{\frac 1 2}i(\epsilon_x+i\epsilon_y)&&1+{\frac 1 2}i\epsilon_z
\end{array}\right],$$ which includes a conventional factor $-i/2$ and where the $\{\epsilon_x,\epsilon_y,\epsilon_z\}$ now are infinitesimal real numbers. This can be rewritten in terms of the Pauli spin matrices as $$\left[\begin{array}{rcr}
1&&0\\0&&1
\end{array}\right]
-{\frac 1 2}i\epsilon_x
\left[\begin{array}{rcr}
0&&1\\1&&0
\end{array}\right]
-{\frac 1 2}i\epsilon_y
\left[\begin{array}{rcr}
0&&-i\\i&&0
\end{array}\right]
-{\frac 1 2}i\epsilon_z
\left[\begin{array}{rcr}
1&&0\\0&&-1
\end{array}\right].$$ The infinitesimal transformations between a neutron and a proton can thus be written in terms of the three operators $$t_x\equiv
{\frac 1 2}
\left[\begin{array}{rcr}
0&&1\\1&&0
\end{array}\right],
\qquad
t_y\equiv
{\frac 1 2}
\left[\begin{array}{rcr}
0&&-i\\i&&0
\end{array}\right],
\qquad
t_z\equiv
{\frac 1 2}
\left[\begin{array}{rcr}
1&&0\\0&&-1
\end{array}\right],$$ which satisfy [*exactly*]{} the same commutation relations as the angular momentum operators. The action of the $t_\mu$ operators on a nucleon state is easily found from its matrix representation. For example, $$t_z|{\rm n}\rangle\equiv
{\frac 1 2}
\left[\begin{array}{rcr}
1&&0\\0&&-1
\end{array}\right]
\left[\begin{array}{c}
1\\0
\end{array}\right]=
{\frac 1 2}
|{\rm n}\rangle,
\quad
t_z|{\rm p}\rangle\equiv
{\frac 1 2}
\left[\begin{array}{rcr}
1&&0\\0&&-1
\end{array}\right]
\left[\begin{array}{c}
0\\1
\end{array}\right]=
-{\frac 1 2}
|{\rm p}\rangle,$$ which shows that $e(1-2t_z)/2$ is the charge operator. Also, the combinations $t_\pm\equiv t_x\pm it_y$ can be introduced, which satisfy the commutation relations $$[t_z,t_\pm]=\pm t_\pm,
\qquad
[t_+,t_-]=2t_z,$$ and play the role of raising and lowering operators since $$t_-|{\rm n}\rangle=|{\rm p}\rangle,
\qquad
t_+|{\rm n}\rangle=0,
\qquad
t_-|{\rm p}\rangle=0,
\qquad
t_+|{\rm p}\rangle=|{\rm n}\rangle.$$
This proves the formal equivalence between spin and isospin, and all results familiar from angular momentum can now be readily transposed to the isospin algebra. For a many-nucleon system (such as a nucleus) a total isospin $T$ and its $z$ projection $M_T$ can be defined which results from the coupling of the individual isospins, just as this can be done for the nucleon spins. The appropriate isospin operators are $$T_\mu=\sum_{k=1}^At_\mu(k),$$ where the sum is over all the nucleons in the nucleus.
If, in first approximation, the Coulomb interaction between the protons is neglected and, furthermore, if it is assumed that the strong interaction does not distinguish between neutrons and protons, the resulting nuclear Hamiltonian $H$ is isospin invariant. Explicitly, invariance under the isospin algebra ${\rm SU}(2)\equiv\{T_z,T_\pm\}$ follows from $$[H,T_z]=[H,T_\pm]=0.$$ As a consequence of these commutation relations, the many-particle eigenstates of $H$ have good isospin symmetry. They can be classified as $|\eta TM_T\rangle$ where $T$ is the total isospin of the nucleus obtained from the coupling of the individual isospins ${\frac 1 2}$ of all nucleons, $M_T$ is its projection on the $z$ axis in isospin space, $M_T=(N-Z)/2$ and $\eta$ denotes all additional quantum numbers. If isospin were a true symmetry, all states $|\eta TM_T\rangle$ with $M_T=-T,-T+1,\dots,+T$, and with the same $T$ (and identical other quantum numbers $\eta$), would be degenerate in energy; for example, neutron and proton would have exactly the same mass. States with the same $\eta T$ but different $M_T$ (and hence in different nuclei) are referred to as isobaric analogue states.
The isobaric multiplet mass equation {#ss_imme}
------------------------------------
The Coulomb interaction between the protons destroys the equivalence between the nucleons and hence breaks isospin symmetry. The main effect of the Coulomb interaction is a [*dynamical*]{} breaking of isospin symmetry. This can be shown by rewriting the Coulomb interaction, $$V_{\rm C}=
\sum_{k<l}^A
\left({\frac 1 2}-t_z(k)\right)
\left({\frac 1 2}-t_z(l)\right)
{\frac{e^2}{|\bar r_k-\bar r_l|}},$$ as a sum of isoscalar, isovector and isotensor parts $$V_{\rm C}=
\sum_{k<l}^A\sum_{t=0,1,2}
V^{(t)}_0(k,l),$$ with $$\begin{aligned}
V^{(0)}_0(k,l)&=&
\left({\frac 1 4}-\sqrt{\frac 1 3}
\left[\bar t(k)\times\bar t(l)\right]^{(0)}_0\right)
{\frac{e^2}{|\bar r_k-\bar r_l|}},
\nonumber\\
V^{(1)}_0(k,l)&=&
-{\frac 1 2}\left[t_z(k)+t_z(l)\right]
{\frac{e^2}{|\bar r_k-\bar r_l|}},
\nonumber\\
V^{(2)}_0(k,l)&=&
\sqrt{\frac 2 3}\left[\bar t(k)\times\bar t(l)\right]^{(2)}_0
{\frac{e^2}{|\bar r_k-\bar r_l|}},\end{aligned}$$ where the coupling is carried out in isospin. The Wigner–Eckart theorem in isospin space allows to factor out the $M_T$ dependence of any diagonal matrix element according to $$\langle\eta TM_T|
\sum_{k<l}^AV^{(t)}_0(k,l)
|\eta TM_T\rangle=
\langle TM_T\;t0|TM_T\rangle
\langle\eta T\|
\sum_{k<l}^AV^{(t)}(k,l)
\|\eta T\rangle,$$ where $\langle TM_T\;t0|TM_T\rangle$ is a Clebsch–Gordan coefficient associated with ${\rm SU}(2)\supset{\rm SO}(2)$. From the explicit expressions for these coefficients, $$\begin{aligned}
\langle TM_T\;00|TM_T\rangle&=&1,
\qquad
\langle TM_T\;10|TM_T\rangle=
{\frac{M_T}{\sqrt{T(T+1)}}},
\nonumber\\
\langle TM_T\;20|TM_T\rangle&=&
{\frac{3M_T^2-T(T+1)}{\sqrt{T(T+1)(2T-1)(2T+3)}}},\end{aligned}$$ one concludes that the $M_T$ dependence of the diagonal matrix elements of the Coulomb interaction is at most quadratic. If the off-diagonal, isospin mixing matrix elements of $V_{\rm C}$ are neglected, it can then be represented as $$V_{\rm C}\approx\kappa_0+\kappa_1T_z+\kappa_2T_z^2,
\label{e_vcapp}$$ for some particular coefficients $\kappa_0$, $\kappa_1$ and $\kappa_2$ which, according to the preceding discussion, depend on the isospin $T$ and other quantum numbers $\eta$. This can be viewed as a dynamical symmetry breaking of the type $$\begin{array}{ccc}
{\rm SU}(2)&\supset&{\rm SO}(2)\equiv\{T_z\}\\
\downarrow&&\downarrow\\
T&&M_T
\end{array}.$$ The Hamiltonian (\[e\_vcapp\]) splits but does not admix the eigenstates $|\eta TM_T\rangle$ with $M_T=-T,-T+1,\dots,+T$, and has the eigenspectrum $$E(M_T)=\kappa_0+\kappa_1M_T+\kappa_2M_T^2.$$
The expansion in $T_z$ is but an approximation to the true Coulomb interaction; it represents the diagonal part of it, with the $T$-mixing isovector and isotensor parts being neglected. In that approximation isospin remains a good quantum number. The excitation spectra of the different nuclei belonging to the same isospin multiplet (with the same $T$ but different $M_T$) are identical but their ground states do not have the same binding energy. The energy formula in $M_T$ was derived by Wigner [@Wigner57] who introduced the name of isobaric multiplet mass equation (IMME). Many experimental examples of nuclear isospin multiplets are known at present [@Britz98].
The assumption of isospin symmetry is too strong and should be relaxed to one of dynamical symmetry. One cannot expect that isobaric analogue states have the same [*absolute*]{} energy but one can expect them to have, to a good approximation, the same [*relative*]{} energies. As a result, for example, the excitation spectra of two mirror nuclei should be identical although the binding energy of their ground states differs. (Mirror nuclei have the same total number of nucleons and the number of neutrons in one of them equals the number of protons in the other.) This relation has been observed in many cases. An example where the idea has been tested to high angular momentum, is shown in Fig. \[f\_mass49\] [@Leary97].
![Energy spectra of the mirror nuclei $^{49}$Cr and $^{49}$Mn relative to the ground state of the first nucleus. Levels are labeled by their angular momentum and parity $J^\pi$. The inset shows the difference in excitation energy $\Delta E_{\rm x}\equiv
E_{\rm x}(^{49}{\rm Cr};J)-E_{\rm x}(^{49}{\rm Mn};J)$ as a function of $2J$.[]{data-label="f_mass49"}](f_mass49.eps){width="10cm"}
The ground-state energies of the two nuclei of the $T={\frac 1 2}$ isospin doublet ($^{49}{\rm Cr}$ with $M_T=+{\frac 1 2}$ and $^{49}{\rm Mn}$ with $M_T=-{\frac 1 2}$) are shifted with respect to each other but the energies relative to the ground state are indeed very similar. Nevertheless, the spectra are not identical as is clear from the inset in Fig. \[f\_mass49\] where the difference in excitation energy is plotted as a function of the angular momentum $J$. The deviations from zero signal a breakdown of the dynamical-symmetry approximation and, specifically, reveal subtle differences in alignment properties of the neutrons and protons in the two mirror nuclei [@Bentley07].
The equality of excitation spectra of mirror nuclei is sometimes referred to as mirror symmetry. It should be emphasized that mirror symmetry is but a particular manifestation of isospin symmetry which implies a wider relationship between properties of nuclei as illustrated with the example in Fig. \[f\_mass14\].
![Energy spectra of the nuclei $^{14}$C, $^{14}$N and $^{14}$O relative to the ground state of the middle nucleus. States with isospin $T=1$ are drawn in thick lines. In the self-conjugate nucleus $^{14}$N there exist also states with isospin $T=0$ which are drawn in thin lines. Levels with $T=1$ are labeled by their angular momentum and parity $J^\pi$.[]{data-label="f_mass14"}](f_mass14.eps){width="10cm"}
The nuclei shown contain $A=14$ nucleons but differ by their numbers of neutrons and protons, $(N,Z)=(8,6)$, (7,7) and (6,8). This corresponds to eigenvalues of $T_z$ given by $M_T=(N-Z)/2=+1,0,-1$ and, consequently, the isospin of all states in $^{14}$C and $^{14}$O must be $T=1$ or higher. As a consequence of mirror symmetry, the low-energy spectra of both nuclei should be identical. The $T=1$ analogue states should also occur in $^{14}$N, however. This nucleus has $M_T=0$ but this does not preclude the existence of $T=1$ states. In fact, isospin symmetry [*requires*]{} that such states be present somewhere in the spectrum of $^{14}$N. Figure \[f\_mass14\] illustrates that the isobaric analogue levels of those in $^{14}$C and $^{14}$O are indeed found in $^{14}$N.
For $T\geq{\frac 3 2}$ it is possible to test the IMME since the parameters $\kappa_i$ can be fixed from the isobaric analogue states in three nuclei and a prediction follows for the fourth member of the multiplet. As an example consider the $T={\frac 3 2}$ multiplet consisting of isobaric analogue states in $^{13}$B, $^{13}$C, $^{13}$N and $^{13}$O. Figure \[f\_mass13\] shows the binding energies of the nuclei $^{13}$B and $^{13}$O, both of which have $T=|M_T|={\frac 3 2}$ in their ground state.
![ Binding energies of the $T={\frac 3 2}$ isobaric analogue states with $J^\pi={\frac 1 2}^-$ in $^{13}$B, $^{13}$C, $^{13}$N and $^{13}$O. The column on the left is obtained for an exact ${\rm SU}_T(2)$ isospin symmetry, which predicts states with different $M_T$ to be degenerate. The middle column is obtained with the IMME with $\kappa_0=80.59$, $\kappa_1=-2.96$ and $\kappa_2=-0.26$, in MeV.[]{data-label="f_mass13"}](f_mass13.eps){width="11.5cm"}
The isobaric analogue states in $^{13}$C and $^{13}$N are $J^\pi={\frac 1 2}^-$ states at excitation energies of 15.11 and 15.07 MeV, respectively; these energies are substracted from the ground-state binding energies of $^{13}$C and $^{13}$N to give the energies plotted in Fig. \[f\_mass13\]. In this example the energy splitting due to the Coulomb interaction is well accounted for by the IMME, which is perhaps not surprising since four data points are fitted with three parameters. The quality of fits such as the one in Fig. \[f\_mass13\] is, however, not the most important aspect of dynamical symmetries, but rather the existence of good quantum numbers (isospin $T$ in this case).
Isospin selection rules {#ss_sel}
-----------------------
The most important consequence of a symmetry, which remains valid under the process of a dynamical symmetry breaking, is the existence of conserved (or ‘good’) quantum numbers. Frequently, these quantum numbers give rise to selection rules in radiative transition or particle-transfer processes. The measurement of transition or transfer probabilities is thus the method to establish the goodness of labels needed to characterize a quantum state and this in turn indicates to what extent a given (dynamical) symmetry is valid.
The link between symmetries and selection rules can be given a precise quantitative formulation via the (generalized) Wigner–Eckart theorem. Suppose the calculation is required of a transition or transfer matrix element between an initial state $|\Gamma_{\rm i}\gamma_{\rm i}\rangle$ and a final state $|\Gamma_{\rm f}\gamma_{\rm f}\rangle$, where the labeling of Subsect. \[ss\_deg\] is adopted. To compute the matrix element, it is first necessary to determine the tensor character of the operator associated with the transition or transfer by formally writing the operator as $\sum_{\Gamma\gamma}a_{\Gamma\gamma}T^\Gamma_\gamma$ where $a_{\Gamma\gamma}$ are coefficients. Each piece $T^\Gamma_\gamma$ can now be dealt with separately through the generalized Wigner–Eckart theorem. The essential point is that all dependence on the quantum numbers associated with the subalgebra ${\rm G}_2$ is contained in a generalized coupling coefficient. In addition, selection rules now follow from the multiplication rules for irreducible representations of the algebra ${\rm G}_1$: if $\Gamma_{\rm f}$ is not contained in the product $\Gamma_{\rm i}\times\Gamma$, the generalized coupling coefficient is zero and the matrix element of $T^\Gamma_\gamma$ vanishes.
A well-known example of the idea of selection rules concerns electric dipole transitions in self-conjugate nuclei [@Trainor52; @Radicati52], that is, nuclei with an equal number of neutrons and protons ($N=Z$). The E1 operator is, in lowest order of the long-wave approximation, given by $$T_\mu({\rm E1})=\sum_{k=1}^A e_kr_\mu(k).$$ Since the charge $e_k$ of the $k^{\rm th}$ nucleon is zero for a neutron and $e$ for a proton, the E1 operator can be rewritten as $$T_\mu({\rm E1})=
{\frac e 2}\sum_{k=1}^A[1-2t_z(k)]r_\mu(k)=
{\frac e 2}\left[R_\mu-2\sum_{k=1}^At_z(k)r_\mu(k)\right],$$ where $2t_z$ gives $+1$ for a neutron and $-1$ for a proton. The first term $R_\mu$ in the E1 operator is the centre-of-mass coordinate of the total nucleus and does not contribute to an internal E1 transition. The conclusion is that the electric dipole operator is, in lowest order of the long-wave approximation, of pure isovector character. The application of the Wigner–Eckart theorem in isospin space gives $$\langle\eta_{\rm f}T_{\rm f}M_{T_{\rm f}}|
T^{(1)}_0
|\eta_{\rm i}T_{\rm i}M_{T_{\rm i}}\rangle=
\langle T_{\rm i}M_{T_{\rm i}}\;10|
T_{\rm f}M_{T_{\rm f}}\rangle
\langle\eta_{\rm f}T_{\rm f}\|
T^{(1)}
\|\eta_{\rm i}T_{\rm i}\rangle,$$ where the coupling coefficient is associated with ${\rm SU}(2)\supset{\rm SO}(2)$. Self-conjugate nuclei have $M_{T_{\rm i}}=M_{T_{\rm f}}=0$ and exhibit as a consequence a simple selection rule: E1 transitions are forbidden between levels with the same isospin $T_{\rm i}=T_{\rm f}=T$ because of the vanishing Clebsch–Gordan coefficient, $\langle T0\;10|T0\rangle=0$.
This selection rule has been verified to hold approximately in light self-conjugate nuclei [@Freeman66]. Deviations occur because of higher-order terms in the E1 operator but also, and more importantly, because isospin is not an exactly conserved quantum number. Isospin mixing can be estimated in a variety of nuclear models. They all show that the mixing ([*i.e.*]{}, the non-dynamical breaking of isospin symmetry) is maximal in $N=Z$ nuclei. Isospin mixing effects, caused mainly by the Coulomb interaction, should thus be looked for in heavy $N=Z$ nuclei where they are largest. Such nuclei are created and accelerated for study at radioactive-ion beam facilities. The spectrum of an $N=Z$ nucleus studied in this respect, $^{64}$Ge, is shown in Fig. \[f\_mass64\].
![Energy spectra of the nuclei in the $A=64$ isospin triplet $^{64}$Ga, $^{64}$Ge and $^{64}$As relative to the ground state of the first nucleus. The observed $5^-\rightarrow4^+$ E1 transition between $T=0$ states in $^{64}$Ge is explained through mixing with the $T=1$ states, indicated by the arrows. The levels in broken lines are inferred from the isospin analogue levels in $^{64}$Ga.[]{data-label="f_mass64"}](f_mass64.eps){width="10cm"}
The crucial transition is the E1 between the $5^-$ and $4^+$ levels (indicated by the down arrow) which should be strictly forbidden if the isospin dynamical symmetry were exact. A small $B({\rm E1};5^-\rightarrow4^+)$ value is measured nevertheless and this is explained through the mixing with higher-lying $5^-$ and $4^+$ levels in $^{64}$Ge with $T=1$, which are not observed but inferred from their isospin analogue states in $^{64}$Ga. Although an estimate of the isospin mixing can be made in this way, the procedure is difficult as it requires the measurement of the lifetime, the $\delta({\rm E2}/{\rm M1})$ mixing ratio and the relative intensities of the transitions de-exciting the $5^-$ level [@Farnea03]. Given these uncertainties, a reliable measurement of isospin admixtures in nuclei, as a function of $N$ and $Z$, is still a declared goal of the current experimental efforts with radioactive-ion beams.
The nuclear shell model {#s_shell}
=======================
The structure of the atomic nucleus is determined, in first approximation, by the nuclear mean field, the average potential felt by a nucleon through the interactions exerted by all others. This average potential is responsible for the shell structure of the nucleus because the energy spectrum of a particle moving in this mean field shows regions with many levels and others with few. A second important ingredient that determines the structure of nuclei (and generally of many-body quantum systems) is the Pauli principle. Consequently, the nucleus can be viewed as an onion-like construction, with shells determined by the mean-field potential that are being filled in accordance with the Pauli principle. For a description that goes beyond this most basic level, the residual interaction between nucleons must be taken into account and what usually matters most for nuclear structure at low energies is the residual interaction between nucleons in the valence or outer shell. This interaction depends in a complex fashion on the numbers of valence neutrons and protons, and on the valence orbits available to them.
No review is given here of the nuclear shell model which has been the subject of several comprehensive monographs [@Shalit63; @Bohr69; @Lawson80; @Heyde90; @Talmi93]. Instead, after an introductory subsection, describing the model’s essential features and assumptions, emphasis is laid on its symmetry structure. It turns out that the two most important correlations in nuclei, of the pairing and of the quadrupole type, respectively, can be analyzed with symmetry techniques.
The model {#ss_shell}
---------
In a non-relativistic approximation, the wavefunction of any quantum-mechanical state of a nucleus with $A$ nucleons satisfies the Schrödinger equation $$H
\Psi(\xi_1,\xi_2,\dots,\xi_A)=
E\Psi(\xi_1,\xi_2,\dots,\xi_A),
\label{e_schrod}$$ with the Hamiltonian $$H=
\sum_{k=1}^A {{p^2_k}\over{2m_k}}
+\sum_{k<l}^AW_2(\xi_k,\xi_l)
+\sum_{k<l<m}^AW_3(\xi_k,\xi_l,\xi_m)
+\cdots.
\label{e_hamsm}$$ The notation $\xi_k$ is used to denote all coordinates of nucleon $k$, not only its position vector $\bar r_k$ but also its spin $\bar s_k$ and its isospin $\bar t_k$, $\xi_k\equiv\{\bar r_k,\bar s_k,\bar t_k\}$. The term $p^2_k/2m_k$ is the kinetic energy of nucleon $k$ and acts on a single nucleon only. The operator $W_i(\xi_k,\xi_l,\xi_m,\dots)$ is an $i$-body interaction between the nucleons $k,l,m,\dots$, and, as such, acts on $i$ nucleons simultaneously. Since neutron and proton are not elementary particles, it is not [*a priori*]{} clear that the interaction should be of two-body nature. Nevertheless, for a presentation of the elementary nuclear shell model, it can be assumed that the nature between the nucleons is two-body, $W_{i>2}=0$, as will be done in the subsequent discussion.
Under the assumption of at most two-body interactions, one can rewrite (\[e\_hamsm\]) as $$H=
\sum_{k=1}^A
\left({{p^2_k}\over{2m_k}}+V(\xi_k)\right)
+\left(
\sum_{k<l}^AW_2(\xi_k,\xi_l)
-\sum_{k=1}^AV(\xi_k)
\right).
\label{e_hamsm2}$$ The idea is now to choose $V(\xi_k)$ such that the effect of the residual interaction, that is, the second term in (\[e\_hamsm2\]), is minimized. The independent-particle shell model is obtained by neglecting the residual interaction altogether, $$H_{\rm ip}=
\sum_{k=1}^A
\left({{p^2_k}\over{2m_{\rm n}}}+V(\xi_k)\right),
\label{e_hamip}$$ where it is also assumed that all nucleons have the same mass $m_{\rm n}$. The physical interpretation of the approximation (\[e\_hamip\]) is that each nucleon moves independently in a mean-field potential $V(\xi)$ which represents the average interaction with all other nucleons in the nucleus.
The eigenproblem associated with the Hamiltonian (\[e\_hamip\]) is much easier to solve than the original problem (\[e\_schrod\]) because it can be reduced to a one-particle eigenequation. Its solution proceeds as follows. First, one solves the Schrödinger equation of a particle in a potential $V(\xi)$, that is, one finds the eigenfunctions $\phi_i(\xi)$ satisfying $$\left({{p^2}\over{2m_{\rm n}}}+V(\xi)\right)\phi_i(\xi)=
E_i\phi_i(\xi),
\label{e_onepart}$$ where $i$ labels the different eigensolutions. The exact form of the eigenfunctions $\phi_i(\xi)$ depends on the potential $V(\xi)$. For simple potentials ([*e.g.*]{}, the harmonic oscillator) the eigenfunctions can be found in analytic form in terms of standard mathematical functions; for more complicated potentials ([*e.g.*]{}, Woods–Saxon) $\phi_i(\xi)$ must be determined numerically. For all ‘reasonable’ potentials $V(\xi)$ the solutions of (\[e\_onepart\]) can be obtained, albeit in most cases only in numerical form.
The solution of the [*many-body*]{} Hamiltonian $H_{\rm ip}$ is immediately obtained due to its separability, $$\Phi_{i_1i_2\dots i_A}(\xi_1,\xi_2,\dots,\xi_A)=
\prod_{k=1}^A
\phi_{i_k}(\xi_k).
\label{e_onesol}$$ Although this is a genuine, mathematical eigensolution of the Hamiltonian (\[e\_hamip\]), it is not antisymmetric under the exchange of particles as is required by the Pauli principle. The solution (\[e\_onesol\]) must thus be antisymmetrized. For $A=2$ particles the antisymmetrization procedure yields $$\begin{aligned}
\Psi_{i_1i_2}(\xi_1,\xi_2)&=&
\sqrt{1\over2}
[\phi_{i_1}(\xi_1)\phi_{i_2}(\xi_2)
-\phi_{i_1}(\xi_2)\phi_{i_2}(\xi_1)]
\nonumber\\&=&
\sqrt{1\over2}
\left|\begin{array}{cc}
\phi_{i_1}(\xi_1)&\phi_{i_1}(\xi_2)\\
\phi_{i_2}(\xi_1)&\phi_{i_2}(\xi_2)
\end{array}\right|.\end{aligned}$$ In the $A$-particle case, antisymmetrization leads to the replacement of the wave function $\Phi_{i_1i_2\dots i_A}(\xi_1,\xi_2,\dots,\xi_A)$ by a Slater determinant of the form $$\Psi_{i_1i_2\dots i_A}(\xi_1,\xi_2,\dots,\xi_A)=
{1\over\sqrt{A!}}
\left|\begin{array}{cccc}
\phi_{i_1}(\xi_1)&\phi_{i_1}(\xi_2)&\cdots&\phi_{i_1}(\xi_A)\\
\phi_{i_2}(\xi_1)&\phi_{i_2}(\xi_2)&\cdots&\phi_{i_2}(\xi_A)\\
\vdots&\vdots&\ddots&\vdots\\
\phi_{i_A}(\xi_1)&\phi_{i_A}(\xi_2)&\cdots&\phi_{i_A}(\xi_A)
\end{array}\right|.
\label{e_slater}$$ This is the solution of the Schrödinger equation associated with the Hamiltonian (\[e\_hamip\]) that takes account of the Pauli principle.
The following question now arises. How should one choose the potential $V(\xi)$ introduced in (\[e\_hamsm2\])? This choice can be made at several levels of refinement. Ideally one wants to minimize the expectation value of $H$ in the ground state, that is, solve the variational equation $$\delta\int
\Psi^*(\xi_1,\xi_2,\dots,\xi_A)
H
\Psi(\xi_1,\xi_2,\dots,\xi_A)
d\xi_1d\xi_2\dots d\xi_A=0.
\label{e_var}$$ If, in this variational approach, the wave function $\Psi(\xi_1,\xi_2,\dots,\xi_A)$ is allowed to vary freely, the solution of (\[e\_var\]) is equivalent to the ground-state solution of the Schrödinger equation (\[e\_schrod\]). Obviously, one needs to set more modest goals to arrive at a solvable problem! One way to do so is to restrict $\Psi(\xi_1,\xi_2,\dots,\xi_A)$ in (\[e\_var\]) to the form of a Slater determinant, in other words, to minimize the ground-state energy by varying the potential $V(\xi)$ that defines the single-particle wave functions $\phi_{i_1},\phi_{i_2},\dots,\phi_{i_A}$ in (\[e\_slater\]). This is known as the Hartree–Fock method. One determines the form of the potential $V(\xi)$ by requiring the expectation value of the [*complete*]{} Hamiltonian (\[e\_hamsm\]) in the state (\[e\_slater\]) to be minimal.
The ground-state energy determined in Hartree–Fock theory is not the correct one; nevertheless, it is the best procedure at hand to construct an independent-particle model. Its disadvantage is that it can be computationally rather involved. Therefore, often the following simpler approach is preferred. One proposes a phenomenological form of the potential $V(\xi)$, such that the Schrödinger equation associated with $H_{\rm ip}$ in (\[e\_hamip\]) is analytically solvable. The potential which best mimics the nuclear mean-field potential and which can be solved exactly, is the harmonic-oscillator potential $$V(\xi)\equiv V(r)=
{\frac 1 2}m_{\rm n}\omega^2r^2.
\label{e_hopot}$$ The eigensolutions of the Schrödinger equation of a harmonic oscillator in three dimensions can be written as $$\phi_{n\ell m_\ell}(r,\theta,\varphi)
=R_{n\ell}(r)Y_{\ell m_\ell}(\theta,\varphi),
\label{e_howave}$$ where $R_{n\ell}(r)$ are radial wave functions appropriate for the harmonic oscillator and $Y_{\ell m_\ell}(\theta,\varphi)$ are spherical harmonics, already introduced in Eq. (\[e\_hywave\]). The index $i$, used previously to characterize single-particle eigenfunctions, is replaced now by the full set of quantum numbers $n$, $\ell$ and $m_\ell$. The quantized energy spectrum is given by $$E(n,\ell)=\left(2n+\ell+{\textstyle{3\over2}}\right)\hbar\omega,
\label{e_hoener}$$ in terms of the radial quantum number $n$ which has the allowed values $0,1,2,\dots$ and gives the number of nodes \[values of $r$ for which $R_{n\ell}(r)=0$ excluding those at $r=0$ and $r=\infty$\]. Because of the factor $r^\ell$ in the radial part, the wave function always vanishes at $r=0$ except for $\ell=0$ ($s$ state). The energy $E(n,\ell)$ is independent of $m_\ell$, the projection of the orbital angular momentum along the $z$ axis, as should be for a rotationally invariant Hamiltonian. In addition, $E(n,\ell)$ is only dependent on the sum $2n+\ell$. Introducing $N=2n+\ell$, one can rewrite (\[e\_hoener\]) as $$E(N)=\left(N+{\textstyle{3\over2}}\right)\hbar\omega,$$ which shows that $N$ can be interpreted as the number of oscillator quanta, the term ${3\over2}\hbar\omega$ being accounted for by the zero-point motion of an oscillator in three dimensions; $N$ is called the major oscillator quantum number. The allowed values of the orbital angular momentum are (because $\ell=N-2n$ and $n=0,1,\dots$) $$\ell=N,N-2,\dots,0\;{\rm or}\;1.$$ This completely determines the eigenspectrum of a spinless particle in a harmonic-oscillator potential.
The $2\ell+1$ eigensolutions with the same radial quantum number $n$ and the same orbital angular momentum $\ell$ but different $z$ projections $m_\ell$ are degenerate in energy. This degeneracy arises because the harmonic-oscillator Hamiltonian is rotationally invariant. There exists an [*additional*]{} degeneracy, namely the one for levels with the same $2n+\ell$. As in the case of the spectrum of the hydrogen atom, discussed in Sect. \[s\_hydrogen\], this additional degeneracy is also associated with a symmetry of the Hamiltonian which is identified in this case as U(3) [@Moshinsky69]. The U(3) transformations are more general than rotations in three dimensions \[[*i.e.*]{}, U(3) contains SO(3)\] and U(3) invariance can be understood intuitively as a consequence of the equivalence between the excitation of quanta in the $x$, $y$ and $z$ directions. The degeneracies of the harmonic-oscillator energy levels do not occur for a Woods–Saxon potential. In general one finds for a Woods–Saxon potential that, of the orbits with the same major oscillator quantum number $N$, those with high $\ell$ are more strongly bound than those with low orbital angular momentum.
An important quantity appearing in the harmonic-oscillator model is the elementary quantum of excitation $\hbar\omega$. By relating the radius of the nucleus, $R$, to the number of nucleons, $A$, and subsequently deriving a relationship between $R$, $A$ and the oscillator length $b$, one finds the expression [@Bohr69] $$b\approx1.00 A^{1/6}\;{\rm fm},
\label{ip16}$$ and, since $b=\sqrt{\hbar/m_{\rm n}\omega}$, $$\hbar\omega\approx41A^{-1/3}\;{\rm MeV}.$$
The solutions $\phi_{n\ell m_\ell}(r,\theta,\varphi)$ contain the dependence on the spatial coordinates only and not on the intrinsic spin of the particle. Since the intrinsic spin does not appear in the potential (\[e\_hopot\]), the wave functions are simply given by $$\phi_{n\ell m_\ell}(r,\theta,\varphi)
\chi_{sm_s},
\label{e_spin}$$ where $\chi_{sm_s}$ are spinors for particles with intrinsic spin $s={\frac 1 2}$. The energies are independent of $m_s$ and are still given by (\[e\_hoener\]). The eigenstates (\[e\_spin\]) do not have good [*total*]{} angular momentum, that is, they are not eigenstates of $j^2$ where $\bar j$ results from the coupling of the orbital angular momentum $\bar\ell$ and the spin $\bar s$ of the nucleon. States of good angular momentum are constructed from (\[e\_spin\]) with the help of Clebsch–Gordan coefficients, $$\phi_{n\ell jm_j}(r,\theta,\varphi)=
\sum_{m_\ell m_s}
(\ell m_\ell\:sm_s|jm_j)
\phi_{n\ell m_\ell}(r,\theta,\varphi)
\chi_{sm_s}.$$ Again, this state has the same energy eigenvalue (\[e\_hoener\]) since all states appearing in the sum are degenerate.
If the spin degeneracy of the quantum numbers $(n\ell jm_j)$ is taken into account, stable shell gaps are obtained at the nucleon numbers 2, 8, 20, 40, 70, 112,…. These are the magic numbers of the harmonic oscillator.
The existence of nuclear shell structure can be demonstrated in a variety of ways. The most direct way is by measuring the ease with which a nucleus can be excited. If it has a closed shell structure, one expects it to be rather stable and difficult to excite. This should be particularly so for nuclei that are doubly magic, that is, nuclei with a closed-shell configuration for neutrons [*and*]{} protons. The principle is illustrated in Fig. \[f\_twoplus\].
![The energy of the first-excited $2^+$ state in all even–even nuclei with $N,Z\geq8$ (where known experimentally) plotted as a function of neutron number $N$ along the $x$ axis and proton number $Z$ along the $y$ axis. The excitation energy is multiplied by $A^{1/3}$ and subsequently normalized to 1 for $^{208}$Pb where this quantity is highest. The value of $E_{\rm x}(2_1^+)A^{1/3}$ is indicated by the scale shown on the left. To improve the resolution of the plot, the scale only covers part of the range from 0 to 0.5 since only a few doubly magic nuclei ($^{16}$O, $^{40,48}$Ca, $^{132}$Sn and $^{208}$Pb) have values greater than 0.5.[]{data-label="f_twoplus"}](f_twoplus.eps){width="12cm"}
The figure shows the energy $E_{\rm x}(2^+_1)$ of the first-excited $2^+$ state relative to the ground state for all even–even nuclei. This energy is multiplied with $A^{1/3}$ and the result plotted on a normalized scale. (The factor $A^{1/3}$ accounts for the gradual decrease with mass number $A$ of the strength of the nuclear residual interaction which leads a compression of the spectrum with $A$.) Nuclei with particularly high values of $E_{\rm x}(2^+_1)A^{1/3}$ are $^{16}$O ($N=Z=8$), $^{40}$Ca ($N=Z=20$), $^{48}$Ca ($N=28$, $Z=20$), $^{132}$Sn ($N=82$, $Z=50$) and $^{208}$Pb ($N=126$, $Z=82$). Figure \[f\_twoplus\] establishes the stability properties of the isotopes and/or isotones with $N,Z=8$, 20, 28, 50, 82 and 126.
How to explain the differences between the observed magic numbers (2, 8, 20, 28, 50, 82 and 126) and those of the harmonic oscillator? The observed ones can be reproduced in an independent-particle model if to the harmonic-oscillator Hamiltonian $H_{\rm ho}$ a spin–orbit as well as an orbit–orbit term is added of the form $$V_{\rm so}=\zeta_{\rm so}(r)\bar\ell\cdot\bar s,
\qquad
V_{\rm oo}=\zeta_{\rm oo}(r)\bar\ell\cdot\bar\ell.$$ The eigenvalue problem associated with the Hamiltonian $H_{\rm ho}+V_{\rm so}+V_{\rm oo}$ is not, in general, analytically solvable but the dominant characteristics can be found from the expectation values $$\langle n\ell jm_j|V_{\rm so}|n\ell jm_j\rangle=
{\textstyle{1\over2}}
\langle\zeta_{\rm so}(r)\rangle_{n\ell}
\left[j(j+1)-\ell(\ell+1)-{\textstyle{3\over4}}\right],$$ and $$\langle n\ell jm_j|V_{\rm oo}|n\ell jm_j\rangle=
\ell(\ell+1)\langle\zeta_{\rm oo}(r)\rangle_{n\ell},$$ with radial integrals defined as $$\langle\zeta(r)\rangle_{n\ell}
=\int_0^{+\infty}
\zeta(r)R_{n\ell}(r)R_{n\ell}(r)r^2\;dr.$$ Consequently, the degeneracy of the single-particle levels within one major oscillator shell is lifted. Empirically, one finds that the radial integrals approximately satisfy the relations [@Bohr69] $$\langle\zeta_{\rm so}(r)\rangle_{n\ell}
\approx-20A^{-2/3}\;{\rm MeV},
\qquad
\langle\zeta_{\rm oo}(r)\rangle_{n\ell}
\approx-0.1\;{\rm MeV}.
%\label{ip25}$$ The origin of the orbit–orbit coupling can be understood from elementary arguments. The corrections to the harmonic-oscillator potential are repulsive for short and large distances and attractive for intermediate distances. These corrections therefore favor large-$\ell$ over small-$\ell$ orbits. The spin–orbit coupling has a relativistic origin. An important feature is that the radial integral is negative, reflecting the empirical finding that states with parallel spin and orbital angular momentum are pushed down in energy while in the antiparallel case they are pushed up.
The summary of the preceding discussion is that a simple approximation of the nuclear mean-field potential consists of a three-dimensional harmonic oscillator corrected with a spin–orbit and an orbit–orbit term. If, in addition, a two-body residual interaction is included, the many-body Hamiltonian that must be solved acquires the following form: $$H=
\sum_{k=1}^A\left(
\frac{p^2_k}{2m_{\rm n}}+
{\frac1 2}m_{\rm n}\omega^2r_k^2+
\zeta_{\rm oo}\,\bar\ell_k\cdot\bar\ell_k+
\zeta_{\rm so}\,\bar\ell_k\cdot\bar s_k\right)
+\sum_{k<l}V_{\rm res}(\xi_k,\xi_l),
\label{e_hamho}$$ where the indices in the second sum run over a [*restricted*]{} number of particles, usually only the valence nucleons. In spite of the severe simplifications of the original many-body problem (\[e\_schrod\]), the solution of the Schrödinger equation associated with the Hamiltonian (\[e\_hamho\]) still represents a formidable problem since the residual interaction must be diagonalized in a basis of Slater determinants of the type (\[e\_slater\]). Even if one limits oneself to valence-shell excitations, the dimension of the Hilbert space rapidly explodes with increasing mass of the nucleus. The $m$-scheme basis can be used to illustrate this. Because of the antisymmetry of Slater determinants, their number can be computed easily. For $n$ neutrons and $z$ protons distributed over $\Omega_n$ and $\Omega_z$ orbital states, respectively, the dimension of the basis is $${\frac{\Omega_n!}{n!(\Omega_n-n)!}}
{\frac{\Omega_z!}{z!(\Omega_z-z)!}}.$$ Application of this formula to $^{28}$Si (in the $sd$ shell, $\Omega_n=\Omega_z=12,n=z=6$) and to $^{78}$Y (half-way between the magic numbers 28 and 50, $\Omega_n=\Omega_z=22,n=z=11$) illustrates the point since it leads to dimensions of 8.5 $10^5$ and 5.0 $10^{11}$, respectively.
Given the considerable effort it takes to solve the nuclear many-body problem even only approximately, any analytical solution of (\[e\_hamho\]) that can be obtained through symmetry techniques might be of considerable value. In fact, the residual interaction can approximately be written as pairing-plus-quadrupole, $$V_{\rm res}(\xi_k,\xi_l)=
V_{\rm pairing}(\bar r_k,\bar r_l)+
V_{\rm quadrupole}(\bar r_k,\bar r_l),$$ where the exact form of these interactions is defined below. For particular values of the parameters in the mean field and if either the pairing or the quadrupole residual interaction is dominant, the eigenproblem (\[e\_hamho\]) can be solved analytically. Three situations arise, of which two are of interest:
1. [*No residual interaction.*]{} If $V_{\rm res}(\xi_k,\xi_l)=0$, the solution of (\[e\_hamho\]) reduces to a Slater determinant built from harmonic-oscillator eigenstates.
2. [*Pairing interaction.*]{} If the residual interaction has a pure pairing character, Racah’s SU(2) model of pairing results. This model is usually applied in the $jj$-coupling limit of strong spin–orbit coupling.
3. [*Quadrupole interaction.*]{} If the residual interaction has a pure quadrupole character, Elliott’s SU(3) model of rotation results. This model requires an $LS$-coupling scheme which occurs in the absence of spin–orbit coupling.
![Schematic representation of the shell-model parameter space with its three analytically solvable vertices.[]{data-label="f_trif"}](f_trif.eps){width="8cm"}
The situation is represented schematically in Fig. \[f\_trif\]. It should be emphasized that, in contrast to the top vertex, the two bottom vertices, SU(2) and SU(3), represent solutions of the nuclear Hamiltonian which include genuine many-body correlations. These two limits are thus of particular interest. A brief summary of the pairing and quadrupole limits of the nuclear shell model is given in the following subsections. A more detailed review of the use of symmetries in the shell model has been given elsewhere [@Isacker99a].
Pairing correlations {#ss_pair}
--------------------
The pairing interaction is a reasonable first-order approximation to the strong force between identical nucleons. For nucleons in a single-$j$ shell the interaction is defined by the matrix elements $$\langle j^2;JM_J|V_{\rm pairing}|j^2;JM_J\rangle=
-g_0(2j+1)\delta_{J0},
\label{e_pair}$$ where $j$ is the total (orbital+spin) angular momentum of a single nucleon (hence $j$ is half-odd-integer), $J$ results from the coupling of two $j$s and $M_J$ is the projection of $J$ on the $z$ axis. Furthermore, $g_0$ is the strength of the interaction which is attractive in nuclei ($g>0$).
Evidence for the pairing character of the interaction between identical nucleons can be obtained from simple arguments as is illustrated in Fig. \[f\_evenodd\].
![The even–odd effect in nuclear binding energies as evidence for pairing correlations in nuclei. The difference between the experimental binding energies from the atomic-mass compilation of 2003 (AME03) [@Audi03] and a smooth local fit to these data is shown as a function of neutron number $N$ along the $x$ axis and proton number $Z$ along the $y$ axis. The local fit assumes a polynomial in $N$ and $Z$, whose coefficients are determined from about 50 masses in the neighborhood.[]{data-label="f_evenodd"}](f_evenodd.eps){width="11.5cm"}
The figure shows the difference between the experimental nuclear binding energies and a smooth local fit to these data as a function of neutron and proton numbers $N$ and $Z$. The local fit assumes a polynomial in $N$ and $Z$, whose coefficients are determined to about 50 masses in the neighborhood. The details of this fit are unimportant for the present argument, except for the fact that no difference is made between even–even, odd-mass and odd–odd nuclei which are all fitted with the same polynomial. The figure clearly demonstrates the existence of an even–odd effect in the observed binding energies since even–even nuclei are systematically more bound than found in the polynomial fit while odd–odd nuclei are less bound. The simplest interpretation of this empirical finding is that there exists an attractive interaction between two identical nucleons.
The pairing interaction is less realistic than a short-range delta interaction but has the advantage that the corresponding many-body problem can be solved analytically. Furthermore, its analysis is important because it is at the basis of seniority [@Racah43] which has found fruitful application in nuclear physics with considerable empirical evidence in semi-magic nuclei.
The results can be summarized as follows. A state with $n$ identical particles and diagonal in the pairing interaction, is characterized—in addition to the angular momentum $J$ and its projection $M_J$—by a quantum number $\upsilon$. The energy of this state is given by $$E(n,\upsilon)=-{\frac 1 4}g_0(n-\upsilon)(2\Omega_j-n-\upsilon+2),
\label{e_paire}$$ where $2\Omega_j=2j+1$. The quantum number $\upsilon$ [*counts the number of particles not coupled to $J=0$*]{}. Any state $|j^n\upsilon JM_J\rangle$ can be constructed from $|j^\upsilon\upsilon JM_J\rangle$ according to $$|j^n\upsilon JM_J\rangle\propto
\left(S^j_+\right)^{(n-\upsilon)/2}|j^\upsilon\upsilon JM_J\rangle,
\label{e_pairw}$$ where $S^j_+$ is an operator which creates a pair of particles in the $j$ shell with their angular momentum coupled to $J=0$. In other words, $|j^\upsilon\upsilon JM_J\rangle$ acts as a [*parent*]{} state for a whole class of states $|j^n\upsilon JM_J\rangle$ just by the action of the pair state $S^j_+$. For this reason, $\upsilon$ is called seniority.
The above results remain valid if the $n$ identical particles are distributed over several [*degenerate*]{} $j$ shells by making the substitutions $S^j_+\mapsto S_+\equiv\sum_jS^j_+$ and $\Omega_j\mapsto\Omega=\sum_j\Omega_j$. In this form the pairing formalism can be used to make several characteristic predictions: a constant excitation energy (independent of $n$) of the first-excited $2^+$ state in even–even isotopes, the linear variation of two-nucleon separation energies as a function of $n$, the odd–even staggering in nuclear binding energies, the enhancement of two-nucleon transfer.
![The difference $E(n,2)-E(n,0)$ is a function of particle number $n$ (top) and the corresponding observed excitation energies $E_{\rm x}(2^+_1)\equiv E(2^+_1)-E(0^+_1)$ and $E_{\rm x}(4^+_1)\equiv E(4^+_1)-E(0^+_1)$ in the Sn isotopes.[]{data-label="f_sntwop"}](f_sntwop.eps){width="10cm"}
The first of these predictions is illustrated in Fig. \[f\_sntwop\]. The ground state of an even–even nucleus has $\upsilon=0$ and the lowest excited states have $\upsilon=2$. An example of such $\upsilon=2$ states are those in a two-nucleon $j^2$ configuration with $J\neq0$, $J=2,4,\dots,2j-1$. The energy difference between $\upsilon=2$ and $\upsilon=0$ states is given by $$E(n,2)-E(n,0)=g_0\Omega,$$ and is independent of the number of valence nucleons. This prediction is illustrated in Fig. \[f\_sntwop\] where it is compared with the excitation energies of the $2^+_1$ and $4^+_1$ levels in the even–even Sn isotopes.
The discussion of pairing correlations in nuclei traditionally has been inspired by the treatment of superfluidity in condensed matter [@Bardeen57; @Bohr58]. The superfluid phase in the latter systems is characterized by the presence of a large number of identical bosons in a single quantum state, which is called the condensate. In superconductors the bosons are pairs of electrons with opposite momenta that form at the Fermi surface. The character of the bosons in nuclei can be understood by analyzing the ground state of a pairing Hamiltonian. For an even–even nucleus, according to (\[e\_pairw\]), it is given by $$|j^n\upsilon=0,J=M=0\rangle\propto
\left(S_+\right)^{n/2}|{\rm o}\rangle.$$ In nuclei the bosons are thus pairs of valence nucleons with opposite angular momenta.
Quadrupole correlations {#ss_quad}
-----------------------
The second class of analytically solvable shell-model Hamiltonians corresponds to the case of nucleons occupying an entire shell of the harmonic oscillator and interacting through a quadrupole force. In this case the Hamiltonian is of the form $$H=
\sum_{k=1}^A
\left({{p_k^2}\over{2m_{\rm n}}}+
{\frac 1 2}m_{\rm n}\omega^2r_k^2\right)-
g_2Q\cdot Q,
\label{e_su3ham}$$ which contains a quadrupole operator $$Q_\mu=
\sqrt{\frac 3 2}
\left[
\sum_{k=1}^A{\frac{1}{b^2}}
[\bar r_k\times\bar r_k]^{(2)}_\mu+
{\frac{b^2}{\hbar^2}}
\sum_{k=1}^A[\bar p_k\times\bar p_k]^{(2)}_\mu
\right].
\label{e_su3q}$$ Note that $Q\cdot Q\equiv\sum_\mu Q_\mu Q_\mu$ contains one-body ($k=l$) as well as two-body ($k\neq l$) terms.
The proof that the shell-model Hamiltonian (\[e\_su3ham\]) is analytically solvable was given by Elliott [@Elliott58]. The reasons for its solvability are that the five components of the quadrupole operator (\[e\_su3q\]) together with the three components of the angular momentum vector $\bar L=\sum_k(\bar r_k\wedge\bar p_k)$ form a closed algebra SU(3) and, furthermore, that these operators commute with the harmonic-oscillator Hamiltonian \[[*i.e.*]{}, with the one-body term in (\[e\_su3ham\])\]. The quadrupole interaction is in fact a combination of Casimir operators, $$Q\cdot Q=
4C_2[{\rm SU}(3)]-3\bar L^2=
4C_2[{\rm SU}(3)]-3C_2[{\rm SO}(3)],
\label{su3cas}$$ and it follows that the Hamiltonian (\[e\_su3ham\]) has the eigenvalues $$E(\lambda,\mu,L)=E_0-
g_2\left[
4(\lambda^2+\mu^2+\lambda\mu+3\lambda+3\mu)-3L(L+1)
\right],
\label{su3eig}$$ where $E_0$ is a constant energy associated with the first term in the Hamiltonian (\[e\_su3ham\]) and $\lambda$ and $\mu$ label the SU(3) representations. The quadrupole interaction represents an example of symmetry breaking since the degeneracy associated with an entire oscillator shell is lifted by the quadrupole interaction.
The importance of Elliott’s idea is that it gives rise to a rotational classification of states through mixing of spherical configurations. With the SU(3) model it was shown, for the first time, how deformed nuclear shapes may arise out of the spherical shell model. As a consequence, Elliott’s work bridged the gap between the nuclear shell model and the liquid droplet model which up to that time (1958) existed as separate views of the nucleus.
The interacting boson model {#s_ibm}
===========================
Arguably more than any other model of the nucleus, the interacting boson model (IBM) illustrates the power of group-theoretical techniques and the physics insights that can be obtained from them. In this section a brief introduction to the IBM is given with the primary goal to provide an example of the notion of dynamical symmetry which was introduced in Sect. \[s\_dsym\]. It is not the aim here to give a full account of the IBM which can be found in the book of Iachello and Arima [@Iachello87].
The model {#ss_ibm}
---------
The building blocks of the IBM are $s$ and $d$ bosons with angular momenta $\ell=0$ and $\ell=2$. A nucleus is characterized by a constant total number of bosons $N$ which equals half the number of valence nucleons (particles or holes, whichever is smaller). In these lecture notes no distinction is made between neutron and proton bosons, an approximation which is known as .
Since the Hamiltonian of the conserves the total number of bosons, it can be written in terms of the 36 operators $b_{\ell m_\ell}^\dag b_{\ell' m'_\ell}$ where $b_{\ell m_\ell}^\dag$ ($b_{\ell m_\ell}$) creates (annihilates) a boson with angular momentum $\ell$ and $z$ projection $m_\ell$. This set of 36 operators generates the Lie algebra U(6). A Hamiltonian that conserves the total number of bosons is of the generic form $$H=E_0+H_1+H_2+H_3+\cdots,
\label{e_ibmham}$$ where the index refers to the order of the interaction in the generators of U(6). The first term $E_0$ is a constant which represents the binding energy of the core. The second term is the one-body part $$H_1=
\epsilon_s[s^\dag\times\tilde s]^{(0)}+
\epsilon_d\sqrt{5}[d^\dag\times\tilde d]^{(0)}\equiv
\epsilon_sn_s+
\epsilon_dn_d,
\label{e_ibmham1}$$ where $\times$ refers to coupling in angular momentum, $\tilde b_{\ell m_\ell}\equiv(-)^{\ell-m_\ell}b_{\ell,-m_\ell}$ and the coefficients $\epsilon_s$ and $\epsilon_d$ are the energies of the $s$ and $d$ bosons. The third term in the Hamiltonian (\[e\_ibmham\]) represents the two-body interaction $$H_2=
\sum_{\ell_1\leq\ell_2,\ell'_1\leq\ell'_2,L}
\tilde v^L_{\ell_1\ell_2\ell'_1\ell'_2}
[[b^\dag_{\ell_1}\times b^\dag_{\ell_2}]^{(L)}\times
[\tilde b_{\ell'_2}\times\tilde b_{\ell'_1}]^{(L)}]^{(0)}_0,
\label{e_ibmham2}$$ where the coefficients $\tilde v$ are related to the interaction matrix elements between normalized two-boson states, $$\langle\ell_1\ell_2;LM|H_2|\ell'_1\ell'_2;LM\rangle=
\sqrt{\frac{(1+\delta_{\ell_1\ell_2})(1+\delta_{\ell'_1\ell'_2})}{2L+1}}
\tilde v^L_{\ell_1\ell_2\ell'_1\ell'_2}.$$ Since the bosons are necessarily symmetrically coupled, allowed two-boson states are $s^2$ ($L=0$), $sd$ ($L=2$) and $d^2$ ($L=0,2,4$). Since for $n$ states with a given angular momentum one has $n(n+1)/2$ interactions, seven independent two-body interactions $v$ are found: three for $L=0$, three for $L=2$ and one for $L=4$.
This analysis can be extended to higher-order interactions. One may consider, for example, the three-body interactions $\langle\ell_1\ell_2\ell_3;LM|H_3|\ell'_1\ell'_2\ell'_3;LM\rangle$. The allowed three-boson states are $s^3$ ($L=0$), $s^2d$ ($L=2$), $sd^2$ ($L=0,2,4$) and $d^3$ ($L=0,2,3,4,6$), leading to $6+6+1+3+1=17$ independent three-body interactions for $L=0,2,3,4,6$, respectively.
Dynamical symmetries {#ss_ibmds}
--------------------
The characteristics of the most general IBM Hamiltonian which includes up to two-body interactions and its group-theoretical properties are by now well understood [@Castanos79]. Numerical procedures exist to obtain its eigensolutions but the problem can be solved analytically for particular choices of boson energies and boson–boson interactions. For an IBM Hamiltonian with up to two-body interactions between the bosons, three different analytical solutions or limits exist: the vibrational U(5) [@Arima76], the rotational SU(3) [@Arima78] and the $\gamma$-unstable SO(6) limit [@Arima79]. They are associated with the algebraic reductions $${\rm U}(6)\supset
\left\{\begin{array}{c}
{\rm U}(5)\supset{\rm SO}(5)\\
{\rm SU}(3)\\
{\rm SO}(6)\supset{\rm SO}(5)
\end{array}\right\}
\supset{\rm SO}(3).
\label{e_ibmlat}$$ The algebras appearing in the lattice (\[e\_ibmlat\]) are subalgebras of U(6) generated by operators of the type $b^\dag_{\ell m_\ell}b_{\ell'm'_\ell}$, the explicit form of which is listed, for example, in Ref. [@Iachello87]. With the subalgebras U(5), SU(3), SO(6), SO(5) and SO(3) there are associated one linear \[of U(5)\] and five quadratic Casimir operators. The total of all one- and two-body interactions can be represented by including in addition the operators $C_1[{\rm U}(6)]$, $C_2[{\rm U}(6)]$ and $C_1[{\rm U}(6)]C_1[{\rm U}(5)]$. The most general IBM Hamiltonian with up to two-body interactions can thus be written in an [*exactly*]{} equivalent way with Casimir operators. Specifically, the Hamiltonian reads $$\begin{aligned}
H_{1+2}&=&
\kappa_1C_1[{\rm U}(5)]+
\kappa^\prime_1C_2[{\rm U}(5)]+
\kappa_2C_2[{\rm SU}(3)]
\nonumber\\&&+
\kappa_3C_2[{\rm SO}(6)]+
\kappa_4C_2[{\rm SO}(5)]+
\kappa_5C_2[{\rm SO}(3)],
\label{e_ibmhamc}\end{aligned}$$ which is just an alternative way of writing $H_1+H_2$ of Eqs. (\[e\_ibmham1\],\[e\_ibmham2\]) if interactions are omitted that contribute to the binding energy only.
The representation (\[e\_ibmhamc\]) is much more telling when it comes to the symmetry properties of the IBM Hamiltonian. If some of the coefficients $\kappa_i$ vanish such that $H_{1+2}$ contains Casimir operators of subalgebras belonging to a [*single*]{} reduction in the lattice (\[e\_ibmlat\]), then the eigenvalue problem can be solved analytically. Three classes of spectrum generating Hamiltonians can thus be constructed of the form $$\begin{aligned}
{\rm U}(5)&:&H_{1+2}=
\kappa_1C_1[{\rm U}(5)]+
\kappa^\prime_1C_2[{\rm U}(5)]+
\kappa_4C_2[{\rm SO}(5)]+
\kappa_5C_2[{\rm SO}(3)],
\nonumber\\
{\rm SU}(3)&:&H_{1+2}=
\kappa_2C_2[{\rm SU}(3)]+
\kappa_5C_2[{\rm SO}(3)],
\nonumber\\
{\rm SO}(6)&:&H_{1+2}=
\kappa_3C_2[{\rm SO}(6)]+
\kappa_4C_2[{\rm SO}(5)]+
\kappa_5C_2[{\rm SO}(3)].
\label{e_ibmlim}\end{aligned}$$ In each of these limits the Hamiltonian is written as a sum of commuting operators and, as a consequence, the quantum numbers associated with the different Casimir operators are conserved. They can be summarized as follows: $$\begin{aligned}
&&\begin{array}{ccccccccc}
{\rm U}(6)&\supset&{\rm U}(5)&\supset&{\rm SO}(5)&
\supset&{\rm SO}(3)&\supset&{\rm SO}(2)\\
\downarrow&&\downarrow&&\downarrow&&\downarrow&&\downarrow\\[0mm]
[N]&&n_d&&\tau&&\nu_\Delta L&&M_L
\end{array},
\nonumber\\[1ex]
&&\begin{array}{ccccccc}
{\rm U}(6)&\supset&{\rm SU}(3)&\supset&{\rm SO}(3)&
\supset&{\rm SO}(2)\\
\downarrow&&\downarrow&&\downarrow&&\downarrow\\[0mm]
[N]&&(\lambda,\mu)&&K_LL&&M_L
\end{array},
\nonumber\\[1ex]
&&\begin{array}{ccccccccc}
{\rm U}(6)&\supset&{\rm SO}(6)&\supset&{\rm SO}(5)&
\supset&{\rm SO}(3)&\supset&{\rm SO}(2)\\
\downarrow&&\downarrow&&\downarrow&&\downarrow&&\downarrow\\[0mm]
[N]&&\sigma&&\tau&&\nu_\Delta L&&M_L
\end{array}.
\label{e_ibmclas}\end{aligned}$$ Furthermore, for each of the three Hamiltonians in Eq. (\[e\_ibmlim\]) an analytic eigenvalue expression is available, $$\begin{aligned}
{\rm U}(5)&:&E(n_d,\tau,L)=
\kappa_1 n_d+
\kappa^\prime_1 n_d(n_d+4)+
\kappa_4 \tau(\tau+3)+
\kappa_5 L(L+1),
\nonumber\\
{\rm SU}(3)&:&E(\lambda,\mu,L)=
\kappa_2 (\lambda^2+\mu^2+\lambda\mu+3\lambda+3\mu)+
\kappa_5 L(L+1),
\nonumber\\
{\rm SO}(6)&:&E(\sigma,\tau,L)=
\kappa_3 \sigma(\sigma+4)+
\kappa_4 \tau(\tau+3)+
\kappa_5 L(L+1).
\label{e_ibmeig}\end{aligned}$$ One can add Casimir operators of U(6) to the Hamiltonians in Eq. (\[e\_ibmhamc\]) without breaking any of the symmetries. For a given nucleus they reduce to a constant contribution. They can be omitted if one is only interested in the spectrum of a single nucleus but they should be introduced if one calculates binding energies. Note that none of the Hamiltonians in Eq. (\[e\_ibmlim\]) contains a Casimir operator of SO(2). This interaction breaks the SO(3) symmetry (lifts the $M_L$ degeneracy) and would only be appropriate if the nucleus is placed in an external electric or magnetic field.
The dynamical symmetries of the IBM arise if combinations of certain coefficients $\kappa_i$ in the Hamiltonian (\[e\_ibmhamc\]) vanish. The converse, however, cannot be said. Even if all parameters $\kappa_i$ are non-zero, the Hamiltonian $H_{1+2}$ still may exhibit a dynamical symmetry and be analytically solvable. This is a consequence of the existence of unitary transformations which preserve the eigenspectrum of the Hamiltonian $H_{1+2}$ (and hence its analyticity properties) and which can be represented as transformations in the parameter space $\{\kappa_i\}$. A [*systematic*]{} procedure exists for finding such transformations or parameter symmetries [@Shirokov98] which can, in fact, be applied to any Hamiltonian describing a system of interacting bosons and/or fermions.
While a numerical solution of the shell-model eigenvalue problem in general rapidly becomes impossible with increasing particle number, the corresponding problem in the IBM with $s$ and $d$ bosons remains tractable at all times, requiring the diagonalization of matrices with dimension of the order of $\sim10^2$. One of the main reasons for the success of the IBM is that it provides a workable, albeit approximate, scheme which allows a description of transitional nuclei with a few parameters.
Partial dynamical symmetries {#ss_ibmpds}
----------------------------
As argued in Sect. \[s\_qm\], a dynamical symmetry can be viewed as a generalization and refinement of the concept of symmetry. Its basic paradigm is to write a Hamiltonian in terms of Casimir operators of a set of nested algebras. Its hallmarks are (i) solvability of the complete spectrum, (ii) existence of exact quantum numbers for all eigenstates and (iii) pre-determined structure of the eigenfunctions, independent of the parameters in the Hamiltonian. A further enlargement of these ideas is obtained by means of the concept of partial dynamical symmetry. The idea is to relax the conditions of [*complete*]{} solvability and this can be done in essentially two different ways:
1. [*Some of the eigenstates keep all of the quantum numbers.*]{} In this case the properties of solvability, good quantum numbers, and symmetry-dictated structure are fulfilled exactly, but only by a subset of eigenstates [@Alhassid92; @Leviatan96].
2. [*All eigenstates keep some of the quantum numbers.*]{} In this case none of the eigenstates is solvable, yet some quantum numbers (of the conserved symmetries) are retained. In general, this type of partial dynamical symmetry arises if the Hamiltonian preserves some of the quantum numbers in a dynamical-symmetry classification while breaking others [@Leviatan86; @Isacker99b].
Combinations of 1 and 2 are possible as well, for example, if some of the eigenstates keep some of the quantum numbers [@Leviatan02].
It should be emphasized that dynamical symmetry, be it partial or not, is a notion that is not restricted to a specific model but can be applied to any quantal system consisting of interacting particles. Quantum Hamiltonians with a partial dynamical symmetry can be constructed with general techniques and their existence is closely related to the order of the interaction among the particles. Applications of these concepts continue to be explored in all fields of physics.
Microscopy {#ss_mic}
----------
The connection of the IBM with the shell model arises by identifying the $s$ and $d$ bosons with correlated (or Cooper) pairs formed by two nucleons in the valence shell coupled to angular momentum $J=0$ and $J=2$. There exists a rich and varied literature on general procedures to carry out boson mappings in which pairs of fermions are represented as bosons. They fall into two distinct classes. In the first one establishes a correspondence between boson and fermion operators by requiring them to have the same algebraic structure, that is, the same commutation relations. In the second class the correspondence is established rather between state vectors in both spaces. In each case further subclasses exist that differ in their technicalities ([*e.g.*]{}, the nature of the operator expansion or the hierarchy in the state correspondence). In the specific example at hand, namely the mapping between the IBM and the shell model, arguably the most successful procedure has been the so-called OAI mapping [@Otsuka78b] which associates vectors based on a seniority \[U(5)\] hierarchy in fermion (boson) space. It has been used in highly complex situations that go well beyond the simple version of with just identical $s$ and $d$ bosons and which include, for example, neutron–proton $T=1$ and $T=0$ pairs [@Thompson87; @Juillet01].
The classical limit {#ss_climit}
-------------------
The connection of the IBM with the geometric model of the nucleus can be obtained on the basis of coherent-state formalism [@Ginocchio80; @Dieperink80; @Bohr80]. The central outcome of the formalism is that for any Hamiltonian a corresponding potential $V(\beta,\gamma)$ can be constructed where $\beta$ and $\gamma$ parametrize the intrinsic quadrupole deformation of the nucleus [@Bohr75]. This procedure is known as the classical limit of the IBM.
The coherent states used for obtaining the classical limit of the IBM are of the form $$|N;\alpha_\mu\rangle\propto
\left(s^\dag+\sum_\mu\alpha_\mu d^\dag_\mu\right)^N|{\rm o}\rangle,
\label{e_coh}$$ where $|{\rm o}\rangle$ is the boson vacuum and $\alpha_\mu$ are five complex variables. These have the interpretation of (quadrupole) shape variables and their associated conjugate momenta. If one limits oneself to static problems, the $\alpha_\mu$ can be taken as real; they specify a shape and are analogous to the shape variables of the droplet model of the nucleus [@Bohr75]. The $\alpha_\mu$ can be related to three Euler angles which define the orientation of an intrinsic frame of reference, and two intrinsic shape variables, $\beta$ and $\gamma$, that parametrize quadrupole vibrations of the nuclear surface around an equilibrium shape. In terms of the latter variables, the coherent state (\[e\_coh\]) is rewritten as $$|N;\beta\gamma\rangle\propto
\left(s^\dag+
\beta\left[\cos\gamma\,d^\dag_0
+\sqrt{\frac 1 2}\sin\gamma\,(d^\dag_{-2}+d^\dag_{+2})\right]
\right)^N|{\rm o}\rangle.
\label{e_cohb}$$ The expectation value of the Hamiltonian (\[e\_ibmham\]) in this state can be determined by elementary methods [@Isacker81] and yields a functional expression in $\beta$ and $\gamma$ which is identified with a potential $V(\beta,\gamma)$, familiar from the geometric model. The classical limit of the most general Hamiltonian (\[e\_ibmham\]) is found to be of the generic form $$V(\beta,\gamma)=
E_0+\sum_{n\geq1}\frac{N(N-1)\cdots(N-n+1)}{(1+\beta^2)^n}
\sum_{kl}a^{(n)}_{kl}\beta^{2k+3l}\cos^l3\gamma,
\label{e_climit}$$ where the coefficients $a^{(n)}_{kl}$ can be expressed in terms of the single-boson energies and $n$-body interactions between the bosons.
A catastrophe analysis [@Gilmore81] of the potential surfaces in $(\beta,\gamma)$ as a function of the Hamiltonian parameters determines the stability properties of these shapes. This analysis was carried out for the general IBM Hamiltonian with up to two-body interactions by López–Moreno and Castaños [@Lopez96]. The results of this study are confirmed [@Jolie01] if a simplified IBM Hamiltonian is considered of the form $$H_{1+2}=
\epsilon\,n_d +
\kappa\,Q\cdot Q.
\label{e_ibmhamcqf}$$ This Hamiltonian provides a simple parametrization of the essential features of nuclear structural evolution in terms of a vibrational term $n_d$ (the number of $d$ bosons) and a quadrupole interaction $Q\cdot Q$ with $$Q_\mu=
[s^\dag\times\tilde d+d^\dag\times\tilde s]^{(2)}_\mu+
\chi[d^\dag\times\tilde d]^{(2)}_\mu.
\label{e_quad}$$ Besides an overall energy scale, the spectrum of the Hamiltonian (\[e\_ibmhamcqf\]) is determined by two parameters: the ratio $\epsilon/\kappa$ and $\chi$. The three limits of the IBM are obtained with an appropriate choice of parameters: U(5) if $\kappa=0$, ${\rm SU}_\pm(3)$ if $\epsilon=0$ and $\chi=\pm\sqrt{7}/2$, and SO(6) if $\epsilon=0$ and $\chi=0$. One may thus represent the parameter space of the simplified IBM Hamiltonian (\[e\_ibmhamcqf\]) on a triangle with vertices that correspond to the three limits U(5), SU(3) and SO(6), and where arbitrary points correspond to specific values of $\epsilon/\kappa$ and $\chi$. Since there are two possible choices for SU(3), $\chi=-\sqrt{7}/2$ and $\chi=+\sqrt{7}/2$, the triangle can be extended to cover both cases by allowing $\chi$ to take negative as well as positive values.
The geometric interpretation of any IBM Hamiltonian on the triangle can now be found from its expectation value in the coherent state (\[e\_cohb\]) which for the particular Hamiltonian (\[e\_ibmhamcqf\]) gives $$\begin{aligned}
V(\beta,\gamma) &=&
\frac{N\epsilon\beta^2}{1+\beta^2} +
\kappa\left[
\frac{N(5+(1+\chi^2) \beta^2)}{1+\beta^2}
\right.
\nonumber\\
&&\left.
+\frac{N(N-1)}{(1+\beta^2)^2}
\left( \frac{2}{7}\chi^2\beta^4-
4\sqrt{\frac{2}{7}}\chi\beta^3\cos3\gamma+
4 \beta^2 \right)
\right].
\label{potcqf}\end{aligned}$$ The catastrophe analysis of this surface is summarized with the phase diagram shown in Fig. \[f\_trib\].
![ Phase diagram of the Hamiltonian (\[e\_ibmhamcqf\]) and the associated geometric interpretation. The parameter space is divided into three regions depending on whether the corresponding potential has (I) a spherical, (II) a prolate deformed or (III) an oblate deformed absolute minimum. These regions are separated by dashed lines and meet in a triple point (grey dot). The shaded area corresponds to a region of coexistence of a spherical and a deformed minimum. Also indicated are the points on the triangle (black dots) which correspond to the dynamical-symmetry limits of the Hamiltonian (\[e\_ibmhamcqf\]) and the choice of parameters $\epsilon$, $\kappa$ and $\chi$ for specific points or lines of the diagram.[]{data-label="f_trib"}](f_trib.eps){width="7cm"}
Analytically solvable limits are indicated by the dots. Two different SU(3) limits occur corresponding to two possible choices of the quadrupole operator, $\chi=\pm\sqrt{7}/2$. Close to the U(5) vertex, the IBM Hamiltonian has a vibrational-like spectrum. Towards the SU(3) and SO(6) vertices, it acquires rotational-like characteristics. This is confirmed by a study of the character of the potential surface in $\beta$ and $\gamma$ associated with each point of the triangle. In the region around U(5), corresponding to large $\epsilon/\kappa$ ratios, the minimum of the potential is at $\beta=0$. On the other hand, close to the ${\rm SU}_+(3)$–SO(6)–${\rm SU}_-(3)$ axis the IBM Hamiltonian corresponds to a potential with a deformed minimum. Furthermore, in the region around prolate ${\rm SU}_-(3)$ ($\chi<0$) the minimum occurs for $\gamma=0^{\rm o}$ while around oblate ${\rm SU}_+(3)$ ($\chi>0$) it does for $\gamma=60^{\rm o}$. In this way the picture emerges that the IBM parameter space can be divided into three regions according to the character of the associated potential having (I) a spherical minimum, (II) a prolate deformed minimum or (III) an oblate deformed minimum. The boundaries between the different regions (the so-called Maxwell set) are indicated by the dashed lines in Fig. \[f\_trib\] and meet in a triple point. The spherical–deformed border region displays another interesting phenomenon. Since the [*absolute*]{} minimum of the potential must be either spherical, or prolate or oblate deformed, its character uniquely determines the three regions and the dividing Maxwell lines. Nevertheless, this does not exclude the possibility that, in passing from one region to another, the potential may display a second [*local*]{} minimum. This indeed happens for the U(5)–SU(3) transition [@Iachello98] where there is a narrow region of coexistence of a spherical and a deformed minimum, indicated by the shaded area in Fig. \[f\_trib\]. Since, at the borders of this region of coexistence, the potential undergoes a [*qualitative*]{} change of character, the boundaries are genuine critical lines of the potential surface [@Gilmore81].
Although these geometric results have been obtained with reference to the simplified Hamiltonian (\[e\_ibmhamcqf\]) and its associated ‘triangular’ parameter space, they remain valid for the general IBM Hamiltonian with up to two-body interactions [@Lopez96].
Summary {#s_conc}
=======
In these lecture notes an introduction was given to the notions of symmetry and dynamical symmetry (or spectrum generating algebra). Their use in the solution of the (nuclear) many-body problem was described. Two particular examples of these techniques were discussed in detail: (i) SO(4) symmetry of the hydrogen atom and (ii) isospin symmetry in nuclei. A review was given of the shell model and the interacting boson model, with particular emphasis on the application of group-theoretical techniques in the context of these models.
Acknowledgment {#acknowledgment .unnumbered}
==============
This paper is dedicated to the memory of Marcos Moshinsky, the intellectual father of Mexican physics and founder of the [*Escuela Latino Americana de Física*]{}. The two years I have spent in Mexico as a visitor and the many hours with Marcos as a teacher, were crucial to my formation as a physicist. Without him I never could have given these lectures.
[99]{} R. Gilmore, [*Lie Groups, Physics, and Geometry*]{}, Cambridge, Cambridge University Press, 2008.
J.-P. Blaizot and J.-C. Tolédano, [*Symétries et Physique Microscopique*]{}, Ellipses, Paris, 1997.
B.G. Wybourne, [*Classical Groups for Physicists*]{}, Wiley-Interscience, New York, 1974.
M. Moshinsky, [*The Harmonic Oscillator in Modern Physics: From Atoms to Quarks*]{}, Gordon & Breach, New York, 1969.
W. Pauli, [*Z. Phys. *]{} [**36**]{}, 336 (1926).
E.P. Wigner, [*Phys. Rev. *]{} [**51**]{}, 106 (1937).
E.P. Wigner, [*Proceedings of the Robert A Welch Foundation Conferences on Chemical Research.I The Structure of the Nucleus*]{}, edited by W.O. Millikan (Welch Foundation, Houston, 1957) p 86.
J. Britz, A. Pape and M.S. Antony, [*At. Data Nucl. Data Tables*]{} [**69**]{}, 125 (1998).
C.D. O’Leary, M.A. Bentley, D.E. Appelbe, D.M. Cullen, S. Ertürk, R.A. Bark, A. Maj and T. Saitoh, [*Phys. Rev. Lett. *]{} [**79**]{}, 4349 (1997).
M.A. Bentley and S. Lenzi, [*Prog. Part. Nucl. Phys. *]{}[**59**]{}, 497 (2007).
L.E.H. Trainor, [*Phys. Rev. *]{} [**85**]{}, 962 (1952).
L.A. Radicati, [*Phys. Rev. *]{} [**87**]{}, 521 (1952).
J.M. Freeman, J.G. Jenkin, G. Murray and W.E. Burcham, [*Phys. Rev. Lett. *]{} [**16**]{}, 959 (1966).
E. Farnea [*et al.*]{}, [*Phys. Lett. B*]{} [**551**]{}, 56 (2003).
A. de-Shalit and I. Talmi, [*Nuclear Shell Theory*]{}, Academic, New York, 1963.
A. Bohr and B.R. Mottelson, [*Nuclear Structure. I Single-Particle Motion*]{}, Benjamin, Reading, Massachusetts, 1969.
R.D. Lawson, [*Theory of the Nuclear Shell Model*]{}, Clarendon, Oxford, 1980.
K. Heyde, [*The Nuclear Shell Model*]{}, Springer, Berlin, 1994.
I. Talmi, [*Simple Models of Complex Nuclei. The Shell Model and Interacting Boson Model*]{}, Harwood, Chur, 1993.
P. Van Isacker, [*Rep. Prog. Phys. *]{} [**62**]{}, 1661 (1999).
G. Audi, A.H. Wapstra and C. Thibault, [*Nucl. Phys. A*]{} [**729**]{}, 337 (2003).
G. Racah, [*Phys. Rev. *]{} [**63**]{}, 367 (1943).
J. Bardeen, L.N. Cooper and J.R. Schrieffer, [*Phys. Rev. *]{} [**106**]{}, 162 (1957); [**108**]{}, 1175 (1957).
A. Bohr, B.R. Mottelson and D. Pines, [*Phys. Rev. *]{} [**110**]{}, 936 (1958).
J.P. Elliott, [*Proc. Roy. Soc. (London) A*]{} [**245**]{}, 128 (1958); 562 (1958).
F. Iachello and A. Arima, [*The Interacting Boson Model*]{}, Cambridge University Press, Cambridge, 1987.
O. Castaños, E. Chacón, A. Frank and M. Moshinsky, [*J. Math. Phys. *]{} [**20**]{}, 35 (1979).
A. Arima and F. Iachello, [*Ann. Phys. (NY)*]{} [**99**]{}, 253 (1976).
A. Arima and F. Iachello, [*Ann. Phys. (NY)*]{} [**111**]{}, 201 (1978).
A. Arima and F. Iachello, [*Ann. Phys. (NY)*]{} [**123**]{}, 468 (1979).
A.M. Shirokov, N.A. Smirnova and Yu.F. Smirnov, [*Phys. Lett. B*]{} [**434**]{}, 237 (1998).
Y. Alhassid and A. Leviatan, [*J. Phys. A*]{} [**25**]{}, L1265 (1992).
A. Leviatan, [*Phys. Rev. Lett. *]{}[**77**]{}, 818 (1996).
A. Leviatan, A. Novoselski and I. Talmi, [*Phys. Lett. B*]{} [**172**]{}, 144 (1986).
P. Van Isacker, [*Phys. Rev. Lett. *]{}[**83**]{}, 4269 (1999).
A. Leviatan and P. Van Isacker, [*Phys. Rev. Lett. *]{}[**89**]{}, 222501 (2002).
T. Otsuka, A. Arima and F. Iachello, [*Nucl. Phys. A*]{} [**309**]{},1 (1978).
M.J. Thompson, J.P. Elliott and J.A. Evans, Phys. Lett. B [195]{}, 511 (1987).
O. Juillet, P. Van Isacker and D.D. Warner, [*Phys. Rev. C*]{} [**63**]{}, 054312 (2001).
J.N. Ginocchio and M.W. Kirson, [*Phys. Rev. Lett. *]{}[**44**]{}, 1744 (1980).
A.E.L. Dieperink, O. Scholten and F. Iachello, [*Phys. Rev. Lett. *]{}[**44**]{}, 1747 (1980).
A. Bohr and B.R. Mottelson, [*Phys. Scripta*]{} [**22**]{}, 468 (1980).
A. Bohr and B.R. Mottelson, [*Nuclear Structure. II Nuclear Deformations*]{}, Benjamin, Reading, Massachusetts, 1975.
P. Van Isacker and J.-Q. Chen, [*Phys. Rev. C*]{} [**24**]{}, 684 (1981).
R. Gilmore, [*Catastrophe Theory for Scientists and Engineers*]{}, Wiley, New York, 1981.
E. López-Moreno and O. Castaños, [*Phys. Rev. C*]{} [**54**]{}, 2374 (1996).
J. Jolie, R.F. Casten, P. von Brentano and V. Werner, [*Phys. Rev. Lett. *]{}[**87**]{}, 162501 (2001).
F. Iachello, N.V. Zamfir and R.F. Casten, [*Phys. Rev. Lett. *]{}[**81**]{}, 1191 (1998).
[^1]: The notation with a tilde is used to distinguish the radial part of the wave function from the $R_{n\ell}(r)$ for the harmonic oscillator that will be encountered in Sect. \[s\_shell\].
[^2]: Throughout these lecture notes, small letters are normally reserved for operators associated with a single particle and capital letters for operators summed over many particles. This section deals with one-particle operators but for clarity’s sake it is important to distinguish between operators, which shall be denoted in this section by capital letters $L$, $P$,…, and their associated labels, which shall be denoted by corresponding small letters $\ell$, $p$,….
|
---
abstract: 'We use the “higher Hida theory” recently introduced by the second author to $p$-adically interpolate periods of non-holomorphic automorphic forms for $\operatorname{GSp}_4$, contributing to coherent cohomology of Siegel threefolds in positive degrees. We apply this new method to construct $p$-adic $L$-functions associated to the degree 4 (spin) $L$-function of automorphic representations of $\operatorname{GSp}_4$, and the degree 8 $L$-function of $\operatorname{GSp}_4 \times \operatorname{GL}_2$.'
address:
- |
David Loeffler, Mathematics Institute\
University of Warwick\
Coventry CV4 7AL, UK.
- |
Vincent Pilloni, CNRS\
Ecole Normale Superieure de Lyon\
Lyon, France.
- |
Christopher Skinner\
Department of Mathematics\
Princeton University\
Princeton, NJ 08544-1000\
USA.
- |
Sarah Livia Zerbes, Department of Mathematics\
University College London\
London WC1E 6BT, UK.
author:
- David Loeffler
- Vincent Pilloni
- Christopher Skinner
- Sarah Livia Zerbes
title: 'Higher Hida theory and $p$-adic $L$-functions for $\operatorname{GSp}_4$'
---
=1
Introduction
============
Background
----------
Several of the most important open problems in mathematics involve the arithmetic significance of special values of $L$-functions; and a major role in work on these problems is played by $p$-adic $L$-functions. There are, essentially, two main approaches to constructing these objects: “topological” constructions via Betti cohomology of symmetric spaces (such as the theory of modular symbols for $\operatorname{GL}_2$); or constructions of a more “algebro-geometric” nature, using Shimura varieties and sections of coherent sheaves over them, as in Hida and Panchishkin’s construction of $p$-adic $L$-functions for $\operatorname{GL}_2 \times \operatorname{GL}_2$.
The most powerful applications of $p$-adic $L$-functions are in cases where one can relate the $p$-adic $L$-function to a family of classes in Galois cohomology (an Euler system). Results of this kind are known as *explicit reciprocity laws*, and have hugely important consequences, leading to cases of the Birch–Swinnerton-Dyer conjecture and the Bloch–Kato conjecture, as in the work of Kato [@kato04] and more recently [@DR16], [@KLZ17] and others. However, a prerequisite for such an explicit reciprocity law is to have a construction of the $p$-adic $L$-function by algebro-geometric techniques, which can be related to Galois representations in étale cohomology.
At this point, one hits a serious obstacle. There are many integral formulae known which relate values of $L$-functions to cohomology of automorphic sheaves on Shimura varieties. However, these automorphic sheaves can have cohomology in a range of degrees, and the $L$-value formulae that are relevant in Euler system settings always involve cohomology classes near the middle of the range of possible degrees. On the other hand, the established techniques for studying $p$-adic variation of these objects are only applicable to sections, i.e. to cohomology in degree 0 (or, via Serre duality, to cohomology in the highest degree). This incompatibility is a fundamental limitation in the theory as it presently stands: because of this, all the reciprocity laws known so far relate to Shimura varieties which have small dimension, or which factorise as a product of two simpler subvarieties of approximately equal dimension. In particular, the previously-known techniques are not sufficient to prove an explicit reciprocity law for the Euler system constructed in [@loefflerskinnerzerbes17] for automorphic representations of $\operatorname{GSp}_4$; this is the major obstacle that must be solved in order to use this new Euler system to prove the Bloch–Kato conjecture for automorphic motives attached to this group. The same difficulty arises for several other recently-discovered Euler systems, such as those of [@LLZ18; @CLR19; @LSZ-unitary].
Our results
-----------
In this paper, we develop a new algebro-geometric approach to constructing $p$-adic $L$-functions which resolves this difficulty, and apply it to construct $p$-adic spin $L$-functions attached to automorphic representations of the group $\operatorname{GSp}_4$. This is the basis for a forthcoming sequel in which we shall prove an explicit reciprocity law relating the $p$-adic $L$-function of the present paper to the Euler system of [@loefflerskinnerzerbes17]. We expect that the techniques of this paper should also be applicable to $p$-adic interpolation of many other automorphic $L$-functions beyond the examples we study here. For instance, the cases of the Asai $L$-function of a quadratic Hilbert modular form, and the degree 6 $L$-function of an automorphic representation of $\operatorname{GU}(2, 1)$, will be treated in forthcoming work.
Our new approach to constructing $p$-adic $L$-functions relies crucially on the *higher Hida theory* introduced by the second author in [@pilloni17]. This gives a theory which $p$-adically interpolates coherent cohomology of Siegel threefolds in positive degrees, while conventional Hida theory only sees $H^0$. Since Harris has shown in [@harris04] that the critical values of the spin $L$-function can be interpreted as cup-products involving coherent cohomology in degrees 1 and 2, this gives a path by which to approach the $p$-adic interpolation of spin $L$-values.
Let us now state our results a little more precisely. Let $\Pi$ be a cuspidal automorphic representation of $\operatorname{GSp}_4({\mathbf{A}}_{\mathbf{Q}})$ which is non-CAP, globally generic, and cohomological with coefficients in the algebraic representation of highest weight $(r_1, r_2)$, for some integers $r_1 {\geqslant}r_2 {\geqslant}0$. For technical reasons we need to suppose that $r_2 {\geqslant}1$. Let $p$ be a prime such that $\Pi_p$ is unramified and Klingen-ordinary (with respect to some choice of embedding $\overline{{\mathbf{Q}}} {\hookrightarrow}\overline{{\mathbf{Q}}}_p$).
Suppose that either $d = r_1 - r_2 {\geqslant}1$ or that $d = 0$ and Hypothesis \[hyp:non-vanish\] holds. Then there exist two constants $\Omega^+_\Pi$, $\Omega^-_\Pi \in {\mathbf{C}}^\times$, uniquely determined modulo $\overline{{\mathbf{Q}}}{}^\times$, and a $p$-adic measure $\mu_{\Pi}$ on ${{\mathbf{Z}}_p}^\times$, such that for all Dirichlet characters $\chi$ of $p$-power conductor and all integers $0 {\leqslant}a {\leqslant}d$, we have $$\int_{{{\mathbf{Z}}_p}^\times} x^a \chi(x)\, \mathrm{d}\mu_\Pi(x) = (-1)^a R_p(\Pi, a, \chi) \cdot \frac{\Lambda(\Pi \otimes \chi^{-1}, \tfrac{1-d}{2} + a)}{\Omega_\Pi^{\pm}},$$ where $R(\Pi, a, \chi)$ is an explicit non-zero factor, and the sign $\pm$ denotes $(-1)^a \chi(-1)$.
Let $\sigma$ be a cuspidal automorphic representation of $\operatorname{GL}_2({\mathbf{A}}_{\mathbf{Q}})$ generated by a holomorphic modular form of weight $\ell$, with $1 {\leqslant}\ell {\leqslant}r_1 - r_2 + 1$, and level coprime to $p$ and to the primes of ramification of $\Pi$. Let $d' = r_1 - r_2 - \ell + 1 {\geqslant}0$.
Then there is a constant $\Omega_\Pi^W$, uniquely determined modulo $\overline{{\mathbf{Q}}}{}^\times$, and a $p$-adic measure $\mu_{\Pi \times \sigma}$ on ${{\mathbf{Z}}_p}^\times$, such that for all Dirichlet characters $\chi$ of $p$-power conductor and all integers $0 {\leqslant}a {\leqslant}d'$, we have $$\int_{{{\mathbf{Z}}_p}^\times} x^a \chi(x)\, \mathrm{d}\mu_{\Pi \times \sigma}(x) = R_p(\Pi \otimes \sigma, a, \chi) \cdot \frac{\Lambda(\Pi \otimes \sigma \otimes \chi^{-1}, \tfrac{1-d'}{2} + a)}{\Omega_\Pi^{W}},$$ where $R(\Pi \otimes \sigma, a, \chi)$ is an explicit non-zero factor. If the hypotheses of Theorem A are satisfied then we have $\Omega_\Pi^{W} = \Omega_\Pi^+ \cdot \Omega_\Pi^-$.
In fact the exact statements are a little stronger, although more complicated to state – we refer to §9 below for the details. In both theorems, $\Lambda(-, s)$ denotes the completed $L$-function (including its Archimedean $\Gamma$-factors); the factors $R_p(-)$ are local Euler factors at $p$, consistent with those predicted by general conjectures of Panchishkin and Coates–Perrin-Riou; and the range of values for $a$ corresponds to the whole interval of critical values of the $L$-functions concerned.
The first step in our strategy is to extend the higher Hida theory of [@pilloni17], which was developed there for applications to “non-regular” weights (limits of holomorphic discrete series), to cover regular weights. This is carried out in §\[sect:vincent\]. The chief new result here is that for regular weights lying in the relevant Weyl chamber, the perfect complex representing the ordinary part of $p$-adic cohomology is concentrated purely in degree 1.
In §\[sect:p-adic-pushfwd\] we prove our second main technical result: a “functoriality” property for higher Hida theory relating the group $G = \operatorname{GSp}_4$ and its subgroup $H = \operatorname{GL}_2 \times_{\operatorname{GL}_1} \operatorname{GL}_2$. This shows that $p$-adic families of modular forms for the subgroup $H$ can be pushed forward to $p$-adic families in $H^1$ of $G$. This allows us to $p$-adically interpolate period integrals of the form $$\int_{{\mathbf{R}}^\times H({\mathbf{Q}}) \backslash H({{\mathbf{A}}_{\mathrm{f}}})} F(\iota(h)) f_1(h_1) f_2(h_2)\, \mathrm{d}h,$$ where $F$ is an automorphic form for $G$ contributing to $H^2$ of the Siegel threefold, and $f_1$ and $f_2$ are holomorphic modular forms varying in $p$-adic families.
However, for our desired applications, we also need to consider integrals of the above shape in which one of the $f_i$ is not holomorphic but only *nearly-holomorphic* (the image of a holomorphic form under a power of the Maass–Shimura differential operator). Hence the next step in our strategy, carried out in §\[sect:nearly\], is to develop a theory of “nearly” coherent cohomology for $G$, and an analogue of the $p$-adic unit-root splitting to relate these spaces to higher Hida theory. These are analogues for higher Hida theory of results recently proved by Zheng Liu in the setting of classical Hida theory for symplectic groups [@liu-thesis].
Finally, it remains to show that these period integrals can be interpreted as values of $L$-functions. For this we use two integral formulae of Rankin–Selberg type: a 1-parameter integral formula due to Novodvorsky for the degree 8 $L$-function of $\operatorname{GSp}_4 \times \operatorname{GL}_2$, and a 2-parameter integral formula giving the product of two copies of the $\operatorname{GSp}_4$ $L$-function, which is an extension of work of Piatetski-Shapiro. In §§\[sect:localintegrals\] and \[sect:globalintegrals\] we define the local and global versions of these integrals, and evaluate the local integrals at $p$ and at $\infty$ for the specific choices of test data that arise in our construction. Finally, in §\[sect:final\] we put together all of the above pieces to prove our main theorems.
(These formulae should both be seen as “degenerate cases” of a third, presently conjectural, integral formula: the Gan–Gross–Prasad conjecture predicts that if $f_1$ and $f_2$ are both cuspidal, then the above integral should be related to the square root of the central value of the degree 16 $L$-function for $\operatorname{GSp}_4 \times \operatorname{GL}_2 \times \operatorname{GL}_2$. So, assuming the GGP conjecture, our methods give $p$-adic $L$-functions interpolating these square roots as $f_1$ and $f_2$ vary in cuspidal Hida families. However, we shall not treat this case in detail here, for reasons of space.)
Relation to other work
----------------------
We note that Theorem A can be seen as a consequence of a theorem of Dimitrov–Janusewski–Raghuram [@DJR18] (applied to the lifting of $\Pi$ to $\operatorname{GL}_4$). However, our proof of the theorem is very different: their construction is of a topological nature, using Betti cohomology of a symmetric space associated to $\operatorname{GL}_4$, while ours is algebro-geomeric, using coherent cohomology of the Shimura variety associated to $G$. This allows us to prove Theorem A in parallel with Theorem B, which does not seem to be accessible using the methods of [@DJR18]. More importantly, as mentioned above, working on the $\operatorname{GSp}_4$ Shimura variety, and using algebraic rather than topological methods, are vital in order to relate the $p$-adic $L$-functions we construct to the Euler system of [@loefflerskinnerzerbes17]. We shall pursue this further in a forthcoming paper, in which we shall relate the values of the $p$-adic $L$-function of Theorem A (at points outside the range of interpolation) to the syntomic regulators of Euler system classes. One can also use this technique to construct multi-variable $p$-adic $L$-functions. For instance, in the setting of Theorem B, one can allow $f_1$ to vary through a Hida family of cusp forms, giving a 2-variable $p$-adic $L$-function in which both the weight of $f_1$ and the cyclotomic twist are varying. As a special case of this, taking $f_1$ to be a CM-type family, one obtains a measure interpolating the $L$-values of twists of $\Pi$ by Grössencharacters of an imaginary quadratic field. In an alternative direction, one can take both $f_1$ and $f_2$ to be cusp forms varying in Hida families, giving a 2-variable measure interpolating automorphic periods of Gan–Gross–Prasad type for the pair $(\operatorname{SO}_4, \operatorname{SO}_5$); this is a higher-rank analogue of the triple-product $L$-function studied by Harris–Tilouine and Darmon–Rotger [@harristilouine01; @darmonrotger14], which interpolates Gan–Gross–Prasad periods for the pair $(\operatorname{SO}_3, \operatorname{SO}_4)$. However, for reasons of space we shall pursue these generalisations in a subsequent paper.
Acknowledgements {#acknowledgements .unnumbered}
----------------
This project developed out of conversations at the workshop “Motives, Galois Representations and Cohomology” hosted by the Institute of Advanced Study, Princeton, in November 2017. The three of us who attended this workshop (Loeffler, Pilloni and Zerbes) are very grateful to the Institute and the organisers of the workshop for inviting them to participate. We are also grateful for the support of the Centre Bernoulli in Lausanne, and the Morningside Centre in Beijing, who hosted various subsets of the authors for extended visits during the preparation of this paper.
We are also grateful to Michael Harris, Mark Kisin, and Kai-Wen Lan for helpful remarks on the topic of toroidal compactifications; and to Laurent Moret-Bailly for pointing out a valuable reference regarding properties of multiplicative group schemes (in answer to a question of ours on the website “MathOverflow”).
Preliminaries: Groups and Shimura varieties
===========================================
Groups {#sect:groups}
------
We denote by $G = \operatorname{GSp}_4$ the group scheme (over ${\mathbf{Z}}$) associated to the skew-symmetric matrix $J = \left( \begin{smallmatrix} &&& 1\\ && 1 &\\ & -1 &&\\ -1 &&&\end{smallmatrix} \right) $. The standard Siegel and Klingen parabolics are then given by $$\begin{aligned}
P_{\operatorname{S}}&=
\left( \begin{smallmatrix}
\star & \star & \star & \star \\
\star & \star & \star & \star \\
& & \star & \star \\
& & \star & \star
\end{smallmatrix} \right) , &
P_{\operatorname{Kl}}&=
\left( \begin{smallmatrix}
\star & \star & \star & \star \\
& \star & \star & \star \\
& \star & \star & \star \\
& & & \star
\end{smallmatrix} \right) .\end{aligned}$$ We write $M_{\operatorname{S}}$, $M_{\operatorname{Kl}}$ for the standard (block-diagonal) Levi subgroups of $P_{\operatorname{S}}$ and $P_{\operatorname{Kl}}$, and $T$ for the diagonal maximal torus.
Let $H=\operatorname{GL}_2\times_{\operatorname{GL}_1}\operatorname{GL}_2$. We define an embedding $\iota: H\hookrightarrow G$ by $$\left(\begin{pmatrix} a & b \\ c & d\end{pmatrix}, \begin{pmatrix} a' & b'\\ c' & d'\end{pmatrix}\right) \mapsto \left( \begin{smallmatrix} a &&& b\\ & a' & b' & \\ & c' & d' & \\c &&&d \end{smallmatrix} \right) .$$
Shimura varieties
-----------------
### Open varieties over ${\mathbf{Q}}$
Let $K$ be a neat open compact subgroup of $\operatorname{GSp}_4({{\mathbf{A}}_{\mathrm{f}}})$. Denote by $Y_{G, {\mathbf{Q}}}$ the canonical model over ${\mathbf{Q}}$ of the level $K$ Shimura variety. This is a smooth quasiprojective threefold, endowed with an isomorphism of complex manifolds $$\label{eq:cplxunif}
Y_{G, {\mathbf{Q}}}({\mathbf{C}}) \cong G({\mathbf{Q}})_+ \backslash \left[ {\mathcal{H}}_2 \times G({{\mathbf{A}}_{\mathrm{f}}}) / K\right],$$ where ${\mathcal{H}}_2$ is the genus 2 Siegel upper half-space. It can be identified with the moduli space of abelian surfaces endowed with principal polarisations and level $K$ structures.
We write $Y_{H, {\mathbf{Q}}}$ for the canonical ${\mathbf{Q}}$-model of the Shimura variety for $H$ of level $K_H = K \cap H({{\mathbf{A}}_{\mathrm{f}}})$, which is a moduli space for (ordered) pairs of elliptic curves with level structure. (Note that if $K = K_G(N)$ is the principal congruence subgroup, then $Y_{H, {\mathbf{Q}}}$ is the fibre product of two copies of the modular curve $Y(N)$ over $\mu_N$.) There is a morphism of algebraic varieties $$\label{eq:iota1}
\iota: Y_{H, {\mathbf{Q}}} \to Y_{G, {\mathbf{Q}}},$$ with image a closed codimension 1 subvariety of $Y_{G, {\mathbf{Q}}}$ (a *Humbert surface*). If $K$ is contained in the principal congruence subgroup $K_G(N)$ for some $N {\geqslant}3$, then $\iota$ is a closed immersion, by [@loefflerskinnerzerbes17 Proposition 5.3.1].
### Integral models and levels at $p$
Let $p$ be prime, and suppose $K = K^p K_p$ with $K^p \subset G({{\mathbf{A}}_{\mathrm{f}}}^p)$ and $K_p = G({{\mathbf{Z}}_p})$. Then $Y_{H, {\mathbf{Q}}}$ and $Y_{G, {\mathbf{Q}}}$ have canonical smooth models over ${\mathbf{Z}}_{(p)}$ (parametrising abelian surfaces over ${\mathbf{Z}}_{(p)}$-algebras with appropriate additional structures) which we denote simply by $Y_{H}$ and $Y_{G}$. The morphism $\iota$ extends to a morphism $Y_H \to Y_G$; if $K^p$ is contained in $K_G(N)$ for some $N {\geqslant}3$ coprime to $p$, which we shall assume, then by the same arguments used for the generic fibre above, one can check that this is a closed immersion of ${\mathbf{Z}}_{(p)}$-schemes.
For $n {\geqslant}1$, we let $\operatorname{Kl}(p^n) = \{ g \in G({{\mathbf{Z}}_p}): g \pmod{p^n} \in P_{\operatorname{Kl}}\}$, and we denote by $Y_{G, \operatorname{Kl}}(p^n)_{{\mathbf{Q}}}$ the canonical ${\mathbf{Q}}$-model of the Shimura variety of level $K^p \operatorname{Kl}(p^n)$. These do not have natural smooth ${\mathbf{Z}}_{(p)}$-models.
### Toroidal compactifications {#sect:toroidal}
As in [@faltingschai], we may define arithmetic toroidal compactifications of $Y_G$, depending on a choice of combinatorial datum $\Sigma$ (a rational polyhedral cone decomposition, or “rpcd” for short). We shall restrict attention to rpcd’s which are “good” in the sense of [@pilloni17 §5.3.2]. A choice of good rpcd $\Sigma$ gives rise to an open embedding of ${{\mathbf{Z}}_p}$-schemes $Y_G {\hookrightarrow}X_G^\Sigma$ with the following properties:
- $X_G^\Sigma$ is smooth and projective over ${{\mathbf{Z}}_p}$.
- The boundary $D_G^\Sigma = X_G^\Sigma - Y_G$ is a relative Cartier divisor.
Any such compactification $X_G^\Sigma$ maps naturally to the minimal compactification $X_G^{\mathrm{min}}$. If $\Sigma'$ is a refinement of $\Sigma$, then there is a projective morphism $\pi_{(\Sigma', \Sigma)}: X_G^{\Sigma'} \to X_G^\Sigma$ (compatible with the maps from $Y_G$ and to $X_G^{\mathrm{min}}$); and we have $\pi_{(\Sigma', \Sigma)}^*(I_G^\Sigma) = I_G^{\Sigma'}$, where $I_G^\Sigma$ is the ideal sheaf of the boundary $D_G^\Sigma$ in $X_G^\Sigma$.
Over $X_G^\Sigma$ there is a canonical semiabelian variety $A_G^\Sigma$, extending the universal abelian variety over $Y_G$; and a $P_S$-torsor $\mathcal{T}_G^\Sigma$, parametrising trivialisations, as a filtered vector bundle, of the canonical extension to $X_G^\Sigma$ of the first relative de Rham cohomology of $A_G / Y_G$ (see [@madapusipera §4.3]). These are all compatible with refining the cone decomposition $\Sigma$.
The same statements as above hold *mutatis mutandis* for the Shimura varieties of level $K^p \operatorname{Kl}(p^n)$, although the resulting ${{\mathbf{Z}}_p}$-models are no longer smooth. Similarly, we can define toroidal compactifications for $H$ in place of $G$. In this case, there is an “optimal” choice of $\Sigma$, for which the map $X_H^\Sigma \to X_H^{\mathrm{min}}$ is an isomorphism, but this is not the only possible choice.
If $K_H = K_1 \times K_2$ is the fibre product of subgroups of the $\operatorname{GL}_2$ factors, then $Y_H$ is a subset of the components of $Y_1 \times Y_2$, where $Y_i$ is the modular curve of level $K_i$; and $X_H^{\min}$ is the product of their compactifications $X_i$. Any other toroidal compactification $X_H^{\Sigma}$ is obtained from $X_H^{\mathrm{min}}$ by blowing up at some sheaf of ideals supported at points of the form $\{\mathrm{cusp}\} \times \{\mathrm{cusp}\}$.
### Functoriality of the compactifications {#sect:toroidalfunct}
As explained in [@harris89 Proposition 3], any rpcd $\Sigma$ for $G$ uniquely determines an rpcd $\iota^*(\Sigma)$ for $H$. It has recently been shown by Lan [@lan19] that if $K^p$ is sufficiently small, one may choose an rpcd $\Sigma$ for $G$ such that:
- both $\Sigma$ and $\iota^*(\Sigma)$ are good (for $G$ and $H$ respectively),
- the map $\iota$ extends to a closed embedding of ${\mathbf{Z}}_{(p)}$-schemes $\iota_{\Sigma}: X_H^{\iota^*(\Sigma)} {\hookrightarrow}X_G^{\Sigma}$.
We shall fix a choice of $\Sigma$ satisfying this condition. It follows from the construction of the torsors ${\mathcal{T}}_G^{\Sigma}$ and ${\mathcal{T}}_H^{\Sigma}$ that we have an isomorphism $$\label{eq:redstructure}
\iota_{\Sigma}^*\left( {\mathcal{T}}_G^{\Sigma} \right) = P_{\operatorname{S}} \times^{B_H} {\mathcal{T}}_H^{\Sigma},$$ so that ${\mathcal{T}}_H^{\Sigma}$ is a reduction of structure of $\iota_{\Sigma}^*\left( {\mathcal{T}}_G^{\Sigma} \right)$ from a $P_{\operatorname{S}}$-torsor to a $B_H$-torsor. We also note the inclusion of ideal sheaves $$\label{eq:idealsheaf}
\iota_{\Sigma}^*\left(I_G^\Sigma\right) \subseteq I_H^{\Sigma}.$$
We shall frequently omit the superscript $\Sigma$ from the notation $X_G^\Sigma$, $D_G^\Sigma$ etc when there is no risk of ambiguity (and sometimes the subscripts $G, H$ as well).
It seems likely that one can choose $\Sigma$ in such a way that is an equality, but we have not verified this.
Representations and coefficient sheaves
---------------------------------------
### Weights and representations
As in [@pilloni17 §5.1.1], the character group $X^\bullet(T)$ can be identified with the group of triples $(r_1, r_2; c) \in {\mathbf{Z}}^3$ such that $c = r_1 + r_2 \bmod 2$, by defining $\lambda(r_1, r_2; c)$ as the unique character of $T$ such that $${\left( \begin{smallmatrix} st_1\\ &st_2 \\ &&s t_2^{-1} \\ &&& st_1^{-1}\end{smallmatrix} \right) } \mapsto t_1^{r_1} t_2^{r_2} s^c.$$
The weights $\lambda(r_1, r_2; c)$ which are dominant for $M_{\operatorname{S}}$ are those with $r_1 {\geqslant}r_2$; if $(r_1, r_2; c)$ satisfies this, we write $W_G(r_1, r_2; c)$ for the irreducible representation of $M_{\operatorname{S}}$ with this highest weight. Those weights which also satisfy $r_2 {\geqslant}0$ are dominant for $G$, and in this case we write $V_G(r_1, r_2; c)$ for the corresponding $G$-representation.
The torus $T$ is also a maximal torus of $H$, and we write $W_H(r_1, r_2; c)$ for the 1-dimensional representation of $M_{B_H} = T$ on which $T$ acts via $\lambda(r_1, r_2; c)$. If $r_1, r_2 {\geqslant}0$, then we write $V_H(r_1, r_2; c)$ for the representation of $H$ of highest weight $\lambda(r_1, r_2; c)$; concretely, we have $$V_H(r_1, r_2; c) \coloneqq \left(\operatorname{Sym}^{r_1} \boxtimes \operatorname{Sym}^{r_2}\right) \otimes \det{}^{(c - r_1 - r_2)/2}.$$
We have changed notations for $G$-representations relative to [@loefflerskinnerzerbes17]; the $G$-representation denoted $V_{a,b}$ in *op.cit.* is $V(a+b, a; 2a + b)$ in the new notation. The new notation has the advantage of greatly simplifying the formulae for the action of the Weyl group: the Weyl group conjugates of $\lambda(r_1, r_2; c)$ are the weights $\{ \lambda(\pm r_1, \pm r_2, c), \lambda(\pm r_2, \pm r_1, c)\}$.
### Automorphic vector bundles
If $V$ is an algebraic representation of $P_{\operatorname{S}}$ over ${\mathbf{Z}}_{(p)}$, then we have a vector bundle $[V]$ on $X_G$ defined by $$[V] \coloneqq V \times^{P_{\operatorname{S}}} {\mathcal{T}}_G,$$ where ${\mathcal{T}}_G$ is the canonical $P_{\operatorname{S}}$-torsor over $X_G$ defined above (see e.g. [@liu-thesis §2.1]). The same applies, of course, to $G$ in place of $H$. This construction is obviously compatible with direct sums, tensor products, and duals.
Over ${\mathbf{C}}$ the automorphic bundles $[V]$ have a convenient interpretation in terms of the complex uniformisation . We can interpret ${\mathcal{H}}_2$ as a $G({\mathbf{R}})_+$-invariant open subset of ${\mathcal{F}}_{\operatorname{S}}({\mathbf{C}})$, where ${\mathcal{F}}_{\operatorname{S}}$ is the Siegel flag variety $G / P_{\operatorname{S}}$. Any $P_{\operatorname{S}}$-representation $V$ gives rise to a $G({\mathbf{C}})$-equivariant holomorphic vector bundle over ${\mathcal{F}}_{\operatorname{S}}$, and $[V]_{{\mathbf{C}}}$ is the pullback of this to $X_{G, {\mathbf{C}}}$. Note that the real-analytic vector bundle $[V]_{C^\infty}$ obtained from $[V]_{{\mathbf{C}}}$ by forgetting the holomorphic structure depends only on the restriction of $V$ to $M_{\operatorname{S}}$ (but the holomorphic structure genuinely does depend on $V$ as a $P_{\operatorname{S}}$-representation).
By construction, if we take $V = V(1, 0; 1)$ to be the defining 4-dimensional representation of $G$, then $[V]$ is the relative logarithmic de Rham cohomology sheaf $\mathcal{H}^1_{{\mathrm{dR}}}(A_G)$ of the universal semi-abelian surface $A_G / X_G$, and the evident two-step filtration of $V$ as a $P_{\operatorname{S}}$-representation corresponds to the Hodge filtration of $\mathcal{H}^1_{{\mathrm{dR}}}(A_G)$. Note that this is slightly non-standard (it is more usual to send $V$ to the dual ${\mathcal{H}}_1^{{\mathrm{dR}}}$), but is consistent with the conventions used for étale and motivic sheaves in [@loefflerskinnerzerbes17]. As a representation of $M_{\operatorname{S}}$, $V$ splits as a direct sum of 2-dimensional subspaces; this corresponds to the *Hodge splitting* of the bundle $[V]$ in the real-analytic category.
\[not:sheaves\] We write $\omega_G(r_1, r_2; c)$ for the vector bundle $[W_G(r_1, r_2; c)]$ on $X_G$, and similarly for $H$. We write $\omega_G(r_1, r_2) = \omega_G(r_1, r_2; r_1 + r_2 - 6)$, and similarly $\omega_H(r_1, r_2) = \omega_H(r_1, r_2; r_1 + r_2 - 4)$.
The sheaf $\omega_G(r_1, r_2)$ is isomorphic to $\operatorname{Sym}^{r_1 - r_2}(\omega_A) \otimes \det(\omega_A)^{r_2}$, where $\omega_A$ is the conormal bundle at the identity section of the semi-abelian scheme $A_G$ over $X_{G}$; in other words, $\omega_G(r_1, r_2)$ is the sheaf that was denoted by $\Omega^{(k, r)}$ in [@pilloni17], for $(k, r) = (r_1 - r_2, r_2)$. Our choice of “default” normalisation for the central character coincides with that chosen in Remark 5.3.1 of *op.cit.*.
The Kodaira–Spencer construction identifies the sheaves of logarithmic differentials $\Omega^i_{X_G}(\log D_G)$ and $\Omega^i_{X_H}(\log D_H)$ with automorphic vector bundles. These are given by $$\begin{array}{r|cccc}
i & 0 & 1 & 2 & 3\\
\hline
\rule{0pt}{2.5ex} \Omega^i_{X_G}(\log D_G) &
\omega_G(0, 0; 0) &
\omega_G(2, 0; 0) &
\omega_G(3, 1; 0) &
\omega_G(3, 3; 0) \\
\rule{0pt}{2.5ex} \Omega^i_{X_H}(\log D_H) &
\omega_H(0, 0; 0) &
\omega_H(2, 0; 0) \oplus \omega_H(0, 2; 0)&
\omega_H(2, 2; 0)
\end{array}$$
### Pullback and pushforward {#sect:pfwdsheaf}
We shall need to consider pullbacks of automorphic vector bundles from $G$ to $H$. From the compatibility of torsors and the inclusion of ideal sheaves we obtain the relations $$\label{eq:pullback1}
\iota^*\big( [V] \big) = [V |_{B_H}], \qquad \iota^*\big( [V](-D_G) \big) \subseteq [V |_{B_H}](-D_H).$$ for any $P_{\operatorname{S}}$-representation $V$. (See also [@harris90b (2.5.3)] for the generic fibres; it is claimed in *op.cit.* that the latter inclusion is also an inequality, but no proof is given, and we have not been able to locate a proof.) We also consider the “exceptional inverse image” functor $\iota^!$. Since $X_G$ and $X_H$ are smooth and projective over $\operatorname{Spec}{\mathbf{Z}}_{(p)}$, of relative dimensions 3 and 2 respectively, their relative dualising complexes are isomorphic to $\Omega^3_{X_G}[3]$ and $\Omega^2_{X_H}[2]$. The functoriality of the dualising complex gives a canonical isomorphism in the derived category of coherent sheaves on $X_H$, $$\iota^!\left(\Omega^3_{X_G}\right) \cong \Omega^2_{X_H}[-1].$$ Tensoring this with the pullback isomorphism (with $V$ replaced by $V \otimes W_G(3, 3; 0)^*$), we obtain isomorphisms $$[V|_{B_H} \otimes \alpha_{G/H}^{-1}](-D_H) \cong \iota^!\big([V](-D_G)\big)[1]$$ for any $P_{\operatorname{S}}$-representation $V$, where $\alpha_{G/H}$ denotes the character $\lambda(1, 1; 0)$ of $B_H$. Together with the inclusion of ideal sheaves we also obtain maps (which might not be isomorphisms) $$[V|_{B_H} \otimes \alpha_{G/H}^{-1}] \to \iota^!\big([V]\big)[1].$$
The restriction of $[\alpha_{G/H}]$ to the open variety $Y_{H, {\mathbf{Q}}}$ is the conormal bundle of the closed embedding $\iota$, which explains its appearance in the pushforward formulae.
Cohomology
----------
The (Zariski) cohomology groups $H^i(X_G, [V])$ and $H^i(X_G, [V](-D))$ are independent, up to canonical isomorphism, of the choice of cone decomposition $\Sigma$, and have actions of prime-to-$p$ Hecke operators $[K g K]$, for $g \in G({{\mathbf{A}}_{\mathrm{f}}}^{p})$.
Note that if $V$ has central character $c$, and $K$ has level $N$, then for all primes $\ell \ne p$ congruent to 1 modulo $N$, the double coset of $\operatorname{diag}(\varpi_\ell, \dots, \varpi_\ell)$ acts as $\ell^c$, where $\varpi_\ell$ is a uniformizer at $\ell$.
The same is true for $H$ in place of $G$, and the morphisms of sheaves in the previous section gives us maps
$$\begin{aligned}
\label{eq:pwfd1a}
H^i\left(X_G, [V]\right) &\rTo^{\iota^{\star}} H^{i}\left(X_H, [V|_{B_H}]\right), \\
\label{eq:pwfd1b}
H^i\left(X_G, [V](-D_H)\right) &\rTo^{\iota^{\star}} H^{i}\left(X_H, [V|_{B_H}](-D_G)\right),\\
\label{eq:pwfd1c}
H^i\left(X_H, [V|_{B_H} \otimes \alpha_{G/H}^{-1}]\right) &\rTo^{\iota_{\star}} H^{i+1}\left(X_G, [V]\right)\\
\label{eq:pwfd1d}
H^i\left(X_H, [V|_{B_H} \otimes \alpha_{G/H}^{-1}](-D_H)\right) &\rTo^{\iota_{\star}} H^{i+1}\left(X_G, [V](-D_G)\right).
\end{aligned}$$
for $0 {\leqslant}i {\leqslant}2$ and any $P_{\operatorname{S}}$-representation $V$. Serre duality gives canonical trace maps $H^3(X_G, \omega_G(3, 3;0)(-D)) \to {{\mathbf{Z}}_p}$ and $H^2(X_H, \omega_H(2, 2; 0)(-D_H)) \to {{\mathbf{Z}}_p}$, and with respect to the cup product pairings, the morphism is the dual of the morphism , and similarly is dual to .
If we base-extend to ${\mathbf{Q}}$, we may drop the assuption that $K_p$ be hyperspecial and allow arbitrary level groups $K$. The direct limit $$H^*\left(X_{G, {\mathbf{Q}}}(\infty), [V]\right)\coloneqq \varinjlim_K H^*\left(X_{G, {\mathbf{Q}}}(K), [V]\right)$$ is then an admissible smooth ${\mathbf{Q}}$-linear representation of $G({{\mathbf{A}}_{\mathrm{f}}})$. The pullback maps assemble into morphisms of $H({{\mathbf{A}}_{\mathrm{f}}})$-representations $$\iota^*: H^*\left(X_{G, {\mathbf{Q}}}(\infty), [V]\right) \to H^*\left(X_{H, {\mathbf{Q}}}(\infty), [V|_{B_H}]\right).$$ The same statements also hold with $[V]$ replaced by the subcanonical extension $[V](-D)$, using the maps .
The ordinary part of coherent cohomology {#sect:vincent}
========================================
In this section, we’ll explain how to embed the cohomology of automorphic vector bundles on $X_G$ inside larger spaces which vary in $p$-adic families, focussing on the case of $H^1$. This is an analogue for regular weights of the theory developed for singular (non-regular) weights in [@pilloni17]. In this section we shall work solely with objects attached to the group $G$, with the subgroup $H$ playing no role, so we shall drop the subscripts $G$ from the notation; they will reappear in the next section (where we compare the theories for $G$ and $H$).
We have aimed to recall enough of the definitions and notation from [@pilloni17] to give a reasonably self-contained *statement* of the main result of this section (Theorem \[thm:vincent\]). However, in the proof of this theorem we shall assume familiarity with *op.cit.*.
The ordinary part of classical cohomology
-----------------------------------------
Suppose $(k_1, k_2; c)$ is a weight with $k_2 {\leqslant}1$ and $k_1 + k_2 {\geqslant}4$. For $n {\geqslant}1$, we shall consider the Hecke operator on $H^1(X_{\operatorname{Kl}}(p^n)_{{{\mathbf{Q}}_p}}, \omega(k_1, k_2; c)(-D))$ defined by: $${\mathcal{U}}_{p, \operatorname{Kl}} \coloneqq
p^{k_1-3-c} [\operatorname{Kl}(p^n) \operatorname{diag}(p^2, p, p, 1) \operatorname{Kl}(p^n)]$$ where we consider $ \operatorname{diag}(p^2, p, p, 1)$ as an element of $G({{\mathbf{Q}}_p}) \subset G({{\mathbf{A}}_{\mathrm{f}}})$. The normalisation factor implies that all eigenvalues of ${\mathcal{U}}_{p, \operatorname{Kl}}$ are $p$-adically integral, and we denote by $e_{\operatorname{Kl}}$ the projection onto the ordinary part for this operator (the direct sum of the generalised eigenspaces whose eigenvalues are $p$-adic units).
It is clear that this operator is independent of $c$, if we identify the cohomology of $\omega(k_1, k_2; c)$ for different values of $c$ in the obvious fashion. Hence for the remainder of this section, we shall fix $c$ to be equal to $k_1 + k_2 - 6$, so that the normalising factor for ${\mathcal{U}}_{p, \operatorname{Kl}}$ is $p^{3-k_2}$.
\[prop:newtonpoly\] Let $\Pi_p$ be an irreducible subquotient of the $G({{\mathbf{Q}}_p})$-representation $$\varinjlim_{K_p} H^i\left(X(K^p K_p)_{\bar{{\mathbf{Q}}}_p}, \omega(k_1, k_2)(-D)\right),$$ for any $i$, such that $e_{\operatorname{Kl}} \cdot \Pi_p^{\operatorname{Kl}(p)} \ne 0$. Then $e_{\operatorname{Kl}} \cdot \Pi_p^{\operatorname{Kl}(p)}$ has dimension $1$; and if $(\alpha, \beta, \gamma, \delta)$ are the Hecke parameters of $\Pi_p$ (in the sense of [@pilloni17 §5.1.5]), ordered such that $v_p(\alpha) {\leqslant}v_p(\beta)$ and $p^{2-k_2} \alpha \beta$ is the eigenvalue of ${\mathcal{U}}_{p, \operatorname{Kl}}$ on $e_{\operatorname{Kl}} \cdot \Pi_p^{\operatorname{Kl}(p)}$, then we have $$k_2 - 2 {\leqslant}v_p(\alpha) {\leqslant}v_p(\beta) {\leqslant}0, \qquad k_1 + k_2 - 3 {\leqslant}v_p(\gamma) {\leqslant}v_p(\delta) {\leqslant}k_1-1.$$
By assumption, there is a line in $\Pi_p^{\operatorname{Kl}(p)}$ on which ${\mathcal{U}}_{p, \operatorname{Kl}}$ acts as $p^{2-k_2}\alpha \beta$ and the Hecke operator $p^{2-k_2}[\operatorname{Kl}(p) \operatorname{diag}(p, p, 1, 1) \operatorname{Kl}(p)]$ acts as $p^{2-k_2}(\alpha + \beta)$. Since both of these Hecke operators preserve an integral lattice, we deduce that $v_p(\alpha + \beta) {\geqslant}k_2 - 2$ and $v_p(\alpha \beta) = k_2 - 2$. It follows that $v_p(\alpha)$ and $v_p(\beta)$ lie in the interval $[k_2-2, 0]$, and we can order them so that $v_p(\alpha) {\leqslant}v_p(\beta)$. To see that both operators preserve an integral lattice, we first observe that the cohomology we consider is a subquotient of the cohomology of a local system by [@faltingschai chap. VI, Theorem 5.5]. We can produce stable lattices for the action of the Hecke operators on the cohomology of local systems as in [@laff-est]. The inequalities for $v_p(\gamma)$ and $v_p(\delta)$ follow using the fact that $\alpha\delta = \beta\gamma$ is $p^{k_1 + k_2 - 3}$ times a root of unity.
In particular, the other three elements of the multiset $p^{2-k_2}\{ \alpha\beta, \alpha\gamma, \beta\delta, \gamma\delta\}$ have valuation at least 1. Since the multiset of roots of the characteristic polynomial of ${\mathcal{U}}_{p, \operatorname{Kl}}$ on $\Pi_p^{\operatorname{Kl}(p)}$ is a subset of this, we can conclude that $e_{\operatorname{Kl}} \cdot \Pi_p^{\operatorname{Kl}(p)}$ is 1-dimensional.
\[cor:sphericalordprojection\] Suppose $\Pi_p$ is as in Proposition \[prop:newtonpoly\], and $k_1 + k_2 {\geqslant}5$. Then:
(i) $\Pi_p$ is either a representation of Sally–Tadic type I (i.e. an irreducible principal series), or it is of type IIIb, so $\Pi_p \cong \chi \rtimes \sigma 1_{\operatorname{GSp}_2}$, with $\chi$ and $\sigma$ unramified characters. In particular, $\Pi_p$ is spherical.
(ii) The composite map $$\Pi_p^{G({{\mathbf{Z}}_p})} {\hookrightarrow}\Pi_p^{\operatorname{Kl}(p)} \rOnto e_{\operatorname{Kl}} \Pi_p^{\operatorname{Kl}(p)}$$ is an isomorphism of one-dimensional $\overline{{\mathbf{Q}}}_p$-vector spaces.
We know that $\Pi_p$ is a subquotient of the induction of an unramified character of $B_G$, determined (modulo the action of the Weyl group) by the parameters $(\alpha, \beta, \gamma, \delta)$. From the inequalities for the valuations established above, we see that either:
- none of the ratios of the Hecke parameters is $p^{\pm 1}$; or
- we have $\beta = p\alpha$ and $\delta = p\gamma$, and $\tfrac{\gamma}{\alpha} = \tfrac{\beta}{\delta}$ has valuation ${\geqslant}3$.
In the first case, the induced representation from $B_G$ is irreducible, so $\Pi_p$ is of type I. In the second case, the induced representation has exactly two composition factors, so $\Pi_p$ is either $\chi \rtimes \sigma \operatorname{St}_{\operatorname{GSp}_2}$ (type IIIa) or $\chi \rtimes \sigma 1_{\operatorname{GSp}_2}$ (type IIIb) in the notation of Sally–Tadic, where $\chi, \sigma$ are unramified characters with $\chi(p) = \gamma/\alpha$ and $\sigma(p) = \alpha / p$. However, the type IIIa case cannot occur, because the $\operatorname{Kl}(p)$-invariants of $\chi \rtimes \sigma \operatorname{St}_{\operatorname{GSp}_2}$ are 1-dimensional with $[\operatorname{Kl}(p) \operatorname{diag}(p^2, p, p, 1) \operatorname{Kl}(p)]$ acting as $p \chi(p) \sigma(p)^2$, so the ${\mathcal{U}}_{p, \operatorname{Kl}}$-eigenvalue is $p^{2-k_2} \alpha \gamma$, which has valuation ${\geqslant}2$. Thus $\Pi_p$ must be of type IIIb, and in particular is spherical.
If $\Pi_p$ is an irreducible unramified principal series, then the bijectivity of the map (ii) is proved in [@genestiertilouine05 Corollary 3.2.4]. For the type IIIb case, one can argue similarly: one checks that if $\Pi_p = \chi \rtimes \sigma 1_{\operatorname{GSp}_2}$, then there is a basis of the 3-dimensional space $\Pi_p^{\operatorname{Kl}(p)}$ in which the spherical vector is $(1, 1, 1)^T$ and the matrix of $[\operatorname{Kl}(p)\operatorname{diag}(p^2, p, p, 1) \operatorname{Kl}(p)]$ has the form $$\sigma(p)^2 \cdot
\begin{pmatrix}
p^2 \lambda^2 & 0 & 0 \\
(p^2 - p) \lambda^2 & p^3 \lambda & 0 \\
(p^2 - p) \lambda^2 + (p-1)\lambda & (p^3 - p) \lambda & p^2 \\
\end{pmatrix}$$ where $\lambda = \chi(p)$. From this one verifies by explicit computation that the projection of the spherical vector to the ordinary eigenspace, which corresponds to the eigenvalue $p^2 \sigma(p)^2$, is always non-zero.
P-adic coefficient sheaves {#sect:Gsetup}
--------------------------
We recall some notations and constructions from [@pilloni17 §9]. Recall that $X \to \operatorname{Spec}{{\mathbf{Z}}_p}$ is a toroidal compactification of the Shimura variety of prime-to-$p$ level $K^p G({{\mathbf{Z}}_p})$.
### The Klingen and Igusa towers {#the-klingen-and-igusa-towers .unnumbered}
Let ${\mathfrak{X}}$ be the $p$-adic completion of $X$, and ${\mathfrak{X}}^{{\geqslant}1}$ the open subscheme where the $\mu_p$-rank of the universal semiabelian scheme $A$ is ${\geqslant}1$.
Over ${\mathfrak{X}}^{{\geqslant}1}$ we have the tower ${\mathfrak{X}}^{{\geqslant}1}_{\operatorname{Kl}}(p^\infty) = \varprojlim_m{\mathfrak{X}}^{{\geqslant}1}_{\operatorname{Kl}}(p^m)$, where ${\mathfrak{X}}^{{\geqslant}1}_{\operatorname{Kl}}(p^m)$ parametrises choices of subgroup $C_m \subset A[p^m]$ étale-locally isomorphic to $\mu_{p^m}$. Above ${\mathfrak{X}}^{{\geqslant}1}_{\operatorname{Kl}}(p^m)$ there is a $({\mathbf{Z}}/ p^m)^\times$-torsor ${\mathfrak{IG}}(p^m)$, parametrising choices of isomorphism $C_m \cong \mu_{p^m}$.
We can identify the generic fibre ${\mathcal{X}}^{{\geqslant}1}_{\operatorname{Kl}}(p^m)$ of ${\mathfrak{X}}^{{\geqslant}1}_{\operatorname{Kl}}(p^m)$ with an open rigid-analytic subvariety of the analytification of $X_{\operatorname{Kl}}(p^m)_{{{\mathbf{Q}}_p}}$, the compactified Shimura variety of level $K^p \operatorname{Kl}(p^m)$, where $\operatorname{Kl}(p^m) \subset G({{\mathbf{Z}}_p})$ is the preimage of the Klingen parabolic modulo $p^m$. Similarly, the generic fibre of ${\mathfrak{IG}}(p^m)$ is an open in the rigid-analytic Shimura variety of level $$K^p \times \left\{ \left( \begin{smallmatrix}
1 & * & * & * \\ &*&*&* \\ &*&*&* \\ &&&*\end{smallmatrix} \right) \pmod{p^m}.\right\}$$
This is **different** from the normalisations of [@pilloni17] where the Igusa tower corresponds to the subgroups $\left( \begin{smallmatrix}
* & * & * & * \\ &*&*&* \\ &*&*&* \\ &&&1\end{smallmatrix} \right) \pmod{p^m}$. This difference arises because the choices of Shimura cocharacter in use in [@pilloni17] and [@loefflerskinnerzerbes17] differ by a sign (cf. [@loefflerskinnerzerbes17 Remark 5.1.2]), and we have chosen to maintain compatibility with the latter.
### A “big” coefficient sheaf {#a-big-coefficient-sheaf .unnumbered}
Let $\pi$ be the natural map ${\mathfrak{IG}}(p^\infty) \to {\mathfrak{X}}^{{\geqslant}1}_{\operatorname{Kl}}(p)$. For $R$ a $p$-adic ring with a continuous character $\kappa: \Gamma \to R^\times$, and $k_2 \in {\mathbf{Z}}$, we define a sheaf of $R$-modules on ${\mathfrak{X}}^{{\geqslant}1}_{\operatorname{Kl}}(p)$ by $$\mathfrak{F}_{G, R}(\kappa, k_2)
\coloneqq \left( \pi_\star \mathscr{O}_{{\mathfrak{IG}}_{G}(p^\infty)} {\mathop{\hat\otimes}}R\right)[\Gamma = \kappa - k_2] \otimes \omega(k_2, k_2),$$ where $\Gamma = {{\mathbf{Z}}_p}^\times$ acts on $\pi_\star \mathscr{O}_{{\mathfrak{IG}}_{G}(p^\infty)}$ via the $\Gamma$-torsor structure of ${\mathfrak{IG}}_{G}(p^\infty)$. In this section we will omit $G$ from the subscript. We shall also omit $R$ if $R = {{\mathbf{Z}}_p}$ and $\kappa$ is an integer $k_1 \in {\mathbf{Z}}$.
In [@pilloni17] the sheaf is defined as above when $(R, \kappa)$ is the universal object $\Lambda = {{\mathbf{Z}}_p}[[\Gamma]]$ (with its canonical character), and then extended to general $(R, \kappa)$ by base-change; but it is easily checked that this agrees with the definition above.
### Comparison maps {#comparison-maps .unnumbered}
For any integers $k_1 {\geqslant}k_2$ there is a canonical morphism of sheaves on ${\mathfrak{X}}^{{\geqslant}1}_{\operatorname{Kl}}(p)$ ([@pilloni17 Section 9.4]): $$\label{eq:comparison}
\operatorname{comp}: \omega(k_1, k_2) \to \mathfrak{F}(k_1, k_2).$$
Statement of Theorem \[thm:vincent\]
------------------------------------
Let $\Lambda = {{\mathbf{Z}}_p}[[\Gamma]]$ and $\kappa: \Gamma \to \Lambda^\times$ the canonical character. We consider the complex $\mathrm{R}\Gamma \left( \mathfrak{X}^{{\geqslant}1}_{\operatorname{Kl}}(p), {\mathfrak{F}}_\Lambda(\kappa, k_2)(-D)\right)$ in the derived category of $\Lambda$-modules, for some $k_2 \in {\mathbf{Z}}$. This complex has an an action of Hecke operators away from $p$. By [@pilloni17 §10.6], it also has an action of ${\mathcal{U}}_{p, \operatorname{Kl}}$. As we shall see in §\[sect-U-loc-finite\] below, this operator ${\mathcal{U}}_{p, \operatorname{Kl}}$ is locally finite (in the sense of [@pilloni17 Definition 2.3.2]), and hence has an associated ordinary idempotent $e_{\operatorname{Kl}}$. We set $$M^\bullet_{\kappa, k_2} = e_{\operatorname{Kl}} \mathrm{R}\Gamma \left( \mathfrak{X}^{{\geqslant}1}_{\operatorname{Kl}}(p), {\mathfrak{F}}_\Lambda(\kappa, k_2)(-D)\right).$$
The main result of this section is the following:
\[thm:vincent\] If $k_2 {\leqslant}0$, the complex $M^\bullet_{\kappa, k_2}$ is quasi-isomorphic to a finite projective $\Lambda$-module placed in degree $1$, and for all $k_1 \in {\mathbf{Z}}$ such that $k_1 + k_2 {\geqslant}4$, we have a canonical quasi-isomorphism: $$M^\bullet_{\kappa, k_2} \otimes^{\mathbf{L}}_{\Lambda, k_1} {{\mathbf{Q}}_p}= e_{\operatorname{Kl}} \mathrm{R}\Gamma \left(X_{\operatorname{Kl}}(p)_{{{\mathbf{Q}}_p}}, \omega(k_1, k_2)(-D)\right).$$ This isomorphism is compatible with the action of the Hecke algebra away from $p$, and the operator ${\mathcal{U}}_{p, \operatorname{Kl}}$ at $p$.
We expect that this should be true for $k_2 = 1$ as well, but we have not been able to prove this. This contrasts with the case $k_2 = 2$ studied in [@pilloni17], where there exist automorphic representations contributing to both $H^0$ and $H^1$.
Proof of Theorem \[thm:vincent\]
--------------------------------
For simplicity, we simply write $U$ for ${\mathcal{U}}_{p, \operatorname{Kl}}$ in this section.
### Existence of the projector and classicity “along the sheaf” {#sect-U-loc-finite}
Let $(k_1, k_2) \in {\mathbf{Z}}, k_1 {\geqslant}k_2$.
\[lem-control-sheaf\] The $U$-operator acts locally finitely on the complexes $\mathrm{R}\Gamma ( \mathfrak{X}^{{\geqslant}1}_{\operatorname{Kl}}(p), \omega(k_1, k_2)(-D))$ and $\mathrm{R}\Gamma ( \mathfrak{X}^{{\geqslant}1}_{\operatorname{Kl}}(p), {\mathfrak{F}}(k_1, k_2)(-D))$. Moreover, the map $$e\mathrm{R}\Gamma ( \mathfrak{X}^{{\geqslant}1}_{\operatorname{Kl}}(p), \omega(k_1, k_2)(-D)) \rightarrow e\mathrm{R}\Gamma( \mathfrak{X}^{{\geqslant}1}_{\operatorname{Kl}}(p), {\mathfrak{F}}(k_1, k_2)(-D))$$ is a quasi-isomorphism.
We first show the local finiteness for $\mathrm{R}\Gamma ( \mathfrak{X}^{{\geqslant}1}_{\operatorname{Kl}}(p), \omega(k_1, k_2)(-D))$. By [@pilloni17 proposition 2.3.1], it suffices to check that the cohomology $\mathrm{R}\Gamma ( \mathfrak{X}^{{\geqslant}1}_{\operatorname{Kl}}(p), \omega(k_1, k_2)(-D))$ can be represented by a complex of flat, $p$-adically separated and complete ${{\mathbf{Z}}_p}$-modules, that the $U$-operator can be represented by an endomorphism of this complex, and that $U$ is locally finite on the cohomology groups $H^i ({X}^{{\geqslant}1}_{\operatorname{Kl}}(p)_1, \omega{(k_1,k_2)}(-D) )$ (for ${X}^{{\geqslant}1}_{\operatorname{Kl}}(p)_1$ the special fiber of $\mathfrak{X}^{{\geqslant}1}_{\operatorname{Kl}}(p)$). The sheaf $\omega(k_1, k_2)(-D)$ is acyclic relatively to the minimal compactification. Let $j : \mathfrak{X}^{{\geqslant}2}_{\operatorname{Kl}}(p) \hookrightarrow \mathfrak{X}^{{\geqslant}1}_{\operatorname{Kl}}(p)$ be the inclusion. We deduce that $$j_\star j^\star \omega(k_1, k_2)(-D) \longrightarrow
\frac{j_\star j^\star \omega(k_1, k_2)(-D)}{\omega(k_1, k_2)(-D)}$$ is an acyclic resolution of $\omega(k_1, k_2)(-D))$ (because the images of $\mathfrak{X}^{{\geqslant}2}_{\operatorname{Kl}}(p)$ and $\mathfrak{X}^{= 1}_{\operatorname{Kl}}(p)$ are affine in the minimal compactification). Therefore, the cohomology $\mathrm{R}\Gamma ( \mathfrak{X}^{{\geqslant}1}_{\operatorname{Kl}}(p), \omega(k_1, k_2)(-D))$ can be represented by a complex of amplitude $[0,1]$ and the $U$-operator induces an endomorphism of this complex (because the Hecke correspondence respects the $p$-rank stratification). We now check that $U$ is locally finite on $H^i ({X}^{{\geqslant}1}_{\operatorname{Kl}}(p)_1, \omega(k_1, k_2)(-D))$. We will prove this by decreasing induction on $k_2$. We know by [@pilloni17 Theorem 11.2.1, Theorem 11.3.1 ] that this holds true for $k_2 {\geqslant}2$. We consider the following exact sequence, where $\mathrm{Ha}$ is the Hasse invariant: $$0 \rightarrow \omega(k_1,k_2)(-D) \stackrel{ \mathrm{Ha}}\rightarrow \omega(k_1 +p-1, k_2 + p-1)(-D) \rightarrow \omega(k_1 +p-1, k_2 + p-1)(-D)/ \mathrm{Ha} \rightarrow 0.$$ This yields a $U$-equivariant long exact sequence on cohomology by [@pilloni17 lemma 10.5.2.1]. By our inductive hypothesis, $U$ is locally finite on $H^i ({X}^{{\geqslant}1}_{\operatorname{Kl}}(p)_1, \omega(k_1 +p-1, k_2 + p-1)(-D))$. On $H^i({X}^{{\geqslant}1}_{\operatorname{Kl}}(p)_1, \omega(k_1 +p-1, k_2 + p-1)(-D)/\mathrm{Ha}) = H^i({X}^{= 1}_{\operatorname{Kl}}(p)_1, \omega(k_1 +p-1, k_2 + p-1)(-D))$, multiplication by the second Hasse invariant (see [@pilloni17 Section 6.3.2]) induces $U$-equivariant isomorphisms $H^i({X}^{= 1}_{\operatorname{Kl}}(p)_1, \omega(k_1 +p-1, k_2 + p-1)(-D)) = H^i({X}^{= 1}_{\operatorname{Kl}}(p)_1, \omega(k_1 + p-1 + p^2 -1, k_2 + p-1 + p^2-1)(-D))$ ([@pilloni17 Lemma 10.5.3.1]). By the inductive hypothesis, $U$ is locally finite on $H^i({X}^{= 1}_{\operatorname{Kl}}(p)_1, \omega(k_1 + p-1 + p^2 -1, k_2 + p-1 + p^2-1)(-D))$. We can therefore conclude that $U$ is locally finite on $H^i ({X}^{{\geqslant}1}_{\operatorname{Kl}}(p)_1, \omega(k_1, k_2)(-D))$ by [@pilloni17 lemma 2.1.1].
We now prove the local finiteness on $\mathrm{R}\Gamma( \mathfrak{X}^{{\geqslant}1}_{\operatorname{Kl}}(p), {\mathfrak{F}}(k_1, k_2)(-D))$ and the quasi-isomorphism. This cohomology is computed by a complex of amplitude $[0,1]$ and $U$ lifts to an operator on this complex by arguments similar to the ones used for the sheaf $\omega(k_1,k_2)(-D)$. We next show that $U$ is locally finite on $H^i( \mathfrak{X}^{{\geqslant}1}_{\operatorname{Kl}}(p), {\mathfrak{F}}(k_1, k_2)(-D) \otimes {\mathbf{F}}_p)$. As in the proof of [@pilloni17 Theorem 11.3.1 ] we establish at the same time local finiteness and the isomorphism: $ e H^i ({X}^{{\geqslant}1}_{\operatorname{Kl}}(p)_1, \omega(k_1, k_2)(-D)) \rightarrow e H^i( \mathfrak{X}^{{\geqslant}1}_{\operatorname{Kl}}(p), {\mathfrak{F}}(k_1, k_2)(-D) \otimes {\mathbf{F}}_p)$. Details are left to the reader.
The $U$-operator is locally finite on $\mathrm{R}\Gamma ( \mathfrak{X}^{{\geqslant}1}_{\operatorname{Kl}}(p), {\mathfrak{F}}_\Lambda(\kappa, k_2)(-D))$.
This follows from the previous lemma and [@pilloni17 Proposition 2.3.1].
### A vanishing theorem
Recall that $X^{{\geqslant}1}_{\operatorname{Kl}}(p)_1$ denotes the mod $p$ reduction of ${\mathfrak{X}}^{{\geqslant}1}_{\operatorname{Kl}}(p)$.
\[prop-vanish\] If $k_2 {\leqslant}0$, then $e H^0(X^{{\geqslant}1}_{\operatorname{Kl}}(p)_1, \omega(k_1, k_2)(-D)) = 0$.
Over $X^{{\geqslant}1}_{\operatorname{Kl}}(p)_1$ we have a universal multiplicative group $H$. The associated conormal sheaf is denoted by $\omega_H$. This is an invertible sheaf. We have a surjective map $\omega_G \rightarrow \omega_H$ and therefore there is a surjective map : $\omega(k_1,k_2) \rightarrow \det \omega_G^{k_2} \otimes \omega_H^{k_1-k_2}$. It follows from [@pilloni17 Lemma 10.7.1], that the map $eH^0(X^{{\geqslant}1}_{\operatorname{Kl}}(p)_1, \omega(k_1, k_2)) \rightarrow eH^0({X^{{\geqslant}1}_{\operatorname{Kl}}(p)_1}, \det \omega_G^{k_2} \otimes \omega_H^{k_1-k_2}) $ is injective.
The idea of the proof is to evaluate sections of $H^0({X^{{\geqslant}1}_{\operatorname{Kl}}(p)_1}, \det \omega_G^{k_2} \otimes \omega_H^{k_1-k_2}) $ on various modular curves that map to ${X^{{\geqslant}1}_{\operatorname{Kl}}(p)_1}$. We will see that the sections of $H^0({X^{{\geqslant}1}_{\operatorname{Kl}}(p)_1}, \det \omega_G^{k_2} \otimes \omega_H^{k_1-k_2})$ vanish along these modular curves, and that the union of the images of these modular curves is Zariski dense.
The tame level of $X^{{\geqslant}1}_{\operatorname{Kl}}(p)_1$ is the compact open subgroup $K^p \subset G({{\mathbf{A}}_{\mathrm{f}}}^p)$. Let ${K'}^p \subset {G}({{\mathbf{A}}_{\mathrm{f}}}^p) $ be a compact open subgroup and let $g \in {G}({{\mathbf{A}}_{\mathrm{f}}}^p)$ be such $g^{-1} {K'}^p g \subset K^p$. We have a map $g : X^{{\geqslant}1}_{{K'}^p\operatorname{Kl}}(p)_1 \rightarrow X^{{\geqslant}1}_{\operatorname{Kl}}(p)_1$ for suitable choices of polyhedral cone decompositions.
Let $E_0$ be an ordinary elliptic curve defined over a field $k$ of characteristic $p$. For a suitable level structure ${K''}^p \subset \operatorname{GL}_2({{\mathbf{A}}_{\mathrm{f}}}^p)$ and the choice of a suitable level structure on $E_0$, we have a map from the modular curve of level ${K''}^p$ away from $p$ and spherical level at $p$ to $X^{{\geqslant}1}_{{K'}^p\operatorname{Kl}}(p)_1$: $$j : Y_{GL_2, K''^p,1} \times_{\operatorname{Spec}{\mathbf{F}}_p} \operatorname{Spec}k \rightarrow X^{{\geqslant}1}_{{K'}^p\operatorname{Kl}}(p)_1$$ defined by sending the universal elliptic curve $E$ to the abelian surface $E_0 \times E$, equipped with the product polarization, the multiplicative subgroup $H \subset E_0$ and the apropriate level structure away from $p$. This map extends clearly to a map $j : X_{GL_2, K''^p,1} \times_{\operatorname{Spec}{\mathbf{F}}_p} \operatorname{Spec}k \rightarrow X^{{\geqslant}1}_{{K'}^p\operatorname{Kl}}(p)_1$ of the compactified modular curve.
Let $f \in H^0({X^{{\geqslant}1}_{\operatorname{Kl}}(p)_1}, \det \omega_G^{k_2} \otimes \omega_H^{k_1-k_2}(-D)) $. The pull back $j^\star g^\star(f)$ is a cuspidal modular form of weight $k_2 {\leqslant}0$. We therefore find that $j^\star g^\star(f) = 0$.
It follows that $f$ vanishes at all prime to $p$ Hecke translates of points of the form $E_0 \times E$, where $E_0$ is an ordinary elliptic curve, $E$ is any elliptic curve and $H \subset E_0$ is the multiplicative subgroup. We claim that this set is Zariski dense in $X^{{\geqslant}1}_{\operatorname{Kl}}(p)_1$ and therefore that $f=0$.
It suffices to prove that this set is Zariski dense in $X^{{\geqslant}2}_{\operatorname{Kl}}(p)_1$, and using the irreducibility of the étale cover : $X^{{\geqslant}2}_{\operatorname{Kl}}(p)_1 \rightarrow X^{{\geqslant}2}_1$, we are left to prove the Zariski density of prime to $p$ Hecke translates of points in $X^{{\geqslant}2}_1$ (the ordinary part of the reduction modulo $p$ Shimura variety of prime to $p$ level) of the form $E_0 \times E$ for a product of two ordinary elliptic curves. This set is the union of the Hecke translates of the codimension one subscheme $\iota(X_{H,1}^{{\geqslant}2})$ (the image of the ordinary part of the reduction modulo $p$ of the Shimura variety for $H$). It is sufficient to prove that these Hecke translates form an infinite union of codimension one subschemes (indeed, one can easily reduce to the case that $X^{{\geqslant}2}_1$ is connected). We prove this last claim as follows. Given a geometric point $x$ on some prime to $p$ Hecke translate of $X_{H,1}^{{\geqslant}2}$, we let $d(x)$ be the minimal degree of a prime to $p$ isogeny between $x$ and a product of two elliptic curves. It will suffice to prove that there exists a sequence of points $x_n$ with $d(x_n) \rightarrow \infty$. We can produce such points as follows. We consider two non isogenous ordinary elliptic curves $E$ and $F$ over $\bar{{\mathbf{F}}}_p$. Let $\ell \neq p$ be a prime, and let $e_1,e_2, f_1, f_2$ be a basis of the $\ell$-adic Tate modules of $E$ and $F$ (with $\langle e_1, e_2 \rangle = \langle f_1, f_2 \rangle$), and let $H_\ell$ be the subgroup generated by the images of $e_1 + f_1$ and $e_2 - f_2$ in $E[\ell] \times F [\ell]$. This is a totally isotropic subgroup. We let $A_\ell = E \times F/H_\ell$. This is a principally polarized abelian surface and it is easy to see that the minimal degree of an isogeny between $A_\ell$ and a product of elliptic curves is $\ell^2$.
If $k_2 {\leqslant}0$, then $eH^i(\mathfrak{X}^{{\geqslant}1}_{\operatorname{Kl}}(p), \omega(k_1, k_2)(-D))$ is $0$ if $i \neq 1$, and for $i = 1$ it is a flat, $p$-adically complete and separated ${{\mathbf{Z}}_p}$-module.
The sheaf $\omega(k_1, k_2)(-D)$ is acyclic relative to the minimal compactification. Since the image of $\mathfrak{X}^{{\geqslant}1}_{\operatorname{Kl}}(p)$ in the minimal compactification is covered by two affines, we deduce that the cohomology $\mathrm{R}\Gamma(\mathfrak{X}^{{\geqslant}1}_{\operatorname{Kl}}(p), \omega(k_1, k_2)(-D)) $ is represented by a complex of amplitude $[0,1]$ of complete, separated, flat ${{\mathbf{Z}}_p}$-modules. The same holds for $e\mathrm{R}\Gamma(\mathfrak{X}^{{\geqslant}1}_{\operatorname{Kl}}(p), \omega(k_1, k_2)(-D))$. Let us write this complex as $M^0 \stackrel{d}\rightarrow M^1$. We claim that $d$ is injective and that $\operatorname{coker} d$ is flat, $p$-adically complete and separated. The complex $ M^0 \otimes {\mathbf{F}}_p \stackrel{d \otimes 1} \rightarrow M^1 \otimes {\mathbf{F}}_p$ computes $e\mathrm{R}\Gamma({X}^{{\geqslant}1}_{\operatorname{Kl}}(p), \omega(k_1, k_2)(-D))$. By proposition \[prop-vanish\], $\mathrm{ker}( d \otimes 1) = 0$ and it follows that $\ker d = p \ker d$. Since $\ker d$ is a complete and separated ${{\mathbf{Z}}_p}$-module, we deduce that $\ker d = 0$. We also deduce that $M^0 \cap p M^1 = p M^0$ and it follows that $\operatorname{coker} d = M^1/M^0$ is flat, $p$-adically complete and separated.
### Finiteness
Let $\mathcal{X}_{\operatorname{Kl}}(p)$ be the analytic adic space associated to $X_{\operatorname{Kl}}(p)$. Let $\mathcal{X}^{{\geqslant}1}_{\operatorname{Kl}}(p)$ be the locus where $H$ is multiplicative (the generic fiber of $\mathfrak{X}^{{\geqslant}1}_{\operatorname{Kl}}(p)$). We may consider the following cohomology $\mathrm{R}\Gamma ( \mathcal{X}^{{\geqslant}1}_{\operatorname{Kl}}(p), \omega(k_1, k_2)(-D)) = \mathrm{R}\Gamma ( \mathfrak{X}^{{\geqslant}1}_{\operatorname{Kl}}(p), \omega(k_1, k_2)(-D)) \otimes^L_{{{\mathbf{Z}}_p}} {{\mathbf{Q}}_p}$ as well as its ordinary part $$e\mathrm{R}\Gamma ( \mathcal{X}^{{\geqslant}1}_{\operatorname{Kl}}(p), \omega(k_1, k_2)(-D)) = e \mathrm{R}\Gamma ( \mathfrak{X}^{{\geqslant}1}_{\operatorname{Kl}}(p), \omega(k_1, k_2)(-D)) \otimes^L_{{{\mathbf{Z}}_p}} {{\mathbf{Q}}_p}.$$
We may also consider the overconvergent cohomology $\mathrm{R}\Gamma ( \mathcal{X}^{{\geqslant}1}_{\operatorname{Kl}}(p)^{\dag}, \omega(k_1,k_2)(-D))$. One checks that the $U$-operator is compact on this cohomology (see [@pilloni17 sect. 13. 2]). In particular it makes sense to speak of the ordinary part : $e\mathrm{R}\Gamma ( \mathcal{X}^{{\geqslant}1}_{\operatorname{Kl}}(p)^\dag, \omega(k_1,k_2)(-D))$, and this is a perfect complex of ${{\mathbf{Q}}_p}$-vector spaces.
If $k_2 {\leqslant}0$, $e\mathrm{R}\Gamma ( \mathcal{X}^{{\geqslant}1}_{\operatorname{Kl}}(p)^\dag, \omega(k_1,k_2)(-D))$ is concentrated in degree $1$ and the map $$eH^1( \mathcal{X}^{{\geqslant}1}_{\operatorname{Kl}}(p)^\dag , \omega(k_1,k_2)(-D)) \rightarrow eH^1( \mathcal{X}^{{\geqslant}1}_{\operatorname{Kl}}(p), \omega(k_1, k_2)(-D))$$ is surjective.
The image of $\mathcal{X}^{{\geqslant}1}_{\operatorname{Kl}}(p)$ in the minimal compactification is covered by two affinoids, say $U_1$ and $U_2$. Let $\pi$ be the projection from toroidal to minimal compactification. The complex $$H^0(U_1, \pi_\star \omega(k_1, k_2)(-D)) \oplus H^0(U_2, \pi_\star \omega(k_1, k_2)(-D)) \rightarrow H^0(U_1 \cap U_2, \pi_\star \omega(k_1, k_2)(-D))$$ computes $\mathrm{R}\Gamma ( \mathcal{X}^{{\geqslant}1}_{\operatorname{Kl}}(p), \omega(k_1, k_2)(-D))$. The subcomplex of overconvergent sections $$H^0(U_1^\dag, \pi_\star \omega(k_1, k_2)(-D)) \oplus H^0( U_2^\dag, \pi_\star \omega(k_1, k_2)(-D)) \rightarrow H^0(U_1^\dag \cap U_2^\dag, \pi_\star \omega(k_1, k_2)(-D))$$ computes $\mathrm{R}\Gamma ( \mathcal{X}^{{\geqslant}1}_{\operatorname{Kl}}(p)^\dag, \omega(k_1,k_2)(-D)).$ Therefore there is a surjective map $H^0(U_1 \cap U_2, \pi_\star \omega(k_1, k_2)(-D)) \rightarrow H^1( \mathcal{X}^{{\geqslant}1}_{\operatorname{Kl}}(p), \omega(k_1, k_2)(-D))$ and overconvergent sections in $H^0(U_1 \cap U_2, \pi_\star \omega(k_1, k_2)(-D))$ are dense. The map $H^1( \mathcal{X}^{{\geqslant}1}_{\operatorname{Kl}}(p)^\dag , \omega(k_1,k_2)(-D)) \rightarrow H^1( \mathcal{X}^{{\geqslant}1}_{\operatorname{Kl}}(p), \omega(k_1, k_2)(-D))$ induces a continuous map $$eH^1( \mathcal{X}^{{\geqslant}1}_{\operatorname{Kl}}(p)^\dag , \omega(k_1,k_2)(-D)) \rightarrow eH^1( \mathcal{X}^{{\geqslant}1}_{\operatorname{Kl}}(p), \omega(k_1, k_2)(-D))$$ (by functoriality of slope $0$ decomposition) with dense image. The space $eH^1( \mathcal{X}^{{\geqslant}1}_{\operatorname{Kl}}(p)^\dag, \omega(k_1,k_2)(-D))$ is a finite-dimensional ${{\mathbf{Q}}_p}$-vector space, and therefore the map is surjective and the target is also finite dimensional. There is also an injective map $H^0(\mathcal{X}^{{\geqslant}1}_{\operatorname{Kl}}(p)^\dag , \omega(k_1,k_2)(-D)) \rightarrow H^0( \mathcal{X}^{{\geqslant}1}_{\operatorname{Kl}}(p), \omega(k_1, k_2)(-D))$ which induces an injective map $eH^0(\mathcal{X}^{{\geqslant}1}_{\operatorname{Kl}}(p)^\dag , \omega(k_1,k_2)(-D)) \rightarrow eH^0( \mathcal{X}^{{\geqslant}1}_{\operatorname{Kl}}(p), \omega(k_1, k_2)(-D))$. This last module is trivial and therefore $eH^0(\mathcal{X}^{{\geqslant}1}_{\operatorname{Kl}}(p)^\dag , \omega(k_1,k_2)(-D)) =0$.
If $k_2 {\leqslant}0$, then $M^\bullet_{\kappa^{un}, k_2}[1]$ is a finite-rank projective $\Lambda$-module.
By corollary \[lem-control-sheaf\], for any $k_1 {\geqslant}k_2$, we have $M^\bullet_{\kappa^{un}, k_2} \otimes^L_{\Lambda, k_1} {{\mathbf{Z}}_p}= e \mathrm{R}\Gamma( \mathfrak{X}^{{\geqslant}1}_{\operatorname{Kl}}(p), \omega(k_1, k_2)(-D))$. The corollary follows from the previous corollary and Nakayama’s lemma for complexes [@pilloni17 Proposition 2.2.1].
### Classicity
We recall the following control theorem :
[@pilloni17 theorem 14.7.1] \[thm:OCclass\] The map $$e \mathrm{R}\Gamma(\mathcal{X}_{\operatorname{Kl}}(p), \omega(k_1, k_2)(-D)) \rightarrow
e \mathrm{R}\Gamma(\mathcal{X}^{{\geqslant}1}_{\operatorname{Kl}}(p)^\dag, \omega(k_1, k_2)(-D))$$ is a quasi-isomorphism if $k_1 + k_2 > 3$, and similarly without the $(-D)$.
We want to conclude :
The map $$e \mathrm{R}\Gamma(\mathcal{X}_{\operatorname{Kl}}(p), \omega(k_1, k_2)(-D)) \rightarrow
e \mathrm{R}\Gamma(\mathcal{X}^{{\geqslant}1}_{\operatorname{Kl}}(p), \omega(k_1, k_2)(-D))$$ is a quasi-isomorphism if $k_1 + k_2 > 3$ and $k_2 {\leqslant}0$.
We already know that both complexes are concentrated in degree $1$ and that $e H^1(\mathcal{X}_{\operatorname{Kl}}(p), \omega(k_1, k_2)(-D)) \rightarrow e H^1(\mathcal{X}^{{\geqslant}1}_{\operatorname{Kl}}(p), \omega(k_1, k_2)(-D))$ is surjective. It will therefore suffice to prove injectivity.
### Injectivity if $k_1$ is large enough
We will first show that injectivity holds if $k_1$ is very large. The idea is to prove that the ordinary cohomology is isomorphic to ordinary cohomology at spherical level, and then to use the following type of result:
\[thmSGA\] Let $S$ be a Cohen–Macaulay scheme and $\mathcal{L}$ be a locally free sheaf over $S$. Let $U \subset S$ be a codimension $i$ subscheme. Then the map $H^j (S, \mathcal{L}) \rightarrow H^j(U, \mathcal{L})$ is bijective if $j < i$ and injective for $i=j$.
SGA 2, III, lem. 3.1 and prop. 3.3.
We need to define the relevant ordinary projector at spherical level. This is a somewhat involved calculation. Let $Y \rightarrow \operatorname{Spec}{{\mathbf{Z}}_p}$ be the Shimura variety with spherical level at $p$ and tame level $K^p$ away from $p$. Let $Y_p$ be the Shimura variety with paramodular level at $p$ and tame level $K^p$ away from $p$. Let $Y_{\operatorname{Kl}}(p)$ be the Shimura variety with Klingen level at $p$ and tame level $K^p$ away from $p$. Over $Y_{\operatorname{Kl}}(p)$ we have a chain $G \rightarrow G/L_1 \rightarrow G$ where $L_1$ is a subgroup of order $p^3$ of $G[p]$ and the total map is multiplication by $p$. Therefore $G/L_1$ carries a degree $p^2$ polarization $\lambda$ and $\mathrm{ker} (G/L_1 \rightarrow G) $ is a subgroup of order $p$ of the kernel of the polarization. We have morphisms $p_1 : Y_{\operatorname{Kl}}(p) \rightarrow Y$ defined by $(G, L_1) \mapsto G$ and $p_2 : Y_{\operatorname{Kl}}(p) \rightarrow Y_p$ defined by $(G,L_1) \mapsto G/L_1$.
We now consider toroidal compactifications $X$, $X_{\operatorname{Kl}}(p)$ and $X_{p}$ of $Y$, $Y_{\operatorname{Kl}}(p)$ and $Y_p$, such that the maps $p_1$ and $p_2$ extend. We make further specifications. We can choose a smooth cone decomposition $\Sigma$ for $X$, a smooth cone decomposition $\Sigma'$ for $X_p$ and we take for $X_{\operatorname{Kl}}(p)$ the same cone decomposition $\Sigma'$.
We now take formal completion of all these spaces and restrict to the $p$-rank at least one locus. We denote by $\mathfrak{X}^{{\geqslant}1}$, $\mathfrak{X}^{{\geqslant}1}_p$ and $\mathfrak{C}_1$ the resulting spaces (we use $\mathfrak{C}_1$ for the $p$-rank at least one locus in $\mathfrak{X}_{\operatorname{Kl}}(p)$ to avoid any confusion with $\mathfrak{X}^{{\geqslant}1}_{\operatorname{Kl}}(p)$). The spaces $\mathfrak{X}^{{\geqslant}1}$ and $\mathfrak{X}^{{\geqslant}1}_p$ are smooth over $\operatorname{Spf}{{\mathbf{Z}}_p}$. The morphism $\mathfrak{C}_1 \rightarrow \operatorname{Spf}{{\mathbf{Z}}_p}$ is Cohen-Macaulay.
We remark that the maps $p_1$ and $p_2$ are quasi-finite away from the boundary (this is a consequence of the fact that abelian surfaces of $p$-rank at least one over algebraically closed fields have only a finite number of subgroups of order $p$). It follows from miracle flatness that $p_1$ and $p_2$ are finite flat away from the boundary. Actually, by our choice of cone decomposition, $p_2$ is quasi-finite at the boundary and therefore $p_2$ is finite flat.
We let $\mathfrak{C} = \mathfrak{C}_1 \times_{p_2, \mathfrak{X}_p^{{\geqslant}1}, p_2} \mathfrak{C}_1$. Over $\mathfrak{C}$ we have a chain of isogenies $G \rightarrow G/L_1 \rightarrow G/L$ where the first isogeny has degree $p^3$, the second isogeny has degree $p$. We have two projections $q_1, q_2 : \mathfrak{C} \rightarrow \mathfrak{X}^{{\geqslant}1}$, defined by $q_1(G,L) = G$, $q_2(G,L) = G/L$. We have a natural map $q_2^\star \omega(k_1, k_2)[1/p] \rightarrow q_1^\star \omega(k_1, k_2)[1/p]$ arising from the differential of the isogeny $G \rightarrow G/L$ which is étale after inverting $p$.
\[lem-justabove\] The relative dualizing sheaf $q_1^! {\mathcal{O}}_{\mathfrak{X}^{{\geqslant}1}}$ is a Cohen-Macaulay sheaf. We have a fundamental class $q_1^\star {\mathcal{O}}_{\mathfrak{X}^{{\geqslant}1}} \rightarrow q_1^! {\mathcal{O}}_{\mathfrak{X}^{{\geqslant}1}}$ which extends the usual trace map on the complement of the boundary.
Since $\mathfrak{C}_1$ is Cohen-Macaulay, and the map $\mathfrak{C}_1 \rightarrow \mathfrak{X}_p^{{\geqslant}1}$ if finite flat, we deduce from [@MR1011461], corollary on page 181, that the map $\mathfrak{C}_1 \rightarrow \mathfrak{X}_p^{{\geqslant}1}$ is a Cohen-Macaulay morphism (i.e a flat morphism with Cohen-Macaulay fibers). A base change of a Cohen-Macaulay morphism is still a Cohen-Macaulay morphism. Therefore, the morphism : $\mathfrak{C} \rightarrow \mathfrak{C}_1$ is a Cohen-Macaulay morphism. Since $\mathfrak{C}_1$ is Cohen-Macaulay, we deduce that $\mathfrak{C}$ is Cohen-Macaulay and that the structural morphism $ g : \mathfrak{C} \rightarrow \operatorname{Spf}{{\mathbf{Z}}_p}$ is a Cohen-Macaulay morphism. On the other hand the morphism $h : \mathfrak{X}^{{\geqslant}1} \rightarrow \operatorname{Spf}{{\mathbf{Z}}_p}$ is smooth. We find that $q_1^! {\mathcal{O}}_{\mathfrak{X}^{{\geqslant}1}} \otimes h^! {{{\mathbf{Z}}_p}} = g^! {{\mathbf{Z}}_p}$ and it follows that $q_1^! {\mathcal{O}}_{\mathfrak{X}^{{\geqslant}1}}$ is a CM sheaf (see [@Fakhruddin-Pilloni Lemma 2.2]). The usual trace map away from the boundary extends to give a morphism $q_1^\star {\mathcal{O}}_{\mathfrak{X}^{{\geqslant}1}} \rightarrow q_1^! {\mathcal{O}}_{\mathfrak{X}^{{\geqslant}1}}$ by [@Fakhruddin-Pilloni Proposition 2.6].
Therefore we get a map $\Theta : q_2^\star \omega(k_1, k_2)[1/p] \rightarrow q_1^! \omega(k_1, k_2)[1/p]$ obtained by composing the map $q_2^\star \omega(k_1, k_2)[1/p] \rightarrow q_1^\star \omega(k_1, k_2)[1/p]$ and the map $q_1^\star \omega(k_1, k_2)[1/p] \rightarrow q_1^! \omega(k_1, k_2)[1/p]$ (simply obtained by tensoring the map of lemma \[lem-justabove\] with $q_1^\star \omega(k_1, k_2)[1/p]$). By adjunction, this is also a map $(q_1)_\star p_2^\star \omega(k_1, k_2)[1/p] \rightarrow \omega(k_1, k_2)[1/p]$. We now optimize integrality : we determine the optimal integer $s$ for which $p^s \Theta$ extends to a map $q_2^\star \omega(k_1, k_2) \rightarrow q_1^! \omega(k_1, k_2)$. It is enough to determine the value of $s$ over the ordinary locus (and even away from the boundary) because a map from a locally free sheaf to a CM sheaf which is defined up to a codimension $2$ closed subset is defined everywhere.
Let us say that the isogeny $G \rightarrow G/L_1$ is *as étale as possible* if $L_1$ has étale rank $2$ and *as multiplicative as possible* if $L_1$ has multiplicative rank $2$.
We have $\mathfrak{C}^{=2} = \mathfrak{C}^{et,et} \cup \mathfrak{C}^{m,et} \cup \mathfrak{C}^{et,m} \cup \mathfrak{C}^{m,m}$ where the first supscript refers to $G \rightarrow G/L_1$ being as étale or as multiplicative as possible and the second supscript to $G/L_1 \rightarrow G/L$ to be mutliplicative or étale.
We denote by $\Theta^{\star, \star \star}$ the projection of $\Theta$ on the connected components $\mathfrak{C}^{ \star, \star \star}$.
We check that :
- $\Theta^{et , et } : q_2^\star \omega(k_1, k_2) \rightarrow p^{k_2+3} q_1^! \omega(k_1, k_2)$,
- $\Theta^{et , m } : q_2^\star \omega(k_1, k_2) \rightarrow p^{k_2+k_1 + 2} q_1^! \omega(k_1, k_2)$,
- $\Theta^{m, et } : q_2^\star \omega(k_1, k_2) \rightarrow p^{k_2+k_1 + 1} q_1^! \omega(k_1, k_2)$,
- $\Theta^{m, m } : q_2^\star \omega(k_1, k_2) \rightarrow p^{2k_2+k_1} q_1^! \omega(k_1, k_2)$
Let us briefly explain where these numbers arise from.
- $\Theta^{et , et } : q_2^\star \omega(k_1, k_2) \rightarrow p^{k_2} q_1^\star \omega(k_1, k_2) \rightarrow p^{k_2+3} q_1^! \omega(k_1, k_2)$ where the first $p^{k_2}$ factor arises from the differential of the isogeny $q_2^\star \omega_{G/L} \rightarrow p^{k_2} q_1^\star \omega_G$ (the group $L$ has multiplicative rank $1$) and the second $p^3$ factor arises from the trace.
- $\Theta^{m , et } : q_2^\star \omega(k_1, k_2) \rightarrow p^{k_2 + k_1} q_1^\star \omega(k_1, k_2) \rightarrow p^{k_2+k_1 + 1} q_1^! \omega(k_1, k_2)$ where the first factor $p^{k_2+k_1}$ arises from the differential of the isogeny. The group $L$ contains $G[p]^{mult}$. The factor $p$ arises from the trace.
- $\Theta^{et, m } q_2^\star \omega(k_1, k_2) \rightarrow p^{k_2 + k_1} q_1^\star \omega(k_1, k_2) \rightarrow p^{k_2+k_1 + 2} q_1^! \omega(k_1, k_2)$ where the first factor $p^{k_2+k_1}$ arises from the differential of the isogeny. The group $L$ contains $G[p]^{mult}$. The factor $p^2$ arises from the trace.
- $\Theta^{m, m } : q_2^\star \omega(k_1, k_2) \rightarrow p^{2k_2+k_1} q_1^\star \omega(k_1, k_2) \rightarrow p^{3k_2+k_1} q_1^! \omega(k_1, k_2)$ where the first factor $p^{2k_2+k_1}$ arises from the isogeny. The group $L$ contains a subgroup isomorphic (for the étale topology) to $\mu_p \times \mu_{p^2}$.
Therefore, under the hypothesis that $k_1 {\geqslant}2$ and $k_2 + k_1 {\geqslant}3$ we have an operator $T = p^{-k_2-3} \Theta : q_2^\star \omega(k_1, k_2) \rightarrow q_1^! \omega(k_2,k_1)$. We also denote by $T \in \mathrm{End}( \mathrm{R}\Gamma ( \mathfrak{X}^{{\geqslant}1}, \omega(k_1, k_2)))$ the associated Hecke operator. We will especially be interested in the reduction modulo $p$ of this operator $T \in \mathrm{End}( \mathrm{R}\Gamma ({X}_1^{{\geqslant}1}, \omega(k_1, k_2)))$
There is a natural map $\mathrm{R}\Gamma (X^{{\geqslant}1}_1, \omega(k_1, k_2)) \rightarrow \mathrm{R}\Gamma (X_{\operatorname{Kl}}^{{\geqslant}1}(p)_1, \omega(k_1, k_2))$ and we will compare the ordinary parts of these cohomology for $T$ and $U$ respectively.
We first look at the cohomology over the ordinary locus :
\[lem-UTord\] If $k_1 > 2$ and $k_2 + k_1 > 3$, there is an operator $\tilde{T} : H^0( X^{{\geqslant}2}_{\operatorname{Kl}}(p)_1, \omega(k_1, k_2)) \rightarrow H^0( X^{{\geqslant}2}_{\operatorname{Kl}}(p)_1, \omega(k_1, k_2))$ fitting in a commutative diagram : $$\xymatrix{ H^0( X^{{\geqslant}2}_{\operatorname{Kl}}(p)_1, \omega(k_1, k_2)) \ar[r]^{\tilde{T}} \ar[rd] & H^0( X^{{\geqslant}2}_{\operatorname{Kl}}(p)_1, \omega(k_1, k_2)) \\
H^0( X^{{\geqslant}2}_1, \omega(k_1, k_2)) \ar[u] \ar[r]^{{T}} & H^0( X^{{\geqslant}2}_1, \omega(k_1, k_2)) \ar[u]}$$
Moreover, if $k_1-k_2 >0$, $U \circ \tilde{T} = U^2$.
Under the assumptions $k_1 + k_2 +1 > k_2 +3$ and $2k_2 + k_1 > k_2 +3$, only the “étale” part of the correspondence contributes modulo $p$ : the cohomological correspondence $T$ is supported on the component $\mathfrak{C}^{et,et} \times \operatorname{Spec}{\mathbf{F}}_p$. The second projection $q_2 : \mathfrak{C}^{et,et} \rightarrow \mathfrak{X}^{{\geqslant}1}$ lifts to $\tilde{q_2} : \mathfrak{C}^{et,et} \rightarrow \mathfrak{X}^{{\geqslant}1}_{\operatorname{Kl}}(p)$ by sending $(G,L)$ to $(G/L, G[p]+L/L)$. It follows that there is a commutative diagram : $$\xymatrix{ H^0( X^{{\geqslant}2}_{\operatorname{Kl}}(p)_1, \omega(k_1, k_2)) \ar[rd] & \\
H^0( X^{{\geqslant}2}_1, \omega(k_1, k_2)) \ar[u] \ar[r]^{{T}} & H^0( X^{{\geqslant}2}_1, \omega(k_1, k_2))}$$ which can be completed into the diagram of the proposition. It remains to see that $U \circ \tilde{T} = U^2$. We can argue exactly as in the proof of [@pilloni17 Lemma 11.1.3]
We now examine the situation on the $p$-rank $1$ locus. We have two Hecke operators : $$U \in \mathrm{End}(\mathrm{R}\Gamma(X^{=1}_{\operatorname{Kl}}(p)_1, \omega(k_2,k_1)))$$ for all $k_1 {\geqslant}k_2$, and $$T \in \mathrm{End}(\mathrm{R}\Gamma(X^{=1}_1, \omega(k_2+p-1,k_1+p-1)))$$ for $k_1 > 2$ and $k_2 + k_1 > 3$ (see [@pilloni17 section 7.4 and 10.5.2] to see how we can obtain these operators by restriction of the cohomological correspondence).
Observe that the obvious projection $p_1 : X_{\operatorname{Kl}}(p)_1 \rightarrow X_1$ induces an isomorphism $X^{=1}_{\operatorname{Kl}}(p)_1\simeq X^{=1}_1$ (because over the $p$-rank one locus there is a unique choice for a multiplicative $H$). We can therefore compare $U$ and $T$.
In order to do so, let us consider $\mathfrak{X}^{=1}$, $\mathfrak{X}_p^{=1}$, $\mathfrak{C}_1^{=1}$ and $\mathfrak{C}^{=1}$ the completion along the rank one locus of all the formal schemes $\mathfrak{X}^{{\geqslant}1}$, $\mathfrak{X}^{{\geqslant}1}_p$, $\mathfrak{C}_1$ and $\mathfrak{C}$. We have $\mathfrak{X}_p^{=1} = \mathfrak{X}_p^{=1, oo} \cup \mathfrak{X}_p^{=1, m/et}$. This is the decomposition according to wether the kernel of the polarization is a connected group or an extension of étale by multiplicative group.
We have $\mathfrak{C}_1^{=1} = \mathfrak{C}_1^{=1, m} \cup \mathfrak{C}_1^{=1, et} \cup \mathfrak{C}_1^{=1,o}$ according to whether $L_1^\bot$ is multiplicative, étale or bi-infinitesimal.
We have $\mathfrak{C}^{=1} = \mathfrak{C}^{=1, m,et} \cup \mathfrak{C}^{=1, et,et} \cup \mathfrak{C}^{=1,m,m} \cup \mathfrak{C}^{=1,o,o}$ according to the isogeny $G \rightarrow G/L_1 \rightarrow G/L$. Namely :
- in case $m,et$, the group $L_1^\bot$ is multiplicative and $L/L_1$ is étale,
- in case $et,et$, the group $L_1^\bot$ is étale and $L/L_1$ is multiplicative,
- in case $m,m$, the group $L_1^\bot$ is multiplicative and $L/L_1$ is multiplicative,
- in case $o,o$, the group $L_1^\bot$ is bi-infinitesimal and $L/L_1$ is also bi-infinitesimal.
\[lemUTnord\] If If $k_1 > p+1$, $k_2 + k_1 > 1 + 2p$ and $k_1-k_2 > 3(p+1)$, then $T=U$ on $H^0(X^{=1}_1, \omega(k_1, k_2))$.
The proof is very similar to [@pilloni17 Lemma 11.1.2.1]. We let $T^{\star, \star \star} : q_2^\star \omega(k_2,k_1) \rightarrow q_1^! \omega(k_2,k_1)$ the restriction of $T$ to a component of $\mathfrak{C}^{=1, \star, \star \star}$. Our objective is to prove that $T$ reduces to $T^{et,et}=U$ on $X^{=1}_1$.
We have to look at all possible types of the isogeny:
- We have $T^{et,m} : q_2^\star \omega(k_1,k_2) \rightarrow p^{k_2 +k_1 - k_2-3} q_1^! \omega(k_2,k_1)$ because $G[p] \subset L$,
- We have $T^{m,et} : q_2^\star \omega(k_1,k_2) \rightarrow p^{k_2 +k_1-k_2-3} q_1^! \omega(k_2,k_1)$ because $G[p]^0 \subset L$,
- We have $T^{m,m} : q_2^\star \omega(k_1,k_2) \rightarrow p^{k_2 +k_1-k_2-3} q_1^! \omega(k_2,k_1)$ because $G[p] \subset L$.
The case of the component $o,o$ remains. Let $\xi = \operatorname{Spec}\bar{k} \rightarrow X^{=1}_1$ be a generic point. Let $\tilde{\xi} : W(\bar{k}) \rightarrow \mathfrak{X}^{=1}$ be a lift of $\xi$. We may restrict the cohomological correspondence to $\tilde{\xi}$ and we have a map $T^{o,o} : (q_1)_\star q_2^\star \omega(k_1,k_2)_{\tilde{\xi}} \rightarrow \omega(k_1,k_2)_{\tilde{\xi}}$. For a section $f \in (q_1)_\star q_2^\star \omega(k_1,k_2)_{\tilde{\xi}}$, this map writes $T^{oo}f(G,\mu) = \frac{1}{p^{k_2 +3}} \sum_{L} f(G/L, \mu_L)$ where $L$ runs over all subgroups of $G[p^2]$ corresponding to the component $\mathfrak{C}^{=1, o,o}$ (defined over ${\mathcal{O}}_{{\mathbf{C}}_p}$), $\mu$ is a trivialization of $\omega_G$ and $\mu_L$ is a rational trivialization of $\omega_{G/L}$ such that $\pi_L^\star \mu_L = \mu$ for the isogeny $\pi_L : G \rightarrow G/L$. For any $L$, the map $\pi_L^\star : \omega_{G/L} \rightarrow \omega_G$ has elementary divisor $(p, \varpi)$ for an element $\varpi$ with $v(\varpi) {\geqslant}\frac{1}{p+1}$. We find that $f(G/L, \mu_L) \in p^{k_2} \varpi^{k_1-k_2} {\mathcal{O}}_{{\mathbf{C}}_p}$. If $k_1-k_2 > 3(p+1)$ we find that $T^{o,o} ( f(G, \mu)) = 0 \pmod p$.
The local finiteness of $T$ on $\mathrm{R}\Gamma(X,\omega(k_1,k_2)(-D))$ for $k_1$ large enough (depending on $k_2$) follows easily from the previous lemmas. We can now prove :
\[prop-cool\] If $k_1 > 2$ and $k_2 + k_1 > 3$ and $k_1 -k_2 > 3(p-1)$, the map $e(T) H^1(X_1^{{\geqslant}1}, \omega(k_1, k_2)(-D)) \rightarrow e(U) H^1(X^{{\geqslant}1}_{\operatorname{Kl}}(p)_1, \omega(k_1, k_2)(-D))$ is an isomorphism.
The following complexes compute the cohomology of $\omega(k_1, k_2)(-D)$ over $X_1$ and $X^{{\geqslant}1}_{\operatorname{Kl}}(p)_1$
$$\xymatrix{ H^0( X^{{\geqslant}2}_{\operatorname{Kl}}(p)_1, \omega(k_1, k_2)(-D)) \ar[r] & \displaystyle \varinjlim_{n} H^0( X^{{\geqslant}1}_{\operatorname{Kl}}(p)_1, \omega{(k_1+n(p-1),k_2+n(p-1))}(-D)/Ha^n) \\
H^0( X^{{\geqslant}2}_1, \omega(k_1, k_2)(-D)) \ar[r] \ar[u]& \displaystyle \varinjlim_{n} H^0( X^{{\geqslant}1}_1, \omega{(k_1+n(p-1),k_2+n(p-1))}(-D)/Ha^n) \ar[u] }$$
We apply the projector for $U$ on the top and for $T$ on the bottom.
It follows from lemmas \[lem-UTord\] and \[lemUTnord\] and that the left vertical map becomes surjective and the right vertical map an isomorphism after applying the projector. This is enough to conclude.
\[lem-inj1\] If $k_1 > 2$, $k_2 + k_1 > 3$ and $k_1 -k_2 > 3(p-1)$, the map $e(T) H^1(\mathfrak{X}^{{\geqslant}1}, \omega(k_1, k_2)(-D)) \rightarrow e(U) H^1(\mathfrak{X}^{{\geqslant}1}_{\operatorname{Kl}}(p), \omega(k_1, k_2)(-D))$ is an isomorphism.
We first show that the map is surjective. Since $H^2(\mathfrak{X}^{{\geqslant}1}, \omega(k_1, k_2)(-D)) =0$, we deduce that $e(T) H^1(\mathfrak{X}^{{\geqslant}1}, \omega(k_1, k_2)(-D))/p = e(T) H^1({X}_1^{{\geqslant}1}, \omega(k_1, k_2)(-D))$ and by Nakayama’s lemma and proposition \[prop-cool\], we deduce the surjectivity. Let $K$ be the kernel of the map of the proposition. By reduction modulo $p$ we get an exact sequence (since $ e(U) H^1(\mathfrak{X}^{{\geqslant}1}_{\operatorname{Kl}}(p), \omega(k_1, k_2)(-D))$ is flat): $$0 \rightarrow K/p \rightarrow e(T) H^1({X}^{{\geqslant}1}_1, \omega(k_1, k_2)(-D)) \rightarrow e(U) H^1({X}^{{\geqslant}1}_{\operatorname{Kl}}(p)_1, \omega(k_1, k_2)(-D))$$ from which it follows that $K/p=0$ and that $K=0$.
We now prove the analoguous result for classical cohomology.
\[lem-inj2\] If $k_1+k_2 -3{\geqslant}2$ and $k_1 {\geqslant}3$, the map $$e(T)\mathrm{R}\Gamma(\mathcal{X}, \omega(k_1, k_2)(-D)) \rightarrow e (U)\mathrm{R}\Gamma(\mathcal{X}_{\operatorname{Kl}}(p), \omega(k_1, k_2)(-D))$$ is an isomorphism.
We shall consider the smooth admissible $\operatorname{GSp}_4({{\mathbf{Q}}_p})$ representation $\varinjlim_n H^i( \mathcal{X}(p^n), \omega(k_1, k_2)(-D))$. The invariants under $\operatorname{GSp}_4({{\mathbf{Z}}_p})$ are $H^i(\mathcal{X}, \omega(k_1, k_2)(-D))$ and the invariants under the Klingen parahoric are $H^i(\mathcal{X}_{\operatorname{Kl}}(p), \omega(k_1, k_2)(-D))$. Let $\pi$ be an irreducible representation with spherical invariants contributing to $\varinjlim_n H^i( \mathcal{X}(p^n), \omega(k_1, k_2)(-D))$. Then $\pi$ is a quotient of a principal series representation and its Hecke parameters are $(\alpha, \beta, \gamma, \delta)$ such that $\alpha \delta = \beta \gamma$. We may assume that they are in increasing $p$-adic valuation. Morover, by Proposition \[prop:newtonpoly\], the Newton polygon is above the Hodge polygon, which has slopes in increasing order $k_2-2, 0, k_1+k_2-3, k_1-1$ (or $0,k_2-2, k_1-1, k_1+k_2-2$ but the case of interest to us will be $k_2 {\leqslant}0$ so we will assume for simplicity that we are in the first case), and they have the same initial and end point. The $T$ eigenvalue is $p^{2-k_2}( \alpha\beta + \alpha \gamma + \alpha \delta + \beta\delta + \gamma \delta) + p^{1-k_2} (1 + p + p^2) \alpha\delta$. Under the assumption that $k_1 {\geqslant}3$, we find that $p^{1-k_2} (1 + p + p^2) \alpha\delta$ has $p$-adic valuation at least one and that $\pi$ is $T$-ordinary if and only if $\alpha\beta$ has valuation $k_2-2$. Under the assumption that $k_1+k_2 -3{\geqslant}2$ we find that $\xi^{-1} \xi' \neq p$ for all $\xi, \xi' \in \{ \alpha, \beta, \gamma, \delta\}$ or that $\beta = p \alpha$ and $p\gamma = \delta$. Therefore, if $\pi$ is $T$-ordinary it is either an irreducible principal series type I or a type III(b) representation in Schmidt’s classification [@MR2114732]. Comparing with Corollary \[cor:sphericalordprojection\], we see that $e(T) \pi^{\operatorname{GSp}_4({{\mathbf{Z}}_p})}$ is non-zero if and only if $e(U) \pi^{\operatorname{Kl}(p)}$ is non-zero, and the natural map between the two is an isomorphism.
It follows that the map $e(T)\mathrm{R}\Gamma(\mathcal{X}, \omega(k_1, k_2)(-D)) \rightarrow e (U)\mathrm{R}\Gamma(\mathcal{X}_{\operatorname{Kl}}(p), \omega(k_1, k_2)(-D))$ induces an isomorphism on every graded piece of some Jordan–Hölder filtration, and hence is an isomorphism as required.
If $k_1+k_2 -3{\geqslant}2$, $k_1 {\geqslant}3$ and $k_1 -k_2 > 3(p-1)$, we have a commutative diagram : $$\xymatrix{ e (U){H}^1(\mathcal{X}_{\operatorname{Kl}}(p), \omega(k_1, k_2)(-D)) \ar[r] & e(U){H}^1(\mathcal{X}^{{\geqslant}1}_{\operatorname{Kl}}(p), \omega(k_1, k_2)(-D)) \\
e(T){H}^1(\mathcal{X}, \omega(k_1, k_2)(-D)) \ar[r] \ar[u] & e(T) {H}^1(\mathcal{X}^{{\geqslant}1}, \omega(k_1, k_2)(-D)) \ar[u]}$$
and the vertical maps are isomorphisms.
This follows from lemmas \[lem-inj1\] and \[lem-inj2\].
If $k_1+k_2 -3{\geqslant}2$, $k_1 {\geqslant}3$ and $k_1 -k_2 > 3(p-1)$, the map $eH^1(\mathcal{X}, \omega(k_1, k_2)(-D)) \rightarrow eH^1(\mathcal{X}^{{\geqslant}1}, \omega(k_1, k_2)(-D))$ is injective.
This follows from theorem \[thmSGA\].
If $k_1+k_2 -3{\geqslant}2$, $k_1 {\geqslant}3$ and $k_1 -k_2 > 3(p-1)$, the map $eH^1(\mathcal{X}_{\operatorname{Kl}}(p), \omega(k_1, k_2)(-D)) \rightarrow e H^1(\mathcal{X}^{{\geqslant}1}_{\operatorname{Kl}}(p), \omega(k_1, k_2)(-D))$ is injective.
We observe that for a fixed value of $k_2$, the conditions on the weight are satisfied for all $k_1$ large enough.
### Injectivity for all values of $k_1$
The perfect complex $M^\bullet_{\kappa^{un}, k_2}$ admits an overconvergent variant $M^{\dag, \bullet}_{\kappa^{un}, k_2}$. Its main properties are (see section 13 and 14 of [@pilloni17]) :
1. If $k_1 + k_2 > 3$, $k_1-k_2 >2$, then $M^{\dag, \bullet}_{k_1, k_2} = e \mathrm{R}\Gamma ( \mathcal{X}_{\operatorname{Kl}}(p), \omega(k_1, k_2))(-D))$.
2. If $k_1 + k_2 >3$ and $k_1 {\geqslant}k_2$, there is an isomorphism $$H^0(M^{\dag, \bullet}_{k_1, k_2}) = e H^0 ( \mathcal{X}_{\operatorname{Kl}}(p), \omega(k_1, k_2))(-D)) =0$$ and an injective map $$H^1(M^{\dag, \bullet}_{k_1, k_2}) \leftarrow e H^1 ( \mathcal{X}_{\operatorname{Kl}}(p), \omega(k_1, k_2))(-D)).$$
We consider the two functions defined for $k_1 > 3-k_2$ : $d(k_1) = \dim e H^1( \mathcal{X}_{\operatorname{Kl}}(p), \omega(k_1, k_2)(-D))$, $d'(k_1) = \dim H^1(M^{\dag, \bullet}_{k_1, k_2})$, and $d^{ord} (k_1) = \dim e H^1( \mathcal{X}^{{\geqslant}1}_{\operatorname{Kl}}(p), \omega(k_1, k_2)(-D))$.
We have proved that $d(k_1) {\geqslant}d^{ord}(k_1)$ and that equality holds if $k_1$ is large enough. We have $d(k_1) {\leqslant}d'(k_1)$ and equality holds if $k_1$ is large enough. Moreover, $d^{ord}$ and $d'$ are locally constant if we endow ${\mathbf{Z}}_{{\geqslant}4-k_2}$ with the topology of characters of ${{\mathbf{Z}}_p}^\times$. We deduce that $d^{ord} = d'$ and finally that $d=d^{ord}$.
Therefore $d(k_1) = d^{ord}(k_1)$ for all values of $k_1$, completing the proof of Theorem \[thm:vincent\].
Construction of the p-adic pushforward map {#sect:p-adic-pushfwd}
==========================================
In this section, we’ll define morphisms from spaces of $p$-adic modular forms for $H$ to the $p$-adic $H^1$ spaces for $G$ defined in the preceding section, and show that they interpolate pushforward maps for the usual coherent automorphic sheaves.
Level groups
------------
For $m {\geqslant}1$, define subgroups of $H({{\mathbf{Z}}_p})$ by $$\begin{aligned}
K_{H, 1}(p^m) &\coloneqq \{ h \in H({{\mathbf{Z}}_p}): h = \left({\left( \begin{smallmatrix} 1 & * \\ & *\end{smallmatrix} \right) }, {\left( \begin{smallmatrix} 1 & * \\ & *\end{smallmatrix} \right) }\right) \pmod{p^m}\},\\
K_{H, \Delta}(p^m) &\coloneqq \{ h: h = \left({\left( \begin{smallmatrix} x & * \\ & *\end{smallmatrix} \right) }, {\left( \begin{smallmatrix} x & * \\ & *\end{smallmatrix} \right) }\right) \pmod{p^m} \text{ \textup{for some $x$}}\},\\
K_{H, 0}(p^m) &\coloneqq \{ h: h = \left({\left( \begin{smallmatrix} * & * \\ & *\end{smallmatrix} \right) }, {\left( \begin{smallmatrix} * & * \\ & *\end{smallmatrix} \right) }\right) \pmod{p^m}\}.
\end{aligned}$$
The following elementary but important group-theoretic computation underpins our construction. Recall the subgroup $\operatorname{Kl}(p^m) = \left\{ \left( \begin{smallmatrix}
* & * & * & * \\ &*&*&* \\ &*&*&* \\ &&&*\end{smallmatrix} \right) \pmod{p^m}\right\} \subset G({{\mathbf{Z}}_p})$ defined in §\[sect:Gsetup\] above.
Let $\gamma \in G({{\mathbf{Z}}_p})$ be any element whose first column is $(1, 1, 0, 0)^T$. Then we have $$\pushQED{\qed}
H({{\mathbf{Q}}_p}) \cap \gamma \operatorname{Kl}(p^m) \gamma^{-1} = K_{H, \Delta}(p^m).\qedhere
\popQED$$
The group $H$ acts on $G / P_{\operatorname{Kl}} \cong \mathbf{P}^3$ with exactly 3 orbits, two closed and one open, and the element $\gamma$ represents the open orbit.
We write $X_{H, ?}(p^m)_{{\mathbf{Q}}}$ for the Shimura variety for $H$ of level $K_H^p K_{H, ?}(p^m)$ (for our fixed prime-to-$p$ level $K^p$), and ${\mathcal{X}}_{H, ?}(p^m)$ for the associated rigid-analytic spaces over ${{\mathbf{Q}}_p}$. The proposition implies that there is a finite morphism of ${\mathbf{Q}}$-varieties $$\iota_m: X_{H, \Delta}(p^m)_{{\mathbf{Q}}} \to X_{G, \operatorname{Kl}}(p^m)_{{\mathbf{Q}}},$$ given by composing $\iota$ with right-translation by $\gamma$ (and this is a closed embedding away from the boundary if $K^p$ is small enough). Our goal is to interpolate pullback and pushforward maps for these embeddings $\iota_m$.
Classical modular forms
-----------------------
For integers $\ell_1, \ell_2 {\geqslant}0$, $L$ any field extension of ${\mathbf{Q}}$, we define $$\begin{aligned}
M_{(\ell_1, \ell_2)}\big(p^m, L\big) &\coloneqq H^0\big( X_{H, 0}(p^m)_L, \omega_H(\ell_1, \ell_2)\big),\\
S_{(\ell_1, \ell_2)}\big(p^m, L\big) &\coloneqq H^0\big( X_{H, 0}(p^m)_L, \omega_H(\ell_1, \ell_2)(-D_H)\big),\end{aligned}$$ with Hecke operators normalised as in Notation \[not:sheaves\], so that if $K_H^p$ has level $N$ and $q \ne p$ is a prime congruent to 1 modulo $N$, then the double coset of $\operatorname{diag}(q, q, q, q) \in H({\mathbf{Q}}_q)$ acts as multiplication by $q^{\ell_1 + \ell_2 - 4}$.
More generally, let $\chi_1, \chi_2$ be characters of $({\mathbf{Z}}/ p^m)^\times$ with values in $L$. We let $M_{(\ell_1, \ell_2)}(p^m, \chi_1, \chi_2, L)$ be the subspace of $H^0\big( X_{H, 1}(p^m)_L, \omega_H(\ell_1, \ell_2)\big)$ on which the quotient $K_{H, 0}(p^m) / K_{H, 1}(p^m)$ acts via the character $\left({\left( \begin{smallmatrix} a & * \\ & *\end{smallmatrix} \right) }, {\left( \begin{smallmatrix} b & * \\ & *\end{smallmatrix} \right) }\right)\mapsto \chi_1(a) \chi_2(b)$, and similarly $S_{(\ell_1, \ell_2)}(p^m, \chi_1, \chi_2, L)$. Note that for $q$ as above, the action of $\operatorname{diag}(q, \dots, q) \in H({\mathbf{Q}}_q)$ on this space is now via $q^{\ell_1 + \ell_2 - 4} \cdot \chi_1\chi_2(q)^{-1}$.
The Igusa tower for $H$
-----------------------
Recall that $X_H$ denotes a smooth compactification over ${\mathbf{Z}}_{(p)}$ of the modular curve $Y_H$ of prime-to-$p$ level $K_H = K_H^p H({{\mathbf{Z}}_p})$. Over $Y_H$ we have two elliptic curves $E_1, E_2$, which extend to semiabelian schemes over $X_H$; and the direct sum $E_1 \oplus E_2$ is the pullback via $\iota: X_H \to X_G$ of the universal semiabelian scheme $A_G / X_G$.
Let ${\mathfrak{X}}_H / \operatorname{Spf}{{\mathbf{Z}}_p}$ be the $p$-adic formal completion of $X_H$, and ${\mathfrak{X}}_H^{{\mathrm{ord}}}$ the ordinary locus in ${\mathfrak{X}}_H$. Over ${\mathfrak{X}}_H^{{\mathrm{ord}}}$ we have an Igusa tower ${\mathfrak{IG}}_H(p^\infty) = \varprojlim_m {\mathfrak{IG}}_H(p^m)$, parametrising embeddings $\alpha_i: \mu_{p^n} {\hookrightarrow}E_i$ for $i= 1, 2$; this is evidently a $(\Gamma \times \Gamma)$-torsor.
Let $\Delta_m$ denote the diagonally-embedded copy of $({\mathbf{Z}}/ p^m {\mathbf{Z}})^{\times}$ in $\left(({\mathbf{Z}}/ p^m {\mathbf{Z}})^{\times}\right)^2$, and let ${\mathfrak{X}}_{H, \Delta}^{\mathrm{ord}}(p^m)$ denote the quotient ${\mathfrak{IG}}_H(p^m) / \Delta_m$.
The generic fibres of the formal schemes ${\mathfrak{X}}_{H, \Delta}^{\mathrm{ord}}(p^m)$ and ${\mathfrak{IG}}_H(p^m)$ are naturally open subvarieties of the rigid-analytic varieties ${\mathcal{X}}_{H, \Delta}(p^m)$ and ${\mathcal{X}}_{H, 1}(p^m)$ respectively.
$R$-adic sheaves {#sect:bigsheafH}
----------------
In analogy with the theory for $G$, we shall let $R$ be a $p$-adic ring with two characters $\kappa_1, \kappa_2: \Gamma \to R$, and consider the sheaf on ${\mathfrak{X}}_{H, \Delta}^{\mathrm{ord}}(p)$ defined by $$\mathfrak{F}_{H, R}(\kappa_1, \kappa_2) = \left( \pi_\star {\mathcal{O}}_{{\mathfrak{IG}}_H(p^\infty)} {\mathop{\hat\otimes}}R\right)[\Gamma \dot{\times} \Gamma = (\kappa_1, \kappa_2)] \otimes W_H(0, 0; -4)$$ where $\pi: {\mathfrak{IG}}_H(p^\infty) \to {\mathfrak{X}}_{H, \Delta}^{{\mathrm{ord}}}(p)$ is the projection map, and $\Gamma \dot{\times} \Gamma$ denotes the group $\{(x, y) \in \Gamma \times \Gamma: x = y \bmod p\}$. This is a rank 1 locally free sheaf of $R$-modules on ${\mathfrak{X}}_H^{\mathrm{ord}}(p)$; and we define spaces of $p$-adic modular forms (resp. cusp forms) by $$\begin{aligned}
{\mathcal{M}}_{(\kappa_1, \kappa_2)}(R) &\coloneqq H^0\left({\mathfrak{X}}_{H, \Delta}^{\mathrm{ord}}(p), \mathfrak{F}_{H, R}(\kappa_1, \kappa_2)\right)^Q,\\
{\mathcal{S}}_{(\kappa_1, \kappa_2)}(R) &\coloneqq H^0\left({\mathfrak{X}}_{H, \Delta}^{\mathrm{ord}}(p), \mathfrak{F}_{H, R}(\kappa_1, \kappa_2)(-D)\right)^Q\end{aligned}$$ where $Q$ is the quotient group $(\Gamma \times \Gamma) / (\Gamma \dot{\times} \Gamma) \cong ({\mathbf{Z}}/ p)^\times$. These spaces (and their classical analogues) are independent of the choice of rpcd $\Sigma$, and have an action of Hecke operators away from $p$, with $\operatorname{diag}(q, q, q, q) \in H({\mathbf{Q}}_q)$ for $q = 1 \bmod N$ acting as $q^{\kappa_1 + \kappa_2 - 4}$.
It might seem more natural to work with sheaves on ${\mathfrak{X}}_H^{\mathrm{ord}}$, not ${\mathfrak{X}}_{H, \Delta}^{\mathrm{ord}}(p)$, which would obviate the need for the tiresome quotient $Q$; but we choose to work over ${\mathfrak{X}}_{H, \Delta}^{\mathrm{ord}}(p)$ since it is easier to relate to the theory for $G$.
For any $k_1, k_2 \in {\mathbf{Z}}$, there is a comparison isomorphism $$\operatorname{comp}:\omega_H(k_1, k_2) \rTo^\cong \mathfrak{F}_{H}(k_1, k_2).$$ Hence, for any finite extension $L/{{\mathbf{Q}}_p}$ and characters $\chi_1, \chi_2: ({\mathbf{Z}}/ p^m)^\times \to L^\times$, we obtain an injection $$M_{(k_1, k_2)}(p^m, \chi_1, \chi_2, L) {\hookrightarrow}{\mathcal{M}}_{(k_1 + \chi_1, k_2 + \chi_2)}({\mathcal{O}}_L) \otimes L.$$ This is *not* Hecke-equivariant in general (unsurprisingly, because the central characters do not match). Rather, it intertwines the action of a double coset $[K^p_H h K^p_H]$, for $h \in H({{\mathbf{A}}_{\mathrm{f}}}^p)$, on the left-hand side with $\chi_1\chi_2( \|\det h\| ) \cdot [K^p_H h K^p_H]$ on the right-hand side. However, we shall only use this if $\chi_1 \chi_2 = 1$, in which case the map does indeed become Hecke-equivariant.
There are also versions of these comparisons for cusp forms. Crucially, however, a classical modular form may be a $p$-adic cusp form without being a cusp form in the classical sense. These “fake cusp forms” will be vital in our construction of $p$-adic $L$-functions below (as a substitute for our inability to prove an analogue of Theorem \[thm:vincent\] for cohomology of non-cuspidal sheaves on $G$).
Maps between Igusa towers
-------------------------
The morphism ${\mathfrak{X}}_{H, \Delta}^{{\mathrm{ord}}}(p^m) \to {\mathfrak{X}}_H^{{\mathrm{ord}}}$ represents the functor of isomorphisms $E_1[p^m]^\circ \cong E_2[p^m]^\circ$, where $E_i[p^m]^\circ$ is the identity component of $E_i[p^m]$. Note that $E_1 \oplus E_2$ is the pullback (via $\iota_{\Sigma}$) of the universal semiabelian surface $A$ over ${\mathfrak{X}}_G$. We can therefore consider the following diagram of morphisms of formal schemes, for any $m {\geqslant}1$:
\_H(p\^m) & \^[\_m]{} & \_G(p\^m)\
& &\
\_[H, ]{}\^(p\^m) & \^[\_m]{} & \_[G, ]{}\^[1]{}(p\^m)\
& &\
\_H & \^ & \_G
where the vertical arrows are the natural degeneracy maps, and the horizontal arrows are defined as follows:
- $\iota = \iota_{\Sigma}$ is as defined in §\[sect:toroidalfunct\] above;
- $\iota_m$ maps an isomorphism $E_1[p^m]^\circ \cong E_2[p^m]^\circ$ to its graph, considered as a multiplicative subgroup of $(E_1 \oplus E_2)[p^m]$;
- $\tilde{\iota_m}$ maps a pair of embeddings $(\alpha_1, \alpha_2)$ to $\alpha_1 + \alpha_2: \mu_{p^m} {\hookrightarrow}(E_1 \oplus E_2)[p^m]$.
The morphisms $\iota_m$ and $\tilde{\iota}_m$ are closed embeddings.
We first treat $\tilde\iota_m$. The commutativity of the above diagram implies that $\tilde{\iota}_m$ factors through the fibre product $\mathfrak{Z}_m = {\mathfrak{X}}_H \mathop{\times}_{{\mathfrak{X}}_G} {\mathfrak{IG}}_G(p^m)$. Since ${\mathfrak{X}}_H {\hookrightarrow}{\mathfrak{X}}_G$ is a closed embedding, so is the second projection $\mathfrak{Z}_m {\hookrightarrow}{\mathfrak{IG}}_G(p^m)$. Hence it suffices to show that the morphism ${\mathfrak{IG}}_H(p^m) \to \mathfrak{Z}_m$ is a closed embedding.
We shall show that ${\mathfrak{IG}}_H(p^m)$ is in fact a component of $\mathfrak{Z}_m$. The space $\mathfrak{Z}_m$, by construction, classifies embeddings $\alpha: \mu_{p^n} {\hookrightarrow}(E_1 \oplus E_2)[p^m]$. Composing $\alpha$ with the first and second projections, we obtain morphisms of group schemes $\alpha_i: \mu_{p^m} \to E_i[p^m]$ over $\mathfrak{Z}_m$. By Theorem IX.6.8 of SGA III, there is an open-and-closed formal subscheme $\mathfrak{Z}_m^\circ \subseteq \mathfrak{Z}_m$ over which the maps $\alpha_i$ are both embeddings[^1]. The map $\tilde\iota_m$ clearly factors through $\mathfrak{Z}_m^\circ$, and the projections $\alpha_i$ determine a map $\mathfrak{Z}_m^\circ \to {\mathfrak{IG}}_H(p^m)$ which is an inverse to $\tilde\iota_m$. Hence $\tilde\iota_m$ defines an isomorphism between ${\mathfrak{IG}}_H(p^m)$ and $\mathfrak{Z}_m^\circ$, and in particular it is a closed embedding into $\mathfrak{Z}_m$ and hence also into ${\mathfrak{IG}}_G(p^m)$.
To obtain the statement for $\iota_m$, we note that $\tilde\iota_m$ intertwines the action of $\Delta_m \subset (({\mathbf{Z}}/ p^m {\mathbf{Z}})^\times)^2$ on ${\mathfrak{IG}}_H(p^m)$ with the natural $({\mathbf{Z}}/ p^m {\mathbf{Z}})^\times$-action on ${\mathfrak{IG}}_G(p^m)$, so we can recover $\iota_m$ by passage to the quotient.
\[lem:cartesian\] For any $m {\geqslant}1$, the diagrams $$\begin{diagram}[small]
{\mathfrak{IG}}_H(p^m) & \rTo^{\tilde\iota_m} & {\mathfrak{IG}}_G(p^m)\\
\dTo & & \dTo \\
{\mathfrak{X}}_{H, \Delta}^{\mathrm{ord}}(p) & \rTo^{\iota_1} & {\mathfrak{X}}_{G, \operatorname{Kl}}^{{\geqslant}1}(p)
\end{diagram}
\quad\text{and}\quad
\begin{diagram}[small]
{\mathfrak{X}}_{H, \Delta}^{\mathrm{ord}}(p^m) & \rTo^{\iota_m} & {\mathfrak{X}}_{G, \operatorname{Kl}}^{{\geqslant}1}(p^m)\\
\dTo & & \dTo \\
{\mathfrak{X}}_{H, \Delta}^{\mathrm{ord}}(p) & \rTo^{\iota_1} & {\mathfrak{X}}_{G, \operatorname{Kl}}^{{\geqslant}1}(p)
\end{diagram}$$ are Cartesian.
The first statement follows from the fact that if $\alpha: \mu_{p^m} {\hookrightarrow}(E_1 \oplus E_2)[p^m]$ is an embedding and the projections $\alpha_i$ of $\alpha$ to the $E_i$ are injective restricted to $\mu_p$, then the $\alpha_i$ are in fact injective, so $\alpha$ comes from a point of ${\mathfrak{IG}}_H(p^m)$. The second statement is similar (replacing the embeddings with their images).
We can restate this in terms of a compatibility of “big” sheaves on ${\mathfrak{X}}_{H, \Delta}^{\mathrm{ord}}(p)$. We let $\pi_G$ denote the projection ${\mathfrak{IG}}_G(p^\infty) \to {\mathfrak{X}}_{G, \operatorname{Kl}}^{{\geqslant}1}(p)$; and we let $\pi_H$ denote the projection ${\mathfrak{IG}}_H(p^\infty) \to {\mathfrak{X}}_{H, \Delta}^{{\mathrm{ord}}}(p)$. Then the Cartesian diagram above translates into the following equality of sheaves on ${\mathfrak{X}}_{H, \Delta}^{{\mathrm{ord}}}(p)$: $$\iota_1^\star\left( \pi_{G, \star} {\mathcal{O}}_{{\mathfrak{IG}}_G(p^\infty)} \right) = \pi_{H, \star} {\mathcal{O}}_{{\mathfrak{IG}}_H(p^\infty)}.$$ We note that $\pi_{G, \star} {\mathcal{O}}_{{\mathfrak{IG}}_G(p^\infty)}$ is a sheaf of $\Gamma$-modules. On the other hand, $\pi_{H, \star} {\mathcal{O}}_{{\mathfrak{IG}}_H(p^\infty)}$ has an action of the larger group $\Gamma \dot{\times} \Gamma \coloneqq \{ (a, b) \in \Gamma \times \Gamma: a = b \bmod p\}$; if we regard it as a sheaf of $\Gamma$-modules by restriction to the diagonal $\Delta_\infty \cong \Gamma$, then the above isomorphism is $\Gamma$-equivariant.
Pullback of $R$-adic sheaves
----------------------------
We now let $R$ be any $p$-adic ring equipped with two continuous characters $\lambda_1, \lambda_2: \Gamma \to R^\times$; and let $k_2 \in {\mathbf{Z}}$. If we define[^2] $\kappa = \lambda_1 + \lambda_2 - k_2$, then we have defined above the sheaf ${\mathfrak{F}}_{G, R}(\kappa, k_2)$ of $R$-modules on ${\mathfrak{X}}_{G, \operatorname{Kl}}^{{\geqslant}1}(p)$ by $${\mathfrak{F}}_{G, R}(\kappa, k_2) = \left( \pi_{G, \star} {\mathcal{O}}_{{\mathfrak{IG}}_G(p^\infty)} \hat\otimes R\right)[\Gamma = \kappa - k_2] \otimes \omega_G(k_2, k_2; 2k_2- 6).$$ The Cartesian property of the first diagram of Lemma \[lem:cartesian\] gives an isomorphism of $R$-module sheaves on ${\mathfrak{X}}_H^{\mathrm{ord}}(p)$, $$\iota_1^\star\left({\mathfrak{F}}_{G, R}(\kappa, k_2)\right) = \left( \pi_{H, \star} {\mathcal{O}}_{{\mathfrak{IG}}_H(p^\infty)} \hat\otimes R\right)[\Delta_\infty = \kappa-k_2] \otimes \omega_H(k_2, k_2; 2k_2 - 6).$$ Since the character $\kappa-k_2 = \lambda_1 + \lambda_2-2k_2$ of $\Delta_\infty$ is the restriction of the character $(\lambda_1 - k_2, \lambda_2 - k_2)$ of $\Gamma \dot{\times} \Gamma$, we have a canonical inclusion $$\left( \pi_{H, \star} {\mathcal{O}}_{{\mathfrak{IG}}_H(p^\infty)} \hat\otimes R\right)[\Gamma \dot{\times} \Gamma = (\lambda_1 - \kappa_1, \lambda_2 - \kappa_2)] {\hookrightarrow}\left( \pi_{H, \star} {\mathcal{O}}_{{\mathfrak{IG}}_H(p^\infty)} \hat\otimes R\right)[\Delta_\infty = \kappa-k_2],$$ and hence an inclusion of sheaves $${\mathfrak{F}}_{H, R}(\lambda_1, \lambda_2) \{-2\} {\hookrightarrow}\iota_1^\star\left({\mathfrak{F}}_{G, R}(\kappa, k_2)\right),$$ where $\{-2\}$ indicates that the Hecke action has been twisted by the character $\det^{-2}$.
\[prop:pullbackcomp\] For any integers $\ell_1, \ell_2, k_1, k_2$ with $\min(\ell_1, \ell_2, k_1) {\geqslant}k_2$ and $\ell_1 + \ell_2 = k_1 + k_2$, we have the following commutative diagram of sheaves on ${\mathfrak{X}}_{H, \Delta}^{{\mathrm{ord}}}(p)$:
\_H(\_1, \_2){-2} & & \_1\^\_G(k\_1, k\_2)\
& &\
\_H(\_1, \_2){-2} & & \_1\^(\_G(k\_1, k\_2))
Here the right-hand vertical arrow is the pullback via $\iota_1$ of the comparison morphism for $G$, and the left-hand arrow the analogous comparison morphism for $H$.
We may assume without loss of generality that $k_2 = 0$. It suffices to prove the statement modulo $p^n$ for each $n$. This arises (via adjunction) from the following diagram of sheaves on the mod $p^n$ reduction of ${\mathfrak{IG}}_H(p^n)$ (where we omit the central character terms for brevity)
\_H(\_1, \_2) & & \_n\^\_G(k\_1, 0)\
& &\
\_[\_H(p\^n)\_n]{} & & \_n\^(\_[\_G(p\^n)\_n]{})
Here the right vertical arrow is given by pullback to the universal $\mu_{p^n}$-subgroup composed with the Hodge–Tate period map. By construction, this restricts to the standard trivialisation of either $\omega_H(1, 0)$ or of $\omega_H(0, 1)$ modulo $p^n$.
As noted in [@pilloni17 Remark 9.4.1], the comparison morphism $\omega_G(k_1, k_2) \to {\mathfrak{F}}_G(k_1, k_2)$ can be interpreted as projection onto the highest weight vector in the representation $W_G(k_1, k_2)$ of $\operatorname{GL}_2$ (the Levi factor of the Siegel parabolic in $\operatorname{Sp}_4$). In this optic, our choice of embedding of ${\mathfrak{IG}}_H(p^\infty)$ into ${\mathfrak{IG}}_G(p^\infty)$ corresponds to acting on this highest weight vector by the element ${\left( \begin{smallmatrix} 1 & \\ 1 & 1\end{smallmatrix} \right) }$ of $\operatorname{GL}_2$, so that it projects nontrivially to every direct summand for the action of the subgroup $\operatorname{GL}_1 \times \operatorname{GL}_1$ (the Levi factor of the Borel in the derived subgroup of $H$). This is an analogue for coherent automorphic sheaves of the branching computations for étale sheaves in [@loefflerskinnerzerbes17].
Pushforward of $R$-adic sheaves
-------------------------------
Let $X^{{\geqslant}1}_{G, \operatorname{Kl}}(p)_n$ denote the mod $p^n$ reduction of the formal scheme ${\mathfrak{X}}^{{\geqslant}1}_{G, \operatorname{Kl}}(p)$. Note that this is a smooth and quasi-projective $({\mathbf{Z}}/ p^n)$-scheme, and thus its relative dualising complex is defined, and is isomorphic to a shift of the sheaf of top-degree differentials; and similarly for ${\mathfrak{X}}_{H, \Delta}^{{\mathrm{ord}}}(p)$. Carrying out the same construction as in §\[sect:pfwdsheaf\] for these truncated schemes, we obtain homomorphisms $$H^0\left( X_{H, \Delta}^{{\mathrm{ord}}}(p)_n,\iota_1^\star(M) \otimes \omega_H(-1, -1;0)(-D_H)\right)
\rTo
H^1\left(X_{G, \operatorname{Kl}}^{{\geqslant}1}(p)_n, M(-D_G) \right),$$ for $M$ any flat quasi-coherent sheaf on $X^{{\geqslant}1}_{G, \operatorname{Kl}}(p)_n$ (not necessarily of finite rank). If $\mathfrak{M} = (M_n)_{n {\geqslant}1}$ is a flat formal Banach sheaf on ${\mathfrak{X}}_{G, \operatorname{Kl}}(p)$ in the sense of [@pilloni17 §12.6], then the above morphisms are compatible as $n$ varies and assemble into a map $$\iota_{\star}: H^0\left( {\mathfrak{X}}_{H, \Delta}^{{\mathrm{ord}}}(p), \iota_1^\star(\mathfrak{M}) \otimes \omega_H(-1, -1;0)(-D_H)\right)
\rTo
H^1\left({\mathfrak{X}}_{G, \operatorname{Kl}}^{{\geqslant}1}(p), \mathfrak{M}(-D_G) \right).$$ In particular, this applies to the sheaf ${\mathfrak{F}}_{G, R}(\kappa, k_2)$ defined above (assuming the coefficient ring $R$ to be ${{\mathbf{Z}}_p}$-flat). If we choose characters $\lambda_1, \lambda_2: {{\mathbf{Z}}_p}^\times \to R^\times$ with $\lambda_1 + \lambda_2 = \kappa + k_2 - 2$, then the preceding section gives a morphism $${\mathfrak{F}}_H(\lambda_1+1, \lambda_2+1) \otimes \omega_H(0, 0; -2) \to \iota_1^\star \left({\mathfrak{F}}_G(\kappa, k_2)\right).$$ Composing this with the above coboundary morphism, and noting that ${\mathfrak{F}}_H(\lambda_1+1, \lambda_2+1) \otimes \omega_H(-1, -1; -2) = {\mathfrak{F}}_H(\lambda_1, \lambda_2)$, gives a map of $R$-modules $$\label{eq:pwfdR}
\iota_\star: {\mathcal{S}}_{(\lambda_1, \lambda_2)}(R) \rTo
H^1\left({\mathfrak{X}}_{G, \operatorname{Kl}}^{{\geqslant}1}(p), {\mathfrak{F}}_G(\kappa, k_2)(-D_G) \right).$$ By construction, this map is compatible with base-change in $(R, \lambda_1, \lambda_2)$.
Comparison with classical pushforward
-------------------------------------
We want to compare the map $\iota_\star$ of with the analogous construction for algebraic varieties. The comparison is most conveniently carried out in the language of rigid-analytic spaces. We choose integers $k_1, k_2, \ell_1, \ell_2$ such that $\min(\ell_1+1, \ell_2 + 1, k_1) {\geqslant}k_2$ and $\ell_1 + \ell_2 = k_1 + k_2 - 2$.
Let ${\mathcal{X}}^{{\geqslant}1}_{G, \operatorname{Kl}}(p^m)$ be the rigid-analytic generic fibre of the formal scheme ${\mathfrak{X}}^{{\geqslant}1}_{G, \operatorname{Kl}}(p^m)$. On the algebraic side, let $X_{G, \operatorname{Kl}}(p^m)_{{{\mathbf{Q}}_p}}$ be the base-change to ${{\mathbf{Q}}_p}$ of the toroidal compactification of the Shimura variety of level $K_{G, \operatorname{Kl}}(p^m)$ (using the same rpcd $\Sigma$ we used at spherical level; this may not be smooth for level $K_{G, \operatorname{Kl}}(p^m)$, but this does not matter). If ${\mathcal{X}}_{G, \operatorname{Kl}}(p^m)$ is the associated rigid space, then we have a natural open immersion $${\mathcal{X}}^{{\geqslant}1}_{G, \operatorname{Kl}}(p^m) {\hookrightarrow}{\mathcal{X}}_{G, \operatorname{Kl}}(p^m).$$
We can argue similarly with $H$ in place of $G$ to obtain an open immersion $${\mathcal{X}}_{H, \Delta}^{{\mathrm{ord}}}(p^m) {\hookrightarrow}{\mathcal{X}}_{H, \Delta}(p^m).$$ There is a finite morphism of ${\mathbf{Q}}$-varieties $X_{H, \Delta}(p^m)_{{\mathbf{Q}}} \to X_{G, \operatorname{Kl}}(p^m)_{{\mathbf{Q}}}$, injective away from the boundary, given by composing the standard embedding $\iota: H {\hookrightarrow}G$ with translation by the element $\gamma$ above. We denote this morphism also by $\iota_m$. We can therefore obtain two finite maps of rigid spaces ${\mathcal{X}}_H^{\mathrm{ord}}(p^m) \to {\mathcal{X}}_{G, \operatorname{Kl}}^{{\geqslant}1}(p^m)$: either as the generic fibre of the formal-scheme morphism $\iota_m$ of the previous section, or as the restriction of the analytification of the map of ${\mathbf{Q}}$-varieties we have just constructed. A simple check using the explicit description of the action of $G(\hat{{\mathbf{Z}}})$ on the moduli problem shows that these two morphisms coincide.
From this equality (and the compatibility of the formal-analytic and rigid-analytic dualizing complexes) we obtain the following commutative diagram. Suppose ${\mathcal{O}}$ is the ring of integers of a finite extension $L / {{\mathbf{Q}}_p}$, and $\chi_1, \chi_2$ are characters $({\mathbf{Z}}/ p^m)^\times \to {\mathcal{O}}^\times$ (not necessarily primitive) with $\chi_1 \chi_2 = \mathrm{id}$, we have a commutative diagram
S\_[(\_1, \_2)]{}(p\^m, \_1, \_2, L) &\^[\_[m, ]{}]{} & H\^1(X\_[G, ]{}(p\^m)\_[L]{}, \_G(k\_1, k\_2)(-D\_G) )\
& &\
\_[(\_1 + \_1, \_2 + \_2)]{}(\_L) L &\^[\_]{} & H\^1(\_[G, ]{}\^[1]{}(p), \_G(k\_1, k\_2)(-D\_G) )L.
where the lower horizontal arrow is the map of .
We could also relax the assumption that $\chi_1 \chi_2$ be the identity, and replace ${\mathfrak{F}}_G(k_1, k_2)$ on the right-hand side with ${\mathfrak{F}}_G(k_1 + \chi, k_2)$ where $\chi = \chi_1\chi_2$. However, we shall not use this additional generality in the present paper.
Automorphic cohomology and periods {#sect:coherentcoh}
==================================
We now use the $p$-adic theory of the previous sections to study the $p$-adic variation of period integrals, given by pairing a class in $H^0$ of an automorphic vector bundle over $H$ with a class in $H^2$ over $G$.
Discrete-series automorphic representations
-------------------------------------------
Let $r_1 {\geqslant}r_2 {\geqslant}0$ be integers. Then there exists a unique *generic* discrete series representation $\Pi_\infty$ of $\operatorname{GSp}_4({\mathbf{R}})$ whose central character has finite order, and which has non-zero $(\mathfrak{g}, K)$-cohomology with coefficients in the appropriate twist of $V(r_1, r_2)$. This is the representation denoted $\Pi_{k_1, k_2}^W$ in [@loefflerskinnerzerbes17 §10], taking the parameters $(k_1, k_2)$ of *op.cit.* to be $(r_1 + 3, r_2 + 3)$.
We fix – for the remainder of this paper – a globally generic automorphic representation $\Pi$ of $\operatorname{GSp}_4({\mathbf{A}}_{\mathbf{Q}})$ whose component at $\infty$ is this discrete series representation. We also assume that $\Pi$ is not a Saito–Kurokawa lifting. Via the classification of cuspidal automorphic representations of $\operatorname{GSp}_4$ announced in [@arthur04] and proved in [@geetaibi18], this leaves exactly two possibilities:
- (“Yoshida type”) $\Pi$ lifts to a non-cuspidal automorphic representation of $\operatorname{GL}_4$ of the form $\pi_1 \boxplus \pi_2$, where the $\pi_i$ are cuspidal automorphic representations of $\operatorname{GL}_2$ generated by holomorphic modular forms of weights $r_1 + r_2 + 4$ and $r_1 - r_2 + 2$ respectively.
- (“General type”) $\Pi$ lifts to a cuspidal automorphic representation of $\operatorname{GL}_4$.
Note that in the present work, unlike its predecessor [@loefflerskinnerzerbes17], we *shall* allow the case of Yoshida lifts, since it presents no additional difficulties to do so (although in this special case there exist other, simpler proofs of our main results).
The “automorphic” normalisations are such that $\Pi$ is unitary, and its central character $\chi_\Pi$ is of finite order. Note that the sign of this character is $(-1)^{r_1 - r_2}$. We are most interested in the “arithmetic” normalisation of the finite part, ${\Pi_{\mathrm{f}}}' \coloneqq {\Pi_{\mathrm{f}}}\otimes \|\nu\|^{-(r_1 + r_2)/2}$; this is definable over a number field (while ${\Pi_{\mathrm{f}}}$ in general is not).
Coherent cohomology
-------------------
We define $L_i$, for $0 {\leqslant}i {\leqslant}3$, to be the irreducible $M_{\operatorname{S}}$-representations with the following highest weights: $$\begin{aligned}
L_0 &: \lambda(r_1 + 3, r_2 + 3; m) & L_1 &: \lambda(r_1 + 3, 1-r_2; m) \\
L_2 &: \lambda(r_2 +2, -r_1; m) & L_3 &: \lambda(-r_2, -r_1; m),
\end{aligned}$$ where $m = r_1 + r_2$.
Note that the highest weights of the representations $L_i$ lie in a Weyl-group orbit, for a suitably shifted action; geometrically, they are vertices of an octagon centred at $(1, 2)$.
The following result follows by standard methods from Arthur’s classification:
\[thm:cohcoh\] If $\Pi$ is of general type, then for each $0 {\leqslant}i {\leqslant}3$, ${\Pi_{\mathrm{f}}}'$ appears with multiplicity 1 as a Jordan–Hölder factor of the $G({{\mathbf{A}}_{\mathrm{f}}})$-representations $$H^i\left(X_{G, {\mathbf{Q}}}, [L_i](-D)\right) \otimes {\mathbf{C}}\quad\text{and}\quad H^i\left(X_{G, {\mathbf{Q}}}, [L_i]\right) \otimes {\mathbf{C}};$$ moreover, it appears as a direct summand of both representations, and the map between the two is an isomorphism on this summand. If $\Pi$ is of Yoshida type, then the preceding statements apply for $i = 1,2$, while for $i = 0, 3$, ${\Pi_{\mathrm{f}}}'$ does not appear as a Jordan–Hölder factor of either representation.
If $0 {\leqslant}i {\leqslant}3$ and $L$ is any irreducible representation of $M_{\operatorname{S}}$ which is *not* isomorphic to $L_i$, then the localisations of $H^i\left(X_{G, {\mathbf{Q}}}, [L](-D)\right)$ and $H^i\left(X_{G, {\mathbf{Q}}}, [L]\right)$ at the maximal ideal of the spherical Hecke algebra associated to the $L$-packet of $\Pi$ are zero for all $i$.
We shall write $H^i({\Pi_{\mathrm{f}}})$ for the ${\Pi_{\mathrm{f}}}'$-isotypical subspace of $H^i\left(X_{G, E}, [L_i](-D)\right)$, for some number field $E$ over which ${\Pi_{\mathrm{f}}}'$ is definable.
By Serre duality, we have $\operatorname{GSp}_4({{\mathbf{A}}_{\mathrm{f}}})$-equivariant perfect pairings $$\label{eq:serredual}
H^{3-i}\left(X_{G, {\mathbf{Q}}}, [L_{3-i}]\right) \times H^i\left(X_{G, {\mathbf{Q}}}, [L_i](-D)\right) \to {\mathbf{Q}}\{r_1 + r_2\},$$ for each $i$, where $\{m\}$ denotes the character mapping a uniformizer at a prime $\ell$ to $\ell^m$. In particular, if we choose a vector $\eta \in H^2({\Pi_{\mathrm{f}}})$, then we may regard it as a linear functional on $H^1\left(X_{G, E}, [L_1]\right)$, factoring through projection to $H^1({\Pi_{\mathrm{f}}}^\vee)$, where $\Pi^\vee$ is the contragredient of $\Pi$.
Definition of the period pairing {#sect:periodpairing}
--------------------------------
Let $(t_1, t_2)$ be integers ${\geqslant}0$ such that $t_1 + t_2 = r_1 - r_2$. Then there exists a non-zero $T$-homomorphism $$W_H(1+t_1, 1+t_2; r_1+r_2) \to L_1 |_{B_H} \otimes \alpha_{G/H}^{-1},$$ unique up to scalars, and hence a pushforward map $$\iota_\star: H^0\left(X_{H, {\mathbf{Q}}}, \omega_H(1+t_1, 1+t_2; r_1 + r_2)\middle)
\to
H^1\middle(X_{G, {\mathbf{Q}}}, [L_1]\right).$$ Note that the source of this map is simply the space of modular forms of weight $(1+t_1, 1+t_2)$ for $H$ (up to a twist by the norm character). Combining this with we obtain a bilinear, $H({{\mathbf{A}}_{\mathrm{f}}})$-equivariant *period pairing* $$H^0\left(X_{H, E}, \omega_H(1+t_1, 1+t_2)\right) \otimes H^2({\Pi_{\mathrm{f}}}) \to E\{r_1 - 2\},$$ mapping $f \otimes \eta$ to $\langle \iota_\star(f), \eta \rangle$.
Real-analytic comparison {#sect:archimedean}
------------------------
We now recall the comparison between the cohomological cup products above, and period integrals for automorphic forms, justifying the terminology “period pairing”. (We shall prove a more general statement in §\[sect:hodgesplitting\] below, allowing our modular forms to be nearly-holomorphic rather than holomorphic, but we give the “base case” here for ease of reading.)
Let $K_\infty = {\mathbf{R}}^\times \cdot U_2({\mathbf{R}})$ denote the maximal compact-mod-centre subgroup of $G({\mathbf{R}})_+$. The representation $\Pi_\infty$ has two direct summands as a $G({\mathbf{R}})_+$-representation, $\Pi_\infty = \Pi_{\infty, 1} \oplus \Pi_{\infty, 2}$, which have minimal $K_\infty$-types (Blattner parameters) $\tau_1 = (r_1 + 3, -r_2 - 1)$ and $\tau_2 = (r_2 + 1, -r_1 - 3)$ respectively. Since the minimal $K_\infty$-type in an irreducible discrete series has multiplicity 1, we have $\dim \operatorname{Hom}_{K_\infty}(\tau_i, \Pi_\infty) = 1$ for $i = 1, 2$.
\[thm:harris\]
1. For $j = 1, 2$ there is a canonical isomorphism of irreducible smooth $G({{\mathbf{A}}_{\mathrm{f}}})$-representations $$\operatorname{Hom}_{K_\infty}( \tau_j, \Pi)\left\{\tfrac{r_1 + r_2}{2}\right\} \rTo^\cong H^j\Big(X_{G, {\mathbf{C}}}, [L_j](-D)\Big)[{\Pi_{\mathrm{f}}}].$$
2. If $\eta \in H^2\Big(X_{G, {\mathbf{C}}}, [L_2](-D)\Big)[{\Pi_{\mathrm{f}}}]$, corresponding to some $K_\infty$-homomorphism $F_\eta: \tau_2 \to \Pi$, then for any integers $t_1, t_2 {\geqslant}0$ with $t_1 + t_2 = r_1 - r_2$, and any holomorphic modular forms $f, g$ of weights $1+t_1, 1+t_2$ respectively, then we have $$\langle \iota_\star\left( f \boxtimes g\right), \eta \rangle = \frac{1}{(2\pi i)^3} \int_{{\mathbf{R}}^\times H({\mathbf{Q}}) \backslash H({\mathbf{A}})} F_{\eta}(v_{t_1, t_2}) f(h_1) g(h_2)\, \mathrm{d}h,$$ where $v_{t_1, t_2}$ is the standard basis vector of $\tau_2$ of weight $(-t_1-1, -t_2-1)$. Here we identify $f$ and $g$ with functions on $\operatorname{GL}_2({\mathbf{Q}}) \backslash \operatorname{GL}_2({\mathbf{A}})$.
For weights $(r_1, r_2)$ is sufficiently far from the walls of the Weyl chamber, this is proved in [@harriskudla92], as an application of general results in §§3.5 and 3.8 of [@harris90b]. It follows from the results of [@su-preprint] that the result in fact applies for all regular weights.
Hecke eigenvalues at $p$ {#sect:hecke}
------------------------
Let $p$ be a prime such that $\Pi_p$ is unramified. Recall that we chose above a number field $E$ such that ${\Pi_{\mathrm{f}}}' = {\Pi_{\mathrm{f}}}\otimes \|\cdot\|^{-(r_1 + r_2)}$ is definable over $E$. We let $\{\alpha, \beta, \gamma, \delta\}$ be the elements of $\overline{E}$ (unique modulo the action of the Weyl group) such that the spin $L$-factor is given by $$L(\Pi_p', s - \tfrac{3}{2}) = L(\Pi_p, s - \tfrac{r_1+r_2+3}{2}) = \left[(1 - \alpha p^{-s}) \dots (1 - \delta p^{-s})\right]^{-1}.$$ We order these such that $\alpha \delta = \beta \gamma = p^{(r_1+r_2+3)} \chi_{\Pi}(p)$. As we have already noted above, $(\Pi_p')^{\operatorname{Kl}(p)}$ is 4-dimensional, and the two operators $$\begin{aligned}
{\mathcal{U}}_{p, \operatorname{Kl}} &= p^{-r_2} \cdot \left[ \operatorname{Kl}(p) \operatorname{diag}(p^2, p, p, 1) \operatorname{Kl}(p)\right], &
{\mathcal{U}}_{p, \operatorname{Kl}}' &= p^{-r_2} \cdot \left[ \operatorname{Kl}(p) \operatorname{diag}(1, p, p, {p^2}) \operatorname{Kl}(p)\right]
\end{aligned}$$ acting on this space both have eigenvalues $\left\{ \tfrac{\alpha\beta}{p^{r_2 + 1}}, \tfrac{\alpha\gamma}{p^{r_2 + 1}}, \tfrac{\beta \delta}{p^{r_2 + 1}}, \tfrac{\gamma \delta}{p^{r_2 + 1}}\right\}$.
More generally, we may consider the Hecke operator ${\mathcal{U}}_{p, \operatorname{Kl}}$ on the $\operatorname{Kl}(p^m)$-invariants, for any $m {\geqslant}1$. One checks easily that the operators ${\mathcal{U}}_{p, \operatorname{Kl}}$ at different levels are compatible with respect to the natural inclusion maps (since they admit a common set of coset representatives), and one has the following compatibility:
For $m {\geqslant}2$, the endomorphism ${\mathcal{U}}_{p, \operatorname{Kl}}$ at level $\operatorname{Kl}(p^m)$ factors through the canonical inclusion of the $\operatorname{Kl}(p^{m-1})$ invariants in the $\operatorname{Kl}(p^m)$ invariants.
It follows that the space $(\Pi_p')^{\operatorname{Kl}(p^m)}$ has a ${\mathcal{U}}_{p, \operatorname{Kl}}$-equivariant direct-sum decomposition as the sum of $(\Pi_p^{\prime})^{\operatorname{Kl}(p)}$, on which ${\mathcal{U}}_{p, \operatorname{Kl}}$ is invertible, and a complementary subspace on which ${\mathcal{U}}_{p, \operatorname{Kl}}$ is nilpotent.
Similar statements hold for the dual $\Pi^\vee$ in place of $\Pi$, and the transpose of ${\mathcal{U}}_{p, \operatorname{Kl}}'$ with respect to the natural pairing $(\Pi_p')^{\operatorname{Kl}(p^m)} \times (\Pi_p^{\prime \vee})^{\operatorname{Kl}(p^m)}$ is given by $\chi_\Pi(p) \cdot {\mathcal{U}}_{p, \operatorname{Kl}}$.
If $\eta \in H^2({\Pi_{\mathrm{f}}})^{\operatorname{Kl}(p)}$, and $m {\geqslant}1$, then we define $\eta_m \in H^2({\Pi_{\mathrm{f}}})^{\operatorname{Kl}(p^m)}$ to be the unique vector such that the linear functional $\langle -, \eta_m\rangle$ on $H^1({\Pi_{\mathrm{f}}}^{\vee})^{\operatorname{Kl}(p^m)}$ has the following properties:
- it vanishes on the ${\mathcal{U}}_{p, \operatorname{Kl}} = 0$ generalised eigenspace,
- it agrees with $\eta$ on the image of the $\operatorname{Kl}(p)$-invariants.
It is clear that the $\eta_m$ are compatible under the “normalised trace” maps, $$\eta_m = \tfrac{1}{p^3}\sum_{\gamma \in \operatorname{Kl}(p^m) / \operatorname{Kl}(p^{m+1})} \gamma \eta_{m+1}.$$
We have the following reasonably concrete formula:
If $\eta_1 = \eta$ lies in the ${\mathcal{U}}_{p, \operatorname{Kl}}' = \lambda$ eigenspace, then we have $$\eta_m = \left(\frac{p^{3-r_2}}{\lambda}\right)^{m-1} \sum_{a \in {\mathbf{Z}}/ p^{m-1}} \left( \begin{smallmatrix} 1 \\ &1 \\ &&1 \\ p^ma&&&1 \end{smallmatrix} \right) {{\left( \begin{smallmatrix} 1\\ &p \\ &&p \\ &&& p^2\end{smallmatrix} \right) }}^{m-1} \eta.$$
Choose a place $v$ of $E$ above $p$. We say that $\Pi$ is *Klingen-ordinary* at $p$ (with respect to $v$) if the operator ${\mathcal{U}}_{p, \operatorname{Kl}}$ on $(\Pi_p')^{\operatorname{Kl}(p)}$ has an eigenvalue $\lambda$ which is a $p$-adic unit.
It follows from the proof of Lemma \[lem-inj2\] above that the unit eigenvalue $\lambda$ is unique if it exists (and the corresponding eigenspace is 1-dimensional). In particular, $\lambda$ lies in $E_v$, rather than in some finite extension.
Interpolation of cup-products
-----------------------------
We now apply the above constructions to interpolate cup-products in families. Let $L$ be the completion of $E$ at our place $v \mid p$, and ${\mathcal{O}}$ its ring of integers. We shall assume that $r_2 {\geqslant}1$ and $\Pi$ is Klingen-ordinary at $p$ (with respect to the place $v$); and we choose some class $\eta \in H^2\left({\Pi_{\mathrm{f}}}\right)$, lying in the ordinary eigenspace for ${\mathcal{U}}_{p, \operatorname{Kl}}'$. Thus $\eta$ defines a linear functional $$e_{\operatorname{Kl}}\cdot H^1\left(X_{G, \operatorname{Kl}}(p)_L, [L_1]\right) \to L.$$
We also choose a flat $p$-adic ${\mathcal{O}}$-algebra $R$, and $\tau_1, \tau_2 : {{\mathbf{Z}}_p}^\times \to R^\times$ continuous characters such that $\tau_1 + \tau_2 = r_1 - r_2 + 2$. With these notations, gives a map $$\begin{aligned}
\iota_\star: {\mathcal{S}}_{(\tau_1, \tau_2)}(R) &\to H^1\left( {\mathfrak{X}}^{{\geqslant}1}_{G, \operatorname{Kl}}(p), {\mathfrak{F}}_{R}(3+r_1, 1-r_2)(-D)\right)\\
&= R {\mathop{\hat\otimes}}_{{{\mathbf{Z}}_p}} H^1\left( {\mathfrak{X}}^{{\geqslant}1}_{G, \operatorname{Kl}}(p), {\mathfrak{F}}_{{{\mathbf{Z}}_p}}(3+r_1, 1-r_2)(-D)\right).
\end{aligned}$$
Given ${\mathcal{E}}\in {\mathcal{S}}_{(\tau_1, \tau_2)}(R)$, we define an element $\left\langle \iota_\star\left({\mathcal{E}}\right), \eta \right\rangle \in R[1/p]$ as follows: it is the image of $\iota_\star({\mathcal{E}})$ under the composition of maps $$\begin{aligned}
R[1/p] {\mathop{\hat\otimes}}_{{{\mathbf{Z}}_p}} H^1\left( {\mathfrak{X}}^{{\geqslant}1}_{G, \operatorname{Kl}}(p), {\mathfrak{F}}(3+r_1, 1-r_2)(-D)\right)
&\rTo^{e_{\operatorname{Kl}}} R[1/p] {\mathop{\hat\otimes}}_{{{\mathbf{Z}}_p}} e_{\operatorname{Kl}}\cdot H^1\left( {\mathfrak{X}}^{{\geqslant}1}_{G, \operatorname{Kl}}(p), {\mathfrak{F}}(3+r_1, 1-r_2)(-D)\right) \\
&\rTo^{\cong} R[1/p] \otimes_{{{\mathbf{Q}}_p}} e_{\operatorname{Kl}} \cdot H^1\left( X_{G, \operatorname{Kl}}(p)_{{{\mathbf{Q}}_p}}, [L_1](-D)\right) \\
&\rTo^{\varpi} R[1/p] \otimes_{{{\mathbf{Q}}_p}} e_{\operatorname{Kl}} \cdot H^1\left( X_{G, \operatorname{Kl}}(p)_{{{\mathbf{Q}}_p}}, [L_1]\right)\\
&\rTo^{\langle -, \eta\rangle} R[1/p].
\end{aligned}$$ Here the second map is the isomorphism of Theorem \[thm:vincent\], and the map $\varpi$ is the“forget cuspidality” map, given by the natural inclusion of sheaves $[L_1](-D) {\hookrightarrow}[L_1]$.
\[prop:interp1\] Let $\phi: R \to L$ be a continuous ring homomorphism such that
- the composites $\phi \circ \tau_i: {{\mathbf{Z}}_p}^\times \to L^\times$ are the algebraic characters $x \mapsto x^{t_i}$, for some integers $t_i {\geqslant}1$ with $t_1 + t_2 = r_1 - r_2 + 2$;
- the $p$-adic modular form ${\mathcal{E}}_\phi \in {\mathcal{S}}_{(t_1, t_2)}(K^p_H; L)$ given by specialising ${\mathcal{E}}$ is the restriction to ${\mathcal{X}}_H^{{\mathrm{ord}}}$ of a classical holomorphic form $E_\phi \in M_{(t_1, t_2)}(p, L)$.
Then we have $$\phi\left( \left\langle \iota_{\star}\left({\mathcal{E}}\right), \eta \right\rangle \right) = \left\langle \iota_{1, \star}(E_\phi), \eta \right\rangle.$$
The construction of the pairing $\langle -, - \rangle$ is compatible with base-change in $R$, so it suffices to assume that $R = {\mathcal{O}}$, $\phi$ is the identity map, and ${\mathcal{E}}$ is the $p$-adic modular form associated to a classical modular form $E$.
If $E$ is cuspidal as a classical modular form, then the result is essentially a restatement of the commutativity of the diagrams in the previous section (in the simple case where $m = 1$ and the $\chi_i$ are trivial). However, we are working in a slightly greater degree of generality, where we allow $E$ to be a “fake cusp form” in the sense of §\[sect:bigsheafH\] above, so it can be non-vanishing at some components of $D_H$ lying outside the ordinary locus; and this is rather more delicate.
We first summarise why the obvious argument does not work. Let $(k_1, k_2) = (r_1 + 3, 1-r_2)$. From $E$ we can form the following two cohomology classes:
- An “algebraic” class $z^{\mathrm{alg}} = \iota_{1, \star}(E) \in H^1\left(X_{G, \operatorname{Kl}}(p)_{L}, \omega_G(k_1, k_2)\right)$.
- A “analytic” class $z^{\mathrm{an}} = \iota_{\star}({\mathcal{E}}) \in H^1\left({\mathcal{X}}_{G, \operatorname{Kl}}^{{\geqslant}1}(p), {\mathcal{F}}_G(k_1, k_2)(-D_G) \right)$.
These are compatible, in the sense that the restriction of $z^{\mathrm{alg}}$ to ${\mathcal{X}}_{G, \operatorname{Kl}}^{{\geqslant}1}(p)$ coincides with the image of $z^{\mathrm{an}}$ in $H^1\left({\mathcal{X}}_{G, \operatorname{Kl}}^{{\geqslant}1}(p), {\mathcal{F}}_G(k_1, k_2) \right)$ under the “forget cuspidality” map $\varpi$.
The classicity theorem for ordinary cohomology, used in the construction of our pairing, shows that there is some $z^{\mathrm{alg}}_{\mathrm{cusp}} \in H^1\left(X_{G, \operatorname{Kl}}(p)_{L}, \omega_G(k_1, k_2)(-D)\right)$ whose restriction to ${\mathcal{X}}_{\operatorname{Kl}}^{{\geqslant}1}(p)$ is $e_{\operatorname{Kl}} \cdot z^{\mathrm{an}}$. If we can show that the image of $z^{\mathrm{alg}}_{\mathrm{cusp}}$ under the “forget cuspidality” map $\varpi$ coincides with $e_{\operatorname{Kl}} \cdot z^{\mathrm{alg}}$, then we are done. However, all that we know is that $\varpi(z^{\mathrm{alg}}_{\mathrm{cusp}})$ and $e_{\operatorname{Kl}} \cdot z^{\mathrm{alg}}$ have the same image in $H^1\left({\mathcal{X}}_{G, \operatorname{Kl}}^{{\geqslant}1}(p), {\mathcal{F}}_G(k_1, k_2) \right)$, and since we have no analogue of Theorem \[thm:vincent\] for the non-cuspidal $p$-adic cohomology, we do not know that the restriction map $H^1\left(X_{G, \operatorname{Kl}}(p)_L, \omega_G(k_1, k_2) \right) \to H^1\left({\mathcal{X}}_{G, \operatorname{Kl}}^{{\geqslant}1}(p),{\mathcal{F}}_G{(k_1, k_2)} \right)$ is injective on the ordinary part.
We work around this by using *overconvergent* cohomology. Theorem \[thm:OCclass\] gives a classicity result for both of the spaces $$H^1\left({\mathcal{X}}^{{\geqslant}1}_{G, \operatorname{Kl}}(p)^\dagger, \omega_G(k_1, k_2) \right)\quad\text{and}\quad H^1\left({\mathcal{X}}^{{\geqslant}1}_{G, \operatorname{Kl}}(p)^\dagger, \omega_G(k_1, k_2)(-D) \right).$$ Performing the same construction as before for the overconvergent spaces, we obtain a class $z^{\dag} \in H^1\left({\mathcal{X}}^{{\geqslant}1}_{G, \operatorname{Kl}}(p)^\dagger, \omega_G(k_1, k_2)(-D) \right)$ with the following properties: the restriction of $z^{\dag}$ to the $p$-rank ${\geqslant}1$ locus (forgetting the overconvergence) is $z^{\mathrm{an}}$; and the image of $z^{\dag}$ in $H^1\left({\mathcal{X}}^{{\geqslant}1}_{G, \operatorname{Kl}}(p)^\dagger, \omega_G(k_1, k_2)\right)$ (forgetting the cuspidality) coincides with the restriction of $z^{\mathrm{alg}}$.
The classes $\varpi(z^{\mathrm{alg}}_{\mathrm{cusp}})$ and $e_{\operatorname{Kl}} \cdot z^{\mathrm{alg}}$ have the same image in $H^1\left({\mathcal{X}}^{{\geqslant}1}_{G, \operatorname{Kl}}(p)^\dagger, \omega_G(k_1, k_2)\right)$ (namely $e_{\operatorname{Kl}} \cdot \varpi(z^{\dag})$). Using the classicity theorem for this overconvergent cohomology, we conclude that $\varpi(z^{\mathrm{alg}}_{\mathrm{cusp}})$ and $e_{\operatorname{Kl}} \cdot z^{\mathrm{alg}}$ coincide.
Higher level specialisations
----------------------------
We also have a version of this theorem for specialisations that are classical of higher $p$-power levels, using the vectors $\eta_m$ at level $\operatorname{Kl}(p^m)$ constructed from $\eta$ as in §\[sect:hecke\].
\[thm:interp2\] Let $\phi: R \to L$ be a continuous ring homomorphism such that $\phi \circ \tau_i$ has the form $x \mapsto x^{t_i} \chi_i(x)$, for some $t_i$ as before and finite-order characters $\chi_1, \chi_2$ with $\chi_1 \chi_2 = 1$. Suppose that $\phi({\mathcal{E}})$ is the image of a classical modular form $E_\phi \in M_{(t_1 + 1, t_2 + 1)}(K_H(p^m), L)$, for some $m \gg 0$.
Then we have $$\phi\left(\langle \iota_\star({\mathcal{E}}), \eta \rangle\right) = \left\langle \iota_{m, \star}(E_\phi), \eta_m \right\rangle.$$
By the same arguments as above, $\phi\left(\langle \iota_\star({\mathcal{E}}), \eta \rangle\right)$ is the image of $\iota_{m, \star}(E_\phi)$ under the unique linear functional on $H^1(X_{G, \operatorname{Kl}}(p^m)_{L}, [L_1])$ which factors through the ordinary idempotent $e_{\operatorname{Kl}}$ and agrees with $\langle -, \eta\rangle$ on the image of $H^1(X_{G, \operatorname{Kl}}(p)_{L}, [L_1])$. This is exactly the definition of $\eta_m$.
“Nearly” coherent cohomology {#sect:nearly}
============================
It is well-known that nearly-holomorphic modular forms can be considered as $p$-adic modular forms. In this section, we shall formulate an analogous statement for our $p$-adic cohomology spaces for $\operatorname{GSp}_4$, and show that the $\Lambda$-adic pushforward map constructed in the previous section is compatible with this additional structure.
“Nearly” sheaves
----------------
In this section, we’ll consider coherent cohomology with coefficients in certain indecomposable representations of $P_{\operatorname{S}}$. This is needed in order to study pushforwards of non-holomorphic Eisenstein series; we regard it as an analogue for $H^1$ of Siegel varieties of Shimura’s theory of nearly-holomorphic modular forms.
If $V$ is a finite-dimensional algebraic representation of $P_{\operatorname{S}}$, we let $\operatorname{Fil}^n V$ denote the direct sum of the eigenspaces of $W$ on which the torus ${\left( \begin{smallmatrix} x\\ &x \\ &&1 \\ &&& 1\end{smallmatrix} \right) }$ acts with weights ${\geqslant}n$ (i.e. the weight spaces of weight $\lambda(r_1, r_2; c)$ with $r_1 + r_2 + c {\geqslant}2n$).
This filtration is stable under $P_{\operatorname{S}}$, and the $P_{\operatorname{S}}$-action on the graded pieces $\operatorname{Gr}^n V = \operatorname{Fil}^n V / \operatorname{Fil}^{n+1} V$ factors through $M_{\operatorname{S}}$. We shall apply this to $V = V(r_1, r_2; r_1 + r_2)$, for some integers $r_1 {\geqslant}r_2 {\geqslant}0$; in this case the non-zero graded pieces are in degrees $0 {\leqslant}n {\leqslant}r_1 + r_2$.
- Let $\tilde L_1$ denote the representation $V / \operatorname{Fil}^{r_1 + 1} V \otimes W(3, 1; 0)$, and $L_1'$ the smallest filtration subspace $\operatorname{Gr}^{r_1} V \otimes W(3, 1;0)$ of $\tilde L_1$.
- Dually, let $\tilde L_2$ denote the $P_{\operatorname{S}}$-representation $\operatorname{Fil}^{r_2} V \otimes W(2, 0; 0)$, and $L_2'$ the top filtration quotient $\operatorname{Gr}^{r_2} V \otimes W(2, 0; 0)$ of $\tilde L_2$.
There is a natural projection $L_1' \twoheadrightarrow L_1$, and a natural inclusion $L_2 {\hookrightarrow}L_2'$ (both of which are split, but we do not need this).
The natural maps $$H^2\left(X_{G, E}, [L_2](-D)\right) \to H^2\left(X_{G, E}, [L'_2](-D)\right)$$ and $$H^2\left(X_{G, E}, [\tilde L_2](-D)\right) \to H^2\left(X_{G, E}, [L'_2](-D)\right),$$ induced by the maps of $P_{\operatorname{S}}$-representations $\tilde L_2 \twoheadrightarrow L'_2 \hookleftarrow L_2$, are both isomorphisms on the ${\Pi_{\mathrm{f}}}'$ generalised eigenspace (and similarly without $(-D)$).
This follows readily from the final statement of Theorem \[thm:cohcoh\], since the kernel of $\tilde L_2 \twoheadrightarrow L'_2$ and the cokernel of $L_2 {\hookrightarrow}L'_2$ do not contribute to the ${\Pi_{\mathrm{f}}}'$-eigenspace of $H^2$.
We write $\tilde\eta$ for the unique class in $H^2\left(X_{G, E}, [\tilde L_2](-D)\right)[{\Pi_{\mathrm{f}}}]$ corresponding to $\eta$ under the above isomorphisms. The linear functional $\langle -, \tilde\eta\rangle$ is thus a homomorphism $$H^1\left(X_{G, E}, [\tilde L_1]\right) \to E,$$ characterised as follows:
- it factors through the ${\Pi_{\mathrm{f}}}^\vee$-isotypical part;
- its restriction to $H^1\left(X_{G, E}, [L_1^{\prime}]\right)$ is the composite of the projection $L_1^{\prime} {\twoheadrightarrow}L_1$ and pairing with $\eta$.
This extended homomorphism will play a role analogous to Shimura’s “holomorphic projection” operator in the $\operatorname{GL}_2$ theory.
Pullback and pushforward of “nearly” sheaves
--------------------------------------------
Suppose $(r_1, r_2; c)$ is a $G$-dominant weight. The restriction to $H$ of the irreducible representation $V_G(r_1, r_2; c)$ of $G$ is a direct sum of distinct irreducible $H$-representations. The representation $V_H(t_1, t_2; c)$ appears as a direct summand if and only if the integers $t_1, t_2$ satisfy $t_1 + t_2 = r_1 + r_2 \bmod 2$ and $$\pushQED{\qed}
r_1 - r_2 {\leqslant}t_1 +t_2 {\leqslant}r_1 + r_2,\quad |t_1 - t_2| {\leqslant}r_1 - r_2.\qedhere
\popQED$$
Since the cocharacter used to define the filtration is strictly dominant with respect to $B_H$, any representation of $B_H$ has a canonical $B_H$-invariant filtration, with the action on the graded pieces factoring through $T$; and this is compatible with the above branching from $G$ to $H$.
We shall be particularly interested in $H$-subrepresentations of the form $V_H(t_1, t_2; c)$ with $t_1 + t_2 = r_1 - r_2$; note that there are precisely $r_1 - r_2 + 1$ such subrepresentations. We shall call these subrepresentations *small*.
Recall now the larger sheaves $\tilde L_2 {\twoheadrightarrow}L_2' \hookleftarrow L_2$ defined above. Since $V_H(t_1, t_2; r_1 + r_2)$ is a direct summand of $V$, we have a projection map $$\tilde L_2 |_{B_H} = \left(\operatorname{Fil}^{r_2} V \otimes W(2, 0; 0)\right)|_{B_H} \to V_H \otimes \lambda(1, 1; 0)$$ given by the tensor product of the projections $V |_H \to V_H$ and $W(2, 0; 0)|_H \to \lambda(1, 1;0 )$. This determines a homomorphism $$\iota_{\text{nearly}}^\star:
H^2\left(X_{G, {\mathbf{Q}}}, [\tilde L_2](-D)\middle)
\to H^2\middle(X_{H, {\mathbf{Q}}}, [V_H] \otimes \omega_H(1, 1; 0)(-D)\right).$$
Note that the composition of $\iota_{\text{nearly}}^\star$ with the projection $V_H \to V_H / \operatorname{Fil}^{r_2 + 1} V_H \cong W_H(1-t_1, 1-t_2; r_1 + r_2)$ factors through $ H^2\left(X_{G, {\mathbf{Q}}}, [L_2'](-D)\right)$, and we can (and do) normalise such that this composite agrees with $\iota^\star$ on the image of $L_2 {\hookrightarrow}L_2'$.
Dually, we obtain a map $$\iota^{\text{nearly}}_\star: H^0\left(X_{H, {\mathbf{Q}}}, [V_H] \otimes \omega_H(1, 1; 0)\middle) \to
H^1\middle(X_{G, {\mathbf{Q}}}, [\tilde L_1]\right).$$ The source of this map is the space of *nearly-holomorphic* modular forms for $H$ of weight $(1+t_1, 1+t_2)$. We thus have an extended period pairing $$H^0\big(X_{H, E}, [V_H] \otimes \omega_H(1, 1; 0)\big) \otimes H^2({\Pi_{\mathrm{f}}}) \to E,\quad (f, \eta) \mapsto \langle \iota^{\text{nearly}}_\star(f), \tilde\eta \rangle.$$ From the construction of $\tilde\eta$, we see that the restriction of this pairing to the space of holomorphic forms agrees with the period pairing of §\[sect:periodpairing\] above.
Archimedean theory: the Hodge splitting {#sect:hodgesplitting}
---------------------------------------
We can compute $H^2(X_{G, {\mathbf{C}}}, [\tilde L_2](-D))$ in terms of automorphic forms, using results of Su [@su-preprint]. As above, let $K_\infty$ be the standard maximal compact subgroup of $G({\mathbf{R}})_+$, fixing the point $h = i I_2 \in {\mathcal{H}}_2$. The Shimura cocharacter determines a decomposition of $\operatorname{Lie}(G)_{{\mathbf{C}}}$ as $\mathfrak{k} \oplus \mathfrak{p}^+ \oplus \mathfrak{p}^-$, where $\mathfrak{k} = \operatorname{Lie}(K_\infty)_{{\mathbf{C}}}$, and $\mathfrak{p}^+$ (resp. $\mathfrak{p}^-$) is identified with the holomorphic (resp. antiholomorphic) tangent space of $\mathcal{H}_2$ at $h$. The parabolic $P_h \subset G({\mathbf{C}})$ with Lie algebra $\mathfrak{p} = \mathfrak{k} \oplus \mathfrak{p}^-$ is a conjugate of the Siegel parabolic $P_{\operatorname{S}}$, so we may identify any $P_{\operatorname{S}}$-representation $V$ with a representation of $P_\infty$.
The main theorem of *op.cit.* gives a canonical and Hecke-equivariant isomorphism $$H^*(X_{G, {\mathbf{C}}}, [V]) \cong H^*\!\left(\mathfrak{p}, K_\infty; \mathcal{A}(G)^K \otimes V\right),$$ for any level $K$ and any algebraic representation $V$ of $P_{\operatorname{S}}$, where $\mathcal{A}(G)$ is the space of automorphic forms on $G$ (twisted by an appropriate power of the norm character so that the central characters match). The relative Lie algebra cohomology on the right-hand side can be computed as the cohomology of the complex with $j$-th term $$\operatorname{Hom}_{K_\infty}\left( \textstyle{\bigwedge^j}(\mathfrak{p}^-) \otimes V^\vee, \mathcal{A}(G)^K\right).$$
Let $\Pi_{\infty, 2}$ be the $G({\mathbf{R}})_+$-submodule of $\Pi_\infty$ described in §\[sect:archimedean\] above. Then we have $$\operatorname{Hom}_{K_\infty}\left(\textstyle{\bigwedge^j}(\mathfrak{p}^-) \otimes (\tilde L_2)^\vee,\ \Pi_{\infty, 2}\right) = 0$$ for all $j \ne 2$, and for $j = 2$ this space is 1-dimensional and maps isomorphically to its image with $L_2$ in place of $\tilde L_2$.
An explicit description of the $K_\infty$-types appearing in $\Pi_{\infty, 2}$ is given in [@schmidt17]. The minimal $K_\infty$-type is $\tau_2 = (r_2 + 1, -r_1-3)$, and the other $K_\infty$-types lie in the convex cone $\{ \tau_2 + m \cdot (1, 0) + n \cdot (-1, -1): m, n {\geqslant}0\}$.
On the other hand, the $K_\infty$-types appearing in $(\tilde L_2)^\vee$ are all contained in a different convex cone, $\{ (r_2 + 2, -r_1) + m \cdot (1, 1) + n \cdot (-1, 1): m, n {\geqslant}0\}$. So the weights of $(\tilde L_2)^\vee \otimes \bigwedge^j(\mathfrak{p}^-)$ are contained in the translate of this cone by the highest weight of $\bigwedge^j(\mathfrak{p}^-)$; this translation is by $(0, -2)$ if $j = 1$, by $(-1, -3)$ if $j = 2$, and by $(-3, -3)$ if $j = 3$. So if $j \ne 2$, these regions have empty intersection, whereas if $j = 2$ the intersection consists only of $\tau_2$, which appears in both representations with multiplicity 1.
Thus the $\Pi_f$-isotypical part of the $(\mathfrak{p}, K_\infty)$-cohomology is represented by a complex concentrated in degree 2, giving a a canonical space of Dolbeault differential forms representing $H^2\big(X_{G, {\mathbf{C}}}, [\tilde L_2]\big)[{\Pi_{\mathrm{f}}}]$: the vector-valued automorphic forms whose coordinate projections lie in the minimal $K_\infty$-type of ${\Pi_{\mathrm{f}}}\otimes \Pi_{\infty, 2}$. Moreover, these differentials in fact take values in $[L_2]$, regarded as a subsheaf of $[\tilde L_2]$ via the Hodge splitting: the projections of these differentials to the other graded pieces of $[\tilde L_2]$ are trivial, since the corresponding $K_\infty$-types do not appear in $\Pi_{\infty, 2}$. Since $\Pi$ is cuspidal, these differential forms are rapidly-decreasing.
Now let $t_1, t_2$ be integers with $t_1 + t_2 = r_1 - r_2$, as before. The analogue of this $K_\infty$-equivariant splitting for $H$ in place of $G$ is the *Hodge splitting* in the category of $C^\infty$ vector bundles, $$[V_H] \otimes \omega_H(1, 1; 0) \to \omega_H(1 + t_1, 1 + t_2),$$ which allows nearly-holomorphic modular forms to be interpreted as scalar-valued real-analytic functions on $H({\mathbf{Q}}) \backslash H({{\mathbf{A}}_{\mathrm{f}}})$, transforming under $K_{H,\infty}$ via the character $(1 + t_1, 1 + t_2)$. Evidently these two splittings are compatible (since they are both given by projection to eigenspaces for the action of the centre of $K_\infty$, which is contained in $K_{H, \infty}$), so we deduce the following theorem:
Let $E \in H^0\left(X_{H, {\mathbf{C}}}, [V_H] \otimes \omega_H(1, 1; 0)\right)$ and let $E_\infty \in H^0\left(X_{H, {\mathbf{C}}}, \omega(t_1+1, t_2+1)_{C^\infty}\right)$ be its image under the Hodge splitting. If $\eta$ and $F_{\eta}$ are as in Theorem \[thm:harris\], then we have $$\langle \iota^{\text{nearly}}_\star(E), \tilde\eta \rangle = \frac{1}{(2\pi i)^3}\int_{{\mathbf{R}}^{\times} H({\mathbf{Q}}) \backslash H({\mathbf{A}})} F_\eta(v_{t_1, t_2}) E_\infty(h) \, \mathrm{d}h.$$
We know that the $[\tilde L_1]$-valued current representing $\iota^{\text{nearly}}_\star(E)$ is the direct sum of $\iota_\star(E_\infty)$ and some other terms lying in other graded pieces of the sheaf $[\tilde L_1]$. By the previous lemma, $\tilde \eta$ pairs to zero with the latter; on the other hand, we clearly have $\langle \iota^{\text{nearly}}_\star(E), \tilde\eta \rangle = \langle \iota_\star(E_\infty), \eta \rangle$ since $\tilde\eta$ maps to $\eta$ in $[L_2]$.
This construction is strongly analogous to one appearing in Harris’ work [@harris04], which also considers period integrals of automorphic forms for $G$ multiplied by nearly-holomorphic Eisenstein series on $H$. However, our treatment differs from Harris’ in the following point: in defining $\tilde{\eta}$, we used $G({{\mathbf{A}}_{\mathrm{f}}})$-equivariance to split a cohomology exact sequence on $G$ (relating the cohomology of $[L_2]$ and $\tilde L_2]$). Harris uses instead a splitting on $H$, characterised by equivariance for the action of $H({{\mathbf{A}}_{\mathrm{f}}})$ (see the proof of Proposition 1.10.3 of *op.cit*). It is not clear to us whether these constructions agree in general.
A $p$-adic splitting
--------------------
We now define a partial $p$-adic analogue of the Hodge splitting. Let us briefly recall the definition of the torsor ${\mathcal{T}}_G$ defined in §\[sect:toroidal\] above. Over the integral model of the open Shimura variety $Y_G$ of prime-to-$p$ level $K^p \cdot G({{\mathbf{Z}}_p})$, we have a locally free sheaf ${\mathcal{H}}^1_{{\mathrm{dR}}}(A)$ of rank 4, with a rank 2 locally free subsheaf $\operatorname{Fil}^1 {\mathcal{H}}^1_{{\mathrm{dR}}}(A)$. There is a canonical extension of ${\mathcal{H}}^1_{{\mathrm{dR}}}(A)$ to a locally free sheaf ${\mathcal{H}}^1_{{\mathrm{dR}}}(A)^{\mathrm{can}}$ on $X_G$, fitting into a short exact sequence of locally free sheaves $$0 \to \omega_{A^\Sigma} \to {\mathcal{H}}^1_{\mathrm{dR}}(A)^\mathrm{can} \to \omega_{A^\Sigma}^\vee \to 0,$$ where $A^\Sigma$ is the semiabelian variety over $X_G$ extending $A$, and $\omega_{A^\Sigma}$ its conormal sheaf at the identity section. Moreover, if ${\mathfrak{IG}}_G(p^\infty)^{{\mathrm{ord}}}$ denotes the preimage in ${\mathfrak{IG}}_G(p^\infty)$ of the ordinary locus ${\mathfrak{X}}_G^{{\mathrm{ord}}} \subset {\mathfrak{X}}_G$, then over ${\mathfrak{X}}_G^{{\mathrm{ord}}}$ there is a splitting $${\mathcal{H}}^1_{\mathrm{dR}}(A)^\mathrm{can} \cong \omega_{A^\Sigma} \oplus \mathcal{U},$$ where $\mathcal{U}$ is the *unit root subsheaf* [@liu-thesis §3.12].
Over the Igusa tower ${\mathfrak{IG}}_G(p^\infty)$, we have a morphism of $p$-divisible groups $\alpha: \mu_{p^\infty} {\hookrightarrow}A^\Sigma[p^\infty]$, and hence a canonical map $\omega_{A_G^\Sigma} \to {\mathcal{O}}_{{\mathfrak{IG}}_G(p^\infty)}$, i.e. a class $[\alpha] \in H^0({\mathfrak{IG}}_G(p^\infty), \omega_{A^\Sigma}^\vee)$.
There exists a unique lifting of $[\alpha]$ to a class $[\alpha]^{\mathrm{ur}} \in H^0({\mathfrak{IG}}_G(p^\infty), {\mathcal{H}}^1_{{\mathrm{dR}}}(A)^{{\mathrm{can}}})$ with the following property: its restriction to ${\mathfrak{IG}}_G(p^\infty)^{{\mathrm{ord}}}$ takes values in the subsheaf ${\mathcal{U}}$.
By construction, there is a unique class $[\alpha]^{{\mathrm{ord}}} \in H^0({\mathfrak{IG}}_G(p^\infty)^{{\mathrm{ord}}}, {\mathcal{H}}^1_{{\mathrm{dR}}}(A)^{{\mathrm{can}}})$ lifting the restriction of $[\alpha]$ and taking values in the unit-root subsheaf ${\mathcal{U}}$. Since ${\mathfrak{IG}}_G(p^\infty)^{{\mathrm{ord}}}$ is dense in ${\mathfrak{IG}}_G(p^\infty)$, it follows that $[\alpha]^{ur}$ is unique if it exists.
Let ${\mathfrak{IG}}_G(p^\infty)^{\circ}$ denote the preimage in ${\mathfrak{IG}}_G(p^\infty)$ of the open Shimura variety $Y_G \pmod p$. Since the complement of ${\mathfrak{IG}}_G(p^\infty)^{\circ} \cup {\mathfrak{IG}}_G(p^\infty)^{{\mathrm{ord}}}$ has codimension ${\geqslant}2$ in ${\mathfrak{IG}}_G(p^\infty)$ (and the sheaf is locally free), it suffices to find a second section $[\alpha]^{\circ}$ over ${\mathfrak{IG}}_G(p^\infty)^{\circ}$ which coincides with $[\alpha]^{{\mathrm{ord}}}$ where both are defined.
We shall carry this out using a generalisation of the construction of the unit-root splitting given in [@iovita00]. Recall that if $A/S$ is an abelian scheme over an arbitrary base $S$, then there exists a *universal vectorial extension* of $A$, which is universal among short exact sequences of $S$-group schemes $$0 \to V \to I \to A \to 0$$ where $V$ is a vector group. Moreover, the space of invariant differentials $\operatorname{Inv}(I/S)$ is isomorphic to $H^1_{{\mathrm{dR}}}(A/S)$, with the subspace $\operatorname{Inv}(A/S)$ corresponding to $\operatorname{Fil}^1$. Taking associated formal groups, we have a short exact sequence $0 \to \hat V \to \hat I \to \hat A \to 0$.
If we are given a morphism $\alpha: \hat{\mathbf{G}}_m {\hookrightarrow}\hat A$, and $\hat I_\alpha$ denotes the pullback of this extension along $\alpha$, we have a diagram of exact sequences
0 &&V& & I &&A& & 0\
& & & & & &\
0 && V && I\_& & \_m &&0.
However, any extension of a formal multiplicative group by an additive one must be split, so there is a (necessarily unique) map $\hat{\mathbf{G}}_m \to \hat I_\alpha$ splitting the lower sequence. Thus the composite $$v_\alpha: \operatorname{Inv}(I/S) \to \operatorname{Inv}(I_\alpha/S) \to \operatorname{Inv}\left(\hat{\mathbf{G}}_m/S\right)$$ is an extension of the pullback map $\operatorname{Inv}(A/S) \to \operatorname{Inv}\left(\hat{\mathbf{G}}_m/S\right) \cong {\mathcal{O}}_S$ to $H^1_{{\mathrm{dR}}}(A/S)$. Moreover, if $p$ is topologically nilpotent on $S$ and $A$ is ordinary, then $\hat A$ is itself of multiplicative type. Hence we have a splitting of the top row, which is clearly compatible with the splitting of the bottom row; and the main result of [@iovita00] shows that this construction recovers the unit-root splitting. This gives the required section $[\alpha]^\circ$.
Let $Q$ be the subgroup $P_{\operatorname{S}} \cap \overline{P}_{\operatorname{Kl}}$ of $G$, where $P_{\operatorname{S}}$ is the standard Siegel parabolic subgroup and $\overline{P}_{\operatorname{Kl}} = J^{-1} P_{\operatorname{Kl}} J$ is the lower-triangular Klingen.
Note that $T$ is a Levi subgroup of $Q$, so any weight $\lambda(r_1, r_2; c)$ can be regarded as a representation of $Q$.
Let ${\mathcal{T}}^{\operatorname{Kl}}$ denote the sheaf over ${\mathfrak{X}}_{G, \operatorname{Kl}}^{{\geqslant}1}(p^\infty)$ parametrising bases $f_1, \dots, f_4$ of ${\mathcal{H}}^1_{{\mathrm{dR}}}(A)^{{\mathrm{can}}}$ compatible with the filtration and polarisation, and with the additional property that the pullback of $f_4$ to ${\mathfrak{IG}}_{G, \operatorname{Kl}}^{{\geqslant}1}(p^\infty)$ is a scalar multiple of $[\alpha]^{\mathrm{ur}}$. This is is a reduction of structure of the $P_{\operatorname{S}}$-torsor ${\mathcal{T}}$ to a $Q$-torsor over ${\mathfrak{X}}_{G, \operatorname{Kl}}^{{\geqslant}1}(p^\infty)$.
One can interpret the comparison morphism of §\[sect:Gsetup\] using this torsor. Let $k_1 {\geqslant}k_2$ be integers. We shall see in Lemma \[lemma:HWtheory\](i) below that the inclusion of the highest weight space $\lambda(k_1, k_2)$ into $W_G(k_1, k_2)$ has a $Q$-equivariant splitting. If we pull back further to ${\mathfrak{IG}}_G(p^\infty)$, the sheaf $[\lambda(k_1 - k_2, 0)]$ has a canonical trivialisation, so we obtain a morphism of sheaves $$[ W_G(k_1, k_2) ] \to [W_G(k_2, k_2)] = (\det \omega_A)^{k_2}.$$ Pushing back down to ${\mathfrak{X}}_{G, \operatorname{Kl}}^{{\geqslant}1}(p)$, this gives us a morphism $$[ W(k_1, k_2) ] \to \left( \pi_\star {\mathcal{O}}_{{\mathfrak{IG}}_G(p^\infty)} \right)^{\Gamma = k_1 - k_2} \otimes (\det \omega_A)^{k_2},$$ which is the sought-after comparison morphism. The advantage of this new interpretation is that we can easily see how to extend the comparison morphism to larger coefficient sheaves, using some simple Lie-theoretic computations.
\[lemma:HWtheory\]
(i) Let $k_1 {\geqslant}k_2$. Then there is a unique (up to scalars) $Q$-equivariant map $$W(k_1, k_2; c) \to \lambda(k_1, k_2; c).$$
(ii) Let $r_1 {\geqslant}r_2 {\geqslant}0$. Then there is a unique (up to scalars) $Q$-equivariant map $$V(r_1, r_2; r_1 + r_2) \to \lambda(r_1, -r_2; r_1 + r_2).$$ This map factors through $V / \operatorname{Fil}^{r_1 + 1} V$.
(iii) The restriction of the homomorphism (ii) to $\operatorname{Gr}^{r_1} V$ is non-trivial, and factors as the composite of the projection from $\operatorname{Gr}^{r_1} V$ to its unique $M_{\operatorname{S}}$-summand isomorphic to $W(r_1, -r_2; r_1 + r_2)$, composed with the map from (i) for this subrepresentation.
Part (i) is clear, since the image of $Q$ in the Levi quotient of $P_{\operatorname{S}}$ is the lower-triangular Borel. For part (ii), we argue analogously, using the fact that $Q$ is contained in a Weyl-group conjugate of the standard Borel subgroup of $G$, and $\lambda(r_1, -r_2; c)$ is the lowest-weight vector for this conjugate Borel. The compatibility (iii) is clear by considering the weights appearing in each factor.
\[prop:Qmorphism\] The morphism of $Q$-representations $$\operatorname{Gr}^{r_1} V \otimes W(3, 1; 0) \rTo W(r_1 + 3, 1-r_2; r_1 + r_2) \rTo \lambda(r_1 + 3, 1-r_2; r_1 + r_2)$$ has a unique $Q$-equivariant extension to the whole of $\tilde L_1$.
This is an easy consequence of Lemma \[lemma:HWtheory\](iii).
In terms of sheaves, this gives the following.
For $r_1 {\geqslant}r_2 {\geqslant}0$, we define a map $$[\widetilde L_1] \to \left( \pi_\star {\mathcal{O}}_{{\mathfrak{IG}}_G(p^\infty)} \right)^{\Gamma = r_1 + r_2 + 2} \otimes (\det \omega_A)^{1-r_2}$$ using the $Q$-equivariant morphism of Proposition \[prop:Qmorphism\].
If $r_2 = 0$, so that $[V] = \operatorname{Sym}^{r_1} {\mathcal{H}}^1_{{\mathrm{dR}}}(A)$, this morphism has a simple explicit description: it is given by restriction to $\operatorname{Sym}^{r_1} {\mathcal{H}}^1_{{\mathrm{dR}}}({\mathcal{G}}_1)$ and the trivialisation ${\mathcal{G}}_1 \cong \mu_{p^\infty}$ over ${\mathfrak{IG}}_G(p^\infty)$.
\[cor:unitroot\] For any $r_1 {\geqslant}r_2 {\geqslant}0$, we have a diagram of cohomology groups
H\^1(\_[G, ]{}(p)\^[1]{}, \[L\_1\](-D))\
&\
H\^1(\_[G, ]{}(p)\^[1]{}, \[L\_1\](-D)) & & H\^1(\_[G, ]{}(p)\^[1]{}, (r\_1 + 3, 1-r\_2)(-D)).
Branching to $H$
----------------
Recall that we have defined a homomorphism of formal schemes $$\iota_\infty: {\mathfrak{X}}_{H}^{{\mathrm{ord}}}(p^\infty) \to {\mathfrak{X}}_{G, \operatorname{Kl}}^{{\geqslant}1}(p^\infty).$$ The space ${\mathfrak{X}}_{H}^{{\mathrm{ord}}}(p^\infty)$ classifies pairs of *ordinary* elliptic curves $(E_1, E_2)$ with prime-to-$p$ level structure and isomorphisms $$\gamma: E_1[p^\infty]^\circ \rTo^\cong E_2[p^\infty]^\circ.$$ Hence there is a reduction of $\iota_\infty^*({\mathcal{T}}^{\operatorname{Kl}})$ to a torsor for the group $\{\operatorname{diag}(x, x, y, y) : x, y \in {\mathcal{O}}^\times\}$.
Let $V_H$ be a small subrepresentation of $V$ given by a pair $(t_1, t_2)$ as above. Then the composite $$H^0({\mathfrak{X}}_H^{{\mathrm{ord}}}, \omega^{1, 1} \otimes [V_H](-D)) \to H^1({\mathfrak{X}}_G^{{\geqslant}1}, \tilde L_1(-D)) \to H^1\left({\mathfrak{X}}_G^{{\geqslant}1}, {\mathfrak{F}}(r_1 + 3, 1-r_2)(-D)\right)$$ coincides with the composite of the unit-root splitting $$H^0({\mathfrak{X}}_H^{{\mathrm{ord}}}, \omega^{1, 1} \otimes [V_H](-D)) \to H^0({\mathfrak{X}}_H^{{\mathrm{ord}}}, \omega^{(t_1 + 1, t_2 + 1)}(-D))$$ and the pushforward $$H^0({\mathfrak{X}}_H^{{\mathrm{ord}}}, \omega^{(t_1 + 1, t_2 + 1)}(-D)) \to H^1({\mathfrak{X}}_G^{{\geqslant}1}, L_1(-D)) \to H^1\left({\mathfrak{X}}_G^{{\geqslant}1}, {\mathfrak{F}}(r_1 + 3, 1-r_2)(-D)\right).$$
We need to compare two maps of sheaves on ${\mathfrak{X}}_H^{{\mathrm{ord}}}(p^\infty)$. The first is the composite $$V_H(t_1, t_2) \to \iota_\infty^*(V / \operatorname{Fil}^{r_1 + 1}) \to \iota_\infty^*\left( [\lambda(r_1, -r_2)]\right),$$ where the second map is given by the splitting of Corollary \[cor:unitroot\]. The second is the map $$V_H(t_1, t_2) \to \omega^{t_1, t_2} \to \iota^*\left( [\lambda(r_1, -r_2)]\right),$$ where the first map comes from the unit-root splitting on $H$. Via our torsor formalism we are reduced to checking the equality of two morphisms of representations of the group $\{{\left( \begin{smallmatrix} x\\ &x \\ &&y \\ &&& y\end{smallmatrix} \right) } : x, y \in {\mathcal{O}}^\times\}$, and this is obvious.
Theorem \[thm:interp2\] holds as stated if we assume only that $\phi({\mathcal{E}})$ is the image under the $p$-adic unit-root splitting of a classical nearly-holomorphic modular form.
If $V_H$ is any irreducible $H$-subrepresentation of $V$ (not necessarily small), then the same argument as above extends to show that the composite $H^0({\mathfrak{X}}_H^{{\mathrm{ord}}}, \omega^{1, 1} \otimes [V_H](-D)) \to H^1({\mathfrak{X}}_G^{{\geqslant}1}, \tilde L_1(-D)) \to H^1\left({\mathfrak{X}}_G^{{\geqslant}1}, {\mathfrak{F}}(r_1 + 3, 1-r_2)(-D)\right)$ factors through projection to $\omega^{1, 1} \otimes [\operatorname{Gr}^{r_1} V_H]$, regarded as a subsheaf of $[V_H]$ over ${\mathfrak{X}}_H^{{\mathrm{ord}}}$ via the unit root splitting. The small subrepresentations are precisely those where $\operatorname{Gr}^{r_1} V_H$ is the smallest nonzero filtration step.
Families of Eisenstein series
=============================
The theory developed in the previous sections allows us to define $p$-adic measures interpolating the cohomological periods $\langle \iota_{m, \star}(E), \eta\rangle$, as $E$ varies over the specialisations of some family of cuspidal $p$-adic modular forms ${\mathcal{E}}$ on $H$. In this section, we shall specify the particular families ${\mathcal{E}}$ which we shall consider, which will be built up from Eisenstein series and cusp forms for $\operatorname{GL}_2$. We shall prove in the remaining sections that the resulting periods $\langle \iota_{m, \star}(E), \eta\rangle$ are special values of $L$-functions.
Non-holomorphic Eisenstein series {#sect:nonholoEis}
---------------------------------
Let $\Phi_f \in {\mathcal{S}}({{\mathbf{A}}_{\mathrm{f}}}^2, {\mathbf{C}})$ be a Schwartz function.
For $k {\geqslant}1$, $\tau = x + iy$ in the upper half-plane, and $s \in {\mathbf{C}}$ with ${\operatorname{Re}}(s) {\geqslant}1$, we define $$E^{(k, \Phi_f)}(\tau; s) \coloneqq \frac{\Gamma(s + \tfrac{k}{2})}{(-2\pi i)^k \pi^{s- \tfrac{k}{2}}} \sum_{(m, n) \in {\mathbf{Q}}^2 - (0, 0)} \frac{\Phi_f(m, n) y^{(s - \tfrac{k}{2})}}{(m\tau + n)^k |m\tau + n|^{2s-k}}.$$ This can be extended to all $s \in {\mathbf{C}}$ by analytic continuation in $s$.
If $\Phi_f$ is the indicator function of $(0, \alpha) + \hat{\mathbf{Z}}^2$, this series is $E^{(k)}_\alpha(\tau, s - \tfrac{k}{2})$ in the notation of [@LLZ14].
If $\chi$ is a finite-order character of ${\mathbf{A}}^\times / {\mathbf{Q}}^\times$, we define $$R_\chi(\Phi_f) \coloneqq \int_{a \in \hat{{\mathbf{Z}}}^\times} \chi(a) \left({\left( \begin{smallmatrix} a & 0 \\ 0 & a\end{smallmatrix} \right) } \cdot \Phi_f\right)\, \mathrm{d}^\times a,$$ the projection of $\Phi_f$ to the $\chi^{-1}$-isotypical subspace for $\hat{{\mathbf{Z}}}^\times$, and we set $
E^{(k, \Phi_f)}(\tau; \chi, s) \coloneqq E^{(k, R_\chi(\Phi_f))}(\tau; s).
$ Note that $E^{(k, \Phi_f)}(-; \chi, s)$, vanishes if $(-1)^k \chi(-1) \ne 1$.
We can interpret $E^{(k, \Phi_f)}(-; s)$ and $E^{(k, \Phi_f)}(-; \chi, s)$ as $C^\infty$ sections of a line bundle on the $\operatorname{GL}_2$ Shimura variety. More precisely, the $C^\infty$ sections of the automorphic line bundle $\omega(k)$ are the smooth functions $f: \operatorname{GL}_2({{\mathbf{A}}_{\mathrm{f}}}) \times {\mathcal{H}}\to {\mathbf{C}}$ satisfying $$f(g, \tau) = (ad - bc) (c\tau + d)^{-k} f\left({\left( \begin{smallmatrix} a & b \\ c & d\end{smallmatrix} \right) } g, \tfrac{a\tau + b}{c\tau + d}\right)\quad\text{for all} {\left( \begin{smallmatrix} a & b \\ c & d\end{smallmatrix} \right) } \in \operatorname{GL}_2^+({\mathbf{Q}}).$$ With these notations, $E^{(k, \Phi_f)}(s) \coloneqq (g, \tau) \mapsto \|\det g\|^{s + 1 - k/2} E^{(k, g \cdot \Phi)}(\tau; s)$ is a $C^\infty$ section of $\omega(k)$, and similarly with $\chi$; and $E^{(k, \Phi_f)}(\chi, s)$ transforms under the centre of $\operatorname{GL}_2({{\mathbf{A}}_{\mathrm{f}}})$ by $\chi^{-1}\|\cdot\|^{2-k}$.
Nearly holomorphic specialisations
----------------------------------
For integers $j \in [0, k-1]$, the $C^\infty$ section $E^{(k, \Phi_f)}(\tfrac{k}{2} - j)$ of $\omega(k)$ is nearly-holomorphic; and if $\Phi_f$ takes values in a number field $E$, this section is defined over $E$.
We let $\psi$ denote the unique additive character ${\mathbf{A}}/ {\mathbf{Q}}\to {\mathbf{C}}^\times$ satisfying $\psi(x_\infty) = \exp(-2\pi i x_\infty)$ for $x_\infty \in {\mathbf{R}}$. For $\Phi_f \in {\mathcal{S}}({{\mathbf{A}}_{\mathrm{f}}})$, we write $\Phi'(u, v)$ for the Fourier transform in the second variable only: $$\Phi'(u, v) \coloneqq \int_{{{\mathbf{A}}_{\mathrm{f}}}} \Phi(u, w) \psi(vw) \, \mathrm{d}w.$$
If we define the “$q$-expansion” of a nearly-holomorphic form to be the Fourier expansion of its holomorphic part, then for $n > 0$ we have $$a_n\left(E^{(k, \Phi_f)}(\tfrac{k}{2}-j)\right) = \sum_{\substack{(u, v) \in ({\mathbf{Q}}^\times)^2 \\ uv = n}} u^j v^{(k-1-j)} \operatorname{sgn}(u) \grave{\Phi}(u, v).$$ The constant term $a_0\left(E^{(k)}_{\Phi}(\tfrac{k}{2}-j)\right)$ is zero unless $j = 0$ or $j = k-1$, in which case it is given by a special value of a linear combination of Hurwitz zeta functions. For our purposes it suffices to note that if $\Phi(-, 0) = \Phi(0, -) = 0$, then the constant term is zero for all $j$.
Families of p-adic Eisenstein series
------------------------------------
We shall now define a family of local Schwartz functions $\Phi_{p, \mu, \nu}$, depending on a choice of two finite-order characters $\mu$ and $\nu$ of ${{\mathbf{Z}}_p}^\times$.
We define $\Phi_{p, \mu, \nu} \in {\mathcal{S}}({{\mathbf{Q}}_p}^2, {\mathbf{C}})$ as the unique function such that $$\Phi'_{p, \mu, \nu}(x, y) =
\begin{cases}
\mu(x) \nu(y) &\text{if $x, y \in {{\mathbf{Z}}_p}^\times$,}\\
0 &\text{otherwise.}
\end{cases}$$
Note that $\Phi_{p, \mu, \nu}$ transforms under ${\left( \begin{smallmatrix} a & 0 \\ 0 & d\end{smallmatrix} \right) }$, $a,d \in {{\mathbf{Z}}_p}^\times$, by $\mu(a) \nu(d)^{-1}$. (The values of $\Phi_{p, \mu, \nu}$ can be made explicit in terms of Gauss sums, but we do not need this.) Now let $\Phi^{(p)}$ be a Schwartz function on $({{\mathbf{A}}_{\mathrm{f}}}^p)^2$ with values in a number field $K$, and $\chi^{(p)}$ a finite-order character of ${\mathbf{A}}^\times / {\mathbf{Q}}^\times$ of conductor coprime to $p$ such that ${\left( \begin{smallmatrix} a & 0 \\ 0 & a\end{smallmatrix} \right) } \cdot \Phi^{(p)} = \chi^{(p)}(a)^{-1} \Phi^{(p)}$ for $a \in (\widehat{{\mathbf{Z}}}^{(p)})^\times$. Let $L$ be the completion of $K$ at some prime above $p$. If $\mu, \nu$ are finite-order characters of ${{\mathbf{Z}}_p}^\times$ as above, we may extend $\mu$ and $\nu$ to characters of ${\mathbf{A}}^\times$ trivial on ${\mathbf{R}}_{>0}^\times \cdot {\mathbf{Q}}^\times \cdot (\widehat{{\mathbf{Z}}}^{(p)})^\times$; in particular, $\chi = \chi^{(p)} \mu^{-1} \nu$ is well-defined as an adelic character, and $\Phi_{\mu, \nu} = \Phi^{(p)} \Phi_{p, \mu, \nu}$ is in the image of the idempotent $R_\chi$.
Let $R = \Lambda_L({{\mathbf{Z}}_p}^\times \times {{\mathbf{Z}}_p}^\times)$, with its two canonical characters $\kappa_1, \kappa_2$. For each $\Phi^{(p)}$ taking values in $L$, there exists an element $\mathcal{E}^{\Phi^{(p)}}(\kappa_1, \kappa_2; \chi^{(p)}) \in \mathcal{S}_{\kappa_1 + \kappa_2 + 1}(R)$, whose specialisation at $(a + \mu, b + \nu)$, for integers $a, b{\geqslant}0$, is the $p$-adic modular form associated to the algebraic nearly-holomorphic form $$(g, \tau) \mapsto \nu(\det g) \cdot E^{(a+b+1, \Phi_{\mu, \nu})}(g, \tau; \tfrac{b-a+1}{2})\in M_{a+b+1}^{\mathrm{nh}}.$$ For $g \in \operatorname{GL}_2({{\mathbf{A}}_{\mathrm{f}}}^p)$, we have $g \cdot \mathcal{E}^{\Phi^{(p)}} =\|\det g\|^{1-\kappa_1} \cdot \mathcal{E}_{g \cdot \Phi^{(p)}}$. The $q$-expansion of $\mathcal{E}^{\Phi^{(p)}}$ at $\infty$ is $$\sum_{\substack{u, v \in ({\mathbf{Z}}_{(p)}^\times)^2 \\ uv > 0}} \operatorname{sgn}(u) u^{\kappa_1} v^{\kappa_2} (\Phi^{(p)})'(u, v) q^{uv}.$$ The Eisenstein series $\mathcal{E}_{\Phi^{(p)}}(-; \chi^{(p)})$ is identically 0 on the components of $\operatorname{Spec}R$ where $\kappa_1(-1) \kappa_2(-1) \ne -\chi^{(p)}(-1)$.
This result is essentially a restatement of Katz’s theory of the Eisenstein measure; we have stated it in a slightly unusual form in order to spell out precisely the $\operatorname{GL}_2\left({{\mathbf{A}}_{\mathrm{f}}}^p\right)$-equivariance properties of the construction. Note that the factor $\nu(\det g)$ is required in order that the right-hand side be invariant under ${\left( \begin{smallmatrix} 1 & * \\ & *\end{smallmatrix} \right) } \subset \operatorname{GL}_2({{\mathbf{Z}}_p})$, which is a prerequisite for it to be the specialisation of a $p$-adic modular form.
Input to the machine: the case of $\operatorname{GSp}_4$ {#sect:input1}
--------------------------------------------------------
Suppose we are given an automorphic representation $\Pi$ as before, which is globally generic and cohomological with coefficients in $V(r_1, r_2)$, and unramified and Klingen-ordinary at $p$. We set $d = r_1 - r_2 {\geqslant}0$.
Let $R = \Lambda_L({{\mathbf{Z}}_p}^\times \times {{\mathbf{Z}}_p}^\times)$ with its two canonical characters $\mathbf{j}_1, \mathbf{j}_2$. Given $\Phi_1^{(p)}, \Phi_2^{(p)}$, we consider the family of $p$-adic modular forms for $H$ given by $${\mathcal{E}}^{\Phi_1^{(p)}}(d-\mathbf{j}_1, \mathbf{j}_2; \chi_\pi) \boxtimes {\mathcal{E}}^{\Phi_2^{(p)}}(0, \mathbf{j}_1- \mathbf{j}_2; \mathrm{id}).$$ By construction, the specialisation of this family at $(a_1 + \rho_1, a_2 + \rho_2)$, for integers $a_1, a_2$ such that $d {\geqslant}a_1 {\geqslant}a_2 {\geqslant}0$ and finite-order characters $\rho_i$, is the product of two Eisenstein series of the form $$\nu_i(\det g) E^{k_i, \Phi_i}(\chi_i, s_i),$$ where the parameters are given by $$\begin{aligned}
k_1 &= 1+d -a_1 + a_2, & k_2 &= 1 + a_1 - a_2,\\
\chi_1 &= \rho_1 \rho_2 \chi_\pi, & \chi_2 &= \rho_1 \rho_2^{-1}, \\
s_1 &= \tfrac{1-d+a_1 + a_2}{2}, & s_2 &= \tfrac{1 + a_1 - a_2}{2}.
\end{aligned}$$ The Schwartz functions $\Phi_{f, i}$ are given by $\Phi_{f, i} = \Phi_i^{(p)} \times \Phi_{p, \mu_i, \nu_i}$, where $\Phi_i^{(p)}$ are chosen arbitrarily, and the characters $\mu_i, \nu_i$ are given by $$\begin{aligned}
\mu_1 &= \rho_1^{-1}, & \mu_2 &= \mathrm{id},\\
\nu_1 &= \rho_2, & \nu_2 &= \rho_1 \rho_2^{-1}.
\end{aligned}$$
Note that our choices are made such that the Eisenstein series on the second factor of $H$ is holomorphic, although the one on the first factor is only nearly-holomorphic in general.
We also choose a vector $\eta \in H^2({\Pi_{\mathrm{f}}})$, lying in the ordinary ${\mathcal{U}}_{p, \operatorname{Kl}}'$-eigenspace at level $\operatorname{Kl}(p)$. From the theory of §\[sect:p-adic-pushfwd\], we obtain a measure $\mathcal{L} \in R$, whose specialisation at $(a_1 + \rho_1, a_2 + \rho_2)$ interpolates the period integral of $\eta$ against the specialisation of the above family.
We shall see in the following chapters that if $(-1)^{a_1}\rho_1(-1) \ne (-1)^{a_2}\rho_2(-1)$, this period integral will be (up to various elementary factors) equal to the product of $L$-values $$L\left(\pi \otimes \rho_1, \frac{1-d}{2} + a_1\right) \cdot L\left(\pi \otimes \rho_2, \frac{1-d}{2} + a_2\right).$$ On the other hand, if $(-1)^{a_1}\rho_1(-1) = (-1)^{a_2}\rho_2(-1)$, then both of the Eisenstein series will be identically 0. This shows that we can only interpolate products of twisted $L$-values of opposite parity, a condition which is familiar from the setting of Kato’s $\operatorname{GL}_2$ Euler system.
Input to the machine: the case of $\operatorname{GSp}_4 \times \operatorname{GL}_2$ {#sect:input2}
-----------------------------------------------------------------------------------
Now let us suppose we are given an auxiliary automorphic representation $\sigma$ of $\operatorname{GL}_2$, corresponding to a holomorphic modular form of weight $\ell {\geqslant}1$, and suppose that $$d' = r_1 - r_2 + 1 - \ell {\geqslant}0.$$ For $\lambda$ a holomorphic modular form in the space of $\sigma$, we consider the family of $p$-adic modular forms over $R = \Lambda_L({{\mathbf{Z}}_p}^\times)$ given by $${\mathcal{E}}^{\Phi^{(p)}}(d' - \mathbf{j}, \mathbf{j}; \chi_\pi \chi_\sigma)\boxtimes \lambda .$$ Note that at $\mathbf{j} = a + \rho$ with $0 {\leqslant}a {\leqslant}d'$, this specialises to $$\rho(\det h) E^{(\Phi_f, d' + 1)}\left(h_1; \rho^2 \chi_\pi \chi_\sigma, \tfrac{1-d'}{2} + a\right) \lambda(h_2)$$ where $\Phi_f = \Phi^{(p)} \Phi_{(p, \rho^{-1}, \rho)}$. We shall see in the next chapter that the cup-product of this series with $\eta$ gives a zeta-integral computing $$L\left(\pi \otimes \sigma \otimes \rho, \tfrac{1-d'}{2} + a\right).$$
One checks that the critical values of $L(\pi \otimes \mu, s)$ are exactly the values $s = \tfrac{1-d}{2} + a$, for integers $0 {\leqslant}a {\leqslant}d$. Similarly, the critical values of $L(\pi \otimes \sigma \otimes \mu, s)$ are the $s = \tfrac{1-d'}{2} + a$ for $0 {\leqslant}a {\leqslant}d'$. So in each case we hit the full interval of critical values of the relevant $L$-function (and not any others!).
If we have $\ell {\geqslant}r_1 - r_2 + 2$, then the $L$-function for $\pi \otimes \sigma$ may still have some critical values; but we do not see them by this method.
Integral formulae for L-functions I: local theory {#sect:localintegrals}
=================================================
In this section and the next, we recall a general formula which will be used to relate the cohomological periods studied above to critical values of $L$-functions, based on work of Novodvorsky and others. In this section, we let $F$ be an arbitrary local field, and $\psi_F$ a non-trivial additive character $F \to {\mathbf{C}}^\times$. We use $\psi_F$ to define a character $\psi: N(F) \to {\mathbf{C}}$ by by $$\psi(n) = \psi_F(x+y), \ \ n=\left(\left( \begin{smallmatrix} 1 & x & * & * \\ & 1 & y & * \\ & & 1 & -x \\ &&&1\end{smallmatrix} \right) \right).$$
Siegel sections
---------------
If $\Phi$ is a Schwartz function on $F^2$, $\chi$ a smooth unitary character of $F^\times$, and $g \in \operatorname{GL}_2(F)$, we define $$f^{\Phi}(g; \chi, s) \coloneqq |\det g|^s \int_{F^\times} \Phi(( 0, a)g) \chi(a) |a|^{2s} \mathrm{d}^\times a.$$ This integral converges absolutely for ${\operatorname{Re}}(s)>0$ and defines an element of the principal series representation $I(|\cdot|^{s-{\frac{1}{2}}},\chi^{-1}|\cdot|^{{\frac{1}{2}}-s})$ of $\operatorname{GL}_2(F)$, the normalized induction of the representation ${\left( \begin{smallmatrix} x & * \\ & y\end{smallmatrix} \right) } \mapsto |x/y|^{s-{\frac{1}{2}}} \chi(y)^{-1}$ of the Borel. It has meromorphic continuation in $s$; if $v$ is a finite place, it is even a rational function of $q^{s}$ (where $q$ is the cardinality of the residue field, as usual).
Note that this is an *un-normalised* Siegel section, and hence is not necessarily entire.
We define the *Whittaker transform* $W^{\Phi}(g; \chi, s)$ of $f^{\Phi}(g; \chi, s)$ to be the function $$W^{\Phi}(g;\chi,s) = \int_ {F} f^{\Phi}\left(\eta {\left( \begin{smallmatrix} 1 & x \\ 0 & 1\end{smallmatrix} \right) } g; \chi, s\right) \psi_{F}(x) \mathrm{d}x, \ \ \eta = \left(\smallmatrix 0 & 1 \\ -1 & 0 \endsmallmatrix\right).$$ If $s$ is such that $I(|\cdot|^{s-{\frac{1}{2}}}, \chi^{-1}|\cdot|^{{\frac{1}{2}}-s})$ is irreducible, then every vector in this representation is $f^{\Phi}$ for some $\Phi$, and the map $f^{\Phi} \mapsto W^{\Phi}$ gives the isomorphism from $I(|\cdot|^{s-{\frac{1}{2}}}, \chi^{-1}|\cdot|^{{\frac{1}{2}}-s})$ to its Whittaker model (with respect to the inverse character $\bar{\psi}_{F}$).
For fixed $g$ and $\Phi$, $W^{\Phi}(g;\chi,s)$ is entire as a function of $s$; if $v$ is nonarchimedean, $\chi$ and $\psi_{F}$ are unramified, and $\Phi$ is the characteristic function of ${\mathcal{O}}_{F}^2$, then $W^{\Phi}(1; \chi,s)$ is identically 1. These functions satisfy a functional equation: if $\hat{\Phi}$ is the 2-variable Fourier transform defined by $$\hat{\Phi}(x, y) = \int_{F^2} \Phi(u, v) \psi_F(xv - yu)\, \mathrm{d}u \, \mathrm{d}v,$$ then we compute that $$\label{eq:whittaker-fcl-eq}
W^{\hat\Phi}(g; \chi^{-1}, 1-s) = \chi(\det g) W^{\Phi}(g; \chi, s).$$
The values of $W^{\Phi}$ on the diagonal torus can be given in terms of the “partial” Fourier transform $$\label{eq:partialfourier}
\Phi'(x, y) \coloneqq \int_{F} \Phi(x, w) \psi_{F}(yw) \, \mathrm{d}w.$$ With this notation, we have $$W^{\Phi}({\left( \begin{smallmatrix} x & \\ & y\end{smallmatrix} \right) }; \chi,s) = \chi(-1) |x|^s|y|^{s-1}\int_a \Phi'\left(xa, \tfrac{1}{ya}\right) \chi(a) |a|^{2s-1}\, \mathrm{d}^\times a. \qedhere$$
Definition of the local integrals {#local-zeta-int}
---------------------------------
Let $\pi$ be an irreducible smooth representation of $G(F)$, with unitary central character $\chi_\pi$. We assume $\pi$ is *generic*, i.e. that $\pi$ is isomorphic to a space of functions $G(F) \to {\mathbf{C}}$ satisfying $W(ng) = \psi(n) W(g)$ for $n \in N(F)$. Such a model is unique (see [@Rodier-padic Thm. 3] if $F$ is nonarchimedean, and [@Wallach-real Thm. 8.8(1)] if $F$ is archimedean). We fix a choice of isomorphism between $\pi$ and its Whittaker model, and for $\varphi \in \pi$, we write $W_\varphi$ for the corresponding Whittaker function.
### The two-parameter $\operatorname{GSp}_4$ integral {#the-two-parameter-operatornamegsp_4-integral .unnumbered}
Let $\chi_1, \chi_2$ be smooth characters of $F^\times$ such that $\chi_1 \chi_2 = \chi_{\pi}$.
We define $$Z(\varphi, \Phi_1, \Phi_2, s_1, s_2) \coloneqq
\int_{Z_G(F)N_H(F)\backslash H(F)}
W_{\varphi}(h)
f^{\Phi_1}(h_1;\chi_1,s_1)
W^{\Phi_2}(h_2;\chi_2,s_2)
\, \mathrm{d} h,$$ where $\varphi \in \pi$ and each $\Phi_{i}$ is a Schwartz function on $F^2$.
Substituting in the definition of $W^{\Phi_2}(-)$ as an integral, we obtain the following alternative formula:
For $\varphi \in \pi$, the integral $$B_{\varphi}(g, s) \coloneqq \int_{F^\times} \int_{F} W_{\varphi}\left(\left( \begin{smallmatrix} a \\ &a \\ &x&1\\ &&&1 \end{smallmatrix} \right) w_2 g\right) |a|^{s-\tfrac{3}{2}} \chi_2^{-1}(a)\, \mathrm{d}x\, \mathrm{d}^\times a,\qquad w_2 = \left( \begin{smallmatrix} 1\\&&1 \\ &-1 \\&&&1\end{smallmatrix} \right) $$ converges for ${\operatorname{Re}}(s) \gg 0$ and has meromorphic continuation to all values of $s$; and we have $$Z(\varphi, \Phi_1, \Phi_2, s_1, s_2) = \int_{D N_H \backslash H}B_\varphi(h; s_1 - s_2 + \tfrac{1}{2}) f^{\Phi_1}(h_1;\chi_1,s_1) f^{\Phi_2}(h_2;\chi_2,s_2)\, \mathrm{d}h,$$ where $D$ denotes the torus $\{\operatorname{diag}(p, q, p, q): p, q\in F^\times\}$.
The function $B_{\varphi}(g; s)$ is an element of the *Bessel model* of the representation $\pi$: it transforms by a character under left-translation by the Bessel subgroup $D N_{\operatorname{S}}$. On the other hand, the integral defining $B_\varphi$ in terms of $W_\varphi$ is an instance of Novodvorsky’s local zeta integral for $L(\pi \otimes \chi_2^{-1}, s)$ [@novodvorsky79]. These two facts will be crucial in our analysis of the two-parameter integral $Z(\varphi, \Phi_1, \Phi_2, s_1, s_2)$.
### The $\operatorname{GSp}_4 \times \operatorname{GL}_2$ integral {#the-operatornamegsp_4-times-operatornamegl_2-integral .unnumbered}
Similarly, if $\pi$ is as before and $\sigma$ is a generic representation of $\operatorname{GL}_2(F)$, we let $\chi = \chi_{\pi} \chi_{\sigma}$ and define $$Z(\varphi, \lambda, \Phi, s) \coloneqq
\int_{Z_G(F)N_H(F)\backslash H(F)}
W_{\varphi}(h)
f^{\Phi}(h_1;\chi, s)
W_{\lambda}(h_2)
\, \mathrm{d} h,$$ where $\varphi \in \pi$ and $\lambda \in \sigma$, and $W_{\lambda}$ denotes the image of $\lambda$ in the Whittaker model of $\sigma$, again with respect to the opposite character $\bar{\psi}_{F}$.
Note that if $\chi_2 |\cdot|^{2s_2 - 1} \notin \{ |\cdot|^{\pm 1}\}$ (so that $\sigma = I(|\cdot|^{s_2 - 1/2}, |\cdot|^{1/2 - s_2} \chi_2^{-1})$ is irreducible), we can regard the first integral as a special case of the second, taking $\lambda = f^{\Phi_2}(-;\chi_2, s_2)$. However, it is convenient to treat the first integral separately in order to understand the reducible cases, and the variation in $s_2$.
Essentially the same analysis of the local zeta integrals as in [@Soudry-GSpGL] and [@Gel-PS-explicit] yields the following:
\[local-zeta-prop\]
- There exists a positive real number $R > 0$, depending on $\pi$ and $\sigma$, such that the local zeta integral $Z(\varphi, \lambda, \Phi, s)$ converges absolutely for ${\operatorname{Re}}(s) > R$, for all choices of $(\varphi, \lambda, \Phi)$. Furthermore, each $Z(\varphi, \lambda, \Phi, s)$ has a meromorphic continuation in $s$, and is a rational function of $q^{s}$ if $F$ is nonarchimedean.
- There exists a positive real number $R > 0$, depending on $\pi$ and $s_2$, such that the local zeta integral $Z(\varphi, \Phi_1, \Phi_2, s_1, s_2)$ converges absolutely for ${\operatorname{Re}}(s_1) > R$, for all choices of $(\varphi, \Phi_1, \Phi_2)$. Furthermore, each $Z(\varphi, \Phi_1, \Phi_2, s_1, s_2)$ extends to a meromorphic function of $(s_1, s_2) \in {\mathbf{C}}^2$, and is a rational function of $(q^{s_1}, q^{s_2})$ if $F$ is nonarchimedean.
Nonarchimedean $L$-factors
--------------------------
In this section we assume $F$ is nonarchimedean.
Let $L(\pi, s)$ and $L(\pi \otimes \sigma, s)$ denote the local $L$-factors associated to $\pi$ and to $\pi \otimes \sigma$ via Shahidi’s method, as in [@gantakeda11 §4].
By construction, these $L$-factors coincide with the Artin $L$-factors of the Weil–Deligne representations $\mathrm{rec}(\pi)$ and $\mathrm{rec}(\pi) \otimes \mathrm{rec}(\sigma)$ respectively, where “$\mathrm{rec}$” denotes the local Langlands correspondence of *op.cit.*.
\(i) The vector space of functions on ${\mathbf{C}}^2$ spanned by the $Z(\varphi, \Phi_1, \Phi_2, s_1, s_2)$, as the data $(\varphi, \Phi_1, \Phi_2)$ vary, is a fractional ideal of ${\mathbf{C}}[q^{\pm s_1}, q^{\pm s_2}]$ containing the constant functions. This fractional ideal of ${\mathbf{C}}[q^{\pm s_1}, q^{\pm s_2}]$ is generated by the product of $L$-factors $$L\left(\pi, s_1 + s_2 - \tfrac12\middle) L\middle(\pi \otimes \chi_2^{-1}, s_1 - s_2 + \tfrac12\right).$$ (ii) If $\pi$, $\chi_{i}$ and $\psi_{F}$ are unramified, $\varphi^0 \in \pi$ is the unique spherical vector such that $W_{\varphi^0}(1) = 1$, and $\Phi^0_1 = \Phi^0_2 = {\operatorname{ch}}({\mathcal{O}}_{F}^2)$, then $$Z(\varphi^0, \Phi^0_1, \Phi^0_2, s_1, s_2) =
L\left(\pi, s_1 + s_2 - \tfrac12\middle)
L\middle(\pi \otimes \chi_2^{-1}, s_1 - s_2 + \tfrac12\right).$$
It suffices to prove the analogous result result for fractional ideals in ${\mathbf{C}}[q^{\pm s_1}]$, for a fixed value of $s_2$. It follows from the computations of Takloo-Bighash [@takloobighash00] that the “lowest common denominator” of Novodvorsky’s zeta integrals agrees with Shahidi’s definition of the $L$-factor. That is, if we define $$\widetilde B_\varphi(g) = L(\pi \otimes \chi_2^{-1}, s_1 - s_2 + \tfrac12)^{-1} \cdot B_\varphi(g),$$ then $\widetilde B_\varphi(g) \in {\mathbf{C}}[q^{\pm s_1}]$ for all $g$, and there is no $s_1$ such that $\widetilde B_\varphi(g)$ vanishes for all $g$ and $\varphi$.
It follows from the theory summarized in [@robertsschmidt16] that the space of functions $\{ \widetilde B_\varphi(-) : \varphi \in \pi\}$, is the (unique) *split Bessel model* of $\pi$, with respect to the character of the Bessel torus given by $( |\cdot|^{s_2 - s_1}\chi_2, |\cdot|^{s_1 - s_2} \chi_1)$. The integral $\int_{D N \backslash H}\widetilde{B}_\varphi(h) f^{\Phi_1}(h_1) f^{\Phi_2}(h_2)\, \mathrm{d}h$ is precisely Piatetski-Shapiro’s zeta integral [@piatetskishapiro97] for the local $L$-factor $L(\pi, s)$, with $s = s_1 + s_2 - \tfrac12$. So to prove (i) we are reduced to the question of whether the Piaetski-Shapiro’s zeta integral always agrees with the Shahidi and Novodvorsky definitions of $L(\pi, s)$, independently of the choice of character used in the definition of the Bessel model. This is exactly the main result of [@roesnerweissauer17].
The unramified computation (ii) now follows from the corresponding computation of Novodvorsky and Piatetski-Shapiro’s integrals in the unramified case.
For the $\operatorname{GSp}_4 \times \operatorname{GL}_2$ integral we have a less complete result:
\(i) The vector space of functions on ${\mathbf{C}}$ spanned by the $Z(\varphi, \lambda, \Phi, s)$, as the data $(\varphi,\lambda, \Phi)$ vary, is a fractional ideal of ${\mathbf{C}}[q^{\pm s}]$ containing the constant functions. If *either*
- $\sigma$ is principal series, or
- $\pi$ arises from a pair $(\tau_1, \tau_2)$ of generic irreducible representations of $\operatorname{GL}_2$ with the same central character (via the theta-lifting from $\operatorname{GSO}_{2, 2}$), and if $\sigma$ is supercuspidal, then neither of the $\tau_i$ is isomorphic to an unramified twist of $\sigma^\vee$,
then this fractional ideal is generated by the $L$-factor $L(\pi \otimes \sigma, s)$. In particular, this holds if at least one of $\pi$ and $\sigma$ is unramified.
\(ii) If $\pi$, $\sigma$ and $\psi_{F}$ are all unramified, $\varphi^0 \in \pi$ and $\lambda^0 \in \sigma$ are the spherical vectors normalised such that $W_{\varphi^0}(1) = W_{\lambda^0}(1)=1$, and $\Phi^0 = {\operatorname{ch}}({\mathcal{O}}_{F}^2)$, then $$Z(\varphi^0, \lambda^0, \Phi^0, s) = L(\pi \otimes \sigma, s).$$
The representations of $\operatorname{GSp}_4(F)$ which are theta-lifts from the split orthogonal group $\operatorname{GSO}_{2, 2}(F) \cong (\operatorname{GL}_2(F) \times \operatorname{GL}_2(F)) / \{ (z, z^{-1}): z \in F^\times\}$ are tabulated in [@gantakeda11b]. Note that this class of representations includes all irreducible principal series, and all representations irreducibly induced from supercuspidal representations of the Siegel Levi subgroup.
Firstly, we note that if $\sigma$ is an irreducible principal series representation, then after replacing $\pi$ and $\sigma$ with $\pi \otimes \beta$ and $\sigma \otimes \beta^{-1}$ for a suitable character $\beta$, we can arrange that $\sigma = I(|\cdot|^{s_2 - 1/2}, |\cdot|^{1/2 - s_2} \chi_2^{-1})$ for some $s_2$. Then we have $L(\pi \otimes \sigma, s) = L\left(\pi, s_1 + s_2 - \tfrac12\middle)
L\middle(\pi \otimes \chi_2^{-1}, s_1 - s_2 + \tfrac12\right)$ and part (i) in this case follows from the previous theorem. Part (ii) follows similarly, noting that the Whittaker transform of $\Phi^0 = {\operatorname{ch}}({\mathcal{O}}_F^2)$ satisfies $W^{\Phi^0}(1) = 1$, so the two natural normalisations of the spherical test data coincide.
The assertion concering $\pi$ lifted from $\operatorname{GSO}_{2, 2}$ follows from the results of [@Soudry-GSpGL]. Soudry shows that in this case the fractional ideal of values of $Z(\varphi, \lambda, \Phi, s)$ is generated by the product of $\operatorname{GL}_2 \times \operatorname{GL}_2$ Rankin–Selberg $L$-factors, $L(\tau_1 \otimes \sigma, s) L(\tau_2 \otimes \sigma, s)$; since the Gan–Takeda local Langlands correspondence is (by construction) compatible with theta-lifting from $\operatorname{GSO}_{2, 2}$, this Rankin–Selberg $L$-factor coincides with the Shahidi $L$-factor.
It remains to check that the fractional ideal contains 1 in all cases. This follows from the “$P_3$ theory” of [@robertsschmidt07 §2.5], which can be used to construct $\varphi$ such that the restriction of $W_{\varphi}$ to the Klingen parabolic is non-zero, but has arbitrarily small support modulo the centre and the unipotent radical of the Borel. (Compare the proof of Proposition 2.6.4 of *op.cit.*.) We can then choose the Schwartz function $\Phi$ such that $f^{\Phi}(h_1)$ is supported (modulo the Borel of $H$) in a small neighbourhood of the identity in $\mathbf{P}^1$, cf. the proof of Lemma 14.7.5 of [@jacquet72].
Local calculations at $p$
-------------------------
We carry out an evaluation of the local zeta integral for particular choices of (ramified) data. This calculation will be used to identify the Euler factors at $p$ in our final formula for the values of the $p$-adic $L$-function. To simplify notation we assume that $F = {{\mathbf{Q}}_p}$, and that $\psi_F$ is unramified (i.e. trivial on ${{\mathbf{Z}}_p}$ but not on $p^{-1}{{\mathbf{Z}}_p}$). We also assume that the local representation $\pi$ of $G({{\mathbf{Q}}_p})$ is an unramified principal series.
In this section (*only*) we shall deal exclusively with the Klingen parabolic $P_{\operatorname{Kl}}$ and its Levi $M_{\operatorname{Kl}}$, so we shall drop the subscripts and denote them simply by $P$ and $M$. We identify $M$ with $\operatorname{GL}_2\times\operatorname{GL}_1$ by $${\left( \begin{smallmatrix} \lambda\\ &A \\ &&\det(A) / \lambda\end{smallmatrix} \right) }\mapsto (A,\lambda).$$ Under this identification, the modulus character $\delta_P$ is the character of $\operatorname{GL}_2 \times \operatorname{GL}_1$ given by $(A, \lambda) \mapsto |\lambda^4 / \det(A)^2|$. (See e.g. [@robertsschmidt07 §2.2]). In particular, we have $$\delta_P(\left( \begin{smallmatrix} \det(A) \\ & A \\ &&1 \end{smallmatrix} \right) ) = |\det A|^2.$$
### The vectors $\phi_r \in \pi_p$
Since $\pi$ is a principal series representation, it is in particular induced from the Klingen parabolic. Let us choose an unramified principal series $\tau$ of $\operatorname{GL}_2({{\mathbf{Q}}_p})$, and a character $\theta: {{\mathbf{Q}}_p}^\times \to {\mathbf{C}}^\times$, such that $\pi$ is isomorphic to the induced representation $\operatorname{Ind}_P^G(\tau \boxtimes \theta)$ (normalised induction from the Klingen parabolic). With these notations, the central character of $\pi$ is given by $\chi_\pi = \theta \chi_\tau$ (where $\chi_\tau$ is the central character of $\tau$), and the spin $L$-factor of $\pi$ is given by $$L(\pi, s) = L(\tau, s) L(\tau \otimes \theta, s).$$
For $r {\geqslant}0$, let $K_r\subset G({{\mathbf{Z}}_p})$ be the depth $r$ Klingen parahoric subgroup: $$K_r \coloneqq \{ g\in G({{\mathbf{Z}}_p}) \ : \ g\bmod{p^r} \in P({{\mathbf{Z}}_p}/p^r{{\mathbf{Z}}_p})\}.$$
If we identify the quotient $P({{\mathbf{Q}}_p}) \backslash G({{\mathbf{Q}}_p})$ with $\mathbf{P}^3$, then $K_r$ is precisely the stabiliser of the point $(0:0:0:1) \in \mathbf{P}^3({{\mathbf{Z}}_p}/p^r{{\mathbf{Z}}_p})$, so we have the following characterisation of the coset $P({{\mathbf{Q}}_p}) K_r$:
\[support-lem\] Suppose $r {\geqslant}1$ and let $g = \left(\smallmatrix * & * & * & * \\ b' & * & * & * \\ c' & * & * & * \\ a & b & c & d\endsmallmatrix\right) \in G({{\mathbf{Q}}_p})$. Then $g \in P({{\mathbf{Q}}_p})K_r$ if and only if $d\in{{\mathbf{Q}}_p}^\times$ and $d^{-1}a,d^{-1}b,d^{-1}c\in p^r{{\mathbf{Z}}_p}$.
For $r{\geqslant}1$, let $\phi_r \in \pi_p^{K_r}$ be the function with support $P({{\mathbf{Q}}_p})K_r$, and such that $\phi_r(1) = p^{3r}$.
Note that the functions $\phi_r$ are trace-compatible, i.e. we have $$\frac{1}{[K_r: K_{r+1}]}\sum_{k \in K_{r}/ K_{r+1}} k \phi_{r+1} = \phi_r.$$ Moreover, the restriction of $\phi_r$ to $M({{\mathbf{Q}}_p})$ is independent of $r$ up to scaling: it is given by $(A, \lambda) \mapsto p^{3r} |\lambda^2/\det(A)| \xi(A) \theta(\lambda)$, where $\xi$ is the normalised spherical vector of the unramified $\operatorname{GL}_2({{\mathbf{Q}}_p})$-representation $\tau$.
A straightforward explicit computation gives the following:
\[phi1-phir-lem\] Let $r {\geqslant}1$, and let $t_P = \operatorname{diag}(1, p, p, p^2)$. Then
(i) The vector $\phi_r$ is an eigenvector for the operator $[K_r t K_r]$ on $\pi_p^{K_r}$, with eigenvalue $p^2 \chi_\tau(p)$.
(ii) We have $$\phi_r = \left(\tfrac{p}{\chi_\tau(p)}\right)^{r-1} \sum_{a\in p^r{{\mathbf{Z}}_p}/p^{2r-1}{{\mathbf{Z}}_p}} \left(\smallmatrix 1 & & & \\ & 1 & & \\ & & 1 & \\ a & & & 1\endsmallmatrix\right) t_P^{r-1} \cdot \phi_1.\qed$$
Conversely, given any eigenvalue of $[K_1 t K_1]$ on $\pi_p^{K_1}$, we can always write $\pi_p$ in the form $\operatorname{Ind}_P^G(\tau \boxtimes \theta)$ such that $p^2 \chi_\tau(p)$ is the given eigenvalue.
### Statement of the formula
Let $\sigma$ be any generic representation[^3] of $\operatorname{GL}_2({{\mathbf{Q}}_p})$, and ${\mathcal{W}}(\sigma)$ its Whittaker model with respect to $\psi^{-1}$. Let $\chi = \chi_{\pi} \chi_{\sigma}$. The goal of this section is the following computation:
\[prop:local-zeta-evaluation\] Let $\Phi_1 \in {\mathcal{S}}({{\mathbf{Q}}_p}^2)$ and $\lambda \in \sigma$. Then there is some $R$ (depending on $\Phi_1$ and $\lambda$) such that the zeta integral $$\label{eq:localintegral}
Z_p(\gamma \cdot \phi_r, \lambda, \Phi_1, s) =
\int_{(Z_G N_H \backslash H)({{\mathbf{Q}}_p})}
W_{\phi_r}(h \gamma) f^{\Phi_1}(h_1; \chi, s) W_\lambda(h_2)\, \mathrm{d}h,
\quad \gamma = \left( \begin{smallmatrix} 1\\ 1 & 1\\ & & 1 \\ & & -1 & 1\end{smallmatrix} \right) $$ is independent of $r {\geqslant}R$. If $\Phi_1'(0, 0) = 0$, where $\Phi_1'$ denotes the partial Fourier transform as in , then this limiting value is given by $$\frac{L(\tau \times \sigma, s)}{L(\tau^\vee \times \sigma^\vee, 1-s) \epsilon(\tau\times\sigma, s)}
\int_{{{\mathbf{Q}}_p}^\times} W_\xi({\left( \begin{smallmatrix} x & 0 \\ 0 & 1\end{smallmatrix} \right) }) W^{\Phi_1}\left({\left( \begin{smallmatrix} x & 0 \\ 0 & 1\end{smallmatrix} \right) }; \chi, s\right)W_\lambda({\left( \begin{smallmatrix} x & 0 \\ 0 & 1\end{smallmatrix} \right) }) \frac{\theta(x)}{|x|} \,\mathrm{d}^\times x.$$
(i) The assumption on $\Phi_1$ is probably not needed, but it is true in the cases of interest, and it simplifies the proof.
(ii) In the present paper, we shall apply this formula with the data $\Phi_1$ and $\lambda$ chosen such that the integrand is a multiple of the characteristic function of ${{\mathbf{Z}}_p}^\times$, so the integral is just the product of the three Whittaker functions at the identity. However, we shall need to consider more general choices of the local data in a sequel to the present paper (in preparation).
(iii) Since $\tau$ is assumed to be unramified, we have $$\epsilon(\tau \otimes \sigma, s) = \epsilon(\sigma)^2 \cdot (p^{-2s} \chi_\tau(p))^{\mathfrak{f}(\sigma)}$$ where $\mathfrak{f}(\sigma)$ is the conductor of $\sigma$, and $\epsilon(\sigma) = \epsilon(\sigma, 0)$. In our applications, $\sigma$ will be a principal-series representation, so $\epsilon(\sigma)$ will be a product of Gauss sums. However, $\chi_\tau(p)$ is a more sophisticated invariant: we shall choose $\tau$ such that $\chi_\tau(p) = p^{-(r_1 + 2)}\lambda$, where $\lambda$ is the $p$-adic unit eigenvalue of the operator ${\mathcal{U}}_{p, \operatorname{Kl}}$ acting on $\Pi_p$.
We shall first give the argument assuming the slightly stronger condition that $\Phi_1(0, x) = 0$ for all $x$.
### Reduction to a $\operatorname{GL}_2 \times \operatorname{GL}_2$ integral
In this section, we shall use explicit formulae for the Whittaker transform to express the integral as a Rankin–Selberg integral for $\operatorname{GL}_2 \times \operatorname{GL}_2$. For brevity we shall write $f_1(-) = f^{\Phi_1}(-; \chi, s)$. Recall that $\Phi_1$ is such that $f_1(1) = 0$, so $f_1$ is supported in the “big cell” $B_{\operatorname{GL}_2} \eta N_{\operatorname{GL}_2}$, where $\eta = \left(\smallmatrix 0 & 1 \\ -1 & 0 \endsmallmatrix\right)$ as before. Moreover, the function $x \mapsto f_1\left( \eta {\left( \begin{smallmatrix} 1 & x \\ 0 & 1\end{smallmatrix} \right) }\right)$ is compactly supported.
The integral equals $$\int_{a \in {{\mathbf{Q}}_p}} f_1(\eta {\left( \begin{smallmatrix} 1 & a \\ 0 & 1\end{smallmatrix} \right) }) \int_{h \in N_{\operatorname{GL}_2} \backslash \operatorname{GL}_2}W_{\phi_r}\left( \left( \begin{smallmatrix} \det(h) & & \\ & h & \\ & & & 1\end{smallmatrix} \right) u_a \gamma\right) |\det(h)|^{s-1} W_\lambda(h)\, \mathrm{d}h\, \mathrm{d}a,$$ where $u_a = \iota( \eta {\left( \begin{smallmatrix} 1 & a \\ 0 & 1\end{smallmatrix} \right) }, 1)$.
Note that if $(h_1, h_2) \in H$ with $h_1$ in the big Bruhat cell, then there exists $h \in \operatorname{GL}_2$, uniquely determined modulo $N_{\operatorname{GL}_2}$, and $a \in {{\mathbf{Q}}_p}$ such that $$N_H Z_G \cdot (h_1, h_2) = N_H Z_G \cdot \left( {\left( \begin{smallmatrix} \det h & \\ & 1\end{smallmatrix} \right) }, h\right) \cdot \left( \eta {\left( \begin{smallmatrix} 1 & a \\ & 1\end{smallmatrix} \right) }, 1\right).$$ Making this change of variables in the integral, and using the fact that $f_1$ transforms on the left under ${\left( \begin{smallmatrix} \det h & \\ & 1\end{smallmatrix} \right) }$ by $(\det h)^{s}$, the lemma follows. (The $s-1$ in the exponent comes from the modulus character, because we are writing $N_H Z_G \backslash H$ as the product of two factors that do not commute.)
The map from the model of $\pi_p = \operatorname{Ind}_B^G (\alpha)$ as an induced representation to the Whittaker model ${\mathcal{W}}(\pi_p)$ is given explicitly as follows: for $\phi \in \pi_p$ we have $$W_\phi(g) = \int_{N_B({{\mathbf{Q}}_p})} \phi_r(J ng) \psi(-m-x)\, \mathrm{d}n, \ \ n = \left(\smallmatrix 1 & & & \\ & 1 & m & \\ & & 1 & \\ & & & 1\endsmallmatrix\right)
\left(\smallmatrix 1 & x & y & z \\ & 1 & & y \\ & & 1 & -x \\ & & & 1\endsmallmatrix\right) = n(m)n(x,y,z).$$ Here $J$ is as in §\[sect:groups\]. We have $n(x,y,z) {\left( \begin{smallmatrix} \det h\\ &h \\ &&1\end{smallmatrix} \right) } = {\left( \begin{smallmatrix} \det h\\ &h \\ &&1\end{smallmatrix} \right) } n(u, v, w)$, where $(u, v) = (x, y) \cdot h (\det h)^{-1}$, $w = z/\det h$; thus $x = \langle (u, v), (0, 1)h \rangle$, where $\langle,\rangle : {{\mathbf{Q}}_p}^2\times{{\mathbf{Q}}_p}^2\rightarrow{{\mathbf{Q}}_p}$ is the pairing $\langle (a,b), (c,d) \rangle = ad-bc$. Hence we have $$\begin{split}
W_{\phi_r} & (\left(\smallmatrix \det(h) & & \\ & h & \\ & & & 1\endsmallmatrix\right) u_a \gamma) \\
& = |\det(h)|^2 \int_{N_{B}}\phi_r\left(J n(m)
\left(\smallmatrix \det(h) & & \\ & h & \\ & & & 1\endsmallmatrix\right) n(u,v,w) u_a\gamma\right)
\psi(-m-\langle (u, v), (0, 1)h \rangle)\, \mathrm{d}m\, \mathrm{d}u\, \mathrm{d}v\, \mathrm{d}w.
\end{split}$$
We note the following matrix identity. Let $u(m) = {\left( \begin{smallmatrix} 1 & m \\ 0 & 1\end{smallmatrix} \right) }$. Then $$\begin{gathered}
J n(m) \left(\smallmatrix \det(h) & & \\ & h & \\ & & & 1\endsmallmatrix\right) n(u,v,w) u_a\gamma = \\
=\iota(\eta, \eta) \cdot \iota(1, u(m)) \cdot \iota({\left( \begin{smallmatrix} \det h & \\ & 1\end{smallmatrix} \right) }, h) \cdot n(u, v, w) \cdot \iota(\eta u(a), 1) \cdot \gamma \\
= \left(\smallmatrix -1 & & \\ & \eta u(m)h & \\ & & & -\det(h)\endsmallmatrix\right) \cdot \iota(\eta^{-1}, 1)\cdot n(u, v, w) \cdot \iota(\eta u(a), 1) \gamma.
\end{gathered}$$ One verifies that $$\iota(\eta^{-1}, 1)\cdot n(u, v, w) \cdot \iota(\eta u(a), 1) \cdot \gamma
=
\left( \begin{smallmatrix}
1& & -a& a \\ &1 &a &-a\\&&1 \\&&&1
\end{smallmatrix} \right)
\cdot
\left( \begin{smallmatrix}
wa + 1 & -ua & -(u+w)a^2 + (1-v)a & (u + w)a^2\\
-wa + (-v + 1) &ua + 1& (u + w)a^2 + (2v - 2)a& (-u - w)a^2 + (-v + 1)a\\
u &0& -ua + 1& ua\\
u - w &u &wa + (v - 1) &-wa + 1\\
\end{smallmatrix} \right) $$
If $k_a(u, v, w)$ denotes the last matrix above, then we deduce that $$J n(m)\left(\smallmatrix \det(h) & & \\ & h & \\ & & & 1\endsmallmatrix\right) n(u,v,w)
u_a\gamma \in P({{\mathbf{Q}}_p}) k_a(u, v, w).$$ It follows from the definition of $\phi_r$ that $\phi_r(J n(m)\left(\smallmatrix \det(h) & & \\ & h & \\ & & & 1\endsmallmatrix\right) n(u,v,w) u_a\gamma) = 0$ unless $k_a(u,v,w) \in PK_r$.
Let $C = \max(-v_p(a), 0)$. If $r {\geqslant}\max(2C, C + 1)$, then $k_a(u,v,w) \in P({{\mathbf{Q}}_p}) K_r$ implies $k_a(u, v, w) \in K_r$.
Suppose that $k_a(u, v, w) \in P K_r$. By Lemma \[support-lem\], we must have $$1-aw\in {{\mathbf{Q}}_p}^\times\quad\text{and}\quad
\tfrac{u}{1-aw}, \tfrac{w}{1-aw}, \tfrac{v-1+aw}{1-aw}\in p^r{{\mathbf{Z}}_p}.$$ If $\tfrac{w}{1-aw} = p^r b$ for $b \in {{\mathbf{Z}}_p}$, then $w(1 + p^r a b) = p^rb$. Since $p^r a \in p {{\mathbf{Z}}_p}$ by assumption, we can conclude that $(1 + p^r a b)$ is a unit and hence $w \in p^r {{\mathbf{Z}}_p}$. Thus $1 - aw$ is also a unit, and we deduce that $u \in p^r {{\mathbf{Z}}_p}$. Since $v = 1 -aw \bmod p^r$ we have $v = 1 \bmod p^{r-C}$. In particular, $ua^2$,$wa^2$ and $(v-1)a$ all have valuation at least $r - 2C {\geqslant}0$; hence $k_a(u, v, w)$ is integral (and even congruent to the identity modulo $p^{r-2C}$).
We suppose that $r {\geqslant}\max(2C, C + 1)$, and let $R(a,r)$ denote the set of $(u, v, w)$ such that $k_a(u,v,w) \in K_r$. Then we obtain the formula $$\begin{gathered}
W_{\phi_r} (\left(\smallmatrix \det(h) & & \\ & h & \\ & & & 1\endsmallmatrix\right) u_a \gamma) \\
= |\det(h)|^2 \int_{R(a,r)}\int_{m} \phi_r\left( \left(\smallmatrix -1 & & \\ & \eta u(m)h & \\ & & & -\det(h)\endsmallmatrix\right)\left( \begin{smallmatrix}
1& & -a& a \\ &1 &a &-a\\&&1 \\&&&1
\end{smallmatrix} \right) \right) \psi(-m-\langle (u, v), (0, 1)h \rangle)\, \mathrm{d}m\, \mathrm{d}u\, \mathrm{d}v\, \mathrm{d}w.
\end{gathered}$$
Since the argument of $\phi_r$ lies in $P({{\mathbf{Q}}_p})$, we have $$\phi_r\left( \left(\smallmatrix -1 & & \\ & \eta u(m)h & \\ & & & -\det(h)\endsmallmatrix\right)\left( \begin{smallmatrix}
1& & -a& a \\ &1 &a &-a\\&&1 \\&&&1
\end{smallmatrix} \right) \right) = p^{3r}|\det h|^{-1} \xi\Big(\eta u(m) h u(a)\Big)$$ where $\xi$ is the spherical vector of $\tau$, as before. (Note that $\theta(-1) = 1$, since $\theta$ is assumed to be unramified.) Thus the inner integral over $m$ reduces to the Whittaker transform of $\xi$, evaluated at $h u(a)$; so we can write this as $$W_{\phi_r} (\left(\smallmatrix \det(h) & & \\ & h & \\ & & & 1\endsmallmatrix\right) u_a \gamma) =
p^{3r} |\det(h)| \int_{(u,v,w) \in R(a,r)}W_\xi(hu(a)) \psi(-\langle (u, v), (0, 1)h \rangle)\, \mathrm{d}u\, \mathrm{d}v\, \mathrm{d}w.$$ If $\Phi_{a,r}$ denotes the Schwartz function $\Phi_{a,r}(u, v) = \operatorname{vol}\{ w: (u, v, w) \in R(a,r)\}$, then we can collapse this down to $$W_{\phi_r} (\left(\smallmatrix \det(h) & & \\ & h & \\ & & & 1\endsmallmatrix\right) u_a \gamma) =
p^{3r} |\det(h)|\cdot W_\xi(hu(a))\cdot \hat{\Phi}_{a, r}((0,1)h),$$ where the Fourier transform for Schwartz functions is defined as above.
The function $\Phi_{a,r}(u, v)$ is rather explicit, and depends only on the valuation of $a$. If $C$ is as above, then the set $R(a, r)$ is given by $\{u = 1 \bmod p^r, v = 1 \bmod p^{r-C}, w = a^{-1}(1-v) \bmod p^{r+C}\}$, so $\Phi_{a,r} = p^{-(r+C)} {\operatorname{ch}}( (p^r{{\mathbf{Z}}_p}) \times (1 + p^{r-C} {{\mathbf{Z}}_p}))$.
Putting things together we have:
Suppose $\Phi_1(1, -)$ is supported in $p^{-N}{{\mathbf{Z}}_p}$, and let $r {\geqslant}\max(2N, N+1)$. Then we have $$\eqref{eq:localintegral} = p^{3r}\int_{a \in {{\mathbf{Q}}_p}} f_1(\eta {\left( \begin{smallmatrix} 1 & a \\ 0 & 1\end{smallmatrix} \right) })\int_{h \in N_{\operatorname{GL}_2} \backslash \operatorname{GL}_2} W_{u(a)\xi} (h) \cdot w_2(h)\cdot \hat{\Phi}_{a, r}((0,1)h) |\det(h)|^s \, \mathrm{d}h\, \mathrm{d}a,$$ a finite linear combination of Rankin–Selberg integrals.
### Application of the functional equation
By the functional equation for Rankin–Selberg integrals, due to Jacquet [@jacquet72 Theorem 14.7 (3)], for any $w_1 \in {\mathcal{W}}(\xi)$ and $w_2 \in {\mathcal{W}}(\sigma)$, and any $\Phi \in {\mathcal{S}}({{\mathbf{Q}}_p}^2)$, we have $$\begin{gathered}
\int_{N_2 \backslash \operatorname{GL}_2} w_1(h) w_2(h) \hat{\Phi}((0,1)h) |\det(h)|^{s} \, \mathrm{d}h =
\\ \frac{1}{\gamma(\tau \times \sigma, s)}\int_{N_2 \backslash \operatorname{GL}_2} \frac{w_1(h) w_2(h)}{ (\chi_\tau\chi_\sigma)(\det h)} \Phi((0,1)h) |\det(h)|^{1-s} \, \mathrm{d}h,
\end{gathered}$$ where $$\gamma(\tau \times \sigma, s) = \frac{L(\tau^\vee \times \sigma^\vee, 1-s)}{L(\tau \times \sigma, s)} \epsilon(\tau\times\sigma, s)$$ is Jacquet’s local $\gamma$-factor. (More precisely, the integrals on the left and right sides are convergent for ${\operatorname{Re}}(s) \gg 0$, resp ${\operatorname{Re}}(s) \ll 0$, and both have meromorphic continuation to all $s$ and are equal as meromorphic functions.) Noting that $\chi_\tau \chi_\sigma = \chi / \theta$, and taking $\Phi = \Phi_{a, r}$, we have $$\eqref{eq:localintegral} = \frac{p^{3r}}{\gamma(\tau \times \sigma, s)} \int_{a \in {{\mathbf{Q}}_p}} f_1(\eta {\left( \begin{smallmatrix} 1 & a \\ 0 & 1\end{smallmatrix} \right) })\int_{h \in N_2 \backslash \operatorname{GL}_2} W_{u(a)\xi} (h) w_2(h) \Phi_{a, r}((0,1)h) (\theta/\chi)(\det h)|\det(h)|^{1-s} \, \mathrm{d}h\, \mathrm{d}a.$$
The Schwartz function $\Phi_{a, r}$ has total integral $p^{-3r}$, and as $r \to \infty$, its support becomes concentrated in smaller and smaller neighbourhoods of $(0, 1)$. Hence the limiting value is given by $$\eqref{eq:localintegral} = \tfrac1{\gamma(\tau \times \sigma, s)}\int_{a \in {{\mathbf{Q}}_p}} f_1(\eta {\left( \begin{smallmatrix} 1 & a \\ 0 & 1\end{smallmatrix} \right) })\int_{x \in {{\mathbf{Q}}_p}^\times}W_{u(a)\xi} ({\left( \begin{smallmatrix} x & 0 \\ 0 & 1\end{smallmatrix} \right) }) w_2({\left( \begin{smallmatrix} x & 0 \\ 0 & 1\end{smallmatrix} \right) }) \frac{\theta(x)}{|x|^s \chi(x)} \, \mathrm{d}^\times x\, \mathrm{d}a.$$ We have $W_{u(a)\xi} ({\left( \begin{smallmatrix} x & 0 \\ 0 & 1\end{smallmatrix} \right) }) = \psi(xa) W_{\xi} ({\left( \begin{smallmatrix} x & 0 \\ 0 & 1\end{smallmatrix} \right) })$, and we compute $$\int_a f_1(\eta {\left( \begin{smallmatrix} 1 & a \\ 0 & 1\end{smallmatrix} \right) }) \psi(xa)\, \mathrm{d}a = \chi(x) |x|^{s-1} W^{\Phi_1}({\left( \begin{smallmatrix} x & 0 \\ 0 & 1\end{smallmatrix} \right) }).$$
This concludes the proof of Proposition \[prop:local-zeta-evaluation\] assuming that $\Phi_1(0, -)$ is identically 0.
### Conclusion of the proof
We have proved the formula of Proposition \[prop:local-zeta-evaluation\] for functions $\Phi_1$ satisfying $\Phi_1(0, x) = 0$ for all $x$, which is a more restrictive condition than stated in the proposition. We now reduce the general case to this.
Consider the auxiliary zeta integral $$\widetilde{Z}_p(\varphi, \lambda, \Phi, s) \coloneqq
\int_{(Z_GN_H\backslash H)({{\mathbf{Q}}_p})}
W_{\varphi}(h)
f^{\Phi_1}(h_1;\chi^{-1}, s)
W_{\lambda_v}(h_2)
\chi(\det h)^{-1}
\, \mathrm{d} h.$$ As before, this has meromorphic continuation to all $s$; and the functions $Z(\dots)$ and $\widetilde{Z}(\dots)$ are related by the following local functional equation [@Soudry-GSpGL]: $$Z_p(\varphi, \lambda, \Phi, s) = \gamma(\pi \times \sigma, s)^{-1} \widetilde{Z}_p(\varphi, \lambda, \hat\Phi, 1-s),$$ where $$\gamma(\pi \times \sigma, s) = \gamma(\tau \times \sigma,s)\gamma(\tau\times\sigma \times \theta, s).$$ Since $\widetilde{Z}_p$ can be obtained from $Z_p$ by replacing the representation $\sigma$ with $\sigma \otimes \chi^{-1}$, we deduce that if $\hat\Phi(0, x) = 0$ for all $x$, then $$\begin{aligned}
\widetilde{Z}_p(\varphi, \lambda, \hat\Phi, 1-s) &=
\frac{1}{\gamma(\tau \times \sigma \times \chi^{-1}, 1-s)}
\int_{{{\mathbf{Q}}_p}^\times} W_\xi({\left( \begin{smallmatrix} x & 0 \\ 0 & 1\end{smallmatrix} \right) }) W^{\hat\Phi_1}\left({\left( \begin{smallmatrix} x & 0 \\ 0 & 1\end{smallmatrix} \right) }; \chi^{-1}, 1-s\right)W_\lambda({\left( \begin{smallmatrix} x & 0 \\ 0 & 1\end{smallmatrix} \right) }) \tfrac{\theta(x)}{\chi(x) |x|} \,\mathrm{d}^\times x\\
&= \frac{1}{\gamma(\tau \times \sigma \times \chi^{-1}, 1-s)}
\int_{{{\mathbf{Q}}_p}^\times} W_\xi({\left( \begin{smallmatrix} x & 0 \\ 0 & 1\end{smallmatrix} \right) }) W^{\Phi_1}\left({\left( \begin{smallmatrix} x & 0 \\ 0 & 1\end{smallmatrix} \right) }; \chi, s\right)W_\lambda({\left( \begin{smallmatrix} x & 0 \\ 0 & 1\end{smallmatrix} \right) }) \tfrac{\theta(x)}{|x|} \,\mathrm{d}^\times x
\end{aligned}$$ where the second equality comes from the functional equation relating $W^{\Phi_1}$ and $W^{\hat\Phi_1}$. Since $\chi = \chi_\tau \chi_\sigma \theta$, we have $\gamma(\tau\times\sigma\times \theta, s) \gamma(\tau\times\sigma\times \chi^{-1}, 1-s) = 1$. So we have shown that the formula of Proposition \[prop:local-zeta-evaluation\] holds for all $\Phi_1$ such that $\hat\Phi_1(0, x) = 0$ for all $x$. Since $$(\hat\Phi_1)'(x, y) = \Phi_1'(y, x),$$ this is equivalent to requiring that $\Phi_1'(x, 0) = 0$ for all $x$. Since any Schwartz function $h$ on ${{\mathbf{Q}}_p}^2$ such that $h(0, 0) = 0$ is a linear combination of functions vanishing identically along $\{0\} \times {{\mathbf{Q}}_p}$ and ${{\mathbf{Q}}_p}\times \{0\}$, this shows that our formula holds for all $\Phi_1$ such that $\Phi_1'(0, 0) = 0$, completing the proof of Proposition \[prop:local-zeta-evaluation\].
Local integrals at $\infty$
---------------------------
We now consider the local integral at a real infinite place $v = \infty$. For simplicity, we shall now assume that $\psi_F(x) = \exp(-2\pi i x)$ for $x \in {\mathbf{R}}$.
### Siegel sections
We shall need to compute the functions $f^{\Phi}$ where $\Phi = \Phi_\infty^{(k)} \in {\mathcal{S}}({\mathbf{R}}^2, {\mathbf{C}})$ is the function given by $$\Phi_{\infty}^{(k)}(x, y) \coloneqq 2^{1-k} (x + iy)^k e^{-\pi(x^2 + y^2)}$$ for some $k \in {\mathbf{Z}}_{{\geqslant}1}$. (The factor $2^{1-k}$ is convenient for comparisions with algebraic Eisenstein series, as we shall shortly see.) We find readily that $$f^{\Phi}\left( {\left( \begin{smallmatrix} a & \\ & a^{-1}\end{smallmatrix} \right) }; \chi, s\right) =
\begin{cases}
0 & \text{if $\chi(-1) \ne (-1)^k$},\\
2^{1-k} i^k \Gamma(s + \tfrac{k}{2}) \pi^{-(s + k/2)} a^{2s} & \text{otherwise}.
\end{cases}$$
### Moriyama’s formula
A formula for the archimedean Whittaker function of a vector in the lowest $K$-type subspace of a generic discrete-series representation $\Pi_\infty$ is given in [@moriyama04 §3.2]. Let $(\lambda_1, \lambda_2)$ be integers such that $1-\lambda_1 < \lambda_2 < 0$. Then there is a unique generic discrete-series representation of $\operatorname{GSp}_4({\mathbf{R}})$ whose central character has finite order and which is isomorphic as an $\operatorname{Sp}_4({\mathbf{R}})$-representation to the direct sum $D_{(\lambda_1, \lambda_2)} \oplus D_{(-\lambda_2, -\lambda_1)}$, in Moriyama’s notation. For $0 {\leqslant}k {\leqslant}d$, we let $v_k$ be the $k$-th standard basis vector of the lowest $K$-type of $D_{(-\lambda_2, -\lambda_1)}$.
What we shall need here is Proposition 8 of *op.cit.*, which gives a formula for the image of $v_k$ under the canonical transformation from the Whittaker model to the Bessel model of $\Pi_\infty^W$. Recall that we have defined $$B_{v_k}(g, s) \coloneqq \int_{{\mathbf{R}}^\times} \int_{{\mathbf{R}}} W_{v_k}\left(\left( \begin{smallmatrix} a \\ &a \\ &x&1\\ &&&1 \end{smallmatrix} \right) w_2 g\right) |a|^{s-\tfrac{3}{2}}\chi_2(a)^{-1}\, \mathrm{d}x\, \mathrm{d}^\times a,\qquad w_2 = \left( \begin{smallmatrix} 1\\&&1 \\ &-1 \\&&&1\end{smallmatrix} \right) .$$
For ${\operatorname{Re}}(t)$ sufficiently large, we have $$\int_0^\infty B_{v_k}\left({\left( \begin{smallmatrix} y\\ &y \\ &&1 \\ &&& 1\end{smallmatrix} \right) }, s\right) y^{t-\tfrac{3}{2}} \, \mathrm{d}^\times y
\\=
C \cdot
\frac{(-1)^k L(\Pi_\infty, s) L(\Pi_\infty, t)}
{\pi^t \Gamma(\tfrac{t - s+ \lambda_2 + k + 1}{2})\Gamma(\tfrac{t + s + \lambda_1 - k}{2})},$$ where $C$ is a constant depending on $\lambda_1$ and $\lambda_2$ (but not on $k, s, t$), and $L(\Pi_\infty, s) = \Gamma_{\mathbf{C}}(s + \tfrac{\lambda_1 +\lambda_2 - 1}{2})\Gamma_{\mathbf{C}}(s + \tfrac{\lambda_1 -\lambda_2 - 1}{2})$, $\Gamma_{\mathbf{C}}(s) = 2(2\pi)^{-s} \Gamma(s)$.
Moriyama states his formula slightly differently, in the form of an inverse Mellin transform. The result above follows by applying the “forward” Mellin transform to both sides of Moriyama’s formula, in the same way as the proof of Proposition 5.12 of [@lemma17] (which is essentially the same argument as ours but with a different normalisation of the Bessel function). Moriyama also has a factor $\binom{d}{k}$, which does not appear in our formulae, owing to a different convention for the standard basis of the lowest $K$-type.
We shall apply this with $(\lambda_1, \lambda_2) = (r_1 + 3, -r_2-1)$, so that $d = r_1 + r_2 + 4$. The vector $v_k$ is then an eigenvector for the action of the diagonal torus in $K_{G, \infty}^\circ$, which is just $K_{H, \infty}^\circ$, with weight $(r_2 + 1 - k, -r_1-3 + k)$. We shall let $(k_1, k_2)$ be integers ${\geqslant}1$ summing to $r_1 - r_2 + 2$, and take $k = r_1 + 3 - k_2 = r_2 + 1 + k_1$; thus $v_k$ has $K_H^\circ$-type $(-k_1, -k_2)$, meaning it can pair non-trivially against a pair of holomorphic modular forms of weight $(k_1, k_2)$. Finally, we shall take $s = s_1 - s_2 + \tfrac{1}{2}$ and $t = s_1 + s_2 - \tfrac{1}{2}$. Note that the right-hand side of Moriyama’s formula is now $$(-1)^{k_1 + r_2 + 1} C \cdot
\frac{L(\Pi_\infty, s_1 - s_2 + \tfrac{1}{2}) L(\Pi_\infty, s_1 + s_2 - \tfrac{1}{2})}
{\pi^{s_1 + s_2 + \tfrac12} \Gamma(s_1 + \tfrac{k_1}{2})\Gamma(s_2 + \tfrac{k_2}{2})}.$$
On the other hand, if one chooses $\Phi_{i, \infty} = \Phi_{\infty}^{(k_i)}$, then the integrand in the local Bessel-model integral is right $K_{H, \infty}$-invariant; so by the Iwasawa decomposition, $H = N_H T_H K_H$ where $T_H$ is the diagonal torus, we conclude that the left-hand side of Moriyama’s formula is exactly the integral $Z_\infty\left(v_k, \Phi_{\infty}^{(k_1)}, \Phi_{\infty}^{(k_2)}, s_1, s_2\right)$ up to a sign and the factor $f^{\Phi_1}(1) f^{\Phi_2}(1)$, which corresponds precisely to the Gamma-factors in the denominator above.
If we instead consider the $\operatorname{GSp}_4 \times \operatorname{GL}_2$ integral, then the Whittaker function associated to the holomorphic vector $\lambda \in \sigma_\infty$ coincides with the Whittaker function of $\Phi_{\infty}^{(\ell)}$ at $s_2 = \tfrac{\ell}{2}$. So we see that in this case the local integral at $\infty$ is again equal to the expected value, namely $L(\Pi_\infty \times \sigma_\infty, s)$.
Integral formulae for L-functions II: global theory {#sect:globalintegrals}
===================================================
We now let $F$ be an arbitrary number field, with adèle ring ${\mathbf{A}}$. Let $\psi_F = \prod_v \psi_{F_v}$ be a non-trivial additive character of ${\mathbf{A}}/F$, which we regard as a character of $N({\mathbf{A}})$ as before.
Globally generic $\pi$
----------------------
Let $\pi$ be a cuspidal automorphic represenation of $G({\mathbf{A}})$. For each $\varphi\in\pi$ we define its *Whittaker transform* with respect to $\psi$: $$W_\varphi(g) = \int_{N_G(F)\backslash N_G({\mathbf{A}})} \varphi(ng) \psi(n^{-1}) \mathrm{d}n.$$ The representation $\pi$ is [*globally generic*]{} if some $W_\varphi(g)$ is non-zero. If $\pi$ is globally generic, it can thus be modeled as a space of functions $W: G({\mathbf{A}}) \to {\mathbf{C}}$ satisfying $W(ng) = \psi(n) W(g)$ for all $n \in N_G({\mathbf{A}})$.
We fix an identification $\pi = \sideset{}{'}{\bigotimes_v} \pi_v$ of $\pi$ with a restricted tensor product of irreducible local representations. If $\pi$ is globally generic then each $\pi_v$ is clearly also generic, that is, $\pi_v$ can also be modelled in a space of functions $W: G(F_v)\to {\mathbf{C}}$ satisfying $W(ng) = \psi(n) W(g)$ for all $n\in N_G(F_v)$.
Note that our definition of “globally generic” is strictly stronger than requiring that each $\pi_v$ be generic, and depends on the realisation of $\pi$ as a space of functions on $G(F) \backslash G({\mathbf{A}})$ rather than on the abstract isomorphism class of $\pi$.
Assuming $\pi$ is globally generic, we fix Whittaker models $\pi_v\stackrel{\sim}{\to}\mathcal{W}_v$, $\phi_v\mapsto W_{\phi_v}$ for all places $v$. These choices can and shall be made in such a way that:
- If $\varphi = \otimes_v \varphi_v$, then we have $W_\varphi(g) = \prod_v W_{\phi_v}(g_v)$.
- If $v$ is a finite place such that $\pi_v$ and $\psi_{F_v}$ are unramified, and $\phi^0_v \in \pi_v$ is the basis of the unramified vectors that was implicitly fixed in the definition of the restricted tensor product $\pi = \sideset{}{'}{\bigotimes_v} \pi_v$, then $W_{\phi^0_v}(1) = 1$.
Global Eisenstein series {#Eis-GL2}
------------------------
For a Schwartz function $\Phi: {\mathbf{A}}^2 \to {\mathbf{C}}$, and $\chi$ a unitary Grössencharacter, we define a global Siegel section by $$f^{\Phi}(g; \chi, s) \coloneqq \|\det g\|^s \int_{{\mathbf{A}}^\times} \Phi_v((0, a)g) \chi(a) \|a\|^{2s}\, \mathrm{d}^\times a,$$ We may then form the series $$E^\Phi(g; \chi, s) : = \sum_{\gamma \in B(F) \backslash \operatorname{GL}_2(F)} f^\Phi(\gamma g; \chi, s).$$ (cf. [@jacquet72 §19]). The sum converges absolutely and uniformly on any compact subset of $\{ s: {\operatorname{Re}}(s) > 1\}$, and defines a function on the quotient $\operatorname{GL}_2(F) \backslash \operatorname{GL}_2({\mathbf{A}})$ that transforms under the center by $\chi^{-1}$. It has a meromorphic continuation in $s$.
Note that $g \cdot E^{\Phi}(-; \chi, s) = \|\det g\|^s E^{g \cdot \Phi}(-; \chi, s)$.
We define the Whittaker transform of $E^\Phi(g; \chi, s)$ with respect to $\bar\psi_F$ to be $$W^\Phi(g;\chi,s): = \int_{N_{\operatorname{GL}_2}(F)\backslash N_{\operatorname{GL}_2}({\mathbf{A}})} E^\Phi(ng; \chi, s) \psi_F(x) \mathrm{d}n, \ \ n= \left(\smallmatrix 1 & x \\ 0 & 1 \endsmallmatrix\right).$$ If ${\operatorname{Re}}(s)>1$ this unfolds to $$W^\Phi(g;\chi,s) = \int_ {N_{\operatorname{GL}_2}({\mathbf{A}})} f^\Phi(\eta ng; \chi, s) \psi_F(x) \mathrm{d}n, \ \ \eta = \left(\smallmatrix 0 & 1 \\ -1 & 0 \endsmallmatrix\right).$$
If $\Phi(g)= \prod_v\Phi_{v}(g_v)$ with $\Phi_{v}$ a Schwartz function on $F_v^2$ and $\Phi_{v}$ the characteristic function $\Phi_{v}^0$ of $\mathcal{O}_{F_v}^2$ for almost all finite $v$, then we have factorisations $$f^\Phi(g;\chi,s) = \prod_v f^{\Phi_v}(g_v;\chi_v,s),\qquad
W^\Phi(g;\chi,s) = \prod_v W^{\Phi_v}(g_v;\chi_v,s)$$ where the local integrals $f^{\Phi_v}$ and $W^{\Phi_v}$ are as defined above. Furthermore, the local Whittaker transforms $W^{\Phi_v}(g;\chi_v,s)$ converge for all values of $s$ and are holomorphic as functions of $s$, and (for a given $g$ and $\Phi$) all but finitely many of these are 1.
The global integrals {#global-zeta-int}
--------------------
Let $\chi_1, \chi_2$ be two unitary Grössencharacters of ${\mathbf{A}}^\times$ such that $\chi_1 \chi_2 = \chi_\pi$. For $\varphi \in \pi$ and $\Phi_1, \Phi_2 \in \mathcal{S}({\mathbf{A}}^2)$, define $$Z(\varphi, \Phi_1, \Phi_2, s_1, s_2) \coloneqq
\int_{Z_G({\mathbf{A}})H(F) \backslash H({\mathbf{A}})} \varphi(\iota(h)) E^{\Phi_1}(h_1; \chi_1, s_1) E^{\Phi_2}(h_2; \chi_2, s_2)\, \mathrm{d}h,$$ where $H = \operatorname{GL}_2 \times_{\operatorname{GL}_1} \operatorname{GL}_2$ and the $E^{\Phi_i}(h_i; \chi_i, s_i)$ are Eisenstein series as in Section \[Eis-GL2\]. This integral converges absolutely and is holomorphic at all values of $s_1$ and $s_2$ such that neither Eisenstein series has a pole. This is because the restriction of the cuspform $\varphi$ to $H({\mathbf{A}})$ is rapidly decreasing.
We also consider a second integral associated to $\pi$ and an auxiliary cuspidal automorphic representation $\sigma$ of $\operatorname{GL}_2({\mathbf{A}})$. We now let $\chi = \chi_\pi \chi_\sigma$, and for $\varphi \in \pi$ and $\lambda \in \sigma$, we set $$Z(\varphi, \lambda, \Phi, s) \coloneqq
\int_{Z_G({\mathbf{A}})H(F) \backslash H({\mathbf{A}})} \varphi(\iota(h)) E^{\Phi}(h_1; \chi, s)\lambda(h_2)\, \mathrm{d}h.$$ As before, this is holomorphic away from the poles of the Eisenstein series $E^{\Phi}$.
We now suppose that the test data $(\varphi, \Phi_1, \Phi_2)$ for the first integral, and $(\varphi, \lambda, \Phi)$ for the second, are products of local data $\varphi = \bigotimes_v \varphi_v$ etc; and for each place $v$ of $F$, we write $Z_v(\dots)$ for the local integrals of the previous section at the place $v$. As a straightforward consequence of Proposition \[local-zeta-prop\] and Langlands’ general results on the convergence of automorphic $L$-functions, we have the following Euler product factorisation:
\[global-zeta-prop\] (i) Let $s_2$ be fixed. There exists a positive real number $R'$, possibly depending on $\pi$ and $s_2$, such that for ${\operatorname{Re}}(s_1) > R'$, the product $\prod_v Z_v(\varphi_v, \Phi_{1, v}, \Phi_{2, v}, s_1, s_2)$ converges absolutely. For such $s_1$ we have $$\begin{split}
Z(\varphi, \Phi_1, \Phi_2, s_1, s_2) & = \prod_v Z_v(\varphi_v, \Phi_{1, v}, \Phi_{2, v}, s_1, s_2) \\
& = L^S(\pi_v, s_1 + s_2 - \tfrac12) L^S(\pi_v \otimes \chi_2^{-1}, s_1 - s_2 + \tfrac12) \prod_{v\in S} Z_v(\varphi_v, \Phi_{1, v}, \Phi_{2, v}, s_1, s_2),
\end{split}$$ where $S$ is any finite set of places of $F$ including all the archimedean places and all finite places where $\pi$ or the $\chi_i$ are ramified, and is such that $\Phi_{i, v} = \Phi_{i, v}^0$, $\varphi_v = \varphi^0_v$ for all $v\notin S$.
\(ii) There exists a positive real number $R'$, possibly depending on $\pi$ and $\sigma$, such that for ${\operatorname{Re}}(s) > R'$ the product $\prod_v Z_v(\varphi_v, \lambda_v, \Phi_v, s)$ converges absolutely, and for such $s$ we have $$\begin{split}
Z(\varphi, \lambda, \Phi, s) & = \prod_v Z_v(\varphi_v,\lambda_v, \Phi_v, s) \\
& = L^S(\pi_v \otimes \sigma_v, s) \prod_{v\in S} Z_v(\varphi_v, \lambda_v, \Phi_v, s),
\end{split}$$ for $S$ any sufficiently large finite set of primes as before.
For part (ii), see [@novodvorsky79 Theorem 7], or the introduction of [@Soudry-GSpGL]. Part (i) follows similarly, taking an Eisenstein series in place of the cusp form $\lambda$.
As $Z(\varphi, \Phi_1, \Phi_2, s_1, s_2)$, $L^S(\pi_v, s_1 + s_2 - \tfrac12)$ and $L^S(\pi_v \otimes \chi_2^{-1}, s_1 - s_2 + \tfrac12)$ have meromorphic continuations in $s_1$ (and are even meromorphic in both $s_1$ and $s_2$), for any fixed $s_2$ the equality $$Z(\varphi, \Phi_1, \Phi_2, s_1, s_2) = L^S(\pi_v, s_1 + s_2 - \tfrac12) L^S(\pi_v \otimes \chi_2^{-1}, s_1 - s_2 + \tfrac12) \prod_{v\in S} Z_v(\varphi_v, \Phi_{1, v}, \Phi_{2, v}, s_1, s_2)$$ is an equality of meromorphic functions in $s_1$ (and similarly for the $\operatorname{GSp}_4 \times \operatorname{GL}_2$ integral).
The proof of the preceding proposition also shows that if $\pi$ is not globally generic, then the integrals $Z(\varphi, \Phi_1, \Phi_2, s_1, s_2)$ and $Z(\varphi, \lambda, \Phi, s)$ are identically zero, for all choices of the test data.
Construction of the p-adic L-function {#sect:final}
=====================================
Comparing algebraic and real-analytic Eisenstein series
-------------------------------------------------------
We now assume the number field $F$ is ${\mathbf{Q}}$. A routine unravelling of notations gives the following:
If $\Phi \in {\mathcal{S}}({\mathbf{A}}^2, {\mathbf{C}})$ has the form $\Phi = \Phi_f \cdot \Phi_\infty^{(k)}$, for some $k {\geqslant}1$ and $\Phi_f \in {\mathcal{S}}({{\mathbf{A}}_{\mathrm{f}}}^2, {\mathbf{C}})$, then the Eisenstein series $E^{\Phi}(-; \chi, s)$ on $\operatorname{GL}_2({\mathbf{Q}}) \backslash \operatorname{GL}_2({\mathbf{A}})$ of §\[Eis-GL2\] is related to the Eisenstein series $E^{k, \Phi_f}(-; \chi, s)$ on $\operatorname{GL}_2({{\mathbf{A}}_{\mathrm{f}}}) \times {\mathcal{H}}$ of §\[sect:nonholoEis\] by $$E^{\Phi}\left(g_f {\left( \begin{smallmatrix} y & x \\ 0 & 1\end{smallmatrix} \right) }; \chi, s\right) = y^{k/2} \|\det g_f\|^{s} \cdot E^{k, \Phi_f}(g_f, x + iy; \chi, s)$$ for $x + iy \in {\mathcal{H}}$ and $g_f \in \operatorname{GL}_2({{\mathbf{A}}_{\mathrm{f}}})$.
Periods and algebraicity
------------------------
Let $E$ be a subfield of $\overline{{\mathbf{Q}}}$. An element of $\operatorname{Ind}_{N({{\mathbf{A}}_{\mathrm{f}}})}^{G({{\mathbf{A}}_{\mathrm{f}}})}(\psi)$ (i.e. a $\psi$-Whittaker function on $G$) is said to be *defined over $E$* if it takes values in $E(\mu_\infty)$ and satisfies $$\psi(g)^\sigma = \psi\left( w(\chi(\sigma)) g\right)$$ for all $g \in G({{\mathbf{A}}_{\mathrm{f}}})$ and $\sigma \in \operatorname{Gal}(\overline{{\mathbf{Q}}}/E)$, where $w(x) = \operatorname{diag}(1, x, x^2, x^3)$ for $x \in \hat{{\mathbf{Z}}}^\times$, and $\chi$ is the adelic cyclotomic character.
Given $\eta \in H^2({\Pi_{\mathrm{f}}}) \otimes {\mathbf{C}}$, we say $\eta$ is *Whittaker $E$-rational* if the Whittaker functions of the coordinate projections of the corresponding element of Harris’ space $\mathscr{H}_{L_2}$ are the product of an $E$-defined Whittaker function on $G({{\mathbf{A}}_{\mathrm{f}}})$ and the *standard Whittaker function* at $\infty$, which is the unique function that makes the constant $C$ in Moriyama’s formula equal 1.
It follows easily from the definitions that the space of Whittaker-$E$-rational classes is exactly $\Omega^W(\Pi) \cdot \left( H^2({\Pi_{\mathrm{f}}}) \otimes E\right)$, for some nonzero constant $\Omega^W(\Pi) \in {\mathbf{C}}^\times$, the *Whittaker period* of $\Pi$, well-defined modulo $E^\times$.
The $\operatorname{GSp}_4$ $L$-function
---------------------------------------
\[prop:Lfcn1\] Let $R = \Lambda_L({{\mathbf{Z}}_p}^\times \times {{\mathbf{Z}}_p}^\times)$. There is an element $\mathcal{L}_p(\Pi, \mathbf{j}_1, \mathbf{j}_2) \in R$ whose specialisation at a locally-algebraic point $x = (a_1 + \rho_1, a_2 + \rho_2)$ of $\operatorname{Spec}R$, such that $d = r_1 - r_2 {\geqslant}a_1 {\geqslant}a_2 {\geqslant}0$ and $(-1)^{a_1 + a_2} \rho_1(-1) \rho_2(-1) = -1$, is given by $$\frac{R_p(\Pi, \rho_1, a_1) R_p(\Pi, \rho_2, a_2) \Lambda(\Pi \otimes \rho_1, \tfrac{1-d}{2} + a_1) \Lambda(\Pi \otimes \rho_2, \tfrac{1-d}{2} + a_2)}{\Omega^W(\Pi)},$$ where $$R_p(\Pi, a, \rho) =
(-1)^a \cdot \begin{cases}
\left(1 - \tfrac{p^{a+r_2 + 1}}{\alpha} \right) \left(1 - \tfrac{p^{a+r_2 + 1}}{\beta}\right) \left(1 - \frac{\gamma}{p^{a + r_2 + 2}} \right) \left(\frac{\delta}{p^{a + r_2 + 2}} \right) & \text{if $\rho_i$ is trivial},\\[2ex]
\frac{1}{ \epsilon(\rho)^2} \left(\frac{p^{2a + 2r_2 + 4}}{\alpha\beta} \right)^{m} & \text{if $\rho$ has conductor $p^m > 1$},
\end{cases}$$ where $\alpha, \beta, \gamma, \delta$ are the roots of the Hecke polynomial such that $v_p(\alpha\beta) = r_2 + 1$.
If we choose Schwartz functions $\Phi_1^{(p)}$ and $\Phi_2^{(p)}$ on $({{\mathbf{A}}_{\mathrm{f}}}^p)^2$, then we can use these to build $p$-adic families $\mathcal{E}(\Phi_1^{(p)}, \Phi_2^{(p)})$ of Eisenstein series on $H$, as in §\[sect:input1\] above. If we also choose $\eta \in H^2({\Pi_{\mathrm{f}}})$ which is invariant under $\operatorname{Kl}(p)$ and lands in the ordinary eigenspace, then we can form a $p$-adic analytic function interpolating the cup-products of $\eta$ with specialisations of the family $\mathcal{E}(\Phi_1^{(p)}, \Phi_2^{(p)})$.
Specialising at $(a_1 + \rho_1, a_2 + \rho_2)$, the cup-product is given by a zeta-integral $Z(\dots)$, but with $\pi$ replaced by the twisted representation $\pi \otimes \rho_1$. This factorises as a product of local integrals as above. For finite primes $\ell \ne p$ we may choose a finite linear combination of triples ($\phi_v, \Phi_{1, v}, \Phi_{2, v})$ such that the corresponding local zeta integral is the $L$-factor. This is also true at $\infty$ (for the specific Archimedean data we have chosen), by Moriyama’s results, up to the sign $(-1)^{a_1 - a_2}$. For the place $p$, we have chosen specific local data and evaluated the local integrals explicitly, which gives the local correction factors $R_p(\Pi, \rho, a)$.
If $r_1 - r_2 {\geqslant}1$, then for all finite-order characters $\rho$ of ${{\mathbf{Z}}_p}^\times$, we have $\Lambda(\Pi \otimes \rho, \tfrac{1+d}{2}) \ne 0$.
If $r_1 - r_2 {\geqslant}2$ this is obvious from the convergence of the Euler product. The case $r_1 - r_2 = 1$ follows from results of Jacquet–Shalika [@jacquetshalika76] on non-vanishing of $L$-functions for $\operatorname{GL}_n$ on the abcissa of convergence (the “prime number theorem” for $\operatorname{GL}_n$ $L$-functions), applied to the automorphic representation of $\operatorname{GL}_4$ lifted from $\Pi$.
If $r_1 = r_2$ then we cannot directly establish the non-vanishing, so we must impose it as a hypothesis:
\[hyp:non-vanish\] There exist finite-order characters $\rho_+, \rho_-$ of ${{\mathbf{Z}}_p}^\times$, with $\rho_+(-1) = 1$ and $\rho_-(-1) = -1$, such that $\Lambda(\Pi \otimes \rho_+, \tfrac{1}{2})$ and $\Lambda(\Pi \otimes \rho_-, \tfrac{1}{2})$ are non-zero.
(Theorem A) Suppose that $r_1 - r_2 {\geqslant}1$ or that Hypothesis \[hyp:non-vanish\] holds. Then there exist constants $\Omega^+_{\Pi}$ and $\Omega^-_\Pi$, uniquely determined modulo $E^\times$ and satisfying $\Omega^+_\Pi \Omega^-_\Pi = \Omega_\Pi^W \pmod{E^\times}$; and an element $\mathcal{L}(\Pi, \mathbf{j}) \in \Lambda_L({{\mathbf{Z}}_p}^\times)$, such that $$\mathcal{L}(\Pi, a + \rho) = (-1)^a R_p(\Pi, \rho, a) \frac{\Lambda(\Pi \otimes \rho, \tfrac{1-d}{2} + a)}{\Omega_{\Pi}^{\pm}}$$ for all $(a, \rho)$ with $0 {\leqslant}a {\leqslant}d$, where the sign $\pm$ is given by $(-1)^a \rho(-1)$.
Let $\rho^+$ and $\rho^-$ be any two Dirichlet characters of $p$-power conductor such that $\rho^{+}(-1) = (-1)^d$, $\rho^-(-1) = -(-1)^d$. From the preceding lemma, we see that $\Lambda(\Pi \otimes \rho^{+}, \tfrac{1+d}{2})$ and $\Lambda(\Pi \otimes \rho^{-}, \tfrac{1+d}{2})$ are both nonzero. If $d = 1$ then we assume also that both characters $\rho^{\pm}$ are ramified at $p$, which ensures that the local Euler factors $R(\rho^{\pm}, d)$ are non-zero.
Then we may define $$\begin{aligned}
\Omega^{+}(\Pi) &= \Omega^W(\Pi) / [ R_p(\Pi, \rho^-, d)\Lambda(\Pi \otimes \rho^-, \tfrac{1+d}{2}) ] \\
\Omega^-(\Pi) &= \Omega^W(\Pi) / [ R_p(\Pi, \rho^+, d)\Lambda(\Pi \otimes \rho^+, \tfrac{1+d}{2})].
\end{aligned}$$ We can then define $\mathcal{L}(\Pi, \mathbf{j}) = \mathcal{L}(\Pi, \mathbf{j}, d + \rho^-)$ on the component of weight space where $(-1)^\mathbf{j} = +1$, and $\mathcal{L}(\Pi, \mathbf{j}, d + \rho^+)$ on the other component; and the interpolation formula follows easily from the corresponding property of the two-variable $L$-function $\mathcal{L}(\Pi, \mathbf{j}_1, \mathbf{j}_2)$.
It would be interesting to attempt to define an optimal integral normalisation for the periods $\Omega^{\pm}(\Pi)$ (up to $p$-adic units), but we have not attempted to do this.
The $\operatorname{GSp}_4 \times \operatorname{GL}_2$ $L$-function
------------------------------------------------------------------
A similar analysis for the $\operatorname{GSp}_4 \times \operatorname{GL}_2$ integral formula gives the following:
(Theorem B) Let $R = \Lambda_L({{\mathbf{Z}}_p}^\times)$. There is an element $\mathcal{L}_p(\Pi \times \sigma, \mathbf{j}) \in R$ whose specialisation at a locally-algebraic point $x = a + \rho$ of $\operatorname{Spec}R$, with $0 {\leqslant}a {\leqslant}d' = r_1 - r_2 - \ell + 1$, is given by $$R_p(\Pi \times \sigma, a, \rho) \frac{\Lambda^{S}(\Pi \otimes \sigma \otimes \rho, \tfrac{1-d'}{2} + a)}{\Omega_\Pi^W},$$ where $S$ is the set of finite places $\ell$ such that neither $\pi_\ell$ nor $\sigma_\ell$ is principal series, and $$R_p(\Pi \times \sigma, a, \rho) = \frac{1}{\epsilon(\tau \otimes \sigma, \tfrac{1-d'}{2} + a)^2} L(\tau_p^\vee \otimes \sigma_p^\vee \otimes \rho^\vee, \tfrac{1+d'}{2} - a)^{-1} L(\tau_p \otimes \theta \otimes \sigma \otimes \rho, \tfrac{1-d'}{2} + a)^{-1}$$ where we have written $\Pi_p = \operatorname{Ind}_{P_{\operatorname{Kl}}}^{G}(\tau \boxtimes \theta)$ for some $\tau$ and $\theta$ as above.
[GPSR87]{}
J. Arthur, *Automorphic representations of [$\operatorname{GSp}(4)$]{}*, Contributions to automorphic forms, geometry, and number theory, Johns Hopkins Univ. Press, 2004, pp. 65–81. [ MR [2058604]{}. ]{}
A. Cauchi, F. Lemma, and J. Rodrigues Jacinto, [[*On higher regulators of siegel varieties*]{}]{}, preprint, 2019, .
H. Darmon and V. Rotger, [[*Diagonal cycles and [E]{}uler systems [I]{}: a [$p$]{}-adic [G]{}ross–[Z]{}agier formula*]{}]{}, Ann. Sci. [É]{}cole Norm. Sup. (4) **47** (2014), no. 4, 779–832. [ MR [3250064]{}. ]{}
[to3em]{}, [[*Diagonal cycles and [E]{}uler systems [II]{}: the [B]{}irch and [S]{}winnerton-[D]{}yer conjecture for [H]{}asse–[W]{}eil–[A]{}rtin [$L$]{}-series*]{}]{}, J. Amer. Math. Soc. (2016), in press.
M. Dimitrov, F. Januszewski, and A. Raghuram, [[*[$L$]{}-functions of [${\operatorname{GL}}(2n)$]{}: [$p$]{}-adic properties and non-vanishing of twists*]{}]{}, preprint, 2018, .
N. Fakhruddin and V. Pilloni, [[*Hecke operators and the coherent cohomology of [S]{}himura varieties*]{}]{}, preprint, 2019, .
G. Faltings and C.-L. Chai, [[*Degeneration of abelian varieties*]{}]{}, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 22, Springer, 1990. [ MR [1083353]{}. ]{}
W. T. Gan and S. Takeda, [[*The local [L]{}anglands conjecture for [$\operatorname{GSp}(4)$]{}*]{}]{}, Ann. of Math. (2) **173** (2011), no. 3, 1841–1882. [ MR [2800725]{}. ]{}
[to3em]{}, [[*Theta correspondences for [$\operatorname{GSp}(4)$]{}*]{}]{}, Represent. Theory **15** (2011), 670–718. [ MR [2846304]{}. ]{}
T. Gee and O. Taïbi, [[*Arthur’s multiplicity formula for [$\operatorname{GSp}_4$]{} and restriction to [$\operatorname{Sp}_4$]{}*]{}]{}, preprint, 2018, .
S. Gelbart, I. I. Piatetski-Shapiro, and S. Rallis, [[*Explicit constructions of automorphic [$L$]{}-functions*]{}]{}, Lecture Notes in Mathematics, vol. 1254, Springer-Verlag, Berlin, 1987. [ MR [892097]{}. ]{}
A. Genestier and J. Tilouine, *Systèmes de [T]{}aylor-[W]{}iles pour [$\operatorname{GSp}_4$]{}*, Astérisque (2005), no. 302, 177–290, Formes automorphes. II. Le cas du groupe $\operatorname{GSp}(4)$. [ MR [2234862]{}. ]{}
M. Harris, [[*Functorial properties of toroidal compactifications of locally symmetric varieties*]{}]{}, Proc. London Math. Soc. (3) **59** (1989), no. 1, 1–22. [ MR [997249]{}. ]{}
[to3em]{}, [[*Automorphic forms of [$\overline\partial$]{}-cohomology type as coherent cohomology classes*]{}]{}, J. Differential Geom. **32** (1990), no. 1, 1–63. [ MR [1064864]{}. ]{}
[to3em]{}, *Occult period invariants and critical values of the degree four [$L$]{}-function of [$\operatorname{GSp}(4)$]{}*, Contributions to automorphic forms, geometry, and number theory (in honour of J. Shalika), Johns Hopkins Univ. Press, 2004, pp. 331–354. [ MR [2058613]{}. ]{}
M. Harris and S. S. Kudla, [[*Arithmetic automorphic forms for the nonholomorphic discrete series of [$\operatorname{GSp}(2)$]{}*]{}]{}, Duke Math. J. **66** (1992), no. 1, 59–121. [ MR [1159432]{}. ]{}
M. Harris and J. Tilouine, [[*[$p$]{}-adic measures and square roots of special values of triple product [$L$]{}-functions*]{}]{}, Math. Ann. **320** (2001), no. 1, 127–147. [ MR [1835065]{}. ]{}
A. Iovita, [[*Formal sections and de [R]{}ham cohomology of semistable abelian varieties*]{}]{}, Israel Journal of Mathematics **120** (2000), no. 2, 429–447.
H. Jacquet, *Automorphic forms on [$\operatorname{GL}(2)$]{}. [P]{}art [II]{}*, Lecture Notes in Mathematics, Vol. 278, Springer, Berlin, 1972. [ MR [0562503]{}. ]{}
H. Jacquet and J. Shalika, [[*A non-vanishing theorem for zeta functions of $\operatorname{GL}_n$*]{}]{}, Invent. Math. **38** (1976), 1–16.
K. Kato, [[*[$P$]{}-adic [H]{}odge theory and values of zeta functions of modular forms*]{}]{}, Ast[é]{}risque **295** (2004), 117–290, Cohomologies $p$-adiques et applications arithm[é]{}tiques. III. [ MR [2104361]{}. ]{}
G. Kings, D. Loeffler, and S. L. Zerbes, [[*[R]{}ankin–[E]{}isenstein classes and explicit reciprocity laws*]{}]{}, Cambridge J. Math. **5** (2017), no. 1, 1–122. [ MR [3637653]{}. ]{}
V. Lafforgue, *Estimées pour les valuations [$p$]{}-adiques des valeurs propres des opérateurs de [H]{}ecke*, Bull. Soc. Math. France **139** (2011), no. 4, 455–477. [ MR [2869300]{}. ]{}
K.-W. Lan, [[*Closed immersions of toroidal compactifications of [S]{}himura varieties*]{}]{}, preprint, 2019.
A. Lei, D. Loeffler, and S. L. Zerbes, [[*Euler systems for [R]{}ankin–[S]{}elberg convolutions of modular forms*]{}]{}, Ann. of Math. (2) **180** (2014), no. 2, 653–771. [ MR [3224721]{}. ]{}
[to3em]{}, [[*Euler systems for [H]{}ilbert modular surfaces*]{}]{}, Forum Math. Sigma **6** (2018), no. e23.
F. Lemma, [[*On higher regulators of [S]{}iegel threefolds [II]{}: [T]{}he connection to the special value*]{}]{}, Compositio Mathematica **153** (2017), no. 5, 889–946.
Z. Liu, [[*Nearly overconvergent forms and [$p$]{}-adic [L]{}-functions for symplectic groups*]{}]{}, Ph.D. thesis, Columbia University, 2016.
D. Loeffler, C. Skinner, and S. L. Zerbes, [[*Euler systems for [$\operatorname{GSp}(4)$]{}*]{}]{}, preprint, 2017, .
[to3em]{}, *Euler systems for [$\operatorname{GU}(2, 1)$]{}*, in preparation, 2019.
K. Madapusi Pera, [[*Toroidal compactifications of integral models of [S]{}himura varieties of [H]{}odge type*]{}]{}, preprint, 2012, .
H. Matsumura, *Commutative ring theory*, second ed., Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge Univ. Press, 1989. [ MR [1011461]{}. ]{}
T. Moriyama, [[*Entireness of the spinor [$L$]{}-functions for certain generic cusp forms on [$\operatorname{GSp}(2)$]{}*]{}]{}, Amer. J. Math. **126** (2004), no. 4, 899–920. [ MR [2075487]{}. ]{}
M. Novodvorsky, *Automorphic [$L$]{}-functions for the symplectic group [$\operatorname{GSp}(4)$]{}*, Automorphic forms, representations and $L$-functions (Corvallis, 1977) (A. Borel and W. Casselman, eds.), Proc. Sympos. Pure Math, vol. 33 Part 2, Amer. Math. Soc., 1979, pp. 87–95.
I. I. Piatetski-Shapiro, [[*[$L$]{}-functions for [$\operatorname{GSp}_4$]{}*]{}]{}, Pacific J. Math. **181** (1997), no. 3, 259–275, Olga Taussky-Todd: in memoriam. [ MR [1610879]{}. ]{}
V. Pilloni, [[*Higher coherent cohomology and [$p$]{}-adic modular forms of singular weight*]{}]{}, to appear in Duke Math J., 2017.
B. Roberts and R. Schmidt, [[*Local newforms for [$\operatorname{GSp}(4)$]{}*]{}]{}, Lecture Notes in Mathematics, vol. 1918, Springer, Berlin, 2007. [ MR [2344630]{}. ]{}
[to3em]{}, [[*Some results on [B]{}essel functionals for [$\operatorname{GSp}(4)$]{}*]{}]{}, Doc. Math. **21** (2016), 467–553. [ MR [3522249]{}. ]{}
F. Rodier, *Whittaker models for admissible representations of reductive [$p$]{}-adic split groups*, Harmonic analysis on homogeneous spaces ([W]{}illiamstown, 1972), [P]{}roc. [S]{}ympos. [P]{}ure [M]{}ath., vol. 26, Amer. Math. Soc., 1973, pp. 425–430. [ MR [0354942]{}. ]{}
M. Rösner and R. Weissauer, [[*Regular poles for spinor [$L$]{}-series attached to split [B]{}essel models of $\operatorname{GSp}(4)$*]{}]{}, preprint, 2017, .
R. Schmidt, [[*Iwahori-spherical representations of [$\operatorname{GSp}(4)$]{} and [S]{}iegel modular forms of degree 2 with square-free level*]{}]{}, J. Math. Soc. Japan **57** (2005), no. 1, 259–293. [ MR [2114732]{}. ]{}
[to3em]{}, [[*Archimedean aspects of [S]{}iegel modular forms of degree 2*]{}]{}, Rocky Mountain J. Math. **47** (2017), no. 7, 2381–2422. [ MR [3748235]{}. ]{}
D. Soudry, [[*The [$L$]{} and [$\gamma $]{} factors for generic representations of [$\operatorname{GSp}(4,\,k)\times \operatorname{GL}(2,\,k)$]{} over a local non-[A]{}rchimedean field [$k$]{}*]{}]{}, Duke Math. J. **51** (1984), no. 2, 355–394. [ MR [747870]{}. ]{}
J. Su, [[*Coherent cohomology of [S]{}himura varieties and automorphic forms*]{}]{}, preprint, 2019.
R. Takloo-Bighash, [[*[$L$]{}-functions for the [$p$]{}-adic group [$\operatorname{GSp}(4)$]{}*]{}]{}, Amer. J. Math. **122** (2000), no. 6, 1085–1120. [ MR [1797657]{}. ]{}
N. R. Wallach, [[*Asymptotic expansions of generalized matrix entries of representations of real reductive groups*]{}]{}, Lie group representations, [I]{} ([C]{}ollege [P]{}ark, [M]{}d., 1982/1983), Lecture Notes in Math., vol. 1024, Springer, Berlin, 1983, pp. 287–369. [ MR [727854]{}. ]{}
[^1]: Note that this is a specific property of homomorphisms $G \to H$ of group schemes with $G$ of multiplicative type; it is not true for more general finite flat group schemes $G$. We are grateful to Laurent Moret-Bailly for pointing out this reference, in response to a MathOverflow question of ours ([link](https://mathoverflow.net/questions/309837/kernels-of-homomorphisms-of-group-schemes)).
[^2]: We find it convenient to use additive notation for the group law on characters, since we shall frequently use the embedding of ${\mathbf{Z}}$ into the character group.
[^3]: We allow the case where $\sigma$ is a reducible principal series, in which case we simply *define* ${\mathcal{W}}(\sigma)$ to be the image of the Whittaker transform.
|
---
abstract: 'We report here on the outburst onset and evolution of the new Soft Gamma Repeater SGR0501+4516. We monitored the new SGR with [[*XMM–Newton*]{}]{} starting on 2008 August 23, one day after the source became burst-active, and continuing with 4 more observations in the following month, with the last one on 2008 September 30. Combining the data with the [[*Swift*]{}]{}-XRT and [[*Suzaku*]{}]{} data we modelled the outburst decay over a three months period, and we found that the source flux decreased exponentially with a timescale of $t_c=23.8$days. In the first [[*XMM–Newton*]{}]{} observation a large number of short X-ray bursts were observed, the rate of which decayed drastically in the following observations. We found large changes in the spectral and timing behavior of the source during the first month of the outburst decay, with softening emission as the flux decayed, and the non-thermal soft X-ray spectral component fading faster than the thermal one. Almost simultaneously to our second and fourth [[*XMM–Newton*]{}]{} observations (on 2008 August 29 and September 2), we observed the source in the hard X-ray range with [[*INTEGRAL*]{}]{}, which clearly detected the source up to $\sim$100keV in the first pointing, while giving only upper limits during the second pointing, discovering a variable hard X-ray component fading in less than 10days after the bursting activation. We performed a phase-coherent X-ray timing analysis over about 160days starting with the burst activation and found evidence of a strong second derivative period component ($\ddot{P}$ = -1.6(4)$\times$ 10$^{-19}$s s$^{-2}$). Thanks to the phase-connection, we were able to study the the phase-resolved spectral evolution of in great detail. We also report on the [[*ROSAT*]{}]{} quiescent source data, taken back in 1992 when the source exhibits a flux $\sim$80 times lower than that measured during the outburst, and a rather soft, thermal spectrum.'
author:
- |
N. Rea$^{1}$[^1], G. L. Israel$^{2}$, R. Turolla$^{3,4}$, P. Esposito$^{5,6}$, S. Mereghetti$^{5}$, D. Götz$^{7}$, S. Zane$^{4}$, A. Tiengo$^{5}$, K. Hurley$^{8}$, M. Feroci$^{9}$, M. Still$^{4}$, V. Yershov$^{4}$, C. Winkler$^{10}$, R. Perna$^{11}$, F. Bernardini$^{2}$, P. Ubertini$^{8}$, L. Stella$^{2}$, S. Campana$^{12}$, M. van der Klis$^{1}$, P. Woods$^{13}$\
$^{1}$ Astronomical Institute “Anton Pannekoek”, University of Amsterdam, Kruislaan 403, 1098SJ, Amsterdam, The Netherlands\
$^{2}$ INAF – Osservatorio Astronomico di Roma, via Frascati 33, 00040, Monte Porzio Catone (RM), Italy\
$^{3}$ Università di Padova, Dipartimento di Fisica, via Marzolo 8, I-35131 Padova, Italy\
$^{4}$ Mullard Space Science Laboratory, University College London, Holmbury St. Mary, Dorking, Surrey, RH5 6NT, UK\
$^{5}$ INAF – Istituto di Astrofisica Spaziale e Fisica Cosmica, via E. Bassini 15, I-20133, Milano, Italy\
$^{6}$ INFN – Istituto Nazionale di Fisica Nucleare, Sezione di Pavia, via A. Bassi 6, 27100 Pavia, Italy\
$^{7}$ CEA Saclay, DSM/Irfu/Service d’Astrophysique, Orme des Merisiers, Bât. 709, 91191 Gif sur Yvette, France\
$^{8}$ University of California, Space Sciences Laboratory, 7 Gauss Way, 94720-7450 Berkeley, USA\
$^{9}$ INAF – Istituto di Astrofisica Spaziale e Fisica Cosmica, via Fosso del Cavaliere 100, I-00133 Rome, Italy\
$^{10}$ Astrophysics Division, Research and Scientific Support Department, ESA-ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands\
$^{11}$ JILA, University of Colorado, Boulder, CO 80309-0440, USA\
$^{12}$ INAF – Osservatorio Astronomico di Brera, Via Bianchi 46, I-23807 Merate (Lc), Italy\
$^{13}$ Dynetics, Inc., 1000 Explorer Boulevard, Huntsville, AL 35806, USA
title: The first outburst of the new magnetar candidate SGR0501+4516
---
\[firstpage\]
stars: pulsars: general — pulsar: individual:
Introduction
============
Over the last few years, a number of observational discoveries have placed “magnetars” (ultra-magnetized isolated neutron stars) in the limelight again. These extreme objects comprise the Anomalous X-ray Pulsars (AXPs; 10 objects), and the Soft Gamma-ray Repeaters (SGRs; 4 objects), which are observationally very similar classes in many respects (for a recent review see Mereghetti et al. 2008). They are all slow X-ray pulsars with spin periods clustered in a narrow range ($P\sim$ 2–12s), relatively large period derivatives ($\dot P \sim
10^{-13}-10^{-10}$ss$^{-1}$), spin-down ages of $10^3-10^4$yr, and magnetic fields, as inferred from the classical magnetic dipole spin-down formula, of $10^{14}-10^{15}$G, much higher than the electron quantum critical field ($B_{cr}\simeq4.4\times10^{13}$G). About a dozen AXPs and SGRs are strong persistent X-ray emitters, with X-ray luminosities of about $10^{34}-10^{36}$, and a few transient ones have been discovered in recent years. A peculiarity of these neutron stars is that their X-ray energy output is much larger than their rotational energy losses, so they can not be only rotationally powered. Furthermore, they lack a companion, so they can not be accretion-powered either. Rather, the powering mechanism of AXPs and SGRs is believed to reside in the neutron star ultra-strong magnetic field (Duncan & Thompson 1992; Thompson & Duncan 1993). Other scenarios, beside the “magnetar” model, were proposed to explain AXP and SGR emission, such as the fossil disk (Chatterjee, Hernquist & Narayan 2000; Perna, Hernquist, & Narayan 2000) and the quark-star model (Ouyed, Leahy, & Niebergal 2007a,b).
In the 0.1–10keV energy band, magnetars spectra are relatively soft and empirically modeled by an absorbed blackbody ($kT\sim$0.2–0.6 keV) plus a power-law ($\Gamma\sim$2–4). Thanks to [[*INTEGRAL*]{}]{}–ISGRI and [[*RXTE*]{}]{}–HEXTE, hard X-ray emission up to $\sim$200keV has recently been detected from some sources (Kuiper et al. 2004, 2006; Mereghetti et al. 2005; Götz et al. 2006). This discovery has opened a new window on magnetars studies and has shown that their energy output may be dominated by hard, rather than soft emission.
At variance with other isolated neutron stars, AXPs and SGRs exhibit spectacular episodes of bursting and flaring activity, during which their luminosity may change up to 10 orders of magnitude on timescales down to few milliseconds. Different types of X-ray flux variability have been observed, ranging from slow and moderate flux changes up to a factor of a few on timescales of years (shown by virtually all members of the class), to more intense outbursts with flux variations up to $\sim$100 lasting for $\sim$1-3 years, and to short and intense X-ray burst activity on sub-second timescales (see Kaspi 2007 and Mereghetti 2008 for reviews of X-ray variability).
--------------------- ----------------- --------------- --------------- --------------- ---------------
Start (UT) 01:07:36 07:10:28 12:09:45 10:00:38 02:18:44
End (UT) 14:35:33 13:58:20 14:59:58 15:41:49 11:22:15
Exposure (ks) 48.9 24.9 10.2 20.5 31.0
Counts/s (pn) $8.520\pm0.016$ $7.08\pm0.02$ $6.60\pm0.03$ $6.05\pm0.02$ $3.23\pm0.01$
Pulse Period (s) 5.7620694(1) 5.7620730(1) 5.7620742(1) 5.7620754(1) 5.7620917(1)
Pulsed Fraction (%) 41(1) 35(1) 38(1) 38(1) 43(1)
N. bursts 80 2 0 0 0
--------------------- ----------------- --------------- --------------- --------------- ---------------
In particular, SGRs are characterized by periods of activity during which they emit numerous [*short bursts*]{} in the hard X-ray / soft gamma-ray energy range ($t\sim0.1-0.2$s; $L\sim10^{38}-10^{41}$ erg/s). This is indeed the defining property that led to the discovery of this class of sources. In addition, they have been observed to emit [*intermediate flares*]{}, with typical durations of $t\sim1-60$s and luminosities of $L\sim10^{41}-10^{43}$ erg/s, and spectacular [ *Giant Flares*]{}. The latter are rare and unique events in the X-ray sky, by far the most energetic ($\sim10^{44}-10^{47}$ erg/s) Galactic events currently known, second only to Supernova explosions. Indeed, the idea that SGRs host an ultra-magnetized neutron star was originally proposed to explain the very extreme properties of their bursts and flares: in this model the frequent short bursts are associated with small cracks in the neutron star crust, driven by magnetic diffusion, or, alternatively, with the sudden loss of magnetic equilibrium through the development of a tearing instability, while the giant flares would be linked to global rearrangements of the magnetic field in the neutron stars magnetosphere and interior (Thompson & Duncan 1995; Lyutikov 2003).
Bursts and flares do not seem to repeat with any regular, predictable pattern. Giant flares have been so far observed only three times from the whole sample of SGRs (from in 1979, Mazets et al. 1979; from in 1998, Hurley et al. 1999; and from in 2004, e.g. Hurley et al. 2005, Palmer et al. 2005), and never twice from the same source. As far as short bursts and intermediate flares are concerned, while some SGRs (such as ) are extremely active sources, in other cases no bursts have been detected for many years (as in the case of SGR 1627-41, that re-activated in May 2008 after a 10-yr long stretch of quiescence; Esposito et al. 2008). This suggests that a relatively large number of members of this class has not been discovered yet, and may manifest themselves in the future.
On 2008 August 22, a new SGR, namely , was discovered (the first in ten years), thanks to the [[*Swift*]{}]{}-BAT detection of a series of short X-ray bursts and intermediate flares (Holland et al. 2008; Barthelmy et al. 2008). X-ray pulsations were observed by [[*RXTE*]{}]{} at a period of 5.7s, confirming the magnetar nature of this source (G[ö]{}[ğ]{}[ü]{}[ş]{} et al. 2008), and its counterpart was identified in the infrared and optical bands (Tanvir et al. 2008; Rea et al. 2008b; Fatkhullin et al. 2008; Rol et al. 2008). Prompt radio observations to search for the on-set of radio pulsation and of a persistent counterpart failed to reveal any emission in this band in the first days after the outburst activation (Hessels et al. 2008; Kulkarni & Frail 2008b; Gelfand et al. 2008).
In this paper, we present a series of 5 [[*XMM–Newton*]{}]{} observations of ; the first one was performed only 1 day after the SGR activation, and the last one after 38 days. We also report on two [[*INTEGRAL*]{}]{} observations; the first was performed almost simultaneously with the second [[*XMM–Newton*]{}]{} observation, while the other one was performed soon after the fourth [[*XMM–Newton*]{}]{} pointing. We used the [[*Swift*]{}]{}-XRT monitoring to model the outburst decay and the spin period evolution of the source until $\sim$160days after the onset of the bursting activity. We also report on the 1992 [[*ROSAT*]{}]{} observation of its quiescent counterpart. We present details of the observation and analysis in §\[obs\], and results in §\[timing\] and \[spectra\]. Discussion follows in §\[discussion\].
\[phasecorr\] \[phase\]
Observations and analysis {#obs}
=========================
XMM-Newton {#xmm}
----------
The [[*XMM–Newton*]{}]{} Observatory (Jansen et al. 2001) observed on August/September 2008 (see Tab.\[obslog\]) with the EPIC instruments (pn and MOSs; Turner et al. 2001; Strüder et al. 2001), the Reflecting Grating Spectrometer (RGS; den Herder et al. 2001), and the Optical Monitor (OM; Mason et al. 2001).
Data were processed using SAS version 7.1.0 with the most up to date calibration files (CCF) available at the time the reduction was performed (October 2008). Standard data screening criteria were applied in the extraction of scientific products. Soft proton flares were not observed in any of the observations, resulting in the total on-source exposure times listed in Tab\[obslog\].
### EPIC and RGS
For four of the observations the pn camera was set in [Small Window]{} mode in order to reduce pile–up, while for the 2008 September 30th observation it was in [Large Window]{} mode. The MOS1 camera was in [Full Frame]{} for the first observation, and in [ Small Window]{} for all the other pointings. On the other hand, the MOS2 was in [Timing]{} mode, except for the last observation where it was set in [Small Window]{} mode. All other MOS CCDs were in [ Prime Full Window]{} mode. Thick filters were used for all the instruments, and pile-up was present only in the first MOS1 observation, which we ignored in the rest of the analysis. No transients were present in any imaging camera, so we are confident that the MOS2 in non-imaging mode did not collect photons from anything else than our target.
We performed a 2-dimensional or 1-dimensional PSF fitting, for the data obtained with the EPIC cameras in imaging mode or timing mode, respectively. The extraction radius was chosen in such a way as to obtain more than 90% of the source counts.
We then extracted the source photons, for the cameras set-up in imaging mode, from a circular region with 30radius, centered at the source position (RA 05:01:06.607, Dec $+$45:16:33.47 at J2000, with a 1$\sigma$ error of 15 which refers to the absolute astrometric [[*XMM–Newton*]{}]{} accuracy (Kirsh et al. 2004))[^2]. The background was obtained from a similar region as far away as possible from the source location in the same CCD. For the MOS2 camera in timing mode we extracted the photons from RAWX 274-334, and a similar region was used for the background extraction, although as far as possible from the source position. Only photons with PATTERN$\leq 4$ were used for the pn, with PATTERN$\leq 12$ for the MOS2 when in imaging mode, and with PATTERN$=0$ were used for MOS2 observations in timing mode. All the photon arrival times have been corrected to refer to the barycenter of the Solar System.
Thanks to the high timing and spectral resolution[^3] of the pn and MOS cameras, and to the high spectroscopic accuracy of the RGS, we were able to perform timing and spectral analysis, as well as pulse phase spectroscopy. Both the MOSs and pn cameras gave consistent timing and spectral results, and we report only on the pn results (see Tab.\[obslog\] for the pn source count rates for all five observations), and the RGS is used only to constrain the presence of narrow lines (see §\[spectra\]).
For the timing (§\[timing\]) and spectral analysis (§\[spectra\]) we removed the bursts observed in the first two observations (August 23rd and 29th) discarting all the photons corresponding to intervals where the source count rate exceeded 35 countss$^{-1}$ (a detailed analysis of the bursts themselves will be reported elsewhere).
### Optical Monitor {#obsom}
Twenty five OM images of the field were obtained simultaneously to the X-ray observations through the UVW1 lenticular filter. One further image was obtained through the U filter. The UVW1 has an effective transmission range of $\lambda=$2410–3565A, peak efficiency at $\lambda$2675A, full-width half-maximum image resolution of 2and a Vega-spectrum zeropoint of $m$=17.20. The U has an effective transmission range of $\lambda=$3030–3890A, peak efficiency at $\lambda=$3275A, full-width half-maximum image resolution of 1.55 and a Vega-spectrum zeropoint of $m$=18.26. Modulo-8 fixed photon pattern and scattered background light were removed from individual images before correcting optical distortion and converting images to J2000 celestial coordinates. The [[*XMM–Newton*]{}]{} star trackers provide absolute pointing accurate to 18. To refine astrometry, a correction is performed to individual images by cross-correlating source positions in the OM with counterparts in the USNO-B1.0 catalogue (Monet 2003). The UVW1 images were mosaicked to produce a 70ks summed exposure. The U band image was accumulated over an exposure time of 4ks. Aperture photometry was performed on the source position of using a standard 175 radius circular aperture for the UVW1 image and 3 for the U image, consistent with the calibrated zeropoint.
No XMM-OM source is detected within this aperture to 3$\sigma$ magnitude upper limits of $m_{\rm U}>$ 22.1 and $m_{\rm UVW1}>$ 23.7 (see Fig. \[figom\]). We also searched for possible counterparts to the X-ray bursts in the XMM-OM exposures in the UVW1 filter during the first [[*XMM–Newton*]{}]{} observation. We did not find any signature for such bursts in the UVW1 filter with a 3$\sigma$ upper limit on each 4ks image of $m_{\rm UVW1}>$ 22.05 .
INTEGRAL
--------
[[*INTEGRAL*]{}]{} (Winkler et al. 2003) observed twice, soon after its discovery: the first observation (orbit 717), soon after its discovery, started on 2008 August 27 at 00:31 (UT) as a ToO observation (ending on August 28th 08:36 UT), and the second observation in the framework of the Core Programme observations of the Perseus Arm region starting on 2008 September 5 at 05:48, and ending at 07:40 (UT) on September 10th (orbits 720 and 721). We analyzed the IBIS/ISGRI data of both observations. IBIS (Ubertini et al. 2003) is a coded mask telescope with a wide (29$^{\circ}\times$29$^{\circ}$) field of view, sensitive in the 15 keV–10 MeV energy range. We restricted our analysis to the ISGRI (Lebrun et al. 2003) data, taken by the IBIS low energy (15keV–1MeV) CdTe detector layer, since ISGRI the most sensitive instrument on board [[*INTEGRAL*]{}]{} at energies $<$ 300keV.
For the first observation an effective exposure of 204ks was accumulated at the source position. During this observation, the source was still burst-active and indeed at least 4 weak bursts were detected in the ISGRI data (Hurley et al. 2008). In the 18–60keV image the source is detected at a $\sim$4.2$\sigma$ confidence level, corresponding to a count rate of 0.31$\pm$0.08 counts s$^{-1}$, while in the 60–100 keV band the source was detected at a $\sim$3.5 sigma level (0.25$\pm$0.07 countss$^{-1}$). Above 100 keV the source is not detected and the 3$\sigma$ upper limit is 0.2 countss$^{-1}$ (100–200keV). The ISGRI response matrices were rebinned to match the above two channels and the detected flux values were used in the broad band spectral analysis (see below §\[spectra\]).
We performed the same analysis on the Core Programme data. In this case the exposure time was 361ks at the position of the source. No persistent or burst emission was detected in this second observation. We could infer a 3$\sigma$ upper limit in the 18–60 keV energy band of 0.18 countss$^{-1}$, implying a decrease of the hard X-ray flux in about 10days of a factor of $\sim$2.
Swift–XRT {#swift}
---------
The [[*Swift*]{}]{} satellite (Gehrels et al. 2004) includes a wide-field instrument, the Burst Alert Telescope (BAT; Barthelmy et al. 2005), and two narrow-field instruments, the X-Ray Telescope (XRT; Burrows et al. 2005) and the Ultraviolet/Optical Telescope (UVOT; Roming et al. 2005), and discovered the bursting activity of thanks to the large field of view of the BAT camera (Holland et al. 2008; Barthelmy et al. 2008). We briefly report here on the [[*Swift*]{}]{}-XRT monitoring of , and we refer to Palmer et al. and G[ö]{}[u g]{}[ü]{}[ş]{} et al. (2009, in preparation) for further details on the [[*Swift*]{}]{} observations.
Starting a few hours after the burst activation, the [[*Swift*]{}]{}-XRT camera monitored , collecting a few tens of observations in the following 160days. The XRT instrument was operated in photon counting (PC) mode for the first two observations, and in window timing (WT) mode for all the following observations, which ensures enough timing resolution (1.766ms) to monitor the period changes of the source. In our analysis we ignored the first two observations in PC mode because the were highly affected by photon pile-up.
The data were processed with standard procedures using the <span style="font-variant:small-caps;">ftools</span> task <span style="font-variant:small-caps;">xrtpipeline</span> (version 0.12.0) and events with grades 0–2 were selected for the WT data. For the timing and spectral analysis, we extracted events in a region of 40$\times$40 pixels. To estimate the background, we extracted the WT events within a similar box far from the target. The event files were used to study the timing properties of the pulsar after correcting the photon arrival times to the barycenter of the Solar System. For the spectral fitting (aimed at having a reliable flux measurement over the entire outburst) the data were grouped so as to have at least 20 counts per energy bin. The ancillary response files were generated with <span style="font-variant:small-caps;">xrtmkarf</span>, and they account for different extraction regions, vignetting and point-spread function corrections. We used the latest available spectral redistribution matrix (v011) in <span style="font-variant:small-caps;">caldb</span>. We removed the bursts from the XRT observations taking out all the photons corresponding to intervals where the source count rate exceeded 5 countss$^{-1}$.
ROSAT
-----
The [*Röntgensatellit*]{} (ROSAT; Voges 1992; Snowden & Schmitt 1990) Position Sensitive Proportional Counter (PSPC) serendipitously observed the region of the sky including the position of between 1992 September 21 and 24, for an effective exposure time of 4.2ks. An off-axis point source, 2RXPJ050107.7+451637, was clearly detected in the observation, the position of which is consistent, within uncertainties, with that of as inferred by [[*Chandra*]{}]{}(Woods et al. 2008).
The ROSAT event list and spectrum of 2RXPJ050107.7+451637 included about 260 background-subtracted photons accumulated from a circle of about 17 radius (corresponding to an encircled energy of $\sim$90%). The source count rate is estimated to be (6.6$\pm$ 0.5) $\times$ 10$^{-2}$ countss$^{-1}$ after correction for the point-spread function and vignetting.
---------------- --------------- --------------- --------------- --------------- ---------------
kT(keV) $0.70\pm0.01$ $0.69\pm0.01$ $0.70\pm0.01$ $0.69\pm0.01$ $0.66\pm0.01$
BB Radius (km) $1.41\pm0.05$ $1.49\pm0.05$ $1.42\pm0.06$ $1.39\pm0.04$ $1.04\pm0.06$
BB flux $2.1\pm0.1$ $2.3\pm0.1$ $2.15\pm0.13$ $1.93\pm0.07$ $0.86\pm0.11$
$\Gamma$ $2.75\pm0.02$ $2.92\pm0.04$ $2.90\pm0.06$ $2.96\pm0.08$ $3.01\pm0.04$
PL flux $7.7\pm0.1$ $5.8\pm0.1$ $5.3\pm0.2$ $4.9\pm0.2$ $3.3\pm0.1$
Abs. Flux $4.1\pm0.1$ $3.4\pm0.2$ $3.14\pm0.23$ $2.8\pm0.1$ $1.4\pm0.1$
Unab. Flux $9.6\pm0.1$ $8.1\pm0.2$ $7.5\pm0.3$ $7.0\pm0.3$ $4.17\pm0.11$
---------------- --------------- --------------- --------------- --------------- ---------------
X-ray Timing Analysis {#timing}
=====================
We started the timing analysis by performing a power spectrum of the first [[*XMM–Newton*]{}]{} observation (after having cleaned the data for the bursts; see above), and we found a strong coherent signal at $\sim$5.76s, followed by 8 significant harmonics. We then refined our period measurement studying the phase evolution within the observation by means of a phase-fitting technique (see Dall’Osso et al. 2003 for details). The resulting best-fit period is P$=5.762070(3)$s (1$\sigma$ confidence level; epoch 54701.0 MJD). The accuracy of 3$\mu s$ is enough to phase-connect coherently the first two [[*XMM–Newton*]{}]{} pointings which are about 6 days apart. The procedure was repeated by adding, each time, a single [[*XMM–Newton*]{}]{}pointing. The relative phases were such that the signal phase evolution could be followed unambiguously in the 5 [[*XMM–Newton*]{}]{} observations, and the preliminary phase-coherent solution for these observations had a best-fit period of P$=5.7620692(2)$s and $\dot{\rm P}$ = 6.8(8) $\times$ 10$^{-12}$s s$^{-1}$ (MJD 54701.0 was used as reference epoch; $\chi^2\sim4$ for 3 degrees of freedom, hereafter d.o.f.).
To better sample the pulsations in the time intervals not covered by [[*XMM–Newton*]{}]{} data, and increase the accuracy of our timing solution, we also included the [[*Suzaku*]{}]{}-XIS observation (Enoto et al. 2009) and part of the [[*Swift*]{}]{}-XRT monitoring dataset. A quadratic term in the phase evolution is required starting about one month after the [[*Swift*]{}]{}-BAT onset, when the pulse phases increasingly deviate from the extrapolation of the above P$-\dot{\rm P}$ solution (see Fig.\[phase\]), resulting in an unacceptable fit ($\chi^2 \sim$ 110 for 16 d.o.f.). Therefore, we added a higher order component to the above solution to account for the possible presence of a temporary or secular $\ddot{\rm P}$ term. The resulting new phase-coherent solution had a best-fit for $P=5.7620695(1)$s, $\dot{\rm P}$ = 6.7(1) $\times$ 10$^{-12}$s s$^{-1}$, and $\ddot{\rm P}$ = -1.6(4) $\times$10$^{-19}$s s$^{-2}$ (MJD 54701.0 was used as reference epoch; 1$\sigma$ c.l.; $\chi^2 =
58$ for 45 d.o.f.), or $\nu$ = 0.173548754(4) Hz, $\dot{\nu}$ = -2.01(3) $\times$ 10$^{-13}$Hz s$^{-1}$, and $\ddot{\nu}$ = 5(1)$\times$ 10$^{-21}$Hz s$^{-2}$. The time residuals with respect to the new timing solution are reported in Fig.\[phase\] (central panel; empty squares). The significance of the inclusion of the cubic term is 5.3$\sigma$. Moreover, the new timing solution implies a root mean square variability of only 0.04s. We note that the new timing solution is in agreement with that reported by Israel et al. (2008a).
The negative sign of $\ddot{\rm P}$ implies that the spin-down is decreasing on a characteristic timescale of about half a year. This might imply that a transient increase of the spin-down above the secular trend occurred in connection with the outburst onset, and that the source might now be recovering toward its secular spin-down. We note that timing components of similar strengths and with similar evolution timescales were detected in other AXPs and SGRs following the occurrence of glitches (Dall’Osso et al. 2003; Dib et al. 2008). This finding suggests that a similar event might have occurred connected to the burst and/or outburst behavior displayed by in August 2008. Correspondingly, assuming that the secular spin-down was an order of magnitude smaller than the one we measured during the outburst, our findings imply a magnetic field strength of the dipolar component in the range $7 \times 10^{13} < B_d < 2 \times
10^{14}$Gauss (assuming a neutron star moment of inertia of $10^{45}$gcm$^{2}$).
The 0.3-11keV pulse profiles are relatively complex, with several sub-peaks, though dominated by the sinusoidal fundamental component (see Fig.\[pfenergy\] and top panels of Fig.\[pps\]). The fundamental pulsed fraction calculated as $(max-min)/(max+min)$ is fairly constant in time (although with some oscillations) changing from 41%$\pm$1% during the first [[*XMM–Newton*]{}]{}pointing, to 35%$\pm$1% (2$^{nd}$ pointing), to 38%$\pm$1% (3$^{rd}$ and 4$^{th}$ pointings), and finally to 43%$\pm$1% (last pointing; see also Tab. \[obslog\]). At the same time both the shape and the pulsed fraction change as a function of energy within each pointing (see Fig.\[pulsenergy\] and Fig.\[3Dpulsenergy\]).
The [[*ROSAT*]{}]{} photon arrival times were corrected to the barycenter of the Solar System and a search for coherent periodicities was performed in a narrow range of trial periods (6.1–5.5s; we assumed a conservative value of $|\dot{P}|$=6$\times$10$^{-10}$ ss$^{-1}$) centered around the 2008 August period. No significant peaks were found above the 3$\sigma$ detection threshold. The corresponding upper limit to the pulsed fraction is about 50%.
Spectral Analysis {#spectra}
=================
For the spectral analysis we used source and background photons extracted as described in §\[obs\]. The response matrices were built using ad-hoc bad-pixel files built for each observation. We use the [XSPEC]{} package (version 11.3, and as a further check also the 12.1) for all fittings, and used the [phabs]{} absorption model with the Anders & Grevesse (1989) solar abundances and Balucinska-Church & McCammon (1998) photoelectric cross-sections. We restricted our spectral modeling to the EPIC-pn camera and used only the best calibrated energy range[^4], namely 0.5–10keV.
Phase-averaged spectroscopy {#specave}
---------------------------
We started the spectral analysis by fitting simultaneously the spectra of all the [[*XMM–Newton*]{}]{} observations with the standard blackbody (BB) plus power-law (PL) model, leaving all parameters free to vary except for the which was constrained to be the same in all observations. The values for the simultaneous modeling are reported in Tab.\[tabspec\], with a final reduced $=$1.14 for 838 d.o.f. (see also Fig.\[figspecall\]). The values of the spectral parameters were not significantly different when modelling each observation separately. The measured hydrogen column density is $N_{H}=0.89\times 10^{22}$2 , and the absorbed flux in the 0.5–10keV band varied from 4.1 to 1.4$\times10^{-11}$2 , corresponding to a luminosity range of 1.2 to 0.42$\times10^{35} \,
d_{5}^2$ (where $d_{5}$ is the source distance in units of 5kpc; see § 5.1 for further discussion on the source distance).
In the 0.5–10 keV band, the blackbody component accounts for $\sim$15% of the total absorbed flux throughout the outburst. The blackbody radius, as derived from its normalization, is smaller than the neutron star size, being compatible with a constant of $\sim1.4$km during the first month of the outburst decay (although hints for a decrease can be seen in the last observation). If the blackbody emission originates from the star surface this would imply that only a small fraction of the surface is emitting.
There is evidence that as the flux decreased, the 0.5-10keV spectrum softened during the first month after the bursting activation (see Tab.\[tabspec\] and Fig.\[figspecall\]). Interestingly, the BB flux decreased much slower than the PL flux, remaining almost constant for the first 10 days, and significantly decreasing only in the last observation more than a month after the burst activation (see also §\[discussion\], Fig.\[3Dpulsenergy\] and Fig.\[figspecall\]).
Since the [[*INTEGRAL*]{}]{} observation of was almost simultaneous to our second [[*XMM–Newton*]{}]{} observation, we then extended our spectral modelling to the entire 0.5–100keV spectrum of the 2008 August 29 observation. We found that the BB+PL model was no longer statistically acceptable (=1.29 for 174 d.o.f.), and that the PL used to model the soft X-ray spectrum could not account for the emission above 10keV (as it is usually the case for SGRs; Götz et al. 2006). We then tried more complex models. In line with other magnetar spectra (Kuiper et al. 2006; Götz et al. 2006), we added a second PL to the data to account for the hard X-ray emission.The results are reported in Tab.\[tabspecxmmintegral\] (see also Fig.\[figspecxmmintegral\]), where we also report the F-test probability for the addition of a further component to the fit. We also note that an excess in the residuals at energies larger than 8keV was present in the first [[*XMM–Newton*]{}]{} observation when fit with a BB+PL model (see Fig.\[figspecall\]), probably due to the presence of the same hard X-ray component detected by [[*INTEGRAL*]{}]{}, which might have been present from the beginning of the outburst. The subsequent [[*INTEGRAL*]{}]{} observation close to the fourth [[*XMM–Newton*]{}]{} observation almost a week later, did not show any hard X-ray emission. Assuming (although unlikely) that the hard X-ray spectral index did not change during the flux decay, we can translate our non-detection in a 3$\sigma$ flux upper limit in the 18–60keV band of $< 9.7\times10^{-12}$.
To take into account the presence of this hard X-ray component we also fit the first [[*XMM–Newton*]{}]{} observation with a BB plus two PLs, fixing the power-law index of the hard PL at the value inferred from the [[*XMM–Newton*]{}]{} plus [[*INTEGRAL*]{}]{} modelling of the second observation (namely $\Gamma =
0.8$; see the first and second columns of Tab. \[tabspecxmmintegral\]). The addition of this component was barely significant, less than in the 2008 August 29, although in the latter case the [[*INTEGRAL*]{}]{} data were crucial in the spectral modeling. We similarly tried to model the third [[*XMM–Newton*]{}]{} observation adding this PL component but in this case the addition of this further component was not significant. As in the case of the soft X-ray component, we found that the hard X-ray flux decreased significantly during the outburst decay, being undetectable by [[*INTEGRAL*]{}]{} only 10days after the burst activation.
Simultaneously with the second [[*INTEGRAL*]{}]{} observation, an [*AGILE*]{} observation was reported in the energy range $>$100 MeV, starting on August 31st and ending on September 10th (Feroci et al. 2008). During the [*AGILE*]{} observation the source was marginally burst-active. The [*AGILE*]{}-GRID gamma-ray experiment did not detected the source, with a reported 2$\sigma$ upper limit of 13$\times$10$^{-8}$ photon cm$^{-2}$ s$^{-1}$. Assuming an average photon energy of 500MeV, this value corresponds to $\sim$6$\times$10$^{-2}$ keV(keV cm$^{-2}$ s$^{-1}$ keV$^{-1}$), well below the extrapolation at this energy of the [[*INTEGRAL*]{}]{} power-law detected during the August 29th observation (prior to the [*AGILE*]{} observation), that would predict a flux at 500 MeV of $\sim$10$^{3}$ keV(keV cm$^{-2}$ s$^{-1}$ keV$^{-1}$). This indicates that as in the AXP cases (Kuiper et al. 2006), also in this SGR the presence of a spectral cut-off at energies between 100keV and 100MeV should be present spectrum during outburst.
------------------------- -------------------- --------------------
$N_{H}$ $0.91\pm0.02$ $0.93\pm0.03$
kT (keV) $0.70\pm0.02$ $0.69\pm0.04$
BB$_1$ Radius (km) $1.4\pm0.1$ $1.5\pm0.3$
BB$_1$ flux $2.2\pm0.1$ $2.7\pm0.1$
$\Gamma_{\rm soft}$ $2.92\pm0.07$ $3.2\pm0.1$
PL$_{\rm soft}$ flux $8.3\pm0.1$ $7.1\pm0.2$
$\Gamma_{\rm hard}$ 0.8 frozen $0.8\pm0.2$
PL$_{\rm hard}$ flux $3.5\pm0.1$ $3.9\pm0.2$
Abs. flux $7.9\pm0.1$ $6.8\pm0.3$
Unab. flux $14.3\pm0.1$ $12.6\pm0.3$
$\chi_{\nu}^2$ (d.o.f.) 1.17 (204) 1.18 (175)
F-test prob. $3.1\times10^{-5}$ $4.1\times10^{-8}$
------------------------- -------------------- --------------------
: Parameters of the spectral modelling of the phase-averaged spectra of the first two [[*XMM–Newton*]{}]{} observations of with a blackbody plus two power-laws. For the second observation we used the quasi-simultaneous [[*INTEGRAL*]{}]{} data (see also Fig. \[figspecxmmintegral\] and §\[integral\]). $N_{H}$ is in units of $10^{22}$cm$^{-2}$, and the blackbody radius is calculated at infinity, assuming a distance of 5kpc (uncertainties on the distance have not been included). The blackbody and power-law fluxes are calculated in the 0.5–100keV band. Unless otherwise specified, fluxes are all unabsorbed and in units of $10^{-11}$ergcm$^{-2}$s$^{-1}$. Errors are at the 90% confidence level.\[tabspecxmmintegral\]
We then studied the pre-outburst quiescent spectrum of as observed by [[*ROSAT*]{}]{}. The quiescent spectrum was well fit by either a BB or PL single-component model (see Fig. \[figspecall\]). The best-fit parameters are N$_H$=6$^{+5}_{-3}\times$ 10$^{21}$ cm$^{-2}$ and kT=0.38$^{+0.36}_{-0.15}$keV for the BB, and N$_H$=8$^{+11}_{-4}\times$ 10$^{21}$ cm$^{-2}$ and $\Gamma >$0.6 for the PL (reduced $\chi^2$=1.08 and $\chi^2$=1.13 for 17 d.o.f., respectively). The 0.1–2.4 keV observed flux is F$_X
\sim$1.4$\times$10$^{-12}$erg cm$^{-2}$ s$^{-1}$, corresponding to an extrapolated 1–10keV fluxes of 1.3 and 4.2 $\times$10$^{-12}$erg cm$^{-2}$ s$^{-1}$ for the BB and PL models, respectively. In analogy with the quiescent spectra of other magnetars, and given the slightly better reduced $\chi^2$ we assume that the BB spectral modeling is more correct.
No spectral features were detected in the phase-averaged [[*XMM–Newton*]{}]{} spectra, with 3$\sigma$ upper limits to the equivalent width of 45 and 65eV, for a Gaussian absorption line with $\sigma_{\rm line}$=5eV (using the RGS spectra) and $\sigma_{\rm line}$=100eV (using the pn spectra), respectively.
Phase-resolved spectroscopy {#secpps}
---------------------------
We performed a phase-resolved spectroscopy (PRS) for all the [[*XMM–Newton*]{}]{} observations. We generated 10 phase-resolved spectra for each observation using the ephemeris reported in §\[timing\]. The choice of the number of intervals was made [*a priori*]{} in order to have enough statistics in each phase–resolved spectrum to detect, at a 3$\sigma$ confidence level, a spectral line with an equivalent width $>$ 30 eV (although none was detected). Note that given the phase-connection of all the 5 [[*XMM–Newton*]{}]{} observations (see §\[timing\]), we can reliably follow each phase-resolved spectrum in time.
The absorbed BB plus PL model provides excellent fits for all ten phase-resolved spectra in all the observations, both when leaving free and when fixing it to the most accurate value derived in the phase-averaged fitting of all five [[*XMM–Newton*]{}]{} observations (see Tab.\[tabspec\]). In Fig.\[pps\] we have plotted the parameters derived from the PRS analysis and compared them to the pulse profile in each observation. All the observations showed significant spectral variability with phase, as well as a general softening in time. In particular, the blackbody temperature and normalization follow the pulse profile shape rather well, and remaining on average rather constant throughout the outburst, with a slightly decrease in the last [[*XMM–Newton*]{}]{} observation. On the other hand, the power-law parameters vary in phase and follow a more complex behaviour, with a double-peaked change of the photon index (see also Fig.\[3Dpulsenergy\], and §\[discussion\] for further discussion).
Discussion
==========
In the last few years, thanks to the availability of wide field X-ray instruments, as [[*Swift*]{}]{}-BAT, several outbursts from known AXP and SGR have been observed, and monitored in great detail. The detection of an outburst from has a special significance since this is the first new SGR discovered over a decade. In this paper we presented a comprehensive study of the spectral and timing properties of the source in the X-rays during the entire evolution of the outburst, starting from $\sim 1$ day after the activation and up to $\sim 160$ days later. Our investigation is based on [[*XMM–Newton*]{}]{}, [[*Swift*]{}]{}-XRT, and [[*INTEGRAL*]{}]{}data and we also re-examined [[*ROSAT*]{}]{} archival data in which the quiescent emission of was detected.
The outburst evolution and timescale
------------------------------------
Thanks to the [[*XMM–Newton*]{}]{} and [[*Swift*]{}]{}-XRT quasi-continuous monitoring (see §\[xmm\] and §\[swift\]), we could study in detail the flux decay of and give an estimate of its typical timescale. Fitting the flux evolution in the first 160days after the onset of the bursting activity, we found that an exponential function of the form Flux(t) = $K_1 + K_2\exp{-(t/t_c)}$ provides a good representation of the data (=1.2); the best values of the parameters are $K_1 = (0.66\pm0.03)\times10^{-11}$2 , $K_2
=(3.52\pm0.02)\times10^{-11}$2 , and $t_c = 23.81\pm0.05$ days (see Fig.\[figfluxdecay\]). A fit with a power-law was not found to be satisfactory (=12). Comparing the outburst decay timescale of with other magnetars (see Fig.\[figoutbursts\]), there is a clear difference in timescales. In particular, the outburst decays of other magnetars are usually fitted by two components: an initial exponential or power-law component accounting for the very fast decrease in the first day or so (successfully observed only in a very few cases), followed by a much flatter power-law with an index of $\delta\sim0.2-0.5$, where Flux(t) $ = (t-t_0)^{\delta}$ (see Woods et al. 2004; Israel et al. 2007; Esposito et al. 2008). A pure exponential flux decay with a timescale of about 24 days is unusual and has been never observed before. However, we caveat that the source did not reach the quiescent level yet, hence a second component (e.g. a power-law) in the flux decay can still appear at later times. Further monitoring observations will allow in the future a complete modeling of the outburst decay until the quiescent source level.
From Tab.\[tabspec\] and Fig.\[3Dpulsenergy\] it is apparent that, at least in the first ten days of the outburst, the flux of the blackbody component decayed more slowly than that of the power-law one, both in the phase-average and the phase-resolved spectra. In particular, fitting the phase-average BB and PL fluxes of the first 4 [[*XMM–Newton*]{}]{} observations (see Tab.\[tabspec\]) with a linear function of the form Flux(t)=$A_1+A_2t$ we found a good fit for $A_1(PL)=7.9(1)\times10^{-11}$2 and $A_1(BB) =
2.2(1)\times10^{-11}$2, and with $A_2(PL) =
-0.29(1)\times10^{-11}$2 and $A_2(BB) =
-0.018(3)\times10^{-11}$2. While the PL flux decreased by $\sim 25\%$ from the first to the second observation (and kept decreasing at a reduced rate in observations three and four), the BB flux stayed approximately constant during the first four observations. Both fluxes then substantially decreased in observation five (see also §\[secpps\] and next section for the evolution of the phase-resolved spectra). The relative decays of the thermal and non-thermal components observed here are reminiscent of those of after its intense burst of 2006 September 21 (Muno et al. 2007; Israel et al. 2007). Even in that case, the PL component decayed more rapidly than the BB flux (Israel et al. 2007). The faster decay of the non-thermal emission from is also corroborated by the non-detection of the source in the second [[*INTEGRAL*]{}]{} pointing (see §\[spectra\]).
The transient character of the hard component we detected at the beginning of ’s outburst implies that, whatever the mechanism is, thermal bremsstrahlung in the surface layers heated by returning currents, synchrotron emission from pairs created higher up ($\sim
100$ km) in the magnetosphere (Thompson & Beloborodov 2005), or resonant up-scattering of seed photons on a population of highly relativistic electrons (Baring & Harding 2007), it has to be triggered by the source activity and quickly fade in a few days. All the previous scenarios are indeed compatible with the observed behaviour provided that a flow of highly relativistic particles is injected into the magnetosphere during the outburst. Note that this is the first time that a variable hard X-ray emission is detected for a magnetar during an outburst. Of course, our observations did not allow us to distinguish between a rapid spectral softening (as expected if the particles responsible for the emission becomes less and less energetic) and/or an overall fading of the hard component due to a decrease in its normalization (as expected if the spatial region occupied/heated by such particles shrinks or if their local density decreases).
Several investigations have suggested that the observed magnetar spectra form in the magnetosphere, where thermal photons emitted from the neutron star’s surface undergo repeated resonant scatterings (Thompson, Lyutikov & Kulkani 2002; Lyutikov & Gavriil 2006; Fernandez & Thompson 2007; Rea et al. 2008; Nobili, Turolla & Zane 2008a). In this scenario, the spectral shape of the non-thermal component in the $\sim 0.1$–10 keV band (and possibly also that at [[*INTEGRAL*]{}]{} energies; see Baring & Harding 2007, 2008; Nobili, Turolla & Zane 2008b) is governed by the amount of twist which is implanted in the magnetosphere as a consequence of large scale crustal motions (star-quakes). The twist must decay, due to resistive ohmnic dissipation, in order to support its own currents (Beloborodov & Thompson 2007; Beloborodov 2009) and this, in turn, implies that the high-energy component of the spectrum has to fade. If either the initial twist is global or, as it seems more likely, it affects only a bundle of (closed) field lines (e.g. near a magnetic pole), the magnetosphere evolves in such a way as to confine the current-carrying ($\nabla\times\mathbf B\neq 0$) field lines closer to the magnetic axis (Beloborodov 2009). This necessarily quenches resonant up-scattering because the value of the cyclotron energy in most of the region occupied by the current-carrying field lines (which now extend to large radii) drops below $\approx 1$ keV, the typical energy of thermal photons.
Thompson, Lyutikov & Kulkani (2002) and Beloborodov & Thompson (2007) pointed out that the surface of a magnetar with a twisted magnetosphere is heated by the returning currents. If the twist decays, the luminosity and the area of the heated surface decrease in time. However, while the thermal component is expected to survive over the timescale necessary to dissipate the twist energy, the non-thermal component is more short-lived, since resonant scattering is no longer possible when the current-carrying bundle becomes too small. By comparing the theoretical expectations for a typical twist duration and luminosity, Beloborodov (2009) found an overall agreement with the observed properties of the transient AXP , provided that the twist was localized. In the case of , the typical derived evolution time ($\sim 1$ month) requires both a twist confined to a small volume (angular extent $\sin^2\theta\sim 0.1$) and a modest twist angle ($\psi\sim 0.1$). The distance of is not known yet, but it has recently been estimated to be $\sim 1.5$ kpc at the lowest (Aptekar et al. 2009), which implies a minimum source peak luminosity $L \ga 2.5\times 10^{34}$. under this case the values of the magnetospheric parameters derived above from the timescale of the outburst evolution are too small to explain the observed luminosity in terms of dissipation of the twist energy alone ($L_{twist}\sim
10^{33}$), and the problem worsens if the source distance is larger (unless the emission has a beaming factor $\la 0.1$). One possibility is that part of the energy has been released impulsively in the crust because of the dissipation of the toroidal field following the star-quake, as suggested to explain the decay of and (Lyubarsky, Eichler & Thompson 2003; Kouveliotou et al. 2003). However, this scenario predicts a power-law luminosity decline, $L\sim (t-t_0)^\delta$, which is not observed in . We note that the flux decay may follow different laws in the untwisting magnetosphere model of Beloborodov (2009), and the observed different decay timescales of the thermal and non-thermal components fits in the latter scenario.
Spectral variability with phase {#ppsdisc}
-------------------------------
To study the pulse profiles and the spectral changes in phase and time as a whole, we produced what we define hereafter as Dynamic Spectral Profiles (DSPs), which are shown in Fig.\[dsp\]. Each column in Fig.\[dsp\] is for one of the 5 [[*XMM–Newton*]{}]{} observations (epoch increases from left to right). Each panel shows a contour plot of the $\nu F_{\nu}$ flux as a function of phase and energy, and has been derived from the 10 phase-resolved spectra extracted as explained above. The second row refers to the total flux, as derived from the BB+PL model, while the third and the last rows show, respectively, the flux of the PL and BB components. The plots illustrate well how the source spectrum changes as phase and time, and show a clear evolution of the phase-dependent spectrum during the outburst. At energies above $\sim$5keV the PL dominates the emission at all times. From the DSPs, and by comparing the DSPs with the pulse profiles (see Fig.\[dsp\] top panel and also Fig.\[pulsenergy\]), it is also evident that most of the sub-peaks of the pulse profiles are related to the PL component (this is particularly evident in the third and fourth [[*XMM–Newton*]{}]{} observations). On the other hand, the main component of the profiles is dominated by the BB component, which is always in phase with the main peak. Moreover, by looking at Fig.\[dsp\] it is again evident how the PL component decreases in intensity on a faster timescale than the BB component in all phases. Actually the BB component is not only rather constant over the first four observations (covering the first 10days after the bursting activation), but in some phases shows a re-brightening (see Fig.\[3Dpulsenergy\], and the third panel in the last row of Fig.\[dsp\]). This is likely due to some late heating of the surface, e.g. by returning currents.
The strong phase dependence of the non-thermal component may be explained by the fact that, in the twisted magnetosphere model, both the spatial distributions of the magnetospheric currents (which act as a “scattering medium”) and the surface emission induced by the returning currents (which acts as source of seed photons for the resonant scattering) are substantially anisotropic. Even under the simple assumption where the magnetosphere is dipolar and globally twisted, the heated part of the surface and the magnetospheric charges cover two different ranges of magnetic colatitude. If the twist angle varies during the outburst evolution, both distributions would move away or toward the poles but at different rates. Of course, the situation is more complicated if the magnetospheric twist affects a limited bundle of field lines, as observations seem to indicate in (Woods et al. 2007) and in the transient AXP (Perna & Gotthelf 2008; Bernardini et al. 2009). Recent spectral calculations have shown the resonant comptonization in locally twisted multipolar fields can give rise to a hard tail which is highly phase dependent (Pavan et al. 2009). The phase-resolved spectral evolution of is very complicated, but a possible explanation for the variations of the PL component in terms of a magnetic field which is locally sheared, and the shear evolves in time, seems promising.
: AXP or SGR?
-------------
For about 20 years after their discovery, SGRs and AXPs were thought to be two distinct manifestations of highly magnetic neutron stars: the first mainly discovered and characterized by their powerful bursting activity, and the second recognized as bright persistent soft X-ray emitters with spectra empirically modelled by a BB+PL, and with little or no bursting activity. Furthermore, the discovery of hard X-ray emission (up to about 200keV; Kuiper et al. 2006; Götz et al. 2006) from a few members of both classes, added a further distinction, with AXPs having hard X-ray emission modelled by a second PL component (in addition to the BB+PL describing the soft X-ray emission) with $\Gamma_{\rm hard}\sim 0.8-1$, while the SGR emission was the natural extrapolation at higher energies of the PL component modelling their soft X-ray emission ($\Gamma_{\rm hard}\sim
1.5-2.0$). Over the past 6 years, the discovery of X-ray bursts from AXPs (Kaspi et al. 2003; Woods et al. 2004), and of BB components in the persistent spectrum of SGRs (Mereghetti et al. 2005, 2006a), initiated a revision of this distinction between these two classes.
In this context and e can be considered the Rosetta stone for a final unification of SGRs, AXPs and the so called “transient AXPs (TAXPs)”, into a single class of “magnetars candidates”. In fact the properties of this new SGR, as well as the characteristics of the 2009 January 22 outburst of the AXP e (Gelfand & Gaensler 2007; Halpern et al. 2008; Mereghetti et al. 2009; Israel et al. 2009 in prep), argue for a revision of our definition of SGRs and AXPs. In particular, ’s 0.5-10keV spectrum during outburst, is extremely soft ($\Gamma\sim$2.8-3.0) compared to other SGRs ($\Gamma\sim$1.5-2.0). Such a soft spectrum has been observed in the persistent emission of SGRs only during the “quiescent” (burst-quiet) phases of and (Kouvelioutou et al. 2003; Kulkarni et al. 2003; Mereghetti et al. 2006b). Furthermore, the spectrum of the quiescent X-ray counterpart of (see §\[rosat\] and §\[spectra\]) is far too soft for an SGR, while resemble the pre-outburst spectrum of the transient AXP (Gotthelf et al. 2004).
The name came from the strong bursting activity (see e.g. Enoto et al. 2009; Aptekar et al. 2009) which led to its discovery. However, bursts as bright and numerous as those observed from this source and other SGRs, have recently been observed from the AXP e in January 2009 (Gronwall et al. 2009; Savchenko et al. 2009; von Kienlin & Connaughton 2009), which emitted bursts as powerful as a typical SGR intermediate flares (Mereghetti et al. 2009).
Another piece of evidence for the AXP-like behaviour of , and the SGR-like behaviour of e is the photon index of the variable hard X-ray component. As shown in §\[spectra\] the photon index we measure from the [[*INTEGRAL*]{}]{} spectrum is $\Gamma\sim0.8$, which is close to the one reported for AXPs, while the variable hard X-ray emission during the January 2009 outburst of e has a photon index of $\Gamma\sim1.4-1.6$ (den Hartog et al. 2009), typical of SGRs.
Summary {#conclusion}
=======
Thanks to the unprecedented prompt observational campaigns of [[*XMM–Newton*]{}]{}, [[*INTEGRAL*]{}]{}, and [[*Swift*]{}]{}, we were able to study in great detail the evolution of the first recorded outburst from the first new SGR discovered in a decade, . Furthermore, we could compare its outburst properties with its quiescent emission as seen by [[*ROSAT*]{}]{}. We found the following.
- Phase-connected timing analysis of the entire X-ray outburst of , strongly argue that this source is a magnetar candidate with a magnetic field of $B \sim 2
\times10^{14}$Gauss. Furthermore, we identified a negative second period derivative of $\ddot{\rm P}$ = -1.6(4)$\times$10$^{-19}$s s$^{-2}$ which implies that the spin-down rate is decreasing with time, possibly in its way to recovering to its secular pre-outburst spin-down.
- A variable hard X-ray component was detected at the beginning of the outburst (see Fig.\[figspecxmmintegral\]), and became undetectable by [[*INTEGRAL*]{}]{} some time within 10days after the on-set of the bursting activity. This represent the first detection of a variable hard X-ray component in a magnetar over such a short timescale.
- The phase-connection of all the observations allowed us to study the evolution in time of the phase-resolved spectra. We found that on top of a phase-averaged spectral softening during the outburst decay, with the BB component decaying on a slower timescale than the PL component (see Fig.\[3Dpulsenergy\]), the spectral evolution also changes from phase to phase. The main peak of the pulse profile is dominated by the thermal component, while many other sub-peaks are present in the profiles, which are dominated instead by the non-thermal component (see Fig.\[dsp\]).
- No transient optical/ultraviolet source was detected by the Optical Monitor on board of [[*XMM–Newton*]{}]{} (see §\[obsom\]). Note that the optical counterpart to this source (Tanvir et al. 2008; Fatkhullin et al. 2008) is too faint to be observable by the OM, but we could constrain that no counterpart to the X-ray bursts have been observed with $m_{\rm UVW1}>$ 22.05.
- From a comparison with other outbursts recently detected from SGRs and AXPs (see Fig.\[figoutbursts\]), we show that contrary to other sources, in the first 160days of its outburst, shows a clear exponential decay on a rather slow timescale of about 24days (see Fig.\[phase\]).
- The discovery of , and its AXP-like characteristics, represents another piece of evidence in the unification of the magnetar candidate class, weakening further the differences between AXPs, TAXPs, and SGRs.
Acknowledgements {#acknowledgements .unnumbered}
================
We wish to thank Norbert Schartel for promptly approving our ToO request for the first [[*XMM–Newton*]{}]{} observation, the [[*XMM–Newton*]{}]{} team for the crucial help during the scheduling process of this monitoring program, and the [[*INTEGRAL*]{}]{} mission operations team at ISOC and ESOC for their support during the ToO observations. We also thank Neil Gehrels, the [[*Swift*]{}]{} duty scientists and science planners for making the [[*Swift*]{}]{} observations possible. This paper is based on observations obtained with [[*XMM–Newton*]{}]{} and [[*INTEGRAL*]{}]{}, which are both ESA science missions with instruments and contributions directly funded by ESA Member States and the USA (through NASA), and on observations with the NASA/UK/ASI [[*Swift*]{}]{} mission. NR is supported by an NWO Veni Fellowship and thanks T. Enoto and K. Makishima for useful discussions on this source. PE thanks the Osio Sotto city council for support with a G. Petrocchi Fellowship, SZ acknowledges STFC for support through an Advanced Fellowship, KH is grateful to the U.S. [[*INTEGRAL*]{}]{} Guest Investigator program for support under NASA Grant NNX08AC89G, and PU has been supported by the Italian Space Agency through the [[*INTEGRAL*]{}]{} grant I/008/07/0.
[99]{}
Anders, E., & Grevesse, N. 1989, 53, 197
Aptekar, R.L., Cline, T.L., Frederiks, D.D., Golenetskii, S.V., Mazets, E.P., Pal’shin, V.D., 2009, ApJ, submitted (arXiv:0902.3391)
Balucinska-Church, M. & McCammon, D. 1998, ApJ, 496, 1044
Baring, M.G. & Harding, A.K., 2007, Ap&SS, 308, 109
Baring, M.G. & Harding, A.K., 2008, AIP Conference Proceedings, 968, 93
Barthelmy, S. D., et al. 2005, Space Science Reviews, 120, 143
Barthelmy, S. D., et al. 2008, The Astronomer’s Telegram, 1676
Beloborodov A. M., Thompson C. 2007, ApJ, 657, 967
Beloborodov A. M., 2009, ApJ, in press \[arXiv:0812.4873\]
Bernardini, F., et al. 2009, A&A in press., arXiv:0901.2241
Burrows, D. N., et al. 2005, Space Science Reviews, 120, 165
Chatterjee, P., Hernquist, L., & Narayan, R. 2000, ApJ, 534, 373
Dall’Osso, S., Israel, G. L., Stella, L., Possenti, A., & Perozzi, E. 2003, ApJ, 599, 485
den Hartog, P. R., Kuiper, L., & Hermsen, W. 2009, The Astronomer’s Telegram, 1922
den Herder, J. W., et al. 2001, ApJ, 365, L7
Dib, R., Kaspi, V. M., & Gavriil, F. P. 2008, ApJ, 673, 1044
Duncan, R., Thompson, C. 1992, ApJ, 392, L9
Eichler, D., et al., 2006, arXiv:astro-ph/0611747
Enoto, T., et al. 2009, ApJ, 693, L122
Esposito, P., et al., 2008, MNRAS, 390, L34
Fatkhullin, T., et al. 2008, GRB Coordinates Network, 8160
Fernandez R., & Thompson C., 2007, ApJ, 660, 615
Feroci, M., et al. 2008, The Astronomer’s Telegram, 1705
Gelfand, J. D., & Gaensler, B. M. 2007, ApJ, 667, 1111
Gelfand, J. D., Taylor, G., Kouveliotou, C., Gaensler, B., & van der Horst, A. J. 2008, GRB Coordinates Network, 8168
Gehrels, N., & Swift Team 2004, Gamma-Ray Bursts: 30 Years of Discovery, 727, 637
Gogus, E., Woods, P., & Kouveliotou, C. 2008, The Astronomer’s Telegram, 1677
Gotthelf, E. V., Halpern, J. P., Buxton, M., & Bailyn, C. 2004, ApJ, 605, 368
Götz, D., et al., 2006, A&A, 449, L31
Gronwall, C., et al. 2009, GRB Coordinates Network, 8833
Halpern, J.P., Gotthelf, E.V., Reynolds, J., Ransom, S.M., Camilo, F. 2008, ApJ, 676, 1178
Hessels, J., Rea, N., Ransom, S., & Stappers, B. 2008, GRB Coordinates Network, 8134
Holland, S. T., et al. 2008, GRB Coordinates Network, 8112
Hurley, K., et al., 1999, Nature, 397, 41
Hurley, K., et al., 2005, Nature, 434, 1098
Israel, G.L., et al., 2007, ApJ, 664, 448
Israel, G. L., et al. 2008a, The Astronomer’s Telegram, 1837
Jansen, F., et al. 2001, ApJ, 365, L1
Kaspi, V. M., Gavriil, F. P., Woods, P. M., Jensen, J. B., Roberts, M. S. E., & Chakrabarty, D. 2003, ApJ, 588, L93
Kaspi, V., 2007, Ap&SS, 308, 1
Kouveliotou, C. et al. 2003, ApJ, 596, L79
Kuiper, L., Hermsen, W., & Mendez, M., 2004, ApJ, 613, 1173
Kuiper, L., et al., 2006, ApJ, 645, 556
Kulkarni, S. R., Kaplan, D. L., Marshall, H. L., Frail, D. A., Murakami, T., & Yonetoku, D. 2003, ApJ, 585, 948
Kulkarni, S. R., & Frail, D. A. 2008, GRB Coordinates Network, 8130
Lebrun, F., Leray, J.P., Lavocat, P., et al. 2003, A&A, 411, L141
Lyubarsky, Y., Eichler, D. & Thompson, C. 2003, ApJ, 580, L69
Lyutikov, M. 2003, MNRAS, 346, 540
Lyutikov M., & Gavriil F.P. 2006, MNRAS, 368, 690
Mason, K. O., et al. 2001, A&A, 365, L36
Mazets, E.P., et al., 1979, Nature, 282, 587
Mereghetti, S., et al., 2005, A&A, 433, L9
Mereghetti, S., et al. 2006a, ApJ, 450, 759
Mereghetti, S., et al. 2006b, ApJ, 653, 1423
Mereghetti, S., 2008, A&A Review, 15, 225
Mereghetti, S., et al. 2009, ApJ submitted
Monet, D. G., et al. 2003, AJ, 125, 984
Muno, M. P., Gaensler, B. M., Clark, J. S., de Grijs, R., Pooley, D., Stevens, I. R., & Portegies Zwart, S. F. 2007, MNRAS, 378, L44
Nobili L., Turolla R., & Zane S. 2008a, MNRAS, 386, 1527
Nobili L., Turolla R., & Zane S. 2008b, MNRAS, 389, 989
Ouyed, R., Leahy, D., & Niebergal, B. 2007a, A&A, 473, 357
Ouyed, R., Leahy, D., & Niebergal, B. 2007b, A&A, 475, 73
Palmer, D.M., et al., 2005, Nature, 434, 1107
Pavan, L., et al., 2009, MNRAS in press, arXiv:0902.0720
Perna, R., Hernquist, L., & Narayan, R. 2000, ApJ, 541, 344
Rea, N., Zane, S., Turolla, R., Lyutikov, M., Götz, D. 2008, ApJ, 686, 1245
Rea, N., Rol, E., Curran, P. A., Skillen, I., Russell, D. M., & Israel, G. L. 2008b, GRB Coordinates Network, 8159
Rol, E., Tanvir, N., Rea, N., Wiersema, K., Skillen, I., & Curran, P. A. 2008, GRB Coordinates Network, 8164
Roming, P. W. A., et al. 2005, Space Science Reviews, 120, 95
Savchenko, V., et al. 2009, GRB Coordinates Network 8837
Snowden, S. L., & Schmitt, J. H. M. M. 1990, ApJS, 171, 207
Str[ü]{}der, L., et al. 2001, ApJ, 365, L18
Tanvir, N. R., & Varricatt, W. 2008, GRB Coordinates Network, 8126
Thompson C., & Beloborodov, A.M., 2005, ApJ, 634, 565
Thompson C., & Duncan, R.C., 1993, ApJ, 408, 194
Thompson, C., & Duncan, R. C. 1995, MNRAS, 275, 255
Thompson C., Lyutikov M., & Kulkarni S.R., 2002, ApJ, 274, 332
Turner, M. J. L., et al. 2001, ApJ, 365, L27
Ubertini, P., Lebrun, F., Di Cocco, G., et al. 2003, A&A, 411, L131
Voges, W., et al. 1992, Environment Observation and Climate Modelling Through International Space Projects, 223
von Kienlin, A. & Connaughton, V. 2009, GRB Coordinates Network 8838
Winkler, C., Courvoisier, T.J.-L., Di Cocco G., et al. 2003 A&A, 411, L1
Woods P.M. et al. 2004, ApJ, 605, 378
Woods P.M. et al. 2007, ApJ, 654, 470
Woods, P., Gogus, E., & Kouveliotou, C. 2008, The Astronomer’s Telegram, 1824
[^1]: E-mail: [email protected]
[^2]: Consistent with the more accurate [[*Chandra*]{}]{} determination: RA 05:01:06.756, Dec $+$45:16:33.92 (0.11error circle; Woods et al. 2008)
[^3]: see http://xmm.esac.esa.int/ for details.
[^4]: Note that in all our fittings there is a weak spurious absorption feature at $\sim2.2$keV, which is of instrumental nature, and due to the Au edge.
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abstract: 'In this paper, we study the relation between the zeta function of a Calabi-Yau hypersurface and the zeta function of its mirror. Two types of arithmetic relations are discovered. This motivates us to formulate two general arithmetic mirror conjectures for the zeta functions of a mirror pair of Calabi-Yau manifolds.'
author:
- 'Daqing Wan[^1]'
title: Mirror Symmetry For Zeta Functions
---
Institute of Mathematics, Chinese Academy of Sciences, Beijing, P.R. China
Department of Mathematics, University of California, Irvine, CA 92697-3875
[email protected]
Introduction
============
In this section, we describe two mirror relations between the zeta function of a Calabi-Yau hypersurface in a projective space and the zeta function of its mirror manifold. Along the way, we make comments and conjectures about what to expect in the general case.
Let $d$ be a positive integer. Let $X$ and $Y$ be two $d$-dimensional smooth projective Calabi-Yau varieties over ${\mbox{\Bb C}}$. A necessary condition (the topological mirror test) for $X$ and $Y$ to be a mirror pair is that their Hodge numbers satisfy the Hodge symmetry: $$h^{i,j}(X) = h^{d-i, j}(Y), \ 0\leq i, j \leq d.$$ In particular, their Euler characteristics are related by $$e(X) = (-1)^d e(Y).$$ In general, there is no known rigorous algebraic geometric definition for a mirror pair, although many examples of mirror pairs are known at least conjecturally. Furthermore, it does not make sense to speak of “the mirror” of $X$ as the mirror variety usually comes in a family. In some cases, the mirror does not exist. This is the case for rigid Calabi-Yau $3$-fold $X$, since the rigid condition $h^{2,1}(X)=0$ would imply that $h^{1,1}(Y)=0$ which is impossible.
We shall assume that $X$ and $Y$ are a given mirror pair in some sense and are defined over a number field or a finite field. We are interested in how the zeta function of $X$ is related to the zeta function of $Y$. Since there is no algebraic geometric definition for $X$ and $Y$ to be a mirror pair, it is difficult to study the possible symmetry between their zeta functions in full generality. On the other hand, there do have many explicit examples and constructions which at least conjecturally give a mirror pair, most notably in the toric hypersurface setting as constructed by Batyrev [@Ba1]. Thus, we shall first examine an explicit example and see what kind of relations can be proved for their zeta functions in this case. This would then suggest what to expect in general.
Let $n\geq 2$ be a positive integer. We consider the universal family of Calabi-Yau complex hypersurfaces of degree $n+1$ in the projective space ${\mbox{\Bb P}}^n$. Its mirror family is a one parameter family of toric hypersurfaces. To construct the mirror family, we consider the one parameter subfamily $X_{\lambda}$ of complex projective hypersurfaces of degree $n+1$ in ${\mbox{\Bb P}}^n$ defined by $$f(x_1,\cdots, x_{n+1})= x_1^{n+1}+\cdots +x_{n+1}^{n+1}+\lambda x_1\cdots x_{n+1}=0,$$ where $\lambda\in {\mbox{\Bb C}}$ is the parameter. The variety $X_{\lambda}$ is a Calabi-Yau manifold when $X_{\lambda}$ is smooth. Let $\mu_{n+1}$ denote the group of $(n+1)$-th roots of unity. Let $$G=\{(\zeta_1,\cdots, \zeta_{n+1})|\zeta_i^{n+1}=1, \zeta_1\cdots \zeta_{n+1}=1\}/\mu_{n+1}
\cong ({\mbox{\Bb Z}}/(n+1){\mbox{\Bb Z}})^{n-1},$$ where $\mu_{n+1}$ is embedded in $G$ via the diagonal embedding. The finite group $G$ acts on $X_{\lambda}$ by $$(\zeta_1,\cdots, \zeta_{n+1})(x_1,\cdots, x_{n+1})=(\zeta_1x_1, \cdots, \zeta_{n+1}x_{n+1}).$$ The quotient $X_{\lambda}/G$ is a projective toric hypersurface $Y_{\lambda}$ in the toric variety ${\mbox{\Bb P}}_{\Delta}$, where ${\mbox{\Bb P}}_{\Delta}$ is the simplex in ${\mbox{\Bb R}}^n$ with vertices $\{ e_1,\cdots, e_n, -(e_1+\cdots e_n)\}$ and the $e_i$’s are the standard coordinate vectors in ${\mbox{\Bb R}}^n$. Explicitly, the variety $Y_{\lambda}$ is the projective closure in ${\mbox{\Bb P}}_{\Delta}$ of the affine toric hypersurface in ${\mbox{\Bb G}}_m^n$ defined by $$g(x_1,\cdots, x_{n})= x_1+\cdots +x_n+{1\over x_1\cdots x_{n}}+\lambda=0.$$
Assume that $X_{\lambda}$ is smooth. Then, $Y_{\lambda}$ is a (singular) mirror of $X_{\lambda}$. It is an orbifold. If $W_{\lambda}$ is a smooth crepant resolution of $Y_{\lambda}$, then the pair $(X_{\lambda}, W_{\lambda})$ is called a mirror pair of Calabi-Yau manifolds. Such a resolution exists for this example but not unique if $n\geq 3$. The number of rational points and the zeta function are independent of the choice of the crepant resolution. We are interested in understanding how the arithmetic of $X_{\lambda}$ is related to the arithmetic of $W_{\lambda}$, in particular how the zeta function of $X_{\lambda}$ is related to the zeta function of $W_{\lambda}$. Our main concern in this paper is to consider Calabi-Yau manifolds over finite fields, although we shall mention some implications for Calabi-Yau manifolds defined over number fields.
In this example, we see two types of mirror pairs. The first one is the maximally generic mirror pair $\{ X_{\Lambda}, W_{\lambda}\}$, where $X_{\Lambda}$ is the universal family of smooth projective Calabi-Yau hypersurfaces of degree $(n+1)$ in ${\mbox{\Bb P}}^{n}$ and $W_{\lambda}$ is the one parameter family of Calabi-Yau manifolds as constructed above. Note that $X_{\Lambda}$ and $Y_{\lambda}$ are parametrized by different parameter spaces (of different dimensions). The possible zeta symmetry in this case would then have to be a relation between certain generic property of the zeta function for $X\in X_{\Lambda}$ and the corresponding generic property of the zeta function for $W\in W_{\lambda}$.
The second type of mirror pairs is the one parameter family of mirror pairs $\{ X_{\lambda}, W_{\lambda}\}$ parametrized by the same parameter $\lambda$. This is a stronger type of mirror pair than the first type. For $\lambda\in {\mbox{\Bb C}}$, we say that $W_{\lambda}$ is a [**strong mirror**]{} of $X_{\lambda}$. For such a strong mirror pair $\{ X_{\lambda}, W_{\lambda}\}$, we can really ask for the relation between the zeta function of $X_{\lambda}$ and the zeta function of $W_{\lambda}$. If $\lambda_1\not=\lambda_2$, $W_{\lambda_1}$ would not be called a strong mirror for $X_{\lambda_2}$, although they would be an usual [**weak mirror**]{} pair. Apparently, we do not have a definition for a strong mirror pair in general, as there is not even a definition for a generic or weak mirror pair in general.
Let ${\mbox{\Bb F}_{q}}$ be a finite field of $q$ elements, where $q=p^r$ and $p$ is a prime. For a scheme $X$ of finite type of dimension $d$ over ${\mbox{\Bb F}_{q}}$, let $\# X({\mbox{\Bb F}_{q}})$ denote the number of ${\mbox{\Bb F}_{q}}$-rational points on $X$. Let $$Z(X,T) =\exp (\sum_{k=1}^{\infty} {T^k\over k} \# X({\mbox{\Bb F}_{q^{k}}}))\in 1+T{\mbox{\Bb Z}}[[T]]$$ be the zeta function of $X$. It is well known that $Z(X,T)$ is a rational function in $T$ whose reciprocal zeros and reciprocal poles are Weil $q$-integers. Factor $Z(X,T)$ over the $p$-adic numbers ${\mbox{\Bb C}}_p$ and write $$Z(X,T) =\prod_i (1-\alpha_iT)^{\pm 1}$$ in reduced form, where the algebraic integers $\alpha_i \in {\mbox{\Bb C}}_p$. One knows that the slope ${\rm ord}_q(\alpha_i)$ is a rational number in the interval $[0,d]$. For two real numbers $s_1\leq s_2$, we define the slope $[s_1, s_2]$ part of $Z(X,T)$ to be the partial product $$Z_{[s_1,s_2]}(X,T) =\prod_{s_1\leq {\rm ord}_q(\alpha_i) \leq s_2} (1-\alpha_iT)^{\pm 1}.$$ For a half open and half closed interval $[s_1, s_2)$, the slope $[s_1, s_2)$ part $Z_{[s_1, s_2)}(X,T)$ of $Z(X,T)$ is defined in a similar way. These are rational functions with coefficients in ${\mbox{\Bb Z}}_p$ by the $p$-adic Weierstrass factorization. It is clear that we have the decomposition $$Z(X,T)= \prod_{i=0}^{d} Z_{[i, i+1)}(X, T).$$
Our main result of this paper is the following arithmetic mirror theorem.
Assume that $\lambda \in {\mbox{\Bb F}_{q}}$ such that $(X_{\lambda}, W_{\lambda})$ is a strong mirror pair of Calabi-Yau manifolds over ${\mbox{\Bb F}_{q}}$. For every positive integer $k$, we have the congruence formula $$\# X_{\lambda}({\mbox{\Bb F}_{q^{k}}}) \equiv \# Y_{\lambda}({\mbox{\Bb F}_{q^{k}}}) \equiv \# W_{\lambda}({\mbox{\Bb F}_{q^{k}}})
~ ({\rm mod}~ q^k).$$ Equivalently, the slope $[0,1)$ part of the zeta function is the same for the mirror varieties $\{ X_{\lambda}, Y_{\lambda}, W_{\lambda}\}$: $$Z_{[0, 1)}(X_{\lambda},T) = Z_{[0,1)}(Y_{\lambda}, T)= Z_{[0,1)}(W_{\lambda}, T).$$
We now discuss a few applications of this theorem. In terms of cohomology theory, this suggests that the semi-simplification of the DeRham-Witt cohomology ( in particular, the $p$-adic etàle cohomology) for $\{ X_{\lambda}, Y_{\lambda}, W_{\lambda}\}$ are all the same. A corollary of the above theorem is that the unit root parts (slope zero parts) of their zeta functions are the same: $$Z_{[0, 0]}(X_{\lambda},T) = Z_{[0,0]}(Y_{\lambda}, T)= Z_{[0,0]}(W_{\lambda}, T).$$ The $p$-adic variation of the rational function $Z_{[0, 0]}(X_{\lambda},T)$ as $\lambda$ varies is closely related to the mirror map which we do not discuss it here, but see [@Dw] for the case $n\leq 3$. From arithmetic point of view, the $p$-adic variation of the rational function $Z_{[0, 0]}(X_{\lambda},T)$ as $\lambda$ varies is explained by Dwork’s unit root zeta function [@Dw2]. We briefly explain the connection here.
Let $B$ be the parameter variety of $\lambda$ such that $(X_{\lambda}, W_{\lambda})$ form a strong mirror pair. Let $\Phi: X_{\lambda}\rightarrow B$ (resp. $\Psi: W_{\lambda}\rightarrow B$) be the projection to the base by sending $X_{\lambda}$ (resp. $W_{\lambda}$) to $\lambda$. The pair $(\Phi, \Psi)$ of morphisms to $B$ is called a [**strong mirror pair of morphisms to $B$**]{}. Each of its fibres gives a strong mirror pair of Calabi-Yau manifolds. Recall that Dwork’s unit root zeta function attached to the morphism $\Phi$ is defined to be the formal infinite product $$Z_{\rm unit}(\Phi, T) =\prod_{\lambda \in |B|}
Z_{[0,0]}(X_{\lambda}, T^{{\rm deg}(\lambda)}) \in 1+T{\mbox{\Bb Z}}_p[[T]],$$ where $|B|$ denotes the set of closed points of $B$ over ${\mbox{\Bb F}_{q}}$. This unit root zeta function is no longer a rational function, but conjectured by Dwork in [@Dw2] and proved by the author in [@W4][@W5][@W6] to be a $p$-adic meromorphic function in $T$. The above theorem immediately implies
Let $(\Phi, \Psi)$ be the above strong mirror pair of morphisms to the base $B$. Then, their unit root zeta functions are the same: $$Z_{\rm unit}(\Phi, T) =Z_{\rm unit}(\Psi, T).$$
If $\lambda$ is in a number field $K$, then Theorem 1.1 implies that the Hasse-Weil zeta functions of $X_{\lambda}$ and $Y_{\lambda}$ differ essentially by the L-function of a pure motive $M_n(\lambda)$ of weight $n-3$. That is, $$\zeta(X_{\lambda}, s) = \zeta(Y_{\lambda}, s) L(M_n(\lambda), s-1).$$ In the quintic case $n=4$, the pure weight $1$ motive $M_4(\lambda)$ would come from a curve. This curve has been constructed explicitly by Candelas, de la Ossa and Fernando-Rodriquez [@Ca]. The relation between the Hasse-Weil zeta functions of $X_{\lambda}$ and $W_{\lambda}$ are similar, differing by a few more factors consisting of Tate twists of the Dedekind zeta function of $K$.
Theorem 1.1 motivates the following more general conjecture.
Suppose that we are given a strong mirror pair $\{ X,Y\}$ of Calabi-Yau manifolds defined over ${\mbox{\Bb F}_{q}}$. Then, for every positive integer $k$, we have $$\# X({\mbox{\Bb F}_{q^{k}}}) \equiv \# Y({\mbox{\Bb F}_{q^{k}}})
~ ({\rm mod}~ q^k).$$ Equivalently, $$Z_{[0, 1)}(X,T) = Z_{[0,1)}(Y, T).$$ Equivalently (by functional equation), $$Z_{(d-1, d]}(X,T) = Z_{(d-1,d]}(Y, T).$$
The condition in the congruence mirror conjecture is vague since one does not know at present an algebraic geometric definition of a strong mirror pair of Calabi-Yau manifolds, although one does know many examples such as the one given above. Thus, a major part of the problem is to make the definition of a strong mirror pair mathematically precise. For an additional evidence of the congruence mirror conjecture, see Theorem 6.2 which can be viewed as a generalization of Theorem 1.1. As indicated before, this conjecture implies that Dwork’s unit root zeta functions for the two families forming a strong mirror pair are the same $p$-adic meromorphic functions. This means that under the strong mirror family involution, Dwork’s unit root zeta function stays the same.
Just like the zeta function itself, its slope $[0,1)$ part $Z_{[0,1)}(X_{\lambda}, T)$ depends heavily on the algebraic parameter $\lambda$, not just on the topological properties of $X_{\lambda}$. This means that the congruence mirror conjecture is really a continuous type of arithmetic mirror symmetry. This continuous nature requires the use of a strong mirror pair, not just a generic mirror pair.
Assume that $\{ X, Y\}$ forms a mirror pair, not necessarily a strong mirror pair. A different type of arithmetic mirror symmetry reflecting the Hodge symmetry, which is discrete and hence generic in nature, is to look for a suitable quantum version $Z_Q(X,T)$ of the zeta function such that $$Z_Q(X,T)=Z_Q(Y,T)^{(-1)^d},$$ where $\{ X, Y\}$ is a mirror pair of Calabi-Yau manifolds over ${\mbox{\Bb F}_{q}}$ of dimension $d$. This relation cannot hold for the usual zeta function $Z(X,T)$ for obvious reasons, even for a strong mirror pair as it contradicts with the congruence mirror conjecture for odd $d$. No non-trivial candidate for $Z_Q(X,T)$ has been found. Here we propose a $p$-adic quantum version which would have the conjectural properties for most (and hence generic) mirror pairs. We will call our new zeta function to be the slope zeta function as it is based on the slopes of the zeros and poles.
For a scheme $X$ of finite type over ${\mbox{\Bb F}_{q}}$, write as before $$Z(X,T) =\prod_i (1-\alpha_iT)^{\pm 1}$$ in reduced form, where $\alpha_i \in {\mbox{\Bb C}}_p$. Define the slope zeta function of $X$ to be the two variable function $$S_p(X, u, T) =\prod_i (1-u^{{\rm ord}_q(\alpha_i)}T)^{\pm 1}.$$
Note that $$\alpha_i = q^{{\rm ord}_q(\alpha_i)} \beta_i,$$ where $\beta_i$ is a $p$-adic unit. Thus, the slope zeta function $S_p(X, u, T)$ is obtained from the $p$-adic factorization of $Z(X,T)$ by dropping the $p$-adic unit parts of the roots and replacing $q$ by the variable $u$. This is not always a rational function in $u$ and $T$. It is rational if all slopes are integers. Note that the definition of the slope zeta function is independent of the choice of the ground field ${\mbox{\Bb F}_{q}}$ where $X$ is defined. It depends only on $X\otimes {\bar{\mbox{\Bb F}}_{q}}$ and thus is also a geometric invariant. It would be interesting to see if there is a diophantine interpretation of the slope zeta function.
If $X$ is a scheme of finite type over ${\mbox{\Bb Z}}$, then for each prime number $p$, the reduction $X\otimes {\mbox{\Bb F}_{p}}$ has the $p$-adic slope zeta function $S_p(X\otimes {\mbox{\Bb F}_{p}}, u, T)$. At the first glance, one might think that this gives infinitely many discrete invariants for $X$ as the set of prime numbers is infinite. However, it can be shown that the set $\{ S_p(X\otimes {\mbox{\Bb F}_{p}}, u, T) | p \ {\rm prime}\}$ contains only finitely many distinct elements. In general, it is a very interesting but difficult problem to determine this set $\{ S_p(X\otimes {\mbox{\Bb F}_{p}}, u, T) | p \ {\rm prime}\}$.
Suppose that $X$ and $Y$ form a mirror pair of $d$-dimensional Calabi-Yau manifolds over ${\mbox{\Bb F}_{q}}$. For simplicity and for comparison with the Hodge theory, we always assume in this paper that $X$ and $Y$ can be lifted to characteristic zero (to the Witt ring of ${\mbox{\Bb F}_{q}}$). In this good reduction case, the modulo $p$ Hodge numbers equal the characteristic zero Hodge numbers. Taking $u=1$ in the definition of the slope zeta function, we see that the specialization $S_p(X, 1, T)$ already satisfies the desired relation $$S_p(X, 1, T) =(1-T)^{-e(X)}=(1-T)^{-(-1)^de(Y)} =S_p(Y, 1, T)^{(-1)^d}.$$ This suggests that there is a chance that the slope zeta function might satisfy the desired slope mirror symmetry $$S_p(X,u, T)={S_p(Y,u,T)^{(-1)^d}}.$$ In section $7$, we shall show that the slope zeta function satisfies a functional equation. Furthermore, the expected slope mirror symmetry does hold if both $X$ and $Y$ are ordinary. If either $X$ or $Y$ is not ordinary, the expected slope mirror symmetry is unlikely to hold in general.
If $d\leq 2$, the congruence mirror conjecture implies that the slope zeta function does satisfy the expected slope mirror symmetry for a strong mirror pair $\{ X, Y\}$, whether $X$ and $Y$ are ordinary or not. For $d\geq 3$, we believe that the slope zeta function is still a little bit too strong for the expected symmetry to hold in general, even if $\{ X, Y\}$ forms a strong mirror pair. And it should not be too hard to find a counter-example although we have not done so. However, we believe that the expected slope mirror symmetry holds for a sufficiently generic pair of $3$-dimensional Calabi-Yau manifolds.
Suppose that we are given a maximally generic mirror pair $\{ X, Y\}$ of $3$-dimensional Calabi-Yau manifolds defined over ${\mbox{\Bb F}_{q}}$. Then, we have the slope mirror symmetry $$S_p(X,u, T)={1\over S_p(Y,u,T)}$$ for generic $X$ and generic $Y$.
A main point of this conjecture is that it holds for all prime numbers $p$. For arbitrary $d\geq 4$, the corresponding slope mirror conjecture might be false for some prime numbers $p$, but it should be true for all primes $p\equiv 1 ~({\rm mod}~D)$ for some positive integer $D$ depending on the mirror family, if the family comes from the reduction modulo $p$ of a family defined over a number field. In the case $d\leq 3$, one could take $D=1$ and hence get the above conjecture.
Again the condition in the slope mirror conjecture is vague as it is not presently known an algebraic geometric definition of a mirror family, although many examples are known in the toric setting. In a future paper, using the results in [@W1][@W2], we shall prove that the slope mirror conjecture holds in the toric hypersurface case if $d\leq 3$. For example, if $X$ is a generic quintic hypersurface, then $X$ is ordinary by the results in [@IL][@W1] for every $p$ and thus one finds $$S_p(X\otimes {\mbox{\Bb F}_{p}}, u,T) ={(1-T)(1-uT)^{101}(1-u^2T)^{101}(1-u^3T)
\over (1-T)(1-uT)(1-u^2T)(1-u^3T)}.$$ This is independent of $p$. Note that we do not know if the one parameter subfamily $X_{\lambda}$ is generically ordinary for every $p$. The ordinary property for every $p$ was established only for the universal family of hypersurfaces, not for a one parameter subfamily of hypersurfaces such as $X_{\lambda}$. If $Y$ denotes the generic mirror of $X$, then by the results in [@W1] [@W2], $Y$ is ordinary for every $p$ and thus we obtain $$S_p(Y\otimes {\mbox{\Bb F}_{p}}, u,T) ={ (1-T)(1-uT)(1-u^2T)(1-u^3T)\over
(1-T)(1-uT)^{101}(1-u^2T)^{101}(1-u^3T)}.$$ Again, it is independent of $p$. The slope mirror conjecture holds in this example.
For a mirror pair over a number field, we have the following harder conjecture.
Let $\{ X, Y\}$ be two schemes of finite type over ${\mbox{\Bb Z}}$ such that their generic fibres $\{ X\otimes {\mbox{\Bb Q}}, Y\otimes {\mbox{\Bb Q}}\}$ form a usual (weak) mirror pair of $d$-dimensional Calabi-Yau manifolds defined over ${\mbox{\Bb Q}}$. Then there are infinitely many prime numbers $p$ (with positive density) such that $$S_p(X\otimes {\mbox{\Bb F}_{p}}, u, T)=S_p(Y\otimes {\mbox{\Bb F}_{p}}, u,T)^{(-1)^d}.$$
[**Remarks**]{}. If one uses the weight $2\log_q|\alpha_i|$ instead of the slope ${\rm ord}_q\alpha_i$, where $|\cdot |$ denotes the complex absolute value, one can define a two variable weight zeta function in a similar way. It is easy to see that the resulting weight zeta function does not satisfy the desired symmetry as the weight has nothing to do with the Hodge symmetry, while the slopes are related to the Hodge numbers as the Newton polygon (slope polygon) lies above the Hodge polygon.
In practice, one is often given a mirror pair of singular Calabi-Yau orbifolds, where there may not exist a smooth crepant resolution. In such a case, one could define an orbifold zeta function, which would be equal to the zeta function of the smooth crepant resolution whenever such a resolution exists. Similar results and conjectures should carry over to such orbifold zeta functions.
A counting formula via Gauss sums
=================================
Let $V_1,\cdots, V_m$ be $m$ distinct lattice points in ${\mbox{\Bb Z}}^n$. For $V_j=(V_{1j}, \cdots, V_{nj})$, write $$x^{V_j} = x_1^{V_{1j}}\cdots x_n^{V_{nj}}.$$ Let $f$ be the Laurent polynomial in $n$ variables written in the form: $$f(x_1,\cdots, x_n)=\sum_{j=1}^m{a_j}x^{V_j}, { a_j}\in {\mbox{\Bb F}_{q}},$$ where not all $a_j$ are zero. Let $M$ be the $n\times m$ matrix $$M=(V_1, \cdots, V_m),$$ where each $V_j$ is written as a column vector. Let $N_f^*$ denote the number of ${\mbox{\Bb F}_{q}}$-rational points on the affine toric hypersurface $f=0$ in ${\mbox{\Bb G}}_m^n$. If each $V_j\in {\mbox{\Bb Z}}_{\geq 0}^n$, we let $N_f$ denote the number of ${\mbox{\Bb F}_{q}}$-rational points on the affine hypersurface $f=0$ in ${\mbox{\Bb A}}^n$. We first derive a well known formula for both $N_f^*$ and $N_f$ in terms of Gauss sums.
For this purpose, we now recall the definition of Gauss sums. Let ${\mbox{\Bb F}_{q}}$ be the finite field of $q$ elements, where $q=p^r$ and $p$ is the characteristic of ${\mbox{\Bb F}_{q}}$. Let $\chi$ be the Teichmüller character of the multiplicative group ${\mbox{\Bb F}_{q}}^{*}$. For $a\in {\mbox{\Bb F}_{q}}^*$, the value $\chi(a)$ is just the $(q-1)$-th root of unity in the $p$-adic field ${{\mbox{\Bb C}}}_p$ such that $\chi(a)$ modulo $p$ reduces to $a$. Define the $(q-2)$ Gauss sums over ${\mbox{\Bb F}_{q}}$ by $$G(k)=\sum_{a\in {\mbox{\Bb F}_{q}}^*}\chi(a)^{-k}\zeta_p^{{\rm Tr}(a)}\ \ (1\leq k\leq q-2),$$ where $\zeta_p$ is a primitive $p$-th root of unity in ${\mbox{\Bb C}}_p$ and ${\rm Tr}$ denotes the trace map from ${\mbox{\Bb F}_{q}}$ to the prime field ${\mbox{\Bb F}_{p}}$.
For all $a\in {\mbox{\Bb F}_{q}}$, the Gauss sums satisfy the following interpolation relation $$\zeta_p^{ {\rm Tr}(a)} =\sum_{k=0}^{q-1} {G(k) \over q-1} \chi(a)^k,$$ where $$G(0)=q-1, \ G(q-1)=-q.$$
[**Proof**]{}. By the Vandermonde determinant, there are numbers $C(k)$ ($0\leq k\leq q-1$) such that for all $a\in {\mbox{\Bb F}_{q}}$, one has $$\zeta_p^{ {\rm Tr}(a)} =\sum_{k=0}^{q-1} {C(k) \over q-1} \chi(a)^k.$$ It suffices to prove that $C(k)=G(k)$ for all $k$. Take $a=0$, one finds that $C(0)/(q-1)=1$. This proves that $C(0)=q-1=G(0)$. For $1 \leq k\leq q-2$, one computes that $$G(k)=\sum_{a\in {\mbox{\Bb F}_{q}}^*}\chi(a)^{-k}\zeta_p^{{\rm Tr}(a)} ={C(k)\over q-1}(q-1) =C(k).$$ Finally, $$0 = \sum_{a\in {\mbox{\Bb F}_{q}}}\zeta_p^{{\rm Tr}(a)} ={C(0)\over q-1}q + {C(q-1)\over q-1}(q-1).$$ This gives $C(q-1)=-q =G(q-1)$. The lemma is proved.
We also need to use the following classical theorem of Stickelberger.
Let $0\leq k\leq q-1$. Write $$k=k_0+k_1p+\cdots +k_{r-1}p^{r-1}$$ in $p$-adic expansion, where $0\leq k_i \leq p-1$. Let $\sigma(k)=k_0+\cdots +k_{r-1}$ be the sum of the $p$-digits of $k$. Then, $${\rm ord}_p G(k) ={\sigma(k)\over p-1}.$$
Now we turn to deriving a counting formula for $N_f$ in terms of Gauss sums. Write $W_j =(1, V_j)\in {{\mbox{\Bb Z}}}^{n+1}$. Then, $$x_0f =\sum_{j=1}^m a_j x^{W_j}=\sum_{j=1}^m a_j x_0x_1^{V_{1j}}\cdots x_n^{V_{nj}},$$ where $x$ now has $n+1$ variables $\{ x_0, \cdots, x_n\}$. Using the formula $$\sum_{t\in {\mbox{\Bb F}_{q}}} t^k =\cases{0, & if $(q-1)\not| k$, \cr
q-1, & if $(q-1)|k$ and $k>0$, \cr
q, & if $k=0$, \cr}$$ one then calculates that $$\begin{aligned}
qN_f&=&\sum_{x_0, \cdots, x_n \in {\mbox{\Bb F}_{q}}}\zeta_p^{{\rm Tr}(x_0f(x))} \cr
&=&\sum_{x_0, \cdots, x_n \in {\mbox{\Bb F}_{q}}}\prod_{j=1}^m
\zeta_p^{ {\rm Tr}(a_j x^{W_j})} \cr
&=&\sum_{x_0, \cdots, x_n\in {\mbox{\Bb F}_{q}}}\prod_{j=1}^m
\sum_{k_j=0}^{q-1}{G(k_j)\over q-1} \chi(a_j)^{k_j}\chi(x^{W_j})^{k_j}\cr
&=&\sum_{k_1=0}^{q-1}\cdots \sum_{k_m=0}^{q-1}(\prod_{j=1}^m{G(k_j)\over q-1}
\chi(a_j)^{k_j})
\sum_{x_0, \cdots, x_n \in {\mbox{\Bb F}_{q}}}\chi(x^{k_1W_1 +\cdots +k_mW_m})
\cr
&=&\sum_{\sum_{j=1}^m k_jW_j\equiv 0({\rm mod}~q-1)}{(q-1)^{s(k)}q^{n+1-s(k)} \over (q-1)^m}
\prod_{j=1}^m\chi(a_j)^{k_j} G(k_j), \end{aligned}$$ where $s(k)$ denotes the number of non-zero entries in $k_1W_1+\cdots +k_mW_m$.
Similarly, one calculates that $$\begin{aligned}
qN_f^*&=&\sum_{x_0\in {\mbox{\Bb F}_{q}}, x_1, \cdots, x_n \in {\mbox{\Bb F}_{q}}^*}\zeta_p^{{\rm Tr}(x_0f(x))} \cr
&=&(q-1)^n + \sum_{x_0, \cdots, x_n \in {\mbox{\Bb F}_{q}}^*}\prod_{j=1}^m
\zeta_p^{ {\rm Tr}(a_j x^{W_j})} \cr
&=&(q-1)^n +\sum_{\sum_{j=1}^m k_jW_j\equiv 0({\rm mod}~q-1)}{(q-1)^{n+1} \over (q-1)^m}
\prod_{j=1}^m\chi(a_j)^{k_j} G(k_j). \end{aligned}$$ We shall use these two formulas to study the number of ${\mbox{\Bb F}_{q}}$-rational points on certain hypersurfaces in next two sections.
Rational points on Calabi-Yau hypersurfaces
===========================================
In this section, we apply formula (7) to compute the number of ${\mbox{\Bb F}_{q}}$-rational points on the projective hypersurface $X_{\lambda}$ in ${\mbox{\Bb P}}^n$ defined by $$f(x_1,\cdots, x_{n+1})= x_1^{n+1}+\cdots +x_{n+1}^{n+1}+\lambda x_1\cdots x_{n+1}=0,$$ where $\lambda$ is an element of ${\mbox{\Bb F}_{q}}^*$. We shall handle the easier case $\lambda=0$ separately. Let $M$ be the $(n+2)\times (n+2)$ matrix $$M= \pmatrix{1 &1 &1 &\cdots &1 &1 \cr
n+1 &0 &0 &\cdots &0 &1 \cr
0 &n+1 &0 &\cdots &0 &1 \cr
\vdots & \vdots &\vdots & \cdots & \vdots & \vdots \cr
0&0 &0 &\cdots &n+1 &1}$$ Let $k=(k_1,\cdots, k_{n+2})$ written as a column vector. Let $N_f$ denote the number of ${\mbox{\Bb F}_{q}}$-rational points on the affine hypersurface $f=0$ in ${{\mbox{\Bb A}}}^{n+1}$. By formula (7), we deduce that $$qN_f =\sum_{Mk\equiv 0 ({\rm mod}~ q-1)} {(q-1)^{s(k)}q^{n+2-s(k)}\over (q-1)^{n+2}}
(\prod_{j=1}^{n+2}G(k_j))\chi(\lambda)^{k_{n+2}},$$ where $s(k)$ denotes the number of non-zero entries in $Mk\in {{\mbox{\Bb Z}}}^{n+2}$. The number of ${\mbox{\Bb F}_{q}}$-rational points on the projective hypersurface $X_{\lambda}$ is then given by the formula $${N_f-1 \over q-1} = {-1\over q-1} +
\sum_{Mk\equiv 0 ({\rm mod}~ q-1)} {q^{n+1-s(k)}\over (q-1)^{n+3-s(k)}}
(\prod_{j=1}^{n+2}G(k_j))\chi(\lambda)^{k_{n+2}}.$$ If $k=(0, \cdots, 0, q-1)$, then $Mk=(q-1,\cdots, q-1)$ and $s(k)=n+2$. In this case, the corresponding term in the above expression is $-(q-1)^n$ which is $(-1)^{n-1}$ modulo $q$. If $k=(0,..., 0)$, then $s(k)=0$ and the corresponding term is $q^{n+1}/(q-1)$ which is zero modulo $q$.
Thus, we obtain the congruence formula modulo $q$: $${N_f-1 \over q-1} \equiv 1 +(-1)^{n-1} +
{\mathop{{\sum}^*}_{Mk\equiv 0 ({\rm mod}~ q-1)}} {q^{n+1-s(k)}\over (q-1)^{n+3-s(k)}}
(\prod_{j=1}^{n+2}G(k_j))\chi(\lambda)^{k_{n+2}},$$ where $\sum^*$ means summing over all those solutions $k=(k_1,\cdots, k_{n+2})$ with $0\leq k_i\leq q-1$, $k\not=(0,\cdots, 0)$, and $k\not=(0,\cdots, 0, q-1)$.
If $k\not=(0,\cdots, 0)$, then $\prod_{j=1}^{n+2}G(k_j)$ is divisible by $q$.
[**Proof**]{}. Let $k$ be a solution of $Mk\equiv 0({\rm mod}~q-1)$ such that $k\not=(0,\cdots, 0)$. Then, there are positive integers $\ell_0, \cdots, \ell_{r-1}$ such that $$k_1+\cdots +k_{n+2} =(q-1)\ell_0,$$ $$<pk_1>+\cdots +<pk_{n+2}> =(q-1)\ell_1,$$ $$\cdots$$ $$<p^{r-1}k_1>+\cdots +<p^{r-1}k_{n+2}> =(q-1)\ell_{r-1},$$ where $<pk_1>$ denotes the unique integer in $[0, q-1]$ congruent to $pk_1$ modulo $(q-1)$ and which is $0$ (resp. $q-1$) if $pk_1=0$ (resp., if $pk_1$ is a positive multiple of $q-1$). By the Stickelberger theorem, we deduce that $${\rm ord}_p\prod_{j=1}^{n+2}G(k_j) ={\sum_j \sigma(k_j) \over p-1}
={1\over q-1}\sum_{i=0}^{r-1}(q-1)\ell_i =\sum_{i=0}^{r-1}\ell_i.$$ Since $\ell_i\geq 1$, it follows that $${\rm ord}_q\prod_{j=1}^{n+2}G(k_j) ={1\over r} \sum_{i=0}^{r-1}\ell_i \geq 1$$ with equality holding if and only if all $\ell_i=1$. The lemma is proved.
Using this lemma and the previous congruence formula, we deduce
Let $\lambda\in {\mbox{\Bb F}_{q}}^*$. We have the congruence formula modulo $q$: $$\# X_{\lambda}({\mbox{\Bb F}_{q}}) \equiv 1 +(-1)^{n-1} +
{\mathop{{\sum}^*}_{Mk\equiv 0 ({\rm mod}~ q-1) \atop s(k)=n+2}} {1\over q(q-1)}
(\prod_{j=1}^{n+2}G(k_j))\chi(\lambda)^{k_{n+2}}.$$
Rational points on the mirror hypersurfaces
===========================================
In this section, we apply formula (8) to compute the number of ${\mbox{\Bb F}_{q}}$-rational points on the affine toric hypersurface in ${\mbox{\Bb G}}_m^n$ defined by the Laurent polynomial equation $$g(x_1,\cdots, x_{n})= x_1+\cdots +x_n+{1\over x_1\cdots x_{n}}+\lambda=0,$$ where $\lambda$ is an element of ${\mbox{\Bb F}_{q}}^*$. Let $N$ be the $(n+1)\times (n+2)$ matrix $$N= \pmatrix{1 &1 &\cdots &1 &1 &1 \cr
1 &0 &\cdots &0 &-1&0 \cr
0 &1 &\cdots &0 &-1&0 \cr
\vdots & \vdots & \cdots & \vdots & \vdots \vdots \cr
0&0 &\cdots &1 &-1 &0}$$ Let $k=(k_1,\cdots, k_{n+2})$ written as a column vector. By formula (8), we deduce that $$qN_g^* =(q-1)^n+\sum_{Nk\equiv 0 ({\rm mod}~ q-1)} {1 \over (q-1)}
(\prod_{j=1}^{n+2}G(k_j))\chi(\lambda)^{k_{n+2}},$$ where $k=(k_1,\cdots, k_{n+2})$ with $0\leq k_i\leq q-1$.
The contribution of those trivial terms $k$ (where each $k_i$ is either $0$ or $q-1$) is given by $${1\over q-1}\sum_{s=0}^{n+2}(-q)^s (q-1)^{n+2-s}{n+2\choose s} ={(-1)^n\over q-1}.$$ Since $$(q-1)^n +{(-1)^n\over q-1} = {(q-1)^{n+1}+(-1)^n \over q-1} \equiv q(n+1)(-1)^{n-1}
({\rm mod} q^2),$$ we deduce
For $\lambda\in {\mbox{\Bb F}_{q}}^*$, we have the following congruence formula modulo $q$: $$N_g^* \equiv (n+1)(-1)^{n-1} +{\mathop{{\sum}'}_{Nk\equiv 0 ({\rm mod}~ q-1)}}
{1 \over q(q-1)}
(\prod_{j=1}^{n+2}G(k_j))\chi(\lambda)^{k_{n+2}},$$ where $\sum'$ means summing over all those non-trivial solutions $k$.
The mirror congruence formula
=============================
For $\lambda\in {\mbox{\Bb F}_{q}}^*$, we have the congruence formula $$\# X_{\lambda}({\mbox{\Bb F}_{q}}) \equiv N_g^* +1-n(-1)^{n-1} ~({\rm mod}~ q).$$
[**Proof**]{}. If $k$ is a non-trivial solution of $Nk \equiv 0 ({\rm mod} q-1)$, then we have $$k_1\equiv k_2 \equiv \cdots \equiv k_n \equiv k_{n+1} ({\rm mod} q-1)$$ and $$k_1+\cdots +k_{n+1}+k_{n+2} \equiv 0 ({\rm mod} q-1).$$ Since $k$ is non-trivial, we must have $$0<k_1=k_2=\cdots =k_{n+1}<q-1,$$ $$k_1+\cdots +k_{n+2} =(n+1)k_1+k_{n+2}=(n+1)k_2+k_{n+2}=\cdots \equiv 0 ({\rm mod} q-1).$$ This gives all solutions of the equation $Mk\equiv 0 ({\rm mod}q-1)$ with $k_1=\cdots =k_{n+1}$, $0<k_1<q-1$ and $s(k)=n+2$. The corresponding terms for these $k$’s in $(N_f-1)/(q-1)$ and $N_g^*$ are exactly the same.
A solution of $Mk\equiv 0 ({\rm mod}~q-1)$ is called [**admissible**]{} if $s(k)=n+2$ and its first $k+1$ coordinates $\{k_1,\cdots, k_{n+1}\}$ contain at least two distinct elements. The above results show that we have $${N_f -1 \over q-1} -1 -(-1)^{n-1} -(N_g^* -(n+1)(-1)^{n-1})$$ $$\equiv \sum_{{\rm admissible}~ k}
{1 \over q(q-1)}
(\prod_{j=1}^{n+2}G(k_j))\chi(\lambda)^{k_{n+2}} ~({\rm mod}~ q).$$ This congruence together with the following lemma completes the proof of the theorem.
If $k$ is an admissible solution of $Mk\equiv 0 ({\rm mod}~q-1)$, then $${\rm ord}_q (\prod_{j=1}^{n+2}G(k_j)) \geq 2.$$
[**Proof**]{}. If $k$ is an admissible solution, then $<pk>,\cdots, <p^{r-1}k>$ are also admissible solutions. For each $1\leq i\leq n+1$, write $$(n+1)k_i +k_{n+2} =(q-1)\ell_i,$$ where $\ell_i$ is a positive integer. Adding these equations together, we get $$(n+1)(k_1+\cdots +k_{n+1}) +(n+1)k_{n+2}=(q-1)(\ell_1+\cdots +\ell_{n+1}).$$ Thus, the integer $${k_1+\cdots +k_{n+2}\over q-1} ={\ell_1+\cdots +\ell_{n+1} \over n+1}
=\ell \in {\bf Z}_{>0}.$$ It is clear that $\ell =1$ if and only if each $\ell_i=1$ which would imply that $k_1=\cdots =k_{n+1}$ contradicting with the admissibility of $k$. Thus, we must have that $\ell \geq 2$.
Similarly, for each $0\leq i\leq r-1$, we have $$<p^ik_1> +\cdots +<p^ik_{n+2}> =(q-1)j_i,$$ where $j_i\geq 2$ is a positive integer. We conclude that $${\rm ord}_q (\prod_{j=1}^{n+2}G(k_j)) ={j_0+\cdots +j_{r-1} \over r} \geq 2.$$ The lemma is proved.
Rational points on the projective mirror
========================================
Let ${\Delta}$ be the convex integral polytope associated with the Laurent polynomial $g$. It is the $n$-dimensional simplex in ${\mbox{\Bb R}}^n$ with the following vertices: $$\{ e_1, \cdots, e_n, -(e_1+\cdots +e_n)\},$$ where the $e_i$’s are the standard unit vectors in ${\mbox{\Bb R}}^n$.
Let ${\mbox{\Bb P}}_{\Delta}$ be the projective toric variety associated with the polytope $\Delta$, which contains ${\mbox{\Bb G}}_m^n$ as an open dense subset. Let $Y_{\lambda}$ be the projective closure in ${\mbox{\Bb P}}_{\Delta}$ of the affine toric hypersurface $g=0$ in ${\mbox{\Bb G}}_m^n$. The variety $Y_{\lambda}$ is then a projective toric hypersurface in ${\mbox{\Bb P}}_{\Delta}$. We are interested in the number of ${\mbox{\Bb F}_{q}}$-rational points on $Y_{\lambda}$.
The toric variety ${\mbox{\Bb P}}_{\Delta}$ has the following disjoint decomposition: $${\mbox{\Bb P}}_{\Delta} = \bigcup_{\tau \in \Delta} {\mbox{\Bb P}}_{\Delta, \tau},$$ where $\tau$ runs over all non-empty faces of $\Delta$ and each ${\mbox{\Bb P}}_{\Delta, \tau}$ is isomorphic to the torus ${\mbox{\Bb G}}_m^{{\rm dim}\tau}$. Accordingly, the projective toric hypersurface $Y_{\lambda}$ has the corresponding disjoint decomposition $$Y_{\lambda} = \bigcup_{\tau \in \Delta} Y_{\lambda, \tau},
\ Y_{\lambda, \tau} = Y_{\lambda}\cap {{\mbox{\Bb P}}}_{\Delta, \tau}.$$ For $\tau=\Delta$, the subvariety $Y_{\lambda, \Delta}$ is simply the affine toric hypersurface defined by $g=0$ in ${\mbox{\Bb G}}_m^n$. For zero-dimensional $\tau$, $Y_{\lambda, \tau}$ is empty. For a face $\tau$ with $1\leq {\rm dim}\tau \leq n-1$, one checks that $Y_{\lambda, \tau}$ is isomorphic to the affine toric hypersurface in ${\mbox{\Bb G}}_m^{{\rm dim}\tau}$ defined by $$1+x_1+\cdots +x_{{\rm dim}\tau}=0.$$ For such a $\tau$, the inclusion-exclusion principle shows that $$\# Y_{\lambda, \tau}({\mbox{\Bb F}_{q}}) = q^{{\rm dim}\tau-1} -
{{\rm dim}\tau \choose 1} q^{{\rm dim}\tau-2} +\cdots +
(-1)^{{\rm dim}\tau-1}{{\rm dim}\tau \choose {\rm dim}\tau-1}.$$ Thus, $$\# Y_{\lambda, \tau}({\mbox{\Bb F}_{q}}) ={1\over q}((q-1)^{{\rm dim}\tau} +
(-1)^{{\rm dim}\tau +1}).$$ This formula holds even for zero-dimensional $\tau$ as both sides would then be zero.
Putting these calculations together, we deduce that $$\# Y_{\lambda}({\mbox{\Bb F}_{q}}) = N_g^* -{(q-1)^n +(-1)^{n+1}\over q}
+\sum_{\tau \in \Delta}{1\over q}((q-1)^{{\rm dim}\tau} +
(-1)^{{\rm dim}\tau +1}),$$ where $\tau$ runs over all non-empty faces of $\Delta$ including $\Delta$ itself. Since $\Delta$ is a simplex, one computes that $$\sum_{\tau \in \Delta}((q-1)^{{\rm dim}\tau} +
(-1)^{{\rm dim}\tau +1}) ={q^{n+1}-1\over q-1} +(-1)={q(q^n-1)\over q-1}.$$ This implies that $$\# Y_{\lambda}({\mbox{\Bb F}_{q}}) = N_g^* -{(q-1)^n +(-1)^{n+1}\over q}
+{q^n-1\over q-1}.$$ This equality holds for all $\lambda \in {\mbox{\Bb F}_{q}}$, including the case $\lambda=0$. Reducing modulo $q$, we get $$\# Y_{\lambda}({\mbox{\Bb F}_{q}}) \equiv N_g^* +1 -n(-1)^{n-1} ~({\rm mod}~ q).$$ This and Theorem 5.1 prove the case $\lambda \not=0$ of the following theorem.
For every finite field ${\mbox{\Bb F}_{q}}$ with $\lambda \in {\mbox{\Bb F}_{q}}$, we have the congruence formula $$\# X_{\lambda}({\mbox{\Bb F}_{q}}) \equiv \# Y_{\lambda}({\mbox{\Bb F}_{q}})~({\rm mod}~ q).$$ If furthermore, $\lambda\in {\mbox{\Bb F}_{q}}$ such that $g$ is $\Delta$-regular and $W_{\lambda}$ is a mirror manifold of $X_{\lambda}$, then $$\# Y_{\lambda}({\mbox{\Bb F}_{q}}) \equiv \# W_{\lambda}({\mbox{\Bb F}_{q}})~({\rm mod}~ q).$$
[**Proof**]{}. For the first part, it remains to check the case $\lambda=0$. The proof is similar and in fact somewhat simpler than the case $\lambda\not=0$. We give an outline. Since $\lambda=0$, we can take $k_{n+2}=0$ in the calculations of $N_f$ and $N_g^*$. One finds then $$\# X_0({\mbox{\Bb F}_{q}}) \equiv 1 +
{\mathop{{\sum}^*}_{Mk\equiv 0 ({\rm mod}~ q-1) \atop s(k)=n+2}} {1\over q(q-1)}
(\prod_{j=1}^{n+2}G(k_j)),$$ where $\sum^*$ means summing over all those solutions $k=(k_1,\cdots, k_{n+1},0)$ with $0\leq k_i\leq q-1$ and $k\not=(0,\cdots, 0)$.
Similarly, one computes that $$N_g^* \equiv n(-1)^{n-1} +{\mathop{{\sum}'}_{Nk\equiv 0 ({\rm mod}~ q-1)}}
{1 \over q(q-1)}
(\prod_{j=1}^{n+2}G(k_j)),$$ where $\sum'$ means summing over all those non-trivial solutions $k$ with $k_{n+2}=0$. By (12), we deduce $$\# Y_0({\mbox{\Bb F}_{q}}) \equiv 1 +{\mathop{{\sum}'}_{Nk\equiv 0 ({\rm mod}~ q-1)}}
{1 \over q(q-1)}
(\prod_{j=1}^{n+2}G(k_j)).$$ As before, one checks that $${\mathop{{\sum}^*}_{Mk\equiv 0 ({\rm mod}~ q-1) \atop s(k)=n+2}}
(\prod_{j=1}^{n+2}G(k_j)) \equiv {\mathop{{\sum}'}_{Nk\equiv 0 ({\rm mod}~ q-1)}}
(\prod_{j=1}^{n+2}G(k_j)) ({\rm mod}~q^2).$$ The first part of the theorem follows.
To prove the second part of the theorem, let $\Delta^*$ be the dual polytope of $\Delta$. One checks that $\Delta^*$ is the simplex in ${\mbox{\Bb R}}^n$ with the vertices $$(n+1)e_i-\sum_{j=1}^n e_j ~(i=1,..., n), ~ -\sum_{j=1}^n e_j.$$ This is the $(n+1)$-multiple of a basic (regular) simplex in ${\mbox{\Bb R}}^n$. In particular, the codimension $1$ faces of $\Delta^*$ are $(n+1)$-multiples of a basic simplex in ${\mbox{\Bb R}}^{n-1}$. By the parrallel hyperplane decomposition in [@KK], one deduces that the codimension $1$ faces of $\Delta^*$ have a triangulation into basic simplices. Fix such a triangulation which produces a smooth crepant resolution $\phi: W_{\lambda} \rightarrow Y_{\lambda}$. One checks [@Ba2] that for each point $y\in Y_{\lambda}({\mbox{\Bb F}_{q}})$, the fibre $\phi^{-1}(\lambda)$ is stratified by affine spaces over ${\mbox{\Bb F}_{q}}$. Since the fibres are connected, it follows that the number of ${\mbox{\Bb F}_{q}}$-rational points on $\phi^{-1}(\lambda)$ is congruent to $1$ modulo $q$. Thus, modulo $q$, we have the congruence $$\# W_{\lambda}({\mbox{\Bb F}_{q}}) \equiv \sum_{y\in Y_{\lambda}({\mbox{\Bb F}_{q}})} \phi^{-1}(\lambda)({\mbox{\Bb F}_{q}})
\equiv \sum_{y\in Y_{\lambda}({\mbox{\Bb F}_{q}})} 1 = \# Y_{\lambda}({\mbox{\Bb F}_{q}}).$$ The proof is complete.
In terms of zeta functions, the above theorem says that the slope $[0,1)$ part of the zeta function for $X_{\lambda}$ equals the slope $[0,1)$ part of the zeta function for $Y_{\lambda}$.
The above elementary calculations can be used to treat some other examples of toric hypersurfaces and complete intersections. In a forthcoming joint work with Lei Fu, we can prove the following generalization.
Let $X$ be a smooth connected Calabi-Yau variety defined over the ring $W({\mbox{\Bb F}_{q}})$ of Witt vectors of ${\mbox{\Bb F}_{q}}$. Let $G$ be a finite group acting on $X$. Assume that $G$ fixes the non-zero global section of the canonical bundle of $X$. Then, for each positive integer $k$, we have the congruence formula $$\# (X\otimes{\mbox{\Bb F}_{q}})({\mbox{\Bb F}_{q^{k}}}) \equiv \# (X/G \otimes {\mbox{\Bb F}_{q}})({\mbox{\Bb F}_{q^{k}}}) ({\rm mod}~q^k).$$
Strictly speaking, this is not a complete generalization of Theorem 6.1 yet, since Theorem 6.1 includes singular cases as well.
Applications to zeta functions
==============================
In this section, we compare the two zeta functions $Z(X_{\lambda}, T)$ and $Z(Y_{\lambda}, T)$, where $\{ X_{\lambda}, Y_{\lambda}\}$ is our strong mirror pair.
First, we recall what is known about $Z(X_{\lambda}, T)$. Let $\lambda\in {\mbox{\Bb F}_{q}}$ such that $X_{\lambda}$ is smooth projective. By the Weil conjectures, the zeta function of $X_{\lambda}$ over ${\mbox{\Bb F}_{q}}$ has the following form $$Z(X_{\lambda}, T) = {{P(\lambda, T)^{(-1)^n}}\over (1-T)(1-qT)\cdots (1-q^{n-1}T)},$$ where $P(\lambda, T)\in 1+T{\mbox{\Bb Z}}[T]$ is a polynomial of degree $n(n^n-(-1)^n)/(n+1)$, pure of weight $n-1$. By the results in [@IL][@W1], the universal family of hypersurfaces of degree $n+1$ is generically ordinary for every $p$ (Mazur’s conjecture). However, we do not know if the one parameter family $X_{\lambda}$ of hypersurfaces is generically ordinary for every $p$. Thus, we raise
Is the one parameter family $X_{\lambda}$ of degree $n+1$ hypersurfaces in ${\mbox{\Bb P}}^n$ generically ordinary for every prime number $p$ not dividing $(n+1)$?
The answer is yes if $p \equiv 1~({\rm mod}~n+1)$ since the fibre for $\lambda=0$ is already ordinary if $p \equiv 1~({\rm mod}~n+1)$. It is also true if $n\leq 3$. The first unknown case is when $n=4$, the quintic case.
Next, we recall what is known about $Z(Y_{\lambda}, T)$. Let $\lambda\in {\mbox{\Bb F}_{q}}$ such that $g$ is $\Delta$-regular. This is equivalent to assuming that $\lambda^n \not=(n+1)^{n+1}$. Then, the zeta function of the affine toric hypersurface $g=0$ over ${\mbox{\Bb F}_{q}}$ in ${\mbox{\Bb G}}_m^n$ has the following form (see [@W3]) $$Z(g, T) = {{Q(\lambda, T)^{(-1)^n}}}
\prod_{i=0}^{n-1}(1-q^iT)^{(-1)^{n-i}{n\choose i+1}},$$ where $Q(\lambda, T)\in 1+T{\mbox{\Bb Z}}[T]$ is a polynomial of degree $n$, pure of weight $n-1$. The product of the trivial factors in $Z(g,T)$ is simply the zeta function of this sequence $${(q^k-1)^n +(-1)^{n+1} \over q^k}, \ k=1,2,\cdots.$$ From this and (11), one deduces that the zeta function of the projective toric hypersurface $Y_{\lambda}$ has the form $$Z(Y_{\lambda}, T) = {{Q(\lambda, T)^{(-1)^n}}\over (1-T)(1-qT)\cdots (1-q^{n-1}T)}.$$ By the results in [@W1][@W2], this one parameter family $Y_{\lambda}$ of toric hypersurfaces is generically ordinary for every $n$ and every prime number $p$.
Now, we are ready to compare the two zeta functions $Z(X_{\lambda}, T)$ and $Z(Y_{\lambda}, T)$. Let now $\lambda\in {\mbox{\Bb F}_{q}}$ such that $X_{\lambda}$ is smooth and $g$ is $\Delta$-regular. The above description shows that $${Z(X_{\lambda}, T) \over Z(Y_{\lambda}, T)} = ({P(\lambda, T)\over Q(\lambda, T)})^{(-1)^n}.$$ To understand this quotient of zeta functions, it suffices to understand the quotient ${P(\lambda, T)/Q(\lambda, T)}$.
The polynomial $Q(\lambda, T)$ divides $P(\lambda, T)$.
[**Proof**]{}. We consider the finite Galois covering $X_{\lambda} \rightarrow Y_{\lambda}$ with Galois group $G$, where $G=({\mbox{\Bb Z}}/(n+1){\mbox{\Bb Z}})^{n-1}$ is an abelian group. For an $\ell$-adic representation $\rho: G\rightarrow {\rm GL}(V_{\rho})$, let $L(X_{\lambda}, \rho, T)$ denote the corresponding L-function of $\rho$ associated to this Galois covering. Then, we have the standard factorization $$Z(X_{\lambda}, T) = \prod_{\rho} L(X_{\lambda}, \rho, T),$$ where $\rho$ runs over all irreducible (necessarily one-dimensional) $\ell$-adic representations of $G$. If $\rho=1$ is the trivial representation, then $$L(X_{\lambda}, 1, T)=Z(Y_{\lambda}, T).$$ For a prime number $\ell\not=p$, the $\ell$-adic trace formula for $Z(X_{\lambda}, T)$ is $$Z(X_{\lambda}, T)= \prod_{i=0}^{2(n-1)} {\rm det}(I -T {\rm Frob}_q
|H^i(X_{\lambda}\otimes {\bar{\mbox{\Bb F}}_{q}}, {\mbox{\Bb Q}}_{\ell}))^{(-1)^{i-1}},$$ where ${\rm Frob}_q$ denotes the geometric Frobenius element over ${\mbox{\Bb F}_{q}}$. Since $X_{\lambda}$ is a smooth projective hypersurface of dimension $n-1$, one has the more precise form of the zeta function: $$Z(X_{\lambda}, T)= {{\rm det}(I -T {\rm Frob}_q
|H^{n-1}(X_{\lambda}\otimes {\bar{\mbox{\Bb F}}_{q}}, {\mbox{\Bb Q}}_{\ell}))^{(-1)^{n}} \over
(1-T)(1-qT)\cdots (1-q^{n-1}T)}.$$ Similarly, the $\ell$-adic trace formula for the L-function is $$L(X_{\lambda}, \rho, T)= \prod_{i=0}^{2(n-1)} {\rm det}(I -T ({\rm Frob}_q\otimes 1)
|(H^i(X_{\lambda}\otimes {\bar{\mbox{\Bb F}}_{q}}, {\mbox{\Bb Q}}_{\ell}) \otimes V_{\rho})^G)^{(-1)^{i-1}}.$$ For odd $i\not=n-1$, $$H^i(X_{\lambda}\otimes {\bar{\mbox{\Bb F}}_{q}}, {\mbox{\Bb Q}}_{\ell})=0, \
(H^i(X_{\lambda}\otimes {\bar{\mbox{\Bb F}}_{q}}, {\mbox{\Bb Q}}_{\ell}) \otimes V_{\rho})^G=0.$$ For even $i=2k\not=n-1$ with $0\leq k\leq n-1$, $$H^{2k}(X_{\lambda}\otimes {\bar{\mbox{\Bb F}}_{q}}, {\mbox{\Bb Q}}_{\ell})=
{\mbox{\Bb Q}}_{\ell}(-k), \
(H^{2k}(X_{\lambda}\otimes {\bar{\mbox{\Bb F}}_{q}}, {\mbox{\Bb Q}}_{\ell}) \otimes V_{\rho})^G=0$$ for non-trivial irreducible $\rho$. This proves that for irreducible $\rho\not=1$, we have $$L(X_{\lambda}, \rho, T)= {\rm det}(I -T ({\rm Frob}_q\otimes 1)
|(H^{n-1}(X_{\lambda}\otimes {\bar{\mbox{\Bb F}}_{q}}, {\mbox{\Bb Q}}_{\ell}) \otimes V_{\rho})^G)^{(-1)^{n}}.$$ Similarly, taking $\rho=1$, one finds that $$Z(Y_{\lambda}, T)= {{\rm det}(I -T {\rm Frob}_q
|(H^{n-1}(X_{\lambda}\otimes {\bar{\mbox{\Bb F}}_{q}}, {\mbox{\Bb Q}}_{\ell}))^G)^{(-1)^{n}} \over
(1-T)(1-qT)\cdots (1-q^{n-1}T)}.$$ Comparing (13)-(16), we conclude that $$P(\lambda, T)= {\rm det}(I -T {\rm Frob}_q
|H^{n-1}(X_{\lambda}\otimes {\bar{\mbox{\Bb F}}_{q}}, {\mbox{\Bb Q}}_{\ell})),$$ $$Q(\lambda, T)= {\rm det}(I -T {\rm Frob}_q
|(H^{n-1}(X_{\lambda}\otimes {\bar{\mbox{\Bb F}}_{q}}, {\mbox{\Bb Q}}_{\ell}))^G).$$ Furthermore, the quotient $${P(\lambda, T) \over Q(\lambda, T)} =\prod_{\rho\not=1}
{\rm det}(I -T ({\rm Frob}_q\otimes 1)
|(H^{n-1}(X_{\lambda}\otimes {\bar{\mbox{\Bb F}}_{q}}, {\mbox{\Bb Q}}_{\ell}) \otimes V_{\rho})^G)$$ is a polynomial with integer coefficients of degree ${n(n^n-(-1)^n)\over n+1} -n$, pure of weight $n-1$. The lemma is proved.
This lemma together with Theorem 6.1 gives the following result.
There is a polynomial $R_n(\lambda, T)\in 1+T{\mbox{\Bb Z}}[T]$ which is pure of weight $n-3$ and of degree ${n(n^n-(-1)^n)\over n+1} -n$, such that $${P(\lambda, T)\over Q(\lambda, T)} =R_n(\lambda, qT).$$
The polynomial $R_n(\lambda, T)$ measures how far the zeta function of $Y_{\lambda}$ differs from the zeta function of $X_{\lambda}$. Being of integral pure weight $n-3$, the polynomial $R_n(\lambda, T)$ should come from the zeta function of a variety (or motive $M_n(\lambda)$) of dimension $n-3$. It would be interesting to find this variety or motive $M_n(\lambda)$ parametrized by $\lambda$. In this direction, the following is known.
If $n=2$, then $n-3<0$, $M_2(\lambda)$ is empty and we have $R_2(\lambda, T)=1$. If $n=3$, then $n-3=0$ and $$R_3(\lambda, T)=\prod_{i=1}^{18} (1-\alpha_i(\lambda)T)$$ is a polynomial of degree $18$ with $\alpha_i(\lambda)$ being roots of unity. In fact, Dwork [@Dw] proved that all $\alpha_i(\lambda)=\pm 1$ in this case. Thus, $R_3(\lambda, T)$ comes from the the zeta function of a zero-dimensional variety $M_3(\lambda)$ parameterized by $\lambda$. What is this zero-dimensional variety $M_3(\lambda)$? For every $p$ and generic $\lambda$, the slope zeta function has the form $S_p(Y_{\lambda}, u, T)=1$ and $$S_p(X_{\lambda}, u, T)={1\over (1-T)^2(1-uT)^{20}(1-u^2T)^2}.$$ Note that $Y_{\lambda}$ is singular and not a smooth mirror of $X_{\lambda}$ yet. Thus, it is not surprising that the two slope zeta functions $S_p(X_{\lambda}, u, T)$ and $S_p(Y_{\lambda}, u, T)$ do not satisfy the expected slope mirror symmetry.
If $n=4$, then $n-3=1$ and $$R_4(\lambda, T)=\prod_{i=1}^{200}(1-\alpha_i(\lambda)T)$$ is a polynomial of degree $200$ with $\alpha_i(\lambda)=\sqrt{q}$. Thus, $M_4(\lambda)$ should come from some curve parameterized by $\lambda$. This curves has been constructed explicitly in a recent paper by Candelas, de la Ossa and Fernando-Rodriquez [@Ca]. For every $p$ and generic $\lambda$, we know that $S_p(Y_{\lambda}, u, T)=1$, but as indicated at the beginning of this section, we do not know if the slope zeta function of $X_{\lambda}$ for a generic $\lambda$ has the form $$S_p(X_{\lambda}, u, T)={(1-T)(1-uT)^{101}(1-u^2T)^{101}(1-u^3T)
\over (1-T)(1-uT)(1-u^2T)(1-u^3T)}.$$
For general $n$ and $\lambda \in K$ for some field $K$, in terms of $\ell$-adic Galois representations, the pure motive $M_n(\lambda)$ is simply given by $$M_n(\lambda) = (\bigoplus_{\rho\not=1}
(H^{n-1}(X_{\lambda}\otimes {\bar K}, {\mbox{\Bb Q}}_{\ell}) \otimes V_{\rho})^G) \otimes
{\mbox{\Bb Q}}_{\ell}(-1),$$ where ${\mbox{\Bb Q}}_{\ell}(-1)$ denotes the Tate twist. If $\lambda$ is in a number field $K$, this implies that the Hasse-Weil zeta functions of $X_{\lambda}$ and $Y_{\lambda}$ are related by $$\zeta(X_{\lambda}, s) = \zeta(Y_{\lambda}, s) L(M_n(\lambda), s-1).$$
Slope zeta functions
====================
The slope zeta function satisfies a functional equation. This follows from the usual functional equation which in turn is a consequence of the Poincare duality for $\ell$-adic cohomology.
Let $X$ be a connected smooth projective variety of dimension $d$ over ${\mbox{\Bb F}_{q}}$. Then the slope zeta function $S_p(X, u, T)$ satisfies the following functional equation $$S_p(X, u, {1\over u^dT}) = S_p(X, u, T)(-u^{d/2}T)^{e(X)},$$ where $e(X)$ denotes the the $\ell$-adic Euler characteristic of $X$.
[**Proof**]{}. Let $P_i(T)$ denote the characteristic polynomial of the geometric Frobenius acting on the $i$-th $\ell$-adic cohomology of $X\otimes {\bar{\mbox{\Bb F}}_{q}}$. Then, $$Z(X, T) =\prod_{i=0}^{2d}P_i(T)^{(-1)^{i+1}}.$$ Let $s_{ij}$ ($j=1,\cdots, b_i$) denote the slopes of the polynomial $P_i(T)$, where $b_i$ is the degree of $P_i(T)$ which is the $i$-th Betti number. Write $$Q_i(T)=\prod_{j=1}^{b_i}(1-u^{s_{ij}}T).$$ Then, by the definition of the slope zeta function, we have $$S_p(X,u,T) = \prod_{i=0}^{2d}Q_i(T)^{(-1)^{i+1}}.$$ For each $0\leq i\leq 2d$, the slopes of $P_i(T)$ satisfies the determinant relation $$\sum_{j=1}^{b_i} s_{ij} = {i\over 2}b_i.$$ Using this, one computes that $$Q_i({1\over T}) = (-1/T)^{b_i} u^{ib_i/2}\prod_{j=1}^{b_i}(1-u^{-s_{ij}}T).$$ Replacing $T$ by $u^dT$, we get $$Q_i({1\over u^dT}) = ({-1\over u^dT})^{b_i}
u^{ib_i/2}\prod_{j=1}^{b_i}(1-u^{d-s_{ij}}T).$$ The functional equation for the usual zeta function $Z(X,T)$ implies that $d-s_{ij}$ ($j=1,\cdots, b_i$) are exactly the slopes for $P_{2d-i}(T)$. Thus, $$Q_i({1\over u^d T}) = ({-1\over u^dT})^{b_i} u^{ib_i/2}Q_{2d-i}(T).$$ We deduce that $$S_p(X, u, {1\over u^dT}) = \prod_{i=0}^{2d}
\big( Q_{2d-i}(T)({-1\over u^dT})^{b_i}u^{ib_i/2}\big)^{(-1)^{i+1}}.$$ Since $b_i =b_{2d-i}$, it is clear that $$\sum_{i=0}^{2d}(-1)^{i}{i\over 2}b_i ={d \over 2}e(X).$$ We conclude that $$S_p(X,u,{1 \over u^dT})= S_p(X, u,T) (-T)^{e(X)} u^{{d\over 2}e(X)}.$$ The proposition is proved.
From now on, we assume that $X$ is a smooth projective scheme over $W({\mbox{\Bb F}_{q}})$. Assume that the reduction $X\otimes {\mbox{\Bb F}_{q}}$ is ordinary, i.e., the $p$-adic Newton polygon coincides with the Hodge polygon [@Ma]. This means that the slopes of $P_i(T)$ are exactly $j$ ($0\leq j\leq i$) with multiplicity $h^{j, i-j}(X)$. In this case, one gets the explicit formula $$S_p(X\otimes {\mbox{\Bb F}_{q}}, u, T) = \prod_{j=0}^{d} (1-u^jT)^{e_j(X)},$$ where $$e_j(X) = (-1)^j \sum_{i=0}^d (-1)^{i-1}h^{j,i}(X).$$ If $X$ and $Y$ form a mirror pair over the Witt ring $W({\mbox{\Bb F}_{q}})$, the Hodge symmetry $h^{j,i}(X)=h^{j,d-i}(Y)$ implies for each $j$, $$e_j(X)=(-1)^j\sum_{i=0}^d (-1)^{i-1}h^{j,d-i}(Y) =
(-1)^d e_j(Y).$$ We obtain the following result.
Let $X$ and $Y$ be a mirror pair of $d$-dimensional smooth projective Calabi-Yau schemes over $W({\mbox{\Bb F}_{q}})$. Assume that both $X\otimes {\mbox{\Bb F}_{q}}$ and $Y\otimes {\mbox{\Bb F}_{q}}$ are ordinary. Then, we have the following symmetry for the slope zeta function: $$S_p(X\otimes {\mbox{\Bb F}_{q}}, u, T) = S_p(Y\otimes {\mbox{\Bb F}_{q}}, u, T)^{(-1)^d}.$$
The converse of this proposition may not be always true. The slope mirror conjecture follows from the following slightly stronger
Let $d\leq 3$. Suppose that $\{ X, Y\}$ form a maximally generic mirror pair of $d$-dimensional smooth projective Calabi-Yau schemes over $W({\mbox{\Bb F}_{q}})$. Then, both $X\otimes {\mbox{\Bb F}_{q}}$ and $Y\otimes {\mbox{\Bb F}_{q}}$ are generically ordinary.
For $d\leq 3$, this conjecture can be proved in the toric hypersurface case using the results in [@W1][@W2]. For $d\geq 4$, we expect that the same conjecture holds if $p\equiv 1~({\rm mod}~D)$ for some positive integer $D$. This is again provable in the toric hypersurface case using the results in [@W1]. But we do not know if we can always take $D=1$, even in the toric hypersurface case if $d\geq 4$.
[99]{}
V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Algebraic Geom., 3(1994), no. 3, 493-535.
V. Batyrev and D.I. Dais, Strong McKay correspondence, string-theoretic Hodge numbers and mirror symmetry, Topology, 35(1996), no. 4, 901-929.
P. Candelas, X. de la Ossa and F. Rodriguez-Villegas, Calabi-Yau manifolds over finite fields, II, preprint, 2004. “arXiv:hep-th/0402133”.
B. Dwork, $p$-Adic cycles, Publ. Math., IHES, 39(1969), 327-415.
B. Dwork, Normalized period matrices II, Ann. Math., 98(1973), 1-57.
G. Kempf, F. Knudsen, D. Mumford and B. Saint-Donat, Toroidal Embeddings, Lecture Notes in Math., 339, Spriner-Verlag, 1973.
L. Illusie, Ordinarité des intersections complètes générales, in Grothendieck Festschrift, Volume II (1990), 375-405.
B. Mazur, Frobenius and the Hodge filtration, Bull. Amer. Math. Soc., 78(1972), 653-667.
D. Wan, Newton polygons of zeta functions and L-functions, Ann. Math., 137(1993), 247-293.
D. Wan, Dwork’s conjecture on unit root zeta functions, Ann. Math., 150(1999), 867-927.
D. Wan, Higher rank case of Dwork’s conjecture, J. Amer. Math. Soc., 13(2000), 807-852.
D. Wan, Rank one case of Dwork’s conjecture, J. Amer. Math. Soc., 13(2000), 853-908.
D. Wan, Variation of $p$-adic Newton polygons for L-functions of exponential sums, Asian J. Math., Vol. 8, 3(2004), 427-474.
D. Wan, Zeta functions of toric Calabi-Yau hypersurfaces, Course notes given at the Arizona Winter School 2004, 22 pages. “http://swc.math.arizona.edu/oldaws/04GenlInfo.html”.
[^1]: Partially supported by NSF. It is a pleasure to thank V. Batyrev, P. Candelas, H. Esnault, K.F. Liu, Y. Ruan, S.T. Yau for helpful discussions. The paper is motivated by some open questions in my lectures at the 2004 Arizona Winter School.
|
---
abstract: '[In this paper we propose flexible joint longitudinal-survival models in order to test the association between a longitudinally collected biomarker and a time-to-event endpoint. Our proposed models are robust to common parametric and semi-parametric assumptions in that they avoid explicit distributional assumptions on longitudinal measures and allow for subject-specific baseline hazard in the survival component. Fully joint estimation is performed to account for the uncertainty in the estimated time-dependent biomarker covariate in the survival model.]{}'
author:
- |
SEPEHR AKHAVAN MASOULEH$^\ast$, BABAK SHAHBABA, DANIEL L. GILLEN\
*Department of Statistics, University of California Irvine*\
[[email protected]]{}
bibliography:
- 'refs.bib'
title: 'Flexible Joint Longitudinal-Survival Models for Quantifying the Association Between Longitudinal Biomarkers and Survival Outcomes'
---
[S. Akhavan Masouleh and others]{}
[Joint Longitudinal-Survival, Bayesian Semi-parameteric model, Dirchilet Process Mixture Models, Proportional Hazard modelsa]{}
Introduction {#uniJointIntro}
============
Survival analysis often involves evaluating the effects of longitudinally measured biomarkers on mortality. When longitudinal measures are sparsely collected, incomplete, or prone to measurement error, including them directly as a traditional time-varying covariate in a survival model may lead to biased regression estimates ([@prentice1982]). Alternatively, one could apply a two-stage method, where the first stage consists of modeling the longitudinal components via a mixed-effects model, and in the second stage, the modeled values or their summaries (e.g., first-order trends) are included in a survival model ([@dafni1998], [@tsiatis1995]). However, this approach fails to account for uncertainty in the estimated longitudinal summary measures. To overcome these issues, several joint longitudinal-survival models have been proposed ([@prentice1982]; [@bycott1998]; [@hanson2011]; [@wang2001]; [@faucett1996]; [@brown2003]; [@wulfsohn1997]; [@song2002], and [@law2002]), where all these models account for uncertainty in longitudinal measures by modeling them simultaneously with the survival outcome. However, most existing joint models still rely on multiple restrictive parametric and semi-parametric assumptions and generally focus only on associating the first moment of the distribution of the longitudinal covariate with survival.
In this paper, we propose three flexible joint longitudinal-survival models that avoid simple distributional assumptions on longitudinal measures and allow for subject-specific baseline hazard in modeling a survival outcome. Our models are motivated by data on end-stage renal disease (ESRD) patients obtained from the United States Renal Data System (USRDS). Specifically, our interest lies in quantifying the association between the longitudinally measured serum albumin (a leading index of protein-energy malnutrition (PEM)) and time-to-death using a joint survival-longitudinal modeling approach.
Flexibility in our joint model is achieved in the longitudinal component via the use of a Gaussian process prior with a parameter that captures within-subject volatility in the longitudinally sampled albumin. The survival component of our proposed models quantifies the association between the longitudinally measured albumin and the risk of mortality using a Dirichlet process mixture of Weibull distributions. The clustering mechanism of the Dirichlet process provides a platform for borrowing information when estimating subject-specific baseline hazards in the survival component. Estimation for the longitudinal and survival parameters is carried out simultaneously via Bayesian parameter posterior sampling approach.
Methodology {#UniJointMethod}
===========
In this section, we provide the details of our proposed joint models for a longitudinal covariate, $\boldsymbol{X}$, and a survival outcome, $\boldsymbol{Y}$. Throughout this section, we consider $n$ independent subjects where $l_i$ longitudinal measurements, $X_{ij}$, are obtained for subject $i$ at time points $t_{ij}$, $j = 1, \dots, l_i$. Also, associated with each subject, there is an observed survival time, $Y_i \equiv \mbox{min}\{T_i, C_i\}$ and event indicator $\delta_i\equiv 1_{[Y_i=E_i]}$, where $T_i$ and $C_i$ denote the true event and censoring time for subject $i$, respectively. Further, we make the common assumption that $C_i$ is independent of $T_i$ for all $i$, $i=1,\dots,n$.
The Joint Model {#M_Joint}
---------------
Being interested in estimating the effect of longitudinal measures on survival outcomes, in specifying the joint model likelihood, we took a similar approach as [@brown2003], where we define the contribution of each subject to the joint model likelihood as the multiplication of the likelihood function of the longitudinal measures for that subject and her/his time-to-event likelihood that is conditioned on her/his longitudinal measures. Let $f^{(i)}_L$, $f^{(i)}_{S|L}$, and $f^{(i)}_{L, S}$ denote the longitudinal likelihood contribution, the conditional survival likelihood contribution, and the joint likelihood contribution for subject $i$. One can write the joint longitudinal-survival likelihood function as $$\begin{aligned}
f_{L, S} &=& \prod_{i=1}^{n}f^{(i)}_{L, S} = \prod_{i=1}^{n} \big(f^{(i)}_L \times f^{(i)}_{S|L}\big).\label{jointMLikeLiHood} \end{aligned}$$
Longitudinal Component {#M_Long}
----------------------
We motivate the development of the Gaussian process model for the longitudinal biomarker by first considering the following simple linear model for estimating the trend in the biomarker for a single subject $i$ with an $l_i \times 1$ vector of measure biomarkers of $\boldsymbol{X_i}$ which is of the form
$$\begin{aligned}
\boldsymbol{X_i} =
\begin{pmatrix}
X_i(t_{i1})\\
X_i(t_{i2})\\
\vdots\\
X_i(t_{il_i})
\end{pmatrix},\end{aligned}$$
where $$\begin{aligned}
\boldsymbol{X_i} | \beta^{(L)}_{i0} \sim N(\boldsymbol{\beta^{(L)}_{i0}},\boldsymbol{\Sigma_i}).\end{aligned}$$ with $\beta^{(L)}_{i0}$ as the subject-specific intercept, $\boldsymbol{\beta^{(L)}_{i0}}$ is vector of repeated $\beta^{(L)}_{i0}$ value that is of size $l_i \times 1$, and $\boldsymbol{\Sigma_i} = \sigma^2 I_{l_i \times l_i}$.
By adding a stochastic component that is indexed by time in the model, one can extend the model to capture non-linear patterns over time. Specifically, we consider a stochastic vector, $\boldsymbol{W}$, that is a realization from a Gaussian process prior, $W(t)$ with mean zero and covariance function $C(t,t')$. Thus for subject $i$, $\boldsymbol{W_i} \sim N_{l_i}(\boldsymbol{0}, \boldsymbol{C_{l_i \times l_i}})$, where $\boldsymbol{W_i} = (W_{t_{i1}}, \dots, W_{t_{il_i}})'$ and the $(j,j')$ element of $\boldsymbol{C_{l_i \times l_i}}$ is given by $C(t_{ij},t_{ij'})$,$j , j' \in \{1, \dots, l_i\}$. We characterize the covariance function, $\boldsymbol{C_{l_i \times l_i}}$, using the following squared exponential form
$$\begin{aligned}
\boldsymbol{C_{l_i \times l_i}}(j,j') = {\kappa_i}^2 e^{-\rho^2 (t_{ij} - t_{ij'})^2}.\end{aligned}$$
In this setting, the hyperparameter $\rho^2$ controls the correlation length, and $\kappa^2$ controls the height of oscillations ([@banerjee2014hierarchical]), and $t_{ij}$ and $t_{ij'}$ are two different time points. For notational simplicity, we define $\boldsymbol{K_i} = e^{-\rho^2 (t_{ij} - t_{ij'})^2} \text{ ; } j , j' \in \{1, \dots, l_i\}$, and re-write our longitudinal model as $$\begin{aligned}
\boldsymbol{X_i} | \beta^{(L)}_{i0}, {\kappa_i}^2, \rho^2, \sigma^2 \sim N(\boldsymbol{\beta^{(L)}_{i0}}, {\kappa_i}^2 \boldsymbol{K_i} + \sigma^2 I_{l_i \times l_i}), \end{aligned}$$
where $\sigma^2$ is assumed to be common across all subjects. The correlation length parameter $\rho^2$ controls the maximum distance in time between two time-dependent measurements to be still correlated. This distance for GP models is often called the practical range. [@diggle2007springer] defined the practical range for GP as the distance in time between two time-dependent measurements where the correlation between those two measurements is 0.05. With the squared exponential covariance function, that practical range distance is of the form $\sqrt{3/\rho^2}$. At a $\rho^2 = 0.1$, the practical range distance is 5.7 months which is a reasonable range for the real data on end-stage renal disease patients that was obtained from the USRDS. Hence, we fix $\rho^2$ to 0.1, where this value was obtained from the real data on end stage renal disease patients data. By defining our model in this way, subject-specific parameter ${\kappa^2_i}$ will have the role of capturing within-subject volatility of the longitudinal measures. In the context of the motivating USRDS example, ${\kappa^2_i}$ can be of primary scientific interest as it reflects the within-subject volatility (Figure \[Kappa2SimPlot\]) in serum albumin over time, which is hypothesized to be negatively correlated with longer survival time (Holsclaw et al, 2014).
We specify the longitudinal component of our joint model to have a likelihood of the form $$\begin{aligned}
\boldsymbol{X_i} | \boldsymbol{W_i}, \beta^{(L)}_{i0}, {\kappa_i}^2, \rho^2, \sigma^2 & \sim & N(\boldsymbol{\beta^{(L)}_{i0}} + \boldsymbol{W_i}, \sigma^2 I_{l_i \times l_i}),\label{uniJointLong1} \end{aligned}$$ where $\boldsymbol{X_i}$ is a vector of longitudinal measures on subject $i$, $\boldsymbol{W_i}$ is a Gaussian process stochastic vector, $\beta^{(L)}_{i0}$ is subject specific intercept for subject $i$, ${\kappa_i}^2$ is a subject-specific measure of volatility for subject $i$, $\rho^2$ is a fixed correlation length, $\sigma^2$ is a measurement error that is shared across all subjects, and finally $I_{l_i \times l_i}$ represents the identity matrix of size $l_i$ where $l_i$ is the number of longitudinal measures on subject $i$. The Gaussian process stochastic vector $\boldsymbol{W_i}$ is distributed Gaussian process as $$\begin{aligned}
\boldsymbol{W_i} | {\kappa_i}^2, \boldsymbol{t_i} & \overset{ind}{\sim} & GP_{m_i}(\vec{0}, {\kappa_i}^2\boldsymbol{K_i}),\label{uniJointLong2} \end{aligned}$$ where $\boldsymbol{t_i}$ is a vector of the time points at which longitudinal measures on subject $i$ were collected and $\boldsymbol{K_i} = e^{-\rho^2 (t_{ij} - t_{ij'})^2}$, with $t_{ij}$ and $t_{ij'}$ are the $j^{th}$ and ${j'}^{th}$ element of the time vector $\boldsymbol{t_i}$. We assume a Normal prior on the subject-specific random intercepts $\beta^{(L)}_{i0}$ that is of the form $$\begin{aligned}
\beta^{(L)}_{i0} & \overset{i.i.d}{\sim} & N(\mu_{\beta^{(L)}_{0}}, \sigma^2_{\beta^L_{0}}),\end{aligned}$$ where $\mu_{\beta^{(L)}_{0}}$ and $\sigma^2_{\beta^L_{0}}$ are prior mean and prior variance respectively. $\boldsymbol{K_i}$, where $i \in \{1, \dots, n\}$ with $n$ as the number of subjects in the study, are assumed to have a log-Normal prior with the prior mean $\mu_{\kappa^2}$ and the prior variance $\sigma_{\kappa^2}$ that is of the form $$\begin{aligned}
{\kappa_i}^2 & \overset{i.i.d}{\sim} & log-Normal(\mu_{\kappa^2}, \sigma_{\kappa^2}).\end{aligned}$$ The correlation length $\rho^2$ is assumed to be fixed and known in our model. Finally, the measurement error $\sigma^2$ is assumed to have a log-Normal prior of the form $$\begin{aligned}
\sigma^2 & \sim & log-Normal(\mu_{\sigma^2}, \sigma_{\sigma^2}),\end{aligned}$$ where $\mu_{\sigma^2}$ and $\sigma_{\sigma^2}$ are the prior mean and the prior variance respectively.
Survival Component {#M_Surv}
------------------
In order to quantify the association between a longitudinal biomarker and a time-to-event outcome, we define our survival component by using a multiplicative hazard model with the general form of $$\begin{aligned}
\lambda(T_i | \boldsymbol{{Z_i}^{(s)}}, \boldsymbol{{Z_i}^{(L)}}) = \lambda_0(T_i) exp\{\boldsymbol{\zeta^{(s)}} \boldsymbol{{Z_i}^{(s)}} + \boldsymbol{\zeta^{(L)}} \boldsymbol{{Z_i}^{(L)}(t)}\},\end{aligned}$$ where $\boldsymbol{{Z_i}^{(s)}}$ is a vector of baseline covariates, $\boldsymbol{{Z_i}^{(L)}} $ is a vector of longitudinal covariates from the longitudinal component of the model, $\lambda_0(T_i)$ denotes the baseline hazard function, and $\boldsymbol{\zeta^{(S)}}$ and $\boldsymbol{\zeta^{(L)}}$ are regression coefficients for the baseline survival covariates and the longitudinal covariates, respectively.
We consider a Weibull distribution for the survival component to allow for log-linear changes in the baseline hazard function over time. Thus we assume $$\begin{aligned}
T_i & \sim & Weibull(\tau, \lambda_i),\end{aligned}$$ where $T_i$ is the survival time, $\tau$ is the shape parameter of the Weibull distribution, and $exp\{\lambda_i\}$ is the the scale parameter of the Weibull distribution. One can write the density function for the Weibull distribution above for the random variable $T_i$ as
$$\begin{aligned}
\label{eq:WeibDist}
f(T_i | \tau, \lambda_i) & = & \tau {T_i}^{\tau - 1} exp\big(\lambda_i - exp(\lambda_i) {T_i}^{\tau}\big).\end{aligned}$$
In this case, the Weibull distribution is available in closed form providing greater computational efficiency. Under this parameterization, covariates can be incorporated into the model by defining $
\lambda_i = \boldsymbol{\zeta^{(s)}} \boldsymbol{{Z_i}^{(s)}} + \boldsymbol{\zeta^{(L)}} \boldsymbol{{Z_i}^{(L)}}$. In particular, we specify our model as $$\begin{aligned}
\label{eq:survModel}
T_i | \tau, \boldsymbol{\zeta^{(s)}}, \boldsymbol{\zeta^{(L)}}, \boldsymbol{{Z_i}^{(s)}}, \boldsymbol{{Z_i}^{(L)}} & \sim & Weibull(\tau, \lambda_i = \beta^{(s)}_{i0} + \boldsymbol{\zeta^{(s)}} \boldsymbol{{Z_i}^{(s)}} + \boldsymbol{\zeta^{(L)}} \boldsymbol{{Z_i}^{(L)}}),\end{aligned}$$ where $\tau$ is a common shape parameter shared across all subjects. $\beta^{(s)}_{i0}$ is a subject specific coefficient in the model which allows for a subject-specific baseline hazard. $\boldsymbol{{Z_i}^{(s)}}$ and $\boldsymbol{\zeta^{(s)}}$ are baseline covariates and their corresponding coefficients, respectively. Finally, $\boldsymbol{{Z_i}^{(L)}}$ and $\boldsymbol{\zeta^{(L)}}$ are coefficients linking the longitudinal parameters of interest to the hazard for mortality.
In order to avoid an explicit distributional assumption for the survival times, we specify our survival model as an infinite mixture of Weibull distributions that is mixed on the $\beta^{(s)}_{i0}$ parameter. In particular, we use the Dirichlet process mixture of Weibull distributions that is defined as $$\begin{aligned}
\label{eq:SurvDPMprior}
\beta^{(s)}_{i0} | \mu_i, \sigma^2_{\beta^{(s)}_0} &\sim& N(\mu_i, \sigma^2_{\beta^{(s)}_0}), \\
\mu_i | G &\sim& G, \\
G &\sim& DP\big(\alpha^{(S)}, G_0),\end{aligned}$$ where $\sigma^2_{\beta^{(s)}_0}$ is a fixed parameter, $\mu_i$ is a subject-specific mean parameter from a distribution $G$ with a DP prior, $\alpha^{(S)}$ is the concentration parameter of the DP and $G_0$ is the base distribution. By using the Dirichlet process prior on the distribution of $\beta^{(s)}_{i0}$, we allow patients with similar baseline hazards to cluster together which subsequently provides a stronger likelihood to estimate the baseline hazards. For other covariates in the model, we assume a multivariate normal prior of the form $$\begin{aligned}
\big( \boldsymbol{\zeta^{(s)}}, \boldsymbol{\zeta^{(L)}} \big) & \sim & MVN(\boldsymbol{0}, {\sigma_0}^2 I),\end{aligned}$$ where ${\sigma_0}^2$ is a prior variance and $I$ is an identity matrix.
The shared scale parameter $\tau$ is considered to have a Log-Normal prior of the form $$\begin{aligned}
\tau \sim Log-Normal(a_{\tau}, b_{\tau}),\end{aligned}$$ where $a_{\tau}$ and $b_{\tau}$ are fixed prior location and prior scale parameters, respectively.
Finally, we assume that information about the concentration parameter of the Dirichlet process can be specified with the prior $$\begin{aligned}
\alpha^{(S)} \sim \Gamma(a_{\alpha}^{(S)}, b_{\alpha}^{(S)}),\end{aligned}$$ where $a_{\alpha}^{(S)}$ and $b_{\alpha}^{(S)}$ are fixed prior shape and prior scale parameters, respectively.
Linking Summary Measures of the Biomarker to Survival Times
-----------------------------------------------------------
The proposed modeling framework easily allows for associating multiple summaries of the longitudinal biomarker with the time-to-event outcome. Here we consider three models that incorporate various summary measures of the longitudinal trajectory that are easily and flexibly estimated using the GP model presented in Section \[M\_Long\]:
- **Model I**: directly modeling longitudinal outcome at each event time $t$ as a covariate in the survival model: $${Z_i}^{(L)} = X^i(t)$$
- **Model II**: modeling both the value of the longitudinal covariate and also the average rate at which the biomarker changes for each subject. We define this average rate as a weighted area under the derivative curve of the biomarker trajectory $$\boldsymbol{{Z_i}^{(L)}} = \big(X_i(t), {X'_{AUC}}^{\tau_0-\tau_1}\big)$$ $$\text{where, } {X'_{AUC}}^{\tau_0-\tau_1} = \int_{\tau_0}^{\tau_1} Q(u)X'(u)du$$ where ${X'_{AUC}}^{\tau_0-\tau_1}$ is a time-dependent covariate that is a weighted average of the derivative of the biomarker trajectory, that is denoted by $X'(u)$ from $\tau_0$ to $\tau_1$ where $\tau_1$ is the time of death for each subject. This average area under the derivative curve can be a weighted average with weights $Q(u)$.
- **Model III**: modeling summary measures of the longitudinal trajectory. Motivated by the scientific question of interest, in this paper we consider **random intercepts** and **subject-specific volatility** as summary measures of interest: $$\boldsymbol{{Z_i}^{(L)}} = \big(\beta^{(L)}_{0i},\kappa^2_i \big)$$
Below, we shall explain the three models above in more detail.
### Model I: a Survival model with Longitudinal Biomarker at event time as a covariate
This model quantifies the association between a longitudinal biomarker of interest and the time-to-event outcome by directly adjusting for the biomarker measured values in the survival component. While usually biomarkers are measured on a discrete lab-visit basis, the event of interest happens on a continuous basis. While common frequentist models use the so-called last-observation-carried forward (LOCF) technique where the biomarker value at each even time is assumed to be the same as the last measured value for that biomarker, our joint flexible longitudinal-survival model provides a proper imputation method for the biomarker values at each individual’s event time. In particular, in each iteration of the MCMC, given the sampled parameters for each individual and by using the flexible Gaussian process prior, there exists posterior trajectories of biomarker for that individual. Our method, then, considers the posterior mean of those trajectories as the proposed trajectory for that individual’s biomarker values over time at that iteration. The trajectory, then, can be used to impute time-dependent biomarker covariate value inside the survival component. To be more specific, consider the longitudinal biomarker $\boldsymbol{X_i}$ of the form $$\begin{aligned}
\boldsymbol{X_i} | \beta^{(L)}_{i0}, {\kappa_i}^2, \rho^2, \sigma^2 & \sim & N(\boldsymbol{\beta^{(L)}_{i0}}, {\kappa_i}^2\boldsymbol{K_i} + \sigma^2 I_{l_i \times \boldsymbol{l_i}}),\end{aligned}$$ where $\beta^{(L)}_{i0}$ is subject-specific random intercept for subject $i$, $\boldsymbol{\beta^{(L)}_{i0}}$ is a vector of repeated subject-specific intercept $\beta^{(L)}_{i0}$ that is of size $l_i \times 1$, ${\kappa_i}^2$ is subject-specific measure of volatility in the longitudinal biomarker for individual $i$, $\rho^2$ is a fixed measure of correlation length, $\sigma^2$ is the measurement error shared across all subjects, $\boldsymbol{K_i}$ is a an $l_i \times l_i$ matrix with it’s $jj'$ element as $\boldsymbol{K_i}_{jj'} = e^{-\rho^2 (t_{ij} - t_{ij'})^2}$ where $l_i$ is the number of longitudinal biomarker measures on subject $i$, and $I_{l_i \times l_i}$ is the identity matrix.
For a new time-point $t^{*}$, predicted albumin biomarker for individual $i$ is $X^{*}$ and can be written as $$\begin{aligned}
X^{*} | \boldsymbol{X_i}, \boldsymbol{t}, t^{*} & \sim & N(\mu^{*}, \Sigma^{*}),\end{aligned}$$ where the conditional posterior mean $\mu^{*}$ is $$\begin{aligned}
\label{PostMeanPaper1}
\mu^{*} = \beta^L_{i0} + K(t^*, \boldsymbol{t}) {K_{X}}^{-1} (\boldsymbol{X_i} - \beta^{(L)}_{i0}),\end{aligned}$$ and the conditional posterior variance $\Sigma^{*}$ is $$\begin{aligned}
\label{eq:PostVar}
\Sigma^{*} = K(t^*, t^*) - K(t^*, \boldsymbol{t}) {K_{X}}^{-1} K(t^*, \boldsymbol{t})',\end{aligned}$$ where $K(t^*, \boldsymbol{t})$ is defined as $$\begin{aligned}
K(t^*, \boldsymbol{t}) = {\kappa_i}^2 e^{-\rho^2 (t^* - \boldsymbol{t})^2},\end{aligned}$$ and ${K^{-1}_X}$ is defined as $$\begin{aligned}
{K^{-1}_X} = (K(\boldsymbol{t}, \boldsymbol{t}) + \sigma^2 I_{l_i \times l_i})^{-1}.\end{aligned}$$ In order to relate the biomarker value at each time point $t$ to the risk of the event of interest at that time point, “death”, we form the survival component of the model as $$\begin{aligned}
T_i | \tau, \boldsymbol{\zeta^{(s)}}, \zeta_{X_{i}} & \sim & Weibull(\tau, \lambda_i = \beta^{(s)}_{i0} + \boldsymbol{\zeta^{(s)}} \boldsymbol{{Z_i}^{(s)}} + {\zeta_{X_{i}}} X_{i}(t)),\end{aligned}$$ where $T_i$ is the survival time, $\tau$ is the shape parameter of the Weibull distribution, $\boldsymbol{\zeta^{(s)}}$ is a vector of coefficients relating baseline survival covariates to the risk of the occurrence of the event of interest, $\zeta_{X_{i}}$ is the coefficient that relates the biomarker value at time $t$ and the risk of “death” at that time point, $\lambda_i$ is the log of the scale parameter in the Weibull distribution, $\beta^{(s)}_{i0}$ is the subject-specific baseline hazard for subject $i$, $\boldsymbol{{Z_i}^{(s)}}$ is a vector of survival coefficients, and $X_{i}(t)$ is the biomarker value at time $t$.
### Model II: A Survival model with covariates of Biomarker Value and the Derivative of its Trajectory at event Time
In order to get more precision in quantifying the association between the biomarker value at time $t$ and the risk of death, we can extend our proposed model-I by including a measure of the average slope of the biomarker over time. In particular, we define this average slope from time $\tau_0$ to $\tau_1$ as the area under the derivative of the trajectory curve of the biomarker from $\tau_0$ to $\tau_1$. More generally, this area under the curve can be a weighted sum where weights are chosen according to the scientific question of interest. One may hypothesize that the area under the derivative curve that are closer to the event time should be weighted higher compared to the areas that are farther away from time point $t$. In general, we define a weighted area under the derivative curve of the form $$\begin{aligned}
{X'_{AUC}}^{\tau_0-\tau_1} = \int_{\tau_0}^{\tau_1} Q(u)X'(u)du,\end{aligned}$$ where $\tau_0$ and $\tau_1$ are arbitrary time points chosen according to the scientific question of interest, $Q(t)$ is a weight, and $X'(t)$ represents the derivative of the biomarker over time. In particular, we consider two weighted approaches, where one assumes an equal weight of the form $$\begin{aligned}
Q(t) = \frac{1}{\tau_1 - \tau_0},\end{aligned}$$ and another weight of the form $$Q(t)=
\begin{cases}
1,& \text{if } t = T_i\\
0,& \text{otherwise}.
\end{cases}$$ Under the first weighting scheme, ${X'_{AUC}}$ will be the area under the derivative with equal weights, whereas the second weighting scheme leads to the pointwise derivative value at the the event time. Under this model, the survival component of our joint model will now include two longitudinal covariates, one the biomarker value $X_i(t)$, and another the average derivative of the biomarker trajectory, ${X'_{AUC}}$.
The derivative of the Gaussian process is still a Gaussian process with the same hyper parameters $\rho^2$ and $\kappa^2$. Therefore, using the same idea of modeling the trajectory of the biomarker, we can also model the derivative of that trajectory. In our model-I, we proposed using the posterior mean of all plausible biomarker trajectories as the proposed trajectory for each subject in order to impute biomarker values at any time point $t$ inside the survival component of the model. Similarly, we propose using the posterior mean of all plausible derivative trajectories for each subject in order to compute the average derivative up until time $t$. Given the fact that differentiation is a linear operation, one can easily compute the posterior mean of the derivative curve by simply switching the order of the differentiation and the expectation as $$\begin{aligned}
E(X'_i(t)) & = & E(\frac{\partial X_i(t)}{\partial t}) \\
& = & \frac{\partial \big(E(X_i(t))\big)}{\partial t}.\end{aligned}$$ Hence, by using Formula and by taking the derivative of the posterior mean trajectory of the biomarker with respect to time $t^{*}$, the posterior mean of the derivative of the biomarker trajectory is of the form $$\begin{aligned}
\label{eq:DerPostMean}
\frac{\partial \big(E(X_i(t^*))\big)}{\partial t^*} = -2 \rho^2 (t^{*} - \boldsymbol{t})' \big(K(t^*, \boldsymbol{t}) {K_{X}}^{-1} (\boldsymbol{X_i} - \beta^{(L)}_{i0})\big), \end{aligned}$$ where $E\big(X_i(t^*)\big)$ denotes the posterior mean of the biomarker trajectory as a function of time $t^{*}$, $\rho^2$ is the correlation length, $t^*$ is the time-point at which we desire to impute the biomarker value and the average derivative of the biomarker trajectory, $\beta^{(L)}_{i0}$ is subject-specific random intercept, $K(t^*, \boldsymbol{t})$ is defined as $$\begin{aligned}
K(t^*, \boldsymbol{t}) = {\kappa_i}^2 e^{-\rho^2 (t^* - \boldsymbol{t})^2},\end{aligned}$$ and ${K^{-1}_X}$ is defined as $$\begin{aligned}
{K^{-1}_X} = (K(\boldsymbol{t}, \boldsymbol{t}) + \sigma^2 I_{l_i \times l_i})^{-1}.\end{aligned}$$ Given the the biomarker value $X_i(t)$ and the average derivative value $X'_{AUC, i}(t)$, the survival component of our proposed joint model is of the from $$\begin{aligned}
T_i | \tau, \boldsymbol{\zeta^{(s)}}, \zeta_{X_{i}} , \zeta_{X'_{i}} & \sim & Weibull(\tau, \lambda_i = \beta^{(s)}_{i0} + \boldsymbol{\zeta^{(s)}} \boldsymbol{{Z_i}^{(s)}} + {\zeta_{X_{i}}} X_{i}(t) + {\zeta_{X'_{i}}} X'_{AUC, i}(t)),\end{aligned}$$ where $T_i$ is the survival time, $\tau$ is the shape parameter of the Weibull distribution, $\boldsymbol{\zeta^{(s)}}$ is a vector of coefficients relating baseline survival covariates to the risk of the occurrence of the event of interest, $\zeta_{X_{i}}$ is the coefficient that relates the biomarker value at time $t$ and the risk of “death” at that time point, $\zeta_{X'_{i}}$ is the coefficient that relates the average derivative of the biomarker trajectory up until $t$ and the risk of “death” at that time point, $\lambda_i$ is the log of the scale parameter in the Weibull distribution, $\beta^{(s)}_{i0}$ is the subject-specific baseline hazard for subject $i$, and $\boldsymbol{{Z_i}^{(s)}}$ is a vector of survival coefficients.
### Model III: A Survival Model with Summary Measures of the Longitudinal Curve as Covariates
One may choose to characterize longitudinal trajectories with summary measures instead of using the actual biomarker value. In specific, the longitudinal model we proposed provides a natural parameter for describing the within-subject volatility. Given the nature of our proposed longitudinal model, one can summarize the longitudinal trajectory of biomarker by using $\beta^{(L)}_{0i}$ as a measure of subject-specific intercept of longitudinal biomarker as well as $\kappa^2_i$ as a measure of volatility of those trajectories. The survival component of the model is then of the form $$\begin{aligned}
T_i | \tau, \beta^{(s)}_{i0}, \boldsymbol{\zeta^{(s)}}, {\zeta_{\beta_{0i}}}^{(L)}, {\zeta_{{\kappa_i}^2}}^{(L)} & \sim & Weibull(\tau, \lambda_i = \beta^{(s)}_{i0} + \boldsymbol{\zeta^{(s)}} \boldsymbol{{Z_i}^{(s)}} + {\zeta_{\beta_{0i}}}^{(L)} {\beta_{0i}}^{(L)} + {\zeta_{{\kappa_i}^2}}^{(L)} {\kappa_{i}}^2),\end{aligned}$$ where $T_i$ is the survival time, $\tau$ is the shape parameter of the Weibull distribution, $\boldsymbol{\zeta^{(s)}}$ is a vector of coefficients relating baseline survival covariates to the risk of the occurrence of the event of interest, ${\zeta_{\beta_{0i}}}^{(L)}$ is the coefficient that relates the subject specific random intercept ${\beta_{0i}}^{(L)}$ and the risk of “death”, ${\zeta_{{\kappa_i}^2}}^{(L)}$ is the coefficient that relates the subject-specific measure of volatility of the biomarker measure and the risk of “death”, $\lambda_i$ is the log of the scale parameter in the Weibull distribution, $\beta^{(s)}_{i0}$ is the subject-specific baseline hazard for subject $i$, and $\boldsymbol{{Z_i}^{(s)}}$ is a vector of survival coefficients.
The Posterior Distribution
--------------------------
Consider the joint longitudinal-survival likelihood function, $f_{\boldsymbol{L}, \boldsymbol{S}}$, introduced in equation \[jointMLikeLiHood\]. Let $\boldsymbol{\omega}$ be a vector of all model parameters with the joint prior distribution $\pi(\boldsymbol{\omega})$. The posterior distribution of the parameter vector $\boldsymbol{\omega}$ can be written as $$\begin{aligned}
\pi(\boldsymbol{\omega} | \boldsymbol{X}, \boldsymbol{Y}) &\propto& f_{L, S} \times \pi(\boldsymbol{\omega}), \label{jointPostDist}\end{aligned}$$ where $\boldsymbol{X}$ and $\boldsymbol{Y}$ denote longitudinal and time-to-event data respectively, and $f_{L,S}$ is the joint model likelihood function (equation \[jointMLikeLiHood\]).
The posterior distribution of the parameters in our proposed joint model is not available in closed form. Hence, samples from the posterior distribution of the model parameters are obtained via Markov Chain Monte Carlo (MCMC) methods. We use a hybrid sampling technique where in each iteration of the MCMC, we first sample subject-specific frailty terms in the survival model using Neal’s algorithm 8. Then given the sampled frailty terms, we use the Hamiltonian Monte Carlo ([@HMC]) to draw samples from the posterior distribution. Prior distributions on parameters of the joint model were explained in details in Sections \[M\_Long\] and \[M\_Surv\], and we assume independence among model parameters in the prior (ie. $\pi(\boldsymbol{\omega})$ is the product of the prior components specified previously). We provide further detail on less standard techniques for sampling from the posterior distribution when using a GP prior and issues in evaluating the survival portion of the likelihood function when time-varying covariates are incorporated into the model.
### Evaluation of the Longitudinal Likelihood {#fastUniGP}
The longitudinal component of our model uses the Gaussian process technique. Gaussian process models are typically computationally challenging to fit because in each iteration of the MCMC the evaluation of the log-posterior probability becomes computationally challenging as the number of measurements increases. In particular, consider our proposed longitudinal model introduced in Section \[M\_Long\] where $$\begin{aligned}
\boldsymbol{X_i} | \boldsymbol{W_i}, \beta^{(L)}_{i0}, {\kappa_i}^2, \rho^2, \sigma^2 & \sim & N(\boldsymbol{\beta^{(L)}_{i0}} + \boldsymbol{W_i}, \sigma^2 I_{l_i \times l_i}),\nonumber \\
\boldsymbol{W_i} | {\kappa_i}^2, \boldsymbol{t_i} & \sim & GP_{m_i}(\vec{0}, {\kappa_i}^2\boldsymbol{K_i}),\nonumber\end{aligned}$$ with $\boldsymbol{X_i}$ denoting a vector of longitudinal measures on subject $i$, $\boldsymbol{W}_i$ a Gaussian process stochastic vector, $\beta^{(L)}_{i0}$ a subject specific intercept for subject $i$, ${\kappa_i}^2$ a subject-specific measure of volatility for subject $i$, $\rho^2$ a fixed correlation length, $\sigma^2$ a measurement error that is shared across all subjects, $I_{l_i \times l_i}$ denoting the identity matrix of size $l_i$ with $l_i$ the number of longitudinal measures on subject $i$, $\boldsymbol{t_i}$ a vector of the time points at which longitudinal measures on subject $i$ were collected, and $\boldsymbol{K_i} = e^{-\rho^2 (t_{ij} - t_{ij'})^2}$, where $t_{ij}$ and $t_{ij'}$ are the $j^{th}$ and ${j'}^{th}$ element of the time vector $\boldsymbol{t_i}$.
In order to sample from the posterior distribution of ${\kappa_i}^2$ and $\sigma^2$ parameters, one can consider a marginal distribution of the following form $$\begin{aligned}
\label{uniGPlongMargLike}
\boldsymbol{X_i} | \beta^{(L)}_{i0}, {\kappa_i}^2, \rho^2, \sigma^2 & \sim & N(\beta^{(L)}_{i0}, {\kappa_i}^2\boldsymbol{K_i} + \sigma^2 I_{l_i \times l_i})\end{aligned}$$ The marginal distribution above has log-density of the form $$\begin{aligned}
\label{uniGPlongMargLikeDens}
log(f(\boldsymbol{X_i} | \beta^{(L)}_{i0}, {\kappa_i}^2, \rho^2, \sigma^2)) &=& \text{constant} \nonumber\\
&-& \frac{1}{2}log|{\kappa_i}^2\boldsymbol{K_i} + \sigma^2 I_{l_i \times l_i}| \\
&-& \frac{1}{2} (\boldsymbol{X_i} - \beta^{(L)}_{i0})^{T} ({\kappa_i}^2\boldsymbol{K_i} + \sigma^2 I_{l_i \times l_i})^{-1}(\boldsymbol{X_i} - \beta^{(L)}_{i0}),\nonumber\end{aligned}$$ that is the log contribution of subject $i$ to the longitudinal likelihood ( i.e. $log(f^{(i)}_L)$ ).
sampling from the posterior distribution of ${\kappa_i}^2$ and $\sigma^2$ requires evaluation of the log-density in equation (\[uniGPlongMargLikeDens\]) that involves evaluation of the determinant and the computation of the inverse of the covariance matrix at each iteration of the MCMC. This process requires $O({l_i}^2)$ memory space and a computation time of $O({l_i}^3)$ per subject, with $l_i$ as the number of within subject measurements.
In our model setting, we defined $\boldsymbol{K_i} = e^{-\rho^2 (t_{ij} - t_{ij'})}$ with a fixed $\rho^2$ parameter. This means $K_i$ can be pre-computed before starting posterior sampling using MCMC. Furthermore, we propose using the eigenvalue decomposition technique for a faster log-posterior probability computation. Our proposed method was motivated by [@flaxman2015fast] and is as follows.
Consider the covariance matrix ${\kappa_i}^2\boldsymbol{K_i} + \sigma^2 I_{l_i \times l_i}$ in the marginal log-density in equation (\[uniGPlongMargLikeDens\]). Our goal is now to propose a method that makes computation of the inverse and the determinant of this covariance function as efficient as possible. As shown earlier, $\boldsymbol{K_i}$ can be pre-computed before starting the MCMC process as it does not involve any parameter. Consider the eigenvalue decomposition of $\boldsymbol{K_i} = Q \Lambda Q^{T}$, where $\Lambda$ is a diagonal matrix with the eigenvalues of $\boldsymbol{K_i}$ as the diagonal elements, and $Q$ is the corresponding matrix of eigenvectors. ${\kappa_i}^2$ is a scalar parameter that is sampled in each iteration of the MCMC. Multiplication of ${\kappa_i}^2$ times the matrix $\boldsymbol{K_i}$ implies the eigenvalues of this matrix will be ${\kappa_i}^2$ times bigger where the eigenvectors remain the same. Hence, we can conclude that the eigenvalue decomposition of the matrix ${\kappa_i}^2\boldsymbol{K_i}$ is of the form ${\kappa_i}^2\boldsymbol{K_i} = Q ({\kappa_i}^2\Lambda) Q^{T}$, where $Q$ and $\Lambda$ are elements of the eigenvalue decomposition of the pre-computed matrix $\boldsymbol{K_i}$. Given the pre-computed eigenvalue decomposition of the matrix $\boldsymbol{K_i}$, at each iteration of the MCMC, the determinant of the covariance function of the marginal log-density in equation (\[uniGPlongMargLikeDens\]) can be computed as $$\begin{aligned}
\label{uniGPlongMargLikeDeterminant}
log|{\kappa_i}^2\boldsymbol{K_i} + \sigma^2 I_{l_i \times l_i}| &=& log|Q ({\kappa_i}^2\Lambda) Q^{T} + \sigma^2 I_{l_i \times l_i}| \nonumber \\
&=& log|Q({\kappa_i}^2\Lambda + \sigma^2 I_{l_i \times l_i})Q^{T}| \nonumber \\
&=& log\big(\prod_{k=1}^{l_i}({\kappa_i}^2\lambda_{ik} + \sigma^2)\big) \nonumber \\
&=& \sum_{k=1}^{l_i}log\big({\kappa_i}^2\lambda_{ik} + \sigma^2 \big).\end{aligned}$$ In equation (\[uniGPlongMargLikeDeterminant\]), $\lambda_{ik}$’s are pre-computed eigenvalues of the matrix $\boldsymbol{K_i}$ whereas $\kappa_i$ and $\sigma^2$ are parameters sampled at each iteration of the MCMC.
Similarly and by using the same trick, we can compute the term $(\boldsymbol{X_i} - \beta^{(L)}_{i0})^{T} ({\kappa_i}^2\boldsymbol{K_i} + \sigma^2 I_{l_i \times l_i})(\boldsymbol{X_i} - \beta^{(L)}_{i0})$ in a more computationally efficient as [$$\begin{aligned}
\label{uniGPlongMargLikeInvMatrix}
(\boldsymbol{X_i} - \beta^{(L)}_{i0})^{T} ({\kappa_i}^2\boldsymbol{K_i} + \sigma^2 I_{l_i \times l_i})^{-1}(\boldsymbol{X_i} - \beta^{(L)}_{i0}) &=& (\boldsymbol{X_i} - \beta^{(L)}_{i0})^{T} \big(Q({\kappa_i}^2 \Lambda) Q^{T} + \sigma^2 I_{l_i \times l_i}\big)^{-1} (\boldsymbol{X_i} - \beta^{(L)}_{i0}) \nonumber \\
&=& (\boldsymbol{X_i} - \beta^{(L)}_{i0})^{T} \big(Q({\kappa_i}^2 \Lambda + \sigma^2 I_{l_i \times l_i})Q^{T}\big)^{-1} (\boldsymbol{X_i} - \beta^{(L)}_{i0}) \nonumber \\
&=&(\boldsymbol{X_i} - \beta^{(L)}_{i0})^{T} \big(Q({\kappa_i}^2 \Lambda + \sigma^2 I_{l_i \times l_i})^{-1}Q^{T}\big) (\boldsymbol{X_i} - \beta^{(L)}_{i0}).\nonumber \\\end{aligned}$$ ]{} In equation (\[uniGPlongMargLikeInvMatrix\]), $\boldsymbol{X_i}$ is the data matrix and is fixed, $Q$ and $\Lambda$ are pre-computed eigenvector and diagonal eigenvalue matrices corresponding to the eigenvalue decomposition of the matrix $\boldsymbol{K_i}$. Finally, by utilizing an eigenvalue decomposition, instead of evaluating the term $({\kappa_i}^2\boldsymbol{K_i} + \sigma^2 I_{l_i \times l_i})^{-1}$, one can simply evaluate $\big(Q({\kappa_i}^2 \Lambda + \sigma^2 I_{l_i \times l_i})^{-1}Q^{T}\big)$, where the term $({\kappa_i}^2 \Lambda + \sigma^2 I_{l_i \times l_i})^{-1}$ in the middle is simply the inverse of a diagonal matrix.
### Evaluation of the Survival Likelihood {#UniGPJointSurvEval}
Here we consider evaluation of the survival component of the decomposed joint likelihood. Consider the survival time for subject $i$ that is denoted by $t_i$ and is distributed according to a Weibull distribution with shape parameter $\tau$ and scale parameter $exp(\lambda_i)$, where $\lambda_i = \boldsymbol{\zeta^{(S)}} \boldsymbol{Z}^{(S)}_i + \boldsymbol{\zeta^{(L)}} \boldsymbol{Z}^{(L)}_i(t)$, where $\boldsymbol{Z}^{(S)}_i$ and $\boldsymbol{Z}^{(L)}_i(t)$ are vectors of covariates for subject $i$, with potentially time-varying covariates, corresponding to the survival and the longitudinal covariates respectively, and $\boldsymbol{\zeta^{(S)}}$ and $\boldsymbol{\zeta^{(L)}}$ are vectors of survival and longitudinal coefficients respectively. One can write the hazard function $h_i(t)$ as $$\begin{aligned}
h_i(t) &= \tau t^{\tau - 1} exp(\lambda_i - exp(\lambda_i) t^\tau). \label{UniGPSurLikeForm}\end{aligned}$$ The survival function $S_i(t)$ can be written as $$\begin{aligned}
S_i(t) &= exp\{-\int_{0}^{t} h_i(w)dw\}.\end{aligned}$$ Consider survival data on $n$ subjects, some of whom may have been censored. Let event indicator $\delta_i$ that is $1$ if the event is observed, and $0$ otherwise. The survival likelihood contribution of subject $i$ can be written in terms of the the hazard function $h_i(t)$ and the survival function $S_i(t)$ as $$\begin{aligned}
f^{(i)}_{S|L} &= h_i(t_i)^{\delta_i} S_i(t_i) \\
&= h_i(t_i)^{\delta_i} e^{-\int_{0}^{t_i}h_i(w)dw}.\end{aligned}$$ The overall survival log-likelihood can be written as $$\begin{aligned}
log(L) &= \sum_{i = 1}^{n} log(f^{(i)}_{S|L}) \\
&= \sum_{i = 1}^{n}\big(\delta_i log(h_i(t_i)) - \int_{0}^{t_i}h_i(w)dw\big).\end{aligned}$$ The hazard function in the equation (\[UniGPSurLikeForm\]) includes some time-varying covariates which often makes the integral of the hazard function non-tractable. In this case, one can estimate the integral using the rectangular integration as follows:
1\. Set a fixed number of rectangles $m$ and set $A = 0$ 2. Divide $(0, t_i)$ interval into $m$ equal pieces each of length $L = t_i/m$ $t_{mid} \leftarrow L/2 + (i - 1)*L$ $A_{temp} \leftarrow L*h_i(t_{mid})$ $A \leftarrow A + A_{temp}$
Simulation Studies {#UniJointSim}
------------------
In this section, we evaluate our proposed models using a simulation study. We simulated 200 datasets that resembled the real data on end stage renal disease patients that was obtained from the United States Renal Data System (USRDS). To this end, we first simulated longitudinal trajectories with $\kappa^2$’s which are sampled from a uniform distribution from $0$ to $1$. We fixed $\rho^2 = 0.1$ for all subjects. The subject-specific intercepts for albumin trajectories were randomly sampled from the Normal distribution $N(\mu = 5.0, \sigma^2 = 0.5)$. We simulate 9 to 12 longitudinal albumin values per subject. Using the simulated albumin trajectories, we generated survival times from the Weibull distribution in equation (\[eq:survModel\]) that is of the following form for each of the proposed models
- **Model I**: $$\begin{aligned}
\label{eq:SimSurvModel1}
T_i | \tau, \beta_1, X_i(t) & \sim & Weibull(\tau, \lambda_i = \beta^{(s)}_{i0} + \beta_1 X_i(t) ),\end{aligned}$$
- **Model II**: $$\begin{aligned}
\label{eq:SimSurvModel2}
T_i | \tau, \beta_1, X_i(t) & \sim & Weibull(\tau, \lambda_i = \beta^{(s)}_{i0} + \beta_1 X_i(t) + \beta_2 X'_{AUC, i}(t) ),\end{aligned}$$
- **Model III**: $$\begin{aligned}
\label{eq:SimSurvModel3}
T_i | \tau, \beta_1, X_i(t) & \sim & Weibull(\tau, \lambda_i = \beta^{(s)}_{i0} + \beta_1 Gender + \beta_2 {{\beta_0}^{(L)}} + \beta_3 {{\kappa_i}^2}).\end{aligned}$$
The true values of the coefficients are set as follows
- model I: $\beta_1 = -0.5$,
- model II: $\beta_1 = 0.5$,
- model III: $\beta_1 = 0.5$,
where in all simulations, $\beta^{(s)}_{i0}$ are simulated from a mixture of two Normal distributions of the form $\theta_{i} N(\mu = -1.5, \sigma = 1) + (1-\theta_{i}) N(\mu = 1.5, \sigma = 1)$, where $\theta_i$ is distributed Bernoulli with parameter $p = 0.5$.
Finally, the censoring times were sampled from a uniform distribution and independently from the simulated event times with an overall censoring rate of 20%.
All results are from 200 simulated datasets of size $n=300$ subjects each. For each dataset, we fit our proposed joint models with 10,000 draws where the first 5,000 considered as a burn-in period. Relatively diffuse priors were considered for all parameters. Details of the priors used in the simulations ass well as the results are as follow.
Model I - Simulation Results
----------------------------
In order to compare our proposed joint longitudinal-survival model that is capable of flexibly modeling longitudinal trajectories with simpler models with explicit functional assumptions on the longitudinal trajectories, we simulated longitudinal data once from quadratic polynomial longitudinal trajectory curves and another time from random non-linear curves. We then fit our joint model with a Gaussian Process longitudinal component as well as a joint model with the explicit assumption that the longitudinal trajectories are from a quadratic polynomial curve. As a comparison model, we also fit a two-stage Cox model where in stage one longitudinal data are modeled using our proposed Gaussian process longitudinal model and in the second stage, given the posterior mean parameters from the longitudinal fit, a Cox proportional hazard will fit the survival data.
In particular, we generate synthetic longitudinal and survival data on 300 subjects, each with 9 to 12 within subject longitudinal albumin measures. Under the scenario where the longitudinal data are generated from quadratic polynomial longitudinal trajectories, we consider quadratic polynomial curves of the form $$\begin{aligned}
X_{ij} = \beta^{(L)}_{0i} + \beta_{1i} t + \beta_{2} t^2 + \epsilon_{ij},\end{aligned}$$ where the true value of $\beta_{0i}$ are simulated from the Normal distribution $N(\mu = 5, \sigma = 1)$, $\beta_{1i}$ are simulated from the Normal distribution $N(\mu = -0.5, \sigma = 0.1)$, $\beta_{2}$ is set to be equal to -0.1, and finally $\epsilon_{ij}$ is the measurement error that is independent across measures and across subjects and are simulated from the Normal distribution $N(\mu = 0, \sigma = 0.1)$.
Under the second scenario, longitudinal albumin values are generated from random non-linear curves. In particular, we generate random non-linear albumin trajectories that are realizations of a Gaussian process that are centered around the subject-specific random intercepts $\beta^{(L)}_{0i}$ that are generated from the Normal distribution $N(\mu = 5, \sigma = 1)$. We consider a Gaussian process with the squared exponential covariance function with the correlation length of $\rho^2 = 0.1$ and the subject-specific measures of volatility $\kappa^2_i$ that are generated from the uniform distribution $U(0, 1)$.
For each simulation scenario, once longitudinal measures are generated, we generate survival data where survival times are distributed according to the Weibull distribution $Weibull(\tau, \lambda_i)$, where the shape parameter $\tau$ is set to 1.5 and $\lambda_i$, which is the log of the scale parameter of the Weibull distribution, is set to $\beta_{i0}^{(S)} + \beta_1 X_i(t)$, where $\beta_{i0}^{(S)}$ are generated from an equally weighted mixture of two Normal distributions of $N(\mu = -1.5, \sigma = 1)$ and $N(\mu = 1.5, \sigma = 1)$, $\beta_1$ is fixed to -0.5, and $X_i(t)$ is the longitudinal value for subject $i$ at time $t$ that is already simulated in the longitudinal step of the data simulation.
Our proposed joint longitudinal-survival model assumes the Normal prior $N(\mu = 5, \sigma = 2)$ on the random intercepts $\beta^i_0$, the log-Normal prior $log-Normal(-1, 2)$ on $\kappa^2_i$, the log-Normal prior $log-Normal(-1, 1)$ on $\sigma^2$, the log-Normal prior $log-Normal(0, 1)$ on $\tau$, the Normal prior $N(\mu = 0, \sigma = 5)$ on the survival shared intercept $\beta_0$, the Normal prior $N(\mu = 0, \sigma = 5)$ on the survival coefficient $\beta_1$, the Gamma prior $\Gamma(3, 3)$ on the concentration parameter of the Dirichlet distribution, and the Normal prior $N(\mu = 0, \sigma = 5)$ as the base distribution of the Dirichlet distribution.
As the results in Table \[SimTableModel1Ch4\] show, when data are simulated with a longitudinal trajectories that are quadratic polynomial curves, the joint polynomial model performs better in terms of estimating the albumin coefficient in the survival model with a smaller mean squared error compared to our proposed joint longitudinal-survival. In real world, however, the true functional forms of the trajectories of the biomarkers are not known. Under a general case where the biomarker trajectories can be any random non-linear curve (scenario 2), our proposed joint model outperforms the joint polynomial model. Further, our joint modeling framework that is capable of estimating differential subject-specific log baseline hazards provides significantly better coefficient estimates compared to the proportional hazard Cox model. Estimates under the Cox model are marginalized over all subjects and due to the non-collapsibility aspect of this model ([@struthers1986], [@martinussen2013]), coefficient estimates shrink toward 0.
[lcccccccccccc]{} &True Conditional& && &&\
& Estimand $$& Mean & SD & MSE$$ && Mean & SD & MSE$$&& Mean & SD & MSE$$\
\
\
Albumin(t) & -0.5 & -0.273 & 0.056 & 0.119 && -0.495 & 0.019 & 0.003 && -0.441 & 0.105 & 0.012\
\
\
Albumin(t) & -0.5 & -0.258 & 0.080 & 0.125 && -0.380 & 0.080 & 0.034 && -0.462 & 0.110 & 0.010\
Model II - Simulation Results
-----------------------------
In model II, not only do we adjust for the albumin value at time $Y_i$, but we also adjust for a weighted average slope of albumin from time $\tau_1 = 0$ up until the time $\tau_2 = Y_i$, where $Y_i$ is either the event time for subject $i$ or is the time that the subject got censored. This new model differentiates between the risk of death for a patient whose albumin value is improving compared to another patient with the same albumin level whose albumin is deteriorating. In particular, we consider weighted average slope of albumin once under the weighting scheme of the form $$\begin{aligned}
Q(t) = \frac{1}{\tau_1 - \tau_0},\end{aligned}$$ and another time under the weighting scheme of $$Q(t)=
\begin{cases}
1,& \text{if } t = T_i\\
0,& \text{otherwise}.
\end{cases}$$ The first weighting scheme leads to the area under the derivative curve. The second weighting scheme will result in a point-wise derivative of albumin at time $Y_i$.
We generate synthetic data for 300 subjects each with 9 to 12 longitudinal measurements where longitudinal albumin values are generated from a Gaussian process that is centered around the subject-specific random intercepts $\beta^{(L)}_{0i}$ which are generated from the Normal distribution $N(\mu = 5, \sigma = 1)$. We consider a Gaussian process with the squared exponential covariance function with the correlation length of $\rho^2 = 0.1$ and the subject-specific measures of volatility $\kappa^2_i$ that are generated from the uniform distribution $U(0, 1)$. Once longitudinal measures are generated, we generate survival data where survival times are distributed according to the Weibull distribution $Weibull(\tau, \lambda_i)$, where the shape parameter $\tau$ is set to 1.5 and $\lambda_i$, which is the log of the scale parameter in Weibull distribution, is set to $\beta_{i0}^{(S)} + \beta_1 X_i(t) + \beta_2 X'_{AUC, i}(t)$, where $\beta_{i0}^{(S)}$ are generated from an equally weighted mixture of two Normal distributions of $N(\mu = -1.5, \sigma = 1)$ and $N(\mu = 1.5, \sigma = 1)$, $\beta_1$ is fixed to 0.3, $\beta_2$ is fixed to 0.5, $X_i(t)$ is the longitudinal value for subject $i$ at time $t$ and $X'_{AUC, i}(t)$ is the average slope of albumin.
Our proposed joint longitudinal-survival model assumes the Normal prior $N(\mu = 5, \sigma = 2)$ on the random intercepts $\beta^i_0$, the log-Normal prior $log-Normal(-1, 2)$ on $\kappa^2_i$, the log-Normal prior $log-Normal(-1, 1)$ on $\sigma^2$, the log-Normal prior $log-Normal(0, 1)$ on $\tau$, the Normal prior $N(\mu = 0, \sigma = 5)$ on the survival shared intercept $\beta_0$, the Normal prior $N(\mu = 0, \sigma = 5)$ on the survival coefficient $\beta_1$, the Normal prior $N(\mu = 0, \sigma = 5)$ on the survival coefficient $\beta_2$, the Gamma prior $\Gamma(3, 3)$ on the concentration parameter of the Dirichlet distribution, and the Normal prior $N(\mu = 0, \sigma = 5)$ as the base distribution of the Dirichlet distribution.
We fit our proposed joint longitudinal-survival model. As a comparison, we also fit a two-stage Cox model where the longitudinal curve of albumin and its derivative curve are estimated using hyper-parameters set as the posterior median of a Bayesian Gaussian Process model. As we can see from table \[SimTableModel2AUC\], our joint model provides closer estimates to the coefficient values with a smaller mean squared error compared with the two-stage Cox model. Our proposed model is capable of detecting differential subject-specific baseline hazards whereas the Cox model is not capable of differentiating between subjects and provides estimates that are marginalized across all subjects. Further, the simulation results show the capability of our method in detecting the true underlying longitudinal curves and the ability of our method on properly estimating the average derivative of those curves.
[lccccccccc]{} &True Conditional& &&\
& Estimand $$& Mean & SD & MSE$$ && Mean & SD & MSE$$\
\
\
Albumin(t) & 0.3 & 0.191 & 0.099 & 0.022 && 0.303 & 0.109 & 0.008\
Area under the derivative curve(t) & 0.5 & 0.346 & 0.179 & 0.053 && 0.449 & 0.188 & 0.030\
\
\
Albumin(t) & 0.3 & 0.142 & 0.095 & 0.033 && 0.261 & 0.104 & 0.009\
$\frac{d(Albumin(t))}{dt}$ & 0.5 & 0.412 & 0.123 & 0.022 && 0.477 & 0.152 & 0.013\
\
Model III - Simulation Results
------------------------------
In model III, we test the association between the summary measures of the longitudinal biomarker trajectories and the survival outcomes. In particular, we consider the relation between the summary measures of subject-specific random intercept $\beta^{(L)}_{i0}$ and subject-specific measure of volatility $\kappa^2_i$ and survival times.
We generate synthetic data for $N=300$ subjects each with 9 to 12 longitudinal measurements where longitudinal albumin values are generated from a Gaussian process that is centered around the subject-specific random intercepts $\beta^{(L)}_{0i}$ which are generated from the Normal distribution $N(\mu = 5, \sigma = 1)$. We consider a Gaussian process with the squared exponential covariance function with the correlation length of $\rho^2 = 0.1$ and the subject-specific measures of volatility $\kappa^2_i$ that are generated from the uniform distribution $U(0, 1)$. Once longitudinal measures are generated, we generate survival data where survival times are distributed according to the Weibull distribution $Weibull(\tau, \lambda_i)$, where the shape parameter $\tau$ is set to 1.5 and $\lambda_i$, which is the log of the scale parameter in Weibull distribution, is set to $\beta_{i0}^{(S)} + \beta_1 Age + \beta_2 {\beta_{i0}}^{(L)} + \beta_3 {\kappa_i^{2}}^{(L)}$, where $\beta_{i0}^{(S)}$ are generated from an equally weighted mixture of two Normal distributions of $N(\mu = -1.5, \sigma = 1)$ and $N(\mu = 1.5, \sigma = 1)$, $\beta_1$ is fixed to 0.5, $\beta_2$ is fixed to -0.3, $\beta_3$ is fixed to 0.7, $Age$ is a standardized covariate that is generated from the Normal distribution $N(\mu = 0, \sigma = 1)$, ${\beta_{i0}}^{(L)}$ is subject-specific random intercepts of the longitudinal trajectories, and ${\kappa_i^{2}}^{(L)}$ are subject specific measure of volatility of the longitudinal trajectories.
Our proposed joint longitudinal-survival model assumes the Normal prior $N(\mu = 5, \sigma = 2)$ on the random intercepts $\beta^i_0$, the log-Normal prior $log-Normal(-1, 2)$ on $\kappa^2_i$, the log-Normal prior $log-Normal(-1, 1)$ on $\sigma^2$, the log-Normal prior $log-Normal(0, 1)$ on $\tau$, the Normal prior $N(\mu = 0, \sigma = 5)$ on the survival shared intercept $\beta_0$, the Normal prior $N(\mu = 0, \sigma = 5)$ on the survival coefficient $\beta_1$, the Normal prior $N(\mu = 0, \sigma = 5)$ on the survival coefficient $\beta_2$, the Gamma prior $\Gamma(3, 3)$ on the concentration parameter of the Dirichlet distribution, and the Normal prior $N(\mu = 0, \sigma = 5)$ as the base distribution of the Dirichlet distribution.
We fit our proposed joint survival-longitudinal model (model III) as well as a two-stage Cox proportional hazard model as a comparison model. The two-stage Cox model is a simple Cox proportional hazard model with covariate $\beta_{0i}^{(L)}$ and ${\kappa_i^2}^{(L)}$ that are posterior medians from a separate longitudinal Gaussian process model. As the results in Table \[SimTableModel3\] show, our proposed joint model provides closer estimates to the true coefficients that also have significantly smaller mean squared error compared to the two-stage Cox model. Our proposed joint model is capable of detecting the differential subject-specific baseline hazards. Unlike our model, Cox model is blind to the subject-specific baseline hazards and hence, provides coefficient estimates that are marginalized over all subjects. These marginalized estimates from the Cox model shrink toward 0 as the Cox model with a multiplicative hazard function is non-collapsible.
As one can see in the joint model results in Table \[SimTableModel3\], the coefficient estimate for ${\kappa^2}^{(L)}$ is not as close to the true coefficient value compared with other coefficient estimates. This is due to the fact that only 9 to 12 longitudinal measures per subject, there exists many plausible ${\kappa^2_i}^{(L)}$ values that flexibly characterize the trajectory of the measured albumin values. This additional variability in plausible ${\kappa^2_i}^{(L)}$ values has caused the coefficient estimate to shrink toward 0. Larger number of within subject longitudinal measures will provide more precision in estimating the true underlying ${\kappa^2_i}^{(L)}$ and will lead to a coefficient estimate closer to the true value. In order to confirm this fact, we simulated additional data once with 36 within subject measures and another time with 72 within subject measures. Table \[SimTableModel3Bonus\] shows the results of fitting our proposed joint longitudinal-survival model to datasets that include subjects with 9 to 12 within subject measurements, to datasets with subjects with 36 within subject measurements, and to datasets with subjects with 72 within subject measurements. As the results show, with larger number of within subject measurements, coefficient estimate for ${\kappa^2_i}^{(L)}$ is closer to the true value. This is due to the fact that with larger number of within subject albumin measurements, there exists a stronger likelihood to estimate the subject-specific volatility measures $\kappa^2_i$, and hence, there is less uncertainty about the estimated value of volatility measures.
[lccccccccc]{} &True Conditional& &&\
& Estimand $$& Mean & SD & MSE$$ && Mean & SD & MSE$$\
\
Age (scaled) & 0.5 & 0.262 & 0.124 & 0.070 && 0.492 & 0.149 & 0.013\
Baseline Albumin (${\beta_{0i}}^{(L)}$) & -0.3 & -0.141 & 0.118 & 0.040 && -0.284 & 0.116 & 0.008\
${\kappa^2_i}^{(L)}$ & 0.7 & 0.414 & 0.212 & 0.127 && 0.595 & 0.271 & 0.042\
[lcccccccccccc]{} &True Conditional& && &&\
& Estimand $$& Mean & SD & MSE$$ && Mean & SD & MSE$$&& Mean & SD & MSE$$\
\
Age (scaled) & 0.5 & 0.492 & 0.149 & 0.013 && 0.493 & 0.144 & 0.015 && 0.495 & 0.145 & 0.016\
${\beta_{i0}}^{(L)}$ & -0.3 & -0.284 & 0.116 & 0.008 && -0.308 & 0.116 & 0.006 && -0.295 & 0.115 & 0.007\
${\kappa^2_i}^{(L)}$ & 0.7 & 0.595 & 0.271 & 0.042 && 0.639 & 0.284 & 0.043 && 0.651 & 0.293 & 0.039\
Application of the Proposed Joint Longitudinal-Survival Models to Data from the United States Renal Data System {#uniJointReal}
===============================================================================================================
In this section, we apply our proposed joint longitudinal-survival models to data on $n = 1,112$ end stage renal disease patients participating in the Dialysis Morbidity and Mortality Studies (DMMS) nutritional study that is obtained from the United States Renal Data System. For every participating patient in the study, up to 12 albumin measurements were taken uniformly over two years of followup. The presented analyses are restricted to only the patients who had at least nine albumin measurements in order to provide sufficient data for modeling the trajectory and the volatility of albumin. The censoring rate in the data is at 43% over a maximal follow-up time of 4.5 years.
Using the same data, [@fung02] showed that both baseline albumin level and the slope of albumin over time are significant predictors of mortality among ESRD patients. While our models are capable of replicating Fung et al’s findings, our models are also capable of:
- model 1: testing the association between albumin value at the time of death and the risk of death
- model 2: testing the association between albumin value and an average derivative of albumin up until time t and the risk of death.
- model 3: Testing the association between risk of mortality and the two summary measures of the baseline and the volatility of albumin measures
In order to adjust for other potential confounding factors, our proposed models also include patient’s age, gender, race, smoking status, diabetes, an indicator of whether the patient appeared malnourished at baseline, BMI at baseline, baseline cholesterol, and baseline systolic blood pressure. The adjusted covariates are consistent with those originally presented in [@fung02].
Table \[AppRslt1\] and Table \[AppRslt2auc\] provide the results of fitting our proposed model I, Model II, and Model III to the USRDS data. All joint models were run for 10,000 posterior samples where the initial 5,000 samples are discarded as burn-in samples.
Model I - Application to USRDS data
-----------------------------------
We fit our proposed joint model I to the data. As a comparison model, we also fit as last-observation carried forward (LOCF) Cox model. Table \[AppRslt1\] shows the estimated coefficients from both models. Between the two models, the estimated relative risk associated with all time-invariant baseline survival covariates are similar between the two models. However, the relative risk associated with every one unit decrement in serum albumin is much larger under our proposed joint model compared to the last-observation carried forward Cox model. This is quite expected as our model is capable of estimating subject-specific albumin trajectories over time and is capable of accurately testing the association between albumin value at time of death and risk of death. Unlike our model,the LOCF Cox model uses the most recent albumin measure which in reality might be quite different than the albumin value at the time of death. In both models, albumin is identified as a significant risk factor of mortality. In particular, based on the results from our proposed joint Model I, it is estimated that every 1 $g/dL$ decrement in albumin is associated with a 4.5 times higher risk of death.
------------------------------------- -------- -------- ------------------ ---------- ------------------ --
No. of No. of
Covariates Cases Deaths (95% CI) P-Value (95% CR)
Age (10y) 1,112 630 1.44 (1.35-1.53) $<$.001 1.45 (1.36,1.55)
Sex
Men 560 312 1.0 1.0
Women 552 318 0.96 (0.81,1.13) 0.60 0.97 (0.82,1.16)
Race
White 542 350 1.0 1.0
Black 482 243 0.81 (0.68,0.96) 0.01 0.79 (0.67,0.94)
Other 88 37 0.52 (0.37,0.74) $<$.001 0.49 (0.34,0.69)
Smoking
Nonsmoker 645 337 1.0 1.0
Former 307 197 1.17 (0.98,1.41) 0.09 1.20 (0.99,1.44)
Current 160 96 1.52 (1.19,1.94) $<$.001 1.53 (1.21,1.95)
Diabetes
No 716 363 1.0 1.0
Yes 396 267 1.66 (1.40,1.97) $<$.001 1.69 (1.43,2.00)
Undernourished
No 958 517 1.0 1.0
Yes 154 113 1.39 (1.12,1.72) 0.003 1.35 (1.08,1.66)
BMI (per-5 kg/m$^2$
decrement) 1,112 630 1.08 (1.00,1.17) 0.07 1.08 (1.00,1.17)
Cholesterol (per 20
mg/dL) 1,112 630 0.97 (0.93,1.00) 0.08 0.96 (0.93,1.00)
Systolic blood pressure
(per 10mm Hg) 1,112 630 0.98 (0.95,1.02) 0.38 0.98 (0.95,1.02)
Serum albumin(t) (1-g/dL decrement) 1,112 630 2.48 (2.00,3.07) $<$0.001 4.54 (3.03,5.55)
------------------------------------- -------- -------- ------------------ ---------- ------------------ --
: Estimated Relative Risk and corresponding 95% credible region from our proposed joint model where we adjust for time-dependent albumin value that is imputed from the longitudinal component of the model. We also fit a last-observation carried forward Cox proportional hazards model with last albumin value carried forward where we report coefficients estimates, 95% confidence interval, and p-value for the estimated coefficients. In both models, we adjust for potential confounding factors as reported by [@fung02].[]{data-label="AppRslt1"}
Model II - Application to USRDS data
------------------------------------
Other than the albumin value at the time of death, the average slope of albumin over time might also be a risk factor of mortality in end-stage renal disease patients. In our proposed joint Model II, we also adjust for the area under the derivative curve of the albumin trajectory from the time that the follow-up starts until the survival time which is either the time of death or the censoring time. Table \[AppRslt2auc\] shows the results from our proposed model. Based on the results, every one g/dL decrement in albumin is associated with 3.95 times higher risk of death. Also, higher average slope of albumin, that is every 1 g/dL/month increase in the average slope, is associated with 2.3 times higher risk of death. This is consistent with [@fung02] results on the association between the slope of albumin and the risk of death. Our proposed method is also capable of adjusting for the local effect of the slope of albumin. For instance, instead of averaging the slope of the follow-up time, one may only integrate over the 6 months prior to the time of death.
Model III - Application to USRDS data
-------------------------------------
[@fung02] showed that the baseline albumin and the slope of albumin over time are two independent risk factors of mortality among the end-stage renal disease patients. It is quite natural to hypothesize that the volatility of albumin could also be a risk factor of mortality among these patients. In our proposed joint longitudinal-survival Model III, we consider two summary measures of the trajectories of the longitudinal albumin values, one the baseline albumin measures (${\beta_{0i}}^{(L)}$), and another the subject-specific volatility measure of albumin (${\kappa^2_i}^{(L)}$). Table \[AppRslt2auc\] also shows the results from our proposed Model III. The results from our model confirms that the baseline serum albumin is a risk factor of mortality. Further, the results from our model indicate that the volatility of albumin is also a significant risk factor of mortality, where every one unit increase in $\kappa^2$, which indicates a higher volatility, is associated with 1.2 times higher risk of death.
[lrrcc]{} & & &\
& No. of& No. of&\
Covariates & Cases & Deaths & (95% CI) &\
\
\
Serum Albumin(t) (1-g/dL decrement) &1,112&630&3.95 (3.18,4.71)&\
Average Derivative of Serum Albumin$^1$ (1-g/dL/month decrement) &1,112&630&2.33 (1.40,3.73)&\
\
\
Baseline Albumin$({\beta_{0i}}^{(L)})$ (1-g/dL decrement) &1,112&630&5.54 (4.19,6.94)&\
${\kappa^2_i}^{(L)}$ (increase in volatility)$^2$ &1,112&630&1.23 (1.02,1.41)&\
\
\
\
\
Figure \[ActualUSRDSLong\] shows albumin trajectories of 10 randomly sampled individuals. Hollow circles are the actual albumin measures for each subject. Also, plots show the posterior Gaussian process trajectory fit along with it’s 95% credible region.
![Actual longitudinal albumin trajectories of 10 randomly selected individuals with end-stage renal disease that were selected from the USRDS data. Hollow circles are the actual measured albumin values, red lines are the posterior median fitted curves from our proposed Model III, and the dashed blue lines are the corresponding 95% posterior prediction intervals for the fitted trajectories. The title of each plot shows the posterior median of the volatility measure $\kappa^2$ for the subject whose albumin measures are shown in the plot.[]{data-label="ActualUSRDSLong"}](USRDS_LongCurve_STEP1.png){width="\linewidth"}
Discussion {#uniJointDisc}
==========
Monitoring the health of patients often involves recording risk factors over time. In such situations, it is essential to evaluate the association between those longitudinal measurements and survival outcome. To this end, joint longitudinal-survival models provide an efficient inferential framework.
We proposed a joint longitudinal-survival framework that avoids some of the restrictive assumptions commonly used in the existing models. Further, our methods propose a stronger link between longitudinal and survival data through an introduction of new ways of adjusting for the biomarker value at time t, adjusting for the average derivative of the biomarker over time, and moving beyond the first-order trend and accounting for volatility of biomarker measures over time.
A two-stage approach to associating biomarker volatility with the survival outcome has been proposed by [@holsclaw2014]. Our work here extends this approach by simultaneously estimating longitudinal and survival parameters, thus accounting for uncertainty in the longitudinal measures. Our proposed models can also be considered as an extension of the joint model proposed by [@brown2003] in that we use the same idea of dividing the joint likelihood into a marginal longitudinal likelihood and conditional survival likelihood. However, instead of fitting quadratic trajectories, we use a flexible longitudinal model based on the Gaussian processes. Further, for the survival outcome, instead of assuming a piecewise exponential model, we use a flexible survival model by incorporating the Dirichlet process mixture of Weibull distributions. Our proposed modeling framework is capable of modeling additional summary measures of longitudinally measured biomarkers and relating them to the survival outcome in a time-dependent fashion.
Our proposed models, despite their flexibility and novelty, have some limitations. By using th Bayesian non-parameteric Dirichlet process and the Gaussian process techniques, while we provide a flexible modeling framework that avoids common distributional assumptions, however, these techniques are generally not scalable when the number of subjects and the number of within subject measurements increase. Furthermore, the survival component of our model still relies on the proportional hazard assumption. In future, our modeling framework can be extended to include a more general non-proportional hazard survival models that can also include time-dependent coefficients inside the survival model. By using some alternatives to the common MCMC techniques, including parallel-MCMC methods and variational methods, our method can become more computationally efficient and scalable for larger datasets.
Often times in monitoring the health of patients, multiple longitudinal risk factors are measured. One can use our introduced modeling framework in this chapter in order to build a joint longitudinal-survival model with multiple longitudinal processes each process modeled independently from other longitudinal processes. In reality, however, one expects that patients longitudinal risk factors to be correlated. A methodology that is capable of modeling multiple biomarkers simultaneously by taking the correlation between biomarkers into account can be beneficial specially when there exists differential densities between different longitudinal processes. In a separate work, we introduce a joint modeling framework that is capable of modeling multiple longitudinal processes simultaneously by taking the correlation between those processes into account.
|
---
abstract: 'This report details our solution to the Google AI Open Images Challenge 2019 Object Detection Track. Based on our detailed analysis on the Open Images dataset, it is found that there are four typical features: large-scale, hierarchical tag system, severe annotation incompleteness and data imbalance. Considering these characteristics, many strategies are employed, including larger backbone, distributed softmax loss, class-aware sampling, expert model, and heavier classifier. In virtue of these effective strategies, our best single model could achieve a mAP of 61.90. After ensemble, the final mAP is boosted to 67.17 in the public leaderboard and 64.21 in the private leaderboard, which earns 3rd place in the Open Images Challenge 2019.'
author:
- |
Xingyuan Bu\
Beijing Institute of Technology\
[[email protected]]{}\
- |
Junran Peng\
University of Chinese Academy of Sciences\
[[email protected]]{}
- |
Changbao Wang\
Beihang University\
[[email protected]]{}
- |
Cunjun Yu\
Nanyang Technological University\
[[email protected]]{}
- |
Guoliang Cao\
Peking University\
[[email protected]]{}
bibliography:
- 'egbib.bib'
title: 'Learning an Efficient Network for Large-Scale Hierarchical Object Detection with Data Imbalance: 3rd Place Solution to Open Images Challenge 2019'
---
Introduction
============
Object detection has been a challenging issue in the field of computer vision for a long time. It is a basic and important task for a variety of industrial applications, such as autonomous driving and sense analysis. Due to the blooming development of deep learning and large-scale dataset, great progresses have been achieved in object detection in recent years [@girshick2014rich; @he2014spatial; @girshick2015fast; @ren2015faster; @lin2017feature; @he2017mask; @dai2017deformable; @liu2018path; @cai2017cascade; @lin2018focal; @redmon2018yolov3]. Open Images Dataset V5 [@openimages; @OpenImagesSegmentation] is currently the largest object detection dataset. Different from its predecessors, such as Pascal VOC [@everingham2010pascal], MS COCO [@lin2014microsoft], and Objects365 [@obj365], Open Images dataset consists of extremely large-scale annotations including 12M bounding boxes for 500 categories on 1.7M images, as shown in Table \[tab:dataset\].
Dataset Pascal VOC COCO Objects365 Open Images
------------- ------------ --------- ------------ -------------
Categories 20 80 365 500
Images 11,540 123,287 638,630 1,784,662
Boxes 27,450 886,287 10,101,056 12,421,955
Boxes/image 2.4 7.2 15.8 7.0
![ Example images of Open Images. There are severe missed annotations for the bounding box. In the example images, all human bounding boxes are missing. []{data-label="fig:weakly_human"}](fig/weakly_human.pdf){width="\linewidth"}
![ The data statistics of the COCO and the Open Images datasets. The x-axis and the y-axis are with the categories and the log transformed instance counts, respectively. It should be noted that the category number of the Open Images and the COCO datasets are different, which is 500 and 80, respectively. For better visualization, we have aligned the statistics of COCO and Open Images datasets in x-axis. []{data-label="fig:imbalance"}](fig/imbalance.pdf){width="\linewidth"}
In the era of deep learning, more training data always benefits the generalizability of model [@mahajan2018exploring]. With the help of large-scale Open Images detection dataset, the frontier of object detection would be pushed forward a great step. To take full advantage of the large-scale data, we employ EfficientNet [@tan2019efficientnet] as our backbone. We first grid search an EfficientNet-B1 on Open Images Detection Dataset and then scale it up based on compound scaling method. However, it is found that the standard compound scaling method [@tan2019efficientnet] is not optimal in the setting of multi-scale training and testing pipeline which is a common data augmentation strategy for object detection. We argue that it is because that the standard EfficientNet scale-up the resolution to achieve better performance in single-scale training, but it is harmful when training and testing model in multi-scale input. To remedy this problem, we fix the resolution and re-assign the stage of EfficientNet-B7 to mimic the scheme how ResNeXt [@xie2017aggregated] assigns different amount parameters to different stages architecture.
The hierarchical tag system also should be taken into consideration. Open Images dataset contains 500 categories, consisting of 5 different levels and 57 parent nodes. And in the evaluation procedure, a parent category is asked to be output if its child present. Besides, It is observed that some categories are ambiguous. For example, almost every *Torch* instance highly overlaps with a *Flashlight* bounding box, although there is not a hierarchical relation between them. Considering these fact, We propose distributed softmax loss to replace the standard softmax loss.
The large volume of images and categories of Open Images dataset limits its labeling style. As the enormous numbers of images and categories, it is practical impossible to annotate every instance in every image exhaustively, there are severe missed annotations for the bounding box, as demonstrated in Figure \[fig:weakly\_human\]. Hence, the missed annotations turns the Open Images Detection task into weakly supervised scenario with label noise. We also observe that Open Images dataset suffers from severe data imbalance, as shown in Figure \[fig:imbalance\]. For example, the category of *Person* has 1.4 million instances, which is 105 times larger than that of the *Pressure cooker* which only has 14 instances. we propose the class-aware sampling to make a balance between major and rare categories. Furthermore, to alleviate the overfitting caused by class-aware sampling, we apply auto augmentation [@zoph2019learning] on both image-level and box-level data. Alternatively, we also train expert model [@akiba2018pfdet; @niitani2019sampling] on rare categories then ensemble them to solve the data imbalance problem.
In conclusion, our method has the following advantages:
- We present an variant of EfficientNet, which overcomes the ineligibility in multi-scale training and testing for object detection.
- We propose a distributed softmax loss that generalizes the standard softmax loss by exploiting category relationship. It is more friendly to the hierarchical tag system and more robust to the label noise.
- We propose class-aware sampling along with auto augmentation to solve the severe data imbalance. We also present an effective method to train expert model for data imbalance.
Methods
=======
In this section, we will detailedly demonstrate our methods used in this challenge.
EfficientNet for Object Detection
---------------------------------
In this work, the Faster R-CNN framework is adopted as our baseline, where the backbone is EfficientNet. EfficientNet is recent state-of-the-art in many classification tasks. But there is still limited research about successfully applying EfficientNet to the field of object detection.
EfficientNet is proposed as a paradigm to scale-up Convolutional Neural Networks (CNN) in three dimensions of network, , depth, width, and resolution. In contrary to previous work, EfficientNet balances these three dimensions by grid search their scale-up coefficient simultaneously. The grid search is applied in a small search space, then the final network which lays in a larger search space will be sampled by using the compound scaling method. As there is no pretrained EfficientNet for object detection, we search it directly on a subset of Open Images dataset. Firstly, we trained a basic model which is so-called EfficientNet-B0. Then we grid search the next level EfficientNet-B1 whose scale-up coefficient equals to 2. This depth, width, and resolution of EfficientNet-B1 can be generated by: $$\label{eq:b2}
\begin{aligned}
d, w, r & = \mathop{\arg\max}_{\alpha, \beta, \gamma}( mAP(model(\alpha, \beta, \gamma))) \\
& \text{s.t.} \ \ \ {\alpha} \cdot {\beta}^{2} \cdot {\gamma}^{2} \approx 2
\end{aligned},$$ where $d, w, r$ are the optimal depth, width, and resolution of EfficientNet-B1, respectively. And the $\alpha, \beta, \gamma$ are temporary depth, width, and resolution in grid search, respectively. The $model$ is the function that generates and trains detection model according to its input depth, width, and resolution, which is then evaluated by $mAP$ function. After building EfficientNet-B1, we further obtain EfficientNet-B7 by using the compound scaling method: $$\label{eq:b7}
\begin{aligned}
\text{EfficientNet-B7} = model({d}^{7}, {w}^{7}, {r}^{7})
\end{aligned}.$$
Although EfficientNet-B7 achieve better performance in single-scale training and testing. But under the multi-scale training and testing, it is found that EfficientNet-B7 is slightly inferior to the ResNeXt-152. We argue that it is because that the standard grid search and compound scaling method is sub-optimal in multi-scale input setting. EfficientNet carefully assigns depth, width, and resolution to achieve better for objects at all sizes. However, improving performance for objects at different size is unnecessary even harmful for multi-scale input, because different size objects could be detected better with different input scale. Thus, we propose to fix the resolution coefficient and only adjust the width and depth. We further add more blocks into stage four which mimics the ResNeXt-152 to gain the advantage to a specific input scale. The proposed variant of EfficientNet-B7 is illuminated in Figure \[fig:nb7\].
![ Left: The standard EfficientNet-B7. Right: The proposed variant of EfficientNet-B7. We put more convolutional layers in stage four, so that this architecture could achieve better performance in a specific input scale. []{data-label="fig:nb7"}](fig/NB7.pdf){width="0.95\linewidth"}
Distributed Softmax Loss
------------------------
Open Images dataset contains 500 categories, consisting of 5 different levels and 57 parent nodes. mAP is evaluated for each of the 500 categories. For a leaf category in the hierarchy, AP is computed normally. However, in order to be consistent with the meaning of a non-leaf class, its AP is computed involving all its ground-truth object instances and all instances of its children. Thus, detector are usually asked to output two even more detected boxes for a single region proposal.
Besides, there are many data noise caused by ambiguous categories in Open Images dataset, such as *Torch* and *Flashlight*, they are so similar in semantics or appearance that the annotators hardly discriminate them and label them as each other. It is benefit to boost performance if we can output both ambiguous categories when one of them presents.
Unfortunately, the standard softmax cross-entropy loss is arduous to deal with these hierarchical tag system and the data noise. The standard softmax cross-entropy loss is presented as: $$\label{eq:standardsoftmax}
\begin{aligned}
\mathcal{L}_{cls} & = \sum_{c=1}^{C} \mathbbm{1}_{y_c=1} log(\frac{e^{x_c}}{\sum_{i=1}^{C}e^{x_i}}) \\
\end{aligned},$$ where $C$ is the number of category, $y_c$ is the c-th elements in label vector $\textbf{y}$, and $x_i$ is the i-th elements in logit vector $\textbf{x}$. It should be noted that the label $\textbf{y}$ is an one-hot vector which only allows one positive output, thus, is not suitable for multi-label classification. An intuitive solution is to use the binary cross-entropy loss [@durand2019learning], but in our experiments it can not achieve comparable results. We also try focal loss [@lin2018focal] upon the binary cross-entropy loss, but still worse than standard softmax loss. To overcome this problem we propose the distributed softmax loss. It is formulated as: $$\label{eq:distributedsoftmax}
\begin{aligned}
\mathcal{L}_{cls} & = \sum_{c=1}^{C} y_c log(\frac{e^{x_c}}{\sum_{i=1}^{C}e^{x_i}}) \\
\end{aligned},$$ where $y_c$ belongs to the label vector $\textbf{y}$ with $k$ non-zero elements each set to $1/k$ corresponding to the $k \geq 1$. The proposed distributed softmax loss not only allows multi-label training as binary cross-entropy loss, but also persists the suppression between categories as standard softmax loss. In our experiments, it improves the baseline by 1 points of mAP.
Operation 1 P M Operation 2 P M
-------------- -------------------------- ----- ---- -------------------------- ----- ----
Sub-policy 1 TranslateX\_BBox 0.6 4 Equalize 0.8 10
Sub-policy 2 TranslateY\_Only\_BBoxes 0.2 2 Cutout 0.8 8
Sub-policy 3 Sharpness 0.0 8 ShearX\_BBox 0.4 0
Sub-policy 4 ShearY\_BBox 1.0 2 TranslateY\_Only\_BBoxes 0.6 6
Sub-policy 5 Rotate\_BBox 0.6 10 Color 1.0 6
Class-aware Sampling and Expert Model
-------------------------------------
In Figure \[fig:imbalance\], we illuminate the data statistics of the COCO and Open Images datasets. It can be seen that Open Images dataset exhibits much severe data imbalance compared to the its counterpart. Following Gao [@gao2018solution], we apply class-aware sampling to alleviate this problem. In details, the class-aware sampling firstly samples one category out of 500 uniformly. And then an image containing objects with the same category of the first step is sampled. This strategy could assure all the category has the same chance to be trained.
However, overfitting is introduced unavoidably by this class-aware sampling. We employ auto augmentation in both image level and box level to remedy this problem. The auto augmentation strategies is listed in Table \[tab:autoaug\].
Expert model is another method that can overcome the data imbalance problem. In this challenge, we fine-tune the baseline model on a small subset of the full category space to enhance the recall of rare categories. The baseline model is trained on the entire dataset. Each expert model is concentrated to a expert subset, for instance, 50 categories out of 500. Expert model is useful to increase the recall of rare categories, but tends to bring more false positives. We further develop three strategies to solve the unwanted false positives. First, we build a confusion matrix to find those categories that is easy to be classified into expert subset. In next turn training, those confusion categories is added into the training set, which helps the expert model to discriminate the false positives. Second, we train many expert models with different sizes of expert subset, and there is an overlap between each expert subset. Only the detected boxes of overlapped categories is considered as final results. Third, we train a classifier to re-weight the confidence of detected boxes by multiple the detector score and classifier score. It should be noted that there is not an absolute positive correlation between the mean accuracy of classifier and the final mAP of re-weighted detection results. We also find that the classifier is beneficial to the normal model that trained on the entire dataset.
Ensemble
--------
For the final results, we have used normal models and expert models for ensemble. Since the distribution of the performance over categories is different among models, we re-weight each categories model by model according to its mAP on validation set. In details, we set a weight $w_{c}^{m}$ for model $m$ and its categories $c$. The $w_{c}^{m}$ is defined as: $$\label{eq:nms1}
\begin{aligned}
w_{c}^{m} & = \frac{s_{c}^{m} - {\mu}_{c}}{t_c - {\mu}_c} + \alpha \frac{t_{c} - s_{c}^{m}}{t_c - {\mu}_c} \\
\end{aligned},$$ where $s_{c}^{m}$ is the validation mAP of the model $m$ for category $c$, ${\mu}_{c}$ and $t_{c}$ is the mean and max mAP of the for category $c$ among all models, respectively. $\alpha$ is the lower bound of $w_{c}^{m}$ so that if $s_{c}^{m}$ is lower than ${\mu}_{c}$, then $w_{c}^{m}$ is set to $\alpha$.
We also search optimal NMS threshold for different categories by: $$\label{eq:nms2}
\begin{aligned}
H_c & = \mathop{\arg\max}_{h_c \in (0,1)} (mAP(h_c) + \frac{1}{(h_c - d)^2}) \\
\end{aligned},$$ where $h_c$ is the NMS threshold for category $c$, and the $d$ is the default NMS threshold. $mAP$ is a function mapping NMS threshold to ensemble mAP. We expect to maximize the validation mAP by searching the NMS threshold, meanwhile, minimize the variance of the NMS threshold.
Experiments
===========
Data
----
We train our model on Open Images dataset. We used the recommended train and validation splits of Open Images Challenge 2018 as its validation set is labelled denser than that of Challenge 2019. In addition to the Open Images dataset, Objects365 [@obj365] is used to train the final models for categories that intersects with Open Images dataset.
Results
-------
Method Public Leader board
--------------------------------- ---------------------
Baseline (FPN with ResNeXt-152) 53.88
+EfficientNet-B7 55.59
+Distributed Softmax Loss 56.43
+Class-aware Sampling 61.09
+Auto Augmentation 61.84
+Classifier 62.29
+Ensemble 67.17
: Performance improvement by adding different strategies step by step. []{data-label="tab:results"}
During the training process, we follow the typical hyper-parameter settings. We use SGD with momentum 0.9 and weight decay 0.0001. The initial learning rate is set to 0.00125 per batch, and warm-up [@goyal2017accurate] phase is adopted. All models are trained from scratch [@he2018rethinking].
Table \[tab:results\] demonstrates the detailed ablation with our method. As can be seen, the variant of EfficientNet-B7 surpasses the ResNeXt-152 by 1.71%. After solving the hierarchical tag system and data noise by distributed softmax loss, the performance is boosted by 0.84 points. The class-aware sampling balances the dataset and leads to a heavy gain, , 4.67 points. By further using auto augmentation and classifier, the model is improved by 0.75% and 0.45%, respectively, achieving the best single model with a mAP of 62.29%. With a final ensemble of 12 different models, we achieve the 67.17% mAP in the public leaderboard and 64.21% mAP in the private leaderboard, ranking the 3rd place.
Conclusions
===========
In this work, we present an variant of EfficientNet for detection which is suitable for multi-scale training and testing. On the top of efficient backbone, we introduces a novel distributed softmax loss for the hierarchical label and data noise, and use class-aware sampling, auto augmentation, expert model and classifier to solve the data imbalance. The experiments demonstrate the effectiveness of our method.
|
---
abstract: 'An approach to the model-independent searching for the $Z''$ gauge boson as a virtual state in scattering processes is developed. It accounts for as a basic requirement the renormalizability of underlying unspecified in other respects model. This results in a set of relations between low energy couplings of $Z''$ to fermions that reduces in an essential way the number of parameters to be fitted in experiments. On this ground the observables which uniquely pick out the $Z''$ boson in leptonic processes are introduced and the data of LEP experiments analyzed. The $Z''$ couplings to leptons and quarks are estimated at 95% confidence level. These estimates may serve as a guide for experiments at the Tevatron and/or LHC. A comparison with other approaches and results is given.'
author:
- |
A. V. Gulov, V. V. Skalozub\
*Dnipropetrovsk National University, Dnipropetrovsk, Ukraine*
title: 'Model independent search for $Z''$-boson signals'
---
Introduction
============
The precision test of the standard model (SM) at the LEP gave a possibility not only to determine all the parameters and particle masses at the level of radiative corrections but also afforded an opportunity for searching for signals of new heavy particles beyond the energy scale of it. On the base of the LEP2 experiments the low bounds on parameters of various models extending the SM have been estimated and the scale of new physics was obtained [@EWWG; @OPAL; @DELPHI]. Although no new particles were discovered, a general believe is that the energy scale of new physics to be of order 1 TeV, that may serve as a guide for experiments at the Tevatron and LHC. In this situation, any information about new heavy particles obtained on the base of the present day data is desirable and important.
Numerous extended models include the $Z'$ gauge boson – massive neutral vector particle associated with the extra $U(1)$ subgroup of an underlying group. Searching for this particle as a virtual state is widely discussed in the literature (see for references [@Leike; @Lang08]). In the content of searching for $Z'$ at the LHC and the ILC an essential information and prospects for future investigations are given in lectures [@Rizzo06]. Such aspects as the mass of $Z'$, couplings to the SM particles, $Z - Z'$ mixing and its influence in various processes and particles parameters, distinctions between different models are discussed in details. We shall turn to these papers in what follows. As concerned a searching for $Z'$ in the LEP experiments and the experiments at Tevatron [@Ferroglia], it was carried out mainly in a model-dependent way. A wide class of popular models has been investigated and low bounds on the mass $m_{Z'}$ were estimated (see, [@EWWG; @OPAL; @DELPHI]). As it is occurred, the low masses are varying in a wide energy interval 400-1800 GeV dependently on a specific model. These bounds are a little bit different in the LEP and Tevatron experiments. In this situation a model-independent analysis is of interest.
In the papers [@EPJC2000; @PRD2000; @YAF2004; @PRD2004] of the present authors a new approach for the model-independent search for $Z'$-boson was proposed which, in contrast to other model-independent searches, gives a possibility to pick out uniquely this virtual state and determine its characteristics. The corresponding observables have also been introduced and applied to analyze the LEP2 experiment data. Our consideration is based on two constituents: 1) The relations between the effective low-energy couplings derived from the renormalization group (RG) equation for fermion scattering amplitudes. We called them the RG relations. Due to these relations, a number of unknown $Z'$ parameters entering the amplitudes of different scattering processes considerably decreases. 2) When these relations are accounted for, some kinematics properties of the amplitude become uniquely correlated with this virtual state and the $Z'$ signals exhibit themselves.
The RG relations allow to introduce observables correlated uniquely with the $Z'$-boson. Comparing the mean values of the observables with the necessary specific values, one could arrive at a conclusion about the $Z'$ existence. The confidence level (CL) of these values has been estimated and adduced in addition. Without taking into consideration the RG relations the determination of $Z'$-boson requires a supplementary specification due to a larger number of different couplings contributing to the observables. Similar situation takes place in the “helicity model fits” of LEP Collaborations [@EWWG; @OPAL; @DELPHI] when different virtual states contribute to each of the specific models (AA, VV, and so on). Therefore these fits had the goal to discover any signals of new physics independently of the particular states which may cause deviations from the SM. Note that the LEP Collaborations saw no indications of new contact four fermion interactions in these fits.
In Refs. [@EPJC2000; @YAF2004; @PRD2004] the one-parametric observables were introduced and the signals (hints in fact) of the $Z'$ have been determined at the 1$\sigma$ CL in the $e^+e^-\to\mu^+\mu^-$ process, and at the 2$\sigma$ CL in the Bhabha process. The $Z'$ mass was estimated to be 1–1.2 TeV. An increase in statistics could make these signals more pronounced and there is a good chance to discover this particle at the LHC.
In Ref. [@PRD2007] the updated results of the one-parameter fit and the complete many-parametric fit of the LEP2 data were performed with the goal to estimate a possible signal of the $Z'$-boson with accounting for the final data of the LEP collaborations DELPHI and OPAL [@OPAL; @DELPHI]. Usually, in a many-parametric fit the uncertainty of the result increases drastically because of extra parameters. On the contrary, in our approach due to the RG relations between the low-energy couplings there are only 2-3 independent parameters for the investigated leptonic scattering processes. As it was showed in Ref. [@PRD2007], an inevitable increase of confidence areas in the many-parametric space was compensated due to accounting for all accessible experimental information. Therefore, the uncertainty of the many-parametric fit was estimated as the comparable with previous one-parametric fits in Refs. [@YAF2004; @PRD2004]. In this approach the combined data fit for all lepton processes is also possible.
From the results obtained on the searching for Abelian $Z'$ within the LEP experiment data set we conclude that it is insufficient for convincing discovery of this particle as the virtual state. In this situation it is reasonable to analyze the data by using the neural network approach which is able to make a realistic prognoses for the parameters of interest. This investigation was done within the two parametric global fit of the LEP2 data on the Bhabha scattering process. As the result of all these considerations we derive at the 2$\sigma$ CL the characteristics of the $Z'$ (the vector $v$ and axial-vector $a$ couplings of the $Z'$ with SM leptons and the $Z-Z'$ mixing). The $Z'$ mass is also estimated. Due to the universality of the $a$ we also derived the model independent estimate of the $Z'$ axial-vector couplings to quarks, $a_q = a$. Note that the hints for the $Z'$ have been determined in all the processes considered that increases the reliability of the signal. These results may serve as a good input into the future LHC and ILC experiments and used in various aspects. To underline the importance of them we mention that there are many tools at the LHC for the identification of $Z'$. But many of them are only applicable if $Z'$ is relatively light. The knowledge of the $Z'$ couplings to SM fermions also have important consequences.
The paper is organized as follows. In sect. 2 we give a necessary information about the description of $Z'$ at low energies. In sects. 3-5 we discuss the origin of the RG relations, their explicit forms for the case of the heavy $Z'$ and consequences of the relations for scattering processes investigated. In sect. 6 the cross sections and the observables to pick out uniquely the virtual $Z'$ in the $e^+ e^- \to \mu^+ \mu^-, \tau^+ \tau^-$ processes are given. The fits of data are described and discussed. Then in sect. 7 the same is present for the Bhabha process $e^+ e^- \to e^+ e^-$. The one parametric and two parametric fits are discussed. In sect. 8 the analysis of this process is carried out by using the neuron network approach. The criteria for training the network are introduced which guarantee the 2$\sigma$ CL deviations of the data from the model containing the SM with extra $ Z'$. The obtained parameters of the $Z'$ practically coincide with that of derived in the one parameter analysis. In this way we determine the characteristics of the $Z'$ coming from the LEP experiments. In sect. 9 we discuss the role of the present model-independent analysis for the LHC experiments. The discussion and comparison with results of other approaches are given in sect. 10. In the Appendix we describe the two-mass-scale Yukawa model and analyze in detail how the decoupling of the loop contributions due to heavy virtual states is realized when the mixing of fields is taken into consideration. This point is an essential element of the approach developed.
The Abelian $Z'$ boson at low energies
======================================
Let us adduce a necessary information about the Abelian $Z'$-boson. This particle is predicted by a number of grand unification models. Among them the $E_6$ and $SO(10)$ based models [@Hewett] (for instance, LR, $\chi-\psi$ and so on) are often discussed in the literature. In all the models, the Abelian $Z'$-boson is described by a low-energy $\tilde{U}(1)$ gauge subgroup originated in some symmetry breaking pattern.
At low energies, the $Z'$-boson can manifest itself by means of the couplings to the SM fermions and scalars as a virtual intermediate state. Moreover, the $Z$-boson couplings are also modified due to a $Z$–$Z'$ mixing. In principle, arbitrary effective $Z'$ interactions to the SM fields could be considered at low energies. However, the couplings of non-renormalizable types have to be suppressed by heavy mass scales because of decoupling. Therefore, significant signals beyond the SM can be inspired by the couplings of renormalizable types. Such couplings can be derived by adding new $\tilde{U}(1)$-terms to the electroweak covariant derivatives $D^\mathrm{ew}$ in the Lagrangian [@cvetic87; @degrassi89] (review, [@Leike; @Lang08]) $$\begin{aligned}
\label{Lf}
L_f &=& i \sum\limits_{f_L} \bar{f}_L \gamma^\mu \left(
\partial_\mu - \frac{i g}{2} \sigma_a W^a_\mu - \frac{i g'}{2}
B_\mu Y_{f_L} -
\frac{i \tilde{g}}{2}\tilde{B}_\mu \tilde{Y}_{f_L}\right) f_L \\
\nonumber &+& i \sum\limits_{f_R} \bar{f}_R \gamma^\mu \left(
\partial_\mu - i
g' B_\mu Q_{f} - \frac{i \tilde{g}}{2}\tilde{B}_\mu
\tilde{Y}_{f_R}\right) f_R,\end{aligned}$$ where summation over all the SM left-handed fermion doublets, leptons and quarks, $f_L = {(f_u)_L, (f_d)_L}$, and the right-handed singlets, $f_R = (f_u)_R, (f_d)_R $, is understood. $Q_f$ denotes the charge of $f$ in positron charge units, $\tilde{Y}_{f_L}=\mathrm{diag}(\tilde{Y}_{f_u}, \tilde{Y}_{f_d})$, and $Y_{f_L}= -1$ for leptons and 1/3 for quarks.
For general purposes we derive the RG relations for the $Z'$ beyond the SM with two light Higgs doublets (THDM) [@EPJC2000]. $Z'$ interactions with the scalar doublets can be parametrized in a model-independent way as follows, $$\begin{aligned}
\label{Lscal}
L_\phi = \sum\limits_{i=1}^{2} \left| \left( \partial_\mu -
\frac{i g}{2} \sigma_a W^a_\mu - \frac{i g'}{2} B_\mu Y_{f_L} -
\frac{i \tilde{g}}{2} \tilde{B}_\mu \tilde{Y}_{\phi_{i}} \right)
\phi_i\right|^2.\end{aligned}$$ In these formulas, $g, g', \tilde{g}$ are the charges associated with the $SU(2)_L, U(1)_Y,$ and the $Z'$ gauge groups, respectively, $\sigma_a$ are the Pauli matrices, $\tilde{Y}_{\phi_{i}} = \mathrm{diag}(\tilde{Y}_{\phi_{i,1}},
\tilde{Y}_{\phi_{i,2}}) $ is the generator corresponding to the gauge group of the $Z'$ boson, and $Y_{\phi_i}$ is the $U(1)_Y$ hypercharge.
The Yukawa Lagrangian can be written in the form $$\begin{aligned}
\label{Lyukawa}
L_\mathrm{Yuk.} &=& - \sqrt{2}
\sum\limits_{f_L}\sum\limits_{i=1}^{2} \left(G_ {f_{d,i}}
[\bar{f}_L \phi_i (f_d)_R + (\bar{f}_d)_R \phi^{+}_i f_L]
\right.\nonumber\\&&\left. +G_{f_{u,i}} [\bar{f}_L \phi_i^c
(f_u)_R + (\bar{f}_u)_R \phi^{c+}_i f_L]\right),\end{aligned}$$ where $\phi^c_i = i \sigma_2 \phi^*_i $ is the charge conjugated scalar doublet.
The Lagrangian (\[Lscal\]) leads to the $Z$–$Z'$ mixing. The $Z$–$Z'$ mixing angle $\theta_0$ is determined by the coupling $\tilde{Y}_\phi$ as follows $$\label{2}
\theta_0 =
\frac{\tilde{g}\sin\theta_W\cos\theta_W}{\sqrt{4\pi\alpha_\mathrm{em}}}
\frac{m^2_Z}{m^2_{Z'}} \tilde{Y}_\phi
+O\left(\frac{m^4_Z}{m^4_{Z'}}\right),$$ where $\theta_W$ is the SM Weinberg angle, and $\alpha_\mathrm{em}$ is the electromagnetic fine structure constant. Although the mixing angle is a small quantity of order $m^{-2}_{Z'}$, it contributes to the $Z$-boson exchange amplitude and cannot be neglected at the LEP energies.
In what follows we will also use the $Z'$ couplings to the vector and axial-vector fermion currents defined as $$\label{av} v_f =
\tilde{g}\frac{\tilde{Y}_{L,f} + \tilde{Y}_{R,f}}{2}, \qquad a_f =
\tilde{g}\frac{\tilde{Y}_{R,f} - \tilde{Y}_{L,f}}{2}.$$ The Lagrangian (\[Lf\]) leads to the following interactions between the fermions and the $Z$ and $Z'$ mass eigenstates: $$\begin{aligned}
{\cal L}_{Z\bar{f}f}&=&\frac{1}{2} Z_\mu\bar{f}\gamma^\mu\left[
(v^\mathrm{SM}_{fZ}+\gamma^5 a^\mathrm{SM}_{fZ})\cos\theta_0
+\right.\nonumber\\&&\quad\left.
+(v_f+\gamma^5 a_f)\sin\theta_0 \right]f, \nonumber\\
{\cal L}_{Z'\bar{f}f}&=&\frac{1}{2} Z'_\mu\bar{f}\gamma^\mu\left[
(v_f+\gamma^5 a_f)\cos\theta_0 -\right.\nonumber\\&&\quad\left.
-(v^\mathrm{SM}_{fZ}+\gamma^5
a^\mathrm{SM}_{fZ})\sin\theta_0\right]f,\end{aligned}$$ where $f$ is an arbitrary SM fermion state; $v^\mathrm{SM}_{fZ}$, $a^\mathrm{SM}_{fZ}$ are the SM couplings of the $Z$-boson.
Since the $Z'$ couplings enter the cross-section together with the inverse $Z'$ mass, it is convenient to introduce the dimensionless couplings $$\label{6}
\bar{a}_f=\frac{m_Z}{\sqrt{4\pi}m_{Z'}}a_f,\quad
\bar{v}_f=\frac{m_Z}{\sqrt{4\pi}m_{Z'}}v_f,$$ which can be constrained by experiments.
Low energy parameters $\tilde{Y}_{\phi_{i,1}}$, $\tilde{Y}_{\phi_{i,2}}$, $\tilde{Y}_{L,f}$, $\tilde{Y}_{R,f}$ must be fitted in experiments. In most investigations they were considered as independent ones. In a particular model, the couplings $ \tilde{Y}_{\phi_{i,1}}$, $ \tilde{Y}_{\phi_{i,2}},
\tilde{Y}_{L,f}, \tilde{Y}_{R,f}$ take some specific values. In case when the model is unknown, these parameters remain potentially arbitrary numbers. However, this is not the case if one assumes that the underlying extended model is a renormalizable one. In the papers [@EPJC2000; @PRD2000] it was shown that these parameters are correlated due to renormalizability. We called them the RG relations. Since this notion is a key-point of our consideration, we discuss it in detail.
Renormalization group relations
===============================
What is RG relation? Generally speaking, this is a correlation between low energy parameters of interactions of a heavy new particle with known light particles of the SM following from the requirement that full unknown yet theory extending SM is to be renormalizable.
Strictly speaking, RG relations are the consequence of two constituencies:
1. RG equation for a scattering amplitude;
2. Decoupling theorem.
The latter one describes the modification of both the RG operator $$\label{rgo}
{{\cal D}} = \frac{d}{d \log \mu} = \frac{\partial}{\partial \log
\mu} + \sum\limits_a \beta_a \frac{\partial}{\partial
\hat{\lambda}_a} - \sum\limits_{\hat{X}} \gamma_X
\frac{\partial}{\partial \log \hat{X}}$$ and an amplitude at the energy threshold $\Lambda$ of new physics. Here, $\beta_a$- and $\gamma_X$-functions correspond to all the charges $\hat{\lambda}_a$ and fields and masses $\hat{X}$ of the underlying theory.
The RG equation for a scattering amplitude $f$ reads, $$\label{RGe}
{{\cal D}}f = \left(\frac{\partial}{\partial \log \mu} +
\sum\limits_a \beta_a \frac{\partial}{\partial \hat{\lambda}_a} -
\sum\limits_{\hat{X}} \gamma_X \frac{\partial}{\partial \log
\hat{X}}\right)f = 0,$$ where $f$ accounts for as intermediate states either the light or heavy virtual particles of the full theory. The standard usage of the RG equation is to improve the amplitude by solving this equation for the operator ${{\cal D}}$ calculated in a given order of perturbation theory. However, to search for heavy virtual particles, we will use Eq. (\[RGe\]) in another way.
First we note that for any renormalizable theory, the RG equation is just identity, if $f$ and ${{\cal D}}$ are calculated in a given order of loop expansion. In this case Eq. (\[RGe\]) expresses the well known fact that the structure of the divergent term coincides with the structure of the corresponding term in a tree-level Lagrangian.
For example, in massless QED, the tree-level plus one-loop one-particle-irreducible vertex function describing scattering of electron in an external electromagnetic field $\bar{A}$, $ \Gamma
= \Gamma^{(0)} + \Gamma^{(1)}$, is
{width=".5\textwidth"}
If we calculate the RG operator in one-loop order $$\label{rgo1}
{{\cal D}} = \frac{\partial}{\partial \log \mu} + \beta_e^{(1)}
\frac{\partial}{\partial e} - 2 \gamma_\psi^{(1)} - \gamma_A^{(1)}
,$$ where $\beta_e^{(1)}$, $ \gamma_A^{(1)}, \gamma_\psi^{(1)}$ are the beta-function and the anomalous dimensions of electromagnetic and electron fields, respectively, and apply it to $\Gamma$, we obtain $$\label{rge1}
- \frac{\partial}{\partial\log\mu}\Gamma^{(1)} = \left(
\beta_e^{(1)} \frac{\partial}{\partial e} - 2 \gamma_\psi^{(1)} -
\gamma_A^{(1)}\right)~\Gamma^{(0)} + O(e^5).$$ Then, accounting for the values of $$\label{parameter}
\beta_e^{(1)} = \frac{e^3}{12 \pi^2},\quad \gamma_A^{(1)} =
\frac{e^2}{12 \pi^2},\quad \gamma_{\psi}^{(1)} = \frac{e^2}{16
\pi^2}$$ and the factor $e$ in $\Gamma^{(0)}$, we observe that the first and the last terms in the r.h.s. cancel. Since $\mu$-dependent term in $\Gamma^{(1)} $ is $\Gamma^{(1)}_\mu = \frac{e^3 }{16
\pi^2}\log \mu^2$, we see that Eq.(\[rge1\]) is identity in the order $O(e^3)$.
Next important point is that in a theory with different mass scales the decoupling of heavy-loop contributions at the threshold of heavy masses, $\Lambda$, results in the following property: the running of all functions is regulated by the loops of light particles. Therefore, the $\beta$ and $\gamma$ functions at low energies are determined by the SM particles, only. This fact is the consequence of the decoupling theorem [@appelquist75; @collins78].
The decoupling results in the redefinition of parameters at the scale $\Lambda$ and removing heavy-particle loop contributions from RG equation [@bando93; @EPJC2000]: $$\begin{aligned}
\label{redefinition}
\lambda_a &=& \hat{\lambda}_a + a_{\hat{\lambda}_a}\log
\frac{\hat{\Lambda}}{\mu^2} + b_{\hat{\lambda}_a} \log^2
\frac{\Lambda^2}{\mu^2} + \cdots,
\\ \nonumber
X &=& \hat{X}\left( 1 + a_{\hat{\lambda}_a}\log
\frac{\hat{\Lambda}}{\mu^2} + b_{\hat{\lambda}_a} \log^2
\frac{\Lambda^2}{\mu^2} + \cdots\right),\end{aligned}$$ where $\lambda_a$ and $X$ denote the parameters of the SM. They are calculated assuming that no heavy particles are excited inside loops.
The matching between both sets of parameters [$\lambda_a$, $X$]{} and [$\hat{\lambda}_a$, $\hat{X}$]{} is chosen at the normalization point $\mu \sim \Lambda$, $$\label{match}
\lambda_a |_{\mu = \Lambda} = \hat{\lambda}_a |_{\mu =
\Lambda},\quad X_a |_{\mu = \Lambda} = \hat{X}_a |_{\mu = \Lambda}
.$$
The differential operator ${\cal D}$ in the RG equation is in fact unique; the apparently different ${\cal D}$ in both theories are the same!
Note that if a theory with different mass scales is specified one can freely replace the parameters [$\lambda_a$, $X$]{} and [$\hat{\lambda}_a$, $\hat{X}$]{} by each other [@bando93; @EPJC2000].
An example of the derivation and the main features of the RG relations are shown in the Appendix for a simple model with different mass scales.
If underlying theory is not specified, the set of [$\hat{\lambda}_a$, $\hat{X}$]{} is unknown. The low energy theory consists of the SM plus the effective Lagrangian generated by the interactions of light particles with virtual heavy particle states. The low energy parameters $\lambda'_{l.}$ of these interactions are arbitrary numbers which must be constrained by experiments. By calculating the RG operator ${\cal D}$ and the scattering amplitudes of light particles in this ‘external field’ in a chosen order of loop expansion, it is possible to obtain the model-independent correlations between $\lambda'_{l.}$. These are just the RE relations.
The RG relations for $Z'$ boson couplings
=========================================
Let us derive the correlations between $ \tilde{Y}_{\phi_{i,1}}$, $ \tilde{Y}_{\phi_{i,2}}, \tilde{Y}_{L,f}, \tilde{Y}_{R,f}$ appearing due to renormalizability of the underlying theory containing $Z'$.
In our case, the RG invariance of the vertex leads to the equation $$\label{matchA}
{\cal D}\left(\bar{f} \Gamma_{fZ'} f \frac{1}{m_Z'}\right) = 0,$$ where $$\label{rgo1A}
{{\cal D}} = \frac{d}{d \log \mu} = \frac{\partial}{\partial \log
\mu} + \sum\limits_a \beta_a \frac{\partial}{\partial \lambda}_a -
\sum\limits_{X} \gamma_X \frac{\partial}{\partial \log X},$$ and $$\label{betaX}
\beta_{a} = \frac{d \lambda}{d \log \mu}, ~~\gamma_X = - \frac{d
\log X}{d \log \mu}$$ are computed with taking into account the loops of light particles.
Now we derive the RG relations following from the one-loop consideration. In accordance with the previous sections, the one-loop RG equation for the vertex function reads $$\label{rgez'}
\bar{f}\frac{\partial \Gamma^{(1)}_{f Z'} }{\partial \log \mu} f
\frac{1}{m_{Z'}} + {\cal D}^{(1)} \left(\bar{f} \Gamma^{(0)}_{f
Z'} f \frac{1}{m_{Z'}}\right) = 0,$$ where $\Gamma^{(0)}_{f Z'}$ and $\Gamma^{(1)}_{f Z'}$ are the tree-level and one-loop contributions to the fermion-$Z'$ vertex. ${\cal D}^{(1)} = \sum\limits_a \beta^{(1)}_a
\frac{\partial}{\partial \lambda}_a - \sum\limits_{X}
\gamma^{(1)}_X \frac{\partial}{\partial \log X}$ is the one-loop level part of the RG operator.
![The diagrams contributing to the divergent parts of the $Z'ff$ vertex at the one-loop level.[]{data-label="f-2"}](fig2.eps){width=".6\textwidth"}
![The diagrams contributing to the fermion anomalous dimension at the one-loop level.[]{data-label="f-3"}](fig3.eps){width=".6\textwidth"}
To calculate these functions, only the divergent parts of the one-loop vertices are to be calculated. The corresponding diagrams are shown in Fig. \[f-2\]. The fermion anomalous dimensions $\gamma^{(1)}_X$ can be calculated by using the diagrams in Fig. \[f-3\]. Then, Eq.(\[rgez’\]) leads to algebraic equations for the parameters $ \tilde{Y}_{\phi_{i,1}}$, $
\tilde{Y}_{\phi_{i,2}}, \tilde{Y}_{L,f},$ and $\tilde{Y}_{R,f}$ which have two sets of solutions [@EPJC2000]: $$\begin{aligned}
\label{rgr1}
\tilde{Y}_{\phi_{2,1}} &=& \tilde{Y}_{\phi_{1,1}} = -
\tilde{Y}_{\phi_{,2}} \equiv - \tilde{Y}_{\phi_{}}, \\ \nonumber
\tilde{Y}_{L,f}&+& \tilde{Y}_{L,f^*} = 0,~ \tilde{Y}_{R,f} = 0,\end{aligned}$$ and $$\begin{aligned}
\label{rgr2}
\tilde{Y}_{\phi_{1,1}} &=& \tilde{Y}_{\phi_{2,1}} =
\tilde{Y}_{\phi_{,2}} \equiv \tilde{Y}_{\phi_{}}, \\ \nonumber
\tilde{Y}_{L,f}&=& \tilde{Y}_{L,f^*},~~ \tilde{Y}_{R,f} =
\tilde{Y}_{L,f} + 2 T_{3f}~ \tilde{Y}_{\phi_{}}.\end{aligned}$$ Here $f$ and $f^*$ are the partners of the $SU(2)_L$ fermion doublet ($l^* = \nu_l, \nu^* = l, q^*_u = q_d$ and $q^*_d = q_u$), $T_{3f}$ is the third component of weak isospin.
The first of these relations describes the $Z'$ boson analogous to the third component of the $SU(2)_L$ gauge field. The couplings to the right-handed singlet are absent.
The second relation corresponds to the Abelian $Z'$. In this case the SM Lagrangian appears to be invariant with respect to the $\tilde{U}(1)$ group associated with the $Z'$. The last relation in Eq.(\[rgr2\]) ensures the $L_\mathrm{Yuk.}$ Eq.(\[Lyukawa\]) is to be invariant with respect to the $\tilde{U}(1)$ transformations.
Introducing the $Z'$ couplings to the vector and axial-vector fermion currents (\[av\]), the last line in Eq. (\[rgr2\]) yields $$\label{grgav}
v_f - a_f= v_{f^*} - a_{f^*}, \qquad a_f = T_{3f}
\tilde{g}\tilde{Y}_\phi.$$ The couplings of the Abelian $Z'$ to the axial-vector fermion current have a universal absolute value proportional to the $Z'$ coupling to the scalar doublet.
These relations are model independent. In particular, they hold in all the known models containing the Abelian $Z'$. The most discussed models are derived from the ${\rm E}_6$ group (the so called LR, $\chi$-$\psi$ models). The tree-level $Z'$ couplings to the SM fermions in the models are shown in Table 1.
\[zpcoup\]
------- ----------------------------------- ----------------------------------- ---------------------- ---------------------------------------
$f$
$a_f/\tilde{g}$ $v_f/\tilde{g}$ $a_f/\tilde{g}$ $v_f/\tilde{g}$
$\nu$ $-3\frac{\cos{\beta}}{\sqrt{40}} $3\frac{\cos{\beta}}{\sqrt{40}} $-\frac{1}{2\alpha}$ $\frac{1}{2\alpha}$
-\frac{\sin{\beta}}{\sqrt{24}}$ +\frac{\sin{\beta}}{\sqrt{24}}$
$e$ $-\frac{\cos{\beta}}{\sqrt{10}} $2\frac{\cos{\beta}}{\sqrt{10}}$ $-\frac{\alpha}{2}$ $\frac{1}{\alpha}-\frac{\alpha}{2}$
-\frac{\sin{\beta}}{\sqrt{6}}$
$q_u$ $\frac{\cos{\beta}}{\sqrt{10}} 0 $\frac{\alpha}{2}$ $-\frac{1}{3\alpha}+\frac{\alpha}{2}$
-\frac{\sin{\beta}}{\sqrt{6}}$
$q_d$ $-\frac{\cos{\beta}}{\sqrt{10}} $-2\frac{\cos{\beta}}{\sqrt{10}}$ $-\frac{\alpha}{2}$ $-\frac{1}{3\alpha}-\frac{\alpha}{2}$
-\frac{\sin{\beta}}{\sqrt{6}}$
------- ----------------------------------- ----------------------------------- ---------------------- ---------------------------------------
: The $Z^\prime$ couplings to the SM fermions in the most discussed ${\rm E}_6$-based models.
The ${\rm E}_6$-symmetry breaking scheme $${\rm E}_6\to{\rm SO}(10)\times{\rm U}(1)_\psi,\quad {\rm
SO}(10)\to{\rm SU}(3)_c\times{\rm SU}(2)_L \times{\rm
SU}(2)_R\times{\rm U}(1)_{B-L}.$$ leads to the so called left-right (LR) model. Another scheme, $${\rm E}_6\to{\rm SO}(10)\times{\rm U}(1)_\psi\to{\rm SU}(5)\times
{\rm U}(1)_\chi\times{\rm U}(1)_\psi,$$ predicts the Abelian $Z^\prime$, which is a linear combination of the neutral vector bosons $\psi$ and $\chi$, $$Z^\prime=\chi\cos{\beta} +\psi\sin{\beta}$$ with the mixing angle $\beta$. If we suppose only one $Z'$ boson at low energies, the $\psi$ boson should be much heavier than the $\chi$ field. In this case, the field $\psi$ is decoupled and $\beta\to 0$. As it is seen, both the LR and the $\chi$-$\psi$ models (with $\beta=0$ to avoid two $Z'$ bosons with the same scale of masses) satisfy the RG relations (\[rgr2\]) except for neutrinos. Let us explain this discrepancy. It is usually supposed in theories based on the ${\rm E}_6$ group that the Yukawa terms responsible for generation of the Dirac masses of neutrinos must be set to zero [@Hewett]. Therefore, the terms proportional to the Yukawa couplings vanish in the renormalization group equation, and there are no RG relations for the $Z^\prime$ interactions with the neutrino axial-vector currents. In this case the couplings $a_\nu$ given in Table 1 are not restricted by the RG relations.
Implication of the RG relations
===============================
LEP collaborations applied model dependent search for $Z'$ and have obtained low bounds on the mass $ m_{Z'} \geq 400 - 800$ GeV dependently on a specific model [@EWWG; @OPAL; @DELPHI].
In our analysis, we consider the SM with the additional effective $Z'$ interactions (\[Lf\]), (\[Lscal\]), (\[Lyukawa\]) as a low energy theory. The parameters $a_f, v_f$ and $m_{Z'}$ must be fitted in experiments. The RG relations give a possibility:
1. reduce the number of fitted parameters;
2. determine kinematics of the processes;
3. introduce observables which uniquely pick out the $Z'$ signals.
The RG relations (\[rgr2\]) influences the $Z - Z'$ mixing (\[2\]). The axial-vector coupling determines also the coupling to the scalar doublet and, consequently, the mixing angle. As a result, the number of independent couplings is significantly reduced.
In what follows, both types of the RG relations (\[rgr1\]) and (\[rgr2\]) will be used in order to search for signals of the $Z'$ gauge boson.
$Z'$ search in $e^+e^-\to\mu^+\mu^-,\tau^+\tau^-$ processes
===========================================================
The differential cross section
------------------------------
Let us consider the processes $e^+e^-\to l^+l^-$ ($l=\mu,\tau$) with the non-polarized initial- and final-state fermions. In order to introduce the observable which selects the signal of the Abelian $Z'$ boson we need to compute the differential cross-sections of the processes up to the one-loop level.
The lower-order diagrams describe the neutral vector boson exchange in the $s$-channel ($e^+e^-\to V^\ast\to l^+l^-$, $V=A,Z,Z'$). As for the one-loop corrections, two classes of diagrams are taken into account. The first one includes the pure SM graphs (the mass operators, the vertex corrections, and the boxes). The second set of the one-loop diagrams improves the Born-level $Z'$-exchange amplitude by “dressing” the $Z'$ propagator and and the $Z'$–fermion vertices. We assume that $Z'$ states are not excited inside loops. Such an approximation means that the $Z'$-boson is completely decoupled. Then, the differential cross-section consists of the squared tree-level amplitude and the term from the interference of the tree-level and the one-loop amplitudes. To obtain an infrared-finite result, we also take into account the processes with the soft-photon emission in the initial and final states.
In the lower order in $m^{-2}_{Z'}$ the $Z'$ contributions to the differential cross-section of the process $e^+e^-\to l^+l^-$ are expressed in terms of four-fermion contact couplings, only. If one takes into consideration the higher-order corrections in $m^{-2}_{Z'}$, it becomes possible to estimate separately the $Z'$-induced contact couplings and the $Z'$ mass [@rizzo]. In the present analysis we keep the terms of order $O(m^{-4}_{Z'})$ to fit both of these parameters.
Expanding the differential cross-section in the inverse $Z'$ mass and neglecting the terms of order $O(m^{-6}_{Z'})$, we have $$\begin{aligned}
%\label{eq8}
\frac{d\sigma_l(s)}{dz} &=& \frac{d\sigma_l^{\rm SM}(s)}{dz}
+\sum_{i=1}^{7}\sum_{j=1}^{i}
\left[A_{ij}^l(s,z)+B_{ij}^l(s,z)\zeta\right]x_{i}x_{j}
\nonumber\\&&
+\sum_{i=1}^{7}\sum_{j=1}^{i}\sum_{k=1}^{j}\sum_{n=1}^{k}
C_{ijkn}^l(s,z)x_{i}x_{j}x_{k}x_{n},\end{aligned}$$ where the dimensionless quantities $$\begin{aligned}
%\label{5}
\zeta = \frac{m^2_Z}{m^2_{Z'}},\quad
% \epsilon=\frac{\tilde{g}^2 m^2_Z a^2}{4\pi m^2_{Z'}},
%\nonumber\\
(x_1,x_2,x_3,x_4,x_5,x_6,x_7) =
(\bar{a},\bar{v}_e,\bar{v}_\mu,\bar{v}_\tau,\bar{v}_d,\bar{v}_s,\bar{v}_b)\end{aligned}$$ are introduced. Since the axial-vector couplings of the Abelian $Z'$ boson are universal, we use the shorthand notation $\bar{a}=\bar{a}_e$. In what follows the index $l=\mu,\tau$ denotes the final-state lepton.
The coefficients $A$, $B$, $C$ are determined by the SM couplings and masses. Each factor may include the tree-level contribution, the one-loop correction and the term describing the soft-photon emission. The factors $A$ describe the leading-order contribution, whereas others correspond to the higher order corrections in $m^{-2}_{Z'}$.
The observable
--------------
To take into consideration the correlations (\[2\]) we introduce the observable $\sigma_l(z)$ defined as the difference of cross sections integrated in some ranges of the scattering angle $\theta$ [@PRD2000; @YAF2004]: $$\begin{aligned}
\label{eq8}
\sigma_l(z)
&\equiv&\int\nolimits_z^1
\frac{d\sigma_l}{d\cos\theta}d\cos\theta
-\int\nolimits_{-1}^z
\frac{d\sigma_l}{d\cos\theta}d\cos\theta,\end{aligned}$$ where $z$ stands for the cosine of the boundary angle. The idea of introducing the $z$-dependent observable (\[eq8\]) is to choose the value of the kinematic parameter $z$ in such a way that to pick up the characteristic features of the Abelian $Z'$ signals.
The deviation of the observable from its SM value can be derived by the angular integration of the differential cross-section and has the form: $$\begin{aligned}
%\label{eq8}
\Delta\sigma_l(z) &=& \sigma_l(z) - \sigma^{\rm SM}_l(z)
=\sum_{i=1}^{7}\sum_{j=1}^{i}
\left[\tilde{A}_{ij}^l(s,z)+\tilde{B}_{ij}^l(s,z)\zeta\right]x_{i}x_{j}
\nonumber\\&&
+\sum_{i=1}^{7}\sum_{j=1}^{i}\sum_{k=1}^{j}\sum_{n=1}^{k}
\tilde{C}_{ijkn}^l(s,z)x_{i}x_{j}x_{k}x_{n}.\end{aligned}$$
Then let us introduce the quantity $\Delta\sigma\left(z\right)\equiv \sigma\left(z\right)
-{\sigma}_{SM}\left(z\right)$, which owing to the relations (\[rgr2\]) can be written in the form $$\begin{aligned}
\label{obs:6}
\Delta\sigma_f(z)
&=&\frac{\alpha N_f}{8}\frac{g^2_{Z^\prime}}{m^2_{Z^\prime}}
\left[
F^f_0(z,s) \tilde{Y}^2_\phi
+2 F^f_1(z,s) T_{3f}\tilde{Y}_{L,f}\tilde{Y}_{L,e} \right.
\nonumber\\
&&
+\left.
2 F^f_2(z,s) T_{3f}\tilde{Y}_{L,f}\tilde{Y}_\phi
+ F^f_3(z,s) \tilde{Y}_{L,e}\tilde{Y}_\phi \right].\end{aligned}$$ The factor functions $F^f_i(z,s)$ depend on the fermion type through the $|Q_f|$, only. In Fig. \[fig:2f\] they are shown as the functions of $z$ for $\sqrt{s}=500$ GeV. The leading contributions to $F^f_i(z,s)$, $$\begin{aligned}
\label{obs:7}
F^f_0(z,s)&=&
-\frac{4}{3}\left|Q_f\right|
\left(1 -z -z^{2} -\frac{z^{3}}{3}\right)
% \nonumber\\&&
+O\left(\frac{m^2_Z}{s}\right),
\nonumber\\
F^f_1(z,s)&=&\frac{4}{3}
\left[1 -z^{2} -\left|Q_f\right|
\left(3z+z^{3}\right)\right]
% \nonumber\\&&
+O\left(\frac{m^2_Z}{s}\right),
\nonumber\\
F^f_2(z,s)&=&
-\frac{2}{3}\left(1-z^{2}\right)
+\frac{2}{9}\left(3z+z^{3}\right)
% \nonumber\\ &&\times
\left(4\left|Q_f\right|-1\right)
+O\left(\frac{m^2_Z}{s}\right),
\nonumber\\
F^f_3(z,s)&=&\frac{2}{3}\left|Q_f\right|
\left(1-3z-z^{2}-z^{3}\right)
% \nonumber\\&&
+O\left(\frac{m^2_Z}{s}\right),\end{aligned}$$ are given by the $Z^\prime$ exchange diagram $e^-e^+\to
Z'\to\bar{f}f$, since the $Z$-$Z'$ mixing contribution to the $Z$ exchange diagram is suppressed by the factor $m^2_Z/s$.
From Eqs. (\[obs:7\]) one can see that the leading contributions to the leptonic factors $F^l_1$, $F^l_2$, $F^l_3$ are found to be proportional to the same polynomial in $z$. This is the characteristic feature of the leptonic functions $F^l_i$ originating due to the kinematic properties of fermionic currents and the specific values of the SM leptonic charges. Therefore, it is possible to choose the value of $z=z^\ast$ which switches off three leptonic factors $F^l_1$, $F^l_2$, $F^l_3$ simultaneously. Moreover, the quark function $F^q_3$ in the lower order is proportional to the leptonic one and therefore is switched off, too. As is seen from Fig. \[fig:2f\], the appropriate value of $z^\ast$ is about $\sim 0.3$. By choosing this value of $z^\ast$ one can simplify Eq. (\[obs:6\]). It is also follows from Eq. (\[obs:6\]) that neglecting the factors $F^l_1$, $F^l_2$, $F^l_3$ one obtains the sign definite quantity $\Delta \sigma_l
(z^\ast)\sim \tilde{Y}^2_\phi\sim\bar{a}^2$.
There is an interval of values of the boundary angle, at which the factors $\tilde{A}^l_{11}$, $\tilde{B}^l_{11}$, and $\tilde{C}^l_{1111}$ at the sign-definite parameters $\bar{a}^2$, $\bar{a}^2\zeta$, and $\bar{a}^4$ contribute more than 95% of the observable value. It gives a possibility to construct the sign-definite observable $\Delta\sigma_l(z^*)<0$ by specifying the proper value of $z^*$.
In general, one could choose the boundary angle $z^*$ in different schemes. If just a few number of tree-level four-fermion contact couplings are considered, one can specify $z^*$ in order to cancel the factor at the vector-vector coupling. However, if one-loop corrections are taken into account there are a large number of additional contact couplings. So, we have to define some quantitative criterion $F(z)$ to estimate the contributions from sign-definite factors at a given value of the boundary angle $z$. Maximizing the criterion, one could derive the value $z^*$, which corresponds to the sign-definite observable $\Delta\sigma_l(z^*)$. Since the observable is linear in the coefficients $A$, $B$, and $C$, we introduce the following criterion, $$F=\frac{|\tilde{A}_{11}|+\omega_B |\tilde{B}_{11}| + \omega_C
|\tilde{C}_{1111}|}
{\sum\limits_{{\rm all}~\tilde{A}}\left|\tilde{A}_{ij}\right|
+\omega_B\sum\limits_{{\rm all}~\tilde{B}}\left|\tilde{B}_{ij}\right|
+\omega_C\sum\limits_{{\rm all}~\tilde{C}}\left|\tilde{C}_{ijkn}\right|},$$ where the positive ‘weights’ $\omega_B\sim\zeta$ and $\omega_C\sim\epsilon$ take into account the order of each term in the inverse $Z'$ mass.
The numeric values of the ‘weights’ $\omega_B$ and $\omega_C$ can be taken from the present day bounds on the contact couplings [@EWWG]. As the computation shows, the value of $z^*$ with the accuracy $10^{-3}$ depends on the order of the ‘weight’ magnitudes, only. So, in what follows we take $\omega_B\sim
4\times 10^{-3}$ and $\omega_C\sim 4\times 10^{-5}$.
The function $z^\ast(s)$ is the decreasing function of the center-of-mass energy. It is tabulated for the LEP2 energies in Table \[tobsmu\]. The corresponding values of the maximized function $F$ are within the interval $0.95<F<0.96$.
----------------- ------- ------------------ ------------------ -------------------- ------- ------------------ ------------------ --------------------
$\sqrt{s}$, GeV $z^*$ $\tilde{A}_{11}$ $\tilde{B}_{11}$ $\tilde{C}_{1111}$ $z^*$ $\tilde{A}_{11}$ $\tilde{B}_{11}$ $\tilde{C}_{1111}$
130 0.450 -729 -1792 -19636 0.460 -687 -1664 -25782
136 0.439 -709 -1859 -16880 0.442 -688 -1779 -20784
161 0.400 -643 -2183 -6890 0.400 -625 -2097 -10993
172 0.390 -619 -4099 -4099 0.391 -601 -2263 -8382
183 0.383 -599 -2545 -1334 0.385 -571 -2402 -7580
189 0.380 -586 -2635 -495 0.380 -568 -2533 -5135
192 0.380 -579 -2681 -63 0.380 -562 -2578 -4769
196 0.380 -571 -2745 -528 0.379 -554 -2640 -4272
200 0.378 -564 -2811 -1137 0.378 -547 -2704 -3761
202 0.376 -560 -2845 -1448 0.377 -543 -2736 -3501
205 0.374 -555 -2897 -1923 0.374 -548 -2834 -1292
207 0.372 -552 -2932 -2245 0.372 -544 -2868 -1010
----------------- ------- ------------------ ------------------ -------------------- ------- ------------------ ------------------ --------------------
: The boundary angle $z^*$ and the coefficients in the observable $\Delta\sigma_l(z^*)$ for the scattering into $\mu$ and $\tau$ pairs at the one-loop level. []{data-label="tobsmu"}
Since $\tilde{A}^l_{11}(s,z^*)<0$, $\tilde{B}^l_{11}(s,z^*)<0$ and $\tilde{C}^l_{1111}(s,z^*)<0$, the observable $$\label{7dfg}
\Delta\sigma_l(z^*)=
\left[\tilde{A}^l_{11}(s,z^*) +\zeta\tilde{B}^l_{11}(s,z^*)
\right]\bar{a}^2 + \tilde{C}^l_{1111}(s,z^*)\bar{a}^4$$ is negative with the accuracy 4–5%. Since this property follows from the RG relations (\[2\]) for the Abelian $Z'$ boson, the observable $\Delta\sigma_l(z^*)$ selects the model-independent signal of this particle in the processes $e^+e^-\to l^+l^-$. It allows to use the data on scattering into $\mu\mu$ and $\tau\tau$ pairs in order to estimate the Abelian $Z'$ coupling to the axial-vector lepton currents.
Although the observable can be computed from the differential cross-sections directly, it is also possible to recalculate it from the total cross-sections and the forward-backward asymmetries. The recalculation procedure has the proper theoretical accuracy. Nevertheless, it allows to reduce the experimental errors on the observable, since the published data on the total cross-sections and the forward-backward asymmetries are more precise than the data on the differential cross-sections.
The recalculation is based on the fact that the differential cross-section can be approximated with a good accuracy by the two-parametric polynomial in the cosine of the scattering angle $z$: $$%\label{7}
\frac{d\sigma_l(s)}{dz} = \frac{d\sigma_l^{\rm SM}(s)}{dz} +
(1+z^2)\beta_l + z \eta_l +\delta_l(z),$$ where $\delta_l(z)$ measures the difference between the exact and the approximated cross-sections. The approximated cross-section reproduces the exact one in the limit of the massless initial- and final-state leptons and if one neglects the contributions of the box diagrams.
Performing the angular integration, it is easy to obtain the expression for the observable: $$%\label{7}
\Delta\sigma_l(z^*)=\sigma_l(z^*) - \sigma_l^{\rm SM}(z^*) =
(1-z^{*2})\eta_l -\frac{2\beta_l}{9} z^*(3+z^{*2})
+\tilde\delta_l(z^*),$$ and for the total and the forward-backward cross-sections: $$\begin{aligned}
%\label{7}
\Delta\sigma^{\rm T}_l &=&\sigma^{\rm T}_l- \sigma_l^{\rm T,SM}
=\frac{8\beta_l}{9}
+\tilde\delta_l(-1), \nonumber\\ \Delta\sigma^{\rm FB}_l &=&
\sigma^{\rm FB}_l-\sigma_l^{\rm FB,SM} =\eta_l +\tilde\delta_l(0).\end{aligned}$$ Then, the factors $\beta_l$ and $\eta_l$ can be eliminated from the observable: $$\begin{aligned}
%\label{7}
\Delta\sigma_l(z^*) &=& (1-z^{*2})\Delta\sigma^{\rm FB}_l
-\frac{3}{12} z^*(3+z^{*2})\Delta\sigma^{\rm T}_l +\xi_l.\end{aligned}$$ The quantity $\xi_l$, $$\begin{aligned}
%\label{7}
\xi_l&=&\tilde\delta_l(z^*) -(1-z^{*2})\tilde\delta_l(0)
+\frac{3}{12} z^*(3+z^{*2})\tilde\delta_l(-1),\end{aligned}$$ measures the theoretical accuracy of the approximation.
The forward-backward cross-section is related to the total one and the forward-backward asymmetry by means of the following expression $$\begin{aligned}
%\label{7}
\Delta\sigma^{\rm FB}_l &=& \Delta\sigma^{\rm T}_l \,\, A^{\rm
FB}_l +\sigma_l^{\rm T,SM} \,\, \Delta A^{\rm FB}_l.\end{aligned}$$ As the computation shows, $\tilde\delta_l(z^*)\simeq 0.01
\Delta\sigma_{l}(z^*)$, $\tilde\delta_l(0)\simeq 0.007
\Delta\sigma^{\rm FB}_l$, and $\tilde\delta_l(-1)\simeq -0.07
\Delta\sigma^{\rm T}_l$ at the LEP2 energies. Taking into account the experimental values of the total cross-sections and the forward-backward asymmetries at the LEP2 energies ($\Delta\sigma^{\rm T}_l\simeq 0.1$pb, $\sigma_l^{\rm T,SM}\simeq
2.7$pb, $\Delta A_l^{\rm FB}\simeq 0.04$, $A_l^{\rm FB}\simeq
0.5$), one can estimate the theoretical error as $\xi_l\simeq
0.003{\rm pb}$. At the same time, the corresponding statistical uncertainties on the observable are larger than 0.06pb. Thus, the proposed approximation is quite good and can be successfully used to obtain more accurate experimental values of the observable.
Data fit
--------
To search for the model-independent signals of the Abelian $Z'$-boson we will analyze the introduced observable $\Delta\sigma_l (z^\ast)$ on the base of the LEP2 data set. In the lower order in $m^{-2}_{Z'}$ the observable (\[7dfg\]) depends on one flavor-independent parameter $\bar{a}^2$, $$\Delta\sigma^{\rm th}_l(z^*)=
\tilde{A}^l_{11}(s,z^*)\bar{a}^2 + \tilde{C}^l_{1111}(s,z^*)\bar{a}^4,$$ which can be fitted from the experimental values of $\Delta\sigma_\mu (z^\ast)$ and $\Delta\sigma_\tau (z^\ast)$. As we noted above, the sign of the fitted parameter ($\bar{a}^2 >0$) is a characteristic feature of the Abelian $Z'$ signal.
In what follows we will apply the usual fit method based on the likelihood function. The central value of $\bar{a}^2$ is obtained by the minimization of the $\chi^2$-function: $$\chi^2(\bar{a}^2) = \sum_{n} \frac{\left[\Delta\sigma^{\rm
ex}_{\mu,n}(z^*)- \Delta\sigma^{\rm th}_\mu(z^*)\right]^2}
{\delta\sigma^{\rm ex}_{\mu,n}(z^*)^2},$$ where the sum runs over the experimental points entering a data set chosen. The $1\sigma$ CL interval $(b_1,b_2)$ for the fitted parameter is derived by means of the likelihood function ${\cal
L}(\bar{a}^2)\propto\exp[-\chi^2(\bar{a}^2)/2]$. It is determined by the equations: $$\int\nolimits_{b_1}^{b_2}{\cal L}(\epsilon ')d\epsilon ' = 0.68,
\quad
{\cal L}(b_1)={\cal L}(b_2).$$
To compare our results with those of Refs. [@EWWG] we introduce the contact interaction scale $$\Lambda^2 = 4m^2_Z\bar{a}^{-2}.$$ This normalization of contact couplings is admitted in Refs. [@EWWG]. We use again the likelihood method to determine a one-sided lower limit on the scale $\Lambda$ at the 95% CL. It is derived by the integration of the likelihood function over the physically allowed region $\bar{a}^2>0$. The strict definition is $$\Lambda=2m_Z (\epsilon^*)^{-1/2}, \quad \int_{0}^{\epsilon^*}{\cal
L}(\epsilon ')d\epsilon ' = 0.95\int_{0}^{\infty}{\cal L}(\epsilon
')d\epsilon '.$$
We also introduce the probability of the Abelian $Z'$ signal as the integral of the likelihood function over the positive values of $\bar{a}^2$: $$P=\int\nolimits_{0}^{\infty} L(\epsilon ')d\epsilon '.$$
Actually, the fitted value of the contact coupling $\bar{a}^2$ originates mainly from the leading-order term in the inverse $Z'$ mass in Eq. (\[7dfg\]). The analysis of the higher-order terms allows to estimate the constraints on the $Z'$ mass alone. Substituting $\bar{a}^2$ in the observable (\[7dfg\]) by its fitted central value, one obtains the expression $$\Delta\sigma_l(z^*)=
\left[\tilde{A}^l_{11}(s,z^*)
+\zeta \tilde{B}^l_{11}(s,z^*) \right]\bar{a}_\mathrm{fitted}^2 +
\tilde{C}^l_{1111}(s,z^*)\bar{a}_\mathrm{fitted}^4,$$ which depends on the parameter $\zeta=m^2_Z/m^2_{Z'}$. Then, the central value of this parameter and the corresponding 1$\sigma$ CL interval are derived in the same way as those for $\bar{a}^2$.
To fit the parameters $\bar{a}^2$ and $\zeta$ we start with the LEP2 data on the total cross-sections and the forward-backward asymmetries [@EWWG]. The corresponding values of the observable $\Delta\sigma_l(z^\ast)$ with their uncertainties $\delta\sigma_l(z^\ast)$ are calculated from the data by means of the following relations: $$\begin{aligned}
%\label{}
\Delta\sigma_l(z^\ast)
&=&
\left[
A_l^{\rm FB}\left(1-z^{\ast 2}\right)
-\frac{z^\ast}{4}\left(3 +z^{\ast 2}\right)
\right] \Delta\sigma_l^{\rm T}
\nonumber\\&&
+ \left(1 - z^{\ast 2}\right)
\sigma_l^{\rm T,SM} \Delta A_l^{\rm FB},
\nonumber\\
\delta\sigma_l(z^\ast)^2
&=&
{\left[
A_l^{\rm FB}\left(1-z^{\ast 2}\right)
-\frac{z^\ast}{4}\left(3 +z^{\ast 2}\right)
\right]}^2 (\delta\sigma_l^{\rm T})^2
\nonumber\\&&
+{\left[
\left(1 - z^{\ast 2}\right)
\sigma_{l}^{\rm T,SM}
\right]}^2 (\delta A_l^{\rm FB})^2.\end{aligned}$$
We perform the fits assuming several data sets, including the $\mu\mu$, $\tau\tau$, and the complete $\mu\mu$ and $\tau\tau$ data, respectively. The results are presented in Table \[tfitrecalc\].
Data set $\bar{a}^2$ $\Lambda$, TeV $P$ $\zeta$
------------------------- ---------------------------------------- ---------------- ------ -------------------
$\mu\mu$ $0.0000366^{+0.0000489}_{-0.0000486}$ 16.4 0.77 $0.009\pm 0.278$
$\tau\tau$ $-0.0000266^{+0.0000643}_{-0.0000639}$ 17.4 0.34 $-0.001\pm 0.501$
$\mu\mu$ and $\tau\tau$ $0.0000133^{+0.0000389}_{-0.0000387}$ 19.7 0.63 $0.017\pm 0.609$
: The contact coupling $\bar{a}^2$ with the 68% CL uncertainty, the 95% CL lower limit on the scale $\Lambda$, the probability of the $Z'$ signal, $P$, and the value of $\zeta=m^2_Z/m^2_{Z'}$ as a result of the fit of the observable recalculated from the total cross-sections and the forward-backward asymmetries. []{data-label="tfitrecalc"}
As is seen, the more precise $\mu\mu$ data demonstrate the signal of about 1$\sigma$ level. It corresponds to the Abelian $Z'$-boson with the mass of order 1.2–1.5 TeV if one assumes the value of $\tilde\alpha=\tilde{g}^2/4\pi$ to be in the interval 0.01–0.02. No signal is found by the analysis of the $\tau\tau$ cross-sections. The combined fit of the $\mu\mu$ and $\tau\tau$ data leads to the signal below the 1$\sigma$ CL.
Being governed by the next-to-leading contributions in $m^{-2}_{Z'}$, the fitted values of $\zeta$ are characterized by significant errors. The $\mu\mu$ data set gives the central value which corresponds to $m_{Z'}\simeq 1.1$ TeV.
We also perform a separate fit of the parameters based on the direct calculation of the observable from the differential cross-sections. The experimental uncertainties of the data on the differential cross-sections are of one order larger than the corresponding errors of the total cross-sections and the forward-backward asymmetries. These data also provide the larger values of the contact coupling $\bar{a}^2$. As for the more precise $\mu\mu$ data, three of the LEP2 Collaborations demonstrate positive values of $\bar{a}^2$. The combined $\bar{a}^2$ is also positive and remains practically unchanged by the incorporation of the $\tau\tau$ data.
As it was mentioned in the previous section, the indirect computation of the observable from the total cross-sections and the forward-backward asymmetries inspires some insufficient theoretical uncertainty about 2% of the statistical one. It also increases the statistical error because of the recalculation procedure. Nevertheless, the uncertainty of the fitted parameter $\bar{a}^2$ within the recalculation scheme is of one order less than that for the direct computation from the differential cross-sections. This difference is explained by the different accuracy of the available experimental data on the differential and the total cross-sections.
Search for $Z'$ in $e^+e^-\to e^+e^-$ process
=============================================
The differential cross-section
------------------------------
In our analysis, as the SM values of the cross-sections we use the quantities calculated by the LEP2 collaborations [@OPAL; @DELPHI; @ALEPH; @L3]. They account for either the one-loop radiative corrections or initial and final state radiation effects (together with the event selection rules, which are specific for each experiment). As it is reported by the DELPHI Collaboration, there is a theoretical error of the SM values of about 2%. In our analysis this error is added to the statistical and systematic ones for all the Collaborations. As it was checked, the fit results are practically insensitive to accounting for this error.
The deviation from the SM is computed in the improved Born approximation. This approximation is sufficient for our analysis leading to the systematic error of the fit results less than 5-10 per cents.
The deviation from the SM of the differential cross-section for the process $e^+e^-\to\ell^+\ell^-$ can be expressed through various quadratic combinations of couplings $a=a_e$, $v_e$, $v_\mu$, $v_\tau$. For the Bhabha process it reads $$\label{4}
\frac{d\sigma}{dz}-\frac{d\sigma^\mathrm{SM}}{dz} =
f^{ee}_1(z)\frac{a^2}{m_{Z'}^2} +
f^{ee}_2(z)\frac{v_e^2}{m_{Z'}^2} +
f^{ee}_3(z)\frac{a v_e}{m_{Z'}^2},$$ where the factors are known functions of the center-of-mass energy and the cosine of the electron scattering angle $z$ plotted in Fig. \[fig:0\].
The deviation of the cross-section for $e^+e^-\to\mu^+\mu^-$ ($\tau^+\tau^-$) processes has a similar form $$\begin{aligned}
\label{5}
\frac{d\sigma}{dz}-\frac{d\sigma^\mathrm{SM}}{dz} &=&
f^{\mu\mu}_1(z)\frac{a^2}{m_{Z'}^2} +
f^{\mu\mu}_2(z)\frac{v_e v_\mu}{m_{Z'}^2} +
\nonumber\\ &&+
f^{\mu\mu}_3(z)\frac{a v_e}{m_{Z'}^2} +
f^{\mu\mu}_4(z)\frac{a v_\mu}{m_{Z'}^2}.\end{aligned}$$ Eqs. (\[4\])–(\[5\]) are our definition of the $Z'$ signal.
Note again that the cross-sections in Eqs. (\[4\])–(\[5\]) account for the relations (\[3\]) through the functions $f_1(z)$, $f_3(z)$, $f_4(z)$, since the coupling $\tilde{Y}_\phi$ (the mixing angle $\theta_0$) is substituted by the axial coupling constant $a$. Usually, when a four-fermion effective Lagrangian is applied to describe physics beyond the SM [@Pankov], this dependence on the scalar field coupling is neglected at all. However, in our case, when we are interested in searching for signals of the $Z'$-boson on the base of the effective low-energy Lagrangian (\[Lf\])–(\[Lyukawa\]), these contributions to the cross-section are essential.
One-parameter fit
-----------------
The factor $f^{ee}_2(z)$ is positive monotonic function of $z$ (see Fig. 4 for the center-of-mass energies $\sqrt{s}=200$ GeV. The same behavior is observed for higher energies). Such a property allows one to choose $f^{ee}_2(z)$ as a normalization factor for the differential cross section. Then the normalized deviation of the differential cross-section reads [@PRD2004] $$\begin{aligned}
\label{ncs}
\frac{d\tilde\sigma}{dz}&=& \frac{m_Z^2}{{4\pi}f^{ee}_2(z)}
%{\cal F}_{v}^{-1}(\sqrt{s},z)
\Delta\,\frac{d\sigma}{dz} = \nonumber\\&&
\bar{v}^2 +
F_{a}(\sqrt{s},z) \bar{a}^2 + F_{av}(\sqrt{s},z)\bar{a}\bar{v}
+\ldots ,\end{aligned}$$ and the normalized factors are shown in Fig 5.
\
Now these factors are finite at $z\to 1$. Each of them in a special way influences the differential cross-section.
1. The factor at $\bar{v}^2$ is just the unity. Hence, the four-fermion contact coupling between vector currents, $\bar{v}^2$, determines the level of the deviation from the SM value.
2. The factor at $\bar{a}^2$ depends on the scattering angle in a non-trivial way. It allows to recognize the Abelian $Z'$ boson, if the experimental accuracy is sufficient.
3. The factor at $\bar{a}\bar{ v}$ results in small corrections.
Thus, effectively, the obtained normalized differential cross-section is a two-parametric function. In the next sections we introduce the observables to fit separately each of these parameters.
Observables to pick out $\bar{v}^2$
-----------------------------------
To recognize the signal of the Abelian $Z'$ boson by analyzing the Bhabha process the differential cross-section deviation from the SM predictions should be measured with a good accuracy. At present, no such deviations have been detected at more than the 1$\sigma$ CL. In this situation it is resonable to introduce integrated observables allowing to pick out $Z'$ signals by using the most effective treating of available data. The observables should be sensitive to the separate $Z'$ couplings. This admits of searching for the $Z'$ signals in different processes as well as to perform global fits.
The normalized deviation of the differential cross-section (\[ncs\]) is (effectively) the function of two parameters, $\bar{a}^2$ and $\bar{v}^2$. We are going to introduce the integrated observables which determine separately the four-fermion couplings $\bar{a}^2$ and $\bar{v}^2$ [@PRD2004].
Let us first proceed with the observable for $\bar{v}^2$. After normalization the factor at the vector-vector four-fermion coupling becomes the unity. Whereas the factor at $\bar{a}^2$ is a sign-varying function of the cosine of the scattering angle. As it follows from Fig. 5, for the center-of-mass energy 200 GeV it is small over the backward scattering angles. So, to measure the value of $\bar{v}^2$ the normalized deviation of the differential cross-section has to be integrated over the backward angles. For the center-of-mass energy 500 GeV the factor at $\bar{a}^2$ is already a non-vanishing quantity for the backward scattering angles. The curves corresponding to intermediate energies are distributed in between two these curves. Since they are sign-varying ones at each energy point some interval of $z$ can be chosen to make the integral to be zero. Thus, to measure the $Z'$ coupling to the electron vector current $\bar{v}^2$ we introduce the integrated cross-section (\[ncs\]) $$\label{vobs}
\sigma_V = \int_{z_0}^{z_0+\Delta z} (d\tilde\sigma/dz)dz,$$ where at each energy the most effective interval $[z_0,z_0+\Delta
z]$ is determined by the following requirements:
1. The relative contribution of the coupling $\bar{v}^2$ is maximal. Equivalently, the contribution of the factor at $\bar{a}^2$ is suppressed.
2. The length $\Delta z$ of the interval is maximal. This condition ensures that the largest number of bins is taken into consideration.
\
The relative contribution of the factor at $\bar{v}^2$ is defined as $$%\label{}
\kappa_V= \frac{\Delta z}{\Delta z+\left|\int_{z_0}^{z_0+\Delta
z}F_a \,dz\right|+\left|\int_{z_0}^{z_0+\Delta
z}F_{av}\,dz\right|}$$ and shown in Fig. 6 as the function of the left boundary of the angle interval, $z_0$, and the interval length, $\Delta z$. In each plot the dark area corresponds to the observables which values are determined by the vector-vector coupling $\bar{v}^2$ with the accuracy $>95\%$. The area reflects the correlation of the width of the integration interval $\Delta z$ with the choice of the initial $z_0$ following from the mentioned requirements. Within this area we choose the observable which includes the largest number of bins (largest $\Delta z$). The corresponding values of $z_0$ and $\Delta z$ are marked by the white dot on the plots in Fig. 6. As the carried out analysis showed, the point $z_0$ is shifted to the right with increase in energy whereas $\Delta z$ remains approximately the same.
From the plots it follows that the most efficient intervals are $$\begin{aligned}
%\label{}
&& -0.6<z<0.2,\quad\sqrt{s}=200\mbox{ GeV}, \nonumber\\&&
-0.3<z<0.7,\quad\sqrt{s}=500\mbox{ GeV}.\end{aligned}$$ Therefore the observable (\[vobs\]) allows to measure the $Z'$ coupling to the electron vector current $\bar{v}^2$ with the efficiency $>95\%$.
Fitting the LEP2 final data with the one-parameter observable, we find the values of the $Z'$ coupling to the electron vector current together with their 1$\sigma$ uncertainties: $$\begin{aligned}
\mathrm{ALEPH}: &\bar{v}_e^2=& -0.11\pm 6.53 \times 10^{-4}\nonumber\\
\mathrm{DELPHI}: &\bar{v}_e^2=& 1.60\pm 1.46 \times 10^{-4} \nonumber\\
\mathrm{L3}: &\bar{v}_e^2=& 5.42\pm 3.72 \times 10^{-4} \nonumber\\
\mathrm{OPAL}: &\bar{v}_e^2=& 2.42\pm 1.27 \times 10^{-4} \nonumber\\
\mathrm{Combined}: &\bar{v}_e^2=& 2.24\pm 0.92 \times 10^{-4}. \nonumber\end{aligned}$$ As one can see, the most precise data of DELPHI and OPAL collaborations are resulted in the Abelian $Z'$ hints at one and two standard deviation level, correspondingly. The combined value shows the 2$\sigma$ hint, which corresponds to $0.006\le
|\bar{v}_e|\le 0.020$.
Observables to pick out $\bar{a}^2$
-----------------------------------
In order to pick the axial-vector coupling $\bar{a}^2$ one needs to eliminate the dominant contribution coming from $\bar{v}^2$. Since the factor at $\bar{v}^2$ in the $d\tilde\sigma/dz$ equals unity, this can be done by summing up equal number of bins with positive and negative weights. In particular, the forward-backward normalized deviation of the differential cross-section appears to be sensitive mainly to $\bar{a}^2$, $$\begin{aligned}
%\label{}
\tilde\sigma_{\rm FB}&=&\int\nolimits_{0}^{z_{\rm
max}}dz\,\frac{d\tilde\sigma}{dz} -\int\nolimits_{-z_{\rm
max}}^{0}dz\,\frac{d\tilde\sigma}{dz} \nonumber\\ &=&
\tilde{F}_{a,\rm FB} \bar{a}^2 + \tilde{F}_{av,\rm
FB}\bar{a}\bar{v}.\end{aligned}$$ The value $ z_{\rm max}$ is determined by the number of bins included and, in fact, depends on the data set considered. The LEP2 experiment accepted $e^+e^-$ events with $|z|<0.72$. In what follows we take the angular cut $z_{\mathrm{max}}=0.7$ for definiteness.
The efficiency of the observable is determined as: $$%\label{}
\kappa= \frac{|\tilde{F}_{a,\rm FB}|}{|\tilde{F}_{a,\rm
FB}|+|\tilde{F}_{av,\rm FB}|}.$$ It can be estimated as $\kappa=0.9028$ for the center-of-mass energy 200 GeV and $\kappa=0.9587$ for 500 GeV. Thus, the observable $$\begin{aligned}
\label{aobs}
&& \tilde\sigma_{\rm FB}=0.224 \bar{a}^2 - 0.024
\bar{a}\bar{v},\quad\sqrt{s}=200\mbox{ GeV}, \nonumber\\&&
\tilde\sigma_{\rm FB}=0.472 \bar{a}^2 - 0.020
\bar{a}\bar{v},\quad\sqrt{s}=500\mbox{ GeV}\end{aligned}$$ is mainly sensitive to the $Z'$ coupling to the axial-vector current $\bar{a}^2$.
Consider a usual situation when experiment is not able to recognize the angular dependence of the differential cross-section deviation from its SM value with the proper accuracy because of loss of statistics. Nevertheless, a unique signal of the Abelian $Z'$ boson can be determined. For this purpose the observables $\int_{z_0}^{z_0+\Delta z}(d\tilde\sigma/dz)dz$ and $\tilde\sigma_{\rm FB}$ must be measured. Actually, they are derived from the normalized deviation of the differential cross-section. If the deviation is inspired by the Abelian $Z'$ boson both the observables are to be positive quantities simultaneously. This feature serves as the distinguishable signal of the Abelian $Z'$ virtual state in the Bhabha process for the LEP2 energies as well as for the energies of future electron-positron collider ILC ($\geq 500$ GeV). The observables fix the unknown low energy vector and axial-vector $Z'$ couplings to the electron current. Their values have to be correlated with the bounds on $\bar{a}^2$ and $\bar{v}^2$ derived by means of independent fits for other scattering processes.
We estimated the observable (\[aobs\]) related to the value of $\bar{a}^2$. Since in the Bhabha process the effects of the axial-vector coupling are suppressed with respect to those of the vector coupling, we expect much larger experimental uncertainties for $\bar{a}^2$. Indeed, the LEP2 data lead to the huge errors for $\bar{a}^2$ of order $10^{-3}-10^{-4}$ The mean values are negative numbers which are too large to be interpreted as a manifestation of some heavy virtual state beyond the energy scale of the SM.
Thus, the LEP2 data constrain the value of $\bar{v}^2$ at the $2\sigma$ CL which could correspond to the Abelian $Z'$ boson with the mass of the order 1 TeV. In contrast, the value of $\bar{a}^2$ is a large negative number with a significant experimental uncertainty. This can not be interpreted as a manifestation of some heavy virtual state beyond the energy scale of the SM.
Many-parameter fits
-------------------
As the basic observable to fit the LEP2 experiment data on the Bhabha process we propose the differential cross-section $$\label{7}
\left.\frac{d\sigma^\mathrm{Bhabha}}{dz}-\frac{d\sigma^{\mathrm{Bhabha},SM}}{dz}\right|_{z=z_i,\sqrt{s}=\sqrt{s_i}},$$ where $i$ runs over the bins at various center-of-mass energies $\sqrt{s}$. The final differential cross-sections measured by the ALEPH (130-183 GeV, [@ALEPH]), DELPHI (189-207 GeV, [@DELPHI]), L3 (183-189 GeV, [@L3]), and OPAL (130-207 GeV, [@OPAL]) collaborations are taken into consideration (299 bins).
As the observables for $e^+e^-\to\mu^+\mu^-,\tau^+\tau^-$ processes, we consider the total cross-section and the forward-backward asymmetry $$\label{8}
%\left.
\sigma^{\ell^+\ell^-}_T-\sigma_T^{\ell^+\ell^-,\mathrm{SM}},
%\right|_{\sqrt{s}=\sqrt{s_i}},
\quad
\left.A^{\ell^+\ell^-}_{FB}-A_{FB}^{\ell^+\ell^-,\mathrm{SM}}\right|_{\sqrt{s}=\sqrt{s_i}},$$ where $i$ runs over 12 center-of-mass energies $\sqrt{s}$ from 130 to 207 GeV. We consider the combined LEP2 data [@EWWG] for these observables (24 data entries for each process). These data are more precise as the corresponding differential cross-sections. Our analysis is based on the fact that the kinematics of $s$-channel processes is rather simple and the differential cross-section is effectively a two-parametric function of the scattering angle. The total cross-section and the forward-backward asymmetry incorporate complete information about the kinematics of the process and therefore are an adequate alternative for the differential cross-sections.
The data are analysed by means of the $\chi^2$ fit [@PRD2007]. Denoting the observables (\[7\])–(\[8\]) by $\sigma_i$, one can construct the $\chi^2$-function, $$\label{9}
\chi^2(\bar{a}, \bar{v}_e,\bar{v}_\mu,\bar{v}_\tau) =
\sum\limits_i
\left[\frac{\sigma^\mathrm{ex}_i-\sigma^\mathrm{th}_i(\bar{a},
\bar{v}_e,\bar{v}_\mu,\bar{v}_\tau)}{\delta\sigma_i}\right]^{2},$$ where $\sigma^\mathrm{ex}$ and $\delta\sigma$ are the experimental values and the uncertainties of the observables, and $\sigma^\mathrm{th}$ are their theoretical expressions presented in Eqs. (\[4\])–(\[5\]). The sum in Eq. (\[9\]) refers to either the data for one specific process or the combined data for several processes. By minimizing the $\chi^2$-function, the maximal-likelihood estimate for the $Z'$ couplings can be derived. The $\chi^2$-function is also used to plot the confidence area in the space of parameters $\bar{a}$, $\bar{v}_e$, $\bar{v}_\mu$, and $\bar{v}_\tau$. Note that in this way of experimental data treating all the possible correlations are neglected. We believe that at the present stage of investigation this is reasonable, because the Collaborations have never reported on this possibility.
For all the considered processes, the theoretic predictions $\sigma^\mathrm{th}_i$ are linear combinations of products of two $Z'$ couplings $$\begin{aligned}
\label{10}
\sigma^\mathrm{th}_i&=&\sum_{j=1}^{7} C_{ij}A_j,\\
A_j&=&\{\bar{a}^2,\bar{v}_e^2,\bar{a}\bar{v}_e,\bar{v}_e\bar{v}_\mu,\bar{v}_e\bar{v}_\tau,
\bar{a}\bar{v}_\mu,\bar{a}\bar{v}_\tau\},
\nonumber\end{aligned}$$ where $C_{ij}$ are known numbers. In what follows we use the matrix notation $\sigma^\mathrm{th}=\sigma^\mathrm{th}_i$, $\sigma^\mathrm{ex}=\sigma^\mathrm{ex}_i$, $C=C_{ij}$, $A=A_j$. The uncertainties $\delta\sigma_i$ can be substituted by a covariance matrix $D$. The diagonal elements of $D$ are experimental errors squared, $D_{ii}=(\delta\sigma^\mathrm{ex}_i)^2$, whereas the non-diagonal elements are responsible for the possible correlations of observables. The $\chi^2$-function can be rewritten as $$\begin{aligned}
\label{11}
\chi^2(A)&=&(\sigma^\mathrm{ex}-\sigma^\mathrm{th})^\mathrm{T}
D^{-1}
(\sigma^\mathrm{ex}-\sigma^\mathrm{th})
\nonumber\\
&=&(\sigma^\mathrm{ex}-CA)^\mathrm{T}
D^{-1} (\sigma^\mathrm{ex}-CA),\end{aligned}$$ where the upperscript T denotes the matrix transposition.
The $\chi^2$-function has a minimum, $\chi^2_\mathrm{min}$, at $$\label{12}
\hat{A}=(C^\mathrm{T}D^{-1}C)^{-1}C^\mathrm{T}D^{-1}\sigma^\mathrm{ex}$$ corresponding to the maximum-likelihood values of $Z'$ couplings. From Eqs. (\[11\]), (\[12\]) we obtain $$\begin{aligned}
\label{13}
\chi^2(A)-\chi^2_\mathrm{min}&=& (\hat{A}-A)^\mathrm{T}
\hat{D}^{-1}(\hat{A}-A),
\nonumber\\
\hat{D}&=& (C^\mathrm{T}D^{-1}C)^{-1}.\end{aligned}$$
Usually, the experimental values $\sigma^\mathrm{ex}$ are normal-distributed quantities with the mean values $\sigma^\mathrm{th}$ and the covariance matrix $D$. The quantities $\hat{A}$, being the superposition of $\sigma^\mathrm{ex}$, also have the same distribution. It is easy to show that $\hat{A}$ has the mean values $A$ and the covariance matrix $\hat{D}$.
The inverse matrix $\hat{D}^{-1}$ is symmetric and can be diagonalized. The number of non-zero eigenvalues is determined by the rank (denoted $M$) of $\hat{D}^{-1}$. The rank $M$ equals to the number of linear-independent terms in the observables $\sigma^\mathrm{th}$. So, the right-hand-side of Eq. (\[13\]) is a quantity distributed as $\chi^2$ with $M$ degrees of freedom (d.o.f.). Since this random value is independent of $A$, the confidence area in the parameter space ($\bar{a}$, $\bar{v}_e$, $\bar{v}_\mu$, $\bar{v}_\tau$) corresponding to the probability $\beta$ can be defined as [@SMEP]: $$\label{14}
\chi^2\le \chi^2_\mathrm{min}+\chi^2_{\mathrm{CL},\beta}(M),$$ where $\chi^2_{\mathrm{CL,\beta}}(M)$ is the $\beta$-level of the $\chi^2$-distribution with $M$ d.o.f.
In the Bhabha process, the $Z'$ effects are determined by 3 linear-independent contributions coming from $\bar{a}^2$, $\bar{v}_e^2$, and $\bar{a}\bar{v}_e$ ($M=3$). As for the $e^+e^-\to\mu^+\mu^-,\tau^+\tau^-$ processes, the observables depend on 4 linear-independent terms for each process: $\bar{a}^2$, $\bar{v}_e\bar{v}_\mu$, $\bar{v}_e\bar{a}$, $\bar{a}\bar{v}_\mu$ for $e^+e^-\to\mu^+\mu^-$; and $\bar{a}^2$, $\bar{v}_e\bar{v}_\tau$, $\bar{v}_e\bar{a}$, $\bar{a}\bar{v}_\tau$ for $e^+e^-\to\tau^+\tau^-$ ($M=4$). Note that some terms in the observables for different processes are the same. Therefore, the number of $\chi^2$ d.o.f. in the combined fits is less than the sum of d.o.f. for separate processes. Hence, the predictive power of the larger set of data is not drastically spoiled by the increased number of d.o.f. In fact, combining the data of the Bhabha and $e^+e^-\to\mu^+\mu^-$ ($\tau^+\tau^-$) processes together we have to treat 5 linear-independent terms. The complete data set for all the lepton processes is ruled by 7 d.o.f. As a consequence, the combination of the data for all the lepton processes is possible.
The parametric space of couplings ($\bar{a}$, $\bar{v}_e$, $\bar{v}_\mu$, $\bar{v}_\tau$) is four-dimensional. However, for the Bhabha process it is reduced to the plane ($\bar{a}$, $\bar{v}_e$), and to the three-dimensional volumes ($\bar{a}$, $\bar{v}_e$, $\bar{v}_\mu$), ($\bar{a}$, $\bar{v}_e$, $\bar{v}_\tau$) for the $e^+e^-\to\mu^+\mu^-$ and $e^+e^-\to\tau^+\tau^-$ processes, correspondingly. The predictive power of data is distributed not uniformly over the parameters. The parameters $\bar{a}$ and $\bar{v}_e$ are present in all the considered processes and appear to be significantly constrained. The couplings $\bar{v}_\mu$ or $\bar{v}_\tau$ enter when the processes $e^+e^-\to\mu^+\mu^-$ or $e^+e^-\to\tau^+\tau^-$ are accounted for. So, in these processes, we also study the projection of the confidence area onto the plane ($\bar{a},\bar{v}_e$).
The origin of the parametric space, $\bar{a}=\bar{v}_e=0$, corresponds to the absence of the $Z'$ signal. This is the SM value of the observables. This point could occur inside or outside of the confidence area at a fixed CL. When it lays out of the confidence area, this means the distinct signal of the Abelian $Z'$. Then the signal probability can be defined as the probability that the data agree with the Abelian $Z'$ boson existence and exclude the SM value. This probability corresponds to the most stringent CL (the largest $\chi^2_\mathrm{CL}$) at which the point $\bar{a}=\bar{v}_e=0$ is excluded. If the SM value is inside the confidence area, the $Z'$ boson is indistinguishable from the SM. In this case, upper bounds on the $Z'$ couplings can be determined.
The 95% CL areas in the ($\bar{a},\bar{v}_e$) plane for the separate processes are plotted in Fig. \[fig:1\]. As it is seen, the Bhabha process constrains both the axial-vector and vector couplings. As for the $e^+e^-\to\mu^+\mu^-$ and $e^+e^-\to\tau^+\tau^-$ processes, the axial-vector coupling is significantly constrained, only. The confidence areas include the SM point at the meaningful CLs, so the experiment could not pick out clearly the Abelian $Z'$ signal from the SM. An important conclusion from these plots is that the experiment significantly constrains only the couplings entering sign-definite terms in the cross-sections.
\
The combination of all the lepton processes is presented in Fig. \[fig:2\]. There is no visible signal beyond the SM. The couplings to the vector and axial-vector electron currents are constrained by the many-parameter fit as $|\bar{v}_e|<0.013$, $|\bar{a}|<0.019$ at the 95% CL. If the charge corresponding to the $Z'$ interactions is assumed to be of order of the electromagnetic one, then the $Z'$ mass should be greater than 0.67 TeV. For the charge of order of the SM $SU(2)_L$ coupling constant $m_{Z'}\ge 1.4$ TeV. One can see that the constraint is not too severe to exclude the $Z'$ searches at the LHC.
Let us compare the obtained results with the one-parameter fits. As one can see, the most precise data of DELPHI and OPAL collaborations are resulted in the Abelian $Z'$ hints at one and two standard deviation level, correspondingly. The combined value shows the 2$\sigma$ hint, which corresponds to $0.006\le
|\bar{v}_e|\le 0.020$. On the other hand, our many-parameter fit constrains the $Z'$ coupling to the electron vector current as $|\bar{v}_e|\le 0.013$ with no evident signal. Why does the one-parameter fit of the Bhabha process show the 2$\sigma$ CL hint whereas there is no signal in the two-parameter one? Our one-parameter observable accounts mainly for the backward bins. This is in accordance with the kinematic features of the process: the backward bins depend mainly on the vector coupling $\bar{v}^2_e$, whereas the contributions of other couplings are kinematically suppressed (see Fig. 4). Therefore, the difference of the results can be inspired by the data sets used. To clarify this point, we perform the many-parameter fit with the 113 backward bins ($z\le 0$), only. The $\chi^2$ minimum, $\chi^2_\mathrm{min}=93.0$, is found in the non-zero point $|\bar{a}|=0.0005$, $\bar{v}_e= 0.015$. This value of the $Z'$ coupling $\bar{v}_e$ is in an excellent agreement with the mean value obtained in the one-parameter fit. The 68% confidence area in the ($\bar{a},\bar{v}_e$) plane is plotted in Fig. \[fig:3\]. There is a visible hint of the Abelian $Z'$ boson. The zero point $\bar{a}=\bar{v}_e=0$ (the absence of the $Z'$ boson) corresponds to $\chi^2=97.7$. It is covered by the confidence area with $1.3\sigma$ CL. Thus, the backward bins show the $1.3\sigma$ hint of the Abelian $Z'$ boson in the many-parameter fit. So, the many-parameter fit is less precise than the analysis of the one-parameter observables.
\
At LEP1 experiments [@LEP1] the $Z$-boson couplings to the vector and axial-vector lepton currents ($g_V$, $g_A$) were precisely measured. The Bhabha process shows the 1$\sigma$ deviation from the SM values for Higgs boson masses $m_H\ge 114$ GeV (see Fig. 7.3 of Ref. [@LEP1]). This deviation could be considered as the effect of the $Z$–$Z'$ mixing. It is interesting to estimate the bounds on the $Z'$ couplings following from these experiments.
Due to the RG relations, the $Z$–$Z'$ mixing angle is completely determined by the axial-vector coupling $\bar{a}$. So, the deviations of $g_V$, $g_A$ from their SM values are governed by the couplings $\bar{a}$ and $\bar{v}_e$, $$\label{lep1}
g_V-g_V^{\mathrm{SM}}=-49.06 \bar{a}\bar{v}_e,\quad
g_A-g_A^{\mathrm{SM}} = 49.06 \bar{a}^2.$$ Let us assume that the total deviation of theory from experiments follows due to the $Z$–$Z'$ mixing. This gives an upper bound on the $Z'$ couplings. In this way one can estimate whether the $Z'$ boson is excluded by the experiments or not.
The 1$\sigma$ CL area for the Bhabha process from Ref. [@LEP1] is converted into the ($\bar{a},\bar{v}_e$) plane in Fig. \[fig:3\]. The SM values of the couplings correspond to the top quark mass $m_t=178$ GeV and the Higgs scalar mass $m_H=114$ GeV. As it is seen, the LEP1 data on the Bhabha process is compatible with the Abelian $Z'$ existence at the $1\sigma$ CL. The axial-vector coupling is constrained as $|\bar{a}|\le 0.005$. This bound corresponds to $\bar{a}^2\le 2.5\times 10^{-5}$, which agrees with the one-parameter fits of the LEP2 data for $e^+e^-\to\mu^+\mu^-,\tau^+\tau^-$ processes ($\bar{a}^2= 1.3\pm
3.89\times 10^{-5}$ at 68% CL). On the other hand, the vector coupling constant $\bar{v}_e$ is practically unconstrained by the LEP1 experiments.
For the convenience, in Table 4 we collect the summary of the fits of the LEP data in terms of dimensionless contact couplings (\[6\]). From the analysis carried out we come to conclusion that, in principle, the LEP experiments were able to detect the $Z'$-boson signals if the statistics had been sufficient.
[|l|l|l|]{} Data & $\bar{v}^2_e$ & $\bar{a}^2$\
\
$e^-e^+$, 68% CL & - & $(1.25\pm1.25)\times 10^{-5}$\
\
$e^-e^+$, 68% CL & $(2.24\pm 0.92)\times 10^{-4}$ & -\
$\mu\mu$, 68% CL & - & $(3.66^{+4.89}_{-4.86})\times 10^{-5}$\
$\mu\mu$,$\tau\tau$, 68% CL & - & $(1.33^{+3.89}_{-3.87})\times 10^{-5}$\
\
$e^-e^+,\mu\mu,\tau\tau$, 95% CL & $\le 1.69\times 10^{-4}$ & $\le 3.61\times 10^{-4}$\
$e^-e^+$ backward, 68% CL & $(2.25^{+1.79}_{-2.07})\times 10^{-4}$ & $\le 9.49\times 10^{-4}$\
$Z'$ hints within neural network analysis
=========================================
Since the actual LEP2 data set is not too large to detect $Z'$ boson, one needs in the estimate of its parameters which could be used in future experiments. To determine them in a maximally full way we address to the analysis based on the predictions of the neural networks (for applications in high energy physics see, for example, [@dudko]). The main idea of this approach is to constrain a given data set in such a way that an amount of it is considered as an inessential background and omitted. The remaining data are expected to give a more precise fit of the parameters of interest.
We take into consideration the complete set of the differential cross sections for the Bhabha process accumulated by all the LEP Collaborations and apply the following criteria to restrict the data [@Buryk]:
1. As the signal we use the differential cross sections for the Bhabha process accounting for the SM plus $Z'$ and calculated at $0.25 \times 10^{-4} \leq \bar{v}_e^2 \leq 4 \times 10^{-4}$ and $0.25 \times 10^{-4} \leq \bar{a}_e^2 \leq 4 \times 10^{-4}$.
Such a choice of parameters is motivated by the results obtained in the previous sections. The cross sections due to the $Z'$ exchange diagrams were calculated with the RG relations been taken into consideration.
2. As the background we use the deviations of the experimental differential cross sections from calculated for the SM plus $Z'$ ones which are larger than redoubled uncertainties of LEP2 experimental data.
The network trained with these criteria omits the events which correspond to the large deviations from the theoretical cross sections but accounts for the peculiarities proper the $Z'$ existence. To construct and train the neural network we used the program MLPFit [@MLPFit]. The results of the carried out analysis based on the two parametric fit discussed in the previous sections demonstrate the $2\sigma$ CL hint for the $Z'$. For the vector coupling the neural network at the $2\sigma $ CL predicts $\bar{v}_e^2 = 2.4 \pm 1.99 \times 10^{- 4}$ [@Buryk] that is in agreement with the discussed above result derived in the one parametric fit. The obtained values of $\bar{v}_e^2$ correspond to the value of the mass $m_{Z'} = 0.53 - 1.05 $ TeV, if the coupling $\tilde{g}$ is of the order of the SM gauge couplings, $
g^2/(4\pi)\sim 0.01 - 0.03$. Thus, the carried out analysis demonstrates the hint of $Z'$ boson which can be not too heavy. We conclude once again that the data set of the LEP experiments is not sufficient to detect the pronounced signal of this virtual particle.
Search for Chiral $Z'$ in Bhabha process
========================================
Let us turn to the analysis of the Bhabha process with the aim to search for the Chiral $Z'$ gauge boson [@YAF2007]. The Chiral $Z'$ interacts with the SM doublets only that can be described by one parameter for each doublet ($\tilde{Y}_{fL}$ and $\tilde{Y}_\phi$). It is characterized by the constraints (\[rgr1\]) $$\label{3}
\tilde{Y}_{L,f}=-\tilde{Y}_{fL}\,\sigma_3,\quad
\tilde{Y}_{R,f}=0,\quad
\tilde{Y}_{\phi_i}=-\tilde{Y}_{\phi}\,\sigma_3$$ where $\sigma_3$ is the Pauli matrix.
Remind that in the Bhabha process it is convenient to use the normalized cross-sections (\[ncs\]): $$\frac{d\tilde\sigma}{dz}=\frac{m_Z^2}{{4\pi}f^{ee}_2(z)}\Delta
\frac{d\sigma}{dz} = \bar{v}_e^2 + F_a \bar{a}_e^2 + F_{av}
\bar{a}_e\bar{v}_e +F_{v\phi} \bar{v}_e\bar\phi +F_{a\phi}
\bar{a}_e\bar\phi.$$ Since the Chiral $Z'$ boson does not interact with the right-handed species, the normalized deviation of the differential cross-section from its SM prediction is determined by two factors, $F_L$ and $F_{L\phi}$, $$\frac{d\tilde\sigma}{dz}=\frac{m_Z^2}{{4\pi}f^{ee}_2(z)}\Delta
\frac{d\sigma}{dz} = F_L \bar{l}_e^2 + F_{L\phi}
\bar{l}_e\bar\phi,$$ where we define the dimensionless constants $$\bar{l}_f=\frac{m_Z}{\sqrt{4\pi}m_{Z'}}\tilde{g}\tilde{Y}_{fL},\quad
\bar{\phi}=\frac{m_Z}{\sqrt{4\pi}m_{Z'}}\tilde{g}\tilde{Y}_\phi.$$ The normalization gives us two benefits. First, the obtained factors $F(\sqrt{s},z)$ are finite for all values of the scattering angle $z$. Second, the experimental uncertainties for different bins become equalized that provides the statistical equivalence of different bins. The latter is important for the construction of integrated cross-sections.
The $Z$-$Z'$ mixing angle is determined by $\bar\phi$ as follows, $$\theta_0\simeq\frac{m_W
\sin\theta_W}{\sqrt{\alpha_\mathrm{em}}m_{Z'}}\bar\phi,$$ where $\alpha_\mathrm{em}$ is the fine structure constant.
One-parameter fit for Chiral $Z'$
---------------------------------
The Chiral $Z'$ boson does not interact with the right-handed species. The normalized deviation of the differential cross-section from its SM prediction, $$d\tilde\sigma/dz = F_L \bar{l}_e^2 + F_{L\phi} \bar{l}_e\bar\phi,$$ is determined by two finite factors, $F_L$ and $F_{L\phi}$, which are shown in Fig. 10.
\[fig-dcs-ch-norm\]
As it is seen, the four-fermion contact coupling $\bar{l}_e^2$ contributes mainly to the forward scattering angles, whereas the $Z$-$Z'$ mixing term affects the backward angles. At the LEP energies they can be of the same order of magnitude. The contribution of the mixing vanishes with the energy growth. We have a possibility to derive the effective experimental constraints on them without any additional restrictions.
First, let us construct a one-parametric observable which is most preferred by the statistical treatment of data. As is clear, it is impossible to separate the couplings $\bar{l}_e^2$ and $\bar{l}_e\bar\phi$ in any observable which is an integrated cross-section over some interval of $z$. However, the mixing contribution can be eliminated in the cross-section of the form (which is inspired by the forward-backward asymmetry) $$\Delta\sigma(z^*) = \int_{z^*}^{z_\mathrm{max}}
\frac{d\tilde\sigma}{dz} \, dz - \int_{-z_\mathrm{max}}^{z^*}
\frac{d\tilde\sigma}{dz} \, dz,$$ where the boundary value $z^*$ should be chosen to suppress the coefficient at $\bar{l}_e\bar\phi$. The maximal value of the scattering angle $z_\mathrm{max}$ is determined by a particular experiment. In this way we introduce the one-parametric sign-definite observable sensitive to $\bar{l}_e^2$.
The LEP Collaborations DELPHI and L3 measured the differential cross-sections with $z_\mathrm{max}=0.72$ [@DELPHI; @L3]. The set of boundary angles $z^*$ as well as the theoretic and experimental values of the observable are collected in Table 5. The other LEP Collaborations – ALEPH and OPAL – used $z_\mathrm{max}=0.9$ [@OPAL; @ALEPH]. The corresponding data are presented in Table 6.
\[tab-1\]
------------------------------------------------------------------------
\
$\sqrt{s}$, GeV $z^*$ $\Delta\sigma(z^*)$ $\Delta\sigma^\mathrm{ex}(z^*)$, DELPHI $\Delta\sigma^\mathrm{ex}(z^*)$, L3
----------------- -------- --------------------- ----------------------------------------- -------------------------------------
183 -0.245 1742 $\bar{l}_e^2$ - $-2.38 \pm 7.03$
189 -0.252 1775 $\bar{l}_e^2$ $-4.28 \pm 3.36$ $3.05 \pm 3.65$
192 -0.255 1788 $\bar{l}_e^2$ $4.57 \pm 7.32$ -
196 -0.259 1806 $\bar{l}_e^2$ $3.77 \pm 4.33$ -
200 -0.263 1823 $\bar{l}_e^2$ $-1.95 \pm 4.05$ -
202 -0.265 1831 $\bar{l}_e^2$ $1.31 \pm 5.54$ -
205 -0.267 1843 $\bar{l}_e^2$ $-4.09 \pm 3.89$ -
207 -0.269 1851 $\bar{l}_e^2$ $0.40 \pm 3.33$ -
: The boundary angles $z^*$ and the theoretic and experimental values of the observable $\Delta\sigma(z^*)$ at $z_\mathrm{max}=0.72$.
\[tab-2\]
------------------------------------------------------------------------
\
$\sqrt{s}$, GeV $z^*$ $\Delta\sigma(z^*)$ $\Delta\sigma^\mathrm{ex}(z^*)$, ALEPH $\Delta\sigma^\mathrm{ex}(z^*)$, OPAL
----------------- -------- --------------------- ---------------------------------------- ---------------------------------------
130 -0.217 2017 $\bar{l}_e^2$ $-12.40 \pm 19.24$ $-4.13 \pm 29.29$
136 -0.266 2092 $\bar{l}_e^2$ $-50.21 \pm 16.64$ $-34.18 \pm 31.58$
161 -0.370 2311 $\bar{l}_e^2$ $-15.90 \pm 13.24$ $-14.02 \pm 22.32$
172 -0.400 2398 $\bar{l}_e^2$ $-12.11 \pm 12.50$ $13.71 \pm 17.84$
183 -0.424 2474 $\bar{l}_e^2$ $-1.51 \pm ~5.18$ $11.04 \pm ~5.57$
189 -0.435 2512 $\bar{l}_e^2$ - $-0.63 \pm ~3.28$
192 -0.441 2531 $\bar{l}_e^2$ - $-3.48 \pm 9.85$
196 -0.447 2554 $\bar{l}_e^2$ - $2.96 \pm ~5.09$
200 -0.454 2577 $\bar{l}_e^2$ - $0.35 \pm ~4.68$
202 -0.457 2587 $\bar{l}_e^2$ - $-2.87 \pm ~9.00$
205 -0.461 2604 $\bar{l}_e^2$ - $5.88 \pm ~4.67$
207 -0.464 2614 $\bar{l}_e^2$ - $-1.42 \pm ~3.46$
: The boundary angles $z^*$ and the theoretic and experimental values of the observable $\Delta\sigma(z^*)$ at $z_\mathrm{max}=0.9$.
The standard $\chi^2$-fit gives the following constraints for the coupling $\bar{l}_e^2$ at the 68% CL: $$\begin{aligned}
\mathrm{ALEPH:} && \bar{l}_e^2= -0.00304\pm 0.00176 \nonumber\\
\mathrm{DELPHI:} && \bar{l}_e^2= -0.00054\pm 0.00086\nonumber\\
\mathrm{L3:} && \bar{l}_e^2= 0.00109\pm 0.00184 \nonumber\\
\mathrm{OPAL:} && \bar{l}_e^2= 0.00051\pm 0.00064\nonumber\\
\mathrm{Combined:} && \bar{l}_e^2= -0.00004\pm 0.00048\nonumber\end{aligned}$$ Hence it is seen that the most precise data of DELPHI and OPAL collaborations give no signal of the Chiral $Z'$ at the 1$\sigma$ CL. The combined value also shows no signal at the 1$\sigma$ CL. From the combined fit the 95% CL bound on the value of $\bar{l}_e^2$ can be derived, $\bar{l}_e^2<9\times 10^{-4}$. Supposing the $Z'$ coupling constant $\tilde{g}$ to be of the order of the electroweak one, $\tilde{g}\simeq 0.6$, the corresponding $Z'$ mass has to be larger than $0.5$ TeV.
Two parametric fit for Chiral $Z'$
----------------------------------
A complete two parametric fit of experimental data based directly on the differential cross-sections has been carried out in Ref. [@YAF2007]. Due to only two independent couplings this fit is efficient. In the fitting the available final data for the differential cross-sections of the Bhabha process were used. The data set consists of 299 bins including the data of ALEPH at 130-183 GeV, DELPHI at 189-207 GeV, L3 at 183-189 GeV, and OPAL at 130-207 GeV [@DELPHI; @OPAL; @ALEPH; @L3]. The fitting procedure is similar to that of discussed above for the Abelian $Z'$. The results can be summarized as follows.
The parameter space of the Chiral $Z'$ is the plane ($\bar{l}_e$, $\bar\phi$). The minimum of the $\chi^2$-function, $\chi^2_\mathrm{min}=237.29$, is reached at zero value of $\bar{l}_e$ ($\simeq 10^{-4}$) and almost independent of the value of $\bar\phi$ (the maximal-likelihood values of the couplings). The 95% CL area ($\chi^2_\mathrm{CL}=5.99$) is shown in Figure 3.
\[fig-ch-many\]
As one can see, the zero point, $\bar{l}_e = \bar\phi= 0$ (the absence of the Chiral $Z'$ boson) is inside the confidence area. The value of $\chi^2$ in this point (238.62) is indistinguishable from the $\chi^2_\mathrm{min}$. In other words, the set of experimental data cannot determine the signal of the Chiral $Z'$-boson.
As also is seen, the value of $\bar{l}_e$ is constrained as $\bar{l}_e<0.02$ at the 95% CL. This upper bound is in an agreement with the corresponding result of the one-parameter fit ($\bar{l}_e<0.03$). Thus, the $Z'$ mass has to be larger than $0.75$ TeV, if the $Z'$ coupling constant $\tilde{g}$ is again supposed to be of the order of the electroweak one, $\tilde{g}\simeq 0.6$.
The fit of the differential cross-sections leads to a better accuracy for $\bar{l}_e$ than the fit of the integrated cross-sections based on the same data. Without accounting for the model-independent relations between the $Z'$ couplings it is impossible to obtain such results.
Model independent results and search for $Z'$ at the LHC
========================================================
In this section we discuss all the assumptions giving a possibility to pick out the $Z'$ signal and determine its characteristics in a model independent way. We also note the role of the present results for the future LHC and ILC experiments. As it was already stressed, in searching for this particle at the LEP and Tevatron a model dependent analysis was mainly used. The motivation for this was the different number of chiral fermions involved in different models (see, for example, [@Lang08]). In this approach, the low bounds on $m_{Z'}$ have been estimated and a smallness of the $Z - Z'$ mixing was also observed.
On the contrary, in our model independent approach the RG relations between the parameters of the effective low energy Lagrangians have been accounted for that gave a possibility to determine not only the bounds but also the mass and other parameters of the $Z'$.
First we note all the assumptions used in our considerations. We analyzed the four-fermion scattering amplitudes of order $\sim
m_{Z'}^{-2}$ generated by the $Z'$ virtual states. The vertices linear in $Z'$ were included into the effective low-energy Lagrangian. We also impose a number of natural conditions. The interactions of a renormalizable type are dominant at low energies $\sim m_W$. The non-renormalizable interactions generated at high energies due to radiation corrections are suppressed by the inverse heavy mass $\sim 1/m_{Z'}$ and neglected. We also assumed that the $SU(2)_L \times U(1)_Y$ gauge group of the SM is a subgroup of the GUT group. As a consequence, all structure constants connecting two SM gauge bosons with $Z'$ are to be zero. Hence, the interactions of gauge fields of the types $Z' W^+ W^-,
Z' Z Z$, and other are absent at the tree level. Our effective Lagrangian is also consistent with the absence of the tree-level flavor-changing neutral currents (FCNCs) in the fermion sector. The renormalizable interactions of fermions and scalars are described by the Yukawa Lagrangian (\[Lyukawa\]) that accounts for different possibilities of the Yukawa sector without the tree-level FCNCs. These assumptions are quite general and satisfied in a wide class of $E_6$ inspired models.
Within these constraints for the low energy effective Lagrangian the RG correlations have been derived. Correspondingly, the model independent estimates of the mass $m_{Z'}$ and other parameters are regulated by the noted requirements. Therefore, the extended underlying model has also to accept them.
In this regard, let us discuss the role of the obtained estimates for the LHC. As it is well known (see, for example, [@Lang08; @Rizzo06]), there are many tools at the LHC for $Z'$ identification. But many of them are only applicable if this particle is relatively light. Our results are in favor to this case.
Next important point is the determination of $Z'$ couplings to the various SM fermions. As we have shown, the axial-vector couplings of the $Z'$ to the SM fermions are universal and proportional to its coupling to the Higgs field. Hence we have obtained an estimate of the $a = a_f$ couplings for both leptons and quarks. This is an essential input because experimental analyses for the LHC have mainly concentrated on being able to distinguish models and not on actual couplings. The vector coupling $v_e$ was also estimated that, in particular, may help to distinguish the decay of the $Z'$ resonance state to $ e^+ e^-$ pairs. Since the couplings $a_e$ and $v_e$ were estimated there is a possibility to distinguish this process from the decay of the $K K $ system. In the literature [@Rizzo06; @Dittmar] on searching for the $Z'$ it is also mentioned that the determination of the $Z'$ couplings to fermions could be fulfilled channel by channel, $a_e, v_e,
v_{e,b}, a_{e,b}$, …. In almost all these considerations the relations between the parameters have not been taken into account. But this is very essential for treating of experimental data and introducing relevant observables to measure.
Other parameter is the $Z - Z'$ mixing, which is responsible for different decay processes and the effective interaction vertices generated at the LHC [@Lang08; @Rizzo06]. It is also determined by the axial-vector coupling (see Eq. (\[grgav\])) and estimated in a model independent way. Remind that in our analysis (in contrast to the approaches of the LEP Collaborations) the mixing was systematically taken into account. Its value is of the same order of magnitude as the parameters that were fitted in experiments. Note also that the existence of other heavy particles with masses $m_X \ge m_{Z'}$ does not influence the RG relations which are the consequences of the necessary condition for renormalizability. In fact, this condition (the structure of a divergence generated by radiation corrections coincides with that of the tree-level vertex) holds for each renormalizable type interaction.
An important role of the model independent results for searching for $Z'$ at the LHC and ILC consists, in particular, in possibility to determine the particle as a virtual state due to a large amount of relevant events. We mentioned already that, in principle, LEP experiments were able to determine it if the statistics was sufficiently large. Experiments at the ILC will increase numerously the data set of interest. In fact, the observables, introduced in sects. 6 and 7 for picking out uniquely $a_f^2$ and $v_e^2$ couplings in the leptonic scattering process, are also effective at energies $\sqrt{s}\ge 500$ GeV and could be applied in future experiments at ILC.
Other model independent methods of searching for the $Z'$ as a resonance state are proposed in the literature (see Refs. [@Dittmar; @zplhc1; @zplhc2]). We do not discuss them here because they take no relations between the parameters into consideration. Besides, the main goal of the present paper is to adduce model independent information about the $Z'$ followed from experiments at low energies. Different aspects of $Z'$ physics at the LHC are out of the scope of it.
Discussion
==========
In this section we collect in a convenient form all the results obtained and make a comparison with other investigations on searching for $Z'$ at low energies. In fact, this is a large area to discuss. References to numerous results obtained in either model dependent or model independent approaches can be found in the surveys [@Lang08; @Rizzo06]. Further subdivision can be done into the considerations accounting for any type correlations between the parameters of the low energy effective interactions and that of assuming complete independence of them. Because of a large amount of fitting parameters the latter are less predictable.
Now, for a convenience of readers let us present the results of fits of the $Z'$ parameters in terms of the popular notations [@Leike; @Lang08]. The Lagrangian reads $$\begin{aligned}
\label{avstandard}
{\cal L}_{Z\bar{f}f}&=&\frac{1}{2} Z_\mu\bar{f}\gamma^\mu\left[
(v^{\mathrm{SM}}_f+ \Delta^V_f) - \gamma^5
(a^{\mathrm{SM}}_f+\Delta^A_f) \right]f, \nonumber\\
{\cal L}_{Z'\bar{f}f}&=&\frac{1}{2} Z'_\mu\bar{f}\gamma^\mu\left[
(v'_f-\gamma^5 a'_f)\right]f,\end{aligned}$$ with the SM values of the $Z$ couplings $$v^{\mathrm{SM}}_f = \frac{e\left(T_{3f}-2Q_f
\sin^2\theta_W\right)}{\sin\theta_W\cos\theta_W},\qquad
a^{\mathrm{SM}}_f = \frac{e\,T_{3f}}{\sin\theta_W\cos\theta_W},$$ where $e$ is the positron charge, $Q_f$ is the fermion charge in the units of $e$, $T_{3f}=1/2$ for the neutrinos and $u$-type quarks, and $T_{3f}=-1/2$ for the charged leptons and $d$-type quarks.
[|l|c|c|c|c|]{} Data & $|\theta_0|$, $\times 10^{-3}$ & $|v'_e|$, $\times 10^{-1}$ & $|a'_f|$, $\times 10^{-1}$ & $\Delta^A_e$, $\times 10^{-3}$\
\
$e^-e^+$ & ${3.17}{M}^{-1}$ & - & $1.38M$ & 0.437\
\
$e^-e^+$ & - & $5.83M$ & - & -\
$\mu^-\mu^+$ & $5.42{M}^{-1}$ & - & $2.36M$ & 1.278\
$\mu^-\mu^+,\tau^-\tau^+$ & $3.27{M}^{-1}$ & - & $1.42M$ & 0.464\
\
$e^-e^+$, $z<0$ & - & $5.84M$ & - & -\
[|l|c|c|c|c|c|]{} Data & CL & $|\theta_0|$, $\times 10^{-3}$ & $|v'_e|$, $\times 10^{-1}$ & $|a'_f|$, $\times 10^{-1}$ & $\Delta^A_e$, $\times 10^{-3}$\
\
$e^-e^+$ & 68% & $(0;4.48){M}^{-1}$ & - & $(0;1.95)M$ & $(0;0.873)$\
\
$e^-e^+$ & 95% & - & $(2.46;7.87)M$ & - & -\
$\mu^-\mu^+$ & 95% & $(0;10.39){M}^{-1}$ & - & $(0;4.52)M$ & $(0;4.694)$\
$\mu^-\mu^+$,&&&&&\
$\tau^-\tau^+$ & 95% & $(0;8.64){M}^{-1}$ & - & $(0;3.75)M$ & $(0;3.244)$\
\
$e^-e^+$, &&&&&\
$\mu^-\mu^+$, & 95% & $(0;17.03){M}^{-1}$ & $(0;5.06)M$ & $(0;7.40)M$ & $(0;12.607)$\
$\tau^-\tau^+$ &&&&&\
$e^-e^+$, &&&&&\
$z<0$ & 68% & $(0;27.61){M}^{-1}$ & $(1.68; 7.83)M$ & $(0;12.00)M$ & $(0;33.1288)$\
The results of the fits of the $Z'$ couplings to the SM leptons obtained from the analysis of the LEP experiments are adduced in the Tables \[fitMLV\]-\[fitCLI\] and Fig. \[fig:fitFigs\]. Remind that due to the universality of the axial-vector coupling $a_f$ the same estimates take also place for quarks. First of all, one parameter fits of LEP experiments as well as the many-parameter fit for the $e^+e^-$ backward bins show the hints of the $Z'$ boson at the 1-2$\sigma$ CL. Due to this fact, the fits allow to determine the maximum likelihood values of $Z'$ parameters. In spite of uncertainties, these values can be used as a guiding line for the estimation of possible $Z'$ effects in the LHC experiments. The maximum likelihood values are given in Table \[fitMLV\]. As is seen, different fits and processes lead to the comparable values of the $Z'$ parameters.
In Table \[fitCLI\] we present the confidence intervals for the fitted parameters. With this Table one is able to estimate the uncertainty of the $Z'$ couplings as well as the lower bounds on the parameters.
\
\
In Fig. \[fig:fitFigs\] the maximum likelihood values and the CL intervals are drawn for the different values of the $Z'$ mass. All the plots exploit the same color scheme. The values excluded at 95% CL by the many-parameter fit of all the LEP2 leptonic processes $e^+e^-\to l^+l^-$ are shown in gray. The 95% confidence intervals from the one-parameter fit of LEP2 $e^+e^-\to\mu^+\mu^-,\tau^+\tau^-$ are drawn in pink with the maximum likelihood values as the dashed red line. The corresponding results with taking into account the $\mu^+\mu^-$ process only are shown in yellow with solid red line for the maximum likelihood values. The maximum likelihood values from the LEP1 experiments are represented as dotted blue line. The 95% confidence interval from the one-parameter fit of the LEP2 Bhabha scattering is shown as the blue crosshatched region with the maximum likelihood values as the dashed blue line. Finally, the 68% confidence interval and the maximum likelihood values from the many-parameter fit of the backward bins of LEP2 Bhabha scattering are shown in green.
Now we compare the above results with the ones of other fits accounting for the $Z'$ presence. As it was mentioned in Introduction, LEP collaborations have determined the model dependent low bounds on the $Z'$ mass which vary in a wide energy interval dependently on a model. The same has also been done for Tevatron experiments. The modern low bound is $m_{Z'} \ge 850$ GeV. It is also well known that though almost all the present day data are described by the SM [@EWWG; @OPAL; @DELPHI; @LEP1], the overall fit to the standard model is not very good. In Ref. [@Ferroglia] it was showed that the large difference in $\sin^2 \theta^\mathrm{lept}_\mathrm{eff}$ from the forward-backward asymmetry $A^b_{fb}$ of the bottom quarks and the measurements from the SLAC SLD experiment can be explained for physically reasonable Higgs boson mass if one allows for one or more extra $U(1)$ fields, that is $Z'$. A specific model to describe $Z'$ physics of interest was proposed which introduces two type couplings to the hyper charge $Y$ and to the baryon-minus-lepton number $B-L$. Within this model by using a number of precision measurements from LEP1, LEP2, SLD and Tevatron experiments the parameters $a_Y$ and $a_{B-L}$ of the model were fitted. The presence of $Z'$ was not excluded at 68% CL. The value of $a_Y$ was estimated to be of the same order of magnitude as in our analysis and is comparable with values of other parameters detected in the LEP experiments. The erroneous claim that $a_Y$ is two order less then the value derived from our Table 4 is, probably, a consequence of some missed factors. The upper limit on the mass was also obtained $m_{Z'} \le 2.6 $ TeV .
These two analyzes are different but complementary. A common feature of them is an accounting for the $Z'$ gauge boson as a necessary element of the data fits. The results are in accordance at 68-95% CL with the existence of the not heavy $Z'$ which has a good chance to be discovered at Tevatron and/or LHC.
Appendix. RG relations in a theory with different mass scales {#appendix.-rg-relations-in-a-theory-with-different-mass-scales .unnumbered}
=============================================================
In this Appendix we are going to investigate the Yukawa model with a heavy scalar field $\chi$ and a light scalar field $\varphi$ [@YAF2001]. The goals of our investigation are two fold: 1) to derive the one-loop RG relation for the four-fermion scattering amplitude in the decoupling region and 2) to find out the possibility of reducing this relation in the equation for vertex describing the scattering of light particles on the external field when the mixing between heavy and light virtual states takes place.
The Lagrangian of the model reads $$\begin{aligned}
\label{app1}
{\cal L}&=&\frac{1}{2}{\left( \partial_{\mu}\varphi \right) }^{2}-
\frac{m^{2}}{2}{\varphi}^{2}-\lambda{\varphi}^{4}+
\frac{1}{2}{\left(
\partial_{\mu}\chi \right) }^{2}- \frac{{\Lambda}^{2}}{2}{\chi}^{2}-
\xi{\chi}^{4}\nonumber \\ &&+\rho
{\varphi}^{2}{\chi}^{2}+{\bar\psi}\left(
i\partial_{\mu}\gamma_{\mu}-M-G_{\varphi}\varphi- G_{\chi}\chi
\right) \psi,\end{aligned}$$ where $\psi$ is a Dirac spinor field, and $\Lambda\gg m,M$.
Consider the four-fermion scattering $\bar\psi\psi\to\bar\psi\psi$. The $S$-matrix element at the one-loop level is given by $$\begin{aligned}
\label{app2}
{\hat S}&=&-\frac{i}{2}\int\frac{dp_{1}}{{\left( 2\pi\right)
}^{4}}\ldots\frac{dp_{4}}{{\left( 2\pi\right) }^{4}}{\left(
2\pi\right) }^{4}\delta\left( p_{1}+...+p_{4}\right)
%\nonumber\\&&\times
\left[ S_\mathrm{1PR}+ S_\mathrm{box}\right],\nonumber \\
S_\mathrm{1PR}&=&\sum\limits_{{\phi}_{1},{\phi}_{2}=\varphi,
\chi}G_{{\phi}_{1}} G_{{\phi}_{2}}\left(
\frac{{\delta}_{{\phi}_{1}{\phi}_{2}}}{s-m^2_{{\phi}_{1}}}
+\frac{\Pi_{\phi_1\phi_2}(s)}{(s- m^2_{\phi_1})(s-
m^2_{\phi_2})}\right)
\nonumber\\&&
\times {\bar\psi}\left(p_{4}\right)\left[1+\Gamma\left(
p_{3}, -p_{4}- p_{3}\right) \right] \psi\left(p_{3}\right)\nonumber \\
&&\times {\bar\psi}\left( p_{1}\right) \left[1+\Gamma\left( p_{2},
-p_{1}- p_{2}\right) \right] \psi\left( p_{2}\right),\end{aligned}$$ where $s={\left( p_{1}+p_{2}\right) }^{2}$, $S_\mathrm{1PR}$ is the contribution from the one-particle reducible diagrams shown in Figs. \[fig:tree\]-\[fig:loop\], and $S_\mathrm{box}$ is the contribution from box diagrams.
The one-loop polarization operator of scalar fields ${\Pi}_{{\phi}_{1}{\phi}_{2}}$ and the one-loop vertex function $\Gamma$ are related to the Green functions as $$\begin{aligned}
\label{app3}
D_{{\phi}_{1}{\phi}_{2}}\left( s \right)
&=&\frac{{\delta}_{{\phi}_{1}{\phi}_{2}}}{s-m^2_{
{\phi}_{1}}}+\frac{1}{s-
m^2_{{\phi}_{1}}}{\Pi}_{{\phi}_{1}{\phi}_{2}}\left( s \right)
\frac{1}{s- m^2_{{\phi}_{2}}},\nonumber \\ G_{\phi\phi\psi}\left(
p,q \right)&=&-\sum\limits_{{\phi}_{1} }G_{{\phi}_{1}}
D_{{\phi}_{1}\phi}\left( q^{2} \right) S_{\psi}\left( p \right)
\left(1+\Gamma\left( p, q\right) \right) S_{\psi} \left(
-p-q\right),\end{aligned}$$ where $ S_{\psi}$ is the spinor propagator in the momentum representation.
Renormalization of the model {#renormalization-of-the-model .unnumbered}
----------------------------
The renormalized fields, masses and charges are defined as follows $$\begin{aligned}
\label{app4}
&&\left( \begin{array}{c}\varphi \\ \chi
\end{array}\right)= Z_\phi^{-1/2}\left( \begin{array}{c}\varphi_0 \\
\chi_0 \end{array}\right),\qquad \left(
\begin{array}{c}G_\varphi
\\ G_\chi
\end{array}\right)= Z_G^{-1}\left( \begin{array}{c}G_{\varphi, 0} \\
G_{\chi, 0} \end{array}\right),
\nonumber\\&&
\psi=Z_\psi^{-1/2}\psi_0,\qquad M^2= M_0^2-\delta M^2,
\nonumber\\&&
m^2=m_0^2-\delta m^2,\qquad\Lambda^2=\Lambda_0^2-\delta\Lambda^2,\end{aligned}$$ where subscript 0 marks the corresponding bare quantities.
Using the dimensional regularization (the dimension of the momentum space is $D=4-\varepsilon$) and the $\overline{\mathrm{MS}}$ renormalization scheme one can compute the renormalization constants $$\begin{aligned}
\label{app5}
Z_\psi&=&1-\frac{1}{16\pi^2\varepsilon}\left(G_\varphi^2+G_\chi^2\right),
\nonumber\\ \delta
M^2&=&\frac{3}{8\pi^2\varepsilon}\left(G_\varphi^2+G_\chi^2\right)M^{2}, \nonumber\\
Z_\phi^{1/2}&=&1-\frac{1}{8\pi^2\varepsilon}\left(\begin{array}{cc}
G_\varphi^2& 2G_\varphi
G_\chi\frac{\Lambda^2-6M^2}{\Lambda^2-m^2}\\
-2G_\varphi G_\chi\frac{m^2-6M^2}{\Lambda^2-m^2}&
G_\chi^2\end{array}\right), \nonumber\\ \delta
m^2&=&\frac{1}{4\pi^2\varepsilon}
\left[\left(G_\varphi^2+6\lambda\right)m^2-6G_\varphi^2M^2-\rho\Lambda^2\right],
\nonumber\\
\delta\Lambda^2&=&\frac{1}{4\pi^2\varepsilon}
\left[\left(G_\chi^2+6\xi\right)\Lambda^2-6G_\chi^2M^2-\rho
m^2\right], \nonumber\\
Z_G^{-1}&=&\left[1-\frac{3}{16\pi^2\varepsilon}\left(G_\varphi^2+G_\chi^2\right)\right]
{\left(Z_\phi^{1/2}\right)}^\mathrm{T}.\end{aligned}$$
From Eq. (\[app5\]) we obtain the appropriate $\beta$ and $\gamma$ functions at the one-loop level: $$\begin{aligned}
\label{app6}
\beta_\varphi&=&\frac{dG_\varphi}{d\log\kappa}=
\frac{G_\varphi}{16\pi^2}\left(5G_\varphi^2+3G_\chi^2-4\frac{m^2-6M^2}{\Lambda^2-
m^2}G_\chi^2\right),
\nonumber\\
\beta_\chi&=&\frac{dG_\chi}{d\log\kappa}=
\frac{G_\chi}{16\pi^2}\left(5G_\chi^2+3G_\varphi^2+4\frac{\Lambda^2-6M^2}{\Lambda^2-m^2}G_\varphi^2\right),
\nonumber\\
\gamma_m&=&-\frac{d\log m^2}{d\log\kappa}=
-\frac{1}{4\pi^2}\left(G_\varphi^2\frac{m^2-6M^2}{m^2}+6\lambda-\rho\frac{\Lambda^2}{m^2}\right),
\nonumber\\
\gamma_\Lambda&=&-\frac{d\log\Lambda^2}{d\log\kappa}=
-\frac{1}{4\pi^2}\left(G_\chi^2\frac{\Lambda^2-6M^2}{\Lambda^2}+6\xi-\rho\frac{m^2}{\Lambda^2}\right),
\nonumber\\
\gamma_\psi&=&-\frac{d\log\psi}{d\log\kappa}=
\frac{1}{32\pi^2}\left(G_\varphi^2+G_\chi^2\right).\end{aligned}$$ Then, the $S$-matrix element can be expressed in terms of the renormalized quantities. The contribution from the one-particle reducible diagrams becomes $$\begin{aligned}
\label{app7}
S_\mathrm{1PR}&=&\sum\limits_{\phi_1,\phi_2}G_{\phi_1}G_{\phi_2}
\left(\frac{\delta_{\phi_1\phi_2}}{s-m^2_{\phi_1}}
+\frac{\Pi_{\phi_1\phi_2}^\mathrm{(fin)}(s)}{(s-m^2_{\phi_1})(s-m^2_{\phi_2})}\right)
\nonumber\\&&\times
\bar\psi(p_4)\left[1+\Gamma^\mathrm{(fin)}\left(p_3,-p_4-p_3\right)\right]\psi(p_3)
\nonumber\\&&\times
\bar\psi(p_1)\left[1+\Gamma^\mathrm{(fin)}\left(p_2,-p_1-p_2\right)\right]\psi(p_2),\end{aligned}$$ where the functions $\Pi_{\phi_1\phi_2}^\mathrm{(fin)}$ and $\Gamma^\mathrm{(fin)}$ are the expressions $\Pi_{\phi_1\phi_2}$ and $\Gamma$ without the terms proportional to $1/\varepsilon$. Since the quantity $S_\mathrm{box}$ is finite, the renormalization leaves it without changes.
Introducing the RG operator at the one-loop level [@bando93] $$\begin{aligned}
\label{app8}
{\cal
D}&=&\frac{d}{d\log\kappa}=\frac{\partial}{\partial\log\kappa}+
{\cal D}^{(1)} =\frac{\partial}{\partial\log\kappa} \nonumber\\&&
+\sum\limits_\phi\beta_\phi\frac{\partial}{\partial G_\phi}
-\gamma_m\frac{\partial}{\partial\log m^2}
-\gamma_\Lambda\frac{\partial}{\partial\log\Lambda^2}
-\gamma_\psi\frac{\partial}{\partial\log\psi}\end{aligned}$$ we determine that the following relation holds for the $S$-matrix element $$\label{app9}
{\cal D}\left( S_\mathrm{1PR}+S_\mathrm{box}\right)
=\frac{\partial S_\mathrm{1PR}^{(1)}}{\partial\log\kappa}+{\cal
D}^{(1)} S_\mathrm{1PR}^{(0)}=0,$$ where the $S_\mathrm{1PR}^{(0)}$ and the $S_\mathrm{1PR}^{(1)}$ are the contributions to the $S_\mathrm{1PR}$ at the tree level and at the one-loop level, respectively: $$\label{app10}
S_\mathrm{1PR}^{(0)}=
\left(\frac{G_\varphi^2}{s-m^2}+\frac{G_\chi^2}{s-\Lambda^2}\right)
\bar\psi\psi\times\bar\psi\psi,$$ $$\begin{aligned}
\label{app11}
\frac{\partial S_\mathrm{1PR}^{(1)}}{\partial\log\kappa}&=&
\frac{\bar\psi\psi\times\bar\psi\psi}{4\pi^2}
%\nonumber\\&&\times
\left[ -\left(G_\varphi^2+G_\chi^2\right)
\left(\frac{G_\varphi^2}{s-m^2}+\frac{G_\chi^2}{s-\Lambda^2}\right)
\right.\nonumber\\&&
+\frac{G_\varphi^2\left(\rho\Lambda^2-6\lambda
m^2+G_\varphi^2\left(6M^2-s\right)\right)}{\left(s-m^2\right)^2}
\nonumber\\&&
+\frac{2G_\varphi^2G_\chi^2\left(6M^2-s\right)}{\left(s-m^2\right)\left(s-\Lambda^2\right)}
\nonumber\\&&\left. +\frac{G_\chi^2\left(\rho
m^2-6\xi\Lambda^2+G_\chi^2\left(6M^2-s\right)\right)}{\left(s-\Lambda^2\right)^2}\right].\end{aligned}$$
The first term in Eq. (\[app11\]) is originated from the one-loop correction to the fermion-scalar vertex. The rest terms are connected with the polarization operator of scalars. The third term describes the one-loop mixing between the scalar fields. It is canceled in the RG relation (\[app9\]) by the mass-dependent terms in the $\beta$ functions produced by the non-diagonal elements in $Z_\phi$.
Eq. (\[app9\]) is the consequence of the renormalizability of the model. It insures the leading logarithm terms of the one-loop $S$-matrix element to reproduce the appropriate tree-level structure. In contrast to the familiar treatment we are not going to improve scattering amplitudes by solving Eq. (\[app9\]). We will use it as an algebraic identity implemented in the renormalizable theory. Naturally if one knows the explicit couplings expressed in terms of the basic set of parameters of the model, this RG relation is trivially fulfilled. But the situation changes when the couplings are represented by unknown arbitrary parameters. In this case the RG relations are the algebraic equations dependent on these parameters and appropriate $\beta$ and $\gamma$ functions. In the presence of a symmetry the number of $\beta$ and $\gamma$ functions is less than the number of RG relations. So, one has non trivial system of equations relating the unknown couplings. For example, such a scenario is realized for the gauge coupling. Although the considered simple model has no gauge couplings, we are able to demonstrate the general procedure of deriving the RG relations.
Decoupling of the heavy field {#decoupling-of-the-heavy-field .unnumbered}
-----------------------------
At energies $s\ll\Lambda^2$ the heavy scalar field $\chi$ is decoupled. This means, that the four-fermion scattering amplitude is described by the model with no heavy field $\chi$ plus terms of the order $s/\Lambda^2$. At the tree level, this is the obvious consequence of the expansion of the heavy scalar propagator $$\label{app12}
\frac{1}{s-\Lambda^2}\to
-\frac{1}{\Lambda^2}\left[1+O\left(\frac{s}{\Lambda^2}\right)\right],$$ which is resulted in the effective contact four-fermion interaction in Eq. (\[app10\]) $$\label{app13}
{\cal L}_\mathrm{eff}=-\alpha\,\bar\psi\psi\times\bar\psi\psi,
\quad\alpha=\frac{G_\chi^2}{\Lambda^2}.$$ So, the tree level contribution to the scattering amplitude becomes $$\label{app14}
S_\mathrm{1PR}^{(0)}= \left[
\frac{G_\varphi^2}{s-m^2}-\alpha+O\left(\frac{s}{\Lambda^4}\right)
\right] \bar\psi\psi\times\bar\psi\psi,$$ and the lowest order effects of the heavy scalar in the decoupling region are described by the parameter $\alpha$, only.
Decoupling of heavy particles is present also at the level of radiative corrections. The radiative corrections are generally described by various loop integrals in the momentum space (the Passarino–Veltman functions). Considering a Passarino–Veltman function with at least one heavy mass $Lambda$ inside loop in the low-energy limit, one can see the following asymptotic behavior: the function splits into 1) possible energy-independent divergent part (including also $\log\Lambda$) and 2) energy-dependent finite part which can be expanded by inverse powers of $\Lambda$ and vanishes at the small energies. The important property is that the $\log\Lambda$-term in the divergent part reproduces the logarithm of the cut-off scale. So, such a potentially large term has to be automatically absorbed by the renormalization at low energies and leads to no observable effects. However, if the renormalization is actually performed at high energies (as in the $\overline{\mathrm{MS}}$ renormalization scheme) the potentially large $\log\Lambda$-terms should be re-summed manually by the redefinition of the physical couplings and masses at the scale $\Lambda$.
What is the form of the RG relations in the limit of large $\Lambda$? The method of constructing the RG equation in the decoupling region was proposed in [@bando93]. It introduces the redefinition of the parameters of the model allowing to remove all the heavy particle loop contributions to Eq. (\[app11\]). Let us define a new set of fields, charges and masses $\tilde\psi$, $\tilde{G}_\varphi$, $\tilde{G}_\chi$, $\tilde\Lambda$, $\tilde{m}$, $\tilde{M}$ $$\begin{aligned}
\label{app15}
G_\varphi^2&=&\tilde{G}_\varphi^2
\left(1+\frac{3\tilde{G}_\chi^2}{16\pi^2}
\log\frac{\kappa^2}{\tilde\Lambda^2}+\ldots\right), \nonumber\\
G_\chi^2&=&\tilde{G}_\chi^2
\left(1+\frac{3\tilde{G}_\chi^2}{16\pi^2}
\log\frac{\kappa^2}{\tilde\Lambda^2}+\ldots\right), \nonumber\\
m^2&=&\tilde{m}^2\left(1-\frac{\tilde\rho}{8\pi^2}\frac{\tilde\Lambda^2}{\tilde{m}^2}
\log\frac{\kappa^2}{\tilde\Lambda^2}+\ldots\right), \nonumber\\
\Lambda^2&=&\tilde\Lambda^2 \left(1+\frac{3\tilde\xi}{4\pi^2}
\log\frac{\kappa^2}{\tilde\Lambda^2}+\ldots\right), \nonumber\\
\psi&=&\tilde\psi\left(1-\frac{\tilde{G}_\chi^2}{64\pi^2}
\log\frac{\kappa^2}{\tilde\Lambda^2}+\ldots\right),\end{aligned}$$ where dots stand for the higher powers of $\log\Lambda$ responsible for the decoupling at higher loop orders.
The differential operator (\[app8\]) ban be rewritten in terms of these new low-energy parameters: $$\begin{aligned}
\label{app16}
{\cal D}&=&\frac{\partial}{\partial\log\kappa}+{\tilde{\cal
D}}^{(1)}= \frac{\partial}{\partial\log\kappa}
+\sum\limits_\phi\tilde\beta_\phi\frac{\partial}{\partial\tilde{G}_\phi}
\nonumber\\&&
-\tilde\gamma_m\frac{\partial}{\partial\log\tilde{m}^2}
-\tilde\gamma_\Lambda\frac{\partial}{\partial\log\tilde\Lambda^2}
-\tilde\gamma_\psi\frac{\partial}{\partial\log\tilde\psi}\end{aligned}$$ where $\tilde\beta$ and $\tilde\gamma$ functions are obtained from the one-loop relations (\[app6\]) and (\[app15\]) $$\begin{aligned}
\label{app17}
&& \tilde\beta_\varphi=\frac{1}{16\pi^2}
\left(5\tilde{G}_\varphi^{3}-4\frac{\tilde{m}^2-6\tilde{M}^2}{\tilde\Lambda^2-\tilde{m}^2}\tilde{G}_\varphi\tilde{G}_\chi^2\right),
\nonumber\\
&& \tilde\beta_\chi=\frac{1}{16\pi^2}
\left(2\tilde{G}_\chi^3+\left(3+4\frac{\tilde\Lambda^2-6\tilde{M}^2}{\tilde\Lambda^2-\tilde{m}^2}\right)
\tilde{G}_\chi\tilde{G}_\varphi^2\right),
\nonumber\\
&& \tilde\gamma_m=-\frac{1}{4\pi^2}
\left(\tilde{G}_\varphi^2\frac{\tilde{m}^2-6\tilde{M}^2}{\tilde{m}^2}+6\tilde\lambda\right),
\nonumber\\
&& \tilde\gamma_\Lambda=-\frac{1}{4\pi^2}
\left(\tilde{G}_\chi^2\left(1-6\frac{\tilde{M}^2}{\tilde\Lambda^2}\right)
-\tilde\rho\frac{\tilde{m}^2}{\tilde\Lambda^2}\right),
\nonumber\\
&& \tilde\gamma_\psi=\frac{1}{32\pi^2}\tilde{G}_\varphi^2.\end{aligned}$$ Hence, one immediately notices that $\tilde\beta$ and $\tilde\gamma$ functions contain only the light particle loop contributions, and all the heavy particle loop terms are completely removed from them. The $S$-matrix element expressed in terms of new parameters satisfies the following RG relation $$\label{app18}
{\cal D}\left(S_\mathrm{1PR}+S_\mathrm{box}\right)
=\frac{\partial\tilde{S}_\mathrm{1PR}^{(1)}}{\partial\log\kappa}
+{\tilde{\cal D}}^{(1)}\tilde{S}_\mathrm{1PR}^{(0)}=0,$$ $$\label{app19}
\tilde{S}_\mathrm{1PR}^{(0)}=
\left(\frac{\tilde{G}_\varphi^2}{s-\tilde{m}^2} -\tilde\alpha
+O\left(\frac{s^2}{\tilde\Lambda^4}\right)\right)
{\bar{\tilde\psi}}\tilde\psi\times{\bar{\tilde\psi}}\tilde\psi,$$ $$\begin{aligned}
\label{app20}
\frac{\partial\tilde{S}_\mathrm{1PR}^{(1)}}{\partial\log\kappa}&=&
\frac{{\bar{\tilde\psi}}\tilde\psi\times{\bar{\tilde\psi}}\tilde\psi}{4\pi^2}
\left(-\frac{\tilde{G}_\varphi^4}{s-\tilde{m}^2}
\right.\nonumber\\&&\left.
+\frac{\tilde{G}_\varphi^2\left(-6\tilde\lambda\tilde{m}^2
+\tilde{G}_\varphi^2\left(6\tilde{M}^2-s\right)\right)}{\left(s-\tilde{m}^2\right)^2}
+\tilde\alpha\tilde{G}_\varphi^2 \right.\nonumber\\&&\left.
-\frac{2\tilde{G}_\varphi^2\tilde\alpha\left(6\tilde{M}^2-s\right)}{s-\tilde{m}^2}
+O\left(\frac{s^2}{\tilde\Lambda^4}\right)\right),\end{aligned}$$ where $\tilde\alpha=\tilde{G}_\chi^2/\tilde\Lambda^2$ is the redefined effective four-fermion coupling. As one can see, Eq. (\[app20\]) includes all the terms of Eq. (\[app11\]) except for the heavy particle loop contributions. It depends on the low energy quantities $\tilde\psi$, $\tilde{G}_\varphi$, $\tilde\alpha$, $\tilde\lambda$, $\tilde{m}$, $\tilde{M}$. The first and the second terms in Eq. (\[app20\]) are just the one-loop amplitude calculated within the model with no heavy particles. The third and the fourth terms describe the light particle loop correction to the effective four-fermion coupling and the mixing of heavy and light virtual fields.
Elimination of the one-loop scalar field mixing {#elimination-of-the-one-loop-scalar-field-mixing .unnumbered}
-----------------------------------------------
Due to the mixing term it is impossible to split the RG relation (\[app18\]) for the $S$-matrix element into the one for vertices. Hence, we are not able to consider Eq. (\[app18\]) in the framework of the scattering of light particles on an external field induced by the heavy virtual scalar. But this is an important step in deriving the RG relation for EL parameters. Fortunately, there is a simple procedure allowing to avoid the mixing in Eq. (\[app20\]). The way is to incorporate the diagonalization of the leading logarithm terms of the scalar polarization operator into the redefinition of the $\tilde\varphi$, $\tilde\chi$, $\tilde{G}_\varphi$, $\tilde{G}_\chi$: $$\begin{aligned}
\label{app21}
\left( \begin{array}{c}\varphi \\ \chi \end{array}\right)&=&
{\zeta}^{1/2}\left( \begin{array}{c}{\tilde\varphi} \\
{\tilde\chi}
\end{array}\right),\nonumber\\ \left( \begin{array}{c}G_\varphi \\ G_\chi
\end{array}\right)&=&\left[1+\frac{3\tilde{G}_\chi^2}{32\pi^2}
\log\frac{\kappa^2}{\tilde\Lambda^2}\right]
\left(\zeta^{-1/2}\right)^\mathrm{T}
\left(\begin{array}{c}\tilde{G}_\varphi
\\ \tilde{G}_\chi \end{array}\right) , \\ \nonumber\zeta^{1/2}&=&
1-\frac{\tilde{G}_\varphi\tilde{G}_\chi}{8\pi^2\left(\tilde\Lambda^2-\tilde{m}^2\right)}
\log\frac{\kappa^2}{\tilde\Lambda^2}
%\nonumber\\&&\times
\left(\begin{array}{cc}0 &
\tilde\Lambda^2-6\tilde{M}^2\\-\tilde{m}^2-6\tilde{M}^2&
0\end{array}\right).\end{aligned}$$ The appropriate $\tilde\beta$ functions $$\label{app22}
\tilde\beta_\varphi=\frac{5\tilde{G}_\varphi^3}{16\pi^2},\quad
\tilde\beta_\chi=\frac{1}{16\pi^2}\left(2\tilde{G}_\chi^3+3\tilde{G}_\chi\tilde{G}_\varphi^2\right)$$ contain no terms connected with mixing between light and heavy scalars. So, the fourth term in Eq. (\[app20\]) is removed, and the RG relation for the $S$-matrix element becomes
$$\label{app23}
{\cal D}\left(S_\mathrm{1PR}+S_\mathrm{box}\right)
=\frac{\partial\tilde{S}_\mathrm{1PR}^{(1)}}{\partial\log\kappa}
+{\tilde{\cal D}}^{(1)}\tilde{S}_\mathrm{1PR}^{(0)}=0,$$
$$\label{app24}
\tilde{S}_\mathrm{1PR}^{(0)}=\left(
\frac{\tilde{G}_\varphi^2}{s-\tilde{m}^2} -\tilde\alpha
+O\left(\frac{s^2}{\tilde\Lambda^4}\right)\right)
\bar{\tilde\psi}\tilde\psi\times\bar{\tilde\psi}\tilde\psi,$$
$$\begin{aligned}
\label{app25}
\frac{\partial {\tilde
S}_\mathrm{1PR}^{(1)}}{\partial\log\kappa}&=&
\frac{\bar{\tilde\psi}\tilde\psi\times\bar{\tilde\psi}\tilde\psi}{4\pi^2}
\left(-\frac{\tilde{G}_\varphi^4}{s-\tilde{m}^2}
\right.\nonumber\\&&\left.
+\frac{\tilde{G}_\varphi^2\left(-6\tilde\lambda\tilde{m}^2+\tilde{G}_\varphi^2\left(6\tilde{M}^2-s\right)\right)}
{\left(s-\tilde{m}^2\right)^2} \right.\nonumber\\&&\left.
+\tilde\alpha\tilde{G}_\varphi^2
+O\left(\frac{s^2}{\tilde\Lambda^4}\right)\right).\end{aligned}$$
At $\tilde\alpha=0$ Eq. (\[app23\]) is just the RG identity for the scattering amplitude calculated in the absence of the heavy particles. The terms of order $\tilde\alpha$ describe the RG relation for the effective low-energy four-fermion interaction in the decoupling region. The last one can be reduced in the RG relation for the vertex describing the scattering of the light particle (fermion) on the external field $\sqrt{\tilde\alpha}$ substituting the virtual heavy scalar: $$\label{app26}
{\cal D}\left(\sqrt{\tilde\alpha}\bar{\tilde\psi}\tilde\psi\right)
=\frac{\tilde{G}_\varphi^2}{8\pi^2}\sqrt{\tilde\alpha}\bar{\tilde\psi}\tilde\psi
+{\tilde{\cal D}}^{(1)}
\left(\sqrt{\tilde\alpha}\bar{\tilde\psi}\tilde\psi\right) =0,$$ where $$\begin{aligned}
\label{app27}
&&{\tilde{\cal D}}^{(1)}=
\tilde\beta_\varphi\frac{\partial}{\partial\tilde{G}_\varphi}
-\tilde\gamma_\alpha\frac{\partial}{\partial\log\tilde\alpha}
-\tilde\gamma_m\frac{\partial}{\partial\log\tilde{m}^2}
-\tilde\gamma_\psi\frac{\partial}{\partial\log\tilde\psi},
\nonumber\\&& \tilde\gamma_\alpha=-{\cal
D}\tilde\alpha=-\frac{1}{8\pi^2}\left(3\tilde{G}_\varphi^2+O\left(\tilde\alpha\right)\right).\end{aligned}$$
Eqs. (\[app23\])-(\[app27\]) is the main result of our investigation. One can derive them with only the knowledge about the low-energy couplings of heavy particle (\[app13\]) and the Lagrangian of the model with no heavy particles. One also has to ignore all the heavy particle loop contributions to the RG relation and the one-loop mixing between the heavy and the light fields. Eqs. (\[app23\])-(\[app27\]) depend on the effective low-energy parameters, only. But as the difference between the original set of parameters and the low-energy one is of one-loop order, one may freely substitute them in Eqs. (\[app23\])-(\[app26\]). It is also possible to reduce the RG relation for scattering amplitudes to the one for vertex describing the scattering of light particles on the ‘external’ field induced by the heavy virtual particle. In fact, this result is independent on the specific features of the considered model.
[99]{} The LEP Collaborations ALEPH, DELPHI, L3, OPAL, and the LEP Electroweak Working Group, hep-ex/0612034. G. Abbiendi [*et al.*]{} \[OPAL Collaboration\], Eur. Phys. J. [**C33**]{} (2004) 173 \[hep-ex/0309053\]; Eur. Phys. J. [**C6**]{} (1999) 1; K. Ackerstaff [*et al.*]{} \[OPAL Collaboration\], Eur. Phys. J. [**C2**]{} (1998) 441. J. Abdallah [*et al.*]{} \[DELPHI Collaboration\], Eur. Phys. J. [**C45**]{} (2006) 589 \[hep-ex/0512012\]. A. Leike, Phys. Rep. [**317**]{} (1999) 143. P. Langacker, arXiv:0801.1345 \[hep-ph\]. T. Rizzo, hep-ph/0610104. A. Ferroglia, A. Lorca, and J.J. van der Bij, Annalen Phys [**16**]{} (2007) 563-578 \[hep-ph/0611174\]. A. Gulov and V. Skalozub, Eur. Phys. J. [**C17**]{} (2000) 685. A. Gulov and V. Skalozub, Phys. Rev. [**D61**]{} (2000) 055007. V. Demchik, A. Gulov, V. Skalozub, and A. Tishchenko, Phys. Atom. Nucl. [**67**]{} (2004) 1312 \[Yad. Fiz. [**67**]{} (2004) 1335\]. A. Gulov and V. Skalozub, Phys. Rev. [**D70**]{} (2004) 115010. A. Gulov and V. Skalozub, Phys. Rev. [**D76**]{} (2007) 075008. J.Hewett and T.Rizzo, Phys. Rep. [**183**]{} (1989) 193. M. Cvetic and B.W. Lynn, Phys. Rev. [**D35**]{} (1987) 51. G. Degrassi and A. Sirlin, Phys. Rev. [**D40**]{} (1989) 3066. T. Appelquist and J. Carazzone, Phys. Rev. [**D11**]{} (1975) 2856. J. Collins, F. Wilczek, and A. Zee, Phys. Rev. [**D18**]{} (1978) 242. M. Bando, T. Kugo, N. Maekawa, and H. Nakano, Progress of Theor. Phys. [**90**]{} (1993) 405. T. Rizzo, Phys. Rev. [**D55**]{} (1997) 5483. R. Barate [*et al.*]{} \[ALEPH Collaboration\], Eur. Phys. J. [**C12**]{} (2000) 183 \[hep-ex/9904011\]. M. Acciarri [*et al.*]{} \[L3 Collaboration\], Phys. Lett. [**B479**]{} (2000) 101 \[hep-ex/0002034\]. A. Babich, G. Della Ricca, J. Holt, P. Osland, A. Pankov, and N. Paver, Eur. Phys. J. [**C29**]{} (2003) 103. W. Eadie, D. Dryard, F. James, M. Roos, and B. Sadoulet, [*Statistical methods in experimental physics*]{}, Amsterdam, North-Holland, 1971. ALEPH Collaboration, DELPHI Collaboration, L3 Collaboration, OPAL Collaboration, SLD Collaboration, LEP Electroweak Working Group, SLD Electroweak Group, and SLD Heavy Flavour Group, Phys. Rept. [**427**]{} (2006) 257 \[hep-ex/0509008\]. V. Abazov [*et al.*]{} \[D0 Collaboration\], Phys. Lett. [**B517**]{} (2001) 282. A. Buryk and V. Skalozub, arXiv:0802.1486 \[hep-ph\]. http://home.cern.ch/ schwind/MLPfit.html A. Gulov and V. Skalozub, Phys. Atom. Nucl. [**70**]{} (2007) 1100-1106 \[hep-ph/0510354\]. M. Dittmar, A.-S. Nicollerat, and A. Djouadi, Phys. Lett. [**B583**]{} (2004) 111-120 \[hep-ph/0307020\]. C. Coriano, A. Faraggi, and M. Guzzi, Phys. Rev. [**D78**]{} (2008) 015012 \[arXiv:0802.1792\]. F. Petriello and S. Quackenbush, Phys. Rev. [**D77**]{} (2008) 115004 \[arXiv:0801.4389\]. A. Gulov and V. Skalozub, Phys. Atom. Nucl. [**63**]{} (2000) 139-143 \[Yad. Fiz. [**63**]{} (2000) 152-157\].
|
---
abstract: 'In the study of reaction networks and the polynomial dynamical systems that they generate, special classes of networks with important properties have been identified. These include *reversible*, *weakly reversible*, and, more recently, *endotactic* networks. While some inclusions between these network types are clear, such as the fact that all reversible networks are weakly reversible, other relationships are more complicated. Adding to this complexity is the possibility that inclusions be at the level of the dynamical systems generated by the networks rather than at the level of the networks themselves. We completely characterize the inclusions between reversible, weakly reversible, endotactic, and strongly endotactic network, as well as other less well studied network types. In particular, we show that every strongly endotactic network in two dimensions can be generated by an extremally weakly reversible network. We also introduce a new class of *source-only* networks, which is a computationally convenient property for networks to have, and show how this class relates to the above mentioned network types.'
author:
- 'David F. Anderson[^1], James D. Brunner[^2], Gheorghe Craciun[^3], and Matthew D. Johnston[^4]'
bibliography:
- 'myrefs.bib'
title: On classes of reaction networks and their associated polynomial dynamical systems
---
[Keywords: Reaction Networks, Polynomial Dynamical Systems]{}\
Introduction
============
*Chemical reaction networks* model the behavior of sets of reactants, usually termed *species*, that interact at specified rates to form sets of products. Under simplifying assumptions such as (i) a well-mixed reaction vessel, (ii) a sufficiently large number of reactants, and (iii) mass action kinetics, the dynamics of the concentrations of the species can be modeled by a system of autonomous polynomial ordinary differential equations. Such systems are known as *mass action systems*.
With the increased recent interest in systems biology, significant attention has been given to the question of how dynamical properties of a mass action system can be inferred from the structure of the network of interactions that it models. In particular, it is of great interest to identify network structures that inform the dynamics regardless of the choice of parameters for the model (which are often unknown, or only known up to order of magnitude). This is not a trivial endeavor, as kinetic systems based on reaction networks are known to permit a wide variety of dynamical behaviors, including asymptotic convergence to a unique steady state [@H-J1], multistationarity [@C-F1; @C-F2], periodicity and Hopf bifurcations [@W-H1; @W-H2], and chaotic behavior [@E-T]. Nevertheless, structural properties that have strong implications for the corresponding dynamical systems have been found. In the papers [@F1; @H; @H-J1], Feinberg, Horn, and Jackson introduced the now-classical notion of network deficiency and proved that weak reversibility and a deficiency of zero suffice for characterizing the steady state and local convergence properties of the corresponding mass action system. Moreover, their results hold regardless of the choice of parameters. These papers are commonly credited as providing the framework for so-called *chemical reaction network theory* [@Fe2; @Fe4; @Sh-F; @F2; @F3]. Chemical reaction network theory remains an active area of research to date, with a focus on both deterministic [@A; @A4; @ACKN2018; @A2011bounded; @A-S; @yu2018; @D-B-M-P; @C-D-S-S; @J-S7; @B-P; @Conradi19175; @Hell2015DynamicalFO; @craciun2019realizations] and stochastic models [@AK2018; @AN2019; @ACKN2018; @ACKK2018; @AY2019; @A-C-K; @AC2016; @ACGW:lyapunov; @dembo2018; @CW:product; @AM2018; @AEJ2014].
Chemical reaction network theory can be applied either through direct knowledge or hypothesis of a network of interactions, or by constructing such a network from a system of ordinary differential equations with polynomial right hand sides. In either case, the network in question may have no special properties that can be used to draw conclusions. However, considering *dynamical equivalence* may allow the application of a result that is not obviously relevant [@C-P; @J-S2; @Sz2; @craciun2018]. Dynamical equivalence concerns the case of two distinct chemical reaction networks taken with mass action kinetics having identical governing systems of differential equations. It is simple to observe that not all dynamically equivalent network representations share the same structural properties. For example, consider the network $$\label{eq:9807070}
\xymatrix{
\emptyset \ar[r]^{k_1} & X & \ar[l]_{k_2} 2X.
}$$ This network is not weakly reversible (see \[classificationssection\]). Under mass action kinetics, however, we may easily check that the network $$\label{eq:98}
\xymatrix{
\emptyset \ar@<0.5ex>[r]^{\nicefrac{k_1}{2}} & \ar@<0.5ex>[l]^{\nicefrac{k_2}{2}} 2X
}$$ generates exactly the same differential equation, $\dot{x} = k_2 - k_1x^2$, and therefore is a dynamically equivalent representation. This network, however, is reversible and therefore weakly reversible. Thus, existing theory may be used to immediately characterize the long-time behavior of the dynamical system that is associated with both systems.
In the simple example introduced above, the notion of dynamical equivalence allowed us to make a conclusion about the long term behavior of a dynamical system associated with a network that did not appear to fit the hypothesis of the classical theorems of chemical reaction network theory. Furthermore, any model development and fitting from data must account for dynamical equivalence [@craciun2013statistical; @C-P]. The notion of dynamical equivalence therefore plays an essential role in the study of mass action reaction networks. We therefore ask the following question:
\[q1\] Are there easily checkable (geometric) conditions under which two networks are dynamically equivalent (in that they generate the same system of differential equations)?
A recent addition to the class of networks for which results can be obtained is the class of *endotactic networks*, which include reversible and weakly reversible networks as subclasses. Endotactic networks were first introduced in Craciun et al. [@C-N-P] and, roughly speaking, a network is endotactic if the reactions of the network are “inward-pointing” in relation to the convex hull of the source nodes when the network is embedded in ${\mathbb{R}}^d_{\geq 0}$. See \[classificationsEG\] for a precise formulation. The deterministic dynamical systems corresponding to endotactic networks are conjectured to have positive solutions which are bounded and strictly positive for all time under mild conditions on the reaction kinetics [@C-N-P]. This conjecture is known to be true in special cases, including when the network’s stoichiometric subspace is two-dimensional or less [@Pa] and when the network satisfies an additional condition to make it *strongly endotactic* [@ACKN2018; @Gopal2014].
What is not known is exactly how endotactic networks fit into the hierarchy of well-studied network classifications such as reversible networks, weakly reversible networks, single linkage class networks, networks with a single terminal linkage class, and consistent networks. This is a non-trivial question in the context of mass action kinetics given recent work on dynamical equivalence. In fact there are many endotactic networks, including that shown in \[eq:9807070\] for which we can find a dynamically equivalent weakly reversible network. We therefore ask the following question:
\[q2\] Given the flexibility afforded by dynamical equivalence, how closely related are endotactic networks to the well-studied classifications of reversible, weakly reversible, and consistent networks?
Additionally, we introduce the notion of “source-only networks" and show how endotactic and strongly endotactic networks relate to this class of networks.
To answer both questions, we introduce a general framework in which to consider dynamical equivalence, including defining a reaction network as an embedded graph, called a Euclidean embedded graph (E-graph). This definition is equivalent to the classical definition found in the literature, notably Feinberg [@F3], in the sense that it models the same dynamical system.
In this paper, we show that, although significant overlaps exists, endotacticity is indeed distinct from weak reversibility. \[figure1\], \[figure2\], and \[figure3\] give examples of endotactic and even strongly endotactic networks which cannot be realized as weakly reversible networks. We characterize overlap between these types of networks by analyzing the notion of “dynamical equivalence" (\[dyneqdef\], \[overlap\], and \[includedyn\]) under which distinct networks may give rise to the same dynamical systems. We also give checkable conditions for dynamical equivalence (\[containcondit\]). We show that in two dimensions, strong endotacticity is equivalent to weak reversibility on an important subset of the nodes of the network (\[wkrev\]), but \[figure3\] gives a counterexample in three dimensions.
\[relationships\] summarizes our results by giving a succinct summary of the relationships between classifications of networks. Moreover, \[relationships\] is complete in the sense that any additional paths would be false.
Background
==========
In this section, we introduce background notation and results related to *chemical reaction network theory* and *mass action systems*, in particular.
Chemical Reaction Networks
--------------------------
Classically, a reaction network has been defined as below [@F3]:
\[SCR\] A **chemical reaction network** is a triple of finite sets $(\mathcal{S},\mathcal{C},\mathcal{R})$ where:
1. The **species set** $\mathcal{S} = \{ X_1, \ldots, X_d \}$ consists of the basic species/reactants capable of undergoing chemical change.
2. The **complex set** $\mathcal{C} = \{ C_1, \ldots, C_n \}$ consists of linear combinations of species of the form $$C_i = \sum_{j=1}^d y_{ij} X_j, \; \; \; \; \; i=1,\ldots,n.$$ The constants $y_{ij} \in \mathbb{R}_{\geq 0}$ are called *stoichiometric coefficients* and determine the multiplicity of each species within each complex. We define the complex support vectors $\bm{y}_i = (y_{i1},y_{i2},\ldots,y_{id})$ and assume that each complex is stoichiometrically distinct, i.e. $\bm{y}_i \not= \bm{y}_j$ for $i\not= j$. For simplicity, we will allow the support vector $\bm{y}_i$ to represent the complex $C_i$.
3. The **reaction set** $\mathcal{R} = \{ R_1, \ldots, R_r \}$ consists of elementary reactions of the form $$R_k: \; \; \; \bm{y}_{\rho(k)} \longrightarrow \bm{y}_{\rho'(k)}, \; \; \; k=1, \ldots, r$$ where $\rho(k)=i$ if $\bm{y}_i$ is the reactant complex of the $k^{th}$ reaction, and $\rho'(k) = j$ if $\bm{y}_j$ is the product complex of the $k^{th}$ reaction. We require that $\rho(k) \ne \rho'(k)$ for each $k = 1,\ldots, r$. Reactions may alternatively be represented as ordered pairs of complexes, e.g. $R_k = (\bm{y}_i,\bm{y}_j)$ if $\bm{y}_i \to \bm{y}_j$ is in the network.
We present the preceding classical definition (\[SCR\]) in order to connect our results to the bulk of the literature in chemical reaction network theory. In some recent work (see [@rrobust; @Cr; @gheorgheToricDI; @boros2019weakly; @craciun2019endotactic]), chemical reaction networks have been defined in terms of a *Euclidean Embedded Graph (E-graph)*. In this paper, we prefer to use this newer formulation, given below in \[egraphCRN\], due to its convenient geometric properties.
A *Euclidean embedded graph* (E-graph) ${\mathcal{G}}= ({\mathcal{V}},{\mathcal{E}})$ is a finite directed graph whose nodes ${\mathcal{V}}$ are distinct elements of a finite set $Y \subset \mathbb{R}^d$.
It is convenient to define for each edge $e\in {\mathcal{E}}$ a *source vector* $\bm{s}(e)\in Y$, the label of the source node of $e$, the *target vector* $\bm{t}(e)\in Y$, the label of the target node, and the *reaction vector* $\bm{v}(e) = \bm{t}(e) - \bm{s}(e)$. We may regard $\bm{s}(e)$ as the source complex of some reaction while $\bm{t}(e)$ is the product complex of that same reaction.
Now we define a chemical reaction network to simply be an E-graph for which a set of simple conditions hold.
\[egraphCRN\] A *reaction network* is a Euclidean embedded graph, $({\mathcal{V}},{\mathcal{E}})$, whose nodes ${\mathcal{V}}$ are labeled with distinct elements of a finite set $Y \subset \mathbb{R}^d_{\geq 0}$, and for which the following conditions hold:
1. ${\mathcal{V}}\ne \emptyset$;
2. for each $y \in {\mathcal{V}}$ there exists $e\in {\mathcal{E}}$ for which $\bm t(e) = y$ or $\bm s(e) = y$;
3. $\bm{t}(e) \ne \bm{s}(e)$ for each $e \in {\mathcal{E}}$. That is, we never have ${\bm}v(e) = {\bm}0$.
\[SCR\] and \[egraphCRN\] of a chemical reaction network are equivalent in the following sense: if we regard the set ${\mathcal{S}}$ as the standard basis in ${\mathbb{R}}^d$, then the set of vertices ${\mathcal{V}}$ and edges ${\mathcal{E}}$ in \[egraphCRN\] can be chosen to be the set of complexes ${\mathcal{C}}$ and reactions ${\mathcal{R}}$ in \[SCR\]. It is most common to assume that $Y \subset \mathbb{Z}^d_{\ge 0}$. Further, it is convenient to enumerate the elements of ${\mathcal{E}}$, so that ${\mathcal{E}}= \{e_1,\dots, e_{|{\mathcal{E}}|}\}$.
Mass Action Systems {#mass_action_sec}
-------------------
In this paper, we will focus on dynamical systems that are generated by reaction networks according to *mass action kinetics* [@H-J1; @F3]. We will denote the vector whose $i$th component gives the concentration of the $i$th species at time $t$ by $\mathbf{x}(t) \in \mathbb{R}_{\geq 0}^d$. As is usual, we will often drop the $t$ in the notation and simply denote the concentration by $\mathbf{x}.$ Also, for two vectors ${\bm}{u},{\bm}{v} \in \mathbb{R}_{\ge 0}^d,$ we will denote $${\bm}{u}^{{\bm}{v}} = \prod_{i=1}^d u_i^{v_i}$$ where we take $0^0=1$. A system is said to have mass action kinetics if the rate associated to reaction $i$ is $$k_i \mathbf{x}^{\bm{s}(e_i)}$$ for some constant $k_i >0$, called the *rate constant* of the reaction. That is, the rate of each reaction is assumed to be proportional to the product of the concentrations of the constituent reactants, counted according to multiplicity. For example, a reaction of the form $X_1+ X_2 \to \cdots$ would have rate equal to $k \mathbf{x}_1 \mathbf{x}_2$ for some $k>0$, and a reaction of the form $X_1 + 2X_2 \to \cdots$ would have rate equal to $\tilde k \mathbf{x}_1 \mathbf{x}_2^2$, for some $\tilde k > 0$. Other common kinetic assumptions, especially in systems biology, are Michaelis-Menten kinetics [@M-M] and Hill kinetics [@Hi]. Since reaction $i$ pushes the system in the direction $\bm{v}(e_i)$, we have the following.
\[def:generates\] Given a reaction network ${\mathcal{G}}= ({\mathcal{V}},{\mathcal{E}})$ as in \[egraphCRN\] and, after enumerating ${\mathcal{E}}$, a choice of rate constants ${\mathcal{K}}= \{k_1,...,k_{|{\mathcal{E}}|}\}\subset {\mathbb{R}}_{>0}$, we say that ${\mathcal{G}}$ *generates* the dynamical system ${\mathcal{G}}({\mathcal{K}})$ $$\label{gener}
\frac{d\mathbf{x}}{dt} = \sum_{i = 1}^{|{\mathcal{E}}|} k_{i} \mathbf{x}^{\bm{s}(e_i)}\bm{v}(e_i).$$
We will use the notation $\bm{f}_{{\mathcal{G}}({\mathcal{K}})}(\mathbf{x})$ to refer to the right hand side of the dynamical system in \[gener\]. It is clear from \[gener\] that every mass action system has the properties that $\frac{d\mathbf{x}}{dt} \in S = \mathit{Span}\{\bm{v}(e)|e\in {\mathcal{E}}\}$. Consequently, solutions of \[gener\] are restricted to *stoichiometric compatibility classes* $(\mathbf{x}_0 + S) \cap \mathbb{R}_{\geq 0}^d$ [@V-H].
As we noted in the introduction, different Euclidean embedded graphs (combined with choices of rate constants) can generate the same polynomial dynamical system.
Network Classifications {#classificationssection}
-----------------------
A key feature of chemical reaction network theory is the attempt to relate dynamical properties of kinetic systems, and in particular mass action systems, to structural properties of the underlying reaction graphs. We therefore introduce the following foundational structural properties of chemical reaction networks.
\[classificationsEG\] Consider a reaction network ${\mathcal{G}}= ({\mathcal{V}},{\mathcal{E}})$, where ${\mathcal{E}}$ has been enumerated. The graph ${\mathcal{G}}$ is said to be:
1. **consistent** if there is some choice of $a_1,a_2,...,a_{|{\mathcal{E}}|} \in \mathbb{R}_{>0}$ such that $0 = \sum_{i=1}^{|{\mathcal{E}}|} a_i \bm{v}(e_i)$.
2. **weakly reversible** if each connected component of the graph is strongly connected, or, equivalently, each edge $e \in {\mathcal{E}}$ is contained in a cycle.
3. **endotactic** if, for every $\bm{w} \in \mathbb{R}^d$ and every $e_i \in \mathcal{E}$, $\bm{w} \cdot \bm{v}(e_i) < 0$ implies that there exists $e_j \in {\mathcal{E}}$ such that $\bm{w} \cdot (\bm{s}(e_j)-\bm{s}(e_i)) < 0$ and $\bm{w} \cdot \bm{v}(e_j) > 0$.
4. **strongly endotactic** if, for every $\bm{w} \in \mathbb{R}^d$ and every $e_i \in \mathcal{E}$, $\bm{w} \cdot \bm{v}(e_i) < 0$ implies that there exists $e_j\in{\mathcal{E}}$ such that $\bm{w} \cdot (\bm{s}(e_j)-\bm{s}(e_i)) < 0$ and $\bm{w} \cdot \bm{v}(e_j) > 0$ and furthermore $\bm{w} \cdot (\bm{s}(e_j)-\bm{s}(e_k)) \leq 0$ for all $e_k \in {\mathcal{E}}$.
Consistency is closely related to a mass action system’s capacity to admit positive steady states [@angeli2009tutorial]. Weak reversibility was introduced as a generalization of reversibility in Horn & Jackson [@H-J1]. Endotactic networks were introduced as a generalization to weak reversibility in Craciun et al. [@C-N-P], where it is shown that any weakly reversible network is endotactic.
Notice that consistency is a necessary but not sufficient condition on the network structure for the corresponding mass action system to admit positive steady states. For example, consider the network $\emptyset \longleftarrow X \longrightarrow 2X$. This network is consistent, which can be observed by selecting rate constants $a_1 = a_2 = 1$. However, if the rate constants are selected as $\emptyset \; \stackrel{1}{\longleftarrow} \; X \; \stackrel{2}{\longrightarrow} \; 2X$, then the generated dynamics are $\dot{x} = x$, which has solution $x(t) = x(0)e^t$, and there is no positive steady state. $\diamond$
Weak reversibility may also be understood using \[egraphCRN\] as the property that every edge in ${\mathcal{G}}$ is in a directed cycle. This is distinct, but similar, to the stronger requirement of *reversibility*, i.e. that for every edge $e$ there is some edge $e^*$ such that $\bm{s}(e) = \bm{t}(e^*)$ and $\bm{t}(e) = \bm{s}(e^*)$. For instance, the network $$\xymatrix{
X_2 \ar[r]\ar@<0.5ex>[dr] & X_1 + X_2 \ar[d]\\
&\ar@<0.5ex>[ul] X_1}$$ is not reversible, since there is no reaction $X_1 + X_2 \to X_2$, but is weakly reversible since there is a path from $X_1+X_2$ to $X_2$ through $X_1$. $\diamond$
\[sweep\] Intuitively, a Euclidean embedded graph is endotactic if no reaction “points outward." This can be tested using the so-called “parallel sweep test" (see Craciun et al. [@C-N-P]). In \[dyneqEgraphs\] (a), we can tell that the network is endotactic because any direction ${\bm}{w}$ which is not perpendicular to the two reactions has the property that if ${\bm}{w}\cdot {\bm}{v}(e_1) < 0$ then ${\bm}{w} \cdot ({\bm}{s}(e_2)-{\bm}{s}(e_1)) = {\bm}{w} \cdot {\bm}{v}(e_1) < 0$ and ${\bm}{w} \cdot {\bm}{v}(e_2) = {\bm}{w} \cdot (-{\bm}{v}(e_1)) >0$. In \[dyneqEgraphs\] (b), we can see from ${\bm}{w} = (-1,-1)$ that this is not endotactic. We have that ${\bm}{w} \cdot {\bm}{v}(e_2) <0$, but ${\bm}{w} \cdot ({\bm}{s}(e_1) - {\bm}{s}(e_2)) = {\bm}{w} \cdot {\bm}{v}(e_1) = 0$, and while ${\bm}{w} \cdot {\bm}{v}(e_3)>0$, ${\bm}{w}\cdot({\bm}{s}(e_2) -{\bm}{s}(e_3)) = 0$. $\diamond$
The properties above are intrinsically properties of E-graphs. However, we can define the same notions for dynamical systems using the relationship between E-graphs and dynamical systems established in \[def:generates\].
\[dyneq2\] We will call a dynamical system $\frac{d\mathbf{x}}{dt} = {\bm}{f}(\mathbf{x})$:
1. **consistent** if $\frac{d\mathbf{x}}{dt} ={\bm}{f}(\mathbf{x})$ can only be generated by a consistent Euclidean embedded graph;
2. **weakly reversible** if $\frac{d\mathbf{x}}{dt} = {\bm}{f}(\mathbf{x})$ can be generated by some weakly reversible Euclidean embedded graph ${\mathcal{G}}$;
3. **endotactic** if $ \frac{d\mathbf{x}}{dt} ={\bm}{f}(\mathbf{x})$ can be generated by some endotactic Euclidean embedded graph ${\mathcal{G}}$;
4. **strongly endotactic** if $\frac{d\mathbf{x}}{dt} ={\bm}{f}(\mathbf{x})$ can be generated by some strongly endotactic Euclidean embedded graph ${\mathcal{G}}$.
\[consistent\] In \[dyneq2\] part 1., we have that if a dynamical system is consistent, then every graph which generates the system must be consistent. This is in contrast to parts 2., 3. and 4. of this definition. This is because any polynomial dynamical system can be generated by some consistent network. To see this, we simply add a set of edges $e_1^*,...,e_p^*$ which share a source ${\bm}{s}^*$ to some graph ${\mathcal{G}}= ({\mathcal{V}},{\mathcal{E}})$ which generates the system, requiring that the cone generated by ${\bm}{v}(e_1^*),...,{\bm}{v}(e_p^*)$ is equal to the span of the original reaction vectors. This implies that $0\in \mathit{Cone}(\{{\bm}{v}(e_1^*) ,...,{\bm}{v}(e_p^*)\})$ and so the new graph also generates the polynomial. Also, for any choice of $a_1,...,a_{|{\mathcal{E}}|} > 0$, we have chosen the ${\bm}{v}(e^*_i)$ such that $-\sum_{i=1}^{|{\mathcal{E}}|}a_i {\bm}{v}(e_i)\in \mathit{Cone}(\{{\bm}{v}(e_1^*) ,...,{\bm}{v}(e_p^*)\})$, and furthermore is in the relative interior of that cone. Therefore, there is a choice of $b_1,...,b_p > 0$ such that $\sum_{i=1}^p b_i{\bm}{v}(e_i^*) = -\sum_{i=1}^{|{\mathcal{E}}|}a_i {\bm}{v}(e_i)$ and so the new graph is consistent. $\diamond$
Dynamical Equivalence {#desection}
---------------------
As already noted, it is well known that the dynamical representations of mass action systems (i.e., \[gener\]) are not uniquely determined by the network structure. For another example, consider the following networks, whose E-graphs are shown in \[dyneqEgraphs\]: $$\label{system1}
\xymatrix{
X_1 \ar@<0.5ex>[r]^{1} & \ar@<0.5ex>[l]^{1} X_2
}$$ and $$\label{system2}
\xymatrix@!0@R=6mm@C=10mm{
& &**[r] X_1 + X_2\\
X_2 \ar[r]^1 & X_1 \ar[ur]^1 \ar[dr]_1& \\
& &**[r] \emptyset
}$$ It can be easily seen that \[system1\] and \[system2\] are both governed by the mass action dynamics $\dot{x}_1 = -\dot{x}_2 = -x_1+x_2$. We therefore introduce the following definition [@C-P].
\[dyneqdef\] Consider two chemical reaction networks ${\mathcal{G}}= ({\mathcal{V}},{\mathcal{E}})$ and $\tilde{{\mathcal{G}}} = (\tilde{{\mathcal{V}}},\tilde{{\mathcal{E}}})$, combined with rate constants $\mathcal{K} = \{ k_i \; | \; i=1,\ldots, r\}$ and $\tilde{\mathcal{K}} = \{ \tilde{k}_i \; | \; i=1,\ldots, \tilde{r}\}$, respectively. We will say that the mass action systems ${\mathcal{G}}(\mathcal{K})$ and $\tilde{{\mathcal{G}}}(\tilde{\mathcal{K}})$ are **dynamically equivalent** if the generated functions $\bm{f}_{{\mathcal{G}}({\mathcal{K}})}$ and $\bm{f}_{\tilde{{\mathcal{G}}}(\tilde{{\mathcal{K}}})}$ coincide (i.e. $\bm{f}_{{\mathcal{G}}({\mathcal{K}})}(\mathbf{x}) = \bm{f}_{\tilde{{\mathcal{G}}}(\tilde{{\mathcal{K}}})}(\mathbf{x})$, for all $\mathbf{x}$).
We can see that the dynamical systems $\mathcal{G}(\mathcal{K})$ and $\tilde{\mathcal{G}}(\tilde{\mathcal{K}})$ associated with \[system1\] and \[system2\] are dynamically equivalent. We may furthermore observe that ${\mathcal{G}}$ and $\tilde{{\mathcal{G}}}$ fail to share the same structural properties: ${\mathcal{G}}$ is weakly reversible while $\tilde{{\mathcal{G}}}$ is not. As in \[sweep\], we can also easily verify that ${\mathcal{G}}$ is endotactic, while $\tilde{{\mathcal{G}}}$ is not (see \[dyneqEgraphs\]). Notice, however, that in the classifications given in \[dyneq2\], the polynomial dynamical system *is* said to be weakly reversible (and endotactic) because the network ${\mathcal{G}}= ({\mathcal{V}},{\mathcal{E}})$ is. This example shows that it is possible for a mass action system to behave as though the generating network has a particular desirable network property, even when the generating network does not itself have it. We might therefore say that the known generating E-graph behaves as though it has that property for some (or perhaps even all) choices of rate constants.
In order to formalize this notion, we inspect the case of E-graphs which may generate the same polynomial dynamical system.
\[overlap\] Let ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ be Euclidean embedded graphs. We say that ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ [**have the capacity for dynamical equivalence**]{}, and write ${\mathcal{G}}_1 \sqcap {\mathcal{G}}_2$, if there exists a system $\frac{d\mathbf{x}}{dt} = \bm{f}(\mathbf{x})$ that can be generated by both ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ (i.e. there exists ${\mathcal{K}}_1$ and ${\mathcal{K}}_2$ such that $\bm{f}_{{\mathcal{G}}_1({\mathcal{K}}_1)}(\mathbf{x}) = \bm{f}_{{\mathcal{G}}_2({\mathcal{K}}_2)}(\mathbf{x}) = \bm{f}(\mathbf{x})$, for all $\mathbf{x}$).
Let ${\mathcal{G}}$ be a Euclidean embedded graph. We say that ${\mathcal{G}}$ [**has the capacity for weak reversibility**]{} if there exists a weakly reversible Euclidean embedded graph $\tilde{{\mathcal{G}}}$ such that ${\mathcal{G}}\sqcap \tilde{{\mathcal{G}}}$. Likewise, we say that ${\mathcal{G}}$ [**has the capacity to be endotactic (strongly endotactic)**]{} if there exists some endotactic (strongly endotactic) Euclidean embedded graph $\tilde{{\mathcal{G}}}$ such that ${\mathcal{G}}\sqcap \tilde{{\mathcal{G}}}$.
The following theorem asserts that the above definitions are meaningful. In particular, it shows that there are networks which can generate weakly reversible (respectively, endotactic) networks which are not themselves weakly reversible (respectively, endotactic).
\[inclusions\] The following inclusions hold and are strict.
1. The set of networks with the capacity for weak reversibility contains the set of weakly reversible networks.
2. The set of networks with the capacity to be endotactic contains the set of endotactic networks.
That these inclusions hold follows directly from the definitions. We now simply need to demonstrate that the reverse inclusion does not hold.
Consider the network ${\mathcal{G}}$, below $$\label{gbiggerone}
\xymatrix{
X_2 \ar[r]^{k_3} \ar[d]_{k_4} & X_1 + X_2 \\
\emptyset & \ar[l]_{k_2} \ar[u]_{k_1} X_1
}.$$ The network $\mathcal{G}$ is neither weakly reversible nor endotactic. However, if (and only if) $k_1=k_2$ and $k_3=k_4$, then the generated mass action system may also be generated by the weakly reversible and endotactic network $\tilde{{\mathcal{G}}}$ with correctly chosen rate constants: $$\label{gsmallerone}
\xymatrix{
X_1 \ar@<0.5ex>[r]^{{k}_1} & \ar@<0.5ex>[l]^{{k}_3} X_2
}.$$ Therefore, ${\mathcal{G}}\sqcap \tilde{{\mathcal{G}}}$, completing the proof. $\square$
Notice that we may also define the notion of *having the capacity to be consistent* in the same way. However, by \[consistent\] we see that every E-graph has the capacity to be consistent.
We are interested in the stronger case in which every system generated by an E-graph can also be generated by another E-graph with a desired property. We therefore define the following.
Let ${\mathcal{G}}$ be a Euclidean embedded graph. We say that ${\mathcal{G}}$ is [**effectively weakly reversible**]{} if every dynamical system generated by ${\mathcal{G}}$ is weakly reversible. Likewise, we say that ${\mathcal{G}}$ is [**effectively endotactic (strongly endotactic)**]{} if every polynomial dynamical system generated by ${\mathcal{G}}$ is endotactic (strongly endotactic).
It is possible for a network to be effectively weakly reversible, but not weakly reversible as the example \[eq:9807070\], given in the introduction, demonstrates.
In order to show that a network is effectively weakly reversible or effectively endotactic, it is of course helpful to characterize when one network can generate any system that can be generated by some other system.
\[includedyn\] Let ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ be Euclidean embedded graphs. We say that ${\mathcal{G}}_1$ includes the dynamics of ${\mathcal{G}}_2$, and write ${\mathcal{G}}_2 \sqsubseteq {\mathcal{G}}_1$, if any system $\frac{d\mathbf{x}}{dt} = \bm{f}(\mathbf{x})$ generated by ${\mathcal{G}}_2$ can also be generated by ${\mathcal{G}}_1$ (i.e., for any ${\mathcal{K}}_2$ there is some ${\mathcal{K}}_1$ such that ${\bm}{f}_{{\mathcal{G}}_1({\mathcal{K}}_1)}(\mathbf{x}) = {\bm}{f}_{{\mathcal{G}}_2({\mathcal{K}}_2)}(\mathbf{x})$ for all $\mathbf{x}$).
We have already encountered several examples of \[includedyn\] in this manuscript. For example, the network shown in \[system2\] and \[dyneqEgraphs\] (b) contains the dynamics of the network shown in \[system1\] and \[dyneqEgraphs\] (a). In \[dyneqEgraphs\], it is noted that these networks generate the same dynamical system when every rate constant is taken to be $1$. Now, note that for any choice of rate constants ${k}_{1}$ for edge $e_1$ and ${k}_{2}$ for edge $e_2$ chosen to generate a dynamical system using the network shown in \[dyneqEgraphs\] (a), we can generate the same network using \[dyneqEgraphs\] (b) by choosing $\tilde{{k}}_1 = {k}_1$ for edge $e_1$, and $\tilde{{k}}_2 = \tilde{{k}}_3 = {k}_2$ for edges $e_2$ and $e_3$.
An obvious sufficient condition for a network ${\mathcal{G}}$ to be effectively weakly reversible is then that there exists some weakly reversible $\tilde{{\mathcal{G}}}$ such that ${\mathcal{G}}\sqsubseteq \tilde{{\mathcal{G}}}$.
Main Results
============
In this section, we prove the main correspondences of this paper. In the first subsection, we address \[q1\]. We then apply this result in the subsequent subsections in order to investigate \[q2\].
We will frequently require the following known results [@mangasarian1994nonlinear].
\[lemma01\] Let $\{ \bm{v}_i \}$, $i=1, \ldots, m$, denote a family of vectors in $\mathbb{R}^n$. Then, for any $\bm{b} \in \mathbb{R}^n$ exactly one of the following is true:
1. There exist constants $\lambda_i \geq 0$, $i=1, \ldots, m$, such that $\displaystyle{\bm{b} = \sum_{i=1}^m \lambda_i \bm{v}_i}$; or
2. There is a vector $\bm{w} \in \mathbb{R}^n$ such that $\bm{w} \cdot \bm{v}_i \geq 0$ for $i=1,\ldots, m$, and $\bm{w} \cdot \bm{b} < 0.$
\[stiemke\] Let $\{ \bm{v}_i \}$, $i=1, \ldots, m$, denote a family of vectors in $\mathbb{R}^n$. Then exactly one of the following is true:
1. There exist constants $\lambda_i \in \mathbb{R}_{>0}$, $i=1, \ldots, m$, so that $\displaystyle{\sum_{i=1}^m \lambda_i \bm{v}_i = \bm{0}}$; or
2. There exists a vector $\bm{w} \in \mathbb{R}^n$ so that $\displaystyle{\bm{w} \cdot \bm{v}_i \leq \bm{0}}$, $i=1,\ldots, m$, with the inequality strict for at least one $i_0 \in \{ 1, \ldots, m\}$.
A condition for dynamical equivalence
-------------------------------------
Here, we provide necessary and sufficient conditions under which one network contains the dynamics of another. Furthermore, as a corollary we provide necessary and sufficient conditions under which two networks have the capacity for dynamical equivalence. These conditions are geometric in nature and easily checkable.
The appearance of the source complexes $\bm{s}(e)$ as exponents in \[gener\] suggests that we need to consider this subset of the complexes. We will also need the notion of a *cone*. Given a finite set of vectors $S \subseteq {\mathbb{R}}^d$ we define the set $K = \mathit{Cone}(S)$, the cone *generated* by $S$, as the closed, convex set of all finite, nonnegative linear combinations of the elements of $S$ [@nonneg]. We denote the interior of a cone $K = \mathit{Cone}(S)$ relative to the span of $S$ (i.e. the relative interior of $K$) by $\mathit{RelInt}({K})$.
If ${\mathcal{G}}= ({\mathcal{V}},{\mathcal{E}})$ is a Euclidean embedded graph, let ${\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}} = \{\bm{s}(e)|e\in {\mathcal{E}}\}$ be the source complexes/vectors of ${\mathcal{G}}$, and for $\bm{s}\in {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$ let $V^{{\mathcal{G}}}(\bm{s}) = \mathit{Cone}(\{\bm{v}(e_i)|\bm{s}(e_i) = \bm{s}, e_i \in {\mathcal{E}}\})$ be the cone generated by those reaction vectors with source vector equal to $\bm{s}$. If a vector $\bm{s} \not \in {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$, we define $V^{{\mathcal{G}}}(\bm{s}) = \{\bm{0}\}$.
\[containcondit\] ${\mathcal{G}}_2 \sqsubseteq {\mathcal{G}}_1$ if and only if (i) ${\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}_2} \subset {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}_1}$, and (ii) for every $\bm{s} \in {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}_1}$ $$\mathit{RelInt}(V^{{\mathcal{G}}_2}(\bm{s})) \subseteq \mathit{RelInt}(V^{{\mathcal{G}}_1}(\bm{s})),$$ where we take $V^{{\mathcal{G}}}(\bm{s}) = \{\bm{0}\}$ if $\bm{s} \not \in {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$.
We begin by proving that (i) and (ii) imply that ${\mathcal{G}}_2 \sqsubseteq {\mathcal{G}}_1$. Let ${\mathcal{K}}^2 \in {\mathbb{R}}^{|{\mathcal{E}}_{{\mathcal{G}}_2}|}_{>0}$. We must show that there exists ${\mathcal{K}}^1 \in {\mathbb{R}}^{|{\mathcal{E}}_{{\mathcal{G}}_1}|}_{>0}$ such that $$\bm{f}_{{\mathcal{G}}_2({\mathcal{K}}^2)} = \bm{f}_{{\mathcal{G}}_1({\mathcal{K}}^1)}.$$ We use superscript $1$ or $2$ to differentiate edges, source vectors, and reaction vectors of ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$, respectively. Note that for any choice of ${\mathcal{K}}^1$ and ${\mathcal{K}}^2$ we have $$\bm{f}_{{\mathcal{G}}_2({\mathcal{K}}^2)}(\mathbf{x}) - \bm{f}_{{\mathcal{G}}_1({\mathcal{K}}^1)}(\mathbf{x}) = \sum_{e^2 \in {\mathcal{E}}_{{\mathcal{G}}_2}} k_{e^2} \mathbf{x}^{\bm{s}(e^2)}\bm{v}(e^2) - \sum_{e^1 \in {\mathcal{E}}_{{\mathcal{G}}_1}} k_{e^1} \mathbf{x}^{\bm{s}(e^1)}\bm{v}(e^1).$$ We wish to rewrite these sums in terms of the source complexes, which we enumerate via $${\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}_1} \cup {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}_2} = {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}_1} = \{s_1,s_2,\dots,s_{|{\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}_1}|}\}.$$ Then, for each ${\bm}{s}_i \in {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}_1}$ we let $m_i = |\{e^1 \in {\mathcal{E}}_{{\mathcal{G}}_1}| \bm{s}(e^1) = \bm{s}_i\}|$, and $n_i = |\{e^2 \in {\mathcal{E}}_{{\mathcal{G}}_2}| \bm{s}(e^2) = \bm{s}_i\}|$ be the number of edges out of complex ${\bm}{s}_i$ for networks ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$, respectively. Furthermore, let $\{k_{ij}^l\} = \{k_{e^l} | \bm{s}(e^l) = \bm{s}_i\}$, and $\{\bm{v}_{ij}^l\} = \{\bm{v}(e^l) | \bm{s}(e^l) = \bm{s}_i\}$, $l = 1,2$, where the sizes of the sets are $m_i$ and $n_i$ for $\ell = 1$ and $\ell = 2$, respectively. Then $$\bm{f}_{{\mathcal{G}}_2({\mathcal{K}}^2)}(\mathbf{x}) - \bm{f}_{{\mathcal{G}}_1({\mathcal{K}}^1)}(\mathbf{x}) = \sum_{\bm{s}_i \in {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}_2}}\mathbf{x}^{\bm{s}_i}\sum_{j=1}^{n_i}k_{ij}^2 \bm{v}_{ij}^2 - \sum_{\bm{s}_i \in {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}_1}}\mathbf{x}^{\bm{s}_i}\sum_{j=1}^{m_i}k_{ij}^1\bm{v}_{ij}^1.$$ Then, because ${\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}_2} \subseteq {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}_1}$, $$\bm{f}_{{\mathcal{G}}_2({\mathcal{K}}^2)}(\mathbf{x}) - \bm{f}_{{\mathcal{G}}_1({\mathcal{K}}^1)}(\mathbf{x}) = \sum_{\bm{s}_i \in {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}_2}}\mathbf{x}^{\bm{s}_i}\left(\sum_{j=1}^{n_i}k_{ij}^2 \bm{v}_{ij}^2 - \sum_{j=1}^{m_i}k_{ij}^1 \bm{v}_{ij}^1\right) - \sum_{\bm{s}_i \in {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}_1}\setminus {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}_2}}\mathbf{x}^{\bm{s}_i}\sum_{j=1}^{m_i}k^1_{ij}\bm{v}_{ij}^1.$$ We have that ${\bm}{0} \in \mathit{RelInt}(V^{{\mathcal{G}}_1}(\bm{s}_i))$ if $\bm{s}_i \in {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}_1}\setminus {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}_2}$, so we can choose ${\mathcal{K}}^1$ with $k^1_{ij} >0$ so that each term of the second sum is ${\bm}{0}$. Furthermore, for each $\bm{s}_i\in {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}_2}$, $$\sum_{j=1}^{n_i}k_{ij}^2 \bm{v}_{ij}^2 - \sum_{j=1}^{m_i}k_{ij}^1\bm{v}_{ij}^1 = \bm{w} - \sum_{j=1}^{m_i}k_{ij}^1 \bm{v}_{ij}^1,$$ where $\bm{w} \in \mathit{RelInt}(V^{{\mathcal{G}}_2}(\bm{s}_i))$. Because of condition (ii) we may also conclude that $\bm{w} \in \mathit{RelInt}(V^{{\mathcal{G}}_1}(\bm{s}_i))$. Hence, we can choose $k_{ij}^1>0$ so that $$\bm{w} - \sum_{j=1}^{m_i}k_{ij}^1\bm{v}_{ij}^1 = 0.$$ With these choice of parameters we have $\bm{f}_{{\mathcal{G}}_2({\mathcal{K}}^2)} = \bm{f}_{{\mathcal{G}}_1({\mathcal{K}}^1)}$.
Next, we show that ${\mathcal{G}}_2 \sqsubseteq {\mathcal{G}}_1$ imply (i) and (ii) hold. First, if there is some source $\bm{s}_i \in {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}_2}$ but $\bm{s}_i \not \in {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}_1}$, then for general choice ${\mathcal{K}}^2$ there will be a monomial $\mathbf{x}^{\bm{s}_i}$ in $\bm{f}_{{\mathcal{G}}_2({\mathcal{K}}^2)}$ which cannot appear in $\bm{f}_{{\mathcal{G}}_1({\mathcal{K}}^1)}$. Therefore, (i) must hold.
Next, in order to find a contradiction suppose, that for some $\bm{s}_i \in {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}_1}$ there is a vector ${\bm}{w} \in \mathit{RelInt}(V^{{\mathcal{G}}_2}(\bm{s}_i))$ but $\bm{w} \not \in \mathit{RelInt}(V^{{\mathcal{G}}_1}(\bm{s}_i))$, so that (ii) does not hold. Then, since ${\bm}{w} \in \mathit{RelInt}(V^{{\mathcal{G}}_2}(\bm{s}_i))$, there is a choice ${\mathcal{K}}^2$ so that $\bm{f}_{{\mathcal{G}}_2({\mathcal{K}}^2)}$ contains a term of the form $$\mathbf{x}^{\bm{s}_i}\sum_{j=1}^{n_i}k_{ij}^2\bm{v}_{ij}^2 = \mathbf{x}^{\bm{s}_i}\bm{w}.$$ Then, for any choice of ${\mathcal{K}}^1 \in {\mathbb{R}}^{|{\mathcal{E}}_{{\mathcal{G}}_1}|}_{>0}$ the function $\bm{f}_{{\mathcal{G}}_2({\mathcal{K}}^2)}(\mathbf{x}) - \bm{f}_{{\mathcal{G}}_1({\mathcal{K}}^1)}(\mathbf{x})$ will include the term $$\left(\bm{w} - \sum_{j=1}^{m_i}k_{ij}^1\bm{v}_{ij}^1\right)\mathbf{x}^{\bm{s}_i}.$$ Because $\bm{w} \not \in \mathit{RelInt}(V^{{\mathcal{G}}_1}(\bm{s}_i))$, this term is non-zero for any $\mathbf{x} \neq 0$. Moreover, no other term could cancel it since no other monomials of the form $\mathbf{x}^{\bm{s}_i}$ can appear. Thus, $\bm{f}_{{\mathcal{G}}_2({\mathcal{K}}^2)}(\mathbf{x}) \ne \bm{f}_{{\mathcal{G}}_1({\mathcal{K}}^1)}(\mathbf{x})$. Since this conclusion holds for *any* choice of ${\mathcal{K}}^1$, we conclude that we do not have ${\mathcal{G}}_2 \sqsubseteq {\mathcal{G}}_1$, which is a contradiction. $\square$
\[splitting\] For any edge $e$ of a network ${\mathcal{G}}$, we may generate a new network ${\mathcal{G}}_e$ by removing $e$ and adding edges $e_1,...,e_d$ such that ${\bm}{s}(e_1) =\cdots ={\bm}{s}(e_d) = {\bm}{s}(e)$ and ${\bm}{v}(e) \in \mathit{RelInt}(\mathit{Cone}(\{{\bm}{v}(e_1),...,{\bm}{v}(e_d)\}))$. Then, \[containcondit\] implies that ${\mathcal{G}}_e$ contains the dynamics of ${\mathcal{G}}$. We often refer to this as “splitting" the reaction vector ${\bm}{v}(e)$. $\diamond$
\[containcondit\] shows when the dynamical systems generated from one network are contained within those generated by another. It is reminiscent of Theorem 4.4 of [@C-P], which gives conditions for when the two sets of dynamical systems intersect (i.e., when they have the capacity for dynamical equivalence). Moreover, their proofs are similar. However, a missing case in the proof of Theorem 4.4 of [@C-P] was noted by Gábor Szederkényi in [@sz2009]. A complete statement and proof of Theorem 4.4 of [@C-P] appear below.
Under the mass-action kinetics assumption, two chemical reaction networks represented by the graphs ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ have the capacity for dynamical equivalence, i.e., ${\mathcal{G}}_1 \sqcap {\mathcal{G}}_2$, if and only if for every $\bm{s} \in {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}_1} \cup {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}_2}$ $$\mathit{RelInt}(V^{{\mathcal{G}}_2}(\bm{s})) \cap \mathit{RelInt}(V^{{\mathcal{G}}_1}(\bm{s})) \neq \emptyset$$ where we take $V^{{\mathcal{G}}}(\bm{s}) = \bm{0}$ if $\bm{s} \not \in {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$.
We begin by showing that the above conditions imply that ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ have the capacity for dynamical equivalence. We enumerate the set ${\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}_1} \cup {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}_2}$, and for each $\bm{s}_i$ in that set, we choose $$\bm{w}_i \in \mathit{RelInt}(V^{{\mathcal{G}}_2}(\bm{s})) \cap \mathit{RelInt}(V^{{\mathcal{G}}_1}(\bm{s})).$$ Let $$f(\mathbf{x}) = \sum_i \mathbf{x}^{\bm{s}_i} \bm{w}_i.$$ Then both ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ generate $f$ and we conclude that ${\mathcal{G}}_1 \sqcap {\mathcal{G}}_2$,.
Next suppose that ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ have the capacity for dynamical equivalence. Hence, by \[overlap\], there exists some polynomial $$\label{adkjfa;jf}
f(\mathbf{x}) = \sum_{i = 1}^n k_i \mathbf{x}^{\bm{s}_i}{\bm}{w}_i,$$ with $\bm{s}_i \ne \bm{s}_j$ for $i \ne j$, that can be generated by both ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$. Let $\mathcal{T} = \{{\bm}{s}_1,\dots, {\bm}{s}_n\}$ and note that $\mathcal{T} \subset {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}_1} \cup {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}_2}$. For any ${\bm}{s} \in ( {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}_1} \cup {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}_2}) \setminus \mathcal{T}$, implying no term in the polynomial corresponds with ${\bm}{s}$, we immediately have that $${\bm}{0} \in \mathit{RelInt}(V^{{\mathcal{G}}_2}(\bm{s})) \cap \mathit{RelInt}(V^{{\mathcal{G}}_1}(\bm{s})),$$ as desired. Furthermore, for each ${\bm}{s}_i \in \mathcal{T}$, we have $$\emptyset \ne \{\alpha {\bm}{w}_i : \alpha > 0\} \subseteq \mathit{RelInt}(V^{{\mathcal{G}}_2}(\bm{s_i})) \cap \mathit{RelInt}(V^{{\mathcal{G}}_1}(\bm{s_i}))$$ where containment follows by the fact that each of ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ generated $f$ in \[adkjfa;jf\]. Thus, we have shown the condition holds for all ${\bm}{s} \in {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}_1} \cup {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}_2}$ and the proof is complete. $\square$
Endotactic Networks
-------------------
We turn to our study of endotactic and strongly endotactic networks.
\[effectively\_endo\_endo\] Let ${\mathcal{G}}= ({\mathcal{V}},{\mathcal{E}})$ and $\tilde{{\mathcal{G}}}= (\tilde{{\mathcal{V}}},\tilde{{\mathcal{E}}})$ be reaction networks such that ${\mathcal{G}}\sqsubseteq \tilde{{\mathcal{G}}}$. If $\tilde{{\mathcal{G}}}$ is endotactic, then ${\mathcal{G}}$ is endotactic. Moreover, if $\tilde{{\mathcal{G}}}$ is strongly endotactic, then ${\mathcal{G}}$ is strongly endotactic.
Let $e_i \in {\mathcal{E}}$ and ${\bm}{w}$ be such that ${\bm}{w}\cdot \bm{v}(e_i) < 0$. We must show that there exists $e_j \in {\mathcal{E}}$ such that $\bm{w} \cdot (\bm{s}(e_j)-\bm{s}(e_i)) < 0$ and $\bm{w} \cdot \bm{v}(e_j) > 0$.
Because ${\mathcal{G}}\sqsubseteq \tilde{{\mathcal{G}}}$, we know from \[containcondit\] that $\bm{s}(e_i) \in {\mathcal{S}}{\mathcal{C}}_{\tilde{{\mathcal{G}}}}$, and $\bm{v}(e_i) \in V^{\tilde{{\mathcal{G}}}}(\bm{s}(e_i))$. There then exists $\tilde{e}_i\in \tilde{{\mathcal{E}}}$ such that $\bm{s}(\tilde{e}_i) = \bm{s}(e_i)$ and $\bm{v}(\tilde{e}_i) \cdot {\bm}{w}<0$, since ${\bm}{w}\cdot \bm{v}(e_i) < 0$ $$\bm{v}(e_i) \in V^{\tilde{{\mathcal{G}}}}(\bm{s}(e_i))\quad \Rightarrow \quad {\bm}{v}(e_i) = \sum_{\tilde{e}_i|\bm{s}(\tilde{e}_i) = \bm{s}(e_i)} \lambda_i {\bm}{v}(\tilde{e}_i)\quad \lambda_i \geq 0$$ Because $\tilde{{\mathcal{G}}}$ is endotactic, there is some $\tilde{e}_j$ such that ${\bm}{w}\cdot (\bm{s}(\tilde{e}_j)-\bm{s}(\tilde{e}_i))<0$ and $\bm{v}(\tilde{e}_j)\cdot {\bm}{w} > 0$. Moreover, we can choose $\tilde e_j$ so that ${\bm}w\cdot ({\bm}s(\tilde e_j) - {\bm}s(\tilde e_i))$ is minimal over edges satisfying $\bm{v}(\tilde{e}_j)\cdot {\bm}{w} > 0$.
We can complete the proof by proving two statements.
(1) ${\bm}s(\tilde e_j) \in {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$ (the source complexes for ${\mathcal{G}}$).
(2) If ${\bm}u \in \mathit{RelInt}(V^{{{\mathcal{G}}}}(\bm{s}(\tilde{e}_j))$, then ${\bm}u \cdot {\bm}w > 0$.
Combining the above gives the existence of the necessary edge in ${\mathcal{E}}$.
We prove (1) by showing that ${\bm}{0} \not \in \mathit{RelInt}(V^{\tilde{G}}({\bm}{s}(\tilde{e}_j))$. Then, combined with \[containcondit\], ${\bm}s(\tilde e_j) \in {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$. Suppose that ${\bm}{0} \in \mathit{RelInt}(V^{\tilde{G}}({\bm}{s}(\tilde{e}_j))$. Then, by \[stiemke\], there is some $\tilde{e}_k \in \tilde{{\mathcal{E}}}$ such that ${\bm}{s}(\tilde{e}_k) = {\bm}{s}(\tilde{e}_j)$, and ${\bm}{v}(\tilde{e}_k) \cdot {\bm}{w} <0$. However, the minimality of ${\bm}w\cdot ({\bm}s(\tilde e_j) - {\bm}s(\tilde e_i))$ over edges satisfying $\bm{v}(\tilde{e}_j)\cdot {\bm}{w} > 0$ then implies that there is no edge $\tilde{e}_l$ with ${\bm}{v}(\tilde{e}_l)\cdot {\bm}{w} > 0$ and ${\bm}{w}\cdot({\bm}{s}(\tilde{e}_l){\bm}{w}(\tilde{e}_k)) < 0$. This contradicts the condition that $\tilde{{\mathcal{G}}}$ is endotactic.
To prove (2), let ${\bm}u \in \mathit{RelInt}(V^{\tilde{{\mathcal{G}}}}(\bm{s}(\tilde{e}_j))$. Then, there are $\lambda_k>0$ and $\tilde e_k\in \tilde {\mathcal{G}}$, each with source ${\bm}s(\tilde e_j)$, for which $${\bm}u = \sum_{k} \lambda_k {\bm}v(\tilde e_k) = \lambda_j {\bm}v(\tilde e_j) + \sum_{i\ne k} \lambda_k {\bm}v(\tilde e_k).$$ Dotting with ${\bm}w$ yields $${\bm}u \cdot {\bm}w = \lambda_j {\bm}v(\tilde e_j) \cdot {\bm}w + \sum_{k\ne j} \lambda_{k} {\bm}v(\tilde e_k) \cdot {\bm}w.$$ If ${\bm}u \cdot {\bm}w \le 0$, then, because ${\bm}v (\tilde e_j ) \cdot {\bm}w > 0$, we would be forced to conclude ${\bm}v(\tilde e_i) \cdot {\bm}w < 0$ for some $i \ne j$. Since $\tilde {\mathcal{G}}$ is endotactic, this would contradict the minimality of ${\bm}w\cdot ({\bm}s(\tilde e_j) - {\bm}s(\tilde e_i))$. Hence, we can conclude that ${\bm}u \cdot {\bm}w >0$. \[containcondit\] implies that there is such a ${\bm}{u} \in \mathit{RelInt}(V^{{\mathcal{G}}})({\bm}{s}(\tilde{e}_j))$.
Turning to the case of $\tilde {\mathcal{G}}$ being strongly endotactic. The proof that ${\mathcal{G}}$ is strongly endotactic is identical except the line “... is minimal over edges satisfying $\bm{v}(\tilde{e}_j)\cdot {\bm}{w} > 0$” is changed to “... is minimal over all edges of $\tilde {\mathcal{G}}$” and by noting that the source complexes of ${\mathcal{G}}$ are a subset of the source complexes of $\tilde {\mathcal{G}}$.$\square$
Endotactic and Source-Only Networks
-----------------------------------
We introduce a new concept, that of “source-only” networks, which we will demonstrate is a useful framework.
A chemical reaction network ${\mathcal{G}}= ({\mathcal{V}},{\mathcal{E}})$ is said to be **source-only** if, for any $e\in {\mathcal{E}}$, $\bm{t}(e) = {\bm}{y}$ implies that $\bm{s}(e^*) = {\bm}{y}$ for some $e^* \in E$. A mass action system $\frac{d\mathbf{x}}{dt} = {\bm}{f}(\mathbf{x})$ is said to be **source-only** if it can be generated by a source-only E-graph. A chemical reaction network ${\mathcal{G}}$ is said to be **effectively source-only** if every system generated by ${\mathcal{G}}$ is source-only.
That is, a network is source-only if ${\mathcal{V}}$ does not contain any nodes that are *only* product nodes.
\[example1\] Consider the chemical reaction network $$\label{ex1}
\xymatrix@R=1mm@C=3mm{
3X_1 \ar[rr]^{k_1} & & 3X_2 \ar[ddl]^{k_2} & & & & \\
& & & & X_1 + X_2 \ar[rr]^{k_4}&& 2X_1 + 2X_2 .\\
& \emptyset \ar[uul]_{k_3} & & & & &
}$$ The network can be defined by the E-graph in \[figure1\](a). We can see that the network is strongly endotactic. We next seek to represent the network as a source-only network. To do so we must dispose of the product complex $2X_1 + 2X_2$. We see that the fourth reaction can be split (in the sense of \[splitting\]) to give the following dynamically equivalent reaction network $$\label{ex12}
\xymatrix@R=3mm@C=3mm{
& \ar[dl]_{k_4} X_1 + X_2 \ar[dr]^{k_4} & \\
3X_1 \ar[rr]^{k_1} & & 3X_2 \ar[dl]^{k_2}\\
& \ar[ul]^{k_3} \emptyset &
}$$ This network can be defined by the E-graph in \[figure1\](b). Thus we see that the network is effectively source-only. $\triangle$
We now prove the following.
\[sourceonly\] Let ${\mathcal{G}}= ({\mathcal{V}},{\mathcal{E}})$ be an endotactic network. Then, there is exists a source-only network $\tilde{{\mathcal{G}}}= (\tilde{{\mathcal{V}}},\tilde{{\mathcal{E}}})$ such that ${\mathcal{G}}\sqsubseteq \tilde{{\mathcal{G}}}$, and the nodes of $\tilde{{\mathcal{G}}}$ are the source nodes of ${\mathcal{G}}$. Therefore, every endotactic network is effectively source only.
We assume ${\mathcal{G}}= ({\mathcal{V}},{\mathcal{E}})$ is endotactic and will construct a source-only network $\tilde{{\mathcal{G}}}=(\tilde{{\mathcal{V}}},\tilde{{\mathcal{E}}})$ such that ${\mathcal{G}}\sqsubseteq \tilde{{\mathcal{G}}}$.
We take the nodes of our new network to be the source complexes of the original network. That is, $\tilde {\mathcal{V}}= {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$. Next, we define $\tilde {\mathcal{E}}$ in the following way. First, we index the edges of ${\mathcal{E}}$ as $e_1,...,e_k$. For each edge $e_i \in {\mathcal{E}}$, if ${\bm}{t}(e_i) \in {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$, we include $e_i\in \tilde{{\mathcal{E}}}$. Otherwise, let $C_i^*$ be the set of edges of the complete graph on ${\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$ with source ${\bm}{s}(e_i)$, so that $\{{\bm}{v}(e^*)|e^*\in C_i^*\} = \{{\bm}{s}^*-{\bm}{s}(e_i)|{\bm}{s}^* \in ({\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}} \setminus \{{\bm}{s}(e_i)\})\}$.
We next show that ${\bm}{v}(e_i) \in \mathit{Cone}(\{{\bm}{v}(e^*)|e^*\in C_i^*\}) = \mathit{Cone}(\{{\bm}{s}^*-{\bm}{s}(e_i)|{\bm}{s}^* \in ({\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}} \setminus \{{\bm}{s}(e_i)\})\})$. If this was not the case, then by \[lemma01\] there exists some ${\bm}{w}\in {\mathbb{R}}^d$ such that ${\bm}{w}\cdot {\bm}{v}(e_i) < 0$ and ${\bm}{w}\cdot ({\bm}{s}^* - {\bm}{s}(e_i)) \geq 0$ for all ${\bm}{s}^*\in {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$. However, if there exists any ${\bm}{w}\in {\mathbb{R}}^d$ such that ${\bm}{w}\cdot {\bm}{v}(e_i) < 0$, the condition that ${\mathcal{G}}$ is endotactic implies that for some source ${\bm}{s}^* \in {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$, and therefore some $e^*\in C_i^*$ with ${\bm}{v}(e^*) = {\bm}{s}^*-{\bm}{s}(e_i)$, ${\bm}{w}\cdot {\bm}{v}(e^*) < 0$, because ${\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}} = \{{\bm}{s}(e^*)|e^*\in C_i^*\} \cup \{{\bm}{s}(e_i)\}$.
Let $C_i\subset C_i^*$ be the set so that $\mathit{Cone}(\{{\bm}{v}(e)|e\in C_i\})$ is minimal in over subsets of $C_i^*$ in the sense of inclusion while still satisfying ${\bm}{v}(e_i) \in \mathit{Cone}(\{{\bm}{v}(e)|e\in C_i\})$. Therefore, ${\bm}{v}(e_i) \in \mathit{RelInt}(\mathit{Cone}(\{{\bm}{v}(e)|e\in C_i\}))$. Then, we add the edges $\tilde{e}\in C_i$ to $\tilde{{\mathcal{E}}}$. By this construction, we have for each ${\bm}{s} \in {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$, $$\mathit{RelInt}(V^{{\mathcal{G}}}(\bm{s})) \subseteq \mathit{RelInt}(V^{\tilde {\mathcal{G}}}(\bm{s}))$$ and we can conclude that ${\mathcal{G}}\sqsubseteq \tilde{{\mathcal{G}}}$, where $\tilde{{\mathcal{G}}}$ is clearly source only. $\square$
Notice that while \[sourceonly\] guarantees that any endotactic dynamical system can be generated by source-only network, this source only network is not necessarily endotactic. The following example shows that we cannot guarantee that any endotactic dynamical system can be generated by a source-only endotactic network.
In \[figure:endocounter\] (a), we give an example of a strongly endotactic network ${\mathcal{G}}$ such that if $\tilde{{\mathcal{G}}}$ is source only and ${\mathcal{G}}\sqsubseteq \tilde{{\mathcal{G}}}$, then $\tilde{{\mathcal{G}}}$ is not endotactic. Splitting the edge labeled $e$ in the sense of \[splitting\] requires adding a new edge $e^1$ such that ${\bm}{v}(e^1) \cdot (1,0) <0$ and an edge $e^2$ such that ${\bm}{v}(e^2) \cdot (0,-1) <0$. Then, $e^1$ can be used to show that the resulting network is not endotactic. Therefore, we cannot split any edges to maintain the endotactic property. It follows that to make the network source-only and endotactic, we must add a source node.
Consider again the node labeled $e$ in \[figure:endocounter\] (a). To make the network source-only without splitting any edges, we must add source node at some point ${\bm}{s}^* = {\bm}{s}(e) + \alpha {\bm}{v}(e)$ for $\alpha >0$. To maintain the endotactic property and insure that the new network $\tilde{{\mathcal{G}}}$ contains the dynamics of ${\mathcal{G}}$, we must add edges with source ${\bm}{s}^*$ such that ${\bm}{0} \in V^{\tilde{G}}({\bm}{s}^*)$. However, this requires either adding a new node which is only a target or adding a vector $e^1$ such that ${\bm}{v}(e^1) \cdot (1,0) <0$ which can be used to show the resulting network is not endotactic. To make this target into a source, we have the same requirements. We conclude that there is no way to construct endotactic and source-only $\tilde{{\mathcal{G}}}$ such that ${\mathcal{G}}\sqsubseteq \tilde{{\mathcal{G}}}$.
$\triangle$
Notice that the set of monomials $\{\mathbf{x}^{\bm{s}_i}\}$ of a generated polynomial $\bm{f}_{{\mathcal{G}}({\mathcal{K}})}$ corresponds to a subset of the source complexes ${\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$, and including (but not limited to) the set of sources $s\in {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$ that have ${\bm}{0} \not\in \mathit{RelInt}(V^{{\mathcal{G}}}(s))$. To limit networks that must be considered given a polynomial, we wish to exclude sources from a network which do not appear as monomials. For the case of weakly reversible systems, the following theorem allows us to do this (see also Theorem 4.8 in [@craciun2018]).
\[noaddedwkrev\] If a polynomial dynamical system $\frac{d\mathbf{x}}{dt} = f(\mathbf{x})$ is weakly reversible, then it is generated by a weakly reversible network that has as its sources the exponent vectors of the monomials of $f$.
Let ${\mathcal{G}}= ({\mathcal{V}},{\mathcal{E}})$ be a weakly reversible network which generates $f(\mathbf{x})$. If there is a node ${\bm}{s}^* = \bm{s}(e)$, $e \in {\mathcal{E}}$ such that the monomial $\mathbf{x}^{{\bm}{s}^*}$ does not appear in $f$, we first introduce a term $0\mathbf{x}^{{\bm}{s}^*}$ in $f$. It is now convenient to order such nodes ${\bm}{s}^*_1,{\bm}{s}^*_2,...,{\bm}{s}^*_k$. Consider first ${\bm}{s}^*_1$. We index the edges $e\in {\mathcal{E}}$ such that ${\bm}{s}(e) = {\bm}{s}^*_1$ as $e^1_1,...,e^1_p$. Because the coefficient of the monomial $\mathbf{x}^{{\bm}{s}^*_1}$ in $f$ is $0$, the vectors ${\bm}{v}(e)$ and rate constants $k_e$ must satisfy $ \sum_{i=1}^p k_{e_i^1} \bm{v}(e_i^1) = {\bm}{0}$. For any edge $e^{\dagger}$ which has $\bm{t}(e^{\dagger}) = {\bm}{s}^*_1$, we have in $f$ a term of the form $k^{\dagger} \mathbf{x}^{\bm{s}(e^{\dagger})}\bm{v}(e^{\dagger})$. We let $$\tilde{k}_{e_i^1}= \frac{k_{e_i^1}}{ \sum_{j=1}^p k_{e_j^1}} k^{\dagger}$$ so that $k^{\dagger} = \sum_{i=1}^p \tilde{k}_{e_i^1}$, $$\sum_{i=1}^p \tilde{k}_{e_i^1} \bm{v}(e_i^1) = \frac{k^{\dagger}}{\sum_{i=1}^p k_{e_i^1}}\sum_{i=1}^p k_{e_i^1}{\bm}{v}(e_i^1) = 0$$ and finally $$k^{\dagger}\bm{v}(e^{\dagger}) = \sum_{i=1}^p \tilde{k}_{e_i^1} \bm{v}(e^{\dagger}).$$ Then, we can write $$k^{\dagger}\bm{v}(e^{\dagger}) = \sum_{i=1}^p \tilde{k}_{e_i^1} \bm{v}(e^{\dagger})+ \sum_{i=1}^p \tilde{k}_{e_i^1} \bm{v}(e_i^1)= \sum_{i=1}^p\tilde{k}_{e_i^1} (\bm{v}(e^{\dagger}) + \bm{v}(e_i^1)).$$ Therefore, we can replace the term $k^{\dagger} \mathbf{x}^{\bm{s}(e^{\dagger})}\bm{v}(e^{\dagger})$ in $f$ with $$\sum_{i=1}^p\tilde{k}_{e_i^1} (\bm{v}(e^{\dagger}) + \bm{v}(e_i^1))\mathbf{x}^{\bm{s}(e^{\dagger})}.$$ This implies that $f$ is generated by a network which is built from ${\mathcal{G}}$ in the following way:
1. All of the edges $e$ which have $\bm{s}(e) = {\bm}{s}^*$ or $\bm{t}(e) = {\bm}{s}^*$ are removed. Let $E_1$ be the edges with $\bm{s}(e) = {\bm}{s}^*$ and let $E_2$ be the edges with $\bm{t}(e) = {\bm}{s}^*$.
2. Edges are added from each source node of an edge in $E_2$ to each target node of an edge in $E_1$.
The resulting network is weakly reversible because ${\mathcal{G}}$ was weakly reversible, and the only paths removed consisted of an edge in $E_2$ followed by an edge in $E_1$. These were then replaced with a single edge (or no edge if the path went from a node to itself). We now have a network which is weakly reversible and generates $f(\mathbf{x})$, but does not include ${\bm}{s}_1^*$ as a source. We then simply repeat the argument for ${\bm}{s}_2^*,...,{\bm}{s}_k^*$ to eliminate all nodes that do not appear as non-zero terms in $f(\mathbf{x})$.$\square$
The class of source-only networks is useful because they provide an upper bound on the networks we must consider when we attempt to represent polynomial dynamical systems using various kinds of networks. Furthermore, \[noaddedwkrev\] shows that in some cases one need only consider networks without adding new nodes. Therefore, source-only representations of networks provide *finite* descriptions of reversible, weakly reversible, and endotactic networks, which is useful in computations. For example, in [@craciun2018], it is shown that to find a complex balanced realization of a polynomial system, one need only consider the complexes that appear as exponent vectors. Furthermore, knowing that one can write a network as a source-only network is important in dynamical equivalence and network translation-based computational methods. In these settings, it is often required to know the number and/or stoichiometry of required complexes [@J1; @johnston2013computing].
Endotactic and Consistent Networks
----------------------------------
We continue with results related to endotactic networks.
\[consis\] Every endotactic network is consistent.
Suppose, in order to find a contradiction, that there is a network $({\mathcal{V}},{\mathcal{E}})$ that is endotactic but not consistent. Since the network is not consistent, there does not exist a set of constants $\lambda_e > 0$ for which $$\sum_{e\in {\mathcal{E}}} \lambda_e \bm{v}(e) = \mathbf{0}.$$ It follows that condition 1. of \[stiemke\] is not satisfied, so that condition 2. must be satisfied. That is, there is a ${\bm}{w} \in \mathbb{R}^d$ such that ${\bm}{w} \cdot \bm{v}(e) \leq 0$, with at least one inequality strict. It follows immediately from the definition of endotactic in \[classificationsEG\] that the network is not endotactic, which is a contradiction. It follows that every endotactic network is consistent.$\square$
Endotactic and Weakly Reversible Networks
-----------------------------------------
We have seen that every endotactic network may be represented in a dynamically equivalent form as a source-only network. Since weakly reversible networks are source-only by definition, it is tempting to suppose that every endotactic network is effectively weakly reversible. However, this is not the case, as we will show. In this section, we introduce the concept of *extremal reactions* which helps to bridge the gap between endotactic and weakly reversible networks.
Considering \[example1\] again, we see that this network is endotactic and effectively source-only, but that it is not weakly reversible. In order to make it weakly reversible, we must be able to reconfigure the other reactions so that $X_1 + X_2$ is also the product of some reaction. We observe, however, that we cannot “split” any of the other three reactions as they lie on the outer hull of the source complexes. For instance, to split $3X_1 \to 3X_2$ to connect to $X_1 + X_2$, we must necessarily introduce a balancing reaction which points away from the convex hull of the source complexes, and therefore introduces a strictly product complex. This network, therefore, is not effectively weakly reversible.
We can quickly identify that the reason there is no weakly reversible dynamically equivalent network is that there is a complex in the interior of the convex hull of the complexes which cannot be reached. In this example, however, we might observe that the restriction of the network to just the boundary complexes $3X_1$, $3X_2$, and $\emptyset$ *is* weakly reversible. This is perhaps not surprising; after all, we observed that the reactions which could not be “split” were exactly those which were on the boundary of the convex hull. We therefore introduce the following.
Consider a chemical reaction network ${\mathcal{G}}= ({\mathcal{V}},{\mathcal{E}})$. We define the *extreme source complexes* ${\mathcal{E}}{\mathcal{C}}_{{\mathcal{G}}} \subseteq {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}} $ to be the set of source nodes $\bm{s}(e)$ which are on the border of the convex hull of ${\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$. The **extremal reaction set** ${\mathcal{E}}{\mathcal{E}}_{{\mathcal{G}}}$ is defined to be the set $\{e\in {\mathcal{E}}| \bm{s}(e) \in {\mathcal{E}}{\mathcal{C}}_{{\mathcal{G}}} \}$, and we define ${\mathcal{E}}{\mathcal{V}}_{{\mathcal{G}}}$ to be the subset of ${\mathcal{V}}$ that are the sources and/or targets of the edges in ${\mathcal{E}}{\mathcal{E}}_{{\mathcal{G}}}$. Then the network ${\mathcal{G}}$ is said to be **extremally weakly reversible** if the reduced network $({\mathcal{E}}{\mathcal{V}}_{{\mathcal{G}}},{\mathcal{E}}{\mathcal{E}}_{{\mathcal{G}}} )$ is weakly reversible.
A mass action system is said to be **extremally weakly reversible** if it is generated by an extremally weakly reversible E-graph. A chemical reaction network ${\mathcal{G}}$ is said to be **effectively extremally weakly reversible** if any system generated by ${\mathcal{G}}$ is extremally weakly reversible.
It is clear that for \[example1\] we have that $\mathcal{E}{\mathcal{C}}_{{\mathcal{G}}} = {\mathcal{E}}{\mathcal{V}}_{{\mathcal{G}}} = \{ 3X_1, 3X_2, \emptyset \}$ and that $(\mathcal{E}{\mathcal{V}}_{{\mathcal{G}}} ,{\mathcal{E}}{\mathcal{E}}_{{\mathcal{G}}})$ is weakly reversible. Hence, the original network is extremally weakly reversible.
We wish to determine how robust this property is among endotactic networks. Consider the following example.
\[example2\] Consider the network $$\label{ex3}
\xymatrix@R=5mm@C=7mm{
X_2 \ar[r]^{k_1} & X_1 + 3X_2 \ar@<0.5ex>[r]^{k_2} & \ar@<0.5ex>[l]^{k_3} 2X_1 + 3X_2\\
3X_1 + 2X_2 \ar[r]^{k_4} &2X_1 \ar@<0.5ex>[r]^{k_5} & \ar@<0.5ex>[l]^{k_6} X_1
}$$ The network can also be represented by the E-graph in \[figure2\]. It can be visually checked that the network is endotactic. Furthermore, we can see that every complex is an extremal complex, so that $\mathcal{E} {\mathcal{C}}_{{\mathcal{G}}}= {\mathcal{S}}\mathcal{C}_{{\mathcal{G}}}$, and ${\mathcal{E}}{\mathcal{E}}_{{\mathcal{G}}} = {\mathcal{E}}$. Nevertheless, no reaction may be “split” while preserving the property that the network is source-only. It follows that the network is neither effectively weakly reversible nor effectively extremally weakly reversible. $\triangle$
While the network in \[example2\] is endotactic, it is not strongly endotactic. We now consider whether the property of being effectively extremally weakly reversible holds for strongly endotactic networks. Consider the following example.
\[example3\] Consider the network given in the three-dimensional complex space by \[figure3\]. It can be verified by visual inspection that the network is strongly endotactic. We also have that ${\mathcal{E}}{\mathcal{C}}_{{\mathcal{G}}} = {\mathcal{S}}{\mathcal{E}}_{{\mathcal{G}}}$; however we again may not “split” any reaction from these complexes while maintaining the property of being source-only. It follows that the network is not effectively weakly reversible. $\triangle$
This example is three-dimensional. The following result considers strongly endotactic networks which have a two-dimensional stoichiometric subspace.
\[wkrev\] Let ${\mathcal{G}}= ({\mathcal{V}},{\mathcal{E}})$ be a strongly endotactic two-dimensional network with two-dimensional stoichiometric subspace and assume that the source complexes only reside on the boundary of the convex hull generated from the source complexes. Then there exists a weakly reversible Euclidean embedded graph $\tilde{{\mathcal{G}}}$ such that ${\mathcal{G}}\sqsubseteq \tilde{{\mathcal{G}}}$. Therefore, every two dimensional strongly endotactic network is effectively extremally weakly reversible.
Let $\mathcal{G} = ({\mathcal{V}},{\mathcal{E}})$ be a strongly endotactic network with sources only on the convex hull of source complexes. We will build $\tilde{{\mathcal{G}}}$ in three stages, constructing “intermediate" networks ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$, and finally $\tilde{{\mathcal{G}}}$.
Consider the Euclidean embedded graph ${\mathcal{G}}_1=({\mathcal{V}}_1,{\mathcal{E}}_1)$ with ${\mathcal{V}}_1= {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$, and as its edges ${\mathcal{E}}_1$ all possible edges which lie along the sides of the convex hull of ${\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$. ${\mathcal{G}}_1$ is clearly weakly reversible, and while ${\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}_1} = {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$, ${\mathcal{G}}_1$ does not necessarily contain the dynamics of ${\mathcal{G}}$.
We now prove, however, that for each ${\bm}{s}\in {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$, we have that either $V^{{\mathcal{G}}}(\bm{s}_i) \subseteq V^{{\mathcal{G}}_1}(\bm{s}_i)$ or $V^{{\mathcal{G}}_1}(\bm{s}_i)$ is the border of a half space which contains $V^{{\mathcal{G}}}(\bm{s}_i)$. Let ${\mathcal{W}}_i$ be the set such that for ${\bm}{w} \in {\mathcal{W}}_i$, ${\bm}{w} \cdot ({\bm}{s}_i-{\bm}{s}^*) \leq 0$ for any ${\bm}{s}^* \in {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$. This set contains some non-zero vector because ${\bm}{s}_i$ is in the convex hull of ${\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$. Let ${\bm}{w}\in {\mathcal{W}}$ and $e\in {\mathcal{E}}$ be such that ${\bm}{s}(e) = {\bm}{s}_i$. Because ${\mathcal{G}}$ is endotactic and ${\bm}{w} \cdot ({\bm}{s}_i-{\bm}{s}^*) \leq 0$, it must be that ${\bm}{v}(e) \cdot {\bm}{w} \geq 0$. If $V^{{\mathcal{G}}_1}(\bm{s}_i)$ is pointed, then ${\mathcal{W}}_i$ is the dual cone to $V^{{\mathcal{G}}_1}(\bm{s}_i)$ and by \[lemma01\] we can conclude that $V^{{\mathcal{G}}}(\bm{s}_i) \subseteq V^{{\mathcal{G}}_1}(\bm{s}_i)$. If $V^{{\mathcal{G}}_1}(\bm{s}_i)$ is a line (note that by construction $V^{{\mathcal{G}}_1}(\bm{s}_i)$ cannot be a ray), then ${\mathcal{W}}_i$ is perpendicular to $V^{{\mathcal{G}}_1}(\bm{s}_i)$. Then, we can conclude that $V^{{\mathcal{G}}_1}(\bm{s}_i)$ is the border of a half space which contains $V^{{\mathcal{G}}}(\bm{s}_i)$, and that this half space also contains ${\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$, because ${\bm}{w} \cdot ({\bm}{s}-{\bm}{s}^*) \leq 0$ for any ${\bm}{s}^* \in {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$.
We next construct ${\mathcal{G}}_2 = ({\mathcal{V}}_2,{\mathcal{E}}_2)$ from ${\mathcal{G}}_1$ such that ${\mathcal{G}}_2$ contains the dynamics of ${\mathcal{G}}$ by modifying the edge set ${\mathcal{E}}_1$. We index the set ${\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$ of source complexes as ${\bm}{s}_1,...,{\bm}{s}_{|{\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}|}$ and build ${\mathcal{E}}_2$ by adding edges $e$ with ${\bm}{s}(e) = {\bm}{s}_i$ for $i=1,...,|{\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}|$. For each source $\bm{s}_i$ of ${\mathcal{G}}_1$, there are three possibilities we must consider:
1. If $\mathit{RelInt}(V^{{\mathcal{G}}}(\bm{s}_i)) \subseteq \mathit{RelInt}(V^{{\mathcal{G}}_1}(\bm{s}_i))$, we simply include the edges $e$ of ${\mathcal{G}}_1$ with ${\bm}{s}(e) = {\bm}{s}_i$ in ${\mathcal{E}}_2$. Then, clearly $\mathit{RelInt}(V^{{\mathcal{G}}}(\bm{s}_i)) \subseteq \mathit{RelInt}(V^{{\mathcal{G}}_2}(\bm{s}_i))$.
2. If $V^{{\mathcal{G}}_1}(\bm{s}_i)$ is a line and $V^{{\mathcal{G}}}(\bm{s}_i)$ is not contained in that line, we again include each edge $e$ of ${\mathcal{G}}_1$ with ${\bm}{s}(e) = {\bm}{s}_i$ in ${\mathcal{E}}_2$, but must also include additional edges. We know that, in this case, $V^{{\mathcal{G}}_1}(\bm{s}_i)$ is the border of a half space which contains $V^{{\mathcal{G}}}(\bm{s}_i)$ and ${\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$. We add an edge with source ${\bm}{s}_i$ and a target in ${\mathcal{V}}_1= {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$ that does not lie in the line $V^{{\mathcal{G}}_1}(\bm{s}_i)$. Thus, $V^{{\mathcal{G}}_2}(\bm{s}_i)$ is the appropriate half space. Then, $\mathit{RelInt}(V^{{\mathcal{G}}}({\bm}{s}_i)) \subseteq \mathit{RelInt}(V^{{\mathcal{G}}_2}({\bm}{s}_i))$.
3. If $V^{{\mathcal{G}}}(\bm{s}_i) \subset \partial V^{{\mathcal{G}}_1}(\bm{s}_i)$, then $V^{{\mathcal{G}}}(\bm{s}_i)$ is a ray along one face of the convex hull of ${\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$. We then take only the edges of ${\mathcal{G}}_1$ which lie along this ray to be edges in ${\mathcal{E}}_2$. Then, clearly $\mathit{RelInt}(V^{{\mathcal{G}}}(\bm{s}_i)) \subseteq \mathit{RelInt}(V^{{\mathcal{G}}_2}(\bm{s}_i))$ because $V^{{\mathcal{G}}}(\bm{s}_i) = V^{{\mathcal{G}}_2}(\bm{s}_i)$.
We have now constructed ${\mathcal{G}}_2 = ({\mathcal{V}}_2,{\mathcal{E}}_2)$ such that ${\mathcal{G}}\sqsubseteq {\mathcal{G}}_2$. Furthermore, ${\mathcal{G}}_2$ is strongly endotactic, as we now show. Let ${\bm}{w} \in {\mathbb{R}}^d$ and $e \in {\mathcal{E}}_2$ be such that ${\bm}{w}\cdot {\bm}{v}(e) < 0$. Let $\{{\bm}{s}^*\}\subset {\mathcal{E}}_2$ be the sources such that ${\bm}{w}\cdot ({\bm}{s}^* - {\bm}{s}(e_j)) \leq 0$ for all $e_j \in {\mathcal{E}}_2$, and $\{e^*\}$ the corresponding edges. Any edges in ${\mathcal{G}}_2$ correspond to reaction vectors which do not point out of the convex hull of ${\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$, so we know that ${\bm}{w}\cdot({\bm}{s}^* - {\bm}{s}(e)) <0$. Also, $\{{\bm}{s}^*\}$ is the same as the set of sources of ${\mathcal{G}}$ which have ${\bm}{w}\cdot ({\bm}{s}(e_j) - {\bm}{s}^*) \leq 0$ for all $e_j \in {\mathcal{E}}$. If none of the ${\bm}{v}(e^*)\cdot {\bm}{w} >0$, then our construction implies this is true of the edges of ${\mathcal{G}}$ as well. This contradicts the assumption that ${\mathcal{G}}$ is strongly endotactic. We conclude that ${\mathcal{G}}_2$ is strongly endotactic.
It is possible that ${\mathcal{G}}_2$ is not weakly reversible, so we finally construct $\tilde{{\mathcal{G}}}$ such that $\tilde{{\mathcal{V}}} = {\mathcal{V}}_2$, ${\mathcal{E}}_2 \subseteq \tilde{{\mathcal{E}}}$, and both $\tilde{{\mathcal{G}}} \sqsubseteq {\mathcal{G}}_2$ and ${\mathcal{G}}_2 \sqsubseteq \tilde{{\mathcal{G}}}$ hold. To complete the construction, we must first establish the following about the structure of ${\mathcal{G}}_2$:
1. For each ${\bm}{s}\in {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$, either $V^{{\mathcal{G}}_2}({\bm}{s})$ is one dimensional and intersects a (one dimensional) face of the convex hull of ${\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$, or $V^{{\mathcal{G}}_2}({\bm}{s})$ is solid (meaning it has two-dimensional span) and $({\bm}{s}_i-{\bm}{s}) \in V^{{\mathcal{G}}_2}({\bm}{s})$ for all ${\bm}{s}_i \in {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$.
2. On every face of the convex hull of ${\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$, at least one of the following is true: there is some ${\bm}{s}$ such that $({\bm}{s}_i-{\bm}{s}) \in V^{{\mathcal{G}}_2}({\bm}{s})$ for all ${\bm}{s}_i \in {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$ (and $V^{{\mathcal{G}}_2}({\bm}{s})$ is solid), or there is some ${\bm}{s}$ such that ${\bm}{s}$ is a corner of the convex hull of ${\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$ and $V^{{\mathcal{G}}_2}({\bm}{s})$ is a ray pointing along an adjacent face of the convex hull of ${\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$.
3. We may add a path from any source on some face of the convex hull of ${\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$ to a source as in (ii) to create a network ${\mathcal{G}}^*$ such that ${\mathcal{G}}^*\sqsubseteq {\mathcal{G}}_2$ and ${\mathcal{G}}_2 \sqsubseteq {\mathcal{G}}^*$.
To establish (i), suppose that we have some ${\bm}{s}$ and $V^{{\mathcal{G}}_2}({\bm}{s})$ is not one dimensional. Then, either $V^{{\mathcal{G}}_2}({\bm}{s}) = V^{{\mathcal{G}}_1}({\bm}{s})$ or $V^{{\mathcal{G}}_2}({\bm}{s})$ is a half space such that $V^{{\mathcal{G}}_1}({\bm}{s}) = \partial V^{{\mathcal{G}}_2}({\bm}{s})$. In either case, the convexity of the convex hull of ${\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$ implies (i). By our construction possibility (c), if $V^{{\mathcal{G}}_2}({\bm}{s})$ is one dimensional, it must intersect a (one dimensional) face of the convex hull of ${\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$.
To establish (ii), let ${\bm}{w}$ be such there is some set $S \subset {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$ with at least two distinct elements and ${\bm}{w} \cdot ({\bm}{s}_i-{\bm}{s}_j) = 0$ for ${\bm}{s}_i,{\bm}{s}_j \in S$ and ${\bm}{w} \cdot ({\bm}{s}_i-{\bm}{s}_k) < 0$ for ${\bm}{s}_i \in S$, ${\bm}{s}_k \not\in S$ (i.e., ${\bm}{w}$ is the inward pointing normal to a one dimensional face of the convex hull of ${\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$). ${\mathcal{G}}_2$ is strongly endotactic, so for some ${\bm}{s}\in S$, there is some $e\in {\mathcal{E}}_2$ with ${\bm}{s}(e) = {\bm}{s}$ and ${\bm}{v}(e) \cdot {\bm}{w} >0$. We can conclude using fact (a) that on every face of the convex hull of ${\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$, at least one of the following is true: there is some ${\bm}{s}$ such that $({\bm}{s}_i-{\bm}{s}) \in V^{{\mathcal{G}}_2}({\bm}{s})$ for all ${\bm}{s}_i \in {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$ (and $V^{{\mathcal{G}}_2}({\bm}{s})$ is solid), or there is some ${\bm}{s}$ such that ${\bm}{s}$ is a corner of the convex hull of ${\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$ and $V^{{\mathcal{G}}_2}({\bm}{s})$ is a ray pointing along an adjacent face of the convex hull of ${\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$.
To establish (iii), let ${\mathcal{S}}$ be a face of the convex hull of ${\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$, and let ${\bm}{s} \in {\mathcal{S}}$ be a source such that either $({\bm}{s}_i-{\bm}{s}) \in V^{{\mathcal{G}}_2}({\bm}{s})$ for all ${\bm}{s}_i \in {\mathcal{S}}{\mathcal{C}}$ or $V^{{\mathcal{G}}_2}({\bm}{s})$ is a ray pointing along an adjacent face of the convex hull of ${\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$. Let ${\bm}{s}^* \in {\mathcal{S}}$ be some other source on the same face of the convex hull of ${\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$. If $V^{{\mathcal{G}}_2}({\bm}{s}^*)$ is solid or a full line, then ${\bm}{s}-{\bm}{s}^*\in V^{{\mathcal{G}}_2}({\bm}{s}^*)$, and so if ${\mathcal{G}}^*$ is the network with an edge added from ${\bm}{s}^*$ to ${\bm}{s}$, then $V^{{\mathcal{G}}^*}({\bm}{s}^*) = V^{{\mathcal{G}}_2}({\bm}{s}^*)$. If $V^{{\mathcal{G}}_2}({\bm}{s}^*)$ is a ray and ${\bm}{s}-{\bm}{s}^* \not \in V^{{\mathcal{G}}_2}({\bm}{s}^*)$, there must be some ${\bm}{s}^{**}$ such that $({\bm}{s}^{**}-{\bm}{s}^*) \in V^{{\mathcal{G}}_2}({\bm}{s}^*)$ and either ${\bm}{s}^{**}$ also has either $({\bm}{s}_i-{\bm}{s}^{**}) \in V^{{\mathcal{G}}_2}({\bm}{s}^{**})$ for all ${\bm}{s}_i \in {\mathcal{S}}{\mathcal{C}}$ or $V^{{\mathcal{G}}_2}({\bm}{s}^{**})$ is a ray pointing along an adjacent face of the convex hull of ${\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$, or $V^{{\mathcal{G}}_2}({\bm}{s}^{**}) = - V^{{\mathcal{G}}_2}({\bm}{s}^{*})$. In the last case, $({\bm}{s}-{\bm}{s}^{**})\in V^{{\mathcal{G}}_2}({\bm}{s}^{**})$, and so if ${\mathcal{G}}^*$ is the network to which we added edges to form a path from ${\bm}{s}^*$ to ${\bm}{s}$ through ${\bm}{s}^{**}$, we have that $V^{{\mathcal{G}}^*}({\bm}{s}^*) = V^{{\mathcal{G}}_2}({\bm}{s}^*)$ and $V^{{\mathcal{G}}^*}({\bm}{s}^{**}) = V^{{\mathcal{G}}_2}({\bm}{s}^{**})$ .
The above arguments show that for any source ${\bm}{s}^* \in {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$, there is a source ${\bm}{s}$ in the same face of the convex hull of ${\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$ such that either $({\bm}{s}_i-{\bm}{s}) \in V^{{\mathcal{G}}_2}({\bm}{s})$ for all ${\bm}{s}_i \in {\mathcal{S}}{\mathcal{C}}$ or $V^{{\mathcal{G}}_2}({\bm}{s})$ is a ray pointing along an adjacent face of the convex hull of ${\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$, and furthermore that if ${\mathcal{G}}^*$ is the network to which we added edges to form a path from ${\bm}{s}^*$ to ${\bm}{s}$, then ${\mathcal{G}}^*\sqsubseteq {\mathcal{G}}_2$ and ${\mathcal{G}}_2 \sqsubseteq {\mathcal{G}}^*$.
We may now complete the construction of $\tilde{{\mathcal{G}}}$. Recalling that ${\mathcal{G}}_1$ is weakly reversible, $\tilde{{\mathcal{G}}}$ is weakly reversible if $\tilde{{\mathcal{E}}}$ includes edges which replace any paths present in ${\mathcal{G}}_1$ that were not included in ${\mathcal{G}}_2$. Let $e_j$ be any edge in ${\mathcal{E}}_1$ but not in ${\mathcal{E}}_2$. Note that ${\bm}{t}(e_j) = {\bm}{s}(e_k)$ for some $e_k \in \tilde{{\mathcal{E}}}$ (because $\tilde{{\mathcal{V}}} = {\mathcal{V}}_2={\mathcal{V}}_1 = {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$). We must add a path of edges in $\tilde{{\mathcal{E}}}$ from ${\bm}{s}(e_j)$ to ${\bm}{s}(e_k)$, or prove that such a path is already present in $\tilde{{\mathcal{E}}}$. Let ${\bm}{s}(e_j) = {\bm}{s}$. We have seen that we may add a path of edges to some source ${\bm}{s}^*$ in the same of face of the convex hull of ${\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$ as ${\bm}{s}$ such that one of the following is true:
1. $({\bm}{s}_i-{\bm}{s}^*) \in V^{{\mathcal{G}}_2}({\bm}{s}^*)$ for all ${\bm}{s}_i \in {\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$, or
2. ${\bm}{s}^*$ is is a corner of the convex hull of ${\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$ and $V^{{\mathcal{G}}_2}({\bm}{s}^*)$ is a ray pointing along an adjacent face of the convex hull of ${\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$.
If (a) holds, then we can add an edge to $\tilde{{\mathcal{E}}}$ with source ${\bm}{s}^*$ and target ${\bm}{s}(e_k)$ and still have both $\tilde{{\mathcal{G}}} \sqsubseteq {\mathcal{G}}_2$ and ${\mathcal{G}}_2 \sqsubseteq \tilde{{\mathcal{G}}}$ , and a path from ${\bm}{s}(e_j)$ to ${\bm}{s}(e_k)$.
If (b) holds but (a) does not, we repeat the argument on the adjacent face of the convex hull of sources for ${\bm}{s}^*$, letting ${\bm}{s}^{**}$ be the new source which satisfies one of (a) or (b). Note that $ V^{{\mathcal{G}}_2}({\bm}{s}^*)$ is not solid, so ${\bm}{s}^{**} \neq {\bm}{s}^*$ (otherwise (a) was originally satisfied). If again only (b) holds, we may continue the argument until (a) holds for some ${\bm}{s}$ in a face of the convex hull of ${\mathcal{S}}{\mathcal{C}}_{{\mathcal{G}}}$, or (b) holds for some ${\bm}{s}$ and ${\bm}{s}(e_k)$ is in the face of the convex hull that $V^{{\mathcal{G}}_2}({\bm}{s})$ points along.
We conclude that if $\tilde{{\mathcal{G}}}$ is the weakly reversible network with paths added to replace any edges in ${\mathcal{E}}_1$ that are missing from ${\mathcal{E}}_2$, then $\tilde{{\mathcal{G}}} \sqsubseteq {\mathcal{G}}_2$. Then, \[effectively\_endo\_endo\] implies that $\tilde{{\mathcal{G}}}$ is strongly endotactic. Furthermore, ${\mathcal{G}}\sqsubseteq {\mathcal{G}}_2\sqsubseteq \tilde{{\mathcal{G}}}$.$\square$
Additional Examples
-------------------
We now present counterexamples to various possible inclusions of network types in the sense of dynamical equivalence. This will allow us to conclude that any arrow added to our \[relationships\] would be false.
\[EWRnotWR\] Consider the E-graph shown in \[counters\] (a). This network is extremally weakly reversible, but it is not effectively weakly reversible. According to \[noaddedwkrev\], if there is a weakly reversible network which generates a system generated by this network, it needs no added sources. Therefore, there must be a path from at least one extremal source to the interior source (shown in red). However, splitting (as in \[splitting\]) any extremal reaction will result in a new reaction which points out of the convex hull of sources. The resulting network cannot be endotactic, and so is not weakly reversible. $\triangle$
\[WRnotEWR\] Consider the E-graph shown in \[counters\] (b). This network is weakly reversible, while it is not effectively extremally weakly reversible. The extremal reaction set consists of two irreversible reactions which form a path (labeld $e_1$ and $e_2$) and one reversible reaction pair. Again, \[noaddedwkrev\] implies that if there is a weakly reversible network generates a system generated by the extremal reaction set then it needs no added sources. While this does allow us to reverse edge $e_2$ in \[counters\] (b) by splitting (as in \[splitting\]) edge $e_1$, the result is a new irreversible path into the reversible reaction pair. Neither reversible reaction can be split without introducing a new reaction which points out of the convex hull of sources. Thus, there is no weakly reversible network which contains the dynamics of this extremal reaction set. We conclude that the network is not effectively extremally weakly reversible. $\triangle$
\[EWRnotSE\] Consider the E-graph shown in \[counters\] (c). This network is reversible, weakly reversible, and extremally weakly reversible, while it is not effectively strongly endotactic. No reaction present can be split (as in \[splitting\]) while preserving the endotactic property. By \[containcondit\], any node added to create a new network $\tilde{{\mathcal{G}}}$ must have ${\bm}{0} \in \mathit{RelInt}(V^{\tilde{{\mathcal{G}}}}({\bm}{s}))$. Therefore, any direction ${\bm}{w}$ which violated the strongly endotactic conditions must still do so. $\triangle$
\[SOnotE\] Consider the E-graph shown in \[counters\] (d). This network is source-only, while it is not effectively endotactic. No reaction present can be split (as in \[splitting\]) to gain the endotactic property. By \[containcondit\], any node added to create a new network $\tilde{{\mathcal{G}}}$ must have ${\bm}{0}\in \mathit{RelInt}(V^{\tilde{{\mathcal{G}}}}({\bm}{s}(e)))$. Therefore, any direction ${\bm}{w}$ which violated the endotactic conditions must still do so. $\triangle$
\[CnotE\] Consider the E-graph shown in \[counters\] (e). This network is consistent. However, for generic choices of rate constants, the polynomial dynamical systems generated by this network are also generated by a network with a single irreversible reaction. Hence, the network is not effectively endotactic. Note that the same is true for the E-graph shown in \[counters\] (f). $\triangle$
\[CnotSO\] Consider the E-graph shown in \[counters\] (f). This one-dimensional network is consistent, but it is not effectively source-only. Any system generated by this network has only a single term. Any other network $\tilde{{\mathcal{G}}}$ which also generates such a system and contains more than one source must have additional sources which are extremal sources. However, \[containcondit\] implies that these must have ${\bm}{0}\in \mathit{RelInt}(V^{\tilde{{\mathcal{G}}}}({\bm}{s}))$, and so must have target nodes outside of the convex hull of sources, which could therefore not be sources. $\triangle$
Conclusion
==========
We have determined the extent to which different reaction networks may represent the same dynamical system when modeled with mass action kinetics. This allows us to investigate the overlap between classes of reaction networks, in the sense of dynamical equivalence and “effective" properties. \[relationships\] provides a summary of the relationships between classifications of networks, giving an answer to \[q2\]. Furthermore, the graph in \[relationships\] is complete in the sense that any additional arrows would be false, with the exception of arrows that are already implied by directed paths.
Our answers to \[q1\] and \[q2\] provide a framework for the study of generic interaction networks, and indeed systems of ODEs with polynomial right hand sides, in the context of reaction network theory. Reaction network theory provides useful tools for the analysis of dynamical systems [@H-J1; @C-F1; @C-F2; @W-H1; @W-H2; @E-T], and an answer to \[q1\] provides a way to extend these results to systems for which they are not immediately applicable. Our work on \[q2\] organizes the hierarchy of the various results in reaction network theory, allowing them to be extended where appropriate.
Acknowledgments
===============
David F. Anderson was supported by Army Research Office grant W911NF-18-1-0324. James D. Brunner was supported by the DeWitt & Curtiss Family Foundation and the Mayo Clinic Center for Individualized Medicine. Gheorghe Craciun was partially supported by the National Science Foundation under grants DMS-1412643 and DMS-1816238. Matthew D. Johnston was supported by the Henry Woodward Fund.
[^1]: Department of Mathematics, University of Wisconsin-Madison, [email protected]
[^2]: Division of Surgical Research, Department of Surgery, Mayo Clinic, [email protected]
[^3]: Departments of Mathematics and Biomolecular Chemistry, University of Wisconsin-Madison, [email protected]
[^4]: Department of Mathematics & Computer Science, Lawrence Technological University, [email protected]
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abstract: 'In a general-relativistic spacetime (Lorentzian manifold), gravitational lensing can be characterized by a lens map, in analogy to the lens map of the quasi-Newtonian approximation formalism. The lens map is defined on the celestial sphere of the observer (or on part of it) and it takes values in a two-dimensional manifold representing a two-parameter family of worldlines. In this article we use methods from differential topology to characterize global properties of the lens map. Among other things, we use the mapping degree (also known as Brouwer degree) of the lens map as a tool for characterizing the number of images in gravitational lensing situations. Finally, we illustrate the general results with gravitational lensing (a) by a static string, (b) by a spherically symmetric body, (c) in asymptotically simple and empty spacetimes, and (d) in weakly perturbed Robertson-Walker spacetimes.'
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$\,$
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Permanent address: TU Berlin, Sekr. PN 7-1, 10623 Berlin, Germany\
[*email*]{}: [email protected]
Introduction {#sec:intro}
============
Gravitational lensing is usually studied in a quasi-Newtonian approximation formalism which is essentially based on the assumptions that the gravitational fields are weak and that the bending angles are small, see Schneider, Ehlers and Falco [@SEF] for a comprehensive discussion. This formalism has proven to be very powerful for the calculation of special models. In addition it has also been used for proving general theorems on the qualitative features of gravitational lensing such as the possible number of images in a multiple imaging situation. As to the latter point, it is interesting to inquire whether the results can be reformulated in a Lorentzian manifold setting, i.e., to inquire to what extent the results depend on the approximations involved.
In the quasi-Newtonian approximation formalism one considers light rays in Euclidean 3-space that go from a fixed point (observer) to a point that is allowed to vary over a 2-dimensional plane (source plane). The rays are assumed to be straight lines with the only exception that they may have a sharp bend at a 2-dimensional plane (deflector plane) that is parallel to the source plane. (There is also a variant with several deflector planes to model deflectors which are not “thin”.) For each concrete mass distribution, the deflecting angles are to be calculated with the help of Einstein’s field equation, or rather of those remnants of Einstein’s field equation that survive the approximations involved. Hence, at each point of the deflector plane the deflection angle is uniquely determined by the mass distribution. As a consequence, following light rays from the observer into the past always gives a unique “lens map” from the deflector plane to the source plane. There is “multiple imaging” whenever this lens map fails to be injective.
In this article we want to inquire whether an analogous lens map can be introduced in a spacetime setting, without using quasi-Newtonian approximations. According to the rules of general relativity, a spacetime is to be modeled by a Lorentzian manifold $({{\mathcal{M}}},g)$ and the light rays are to be modeled by the lightlike geodesics in ${{\mathcal{M}}}$. We shall assume that $({{\mathcal{M}}},g)$ is time-oriented, i.e., that the timelike and lightlike vectors can be distinguished into future-pointing and past-pointing in a globally consistent way. To define a general lens map, we have to fix a point $p \in {{\mathcal{M}}}$ as the event where the observation takes place and we have to look for an analogue of the deflector plane and for an analogue of the source plane. As to the deflector plane, there is an obvious candidate, namely the [*celestial sphere*]{} ${{\mathcal{S}}}_p$ at $p$. This can be defined as the the set of all one-dimensional lightlike subspaces of the tangent space $T_p{{\mathcal{M}}}$ or, equivalently, as the totality of all light rays issuing from $p$ into the past. As to the source plane, however, there is no natural candidate. Following Frittelli, Newman and Ehlers [@FN; @EFN; @Eh], one might consider any timelike 3-dimensional submanifold ${{\mathcal{T}}}$ of the spacetime manifold as a substitute for the source plane. The idea is to view such a submanifold as ruled by worldlines of light sources. To make this more explicit, one could restrict to the case that ${{\mathcal{T}}}$ is a fiber bundle over a two-dimensional manifold ${{\mathcal{N}}}$, with fibers timelike and diffeomorphic to ${{\mathbb{R}}}$. Each fiber is to be interpreted as the worldline of a light source, and the set ${{\mathcal{N}}}$ may be identified with the set of all those worldlines. In this situation we wish to define a lens map $f_p : {{\mathcal{S}}}_p \longrightarrow {{\mathcal{N}}}$ by extending each light ray from $p$ into the past until it meets ${{\mathcal{T}}}$ and then projecting onto ${{\mathcal{N}}}$. In general, this prescription does not give a well-defined map since neither existence nor uniqueness of the target value is guaranteed. As to existence, there might be some past-pointing lightlike geodesics from $p$ that never reach ${{\mathcal{T}}}$. As to uniqueness, one and the same light ray might intersect ${{\mathcal{T}}}$ several times. The uniqueness problem could be circumvented by considering, on each past-pointing lightlike geodesic from $p$, only the first intersection with ${{\mathcal{T}}}$, thereby willfully excluding some light rays from the discussion. This comes up to ignoring every image that is hidden behind some other image of a light source with a worldline $\xi \in {{\mathcal{N}}}$. For the existence problem, however, there is no general solution. Unless one restricts to special situations, the lens map will be defined only on some subset ${{\mathcal{D}}}_p$ of ${{\mathcal{S}}}_p$ (which may even be empty). Also, one would like the lens map to be differentiable or at least continuous. This is guaranteed if one further restricts the domain ${{\mathcal{D}}}_p$ of the lens map by considering only light rays that meet ${{\mathcal{T}}}$ transversely.
Following this line of thought, we give a precise definition of lens maps in Section \[sec:deff\]. We will be a little bit more general than outlined above insofar as the source surface need not be timelike; we also allow for the limiting case of a lightlike source surface. This has the advantage that we may choose the source surface “at infinity” in the case of an asymptotically simple and empty spacetime. In Section \[sec:regular\] we briefly discuss some general properties of the caustic of the lens map. In Section \[sec:degree\] we introduce the mapping degree (Brouwer degree) of the lens map as an important tool from differential topology. This will then give us some theorems on the possible number of images in gravitational lensing situations, in particular in the case that we have a “simple lensing neighborhood”. The latter notion will be introduced and discussed in Section \[sec:sln\]. We conclude with applying the general results to some examples in Section \[sec:examples\].
Our investigation will be purely geometrical in the sense that we discuss the influence of the spacetime geometry on the propagation of light rays but not the influence of the matter distribution on the spacetime geometry. In other words, we use only the geometrical background of general relativity but not Einstein’s field equation. For this reason the “deflector”, i.e., the matter distribution that is the cause of gravitational lensing, never explicitly appears in our investigation. However, information on whether the deflectors are transparent or non-transparent will implicitly enter into our considerations.
Definition of the lens map {#sec:deff}
==========================
As a preparation for precisely introducing the lens map in a spacetime setting, we first specify some terminology.
By a [*manifold*]{} we shall always mean what is more fully called a “real, finite-dimensional, Hausdorff, second countable (and thus paracompact) $C^{\infty}$-manifold without boundary”. Whenever we have a $C^{\infty}$ vector field $X$ on a manifold ${{\mathcal{M}}}$, we may consider two points in ${{\mathcal{M}}}$ as [*equivalent*]{} if they lie on the same integral curve of $X$. We shall denote the resultant quotient space, which may be identified with the set of all integral curves of $X$, by ${{\mathcal{M}}}/X$. We call $X$ a [*regular*]{} vector field if ${{\mathcal{M}}}/X$ can be given the structure of a manifold in such a way that the natural projection $\pi_X : {{\mathcal{M}}}\longrightarrow {{\mathcal{M}}}/X$ becomes a $C^{\infty}$-submersion. It is easy to construct examples of non-regular vector fields. E.g., if $X$ has no zeros and is defined on ${{\mathbb{R}}}^n \setminus \{0\}$, then ${{\mathcal{M}}}/X$ cannot satisfy the Hausdorff property, so it cannot be a manifold according to our terminology. Palais [@Pa] has proven a useful result which, in our terminology, can be phrased in the following way. If none of $X$’s integral curves is closed or almost closed, and if ${{\mathcal{M}}}/X$ satisfies the Hausdorff property, then $X$ is regular.
We are going to use the following terminology. A [ *Lorentzian manifold*]{} is a manifold ${{\mathcal{M}}}$ together with a $C^{\infty}$ metric tensor field $g$ of Lorentzian signature $(+ \dots + -)$. A Lorentzian manifold is [*time-orientable*]{} if the set of all timelike vectors $\{ Z \in T {{\mathcal{M}}}\, | \,
g(Z,Z) < 0\}$ has exactly two connected components. Choosing one of those connected components as [*future-pointing*]{} defines a [*time-orientation*]{} for $({{\mathcal{M}}}, g)$. A [*spacetime*]{} is a connected 4-dimensional time-orientable Lorentzian manifold together with a time-orientation.
We are now ready to define what we will call a “source surface” in a spacetime. This will provide us with the target space for lens maps.
\[def:TW\] $({{\mathcal{T}}},W)$ is called a [*source surface*]{} in a spacetime $({{\mathcal{M}}},g)$ if\
(a) ${{\mathcal{T}}}$ is a 3-dimensional $C^{\infty}$ submanifold of ${{\mathcal{M}}}$;\
(b) $W$ is a nowhere vanishing regular $C^{\infty}$ vector field on ${{\mathcal{T}}}$ which is everywhere causal, $g(W,W) \le 0$, and future-pointing;\
(c) $\pi _W : {{\mathcal{T}}}\longrightarrow {{\mathcal{N}}}= {{\mathcal{T}}}/W$ is a fiber bundle with fiber diffeomorphic to ${{\mathbb{R}}}$ and the quotient manifold ${{\mathcal{N}}}= {{\mathcal{T}}}/W$ is connected and orientable.
We want to interpret the integral curves of $W$ as the worldlines of light sources. Thus, one should assume that they are not only causal but even timelike, $g(W,W) < 0$, since a light source should move at subluminal velocity. For technical reasons, however, we allow for the possibility that an integral curve of $W$ is lightlike (everywhere or at some points), because such curves may appear as ($C^1$-)limits of timelike curves. This will give us the possibility to apply the resulting formalism to asymptotically simple and empty spacetimes in a convenient way, see Subsection \[subsec:asy\] below. Actually, the causal character of $W$ will have little influence upon the results we want to establish. What really matters is a transversality condition that enters into the definition of the lens map below.
Please note that, in the situation of Definition \[def:TW\], the bundle $\pi _W : {{\mathcal{T}}}\longrightarrow
{{\mathcal{N}}}$ is necessarily trivializable, i.e., ${{\mathcal{T}}}\simeq {{\mathcal{N}}}\times
{{\mathbb{R}}}$. To prove this, let us assume that the flow of $W$ is defined on all of ${{\mathbb{R}}}\times {{\mathcal{T}}}$, so it makes $\pi _W : {{\mathcal{T}}}\longrightarrow
{{\mathcal{N}}}$ into a principal fiber bundle. (This is no restriction of generality since it can always be achieved by multiplying $W$ with an appropriate function. This function can be determined in the following way. Owing to a famous theorem of Whitney [@Whi], also see Hirsch [@Hi], p.55, paracompactness guarantees that ${{\mathcal{T}}}$ can be embedded as a closed submanifold into ${{\mathbb{R}}}^n$ for some $n$. Pulling back the Euclidean metric gives a complete Riemannian metric $h$ on ${{\mathcal{T}}}$ and the flow of the vector field $h(W,W)^{-1/2}W$ is defined on all of ${{\mathbb{R}}}\times {{\mathcal{T}}}$, cf. Abraham and Marsden [@AM], Proposition 2.1.21.) Then the result follows from the well known facts that any fiber bundle whose typical fiber is diffeomorphic to ${{\mathbb{R}}}^n$ admits a global section (see, e.g., Kobayashi and Nomizu [@KN], p.58), and that a principal fiber bundle is trivializable if and only if it admits a global section (see again [@KN], p.57).
Also, it is interesting to note the following. If ${{\mathcal{T}}}$ is any 3-dimensional submanifold of ${{\mathcal{M}}}$ that is foliated into timelike curves, then time orientability guarantees that these are the integral curves of a timelike vector field $W$. If we assume, in addition, that ${{\mathcal{T}}}$ contains no closed timelike curves, then it can be shown that $\pi _W : {{\mathcal{T}}}\longrightarrow {{\mathcal{N}}}$ is necessarily a fiber bundle with fiber diffeomorphic to ${{\mathbb{R}}}$, providing ${{\mathcal{N}}}$ satisfies the Hausdorff property, see Harris [@Ha], Theorem 2. This shows that there is little room for relaxing the conditions of Definition \[def:TW\].
Choosing a source surface in a spacetime will give us the target space ${{\mathcal{N}}}= {{\mathcal{T}}}/W$ for the lens map. To specify the domain of the lens map, we consider, at any point $p \in {{\mathcal{M}}}$, the set ${{\mathcal{S}}}_p$ of all lightlike directions at $p$, i.e., the set of all one-dimensional lightlike subspaces of $T_p {{\mathcal{M}}}$. We shall refer to ${{\mathcal{S}}}_p$ as to the [*celestial sphere*]{} at $p$. This is justified since, obviously, ${{\mathcal{S}}}_p$ is in natural one-to-one relation with the set of all light rays arriving at $p$. As it is more convenient to work with vectors rather than with directions, we shall usually represent ${{\mathcal{S}}}_p$ as a submanifold of $T_p {{\mathcal{M}}}$. To that end we fix a future-pointing timelike vector $V_p$ in the tangent space $T_p{{\mathcal{M}}}$. The vector $V_p$ may be interpreted as the 4-velocity of an observer at $p$. We now consider the set $$\label{eq:Sp}
{{\mathcal{S}}}_p = \big\{ Y_p \in T_p {{\mathcal{M}}}\, \big| \, g(Y_p,Y_p)=0 \;
{\mathrm{and}} \; g(Y_p,V_p) = 1 \; \big\} \, .$$ It is an elementary fact that (\[eq:Sp\]) defines an embedded submanifold of $T_p {{\mathcal{M}}}$ which is diffeomorphic to the standard 2-sphere $S^2$. As indicated by our notation, the set (\[eq:Sp\]) can be identified with the celestial sphere at $p$, just by relating each vector to the direction spanned by it.
Representation (\[eq:Sp\]) of the celestial sphere gives a convenient way of representing the light rays through $p$. We only have to assign to each $Y_p \in {{\mathcal{S}}}_p$ the lightlike geodesic $s \longmapsto {\mathrm{exp}}_p ( s Y_p)\,$ where exp$_p : W_p \subseteq T_p {{\mathcal{M}}}\longrightarrow {{\mathcal{M}}}$ denotes the exponential map at the point $p$ of the Levi-Civita connection of the metric $g$. Please note that this geodesic is past-pointing, because $V_p$ was chosen future-pointing, and that it passes through $p$ at the parameter value $s=0$.
The lens map is defined in the following way. After fixing a source surface $({{\mathcal{T}}},W)$ and choosing a point $p \in {{\mathcal{M}}}$, we denote by ${{\mathcal{D}}}_p \subseteq {{\mathcal{S}}}_p$ the subset of all lightlike directions at $p$ such that the geodesic to which this direction is tangent meets ${{\mathcal{T}}}$ (at least once) if sufficiently extended to the past, and if at the first intersection point $q$ with ${{\mathcal{T}}}$ this geodesic is transverse to ${{\mathcal{T}}}$. By projecting $q$ to ${{\mathcal{N}}}= {{\mathcal{T}}}/W$ we get the lens map $f_p : {{\mathcal{D}}}_p \longrightarrow
{{\mathcal{N}}}= {{\mathcal{T}}}/W$, see Figure \[fig:fp\]. If we use the representation (\[eq:Sp\]) for ${{\mathcal{S}}}_p$, the definition of the lens map can be given in more formal terms in the following way.
\[def:lensmap\] Let $({{\mathcal{T}}},W)$ be a source surface in a spacetime $({{\mathcal{M}}},g)$. Then, for each $p \in {{\mathcal{M}}}$, the lens map $f_p : {{\mathcal{D}}}_p \longrightarrow
{{\mathcal{N}}}= {{\mathcal{T}}}/W$ is defined in the following way. In the notation of equation (\[eq:Sp\]), let ${{\mathcal{D}}}_p$ be the set of all $Y_p \in {{\mathcal{S}}}_p$ such that there is a real number $w_p(Y_p) > 0$ with the properties\
(a) $sY_p$ is in the maximal domain of the exponential map for all $s \in [\, 0 \, ,w_p(Y_p)]$;\
(b) the curve $s \longmapsto {\mathrm{exp}} (sY_p)$ intersects ${{\mathcal{T}}}$ at the value $s = w_p(Y_p)$ transversely;\
(c) ${\mathrm{exp}}_p ( s Y_p) \notin {{\mathcal{T}}}$ for all $s \in [\, 0 \, , w_p(Y_p)[\;$.\
This defines a map $w_p : {{\mathcal{D}}}_p \longrightarrow {{\mathbb{R}}}$. The [*lens map*]{} at $p$ is then, by definition, the map $$\label{eq:lensmap}
f_p : {{\mathcal{D}}}_p \longrightarrow {{\mathcal{N}}}= {{\mathcal{T}}}/X \, , \quad
f_p(Y_p) = \pi_W \big( {\mathrm{exp}}_p ( w_p(Y_p )Y_p)\big) \, .$$ Here $\pi_W : {{\mathcal{T}}}\longrightarrow {{\mathcal{N}}}$ denotes the natural projection.
(10,10) (6,0.5, 5,1, 4,1.2, 3,1.2) (6,3.5, 5,4, 4,4.2, 3,4.2) (6,8.5, 5,9, 4,9.2, 3,9.2) (6,3.5, 6,8.5) (5,4, 5,9) (4,4.2, 4,9.2) (3,4.2, 3,9.2) (1.5,8, 4.5,5.2) (4.7,7.5)[(0,1)[1]{}]{} (1.5,8) (1.5,8)[(1,-1)[0.5]{}]{} (4.5,5.2) (4.5,1.13) (6.4,7)[${\mathcal{T}}$]{} (6.4,0.5)[${\mathcal{N}}$]{} (1.2,8.3)[$p$]{} (1.4,7.2)[$Y_p$]{} (4.1,8.)[$W$]{} (3.9,0.6)[$f_p(Y_p)$]{} (4.5,3.2)[(0,-1)[1]{}]{} (4.6,2.6)[$\pi _W$]{}
The transversality condition in part (b) of Definition \[def:lensmap\] guarantees that the domain ${{\mathcal{D}}}_p$ of the lens map is an open subset of ${{\mathcal{S}}}_p$. The case ${{\mathcal{D}}}_p = \emptyset$ is, of course, not excluded. In particular, ${{\mathcal{D}}}_p = \emptyset$ whenever $p \in {{\mathcal{T}}}$, owing to part (c) of Definition \[def:lensmap\].
Moreover, the transversality condition in part (b) of Definition \[def:lensmap\], in combination with the implicit function theorem, makes sure that the map $w_p :
{{\mathcal{D}}}_p \longrightarrow {{\mathbb{R}}}$ is a $C^{\infty}$ map. As the exponential map of a $C^{\infty}$ metric is again $C^{\infty}$, and $\pi_W$ is a $C^{\infty}$ submersion by assumption, this proves the following.
\[prop:diff\] The lens map is a $C^{\infty}$ map.
Please note that without the transversality condition the lens map need not even be continuous.
Although our Definition \[def:lensmap\] made use of the representation (\[eq:Sp\]), which refers to a timelike vector $V_p$, the lens map is, of course, independent of which future-pointing $V_p$ has been chosen. We decided to index the lens map only with $p$ although, strictly speaking, it depends on ${{\mathcal{T}}}$, on $W$, and on $p$. Our philosophy is to keep a source surface $({{\mathcal{T}}},W)$ fixed, and then to consider the lens map for all points $p \in {{\mathcal{M}}}$.
In view of gravitational lensing, the lens map admits the following interpretation. For $\xi \in {{\mathcal{N}}}$, each point $Y_p \in {{\mathcal{D}}}_p$ with $f_p (Y_p)$ corresponds to a past-pointing lightlike geodesic from $p$ to the worldline $\xi$ in ${{\mathcal{M}}}$, i.e., it corresponds to an image at the celestial sphere of $p$ of the light source with worldline $\xi$. If $f_p$ is not injective, we are in a multiple imaging situation. The converse need not be true as the lens map does not necessarily cover all images. There might be a past-pointing lightlike geodesic from $p$ reaching $\xi$ after having met ${{\mathcal{T}}}$ before, or being tangential to ${{\mathcal{T}}}$ on its arrival at $\xi$. In either case, the corresponding image is ignored by the lens map. The reader might be inclined to view this as a disadvantage. However, in Section \[sec:examples\] below we discuss some situations where the existence of such additional light rays can be excluded (e.g., asymptotically simple and empty spacetimes) and situations where it is desirable, on physical grounds, to disregard such additional light rays (e.g., weakly perturbed Robertson-Walker spacetimes with compact spatial sections).
It was already mentioned that the domain ${{\mathcal{D}}}_p$ of the lens map might be empty; this is, of course, the worst case that could happen. The best case is that the domain is all of the celestial sphere, ${{\mathcal{D}}}_p = {{\mathcal{S}}}_p$. We shall see in the following sections that many interesting results are true just in this case. However, there are several cases of interest where ${{\mathcal{D}}}_p$ is a proper subset of ${{\mathcal{S}}}_p$. If the domain of the lens map $f_p$ is the whole celestial sphere, none of the light rays issuing from $p$ into the past is blocked or trapped before it reaches ${{\mathcal{T}}}$. In view of applications to gravitational lensing, this excludes the possibility that these light rays meet a non-transparent deflector. In other words, it is a typical feature of gravitational lensing situations with non-transparent deflectors that ${{\mathcal{D}}}_p$ is not all of ${{\mathcal{S}}}_p$. Two simple examples, viz., a non-transparent string and a non-transparent spherical body, will be considered in Subsection \[subsec:string\] below.
Regular and critical values of the lens map {#sec:regular}
===========================================
Please recall that, for a differentiable map $F : {{\mathcal{M}}}_1
\longrightarrow {{\mathcal{M}}}_2$ between two manifolds, $Y \in {{\mathcal{M}}}_1$ is called a [*regular point*]{} of $F$ if the differential $T_{Y} F : T_{Y} {{\mathcal{M}}}_1 \longrightarrow T_{F(Y)} {{\mathcal{M}}}_2$ has maximal rank, otherwise $Y$ is called a [*critical point*]{}. Moreover, $\xi \in {{\mathcal{M}}}_2$ is called a [*regular value*]{} of $F$ if all $Y \in F^{-1} (\xi )$ are regular points, otherwise $\xi$ is called a [*critical value*]{}. Please note that, according to this definition, any $\xi \in {{\mathcal{M}}}_2$ that is not in the image of $F$ is regular. The well-known ([*Morse*]{}-)[*Sard theorem*]{} (see, e.g., Hirsch [@Hi], p.69) says that the set of regular values of $F$ is residual (i.e., it contains the intersection of countably many sets that are open and dense in ${{\mathcal{M}}}_2$) and thus dense in ${{\mathcal{M}}}_2$ and the critical values of $F$ make up a set of measure zero in ${{\mathcal{M}}}_2$.
For the lens map $f_p : {{\mathcal{D}}}_p \longrightarrow {{\mathcal{N}}}$, we call the set $$\label{eq:caustic}
{\mathrm{Caust}}(f_p) = \big\{ \, \xi \in {{\mathcal{N}}}\, \big|
\, \xi {\text{ is a critical value of }} f_p \, \big\}$$ the [*caustic*]{} of $f_p$. The Sard theorem then implies the following result.
\[prop:sard\] The caustic ${\mathrm{Caust}}(f_p)$ is a set of measure zero in ${{\mathcal{N}}}$ and its complement ${{\mathcal{N}}}\setminus {\mathrm{Caust}}(f_p)$ is residual and thus dense in ${{\mathcal{N}}}$.
Please note that ${\mathrm{Caust}}(f_p)$ need not be closed in ${{\mathcal{N}}}$. Counter-examples can be constructed easily by starting with situations where the caustic is closed and then excising points from spacetime. For lens maps defined on the whole celestial sphere, however, we have the following result.
\[prop:ccompact\] If ${{\mathcal{D}}}_p = {{\mathcal{S}}}_p$, the caustic ${\mathrm{Caust}}(f_p)$ is compact in ${{\mathcal{N}}}$.
This is an obvious consequence of the fact that ${{\mathcal{S}}}_p$ is compact and that $f_p$ and its first derivative are continuous.
As the domain and the target space of $f_p$ have the same dimension, $Y_p \in {{\mathcal{D}}}_p$ is a regular point of $f_p$ if and only if the differential $T_{Y_p}f_p :
T_{Y_p} {{\mathcal{S}}}_p\longrightarrow T_{f_p(Y_p)} {{\mathcal{N}}}$ is an isomorphism. In this case $f_p$ maps a neighborhood of $Y_p$ diffeomorphically onto a neighborhood of $f_p(Y_p)$. The differential $T_{Y_p}f_p$ may be either orientation-preserving or orientation-reversing. To make this notion precise we have to choose an orientation for ${{\mathcal{S}}}_p$ and an orientation for ${{\mathcal{N}}}$. For the celestial sphere ${{\mathcal{S}}}_p$ it is natural to choose the orientation according to which the origin of the tangent space $T_p {{\mathcal{M}}}$ is to the inner side of ${{\mathcal{S}}}_p$. The target manifold ${{\mathcal{N}}}$ is orientable by assumption, but in general there is no natural choice for the orientation. Clearly, choosing an orientation for ${{\mathcal{N}}}$ fixes an orientation for ${{\mathcal{T}}}$, because the vector field $W$ gives us an orientation for the fibers. We shall say that the orientation of ${{\mathcal{N}}}$ is [*adapted*]{} to some point $Y_p \in {{\mathcal{D}}}_p$ if the geodesic with initial vector $Y_p$ meets ${{\mathcal{T}}}$ at the inner side. If ${{\mathcal{D}}}_p$ is connected, the orientation of ${{\mathcal{N}}}$ that is adapted to some $Y_p \in {{\mathcal{D}}}_p$ is automatically adapted to all other elements of ${{\mathcal{D}}}_p$. Using this terminology, we may now introduce the following definition.
\[def:parity\] A regular point $Y_p \in {{\mathcal{D}}}_p$ of the lens map $f_p$ is said to have [*even parity*]{} (or [*odd parity*]{}, respectively) if $T_{Y_p}f_p$ is orientation-preserving (or orientation-reversing, respectively) with respect to the natural orientation on ${{\mathcal{S}}}_p$ and the orientation adapted to $Y_p$ on ${{\mathcal{N}}}$. For a regular value $\xi \in {{\mathcal{N}}}$ of the lens map, we denote by $n_+ (\xi )$ (or $n_- (\xi )$, respectively) the number of elements in $f_p^{-1}(\xi )$ with even parity (or odd parity, respectively).
Please note that $n_+(\xi)$ and $n_-(\xi)$ may be infinite, see the Schwarzschild example in Subsection \[subsec:string\] below. A criterion for $n_{\pm}(\xi)$ to be finite will be given in Proposition \[prop:finite\] below.
Definition \[def:parity\] is relevant for gravitational lensing in the following sense. The assumption that $Y_p$ is a regular point of $f_p$ implies that an observer at $p$ sees a neighborhood of $\xi = f_p(Y_p)$ in ${{\mathcal{N}}}$ as a neighborhood of $Y_p$ at his or her celestial sphere. If we compare the case that $Y_p$ has odd parity with the case that $Y_p$ has even parity, then the appearance of the neighborhood in the first case is the mirror image of its appearance in the second case. This difference is observable for a light source that is surrounded by some irregularly shaped structure, e.g. a galaxy with curved jets or with lobes.
If $\xi$ is a regular value of $f_p$, it is obvious that the points in $f_p^{-1} (\xi )$ are isolated, i.e., any $Y_p$ in $f_p^{-1} (\xi )$ has a neighborhood in ${{\mathcal{D}}}_p$ that contains no other point in $f_p^{-1} (\xi )$. This follows immediately from the fact that $f_p$ maps a neighborhood of $Y_p$ diffeomorphically onto its image. In the next section we shall formulate additional assumptions such that the set $f_p^{-1} ( \xi )$ is finite, i.e., such that the numbers $n_{\pm} (\xi)$ introduced in Definition \[def:parity\] are finite. It is the main purpose of the next section to demonstrate that then the difference $n_+ (\xi ) - n_- ( \xi )$ has some topological invariance properties. As a preparation for that we notice the following result which is an immediate consequence of the fact that the lens map is a local diffeomorphism near each regular point.
\[prop:n\] $n_+$ and $n_-$ are constant on each connected component of $f_p ({{\mathcal{D}}}_p) \setminus {\mathrm{Caust}}(f_p)$.
Hence, along any continuous curve in $f_p({{\mathcal{D}}}_p)$ that does not meet the caustic of the lens map, the numbers $n_+$ and $n_-$ remain constant, i.e., the observer at $p$ sees the same number of images for all light sources on this curve. If a curve intersects the caustic, the number of images will jump. In the next section we shall prove that $n_+$ and $n_-$ always jump by the same amount (under conditions making sure that these numbers are finite), i.e., the total number of images always jumps by an even number. This is well known in the quasi-Newtonian approximation formalism, see, e.g., Schneider, Ehlers and Falco [@SEF], Section 6.
If ${\mathrm{Caust}}(f_p)$ is empty, transversality guarantees that $f_p({{\mathcal{D}}}_p)$ is open in ${{\mathcal{N}}}$ and, thus a manifold. Proposition \[prop:n\] implies that, in this case, $f_p$ gives a $C^{\infty}$ covering map from ${{\mathcal{D}}}_p$ onto $f_p({{\mathcal{D}}}_p)$. As a $C^{\infty}$ covering map onto a simply connected manifold must be a global diffeomorphism, this implies the following result.
\[prop:cover\] Assume that ${\mathrm{Caust}}(f_p)$ is empty and that $f_p({{\mathcal{D}}}_p)$ is simply connected. Then $f_p$ gives a global diffeomorphism from ${{\mathcal{D}}}_p$ onto $f_p({{\mathcal{D}}}_p)$.
In other words, the formation of a caustic is necessary for multiple imaging provided that $f_p({{\mathcal{D}}}_p)$ is simply connected. In Subsection \[subsec:string\] below we shall consider the spacetime of a non-transparent string. This will demonstrate that the conclusion of Proposition \[prop:cover\] is not true without the assumption of $f_p({{\mathcal{D}}}_p)$ being simply connected.
In the rest of this subsection we want to relate the caustic of the lens map to the caustic of the past light cone of $p$. The past light cone of $p$ can be defined as the image set in ${{\mathcal{M}}}$ of the map $$\label{eq:Fp}
F_p : (s, Y_p) \longmapsto {\mathrm{exp}}_p (sY_p)$$ considered on its maximal domain in $]\, 0 \, , \, \infty \, [ \: \times \, {{\mathcal{S}}}_p \,$, and its caustic can be defined as the set of critical values of $F_p$. In other words, $q \in {{\mathcal{M}}}$ is in the caustic of the past light cone of $p$ if and only if there is an $s_0 \in \; ]\, 0 \, , \, \infty\, [ \,$ and a $Y_p \in {{\mathcal{S}}}_p$ such that the differential $T_{(s_0,Y_p)} F_p$ has rank $k < 3$. In that case one says that the point $q = {\mathrm{exp}}_p(s_0 Y_p)$ is [*conjugate*]{} to $p$ along the geodesic $s \longmapsto {\mathrm{exp}}_p(sY_p)$, and one calls the number $m = 3-k$ the [*multiplicity*]{} of this conjugate point. As $F_p ( \, \cdot\, , Y_p)$ is always an immersion, the multiplicity can take the values 1 and 2 only. (This formulation is equivalent to the definition of conjugate points and their multiplicities in terms of [*Jacobi vector fields*]{} which may be more familiar to the reader.) It is well known, but far from trivial, that along every lightlike geodesic conjugate points are isolated. Hence, in a compact parameter interval there are only finitely many points that are conjugate to a fixed point $p$. A proof can be found, e.g., in Beem, Ehrlich and Easley [@BEE], Theorem 10.77.
After these preparations we are now ready to establish the following proposition. We use the notation introduced in Definition \[def:lensmap\].
\[prop:caustic\] An element $Y_p \in {{\mathcal{D}}}_p$ is a regular point of the lens map if and only if the point ${\mathrm{exp}}_p(w_p(Y_p)Y_p)$ is not conjugate to $p$ along the geodesic $s \longmapsto
{\mathrm{exp}}_p (sY_p)$. A regular point $Y_p \in {{\mathcal{D}}}_p$ has even parity $($or odd parity, respectively$\, )$ if and only if the number of points conjugate to $p$ along the geodesic $[\, 0 \, ,w_p(Y_p)] \; \longrightarrow {{\mathcal{M}}}\, , \: s
\longmapsto {\mathrm{exp}}_p (sY_p)$ is even $($or odd, respectively$\, )$. Here each conjugate point is to be counted with its multiplicity.
In terms of the function (\[eq:Fp\]), the lens map can be written in the form $$\label{eq:fF}
f_p(Y_p) = \pi _W \big( F_p ( w_p (Y_p), Y_p ) \big) \; .$$ As $s \longmapsto F_p (s, Y_p)$ is an immersion transverse to ${{\mathcal{T}}}$ at $s = w_p(Y_p)$ and $\pi _W$ is a submersion, the differential of $f_p$ at $Y_p$ has rank 2 if and only if the differential of $F_p$ at $(w_p(Y_p),Y_p)$ has rank 3. This proves the first claim. For proving the second claim define, for each $s \in [0,w_p(Y_p)]$, a map $$\label{eq:Phi}
\Phi_s : T_{Y_p} {{\mathcal{S}}}_p \longrightarrow T_{f_p(Y_p)}{{\mathcal{N}}}$$ by applying to each vector in $T_{Y_p} {{\mathcal{S}}}_p$ the differential $T_{(s,Y_p)} F_p$, parallel-transporting the result along the geodesic $F_p(\, \cdot \, , Y_p )$ to the point $q = F_p\big(
w_p(Y_p), Y_p \big)$ and then projecting down to $T_{f_p(Y_p)} {{\mathcal{N}}}$. In the last step one uses the fact that, by transversality, any vector in $T_q {{\mathcal{M}}}$ can be uniquely decomposed into a vector tangent to ${{\mathcal{T}}}$ and a vector tangent to the geodesic $F_p(\,
\cdot \, , Y_p)$. For $s = 1$, this map $\Phi_s$ gives the differential of the lens map. We now choose a basis in $T_{Y_p} {{\mathcal{S}}}_p$ and a basis in $T_{f_p(Y_p)} {{\mathcal{N}}}$, thereby representing the map $\Phi_s$ as a $(2 \times 2)$-matrix. We choose the first basis right-handed with respect to the natural orientation on ${{\mathcal{S}}}_p$ and the second basis right-handed with respect to the orientation on ${{\mathcal{N}}}$ that is adapted to $Y_p$. Then ${\mathrm{det}}(\Phi_0)$ is positive as the parallel transport gives an orientation-preserving isomorphism. The function $s \longmapsto {\mathrm{det}}(\Phi_s)$ has a single zero whenever $F_p(s,Y_p)$ is a conjugate point of multiplicity one and it has a double zero whenever $F_p(s,Y_p)$ is a conjugate point of multiplicity two. Hence, the sign of ${\mathrm{det}}(\Phi_1)$ can be determined by counting the conjugate points.
This result implies that $\xi \in {{\mathcal{N}}}$ is a regular value of the lens map $f_p$ whenever the worldline $\xi$ does not pass through the caustic of the past light cone of $p$. The relation between parity and the number of conjugate points is geometrically rather evident because each conjugate point is associated with a “crossover” of infinitesimally neighboring light rays.
The mapping degree of the lens map {#sec:degree}
==================================
The mapping degree (also known as Brouwer degree) is one of the most powerful tools in differential topology. In this section we want to investigate what kind of information could be gained from the mapping degree of the lens map, providing it can be defined.
For the reader’s convenience we briefly summarize definition and main properties of the mapping degree, following closely Choquet-Bruhat, Dewitt-Morette, and Dillard-Bleick [@CDD], pp.477. For a more abstract approach, using homology theory, the reader may consult Dold [@Do], Spanier [@Sp] or Bredon [@Br]. In this article we shall not use homology theory with the exception of the proof of Proposition \[prop:sln\].
The definition of the mapping degree is based on the following observation.
\[prop:F\] Let $F : {\overline{{{\mathcal{D}}}}} \subseteq {{\mathcal{M}}}_1 \longrightarrow {{\mathcal{M}}}_2$ be a continuous map, where ${{\mathcal{M}}}_1$ and ${{\mathcal{M}}}_2$ are oriented connected manifolds of the same dimension, ${{\mathcal{D}}}$ is an open subset of ${{\mathcal{M}}}_1$ with compact closure ${\overline{{{\mathcal{D}}}}}$ and $F|_{{{\mathcal{D}}}}$ is a $C^{\infty}$ map. $($Actually, $C^1$ would do.$)$ Then for every $\xi \in {{\mathcal{M}}}_2 \setminus F(\partial {{\mathcal{D}}})$ which is a regular value of $F|_{{{\mathcal{D}}}}$, the set $F^{-1}(\xi)$ is finite.
By contradiction, let us assume that there is a sequence $(y_i)_{i \in {{\mathbb{N}}}}$ with pairwise different elements in $F^{-1} ( \xi )$. By compactness of ${\overline{{{\mathcal{D}}}}}$, we can choose an infinite subsequence that converges towards some point $y_{\infty} \in {\overline{{{\mathcal{D}}}}}$. By continuity of $F$, $F (y_{\infty}) = \xi$, so the hypotheses of the proposition imply that $y_{\infty} \notin \partial {{\mathcal{D}}}$. As a consequence, $y_{\infty}$ is a regular point of $F|_{{{\mathcal{D}}}}$, so it must have an open neighborhood in ${{\mathcal{D}}}$ that does not contain any other element of $F^{-1} ( \xi )$. This contradicts the fact that a subsequence of $(y_i)_{i \in {\mathbb{N}}}$ converges towards $y_{\infty}$.
If we have a map $F$ that satisfies the hypotheses of Proposition \[prop:F\], we can thus define, for every $\xi \in {{\mathcal{M}}}_2
\setminus F (\partial {{\mathcal{D}}})$ which is a regular value of $F|_{{{\mathcal{D}}}}$, $$\label{eq:deg}
{{\mathrm{deg}}}(F, \xi) = \sum_{y \, \in \, F^{-1} (\xi )} {\mathrm{sgn}} (y) \: ,$$ where sgn($y$) is defined to be $+1$ if the differential $T_yF$ preserves orientation and $-1$ if $T_yF$ reverses orientation. If $F^{-1} (\xi )$ is the empty set, the right-hand side of (\[eq:deg\]) is set equal to zero. The number ${{\mathrm{deg}}}(F, \xi )$ is called the [*mapping degree*]{} of $F$ at $\xi$. Roughly speaking, ${{\mathrm{deg}}}(F, \xi )$ tells how often the image of $F$ covers the point $\xi$, counting each “layer” positive or negative depending on orientation.
The mapping degree has the following properties (for proofs see Choquet-Bruhat, Dewitt-Morette, and Dillard-Bleick [@CDD], pp.477).\
[**Property A**]{}: ${{\mathrm{deg}}}(F, \xi ) = {{\mathrm{deg}}}(F, \xi ' )$ whenever $\xi $ and $\xi '$ are in the same connected component of ${{\mathcal{M}}}_2 \setminus F(\partial {{\mathcal{D}}})$.\
[**Property B**]{}: ${{\mathrm{deg}}}(F, \xi ) = {{\mathrm{deg}}}(F', \xi )$ whenever $F$ and $F'$ are homotopic, i.e., whenever there is a continuous map $\Phi : [0,1] \times {\overline{{{\mathcal{D}}}}} \longrightarrow
{{\mathcal{M}}}_2\, , \: (s,y) \longmapsto \Phi_s(y)$ with $\Phi_0 = F$ and $\Phi_1 = F'$ such that ${{\mathrm{deg}}}(\Phi _s , \xi )$ is defined for all $s \in [0,1]$.
Property A can be used to extend the definition of ${{\mathrm{deg}}}(F, \xi )$ to the non-regular values $\xi \in {{\mathcal{M}}}_2
\setminus F(\partial {{\mathcal{D}}})$. Given the fact that, by the Sard theorem, the regular values are dense in ${{\mathcal{M}}}_2$, this can be done just by continuous extension.
Property B can be used to extend the definition of ${{\mathrm{deg}}}(F, \xi )$ to continuous maps $F : {\overline{D}}
\longrightarrow {{\mathcal{M}}}_2$ which are not necessarily differentiable on ${{\mathcal{D}}}$. Given the fact that the $C^{\infty}$ maps are dense in the continuous maps with respect to the $C^0$-topology, this can be done again just by continuous extension.
We now apply these general results to the lens map $f_p : {{\mathcal{D}}}_p \longrightarrow {{\mathcal{N}}}$. In the case ${{\mathcal{D}}}_p \neq
{{\mathcal{S}}}_p$ it is necessary to extend the domain of the lens map onto a compact set to define the degree of the lens map. We introduce the following definition.
\[def:extend\] A map ${\overline{f_p}} : {\overline{{{\mathcal{D}}}_p}} \subseteq {{\mathcal{M}}}_1
\longrightarrow {{\mathcal{M}}}_2$ is called an [*extension*]{} of the lens map $f_p : {{\mathcal{D}}}_p \longrightarrow {{\mathcal{N}}}$ if\
(a) ${{\mathcal{M}}}_1$ is an orientable manifold that contains ${{\mathcal{D}}}_p$ as an open submanifold;\
(b) ${{\mathcal{M}}}_2$ is an orientable manifold that contains ${{\mathcal{N}}}$ as an open submanifold;\
(c) the closure ${\overline{{{\mathcal{D}}}_p}}$ of ${{\mathcal{D}}}_p$ in ${{\mathcal{M}}}_1$ is compact;\
(d) ${\overline{f_p}}$ is continuous and the restriction of ${\overline{f_p}}$ to ${{\mathcal{D}}}_p$ is equal to $f_p$.
If the lens map is defined on the whole celestial sphere, ${{\mathcal{D}}}_p = {{\mathcal{S}}}_p$, then the lens map is an extension of itself, ${\overline{f_p}} = f_p$, with ${{\mathcal{M}}}_1 = {{\mathcal{S}}}_p$ and ${{\mathcal{M}}}_2 = {{\mathcal{N}}}$. If ${{\mathcal{D}}}_p \neq {{\mathcal{S}}}_p$, one may try to continuously extend $f_p$ onto the closure of ${{\mathcal{D}}}_p$ in ${{\mathcal{S}}}_p$, thereby getting an extension with ${{\mathcal{M}}}_1 = {{\mathcal{S}}}_p$ and ${{\mathcal{M}}}_2 = {{\mathcal{N}}}$. If this does not work, one may try to find some other extension. The string spacetime in Subsection \[subsec:string\] below will provide us with an example where an extension exists although $f_p$ cannot be continuously extended from ${{\mathcal{D}}}_p$ onto its closure in ${{\mathcal{S}}}_p$. The spacetime around a spherically symmetric body with $R_o<3m$ will provide us with an example where the lens map admits no extension at all, see Subsection \[subsec:string\] below.
Applying Proposition \[prop:F\] to the case $F =
{\overline{f_p}}$ immediately gives the following result.
\[prop:finite\] If the lens map $f_p : {{\mathcal{D}}}_p \longrightarrow {{\mathcal{N}}}$ admits an extension ${\overline{f_p}} : {\overline{{{\mathcal{D}}}_p}} \subseteq {{\mathcal{M}}}_1
\longrightarrow {{\mathcal{M}}}_2$, then for all regular values $\xi \in
{{\mathcal{N}}}\setminus {\overline{f_p}} (\partial {{\mathcal{D}}}_p)$ the set $f_p^{-1} (\xi)$ is finite, so the numbers $n_+(\xi)$ and $n_-(\xi)$ introduced in Definition $\ref{def:parity}$ are finite.
If ${\overline{f_p}}$ is an extension of the lens map $f_p$, the number ${{\mathrm{deg}}}({\overline{f_p}}, \xi )$ is a well defined integer for all $\xi \in {{\mathcal{N}}}\setminus {\overline{f_p}}(
\partial {{\mathcal{D}}}_p )$, provided that we have chosen an orientation on ${{\mathcal{M}}}_1$ and on ${{\mathcal{M}}}_2$. The number ${{\mathrm{deg}}}({\overline{f_p}},
\xi)$ changes sign if we change the orientation on ${{\mathcal{M}}}_1$ or on ${{\mathcal{M}}}_2$. This sign ambiguity can be removed if ${{\mathcal{D}}}_p$ is connected. Then we know from the preceding section that ${{\mathcal{N}}}$ admits an orientation that is adapted to all $Y_p \in
{{\mathcal{D}}}_p$. As ${{\mathcal{N}}}$ is connected, this determines an orientation for ${{\mathcal{M}}}_2$. Moreover, the natural orientation on ${{\mathcal{S}}}_p$ induces an orientation on ${{\mathcal{D}}}_p$ which, for ${{\mathcal{D}}}_p$ connected, gives an orientation for ${{\mathcal{M}}}_1$.
In the rest of this paper we shall only be concerned with the situation that ${{\mathcal{D}}}_p$ is connected, and we shall always tacitly assume that the orientations have been chosen as indicated above, thereby fixing the sign of ${{\mathrm{deg}}}(f_p)$. Now comparison of (\[eq:deg\]) with Definition \[def:parity\] shows that $$\label{eq:degn}
{{\mathrm{deg}}}({\overline{f_p}} , \xi ) = n_+ (\xi ) - n_- (\xi )$$ for all regular values in ${{\mathcal{N}}}\setminus {\overline{f_p}}
(\partial {{\mathcal{D}}}_p)$. Owing to Property A, this has the following consequence.
\[prop:nn\] Assume that ${{\mathcal{D}}}_p$ is connected and that the lens map admits an extension ${\overline{f_p}} :
{\overline{{{\mathcal{D}}}_p}} \subseteq {{\mathcal{M}}}_1 \longrightarrow {{\mathcal{M}}}_2$. Then $n_+ (\xi ) - n_- (\xi ) = n_+ (\xi' ) - n_- (\xi' )$ for any two regular values $\xi$ and $\xi '$ which are in the same connected component of ${{\mathcal{N}}}\setminus
{\overline{f_p}} (\partial {{\mathcal{D}}}_p)$. In particular, $n_+ (\xi ) + n_- (\xi )$ is odd if and only if $n_+ (\xi' ) + n_- (\xi' )$ is odd.
We know already from Proposition \[prop:n\] that the numbers $n_+$ and $n_-$ remain constant along each continuous curve in $f_p({{\mathcal{D}}}_p)$ that does not meet the caustic of $f_p$. Now let us consider a continuous curve $\alpha \, : \; ]- \varepsilon _0 \, , \, \varepsilon _0 \, [
\; \longrightarrow f_p ({{\mathcal{D}}}_p)$ that meets the caustic at $\alpha (0)$ whereas $\alpha (\varepsilon )$ is a regular value of $f_p$ for all $\varepsilon \neq 0$. Under the additional assumptions that ${{\mathcal{D}}}_p$ is connected, that $f_p$ admits an extension, and that $\alpha (0) \notin {\overline{f_p}} (\partial {{\mathcal{D}}}_p )$, Proposition \[prop:nn\] tells us that $n_+ \big( \alpha
(\varepsilon) \big) - n_- \big( \alpha ( \varepsilon ) \big)$ remains constant when $\varepsilon$ passes through zero. In other words, $n_+$ and $n_-$ are allowed to jump only by the same amount. As a consequence, the total number of images $n_+ + n_-$ is allowed to jump only by an even number.
We now specialize to the case that the lens map is defined on the whole celestial sphere, ${{\mathcal{D}}}_p = {{\mathcal{S}}}_p$. Then the assumption of $f_p$ admitting an extension is trivially satisfied, with ${\overline{f_p}} = f_p$, and the degree ${{\mathrm{deg}}}(f_p , \xi )$ is a well-defined integer for all $\xi \in {{\mathcal{N}}}$. Moreover, ${{\mathrm{deg}}}(f_p , \xi )$ is a constant with respect to $\xi$, owing to Property A. It is then usual to write simply ${{\mathrm{deg}}}(f_p)$ instead of ${{\mathrm{deg}}}(f_p ,
\xi )$. Using this notation, (\[eq:degn\]) simplifies to $$\label{eq:degng}
{{\mathrm{deg}}}(f_p ) = n_+ (\xi ) - n_- (\xi )$$ for all regular values $\xi$ of $f_p$. Thus, the total number of images $$\label{eq:gnum}
n_+ (\xi ) + n_- (\xi ) = {{\mathrm{deg}}}(f_p ) + 2 n_- (\xi )$$ is either even for all regular values $\xi$ or odd for all regular values $\xi$, depending on whether ${{\mathrm{deg}}}(f_p)$ is even or odd.
In some gravitational lensing situations it might be possible to show that there is one light source $\xi \in {{\mathcal{N}}}$ for which $f_p^{-1}(\xi)$ consists of exactly one point, i.e., $\xi$ is not multiply imaged. This situation is characterized by the following proposition.
\[prop:unique\] Assume that ${{\mathcal{D}}}_p = {{\mathcal{S}}}_p$ and that there is a regular value $\xi$ of $f_p$ such that $f_p^{-1}( \xi )$ is a single point. Then $|{{\mathrm{deg}}}(f_p)| = 1$. In particular, $f_p$ must be surjective and ${{\mathcal{N}}}$ must be diffeomorphic to the sphere $S^2$.
The result $|{{\mathrm{deg}}}(f_p)| = 1$ can be read directly from (\[eq:degng\]), choosing the regular value $\xi$ which has exactly one pre-image point under $f_p$. This implies that $f_p$ must be surjective since a non-surjective map has degree zero. So ${{\mathcal{N}}}$ being the continuous image of the compact set ${{\mathcal{S}}}_p$ under the continuous map $f_p$ must be compact. It is well known (see, e.g., Hirsch [@Hi], p.130, Exercise 5) that for $n \ge 2$ the existence of a continuous map $F : S^n
\longrightarrow {{\mathcal{M}}}_2$ with ${{\mathrm{deg}}}(F) = 1$ onto a compact oriented $n$-manifold ${{\mathcal{M}}}_2$ implies that ${{\mathcal{M}}}_2$ must be simply connected. As the lens map gives us such a map onto ${{\mathcal{N}}}$ (after changing the orientation of ${{\mathcal{N}}}$, if necessary), we have thus found that ${{\mathcal{N}}}$ must be simply connected. Owing to the well-known classification theorem of compact orientable two-dimensional manifolds (see, e.g., Hirsch [@Hi], Chapter 9), this implies that ${{\mathcal{N}}}$ must be diffeomorphic to the sphere $S^2$.
In the situation of Proposition \[prop:unique\] we have $n_+(\xi ) + n_- (\xi) = 2 n_- (\xi) \pm 1$, for all $\xi \in
{{\mathcal{N}}}\setminus {\mathrm{Caust}} (f_p)$, i.e., the total number of images is odd for all light sources $\xi \in {{\mathcal{N}}}\simeq S^2$ that lie not on the caustic of $f_p$. The idea to use the mapping degree for proving an odd number theorem in this way was published apparently for the first time in the introduction of McKenzie [@McK]. In Proposition \[prop:unique\] one would, of course, like to drop the rather restrictive assumption that $f_p^{-1} (\xi)$ is a single point for some $\xi$. In the next section we consider a special situation where the result $| {{\mathrm{deg}}}(f_p) | = 1$ can be derived without this assumption.
Simple lensing neighborhoods {#sec:sln}
============================
In this section we investigate a special class of spacetime regions that will be called “simple lensing neighborhoods”. Although the assumption of having a simple lensing neighborhood is certainly rather special, we shall demonstrate in Section \[sec:examples\] below that sufficiently many examples of physical interest exist. We define simple lensing neighborhoods in the following way.
\[def:sln\] $({{\mathcal{U}}},{{\mathcal{T}}}, W)$ is called a [*simple lensing neighborhood*]{} in a spacetime $({{\mathcal{M}}},g)$ if\
(a) ${{\mathcal{U}}}$ is an open connected subset of ${{\mathcal{M}}}$ and ${{\mathcal{T}}}$ is the boundary of ${{\mathcal{U}}}$ in ${{\mathcal{M}}}$;\
(b) $(\, {{\mathcal{T}}}= \partial U , \, W \, )$ is a source surface in the sense of Definition \[def:TW\];\
(c) for all $p \in {{\mathcal{U}}}$, the lens map $f_p : {{\mathcal{D}}}_p \longrightarrow
{{\mathcal{N}}}= \partial {{\mathcal{U}}}/W$ is defined on the whole celestial sphere, ${{\mathcal{D}}}_p = {{\mathcal{S}}}_p$;\
(d) ${{\mathcal{U}}}$ does not contain an almost periodic lightlike geodesic.
Here the notion of being “almost periodic” is defined in the following way. Any immersed curve $\lambda : I \longrightarrow
{{\mathcal{U}}}$, defined on a real interval $I$, induces a curve ${\hat{\lambda}} : I \longrightarrow P{{\mathcal{U}}}$ in the projective tangent bundle $P {{\mathcal{U}}}$ over ${{\mathcal{U}}}$ which is defined by ${\hat{\lambda}} (s) = \{ \, c {\dot{\lambda}} (s) \: |
\: c \in {{\mathbb{R}}}\: \}$. The curve $\lambda$ is called [*almost periodic*]{} if there is a strictly monotonous sequence of parameter values $(s_i)_{i \in {{\mathbb{N}}}}$ such that the sequence $\big( {\hat{\lambda}} (s_i) \big) {}_{i \in {{\mathbb{N}}}}$ has an accumulation point in $P {{\mathcal{U}}}$. Please note that Condition (d) of Definition \[def:sln\] is certainly true if the strong causality condition holds everywhere on ${{\mathcal{U}}}$, i.e., if there are no closed or almost closed causal curves in ${{\mathcal{U}}}$. Also, Condition (d) is certainly true if every future-inextendible lightlike geodesic in ${{\mathcal{U}}}$ has a future end-point in ${{\mathcal{M}}}$.
Condition (d) should be viewed as adding a fairly mild assumption on the future-behavior of lightlike geodesics to the fairly strong assumptions on their past-behavior that are contained in Condition (c). In particular, Condition (c) excludes the possibility that past-oriented lightlike geodesics are blocked or trapped inside ${{\mathcal{U}}}$, i.e., it excludes the case that ${{\mathcal{U}}}$ contains non-transparent deflectors. Condition (c) requires, in addition, that the past-pointing lightlike geodesics are transverse to $\partial {{\mathcal{U}}}$ when leaving ${{\mathcal{U}}}$.
In the situation of a simple lensing neighborhood, we have for each $p \in {{\mathcal{U}}}$ a lens map that is defined on the whole celestial sphere, $f_p: {{\mathcal{S}}}_p \longrightarrow {{\mathcal{N}}}= \partial {{\mathcal{U}}}/W$. We have, thus, equation (\[eq:degng\]) at our disposal which relates the numbers $n_+(\xi)$ and $n_-(\xi)$, for any regular value $\xi \in {{\mathcal{N}}}$, to the mapping degree of $f_p$. (Please recall that, by Proposition \[prop:finite\], $n_+(\xi)$ and $n_-(\xi)$ are finite.) It is our main goal to prove that, in a simple lensing neighborhood, the mapping degree of the lens map equals $\pm 1$, so $n( \xi ) = n_+ ( \xi ) + n_- ( \xi )$ is odd for all regular values $\xi$. Also, we shall prove that a simple lensing neighborhood must be contractible and that its boundary must be diffeomorphic to $S^2 \times {{\mathbb{R}}}$. The latter result reflects the fact that the notion of simple lensing neighborhoods generalizes the notion of asymptotically simple and empty spacetimes, with $\partial {{\mathcal{U}}}$ corresponding to past lightlike infinity $\J ^-$, as will be detailed in Subsection \[subsec:asy\] below. When proving the desired properties of simple lensing neighborhoods we may therefore use several techniques that have been successfully applied to asymptotically simple and empty spacetimes before.
As a preparation we need the following lemma.
\[lem:sln\] Let $({{\mathcal{U}}},{{\mathcal{T}}},W)$ be a simple lensing neighborhood in a spacetime $({{\mathcal{M}}},g)$. Then there is a diffeomorphism $\Psi$ from the sphere bundle ${{\mathcal{S}}}= \big\{ Y_p \in {{\mathcal{S}}}_p \, \big| \, p \in {{\mathcal{U}}}\, \big\}$ of lightlike directions over ${{\mathcal{U}}}$ onto the space $T {{\mathcal{N}}}\times {{\mathbb{R}}}^2$ such that the following diagramm commutes. $$\label{eq:cd}
\begin{matrix}
&\; {{\mathcal{S}}}& {\overset{\Psi}{\longrightarrow}} & T{{\mathcal{N}}}\times {{\mathbb{R}}}^2
\\[0.2cm]
& i_p \uparrow \; \; & \, & \; \; \downarrow {\mathrm{pr}}
\\[0.2cm]
& \; {{\mathcal{S}}}_p & {\overset{f_p}{\longrightarrow}} & {{\mathcal{N}}}\;
\end{matrix}$$ Here $i_p$ denotes the inclusion map and ${\mathrm{pr}}$ is defined by dropping the second factor and projecting to the foot-point.
We fix a trivialization for the bundle $\pi _W : {{\mathcal{T}}}\longrightarrow
{{\mathcal{N}}}$ and identify ${{\mathcal{T}}}$ with ${{\mathcal{N}}}\times {{\mathbb{R}}}$. Then we consider the bundle $ {{\mathcal{B}}}= \big\{ X_q \in {{\mathcal{B}}}_q \, \big| \, q \in
{{\mathcal{T}}}\, \big\}$ over ${{\mathcal{T}}}$, where ${{\mathcal{B}}}_q \subset {{\mathcal{S}}}_q$ is, by definition, the subspace of all lightlike directions that are tangent to past-oriented lightlike geodesics that leave ${{\mathcal{U}}}$ transversely at $q$. Now we choose for each $q \in {{\mathcal{T}}}$ a vector $Q_q \in T_q {{\mathcal{M}}}$, smoothly depending on $q$, which is non-tangent to ${{\mathcal{T}}}$ and outward pointing. With the help of this vector field $Q$ we may identify ${{\mathcal{B}}}$ and $ T {{\mathcal{N}}}\times {{\mathbb{R}}}$ as bundles over ${{\mathcal{T}}}\simeq {{\mathcal{N}}}\times {{\mathbb{R}}}$ in the following way. Fix $\xi \in {{\mathcal{N}}}$, $X_{\xi} \in T _{\xi} {{\mathcal{N}}}$ and $s \in {{\mathbb{R}}}$ and view the tangent space $T_{\xi} {{\mathcal{N}}}$ as a natural subspace of $T_q ({{\mathcal{N}}}\times {{\mathbb{R}}})$, where $q = (\xi , s)$. Then the desired identification is given by associating the pair $(X_{\xi},s)$ with the direction spanned by $Z_q = X_{\xi} + Q_q - \alpha \, W(q)$, where the number $\alpha$ is uniquely determined by the requirement that $Z_q$ should be lightlike and past-pointing. – Now we consider the map $$\label{eq:pi}
\pi : {{\mathcal{S}}}\longrightarrow {{\mathcal{B}}}\simeq T {{\mathcal{N}}}\times {{\mathbb{R}}}$$ given by following each lightlike geodesic from a point $p \in {{\mathcal{U}}}$ into the past until it reaches ${{\mathcal{T}}}$, and assigning the tangent direction at the end-point to the tangent direction at the initial point. As a matter of fact, (\[eq:pi\]) gives a principal fiber bundle with structure group ${{\mathbb{R}}}$. To prove this, we first observe that the geodesic spray induces a vector field without zeros on ${{\mathcal{S}}}$. By multiplying this vector field with an appropriate function we get a vector field whose flow is defined on all of ${{\mathbb{R}}}\times {{\mathcal{S}}}$ (see the second paragraph after Definition \[def:TW\] for how to find such a function). The flow of this rescaled vector field defines an ${{\mathbb{R}}}$-action on ${{\mathcal{S}}}$ such that (\[eq:pi\]) can be identified with the projection onto the space of orbits. Conditions (c) and (d) of Definition \[def:sln\] guarantee that no orbit is closed or almost closed. Owing to a general result of Palais [@Pa], this is sufficient to prove that this action makes (\[eq:pi\]) into a principal fiber bundle with structure group ${{\mathbb{R}}}$. However, any such bundle is trivializable, see, e.g., Kobayashi and Nomizu [@KN], p.57/58. Choosing a trivialization for (\[eq:pi\]) gives us the desired diffeomorphism $\Psi$ from ${{\mathcal{S}}}$ to ${{\mathcal{B}}}\times {{\mathbb{R}}}\simeq
T {{\mathcal{N}}}\times {{\mathbb{R}}}^2$. The commutativity of the diagram (\[eq:cd\]) follows directly from the definition of the lens map $f_p$.
With the help of this lemma we will now prove the following proposition which is at the center of this section.
\[prop:sln\] Let $({{\mathcal{U}}}, {{\mathcal{T}}}, W)$ be a simple lensing neighborhood in a spacetime $({{\mathcal{M}}},g)$. Then\
[*(a)*]{} ${{\mathcal{N}}}= {{\mathcal{T}}}/W$ is diffeomorphic to the standard $2$-sphere $S^2$;\
[*(b)*]{} ${{\mathcal{U}}}$ is contractible;\
[*(c)*]{} for all $p \in {{\mathcal{U}}}$, the lens map $f_p : {{\mathcal{S}}}_p \simeq S^2
\longrightarrow {{\mathcal{N}}}\simeq S^2$ has $ | {{\mathrm{deg}}}(f_p) | = 1$; in particular, $f_p$ is surjective.
In the proof of part (a) and (b) we shall adapt techniques used by Newman and Clarke [@NC; @Ne] in their study of asymptotically simple and empty spacetimes. To that end it will be necessary to assume that the reader is familiar with homology theory. With the sphere bundle ${{\mathcal{S}}}$, introduced in Lemma \[lem:sln\], we may associate the [*Gysin homology sequence*]{} $$\label{eq:Gysin}
{} \dots {} \longrightarrow H_m ({{\mathcal{S}}}) \longrightarrow H_m ({{\mathcal{U}}}) \longrightarrow
H_{m-3} ({{\mathcal{U}}}) \longrightarrow H_{m-1} ( {{\mathcal{S}}}) \longrightarrow \dots$$ where $H_m( {\mathcal{X}} )$ denotes the $m^{\mathrm{th}}$ homology group of the space ${\mathcal{X}}$ with coefficients in a field ${{\mathbb{F}}}$. For any choice of ${{\mathbb{F}}}$, the Gysin sequence is an exact sequence of abelian groups, see, e.g., Spanier [@Sp], p.260 or, for the analogous sequence of cohomology groups, Bredon [@Br], p.390. By Lemma \[lem:sln\], ${{\mathcal{S}}}$ and ${{\mathcal{N}}}$ have the same homotopy type, so $H_m ( {{\mathcal{S}}})$ and $H_m ({{\mathcal{N}}})$ are isomorphic. Upon inserting this into (\[eq:Gysin\]), we use the fact that $H_m ( {{\mathcal{U}}}) = {\boldsymbol{1}} \, ( \, = \,$trivial group consisting of the unit element only) for $m > 4$ and $H_m ({{\mathcal{N}}}) = {\boldsymbol{1}}$ for $m > 2$ because ${\mathrm{dim}} ({{\mathcal{U}}}) = 4$ and ${\mathrm{dim}} ({{\mathcal{N}}}) = 2$. Also, we know that $H_0 ({{\mathcal{U}}}) = {{\mathbb{F}}}$ and $H_0 ({{\mathcal{N}}}) = {{\mathbb{F}}}$ since ${{\mathcal{U}}}$ and ${{\mathcal{N}}}$ are connected. Then the exactness of the Gysin sequence implies that $$\label{eq:Hms}
H_m ({{\mathcal{U}}}) = {\boldsymbol{1}} \quad {\mathrm{for}} \; \; m > 0$$ and $$\label{eq:HmN}
H_1 ({{\mathcal{N}}}) = {\boldsymbol{1}} \; , \quad H_2 ({{\mathcal{N}}}) = {{\mathbb{F}}}\, .$$ From (\[eq:HmN\]) we read that ${{\mathcal{N}}}$ is compact since otherwise $H_2( {{\mathcal{N}}}) = {\boldsymbol{1}}$. Moreover, we observe that ${{\mathcal{N}}}$ has the same homology groups and thus, in particular, the same Euler characteristic as the 2-sphere. It is well known that any two compact and orientable 2-manifolds are diffeomorphic if and only if they have the same Euler characteristic (or, equivalently, the same genus), see, e.g., Hirsch [@Hi], Chapter 9. We have thus proven part (a) of the proposition. – To prove part (b) we consider the end of the exact [*homotopy sequence*]{} of the fiber bundle ${{\mathcal{S}}}$ over ${{\mathcal{U}}}$, see, e.g., Frankel [@Fr], p.600, $$\label{eq:homseq}
{} \dots {} \longrightarrow \pi _1 ({{\mathcal{S}}})
\longrightarrow \pi_1 ({{\mathcal{U}}}) \longrightarrow {\boldsymbol{1}} \, .$$ As ${{\mathcal{S}}}$ has the same homotopy type as ${{\mathcal{N}}}\simeq S^2$, we may replace $\pi_1 ({{\mathcal{S}}})$ with $\pi_1 (S^2 ) = {\boldsymbol{1}}$, so the exactness of (\[eq:homseq\]) implies that $\pi _1 ({{\mathcal{U}}}) =
{\boldsymbol{1}}$, i.e., that ${{\mathcal{U}}}$ is simply connected. If, for some $m > 1$, the homotopy group $\pi _m ({{\mathcal{U}}})$ would be different from ${\boldsymbol{1}}$, the Hurewicz isomorphism theorem (see, e.g., Spanier [@Sp], p.394 or Bredon [@Br], p.479, Corollary 10.10.) would give a contradiction to (\[eq:Hms\]). Thus, $\pi _m ({{\mathcal{U}}}) = {\boldsymbol{1}}$ for all $m \in {{\mathbb{N}}}$, i.e., ${{\mathcal{U}}}$ is contractible. – We now prove part (c). Since ${{\mathcal{U}}}$ is contractible, the tangent bundle $T {{\mathcal{U}}}$ and thus the sphere bundle ${{\mathcal{S}}}$ over ${{\mathcal{U}}}$ admits a global trivialization, $ {{\mathcal{S}}}\simeq {{\mathcal{U}}}\times S^2$. Fixing such a trivialization and choosing a contraction that collapses ${{\mathcal{U}}}$ onto some point $p \in {{\mathcal{U}}}$ gives a contraction ${\tilde{i_p}} :
{{\mathcal{S}}}\longrightarrow {{\mathcal{S}}}_p \,$. Together with the inclusion map $i_p : {{\mathcal{S}}}_p \longrightarrow {{\mathcal{S}}}$ this gives us a homotopy equivalence between ${{\mathcal{S}}}_p$ and ${{\mathcal{S}}}$. $($Please recall that a [*homotopy equivalence*]{} between two topological spaces ${\mathcal{X}}$ and ${\mathcal{Y}}$ is a pair of continuous maps $\varphi: {\mathcal{X}} \longrightarrow {\mathcal{Y}}$ and ${\tilde{\varphi}}: {\mathcal{Y}} \longrightarrow {\mathcal{X}}$ such that $\varphi \circ {\tilde{\varphi}}$ can be continuously deformed into the identity on ${\mathcal{Y}}$ and ${\tilde{\varphi}}
\circ \varphi$ can be continuously deformed into the identity on ${\mathcal{X}}.)$ On the other hand, the projection ${\mathrm{pr}}$ from (\[eq:cd\]), together with the zero section $\tilde{\mathrm{pr}}
: {{\mathcal{N}}}\longrightarrow T {{\mathcal{N}}}\times {{\mathbb{R}}}^2$ gives a homotopy equivalence between $T {{\mathcal{N}}}\times {{\mathbb{R}}}^2$ and ${{\mathcal{N}}}$. As a consequence, the diagram (\[eq:cd\]) tells us that the lens map $f_p = {\mathrm{pr}}
\circ \Psi \circ i_p$ together with the map ${\tilde{f}}{}_p
= {\tilde{i}}{}_p \circ \Psi ^{-1} \circ {\tilde{\mathrm{pr}}}$ gives a homotopy equivalence between ${{\mathcal{S}}}_p \simeq S^2$ and ${{\mathcal{N}}}\simeq S^2$, so $f_p \circ {\tilde{f}}{}_p$ is homotopic to the identity. Since the mapping degree is a homotopic invariant (please recall Property B of the mapping degree from Section \[sec:degree\]), this implies that ${{\mathrm{deg}}}( f_p \circ {\tilde{f}}{}_p ) = 1$. Now the product theorem for the mapping degree (see, e.g., Choquet-Bruhat, Dewitt-Morette, and Dillard-Bleick [@CDD], p.483) yields ${{\mathrm{deg}}}(f_p) \, {{\mathrm{deg}}}({\tilde{f}}{}_p ) = 1$. As the mapping degree is an integer, this can be true only if $ {{\mathrm{deg}}}(f_p ) = {{\mathrm{deg}}}( {\tilde{f}}{}_p ) = \pm 1$. In particular, $f_p$ must be surjective since otherwise ${{\mathrm{deg}}}(f_p ) = 0$.
In all simple examples to which this proposition applies the degree of $f_p$ is, actually, equal to $+1$, and it is hard to see whether examples with ${{\mathrm{deg}}}(f_p) = -1 $ do exist. The following consideration is quite instructive. If we start with a simple lensing neighborhood in a flat spacetime (or, more generally, in a conformally flat spacetime), then conjugate points cannot occur, so it is clear that the case ${{\mathrm{deg}}}(f_p) = -1$ is impossible. If we now perturb the metric in such a way that the simple-lensing-neighborhood property is maintained during the perturbation, then, by Property B of the degree, the equation ${{\mathrm{deg}}}(f_p ) = + 1$ is preserved. This demonstrates that the case ${{\mathrm{deg}}}(f_p) = -1$ cannot occur for weak gravitational fields (or for small perturbations of conformally flat spacetimes such as Robertson-Walker spacetimes).
Among other things, Proposition \[prop:sln\] gives a good physical motivation for studying degree-one maps from $S^2$ to $S^2$. In particular, it is an interesting problem to characterize the caustics of such maps. Please note that, by parts (a) and (c) of Proposition \[prop:sln\], $f_p({{\mathcal{D}}}_p)$ is simply connected for all $p \in {{\mathcal{U}}}$. Hence, Proposition \[prop:cover\] applies which says that the formation of a caustic is necessary for multiple imaging.
Owing to (\[eq:gnum\]), part (c) of Proposition \[prop:sln\] implies in particular that $n (\xi ) = n_+(\xi ) + n_-(\xi )$ is odd for all worldlines of light sources $\xi \in {{\mathcal{N}}}$ that do not pass through the caustic of the past light cone of $p$, i.e., if only light rays within ${{\mathcal{U}}}$ are taken into account the observer at $p$ sees an odd number of images of such a worldline. It is now our goal to prove a similar ’odd number theorem’ for a light source with worldline inside ${{\mathcal{U}}}$. As a preparation we establish the following lemma.
\[lem:boundary\] Let $({{\mathcal{U}}},{{\mathcal{T}}},W)$ be a simple lensing neighborhood in a spacetime $({{\mathcal{M}}},g)$ and $p \in {{\mathcal{U}}}$. Let $J^{-}(p, {{\mathcal{U}}})$ denote, as usual, the [*causal past*]{} of $p$ in ${{\mathcal{U}}}$, i.e., the set of all points in ${{\mathcal{M}}}$ that can be reached from $p$ along a past-pointing causal curve in ${{\mathcal{U}}}$. Let $\partial _{{{\mathcal{U}}}}
J^-(p, {{\mathcal{U}}})$ denote the boundary of $J^-(p, {{\mathcal{U}}})$ in ${{\mathcal{U}}}$. Then\
[*(a)*]{} every point $q \in \partial _{{{\mathcal{U}}}} J^-(p, {{\mathcal{U}}})$ can be reached from $p$ along a past-pointing lightlike geodesic in ${{\mathcal{U}}}$;\
[*(b)*]{} $\partial _{{{\mathcal{U}}}} J^-(p, {{\mathcal{U}}})$ is relatively compact in ${{\mathcal{M}}}$.
As usual, let $I^-(p, {{\mathcal{U}}})$ denote the chronological past of $p$ in ${{\mathcal{U}}}$, i.e., the set of all points that can be reached from $p$ along a past-pointing timelike curve in ${{\mathcal{U}}}$. To prove part (a), fix a point $q \in \partial _{{{\mathcal{U}}}} J^-(p, {{\mathcal{U}}})$. Choose a sequence $(p_i)_{i \in {{\mathbb{N}}}}$ of points in ${{\mathcal{U}}}$ that converge towards $p$ in such a way that $p \in I^- (p_i , {{\mathcal{U}}})$ for all $i \in {{\mathbb{N}}}$. This implies that we can find for each $i \in {{\mathbb{N}}}$ a past-pointing timelike curve $\lambda _i$ from $p_i$ to $q$. Then the $\lambda _i$ are past-inextendible in ${{\mathcal{U}}}\setminus \{ q \}$. Owing to a standard lemma (see, e.g., Wald [@Wa], Lemma 8.1.5) this implies that the $\lambda _i$ have a causal limit curve $\lambda$ through $p$ that is past-inextendible in ${{\mathcal{U}}}\setminus \{ q \}$. We want to show that $\lambda$ is the desired lightlike geodesic. Assume that $\lambda$ is not a lightlike geodesic. Then $\lambda$ enters into the open set $I^- (p, {{\mathcal{U}}})$ (see Hawking and Ellis [@HE], Proposition 4.5.10), so $\lambda _i$ enters into $I^- (p , {{\mathcal{U}}})$ for $i$ sufficiently large. This, however, is impossible since all $\lambda _i$ have past end-point on $\partial _{{{\mathcal{U}}}} J^-(p, {{\mathcal{U}}})$, so $\lambda$ must be a lightlike geodesic. It remains to show that $\lambda$ has past end-point at $q$. Assume that this is not true. Since $\lambda$ is past-inextendible in ${{\mathcal{U}}}\setminus \{ q \}$ this assumption implies that $\lambda$ is past-inextendible in ${{\mathcal{U}}}$, so by condition (c) of Definition \[def:sln\] $\lambda$ has past end-point on $\partial {{\mathcal{U}}}$ and meets $\partial {{\mathcal{U}}}$ transversely. As a consequence, for $i$ sufficiently large $\lambda _i$ has to meet $\partial {{\mathcal{U}}}$ which gives a contradiction to the fact that all $\lambda _i$ are within ${{\mathcal{U}}}$. – To prove part (b), we have to show that any sequence $(q_i)_{i \in {{\mathbb{N}}}}$ in $\partial _{{{\mathcal{U}}}} J^-(p, {{\mathcal{U}}})$ has an accumulation point in ${{\mathcal{M}}}$. So let us choose such a sequence. From part (a) we know that there is a past-pointing lightlike geodesic $\mu _i$ from $p$ to $q_i$ in ${{\mathcal{U}}}$ for all $i \in {{\mathbb{N}}}$. By compactness of ${{\mathcal{S}}}_p
\simeq S^2$, the tangent directions to these geodesics at $p$ have an accumulation point in ${{\mathcal{S}}}_p$. Let $\mu$ be the past-pointing lightlike geodesic from $p$ which is determined by this direction. By condition (c) of Definition \[def:sln\], this geodesic $\mu$ and each of the geodesics $\mu _i$ must have a past end-point on $\partial {{\mathcal{U}}}$ if maximally extended inside ${{\mathcal{U}}}$. We may choose an affine parametrization for each of those geodesics with the parameter ranging from the value 0 at $p$ to the value 1 at $\partial {{\mathcal{U}}}$. Then our sequence $(q_i)_{i \in {{\mathbb{N}}}}$ in ${{\mathcal{U}}}$ determines a sequence $(s_i)_{i \in {{\mathbb{N}}}}$ in the interval $[0,1]$ by setting $q_i = \mu _i (s_i)$. By compactness of $[0,1]$, this sequence must have an accumulation point $s \in [0.1]$. This demonstrates that the $q_i$ must have an accumulation point in ${{\mathcal{M}}}$, namely the point $\mu (s)$.
We are now ready to prove the desired odd-number theorem for light sources with worldline in ${{\mathcal{U}}}$.
\[prop:gamma\] Let $({{\mathcal{U}}}, {{\mathcal{T}}},W)$ be a simple lensing neighborhood in a spacetime $({{\mathcal{M}}},g)$ and assume that ${{\mathcal{U}}}$ does not contain a closed timelike curve. Fix a point $p \in {{\mathcal{U}}}$ and a timelike embedded $C^{\infty}$ curve $\gamma$ in ${{\mathcal{U}}}$ whose image is a closed topological subset of ${{\mathcal{M}}}$. $($The latter condition excludes the case that $\gamma$ has an end-point on $\partial {{\mathcal{U}}}$.$)$ Then the following is true.\
[*(a)*]{} If $\gamma$ does not meet the point $p$, then there is a past-pointing lightlike geodesic from $p$ to $\gamma$ that lies completely within ${{\mathcal{U}}}$ and contains no conjugate points in its interior. $($The end-point may be conjugate to the initial-point.$)$ If this geodesic meets $\gamma$ at the point $q$, say, then all points on $\gamma$ that lie to the future of $q$ cannot be reached from $p$ along a past-pointing lightlike geodesic in ${{\mathcal{U}}}$.\
[*(b)*]{} If $\gamma$ meets neither the point $p$ nor the caustic of the past light cone of $p$, then the number of past-pointing lightlike geodesics from $p$ to $\gamma$ that are completely contained in ${{\mathcal{U}}}$ is finite and odd.
In the first step we construct a $C^{\infty}$ vector field $V$ on ${{\mathcal{M}}}$ that is timelike on ${{\mathcal{U}}}$, has $\gamma$ as an integral curve, and coincides with $W$ on ${{\mathcal{T}}}= \partial {{\mathcal{U}}}$. To that end we first choose any future-pointing timelike $C^{\infty}$ vector field $V_1$ on ${{\mathcal{M}}}$. (Existence is guaranteed by our assumption of time-orientability.) Then we extend the vector field $W$ to a $C^{\infty}$ vector field $V_2$ onto some neighborhood ${{\mathcal{V}}}$ of ${{\mathcal{T}}}$. Since $W$ is causal and future-pointing, $V_2$ may be chosen timelike and future-pointing on ${{\mathcal{V}}}\setminus {{\mathcal{T}}}$. (Here we make use of the fact that ${{\mathcal{T}}}= \partial {{\mathcal{U}}}$ is a closed subset of ${{\mathcal{M}}}$.) Finally we choose a timelike and future-pointing vector field $V_3$ on some neighborhood ${{\mathcal{W}}}$ of $\gamma$ that is tangent to $\gamma$ at all points of $\gamma$. (Here we make use of the fact that the image of $\gamma$ is a closed subset of ${{\mathcal{M}}}$.) We choose the neighborhoods ${{\mathcal{V}}}$ and ${{\mathcal{W}}}$ disjoint which is possible since $\gamma$ is completely contained in ${{\mathcal{U}}}$ and closed in ${{\mathcal{M}}}$. With the help of a partition of unity we may now combine the three vector fields $V_1, V_2, V_3$ into a vector field $V$ with the desired properties.
In the second step we consider the quotient space ${{\mathcal{M}}}/V$. This space contains the open subset ${{\mathcal{U}}}/V$ whose boundary ${{\mathcal{T}}}/V = {{\mathcal{N}}}$ is, by Proposition \[prop:sln\], a manifold diffeomorphic to $S^2$. We want to show that ${{\mathcal{U}}}/V$ is a manifold (which, according to our terminology, in particular requires that ${{\mathcal{U}}}/V$ is a Hausdorff space). To that end we consider the map $j_p :
{\overline{\partial _{{{\mathcal{U}}}} J^-(p, {{\mathcal{U}}})}} \longrightarrow
{\overline{{{\mathcal{U}}}}} /V$ which assigns to each point $q \in
{\overline{\partial _{{{\mathcal{U}}}} J^-(p, {{\mathcal{U}}})}}$ the integral curve of $V$ passing through that point. (Overlining always means closure in ${{\mathcal{M}}}$.) Clearly, $j_p$ is continuous with respect to the topology ${\overline{\partial _{{{\mathcal{U}}}} J^-(p, {{\mathcal{U}}})}}$ inherits as a subspace of ${{\mathcal{M}}}$ and the quotient topology on ${\overline{{{\mathcal{U}}}}} /V$. Moreover, ${\overline{\partial _{{{\mathcal{U}}}} J^- (p, {{\mathcal{U}}})}}$ intersects each integral curve of $V$ at most once, and if it intersects one integral curve then it also intersects all neighbboring integral curves in ${\overline{{{\mathcal{U}}}}}$; this follows from Wald [@Wa], Theorem 8.1.3. Hence, $j_p$ is injective and its image is open in ${\overline{{{\mathcal{U}}}}} /V$. On the other hand, part (b) of Lemma \[lem:boundary\] implies that the image of $j_p$ is closed. Since the image of $j_p$ is non-empty and connected, it must be all of ${\overline{{{\mathcal{U}}}}} /V$. (The domain of $j_p$ and, thus, the image of $j_p$ is non-empty because ${{\mathcal{U}}}$ does not contain a closed timelike curve. The domain and, thus, the image of $j_p$ is connected since ${{\mathcal{U}}}$ is connected.) We have, thus, proven that $j_p$ is a homeomorphism. This implies that the Hausdorff condition is satisfied on ${\overline{{{\mathcal{U}}}}} /V$ and, in particular, on ${{\mathcal{U}}}/V$. Since $V$ is timelike and ${{\mathcal{U}}}$ contains no closed timelike curves, this makes sure that ${{\mathcal{U}}}/V$ is a manifold according to our terminology, see Harris [@Ha], Theorem 2.
In the third step we use these results to prove part (a) of the proposition. Our result that $j_p$ is a homeomorphism implies, in particular, that $\gamma$ has an intersection with $\partial _{{{\mathcal{U}}}} J^- (p, {{\mathcal{U}}})$ at some point $q$. Now part (a) of Lemma \[lem:boundary\] shows that there is a past-pointing lightlike geodesic from $p$ to $q$ in ${{\mathcal{U}}}$. This geodesic cannot contain conjugate points in its interior since otherwise a small variation would give a timelike curve from $p$ to $q$, see Hawking and Ellis [@HE], Proposition 5.4.12, thereby contradicting $q \in \partial _{{{\mathcal{U}}}} J^-(p, {{\mathcal{U}}})$. The rest of part (a) is clear since all past-pointing lightlike geodesics in ${{\mathcal{U}}}$ that start at $p$ are confined to $J^-(p, {{\mathcal{U}}})$.
In the last step we prove part (b). To that end we choose on the tangent space $T_p {{\mathcal{M}}}$ a Lorentz basis $(E^1_p,E^2_p,E^3_p,E^4_p)$ with $E^4_p$ future-pointing, and we identify each $x = (x^1,x^2,x^3) \in {{\mathbb{R}}}^3$ with the past-pointing lightlike vector $Y_p = x^1E^1_p + x^2E^2_p + x^3E^3_p - |x| E^4_p$. With this identification, the lens map takes the form $f_p :
S^2 \longrightarrow {{\mathcal{N}}}= \partial {{\mathcal{U}}}/V \, , \;
x \longmapsto \pi _V \big( {\mathrm{exp}}_p ( w_p (x) x ) \big)$. We now define a continuous map $F: B \longrightarrow {{\mathcal{M}}}/V$ on the closed ball $B = \big\{ x \in {{\mathbb{R}}}^3 \, \big|
\, |x| \le 1 \, \big\}$ by setting $F(x) =
\pi _V \big( {\mathrm{exp}}_p ( w_p (\tfrac{x}{|x|}) \, x ) \big)$ for $x \neq 0$ and $F(0) = \pi _V (p)$. The restriction of $F$ to the interior of ${{\mathcal{B}}}$ is a $C^{\infty}$ map onto the manifold ${{\mathcal{U}}}/V$, with the exception of the origin where $F$ is not differentiable. The latter problem can be circumvented by approximating $F$ in the $C^o$-sense, on an arbitrarily small neighborhood of the origin, by a $C^{\infty}$ map. Then the mapping degree ${{\mathrm{deg}}}(F)$ can be calculated (see, e.g., Choquet-Bruhat, Dewitt-Morette, and Dillard-Bleick [@CDD], pp.477) with the help of the integral formula $$\label{eq:omega}
\int _B F^* \omega = {{\mathrm{deg}}}(F) \, \int_{{{\mathcal{U}}}/V} \omega$$ where $\omega$ is any 3-form on ${{\mathcal{U}}}/V$ and the star denotes the pull-back of forms. For any 2-form $\psi$ on ${{\mathcal{U}}}/V$, we may apply this formula to the form $\omega = d \psi$. With the help of the Stokes theorem we then find $$\label{eq:Stokes}
\int _{S^2} F^* \psi = {{\mathrm{deg}}}(F) \, \int_{{{\mathcal{N}}}} \psi \, .$$ However, the restriction of $F$ to $\partial B = S^2$ gives the lens map, so on the left-hand side of (\[eq:Stokes\]) we may replace $F^* \psi$ by $f_p^* \psi$. Then comparison with the integral formula for the degree of $f_p$ shows that ${{\mathrm{deg}}}(F) =
{{\mathrm{deg}}}(f_p)$ which, according to Proposition \[prop:sln\], is equal to $\pm 1$. For every $\zeta \in {{\mathcal{U}}}/V$ that is a regular value of $F$, the result ${{\mathrm{deg}}}(F) = \pm 1$ implies that the number of elements in $F^{-1} (\zeta )$ is finite and odd. By assumption, the worldline $\gamma \in {{\mathcal{U}}}/V$ meets neither the point $p$ nor the caustic of the past light cone of $p$. The first condition makes sure that our perturbation of $F$ near the origin can be done without influencing the set $F^{-1} (\gamma )$; the second condition implies that $\gamma$ is a regular value of $F$, please recall our discussion at the end of Section \[sec:regular\]. This completes the proof.
If only light rays within ${{\mathcal{U}}}$ are taken into account, then Proposition \[prop:gamma\] can be summarized by saying that, for light sources in a simple lensing neighborhood, the “youngest image” has always even parity and the total number of images is finite and odd.
In the quasi-Newtonian approximation formalism it is a standard result that a transparent gravitational lens produces an odd number of images, see Schneider, Ehlers and Falco [@SEF], Section 5.4, for a detailed discussion. Proposition \[prop:gamma\] may be viewed as a reformulation of this result in a Lorentzian geometry setting. It is quite likely that an alternative proof of Proposition \[prop:gamma\] can be given by using the Morse theoretical results of Giannoni, Masiello and Piccione [@GMP1; @GMP2]. Also, the reader should compare our results with the work of McKenzie [@McK] who used Morse theory for proving an odd-number theorem in certain globally hyperbolic spacetimes. Contrary to McKenzie’s theorem, our Proposition \[prop:gamma\] requires mathematical assumptions which can be physically interpreted rather easily.
Examples {#sec:examples}
========
Two simple examples with non-transparent deflectors {#subsec:string}
---------------------------------------------------
[**(a) Non-transparent string**]{}
As a simple example, we consider gravitational lensing in the spacetime $({{\mathcal{M}}},g)$ where ${{\mathcal{M}}}= {{\mathbb{R}}}^2
\times \big( {{\mathbb{R}}}^2 \setminus \{0\} \big)$ and $$\label{eq:string}
g = -dt^2 + dz^2 + dr^2 + k^2\, r^2 \, d{\varphi}^2$$ with some constant $0 < k < 1$. Here $(t,z)$ denote Cartesian coordinates on ${{\mathbb{R}}}^2$ and $(r,\varphi )$ denote polar coordinates on ${{\mathbb{R}}}^2 \setminus \{0\}$. This can be interpreted as the spacetime around a static non-transparent string, see Vilenkin [@Vi], Hiscock [@Hc] and Gott [@Go]. One should think of the string as being situated at the $z$-axis. Since the latter is not part of the spacetime, it is indeed justified to speak of a [*non-transparent*]{} string.
As $\partial / \partial t$ is a Killing vector field normalized to $-1$, the lightlike geodesics in $({{\mathcal{M}}},g)$ correspond to the geodesics of the space part. The latter is a metrical product of a real line with coordinate $z$ and a cone with polar coordinates $(r,\varphi)$. So the geodesics are straight lines if we cut the cone open along some radius $\varphi = {\mathrm{const.}}$ and flatten it out in a plane. Owing to this simple form of the lightlike geodesics, the investigation of lens maps in this string spacetime is quite easy.
To work this out, choose some constant $R>0$ and let ${{\mathcal{T}}}$ denote the hypercylinder $r = R$ in ${{\mathcal{M}}}$. Let $W$ denote the restriction of the vector field $\partial / \partial t$ to ${{\mathcal{T}}}$. Then $({{\mathcal{T}}},W)$ is a source surface, with ${{\mathcal{N}}}= {{\mathcal{T}}}/W \simeq
S^1 \times {{\mathbb{R}}}$. Henceforth we discuss the lens map $f_p$ for any point $p \in {{\mathcal{M}}}$ at a radius $r < R$. There are no past-pointing lightlike geodesics from $p$ that intersect ${{\mathcal{T}}}$ more than once or touch ${{\mathcal{T}}}$ tangentially, so the lens map $f_p$ gives full information about all images at $p$ of each light source $\xi \in {{\mathcal{N}}}$. The domain ${{\mathcal{D}}}_p$ of the lens map is given by excising a curve segment, namely a meridian including both end-points at the “poles”, from the celestial sphere ${{\mathcal{S}}}_p$, so ${{\mathcal{D}}}_p \simeq {{\mathbb{R}}}^2$ is connected. The boundary of ${{\mathcal{D}}}_p$ in ${{\mathcal{S}}}_p$ corresponds to light rays that are blocked by the string before reaching ${{\mathcal{T}}}$. It is easy to see that the lens map cannot be continuously extended onto ${{\mathcal{S}}}_p$ (= closure of ${{\mathcal{D}}}_p$ in ${{\mathcal{S}}}_p$). Nonetheless, the lens map admits an extension in the sense of Definition \[def:extend\]. We may choose ${{\mathcal{M}}}_1 = S^2$ and ${{\mathcal{M}}}_2 = S^2$. Here ${{\mathcal{D}}}_p$ is embedded into the sphere in such a way that it covers a region $(\theta , \varphi ) \in \; ]0,\pi [ \; \times \;
]\, \varepsilon \, , 2 \pi - \varepsilon [ \,$, i.e., in comparison with the embedding into ${{\mathcal{S}}}_p$ the curve segment excised from the sphere has been “widened” a bit. The embedding of ${{\mathcal{N}}}\simeq S^1 \times {{\mathbb{R}}}$ into $S^2$ is made via Mercator projection.
As the string spacetime has vanishing curvature, the light cones in ${{\mathcal{M}}}$ have no caustics. Owing to our general results of Section \[sec:regular\], this implies that the caustic of the lens map is empty and that all images have even parity, so (\[eq:degn\]) gives ${{\mathrm{deg}}}( {\overline{f_p}} , \xi ) =
n_+ (\xi) = n(\xi)$ for all $\xi \in {{\mathcal{N}}}\setminus
{\overline{f_p}}(\partial {{\mathcal{D}}}_p )$.
The actual value of $n (\xi)$ depends on the parameter $k$ that enters into the metric (\[eq:string\]). If $i = 1/k$ is an integer, ${{\mathcal{N}}}\setminus {\overline{f_p}}(\partial {{\mathcal{D}}}_p)$ is connected and $n (\xi) = i$ everywhere on this set. If $i <
1/k < i+1$ for some integer $i$, ${{\mathcal{N}}}\setminus
{\overline{f_p}}(\partial {{\mathcal{D}}}_p)$ has two connected components, with $n (\xi ) = i$ on one of them and $n (\xi ) = i+1$ on the other. Thus, the string produces multiple imaging and the number of images is (finite but) arbitrarily large if $k$ is sufficiently small.
For all $k \in \; ]0,1[ \,$, the lens map is surjective, $f_p ({{\mathcal{D}}}_p) = {{\mathcal{N}}}\simeq S^1 \times {{\mathbb{R}}}$. So this example shows that the assumption of $f_p({{\mathcal{D}}}_p)$ being simply connected was essential in Proposition \[prop:cover\].
[**(b) Non-transparent spherical body**]{}
We consider the Schwarzschild metric $$\label{eq:ss}
g = \big( \, 1 \, - \tfrac{2m}{r} \, \big)^{-1} dr^2 +
r^2 \big( d \theta ^2 +
{\mathrm{sin}}^2 \theta \, d \varphi ^2 \big) -
\big( \, 1 \, - \tfrac{2m}{r} \, ) \, dt^2$$ on the manifold ${{\mathcal{M}}}= \; ]R_o,\infty [ \; \times \, S^2
\times {{\mathbb{R}}}$. In (\[eq:ss\]), $r$ is the coordinate ranging over $\, ]R_o, \infty [ \;$, $t$ is the coordinate ranging over ${{\mathbb{R}}}$, and $\theta$ and $\varphi$ are spherical coordinates on $S^2$. This gives the static vacuum spacetime around a spherically symmetric body of mass $m$ and radius $R_o$. Restricting the spacetime manifold to the region $r > R_o$ is a way of treating the central body as non-transparent. In the following we keep a value $R_o > 0$ fixed and we allow $m$ to vary between $m = 0 $ (flat space) and $m = R_o/2$ (black hole).
For discussing lens maps in this spacetime we fix a constant $R > 3 R_o /2$. We denote by ${{\mathcal{T}}}$ the set of all points in ${{\mathcal{M}}}$ with coordinate $r = R$ and we denote by $W$ the restriction of $\partial / \partial t$ to $W$. Then $({{\mathcal{T}}},W)$ is a source surface, with ${{\mathcal{N}}}= {{\mathcal{T}}}/W \simeq S^2$. It is our goal to discuss the properties of the lens map $f_p : {{\mathcal{D}}}_p \longrightarrow {{\mathcal{N}}}$ for a point $p \in {{\mathcal{M}}}$ with a radius coordinate $r < R$ in dependence of the mass parameter $m$. To that end we make use of well-known properties of the lightlike geodesics in the Schwarzschild metric, see, e.g., Chandrasekhar [@Ch], Section 20, for a comprehensive discussion. For determining the relevant features of the lens map it will be sufficient to concentrate on qualitative aspects of image positions. For quantitative aspects the reader may consult Virbhadra and Ellis [@VE].
We first observe that, for any $m \in [0,R_o/2]$, there is no past-pointing lightlike geodesic from $p$ that intersects ${{\mathcal{T}}}$ more than once or touches ${{\mathcal{T}}}$ tangentially. This follows from the fact that in the region $r > 3m$ the radius coordinate has no local maximum along any light ray. So the lens map $f_p$ gives full information about all images at $p$ of light sources $\xi \in {{\mathcal{N}}}$.
(10,10) (5,5) (5,5) (5,9) (5,1) (5,8) (5,8, 3,4.4, 5,1) (5,8, 7,4.4, 5,1) (5.4,8.1)[$p$]{} (4.8,9.3)[$\xi_N$]{} (4.8,0.5)[$\xi_S$]{} (4.3,3.6)[$r = R_o$]{} (7.5,1.6)[$r = R$]{}
For $m=0$, the light rays are straight lines. The domain ${{\mathcal{D}}}_p$ of the lens map is given by excising a disc, including the boundary, from the celestial sphere ${{\mathcal{S}}}_p$, i.e., ${{\mathcal{D}}}_p
\simeq {{\mathbb{R}}}^2$. The boundary of ${{\mathcal{D}}}_p$ corresponds to light rays grazing the surface of the central body, so $f_p$ can be continuously extended onto the closure of ${{\mathcal{D}}}_p$ in ${{\mathcal{S}}}_p$, thereby giving an extension of $f_p$, in the sense of Definition \[def:extend\], ${\overline{f_p}} :
{\overline{{{\mathcal{D}}}_p}} \subseteq {{\mathcal{S}}}_p \longrightarrow {{\mathcal{N}}}$. In Figure \[fig:ss\], ${\overline{f_p}}(\partial {{\mathcal{D}}}_p)$ can be represented as a “circle of equal latitude” on the sphere $r=R$, with the image of $f_p$ “to the north” of this circle. With increasing $m$, ${\overline{f_p}}(\partial {{\mathcal{D}}}_p)$ moves “south” until, at some value $m = m_1$, it has reached the “south pole” $\xi_S$. This is the situation depicted in Figure \[fig:ss\]. From now on the lens map is surjective and $\xi_S$ is seen as an [*Einstein ring*]{}, thereby indicating that a caustic has formed. Now ${\overline{f_p}} (\partial {{\mathcal{D}}}_p)$ moves north until, at some value $m = m_1'$, it has reached the “north pole” $\xi_N$. From now on $\xi_N$ is seen as an Einstein ring, in addition to the regular image that exists from the beginning. With further increasing $m$, we find an infinite sequence of values $0 < m_1 < m_1' < m_2 < m_2' <
\dots < m_i < m_i' < \dots $ such that at $m = m_i$ a new Einstein ring of $\xi_S$ and at $m_i'$ a new Einstein ring of $\xi_N$ comes into existence. For all intermediate values of $m$, ${\overline{f_p}}(\partial {{\mathcal{D}}}_p)$ divides ${{\mathcal{N}}}$ into two connected components. All points $\xi$ in the southern component, with the exception of the south pole $\xi_S$, are regular values of the lens map. $f_p^{-1}(\xi )$ consists of exactly $2i$ points where $i$ is the largest integer with $m_i < m$. There are $i$ images of even parity, $n_+(\xi) = i$, and $i$ images of odd parity, $n_-(\xi) = i$, hence ${{\mathrm{deg}}}({\overline{f_p}}, \xi )
= n_+(\xi) - n_-(\xi) = 0$. Similarly, all points $\xi$ in the northern component, with the exception of the north pole $\xi_N$, are regular values of the lens map. $f_p^{-1}(\xi )$ consists of exactly $2i+1$ points, where $i$ is the largest integer with $m_i' < m$. There are $i+1$ images of even parity, $n_+(\xi) = i+1$, and $i$ images of odd parity, $n_-(\xi) = i$, hence ${{\mathrm{deg}}}({\overline{f_p}} , \xi ) = n_+(\xi) - n_-(\xi) = 1$. Both sequences $(m_i)_{i \in {{\mathbb{N}}}}$ and $(m_i')_{i \in {{\mathbb{N}}}}$ converge towards $m = R_o/3$. For $m \ge R_o/3$, the boundary of ${{\mathcal{D}}}_p$ corresponds to light rays that approach the sphere $r = 3m$ asymptotically in a neverending spiral motion, cf. Chandrasekhar [@Ch], Figure 9 and Figure 10. The lens map no longer admits an extension in the sense of Definition \[def:extend\], so we cannot assign a mapping degree to it. There are infinitely many concentric Einstein rings for both poles, and infinitely many isolated images for all other $\xi \in {{\mathcal{N}}}$, with both $n_+(\xi) $ and $n_-(\xi)$ being infinite. These features remain unchanged until the black-hole case $m = R_o/2$ is reached.
The fact that in this case the caustic of the lens map consists of just two points is rather exceptional. After a small perturbation of the spherical symmetry the caustic would show a completely different behavior. For regular $\xi \in {{\mathcal{N}}}$, however, the statements about $n_{\pm}(\xi )$ are stable against small perturbations.
Having studied Schwarzschild spacetimes around non-transparent bodies, the reader might ask what about transparent bodies, i.e., what about matching an interior solution to the exterior Schwarzschild solution at Radius $R_o$, with $R_o > 2m$, and allowing for light rays passing through the interior region. If $R_o > 3m$, and if there are no light rays trapped within the interior region, the resulting spacetime will be asymptotically simple and empty. Qualitative features of lens maps in this class of spacetimes are discussed in the following subsection. For a more explicit discussion of lens maps in the Schwarzschild spacetime of a transparent body, choosing a perfect fluid with constant density for the interior region, the reader is refered to Kling and Newman [@KNm].
Asymptotically simple and empty spacetimes {#subsec:asy}
------------------------------------------
Asymptotically simple and empty spacetimes are considered to be good models for the gravitational fields of transparent gravitating bodies that can be viewed as isolated from all other masses in the universe. The formal definition, which is essentially due to Penrose [@Pn], cf., e.g. Hawking and Ellis [@HE], p. 222, reads as follows.
\[def:asy\] A spacetime $({{\mathcal{M}}},g,)$ is called [*asymptotically simple*]{} if there is a strongly causal spacetime $({\tilde{{{\mathcal{M}}}}},{\tilde{g}})$ with the following properties.\
(a) ${{\mathcal{M}}}$ is an open submanifold of ${\tilde{{{\mathcal{M}}}}}$ with a non-empty boundary $\partial {{\mathcal{M}}}\,$.\
(b) There is a $C^{\infty}$ function $\Omega :
{\tilde{{{\mathcal{M}}}}} \longrightarrow {\mathbb{R}}$ such that ${{\mathcal{M}}}=
\{ \, p \in {\tilde{{{\mathcal{M}}}}} \, | \; \Omega (p) > 0 \; \}$, $\partial {{\mathcal{M}}}=
\{ \, p \in {\tilde{{{\mathcal{M}}}}} \, | \, \Omega (p) = 0 \, \}$, $d \Omega \neq 0$ everywhere on $\partial {{\mathcal{M}}}$ and ${\tilde{g}} = \Omega^2 \, g \,$ on ${{\mathcal{M}}}\,$.\
(c) Every inextendible lightlike geodesic in ${\mathcal{M}}$ has past and future end-point on $\partial {{\mathcal{M}}}\,$.\
$({{\mathcal{M}}},g)$ is called [*asymptotically simple and empty*]{} if, in addition,\
(d) there is a neighborhood ${{\mathcal{V}}}$ of $\partial {{\mathcal{M}}}$ in ${\tilde{{{\mathcal{M}}}}}$ such that the Ricci tensor of $g$ vanishes on ${{\mathcal{V}}}\cap {{\mathcal{M}}}$.
Condition (d) of Definition \[def:asy\] is a way of saying that, sufficiently far away from the gravitating body under consideration, Einstein’s vacuum field equation is satisfied. This assumption is reasonable for the spacetime around an isolated body producing gravitational lensing as long as cosmological aspects can be ignored.
The assumptions (a)–(d) of Definition \[def:asy\] imply that $\partial {{\mathcal{M}}}$ is a $\tilde{g}$-lightlike hypersurface in ${\tilde{{{\mathcal{M}}}}}$ that has two connected components, usually denoted by $\J^+$ and $\J^-$ (cf., e.g., Hawking and Ellis, [@HE], p.222). Every inextendible lightlike geodesic in ${{\mathcal{M}}}$ has future end-point on $\J^+$ and past end-point on $\J^-$.
In the following we concentrate on $\J^-$ which is the relevant quantity in view of gravitational lensing. By construction, $\J^-$ is ruled by the integral curves of the ${\tilde{g}}$-gradient $Z$ of $\Omega$. (In coordinate notation, the vector field $Z$ is defined by $Z^a = {\tilde{g}}{}^{ab} \, \partial _b \Omega$ on $\J^-$.) It is well known that $Z$ is regular, with $\J^- /Z$ being diffeomorphic to $S^2$, and that the natural projection $\pi _Z : \J^- \longrightarrow
\J^- /Z \simeq S^2$ makes $\J^-$ into a trivializable fiber bundle with typical fiber diffeomorphic to ${{\mathbb{R}}}$. For a full proof we refer to Newman and Clarke [@NC; @Ne]. (The argument given in Hawking and Ellis [@HE], Proposition 6.9.4, which is due to Geroch [@Ge], is incomplete.) This result can be translated into our terminology in the following way.
\[prop:scri\] In the case of an asymptotically simple and empty spacetime, $(\J^- , Z)$ is a source surface in the spacetime $({\tilde{{{\mathcal{M}}}}}, {\tilde{g}})$, with ${{\mathcal{N}}}= \J^- /Z$ diffeomorphic to $S^2$.
Each integral curve of $Z$ can be written as the $C^1$-limit of a sequence $(\gamma _i)_{i \in {{\mathbb{N}}}}$ of timelike curves in ${{\mathcal{M}}}$. We may interpret the $\gamma _i$ as a sequence of worldlines of light sources approaching infinity. From the viewpoint of the physical spacetime $({{\mathcal{M}}},g)$, it is thus justified to interpret the integral curves of $Z$ as “light sources at infinity”. With respect to the unphysical metric ${\tilde{g}}$, these worldlines are lightlike. With respect to the physical metric, however, they have no causal character at all, because the metric $g$ is not defined on $\J^-$. It is, thus, a misinterpretation to say that the “light sources at infinity” move at the speed of light.
We shall now show that the formalism of “simple lensing neighborhoods” applies to the situation at hand. To that end, we observe that $\J^-$ is the boundary of ${{\mathcal{M}}}$ in the manifold ${\tilde{M}} \setminus \J^+$. This gives rise to the following result.
\[prop:asln\] In the case of an asymptotically simple and empty spacetime, $({{\mathcal{M}}}, \J^-,Z)$ is a simple lensing neighborhood in the spacetime $({\tilde{{{\mathcal{M}}}}} \setminus \J^+ , {\tilde{g}}|
_{{\tilde{{{\mathcal{M}}}}} \setminus \J^+})$.
Condition (a) of Definition \[def:sln\] is obvious from Definition \[def:asy\] and Condition (b) was just established. The proof of the remaining two conditions is based on the fact that on ${{\mathcal{M}}}$ the $g$-lightlike geodesics coincide with the ${\tilde{g}}$-lightlike geodesics (up to affine parametrization). Condition (d) of Definition \[def:sln\] is satisfied since every lightlike geodesic in ${{\mathcal{M}}}$ has past end-point on $\J^-$ and future end-point on $\J^+$. Moreover, the arrival on $\J^{\pm}$ must be transverse since $\J^{\pm}$ is ${\tilde{g}}$-lightlike. This shows that Condition (c) of Definition \[def:sln\] is satisfied as well.
We can, thus, apply our results on simple lensing neighborhoods to asymptotically simple and empty spacetimes. As a first result, Proposition \[prop:sln\] tells us that every asymptotically simple and empty spacetime ${{\mathcal{M}}}$ must be contractible. This result is not new. It is well known that every asymptotically simple and empty spacetime is globally hyperbolic and, thus, homeomorphic to a product of a Cauchy surface ${{\mathcal{C}}}$ with the real line, ${{\mathcal{M}}}\simeq {{\mathcal{C}}}\times {{\mathbb{R}}}$, and that ${{\mathcal{C}}}$ is contractible. For a full proof we refer again to Newman and Clarke [@NC; @Ne]. The stronger result that ${{\mathcal{C}}}$ must be homeomorphic to ${{\mathbb{R}}}^3$ requires the assumption that the Poincar' e conjecture is true (i.e., that every simply connected and compact 3-manifold is homeomorphic to $S^3$).
In addition, Proposition \[prop:sln\] gives us the following result.
\[prop:asyb\] In the case of an asymptotically simple and empty spacetime, for all $p \in {{\mathcal{M}}}$ the lens map $f_p :
{{\mathcal{S}}}_p \longrightarrow \J^- /Z \simeq S^2$ has $| {{\mathrm{deg}}}(f_p) |= 1$.
The lens map $f_p$ for “light sources at infinity” in an asymptotically simple and empty spacetime was already discussed in Perlick [@Pe1; @Pe2]. In particular, a proof of the result ${{\mathrm{deg}}}(f_p) = 1$ was given in Theorem 6 of [@Pe1]. An equivalent statement, using a different terminology, can be found as Lemma 1 in Kozameh, Lamberti and Reula [@KLR], together with a short proof. However, both these earlier proofs are incomplete. The proof in [@Pe1] is based on the idea to homotopically deform $f_p$ into the identity, but it is not shown that the construction can be made in such a way that the dependence on the deformation parameter is, indeed, continuous. In [@KLR], the authors write the future light cone (or, equivalently, the past light cone) of a point $p \in {{\mathcal{M}}}$ as the image of a map $\Phi : \: ]0, \infty[
\: \, \times S^2 \longrightarrow {{\mathcal{M}}}$, and they assign a [*winding number*]{} to each map $\Phi (s, \cdot )$. Since a winding number has to refer to a “center”, the authors in [@KLR] apparently take for granted that there is a timelike curve through $p$ that has no further intersection with the light cone of $p$. The existence of such a curve, however, is an open question. With our Proposition \[prop:sln\] we have filled these gaps insofar as we have established the result ${{\mathrm{deg}}}(f_p) = \pm 1$. However, we have not shown whether, with our choice of orientations, the occurence of the minus sign can be ruled out.
Proposition \[prop:asyb\] implies that every observer in $p$ sees an odd number of images of each light source at infinity that does not pass through the caustic of the past light cone of $p$. (Here one has to refer to the ${\tilde{g}}$-cone which is an extension of the $g$-cone.) As an immediate consequence of Proposition \[prop:gamma\], we find that a similar statement is true for light sources inside ${{\mathcal{M}}}$, see Figure \[fig:asy\].
\[prop:asygamma\] Fix a point $p$ and a timelike embedded $C^{\infty}$curve $\gamma$ in an asymptotically simple and empty spacetime $({{\mathcal{M}}},g)$. Assume that the image of $\gamma$ is a closed subset of ${\tilde{{{\mathcal{M}}}}} \setminus \J ^+$ and that $\gamma$ meets neither the point $p$ nor the caustic of the past light cone of $p$. Then the number of past-pointing lightlike geodesics from $p$ to $\gamma$ in ${{\mathcal{M}}}$ is finite and odd.
(8,9) (2,1, 2,9) (2,1, 6,5) (6,5, 2,9) (2,1, 3.5,4, 4,7) (3,6.5) (2.5,6.5)[$p$]{} (4,4.5)[$\gamma$]{} (4.3,2.5)[$\J^-$]{} (4.3,7.5)[$\J^+$]{}
Let us conclude this subsection with a few remarks on spacetimes that are asymptotically simple but not empty. For any asymptotically simple spacetime it is easy to verify that $\partial {{\mathcal{M}}}$ has either one or two connected components, and that all lightlike geodesics in ${{\mathcal{M}}}$ have their past end-point in the same connected component of $\partial {{\mathcal{M}}}$. Let us denote this component by $\J^-$ henceforth. In order to apply our formalism of simple lensing neighborhoods the additional assumptions needed are that $\J^-$ is a fiber bundle with ${\tilde{g}}$-causal fibers diffeomorphic to ${{\mathbb{R}}}$ over an orientable basis manifold, and that all past-inextendible lightlike geodesics in ${{\mathcal{M}}}$ meet $\J^-$ transversely. If these assumptions are satisfied, our results on simple lensing neighborhoods apply. In particular, $\J^-$ must be diffeomorphic to $S^2 \times {{\mathbb{R}}}$ and ${{\mathcal{M}}}$ must be contractible.
As an interesting special case, we might modify Condition (d) of Definition \[def:asy\] by requiring the Ricci tensor of $g$ to be equal to $\Lambda \, g$ near $\partial {{\mathcal{M}}}$ with a positive or negative cosmological constant $\Lambda$. The resulting spacetimes are called [*asymptotically deSitter*]{} for $\Lambda > 0$ and [*asymptotically anti-deSitter*]{} for $\Lambda < 0$. It was verified already by Penrose [@Pn] that then $\partial {{\mathcal{M}}}$ is ${\tilde{g}}$-spacelike for $\Lambda > 0$ and ${\tilde{g}}$-timelike for $\Lambda < 0$. Thus, the formalism of simple lensing neighborhoods is inappropriate for investigating asymptotically deSitter spacetimes, but it may be used for the investigation of asymptotically anti-deSitter spacetimes.
Weakly perturbed Robertson-Walker spacetimes {#subsec:RW}
--------------------------------------------
It is a characteristic feature of the lens map, as defined in this paper, that it is constructed by following each past-pointing lightlike geodesic up to its first intersection with the source surface only. Further intersections are ignored, i.e., some images are willfully excluded from the gravitational lensing discussion. In the preceding examples no such further intersections occured. We shall now discuss an example where they do occur but where it is physically well motivated to disregard them.
To that end we start out with a spacetime $({{\mathcal{M}}},g)$ with ${{\mathcal{M}}}= S^3 \times {{\mathbb{R}}}$ and $$\label{eq:RW}
g = R(t)^2 \big(-dt^2 + d \chi ^2 + {\mathrm{sin}}^2 \chi
( d \theta ^2 + {\mathrm{sin}} ^2 \theta \, d \phi ^2 ) \big) \, .$$ Here $\chi \in [0, \pi ]$, $\theta \in [0, \pi ]$ and $\phi \in [0,2\pi]$ denote standard coordinates on $S^3$ (with the usual coordinate singularities), $t$ denotes the projection from ${{\mathcal{M}}}= S^3 \times {{\mathbb{R}}}$ onto ${{\mathbb{R}}}$, and $R : {{\mathbb{R}}}\longrightarrow {{\mathbb{R}}}$ is a strictly positive but otherwise arbitrary $C^{\infty}$ function. This is the general form of a Robertson-Walker spacetime with positive spatial curvature and natural topology which has no particle horizons. (Particle horizons are excluded by the assumption that the “conformal time” $t$ runs over all of ${{\mathbb{R}}}$.)
Now fix a coordinate value $\chi_o \in \; ]\, 0 \, , \pi /2[ \;$ and let ${{\mathcal{U}}}$ denote the set of all points in ${{\mathcal{M}}}$ whose $\chi$-coordinate is smaller than $\chi_o$. Let $W$ denote the restriction of the vector field $\partial / \partial t$ to the boundary $\partial U$. Then $({{\mathcal{U}}}, \partial {{\mathcal{U}}}, W)$ is a simple lensing neighborhood. This is easily verified using the fact that the lightlike geodesics in ${{\mathcal{M}}}$ project to the geodesics of the standard metric on $S^3$. Our assumptions that $t$ ranges over all of ${{\mathbb{R}}}$ and that $\chi_o < \pi /2$ are essential to make sure that, for all $p \in {{\mathcal{U}}}$, the lens map is defined on all of ${{\mathcal{S}}}_p$. In the case at hand, the lens map $f_p : {{\mathcal{S}}}_p
\longrightarrow \partial {{\mathcal{U}}}/W$ is a global diffeomorphism for all points $p \in {{\mathcal{U}}}\,$. Actually, there are infinitely many past-pointing lightlike geodesics from any fixed $p \in {{\mathcal{U}}}$ to any fixed $\xi \in \partial {{\mathcal{U}}}/W$, but only one of them reaches $\xi$ without having left ${{\mathcal{U}}}$. All the other ones make at least a half circle around the whole universe, so they will give rise to rather faint images as a consequence of absorption in the intergalactic medium. It is, thus, reasonable to assume that only the one image which enters into the lens map is actually visible. In this sense, disregarding all the other light rays is physically well motivated. Please note that all the infinitely many images of $\xi$ are situated at just two points of the celestial sphere at $p \,$; the two brightest images cover all the other ones.
Now this example is boring in view of gravitational lensing because the lens map is a global diffeomorphism. However, we can switch to a more interesting situation by choosing a compact subset ${{\mathcal{K}}}\in S^3$ and modifying the metric on the set ${{\mathcal{K}}}\times {{\mathbb{R}}}$. In view of Einstein’s field equation, this can be interpreted as introducing local mass concentrations that act as gravitational lens deflectors. If ${{\mathcal{K}}}\times
{{\mathbb{R}}}$ is completely contained in ${{\mathcal{U}}}$, and if the modification of the metric is sufficiently small to make sure that, even after the modification, no light rays are past- or future-trapped inside ${{\mathcal{U}}}$, then ${{\mathcal{U}}}$ remains a simple lensing neighborhood. We have, thus, Proposition \[prop:sln\] at our disposal. Under the (very mild) additional assumption that, even after the perturbation, there are no closed timelike curves in ${{\mathcal{U}}}$, we may also use Proposition \[prop:gamma\]. This is a line of argument to the effect that, in a Robertson-Walker spacetime of the kind considered here, any transparent gravitational lens deflector produces an odd number of visible images. The assumption that there are no particle horizons was essential since otherwise the lens map would not be defined on the whole celestial sphere for all $p \in {{\mathcal{U}}}$.
A similar argument applies, of course, to Robertson-Walker spacetimes with non-compact spatial sections. Then we don’t have to care about light rays traveling around the whole universe, so there are no additional images which are ignored by the lens map.
[99]{} Schneider, P., Ehlers, J., Falco, E.: [*Gravitational lenses*]{}, Springer, New York (1992) Frittelli, S., Newman E.: [*Phys. Rev. D*]{} [**59**]{}, 124001 (1999) Ehlers, J., Frittelli, S., Newman, E.: in J. Renn (ed.), [ *Festschrift in honor of John Stachel*]{}, to appear 2000 Ehlers, J.: [*Annalen der Physik*]{} (Leipzig), [**9**]{}, 307 (2000) Palais, R.: [*Ann. Math.*]{} [**73**]{}, 295 (1961) Whitney, H.: [*Ann. Math.*]{} [**37**]{}, 645 (1936) Hirsch, M. W.: [*Differential topology*]{}, Springer, New York (1976) Abraham, R., Marsden, J.: [*Foundations of mechanics*]{}, Benjamin-Cummings, Reading, Massachusetts (1978) Kobayashi, S., Nomizu, K.: [*Foundations of differential geometry. Vol.I*]{}, Wiley-Interscience, New York (1963), p.58 Harris, S.: [*Class. Quantum Grav.*]{} [**9**]{}, 1823 (1992) Beem, J., Ehrlich, P., Easley, K.: [*Global Lorentzian geometry*]{}, Dekker, New York (1996) Choquet-Bruhat, Y., Dewitt-Morette, C., Dillard-Bleick, M.: [*Analysis, manifolds and physics*]{}, North-Holland, Amsterdam (1977) p.477 Dold, A.: [*Lectures on algebraic topology*]{}, Springer, Berlin (1980) Spanier, E.: [*Algebraic topology*]{}, McGraw Hill, New York (1966) Bredon, G. E.: [*Topology and geometry*]{}, Springer, New York (1993) McKenzie, R. H.: [*J. Math. Phys.*]{} [**26**]{}, 1592 (1985) Newman, R. P. C., Clarke, C. J. S.: [*Class. Quantum Grav.*]{} [**4**]{}, 53 (1987) Newman, R. P. C.: [*Commun. Math. Phys.*]{} [**123**]{}, 17 (1989) Frankel, T.: [*The geometry of physics*]{}, Cambridge UP (1997) Wald, R.: [*General relativity*]{}, University of Chicago Press, Chicago (1984) Hawking, S., Ellis, G.: [*The large scale structure of space-time*]{}, Cambridge UP (1973) Giannoni, F., Masiello, A., Piccione, P.: [*Commun. Math. Phys.*]{} [**187**]{}, 375 (1997) Giannoni, F., Masiello, A., Piccione, P.: [*Ann. Inst. H. Poincar' e, Physique Theoretique*]{} [**69**]{}, 359 (1998) Vilenkin, A.: [*Phys. Rev. D*]{} [**23**]{} , 852 (1981) Hiscock, W.: [*Phys. Rev. D*]{} [**31**]{}, 3288 (1985) Gott, J. R.: [*Astrophys. J.*]{} [**288**]{}, 422(1985) Virbhadra, K. S., and Ellis, G. F. R.: [*Schwarzschild black hole lensing*]{}, astro-ph/9904193 Chandrasekhar, S.: [*The mathematical theory of black holes*]{}, Oxford UP, Oxford (1983) Kling, T., Newman, E. T.: [*Phys. Rev. D.*]{} [**59**]{}, 124002 (1999) Penrose, R.: in deWitt, C. M., deWitt, B. (Eds.) [*Relativity, groups and topology*]{}, Les Houches Summer School 1963. Gordon and Breach, New York, 565 (1964) Geroch, R.: in Sachs, R. K. (ed.) [*General relativity and cosmology*]{}, Enrico Fermi School, Course XLVII. Academic Press, New York, 71–103 (1971) Perlick, V.: in Schmidt, B. (ed.), [*Einstein’s field equations and their physical implications*]{}, Springer, Heidelberg (2000) Perlick, V.: [*Ann. Physik*]{} (Leipzig) [**9**]{} , SI–139 (2000) Kozameh, C., Lamberti, P. W., Reula, O.: [*J. Math. Phys.*]{} [**32**]{}, 3423 (1991)
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abstract: 'This paper presents a distributed algorithm for finding near optimal dominating sets on grids. The basis for this algorithm is an existing centralized algorithm that constructs dominating sets on grids. The size of the dominating set provided by this centralized algorithm is upper-bounded by $\left\lceil\frac{(m+2)(n+2)}{5}\right\rceil$ for $m\times n$ grids and its difference from the optimal domination number of the grid is upper-bounded by five. Both the centralized and distributed algorithms are generalized for the $k$-distance dominating set problem, where all grid vertices are within distance $k$ of the vertices in the dominating set.'
author:
- 'Elaheh Fata Stephen L. Smith Shreyas Sundaram [^1] [^2]'
title: Distributed Dominating Sets on Grids
---
Introduction {#sec:intro}
============
Significant attention has been devoted in recent years to the study of large-scale sensor and robotic networks due to their promise in fields such as environmental monitoring [@NEL-DP-FL-RS-DMF-RD:07], inventory warehousing [@Guizzo08], and reconnaissance [@RWB-TWM-MAG-EPA:02]. One of the key objectives in such networks is to ensure [*coverage*]{} of a given area, where every point in the space is within the sensing radius of one or more of the agents (i.e., sensors or robots). Various algorithms have been proposed to achieve coverage based on differing assumptions on the mobility and sensing capabilities of the agents [@DistCtrlRobotNetw].
In certain scenarios, the environment may impose restrictions on the feasible locations and motion of the agents [@SML:06]. In such cases, it is natural to model the environment as a [*graph*]{}, where each node represents a feasible location for an agent, and edges between nodes indicate available paths for the agents to follow. The coverage capabilities of any given agent are then related to the shortest-path distance metric on the graph: an agent located on a node can cover all nodes within a certain distance of that node. The goal of selecting certain nodes in a graph so that all other nodes are within a specified distance of the selected nodes is classical in graph theory, and is known as the [*dominating set problem*]{} [@GJ'79]. Versions of this problem appear in settings such as multi-agent security and pursuit [@Abbas12], routing in communication networks [@Wu01], and sensor placement in power networks [@Dorfling06].
Finding the [*domination number*]{} (i.e., the size of a smallest dominating set) of arbitrary graphs is NP-hard [@GJ'79]. In fact, Raz and Safra showed that achieving an approximation ratio better than $c\log n$ for the dominating set problem in general graphs is NP-hard, where $c>0$ is some constant and $n$ is the number of vertices of the graph [@RazSaf'97]. However, there are several algorithms known for the dominating set problem for which the ratio between the size of the resulting dominating set and graph domination number can closely reach the $c\log n$ bound. The simplest of these algorithms is a *greedy algorithm* that at each step adds one vertex to the dominating set. The vertices that are already in the dominating set are marked as ‘black’, the vertices that share edges with black vertices are marked as ‘gray’ and other vertices are ‘white’. At each step, a white vertex that shares the maximum number of edges with other white vertices are added to the dominating set and the colour labels of all vertices are updated according to the aforementioned rules. Another widely used approximation algorithm for the problem uses a *linear programming relaxation*. Both greedy and linear programming approaches for the dominating set problem are known to have $(\ln n+1)$-approximation ratios [@Johnson'74; @lovasz'75], which are in $O(\log n)$.
Even though in general graphs one cannot obtain an approximation ratio in $o(\log n)$, in special types of graphs better approximation ratios are obtainable. One of the most important classes of graphs are *planar graphs*. A *planar graph* is a graph that can be drawn in a plane so that none of its edges intersect except at their ends [@BM'08]. The dominating set problem is still NP-hard for planar graphs; however, the domination number of this type of graphs can be approximated within a factor of $(1+\epsilon)$ for an arbitrarily small $\epsilon>0$ [@Baker'94].
Grid graphs are a special class of graphs that have attracted attention due to their ability to model and discretize rectangular environments [@LBTD'03; @LJDKM'00]. Grids can be used in simplifying the underlying environment and limiting energy consumption by representing a certain area of the environment with only one node in the grid [@BJDMTACX'05]. Moreover, grid graphs, due to their special structure that do not leave any area of environment unrepresented while transferring the problem environment into a tractable domain, can successfully provide efficient area coverage and hence are used very commonly in the network coverage and delectability literature [@LBTD'03; @CW'04]. All these application motivated us in studying the dominating set problem when the underlying graph is a grid.
As discussed above, it is NP-hard to find the domination number of general or even planar graphs. It can be easily observed that grid graphs lie in the class of planar graphs and hence their domination number can be obtained within a small ratio. However, due to the special structure of grids, their domination number can in fact be determined optimally, although the path to obtaining the exact domination number of grids was not straightforward. For $m \times n$ grid graphs, the size of the optimal dominating set was unknown until recently, although an upper bound of $\left\lfloor \frac{(m+2)(n+2)}{5}\right\rfloor-4$ was shown in [@TYC'92] for $8\le m\le n$ using a constructive method. Various attempts have been made in recent years to find a tight lower bound on the size of the optimal dominating set. In [@ACIP'11], the authors used brute-force computational techniques to find optimal dominating sets in grids of size up to $n = m = 29$. The paper [@GPRT'11] showed finally that the lower bound on the domination number is equal to the upper bound for $16\le m\le n$, thus characterizing the domination number in grids.
In this paper, we make two contributions to the study of dominating sets on grids, and their application to multi-agent coverage. First, we provide a distributed algorithm that locates a set of agents on the vertices of an $m\times n$ grid such that they construct a dominating set for the grid and the required number of agents is within a constant error from the optimal. The agents require only limited memory, sensing and communication abilities, and thus the solution is applicable to multi-robot coverage applications where the environment can be discretized as a grid. Our distributed algorithm is based on a simple constructive method to obtain [*near-optimal*]{} dominating sets (i.e., that require no more than $5$ vertices over the optimal number) in grids by Chang ${{\emph{et al.}}}$ [@TYC'92]. This approach is based on a systematic tiling pattern that we call a [*diagonalization*]{}. Second, we generalize Chang’s construction to the $k$-distance dominating set problem, where a given vertex can cover all other vertices within a distance $k$ from it. We show that our distributed algorithm can also be generalized to work in the $k$-distance domination scenario.
In Section \[sec:background\], we introduce the essential models and notation for formulating the dominating set problem on grids. In Section \[sec:center-domin\] we discuss the constructive centralized grid domination algorithm. The materials in Section \[sec:center-domin\] are used in Section \[sec:dist-domin\] to design a distributed algorithm for the dominating set problem. Section \[sec:k-distance\] generalizes the results in Sections \[sec:center-domin\] and \[sec:dist-domin\] to the $k$-distance dominating set problem. Finally, Section \[sec:conclusion\] concludes the paper and discusses the corresponding open problems.
Background {#sec:background}
==========
A graph $G=(V,E)$ is defined as a set of vertices $V$ connected by a set of edges $E\subseteq V\times V$. We assume the graph is *undirected*, i.e., $(v,u)\in E\Leftrightarrow (u,v)\in E,\forall v,u\in V$. A vertex $u\in V$ is defined as a *neighbour* of vertex $v\in V$, if $(u,v)\in E$. The set of all neighbours of vertex $v$ is denoted by $N(v)$. For a set of vertices $U\subseteq V$, we define $N(U)$ as $\bigcup_{u\in U}{N(u)}$. For a set of vertices $U\subseteq V$, we say the vertices in $N(U)$ are *dominated* by the vertices in $U$. For graph $G$, a set of vertices $S\subseteq V$ is a *dominating set* if each vertex $v\in V$ is either in $S$ or is dominated by $S$.
A dominating set with minimum cardinality is called an *optimal dominating set* of a graph $G$; its cardinality is called the *domination number* of $G$ and is denoted by $\gamma(G)$. Note that although the domination number of a graph, $\gamma(G)$, is unique, there may be different optimal dominating sets [@CLRS'01].
Here, we study the dominating set problem on a special class of graphs called *grid graphs*. An $m\times n$ grid graph $G=(V,E)$ is defined as a graph with vertex set $V=\{ v_{i,j}| 1\le i\le m, 1\le j\le n\}$ and edge set $E=\{(v_{i,j},v_{i,j'})|~|j-j'|=1\}\bigcup \{(v_{i,j},v_{i',j})|~|i-i'|=1\}$ [@BM'08]. For ease of exposition, we will fix an orientation and labelling of the vertices, so that vertex $v_{1,1}$ is the lower-left vertex and vertex $v_{m,n}$ is the upper-right vertex of the grid. We denote the domination number of an $m\times n$ grid $G$ by $\gamma_{m,n}=\gamma(G)$.
\[theo:cited-4\] For an $m\times n$ grid with $16\le m\le n$, $\gamma_{m,n}=\left\lfloor \frac{(m+2)(n+2)}{5}\right\rfloor-4$.
Our distributed grid domination algorithm is based on a procedure developed by Chang ${{\emph{et al.}}}$ [@TYC'92]. To obtain the tools needed in our distributed algorithm, we discuss an overview of Chang’s algorithm in Section \[sec:center-domin\]. These tools are used in Sections \[sec:dist-domin\] and \[sec:k-distance\] in the distributed domination algorithm and in developing the $k$-distance dominating set results. Before discussing these tools, we introduce two useful definitions.
(Grid Boundary) \[def:boundaries\] For an $m\times n$ grid $G=(V,E)$, we define the *boundary* of $G$, denoted by $B(G)$, as the set of vertices with less than 4 neighbours.
(Sub-Grids and Super-Grids) \[def:sub-super-grids\] An $m\times n$ grid $G=(V,E)$ is called a *sub-grid* of an $m'\times n'$ grid $G'=(V',E')$ if $G$ is induced by vertices $v'_{i,j}\in V'$, where $2\le i\le m'-1$ and $2\le j\le n'-1$. If $G$ is a sub-grid of $G'$, $G'$ is called the *super-grid* of $G$ (see Figure \[fig:eg\](a)).
Overview of Centralized Grid Domination Algorithm {#sec:center-domin}
=================================================
In [@ACIP'11], Alanko ${{\emph{et al.}}}$ provided examples of optimal dominating sets for $n\times n$ grids with $1\le n\le 29$, obtained via a brute-force computational method. A visual inspection of these examples shows that as the size of the grid increases, the patterns of dominating vertices become more regular in the interior of grids, with irregularities at the boundaries. Figure \[fig:patterns\] demonstrates some examples of patterns that arise in the dominated grids in [@ACIP'11].
![In (a), a $12\times 12$ grid $G'$ is demonstrated and its $10\times 10$ sub-grid $G$ is highlighted by a red dashed square. $G'$ is diagonalized by a set $U'$ of 28 vertices. In (b), vertices in $U'\backslash V$ are projected onto their neighbours in $G$.[]{data-label="fig:eg"}](eg)
Among the patterns used to dominate grids, the one illustrated in Figure \[fig:patterns\](b) is the most efficient, since there is no vertex that is dominated by more than one dominating vertex in this pattern. Hence, this pattern would be useful in obtaining dominating sets with near optimal size. We refer to the structure in Figure \[fig:patterns\](b) as a *diagonal pattern*. Chang ${{\emph{et al.}}}$ in [@TYC'92] used these patterns to provide an upper-bound on the domination number of grids. As the proof on the upper-bound they obtained was constructive, we could derive a centralized algorithm for finding near-optimal dominating sets from their constructions. In this section, we provide an overview of Chang’s construction and the derived algorithm from their results which we will use in the subsequent sections. First, we define the diagonal patterns formally as follows. Note that the $x$ and $y$ axes are as shown in Figure \[fig:patterns\].
(Diagonal Pattern) \[def:diag-patt\] A set of vertices $U\subset V$ constitutes a *diagonal pattern* on grid $G=(V,E)$ if there exists a fixed $r\in \{0,1,2,3,4\}$ such that for any vertex $v_{x,y}\in U$ we have $y-2x\equiv r \pmod{5}$.
(Diagonalization) \[def:diagonalization\] A set of vertices $U\subset V$ *diagonalizes* grid $G=(V,E)$ if it constitutes a diagonal pattern and there exists no vertex $v\in V\backslash U$ that can be added to $U$ so that $U$ remains a diagonal pattern.
An example of a diagonalization is shown in Figure \[fig:eg\](a).[^3] The algorithm derived from Chang’s construction consists of the following two main steps:
1. Diagonalization: At this step, a set of vertices $U$ that diagonalizes the grid is provided.
2. Projection: Using a process called *projection*, the vertices that were not dominated by vertices in $U$ are characterized and new vertices are added to $U$ to dominate those vertices as well.
We know discuss these two steps in more details. Chang ${{\emph{et al.}}}$ showed that if a grid $G=(V,E)$ is diagonalized by a set of vertices $U\subset V$, then for any vertex $v\in (V\backslash U)$ that is not located on the grid’s boundary there exists exactly one vertex in $U$ that shares an edge with $v$. In other words, every node that is not located on the grid’s boundary, $B(G)$, is dominated by exactly one vertex in $U$. Moreover, they proved that if a set of vertices $U\subset V$ diagonalizes an $m\times n$ grid $G=(V,E)$, then $U$ contains at most $\left\lceil \frac{mn}{5}\right\rceil$ vertices. To construct a dominating set for $G$ it only remains to add some vertices to $U$ so that the resulting set dominates the vertices on the boundary as well. The vertices located on $B(G)$ with no neighbour in $U$ are called *orphans* and are defined formally as follows.
(Orphans) \[def:orphan\] Let $U\subset V$ be a set of vertices that diagonalizes grid $G=(V,E)$. A vertex $v\in V$ that has no neighbour in $U$ is called an *orphan* (see Figure \[fig:eg\](a)).
To dominate orphans, Chang ${{\emph{et al.}}}$ used the super-grid of $G$, denoted by $G'=(V',E')$. Since the vertices on the boundary of $G$ lie inside grid $G'$, a set of vertices $U'\subset V'$ that diagonalizes $G'$ dominates all vertices of $G$. Moreover, it can be easily seen that the set of vertices $U=U'\cap V$ is a diagonalization for grid $G$.
Recall that diagonalization results in every vertex being dominated by at most one vertex in the diagonal pattern. Therefore, if a set of vertices $U'\subset V'$ diagonalizes $G'=(V',E')$, that is, the super-grid of $G=(V,E)$, then there are vertices in $B(G)$ that are dominated by vertices in $U'\backslash V$. Hence, the *orphan* of a vertex $v\in U'\backslash V$ is a vertex $u\in B(G)$ such that $u\in N(v)$, and is denoted by $u=\mathrm{orphan}(v)$.
\[cor:orphan-n+m\] For an $m\times n$ grid $G$, the number of orphans is $O(n+m)$.
Since by diagonalizing $G'$ the orphans in $G$, i.e, vertices in $N(U'\backslash U)\cap V$, are dominated by the dominating vertices on the boundary of $G'$, a procedure called *projection* is introduced that projects the dominating vertices in $B(G')$ inside sub-grid $G$. Hence, projection results in having all vertices in $G$ being dominated. This procedure is defined formally as follows.
(Projection) \[def:porjection\] Consider a grid $G=(V,E)$ and its super-grid $G'=(V',E')$. For a set $U'\subseteq V'$, its *projection* is defined as the set $U''=\big(N(U'\backslash V)\cup U'\big)\cap V$. Similarly, we say a vertex $v\in U'\backslash V$ is *projected* if it is mapped to its neighbour in $V$.
![Examples of dominating vertex patterns that appear in optimally dominated grids. The black vertices are the dominating vertices. The red line segments form regions so that in each region there exist one black vertex and at most four white vertices dominated by that black vertex.[]{data-label="fig:patterns"}](patterns)
Figure \[fig:eg\](b) shows an example of a projection. For grid $G=(V,E)$, its super-grid $G'=(V',E')$ and set $U'\subset V'$ that diagonalizes $G'$, by performing projection, the size of the obtained dominating set of $G$ is between $|U'|-4$ and $|U'|$. This is due to the fact that a vertex $v\in U'$ located at any corner of $G'$ has no neighbour in $V$ and hence, after projection it is not mapped into $V$. Since $G'$ has four corners, for $U''$, the result of projection of $U'$, we have $|U'|-4\le |U''|\le |U'|$. Hence $|U'|$, that is, the number of dominating vertices used in diagonalizing the super-grid of $G$, is an upper-bound on the number of dominating vertices used to fully dominate $G$ by diagonalization and projection. Since the size of super-grid of and $m\times n$ grid $G$ is $(m+2)\times (n+2)$, therefore, $|U|'\le \left\lceil \frac{(m+2)(n+2)}{5}\right\rceil$. Hence, $\left\lceil \frac{(m+2)(n+2)}{5}\right\rceil$ is an upper-bound on the number of dominating vertices used to dominate grid $G$ by Chang’s algorithm. The following theorem reflects this upper-bound.
\[theo:upper-bound\] For any $m\times n$ grid $G=(V,E)$ with $m,n\in \mathbb{N}$, a dominating set $S\subset V$ can be constructed in polynomial-time, such that $|S|\le \left\lceil \frac{(m+2)(n+2)}{5}\right\rceil$. Moreover, for grids with $16\le m\le n$ we have $|S|-\gamma_{m,n}\le 5$.
The upper-bound on the difference between the cardinality of the provided dominating set $S$ from the domination number of an $m\times n$ grid $G$ with $16\le m\le n$, $\gamma_{m,n}$, is obtained by virtue of Theorem \[theo:cited-4\]. An example of constructing dominating sets for grids using diagonalization and projection is shown in Figure \[fig:eg\].
In the following lemma we show that although in diagonal patterns no vertex is covered by more than one dominating vertex, using a simple greedy algorithm does not necessarily result in diagonalizing the grid or using at most $\left\lceil \frac{(m+2)(n+2)}{5}\right\rceil$ dominating vertices to dominate the grid.
\[lem:greedy-grid\] The size of the dominating set obtained by a greedy algorithm on an $m\times n$ grid $G$ might be as large as $\left\lceil \frac{m}{3}\right\rceil \left\lceil \frac{n}{3}\right\rceil+2\left\lfloor \frac{m}{3}\right\rfloor \left\lfloor \frac{n}{3}\right\rfloor$.
As discussed in Section \[sec:intro\], after the first vertex $v$ is added to the dominating set $S$, greedy algorithm chooses a vertex that does not share any neighbours with $v$. Although this is also a property of diagonal patterns, the set of all the closest vertices around $v$ that can be added to $S$ using diagonal patterns has size at most four (see Figure \[fig:patterns\](b)). However, there are 12 vertices around $v$ that do not share any neighbours with $v$ and hence candidate to be added to $S$ in a greedy algorithm, Figure \[fig:greedy\](a). At each step of a greedy algorithm one of these 12 vertices is chosen arbitrarily. However, choosing only all red vertices or all blue vertices would start developing a diagonal pattern. Other combinations of candidate vertices would fail to diagonalize the grid and some vertices of the graph would be dominated by more than one dominating vertex. Hence, the size of the constructed dominating set would be greater than $\left\lceil \frac{(m+2)(n+2)}{5}\right\rceil$.
In particular, the algorithm might add all the green vertices to $S$ and repeat the same pattern in the grid, Figure \[fig:greedy\](b). However, using this pattern, between any four green vertices there remains a set of four vertices that are not dominated by any vertex in $S$. These vertices are highlighted by dotted rectangles in Figure \[fig:greedy\](b). To dominate each of these sets of vertices at least two extra dominating vertices should be added to $S$. Therefore, the number of obtained dominating vertices would be at least $\left\lceil \frac{m}{3}\right\rceil \left\lceil \frac{n}{3}\right\rceil+2\left\lfloor \frac{m}{3}\right\rfloor \left\lfloor \frac{n}{3}\right\rfloor$, which is much greater than the size of the dominating set obtained by Chang’s construction, i.e., $\left\lceil \frac{(m+2)(n+2)}{5}\right\rceil$.
![In Figure (a), after adding the black vertex to the dominating set, the next vertex added to that set can be any of the blue, red or green vertices, without dominating any vertex by two dominating vertices. In (b), a dominating set is built by starting from the black vertex and keep adding the green vertices shown in (a) to the set. Each dotted rectangle contains four vertices that are not dominating by the obtained dominating set.[]{data-label="fig:greedy"}](greedy)
Distributed Grid Domination {#sec:dist-domin}
===========================
In the preceding section, a centralized algorithm was discussed that produced a dominating set $S$ for a given $m\times n$ grid $G$ such that $|S|\le \left\lceil \frac{(m+2)(n+2)}{5}\right\rceil$. In this section, we show how to achieve the same upper-bound in a distributed way.
Model and Notation {#subsec:model2}
------------------
Here we assume that the environment is an $m\times n$ grid $G=(V,E)$ with $m,n\in \mathbb{N}$. The goal is to dominate the grid environment in a distributed fashion using several robots (or agents) without any knowledge of environment size. Initially, there exist $k$ agents in the environment, where $k$ can be smaller or greater than the number of agents needed to dominate the grid. The following assumptions are made for the grid and agents.
*Grid Assumptions:* Agents can be located only on the vertices of the grid and are able to move between the grid vertices only on the edges of the grid. At each moment, a vertex can contain more than one agent. We refer to the vertices using the standard Cartesian coordinates defined in Section \[sec:background\].
*Agent Assumptions:* The agents, denoted by $a_1,\ldots, a_{k}$, are initially located at arbitrary vertices on the grid. The agents have three modes: (a) *sleep*, (b) *active*, and (c) *settled*. The mode of an agent $a$ and the vertex it is located at are denoted by $\mathrm{mode}(a)$ and $v(a)$, respectively. Only agents in the active and settled modes are able to communicate. At the beginning of the procedure, all the agents are in the sleep mode. During each *epoch*, that is, a time interval with a specified length, one agent goes to active mode. The activation sequence of agents is arbitrary (e.g., it can be scheduled in advance or it can be random). The active agent can communicate with the settled agents to perform the distributed dominating set algorithm. Once an agent activates and performs its part in the algorithm, it goes to settled mode. Ultimately, all the settled agents go back to sleep mode and will not activate again.
Here, each agent is equipped with suitable angle-of-arrival (bearing) and range sensors. Using these sensors, agent $a$ computes the coordinates of other agents in its own coordinate frame $\Sigma_a$ with its origin at $v(a)$ and an arbitrary orientation, fixed relative to agent $a$. Each agent also has a compass to determine its heading direction. Additionally, agents are equipped with short-ranged proximity sensors to sense the environment boundary. Agents are able to sense the boundary only if they are on a vertex $v$ whose neighbour is a boundary vertex of the grid, i.e., $N(v)\cap B(G)\ne \emptyset$. The compass helps agents to distinguish which of the four boundary edges they are approaching.
Overview of Algorithm {#subsec:dist-alg-summary}
---------------------
The main idea in this algorithm is to implement the diagonal pattern defined in Section \[sec:center-domin\] on grid $G=(V,E)$, using communications among active and settled agents. A special unit called a *module* is defined for the active and settled agents. A module is a cross-like shape consisting of the agent at its center with the associated dominated vertices in the arms of the cross (see Figure \[fig:patterns\](b)). For each module $m$, the vertex that contains the agent, i.e., the center vertex, is referred to as the *module center*, denoted by $c(m)$. As an agent moves on the grid to contribute to the diagonal pattern, its module moves with it as well. Modules $m_1$ and $m_2$ with module centers $c(m_1)=v_{i,j}$ and $c(m_2)=v_{i',j'}$ can connect to each other if $v_{i',j'}\in \{ v_{i+1,j+2},v_{i+2,j-1},v_{i-1,j-2},v_{i-2,j+1}\}$ (see Figure \[fig:steps\](f)). This condition is called the *module connection condition*. The set of centers of the connected modules is called a *cluster*. We will later show that the module connection condition ensures that the module centers are a diagonalization of the vertices covered by the modules in the cluster.
*Valid Slots:* Let $G'=(V',E')$ be the super-grid of $G$. A vertex $v_{a,b}\in V'$ is called a *slot* if there exists a module $m$ in the cluster with center $v_{i,j}$ such that $v_{a,b}\in \{ v_{i+1,j+2},v_{i+2,j-1},v_{i-1,j-2},v_{i-2,j+1}\}$ and $v_{a,b}$ is not already a center for a module in the cluster. For a settled agent $a$ located at $v(a)$, denote the set of all its slots by $\mathrm{slots}(a)$. Recall that the *orphan* of a vertex $v\in V'\backslash V$, i.e., $\mathrm{orphan}(v)$, is a vertex $u\in B(G)$ such that $u\in N(v)$. The set of all *valid slots* for settled agent $a$, denoted by $\mathrm{vslots}(a)$, is defined as $(\mathrm{slots}(a)\cap V) \cup \mathrm{orphan}(\mathrm{slots}(a)\backslash V)$. Newly activated agents can settle only on the valid slots of the settled agents.
*Updating Valid Slots:* When an active agent settles, it creates the list of its valid slots as follows. If a settled agent $a$ cannot sense the boundary (i.e., it has no neighbour on the boundary), $\mathrm{slots}(a)\backslash V=\emptyset$ and hence $\mathrm{vslots}(a)=\mathrm{slots}(a)$. Conversely, a settled agent can also determine which of its slots lie outside the grid boundary (Figure \[fig:orphans\](a)). Each newly settled agent marks the vertices on the grid boundary that are neighbours of $\mathrm{slots}(a)\backslash V$ as *orphans* and so $\mathrm{vslots}(a)=(\mathrm{slots}(a)\cap V) \cup \mathrm{orphan}(\mathrm{slots}(a)\backslash V)$ (Figure \[fig:orphans\](b)). By the definition of valid slots, no valid slot exists in an orphan’s neighbourhood. Therefore, each orphan needs one agent to be located on itself or one of its neighbours to be dominated. For simplicity we always put an agent on the orphan itself.
When an agent activates, it transmits a signal to find the settled agents on the grid and waits for some specified time for a response from them. Since there is no settled agent in the environment when the first agent activates, it receives no signal and concludes it is the first one activated. Thus, the agent stays at its initial location and goes to the settled mode. Subsequently, each active agent translates to the closest settled agent.[^4]
Distributed Grid Domination Algorithm {#susec:dist-alg}
-------------------------------------
During the distributed grid domination algorithm, active agents can either contribute to grid diagonalization by locating on non-orphan valid slots or can settle on orphans. In each epoch, the set of the non-orphan vertices containing the previously settled agents is called the *cluster* and is denoted by $C$, while the set of occupied orphans is denoted by $P$. At the beginning of the algorithm $C=P=\emptyset$. It should be mentioned that $C$ and $P$ are not saved by any agent, and are used only to aid in the presentation of the algorithm. Moreover, we denote the set of all settled agents at each moment by $A_s$, where at the beginning of the algorithm $A_s=\emptyset$. Also if agent $a$ is already settled and is now in sleep mode $\mathrm{done}(a)=1$, otherwise $\mathrm{done}(a)=0$.
The remaining non-activated agents leave the grid.
(Comments on Algorithm)\
1) Since agents can move only on the grid edges, the distance between two vertices can be computed simply by adding their $x$-coordinate and $y$-coordinate differences, i.e., $\Delta x$ and $\Delta y$. There exist many shortest paths between any two vertices and agent $a$ arbitrarily chooses one of them to traverse; for instance it can first traverse on the $x$-coordinate and then on the $y$-coordinate.\
2) In Step 11, agent $a$ locates $s$ in $\Sigma_a$ (i.e., coordinate frame of $a$), while $\mathrm{vslots}(s)$ is computed by $s$ in $\Sigma_s$ in Step 12. In Step 13, agent $a$ converts the coordinates of $\mathrm{vslots}(s)$ from $\Sigma_s$ to $\Sigma_a$ for traversing, using relative sensing techniques [@PSFBA'08].\
3) When an agent settles, all settled agents wait for a specified amount of time for the next agent to activate. If no agent activates, Algorithm \[alg:dist-domin\] halts and the previously settled agents construct a subset of a dominating set of the grid. This happens when the initial number of agents is not sufficient to dominate the grid.\
4) If the agents are equipped with GPS, then they can agree on a fixed diagonalization (i.e., agree on a value of $r$), and move to the vertices $U$ in the diagonalization. At this point, only orphan vertices exist. The remaining agents can move along the boundary to find and cover all orphans and consequently dominate the grid. Hence, in this paper we study the case that agents are not armed with GPS.
![Non-activated agents are marked by black crosses and the already settled agents are shown by black circles. Agents in $C$ have red crosses as their modules. Figure (a) shows the first active agent, as in Step 6. In (b), an active agent is highlighted by a blue square. Step 13 is depicted in (c), where a dashed blue square shows the closest valid slot to the active agent. In (d), the active agent moves to the valid slot and joins $C$, as in Step 15. In (e), the list of valid slots is updated as in Step 4. In (f), the grey circle shows an agent that goes from settled to sleep mode.[]{data-label="fig:steps"}](algorithm)
![Non-activated agents are marked by black crosses and the already settled agents are shown by black circles, with red crosses as their modules. In (a), a settled agent, highlighted by a solid blue square, realizes one of its slots, shown by a dashed blue square, is outside the grid boundary. In (b), the settled agent replaces the slot outside the grid boundary with its orphan and name the resulting set as its valid slots. In (c), the active agent locates at the orphan. Figure (d) shows that an agent on an orphan has no valid slot.[]{data-label="fig:orphans"}](orphans)
Distributed Algorithm Analysis {#subsec:dist-alg-analys}
------------------------------
We now prove that the set of vertices determined by Algorithm \[alg:dist-domin\], i.e., $C\cup P$, creates a dominating set for the grid. Recall that at each epoch, $C$ is the set of non-orphan vertices containing the previously settled agents and $P$ is the set of occupied orphans.
\[lem:modules-are-diag\] During the operation of Algorithm \[alg:dist-domin\], the module connection condition forces the vertices in $C$ to create a diagonal pattern.
This will be proved using induction on the size of $C$ during the operation of the algorithm. According to the module connection condition, the module of agent $a$ located at vertex $v(a)=v_{i',j'}\notin C$ can connect to the module of vertex $v_{i,j}\in C$ if $v_{i',j'}\in \{v_{i+1,j+2},v_{i+2,j-1},v_{i-1,j-2},v_{i-2,j+1}\}$. The base of induction is $|C|=0$, when the first agent is about to be added to $C$. In this case, the first agent settles at its current location $v(a)=v_{i,j}$ and establishes the value $r\equiv j-2i \pmod{5}$.
For $|C|>1$, $C$ already has a diagonal pattern and an active agent $a$ at $v(a)=v_{i',j'}$ aims to join it by connecting to a module centered at $v_{i,j}$. Since $v_{i,j}$ is already in $C$, $j-2i\equiv r \pmod{5}$. It can be seen that for a vertex $v_{i',j'}$ that satisfies the module connection condition with respect to $v_{i,j}$ we have $j'-2i'\equiv r \pmod{5}$. Therefore, the resulting set has a diagonal pattern.
\[theo:error-dist\] The number of agents used to dominate an $m\times n$ grid $G=(V,E)$ by Algorithm \[alg:dist-domin\] is upper-bounded by $\left\lceil \frac{(m+2)(n+2)}{5}\right\rceil$. For grids with $16\le m\le n$, the number of agents used is upper-bounded by $\gamma_{m,n}+5$.
We first prove Algorithm \[alg:dist-domin\] is correct and then show the upper-bound holds. Let $G'=(V',E')$ be the super-grid of $G$ and $C$ denote the non-orphan vertices occupied by previously settled agents when the algorithm finishes. By Lemma \[lem:modules-are-diag\], $C$ constitutes a diagonal pattern and by the condition in Step 10 of the algorithm no other agent can be added to $C$; therefore, $C$ diagonalizes $G$. Moreover, orphans are neighbours of the vertices in $V'\backslash V$ that are initially detected as slots by the settled agents and hence diagonalize $G'$ by Lemma \[lem:modules-are-diag\]. Thus, locating one agent on each orphan is equivalent to the projection process. Hence, if a sufficient number of agents exist in the grid, Algorithm \[alg:dist-domin\] provides a dominating set for $G$ (from Theorem \[theo:upper-bound\]). Consequently, the algorithm is *complete*, meaning it always finds a solution, if one exists.
Furthermore, since Algorithm \[alg:dist-domin\] performs diagonalization and projection on $G$, from Theorem \[theo:upper-bound\] it immediately follows that the number of agents used in the algorithm, $n_a$, is upper-bounded by $\left\lceil \frac{(m+2)(n+2)}{5}\right\rceil$. Also by Theorem \[theo:cited-4\], for $16\le m\le n$ we have $n_a-\gamma_{m,n}\le 5$.
Note that while the agents do not form a dominating set for $G$, an active agent finds a valid slot in at most $n+m$ steps. A step is a specified time duration within which an agent performs its basic operation, such as traversing an edge or transmitting signals. Since the number of agents needed to dominate an $m\times n$ grid is less than $mn$, Algorithm \[alg:dist-domin\] takes at most $mn(m+n)$ steps to construct a dominating set for $G$.
Simulations {#subsec:simulation}
-----------
To augment and examine the results discussed in this section, we simulated Algorithm \[alg:dist-domin\] on various grids and different initial configurations of agents on grid vertices. Figure \[fig:sim1\](a) demonstrates a $10\times 15$ grid graph with 41 agents located randomly on it. The first agent that activates is located on vertex $(5,9)$ and hence stays on that vertex. Figure \[fig:sim1\](b) shows the location of agents when Algorithm \[alg:dist-domin\] is complete. It can be seen that every vertex is dominated. However, there are some agents located at vertices (such as $(6,5), $$(6,12)$ and $(7,15)$) that are never activated in the algorithm. These are the additional agents that are not required to dominate the grid and they are removed in Figure \[fig:sim1\](c).
[0.28]{} ![A $10\times 15$ grid is depicted with agents shown in blue. In (a) the initial configuration of the agents is shown and (b) shows the agents configuration when Algorithm \[alg:dist-domin\] is finished. In (c), all non-settled and non-asleep agents leave the grid[]{data-label="fig:sim1"}](s1_crop "fig:"){width="\linewidth"}
[0.28]{} ![A $10\times 15$ grid is depicted with agents shown in blue. In (a) the initial configuration of the agents is shown and (b) shows the agents configuration when Algorithm \[alg:dist-domin\] is finished. In (c), all non-settled and non-asleep agents leave the grid[]{data-label="fig:sim1"}](s3_crop "fig:"){width="\linewidth"}
[0.28]{} ![A $10\times 15$ grid is depicted with agents shown in blue. In (a) the initial configuration of the agents is shown and (b) shows the agents configuration when Algorithm \[alg:dist-domin\] is finished. In (c), all non-settled and non-asleep agents leave the grid[]{data-label="fig:sim1"}](s4_crop "fig:"){width="\linewidth"}
$k$-Distance Domination on Grids {#sec:k-distance}
================================
In this section we generalize Chang’s algorithm for grid domination, discussed in Section \[sec:center-domin\], to the $k$-distance dominating set problem, where a vertex dominates all the vertices within distance $k$ from it. Before defining the problem formally, let $d(u,v)$ denote the shortest path distance between vertices $v,u\in V$ in $G=(V,E)$. Moreover, vertex $u\in V$ is defined as a *$k$-neighbour* of vertex $v\in V$, if $0<d(u,v)\le k$. The set of all $k$-neighbours of $v$ is denoted by $N^k(v)$. Moreover, for a set of vertices $W\subset V$ and a vertex $v\in V\backslash W$, we have $u=\mathrm{friend}^k(v,W)$ if (a) $u\in W$, (b) $u\in N^k(v)$, and (c) $d(v,u)\le d(v,w),\forall w\in W$.
($k$-Distance Dominating Set Problem) \[def:k-distance\] Given a graph $G=(V,E)$, the $k$-distance dominating set problem is to find a set of vertices $S\subseteq V$ such that for every vertex $v\in V\backslash S$ there exists a vertex $u\in S$ where $u\in N^k(v)$. The cardinality of a smallest $k$-distance dominating set for $G$ is called the *$k$-distance domination number* of $G$ and is denoted by $\gamma^k(G)$ [@Henning'98].
We say that vertex $u\in S$ *$k$-distance dominates* $v\in V\backslash S$ if $d(u,v)\le k$. The regular dominating set problem is a special case of the $k$-distance dominating set problem, where $k=1$. Therefore, $k$-distance domination is also NP-hard on general graphs. However, to the best of our knowledge the $k$-distance domination number of grids is not known and the complexity of the problem is open. In Section \[subsec:upper-bound-k-distance\], we generalize the approaches in Sections \[sec:center-domin\] and \[sec:dist-domin\] to provide a $k$-distance dominating set for an $m\times n$ grid graph $G$.
Centralized $k$-Distance Domination on Grids {#subsec:upper-bound-k-distance}
--------------------------------------------
Before discussing the $k$-distance domination algorithms on grids we introduce the following definitions.
($k$-Sub-Grids and $k$-Super-Grids) \[def:k-super\] An $m\times n$ grid $G=(V,E)$ is called a *$k$-sub-grid* of an $m'\times n'$ grid $G'=(V',E')$ if $G$ is induced by vertices $v'_{i,j}\in V'$, where $k+1\le i\le m'-k$ and $k+1\le j\le n'-k$. If $G$ is a $k$-sub-grid of $G'$, $G'$ is called the *$k$-super-grid* of $G$.
\[lem:k-distance-unit\] For an $m\times n$ grid $G=(V,E)$, $|N^k(v)|\le 2k^2+2k+1$.
Since $G$ is a grid, the $k$-neighbours of $v$ form a diamond around it with a diameter of $2k+1$ (see the red regions in Figure \[fig:k-distance-elem\]). Thus $|N^k(v)|$ is upper-bounded by the area of this region, which is $\left\lceil \frac{(2k+1)^2}{2}\right\rceil=2k^2+2k+1$.
In what follows we define $N^k_{\max}=2k^2+2k+1$.
($k$-Diagonal Pattern) \[def:k-diag-patt\] A set of vertices $U\subset V$ constitutes a *$k$-diagonal pattern* on grid $G=(V,E)$ if there exists a fixed $0\le r< N^k_{\max}, r\in \mathbb{Z}_+$ such that for any vertex $v_{x,y}\in U$ we have $ky-(k+1)x\equiv r \pmod{N^k_{\max}}$ (see Figure \[fig:k-distance-elem\]).
($k$-Diagonalization) \[def:k-diagonalization\] A set of vertices $U\subset V$ *$k$-diagonalizes* grid $G=(V,E)$ if it constitutes a $k$-diagonal pattern and there exists no vertex $v\in V\backslash U$ that can be added to $U$ so that $U$ remains a $k$-diagonal pattern.
Moreover, for a grid $G=(V,E)$ and its $k$-super-grid $G'=(V',E')$, the *$k$-projection* is defined as a special mapping from the vertices in $V'\backslash V$ to their $k$-neighbours in $V$. It is defined formally as follows.
($k$-Projection) \[def:k-porjection\] Consider a grid $G=(V,E)$ and its $k$-super-grid $G'=(V',E')$. The *$k$-projection* for a set $U'\subseteq V'$ is defined as the set $U''=\{u\in V|~\exists v\in U'\backslash V~s.t.~u=\mathrm{friend}^k(v,V) \}\cup \{U'\cap V\}$ (see Figure \[fig:k-grid\]).
\[lem:k-min-2k+1\] Let $U$ be a set of vertices that $k$-diagonalizes a grid $G=(V,E)$. For any two vertices $v_{x,y},v_{x',y'}\in U$ we have $d(v_{x,y},v_{x',y'})\ge 2k+1$.
Since $v_{x,y},v_{x',y'}\in U$, we have $y=\frac{1}{k}((k+1)x+r+qN^k_{\max})$ and $y'=\frac{1}{k}((k+1)x'+r+q'N^k_{\max})$, where $r,q\in \mathbb{Z}$ and $0\le r< N^k_{\max}$. We define $\Delta_q=q'-q$, $\Delta_1=x'-x$ and $\Delta_2=y'-y=\frac{k+1}{k}\Delta_1+\frac{N^k_{\max}}{k}\Delta_q$. The shortest distance between $v_{x,y},v_{x',y'}$ is equal to $|\Delta_1|+|\Delta_2|$. From $\Delta_2=\frac{k+1}{k}\Delta_1+\frac{N^k_{\max}}{k}\Delta_q$ it can be observed that as $\Delta_1$ grows, $\Delta_2$ grows faster compared to $\Delta_1$. Hence $|\Delta_1|+|\Delta_2|$ is minimum when $\Delta_2=0$ and $|\Delta_1|=|\frac{N^k_{\max}}{k+1}\Delta_q|$. Note that the minimum (non-zero) distance occurs for $\Delta_q=1$ and also it is an integer, hence it is lower-bounded by $\left\lceil{\frac{2k^2+2k+1}{k+1}}\right\rceil=2k+1$.
\[lem:k-distane-k-super\] Consider a grid $G=(V,E)$ and its $k$-super-grid $G'=(V',E')$. If $U'\subset V'$ $k$-diagonalizes $G'$, then each vertex in $V$ is $k$-dominated by exactly one vertex from $U'$.
For each vertex $v_{x,y}\in V$ let $r_{v_{x,y}}\equiv ky-(k+1)x\pmod{N^k_{\max}}$. Consider any vertex $v\in V$ and its $k$-neighbourhood $N^k(v)$. The distance between any two vertices in $J=\{v\}\cup N^k(v)$ is at most $2k$. Also, there are exactly $N^k_{\max}$ vertices in this set. Thus, for any two distinct vertices $u,w\in J$ we have $r_u\neq r_w$ by Lemma \[lem:k-min-2k+1\]. Hence each vertex $u\in N^k(v)$ has a distinct value of $r_u$. Consequently, for the value of $r$ that corresponds to the diagonalization $U'$, there is exactly one vertex in the $k$-neighbourhood of $v$ such that $r_v=r$ and thus $v$ is $k$-dominated by exactly one vertex from $U'$.
\[lem:k-diag-cardinal\] If a set of vertices $U\subset V$ $k$-diagonalizes an $m\times n$ grid $G=(V,E)$, then $U$ contains at most $\left\lceil \frac{mn}{ N^k_{\max}}+\frac{N^k_{\max}}{4}\right\rceil$ vertices.
Since $U$ $k$-diagonalizes $G$, it constitutes a $k$-diagonal pattern on $G$ such that no more vertices can be added to it while maintaining a $k$-diagonal pattern. Therefore, among each $N^k_{\max}$ consecutive vertices in any row or column there is exactly one vertex from $U$. Hence, the number of vertices of $U$ in a row/column of $t$ vertices is at most $\left\lceil\frac{t}{N^k_{\max}}\right\rceil$.
Thus, there are at most $N^k_{\max}$ vertices from $U$ in any $N^k_{\max}\times N^k_{\max}$ grid. Hence, in any $N^k_{\max}q\times N^k_{\max}p$ grid with $p,q\in \mathbb{Z_+}$, there are at most $N^k_{\max}pq$ vertices from $U$. For an $m\times n$ grid $G=(V,E)$ with $m=qN^k_{\max}+a$, $n=pN^k_{\max}+b$ and $0\le a,b< N^k_{\max}$, we partition $V$ into the four following sets: $$V_1=\{v_{i,j}|~1\le i\le qN^k_{\max}, 1\le j \le pN^k_{\max} \},$$ $$V_2=\{v_{i,j}|~qN^k_{\max}+1\le i\le m, 1\le j \le pN^k_{\max} \},$$ $$V_3=\{v_{i,j}|~1\le i\le qN^k_{\max}, pN^k_{\max}+1\le j \le n \},$$ and $$V_4=\{v_{i,j}|~qN^k_{\max}+1\le i\le m, pN^k_{\max}+1\le j \le n\}.$$ As stated, $|V_1\cap U|\le pqN^k_{\max}$. Grid $V_2$ has $a$ columns each having $N^k_{\max}p$ vertices, hence $|V_2\cap U|\le pa$. Similarly, we have $|V_3\cap U|\le qb$. In summary, we so far have $|(V_1\cup V_2\cup V_3)\cap U|\le pqN^k_{\max}+qb+pa$. Note that $pqN^k_{\max}+qb+pa=\frac{mn}{N^k_{\max}}-\frac{ab}{N^k_{\max}}$.
It remains to upper-bound $|V_4\cap U|$. Without loss of generality assume that $a\le b$. Since the number of rows and columns in $V_4$ are less than $N^k_{\max}$, in each row/column at most one dominating vertex can exist. Since $a\le b$, then $|V_4\cap U|\le a$. Therefore $|V\cap U|=|U|\le \frac{mn}{N^k_{\max}}-\frac{ab}{N^k_{\max}}+a$. Maximum of $-\frac{ab}{N^k_{\max}}+a$ takes place when $b$ has its minimum value, i.e., $b=a$. Moreover, for $-\frac{a^2}{N^k_{\max}}+a$ we have that the maximum is $\frac{N^k_{\max}}{4}$ and it happens when $a=\frac{N^k_{\max}}{2}$. This results in $|U|\le \left\lceil \frac{mn}{ N^k_{\max}}+\frac{N^k_{\max}}{4}\right\rceil$.
\[theo:upper-bound-k-dist\] For an $m\times n$ grid $G=(V,E)$, a $k$-distance dominating set $S\subset V$ can be constructed using $k$-diagonalization and $k$-projection in polynomial-time such that $|S|\le \left\lceil \frac{(m+2k)(n+2k)}{N^k_{\max}}+\frac{N^k_{\max}}{4}\right\rceil$.
The proof follows from Lemmas \[lem:k-distance-unit\], \[lem:k-distane-k-super\] and \[lem:k-diag-cardinal\] and by replacing the diagonalization and projection operations with the $k$-diagonalization and $k$-projection operations in the proof of Theorem \[theo:upper-bound\] [@TYC'92].
\[lem:lower-bound-k-dist\] If $S\subset V$ is a $k$-distance dominating set for an $m\times n$ grid $G=(V,E)$, $|S|\ge\left\lceil \frac{mn}{N^k_{\max}}\right\rceil$
According to Lemma \[lem:k-distance-unit\], a vertex $v\in V$ $k$-dominates at most $N^k_{\max}$ vertices. Hence, at least $\left\lceil \frac{mn}{N^k_{\max}}\right\rceil$ dominating vertices are needed to $k$-dominate an $m\times n$ grid. Note that we use $|S|\ge\left\lceil \frac{mn}{N^k_{\max}}\right\rceil$ instead of $|S|\ge\left\lfloor \frac{mn}{N^k_{\max}}\right\rfloor$ since dominating vertices in the $k$-neighbourhood of vertices on the grid boundary do not have all their $k$-neighbours in $V$.
\[cor:upper-lower\] Let $S$ be a $k$-distance dominating set for an $m\times n$ grid $G=(V,E)$ obtained by $k$-diagonalization and $k$-projection and let $L$ denote the lower-bound for $S$ from Lemma \[lem:lower-bound-k-dist\]. For any constant $k\in \mathbb{Z_+}$, the approximation ratio $\frac{|S|}{L}$ satisfies $\lim_{n,m\rightarrow\infty} \frac{|S|}{L}=1$.
From Theorem \[theo:upper-bound-k-dist\] and Lemma \[lem:lower-bound-k-dist\], we have $$\frac{|S|}{L}\le\frac{\left\lceil(m+2k)(n+2k)/N^k_{\max}+N^k_{\max}/4\right\rceil}{\left\lceil
mn/N^k_{\max}\right\rceil}.$$ Therefore, $$\frac{(m+2k)(n+2k)/N^k_{\max}+N^k_{\max}/4}{mn/N^k_{\max}+1}\le\frac{|S|}{L},$$ and $$\frac{|S|}{L}\le\frac{(m+2k)(n+2k)/N^k_{\max}+N^k_{\max}/4+1}{mn/N^k_{\max}}.$$ Hence, we have $$\frac{(m+2k)(n+2k)+(N^k_{\max})^2/4}{mn+N^k_{\max}}\le\frac{|S|}{L},$$ and $$\frac{|S|}{L}\le\frac{(m+2k)(n+2k)+(N^k_{\max})^2/4+N^k_{\max}}{mn}.$$ For constant $k$ we have $$\lim_{n,m\rightarrow\infty}\frac{(m+2k)(n+2k)+(N^k_{\max})^2/4}{mn+N^k_{\max}}=$$ $$\lim_{n,m\rightarrow\infty}\frac{(m+2k)(n+2k)+(N^k_{\max})^2/4+N^k_{\max}}{mn}=1.$$ Therefore by the Squeeze Theorem $\lim_{n,m\rightarrow\infty} \frac{|S|}{L}=1$.
![A 2-diagonal pattern and a 3-diagonal pattern are depicted. Observe that the structure is similar to the regular diagonal pattern.[]{data-label="fig:k-distance-elem"}](k-distance-element)
For a graph $G$, its *$k$-th power*, denoted by $G^k=(V',E')$, is a graph with the same vertex set as $G$, i.e., $V=V'$, in which two distinct vertices share an edge if and only if their distance in $G$ is at most $k$ [@BM'08] (see Figure \[fig:k-power\]). Hence, in $G^k$ each vertex is connected to the vertices it $k$-distance dominates in $G$. We finish this section with the following remark that relates the $k$-distance dominating set problem in grids to the regular dominating set problem in their $k$-th power graphs.
\[rem:k-power\] It might seem that a reasonable approach for $k$-distance domination on a grid $G$ is to simply take the $k$-th power of the graph to obtain $G^k$, and then perform regular domination algorithms on $G^k$. Note that by the definition of $G^k$, a regular dominating set in $G^k$ is equivalent to a $k$-distance dominating set in $G$ and hence $\gamma(G^k)=\gamma^k(G)$. Unfortunately, $G^k$ is no longer a grid (e.g., there are diagonal edges connecting $v_{x,y}$ to $v_{x+1,y+1}$ for $k\ge 2$). In fact, it is not even a planar graph for $m\times n$ grids with $m,n\ge 2$. Therefore, as discussed in Section \[sec:intro\], choosing dominating vertices greedily in $G^k$ might obtain a dominating set with size as large as $(\ln(|V|)+1)\gamma^k(G)$.
![Figures (a) and (b) show a $3\times 3$ graph and its second power, respectively. Vertices within distance two are connected to each other in (b).[]{data-label="fig:k-power"}](k-power)
Distributed $k$-Distance Domination on Grids {#subsection:k-distance-dist}
--------------------------------------------
Using the algorithm explained in Section \[subsec:upper-bound-k-distance\], a distributed $k$-distance domination algorithm can be designed for grids. In practice, the $k$-distance dominating set problem corresponds to settings where agents are equipped with longer range sensory equipment and can sense vertices up to distance $k$ from them. Therefore, the goal is to arrange the agents on the grid vertices in a distributed way such that for each vertex there exists at least one agent with distance at most $k$ from it.
This algorithm is similar to Algorithm \[alg:dist-domin\] in Section \[susec:dist-alg\], except for two modifications. The first modification is that module $m_2$ and module $m_1$ with module centers $c(m_1)=v_{i,j}$ and $c(m_2)=v_{i',j'}$ can now connect to each other if $v_{i',j'}\in \{ v_{i+k,j+k+1},v_{i+k+1,j-k},v_{i-k,j-k-1},v_{i-k-1,j+k}\}$ (see Figure \[fig:k-distance-elem\]). These constitute the *slots*. The second modification is the definition of *orphans*. If $U'$ is a set of vertices that $k$-diagonalizes the $k$-super-grid of $G$, vertex $v\in V$ is an orphan if it satisfies the two following conditions: (a) $v$ has no $k$-neighbour in $U' \cap V$, and (b) $v$ is in the $k$-neighbourhood of a vertex $u\in U'\backslash V$ with the same $x$ or $y$ coordinates. Hence, *valid slots* are defined for each settled agent as the union of its slots located inside the grid and the orphans of its slots located outside the grid (see Figure \[fig:k-grid\]).
![A $16\times 16$ grid $G$ and its 2-super-grid $G'$ are shown by solid and dashed squares, respectively. Both grids are 2-diagonalized. The black circles are the vertices that 2-diagonalize $G$. The union of red and black circles 2-diagonalizes $G'$. The green circles are the 2-projections of the red circles onto $G$. Before projection these vertices are called orphans.[]{data-label="fig:k-grid"}](k-grid)
Summary and Open problems {#sec:conclusion}
=========================
In this paper we studied the dominating set and $k$-distance dominating set problems on $m\times n$ grids. We discussed a construction from [@TYC'92] to obtain dominating sets for grids with near optimal size and generalized it to work in the $k$-distance domination scenario. We used these methods in distributed algorithms and showed that the resulting dominating sets are upper-bounded by $\left\lceil\frac{(m+2k)(n+2k)}{2k^2+2k+1}+\frac{2k^2+2k+1}{4}\right\rceil$. The difference between the acquired upper-bound and the domination number of grid is at most five, for $16\le m\le n$ and $k=1$. However, via a more detailed case-based analysis in the grid corners, our distributed procedure can be used to obtain optimal dominating sets for $16\le m\le n$.
There are many open problems in this area. The $k$-domination number of grids is still unknown. It is also of interest to find centralized and distributed algorithms for dominating sub-graphs of grids, that is, grids with some of their vertices or edges missing. Generalizing these algorithms to the cases where the underlying graphs are cubic or hyper-cubic grids is another direction of this research.
[10]{} \[1\][\#1]{} url@rmstyle \[2\][\#2]{}
N. E. Leonard, D. Paley, F. Lekien, R. Sepulchre, D. M. Fratantoni, and R. Davis, “Collective motion, sensor networks and ocean sampling,” *Proceedings of the IEEE*, vol. 95, no. 1, pp. 48–74, 2007.
E. Guizzo, “Three engineers, hundreds of robots, one warehouse,” *IEEE Spectrum*, vol. 45, no. 7, pp. 26–34, July 2008.
R. W. Beard, T. W. McLain, M. A. Goodrich, and E. P. Anderson, “Coordinated target assignment and intercept for unmanned air vehicles,” *IEEE Transactions on Robotics and Automation*, vol. 18, no. 6, pp. 911–922, 2002.
F. Bullo, J. Cortés, and S. Mart[í]{}nez, *Distributed Control of Robotic Networks*, ser. Applied Mathematics Series.1em plus 0.5em minus 0.4emPrinceton University Press, 2009.
S. M. LaValle, *Planning Algorithms*.1em plus 0.5em minus 0.4emCambridge University Press, 2006, available at `http://planning.cs.uiuc.edu`.
M. Garey and D. Johnson, *Computers and Intractability: A Guide to the Theory of NP-Completeness*, ser. A Series of Books in the Mathematical Sciences.1em plus 0.5em minus 0.4emW. H. Freeman, 1979.
W. Abbas and M. B. Egerstedt, “Securing multiagent systems against a sequence of intruder attacks,” in *American Control Conference*, Montreal, Canada, June 2012.
J. Wu, M. Gao, and I. Stojmenovic, “On calculating power-aware connected dominating sets for efficient routing in ad hoc wireless networks,” in *International Conference on Parallel Processing*, 2001, pp. 346 –354.
M. Dorfling and M. A. Henning, “A note on power domination in grid graphs,” *Discrete Applied Mathematics*, vol. 154, no. 6, pp. 1023 – 1027, 2006.
R. Raz and S. Safra, “A sub-constant error-probability low-degree test, and a sub-constant error-probability pcp characterization of np,” in *Proceedings of the twenty-ninth annual ACM symposium on Theory of computing*.1em plus 0.5em minus 0.4emACM, 1997, pp. 475–484.
D. S. Johnson, “Approximation algorithms for combinatorial problems,” *Journal of computer and system sciences*, vol. 9, no. 3, pp. 256–278, 1974.
L. Lov[á]{}sz, “On the ratio of optimal integral and fractional covers,” *Discrete mathematics*, vol. 13, no. 4, pp. 383–390, 1975.
A. Bondy and U. Murty, *Graph Theory*, ser. Graduate Texts in Mathematics.1em plus 0.5em minus 0.4emSpringer, 2008.
B. S. Baker, “Approximation algorithms for np-complete problems on planar graphs,” *Journal of the ACM (JACM)*, vol. 41, no. 1, pp. 153–180, 1994.
B. Liu and D. Towsley, “On the coverage and detectability of large-scale wireless sensor networks,” in *WiOpt’03: Modeling and Optimization in Mobile, Ad Hoc and Wireless Networks*, 2003.
J. Li, J. Jannotti, D. D. Couto, D. Karger, and R. Morris, “A scalable location service for geographic ad hoc routing,” in *International conference on Mobile computing and networking*, Boston, MA, 2000.
J. Blum, M. Ding, A. Thaeler, and X. Cheng, “Connected dominating set in sensor networks and manets,” *Handbook of Combinatorial Optimization*, pp. 329–369, 2005.
M. Cardei and J. Wu, “Coverage in wireless sensor networks,” *Handbook of Sensor Networks*, pp. 422–433, 2004.
T. Y. Chang, “Domination numbers of grid graphs,” Ph.D. dissertation, University of South Florida, 1992.
S. Alanko, S. Crevals, A. Isopoussu, and V. Pettersson, “Computing the domination number of grid graphs,” *The Electronic Journal of Combinatorics*, vol. 18, no. P141, p. 1, 2011.
D. Gonçalves, A. Pinlou, M. Rao, and S. Thomass[é]{}, “The domination number of grids,” *CoRR*, vol. abs/1102.5206, 2011.
T. Cormen, C. Leiserson, R. Rivest, and C. Stein, *Introduction To Algorithms*.1em plus 0.5em minus 0.4emMIT Press, 2001.
G. Piovan, I. Shames, B. Fidan, F. Bullo, and B. D. O. Anderson, “On frame and orientation localization for relative sensing networks,” in *[IEEE]{} Conf. on Decision and Control*, Cancún, México, Dec. 2008, pp. 2326–2331.
M. A. Henning, “Distance domination in graphs,” in *Domination in Graphs: Advanced Topics*, T. W. Haynes, S. T. Hedetniemi, and P. J. Slater, Eds.1em plus 0.5em minus 0.4emMarcel Dekker, New York, 1998, pp. 321–349.
[^1]: This research is partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).
[^2]: The authors are with the Department of Electrical and Computer Engineering, University of Waterloo, Waterloo ON, N2L 3G1 Canada. (email: [email protected]; [email protected]; [email protected])
[^3]: One can also define a diagonal pattern as a set of vertices whose $(x,y)$ coordinates satisfy $x-2y\equiv r \pmod{5}$, for some fixed $r$. This corresponds to swapping the $x$ and $y$ axes. For the proofs we only analyze the case mentioned in Definition \[def:diag-patt\]; the other case can be treated similarly.
[^4]: Note that for completeness of the algorithm, it is not necessary for the active agents to go to the closest settled agents. An active agent can go toward any arbitrary settled agent to occupy its valid slot.
|
---
abstract: 'The high-luminosity LHC (HL-LHC) upgrade is setting now a new challenge for particle detector technologies. The increase in luminosity will produce a particle background in the gas-based muon detectors that is ten times higher than under conditions at the LHC. The detailed knowledge of the detector performance in the presence of such a high background is crucial for an optimized design and efficient operation after the HL-LHC upgrade. A precise understanding of possible aging effects of detector materials and gases is of extreme importance. To cope with these challenging requirements, a new Gamma Irradiation Facility (GIF++) was designed and built at the CERN SPS North Area as successor of the Gamma Irradiation Facility (GIF) during the Long Shutdown 1 (LS1) period. It features an intense source of 662 keV photons with adjustable intensity, to simulate continuous background over large areas, and, combined with a high energy muon beam, to measure detector performance in the presence of the background. The new GIF++ facility has been operational since spring 2015. In addition to describing the facility and its infrastructure, the goal of this work is to provide an extensive characterization of the GIF++ photon field with different configurations of the absorption filters in both the upstream and downstream irradiation areas. Moreover, the measured results are benchmarked with Geant4 simulations to enhance the knowledge of the radiation field. The absorbed dose in air in the facility may reach up to 2.2 Gy/h directly in front of the irradiator. Of special interest is the low-energy photon component that develops due to the multiple scattering of photons within the irradiator and from the concrete walls of the bunker.'
address:
- 'CERN, CH-1211 Geneva 23, Switzerland'
- 'European Spallation Source (ESS ERIC), P.O. Box 176, SE-22100 Lund, Sweden'
- 'Ecole Polytechnique Fédérale de Lausanne(EPFL), CH-1015 Lausanne, Switzerland'
- ' RWTH Aachen University, 52062 Aachen, Germany'
- 'Institute of Experimental and Applied Physics, Czech Technical University in Prague, 16636 Prague 6, Czech Republic'
author:
- Dorothea Pfeiffer
- Georgi Gorine
- Hans Reithler
- Bartolomej Biskup
- Alasdair Day
- Adrian Fabich
- Joffrey Germa
- Roberto Guida
- Martin Jaekel
- Federico Ravotti
title: The radiation field in the Gamma Irradiation Facility GIF++ at CERN
---
Gamma irradiation ,irradiation facility ,detector test ,$^{137}$Cs source ,Geant4 ,simulation
Introduction {#sec:intro}
============
The Gamma Irradiation Facility (GIF) [@Agosteo] at CERN [@CERN], the European Organization for Nuclear Research, was extensively used from 1997 until its closure in 2014 for the characterization of particle detectors. Located in the former CERN SPS West Area, the facility played in particular an important role in testing large area muon detector systems and components for the Large Hadron Collider (LHC) [@LHC]. In this facility, detectors could simultaneously be exposed to the photons from a $^{137}$Cs source and to a high-energy muon beam. Although from 2005 onwards only the caesium source was available, the GIF continued to be fully exploited all year round by a wide community of users. The high-luminosity LHC (HL-LHC) upgrade [@HL-LHC] is setting now a new challenge for particle detector technologies. The increase in luminosity will produce a particle background in the gas-based muon detectors that is an order of magnitude higher than under present conditions at the LHC, hence detector tests at accordingly higher rates are required. The detailed knowledge of the detector performance in the presence of such a high background is crucial for an optimized design and efficient operation after the HL-LHC upgrade. A precise understanding of possible aging effects of detector materials and gases is also of extreme importance. To cope with these challenging requirements, a new Gamma Irradiation Facility (GIF++) was designed and built during the Long Shutdown 1 (LS1) period at the CERN SPS North Area as successor of the Gamma Irradiation Facility (GIF) [@Jaekel]. Whereas CERN was responsible for the construction of the facility and the procurement of the irradiator, a comprehensive user infrastructure was provided within the framework of the FP7 AIDA project [@Aielli]. The new GIF++ facility offers two separated irradiation areas and has been operational since spring 2015.
The goal of this note is to provide an extensive characterization of the GIF++ photon field with different configurations of the absorption filters in both the upstream and downstream irradiation areas. Measured dose rates are benchmarked with Geant4 [@Geant4a] simulations to enhance the knowledge of the radiation field. Of special interest is the low-energy photon component that develops due to the multiple scattering of photons within the irradiator and from the concrete walls of the bunker. A detailed description of the shielding of the GIF++ bunker and its optimization with the help of Monte Carlo simulations is in preparation [@FLUKA].
The GIF++ facility {#sec:facility}
==================
Layout of the facility {#subsec:Layout of the facility}
----------------------
Focused on the characterization and understanding of the long-term behavior of large gas-based particle detectors, GIF++ combines a $^{137}$Cs source[^1] with two sets of adjustable filters to vary the intensity, and a high-energy muon beam (100 GeV/c) from the secondary SPS beam line H4 in EHN1. The $^{137}$Cs isotope was chosen instead of $^{60}$Co due to its long half-life of 30.08 years, leading to a smaller decrease of the photon rate over the expected lifetime of this facility. Further, the typical energy of the neutron-induced background radiation at LHC experiments like CMS [@Mueller] approximately matches the energy spectrum of the $^{137}$Cs source, composed of the primary 662 keV photons and lower energetic scattered photons. The layout of the GIF++ facility and the used coordinate system are shown in Figure \[fig: facility\].
![Floor plan of the GIF++ facility with entrance doors MAD (material access door), PPG (personal protection gate), PPE (personal protection entrance), PPX (personal protection exit). When the facility downstream of the GIF++ takes electron beam, a beam pipe is installed along the beam line (z-axis) between the vertical mobile beam dump (XTDV). The irradiator can be displaced laterally (its center moves from x = 0.65 m to 2.15 m), to increase the distance to the beam pipe.[]{data-label="fig: facility"}](./figures/GIF++_facility.pdf){width="85.00000%"}
High-energy reference beams {#subsec:High-energy reference beams}
---------------------------
Located near the end of the beam line H4, GIF++ is the main user of this beam line for six to eight weeks per year. In addition, the facility receives parasitic muon beam halo for 30-50$\%$ of the SPS operation time. The muon beam is generated as a secondary beam from the primary SPS proton beam on a production target. The spectrometer of the H4 beam line allows the beam line user to choose the nominal momentum of the secondary beam with a maximum momentum of 400 GeV/c. The muon beam in the energy range of 57-100$\%$ of the nominal beam energy is mainly generated by the decay of pions and kaons that are produced in the primary target. Depending on the beam line settings, either the full bandwidth or an energy spread reduced to the percent level is transported to the GIF++ area. The spill structure of the muon beam follows the primary proton beam structure, which has a spill of 4.8 seconds with a close to flat distribution. Depending on the SPS cycling for other users (e.g. LHC), one spill arrives about every 30 seconds on average. With the full acceptance of the beam line, the intensity can be up to 10$^4$ muons per spill, which is limited by radiation protection aspects. The secondary beam can also be adjusted for hadrons and/or electrons. For about five weeks per year an electron beam is granted to main users whose experiments are located downstream of the GIF++ area. During this time, an evacuated pipe is installed all along the beam line in the GIF++ bunker, but the photon source in GIF++ can still be used. In order to increase the distance to the beam pipe when needed, provision is made to shift the whole irradiator laterally. The lateral distribution of the muon beam at the GIF facility depends on several factors like the final focusing, which can be varied within a large range. In case of parasitic use, the settings chosen by the primary beam line user upstream determine the intensity and energy. As a general rule, the core of the muon beam covers a surface of 10 cm x 10 cm, containing half of the muon beam. The remaining part of the muon beam, the beam halo, is spread over a footprint of about 1 m$^2$.
Irradiator and filter system {#subsec:Irradiator and filter system}
----------------------------
Gamma irradiation is however available throughout the whole year, except during short maintenance periods. The 100 m$^{2}$ GIF++ bunker is about 5 m high and has two independent irradiation zones (named as upstream and downstream in Figure \[fig: facility\]), making it possible to test simultaneously several real size detectors, with a size of up to several square meters, as well as a broad range of smaller prototype detectors and electronic components. The GIF++ irradiator and its filter systems are depicted in Figure \[fig: Filters and filter factors\]. The irradiator has been developed in cooperation with the Czech company VF a.s. [@VF]. The caesium source can be moved from the garage position at the bottom of the support tube inside the shielded receptacle to the irradiation position at the top of the tube.
With two $\pm$ 37$^\circ$ panoramic collimators, the irradiation zone covers a large part of the bunker area, both in the downstream and upstream regions. As shown in Figure \[fig: irradiator\], both outlets of the irradiator are equipped with a lens shaped angular correction filter to provide a uniform photon distribution over a plane, as needed for flat large area detectors. Embedded inside a common enclosure, two complete and independent attenuation systems are available (Figure \[fig: filters\]), each consisting of an array of 3x3 convex lead attenuation filters, to fine tune the photon flux for each irradiation field individually. A collaboration with the RWTH Aachen [@RWTH] took care of design and production of the angular correction and attenuation filters. Mounted on aluminum support plates, the filters are positioned inside steel frames, as collimators, and connected to counterweights moving on the side of the irradiator. Three planes of filters (A, B, C), with three filters per plane are installed. The filters have the nominal attenuation factors 1 (A1,B1,C1), 1.5 (B2), 2.2 (C2), 4.6 (C3), 10 (A2) and 100 (A3, B3). The total attenuation factor, from the three filter layers, can be set both via a dedicated control panel, as well as via the GIF++ control system. In total, 24 different nominal attenuation factors between 1 and 46415 can be selected according to the 27 possible combinations. The factors were chosen to be nearly equidistant, on a logarithmic scale, over the first three orders of magnitude, as illustrated in Figure \[fig: filter\_factors\]. The irradiator and the two filter systems are mounted on rails, so that the whole assembly can be moved in the direction transverse to the muon beam.
Angular correction filter and attenuation filters: Adjustment of current {#subsec:modification of current}
------------------------------------------------------------------------
Without the angular correction filter, the GIF++ source would be approximately a point source. The angular correction filter, made of steel, is shaped in such a way, that the $\frac{1}{r^2}$ dependence of the photon current is replaced by a uniform current in each xy plane. The $\pm$ 37$^\circ$ opening of the irradiator collimators allows horizontally and vertically angles $\theta$ (angle between the incoming photon and the surface normal of the filter) from 0$^\circ$ in the center of the filter to 46.8$^{\circ}$ in the corners of the collimator. To reach a uniform current, the photons have to be attenuated with a factor of cos$^{-3}$($\theta$ - 46.8$^\circ$). That means that the photons in the center of the filter are attenuated by a factor of cos$^{-3}$(46.8$^\circ$), whereas the photons in the corners of the collimators are not attenuated. Each angle of incidence corresponds to a different path length traversed in the filter. The actual attenuation of the photon depends on this path length and the linear attenuation coefficient $\mu$ of the material[^2]. The thickness of the angular correction filter varies for each angle of incidence in such a way, that the desired attenuation is reached for all 662 keV photons. Here attenuation can either mean that the photon has lost energy, or that it has been fully absorbed. For photons with lower energies, created due to scattering in the source capsule and the collimator of the irradiator, the angular correction filter leads to a less uniform current.
For the attenuation filters, on the other hand, the attenuation factor does not depend on the angle of incidence of the photons. The convex face of the attenuation filters is shaped in such a way that at every point of the filter, photons from the source traverse the same thickness of material and hence undergo the same attenuation. The nominal attenuation factor is again defined as attenuation of 662 keV photons; photons with lower energy are attenuated to a larger degree. Due to the presence of lower energy photons, the effective total attenuation over the whole spectral range is lower than the nominal attenuation. The exact value will depend on the spectral sensitivity of the detector used, as well as on the presence of other objects in the bunker. Figure \[fig: current\_662keV\_y\] shows the simulated current of 662 keV photons in the GIF++ facility in the yz plane for an attenuation factor 1, i.e. unattenuated. This is thus the highest current obtainable at GIF++. With the help of the angular correction filters, the current depends only on the distance from the irradiator along the z-coordinate and is uniform in all xy planes. The corresponding xz view of the current of 662 keV photons can be found in Figure \[fig: Current\_DS\_US\_x\_600\_667\].
![Photon current in the vertical plane through the source (yz plane) at x = 0.65 m; attenuation filters at factor 1. With angular correction filters, the current of 662 keV photons is made uniform in xy planes. []{data-label="fig: current_662keV_y"}](./figures/Current_gifpp_DS_US_flatSurfaceCurrent_600_662_x_60_70.pdf){width=".785\textwidth"}
Area monitoring {#subsec:monitoring}
---------------
Ambient dose equivalent and ambient dose equivalent rates in the GIF++ zone are monitored by the Radiation Protection Group (RP) of CERN’s Health and Safety unit using the RAdiation Monitoring System for the Environment and Safety[^3] (RAMSES) [@RAMSES]. In the GIF++ facility, two RAMSES Induced Activity Monitors[^4] are installed at fixed locations. One is attached to the bunker wall upstream and the other downstream at a similar height. The monitors are connected to signal conditioning electronics contained within a rack mounted monitoring station. The monitoring stations are designed in such a way that they perform their function independently of their supervisory system. They are equipped with an uninterruptible power supply that allows them to assure their function without external electrical power for at least two hours. In the case of GIF++, RAMSES generates hardwired interlocks to the access control system to block access to the zone whilst the source is exposed. RAMSES allows remote on-line supervision of all the measured variables as well as data logging and long-term archiving for off-line data analysis and reporting. The RAMSES installation is thus part of the safety infrastructure of the zone, but in addition provides radiation monitoring information for the users of the facility.
Within the framework of the EU-funded AIDA project, the Institute For Nuclear Research And Nuclear Energy in Sofia (Bulgaria), provided a Radiation Monitoring (RADMON) readout system for the monitoring of the integrated dose delivered by the photon source. The RADMON sensor was developed at CERN [@RADMON] as a flexible and convenient solution for radiation monitoring, and is currently being used by several experiments at CERN and external facilities. Various dosimeters with different sensitivities and dynamic ranges can be mounted on the RADMON Integrated Sensor Carrier(ISC) to monitor radiation fields with wide ranges of ionizing dose and particle fluence. Starting in October 2015, the RADMON readout system has been fully implemented within the GIF++ bunker. In total 12 RADMON sensors can be freely positioned in the downstream and upstream area of the facility. The sensors are read out automatically by the GIF++ Control System and data are made available via software through the DIP communication protocol. In contrast to the RAMSES monitors, the RADMONs are not a safety system but a dedicated radiation monitoring system for the GIF++ users.
Setup of simulations and measurements {#sec:measurement and simulations}
=====================================
Units {#subsec:units}
-----
In the present paper, the air kerma[^5] $K_a$ is used to quantify the absorbed dose in air. Kerma and absorbed dose have both the unit Gy and are closely related, but the kerma is usually more convenient to calculate [@IAEA]. The ambient dose equivalent[^6] H$^*$(10), the unit used by area radiation monitors, can be converted to air kerma $K_a$ with the help of conversion factors [@ICRP74].
The particle fluence is defined as $\Phi=\frac{dl}{dV}$, the sum of all the particle trajectories $dl$ per unit volume $dV$ [@ICRP119]. The unit of the fluence is m$^{-2}$. The fluence rate or flux is defined as $\phi=\frac{d\Phi}{dt}$ and has the unit of m$^{-2}$s$^{-1}$. The deposited energy of a particle in matter is proportional to the particle flux, therefore flux is the adequate unit when dealing with detectors for calorimetry.
On the other hand, current is a measure of the net number of particles crossing a *flat surface* with a well-defined orientation. The unit of current is m$^{-2}$s$^{-1}$ and thus identical to the unit of flux. Current is meaningful in cases where particles are counted without any interest in their interactions. For example, a fully efficient infinitesimal thin detector would record each photon entering the detector independent of the actual energy deposited by the particle.
For the GIF++ it was decided, that the typical detector tested in the facility comes rather close to this model, and thus current and not flux was used as design quantity for the angular correction filter. In a directed radiation field flux and current are identical for normal incidence to the surface. At all other angles the flux is higher by a factor of cos$^{-1}$($\theta$), with $\theta$ as the angle between incoming particle and surface normal of the scoring surface [@Huhtinen].
Simulation setup {#subsec:simulation setup}
----------------
Fluence and current in the GIF++ photon field were simulated with Geant4 10.0 and the G4EmLivermorePhysics physics list. For this purpose, the whole simulated facility was partitioned using a mesh of 5 cm x 5 cm x 5 cm scoring cubes. In each of the cubes, the photon fluence and photon current binned in 100 keV bins were calculated[^7]. Additionally, for the measurement locations listed in Table \[table: dose\], the spectral distribution of the photons was simulated in 1 keV bins. From the simulated fluence, the air kerma $K_a$ was calculated using the conversion factors published in reference [@Veinot]. For the albedo studies, the interactions of photons of different energies and angles of incidence with the materials present in the facility (lead, steel, concrete, aluminum) were simulated.
Measurement locations {#subsec:measurement locations}
---------------------
Dose measurements were carried out in the GIF++ facility in March 2015, December 2015 and March 2016 in the positions shown in Table \[table: dose\]. D stands for positions downstream, U for positions upstream and I for a position outside the irradiation area. The origin of the GIF++ coordinate system (Figure \[fig: facility\]) is defined in x and y by the position of the beam line and in z, along the beam line, by the center of the source.
RAMSES and Automess gamma probe 6150AD-15 measurements {#subsec:automess and ramses measurements}
------------------------------------------------------
The RAMSES data shown here have been taken in March 2015, prior to the installation of any user equipment. Additionally a gamma probe[^8] was used to measure the ambient dose H$^*$(10). The useful dose rate range of the probe is 1 mSv/h - 9.99 Sv/h. The manufacturer calibrated the probe with a 333 kBq $^{137}$Cs calibration source. Within the useful dose rate range, a linearity deviation of the measured intensity of up to $\pm$ 10$\%$ compared to the calibration is permitted. The energy range extends from 65 keV to 3 MeV. The nominal angular range is $\pm$ 45$^\circ$ around the preferential direction perpendicular to the axis of the tube. The combined energy and directional dependence shows a maximum deviation of up to $\pm$ 40$\%$ for all energies and directions compared to 662 keV photons from $^{137}$Cs arriving in the preferential direction [@gamma; @probe; @probe; @manual]. The ambient dose equivalent H$^*$(10) was converted to air kerma $K_a$ using the factors described in reference [@Units].
RADMON measurements {#subsec:radmon measurements}
-------------------
In December 2015 RADMON [@RADMON] measurements were conducted in the downstream area, and in March 2016 in the upstream area. All the RADMON have been equipped with an on-board temperature sensor and two RADFETs for ionizing dose measurement[^9]. The choice was driven by the need for a device with sensitivity in the range of several mGy, but capable to integrate doses up to the tens of kGy range. As a passive integrating sensor, the RADMON gives as information the total cumulative dose. Normally it serves as device for monitoring radiation levels, but can also be used as a dose rate dosimeter. Due to the very small active area of the sensor, the length of the cables and the precision of the electronics readout, the minimum detectable dose rate at GIF++ was found to be in the order of 10 mGy/h. Consequently, a consistent dose rate measurement at GIF++ can be achieved only by operating the sensors at an adequately high dose rate, or by allowing a sufficient integration time to obtain a measurable dose increase. In order to fully test the RADMON performance in the GIF++ photon field and to check the correct functioning of the readout system, several calibration measurements have been performed. The presence of other experimental equipment in the zone lead to slightly modified RADMON positions compared to the survey in March 2015, and also affected the measurements.
The experimental data have been collected in two runs. The measurements in the downstream area were performed over two weeks of irradiation in December 2015. In that period the attenuation factors were varied between 1 and 100. The source duty cycle and the different attenuation factors were taken into account when calculating the dose rates. Measurements in the upstream area were conducted in March 2016 over three days of operation, with the source always on and the attenuation factor always equal to 1. The measured dose rate was automatically corrected for temperature variations, although the variation during this test stayed within $\pm$ 5$^\circ$C, which is lower than the minimum needed to increase the dose readout by one bit. Additionally, the losses related to annealing were negligible, due to the relatively short exposure time. Since April 2016, the GIF++ bunker has been equipped with an air conditioning system that allows the control of the temperature and thus limits the temperature variations. The annealing on the other hand has to be taken into account in long-term measurements or after periods of long inactivity of the sensor.
Results {#sec:results}
=======
Measurements and simulations of the filter attenuation factors {#subsec:attenuation}
--------------------------------------------------------------
An attenuation of 1 means that photons emerging from the angular correction filter are not further attenuated, and is subsequently called *fully open*. The maximum attenuation factor of 46415 is called *fully closed*. Table \[table: attenuation measured\] shows that for the lower attenuation factors, the dose attenuation measured with the Automess gamma probe 6150AD-15 is comparable to the nominal attenuation of the 662 keV photons. For factors greater than 10 though, the effective dose attenuation is considerably lower than the nominal attenuation factor, since scattered photons with an energy smaller than 662 keV contribute substantially.
-- ---------- --------------- --------
**** **\[mGy/h\]** ****
A1 B1 C1 470.00 -
A1 B2 C1 400.00 1.2
A1 B1 C2 211.00 2.2
A1 B1 C3 105.00 4.5
A2 B1 C1 55.00 8.8
A3 B1 C1 6.50 72.3
A1 B3 C1 6.20 75.8
A1 B3 C3 1.59 295.6
A2 B3 C3 0.22 2156.0
A3 B3 C3 0.05 9400.0
-- ---------- --------------- --------
: Nominal attenuation factors (attenuation of the 662 keV photons) of some filter settings and measured effective attenuation in position D1 (x=0.65m, y=0.00m, z=1.10m).[]{data-label="table: attenuation measured"}
Table \[table: Simulated current attenuation\] displays the nominal and simulated attenuation of the photon current in position U1. The values in the different energy ranges are expressed as ratio of unattenuated over attenuated current. As explained in Section \[subsec:modification of current\], the nominal attenuation factor of the filters is the attenuation of the 662 keV photons. The attenuation for lower energy photons deviates from this factor. In general, photons with energies between 100 keV and 300 keV are attenuated to a larger extent than the attenuation factor implies, whereas photons with energies smaller than 100 keV and between 400 keV and 600 keV are attenuated less. Depending on the energy of the incoming photons, a different percentage of the photons Compton scatters or reacts via the photoelectric effect. Further, detectors to be tested at GIF++ are characterized by a variety of materials, shapes, used gases and conditions of operation. The effective attenuation with regard to different photon energies is thus also detector dependent and hard to predict. The attenuation filters for detector tests should therefore be chosen carefully, also considering the detector specific sensitivity.
-- ------ ------ ------ ------ ------ ------ ------ ------ ------
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
0.74 4.11 2.47 1.41 1.15 1.18 1.41 1.42 1.45
1.19 7.3 5.7 2.10 1.56 1.55 2.02 2.0 2.12
1.63 10.4 8.6 2.89 1.94 1.97 2.87 2.8 3.10
2.3 13.5 13.2 4.28 2.80 2.68 4.09 3.9 4.50
3.3 18.8 18.5 5.9 3.65 3.50 5.78 5.4 6.54
5.0 19.6 19.0 8.8 4.69 4.50 8.22 7.3 9.47
6.3 32 30 11.8 6.2 5.85 11.8 10 13.8
8.7 47 58 17.7 9.3 8.2 16.7 15 19.9
12.6 62 84 24.1 12.0 11.1 24.1 20 29.5
-- ------ ------ ------ ------ ------ ------ ------ ------ ------
: Nominal and simulated attenuation of photon current in position U1. The values in the different energy ranges are expressed as ratio of unattenuated over attenuated current. The intensity of the unattenuated photon current in U1 can be found in Table \[table: Simulated current\].[]{data-label="table: Simulated current attenuation"}
Figure \[fig: Spectra attenuation factors\] shows the spectra of the photon current in location D1 for the attenuation factors 1, 10 and 100. In addition to the narrow 662 keV peak, a broad low energy component is visible. In fact the photons arrive already at the attenuation filters with this low energy component. Before even reaching the angular correction filter, the source capsule and the irradiator collimator already cause an amount of scattering that broadens the spectrum. Scattering in the angular correction filters and the bunker (walls, floor and roof) adds further low energy contributions to the spectrum. As expected, the intensity of the photons with an energy of 662 keV is reduced by a factor of 10 or 100, in agreement with the nominal attenuation of the lead filters. Equivalent to the situation in position U1 described in Table \[table: Simulated current attenuation\], the 400 keV to 600 keV photons are attenuated less, and the 100 keV to 300 keV photons are attenuated more than the nominal attenuation factor. Clearly visible in the spectra are the back-scatter peak at 184 keV and the characteristic $^{82}$Pb K-shell X-ray peaks around 80 keV, which cannot be resolved individually though, due to the energy binning. Both features are explained in more detail in Section \[sec: albedo studies\].
Figure \[fig: crosstalk\] shows three spectra in location D1, one with the downstream filters fully open and the upstream filters fully closed, one with the downstream filters fully closed and the upstream filters fully open and the third one with both sides fully closed. The two spectra with D closed show that when opening U, the rate in D increases significantly. By Compton scattering multiple times, photons from the upstream opening manage to arrive in the downstream zone of the facility. The absolute intensity of this *crosstalk* photon current is nevertheless small. Therefore the field in the measurement zone can be regarded as basically independent from the field in the opposing zone, unless a very large attenuation factor is used in the measurement zone.
Simulations of degraded photons and albedo {#sec: albedo studies}
------------------------------------------
The photon current is modified by the interaction of photons with the materials present in the GIF++ facility. The materials concerned are lead (irradiator, filters), steel (filters, floor), aluminium (filters) and concrete (surrounding bunker enclosure). For the available photon energies of 662 keV or below, the main processes involved are Compton scattering and photoelectric effect. Following the photoelectric effect in lead, there is a high probability that the atom de-excites by emitting fluorescent X-rays. To better understand the photon current modification, these processes were simulated for photons of different energies and angles of incidence. Through collisions with electrons of dense materials, photons may loose a fraction of their energy along their travel. Therefore, in addition to the monochromatic 662 keV photons from $^{137}$Cs, a low energy component of *degraded* photons develops within the irradiator. This low energy component is already present when the photons emerge from the small thin-walled capsule containing the active material[^10]. Interactions in the material subsequently traversed further modify the spectral shape of degraded photons, as illustrated in Figure \[fig: energy distribution degraded photons\] for photons emerging from the capsule, from the angular correction filter, and from an exemplary attenuation filter of nominal factor 100. The peak around 80 keV corresponds to the $K_\alpha$ fluorescence line expected for lead. The vertical scale is normalized to one emerging 662 keV photon. For large attenuation factors, a general trend is visible. In comparison with the nominal attenuation factor for the 662 keV photons, the attenuation is more pronounced below 400 keV, but less pronounced above 400 keV. The fraction of degraded photons is coarsely the same at all stages, that means the number of degraded photons is also reduced when the number of 662 keV photons is reduced by absorption filters. The degraded photons amount to about 46$\%$ of the photons that reach the angular correction filter. Behind the angular correction filter, 39$\%$ of the photons that emerge from the irradiator within its 37$^\circ$ by 37$^\circ$ aperture are degraded. Behind the filter with a nominal attenuation of 100, the final fraction of degraded photons reaches 56$\%$.
All photons emitted by the irradiator will interact with the surrounding material in the bunker and will finally be absorbed. The photons that are not absorbed when impinging on a wall, but manage to re-enter into the bunker, are called *albedo* photons. They are expected to be mainly back-scattered photons from Compton scattering; for lead some fluorescence photons contribute to the albedo. The calculated energy distribution of albedo photons is shown in Figure \[fig: energy distribution albedo\] for 662 keV photons impinging with angles of 0$^\circ$ and 37$^\circ$ with respect to the surface normal on a wall of infinite thickness and made of concrete, steel or lead. The vertical scale is per photon incident on the wall.
While colliding with a nearly free electron, the lowest energy of the scattered photon corresponds to a scattering angle of 180$^\circ$. For 662 keV photons this leads, independent of the wall material, to a back-scattered photon of about 184 keV. Photons back-scattered at other angles will have somewhat higher energy. Inside the wall, before emerging again, the photon may suffer further Compton scatterings, which will lower the energy of the albedo photon. Also incident photons of energies lower than 662 keV will extend the albedo spectrum to lower energies. As visible in Figure \[fig: energy distribution albedo\], the concrete and steel spectra are dominated by this Compton back-scatter peak, and it is also visible in the lead spectrum. The contribution from the photoelectric effect, approximately proportional to Z$^3$ (Z is the nuclear charge of the target material) and to about E$^{-3}$ (E is the energy of the incident photon), is therefore only relevant for lead in terms of probability and of energy. The characteristic K$\alpha$ and K$\beta$ X-ray fluorescence lines [@X-ray] of 72 to 88 keV are prominent in the lead albedo spectra. For all three materials, the spectra at 0$^\circ$ and at 37$^\circ$ incidence are almost identical, except in the tail region above 250 keV. At 0$^\circ$ (37$^\circ$) incidence the fraction of albedo per incident 662 keV photon is 24.6$\%$ (29.5$\%$) for concrete, 14.7$\%$ (18.6$\%$) for steel and 1.9$\%$ (2.8$\%$) for lead.
To give an overview of the amount of albedo under different conditions relevant at GIF++, Figure \[fig: albedo fraction\] illustrates the dependence of albedo on the energy of the incident photon, for incident angles from 0$^\circ$ to 45$^\circ$. The lower(higher) border of the bands is for 0$^\circ$(45$^\circ$). The influence of the angle of incidence on the back-scatter probability is small. Down to about 250 keV the probability of *reflecting* a photon is roughly the same as for 662 keV. In this main region of energies it is an order of magnitude lower for lead than for concrete. Below 250 keV the probability decreases, except for lead which shows a large increase related to the photoelectric effect.
It is interesting to see how deep a photon penetrates into a wall before emerging again. The calculated average depth reached by these photons, shown in Figure \[fig: albedo depth\], is basically independent of the angle of incidence. It decreases slowly from the largest depth at 662 keV down to about 200 keV; the standard deviation has almost the same value as the average. With a wall thickness of twice this average depth the albedo production is thus saturated. Therefore with a 2 mm thick lead layer on walls, ceiling and floor, the amount of albedo photons could be reduced by one order of magnitude with respect to concrete. The large penetration depth in concrete, together with the angular distribution of the scattered photons inside the wall material has as consequence that the photons will emerge at a location, which may be different from the location of the impinging photon. The emission of albedo photons is isotropic in the azimuth angle with regard to the surface normal of the scattering surface. The polar emission angle is almost independent from the energy and the angle of incidence of the incoming photon from the source. At an incidence angle of 0$^\circ$ and 37$^\circ$, the average polar emission angle amounts to 50$^\circ$. At the wall surface the exit point of albedo from 662 keV photons is on average at about 3 cm distance from the impact point, with a 3.5 cm standard deviation. At 100 cm from the wall, the exit point is on average 160 cm away from the impact point, with 68$\%$ of the photons having an exit point that is less than 155 cm away. This shows that albedo is expected to be widespread in the bunker.
The angular distribution of albedo photons, as they are originating at points distributed over the whole GIF++ bunker, is of course very different from the distribution for photons from the irradiator. The final distributions of photon rate, angle, energy in the GIF++ bunker will follow the principles described in this section, while reflecting the complexity of the actual geometry of the bunker and of all the installed equipment like detectors in the facility. Simulations providing a guideline for GIF++ as a whole are described in the following sections.
Measurements and simulations of the dose rate {#sec:dose}
---------------------------------------------
Figure \[fig: Dose\] shows the simulated absorbed dose rate in air in the xz plane of the GIF++ bunker. The measurement locations are marked in black. In Figure \[fig: DS\] the irradiator was fully open downstream and fully closed upstream, whereas in Figure \[fig: US\] the irradiator was fully closed downstream and fully open upstream. A fully closed irradiator outlet means an attenuation factor of 46415. The ratio of the doses in U1 and D1 reflects this attenuation in both figures. They demonstrate that the photons scatter significantly from the bunker walls, floor and roof. A quiet spot in the facility does not exist.[^11] Nevertheless, the two figures confirm that the downstream and upstream zones of the GIF++ facility are basically independent. The intensity of the scattered photons depends on the exact position in the facility. A position closer to the walls like U6 suffers more from albedo than position D5, but both positions see about the same amount of direct photons. Furthermore the figures show that the angular correction filters do not only create a uniform current of 662 keV photons over the xy-planes within the irradiation regions, but also lead to a dose distribution that almost exclusively depends on z.
Measurements, as shown in Table \[table: dose\], are compared with the simulations for the locations indicated in Figures \[fig: Dose\] and \[fig: RADMON measurements\]. For the simulations, in addition to the statistical error, a position uncertainty of $\pm$ 5 cm was used, as taken from the estimated position uncertainty of the measurement probes.
Given the Automess 6150AD-15 probe’s allowed linearity deviation of $\pm$ 10$\%$ and combined directional and energy deviation of $\pm$ 40$\%$, a total dose measurement uncertainty of $\pm$ 40$\%$ was assumed. The simulated dose rates are always larger than the measured dose rates with the Automess 6150AD-15 probe. In locations close to the irradiator, like the locations D1 and U1, where the majority of the dose is caused by 662 keV photons, measurement and simulation agree to about 15$\%$. Even at location I1, outside of the irradiation area, measurement and simulation agree within 40$\%$. Within the irradiation area, in locations with the same absolute value of the z-coordinate but different x coordinate, the measured dose rates vary less than 20$\%$. For the positions U3, U3a and U3b at different y coordinates, the variation is within 10$\%$. Dose rate for positions downstream and upstream, on the axis of the source and with the same distance from the source, agree within 10$\%$. This proves that the downstream and upstream zones of the facility have very similar radiation fields.
The RAMSES measurement took place in positions at the upstream and downstream end wall of the bunker. The value in UR agrees well with the Automess measurement in U6, whereas the RAMSES measurement in DR lies between the Automess measurement and the simulated value.
[ |c|c|c|c|c|c|c|c|]{} &\
& & & & & & &\
\
& 3.65 & 0.0 & 0.0 & 6150AD-15 & 1.3(7) & 2.2(2) & 57\
& 0.65 & 0.0 & 1.1 & 6150AD-15 & - & 410(42) & -\
& 0.65 & 0.0 & 2.9 & 6150AD-15 & - & 56(3) & -\
& 0.65 & 0.0 & 4.9 & 6150AD-15 & 17(7) & 22(1) & 77\
& 3.65 & 0.0 & 4.9 & 6150AD-15 & 15(6) & 22(1) & 69\
& -2.35 & 0.0 & 4.9 & 6150AD-15 & 13(5) & 21(1) & 60\
& -1.26 & 0.09 & 5.86 & RAMSES & 12(5) & 15(1) & 76\
& 0.65 & 0.0 & -1.1 & 6150AD-15 & 379(152)& 412(43) & 92\
& 0.65 & 0.0 & -2.9 & 6150AD-15 & 45(18) & 55(3) & 81\
& 0.65 & 0.0 & -4.9 & 6150AD-15 & 17(7) & 22(1) & 79\
& 0.65 & -0.7 & -4.9 & 6150AD-15 & 20(8) & 23(1) & 88\
& 0.65 & -1.5 & -4.9 & 6150AD-15 & 18(7) & 23(1) & 76\
& -1.35 & 0.0 & -2.9 & 6150AD-15 & 43(17) & 63(4) & 68\
& -1.35 & 0.0 & -4.2 & 6150AD-15 & 25(10) & 32(1) & 77\
& 3.65 & 0.0 & -4.9 & 6150AD-15 & 16(6) & 24(1) & 67\
& 3.8 & 0.05 & -5.90 & RAMSES & 11(4) & 17(1) & 66\
\
& 0.65 & 0.0 & 0.45 & RADMON & 2330(559) & 2213(521) & 105\
& 0.65 & 0.0 & 1.00 & RADMON & 470(60) & 440(47) & 107\
& 2.70 & 1.0 & 5.48 & RADMON & 16(1) & 18(1) & 89\
\
& 0.65 & 0.0 & -0.45 & RADMON & 2251(557) & 2274(536) & 99\
& 0.65 & 0.0 & -1.27 & RADMON & 249(30) & 283(25) & 88\
& 0.65 & 0.07 & -2.95 & RADMON & 40(4) & 55(2) & 73\
& 3.65 & 0.13 & -5.79 & RADMON & 20(2) & 18(1) & 111\
For the RADMON measurement locations in Figure \[fig: RADMON positions\], in addition to the $\pm$ 5 cm position uncertainty, a precision uncertainty of $\pm$7.5 $\%$ has been estimated. Figure \[fig: RADMON fit\] shows a very good agreement ($\le$5$\%$ difference) between measurement and simulation for positions close to the source. The difference between measurement and simulations increases with the distance from the source, but stays below 12$\%$ for all positions with the exception of Uc. The deviation in Uc might be explained with the presence of other experimental equipment in the zone during the measurements. The GIF++ source, due to the angular correction filters, is expected to provide a dose rate dependence mainly on z instead of on r like a point source. Consequently, the comparison in Figure \[fig: RADMON fit\] is plotted as function of z. It shows that the power law is indeed a good approximation for the intensity distribution along the z-axis, especially close to the source.
To summarize, all measured values show within +11$\%$ and -43$\%$ agreement with the simulations.
Measurements and simulations of the photon current {#sec: gamma current}
--------------------------------------------------
As described in Section \[subsec:modification of current\], the GIF++ irradiator and filter system is designed to produce a homogeneous radiation field of $\pm$ 37$^\circ$, with a sharp decrease at the edges. In August 2015, a Drift Tube (DT) chamber[^12] from the experiment CMS [@DT], was placed at 4.9 m distance from the source and at the edge of the 37$^\circ$ irradiation cone. The four solid lines of Figure \[fig: DT\] show the occupancy measured in four layers of 2.5 m long vertical drift cells of the 2 m wide drift chamber. The profiles measured by the four independent layers of 50 drift cells each look identical. Scaling the measured rates permits to compare the shape of the profile with simulation curves and it shows a good agreement for photon energies of at least 150 keV, suggesting that very low energy photons do not contribute significantly to the measured signal.This example demonstrates the capability of the simulation to also provide a description outside the borders of the irradiation cone. The width of this transition region is mainly related to the width of the source and its distance to the border of the collimator.
![Comparison of simulated photon current in air with the occupancy measured in drift cells of a CMS Drift Tube Chamber. The chamber was placed at the border of the irradiation cone. The steep drop of photon current outside the irradiation region is well visible in simulation and in data.[]{data-label="fig: DT"}](./figures/DT_measurement){width="65.00000%"}
To study the energy composition of the radiation field in GIF++, spectra of the photon current were simulated for the 14 measurement locations of March 2015. The spectra in Figure \[fig: Spectra\] confirm the results presented in Sections \[sec: albedo studies\] and \[sec:dose\]. Spectra in positions with the same z coordinate are basically identical. As expected, the spectrum in I1 does not contain any 662 keV photons, but consists of photons that scattered multiple times. In all other spectra the narrow and very high main 662 keV $^{137}$Cs peak is clearly visible at the far right. The concrete and steel back-scatter peak (Section \[sec: albedo studies\]) is present in all spectra. In the spectra of positions U1 and D1, that are close to the irradiator and the filter systems, the characteristic $^{82}$Pb K-shell X-rays are present.
\
Figures \[fig: Simulated current 1\] and \[fig: Simulated current 2\] depict the photon current at the height of the source in the xz plane of the facility. Thanks to the angular correction filters, the current of photons between 600 keV and 662 keV is uniform at given z value within the $\pm$ 37$^\circ$ opening as displayed in Figure \[fig: Current\_DS\_US\_x\_600\_667\]. This does not apply to the current of photons with energies below 500 keV. At a given z coordinate, the further away in x direction from the source a location is, the smaller the current. Due to the contribution from lower energy photons, the total current in Figure \[fig: Current\_DS\_US\_x\] also shows some dependence from the x coordinate. In location I1, photons with an energy between 100 keV and 200 keV form the strongest contribution to the photon current as shown in Figure \[fig: Current\_DS\_US\_x\_100\_200\], whereas photons with an energy above 400 keV do not occur in this location. In the locations U4, U5 and U6, the current is somewhat enhanced (Figures \[fig: Current\_DS\_US\_x\_200\_300\] to \[fig: Current\_DS\_US\_x\_000\_100\]) due to scattering of photons from the nearby walls, since the upstream zone of the GIF++ facility is narrower and has an 80 cm lower roof than the downstream zone. The absolute contribution of these scattered photons to the total current is nevertheless relatively small. Table \[table: Simulated current\] lists the contribution of the different energy ranges to the total photon current. Inside the $\pm$ 37$^\circ$ wide irradiation area unattenuated 662 keV photons contribute between 33$\%$ and 54$\%$ of the total current. Directly in front of the irradiator about half of the photons are unattenuated photons.
\
\
-- ------ ------ ------ ------- ------- -------- ------ ------ ------ ------
0.36 2.92 2.44 3.29 5.18 5.88 23.8 43.9 20.0 45.5
0.17 0.72 0.45 0.35 0.67 0.83 3.34 6.52 2.79 42.8
0.14 0.47 0.22 0.20 0.23 0.29 1.14 2.70 0.95 35.2
0.12 0.38 0.19 0.13 0.14 0.15 1.28 2.39 1.17 48.7
0.11 0.32 0.17 0.13 0.15 0.15 1.29 2.32 1.17 50.3
0.13 0.35 0.08 0.008 0.005 0.0002 0.00 0.57 0.00 0.00
0.39 2.96 2.44 3.30 5.14 5.89 23.8 43.9 20.0 45.5
0.21 0.81 0.41 0.34 0.66 0.83 3.38 6.64 2.82 42.6
0.19 0.57 0.28 0.20 0.23 0.28 1.15 2.91 0.96 33.1
0.18 0.54 0.27 0.21 0.27 0.29 1.15 2.90 0.96 33.3
0.14 0.47 0.26 0.19 0.26 0.31 1.18 2.82 1.01 36.0
0.29 0.95 0.49 0.27 0.33 0.35 3.82 6.51 3.54 54.4
0.34 0.86 0.45 0.32 0.29 0.26 1.71 4.23 1.52 35.9
0.22 0.56 0.29 0.19 0.15 0.15 1.28 2.84 1.16 40.9
-- ------ ------ ------ ------- ------- -------- ------ ------ ------ ------
: Overview of simulated photon current \[10$^6 cm^{-2} s^{-1}$\]: downstream and upstream fully open. These are the highest currents available.[]{data-label="table: Simulated current"}
Conclusions {#sec: Conclusions}
===========
The new GIF++ radiation facility, operational since spring 2015, features an intense source of 662 keV photons. The irradiator has two openings of $\pm$ 37$^\circ$ oriented towards a downstream and an upstream irradiation area. A high-energy muon beam passes close to the irradiator and permits to study detector performance while being irradiated at adjustable high rates. Each irradiation zone is equipped with a versatile filter system permitting to attenuate the current of 662 keV photons in 24 steps by up to a factor 4.6$\times$10$^{4}$. To improve uniformity over large planar detectors, integrated angular correction filters serve to ensure that the current of the unattenuated photons depends predominantly on the z coordinate. Lower energy photons, from interactions in irradiator and in surrounding material, contribute in a complex way to the radiation field. The photon current in the whole bunker has been simulated with Geant4. Photon currents of up to 5$\times$10$^{7}$ photons/(cm$^2$ s) are available at a distance of 1 m from the source. Between 33$\%$ and 54$\%$ of the photon current comes from unattenuated photons with an energy of 662 keV, the remainder are photons that lost energy mostly through Compton scattering in the lead of the filters, in the steel of the facility floor or in the concrete of walls and roof. RADMON measurements of the absorbed dose in air have been compared with simulations. The measurements agree to 12$\%$ with the simulations, and are well within expected uncertainties. The same agreement is also found for the Automess 6150AD-15 measurements in locations close to the source. The maximum absorbed dose rate in air amounts to 2.2 Gy/h directly in front of the irradiator, at a distance of 50 cm from the center of the source.
Acknowledgments {#sec:Acknowledgements}
===============
DP would like to acknowledge the support of the EU Horizon 2020 framework, BrightnESS project 676548.
[00]{}
S. Agosteo et al, *A facility for the test of large area muon chambers at high rates*, Nuclear Instruments and Methods in Physics Research A 452 (2000) 94-104.
European Organization for Nuclear Research (CERN), <http://home.cern/>.
Lyndon Evans and Philip Bryant, *The LHC Machine*, JINST 3 (2008) S08001.
High Luminosity Linear Hadron Collider (HL-LHC), <http://home.cern/topics/high-luminosity-lhc>.
M.R. Jaekel et al, *CERN GIF++: A new irradiation facility to test large-area particle detectors for the high-luminosity LHC program*, International Conference on Technology and Instrumentation in Particle Physics 2014, Amsterdam, Netherlands, 2 - 6 Jun 2014, PoS TIPP2014 (2014) 102, <http://pos.sissa.it/cgi-bin/reader/conf.cgi?confid=213>.
G. Aielli et al, *Detector and detector control system operational: GIF++ Infrastructure Commissioning and Utilization*, AIDA deliverable report AIDA-D8.6, 28 pages, 2015, <http://cds.cern.ch/record/1984406?ln=en>.
S. Agostinelli et al., *GEANT4: A Simulation toolkit*, Nuclear Instruments and Methods in Physics Research A 506 (2003) 250-303.
B. Biskup, E. Efthymiopoulos, A. Fabich, D. Pfeiffer, *Shielding Optimization for the New Gamma Irradiation Facility at CERN*, Manuscript to be submitted for publication.
Müller, Steffen and Boer, W and Müller, T, *The Beam Condition Monitor 2 and the Radiation Environment of the CMS Detector at the LHC*, CERN-THESIS-2011-085, <http://cds.cern.ch/record/1319599>
VF, *VF a.s., Svitavská 588, 679 21 Cerná Hora, Czech Republic*, <http://www.vf.eu>.
Physikalisches Institut IIIA, RWTH Aachen University, O. Blumenthal-Str. 16, D-52074 Aachen, Germany.
G. Segura Millan, D. Perrin, L. Scibile, *RAMSES: The LHC Radiation Monitoring System for the Environment and Safety*, 10th ICALEPCS Int. Conf. on Accelerator & Large Expt. Physics Control Systems. Geneva, 10 - 14 Oct 2005, TH3B.1-3O (2005).
F. Ravotti, *Development and characterisation of radiation monitoring sensors for the high energy physics experiments of the CERN LHC accelerator*, CERN-THESIS-2007-013, <http://cds.cern.ch/record/1014776/files/thesis-2007-013.pdf>.
IAEA, *Dosimetry in Diagnostic Radiology: An International Code of Practice*, <http://www-pub.iaea.org/MTCD/publications/PDF/TRS457_web.pdf>.
ICRP, *Conversion Coefficients for Radiological Protection Quantities for External Radiation Exposures*, ICRP Publication 119, Ann. ICRP 40 (2-5), 2010.
ICRP, *Conversion Coefficients for use in Radiological Protection against External Radiation*, ICRP Publication 74, Ann. ICRP 26 (3-4), 1996.
M. Huhtinen, *The Radiation environment at the CMS experiment at the LHC*, <http://inspirehep.net/record/421028/files/mika.pdf> PhD thesis HU-SEFT-R-1996-14,1996.
Geant4 Primitive Scorers, <http://geant4.web.cern.ch/geant4/G4UsersDocuments/UsersGuides/ForApplicationDeveloper/html/Detector/hit.html>, Geant4 User Guide, 2016.
K.G. Veinot, N.E. Hertel, *Personal Dose Equivalent Conversion Coefficients for Photons to 1 GeV*, Radiation Protection Dosimetry Vol. 145 No. 1 (2011) 28-35.
Automation und Messtechnik GmbH, *Datasheet for Gamma Probes 6150AD-15 and 6150AD-18*, <http://www.automess.de/Download/Prospekt_AD1518_E.pdf>.
Automation und Messtechnik GmbH, *Operating Manual for the Dose Rate Meter 6150AD from 26.10.2005*, <https://wwwcompass.cern.ch/compass/organisation/safety/documents/dosimeter_automess_6150AD.pdf>.
Automation und Messtechnik GmbH, *Radiation quantities and units*, <http://www.automess.de/Messgroessen_E.htm>.
Tables of Physical & Chemical Constants (16th edition 1995), *4.2.1 X-ray absorption edges, characteristic X-ray lines and fluorescence yields*, <http://www.kayelaby.npl.co.uk/atomic_and_nuclear_physics/4_2/4_2_1.html>.
CMS Muon DT group, communication to GIF++ users, <https://indico.cern.ch/event/523267/>.
[^1]: Source activity: 14.9 TBq in November 2011, 13.9 TBq in March 2015, 13.5 TBq in March 2016.
[^2]: After traversing a length of $x$ cm in a material, the intensity $I$ of a beam of mono-energetic photons with original intensity $I_0$ amounts to I=I$_{0}$e$^{-\mu x}$. The following linear attenuation factors for photons with E=662 keV were used for the design of the filters: 0.116 mm$^{-1}$ for Pb, 0.020 mm$^{-1}$ for Al, 0.053 mm$^{-1}$ for Fe.
[^3]: RAMSES comprises more than 400 radiation monitors (not including environmental monitors) and, in conjunction with two other systems, provides radiation monitoring throughout CERN’s accelerators, experimental and service areas.
[^4]: PTW 32006 air filled 3 L ionisation chambers from ptw.de with a calibrated measurement range of 5 $\mu$Sv/h - 500 mSv/h
[^5]: The acronym kerma stands for inetic nergy eleased in per unit ss. The kerma is defined as $K=\frac{dE_{tr}}{dm}$, where dE$_{tr}$ is the sum of the initial kinetic energies of all the charged ionizing particles liberated by uncharged ionizing particles in a material of mass dm. The absorbed dose is defined as $D=\frac{d\bar{\epsilon}}{dm}$, with $d\bar{\epsilon}$ the mean energy imparted to matter of mass $dm$. Absorbed dose and kerma are numerically identical, provided that electron equilibrium is attained and radiative losses are negligible [@ICRP119]. This is basically the case in the air of the radiation zone of GIF++.
[^6]: The ambient dose equivalent H$^*$(10), an operational quantity for area monitoring, is measured in Sv and is used for strongly penetrating radiation like photons above 15 keV.
[^7]: The primitive scorers *G4PSCellFlux* and *G4PSFlatSurfaceCurrent* were used [@PS]. A production cut for gammas of 28.5 keV in lead and 2.14 keV in concrete was used in the simulations.
[^8]: The gamma probe 6150AD-15 [@gamma; @probe], from Automess, contains a Geiger-Müller counting tube as detector and is used in combination with the dose rate meter 6150AD [@probe; @manual].
[^9]: As RADFET a REM 250 from radfet.com and a LAAS 1600 from laas.fr were used.
[^10]: The wall of the capsule is actually 1.75 mm thick, 1 mm thicker than assumed in the simulation.
[^11]: The dose contribution from photons passing through the irradiator shielding is negligible.
[^12]: The Drift Tube chamber consists essentially of 1.5 mm thick Aluminium plates. The 11.5 mm space between them is filled with Ar/CO$_2$ 85/15.
|
---
address:
- 'Vincent Lafforgue: CNRS et Institut Fourier, UMR 5582, Université Grenoble Alpes, 100 rue des Maths, 38610 Gières, France.'
- 'Xinwen Zhu: California Institute of Technology, Pasadena, CA 91125'
author:
- Vincent Lafforgue et Xinwen Zhu
title: 'Décomposition au-dessus des paramètres de Langlands elliptiques'
---
Dans ce texte on étend la remarque 8.5 de [@ICM-text] au cas non déployé, et on justifie le fait que l’heuristique (8.3) de [@ICM-text] est vraie au-dessus des paramètres de Langlands elliptiques $\sigma$ (comme cela est affirmé dans la remarque 8.5 de [@ICM-text]). Ce résultat avait été proposé par le second auteur dans [@zhu-notes].
Nous remercions Cong Xue pour des commentaires sur cet article.
Introduction
============
Comme dans le paragraphe 12 de [@coh], on fixe une courbe projective lisse géométriquement irréductible $X$ sur $\Fq$, de corps des fonctions $F$, et un groupe réductif lisse et géométriquement connexe $G$ sur $F$. On note $U_0$ l’ouvert maximal de $X$ tel que $G$ se prolonge en un schéma en groupes lisse et réductif sur $U_0$. On étend $G$ en un schéma en groupes lisse sur $X$, réductif sur $U_0$, de type parahorique en les points de $X\sm U_0$, et dont toutes les fibres sont géométriquement connexes. On fixe aussi une extension finie $E$ de $\Ql$ contenant une racine carrée de $q$.
Soit $N\subset X$ un sous-schéma fini. On note $\wh N=|N|\cup (X\sm U_0)$. Soit $\Bun_{G,N}$ le champ classifiant les $G$-fibrés principaux sur $X$ avec structure de niveau $N$. On choisit un réseau $\Xi\subset Z(F)\backslash Z(\mathbb A)$. Soit $C_{\cusp}(\Bun_{G,N}(\mathbb F_q)/\Xi,E)$ l’espace des formes automorphes cuspidales $\Xi$-invariantes à valeurs dans $E$. On note $\mathbb T_N=C_c(K_N\backslash G(\mathbb A)/K_N, E)$ l’algèbre de Hecke en niveau $N$, qui agit sur cet espace.
On note $\wt F$ l’extension finie galoisienne de $F$ telle que $\on{Gal}(\wt F/F)$ soit l’image de $\on{Gal}(\ov F/F)$ dans le groupe des automorphismes de la donnée radicielle de $G$. Le $L$-groupe ${}^{L }G$ est le produit semi-direct $\wh G\rtimes \on{Gal}(\wt F/F)$ pour l’action de $\on{Gal}(\wt F/F)$ sur $\wh G$ qui préserve un épinglage.
On note $\Gamma=\on{Gal}(\ov F/F)$ le groupe de Galois de $F$.
Soit $\on{FinS}_*$ la catégorie des ensembles finis pointés. On considère deux catégories additives cofibrées sur $\on{FinS}_*$. La première, notée $\on{Rep}(\hat{G}\times {}^L{G}^\bullet)$ associe à tout ensemble fini pointé $\{0\}\cup I$ la catégorie $\on{Rep}(\hat{G}\times ({}^L{G})^I)$ des représentations algébriques $E$-linéaires de dimension finie de $\hat{G}\times ({}^{L}G)^I$, et le foncteur d’image inverse par une application pointée $\xi: \{0\}\cup I\to \{0\}\cup J$ est donné par la restriction naturelle suivant le morphisme de groupes algébriques $\hat{G}\times ({}^L{G})^J\to \hat{G}\times ({}^L{G})^I$. La seconde, notée $\on{Rep}_{\mathbb T_N}(\Gamma^\bullet)$, associe à $\{0\}\cup I$ la catégorie $\on{Rep}_{\mathbb T_N}(\Gamma^I)$ des représentations continues de $\Gamma^I$ sur des $\mathbb T_N$-modules qui sont de dimension finie en tant que $E$-espaces vectoriels, et le foncteur d’image inverse par une application pointée $\xi: \{0\}\cup I\to \{0\}\cup J$ est donné par la restriction naturelle suivant le morphisme de groupes topologiques $\Gamma^J=\{1\}\times\Gamma^J\to\{1\}\times\Gamma^I=\Gamma^I$.
On rappelle le théorème suivant.
(démontré pour $G$ déployé, conditionnel en général en ce qui concerne la finitude des dimensions [@coh; @cong]). On possède un foncteur $E$-linéaire $H$ de $\on{Rep}(\hat{G}\times {}^L{G}^\bullet)$ vers $\on{Rep}_{\mathbb T_N}(\Gamma^\bullet)$ cofibré sur $\on{FinS}_*$.
Pour tout ensemble fini pointé $\{0\}\cup I$, et toute représentation $W$ de $\hat{G}\times {}^{L}G^I$, on note $H_{\{0\}\cup I,W}$ le $\Gamma^I\times \mathbb T_N$-module correspondant. En particulier on a $H_{\{0\},\mathbf{1}}=C_{\cusp}(\Bun_{G,N}(\mathbb F_q)/\Xi,E)$, muni de la $\mathbb T_N$-action naturelle.
On expliquera dans le paragraphe \[rappels-coho\] la construction du foncteur $H$, d’après [@coh]. En gros $H_{\{0\}\cup I,W}$ est la partie cuspidale de la cohomologie d’un champ de chtoucas associé à $W$. Le théorème signifie plus concrètement que $H_{\{0\}\cup I,W}$ est fonctoriel en $W$ et compatible avec les isomorphismes de coalescence. (Nous nous référons à [@coh Proposition 9.7] pour plus de détails.) La finitude de la dimension des espaces vectoriels $H_{\{0\}\cup I,W}$ est montrée dans [@cong] lorsque $G$ est déployé. On s’attend à ce que la preuve s’étende pour tout $G$ mais cela n’est pas encore écrit. On l’admet dans cet article.
En particulier les espaces $H_{I, W}:=H_{\{0\}\cup I, \mathbf{1}\boxtimes W}$ sont exactement ceux construits dans [@coh]. Comme dans [@coh], on déduit du foncteur $H$ l’algèbre d’excursion $\mathcal B$ qui agit sur chaque $H_{I, W}$, en commutant avec les actions de $\Gamma^I\times \mathbb T_N$. De plus les algèbres $\mathcal B$ et $\mathbb T_N$ contiennent les opérateurs de Hecke aux places non-ramifiées et ceux-ci agissent de la même façon par les deux algèbres (grâce à [@genestier-lafforgue] on a aussi une compatibilité entre les actions de $\mathcal B$ et $\mathbb T_N$ en les places ramifiées). L’action de $\mathcal B$ sur l’espace vectoriel $H_{ I, W}$ de dimension finie fournit une décomposition de cet espace en sous-espaces *généralisés*, $$H_{ I, W}\otimes_E\Qlbar=\bigoplus (H_{I, W})_\sigma,$$ où la somme directe est indexée par des caractères $\nu: \mathcal B\to\Qlbar$. D’après [@coh], tout caractère $\nu $ détermine un morphisme semi-simple $\sigma:\Gamma\to {}^L{G}(\Qlbar)$ à conjugaison près par $\hat{G}$, satisfaisant les conditions suivantes:
- (C1) $\sigma$ prend ses valeurs dans ${}^{L}G(E')$, où $E'$ est une extension finie de $E$, et il est continu,
- (C2) l’adhérence de Zariski de son image est réductive,
- (C3) $\sigma$ se factorise à travers $ \pi_{1}(U, \ov\eta)$ pour un certain ouvert $U\subset U_0$,
- (C4) on a la commutativité du diagramme $$\begin{gathered}
\label{diag-sigma}
\xymatrix{
\on{Gal}(\ov F/F) \ar[rr] ^{\sigma}
\ar[dr]
&& {}^{L} G(\Qlbar) \ar[dl]
\\
& \on{Gal}(\wt F/F) }\end{gathered}$$
- (C5) la composition de $\sigma$ avec $$\begin{gathered}
\label{compo-ab} {}^{L} G(\overline{{\mathbb Q}_\ell})\to
(\wh G^{ab})(\overline{{\mathbb Q}_\ell})\rtimes \on{Gal}(\wt F/F)\end{gathered}$$ a une image finie.
Réciproquement $\sigma$ détermine $\nu$, c’est pourquoi on écrit $(H_{I, W})_\sigma$.
Pour tout $\sigma$ comme ci-dessus, on note $S_\sigma$ le centralisateur de son image dans $\hat{G}$. Un tel $\sigma$ est dit elliptique si $\mathfrak S_\sigma:=S_\sigma/(Z_{\hat{G}})^{\on{Gal}(\wt F/F)}$ est fini.
Pour toute représentation $W$ de $({}^L{G})^I$, on note $W_{\sigma^I}$ la représentation composée $\Gamma^{I}\stackrel{\sigma^I}{\to} ({}^L{G})^I\to GL(W)$. Voici le résultat principal de cet article.
\[prop-ppale\] Pour tout $\sigma$ elliptique, il existe un $\Qlbar$-espace vectoriel de dimension finie $\mathfrak A_\sigma$, muni d’une action de $\mathbb T_N\times\mathfrak S_\sigma$, tel que pour tout $W$, on a l’égalité $$\begin{gathered}
\label{eq-prop-ppale}(H_{I,W})_{\sigma}= \big( \mathfrak A_{\sigma}\otimes_{\overline{{\mathbb Q}_\ell}} W_{\sigma^{I}}\big)^{S_{\sigma}},\end{gathered}$$ équivariante par l’action de $\mathbb T_N\times \Gamma^{I}$, fonctorielle en $W$, compatible avec les isomorphismes de coalescence, et compatible avec le changement de structure de niveau. Autrement dit l’heuristique (8.3) de [@ICM-text] est vraie au-dessus de $\sigma$.
La démonstration sera donnée à partir du paragraphe \[non-deform\].
Le membre de droite de est nul lorsque $W^{ Z_{\hat{G}}^{\Gamma_F}}=0$. Dans ce cas la nullité du membre de gauche est évidente car le champ correspondant de chtoucas est vide. Le cas intéressant de est donc celui où $W$ est une représentation de $ {}^L{G}^\bullet/ Z_{\hat{G}}^{\Gamma_F}$. Dans ce cas on peut remplacer le membre de droite de par $\big( \mathfrak A_{\sigma}\otimes_{\overline{{\mathbb Q}_\ell}} W_{\sigma^{I}}\big)^{\mathfrak S_{\sigma}}$.
\[rem-propre-naif\] La proposition implique que pour $\sigma$ elliptique $(H_{I,W})_{\sigma}$ est en fait le sous-espace (au sens naïf, non généralisé) de $H_{I,W}\otimes_{E}\overline{{\mathbb Q}_\ell}
$ associé à $\sigma$ pour l’action des opérateurs d’excursion, et de plus $(H_{I,W})_\sigma$ est semi-simple comme représentation de $\Gamma^I$.
En général à cause de la déformation possible de certains $\sigma$ non elliptiques il pourrait y avoir des nilpotents, et par exemple on ne sait pas montrer que l’algèbre engendrée par les opérateurs d’excursion agissant sur $H_{I,W}$ (ou même sur $H_{\emptyset,\mbf 1}$) est réduite, donc on ne sait pas montrer la première partie de l’énoncé de la remarque précédente pour tous les $\sigma$ et à fortiori on ne sait pas montrer l’heuristique (8.3) de [@ICM-text].
Le cas de $GL_{r}$
==================
On prend $G=GL_{r}$ et on applique la proposition \[prop-ppale\]. Alors une représentation semi-simple $\sigma$ est elliptique si et seulement si elle est irréductible, et dans ce cas $\mathfrak S_\sigma$ est trivial. Il en résulte que $$\mathfrak A_\sigma=\mathfrak A_\sigma^{\mathfrak S_\sigma}=(H_{\emptyset,\mbf 1})_\sigma.$$ Par le théorème de multiplicité un fort pour l’espace des formes automorphes cuspidales pour $GL_{r}$, on voit que $\mathfrak A_\sigma=\pi^{K_N}$, où $\pi$ est l’unique représentation cuspidale qui correspond à $\sigma$. On a donc le corollaire suivant.
Pour $G=GL_{r}$, si $\sigma$ est irréducible, alors $$(H_{I,W})_\sigma= \pi^{K_N}\otimes (W^{\mb G_m})_{\sigma^I}.$$
Dans le cas des chtoucas de Drinfeld ($G=GL_{r}$, $I=\{1,2\}$, $W=\on{St}\otimes \on{St}^{*}$) l’énoncé obtenu dans le corollaire ci-dessus est exactement celui montré par Laurent Lafforgue dans [@laurent-inventiones] (où les multiplicités sont déterminées aux $r$-négligeables près). Pour $GL_{2}$, cela est dû à Drinfeld [@drinfeld-Petersson].
On conjecture que pour $GL_{r}$ tout $\sigma$ apparaissant dans la décomposition de $H_{I,W}$ est irréductible (cela est connu dans le cas des formes automorphes, c’est-à-dire pour $W=\mbf 1$, mais pas en général). Sous cette hypothèse supplémentaire on obtient la conjecture 2.35 de Varshavsky [@var].
Relation avec les formules de multiplicités d’Arthur-Kottwitz
=============================================================
On revient au cas d’un groupe $G$ réductif arbitraire.
La formule $$(H_{I,W})_\sigma=\big( \mathfrak A_{\sigma}\otimes_{\overline{{\mathbb Q}_\ell}} W_{\sigma^{I}}\big)^{S_{\sigma}}$$ que nous montrons dans cet article pour $\sigma$ elliptique est compatible avec la conjecture de Kottwitz sur la partie cuspidale tempérée du spectre automorphe (voir [@cusp-temper]) et de la cohomologie des variétés de Shimura (see [@KoAnnArbor]). Ici bien sûr on considère $H_{I,W}$ comme un analogue généralisé, dans le cas des corps de fonctions, de la cohomologie des variétés de Shimura. On rappelle que la conjecture de Kottwitz sur la partie cuspidale tempérée du spectre automorphe est un cas particulier des formules de multiplicités d’Arthur. On garde l’hypothèse que $\sigma$ est elliptique.
On rappelle que $\mathfrak A_\sigma$ est muni des actions (commutant entre elles) de l’algèbre de Hecke $\mathbb T_N:=C_c(K_N\backslash G(\mathbb A)/K_N, \overline{{\mathbb Q}_\ell})$ et du groupe fini $\mathfrak S_\sigma: = S_{\sigma}/(Z_{\wh G})^{ \on{Gal}(\wt F/F)}$.
On écrit $$\mathfrak A^{ss}_\sigma=\bigoplus \pi^{K_N}\otimes \rho_\pi,$$ où $\mathfrak A^{ss}_\sigma$ est la semi-simplification de $\mathfrak A_\sigma$ comme $\mathbb T_N$-module, où la somme directe est prise sur les $\mathbb T_N$-modules irréductibles $\pi^{K_N}$, et où $\rho_\pi$ est un espace de multiplicités, qui est en fait une représentation de dimension finie de $\mathfrak S_\sigma$. On conjecture que $\mathfrak A_\sigma$ est semi-simple comme $\mathbb T_N$-module. En fait, on conjecture que les $H_{I,W}$ sont semi-simples comme $\mathbb T_N$-modules, mais on ne sait le montrer que pour $W=\bf 1$ (cas des formes automorphes).
On remarque que si un $\mathbb T_N$-module irréductible $\pi^{K_N}$ apparaît dans $(H_{I,W})_\sigma$, alors $\sigma$ est elliptique si et seulement si $\pi$ est tempéré aux places non-ramifiées et la function $L(s,\pi, \on{Ad}^0)$ partielle (restreinte aux places où $\pi$ est non-ramifié) admet un prolongement méromorphe au plan complexe tout entier et n’a pas de pôle en $s=1$, où $\on{Ad}^0$ est la représentation adjointe de ${}^{L}G$ sur l’algèbre de Lie adjointe $\hat{\mathfrak g}_{ad}$. En effet on a $$\on{Lie} (\mf S_{\sigma})=\on{Lie} (S_{\sigma})/\on{Lie} ((Z_{\wh G})^{ \on{Gal}(\wt F/F)})=(\on{Ad})^{\sigma}/\on{Lie} ((Z_{\wh G})^{ \on{Gal}(\wt F/F)})=(\on{Ad}^{0})^{\sigma}$$ où $(\on{Ad})^{\sigma}$ et $(\on{Ad}^{0})^{\sigma}$ désignent les invariants par $\sigma$. Par conséquent si $\pi^{K_N}$ apparaît dans un $(H_{I,W})_\sigma$ avec $\sigma$ elliptique, il apparaît seulement dans des $(H_{I,W})_{\sigma'}$ avec $\sigma'$ elliptique.
Suivant la littérature sur les formules de multiplicités, on note $\langle \pi, \cdot\rangle$ la fonction trace $\on{tr}\rho_\pi$ sur $\mathfrak S_\sigma$. Il résulte de tout cela que la multiplicité de $\pi^K$ dans $(H_{\emptyset,\mbf 1})_\sigma$ est égale à $$m(\pi,\sigma)= \frac{1}{|\mathfrak S_\sigma|}\sum_{x\in \mathfrak S_\sigma} \langle\pi, x\rangle.$$ Il en résulte que la mulplicité totale de $\pi^{K_N}$ dans l’espace des formes automorphes cuspidales est donnée par $$m(\pi)=\sum_\sigma m(\pi,\sigma).$$ Ceci est en accord avec la formule de multiplicités de Arthur et Kottwitz.
Soit $W$ une représentation arbitraire de $({}^{L} G)^I/(Z_{\wh G})^{ \on{Gal}(\wt F/F)}$. Supposons d’abord que $\sigma$ soit un paramètre elliptique stable, ce qui signifie que $\mathfrak S_\sigma$ est trivial. Alors exactement comme dans le corollaire 2.1,
Pour un paramètre stable $\sigma$, $$(H_{I,W})_\sigma= (H_{\emptyset,\mathbf{1}})_\sigma \otimes W_{\sigma^I}.$$
Ceci a été prouvé par Kazhdan et Varshavsky dans un preprint non publié [@KV], en supposant qu’une composante locale des représentations automorphes dans $(H_{\emptyset,\mathbf{1}})_\sigma$ est supercuspidale. En outre, on s’attend à ce que $(H_{\emptyset,\mathbf{1}})_\sigma=\pi^{K_N}$.
Pour un paramètre non nécessairement stable, $$(H_{I,W})^{ss}_\sigma = \bigoplus_{\pi} \pi^{K_N}\otimes \Hom_{\mathfrak S_\sigma}(\rho_\pi^\vee, W_{\sigma^I}),$$ comme semi-simplifiés de $\mathbb T_N\times \Gamma_F^I$-modules (mais on rappelle que $(H_{I,W})_\sigma$ est déjà semi-simple en tant que $\Gamma_F^I$-module). En particulier la multiplicité de $\pi^K$ dans $(H_{I,W})^{ss}_\sigma$ est $$m_{I,W}(\pi,\sigma)=\frac{1}{|\mathfrak S_\sigma|}\sum_{x\in\mathfrak S_\sigma} \langle \pi,x\rangle\on{tr}(x\mid W),$$ ce qui est similaire à la formule de multiplicités de Kottwitz pour la cohomologie des variétés de Shimura, conjecturée dans [@KoAnnArbor].
On espère la description locale suivante des espaces de multiplicités $\mathfrak A_\sigma$.
Si $v$ est une place de $F$, soit $\sigma_v$ la restriction de $\sigma$ au groupe de décomposition $\on{Gal}(\ov F_v/F_v)$, et soit $S_{\sigma_v}$ le centralisateur dans $\wh G$ de son image dans ${}^{L} G$.
Pour simplifier on suppose $G$ quasi-déployé. On conjecture qu’il existe une factorisation (dépendant d’une normalisation de Whittaker globale) $$\mathfrak A_\sigma=\bigotimes' \mathfrak A_{\sigma_v},$$ où $\mathfrak A_{\sigma_v}$ est un espace vectoriel de dimension finie muni d’actions commutant entre elles de $\mathbb T_{K_v}$ et de $S_{\sigma_v}$, dépendant seulement de $\sigma_v$. On conjecture que l’action de $S_\sigma$ sur $\mathfrak A_\sigma$ est la restriction par $S_\sigma\to \prod_v S_{\sigma_v}$ de l’action de $ \prod_v S_{\sigma_v}$ sur $\bigotimes' \mathfrak A_{\sigma_v}$. Alors $\mathfrak A_{\sigma_v}=\oplus \pi_v^{K_v}\otimes \rho_{\pi_v}$. En notant la fonction trace $\on{tr}\rho_{\pi_v}$ sur $S_{\sigma_v}$ par $\langle \pi_v, \cdot\rangle$, on aurait alors la factorisation $$\langle \pi,x\rangle=\prod_v \langle \pi_v, x\rangle.$$ Si $v$ est une place non ramifiée, on conjecture que $\mathfrak A_{\sigma_{v}}$ est de dimension $1$, et que $\mathbb T_{K_v}$ agit par un caractère déterminé par $\sigma_v$, et que $S_{\sigma_v}$ agit par un caractère (dépendant de la normalisation de Whittaker). En particulier, si $K_N=G(\mathbb O)$ est partout hyperspécial, on s’attend à ce que $\mathfrak A_\sigma$ soit de dimension $1$, sur lequel l’algèbre de Hecke agit par le caractère associé à $\sigma$, et $\mathfrak S_\sigma$ agit trivialement. A priori $\mathfrak S_\sigma$ pourrait agir par un caractère, mais on s’attend à ce qu’on puisse attacher à $\sigma$ une forme automorphe, et donc ce caractère devrait être trivial. Il en résulte que l’on devrait avoir $$(H_{I,W})_\sigma= \pi^{G(\mathbb O)}\otimes W_{\sigma^I}^{\mathfrak S_\sigma}.$$
Non déformabilité des paramètres elliptiques {#non-deform}
============================================
Le reste de l’article est consacré au rappel de la construction du foncteur $H$ et à la preuve de la proposition \[prop-ppale\]. On commence par rappeler un lemme connu.
Soit $\sigma$ un paramètre de Langlands vérifiant les conditions (C1), (C2), (C3), (C4), (C5) ci-dessus. On rappelle que l’on note $S_{\sigma}$ le centralisateur de $\sigma$ dans $\wh G$ et que $\sigma$ est dit elliptique si $\mathfrak S_{\sigma}:=S_{\sigma}/(Z_{\wh G})^{ \on{Gal}(\wt F/F)}$ est fini.
Il est évident que si $\sigma$ n’est pas elliptique il se déforme parmi les paramètres vérifiant (C1), (C2), (C3), (C4), (C5). En effet si $\sigma$ n’est pas elliptique $T:=\Ker( S_{\sigma}\to
(\wh G^{ab})^{ \on{Gal}(\wt F/F)})$ n’est pas fini (car il est isogène à $\mathfrak S_{\sigma}$). Or pour tout $g\in T(\Qlbar)$, $\sigma_{g}
$ défini par $\sigma_{g}(\gamma)=\sigma(\gamma) g^{\deg(\gamma)}$ vérifie (C1), (C2), (C3), (C4), (C5) (et on peut même prendre $g\in T(A)$ où $A$ est une $\Qlbar$-algèbre arbitraire).
Ce qui nous intéresse est la réciproque. Elle est un peu moins évidente, mais bien connue: si $\sigma$ est elliptique, alors il ne peut pas être déformé parmi les morphismes vérifiant des conditions similaires à (C1), (C2), (C3), (C4), (C5). Plus précisément on a le lemme suivant.
\[lem-non-def\] On suppose $\sigma$ elliptique vérifiant (C1), (C2), (C3), (C4), (C5). Pour toute $\Qlbar$-algèbre locale artinienne $\mc A$ d’idéal maximal $\mc I$, pour tout ouvert $U\subset U_0$, tout morphisme continu $\pi_{1}(U, \overline \eta)\to {}^{L} G(\mc A)$ vérifiant les conditions (C1), (C5) (pour $ \mc A$ au lieu de $ \overline{{\mathbb Q}_\ell}$) et dont la réduction modulo $ \mc I$ est $\sigma$, est conjugué à $\sigma$ par un élément de $\wh G(\mc A)$ égal à $1$ modulo $\mc I$.
On rappelle la preuve d’après la proposition 5.12 de [@boeckle-harris...] (qui traite le cas déployé). Il suffit de traiter le cas où $\mc A=\Qlbar[\epsilon]/\epsilon^{2}$. Il suffit de considérer la composition de ces morphismes avec $\wh G \rtimes \on{Gal}(\wt F/F) \to \wh G^{ad} \rtimes \on{Gal}(\wt F/F) $ et nous sommes réduits au cas où $\wh G$ est semi-simple. Le système local associé à la représentation adjointe $\wh {\mathfrak g}_{\sigma}$ est de poids $0$ par la remarque 12.6 de [@coh], donc $H^1(U_{\overline{\mathbb F_{q}}}, \wh {\mathfrak g}_{\sigma})$ a des poids $\geq 1$ par Weil II, donc $$\begin{gathered}
\label{H0H1}H^0(\widehat {\mathbb Z},H^1(U_{\overline{\mathbb F_{q}}}, \wh {\mathfrak g}_{\sigma}))=0.\end{gathered}$$ D’autre part $H^0(\widehat {\mathbb Z} , H^0(U_{\overline{\mathbb F_{q}}}, \wh {\mathfrak g}_{\sigma}))=H^0(U, \wh {\mathfrak g}_{\sigma})$ est nul car $H^0(U, \wh {\mathfrak g}_{\sigma})=\on{Lie}(S_{\sigma})$ et $S_{\sigma}$ est fini par l’ellipticité de $\sigma$. Pour tout $\Qlbar$-espace vectoriel $M$ de dimension finie muni d’une action continue de $\widehat {\mathbb Z}$, on a la suite exacte $$0\to H^{0}(\widehat {\mathbb Z}, M)\to M\xrightarrow{1-u} M\to
H^{1}(\widehat {\mathbb Z}, M)\to 0,$$ où $u$ désigne l’action de $1\in \mathbb Z$, donc $H^{0}(\widehat {\mathbb Z}, M)$ et $H^{1}(\widehat {\mathbb Z}, M)$ ont même dimension. Comme on vient de voir que $H^0(\widehat {\mathbb Z} , H^0(U_{\overline{\mathbb F_{q}}}, \wh {\mathfrak g}_{\sigma}))=0$ on a donc $$\begin{gathered}
\label{H1H0}H^1(\widehat {\mathbb Z} , H^0(U_{\overline{\mathbb F_{q}}}, \wh {\mathfrak g}_{\sigma}))=0.\end{gathered}$$ On déduit de , et de la suite spectrale de Leray que $H^1(U, \wh {\mathfrak g}_{\sigma})=0$.
Cohomologie des champs de chtoucas {#rappels-coho}
==================================
On rappelle que dans 12.3.2 de [@coh] on a défini, pour $I$ un ensemble fini, $k\in \N$, $(I_{1}, ..., I_{k})$ une partition de $I$, et $W$ une représentation de $({}^{L} G)^{I}$, un champ de Deligne-Mumford $$\Cht_{N,I,W}^{(I_{1},...,I_{k})}$$ sur $(X\sm \wh N)^{I}$, réunion d’ouverts $\Cht_{N,I,W}^{(I_{1},...,I_{k}), \leq \mu}$ tels que $\Cht_{N,I,W}^{(I_{1},...,I_{k}), \leq \mu}/\Xi$ soit de type fini. De plus on a défini $\mc F_{N,I,W, \Xi,E}^{(I_{1},...,I_{k})}$ comme le faisceau pervers (à un décalage près) sur $\Cht_{N,I,W}^{(I_{1},...,I_{k})}/\Xi$ égal à l’image inverse d’un faisceau de Satake $\mc S_{I,W,E}^{(I_{1},...,I_{k})}$ par un morphisme lisse $$\epsilon_{N,(I),W,\underline{n}}^{(I_{1},...,I_{k}), \Xi}:
\Cht_{N,I,W} ^{(I_{1},...,I_{k})}/\Xi
\to \mr{Gr}_{I,W}^{(I_{1},...,I_{k})}/G^{\mr{ad}}_{\sum _{i\in I}n_{i}x_{i}}$$ où les entiers $n_{i}$ sont assez grands. Enfin on a posé $$\begin{gathered}
\label{defi-H-alpha-Cht}
\mc H _{ N,I,W}^{\leq\mu,E}= R
\big(\mf p_{N,I}^{(I_{1},...,I_{k}),\leq\mu}\big)_{!}\Big(\restr{\mc F_{N,I,W,\Xi,E}^{(I_{1},...,I_{k})}} {\Cht_{N,I,W}^{(I_{1},...,I_{k}),\leq\mu}/\Xi}\Big)
\end{gathered}$$ qui appartient à $D^{b}_{c}((X\sm \wh N)^{I}, E)$ et ne dépend pas du choix de la partition $(I_{1},...,I_{k})$.
Pour tout point géométrique $\ov x$ dans $(X\sm \wh N)^{I}$, on a défini le $E$-espace vectoriel des éléments Hecke-finis $$\Big(\varinjlim_{\mu }\restr{\mc H _{ N,I,W}^{0,\leq\mu,E}}{\ov x} \Big)^{\mr{Hf}}\subset \varinjlim_{\mu }\restr{\mc H _{ N,I,W}^{0,\leq\mu,E}}{\ov x} .$$ On a montré $$\begin{gathered}
\label{Cccusp-Hf-non-deploye}\Big(\varinjlim_{\mu }\restr{\mc H _{ N,\{0\},\mbf 1}^{0,\leq\mu,E}}{\ov\eta}\Big)^{\mr{Hf}}=
\Big(\varinjlim_{\mu }\restr{\mc H _{ N,\emptyset,\mbf 1}^{0,\leq\mu,E}}{\Fqbar}\Big)^{\mr{Hf}}=C_{c}^{\mr{cusp}}(\Bun_{G,N}(\Fq)/\Xi,E). \end{gathered}$$ On a montré que pour toute flèche de spécialisation $\on{\mf{sp}}$ de $\ov{\eta^{I}}$ vers $\Delta(\ov \eta)$, $$\on{\mf{sp}}^{*}: \Big( \varinjlim _{\mu}\restr{\mc H _{N, I, W}^{0,\leq\mu,E}}{\Delta(\ov{\eta})} \Big)^{\mr{Hf}}\to
\Big( \varinjlim _{\mu}\restr{\mc H _{N, I, W}^{0,\leq\mu,E}}{\ov{\eta^{I}}}\Big)^{\mr{Hf}}$$ est un isomorphisme et que cet espace est muni d’une action de $\pi_{1}(\eta,\ov\eta)^{I}=\on{Gal}(\ov F/F)^{I}$, l’action sur le membre de gauche ne dépendant pas du choix de $\ov{\eta^{I}}$ et de $\on{\mf{sp}}$. Dans [@cong], Cong Xue a montré que ces espaces sont de dimension finie quand $G$ est déployé. Une généralisation non écrite de ce théorème de Cong Xue affirme que ces espaces sont de dimension finie dans le cas général. [*Nous admettons ce résultat*]{}. Les opérateurs d’excursion agissent sur ces espaces par la construction suggérée dans [@coh 12.3.4] et détaillée dans [@cong-finite] et tout $\sigma$ apparaissant dans la décomposition spectrale associée vérifie les conditions (C1), ..., (C5) ci-dessus (la condition (C5) a lieu grâce au quotient par $\Xi$).
On va maintenant introduire une notation nouvelle pour désigner certains sous-espaces de ces espaces de cohomologie.
On considère $I$ de la forme $\{0\}\cup J$, où la réunion est disjointe. On note $(J_{1},...,J_{k})$ une partition de $J$. Soit $W$ une représentation de dimension finie de $\wh G \times ({}^{L} G)^{J}$. Il est clair que $W$ peut être incluse dans une représentation de dimension finie $W'$ de $\wh G \times ({}^{L} G)^{J}$ qui s’étend à $({}^{L} G)^{\{0\}\cup J}$. Une variante du théoreme 12.16 de [@coh] associe à $W$ un faisceau pervers $G^{\mr{ad}}_{\sum \infty x_{i}}$-équivariant $\mc S_{\{0\}\cup J ,W,E}^{(\{0\}, J_{1},...,J_{k})}$ sur $\restr{\mr{Gr}_{\{0\}\cup J }^{(\{0\},J_{1},...,J_{k})}}{\ov\eta\times U_0^{I}}$. En fait on pourrait remplacer $\ov\eta$ par le spectre du corps réflex (de façon tout à fait analogue à ce qu’on fait pour les variétés de Shimura, où $J$ est vide), mais on n’en a pas besoin ici. On définit alors le champ $\Cht_{N,\{0\}\cup J,W}^{(\{0\},J_{1},...,J_{k}) }$ sur $\ov\eta\times (X\sm \wh N)^{J}$ et sur $\Cht_{N,\{0\}\cup J,W}^{(\{0\},J_{1},...,J_{k}) }/\Xi$ on définit un faisceau $\mc F_{N,\{0\}\cup J,W, \Xi,E}^{(\{0\},J_{1},...,J_{k})}$ égal à l’image inverse de $\mc S_{\{0\}\cup J ,W,E}^{(\{0\}, J_{1},...,J_{k})}$ par le morphisme lisse $$\Cht_{N,\{0\}\cup J,W}^{(\{0\},J_{1},...,J_{k}) } /\Xi
\to \restr{\mr{Gr}_{\{0\}\cup J }^{(\{0\},J_{1},...,J_{k})}}{\ov\eta\times U_0^{J}}/G^{\mr{ad}}_{\sum _{i\in I}n_{i}x_{i}}.$$
On pose $$\begin{gathered}
\nonumber
\mc H _{ N,\{0\}\cup J,W}^{\leq\mu,E}= R
\big(\mf p_{N,\{0\}\cup J}^{(\{0\},J_{1},...,J_{k}),\leq\mu}\big)_{!}\Big(\restr{\mc F_{N,\{0\}\cup J,W,\Xi,E}^{(\{0\},J_{1},...,J_{k})}} {\Cht_{N,\{0\}\cup J,W}^{(\{0\},J_{1},...,J_{k}),\leq\mu}/\Xi}\Big)
\end{gathered}$$ qui appartient à $D^{b}_{c}(\ov\eta\times (X\sm \wh N)^{J}, E)$ et ne dépend pas du choix de la partition $(J_{1},...,J_{k})$.
Alors $$H_{\{0\}\cup J,W} := \Big( \varinjlim _{\mu}\restr{\mc H _{N, \{0\}\cup J, W}^{0,\leq\mu,E}}{\Delta(\ov{\eta})} \Big)^{\mr{Hf}}$$ est muni d’une action de $\pi_{1}(\eta,\ov\eta)^{J}=\on{Gal}(\ov F/F)^{J}$. En effet c’est un sous-espace de $ \Big( \varinjlim _{\mu}\restr{\mc H _{N, \{0\}\cup J, W'}^{0,\leq\mu,E}}{\Delta(\ov{\eta})} \Big)^{\mr{Hf}} $ (qui est muni d’une action de $\on{Gal}(\ov F/F)^{\{0\}\cup J}$). En fait, en notant $\check F$ le corps réflex, on pourrait munir $ H_{\{0\}\cup J,W}$ d’une action de $\on{Gal}(\ov F/\check F) \times \on{Gal}(\ov F/F)^{J}$, mais on n’en a pas besoin ici.
Une construction proposée par Drinfeld
======================================
On explique maintenant la construction proposée par Drinfeld, en étendant la remarque 8.5 de [@ICM-text] au cas non déployé.
Soit $\on{Reg}$ la représentation régulière gauche de $\widehat G$ à coefficients dans $E$ (considérée comme une limite inductive de représentations de dimension finie). On va munir le $E$-espace vectoriel $H_{\{0\},\on{Reg}}$
- d’une structure de module sur l’algèbre des fonctions sur “l’espace affine $\mathcal S$ des morphismes $\sigma:\on{Gal}(\overline F/F)\to
{}^{L} G$ à coefficients dans les $E$-algèbres vérifiant des conditions analogues à (C3), (C4), (C5)”,
- d’une action algébrique de $\widehat G$ (venant de l’action à droite de $\widehat G$ sur $\on{Reg}$) qui est compatible avec la conjugaison par $\widehat G$ sur $\mathcal S$.
L’espace $\mathcal S$ n’est pas défini rigoureusement et la définition rigoureuse de la structure a) est la suivante. Pour toute représentation $E$-linéaire de dimension finie $V$ de $ {}^{L} G$, d’espace vectoriel sous-jacent $\underline V$, $H_{\{0\},\on{Reg}}\otimes \underline V$ est muni d’une action de $\on{Gal}(\overline F/F)$, qui en fait une limite inductive de représentations continues de dimension finie de $\on{Gal}(\overline F/F)$, de la façon suivante. On possède l’isomorphisme $\widehat G$-équivariant $$\begin{aligned}
\theta: \on{Reg} \otimes \underline V & \simeq \on{Reg}\otimes V \\
f\otimes x & \mapsto [g\mapsto f(g) g.x]
\end{aligned}$$ où $\widehat G$ agit diagonalement sur le membre de droite, et où pour donner un sens à la formule on identifie ce dernier à l’espace vectoriel des fonctions algébriques $\widehat G\to V$. On en déduit un isomorphisme $$H_{\{0\},\on{Reg}} \otimes \underline V = H_{\{0\},\on{Reg} \otimes \underline V }
\isor{\theta} H_{\{0\},\on{Reg} \otimes V}
\simeq H_{\{0\}\cup\{1\},\on{Reg} \boxtimes V}$$ où la première égalité est tautologique (puisque $ \underline V$ est simplement un espace vectoriel) et le dernier isomorphisme est l’inverse de l’isomorphisme de coalescence. Alors l’action de $ \on{Gal}(\overline F/F)$ sur le membre de gauche est définie comme étant l’action de $ \on{Gal}(\overline F/F)$ sur le membre de droite correspondant à la patte $1$. Si $V_{1}$ et $V_{2}$ sont deux représentations de ${}^{L} G$, les deux actions de $ \on{Gal}(\overline F/F)$ sur $H_{\{0\},\on{Reg}}\otimes \underline {V_{1}}\otimes \underline {V_{2}}$ associées aux actions de $\widehat G$ sur $V_{1}$ et $V_{2}$ commutent entre elles et l’action diagonale de $ \on{Gal}(\overline F/F)$ est égale à l’action associée à l’action diagonale de $\widehat G$ sur $V_{1}\otimes V_{2}$. Cela donne la structure a) car si $V$ est comme ci-dessus, $x\in V$, $\xi\in V^{*}$, $f$ est la fonction sur ${}^{L} G$ définie comme le coefficient de matrice $f(g)=\langle \xi, g.x \rangle$, et $\gamma\in \on{Gal}(\overline F/F)$ on pose que $F_{f,\gamma}: \sigma\mapsto f(\sigma(\gamma))$, considérée comme une “fonction sur $\mathcal S$”, agit sur $H_{\{0\},\on{Reg}}$ par la composée $$\begin{gathered}
H_{\{0\},\on{Reg}} \xrightarrow{\on{Id}\otimes x} H_{\{0\},\on{Reg}} \otimes \underline V \xrightarrow{\gamma} H_{\{0\},\on{Reg}} \otimes \underline V
\xrightarrow{\on{Id}\otimes \xi} H_{\{0\},\on{Reg}}.
\end{gathered}$$ Toute fonction $f$ sur ${}^{L} G$ peut être écrite comme un coefficient de matrice, et les fonctions $F_{f,\gamma}$ quand $f$ et $\gamma$ varient sont supposées “engendrer topologiquement toutes les fonctions sur $\mathcal S$”. La propriété ci-dessus avec $V_{1}$ et $V_{2}$ implique les relations entre les $F_{f,\gamma}$, à savoir que $$\begin{gathered}
\label{formule-coproduit}
F_{f,\gamma_{1}\gamma_{2}}=\sum_{\alpha }F_{f_{1}^{\alpha},\gamma_{1}}
F_{f_{2}^{\alpha},\gamma_{2}}\end{gathered}$$ si l’image de $f$ par le coproduit est $\sum_{\alpha} f_{1}^{\alpha}\otimes f_{2}^{\alpha}$. Dans [@zhu-notes], le second auteur donne une construction équivalente de la structure a). Les structures a) et b) sont compatibles au sens suivant: la conjugaison $g F_{f,\gamma} g^{-1}$ de l’action de $F_{f,\gamma}$ sur $H_{\{0\},\on{Reg}}$ par l’action algébrique de $g\in \widehat G$ est égale à l’action de $F_{f^{g},\gamma} $ où $f^{g}(h)=f(g^{-1}hg)$.
On a $H_{\{0\},\on{Reg}}=\bigoplus H_{\{0\},V} \otimes V^{*}$ où la somme directe est prise sur les représentations irreductibles de $\wh G$. On remarque que $H_{\{0\},V}$ est nul lorsque le centre $(Z_{\wh G})^{ \on{Gal}(\wt F/F)}$ agit non trivialement sur $V$, si bien que $(Z_{\wh G})^{ \on{Gal}(\wt F/F)}$ agit trivialement sur $H_{\{0\},\on{Reg}}$.
Soit $c\in H_{\{0\},\on{Reg}}$ et $f$ une fonction algébrique sur ${}^{L}G$. L’espace engendré par $\{F_{f,\gamma}c\mid \gamma\in\Gamma\}$ est de dimension finie et l’application $\gamma\mapsto F_{f,\gamma}c$ est continue. De plus, pour presque toute place $v$ de $X$, $F_{f,\gamma}c=f(1)c$ pour $\gamma$ dans le sous-groupe d’inertie $I_v$ en cette place.
Sans perdre de généralité, on peut supposer que $f(g)=\langle \xi, gx\rangle$ avec $x\in V$ et $\xi \in V^*$. On écrit $\on{Reg}$ comme une réunion croissante de représentations $W_i$ de $\hat{G}$. Alors $(\on{Id}\otimes x)(c)\in H_{\{0\},W_i}\otimes \underline V$ pour une certaine représentation de dimension finie $W_i$. Le lemme en résulte immédiatement.
Les structures a) et b) fournissent un “$\mathcal O$-module sur le champ $\mathcal S/\widehat G$ des paramètres de Langlands globaux” (tel que l’espace vectoriel de ses “sections globales sur $\mathcal S$” est $H_{\{0\},\on{Reg}} $). Plus précisément l’action de $\widehat G$ sur $\mathcal S$ se factorise par $\widehat G/(Z_{\wh G})^{ \on{Gal}(\wt F/F)}$ et on a un “$\mathcal O$-module sur le champ $\mathcal S/\big(\widehat G/(Z_{\wh G})^{ \on{Gal}(\wt F/F)}\big)$ tel que l’espace vectoriel de ses “sections globales sur $\mathcal S$” est $H_{\{0\},\on{Reg}} $.
Pour tout morphisme $\sigma: \on{Gal}(\overline F/F)\to {}^{L} G(\overline{ \mathbb Q_{\ell}})$, on veut définir $\mathfrak A_{\sigma}$ comme la fibre de ce $\mathcal O$-module au-dessus de $\sigma$ (considéré comme un “$\overline{ \mathbb Q_{\ell}}$-point de $\mathcal S$ dont le groupe des automorphismes dans le champ $\mathcal S/\big(\widehat G/(Z_{\wh G})^{ \on{Gal}(\wt F/F)}\big)$ est $\mathfrak S_{\sigma}=S_{\sigma}/(Z_{\wh G})^{ \on{Gal}(\wt F/F)}$”).
Rigoureusement on définit $\mathfrak A_{\sigma}$ comme le plus grand quotient de $H_{\{0\},\on{Reg}}\otimes_{\mathbb Q_{\ell}} \overline{ \mathbb Q_{\ell}}$ sur lequel toute fonction $F_{f,\gamma}$ comme ci-dessus agit par multiplication par le scalaire $f(\sigma(\gamma))$. On voit que $\mathfrak S_{\sigma}=S_{\sigma}/(Z_{\wh G})^{ \on{Gal}(\wt F/F)}$ agit sur $\mathfrak A_{\sigma}$.
Si l’heuristique (8.3) de [@ICM-text] est vraie c’est le même $\mathfrak A_{\sigma}$ que dans l’heuristique.
Etude du cas elliptique
=======================
On rappelle que tout ensemble fini $I$ et toute représentation $W$ de dimension finie de $({}^{L}G)^{I}$, on a admis que $H_{I,W}$ est de dimension finie (ce qui est une extension non encore écrite de [@cong], où ce résultat est montré lorsque $G$ est déployé). Pour tout paramètre de Langlands $\sigma$, on note alors $(H_{I,W})_{\sigma}$ l’espace propre [*généralisé*]{} dans $H_{I,W}\otimes_{E}\overline{{\mathbb Q}_\ell}
$ associé à $\sigma$ pour l’action des opérateurs d’excursion et le système de valeurs propres de ces opérateurs correspondant à $\sigma$.
La proposition suivante implique la proposition \[prop-ppale\].
Pour tout $\sigma$ elliptique on a l’égalité $$(H_{\{0\},W})_{\sigma}= \big( \mathfrak A_{\sigma}\otimes_{\overline{{\mathbb Q}_\ell}} W_{\sigma^{I}}\big)^{S_{\sigma}} ,$$ équivariante par l’action de $\on{Gal}(\overline F/F)^{I}$, fonctorielle en $W$, et compatible avec les isomorphismes de coalescence. Autrement dit l’heuristique (8.3) de [@ICM-text] est vraie au-dessus de $\sigma$.
[**Démonstration.**]{} On a $H_{\{0\},\on{Reg}}=\bigoplus H_{\{0\},V} \otimes V^{*}$ où la somme directe est prise sur les représentations irreductibles de $\wh G$. On définit $$\begin{gathered}
\label{Hregsigma}(H_{\{0\},\on{Reg}})_{\sigma}=\bigoplus (H_{\{0\},V})_{\sigma} \otimes V^{*}. \end{gathered}$$ On a déjà vu que $(Z_{\wh G})^{ \on{Gal}(\wt F/F)}$ agit trivialement sur $H_{\{0\},\on{Reg}}$ et donc sur $(H_{\{0\},\on{Reg}})_{\sigma}$.
Les fonctions $F_{f,\gamma}$ agissent sur $(H_{\{0\},\on{Reg}})_{\sigma}$ (parce qu’elles commutent avec les opérateurs d’excursion).
On choisit $n$ et $(\gamma_1, ..., \gamma_n)$ tels que
\(H) $\sigma(\gamma_1)$, ..., $\sigma(\gamma_n)$ engendrent un sous-groupe Zariski dense de l’image of $\sigma$.
Par l’hypothèse d’ellipticité de $\sigma$, et grâce à (H), le $n$-uplet $(\sigma(\gamma_1)$, ..., $\sigma(\gamma_n))$ est semi-simple et les résultats de Richardson [@richardson] rappelés dans le lemme 11.9 de [@coh] impliquent que la $\wh G$-orbite par conjugaison de $(\sigma(\gamma_1)$, ..., $\sigma(\gamma_n))$ dans $({}^L G)^n$ est fermée, et égale à $\wh G/S_{\sigma}$. De plus cette orbite est schématiquement égale à l’image inverse de l’image de $(\sigma(\gamma_1), ..., \sigma(\gamma_n))$ par le morphisme de $({}^L G)^n$ dans le quotient grossier de $({}^L G)^n$ par conjugaison par $\wh G$.
Grâce à l’action des fonctions $F_{f,\gamma_{i}}$ pour $i=1,...,n$, $(H_{\{0\},\on{Reg}})_{\sigma}$ est un module sur $\mathcal O(({}^L G)^n)$. Toute fonction $\wh G$-invariante sur $({}^L G)^n$ s’annulant sur l’orbite $\on{Orb}=\wh G (\sigma(\gamma_1), ..., \sigma(\gamma_n))$ agit de façon nilpotente sur chaque élément de $(H_{\{0\},\on{Reg}})_{\sigma}$ (car c’est un opérateur d’excursion s’annulant sur la classe de conjugaison de $\sigma$) et donc $(H_{\{0\},\on{Reg}})_{\sigma}$ est un module sur $\mathcal O(\widehat{\on{Orb}})$ où $\widehat{\on{Orb}}$ est la complétion formelle de $({}^L G)^n$ le long de l’orbite $\on{Orb}$.
Le théorème de la tranche étale de Luna implique qu’il existe une variété affine localement fermée $Z$ dans $({}^L G)^n$, contenant $(\sigma(\gamma_1)$, ..., $\sigma(\gamma_n))$ et stable par conjugaison par $S_{\sigma}$, telle que $Z\times_{S_{\sigma}}\wh G\to ({}^L G)^n$ est (fortement) étale. Soit $\widehat Z$ la complétion formelle de $Z$ en $(\sigma(\gamma_1), ..., \sigma(\gamma_n))$. On a alors $$\begin{gathered}
\label{Orb-Z}\widehat{\on{Orb}}=\widehat Z\widehat{\times}_{S_{\sigma}}\wh G.\end{gathered}$$ On définit $$\begin{gathered}
\label{defi-sigma-Z}
(H_{\{0\},\on{Reg}})_{\sigma, Z}=(H_{\{0\},\on{Reg}})_{\sigma}\otimes_{\mathcal O(\widehat{\on{Orb}})} \mathcal O(\widehat Z).\end{gathered}$$
Il résulte de que pour toute représentation de dimension finie $V$ de $\wh G$ on a $$(H_{\{0\},V})_{\sigma}= \big( (H_{\{0\},\on{Reg}})_{\sigma} \otimes_{\overline{{\mathbb Q}_\ell}} V\big)^{\wh G}$$ De on déduit que $$\begin{gathered}
\label{dec-V-sigma}(H_{\{0\},V})_{\sigma}= \big( (H_{\{0\},\on{Reg}})_{\sigma, Z} \otimes_{\overline{{\mathbb Q}_\ell}} V\big)^{S_{\sigma}}. \end{gathered}$$
On rappelle qu’on a admis que tous les $H_{\{0\},V}$ sont de dimension finie (résultat montré dans [@cong] lorsque $G$ est déployé). Comme $S_{\sigma}$ est fini modulo $(Z_{\wh G})^{ \on{Gal}(\wt F/F)}$ (qui agit trivialement sur $(H_{\{0\},\on{Reg}})_{\sigma, Z}$) il existe une représentation $V$ de dimension finie de $\wh G/(Z_{\wh G})^{ \on{Gal}(\wt F/F)}$ telle que la restriction de $V$ au groupe fini $\mathfrak S_{\sigma}=S_{\sigma}/(Z_{\wh G})^{ \on{Gal}(\wt F/F)}$ contient toutes les représentations irréductibles de ce groupe. Choisissant une telle représentation $V$ on déduit de la finitude de la dimension de $H_{\{0\},V}$ et de que $(H_{\{0\},\on{Reg}})_{\sigma, Z}$ est de dimension finie. Il en résulte aussi que le seul paramètre de Langlands pouvant apparaître dans la décomposition spectrale de $(H_{\{0\},\on{Reg}})_{\sigma, Z}$ par l’action des opérateurs d’excursion est la classe de conjugaison de $\sigma$.
On note $\mathcal C$ la sous-algèbre commutative de $\on{End}((H_{\{0\},\on{Reg}})_{\sigma, Z})$ engendrée par l’action de tous les $F_{f,\gamma}$ quand $f$ et $\gamma$ varient. On a un morphisme $ \mathcal O(( {}^{L}G)^{n})\to \mathcal C$ qui envoie $f_{1}\otimes ... \otimes f_{n}$ sur $F_{f_{1},\gamma_{1}}... F_{f_{n},\gamma_{n}}$. Il résulte de que ce morphisme se factorise à travers le quotient $ \mathcal O(( {}^{L}G)^{n}) \to \mathcal O(\widehat Z)$. On a donc un morphisme $ \mathcal O(\widehat Z)\to \mathcal C$. Comme $\mathcal C$ est de dimension finie, pour $k$ entier assez grand ce morphisme se factorise par $ \mathcal O(Z_{k})$ où $Z_{k}$ désigne le voisinage épaissi d’ordre $k$ de $(\sigma(\gamma_1), ..., \sigma(\gamma_n))$ dans $Z$. On a donc un morphisme $\mathcal O(Z_{k})\to \mathcal C$. Grâce à la structure a) on a (quitte à restreindre l’ouvert $U$ de $X$) un morphisme $$\beta:\pi_{1}(U, \overline \eta)\to {}^L G(\mathcal C)$$ défini par la propriété que pour toute fonction algébrique $f$ sur ${}^L G$ et pour tout $\gamma\in \on{Gal}(\ov F/F)$, on a $f(\beta(\gamma))=F_{f,\gamma}$ dans $ \mathcal C$. On déduit de que $\beta$ est un morphisme de groupes. La composition de $\beta$ avec est d’ordre fini fixé (à cause du quotient par $\Xi$ dans la définition des $H_{I,W}$). Le morphisme $\beta$ est continu parce que pour toute fonction algébrique $f$ sur ${}^L G$, le morphisme $\gamma\mapsto f(\beta(\gamma))$ est une application continue de $\pi_{1}(U, \overline \eta)$ vers $\mathcal C$. On a de plus un diagramme commutatif $$\begin{gathered}
\label{diag-com}
\xymatrix{
\on{Spec} \mathcal C \ar[rr]^{(\beta(\gamma_{1}), ..., \beta(\gamma_{n}))} \ar[rd]
&& ({}^{L}G)^{n}
\\
& \on{Spec} Z_{k}\ar[ur] }\end{gathered}$$
Grâce à (H), le centralisateur de $(\sigma(\gamma_{1}), ..., \sigma(\gamma_{n}))$ est égal à celui de $\sigma$. Il en résulte que pour tout $\sigma'$ tel que
- $\sigma'$ est conjugué à $\sigma$,
- $(\sigma'(\gamma_1), ..., \sigma'(\gamma_n))$ est égal à $(\sigma(\gamma_1), ..., \sigma(\gamma_n))$,
alors $\sigma'$ est égal à $\sigma$.
Pour tout caractère $\chi: \mathcal C\to \Qlbar$, $\chi\circ \beta$ vérifie ces deux conditions donc est égal à $\sigma$. En d’autres termes le réduit du spectre de $\mathcal C$ est égal au seul point $\sigma$. En particulier $\mathcal C$ est une $\Qlbar$-algèbre locale artinienne.
D’après le lemme \[lem-non-def\], $\beta$ est conjugué à $\sigma$. En particulier $(\beta(\gamma_{1}), ..., \beta(\gamma_{n}))$ est conjugué à $(\sigma(\gamma_{1}), ..., \sigma(\gamma_{n}))$, et comme $(\beta(\gamma_{1}), ..., \beta(\gamma_{n}))$ est au-dessus de la tranche $\widehat Z_{k}$ il est égal à $(\sigma(\gamma_{1}), ..., \sigma(\gamma_{n}))$, puisque $\on{Orb}\cap \widehat Z_{k}=\{(\sigma(\gamma_{1}), ..., \sigma(\gamma_{n}))\}$ schématiquement. Comme le centralisateur de $(\sigma(\gamma_{1}), ..., \sigma(\gamma_{n}))$ est égal à celui de $\sigma$ on a alors $\beta=\sigma$ et $\mathcal C=\Qlbar$.
On en déduit $(H_{\{0\},\on{Reg}})_{\sigma, Z} =\mathfrak A_{\sigma}$, ce qui conclut la preuve de la proposition \[prop-ppale\]. En effet pour tout $I$ et pour toute représentation de dimension finie $W$ de $({}^L G)^{I}$, $$(H_{I,W})_{\sigma}= \big( (H_{\{0\},\on{Reg}})_{\sigma, Z} \otimes_{\overline{{\mathbb Q}_\ell}} W_{\sigma^{I}}\big)^{S_{\sigma}}.$$
[boeckle-harris...]{}
G. Böckle, M. Harris, C. Khare, and J. Thorne. $\widehat G$-local systems on smooth projective curves are potentially automorphic. Preprint, arXiv:1609.03491 (2016)
V. G. Drinfeld. Proof of the Petersson conjecture for $GL(2)$ over a global field of characteristic $p$. (1), 28–43 (1988)
A. Genestier et V. Lafforgue. Chtoucas restreints pour les groupes réductifs et paramétrisation de Langlands locale. Preprint, arXiv:1709.00978 (2017)
L. Lafforgue. Chtoucas de Drinfeld et correspondance de Langlands. (1), 1–241 (2002)
V. Lafforgue. Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale. , 719–891 (2018)
V. Lafforgue. Shtukas for reductive groups and Langlands correspondence for functions fields (report for ICM 2018). , arXiv:1803.03791
D. Kazhdan et Y. Varshavsky. On the cohomology of the moduli spaces of $F$-bundles: stable cuspidal Deligne-Lusztig part. (2010)
R. Kottwitz. Stable trace formula: Cuspidal tempered terms. (3) 611–650 (1984)
R. Kottwitz. Shimura varieties and $\lambda$-adic representations. In: [*Automorphic Forms, Shimura Varieties and $L$-functions, part 1*]{}. Perspectives in Mathematics 10. Academic Press, 161–209 (1990)
R. W. Richardson. Conjugacy classes of $n$-tuples in Lie algebras and algebraic groups. , 1–35 (1988)
Y. Varshavsky. Moduli spaces of principal $F$-bundles. (1), 131–166 (2004)
Cong Xue. Cuspidal cohomology of stacks of shtukas. Preprint, arXiv:1802.01362 (2018)
Cong Xue. Finiteness of the cohomologies of stacks of shtukas as modules over Hecke algebras, and applications. Preprint (2018)
Xinwen Zhu. Pseudo-representations for reductive groups. Unpublished notes (2017)
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abstract: 'Even though many point processes have been scrutinized to describe the unique features of emerging wireless networks, the performance of vehicular networks have been largely assessed using mostly the Poisson Point Process (PPP) to model the locations of vehicles along a road. The PPP is not always a realistic model, because it does not account for the physical dimensions of vehicles, and it does not capture the fact that a driver maintains a safety distance from the vehicle ahead. In this paper, we model the inter-vehicle distance equal to the sum of two components: A constant hardcore headway distance, and a random distance following the exponential distribution. We would like to investigate whether a PPP for the locations of interferering vehicles can be used to describe adequately the performance of a link at the origin under the new deployment model. Unfortunately, the probability generating functional (PGFL) of the hardcore point process is unknown. In order to approximate the Laplace transform of interference, we devise simple approximations for the variance and the skewness of interference, and we select suitable probability functions to approximate the interference distribution. It turns out that the PPP (of equal intensity) gives a lower bound for the outage probability under the hardcore point process. When the coefficient of variation and the skewness of interference are high, the bound may become loose at the upper tail. Relevant scenarios are associated with urban street microcells and highway macrocells with low intensity of vehicles. We also show that the performance predictions using the PPP deteriorate with multi-antenna maximum ratio combining receiver and temporal performance indicators related to the performance of retransmission schemes. Our approximations generate good performance predictions in all considered cases.'
author:
- |
Konstantinos Koufos and Carl P. Dettmann [^1]\
[^2]
title: Performance of a Link in a Field of Vehicular Interferers with Hardcore Headway Distance
---
\[PPP\][Poisson Point Process]{} \[PGFL\][Probability Generating Functional]{} \[CDF\][Cumulative Distribution Function]{} \[PDF\][Probability Distribution Function]{} \[PMF\][Probability Mass Function]{} \[PCF\][Pair Correlation Function]{} \[RV\][Random Variable]{} \[SIR\][Signal-to-Interference Ratio]{} \[i.i.d.\][independent and identically distributed]{} \[MRC\][Maximum Ratio Combining]{} \[MAC\][Medium Access Control]{} \[V2I\][Vehicular-to-Infrastructure]{} \[CoV\][coefficient-of-variation]{} \[1D\][one-dimensional]{}
Headway distance models, method of moments, probability generating functional, stochastic geometry.
Introduction
============
Inter-vehicle communication, e.g., dedicated short-range transmission IEEE 802.11p, and/or connected vehicles to roadside units, e.g., LTE-based communication, will be critical for the coordination of road traffic, automated driving and improved safety in emerging vehicular networks [@V2X5G]. In order to analyze the performance of vehicular networks, we need tractable but also realistic models for the locations of vehicles. The theory of point processes [@Veres2002] deals with random spatial patterns and can provide us with the general modeling framework. We have to construct carefully the deployment model for the vehicles along a roadway, balancing between accuracy and complexity, that describes the distribution of headway (or inter-vehicle) distance, i.e., the distance from the tip of a vehicle to the tip of its successor.
With the advent of wireless communication networks with irregular (non-deterministic) structure, e.g., small cell technologies [@FCC2012], elements from the theory of point processes have been employed to study their performance [@Haenggi2009a]. The simplest model is a of some (potentially variable) intensity embedded on a mathematical space [@Kingman1992]. The is characterized by complete randomness; the location of a point does not impose any constraints on the realization of the rest of the process. Due to the lack of inter-point interaction, the is tractable [@Haenggi2013a Theorem 4.9], allowing us to calculate the probabilitistic impact of suitable functions, e.g. an interference field, on the typical point. Because of that, the has been widely adopted for performance evaluation, under interference, of ad hoc, cellular and heterogeneous networks [@Andrews2011; @Dhillon2012; @Haenggi2013b].
Despite its wide acceptance, the has also received a lot of criticism, because it does not capture the repulsive nature of network elements, due to physical constraints and mechanisms. This criticism has sparked network modeling using many more point processes, which however, are not tailored to describe vehicular networks. Stationary determinantal point processes fit better than the the Ripley’s $\text{K}$ function (second-order spatial statistic) of real-world macro-base station datasets [@Andrews2015]. The for some determinantal processes, e.g., Ginibre, Gauss, etc., can be computed and evaluated numerically [@Andrews2015; @Miyoshi2014]. The repulsion induced by collision avoidance protocols is better captured by Mat[è]{}rn rather than softcore processes [@Busson2009; @Haenggi2011]. The locations of users in wireless networks may also exhibit clustering instead of repulsion, due to non-uniform population density and hotspots. The for some Poisson cluster processes is tractable, see [@Veres2002 example 6.3(a)] and [@Ganti2009] for the of the Neyman-Scott process.
Intuitively, a spatial model for vehicular networks should be broken down into two components; a model for the road infrastructure and another for the distribution of vehicles along the roads. The Manhattan Poisson line process can be used to model a vertical and horizontal layout of streets, while the Poisson line process is suitable to describe roads with random orientations. These line processes have been coupled with homogeneous for the distribution of vehicles to study coverage probability [@Dhillon2018; @Baccelli2018]. The Laplace transform of interference from the line containing the typical receiver (defined as the origin) is calculated using the of . The contribution of interference from the rest of the lines requires to work out their distance distribution to the origin and incorporate it into the [@Dhillon2018]. Alternatively, we can map every road to the road containing the origin but with a non-uniform density of vehicles [@Steinmetz2015]. The has also been used in vehicular system set-ups for higher layer performance evaluation [@Blaszczyszyn2013]. Non-homogeneous have been used to model the impact of random waypoint mobility on temporal statistics of interference over finite regions [@Koufos2016].
The distribution of vehicles along a road with few number of lanes, e.g., bidirectional traffic streams with restricted overtaking, will not resemble a . The allows unrealistrically small headways with high probability, while in practice, the follower maintains a safety distance depending on its speed and reaction time plus the length of the vehicle ahead [@Daou1966; @Greenberg1966]. The distribution of headways naturally depends on traffic status. Measurements have revealed that the log-normal distribution is suitable under free flow traffic, while log-logistic distribution is adequate in congestion status [@Yin2009]. These distributions have been used to study the lifetime of inter-vehicle links [@Yan2011], however, without considering interference.
We would like to identify whether the for the locations of vehicular interferers can be used to describe adequately the performance of a link at the origin (not part of the point process generating the interference) and under which conditions. The simplest inter-vehicle distance model which contains the as a special case, but can also be tuned to avoid small headways, consists of a constant hardcore distance plus a random component modeled by an exponential [@Cowan1975]. The inter-vehicle distance follows the shifted-exponential distribution, and the is obtained by setting the shift equal to zero. In [@Koufos2018] we have compared the variance and skewness of interference under two fields of equal intensity $\lambda$; one due to a , and another due to a point process with shifted-exponential inter-arrivals and positive hardcore distance $c$. We have devised a simple formula scaling the variance due to a with a factor, which depends on $\lambda$ and $c$. However, this expression cannot be directly translated into link performance, e.g., outage probability. In addition, the performance of the link over time, e.g., the distribution of local delay, and/or over space, e.g., the outage probability with multi-antenna receiver require the temporal and spatial correlation properties of interference which are different under the two deployment models.
The common methodology to assess the outage probability of a link over all possible network states needs the of the point process generating the interference. With a positive hardcore distance, the locations of vehicles become correlated, and we could not figure out how to use the of as a building block for the calculation of the Laplace transform of interference originated from the hardcore process. Looking at the complications associated with the calculation of the second and third moments of interference [@Koufos2018], it does not seem promising to calculate higher-order terms in the series expansion of [@Westcott1972]. The expansion kernels in factorial moment representation of the are simple only for the [@Baccelli2012]. It has been recently shown that the outage probability in non-Poissonian wireless cellular networks can be well-approximated by shifting horizontally the outage probability due to a [@Haenggi2015]. This result is not applicable in our system set-up because the point process does not impact the distribution of useful signal power but only the distribution of interference level. We will also see that the distance distribution appears quite complicated to convert it into aggregate interference level distribution.
In order to assess the outage probability, we can also calculate few moments of interference, and select suitable distributions, with simple Laplace transform, to approximate it. The method of moments has been widely used for modeling wireless channels, e.g., composite fading [@Halim2010], Signal-to-Noise-Ratio in composite fading [@Atapattu2011], aggregate interference and spectrum sensing channels [@Sousa2008; @Koufos2011].
Complementing our study in [@Koufos2018], we will generate a simple approximation for the skewness of interference, allowing us to capture the impact of different parameters on the skewness and the (the ratio of the standard deviation over the mean). The mean interference levels under the two models are equal. The carries information about the behavior of the interference distribution around the mean, and the skewness about the symmetry between the tails. Note that the right/left tail of interference distribution is associated with the behavior of the outage probability at the low/high reliability regime. In addition, the sign of the skewness would be crucial in selecting appropriate distribution models. The simplicity of the approximations would enable us to deduce under which traffic conditions the fails to approximate closely the and the skewness. Under these conditions, the does not describe accurately the interference distribution and subsequently the outage probability. In order to study the efficacy of with temporal and spatial performance metrics, we will use the mean delay and the outage probability of dual-branch as relevant metrics. A positive hardcore distance reduces the temporal and spatial correlation of interference in comparison with the . Because of that, the performance predictions worsen. The contributions of this paper are listed below.
- We approximate the distance distribution between the nearest interferer and the origin for a point process with hardcore distance $c$ and intensity $\lambda$. Its complexity rules out the possibility to calculate the signal level distribution for the $k-$th nearest interferer, and convert it to aggregate interference level by summing over all $k\!\rightarrow\!\infty$.
- We show that for small hardcore distance as compared to mean inter-vehicle distance $\lambda^{-1}$, the skewness of interference is approximately equal to that due to a of intensity $\lambda$ scaled by $\left(1\!-\!\frac{\lambda c}{2}\right)$. This complements [@Koufos2018], where it is shown that for small $\lambda c$, the variance of interference, and subsequently the , are reduced in comparison with a of equal intensity. Overall, a hardcore distance makes the distribution of interference more concentrated around the mean and less skewed.
- For fixed $\lambda c$, the skewness and the of interference increase for smaller cell size and lower intensity of vehicles. The shifted-gamma distribution with parameters selected using the method of moments (matching the mean, the variance and the skewness) fits well the simulations in all scenarios, including those associated with a high and skewness, e.g., urban microcells and sparse flows of vehicles along macrocells. In these scenarios, the outage probability predicted using the is a loose lower bound in the upper tail.
- Introducing hardcore distance reduces the spatial correlation of interference at the two branches of a receiver. Under Rayleigh fading channels, the Pearson correlation coefficient scales down approximately by $\left(1\!-\!\lambda c\right)$. The outage probability predicted by the is not anymore a bound and worsens in the upper tail. A bivariate gamma approximation for the interference distribution with identical and correlated marginals gives a good fit.
- The makes a pessimistic prediction for the mean delay with simple retransmissions and low mobility. The gamma approximation for the interference distribution can provide a good fit when the fails.
The rest of this paper is organized as follows. In Section \[sec:Model\], we present the system model and the correlation properties of the deployment. In Section \[sec:Distance\], we derive the distance distribution between the nearest interferer and the origin. In Section \[sec:Distribution\], we approximate the skewness of interference, and we select well-known for the distribution of interference. In Section \[sec:Outage\], we illustrate that the approximations fit well the simulations, while the bounds on the probability of outage based on the and the Jensen’s inequality may not be tight. In Section \[sec:Applications\], we test the validity of the approximations using the mean local delay and dual-antenna receiver. In Section \[sec:Conclusions\], we conclude this study and outline relevant topic for future work.
System model {#sec:Model}
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We consider point process of vehicles $\Phi$, where the inter-vehicle distance follows the shifted-exponential . The shift is denoted by $c\!>\!0$, and the parameter of the exponential part by $\mu\!>\! 0$. The average intensity $\lambda$ of vehicles can be calculated from $\lambda^{-1}\!=\!c+\mu^{-1}$, or equivalently $\lambda\!=\!\frac{\mu}{1+\mu c}$. This model has been proposed by Cowan [@Cowan1975], and due to the positive shift $c$, it can avoid small inter-vehicle distances. The penalty paid, in terms of analytical model complexity, is the correlations introduced in the locations of vehicles. The correlation properties have been studied in statistical mechanics, see for instance [@Salsburg1953], where the vehicles are the particles of hardcore fluid, and the shift is equal to the diameter of the rigid disk modeling the identical particles. Conditioning on the location of a particle at $x$, the probability to find another particle at $y$, is [@Salsburg1953 equation (32)] $$\label{eq:rho2}
\rho_k^{\left(2\right)}\!\!\left(y,x\right) \!\!=\!\! \Bigg\{ \!\!\!\! \begin{array}{ccl}\lambda \!\! \sum\limits_{j=1}^k \!\! \frac{\mu^j \left(y-x-jc\right)^{j-1}}{\Gamma\!\left(j\right) e^{\mu\left(y-x-jc\right)}}, \!\!\!\!\!\!& &\!\!\!\!\!\! y\!\in\!\left(x\!\!+\!kc, x\!\!+\!\left(k\!+\!1\right)\!c\right) \\ 0, \!\!\!\!\!\!& &\!\!\!\!\!\! \text{otherwise}, \end{array}$$ where $k\!\geq\! 1$ and $\Gamma\!\left(j\right)\!=\!\left(j\!-\!1\right)!$
![Normalized $\rho^{\left(2\right)}\!\left(x,y\right)\left(\lambda \mu\right)^{-1}$ with respect to the normalized distance $|y-x|c^{-1}$. The dashed lines correspond to $\rho^{\left(2\right)}\!\left(x,y\right)\!=\!\lambda^2$, or, $\rho^{\left(2\right)}\!\left(x,y\right)\left(\lambda \mu\right)^{-1}\!=\!1\!-\!\lambda c$ [@Koufos2018 Fig.2].[]{data-label="fig:CorrFunc"}](CorrFunc.eps){width="3in"}
One can derive equation using basic probability theory. The $k-$th branch of the , $k\!\geq\! 1$, requires to sum over the probabilities of having $\left\{0,1,\ldots, \left(k\!-\!1\right)\right\}$ particles between $x$ and $y$. For $k\!=\!0$, we have $\rho^{\left(2\right)}\!\left(y,x\right)\!=\!0$, as no two particles can be found at distance separation less than $c$. For $y\!<\!x$, one has simply to inter-change $y$ and $x$ in . For more details, see [@Koufos2018 Section III]. The derivation of using thermodynamic equations is available in [@Salsburg1953].
The , $\rho^{\left(2\right)}\!\left(y,x\right)\!=\!\sum\nolimits_{k=1}^\infty \rho_k^{\left(2\right)}\!\left(y,x\right)$, is depicted in Fig. \[fig:CorrFunc\]. We see that for small hardcore distance $c$ as compared to the mean inter-vehicle distance $\lambda^{-1}$, the converges quickly to $\lambda^2$. The locations of vehicles become uncorrelated at few multiples of $c$ when the hardcore distance does not dominate over the random part of the deployment.
![The vehicles outside of the cell (red disks) generate interference at the receiver (black cross) located at the origin. The rest (blue disks) do not generate interference. A transmitter (black square) is paired with the receiver.[]{data-label="fig:SystModel"}](SystModel.eps){width="3.5in"}
The higher-order correlations are naturally more complicated than the . For a stationary determinantal point process, the $n-$th order correlation can be bounded from above by the normalized $\left(n\!-\!1\right)-$th product of using Fan’s inequality [@Baccelli2012 Lemma 4]. Fortunately, due to the nature of our deployment, the inequality is tight [@Salsburg1953 equation (27)]. Let us consider $n$ ordered points on the real line, $x_1,x_2,\ldots x_n$. The $n-$th order correlation is $\rho^{\left(n\right)}\!\left(x_1,x_2,\ldots x_n \right)\!=\! \frac{1}{\lambda^{n-2}}\prod_{i=1}^{n-1}\rho^{\left(2\right)}\!\left(x_{i+1}\!-\!x_i \right)$. For instance, the third-order intensity measure describing the probability to find a triple of distinct vehicles at $x,y$ and $z$, is $$\label{eq:rho3}
\rho^{\left(3\right)}\!\left(x,y,z\right) = \frac{1}{\lambda} \, \rho^{\left(2\right)}\!\left(x,y\right) \rho^{\left(2\right)}\!\left(y,z\right), \, x\!<\!y\!<\!z.$$
We place a receiver at the origin and a transmitter associated to it. The distance-based useful signal level is fixed and known and denoted by $P_r$. The transmitter is not part of the point process generating interference. We assume that only the vehicles outside a guard zone $\left[-r_0,r_0\right]$ contribute to interference, see Fig. \[fig:SystModel\]. For instance, in a communication, the vehicles inside the guard zone might be paired with the receiver (or traffic controller), while vehicles outside the guard zone interfere with it, as they are paired with other controllers. In another scenario, the transmitter-receiver link might be a wireless backhaul link using same spectral resources with vehicles communicating in ad hoc mode. The vehicles are forced to stop their transmissions inside the guard zone.
The transmit power level is normalized to unity. The propagation pathloss exponent is denoted by $\eta\!>\! 2$. The distance-based pathloss for an interferer located at $r$ is $g\!\left(r\right)\!=\!\left|r\right|^{-\eta}$ for $\left|r\right|\!>\!r_0$, and zero otherwise, to filter out vehicles inside the cell. The fading power level over all interfering links, $h$, and over the transmitter-receiver link, $h_t$, is exponential (Rayleigh distribution for the fading amplitudes) with mean unity. The fading is over different links and time slots. The interferers and the transmitter are active in each time slot, and they are equipped with a single antenna. When multiple antennas are employed at the receiver, they are separated at least by half the wavelength and their fading samples are assumed The distance-based useful and interference signal levels at different antennas are assumed equal.
Nearest interferer distance distribution {#sec:Distance}
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The statistics of interference are closely related to the statistics of the distance between the interferers and the reference point. Let us denote by $X_1$ the describing the distance from the nearest interferer to the origin. For the , due to the independence property, it suffices to calculate the distance distribution without the guard zone and shift the distribution by $r_0$. The contact distribution for a is exponential with parameter the intensity $\lambda$. Therefore the $X_1$ is distributed as $f_{X_1}\!\left(x\right)\!=\! 2\lambda e^{-2\lambda \left(x-r_0\right)}, x\!\geq\!r_0$. It is straightforward to verify that the and the skewness of $X_1$ for the are equal to $\frac{1}{1+2\lambda r_0}$ and $2$ respectively.
For the hardcore process, the distribution of $X_1$ follows easily, only if we ignore the guard zone. For $x\!\leq\!\frac{c}{2}$, the point process rules out any other interferer closer than $x$ to the origin. The nearest interferer is located uniformly in $\left[-\frac{c}{2},\frac{c}{2}\right]$. The probability to find a vehicle within an infinitesimal ${\rm d}x$ belonging to this interval is $\lambda {\rm d}x$. As a result, the distance distribution is uniform in $\left[0,\frac{c}{2}\right]$, and the probability to observe any distance of this range is $2\lambda {\rm d}x$. Therefore $\mathbb{P}\!\left(X_1\!\leq\! \frac{c}{2}\right)\!=\!\lambda c$. For $x\!\geq\!\frac{c}{2}$, no interferer must be located within a distance $\left(2x\!-\!c\right)$ from the first interferer, $\mathbb{P}\!\left(X_1\!\geq\! x|x\!\geq\!\frac{c}{2}\right)\!=\!e^{-\mu\left(2x-c\right)}$. After deconditioning, $\mathbb{P}\!\left(X_1\!\geq\! x\right)\!=\!\left(1\!-\!\lambda c\right)e^{-\mu\left(2x-c\right)}, x\!\geq\!\frac{c}{2}$. Finally, the for the $X_1$ takes the following form $$\mathbb{P}\!\left(X_1\!\leq\! x\right) = \Big\{ \begin{array}{ccl} 2\lambda x, \!\!\!\!\!& &\!\!\!\!\! x\!\in\! \left[0,\frac{c}{2}\right) \\ 1\!-\!\left(1\!-\!\lambda c\right) e^{-\mu\left(2x-c\right)}, \!\!\!\!\!& &\!\!\!\!\! x\!\geq\!\frac{c}{2}. \end{array}$$
The guard zone raises the complexity of calculating the distribution of $X_1$ because the correlated locations of vehicles start to have an effect. In a practical system set-up, the locations of vehicles from the two sides of the guard zone are expected to be weakly correlated. In order to give a relevant approximation, we note that a high value for the dimensionless ratio $\frac{\mu}{\lambda}\!=\!\frac{1}{1-\lambda c}\!=\left(1\!+\!\mu c\right)\!>\!1$ indicates that the decorrelates slowly. The point process will decorrelate within $\frac{2 r_0}{c}$ multiples of the hardcore distance if $\frac{2 r_0}{c}\!\gg\!\frac{\mu}{\lambda}\!=\!\left(1\!+\!\mu c\right)\!\approx\! \mu c$, or equivalently, $\mu\!\ll\!\frac{2r_0}{c^2}$. If this condition is true, we introduce minor error by treating as the distances of the nearest interferer from opposite sides of the guard zone. Then, the of $X_1$ would be approximated by the of the minimum of two .
Let us denote by $X_1^{\text{p}}$ the describing the distance between the nearest interferer from the positive half-axis and the origin. For $x\!\in\!\left(r_0,r_0\!+\!c\right)$, $X_1^{\text{p}}$ follows the uniform distribution. For $x\!\geq\!r_0\!+\!c$, no other interferer must be located closer to the cell border, $\mathbb{P}\!\left(X_1^{\text{p}}\!\geq\! x|x\!\geq\!r_0\!+\!c\right)\!=\!e^{-\mu\left(x-r_0-c\right)}$, or $\mathbb{P}\!\left(X_1^{\text{p}}\!\geq\! x\right)\!=\!\left(1\!-\!\lambda c\right)e^{-\mu\left(x-r_0-c\right)}, x\!\geq\!r_0\!+\!c$ after deconditioning. Finally, $$\mathbb{P}\!\left(X_1^{\text{p}}\!\leq\! x\right) = \Big\{ \begin{array}{ccl} \lambda\left(x\!-\!r_0\right), \!\!\!\!\!& &\!\!\!\!\! x\!\in\! \left[r_0,r_0\!+\!c\right) \\ 1\!-\!\left(1\!-\!\lambda c\right) e^{-\mu\left(x-r_0-c\right)}, \!\!\!\!\!& &\!\!\!\!\! x\!\geq\!r_0\!+\!c. \end{array}$$
The approximation for the of $X_1$ follows from the minimum of two $X_1^{\text{p}}$. $$\label{eq:DistanceCDF}
\mathbb{P}\!\left(\!X_1\!\leq\! x\!\right) \!\approx\! \bigg\{\!\! \begin{array}{ccl} \!\! 1\!-\!\left(\!1\!-\lambda\left(x\!-\!r_0\right)\right)^2, \!\!\!\!\!& &\!\!\!\!\! x\!\in\! \left[r_0,r_0\!+\!c\right) \\ \!\! 1\!-\!\left(1\!-\!\lambda c\right)^2 \!\! e^{-2\mu\left(x-r_0-c\right)}, \!\!\!\!\!& &\!\!\!\!\! x\!\geq\!r_0\!+\!c. \end{array}$$
In Fig. \[fig:NearestDistanceA\], we see that the above approximation essentially overlaps with the simulations even if the mean inter-vehicle distance, $\lambda^{-1}\!=\! 40$ m, becomes comparable to the guard zone size, $r_0\!=\!100$ m. This is because for $\lambda c\!=\! 0.4$, the converges to $\lambda^2$ after approximately $4c$, see Fig. \[fig:CorrFunc\], which is equal to $64$ m, roughly one-third of the guard zone length. The condition $\mu\!\ll\!\frac{2 r_0}{c^2}$ obviously holds.
Differentiating , the approximation for the becomes $$\label{eq:DistancePDF}
f_{X_1}\!\!\left(x\right) \approx \Big\{ \begin{array}{ccl} 2\lambda\left(\!1\!-\lambda\left(x\!-\!r_0\right)\right), \!\!\!\!\!& &\!\!\!\!\! x\!\in\! \left[r_0,r_0\!+\!c\right) \\ 2\lambda\left(1\!-\!\lambda c\right) e^{-2\mu\left(x-r_0-c\right)}, \!\!\!\!\!& &\!\!\!\!\! x\!\geq\!r_0\!+\!c. \end{array}$$
It is possible to verify that the and the skewness of are less than $\frac{1}{1+2\lambda r_0}$ and $2$ respectively, the values associated with a of equal intensity. In Fig. \[fig:NearestDistanceB\] we have simulated the distance distribution for the $k-$th nearest interferer, $k\!\leq\! 5$. The distance distributions for $k\!>\!1$ follow the same trend as that proved for $k\!=\!1$. The distributions of the hardcore process have lower and skewness as compared to those of . This complies with the intuition that a hardcore $c$ makes the point process less random. Based on this, we may conjecture that the distribution of interference due to the hardcore process will be more concentrated around the mean and less skewed as compared to that due to a of equal intensity.
Interference distribution {#sec:Distribution}
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From the Campbell’s Theorem, we know that the mean interference for a stationary point processs of intensity $\lambda$ is $\mathbb{E}\!\left\{\mathcal{I}\right\} \!=\!2\lambda\int_{r_0}^\infty \!x^{-\eta}{\rm d}x \!=\! \frac{2\lambda r_0^{1-\eta}}{\eta-1}$. The details for the approximation of the second moment of interference can be found in [@Koufos2018 Section V]. The main idea is to approximate the with the of , $\rho^{(2)}\!\left(x,y\right) \!\approx\!\lambda^2$, for large distance separation $\left|y-x\right|$, and use the exact only for small distances. According to Fig. \[fig:CorrFunc\], this approximation should be valid for $\lambda c\!\rightarrow\! 0$. Since the point process for $c\!>\!0$ becomes less random than the , the variance of inteference should reduce [@Koufos2018 equation (14)]. $$\label{eq:VarApp}
\begin{array}{ccl}
\mathbb{V}\!\left\{\mathcal{I}\right\} \!\!\!&\approx&\!\!\! \displaystyle \frac{4\lambda r_0^{1-2\eta}}{2\eta\!-\!1} \left(1\!-\!\lambda c\!+\!\frac{1}{2}\lambda^2c^2\right),
\end{array}$$ where the term in front of the parenthesis is the variance due to a of intensity $\lambda$.
Some preliminary calculations about the third moment of interference are available in [@Koufos2018 Section IV]. Here, we will derive a simple approximation relating it to that of of intensity $\lambda$, similar to the approximation in for the variance.
\[lemma:1\] The skewness of interference from a hardcore process of intensity $\lambda$ and hardcore distance $c$ can be approximated by the skewness due to a of intensity $\lambda$, scaled by $\left(1\!-\!\frac{\lambda c}{2}\right)$. The approximation is valid for $\lambda c\!\rightarrow\!0$ and $\frac{c}{r_0}\!\rightarrow\!0$. $$\mathbb{S}\!\left\{\mathcal{I}\right\} \approx \frac{12 \lambda r_0^{1-3\eta}}{3\eta\!-\!1} \left( \frac{4\lambda r_0^{1-2\eta}}{2\eta\!-\!1} \right)^{-\frac{3}{2}} \left(1 \!-\! \frac{\lambda c}{2} \right).$$
The proof can be found in the supplementary material (optional reading).
Few properties of the skewness can be drawn based on Lemma \[lemma:1\]: (i) Introducing a small hardcore distance while keeping the intensity of interferers fixed, reduces the skewness but the distribution remains positively-skewed. (ii) The skewness of interference due to a increases for increasing pathloss exponent $\eta$ and decreasing cell size $r_0$. Introducing hardcore distance for fixed $\lambda$ does not change this property. (iii) For a , increasing the intensity $\lambda$ reduces the skewness of interference. This is also true for the hardcore process provided that the product $\lambda c$ is not decreasing.
![Skewness with respect to $\lambda c$ for urban street microcells, $r_0\!=\!100$ m, and large motorway macrocells, $r_0\!=\! 1$ km. In macrocells, we expect higher speeds, thereby lower intensity $\lambda$ and larger tracking distances $c$.[]{data-label="fig:SkewnProp"}](SkewnPropA.eps){width="3in"}
The above properties can be observed in Fig. \[fig:SkewnProp\], where we have simulated the skewness for different cell size $r_0$, pathloss exponent $\eta$ and traffic parameters $\left\{\lambda, c\right\}$, with respect to the product $\lambda c$. We see that for the considered range of $\lambda c$, the approximations for the second- and the third-order correlation, $\rho^{\left(2\right)}\!\left(x,y\right), \rho^{\left(3\right)}\!\left(x,y,z\right)$ do not introduce practically any error as compared to the simulations. In addition, the approximation given in Lemma \[lemma:1\] is quite accurate for small $\lambda c$. While changing from the microcell to macrocell scenario, we have the interplay of two conflicting factors: On one hand, the intensity of vehicles decreases to account for the higher speed of vehicles, and this increases the skewness. On the other hand, the cell size increases which reduces the skewness. For the selected parameter values of Fig. \[fig:SkewnProp\] the skewness reduces because, according to Lemma \[lemma:1\], it is proportional to $\frac{1}{\sqrt{\lambda r_0}}$.
For a bounded pathloss model, the interference distribution strongly depends on the fading process [@Kountouris2014]. In our system set-up we note: (i) the positive skewness of interference, and (ii) the guard zone around the receiver which essentially bounds the pathloss model, along with the exponential for the power fading. The gamma has a positive skewness, and it includes the exponential as a special case. The parameters $k,\beta$ of the gamma , $f_{\mathcal{I}}\!\left(x\right) \approx \frac{x^{k-1} e^{-x/\beta}} {\Gamma\left(k\right) \beta^k}$, can be computed by matching two moments, the mean and the variance approximation in , resulting to $k\!=\!\frac{\mathbb{E}\left\{\mathcal{I}\right\}^2} {\mathbb{V}\left\{\mathcal{I}\right\}}$ and $\beta\!=\!\frac{1}{k}$. The skewness of the gamma distribution is $\frac{2}{\sqrt{k}}$. For practical values of the pathloss exponent $\eta\!\in\!\left[2,6\right]$ and $\lambda c\!<\!\frac{1}{2}$, one can verify that the skewness, $\frac{2}{\sqrt{k}}$, is less than the approximation given in Lemma \[lemma:1\]. The shifted-gamma , which matches also the skewness of interference, is expected to provide better fit than the gamma . $$f_{\mathcal{I}}\!\left(x\right) \approx \frac{\left(x\!-\!\epsilon\right)^{k-1} e^{-\left(x-\epsilon\right)/\beta}} {\Gamma\left(k\right) \beta^k}, \, x\!\geq\! \epsilon,$$ where $k\!=\!\frac{4}{\mathbb{S}\left\{\mathcal{I}\right\}^2}, \beta\!=\!\sqrt{\frac{\mathbb{V}\left\{\mathcal{I}\right\}}{k}}$ and $\epsilon\!=\! \mathbb{E}\!\left\{\mathcal{I}\right\}\!-\!k\beta$.
The approximation accuracy of the gamma and the shifted-gamma is illustrated in Fig. \[fig:InterfDistr\] for two intensities $\lambda$, and $\lambda c\!=\!0.4$. For fixed $\lambda c$, a higher intensity of vehicles paired with a lower tracking distance can be associated with driving at lower speeds. The simulated standard deviation and skewness, along with their approximations, are included in Table \[table1\]. We see in the table that: (i) the variance approximation in is quite accurate, (ii) Lemma \[lemma:1\] estimates the skewness better than the gamma distribution, $\frac{2}{\sqrt{k}}$, and (iii) the has higher variance and skewness than the hardcore process, justifying the approximations in and Lemma \[lemma:1\]. In both cases, the estimates the skewness (in an absolute sense) better than Lemma \[lemma:1\] because the value of $\lambda c\!=\!0.4$ is not close to zero. Nevertheless, the gives much worse estimates for the standard deviation (see Table \[table1\]) and the interference distribution (see Fig. \[fig:InterfDistr\]) than the gamma approximations.
simulations gamma shifted-gamma PPP
------------------------- ------------- ---------- --------------- ----------
sd., $\lambda=0.1$ $0.0024$ $0.0023$ $0.0023$ $0.0028$
skewn., $\lambda=0.1$ $0.60$ $0.46$ $0.53$ $0.66$
sd., $\lambda=0.025$ $0.0012$ $0.0012$ $0.0012$ $0.0014$
skewn., $\lambda=0.025$ $1.27$ $0.93$ $1.06$ $1.32$
: Standard deviation (sd) and skewness of interference for a hardcore process with $\lambda c\!=\!0.4$ obtained by simulations, and estimated using and Lemma \[lemma:1\].[]{data-label="table1"}
For fixed $\lambda c$, the skewness and the are both proportional to $\frac{1}{\sqrt{\lambda}}$. We see in Fig. \[fig:InterfDistr\] that for the higher intensity of vehicles, $\lambda\!=\!0.1$, the interference distribution becomes more concentrated and less skewed, and the gamma approximation provides a very good fit. For a lower intensity of vehicles, $\lambda\!=\!0.025$, the skewness and the of interference increase, and three moments clearly provide a better fit than two. Also, both figures indicate that the will underestimate the of the at the tails.
Probability of outage {#sec:Outage}
=====================
Under Rayleigh fading over the transmitter-receiver link, the probability of outage, ${\text{P}_{\text{out}}}\!\left(\theta\right) \!=\! \mathbb{P}\!\left({\text{SIR}}\!\leq\!\theta\right)$, becomes equal to the complementary Laplace transform of the interference distribution. Even though the interference is unknown, its Laplace transform could be computed provided that the of the hardcore process was available. Unfortunately, this is not the case. Thanks to [@Stucki2014 Theorem 2.1], we deduce that the outage probability due to a of intensity $\lambda$ is actually a lower bound to the outage probability due to the hardcore process of equal intensity$\!\!$ [^3]. $$\label{eq:LowerBound}
\begin{array}{ccl}
{\text{P}_{\text{out}}}\!\left(\theta\right) \!\!\!&\stackrel{(a)}{=}&\!\!\! \displaystyle 1 \!-\! \mathbb{E}_{\rm{x}}\left\{\prod\nolimits_k\frac{1}{1\!\!+\!s\, x_k^{-\eta}} \right\} \\ \!\!\!&\stackrel{(b)}{\geq}&\!\!\! \displaystyle 1\!-\!e^{-2 \lambda\int_{r_0}^\infty\left(1-\frac{1}{1+s x^{-\eta}}\right){\rm d}x} = {\text{P}_{\text{out}}^{\text{ppp}}}\!\left(\theta\right),
%\\ \!\!\!&=&\!\!\! \displaystyle 1-\exp\!\left(-2\lambda\left(\frac{\pi}{\eta}s^{1/\eta}\csc\left(\frac{\pi}{\eta}\right)-r_0 \, {}_2F_1\!\left(1,\frac{1}{\eta},1\!+\!\frac{1}{\eta},-\frac{r_0^\eta}{s}\right)\right)\right),
\end{array}$$ where $\theta$ is the threshold, $s\!=\!\frac{\theta }{P_r}$, $(a)$ follows from exponentially interfering fading channels, $(b)$ from the of along with [@Stucki2014 Theorem 2.1], and the integral in the exponent can be expressed in terms of the ${}_2F_1$ Gaussian hypergeometric function [@Abramo p. 556].
An upper bound to the probability of outage can be obtained using the Jensen’s inequality. This is roughly as tight as the lower bound using the . The upper bound suggested in [@Stucki2014 equation (2.8)], which is essentially a first-order expansion of the around $s\!\rightarrow\! 0$, is tight only for small $\theta$. $$\begin{array}{ccl}
{\text{P}_{\text{out}}}\!\left(\theta\right) \!\!\!&=&\!\!\! \displaystyle 1 -
\mathbb{E}_{\rm{x}}\left\{ e^{-\sum_k\log\left(1+s x_k^{-\eta}\right)}\right\} \\ \!\!\!&\stackrel{(a)}{\leq}&\!\!\! \displaystyle 1 - \exp\!\left(-\mathbb{E}_{\rm{x}}\left\{\sum\nolimits_k\log\left(1\!+\!s x_k^{-\eta}\right)\right\}\right) \\ \!\!\!&\stackrel{(b)}{=}&\!\!\! \displaystyle 1-\exp\!\left(-2\lambda\!\!\int_{r_0}^\infty\!\!\!\log\!\left(1\!+\!s x^{-\eta}\right){\rm d}x\right) \!=\! {\text{P}_{\text{out}}^{\text{Jen}}}\!\left(\theta\right),
%\\ \!\!\!&=&\!\!\! \displaystyle 1 - \exp\!\left(-2\lambda r_0\left(\frac{\eta s r_0^{-\eta}}{\eta-1}{}_2F_1\!\left(\!1,\!1\!-\!\frac{1}{\eta},\!2\!-\!\frac{1}{\eta};\!-s r_0^{-\eta}\!\right) - \log\!\left(1+sr_0^{-\eta}\right) \right) \right),
\end{array}$$ where $(a)$ is due to Jensen’s inequality, $(b)$ follows from the Campbell’s theorem and the integral in the exponent can be expressed in terms of the ${}_2F_1$ function.
The gamma approximations for the of interference studied in the previous section have simple Laplace transforms, and they can be used to generate simple approximations for the outage probability. $$\label{eq:OutageGamma}
\begin{array}{ccl}
{\text{P}_{\text{out}}}\!\left(\theta\right) \!\!\!&\approx&\!\!\! \displaystyle 1 - \left(1+s \beta\right)^{-k} = {\text{P}_{\text{out}}^{\text{g}}}\!\left(\theta\right), \\
{\text{P}_{\text{out}}}\!\left(\theta\right) \!\!\!&\approx&\!\!\! \displaystyle 1-e^{-s \epsilon} \left(1+s\beta\right)^{-k} = {\text{P}_{\text{out}}^{\text{sg}}}\!\left(\theta\right).
\end{array}$$
We see in Fig. \[fig:JensenPPPGammas\] that the bounds, ${\text{P}_{\text{out}}^{\text{ppp}}}\!\left(\theta\right)$ and ${\text{P}_{\text{out}}^{\text{Jen}}}\!\left(\theta\right)$, are tight in the body of the distribution, but they start to fail in the upper tail. Their error is more prominent in microcells and macrocells with a low intensity of vehicles. Recall that smaller cell sizes $r_0$ and lower intensities $\lambda$ are associated with higher and skewness for the interference distribution. According to and Lemma \[lemma:1\], for a fixed $\lambda c$, the absolute prediction error of increases for lower $\left\{\lambda,r_0\right\}$, and subsequently, the induced approximation errors for the interference distribution and the outage probability would be higher. We claim that the cannot always describe accurately the outage probability of a link in a field of interferers with hardcore headway distance. We will illustrate next that for temporal performance metrics and multiple antennas at the receiver the accuracy worsens, while the gamma approximations can be used to generate quite good performance predictions in all cases.
Applications {#sec:Applications}
============
The two deployment models (hardcore vs. ) induce different interference correlation over time and space. We will use the mean local delay to describe the temporal performance of the link, and a dual-branch receiver for the spatial performance. For notational brevity, we will use the gamma approximation for the distribution of interference.
Temporal performance
--------------------
The mean local delay is defined as the average number of transmissions required for successful reception. For a mobility model introducing correlations in the locations of interferers over time, it is challenging to calculate it. For $T$ consecutive transmissions, the joint $T-$th dimensional of interference, with correlated marginals, would be needed. An alternative study that can provide us with some insight, see for instance [@Haenggi2013b], investigates the properties of delay under (i) i.i.d. locations, and (ii) static interferers over time. The performance can be associated with scenarios characterized by very high and very low mobility of interferers respectively.
For locations, the mean delay is equal to the inverse of the probability of successful reception. For the , one may take the complementary of the last line of and invert it. According to , the sets a lower bound to the mean delay with locations of interferers. For the hardcore process, the mean delay would be approximated by $\left(1\!+\!s \beta\right)^k$, see . Since $k,\beta$ are positive, a loose upper bound can be set using the Bernoulli inequality, $\left(1\!+\!s \beta\right)^k\!\leq\!e^{s k \beta}\!=\!e^{s\,\mathbb{E}\!\left\{\mathcal{I}\right\}}$.
In order to calculate the mean delay with static interferers, one has to invert the probability of successful reception conditioned on the realization of interferers, then average over their locations [@Haenggi2013b]. The mean delay with Poisson interferers accepts an elegant form for continuous transmissions, $\mathbb{E}\!\left\{D\right\}\!=\!e^{s\, \mathbb{E}\left\{\mathcal{I}\right\}}$, which follows from substituting $p\!=\!1, q\!=\!0$ in [@Haenggi2013b Lemma 2]. In order to overcome the lack of the for the hardcore process, we use an alternative expression for the mean delay, $\mathbb{E}\!\left\{D\right\}\!=\!\sum\nolimits_{T=1}^\infty \!\! {\text{P}_{\text{out}}}\!\left(T\right)$ [@Haenggi2013c Section V-B], where ${\text{P}_{\text{out}}}\!\left(T\right)$ is the joint outage probability over $T$ consecutive time slots. $$\begin{array}{ccl}
\text{P}_{\text{out}}\!\left(T\right) \!\!\!&=&\!\!\! \displaystyle \mathbb{P}\!\left(\text{SIR}_1\!\leq\!\theta,\text{SIR}_2\!\leq\!\theta,\ldots \text{SIR}_T\!\leq\!\theta\right) \\ \!\!\!&=&\!\!\! \displaystyle \mathbb{E}\left\{\left(1\!-\!e^{-s \mathcal{I}_1}\right)\left(1\!-\!e^{-s \mathcal{I}_2}\right)\ldots \left(1\!-\!e^{-s \mathcal{I}_T}\right)\right\} \\ \!\!\!&=&\!\!\! \displaystyle 1 \!+\! \sum\limits_{t=1}^T \left(-1\right)^t \binom{T}{t} \mathbb{E}\!\left\{ e^{-s \sum\nolimits_{j=1}^t \mathcal{I}_j}\right\} \\ \!\!\!&=&\!\!\! \displaystyle 1 \!+\! \sum\limits_{t=1}^T \left(-1\right)^t \binom{T}{t} \mathbb{E}\!\left\{ e^{-s \sum\limits_{k\in\Phi}\sum\nolimits_{j=1}^t h_{k,j}g\left(x_k\right)}\right\} \\ \!\!\!&=&\!\!\! \displaystyle 1 + \sum\limits_{t=1}^T \left(-1\right)^t \binom{T}{t} \mathbb{E}\!\left\{ e^{-s \sum\limits_{k\in\Phi} h_k\left(t\right)g\left(x_k\right)}\right\},
\end{array}$$ where $\text{SIR}_j$ and $\mathcal{I}_j$ describe the and the instantaneous interference respectively over the $j-$th time slot, and the $h_k\!\left(t\right)\!=\!\sum_{j=1}^t h_{k,j}$, as a sum of exponential follows the gamma distribution.
We deduce that the calculation of the joint Laplace functional over $t$ slots with static interferers is equivalent to the calculation of the Laplace transform of interference for a single time instance, but with a different fading distribution. The first two moments of the $h_k\!\left(t\right)\!=\!h\!\left(t\right)\, \forall k$ are $\mathbb{E}\!\left\{h\!\left(t\right)\right\} \!=\! t$ and $\mathbb{E}\!\left\{h^2\!\left(t\right)\right\} \!=\! t\left(1\!+\!t\right)$. We will still utilize the gamma approximation for the interference, but the fading is now modeled by a gamma instead of an exponential . We have the same simple expression for the Laplace transform $\left(1\!+\!s\beta\!\left(t\right)\right)^{-k\left(t\right)}$, where the parameters $k,\beta$ now depend on $t$. Without showing the derivation details, the mean, and the variance of interference in the presence of Nakagami fading modeled by a gamma with shape $t$ and scale unity are $$\label{eq:MeanVarT}
\begin{array}{ccl}
\mathbb{E}\!\left\{ \mathcal{I}\!\left(t\right) \right\} \!\!\!\!\!&=&\!\!\!\!\! \displaystyle \frac{2\lambda r_0^{1-\eta} t}{\eta-1}\\
\mathbb{V}\!\left\{ \mathcal{I}\!\left(t\right) \right\} \!\!\!\!\!&\approx&\!\!\!\!\! \displaystyle \frac{2\lambda r_0^{1-2\eta} t \left(1\!+\!t\left(1\!-\!\lambda c\right)^2 \right)}{2\eta\!-\!1}.
%\\ \mathbb{S}\!\left\{ \mathcal{I}\!\left(t\right) \right\} \!\!\!&\approx&\!\!\! \displaystyle \frac{2\lambda r_0^{1-3\eta} t}{3\eta-1} \left( \frac{2\lambda r_0^{1-2\eta} t \left(1+t\right)}{2\eta-1}\right)^{-3/2} \left( \left(1+t\right)\left(2+t\right)-3\lambda c \, t^2 \right).
\end{array}$$
Finally, the mean delay can be read as $$\begin{array}{ccl}
\mathbb{E}\!\left\{D\right\} \!\!\!\!\!\!&\approx&\!\!\!\!\!\! \displaystyle \sum_{T=0}^\infty \sum_{t=0}^T \left(-1\right)^t \binom{T}{t} \left(1+s \beta\!\left(t\right) \right)^{-k\left(t\right)},
%\\ \mathbb{E}\!\left\{D\right\} \!\!\!&\approx&\!\!\! \displaystyle \sum_{T=0}^\infty \sum_{t=0}^T \left(-1\right)^t \binom{T}{t} e^{s\, \epsilon\left(t\right)} \left(1+s \beta_{\text{sg}}\!\left(t\right) \right)^{-k_{\text{sg}}\!\left(t\right)}.
\end{array}$$ where $k\!\left(t\right), \beta\!\left(t\right)$ are derived via moment matching using .
The above approximation can be turned into a single sum by changing the order of the summations and setting a sufficient maximum value $T_0$ for the parameter $T$. $$\label{eq:MeanDelayGammas2}
\begin{array}{ccl}
\mathbb{E}\!\left\{D\right\} \!\!\!\!\!&\approx&\!\!\!\!\! \displaystyle \sum_{t=0}^\infty \sum_{T=t}^\infty \left(-1\right)^t \binom{T}{t} \left(1+s \beta\!\left(t\right) \right)^{-k\left(t\right)} \\ \!\!\!\!\!&=&\!\!\!\!\! \displaystyle \lim_{T_0\!\rightarrow\!\infty} \sum_{t=0}^{T_0} \sum_{T=t}^{T_0} \left(-1\right)^t \binom{T}{t} \left(1+s \beta\!\left(t\right) \right)^{-k\left(t\right)} \\ \!\!\!\!\!&=&\!\!\!\!\! \displaystyle \lim_{T_0\!\rightarrow\!\infty} \sum_{t=0}^{T_0} \left(-1\right)^t \binom{T_0+1}{t+1} \left(1+s \beta\!\left(t\right) \right)^{-k\left(t\right)}.
\end{array}$$
Since it is not realistic to assume very low mobility across macrocells, we depict in Fig. \[fig:MeanDelay\] the mean delay for a microcell; the associated outage probabilities are shown in Fig. \[fig:JensenPPPGammasA\]. We observe that the interference field due to the hardcore process induces a much smaller increase in the mean delay in comparison with , as we move from extreme mobile to static interferers. This is because the temporal correlation coefficient of interference due to a static is equal to $\frac{1}{2}$ [@Haenggi2009], while that due to the hardcore process is lower, and approximately $\frac{1}{2}\left(1\!-\!\lambda c\right)$ [@Koufos2018b]. Due to the lower correlation of interference, less retransmissions are needed to meet the target, and the mean delay decreases in comparison with that due to static $\ac{PPP}$. The approximation error of in networks with low mobility blows up in the high reliability regime.
Spatial performance
-------------------
The probability of successful reception for with dual-branch receiver in the presence of spatially correlated interference requires the with respect to the reduced Palm measure [@Tanbourgi2014 equation (25)]. In a Poisson field, due to the Slivnyak’s theorem, this is available, and the performance has been derived in [@Tanbourgi2014 equation (26)]. Unfortunately, in our case, we will need again approximations about the distribution of interference in the two branches and their correlation. We will end up with a simple approximation for the outage probability, while the calculation in [@Tanbourgi2014 equation (26)] requires the numerical computation of three integrals.
Let us denote by $\mathcal{I}_1\!=\!\sum\nolimits_i h_{1,i}g\left(x_i\right)$ and $\mathcal{I}_2\!=\!\sum\nolimits_i h_{2,i}g\left(x_i\right)$, the instantaneous interference, and by $\mathbf{I}$ the vector of $\mathcal{I}_1,\mathcal{I}_2$. Treating the interference as white noise, the post-combining becomes equal to the sum of the at the two branches. $$\mathbb{P}\!\left\{\text{SIR}\geq\theta\right\} = \mathbb{E}_{\mathbf{I}}\!\left\{ \mathbb{P}\left( \frac{h_{t,1}P_r}{\mathcal{I}_1}+\frac{h_{t,2}P_r}{\mathcal{I}_2}\geq\theta|\mathbf{I}\right)\right\}.$$
Let us denote by $W\!=\!\frac{h_{t,2}P_r}{\mathcal{I}_2}$ the describing the at the second branch. Conditioning on the realization $w$, and using that the fading channel is Rayleigh, we have $$\begin{array}{ccl}
\mathbb{P}\!\left\{\text{SIR}\geq\theta\right\} \!\!\!\!\!\!&=&\!\!\!\!\! \displaystyle \mathbb{E}_{\mathbf{I},W}\!\left\{e^{-s_1\mathcal{I}_1}\right\} \!= \mathbb{E}_{\mathbf{I}}\!\left\{\! \int_0^\infty \!\!\!\!\!\! e^{-s_1\mathcal{I}_1} f_{W|\mathcal{I}_2}\!\left(w\right) \!{\rm d}w \right\}\!,
\end{array}$$ where $s_1\!=\!\frac{\max\left\{0,\theta-w\right\}}{P_r}$ and $f_{W|\mathcal{I}_2}$ is the conditional of the at the second branch.
Due to the fact that the fading channel is Rayleigh, $\mathbb{P}\left(W\!\geq\!w|\mathcal{I}_2\right)=e^{-s_2 \mathcal{I}_2}$, where $s_2\!=\!\frac{w}{P_r}$. By differentiation, $f_{W|\mathcal{I}_2}\!\left(w\right)\!=\!\frac{\mathcal{I}_2}{P_r}e^{-s_2 \mathcal{I}_2}$. Therefore $$\label{eq:MRC}
\begin{array}{ccl}
\mathbb{P}\!\left\{\text{SIR}\geq\theta\right\} \!\!\!\!\!\!&=&\!\!\!\!\!\! \displaystyle \frac{1}{P_r} \int_0^\infty \mathbb{E}_{\mathbf{I}}\!\left\{ \mathcal{I}_2 e^{-s_1\mathcal{I}_1} e^{-s_2w \mathcal{I}_2} \right\} {\rm d}w \\ \!\!\!\!\!\!&\stackrel{(a)}{=}&\!\!\!\!\!\! \displaystyle \frac{1}{P_r} \int_0^\theta \mathbb{E}_{\mathbf{I}}\!\left\{ \mathcal{I}_2 e^{-s_1\mathcal{I}_1} e^{-s_2 \mathcal{I}_2} \right\} {\rm d}w \, + \\ & & \displaystyle \frac{1}{P_r} \int_\theta^\infty \mathbb{E}_{\mathbf{I}}\!\left\{ \mathcal{I}_2 e^{-s_2 \mathcal{I}_2} \right\} {\rm d}w,
\end{array}$$ where $(a)$ follows from $s_1\!=\!0$ for $w\!>\!\theta$.
We will assume that the random vector $\mathbf{I}$ follows the bivariate gamma distribution with identical marginals following the gamma distribution with parameters $\left\{k,\beta\right\}$ calculated in Section \[sec:Outage\]. The correlation coefficient is denoted by $\rho$. Using the differentiation property of the Laplace transfom, the first expectation in , $\mathcal{J}\!=\!\mathbb{E}_{\mathbf{I}}\!\left\{ \mathcal{I}_2 e^{-s_1\mathcal{I}_1} e^{-s_2 \mathcal{I}_2} \right\}$, becomes $$\label{eq:LapApp1}
\begin{array}{ccl}
\mathcal{J} \!\!\!&\approx&\!\!\! \displaystyle -\frac{\partial}{\partial s_2}\left\{\left(1\!+\!s_1\beta\!+\!s_2\beta\!+\!s_1s_2\beta^2\left(1\!-\!\rho\right) \right)^{-k}\right\} \\ \!\!\!&=&\!\!\! \displaystyle \frac{k\beta\left(1+\beta s_1\left(1-\rho\right)\right)}{ \left(1+s_1\beta+s_2\beta+s_1s_2\beta^2\left(1-\rho\right) \right)^{k+1}}.
\end{array}$$
The second expectation in can be approximated as $$\label{eq:LapApp2}
\mathbb{E}_{\mathbf{I}}\!\left\{ \mathcal{I}_2 e^{-s_2 \mathcal{I}_2} \right\} \approx k\beta\left(1+s_2\beta\right)^{-k-1}.$$
After substituting and into , cancelling out some terms and carrying out the integration with respect to $w$ for $w\!>\!\theta$, we end up with $$\label{eq:MRC2}
\begin{array}{ccl}
\mathbb{P}\!\left\{\text{SIR}\geq\theta\right\} \!\!\!&=&\!\!\! \displaystyle P_r^k \left(P_r+\theta\beta\right)^{-k} + k\beta P_r^{2k}\, \times \\ \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!& &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle \int_0^\theta \!\!\!\! \frac{1+\beta \left(\theta-w\right)\left(1-\rho\right)\, {\rm d}w}{\left(P_r^2 \!+\! \theta\beta P_r \!+\! \left(\theta-w\right)w\beta^2\left(1\!-\!\rho\right) \right)^{k+1}}.
\end{array}$$
The above integral can be expressed in terms of ${}_2F_1$.
\[lemma:2\] For Rayleigh fading channels, the spatial correlation coefficient of interference $\rho$ between the two antennas can be approximated as $\rho\!\approx\!\frac{1}{2}\left(1-\lambda c\right)$. The approximation is valid for $\lambda c\!\rightarrow\!0$ and $\frac{c}{r_0}\!\rightarrow\!0$.
Under the assumption of Rayleigh fading at the two antennas, the covariance of interference is $$\begin{array}{ccl}
{\text{cov}}\!\left\{\mathcal{I}\right\} \!\!\!\!\!\!\!&=&\!\!\!\!\! \displaystyle \mathbb{E}\!\left\{\!h\!\right\}^{\!2}\! \mathbb{E}\!\left\{\!\sum\limits_{x\in\Phi}\!\! g^2\!\!\left(\!x\!\right)\!\right\} \!\!+\! \mathbb{E}\!\left\{\!h\!\right\}^{\!2}\! \mathbb{E}\!\left\{\!\sum\limits_{x,y\in\Phi}^{x\neq y} \!\!\! g\!\left(\!x\!\right) g\!\left(\!y\!\right)\!\right\} \!-\! \mathbb{E}\!\left\{\mathcal{I}\right\}^{\!2} \\ \!\!\!\!\!\!&=&\!\!\!\!\!\! \displaystyle \frac{2\lambda r_0^{1-2\eta}}{2\eta-1} \!+\! \int \! g\!\left(x\right) g\!\left(y\right) \rho^{\left(2\right)}\!\left(x,y\right) {\rm d}x {\rm d}y \!-\! \mathbb{E}\!\left\{\mathcal{I}\right\}^2.
\end{array}$$
The variance of interference is [@Koufos2018 equation (3)] $$\begin{array}{ccl}
{\text{V}}\!\left\{\mathcal{I}\right\} \!\!\!\!\!\!\!&=&\!\!\!\!\! \displaystyle \mathbb{E}\!\left\{\!h^2\!\right\}\!\mathbb{E}\!\left\{\!\sum\limits_{x\in\Phi}\!\! g^2\!\!\left(\!x\!\right)\!\right\} \!+\! \mathbb{E}\!\left\{\!h\!\right\}^{\!2}\!\mathbb{E}\!\left\{\!\!\sum\limits_{x,y\in\Phi}^{x\neq y} \!\!\! g\!\left(\!x\!\right) g\!\left(\!y\!\right)\!\!\right\} \!-\! \mathbb{E}\!\left\{\mathcal{I}\right\}^{\!2} \\ \!\!\!\!\!\!&=&\!\!\!\!\!\! \displaystyle \frac{4\lambda r_0^{1-2\eta}}{2\eta-1} \!+\! \int \! g\!\left(x\right) g\!\left(y\right) \rho^{\left(2\right)}\!\left(x,y\right) {\rm d}x {\rm d}y \!-\! \mathbb{E}\!\left\{\mathcal{I}\right\}^{\!2}\!.
\end{array}$$
The integral $S\!=\!\int g\!\left(x\right) g\!\left(y\right) \rho^{\left(2\right)}\!\left(x,y\right) {\rm d}x {\rm d}y$ has been approximated in [@Koufos2018 Section V] for $\lambda c\!\rightarrow\! 0$ and $\frac{c}{r_0}\!\rightarrow\! 0$. The first two dominant terms with respect to $r_0$ are $$S \approx \frac{4\lambda^2 r_0^{2-2\eta}}{\left(\eta-1\right)^2} - \frac{4\lambda^2 c r_0^{1-2\eta}}{2\eta-1} + \frac{2\lambda^3 c^2 r_0^{1-2\eta}}{2\eta-1}.$$
After substituting the above approximation for $S$ in the expressions of the covariance and the variance, doing some factorization and cancelling out common terms, the correlation coefficient can be approximated as $$\rho = \frac{{\text{cov}}\!\left\{\mathcal{I}\right\}}{{\text{V}}\!\left\{\mathcal{I}\right\}} \approx \frac{\left(1-\lambda c\right)^2}{2-2\lambda c + \lambda^2 c^2} \stackrel{\lambda c \rightarrow 0}{\approx} \frac{1}{2}\left(1-\lambda c\right),$$ and the Lemma is proved.
In Fig. \[fig:MRC\] we depict the outage probability with dual-branch . The performance prediction of worsens in comparison with single-antenna receiver, and it is expected to deteriorate with more antennas and temporal performance metrics, e.g., mean local delay, in networks with low mobility. Even though the still gives very accurate predictions for the outage probability with dual-antenna in the lower tail, these predictions are not necessarily a bound to the outage probability due to the hardcore process. This is because the lower correlation of interference associated with the hardcore process, see Lemma \[lemma:2\], enhances the performance in comparison with the . This is not visible in Fig. \[fig:MRC\] because only two antennas are employed, but preliminary simulations with $8-$antenna indicated that the hardcore process achieves lower outage than in the lower tail. Overall the use of becomes limited with . Despite the approximations involved in the derivation of , it fits quite well the simulations and can be used to get a quite good performance estimate with low computational complexity.
Conclusions {#sec:Conclusions}
===========
The model for vehicular networks allows small inter-vehicle distances with high probability. This is unrealistic in roads with few number of lanes. A more realistic point process, of equal intensity but with a hardcore distance (shifted-exponential inter-arrivals), changes the properties of interference distribution and increases the outage probability for a link at the origin. The discrepancy in the outage probability predicted by the two models is clear when the coefficient of variation and the skewness of interference are high, e.g., in urban street microcells and motorway macrocells with sparse flows of vehicles. The discrepancy increases if we consider multiple antennas at the receiver because in that case, the spatial correlation of interference, which is different under the two deployment models, starts also to have an effect. Temporal performance indicators associated with the performance of retransmission schemes are affected by the correlation properties of interference too. The may largely overestimate the mean delay for the link at the origin under low mobility of interferers. With high mobility, the temporal correlation of interference under both models vanishes, and the underestimates the mean delay. Even if the of the hardcore point process is not available, a gamma approximation for the interference distribution gives better performance predictions than the in all cases. In this paper, we assumed that the point process impacts the distribution of interferers while the transmitter-receiver link is fixed and known. It would be interesting to use a random link distance, and investigate whether the horizontal deployment gain for non-Poissonian point processes holds, see [@Haenggi2015].
appendix {#appendix .unnumbered}
========
The third moment of interference accepts contributions from a single user, from user pairs and also from triples of users. $$\label{eq:ThirdMomInterf}
\begin{array}{ccl}
\mathbb{E}\!\left\{\mathcal{I}^3\right\} \!\!\!\!\!&=&\!\!\!\!\! \displaystyle \mathbb{E}\!\left\{\!h^3\!\right\}\!\lambda\!\! \int \!\!\!g^3\!\left(x\right) {\rm d}x \, + \\ & & \displaystyle 3\mathbb{E}\!\left\{\!h^2\!\right\}\!\!\int\!\!\! g^2\!\left(x\right) g\!\left(y\right) \rho^{(2)}\!\left(x,y\right) \! {\rm d}x{\rm d}y + \\ & & \displaystyle \int\!\!\! g\!\left(x\right)g\!\left(y\right)g\!\left(z\right) \rho^{(3)}\!\left(x,y,z\right) \! {\rm d}x{\rm d}y{\rm d}z \\ \!\!\!\!\!&=&\!\!\!\!\! \displaystyle 6 \lambda\!\! \int \!\!\!g^3\!\left(x\right) \!{\rm d}x \!+\!6\!\!\int\!\!\! g^2\!\left(x\right)\! g\!\left(y\right) \rho^{(2)}\!\left(x,y\right) \! {\rm d}x{\rm d}y + \\ & & \displaystyle \int\!\!\! g\!\left(x\right)\! g\!\left(y\right) \! g\!\left(z\right) \rho^{(3)}\!\left(x,y,z\right) \! {\rm d}x{\rm d}y{\rm d}z \\ \!\!\!&=&\!\!\! \displaystyle \frac{12\lambda r_0^{1-\eta}}{\eta-1} \!+\!6\!\!\int\!\!\! g^2\!\left(x\right)\! g\!\left(y\right) \rho^{(2)}\!\left(x,y\right) \! {\rm d}x{\rm d}y \, + \\ & & \displaystyle \int\! g\!\left(x\right)\! g\!\left(y\right) \! g\!\left(z\right) \rho^{(3)}\!\left(x,y,z\right) \! {\rm d}x{\rm d}y{\rm d}z,
\end{array}$$ where $\mathbb{E}\!\left\{h^3\right\}\!=\!6$ and $\mathbb{E}\!\left\{h^2\right\}\!=\!2$ for an exponential , and we have scaled the second term by three to count the ways to select a user pair out of a triple of users.
The contributions to the third moment from triples of users involve the third-order correlation in . We will apply in both the approximation adopted in [@Koufos2018]. Similar to [@Koufos2018], we will also assume that the guard zone is much larger than the tracking distance, $r_0\!\gg\!c$. In our approximations, we will keep up to the second order terms, and also the dominant $r_0$ terms with exponents larger or equal to $\left(1\!-\!3\eta\right)$.
Using the approximation for the beyond $2c$, the term $S'\!=\!\iint\!\! x^{-2\eta}y^{-\eta} \rho^{(2)}\!\left(x,y\right) {\rm d}x{\rm d}y$ can be read as $$\label{eq:Sprim0}
\begin{array}{ccl}
S' \!\!\!\!\! &\approx& \!\!\!\!\! \displaystyle 2\lambda\mu\int_{r_0}^\infty\!\int_{x+c}^{x+2c}\!\frac{x^{-2\eta}y^{-\eta}}{e^{\mu\left(y-x-c\right)}} {\rm d}y{\rm d}x \, + \\ & & \displaystyle 2\lambda\mu\int_{r_0}^\infty\!\int_{x-2c}^{x-c}\!\frac{x^{-2\eta} g\!\left(y\right)}{e^{\mu\left(x-y-c\right)}} {\rm d}y{\rm d}x \,\, + \\ & & \displaystyle \,\,\, 2\lambda^2\int_{r_0}^\infty\!\int_{x+2c}^\infty \! x^{-2\eta}y^{-\eta}{\rm d}y{\rm d}x \,\,\,\,\,\, + \\ & & \displaystyle \,\,\,\,\,\, 2\lambda^2\int_{r_0}^\infty\!\int_{-\infty}^{x-2c}\! x^{-2\eta}g\!\left(y\right) {\rm d}y {\rm d}x,
\end{array}$$ where the factor $2$ in front of the integrals accounts for $x\!<\!\!-\!r_0$.
The contribution to $S'$ due to pairs at distances larger than $2c$ is the last two lines of . $$\begin{array}{ccl}
S'_{>2c} \!\!\!\!\!&=&\!\!\!\!\! \displaystyle 2\lambda^2 \!\! \int_{r_0}^\infty\!\!\!\int_{x+2c}^\infty \!\!\! x^{-2\eta}y^{-\eta}{\rm d}y{\rm d}x \,\,\, + \\ \!\!\!\!\!& &\!\!\!\!\! \displaystyle 2\lambda^2 \!\! \int\limits_{r_0}^\infty\!\int\limits_{-\infty}^{-r_0}\!\!\! x^{-2\eta}\left|y\right|^{-\eta}\!{\rm d}y {\rm d}x +\!\!\!\! \int\limits_{r_0+2c}^\infty\!\int\limits_{r_0}^{x-2c}\!\!\! x^{-2\eta}y^{-\eta} {\rm d}y {\rm d}x.
\end{array}$$
The first and the last term in the above expression are not equal due to asymmetry in the exponents of $x$ and $y$. After integrating and adding up the three terms we end up with $$\begin{array}{ccl}
S'_{>2c} \!\!\!\!\! &=& \!\!\!\!\! \displaystyle \frac{2\lambda^2\left(r_0^{2-3\eta}\!+\!r_0^{1-\eta}\left(2c\!+\!r_0\right)^{1-2\eta}\right)}{\left(\eta-1\right)\left(2\eta-1\right)} \,\,\, + \\ \!\!\!\!\!\!\!& &\!\!\!\!\!\!\! \displaystyle \frac{2\lambda^2 r_0^{2-3\eta}}{\left(3\eta\!-\!2\right) \left(\eta\!-\!1\right)} \Bigg({}_2F_1\!\left(\!3\eta\!-\!2,\eta\!-\!1,3\eta\!-\!1,\!-\frac{2c}{r_0} \!\right) \!- \\ & & \displaystyle \,\,\,\,\,\, {}_2F_1\!\left(3\eta\!-\!2,2\eta,3\eta\!-\!1,-\frac{2c}{r_0} \right)\Bigg) \\ \!\!\!\!\!&\stackrel{(a)}{\approx}&\!\!\!\!\! \displaystyle \frac{4\lambda^2 r_0^{2-3\eta}}{\left(2\eta\!-\!1\right)\left(\eta\!-\!1\right)} - \frac{8 r_0^{1-3\eta} \lambda^2 c}{3\eta-1},
\end{array}$$ where $(a)$ follows from expanding $b\!=\!\frac{c}{r_0}\!\rightarrow\! 0$.
The contribution to $S'$ due to pairs of vehicles at distances $\left|y\!-\!x\right|$ smaller than $2c$ is larger for $y\!>\!x$ (there are no vehicles to filter out inside the cell in that case) than it is for $y\!<\!x$. Nevertheless, for a small $\frac{c}{r_0}$, the two integrals in the first line of should be approximately equal. By making this assumption, we can avoid the approximation of the integral for $y\!<\!x$, which is a little more tedious because for $x\!\in\!\left(r_0,r_0\!+\!2c\right)$ we have to exclude the vehicles inside the cell. Finally, we can approximate the term $S'_{<2c}$ as $$\label{eq:Sprim}
\begin{array}{ccl}
S'_{<2c} \!\!\! &\approx& \!\!\! \displaystyle 4\lambda\mu \int_{r_0}^\infty\!\!\int_{x+c}^{x+2c}\!\! \frac{x^{-2\eta}y^{-\eta}}{e^{\mu\left(y-x-c\right)}} {\rm d}y{\rm d}x.
\end{array}$$
After integrating with respect to $y$ we get $$\begin{array}{ccl}
S'_{<2c} \!\!\! &\approx& \!\!\! \displaystyle 4\lambda \mu^\eta\int_{r_0}^\infty\! x^{-2\eta} e^{\mu\left(c+x\right)} \Big(\Gamma\!\left(1\!-\!\eta,\mu\left(c\!+\!x\right)\right) - \\ & & \displaystyle \Gamma\!\left(1\!-\!\eta,\mu\left(2c\!+\!x\right)\right) \Big) {\rm d}x.
\end{array}$$
In order to approximate the above integral, we first expand the integrand for $\mu\left(c\!+\!x\right)\!\rightarrow\!\infty$. Due to the fact that $\mu\!\geq\!\lambda$ and $x\!\geq\!r_0$, the expansion is valid for $\lambda r_0\!\gg\! 1$, which is associated with a large number (on average) of vehicles inside the cell. $$\begin{array}{ccl}
S'_{<2c} \!\!\!&\approx&\!\!\! \displaystyle \frac{4\lambda e^{-c\mu}r_0^{-3\eta} \left(\eta\!-\!c\mu\!+\!c\eta\mu\!+\!e^{c\mu}\left(c\mu\!-\!\eta\right)\right)}{3\eta\mu} \times \\ & & \displaystyle \,\,\,\,\,\, {}_2F_1\!\left(3\eta,\eta\!+\!1,3\eta\!+\!1,-\frac{c}{r_0}\right) + \\ & & \displaystyle \frac{4\lambda r_0^{1-3\eta} \left(1\!-\!e^{-c\mu}\right){}_2F_1\!\left(3\eta\!-\!1,\eta\!+\!1,3\eta,-\frac{c}{r_0}\right)}{\left(3\eta-1\right)}.
\end{array}$$
After substituting $\mu\!=\!\frac{\lambda}{1\!-\!\lambda c}$, the above expression can be further approximated for $\lambda c\!\rightarrow\! 0$ and $\frac{c}{r_0}\!\rightarrow\! 0$. Keeping only the dominant term with respect to $r_0$, we end up with $$S'_{<2c} \approx \frac{4\lambda^2 c \, r_0^{1-3\eta}}{3\eta-1} + \frac{2\lambda^3 c^2 r_0^{1-3\eta}}{3\eta-1}.$$
After summing up $S'_{>2c}$ with $S'_{<2c}$ and scaling the result by six, see equation , we have $$\label{eq:SprimFinal}
6 S'\approx \frac{24\lambda^2 r_0^{2-3\eta}}{\left(2\eta\!-\!1\right)\left(\eta\!-\!1\right)} - \frac{24 \lambda^2 c \, r_0^{1-3\eta}}{3\eta-1} + \frac{12\lambda^3 c^2 r_0^{1-3\eta}}{3\eta-1}.$$
The calculation of $S''\!=\!\int\!\! x^{-\eta}y^{-\eta}z^{-\eta} \rho^{(3)}\!\left(x,y,z\right) {\rm d}x{\rm d}y{\rm d}z$ is more tedious than the calculation of $S'$ because it involves triples instead of pairs of users. It is shown in [@Koufos2018] that the contribution to $S''$ can be split into two parts: (i) $S_1''$ with the vehicles $x,y,z$ located at the same side with respect to the cell, and (ii) $S_2''$ with the location of one vehicle being uncorrelated to the locations of the other two because it is located at the opposite side of the cell. $S''\!=\!S_1''\!+\!S_2''$.
Using the common approximation for the and assuming the order $r_0\!<\!x\!<\!y\!<\!z$ for the three vehicles, the contribution to $S_1''$ can be divided into four terms describing the possible separation of distances between each pair of vehicles $\left\{x,y\right\}$ and $\left\{y,z\right\}$. $$\begin{array}{ccl}
S_1'' \!\!\!&\approx&\!\!\! \displaystyle \underbrace{12\lambda^2\mu \!\! \int_{r_0}^\infty\!\!\!\int_{x+2c}^\infty\!\int_{y+c}^{y+2c}\!\!\frac{z^{-\eta}y^{-\eta}x^{-\eta}}{e^{\mu\left(z-y-c\right)}}{\rm d}z {\rm d}y {\rm d}x}_{S''_{11}} + \\ & & \displaystyle \underbrace{12\lambda^3 \!\! \int_{r_0}^\infty\!\!\!\int_{x+2c}^\infty\!\int_{y+2c}^\infty\!\!\!\!\!\! z^{-\eta}y^{-\eta}x^{-\eta} {\rm d}z {\rm d}y {\rm d}x}_{S''_{12}} \, + \\ \!\!\!& &\!\!\! \displaystyle 12\lambda\mu^2 \!\! \int_{r_0}^\infty\!\!\!\int_{x+c}^{x+2c}\!\!\!\!\int_{y+c}^{y+2c}\!\!\frac{z^{-\eta}y^{-\eta}x^{-\eta}}{e^{\mu\left(z-x-2c\right)}}{\rm d}z {\rm d}y {\rm d}x \,\,\, + \\ & & \displaystyle 12\lambda^2\mu \!\!\int_{r_0}^\infty\!\!\!\int_{x+c}^{x+2c}\!\!\!\!\int_{y+2c}^\infty\!\!\frac{z^{-\eta}y^{-\eta}x^{-\eta}}{e^{\mu\left(y-x-c\right)}}{\rm d}z {\rm d}y {\rm d}x,
\end{array}$$ where the factor $12\!=\!6\times 2$ accounts for the six different orderings of $x,y,z$ and the factor two is added to account for the three users being at the negative half-axis.
The term $S''_{11}$ can be approximated as follows $$\begin{array}{ccl}
S_{11}'' \!\!\!\!\!\!\!&=&\!\!\!\!\!\!\! \displaystyle 12 \lambda^2\mu \!\! \int_{r_0}^\infty\!\!\!\int_{x+2c}^\infty\!\! x^{-\eta}y^{-\eta} e^w \mu^{\eta-1} \Big(\Gamma\!\left(1\!-\!\eta,w\right) - \\ & & \displaystyle \Gamma\!\left(1\!-\!\eta,w\!+\!c\mu\right) {\rm d}y {\rm d}x\Big) \\ \!\!\!\!\!\!\!&\stackrel{(a)}{\approx}&\!\!\!\!\!\!\! \displaystyle 12\lambda^2 \!\!\!\int\limits_{r_0}^\infty\!\! \Bigg( \frac{\left(\eta\!-\!c\mu\!+\!\eta c\mu\!+\!e^{c\mu}\!\left(c\mu\!-\!\eta\right)\right) {}_2F_1\!\!\left(\!2\eta,\!1\!+\!\eta,\!2\eta\!+\!1,\!\frac{-c}{2c+x}\!\right)}{2\mu\, \eta\, e^{c\mu} \, \left(x\!+\!2c\right)^{2\eta}} \\ & & \displaystyle + \frac{\left(1\!-\!e^{-c\mu}\right)\left(x\!+\!2c\right)^{\!1\!-\!2\eta} {}_2F_1\!\left(\eta\!+\!1,\!2\eta\!-\!1,\!2\eta,\!\frac{-c}{2c+x}\right)}{2\eta-1}\Bigg) {\rm d}x \\ \!\!\!\!\!& &\!\!\!\!\! \displaystyle \stackrel{(b)}{\approx} \frac{12\lambda^3 c\, r_0^{2-3\eta}}{\left(2\eta-1\right)\left(3\eta-2\right)},
\end{array}$$ where $w\!=\!\mu\left(c\!+\!y\right)$, $(a)$ follows from expanding at $w\!\rightarrow\!\infty$ before integrating in terms of $y$, and $(b)$ from expanding around $\frac{c}{x}\!\rightarrow\! 0$ before integrating in terms of $x$, then substituting $\mu\!=\!\frac{\lambda}{1-\lambda c}$ and expanding at $\lambda c\!\rightarrow\! 0$.
The term $S''_{12}$ does not involve any exponential but still, it cannot be expressed in semi-closed form, unless approximations are made in the integrand. $$\begin{array}{ccl}
S''_{12} \!\!\!\!\!\!\!& = &\!\!\!\!\!\!\! \displaystyle 12\lambda^3 \!\! \int_{r_0}^\infty\!\!\!\int_{x+2c}^\infty \! \frac{x^{-\eta} y^{-\eta} \left(2c\!+\!y\right)^{1-\eta}}{\eta-1} {\rm d}y {\rm d}x \\ \!\!\!\!\!\!\!& = &\!\!\!\!\!\!\! \displaystyle
12\lambda^3 \!\!\! \int\limits_{r_0}^\infty \!\! \frac{x^{\!-\eta} \! \left(2c\!+\!x\right)^{\!1\!-\!2\eta}}{2\left(\eta\!-\!1\right)} \! \Bigg(\! \frac{\left(2c\!+\!x\right) {}_2F_1\!\!\left(\!2\eta\!-\!2,\!\eta,\!2\eta\!-\!1,\!\frac{-2c}{2c+x}\!\right)}{\eta\!-\!1} + \\ & & \displaystyle \frac{4c}{2\eta-1} \, {}_2F_1\!\left(\!2\eta\!-\!1,\!\eta,\!2\eta,\!\frac{-2c}{2c+x}\!\right) \!\! \Bigg) {\rm d}x \\ \!\!\!\!\!\!\!&\approx&\!\!\!\!\!\!\! \displaystyle \frac{2\lambda^3 r_0^{3-3\eta}}{\left(\eta-1\right)^3} - \frac{24\lambda^3 c\, r_0^{2-3\eta}}{\left(\eta\!-\!1\right)\left(2\eta\!-\!1\right)} \!+\! \frac{12\lambda^3 c^2 r_0^{1-3\eta}\left(9\eta\!-\!7\right)}{\left(\eta\!-\!1\right)\left(3\eta\!-\!1\right)},
\end{array}$$ where the approximation is due to expansion $\frac{c}{x}\!\rightarrow\! 0$ before integrating.
The other two terms of $S''_1$ can be approximated similarly, $$\begin{array}{ccl}
S''_{13} \!\!\!&\approx&\!\!\! \displaystyle \frac{12 \lambda^3 c^2 r_0^{1-3\eta}}{3\eta-1} \\
S''_{14} \!\!\!&\approx&\!\!\! \displaystyle \frac{12 \lambda^3 c r_0^{2-3\eta}}{\left(3\eta-2\right)\left(\eta-1\right)} - \frac{24 \lambda^3 c^2\eta r_0^{1-3\eta}}{\left(3\eta-1\right)\left(\eta-1\right)}.
\end{array}$$
After adding up the approximations for the four terms consisting $S_1''$, we end up with $$\label{eq:S1prim2}
\begin{array}{ccl}
S''_1 \!\!\!\!\!&\approx&\!\!\!\!\! \displaystyle \frac{2\lambda^3 r_0^{3-3\eta}}{\left(\eta-1\right)^3} \!-\! \frac{12\lambda^3 c\, r_0^{2-3\eta}}{\left(\eta\!-\!1\right)\left(2\eta\!-\!1\right)} \!+\! \frac{96\lambda^3 c^2 r_0^{1-3\eta}}{3\eta-1}.
\end{array}$$
Assuming $x\!<\!y\!<\!z$ and the user $x$ at the opposite side of the cell as compared to the users $y,z$, the term $S_2''$ is $$\begin{array}{ccl}
S_2'' \!\!\!&\approx&\!\!\! \displaystyle 12\lambda\int_{-\infty}^{-r_0} \!\! \left|x\right|^{-\eta}{\rm d}x \, \bigg(\lambda^2\!\!\int_{r_0}^\infty\!\!\int_{y+2c}^\infty \!\!\!\! y^{-\eta} z^{-\eta} {\rm dz} {\rm dy} \, + \\ & & \displaystyle \lambda\mu\!\! \int_{r_0}^\infty\!\!\int_{y+c}^{y+2c}\!\! \frac{y^{-\eta} z^{-\eta}}{e^{\mu\left(z-y-c\right)}} {\rm d}z {\rm d}y \bigg),
\end{array}$$ where the factor $12\!=\!6\times 2$ is due to six different orderings of vehicles and the scaling by two is used to describe the case where the sides, with respect to the cell, of the user $x$ and of the pair $\left\{y,z\right\}$ are inter-changed.
The two integrals inside the parenthesis can be approximated similarly to the term $S'$. After integrating the first, and approximating the second for $\mu\left(x\!+\!c\right)\!\rightarrow\!\infty$ and $\lambda c\!\rightarrow\! 0$ we have $$\begin{array}{ccl}
S_2'' \!\!\!\!\!&\approx&\!\!\!\!\! \displaystyle \frac{12\lambda r_0^{1-\eta}}{\eta-1} \Bigg(\frac{\lambda^2 r_0^{1-2\eta}}{2\left(\eta\!-\!1\right)}\bigg( \frac{r_0{}_2F_1\!\left(2\left(\eta\!-\!1\right),\eta,2\eta\!-\!1,-2b\right)}{\eta-1} + \\ \!\!\!\!\!\!\!\!\!\!\!& &\!\!\!\!\!\!\!\!\!\!\! \displaystyle \frac{4c{}_2F_1\!\!\left(\!2\eta\!-\!1,\!\eta,\!2\eta,\!-2b\!\right)}{2\eta-1} \bigg) \!\!+\!\! \frac{\lambda^2 c r_0^{\!1-2\eta}{}_2F_1\!\!\left(\!\eta\!+\!1\!,2\eta\!-\!1,\!2\eta,\!-b\!\right)}{2\eta-1} \\ \!\!\!\!\!& &\!\!\!\!\! \displaystyle + \frac{\lambda^2c^2\, r_0^{-2\eta}}{2} \bigg( \frac{\lambda r_0 \, {}_2F_1\!\left(2\eta\!-\!1,\!1\!+\!\eta,\!2\eta,\!-b\right)}{2\eta-1} \, - \\ & & \displaystyle \frac{\left(\eta\!-\!2\right) {}_2F_1\!\left(2\eta,\!1\!+\!\eta,\!2\eta\!+\!1,\!-b\right)}{2\eta}\bigg) \Bigg).
\end{array}$$
Expanding the above expression for small $b\!=\! \frac{c}{r_0}$ yields $$\label{eq:S2prim}
\begin{array}{ccl}
S_2'' \!\!\!&\approx&\!\!\! \displaystyle \frac{6\lambda^3 r_0^{3-3\eta}}{\left(\eta-1\right)^3} -\frac{12 \lambda^3 c \, r_0^{2-3\eta}}{\left(2\eta-1\right)\left(\eta-1\right)}.
\end{array}$$
Now, we can express the term $S''$ equal to the sum of and $$\label{eq:Sprim2}
\begin{array}{ccl}
S'' \!\!\!\!\!&\approx&\!\!\!\!\! \displaystyle \frac{8\lambda^3 r_0^{3-3\eta}}{\left(\eta-1\right)^3} -\frac{24 \lambda^3 c \, r_0^{2-3\eta}}{\left(2\eta\!-\!1\right)\left(\eta\!-\!1\right)} \!+\! \frac{96\lambda^3 c^2 r_0^{1-3\eta}}{3\eta-1}.
\end{array}$$
Substituting and into , and doing some rearrangement allows us to approximate the third moment of interference as $$\begin{array}{ccl}
\mathbb{E}\!\left\{\mathcal{I}^3\right\} \!\!\!\!\! &\approx& \!\!\!\!\! \displaystyle \frac{12\lambda r_0^{1-3\eta}}{3\eta-1} \left(1\!-\!2\lambda c \!+\! 9\lambda^2 c^2\right) + \\ & & \displaystyle \frac{24\lambda^2 r_0^{2-3\eta}}{\left(2\eta\!-\!1\right)\left(\eta\!-\!1\right)}\left(1-\lambda c\right) + \frac{8\lambda^3 r_0^{3-3\eta}}{\left(\eta\!-\!1\right)^3}.
\end{array}$$
Using the approximation for the variance in , the third central moment, $\mathbb{E}\!\left\{\mathcal{I}^3_c\right\}\!=\!\mathbb{E}\left\{\left(\mathcal{I}-\mathbb{E}\!\left\{\mathcal{I}\right\}\right)^3\right\}$, becomes $$\mathbb{E}\!\left\{\mathcal{I}_c^3\right\} \approx \displaystyle \frac{12\lambda r_0^{1-3\eta}}{3\eta-1} \left(1\!-\!2\lambda c \!+\! 9\lambda^2 c^2\right) - \frac{12 \lambda^4 c^2 r_0^{2-3\eta}}{\left(2\eta-1\right)\left(\eta-1\right)}.$$
Using the approximations for the third central moment above and for the variance in , the skewness is $$\begin{array}{ccl}
\mathbb{S}\!\left\{\mathcal{I}\right\} \!\!\!\!\!&\approx&\!\!\!\!\! \displaystyle \left(\frac{12\lambda r_0^{1-3\eta}\left(1\!-\!2\lambda c \!+\! 9\lambda^2 c^2\right)}{3\eta-1} - \frac{12 \lambda^4 c^2 r_0^{2-3\eta}}{\left(2\eta\!-\!1\right)\left(\eta\!-\!1\right)} \right) \times \\ & & \displaystyle \left(\frac{4\lambda r_0^{1-2\eta}}{2\eta-1} \left(1-\lambda c+\frac{1}{2}\lambda^2c^2\right) \right)^{-\frac{3}{2}}.
\end{array}$$
Expanding the above expression for $\lambda c\!\rightarrow\! 0$ yields $$\label{eq:SkewnessApp}
\mathbb{S}\!\left\{\mathcal{I}\right\} \approx \frac{12 \lambda r_0^{1-3\eta}}{3\eta-1} \left( \frac{4\lambda r_0^{1-2\eta}}{2\eta-1} \right)^{-\frac{3}{2}} \left(1 - \frac{\lambda c}{2} \right),$$ where $\left(1-\frac{\lambda c}{2}\right)$ is the correction as compared to the skewness of interference due to a of intensity $\lambda$.
[1]{} 5G Automotive Association (5GAA), “The case for cellular V2X for safety and cooperative driving”, White Paper, Nov. 2016, available at <http://5gaa.org/wp-content/uploads/2017/10/5GAA-whitepaper-23-Nov-2016.pdf> D.J. Daley, and D. Vere-Jones, An introduction to the theory of point processes: Volume I: Elementary theory and methods. Springer, ISBN 0-387-95541-0, 2002. Federal Communications Commission, FCC proposes innovative small cell use in 3.5 GHz band. Dec. 2012. Available at <https://www.fcc.gov/document/fcc-proposes-innovative-small-cell-use-35-ghz-band> M. Haenggi *et. al.*, “Stochastic geometry and random graphs for the analysis and design of wireless networks”, *IEEE J. Sel. Areas Commun.*, vol. 27, pp. 1029-1046, Sept. 2009. J. Kingman, Poisson processes. vol. 3, Oxford university press, 1992. M. Haenggi, Stochastic geometry for wireless networks. Cambridge University Press, 2013. J.G. Andrews, F. Baccelli, and R.K. Ganti, “A tractable approach to coverage and rate in cellular networks”, *IEEE Trans. Commun.*, vol. 59, pp. 3122-3134, Nov. 2011. H.S. Dhillon, R.K. Ganti, F. Baccelli, and J.G. Andrews, “Modeling and analysis of K-tier downlink heterogeneous cellular networks,” *IEEE J. Sel. Areas Commun.*, vol. 30, pp. 550-560, Apr. 2012. M. Haenggi, “The local delay in Poisson Networks”, *IEEE Trans. Inf. Theory*, vol. 59, pp. 1788-1802, Mar. 2013. Y. Li, F. Baccelli, H.S. Dhillon, and J.G. Andrews, “Statistical modeling and probabilistic analysis of cellular networks with determinantal point processes”, *IEEE Trans. Commun.*, vol. 63, pp. 3405-3422, Sept. 2015. N. Miyoshi and T. Shirai, “A cellular network model with Ginibre configured base stations”, *J. Advances Appl. Probability*, vol. 46, no. 3, pp. 832-845, Sept. 2014. A. Busson, G. Chelius and J.M. Gorce, “Interference modeling in CSMA multi-hop wireless networks”, \[Research Report\] RR-6624, INRIA. pp. 21, 2009. M. Haenggi, “Mean interference in hard-core wireless networks”, *IEEE Commun. Lett.*, vol. 15, pp. 792-794, Aug. 2011. R.K. Ganti, and M. Haenggi, “Interference and outage in clustered wireless ad hoc networks”, *IEEE Trans. Inf. Theory*, vol. 55, pp. 4067-4086, Sept. 2009. V.V. Chetlur and H.S. Dhillon, “Coverage analysis of a vehicular network modeled as Cox Process driven by Poisson Line Process”, *IEEE Trans. Wireless Commun.*, vol. 17, pp. 4401-4416, Jul. 2018. C.-S. Choi and F. Baccelli, “An analytical framework for coverage in cellular networks leveraging vehicles”, *IEEE Trans. Commun.*, to be published. E. Steinmetz, M. Wildemeersch, T. Quek and H. Wymeersch, “A stochastic geometry model for vehicular communication near intersections”, *in Proc. IEEE Globecom Workshops*, San Diego, 2015, pp. 1-6. B. B[ł]{}aszczyszyn, P. M[ü]{}hlethaler and Y. Toor, “Stochastic analysis of Aloha in vehicular ad hoc networks”, *Ann. of Telecommun.*, vol. 68, pp. 95-106, Feb. 2013. K. Koufos and C.P. Dettmann, “Temporal correlation of interference in bounded mobile ad hoc networks with blockage”, *IEEE Commun. Lett.*, vol. 20, pp. 2494-2497, Dec. 2016. A. Daou, “On flow within platoons”, *Australian Road Research*, vol. 2, no. 7, pp. 4-13, 1966. I. Greenberg, “The log-normal distribution of headways”, *Australian Road Research*, vol. 2, no. 7, pp. 14-18, 1966. S. Yin [*[et. al.]{}*]{}, “Headway distribution modeling with regard to traffic status”, *IEEE Intell. Vehicles Symp.*, Xian, 2009, pp. 1057-1062. G. Yan and S. Olariu, “A probabilistic analysis of link duration in vehicular ad hoc networks”, *IEEE Trans. Intell. Transp. Syst.*, vol. 12, pp. 1227-1236, Dec. 2011. R.J. Cowan, “Useful headway models”, *Transportation Research*, vol. 9, no. 6, pp. 371-375, Dec. 1975. K. Koufos and C.P. Dettmann, “Moments of interference in vehicular networks with hardcore headway distance”, *submitted for publication, available at* <http://arxiv.org/abs/1803.00658v2> M. Westcott, “The probability generating functional”, *J. Australian Mathematical Society*, vol. 14, no. 4, pp. 448-466, 1972. R.K. Ganti, F. Baccelli and J.G. Andrews, “Series expansion for interference in wireless networks”, *IEEE Trans. Inf. Theory*, vol. 58, pp. 2194-2205, Apr. 2012. A. Guo and M. Haenggi, “Asymptotic deployment gain: A simple approach to characterize the SINR distribution in general cellular metworks”, *IEEE Trans. Commun.*, vol. 63, pp. 962-976, Mar. 2015. S. Al-Ahmadi and H. Yanikomeroglu, “On the approximation of the Generalized-K distribution by a Gamma distribution for modeling composite fading channels”, *IEEE Trans. Wireless Commun.*, vol. 9, pp. 706-713, Feb. 2010. S. Atapattu, C. Tellambura and H. Jiang, “A mixture gamma distribution to model the SNR of wireless channels”, *IEEE Trans. Wireless Commun.*, vol. 10, pp. 4193-3203, Dec. 2011. A. Ghasemi and E. Sousa, “Interference aggregation in spectrum-sensing cognitive wireless networks”, *IEEE J. Sel. Topics Signal Process.*, vol. 2, pp. 41-56, Feb. 2008. K. Koufos, K. Ruttik and R. J[ä]{}ntti, “Distributed sensing in multiband cognitive networks”, *IEEE Trans. Wireless Commun.*, vol. 10, pp. 1667-1677, May. 2011. Z.W. Salsburg, R.W. Zwanzig and J.G. Kirkwood, “Molecular distribution functions in a one-dimensional fluid”, *J. Chemical Physics*, vol. 21, pp. 1098-1107, Jun. 1953. M. Kountouris and N. Pappas, “Approximating the interference distribution in large wireless networks”, *Int. Symp. Wireless Commun. Systems (ISWCS)*, Barcelona, Aug. 2014, pp. 80-84. K. Stucki and D. Schuhmacher, “Bounds for the probability generating functional of a Gibbs point process”, *J. Advances Appl. Probability*, vol. 46, no. 1, pp. 21-34, Mar. 2014. M. Abramowitz and I.A. Stegun. Handbook of mathematical functions with formulas, graphs and mathematical tables. Washington, DC, USA: GPO, 1972. M. Haenggi and R. Smarandache, “Diversity polynomials for the analysis of temporal correlations in wireless networks”, *IEEE Trans. Wireless Commun.*, vol. 12, pp. 5940-5951, Nov. 2013. Wolfram Research, Inc., Mathematica, Version 10.1, Champaign, IL (2015). R. Ganti and M. Haenggi, “Spatial and temporal correlation of the interference in ALOHA ad hoc networks”, *IEEE Commun. Lett.*, vol. 13, pp. 631-633, Sept. 2009. K. Koufos and C.P. Dettmann, “Temporal correlation of interference in vehicular networks with shifted-exponential time headways”, *IEEE Wireless Commun. Lett.*, to be published. R. Tanbourgi, H.S. Dhillon, J.G. Andrews and F.K. Jondral, “Effect of spatial interference correlation on the performance of maximum ratio combining”, *IEEE Trans. Wireless Commun.*, vol. 13, pp. 3307-3316, Jun. 2014.
[^1]: K. Koufos and C.P. Dettmann are with the School of Mathematics, University of Bristol, BS8 1TW, Bristol, UK. {K.Koufos, Carl.Dettmann}@bristol.ac.uk
[^2]: This work was supported by the EPSRC grant number EP/N002458/1 for the project Spatially Embedded Networks. All underlying data are provided in full within this paper.
[^3]: It follows by setting the local stability constraint $c^*$ in [@Stucki2014 equation (2.8)] equal to the intensity $\lambda$ of the Gibbs process.
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abstract: 'The increase in the number of researchers coupled with the ease of publishing and distribution of scientific papers (due to technological advancements) has resulted in a dramatic increase in astronomy literature. This has likely led to the predicament that the body of the literature is too large for traditional human consumption and that related and crucial knowledge is not discovered by researchers. In addition to the increased production of astronomical literature, recent decades have also brought several advancements in computer linguistics. Especially, the machine aided processing of literature dissemination might make it possible to convert this stream of papers into a coherent knowledge set. In this paper, we present the application of computer linguistics techniques on astronomy literature. In particular, we developed a tool that will find similar articles purely based on text content from an input paper. We find that our technique performs robustly in comparison with other tools recommending articles given a reference paper (known as recommender system). Our novel tool shows the great power in combining computer linguistics with astronomy literature and suggests that additional research in this endeavor will likely produce even better tools that will help researchers cope with the vast amounts of knowledge being produced.'
address: 'European Southern Observatory, Karl-Schwarzschild-Strasse 2, 85748 Garching, Germany'
author:
- 'Wolfgang E. Kerzendorf'
bibliography:
- 'wekerzendorf.bib'
title: 'Knowledge discovery through text-based similarity searches for astronomy literature'
---
Introduction
============
Since the inception of writing, human knowledge has steadily increased as has the number, and size, of published works. The output of the scientific community has doubled every nine years over the past decades [@bornmann2015growth].
The computing and internet revolution has made the publication and dissemination of these works easy and with the advent of open access channels pluralistic. In the physical sciences the public repository [*arXiv*]{}has provided open access to almost the entire corpus of publications since 1992.
Given the rise of publications each year and the fixed capacity of a human to process information either we shall narrow the specialization range in each field to limit the breadth of the necessary knowledge base or have new tools that filter the available publications.
In astronomy, the has provided access (in addition to many other accomplishments such as digitizing old articles) to this large amount of literature with a search interface that captures the traditional way of accessing information (name of first author and year) extremely well. Newer iterations of this system [@2015ASPC..495..401C] have started to branch out and allow not only search algorithms but provide certain bibliometric statistics as well as a recommender system (named “Suggested Articles”). This recommender system is based on citations, text similarity and co-readership [as described on the ADS 2.0 website and suggested in @2010APS..MAR.S1244H; @2011ASSP...24...23K]. Such recommender systems will be a first step to tackle a world in which the scientific literature has massively outgrown the memory capacity of human brains.
In this paper, we present a new method for article recommendations starting from a reference article or text. We employ the techniques of text similarity and specifically avoid citations. This strict abstention from citation was chosen due to the fact that citations are influenced by many factors and may not provide an unbiased link between publications [several examples in @vanWesel2014].
In Section \[sec:data\_proc\], we describe the data acquisition, initial vetting and processing. Section 3, describes the method used and some statistics arising. An overview of the framework used in this work and its application to several example papers in in Section 4. Caveats and possible improvements are discussed in Section 5 and we conclude with an outlook to the future in Section 6.
Data Processing {#sec:data_proc}
===============
For our initial raw corpus, we considered all papers submitted to [*arXiv*]{}. Using the bulk data access[^1], we downloaded the entire corpus. After a series of operations (discarding any non-latex submissions), we arrived at individual source directories (a total of ). This work focuses currently on the field of astronomy. For [*arXiv*]{}entries until 2007-01-01 this can be filtered on the id itself (the id starts with ‘astro-ph’). For later entries we harvested the meta data through the OAI protocol for metadata harvesting (OAI-PMH) and then selected all [*arXiv*]{}entries that had *astro* in the *subject* node of the metadata. This as well as the ‘astro-ph’ selected papers amount to a total corpus of papers.
In each source compilations, we tried to identify the main tex file by requiring a single valid *\\begin{document}* clause and processed this further with <span style="font-variant:small-caps;">latexpand</span>[^2] to a single document that would contain all relevant text content. Not all entries had a uniquely identifiable main *tex-file* resulting in a total of papers.
The resulting *tex-files* were further processed removing the most common environments:
- - - - - - - -
Then we removed any text before the first section command or if this was not present any text before end abstract. If neither of these criteria were met, we discarded the file, resulting in articles.
The final step of the raw reduction process was the removal of latex commands using the <span style="font-variant:small-caps;">opendetex</span> software [^3].
![Kernel density estimate of the distribution of number of authors in each year (using a kernel bandwidth of 0.3). The width of the distribution scales with total number of papers for each year. The black lines indicate the median of 3 and 4 authors (the median changes in 2010).[]{data-label="fig:no_authors"}](plots/noauthors_per_paper.pdf){width="\columnwidth"}
These unprocessed data and metadata already allow for some interesting statistics to study publication behaviour. As [*arXiv*]{}has already shown[^4] there is a roughly linear increase in astronomy papers. The number of authors also is an important metric to understand the culture of a field. Here often people turn to the average number of authors as a measure [e.g. @10.15200/winn.141832.26907]. While this is a useful statistics, it is heavily biased towards the few papers with extreme number of authors. Figure \[fig:no\_authors\] shows that the general number of authors remains relatively low but there is a very small number of papers with increasingly extreme number of authors (only 0.5% of papers have more than one hundred authors).
Natural Language Processing
---------------------------
These raw texts are ready for natural language processing and the following steps use the tools extensively.
The first step in this process is to break the text into individual words using <span style="font-variant:small-caps;">nltk.tokenize\_words</span>, which splits into individual words and removes all punctuation - except the period (which is treated as a single word at this step).
The next process is to remove stop words such as *these, those, am, is, are* (using the english stop words defined in <span style="font-variant:small-caps;">nltk.corpus.stopwords</span>).
The final step of processing is to lemmatize the words, which is the process of grouping together the different inflected forms of a word so it can be analyzed as a single item. For this process we use the tool <span style="font-variant:small-caps;">nltk.wordnet.morphy</span> to bring the words back to their original forms (galaxies maps to galaxy, expanding maps to expand, etc.). We discard words that do not have a corresponding entry in the dictionary provided [<span style="font-variant:small-caps;">WordNet;</span> @fellbaum1998wordnet].
After this final step we are left with a corpus consisting of documents.
Method {#sec:method}
======
In this work, we will entirely rely on the bag-of-words technique [@harris1954distributional], that disregards grammar and word order, treating the document just as a collection of words. The features that we use for our analysis are several statistics based on word frequency. For all feature extraction tasks, we relied on .
The first step for any of these methods is building a vocabulary of unique words. This is helped by the fact that we transformed our document removing any stop words and transforming the words to their simplest form.
This vocabulary consists of $\approx ~\num{32000}$ words. This is only slightly larger than the $\approx\num{20000}$ [@goulden1990large] words a well educated native speaker knows and much lower than the $\approx\num{170000}$ words in the Oxford English dictionary [@simpson1989oxford]. Our vocabulary can then be used to vectorize (feature extract) the documents to vectors the size of the dictionary (in our case $\approx\num{32000}$).
The simplest case is the use of a binary statistics for feature extraction which will only encode if a word is present or not present. This statistics can give a rough overview of the content but will de-weight more frequently used words and thus possible shift the inferred topic of the document in statistical analysis.
The first statistics we have performed on the corpus of documents is a simple count vectorization (using <span style="font-variant:small-caps;">scikit-learn.feature\_extraction.CountVectorizer</span>). This count measures allows us to quantify the growth of literature since the conception of [*arXiv*]{}. Figure \[fig:median\_word\_per\_paper\] shows, that while the growth in document size (using the processed document word counts as a proxy) has been maybe a factor of 1.5 over the last decade, astronomical literature in total has grown exponentially (see Figure \[fig:words\_per\_year\]) over the same period. This given measure of “word count”, however, has several drawbacks, the most important one being that it increases with document length and thus does not give a useful measure for word importance.
![Median Number of words without stopwords for all papers published in each year[]{data-label="fig:median_word_per_paper"}](plots/yr_vs_medwc.pdf){width="\columnwidth"}
![Total Number of words without stopwords for all papers published in each year on [*arXiv*]{}since 1992.[]{data-label="fig:words_per_year"}](plots/yr_vs_wc.pdf){width="\columnwidth"}
The last vectorization technique is a natural extension of the word counts that aims at emphasizing a word’s importance in text - thus ideal for assessing content of a paper. In the following, we will use the moniker “term” and “word” interchangeably as our analysis uses only one-word terms (unigrams). The term frequency $tf(t,d)$ method normalizes the simple word count by the number of words in the document. This relies on the assumption that the importance (or weight) of a term in documents is proportional to this term frequency [@5392697]. In addition to term frequency, we want to quantify the information content a specific term carries. @sparck1972statistical have introduced the concept of inverse document frequency $idf(t, d)$. We use the inverse document frequency given in <span style="font-variant:small-caps;">scikit-learn</span> is $\log{\frac{1 + n_d}{1+df(d, t)}}$, where $n_d$ is the total number of documents and $df(d, t)$ is the number of documents containing the given term. The combination of both measures gives the well established $tf(t, d) \times idf(t,d)$ (henceforth TFiDF) measure which weights terms highly that have a high information content due to their rarity. This measure is used in several machine learning tasks including finding similar texts.
Similarity in papers {#sec:similarity}
====================
We explore how the TFiDF statistics can lead to knowledge discovery.
For this purpose, we first normalize our document vectors using the euclidean norm $\vec{d}_\textrm{norm} = \frac{\vec{d}}{||\vec{d}||_2}$ before proceeding further. Leaving us with the entire sparse matrix (which is available upon request). Here, we present an example of such a matrix (with the different document vectors as rows and the columns representing different words/terms): $$A_\textrm{TFiDF} =
\bordermatrix{~ & star & model & \cdots & galaxy\cr
\textrm{arXiv}-1 & 0.021 & \cdots & 0 \cr
\textrm{arXiv}-2 & 0 & 0.03 & \cdots & 0\cr
\vdots & 0.019 & 0.016 & \cdots & 0 \cr
\textrm{arXiv}-n & 0 & 0 & \cdots & 0.023 \cr}$$ We simply use the cosine distance by choosing a document that we want to compare and multiplying this with the TFiDF matrix $\vec{v}_\textrm{similarity} = A_\textrm{TFiDF} \times \vec{d}_\textrm{norm}$ to measure text similarity. An example service that showcases this technique is available at <http://opensupernova.org:7071>.
Example: SN 1006 companion search
---------------------------------
In the following, we will use this approach on a paper that is well known to the author: “Hunting for the Progenitor of SN 1006: High-resolution Spectroscopic Search with the FLAMES Instrument” [@2012ApJ...759....7K]. This paper describes the failed attempt to find a surviving companion star (often called donor star) to a supernova (likely caused by a white dwarf) searching in one of the supernova remnants in our Galaxy.
We first measure the most important words to the presented algorithm. For this purpose, we inverse sort our $\vec{v}_\textrm{similarity}$ and use the first best 100 matches. We multiply the $\vec{d}_\textrm{norm}$ and look for the highest entries in the resulting vector which should give us the most important words that the algorithm matches. Figure \[fig:example\_papers\] clearly shows that very relevant words that one would expect when writing a paper about searching for a donor star in a supernova remnant likely caused by a white dwarf.
This might also be obtained by doing the typical manual search technique using the citations to the current paper as well as the references of the current paper. However, we aim to find papers that would not be found using this common technique.
![Comparing cumulative similarity to the given TFiDF similarity number. For example, papers that have a higher similarity than the median similarity number only make up 5% of the entire corpus of papers.[]{data-label="fig:group_simil_compare"}](plots/simil_1006_various_groups.pdf){width="\columnwidth"}
All references of our test paper (12 in total found in our [*arXiv*]{}corpus) have a median similarity of 0.38. Only 3% of papers (see Figure \[fig:group\_simil\_compare\]) in the astronomy corpus are more similar than this median which suggests that as expected the citations are highly relevant.
All citations to our test paper (30 in total where found in our [*arXiv*]{}corpus) have a median similarity of 0.26 and 5% of papers are more similar than this suggesting that the citations to this article come from a more varied field than the original references (or that references had been forgotten).
Next we test if the algorithm’s top 30 papers (similar to the citations) are in neither citations nor references and compare this to the relevance of the other papers. Figure \[fig:example\_papers\] shows the comparison between the median cosine distance of the cited papers, the median cosine distance of the references and the median top 30 results excluding papers from both of these groups. This demonstrates that such a system can find relevant papers that could easily be missed otherwise.
ADS has recently implemented a similar system [@2015ASPC..495..401C], however we find that it does not give as relevant matches as the presented algorithm. There are 30% of papers that are more similar to the document in question compared to the median similarity of the suggested papers. Manual inspection also shows that some of the suggested papers (e.g. “Maps of Dust Infrared Emission for Use in Estimation of Reddening and Cosmic Microwave Background Radiation Foregrounds” by @1998ApJ...500..525S) might be very broadly related but not very relevant. The precise algorithm to determine the suggested papers for ADS is not accessible thus prohibiting a quantitative comparison.
Example: Globular cluster
-------------------------
The second test paper we use is “A general abundance problem for all self-enrichment scenarios for the origin of multiple populations in globular clusters” by @2015MNRAS.449.3333B. This paper points out a possible flawed explanation for abundance anomalies in globular clusters. The top matching words (see Figure \[fig:example\_papers\]) are highly relevant to a topic that discusses anomalous abundances that might be caused by different pollution system to varying degrees due to their yields. The references are more similar compared to the first paper suggesting that this paper is more focused. The same is true for the citations that also come from more similar papers when compared to the citations in the first paper. The comparison of all median similarity numbers, however, shows a similar pattern when compared to the first paper including the dissimilarity of the ADS suggestion algorithm (see Figure \[fig:example\_papers\]).
Example: ALMA observations of a disk
------------------------------------
The last test paper is titled “Unveiling the gas-and-dust disk structure in HD 163296 using ALMA observations” . This paper describes observations of structure around young massive stars. The important words (see Figure \[fig:example\_papers\]) again seem highly relevant. Similar to the second paper the citations and references have high similarity numbers that might suggest a narrower focus of this paper. In contrast to the other two example papers in this case the ADS suggestion algorithm also produced a high similarity index. Only three of the papers from the suggestion algorithm could be found the [*arXiv*]{}corpus (compared to the usual six) which might explain this anomaly.
Discussion {#sec:discussion}
==========
This test of the use of natural language processing and machine learning tools , has already shown that even simple techniques result in knowledge discovery. However, there are a number of improvements that can aid in the knowledge discovery part.
Specifically, in the language processing step there are several steps that might be improved in future versions. We remove all words that are not in the english dictionary during our initial run. This already poses some problems in the lemmatization process as the name “Roche” (as in Roche Lobe) is not recognized and is thus removed. This suggests that there is a need to build a domain specific lemmatizer [such as the <span style="font-variant:small-caps;">BioLemmatizer</span> for biology @liu2012biolemmatizer]. In our current approach, we also only consider single words (so called unigrams) but terms like “white dwarf” (bigram) suggest that future iterations of this algorithm might find more relevant results if we treat such bigrams separately. Abbreviations are also commonly used in papers and are most often defined at the beginning of most papers. Thus the expanded word enters the word count only once. However, this leads to a misinterpretation the true importance of the word as all other mentions are discarded.
The next information carrier that is removed are object names which might link papers that are of the same object. However, using this technique, already values a certain type of knowledge above another (any study on the object is valued higher than similar studies on other objects for a given paper).This is especially true in our metric as object names will have a very low document frequency (being only mentioned in few papers) and thus will attain very high values in a TFiDF comparison which might not lead to the desired result.
Conclusion {#sec:conclusion}
==========
We present a new technique for knowledge discovery by using a text similarity approach to find similar papers to a given reference paper. This technique performs robustly and finds relevant papers that are not discovered via either citations, references or suggestions from ADS. This metric also seems to be a useful tool when studying if a paper is relevant to broader field or addresses some detail in a narrower focus. Similar attempts in other fields [e.g. neuroscience; @2016PLoSO..1158423A] also suggest that this can be used to provide a powerful method to disseminate papers.
Currently, this allows an additional method to discover knowledge especially when entering a field that one is unfamiliar with (e.g. using this technique for reviews). Our recommendation method might be further improved by linking our algorithm with citation information and using an algorithm like <span style="font-variant:small-caps;">PageRank</span> [popularized by Google; @page1999pagerank]. This will value highly cited papers more than lower cited ones. While this technique will help in knowledge discovery by finding relevant papers, our future goal is to identify key measurements and statements in each paper. This will allow a scientist to quickly sift through the vast amount of knowledge and identify the relevant paper by the searched quantities (e.g. the most current mass of the proton) before reading the entire paper and critically evaluating the methodology and statistics used.
Such a machinery would in the first instance help scientists to discover sought after knowledge (regardless of bias towards certain authors, etc.) but might also allow for additional services. One of these might be very simple “fact checking” mechanisms that will aid researchers when compiling a paper by providing the most up-to-date quantities and flagging mistakes (similar to a grammar/spelling checker). Such a machinery has uses far beyond astronomy and astrophysics. However, among the many academic fields, astronomy exposes the vast majority of papers and data in machine readable formats (priv. comm. Christine L. Borgman). This suggests that this field is a good start for the development of such a machinery.
Acknowledgements
================
W. E. Kerzendorf was supported by an ESO Fellowship. Especially, we would like to thank the detailed discussions, encouragement and suggestions from Felix Stoehr and Jason Spyromillio. The support from the library team (Uta Grothkopf , Dominic Bordelon & Silvia Measkins) was invaluable to get an insight into the field of knowledge discovery. Christine Borgman and Bernie Randles (at UCLA) gave suggestion from an Information Sciences point of view and the visit would not have been possible if not for the generousity of the UCLA Galactic Center Group (especially Tuan Do and Andrea Ghez). We would also like to thank Bruno Leibundgut, Kathatrina Immer, Ivan Cabrera-Ziri for testing the algorithm on some well known papers.
{width="\columnwidth"} {width="\columnwidth"}
{width="\columnwidth"} {width="\columnwidth"} {width="\columnwidth"} {width="\columnwidth"}
[^1]: e.g. <s3://arxiv/src/arXiv_src_1001_001.tar>
[^2]: <https://www.ctan.org/pkg/latexpand>
[^3]: <https://github.com/pkubowicz/opendetex>
[^4]: see <https://arxiv.org/help/stats/2016_by_area/index>
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---
abstract: 'The dynamo effect is a class of macroscopic phenomena responsible for generation and maintaining magnetic fields in astrophysical bodies. It hinges on hydrodynamic three-dimensional motion of conducting gases and plasmas that achieve high hydrodynamic and/or magnetic Reynolds numbers due to large length scales involved. The existing laboratory experiments modeling dynamos are challenging and involve large apparatuses containing conducting fluids subject to fast helical flows. Here we propose that electronic solid-state materials – in particular, hydrodynamic metals – may serve as an alternative platform to observe some aspects of the dynamo effect. Motivated by recent experimental developments, this paper focuses on hydrodynamic Weyl semimetals, where the dominant scattering mechanism is due to interactions. We derive Navier-Stokes equations along with equations of magneto-hydrodynamics that describe transport of Weyl electron-hole plasma appropriate in this regime. We estimate the hydrodynamic and magnetic Reynolds numbers for this system. The latter is a key figure of merit of the dynamo mechanism. We show that it can be relatively large to enable observation of the dynamo-induced magnetic field bootstrap in experiment. Finally, we generalize the simplest dynamo instability model – Ponomarenko dynamo – to the case of a hydrodynamic Weyl semimetal and show that the chiral anomaly term reduces the threshold magnetic Reynolds number for the dynamo instability.'
author:
- Victor Galitski
- Mehdi Kargarian
- Sergey Syzranov
title: Dynamo Effect and Turbulence in Hydrodynamic Weyl Metals
---
The dynamo effect is a beautiful astrophysical phenomenon, first proposed by Larmor in 1919 [@Larmor], that is believed to be responsible for generating and sustaining magnetic fields in galaxies, stars and planets including the Sun and the Earth [@RSZ]. There exist a large variety of different dynamo mechanisms [@Gilbert; @Gilbert1; @RSZ] that all share the same key ingredient – hydrodynamic motion of an electrically conducting gas, fluid or plasma. The dynamo theory deals with the hydrodynamic motion of a conductive medium focussing on the possibility of self-generating and self-sustaining magnetic fields, whose presence has been observed in astrophysical bodies.
As detailed below, the underlying equations of the theory are the Navier-Stokes equations, describing the hydrodynamic motion of the medium, coupled to the Maxwell equations of electromagnetism. In the non-relativistic limit, they give rise to equations of magneto-hydrodynamics (MHD). These are complicated non-linear equations, and their exact solutions represent a great challenge. However, both the solutions of simplified MHD models \[e.g., kinematic dynamos, with predetermined velocity fields ${\bf u}({\bf r},t)$\] and qualitative arguments [@RSZ] suggest that the dynamo action is possible when the terms enhancing the magnetic field \[e.g. the induction term, ${\bm \nabla} \times ({\bf u} \times {\bf B})$\] overwhelm the magnetic diffusion term, $\eta_m \Delta {\bf B}$ (where $\eta_m = c^2/4 \pi \sigma$ where $c$ is the speed of light and $\sigma$ is the conductivity of the medium), which tend to suppress the self-generation. The respective figure of merit is the [*magnetic Reynolds number*]{} [@LL6] $$\label{Rm}
R_m = \frac{u L}{\eta_m} = u L \frac{4 \pi \sigma}{c^2},$$ where $L$ is the characteristic system size and $u$ is the typical velocity of the medium. The threshold value for a dynamo action to commence (usually lying in in the range $R_m^{\rm(cr)} \sim 10 - 100$, with $R_m^{\rm(cr)} \approx 17.7$ for the simplest Ponomarenko dynamo [@Ponomarenko] discussed below) depends on system’s geometry and is rarely known exactly. It is clear, however, that the larger $R_m$, the more likely and more effective the dynamo action. The conductivity of astrophysical media vary greatly from $10^{-11}{\rm Sm}^{-1}$ for interstellar plasma to $10^{3}{\rm Sm}^{-1}$ for the solar convection shell and $10^{5}{\rm Sm}^{-1}$ for the Earth’s core, but in all of these cases the large magnetic diffusion coefficient is compensated by literally astronomical distances resulting in large magnetic Reynolds numbers, however small the conductivities are. By contrast, laboratory dynamo experiments [@LathropPT] deal naturally limited system size and use the conductivity and the flow velocities as the only potentially tunable parameters.
Apart from large magnetic Reynolds numbers $R_m \gg 1$, the emergence of a dynamo requires a number of other conditions that need to be met. In particular, certain “no-go theorems” [@Jones] have to be overcome, such as the impossibility of a two-dimensional dynamo effect or that in a planar three-dimensional flow (i.e., with one vanishing component of velocity). Finally, it is known the dynamo action is greatly helped by the helicity flow, which may arise either due to the geometry of an imposed flow or due to turbulence. The latter is possible if the second figure of merit, the [*hydrodynamic Reynolds number*]{} $$\label{R}
R = \frac{u L}{\nu},$$ where $\nu$ is the kinematic viscosity. Separating both the velocity and magnetic field into a mean-field and fluctuating component - ${\bm u} = \overline{\bm u} + \delta {\bm u}$ and ${\bm B} = \overline{\bm B} + \delta {\bm B}$, and averaging over the small-scale fluctuations results in the Krause-R[ä]{}dler equations [@KR; @Parker] of mean-field MHD, which in the simplest case of isotropic turbulence is given by $$\label{KR}
\frac{\partial \overline{\bf B}}{\partial t} = {\bm \nabla} \times (\overline{\bm u} \times \overline{\bf B}) + {\bm \nabla} \times (\alpha \overline{\bf B}) + \xi \Delta \overline{\bf B},$$ where the second term in the right-hand-side is the “new” helicity term allowed in turbulent MHD ($\alpha$-effect). If the velocity field is stationary, Eq. (\[KR\]) or a similar MHD equation without helicity for non-turbulent flows becomes an eigenvalue problem for the magnetic field growth ${\bf B}({\bf r},t) \propto {\bf B}({\bf r}) e^{\gamma t}$. The existence of exponentially growing components (${\rm Re}\, \gamma >0$) indicates an instability towards a self-generating magnetic field (were the imaginary part ${\rm Im}\, \gamma >0$ leads to the field oscillations, which have been suggested by one of the authors to lead, e.g., to periodic cycles of solar magnetic activity).
Apart from the astrophysical context, there has been a tremendous interest in testing the predictions of dynamo theory and modeling a planetary-like or solar-like dynamo action in the laboratory [@LathropPT; @DynamoExp; @DynamoExp1; @DynamoExp2]. Several impressive laboratory experiments have been carried out and are currently under way that involve setting in motion a liquid metal – sodium or gallium – with the goal to achieve large Reynolds numbers to enable the dynamo mechanism. As obvious from Eqs. (\[Rm\]) and (\[R\]), this leads to the challenge of ultra-fast mechanical stirring or rotating the liquid metal.
Here we propose that electronic solid-state systems may provide an alternative platform for observing magnetohydrodynamic effects. Firstly, we list several necessary conditions of the dynamo effect in an electronic system: (i) Transport in the electron liquid should be governed by hydrodynamics, i.e. the primary momentum relaxation mechanism should be electron-electron collisions rather than impurity scattering. (ii) The system and the flow must be essentially three-dimensional. (iii) Large magnetic $R_m\gg 1$ and/or hydrodynamic $R\gg 1$ Reynolds numbers are required.
Hydrodynamic transport in solid state \[condition (i)\] has been a subject of intense recent studies [@Kivelson_hydro; @Hartnoll_Lucas; @SDS_hydro; @Balents_SYKs; @Sachdev_book2], both theoretical and experimental. On the experimental side, two widely studied platforms for hydrodynamic phenomena are graphene [@GrapheneHydro] and Weyl semimetals (WSMs) [@Felser1; @Felser2; @Felser3]. Graphene, however, violates a “no-go dynamo theorem” - condition (ii) requiring 3D flows - and is thus of no relevance to the dynamo effect. In what follows, we focus on undoped or weakly doped Weyl semimetals. We note that in systems with the power-law quasiparticle dispersion $\epsilon({\bf p}) \propto \left| {\bf p} \right|^\beta$ with $\beta\leqslant 1$ the creation of electron-hole pairs is suppressed [@FosterAleiner], because the energy and momentum conservation laws cannot be satisfied simultaneously for lowest-order processes. Weyl systems ($\beta=1$) may, therefore, often be considered as electron-hole plasma with a linear particle dispersion.
A WSM generically has an even number of nodes, according to the fermion-doubling theorem [@NielsenNinomiya], and electrons and holes near different nodes often behave as independent liquids. However, simultaneous application of external electric $\bE$ and magnetic $\bB$ fields results in the quasiparticle transfer from one node to another (chiral anomaly[@Adler:anomaly; @BelJackiw:anomaly; @SonSpivak:anomaly; @Burkov:review; @Burkov:AnomalyDiffusive]). For simplicity, we assume in this paper that (a) the system has only two nodes, labeled by $L$ and $R$, with the same quasiparticle dispersion, (b) the entire system is being kept at a constant temperature $T$ and (c) the intranodal equilibration processes are significantly faster than the internodal particle-transfer processes. This allows one to define the chemical potentials $\mu_\alpha$ near each node $\alpha=L, R$ and the hydrodynamic velocity $\bu$ of the Weyl fluid. The distribution function of the linearly-dispersing quasiparticless near each node in the absence of electromagnetic fields is given by [@Narozhny:grapheneHreview] $f_\alpha(\bk)=\left\{\exp\left[\gamma(\bu)\left(\pm v_F|\bk|-\mu_\alpha-\bu\cdot\bk\right)/T\right]+1\right\}^{-1}$, where “+” and “-” refer, respectively, to the conduction and valence bands, and $\gamma(\bu)=\left(1-u^2/v_F^2\right)^\frac{1}{2}$.
The dynamics of charge densities $\rho_\alpha$ near node $\alpha$, where $\alpha=L,R$, are described by the continuity equations $$\begin{aligned}
\partial_t \rho_\alpha+{\bm \nabla\cdot {\bf j}_{\alpha}}-\chi_{\alpha}\frac{g e^3 }{4\pi^2 \hbar^2 c}
\bE\cdot\bB
+\frac{\rho_\alpha-\rho_{\bar{\alpha}}}{\tau_{in}}=0,
\label{ChargeContinuity}\end{aligned}$$ where $\chi_{L}=-1$ and $\chi_R=+1$ are the “chiralities” of quasiparticles near nodes $L$ and $R$ and $g$ accounts for spin and possibly additional valley degeneracy; $\bar{\alpha}$ labels the node other than $\alpha$; hereinafter $e=-|e|$. The first two terms in Eq. (\[ChargeContinuity\]) match the usual continuity equation for a liquid with density $\rho_{\alpha}$; the third term ($\propto \bE\cdot\bB$) accounts [@Burkov:review; @Burkov:AnomalyDiffusive] for the change of the electron concentration at node $\alpha$ due to the chiral anomaly; and the last term in Eq. (\[ChargeContinuity\]) describes internodal scattering, e.g., due to short-range-correlated quenched disorder, with the internodal scattering time $\tau_{in}$. The electric currents $\bj_{L,R}$ of the charge carriers near the two nodes are given by $$\begin{aligned}
\bj_\alpha = \sum_\beta\sigma_{\alpha\beta}
\left[\bE+\frac{1}{c}\bu\times\bB-\frac{1}{e}{\bm \nabla}\mu_\beta\right]
-\chi_\alpha\frac{ge^2}{4\pi^2\hbar^2c}\bB\mu_\alpha,
\label{Currents}\end{aligned}$$ where $\alpha,\beta=L,R$; $\mu_\alpha$ is the chemical potential near node $\alpha$, and ${\bf u}$ is the hydrodynamic velocity of the Weyl fluid. In this paper we assume that the imbalance of the chemical potentials between the nodes, if any, is small $|\mu_L-\mu_R|\ll |\mu_{L,R}|,T$. The diagonal components $\sigma_{LL}=\sigma_{RR}$ of the conductivity tensor $\sigma_{\alpha\beta}$ describe the response of charge carriers near each node to the electromagnetic field; the off-diagonal entries $\sigma_{LR}=\sigma_{RL}$ account for the drag of the quasiparticles near each node by the current near the other node. The last term in Eq. (\[Currents\]) describes the chiral magnetic effect [@Kharzheev:CME; @Kharzheev:CMTlextures], the generation of the charge current by an external magnetic field in the system in the presence of chirality imbalance, $\mu_L-\mu_R\neq 0$.
Equations (\[ChargeContinuity\])-(\[Currents\]), together with the relations [@RodionovSyzranov:impurityWeyl] $$\begin{aligned}
\rho_{R,L}=ge\frac{\mu_{R,L}^3+\pi^2\mu_{R,L}T^2}{6\pi^2 v_F^3 \hbar^3}
\label{RhoMuRelation} \end{aligned}$$ for the charge density at node $\alpha$ and with Maxwell equations, which involve the total charge density $\rho=\rho_L+\rho_R$ and the current $\bj=\bj_L+\bj_R$, constitute a closed system of equations which describes charge and current dynamics of the electron liquid in a WSM which moves with velocity $\bu$ in an external electromagnetic field. The motion of such a liquid may be generated by the electromagnetic field, the temperature and chemical potential gradients, or even fast mechanical rotation of the sample. To determine self-consistently the velocity field $\bu$ (which in practice is a tremendously difficult problem), the system of Eqs. (\[ChargeContinuity\])-(\[RhoMuRelation\]) has to be complemented by the Navier-Stokes equation (derived in Supplemental Material [@suppl_dynamo]) $$\begin{aligned}
\frac{w_\alpha}{v_{F}^2}
\left(\frac{\partial}{\partial t}+\bu\cdot\bnabla \right)\bu=
-\bnabla P_\alpha-\frac{\bu}{v_F^2}\frac{\partial P_\alpha}{\partial t}
+\rho_\alpha\bE+\frac{1}{c}\,\bj_\alpha\times\bB
\nonumber\\
+\frac{\bu}{3}\left(\frac{\partial\varepsilon}{\partial\rho}\right)_\alpha
\left(\chi_\alpha\frac{ge^3}{h^2c}\bE\cdot\bB-\frac{\rho_\alpha-{\rho}_{\bar{\alpha}}}{\tau_{\text{in}}}\right)
+\eta \bnabla^2\bu + \zeta \bnabla\left(\bnabla\cdot\bu\right),
\label{NS}\end{aligned}$$ where $w_\alpha=\varepsilon_\alpha+P_\alpha$ is the the enthalpy of the charge carriers near node $\alpha$ per unit volume, with [@Gorbar:hydroEq] $$\begin{aligned}
\varepsilon_\alpha\approx g\frac{7\pi^4 T^4+30\pi^2\mu_\alpha^2 T^2+15\mu_\alpha^4}{120\pi^2v_F^3\hbar^3}
\label{EpsilonMuRelation}\end{aligned}$$ and $P_\alpha\approx\frac{\varepsilon_\alpha}{3}$ being, respectively, the contributions of node $\alpha$ to the internal energy and pressure; the current $\bj_\alpha$ is given by Eq. (\[Currents\]); $\eta$ and $\zeta$ are the shear and the bulk viscosities; the term $\propto\left(\frac{\partial\varepsilon}{\partial\rho}\right)$ accounts for the change of the energy and pressure of the Weyl liquid near node $\alpha$ due to the internodal scattering and the chiral anomaly, where $\left(\frac{\partial\varepsilon}{\partial\rho}\right)_\alpha=
\frac{3\mu_\alpha}{e}\frac{\mu_\alpha^2+\pi^2 T^2}{3\mu_\alpha^2+\pi^2 T^2}$ for the case of an isothermal flow considered in this paper \[see Supplemental Material [@suppl_dynamo] for the discussion of the assumptions about thermalisation\].
In this paper, we neglect the so-called chiral vortical effect [@Gorbar:hydroEq], i.e. contributions to the current from the interplay of global rotations of the system and chirality imbalance ($\mu_L-\mu_R\neq0$). In the Navier-Stokes equation (\[NS\]) we also neglect terms of higher orders in $u^2/v_F^2$. Equations (\[ChargeContinuity\])-(\[NS\]), together with the Maxwell’s equations and the equations of state, in the form of Eq. (\[EpsilonMuRelation\]) and $P_\alpha=\frac{\varepsilon_\alpha}{3}$, constitute a closed system of equations describing the dynamics of the electromagnetic fields and the electron liquid in a WSM.
Using Eqs. (\[Currents\]), together with the Maxwell’s equations $\bnabla \times\bE=-\frac{1}{c}\frac{\partial\bB}{\partial t}$ and $\bj_L+\bj_R\equiv\bj=\frac{c}{4\pi}\bnabla\times \bB$, where we neglected the displacement current under the assumption of a quasi-stationary flow, we arrive at the equation for the dynamics of the magnetic field: $$\begin{aligned}
\label{Eq:dyn-helical}
\frac{\partial \bB}{\partial t} = {\bm \nabla} \times ({\bm u} \times{\bf B})
+\frac{c^2}{4\pi\sigma}\nabla^2 \bB
+\frac{ge^2}{4\pi^2\hbar^2\sigma}{\bm\nabla}\times\left[(\mu_L-\mu_R)\bB\right],\end{aligned}$$ where $\sigma=2\sigma_{LL}+2\sigma_{LR}$ is the conductivity of the WSM and we have taken into account that the quasiparticles have the same dispersion near the two nodes. [Apart from solid-state WSMs, an equation of the form (\[Eq:dyn-helical\]) with phenomenologically introduced coefficients describes the dynamics of ultrarelativistic chiral particles [@Yamamoto:chiralLiquid]. ]{}
Equation (\[Eq:dyn-helical\]) indicates that Weyl liquids allow for the helicity term for macroscopic fields without turbulence, in contrast with the conventional $\alpha$-dynamo of Krause and R[ä]{}dler [@KR]. However, it can only appear in the presence of an already existing field, and while, as shown below, it can further enhance magnetic field “bootstrap,” it can not lead to generation of the field in and by itself if there is no seed field to begin with. For that, the magnetic Reynolds number (\[Rm\]), $R_m$, has to be large enough, as discussed in the introduction.
To estimate, $R_m$, we use the equation for the Coulomb-interaction dominated conductivity of a Weyl semimetal [@HosurVishwanath] $$\label{sigma}
\sigma \sim \frac{e^2}{\hbar} \frac{k_B T}{\hbar v_F} \frac{1}{\alpha^2},$$ where the Weyl’s “fine-structure constant” is $\alpha = e^2/(\hbar v_F \varkappa)$ and $\varkappa$ is the dielectric constant, which crucially may be rather large. While Eq. (\[sigma\]) has been derived neglecting screening effects [@HosurVishwanath], it should be adequate for estimates. For these purposes, we have also dropped logarithmic renormalisation factors.
![(Colour online) Flow regimes for the electron liquid in a Weyl semimetal on the diagram “fine-structure constant” $\alpha=\frac{e^2}{\varkappa\hbar v_F}$ vs. flow velocity $u$ (log-log scale) for the room temperature $T=T_{\text{room}}=300K$ and the Fermi velocity $v_F=10^8\frac{cm}{s}$. The maximum value of the “fine-structure constant” is $\alpha_{max}=\frac{e^2}{\hbar v_F}\approx 2.2$. \[RegimesPlot\] ](RegimesPlot.png){width="50.00000%"}
Let us emphasise that the dynamo effect is a [*macroscopic classical phenomenon.*]{} The effect if favoured by large system sizes $L$, which lead to large $R_m$. In experiments with solid-state systems the size $L$ is rather limited, with centimetre-size samples being at the upper end of the range accessible for WSMs. Since the effect is not sensitive to quantum interference effects, higher temperatures $T$ are much preferable to maximize $R_m$; the room temperature, $T_{\rm room}$, thus represents a reasonable comparison scale. We emphasise that even at room temperature Weyl semimetals are not Maxwell gases and quantum statistics and quantum nature of the electron-electron scattering are important, but quantum coherence is not essential for the dynamo effect. Using these length and temperature scales, we obtain the following estimate for the main figure of merit in the dynamo theory: $$\begin{aligned}
\label{RmWeyl}
R_m \sim
\frac{1}{\alpha^2}\frac{e^2}{\hbar}\frac{4\pi k_B T}{\hbar v_F c^2 }uL
\sim \frac{10^{-6}}{\alpha^2}
\left( \frac{T}{T_{\rm room}} \right)\times u \left[{\rm \frac{cm}{s}}\right]\, L [{\rm cm}],\end{aligned}$$ where $u$ is the typical velocity of the flow.
Now, we turn to estimates of the hydrodynamic Reynolds number \[R\]. The viscosity of the quasiparticles in a Weyl semimetal at temperature $T$ may be estimated as $\eta\sim n(T)T\tau_{\text{rel}}$, where $n(T)$ is the concentration of the thermally excited quasiparticles and $\tau_{\text{rel}}\sim \hbar(\alpha^2 k_B T)^{-1}$ is the momentum relaxation time. Note that this result follows from second-order perturbation theory in Coulomb interaction and neglects screening effects. This leads to $$\begin{aligned}
\eta\sim\frac{(k_BT)^3}{\alpha^2\hbar^2 v_F^3}.\end{aligned}$$
The motion of a Weyl-semimetal liquid is turbulent in the hydrodynamic sense when the term $\frac{w}{v_F^2}({{\bf u}\cdot\boldsymbol\nabla}){\bf u}$ in the Navier-Stokes equation (\[NS\]) dominates the dissipative terms $\sim \eta\nabla^2{\bf u}$ that come from the viscosity of the Weyl fluid. This yields the following estimate $$\begin{aligned}
R=\frac{wuL}{\eta v_F^2}\sim\alpha^2\frac{k_B T}{\hbar}\frac{uL}{v_F^2}
\sim 4\alpha^2 10^{-3}\left( \frac{T}{T_{\rm room}} \right)
\times u \left[{\rm \frac{cm}{s}}\right]\, L [{\rm cm}],
\label{Restimate}\end{aligned}$$ where we have used the estimate $w\approx\frac{7g\pi^2T^4}{90\hbar^3v_F^3}\sim\frac{(k_B T)^4}{\hbar^3v_F^3}$ for the specific enthalpy at high temperatures.
We note in this context that the viscosity of a Fermi liquid at temperature $T$ may be estimated as $\eta \sim{\varepsilon_F^5}/({T^2 \hbar^2 v_F^3})$, where $\varepsilon_F$ and $v_F$ are the Fermi energy and velocity, respectively. Because the hydrodynamic Reynolds number $R\sim \frac{T^2}{v_F^2\varepsilon_F\hbar}$ gets rapidly suppressed with increasing the Fermi energy $\varepsilon_F$, topological semimetals are indeed a favourable platform for achieving electronic turbulence as compared to “conventional” hydrodynamic metals.
Naturally, the geometry and the magnitude of the velocity field $u$ much depends on the mechanism to stir up hydrodynamic motion and follows from the solution of the Navier-Stoker equations, which is a challenging task in most cases. Furthermore, since observation of a phenomenon of this kind has never been attempted in solid-state materials, it is not clear at the moment what experimental technique would be the most efficient to achieve high hydrodynamic flows – pulsed fields, crossed electric and magnetic fields or just a rapid rotation of the sample are all possibilities to consider. While below we consider in detail one of the standard and simplest dynamo models, we emphasise immediately that the estimates (\[RmWeyl\]) and (\[Restimate\]) are not prohibitive; and it is conceivable that relatively large magnetic Reynolds numbers, necessary for the dynamo to commence, are achievable for realistic flow velocities with $u$ of order one kilometre/second or greater (especially considering that the dielectric constant may be as high as $\varkappa\sim 50$ in WSMs), cf. Fig. \[RegimesPlot\].
Now, we discuss a specific model of dynamo effect – the so-called kinematic Ponomarenko dynamo [@Ponomarenko; @Jones], with an eye on how the terms in MHD equations, descending from the chiral anomaly, change the effect. The Ponomarenko dynamo does not necessarily represent the most experimentally realistic setup, but it does represent the simplest textbook model, which contains the key qualitative features of a dynamo mechanism and is amenable to analytical analysis.
In order for a dynamo action to occur, the magnetic Reynolds number must exceed a critical value $R^{c}_{m}$ [@plasma]. The purpose of the calculation below is to obtain the dependence of the critical Reynolds number, $R^{c}_{m}$, on the helicity term. For simplicity, we neglect the time dependence of the chemical-potential difference $\mu_L-\mu_R$ on the times we consider.
We re-write Eq. (\[Eq:dyn-helical\]) as $$\begin{aligned}
\label{Eq:dyn-helical2}
\frac{\partial \bB}{\partial t} = {\bm \nabla} \times ({\bm u} \times{\bf B})
+\frac{c^2}{4\pi\sigma}\nabla^2 \bB+\xi{\bm\nabla}\times\bB,\end{aligned}$$ where $\xi=ge^2(\mu_L-\mu_R)/(4\pi^2\hbar^2\sigma)$. We consider a cylindrical geometry of the sample with a flow field $\mathbf{u}=(0,r\Omega,u_{0})$, where $\Omega$ and $u_{0}$ are constants, for $r\leq a$, and $\mathbf{u}=0$ for $r>a$ [@plasma]. Plugging the ansatz $\mathbf{B}(r,\theta,z,t)=\mathbf{B}(r) e^{i(n\theta-kz)+\gamma t}$ into (\[Eq:dyn-helical\]), the components of the magnetic field $B_{\pm}=B_{r}\pm iB_{\theta}$ satisfy the equations $$\begin{aligned}
\label{dynamoEq1}
y^2 B''_{\pm}+y B'_{\pm}=\left[q^2y^2+(n\pm1)^2\right]B_{\pm}\nonumber
\\-\delta\left[nyB'_{\mp}\mp n(n\mp1)B_{\mp}\pm k^2a^2y^2B_{\pm}\mp q^2y^2B_r \right] \end{aligned}$$ for $y=r/a\leq1$ and $$\begin{aligned}
\label{dynamoEq2}
y^2 B''_{\pm}+y B'_{\pm}=\left[s^2y^2+(n\pm1)^2\right]B_{\pm} \end{aligned}$$ for $y>1$, where $B'_{\pm}$ ($B''_{\pm}$) is the first (second) derivative with respect to $y$; $\delta=4\pi\sigma \xi/kc^2$, $q^2=k^2a^2+\gamma\tau_{R}+i(n\Omega-ku_{0})$, $s^2=k^2a^2+\gamma\tau_{R}$, where $\tau_{R}=4\pi\sigma a^2/c^2$ is the time scale of the magnetic field diffusion.
![(Colour online) The critical magnetic Reynold number $R_{m}^{c}$ of the $n=1$ kinematic Ponomarenko dynamo as a function of the helicity parameter $\delta=4\pi\sigma \xi/kc^2$, with $k$ being the wave-vector of the dynamo instability. $R^{c}_{m}\simeq 17.7$ is the critical value for the dynamo in the absence of helicity. We note that a self-exciting dynamo will always correspond to the chirality with a lower critical Reynolds number. The chiral anomaly, thus, always aids the dynamo effect.[]{data-label="critical_Rm"}](critical_Rm1.pdf){width="45.00000%"}
For each mode $n$, the magnetic field starts to grow exponentially when $\mathrm{Re}(\gamma)>0$, which occurs if the magnetic Reynolds number exceeds a critical value $R_{m}^{c}$. In the absence of helicity ($\delta=0$), Eq. (\[Eq:dyn-helical2\]) reduces to the conventional dynamo equation and the $n=0$ mode is not excited for an arbitrary intensity of the flow [@plasma]. For non-zero helicity, we solved the inhomogeneous equations (\[dynamoEq1\]) and (\[dynamoEq2\]) with appropriate boundary conditions imposed [@suppl_dynamo] to obtain the dispersion relation for the dynamo mode. The obtained values of $R_{m}^{c}$ [for a dynamo with $n=1$ and a particular direction of wavevector $\bk$ (the $z$ axis)]{} are shown in Fig. \[critical\_Rm\]. The $n=1$ mode is the leading mode, where the dynamo action commences first, and for which the critical magnetic Reynold number is the smallest and potentially within reach for actual Weyl systems. In the absence of helicity (i.e., if $\delta=0$), it is known to be $R^{c}_m\simeq17.7$ [@plasma]. Interestingly enough, the helicity $\delta>0$ reduces the critical value of the magnetic Reynold number for the $n=1$ mode and helps the dynamo action to occur for $R^{c}_m<17.7$. [ Because dynamo flows with various directions of $\bk$ may emerge spontaneously in a turbulent liquid, the presence of helicity (a consequence of the chiral anomaly) would generically aid the dynamo bootstrap in any geometry of the flow.]{}
In conclusion, this paper proposes hydrodynamic Weyl semimetals as a host to electronic turbulence and/or dynamo effect. We derived the Navier-Stokes equations (\[NS\]) and equations of magnetohydrodynamics (\[Eq:dyn-helical\]) and estimated two key figures of merit – the hydrodynamic and magnetic Reynolds numbers. Fig. 1 summarises our findings and shows that both turbulence and dynamo mechanism are in principle experimentally achievable. However, many interesting questions remain, such as experimental signatures of the turbulent electronic motion and the role of “new” terms in the Navier-Stokes equations, descending from the quantum chiral anomaly. Finally, we mention that while three-dimensional Dirac materials are indeed interesting from the perspective of realising the dynamo bootstrap, a number of other electronic materials may also serve as platforms to realize the effect. For example, electronic metals near critical points (e.g., right above a superconducting transition) represent a promising system to look at in this context (both from the perspective of achieving hydrodynamic flows and large Reynolds numbers) and could pave the way to simulating in solid-state materials the effect of magnetic field’s self-excitation – a remarkable phenomenon, usually delegated to the fields of geophysics, astrophysics and cosmology.
[*Acknowledgements.*]{} We are grateful to Matthew Foster, Anton Burkov and Aydin Keser for useful discussions. This research was supported by NSF DMR-1613029 (SS), DOE-BES (DESC0001911) (VG) and the Simons Foundation (VG).
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**Supplemental Material for\
“Dynamo Effect and Turbulence in Hydrodynamic Metals”**
Navier-Stokes equation for a Weyl semimetal
===========================================
In this section we present a microscopic derivation of the Navier-Stokes equation (\[NS\]) for a Weyl semimetal. In the absence of dissipation, electromagnetic fields and internodal scattering processes, the motion of the electronic liquid in a Weyl semimetal is Lorentz-invariant with the Fermi velocity $v_F$ playing the role of the speed of light $c$. Indeed, so long as the crystal lattice of a Weyl semimetal is at rest, Weyl electrons propagate with velocity $v_F$ in all directions regardless of the hydrodynamic velocity $\bu$ of the electron liquid and, thus, obey the relativistic composition law for velocities with the replacement $c\rightarrow v_F$.
The dynamics of such a liquid is described by the relativistic Navier-Stokes equation \[see, for example, Ref. \]. Here, we focus on the interplay of this dynamics with the internodal scattering of electrons (including the chiral anomaly) and electromagnetic fields, which do not transform under representations of the Lorentz group with the speed of light replaced by the Fermi velocity.
In what follows, we set $v_F=1$ and, following Refs. and , introduce the four-position $x^i=(t,\br)$ and the four-velocity $u^i=\gamma(1,\bu)$ of the liquid. The stress-energy tensor of the Weyl electron liquid is given by $$\begin{aligned}
T^{ij}=\gamma w u^i u^j-P g^{ij},
\label{T}\end{aligned}$$ where $w$ is the enthalpy of the liquid per volume; $P$ is the pressure; $\gamma=\left(1-u^2\right)^\frac{1}{2}$; and $g^{ij}=\text{diag}\left(1,-1,-1,-1\right)$ is the metric tensor. The equations of motion of quasiparticles near a Weyl node are given by $$\begin{aligned}
\frac{\partial T^{ij}}{\partial x^i}=\frac{1}{c}F^{jk}j_k+Q^j,
\label{TeqMotion}\end{aligned}$$ where $F^{ij}$ is the effective electromagnetic-field tensor, which we find below, $j_k=(\rho,-\bj)$ is the covariant four-current of Weyl fermions near the node under consideration (in this section we suppress the node index), with $\rho$ being the charge density (cf. Eq. \[RhoMuRelation\]), and $Q^i$ is the effective “force” which comes from the internodal electron dynamics and which we also derive below. In this section, we do not consider dissipation processes due to the viscosity of the electron liquid, as their contribution amounts to the usual viscous force in a relativistic liquid[@LL6S].
[*Lorentz force.*]{} The contribution $S_{em}=-e\int \phi dt+\frac{e}{c}\int\bA d\br$ of the electromagnetic field to the action of a Weyl electron corresponds to the effective electromagnetic four-potential $\tilde{A}^i=(c\phi,\bA)$, defined as $S_{em}=-\frac{e}{c}\int\tilde{A}^i dx_i$, where $c$ is the speed of light (measured in units of the Fermi velocity $v_F$). Using the corresponding electromagnetic-field tensor $$\begin{aligned}
F^{ij}
\equiv \frac{\partial \tilde{A}^j}{\partial x_i}- \frac{\partial \tilde{A}^i}{\partial x_j}
=\left(
\begin{array}{cccc}
0 & -cE_x & -cE_y & -cE_z \\
cE_x & 0 & -B_z & B_y \\
cE_y & B_z & 0 & -B_x \\
cE_z & -B_y & B_x & 0
\end{array}
\right),\end{aligned}$$ we recover the conventional expression $\frac{1}{c}F^{jk}j_k=\rho \bE+\frac{1}{c}\bj\times \bB$ for the Lorenz force acting on a unit volume of the Weyl liquid.
[*Chiral anomaly and internodal scattering.*]{} Simultaneous application of parallel electric and magnetic fields in a Weyl semimetal results in the pumping of charge carriers from one node to the other. Impurities and interactions may also lead to internodal scattering. All these internodal processes contribute to the time derivatives $\frac{\partial T^{0i}}{\partial t}$ (where $i=0,x,y,z$), which corresponds to the four-force \[cf. Eq. (\[TeqMotion\])\] $$Q^i=(\gamma q_w -q_P, \gamma q_w\bu),$$ where $$\begin{aligned}
q_w=\frac{4}{3}\left(\frac{\partial\varepsilon}{\partial \rho}\right)
\left(\chi\frac{ge^3}{h^2c}\bE\cdot\bB-\frac{\rho-\tilde{\rho}}{\tau_{\text{in}}}\right)
\label{qw}
\\
q_P=\frac{1}{3}\left(\frac{\partial\varepsilon}{\partial \rho}\right)
\left(\chi\frac{ge^3}{h^2c}\bE\cdot\bB-\frac{\rho-\tilde{\rho}}{\tau_{\text{in}}}\right)
\label{qP}\end{aligned}$$ are the the rates of change of the enthalpy and the pressure of charge carriers near a given Weyl node due to internodal processes; $\varepsilon$ and $\chi=\pm1$ are the internal energy (per volume) and the chirality of Weyl fermions near this node; $\tilde{\rho}$ is the charge density at the other node. In Eqs. (\[qw\]) and (\[qP\]) we have used that for Weyl fermions $P\approx \varepsilon/3$ and $w\approx 4\varepsilon/3$. The expression $\chi\frac{ge^3}{h^2c}\bE\cdot\bB-\frac{\rho-\tilde{\rho}}{\tau_{\text{in}}}$ in Eqs. (\[qw\]) and (\[qP\]) describes the change of the charge $\rho$ density near a node due to the chiral magnetic effect and internodal scattering \[cf. Eq. (\[Currents\])\].
The change of the internal energy $\varepsilon$ when changing the number of particles near a node depends on the heat transfer between the respective electrons and the environment. In this paper, we focus on isothermal flows, with the system being kept at a constant temperature $T$, and assume that thermal equilibration near the nodes (e.g. due a phonon bath which may flow together with the electron liquid) takes place significantly faster than the internodal particle-transfer processes. Under these conditions, we find from Eqs. (\[RhoMuRelation\]) and (\[EpsilonMuRelation\]) $$\left(\frac{\partial\varepsilon}{\partial\rho}\right)=
\left(\frac{\partial\varepsilon}{\partial\rho}\right)_T=\frac{3\mu}{e}\frac{\mu^2+\pi^2 T^2}{3\mu^2+\pi^2 T^2}.$$ We emphasise, however, that the rate $\left(\frac{\partial\varepsilon}{\partial\rho}\right)$ may, in general, be different under different assumptions about the nature of equilibration processes. For example, if the internodal dynamics is fast compared to internodal equilibration, the former will result in different temperatures or even non-equilibrium distributions of electrons near different nodes.
[*Navier-Stokes equation.*]{} In order to obtain the Navier-Stokes equation, we consider the projection of Eq. (\[TeqMotion\]) on the direction perpendicular to the four-velocity $u^i$: $$\begin{aligned}
\frac{\partial T^{ji}}{\partial x^j}-u^iu_l\frac{\partial T^{lk}}{\partial x^l}
=\frac{1}{c}F^{ik}j_k-\frac{1}{c}u^iu_n F^{nk}j_k
+Q^i-u^iu_jQ^j.
\label{Projection}\end{aligned}$$ Considering the vector components ($i=x,y,z$) of Eq. (\[Projection\]) and using Eqs. (\[T\]), (\[qw\]) and (\[qP\]) and that $u_iu^i=1$, we arrive at the Navier-Stokes equation $$\begin{aligned}
w\left(\frac{\partial}{\partial t}+\bu\cdot\bnabla\right)\bu
=-\bnabla P-\bu\frac{\partial P}{\partial t} +\rho\bE+\frac{1}{c}\,\bj\times \bB
+\frac{\bu}{3}\left(\frac{\partial\varepsilon}{\partial\rho}\right)
\left(\chi\frac{ge^3}{h^2c}\bE\cdot\bB-\frac{\rho-\tilde{\rho}}{\tau_{\text{in}}}\right),\end{aligned}$$ where we neglected corrections of higher orders in $\bu^2$ to each term.
Ponomarenko dynamo and helicity
===============================
In this section we provide the details of derivations of equations governing the kinematic Ponomarenko dynamo for a given flow velocity. For completeness in Sec. \[SI:dynamoconventional\] we present the conventional Ponomarenko dynamo, and then the effects of helicity are examined in Sec. \[SI:dynamohelicity\].
Conventional Ponomarenko Dynamo \[SI:dynamoconventional\]
---------------------------------------------------------
The dynamo equation reads as
=-+ \^2.
[ Following Ref. \[\],]{} we assume that a cylinder is filled up with a conductive liquid with a flow field as
$$\mathbf{u} =
\begin{cases}
(0,r\Omega, u_0) & r\leq a\\
0 & r\geq a
\end{cases},$$ where $a$ is the radius of the cylinder, $\Omega$ is the angular velocity and $u_{0}$ is the velocity along the axis, all taken to be constant. Assuming the ansatz (r,,z,t)=(r) e\^[i(n-kz)+t]{} for the magnetic field and plugging into the dynamo equation, we obtain the $r$- and $\theta$- components of the magnetic fields as
\[Br\_stan\] y\^2 +y =(q\^2y\^2+n\^2+1)B\_r+2inB\_,\
\[Bt\_stan\] y\^2 +y =(q\^2y\^2+n\^2+1)B\_-2inB\_r,
The $z$-component of the magnetic field doesn’t enter the equations above. It can be determined from $\boldsymbol{\nabla}\cdot\mathbf{B}=0$. Letting $B_{\pm}=B_r\pm B_{\theta}$, we get a set of separable equations
\[Bpm\_stan1\] y\^2 +y -\[q\^2y\^2+(n1)\^2\]B\_=0, ya,\
\[Bpm\_stan2\] y\^2 +y -\[s\^2y\^2+(n1)\^2\]B\_=0, y> a, where y=, \_[R]{}=, q\^2=k\^2a\^2+\_[R]{}+i(n-ku\_[0]{})\_[R]{}, s\^2=k\^2a\^2+\_[R]{}.
Equations (\[Bpm\_stan1\]-\[Bpm\_stan2\]) are modified Bessel equations with the solutions \[regBsl\_stan\] B\_(y)=C\_, y1,\
\[sinBsl\_stan\] B\_(y)=D\_, y> 1, where $I_{n}$ and $K_{n}$ are modified Bessel functions. Continuity of the fields across the boundary yields $C_{\pm}=D_{\pm}$. The second matching condition is given due the jumping of angular velocity across the boundary:
\^[y=1\_[+]{}]{}\_[y=1\_[-]{}]{}=i\_[R]{}().
Inserting the solutions (\[regBsl\_stan\]) and (\[sinBsl\_stan\]) in the equations above, we obtain the dispersion relation as \[disp\_stan\] G\_[+]{}G\_[-]{}=\_[R]{}(G\_[+]{}-G\_[-]{}), where G\_=q-s. Here, $'$ denotes derivative with respect to the argument.
Our aim would be to obtain the magnetic Reynold number $R_{m}=\tau_{R}/\tau_{H}$, where $\tau_{H}=a/v$. Here $v$ denotes the typical velocity of the flow. Taking $v=\sqrt{\Omega^2 a^2+u^2_{0}}$, the $R_{m}$ reads as \[Reyld\_mag\] R\_[m]{}=. To find the critical magnetic Reynold number $R^c_m$, beyond which the dynamo action commences, we have to solve equation (\[disp\_stan\]) numerically. We set $\mathrm{Re}(\gamma)=0$, the onset value beyond which the magnetic field grows exponentially for $\mathrm{Re}(\gamma)>0$. We vary $ka$ and $\mathrm{Im}(\gamma \tau_R)$ over a wide range of values and look for unknown variables $\Omega\tau_R$ and $u_0\tau_{R}/a$, all dimensionless, through the imaginary and real parts of the dispersion relation (\[disp\_stan\]). The $n=1$ is the first dynamo mode excited with $R^{c}_{m}\simeq 17.72$.
Ponomarenko dynamo with helicity term\[SI:dynamohelicity\]
----------------------------------------------------------
Now we add the helical term to the dynamo equation as
=-+ \^2+.
In writing down the last term we assumed that the $\xi$ is constant to simplify the subsequent equations. In principle it does depend on the magnetic and electric fields. The last term $\boldsymbol{\nabla}\times \mathbf{B}$ gives rise to new terms involving $B_z$:
e\^[i(n-kz)+t]{} in the $r$-component and
e\^[i(n-kz)+t]{} in the $\theta$-component, which are added to right hand side of Eqs. (\[Br\_stan\]) and (\[Bt\_stan\]), respectively.
Using $\boldsymbol{\nabla}\cdot\mathbf{B}=0$, we write the $z$-component as B\_[z]{}(r)=B\_[r]{}(r)++B\_(r)
\[Br\_chi\] y\^2 +y &=&(q\^2y\^2+n\^2+1)B\_r+2inB\_-(nB\_r+ny+in\^2B\_+ik\^2a\^2y\^2B\_ ) ,\
y\^2 +y &=&(q\^2y\^2+n\^2+1)B\_-2inB\_r-(-ik\^2a\^2y\^2B\_[r]{}-iB\_[r]{}+iy+iy\^2+nB\_-ny )\
\[Bt\_chi\] where $\delta=\tau_R/\tau_c$ with $\tau_c=ka^2/\xi$. In the limit $\xi=0$ ($\delta\rightarrow0$) the equations above reduce to Eqs. (\[Br\_stan\]) and (\[Bt\_stan\]). In (\[Bt\_chi\]) we replace the $y^2\frac{d^2B_{r}}{dy^2}+y\frac{dB_{r}}{dy}$ in the parentheses with the expression (\[Br\_chi\]) and keep the terms up to first order in $\delta$. We get
y\^2 +y &=&(q\^2y\^2+n\^2+1)B\_r+2inB\_-(nB\_r+ny+in\^2B\_+ik\^2a\^2y\^2B\_ ) ,\
y\^2 +y &=&(q\^2y\^2+n\^2+1)B\_-2inB\_r-(-ik\^2a\^2y\^2B\_[r]{}+i(q\^2y\^2+n\^2)B\_r-nB\_-ny )
Rewriting the equations above in terms of $B_{\pm}=B_r\pm iB_{\theta}$, we obtain
\[Bp\] y\^2 +y &=&B\_[+]{}-(ny-n(n-1)B\_[-]{}+k\^2a\^2y\^2B\_[+]{}-q\^2y\^2B\_r ) ,\
y\^2 +y &=&B\_[-]{}-(ny+n(n+1)B\_[+]{}-k\^2a\^2y\^2B\_[-]{}+q\^2y\^2B\_r ). \[Bm\]
In order to examine the effect of helicity $\delta$ on the magnetic Reynold number, in what follows we use equations (\[Bp\]) and (\[Bm\]) to evaluate the $R^{c}_{m}$ discussed in the preceding subsection.
We rewrite the equations as
\[Bp\_m\] y\^2 +y &=&B\_[+]{}-(ny-n(n-1)B\_[-]{}-B\_[-]{} ) ,\
\[Bm\_m\] y\^2 +y &=&B\_[-]{}-(ny+n(n+1)B\_[+]{}+B\_[+]{} ), where \[qpm\] q\_[+]{}\^2=(1+)q\^2-k\^2a\^2, q\_[-]{}\^2=(1-)q\^2+k\^2a\^2.
Without the terms in the parentheses in the right side, we have a set of homogenous equations with helicity included. y\^2 +y &=&B\_[0,+]{},\
y\^2 +y &=&B\_[0,-]{}.
Theses are modified Bessel equations with solutions B\_[0,]{}(y)I\_[n1]{}(q\_y), K\_[n1]{}(q\_y).
Using the above homogenous solutions, we use the perturbation theory to solve the equations in (\[Bp\_m\]-\[Bm\_m\]). We write the solutions up to the first order in $\delta$ as \[ansatz1\] B\_[+]{}(y)=B\_[0,+]{}(y)+A\_[+]{}(y), B\_[-]{}(y)=B\_[0,-]{}(y)+A\_[-]{}(y),Plugging (\[ansatz1\]) into the equations (\[Bp\_m\]-\[Bm\_m\]), we obtain
y\^2 +y &=&A\_[+]{}-(ny-n(n-1)B\_[0,-]{}-B\_[0,-]{} ) ,\
y\^2 +y &=&A\_[-]{}-(ny+n(n+1)B\_[0,+]{}+B\_[0,+]{} ).
We set $n=1$. Using the standard approaches for solving the non-homogeneous differential equations, and after a lengthy but straightforward calculations, we obtain &&A\_[+]{}(y)( y\^4+y\^6 )\
&&A\_[-]{}(y)- y\^4.
Matching the fields across the boundary at $y=1$, we obtain a dispersion relation, which reads as \[disp\_pr\] G\_[+]{}+G(A\^[y=1]{}\_[+]{}s-A’\^[y=1]{}\_[+]{} )=-i\_[R]{}(1-G+A\^[y=1]{}\_[-]{}-G A\^[y=1]{}\_[+]{}), where G=.
Again we solve the dispersion relation (\[disp\_pr\]) numerically to obtain $R_{m}^{c}$; the results are shown in Fig. 2 in the main text.
[99]{} ifxundefined \[1\][ ifx[\#1]{} ]{} ifnum \[1\][ \#1firstoftwo secondoftwo ]{} ifx \[1\][ \#1firstoftwo secondoftwo ]{} ““\#1”” @noop \[0\][secondoftwo]{} sanitize@url \[0\][‘\
12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{} @startlink\[1\] @endlink\[0\] @bib@innerbibempty
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---
abstract: 'We provide a new method to prove and improve the Chemin-Masmoudi criterion for viscoelastic systems of Oldroyd type in [@CM] in two space dimensions. Our method is much easier than the one based on the well-known *losing a priori estimate* and is expected to be easily adopted to other problems involving the losing *a priori* estimate.'
author:
- 'Zhen Lei[^1]'
- 'Nader Masmoudi[^2]'
- 'Yi Zhou[^3]'
title: Remarks on the Blowup Criteria for Oldroyd Models
---
**Keyword**: Losing *a priori* estimate, blowup criteria, Oldroyd model.
Introduction
============
In this paper, we are going to study the non-blowup criteria of solutions of a type of incompressible non-Newtonian fluid flows described by the Oldroyd-B model in the whole 2D space: $$\label{a1}
\begin{cases}
\partial_tv + v\cdot\nabla v + \nabla p = \nu\Delta v + \mu_1\nabla\cdot\tau,\\[-3mm]\\
\partial_t\tau + v\cdot\nabla \tau + a\tau = Q(\tau, \nabla v) + \mu_2 D(v),
\\[-3mm]\\
\nabla\cdot v = 0,
\end{cases}$$ where $v$ is the velocity field, $\tau$ is the non-Newtonian part of the stress tensor and $p$ is the pressure. The constants $\nu$ (the viscosity of the fluid), $a$ (the reciprocal of the relaxation time), $\mu_1$ and $\mu_2$ (determined by the dynamical viscosity of the fluid, the retardation time and $a$) are assumed to be non-negative. The bilinear term $Q$ has the following form: $$\begin{aligned}
\label{a2}
Q(\tau, \nabla v) = W(v)\tau - \tau W(v) + b\big(D(v)\tau + \tau
D(v)\big).\end{aligned}$$ Here $b \in [-1, 1]$ is a constant, $D(v) = \frac{\nabla v +
(\nabla v)^t}{2}$ is the deformation tensor and $W(v) =
\frac{\nabla v - (\nabla v)^t}{2}$ is the vorticity tensor. Fluids of this type have both elastic properties and viscous properties. More discussions and the derivation of Oldroyd-B model can be found in Oldroyd [@Oldroyd] or Chemin and Masmoudi [@CM].
There has been a lot of work on the existence theory of Oldroyd model [@GS-1; @GS-2; @FG; @LM; @CM; @Lei]. In particular, the following Theorem is established by Chemin and Masmoudi in [@CM]:
*Theorem (Chemin and Masmoudi): In two space dimensions, the solutions to the Oldroyd model with smooth initial data do not develop singularities for $t \leq T$ provided that $$\begin{aligned}
\label{a3}
\int_0^{T}\|\tau(t, \cdot)\|_{L^\infty} +
|b| \ \|\tau(t, \cdot)\|_{L^2}^2 dt < \infty.\end{aligned}$$*
To establish the blowup criterion , the authors in [@CM] use a losing *a priori* estimate for solutions of transport equations which was developed by Bahouri and Chemin [@BC] and used later on by a lot of authors (for example, see [@CL; @CM; @ConM; @LZZ; @MZZ08; @LM07; @Masmoudi08cpam] and the references therein). Our purpose of this paper is to provide a simple method which avoids using the complicated losing *a priori* estimate and to improve the blowup criterion for Oldroyd model established by Chemin and Masmoudi [@CM]. To best illustrate our ideas and for simplicity, we will take $a = 0$ and $\nu = \mu_1 = \mu_2 = b = 1$ throughout this paper. More precisely, we study the following system $$\label{a4}
\begin{cases}
\partial_tv + v\cdot\nabla v + \nabla p = \Delta v + \nabla\cdot\tau,\\[-4mm]\\
\partial_t\tau + v\cdot\nabla \tau = \nabla v\tau + \tau(\nabla v)^t + D(v),
\\[-4mm]\\
\nabla\cdot v = 0.
\end{cases}$$ We point out here that the results in this paper are obviously true for general constants $a, \mu_1, \mu_2 \geq 0$, $\nu, b
> 0$ from our proofs.
Our main result concerning system is:
\[thm-criterion-Oldroyd\] Assume that $(v, \tau)$ is a local smooth solution to the Oldroyd model on $[0, T)$ and $\|v(0, \cdot)\|_{L^2 \cap \dot{C}^{1 +
\alpha} (\mathbb{R}^2) } + \|\tau(0, \cdot)\|_{ L^1 \cap \dot{C}^\alpha (\mathbb{R}^2)} < \infty$ for some $\alpha \in (0, 1)$. Then one has $$\begin{aligned}
\nonumber
\|v(t, \cdot)\|_{\dot{C}^{1 + \alpha}} + \|\tau(t,
\cdot)\|_{\dot{C}^\alpha} < \infty\end{aligned}$$ for all $0 \leq t \leq T$ provided that $$\begin{aligned}
\label{a5}
\|\tau(t, \cdot)\|_{L^1_T({\rm BMO})} < \infty \quad {and}\
\|\tau\|_{L^\infty_T(L^1)} < \infty.\end{aligned}$$
This result is in the spirit of the Beale-Kato-Majda [@BKM] non blowup criterion for $3D$ Euler equations. There were many subsequent results improving the criterion, see for instance [@Kozonoaniuchi; @KOT02; @Planchon03cmp]. In particular our result still holds if we replace $BMO$ with the Besov space $B^0_{\infty, \infty}$ used in H. Kozono, T. Ogawa, and Y. Taniuchi. [@KOT02] or if we replace the condition by the one introduced in Planchon [@Planchon03cmp]. In other words, the first condition in in the above theorem can be weakened to $$\begin{aligned}
\nonumber
\int_{0}^{T} \| \tau \|_{B^0_{\infty, \infty}} = \int_{0}^{T}
\sup_q \|\Delta_q\tau(t, \cdot)\|_{L^\infty}dt < \infty,\end{aligned}$$ or to $$\begin{aligned}
\nonumber
\lim_{\delta\rightarrow 0}\sup_q \int_{T - \delta}^{T}
\|\Delta_q\tau(t, \cdot)\|_{L^\infty}dt < \epsilon,\end{aligned}$$ for some sufficiently small $\epsilon > 0$. The second condition in can be replaced by $$\begin{aligned}
\nonumber
\|\tau\|_{L^2_T(L^2)} < \infty,\end{aligned}$$ which was used in [@CM].
It is easy to check that smooth solutions to enjoy the following energy law: $$\begin{aligned}
\label{energy}
\int_{\mathbb{R}^2}|v(t, \cdot)|^2 + {\rm tr}\tau(t, \cdot) dx +
\int_0^t\int_{\mathbb{R}^2}|\nabla v|^2dxds =
\int_{\mathbb{R}^2}|v(0, \cdot)|^2 + {\rm tr}\tau(0, \cdot) dx,\end{aligned}$$ which means that $$\begin{aligned}
\label{aprioriestimate}
v \in L^\infty_T(L^2) \cap L^2_T(\dot{H}^1)\end{aligned}$$ for all $T > 0$ under the second condition of . The *a priori* estimate will be important to apply lemma \[lem-CM\].
Finally, it is well-known that if $A = 2 \tau + I$ is a positive definite symmetric matrix at $t=0$ (which is actually the physical case), then this property is conserved for later times. Indeed, $A$ satisfies the equation $$\partial_t A + v\cdot\nabla A = \nabla v A + A (\nabla v)^t .$$ Also, if at $t=0$, we have $\det(A(0)) > 1$ and $A$ is positive definite, then this will also hold for later times (see [@HL07]). In particular this implies that tr$(\tau) > 0$ (or one has $- 1 < {\rm tr}(\tau) \leq 0$, which contradicts with $\det(A) > 1$). Hence, we have the following corollary where we also use the improved criterion of Planchon.
\[cor-criterion-Oldroyd\] There exists an $\epsilon > 0$, such that if $(v, \tau)$ is a local smooth solution to the Oldroyd model on $[0, T)$, $\|v(0, \cdot)\|_{L^2 \cap \dot{C}^{1 +
\alpha} (\mathbb{R}^2) } + \|\tau(0, \cdot)\|_{ L^1 \cap \dot{C}^\alpha (\mathbb{R}^2)} < \infty$ for some $\alpha \in (0, 1)$ and that det$(I + 2 \tau(0)) > 1, \, A = I + 2 \tau(0) $ is positive definite symmetric. Then one has $$\begin{aligned}
\nonumber
\|v(t, \cdot)\|_{\dot{C}^{1 + \alpha}} + \|\tau(t,
\cdot)\|_{\dot{C}^\alpha} < \infty\end{aligned}$$ for all $0 \leq t \leq T$ provided that $$\begin{aligned}
\label{a5-cor}
\lim_{\delta\rightarrow 0}\sup_q \int_{T - \delta}^{T}
\|\Delta_q\tau(t, \cdot)\|_{L^\infty}dt < \epsilon.
% \|\tau(t, \cdot)\|_{L^1_T({\rm BMO})} < \infty.
% \quad {and}\
% \|\tau\|_{L^\infty_T(L^1)} < \infty.\end{aligned}$$
Our proof is based on careful H${\rm \ddot{o}}$lder estimates of heat and transport equations and the standard Littlewood-Paley theory, which is much easier than the extensively used losing *a priori* estimates (for example, see [@BC; @CL; @CM; @ConM; @LZZ]). In fact, the main innovation of this paper is that our analysis may be viewed as a replacement of the losing *a priori* estimate. Our method is expected to be easily adopted to other problems via the losing *a priori* estimate. Moreover, our criterion slightly improves the one established by Chemin and Masmoudi (see [@CM]).
Finally, let us make a remark on MHD: $$\label{MHD}
\begin{cases}
\partial_tv + v\cdot\nabla v + \nabla p = \nu\Delta v + \nabla\cdot(H\times H),\\[-4mm]\\
\partial_tH + v\cdot\nabla H = H\cdot\nabla v, \\[-4mm]\\
\nabla\cdot v = \nabla\cdot H = 0,
\end{cases}$$ where $H$ denotes the magnetic field. A direct corollary of Theorem \[thm-criterion-Oldroyd\] for MHD is the following:
\[thm-criterion-MHD\] Assume that $(v, H)$ is a local smooth solution to MHD on $[0, T)$ and $\|v(0, \cdot)\|_{L^2\cap\dot{C}^{1 + \alpha}} +
\|H(0, \cdot)\|_{L^2\cap\dot{C}^\alpha} < \infty$ for some $\alpha
\in (0, 1)$. Then one has $$\begin{aligned}
\nonumber
\|v(t, \cdot)\|_{\dot{C}^{1 + \alpha}} + \|H(t,
\cdot)\|_{\dot{C}^\alpha} < \infty\end{aligned}$$ all $0 \leq t \leq T$ provided that $$\begin{aligned}
\label{a9}
\int_0^{T}\|(H\times H)(t, \cdot)\|_{{\rm BMO}}dt < \infty.\end{aligned}$$
The proof of this corollary is given in section 4. Unfortunately, at present we are not able to improve as $$\begin{aligned}
\nonumber
\int_0^{T}\|H(t, \cdot)\|_{{\rm BMO}}^2dt < \infty,\end{aligned}$$ and this is still an open problem.
The paper is organized as follows: Section 2 is devoted to recalling some basic properties of Littlewood-Paley theory and proving two interpolation inequalities. The proof of Theorem \[thm-criterion-Oldroyd\] is given in section 3. In the last section we sketch the proof of Corollary \[thm-criterion-MHD\].
Preliminaries
=============
Let $\mathcal{S}(\mathbb{R}^2)$ be the Schwartz class of rapidly decreasing functions. Given $f \in \mathcal{S}(\mathbb{R}^2)$, its Fourier transform $\mathcal{F}f = \hat{f}$ (inverse Fourier transform $\mathcal{F}^{-1}g = \breve{g}$, respectively) is defined by $\hat{f}(\xi) = \int e^{-ix\cdot\xi}f(x)dx$ ($\breve{g}(x) = \int e^{ix\cdot\xi}g(\xi)d\xi$, respectively). Now let us recall the Littlewood-Paley decomposition (see [@Bony; @Chemin]). Choose two nonnegative radial functions $\psi, \phi \in \mathcal{S}(\mathbb{R}^2)$, supported respectively in $B = \{\xi \in \mathbb{R}^2: |\xi| \leq \frac{4}{3}$ and $C =
\{\xi \in \mathbb{R}^2: \frac{3}{4} \leq |\xi| \leq \frac{8}{3}$ such that $$\nonumber
\psi(\xi) + \sum_{j \geq 0}\phi(\frac{\xi}{2^j}) = 1\ {\rm for}\
\xi \in R^2,\quad \sum_{- \infty \leq j \leq
\infty}\phi(\frac{\xi}{2^j}) = 1\ {\rm for}\ \xi \in R^2/\{0\}.$$ The frequency localization operator is defined by $$\label{b3}
\Delta_qf = \int_{\mathbb{R}^2}\check{\phi}(y)f(x - 2^{-
q}y)dy,\quad S_qf = \int_{\mathbb{R}^2}\check{\psi}(y)f(x - 2^{-
q}y)dy.$$
The following Lemma is well-known (for example, see [@Chemin]).
\[lem2\] For $s \in \mathbb{R}$, $1 \leq p \leq \infty$ and integer $q$, one has $$\label{b2}
\begin{cases}
c2^{qs}\|\Delta_qf\|_{L^p} \leq \|\nabla^s\Delta_qf\|_{L^p} \leq
C2^{qs}\|\Delta_qf\|_{L^p},\\[-4mm]\\
\big\||\nabla|^sS_qf\big\|_{L^p} \leq C2^{qs}\|f\|_{L^p},\\[-4mm]\\
ce^{- C2^{2q}t}\|\Delta_qf\|_{L^\infty} \leq
\|e^{t\Delta}\Delta_qf\|_{L^\infty} \leq Ce^{-
c2^{2q}t}\|\Delta_qf\|_{L^\infty}.
\end{cases}$$ Here $C$ and $c$ are positive constants independent of $s$, $p$ and $q$.
We also need the following Lemma (see also [@MZZ08; @Masmoudi09prep] where similar estimates were used.)
\[lem1\] Assume that $\beta > 0$. Then there exists a positive constant $C
> 0$ such that $$\label{b1}
\begin{cases}
\|f\|_{L^\infty} \leq C\big(1 + \|f\|_{L^2} + \|f\|_{{\rm
BMO}}\ln(e + \|f\|_{\dot{C}^\beta})\big),\\[-4mm]\\
%\|f\|_{L^\infty} \leq C\big(1 + \|f\|_{L^2} + \|\nabla f\|_{L^2}
%\ln^{\frac{1}{2}}(1 + \|f\|_{\dot{C}^\beta})\big),\\[-4mm]\\
\int_0^T\|\nabla g(s, \cdot)\|_{L^\infty}ds \leq
C\Big(1 + \int_0^T\|g(s, \cdot)\|_{L^2}ds\\
\quad +\ \sup_{q}\int_0^T\|\Delta_q\nabla g(s,
\cdot)\|_{L^\infty}ds\ln\big( e + \int_0^T \|\nabla
g(s, \cdot)\|_{\dot{C}^\beta}ds\big)\Big).
\end{cases}$$
The first inequality is well-known. For example, see [@BKM; @Kozonoaniuchi; @LZ]. To prove the second inequality, we use the Littlewood-Paley theory to compute that $$\begin{aligned}
\nonumber
&&\int_0^T\|\nabla g(s, \cdot)\|_{L^\infty}ds \leq
C\int_0^T\|\sum_{q \leq 0}\nabla \Delta_qg(s,
\cdot)\|_{L^\infty}ds\\\nonumber
&&\quad +\ CN\max_{1 \leq q \leq N}\int_0^T\|\Delta_q\nabla g(s,
\cdot)\|_{L^\infty}ds + \int_0^T\sum_{q \geq N +
1}2^{-\beta q}2^{\beta q}\|\Delta_q\nabla g(s, \cdot)\|_{L^\infty}ds\\\nonumber
&&\leq C\Big(\int_0^T\|g(s, \cdot)\|_{L^2}ds + \sup_{1 \leq q \leq
N}\int_0^TN\|\Delta_q\nabla g(s, \cdot)\|_{L^\infty}ds\\\nonumber
&&\quad +\ 2^{- \beta N}\int_0^T\|\nabla g(s, \cdot)
\|_{\dot{C}^\beta}ds\Big).\end{aligned}$$ Then the second inequality in Lemma \[lem1\] follows by choosing $$\nonumber
N = \frac{1}{\beta}\log_2\big(e + \int_0^T \|\nabla g(s,
\cdot)\|_{\dot{C}^\beta}\big) \leq C\ln\big(e + \int_0^T \|\nabla
g(s, \cdot)\|_{\dot{C}^\beta}ds\big).$$
Blowup Criteria for Oldroyd-B Model
===================================
This section is devoted to establishing the blowup criterion for the Oldroyd-B Model and proving Theorem \[thm-criterion-Oldroyd\]. Our analysis is based on careful H${\rm \ddot{o}}$lder estimates of heat and transport equations and the standard Littlewood-Paley theory, which is much easier than the extensively used losing *a priori* estimates (for example, see [@BC; @CL; @CM; @ConM; @LZZ]). Moreover, our criterion slightly improves the one established by Chemin and Masmoudi (see [@CM]). We divide our proof into two steps. The first step is focused on establishing some *a priori* estimates for 2-D Navier-Stokes equations. Then we establish H${\rm \ddot{o}}$lder estimates for the velocity field $v$ and the stress tensor $\tau$ in the second step.
**Step 1. The *a priori* estimates for 2-D Navier-Stokes equations**.
We need the following Lemma which is basically established by Chemin and Masmoudi in [@CM]. For completeness, the proof will be also sketched here.
\[lem-CM\](Chemin-Masmoudi) Let $v$ be a a solution of the Navier-Stokes equations with initial data in $L^2$ and an external force $f \in
\widetilde{L}^1_T(C^{-1}) \cap L^2_T(H^{-1})$: $$\label{c1}
\begin{cases}
\partial_tv + v\cdot\nabla v + \nabla p = \Delta v + f,\\[-4mm]\\
\nabla\cdot v = 0,\\[-4mm]\\
v(0, x) = v_0(x).
\end{cases}$$ Then we have the following a priori estimate: $$\begin{aligned}
\label{c2}
&&\|v\|_{\widetilde{L}^1_T(C^1)} \leq
C\Big(\sup_q\|\Delta_qv_0\|_{L^2}\big(1 - \exp\{-c2^{2q}T\}\big)\\\nonumber
&&\quad +\ \big(\|v_0\|_{L^2} + \|f\|_{L^2_T(\dot{H}^{-1})}\big)
\|\nabla v\|_{L^2_T(L^2)}^2 + \sup_q\int_0^T\|2^{-q}\Delta_qf(s)\|_{L^\infty}ds
\Big).\end{aligned}$$
First of all, applying the operator $\Delta_q$ to the 2-D Navier-Stokes equations and then using Lemma \[lem2\] and the standard energy estimate, we deduce that $$\begin{aligned}
\nonumber
&&\frac{1}{2}\frac{d}{dt}\|\Delta_qv\|_{L^2}^2 +
c2^{2q}\|\Delta_qv\|_{L^2}^2\\\nonumber
&&\leq \|2^q\Delta_qv\|_{L^2}\big(\|2^{-q}\Delta_qf\|_{L^2} +
\|\Delta_q(v \otimes v)\|_{L^2}\big)\\\nonumber
&&\leq c2^{2q}\|\Delta_qv\|_{L^2}^2 + C\big(\|2^{-q}
\Delta_qf\|_{L^2}^2 + \|\Delta_q(v \otimes v)\|_{L^2}^2\big).\end{aligned}$$ Integrating with respect to time and summing over $q$, we get $$\begin{aligned}
\nonumber
&&\sum_q\|\Delta_qv\|_{L^\infty_T(L^2)}^2 \leq
\|v_0\|_{L^2}^2 + C\big(\|f\|_{L^2_T(\dot{H}^{-1})}^2
+ \|v \otimes v\|_{L^2_T(L^2)}^2\big)\\\nonumber
&&\leq \|v_0\|_{L^2}^2 + C\big(\|f\|_{L^2_T(\dot{H}^{-1})}^2 +
\|v\|_{L^\infty_T(L^2)}^2\|\nabla v\|_{L^2_T(L^2)}^2\big),\end{aligned}$$ where we used the standard interpolation inequality $\|v\|_{L^4}^2
\leq C\|v\|_{L^2}\|\nabla v\|_{L^2}$. Recalling the basic energy estimate $$\begin{aligned}
\label{c3}
\|v\|_{L^\infty_T(L^2)}^2 + \|\nabla v\|_{L^2_T(L^2)}^2 \leq
\|v_0\|_{L^2}^2 + \|f\|_{L^2_T(\dot{H}^{-1})}^2,\end{aligned}$$ one has $$\begin{aligned}
\label{c4}
&&\sum_q\|\Delta_qv\|_{L^\infty_T(L^2)}^2 \leq
C\big(\|v_0\|_{L^2}^2 + \|f\|_{L^2_T(\dot{H}^{-1})}^2\big)\\\nonumber
&&\quad \times\big(1 + \|v_0\|_{L^2}^2 +
\|f\|_{L^2_T(\dot{H}^{-1})}^2\big).\end{aligned}$$
Next, let us apply $\Delta_q$ to and use Lemma \[lem2\] to estimate $$\begin{aligned}
\nonumber
&&\|\Delta_qv(t)\|_{L^\infty} \leq C\|\Delta_q
v_0\|_{L^\infty}e^{-c2^{2q}t}\\\nonumber
&&\quad +\ \int_0^t\big(\|\Delta_qf(s)\|_{L^\infty} +
\|\Delta_q\nabla\cdot(v \otimes v)(s)\|_{L^\infty} \big)
e^{-c2^{2q}(t - s)}ds,\end{aligned}$$ which yields $$\begin{aligned}
\label{c5}
&&\|v\|_{\widetilde{L}^1_T(C^1)} \leq C\sup_q\int_0^T\|\Delta_q
v_0\|_{L^\infty}2^qe^{-c2^{2q}t}dt\\\nonumber
&&\quad +\ C\sup_q\int_0^T\int_0^t\|\Delta_qf(s)
\|_{L^\infty}2^qe^{-c2^{2q}(t - s)}dsdt\\\nonumber
&&\quad +\ C\sup_q\int_0^T\int_0^t\|\Delta_q\nabla\cdot(v \otimes
v)(s)\|_{L^\infty}2^qe^{-c2^{2q}(t - s)}dsdt\\\nonumber
&&\leq C\sup_q\|\Delta_q v_0\|_{L^2}\big(1 -
e^{-c2^{2q}T}\big)\\\nonumber
&&\quad +\ C\sup_q\int_0^T\|\Delta_q(v \otimes
v)(s)\|_{L^\infty}ds + \|f\|_{\widetilde{L}^1_T
(C^{-1})}.\end{aligned}$$ Using the Bony’s decomposition, one can write $$\begin{aligned}
\nonumber
&&\|\Delta_q(v \otimes v)(s)\|_{L^\infty} =
\sum_{|p - r| \leq 1}\|\Delta_q(\Delta_pv \otimes
\Delta_rv)(s)\|_{L^\infty}\\\nonumber
&&\quad +\ \sum_{p - r \geq 2}\|\Delta_q(\Delta_pv
\otimes \Delta_rv)(s)\|_{L^\infty}
+ \sum_{r - p \geq 2}\|\Delta_q(\Delta_pv \otimes
\Delta_rv)(s)\|_{L^\infty}.\end{aligned}$$ A straightforward computation gives $$\begin{aligned}
\nonumber
&&\int_0^T\sum_{|p - r| \leq 1}\|\Delta_q(\Delta_pv \otimes
\Delta_rv)(s)\|_{L^\infty}ds\\\nonumber
&&\leq C\int_0^T\sum_{|p - r| \leq 1}2^q\|\Delta_q(\Delta_pv
\otimes \Delta_rv)(s)\|_{L^2}ds\\\nonumber
&&\leq C\int_0^T\sum_{|p - r| \leq 1,\ p \geq q - 3}
2^{q - \frac{p + r}{2}}\|2^p\Delta_pv\|_{L^\infty}^{\frac{1}{2}}
\|\Delta_rv\|_{L^2}^{\frac{1}{2}}\|\Delta_pv\|_{L^\infty}^{\frac{1}{2}}
\|2^r\Delta_rv\|_{L^2}^{\frac{1}{2}}ds\\\nonumber
&&\leq C\int_0^T\sum_{|p - r| \leq 1,\ p \geq q - 3}
2^{q - \frac{p + r}{2}}\|2^p\Delta_pv\|_{L^\infty}^{\frac{1}{2}}
\|\Delta_rv\|_{L^2}^{\frac{1}{2}}\|2^p\Delta_pv\|_{L^2}^{\frac{1}{2}}
\|2^r\Delta_rv\|_{L^2}^{\frac{1}{2}}ds\\\nonumber
&&\leq C\|v\|_{L^\infty_T(L^2)}^{\frac{1}{2}}\|\nabla v
\|_{L^2_T(L^2)}\|v\|_{\widetilde{L}^1_T(C^1)}^{\frac{1}{2}}.\end{aligned}$$ Similarly, one has $$\begin{aligned}
\nonumber
&&\int_0^T\Big(\sum_{p - r \geq 2}\|\Delta_q(\Delta_pv
\otimes \Delta_rv)(s)\|_{L^\infty}
+ \sum_{r - p \geq 2}\|\Delta_q(\Delta_pv \otimes
\Delta_rv)(s)\|_{L^\infty}\Big)ds\\\nonumber
&&\leq C\int_0^T\sum_{p - r \geq 2,\ | p - q| \leq 2}
\|\Delta_pv\|_{L^\infty}\|\Delta_rv\|_{L^\infty}ds\\\nonumber
&&\leq C\int_0^T\sum_{p - r \geq 2,\ | p - q| \leq 2}
\|2^p\Delta_pv\|_{L^\infty}^{\frac{1}{2}}\|2^p\Delta_pv
\|_{L^2}^{\frac{1}{2}}2^{r - \frac{p}{2}}\|\Delta_rv\|_{L^2}ds\\\nonumber
&&\leq C\|v\|_{L^\infty_T(L^2)}^{\frac{1}{2}}\|\nabla v
\|_{L^2_T(L^2)}\|v\|_{\widetilde{L}^1_T(C^1)}^{\frac{1}{2}}.\end{aligned}$$ Using the above two estimate, one can improve as $$\begin{aligned}
\nonumber
\|v\|_{\widetilde{L}^1_T(C^1)} \leq C\Big(\sup_q\|\Delta_q
v_0\|_{L^2}\big(1 - e^{-c2^{2q}T}\big) +
\|v\|_{L^\infty_T(L^2)}\|\nabla v\|_{L^2_T(L^2)}^2 +
\|f\|_{\widetilde{L}^1_T(C^{-1})}\Big).\end{aligned}$$ Consequently, one can deduce from and the basic energy estimate and the above inequality.
Now let us assume that $f \in L^1_T(\dot{C}^{-1}) \cap
L^2_T(H^{-1})$. By Lemma \[lem-CM\], it is easy to see that $$\begin{aligned}
\label{c6}
&&\|v\|_{\widetilde{L}^1_{[t_0, T]}(C^1)}\leq
C\Big(\sup_q\|\Delta_qv(t_0)\|_{L^2}\big(1 - \exp\{-c2^{2q}(T - t_0)\}\\\nonumber
&&\quad +\ \int_{t_0}^T\sup_q\|2^{-q}\Delta_qf(s)\|_{L^\infty}ds +
\big(\|v(t_0)\|_{L^2} + \|f\|_{L^2_{[t_0, T]}(\dot{H}^{-1})}\big)
\|\nabla v\|_{L^2_{[t_0, T]}(L^2)}^2\Big)\end{aligned}$$ holds for any $t_0 \in [0, T)$. By , one can choose some $q_0$ such that $$\begin{aligned}
\nonumber
\sup_{q > q_0}\|\Delta_qv\|_{L^\infty_{[t_0, T]}(L^2)}^2 \leq
\frac{\epsilon}{4C}.\end{aligned}$$ Furthermore, by the basic energy estimate , one can choose some $t_1 \in [t_0, T)$ such that $$\begin{aligned}
\nonumber
&&\sup_{t_1 \leq t \leq T}\sup_{q \leq q_0}
\|\Delta_qv(t)\|_{L^2}\big(1 - \exp\{-c2^{2q}(T - t)\}\\\nonumber
&&\leq \sup_{t_1 \leq t \leq T}\|v(t)\|_{L^2}2c2^{2q_0}(T -
t_1)\\\nonumber
&&\leq C2^{2q_0}\big(\|v_0\|_{L^2} + \|f\|_{L^2_{[0,
T]}(\dot{H}^{-1})}\big)(T - t_1) \leq \frac{\epsilon}{4C}.\end{aligned}$$ Consequently, one has $$\begin{aligned}
\label{c7}
\sup_{t_1 \leq t \leq T}\sup_q\|\Delta_qv(t)\|_{L^2}\big(1 -
\exp\{-c2^{2q}(T - t)\} \leq \frac{\epsilon}{2C}.\end{aligned}$$ On the other hand, it is obvious that one can choose some $t_2 \in
[t_1, T)$ such that $$\begin{aligned}
\label{c8}
&&\big(\sup_{t_2 \leq t \leq T}\|v(t)\|_{L^2} + \|f\|_{L^2_{[t_2,
T]}(\dot{H}^{-1})}\big)\|\nabla v\|_{L^2_{[t_2, T]}(L^2)}^2\\\nonumber
&&\quad +\ \int_{t_2}^T\sup_q\|2^{-q}\Delta_qf(s)\|_{L^\infty}ds
\leq \frac{\epsilon}{2C}.\end{aligned}$$ Combining and with , one arrives at $$\begin{aligned}
\label{c9}
\|v\|_{\widetilde{L}^1_{[t_2, T]}(C^1)} \leq \epsilon.\end{aligned}$$
**Step 2. H${\rm \ddot{o}}$lder estimate for $v$ and $\tau$.**
First of all, by and the assumption , one can choose $t_\star \in [t_2, T)$ such that $$\begin{aligned}
\label{d1}
\|v\|_{\widetilde{L}^1_{[t_\star, T]}(C^1)} \leq \epsilon,\quad
\|\tau\|_{L^1_{[t_\star, T]}({\rm BMO})} \leq \epsilon.\end{aligned}$$ For $0 \leq t < T$, define $$\begin{aligned}
\nonumber
A(t) = \sup_{0 \leq s < t}\|v(t, \cdot)\|_{\dot{C}^{1 +
\alpha}},\quad B(t) = \sup_{0 \leq s < t}\|\tau(t,
\cdot)\|_{\dot{C}^{\alpha}}.\end{aligned}$$ We are about to estimate $A(t)$ and $B(t)$ for $0 \leq t < T$. For this purpose, let us apply $\Delta_q$ to the Oldroyd-B system to get $$\label{d2}
\begin{cases}
\partial_t\Delta_qv - \Delta \Delta_qv + \nabla\Delta_q p
= \nabla\cdot\Delta_q(\tau - v\otimes v),\\[-3mm]\\
\partial_t\Delta_q\tau + v\cdot\nabla \Delta_q\tau = \Delta_q\big(\nabla v\tau
+ \tau(\nabla v)^t + D(v)\big) + [v\cdot
\nabla, \Delta_q]\tau.
\end{cases}$$
Let us first estimate $\|v(t, \cdot)\|_{\dot{C}^{1 + \alpha}}$. By the first equation in and Lemma \[lem2\], one has $$\begin{aligned}
\label{d3}
&&\|\Delta_qv\|_{L^\infty} \leq Ce^{-c2^{2q}t}
\|\Delta_qv(0)\|_{L^\infty}\\\nonumber
&&\quad +\ \int_{0}^te^{-c2^{2q}(t - s)} \big\|
\nabla\cdot\Delta_q(\tau - v\otimes
v)\big\|_{L^\infty}(s)ds.\end{aligned}$$ Multiplying $2^{q(1 + \alpha)}$ to both sides of , we have $$\begin{aligned}
\nonumber
&&\|\Delta_qv(t, \cdot)\|_{\dot{C}^{1 + \alpha}} \leq
C\|v(0, \cdot)\|_{\dot{C}^{1 + \alpha}}\\\nonumber
&&\quad +\ C\int_{0}^t2^{2q}e^{-c2^{2q}(t - s)}
\|\Delta_q\tau\|_{\dot{C}^\alpha}ds\\\nonumber
&&\quad +\ C\int_{0}^t 2^{\frac32 q}e^{-c2^{2q}(t -
s)}\|(v \otimes v)(s, \cdot)\|_{\dot{C}^{1/2 + \alpha}}ds\\\nonumber
&&\leq C\big(\|v(0, \cdot)\|_{\dot{C}^{1 + \alpha}} +
B(t)\big) + C\Big(\int_{0}^t\|v\|_{L^4}^4
\|v\|_{\dot{C}^{1 + \alpha}}^4ds\Big)^{\frac{1}{4}},\end{aligned}$$ where we have used Holder inequality and the fact that $\| v \otimes v \|_{\dot{C}^{1/2 + \alpha}} \leq C
\|v\|_{L^4}
\|v\|_{\dot{C}^{1 + \alpha}}$. Consequently, there holds $$\begin{aligned}
\nonumber
&&\|v(t, \cdot)\|_{\dot{C}^{1 + \alpha}}^4 \leq
C\big(\|v(0, \cdot)\|_{\dot{C}^{1 + \alpha}} +
B(t)\big)^4 + C\int_{0}^t\|v\|_{L^4}^4
\|v\|_{\dot{C}^{1 + \alpha}}^4ds\\\nonumber
&&\leq C\big(\|v(0, \cdot)\|_{\dot{C}^{1 + \alpha}} +
B(\widetilde{t})\big)^4 + C\int_{0}^{t}\|v(s, \cdot)\|_{
L^2}^2\|\nabla v(s, \cdot)\|_{
L^2}^2\|v(s, \cdot)\|_{\dot{C}^{1 + \alpha}}^4ds%\\\nonumber
%&&\quad +\ C\int_{0}^{t}\|\nabla v(s, \cdot)\|_{
% L^2}^2\|v(s, \cdot)\|_{\dot{C}^{1 + \alpha}}^2
% \ln(2 + \|v(s, \cdot)\|_{\dot{C}^{1 + \alpha}})ds\\\nonumber
%&&\leq C\Big(\|v(0, \cdot)\|_{\dot{C}^{1 + \alpha}}^2 +
% B(\widetilde{t})^2 + \int_{0}^{t}\big(1 + \|v(s, \cdot)\|_{
% L^2}^2\\\nonumber
%&&\quad +\ \|\nabla v(s, \cdot)\|_{
% L^2}^2\big)\|v(s, \cdot)\|_{\dot{C}^{1 + \alpha}}^2
% \ln(2 + \|v(s, \cdot)\|_{\dot{C}^{1 + \alpha}})ds\Big)\end{aligned}$$ for any fixed $\widetilde{t}: 0 \leq \widetilde{t} < T$ and $t \leq
\widetilde{t} < T$. Here we used the fact that $B(t)$ is nondecreasing. Consequently, Gronwall’s inequality gives that $$\begin{aligned}
\nonumber
&&A(\widetilde{t})^4 = \sup_{0 \leq t < \widetilde{t}}\|v(t,
\cdot)\|_{\dot{C}^{1 + \alpha}}^4\\\nonumber
&&\leq C\big(\|v(0, \cdot)\|_{\dot{C}^{1 + \alpha}} +
B(\widetilde{t})\big)^4\exp\Big\{C\int_{0}^{t}\|v(s, \cdot)\|_{
L^2}^2\|\nabla v(s, \cdot)\|_{L^2}^2ds\Big\}.\end{aligned}$$ Since $\widetilde{t} \in [0, T)$ is arbitrary, using the basic energy inequality , we in fact have $$\begin{aligned}
\label{d5}
A(t) \leq C\big(\|v(0, \cdot)\|_{\dot{C}^{1 + \alpha}} +
B(t)\big),\quad 0 \leq t < T.\end{aligned}$$
Next, by the second equation in , we have $$\begin{aligned}
\nonumber
&&\|\Delta_q\tau(t, \cdot)\|_{L^\infty} \leq
\|\Delta_q\tau(0, \cdot)\|_{L^\infty} +
\int_{0}^t\Big(2^q\|\Delta_qv(s, \cdot)\|_{L^\infty}\\\nonumber
&& \quad +\ \big\|\Delta_q\big(\nabla v\tau
+ \tau(\nabla v)^t\big)(s, \cdot)\|_{L^\infty} + \|[v\cdot
\nabla, \Delta_q]\tau(s, \cdot)\big\|_{L^\infty}\Big)ds,\end{aligned}$$ which implies that $$\begin{aligned}
\label{d6}
&&\|\Delta_q\tau(t, \cdot)\|_{\dot{C}^\alpha} \leq
C\|\tau(0, \cdot)\|_{\dot{C}^\alpha} +
\int_{0}^t\Big(\|v\|_{\dot{C}^{1 + \alpha}}\\\nonumber
&& \quad +\ \|\nabla v\|_{L^\infty}\|\tau\|_{\dot{C}^\alpha}
+ \|\tau\|_{L^\infty}\|v\|_{\dot{C}^{1 + \alpha}} + 2^{\alpha q}\|[v\cdot
\nabla, \Delta_q]\tau(s, \cdot)\big\|_{L^\infty}\Big)ds.\end{aligned}$$ By Bony’s decomposition, one has $$\begin{aligned}
\nonumber
&&[v\cdot \nabla, \Delta_q]\tau = \sum_{|p - q^\prime| \leq 1}
[\Delta_pv\cdot \nabla, \Delta_q]\Delta_{q^\prime}\tau\\\nonumber
&&\quad +\ \sum_{p \leq q^\prime - 2}[\Delta_pv\cdot \nabla,
\Delta_q]\Delta_{q^\prime}\tau + \sum_{p \leq q^\prime -
2}[\Delta_{q^\prime}v\cdot \nabla, \Delta_q]\Delta_p\tau\\\nonumber
&&= \sum_{|q^\prime - q| \leq 2}\big([S_{q^\prime -
1}v\cdot \nabla, \Delta_q]\Delta_{q^\prime}\tau +
[\Delta_{q^\prime}v\cdot \nabla, \Delta_q]S_{q^\prime
- 1}\tau\big)\\\nonumber
&&\quad + \sum_{|p - q^\prime| \leq 1}
[\Delta_pv\cdot \nabla, \Delta_q]\Delta_{q^\prime}\tau.\end{aligned}$$ Noting that $$\begin{aligned}
\nonumber
[S_{q^\prime - 1}v, \Delta_q]f = \int h(y)\big[(S_{q^\prime -
1}v)(x) - (S_{q^\prime - 1}v)(x - 2^{-q}y)\big] f(x - 2^{-q}y)dy,\end{aligned}$$ one has $$\begin{aligned}
\nonumber
\|[S_{q^\prime - 1}v, \Delta_q]f\|_{L^\infty} \leq
C2^{-q}\|\nabla S_{q^\prime - 1}v\|_{L^\infty}\|f\|_{L^\infty}.\end{aligned}$$ Consequently, we have $$\begin{aligned}
\nonumber
&&\sum_{|q^\prime - q| \leq 2}\int_{0}^t\Big(2^{\alpha q}
\big\|[S_{q^\prime - 1}v\cdot\nabla, \Delta_q]\Delta_{q^\prime}
\tau(s, \cdot)\big\|_{L^\infty}\Big)ds\\\nonumber
&&\leq \sum_{|q^\prime - q| \leq 2}\int_{0}^t2^{\alpha q}
2^{-q}\big\|\nabla S_{q^\prime - 1}v\|_{L^\infty}\|\Delta_{q^\prime}
\nabla\tau\|_{L^\infty}(s, \cdot)ds\\\nonumber
&&\leq C\sum_{|q^\prime - q| \leq 2}\int_{0}^t\big\|\nabla
S_{q^\prime - 1}v\|_{L^\infty}[2^{\alpha q^\prime}
\|\Delta_{q^\prime}\tau\|_{L^\infty}](s, \cdot)ds\\\nonumber
&&\leq C\int_{0}^t\|\nabla v\|_{L^\infty}
\|\tau\|_{\dot{C}^\alpha}ds.\end{aligned}$$ Similarly, we have $$\begin{aligned}
\nonumber
&&\sum_{|q^\prime - q| \leq 2}\int_{0}^t\Big(2^{\alpha q}
\big\|[\Delta_{q^\prime}v\cdot\nabla, \Delta_q]S_{q^\prime - 1}
\tau(s, \cdot)\big\|_{L^\infty}\Big)ds\\\nonumber
&&\leq \sum_{|q^\prime - q| \leq 2}\int_{0}^t2^{\alpha q}
2^{-q}\big\|\Delta_{q^\prime}\nabla v\|_{L^\infty}\|S_{q^\prime - 1}
\nabla\tau\|_{L^\infty}(s, \cdot)ds\\\nonumber
&&\leq C\int_{0}^t\|\tau\|_{L^\infty}
\|v\|_{\dot{C}^{1 + \alpha}}ds.\end{aligned}$$ At last, one computes that $$\begin{aligned}
\nonumber
&&\sum_{|p - q^\prime| \leq 1}\int_{0}^t\Big(2^{\alpha q}
\big\|[\Delta_{p}v\cdot\nabla, \Delta_q]\Delta_{q^\prime}
\tau(s, \cdot)\big\|_{L^\infty}\Big)ds\\\nonumber
&&\leq \sum_{p, q^\prime \backsim q} \int_{0}^t
\Big(2^{(1 + \alpha)q}\big\|[\Delta_{p}v, \Delta_q]\Delta_{q^\prime}
\tau(s, \cdot)\big\|_{L^\infty}\Big)ds\\\nonumber
&&\quad +\ \sum_{p, q^\prime \geq q + 2} \int_{0}^t
\Big(2^{(1 + \alpha)q}\big\|[\Delta_{p}v, \Delta_q]\Delta_{q^\prime}
\tau(s, \cdot)\big\|_{L^\infty}\Big)ds\\\nonumber
&&\leq C\int_{0}^t\|\tau\|_{L^\infty}
\|v\|_{\dot{C}^{1 + \alpha}}ds.\end{aligned}$$ The above inequalities yield an improvement of : $$\begin{aligned}
\label{d7}
&&\|\tau(t, \cdot)\|_{\dot{C}^{\alpha}} \leq
C\|\tau(0, \cdot)\|_{\dot{C}^\alpha}\\\nonumber
&&\quad +\ C\int_{0}^t\big(\|\nabla v\|_{L^\infty} +
\|\tau\|_{L^\infty}\big)\big(\|\tau(s, \cdot)\|_{\dot{C}^{\alpha}}
+ \|v(s, \cdot)\|_{\dot{C}^{1 + \alpha}}\big)ds.\end{aligned}$$
Now let us insert into to get $$\begin{aligned}
\nonumber
&&B(t) = \sup_{0 \leq s < t}\|\tau(t, \cdot)
\|_{\dot{C}^{\alpha}} \leq C\|\tau(0, \cdot)\|_{\dot{C}^\alpha}\\\nonumber
&&\quad +\ C\int_{0}^t\big(\|\nabla v\|_{L^\infty} +
\|\tau\|_{L^\infty}\big)\big(\|v(0,
\cdot)\|_{\dot{C}^{1 + \alpha}} + B(s)\big)ds.\end{aligned}$$ Noting that by the inequalities in Lemma \[lem1\], we can estimate $$\begin{aligned}
\nonumber
&&\int_{0}^t\big(\|\nabla v\|_{L^\infty} +
\|\tau\|_{L^\infty}\big)ds\\\nonumber
&&\leq \int_{0}^{t_\star}\big(\|\nabla v\|_{L^\infty} +
\|\tau\|_{L^\infty}\big)ds + C\int_{t_\star}^t\big(1 + \|v\|_{L^2} +
\|\tau\|_{L^2}\big)ds\\\nonumber
&&\quad +\ C\sup_{q}
\int_{t_\star}^t\|\nabla\Delta_qv\|_{L^\infty}ds\ln\big(e +
\int_0^t\|v\|_{\dot{C}^{1 + \alpha}}ds\big)\\\nonumber
&&\quad +\ C\int_{t_\star}^t
\|\tau\|_{{\rm BMO}}\ln(e +
\|\tau\|_{\dot{C}^{\alpha}})ds\\\nonumber
&&\leq C_\star + C\epsilon\ln\big[e + Ct\big(\|v(0,
\cdot)\|_{\dot{C}^{1 + \alpha}} + B(t)\big)\big] +
C\epsilon\ln\big(e + B(t)\big)\\\nonumber &&\leq C_\star +
C\epsilon\ln\big(e + \|v(0,
\cdot)\|_{\dot{C}^{1 + \alpha}} + B(t)\big).\end{aligned}$$ Here $C_\star$ is a positive constant depending on the solution $(v,
\tau)$ on $[0, t_\star]$. Consequently, we have $$\begin{aligned}
\nonumber
B(t) \leq C_\star\Big(1 + \|\tau(0, \cdot)\|_{\dot{C}^\alpha} +
\|v(0, \cdot)\|_{\dot{C}^{1 + \alpha}}\Big) +
C\int_{0}^t\big(\|\nabla v\|_{L^\infty} +
\|\tau\|_{L^\infty}\big)B(s)ds.\end{aligned}$$ Then Gronwall’s inequality yields that $$\begin{aligned}
\nonumber
&&e + \|v(0, \cdot)\|_{\dot{C}^{1 + \alpha}} + B(t)\\\nonumber
&&\leq C_\star\Big(e + \|\tau(0, \cdot)\|_{\dot{C}^\alpha} +
\|v(0, \cdot)\|_{\dot{C}^{1 + \alpha}}\Big)\exp\Big\{C
\int_{t_\star}^t\big(\|\nabla v\|_{L^\infty} +
\|\tau\|_{L^\infty}\big)ds\Big\}\\\nonumber
&&\leq C_\star\Big(e + \|\tau(0, \cdot)\|_{\dot{C}^\alpha} +
\|v(0, \cdot)\|_{\dot{C}^{1 + \alpha}}\Big)e^{C_\star
+ C\epsilon\ln\big(e + \|v(0, \cdot)\|_{\dot{C}^{1 + \alpha}} + B(t)\big)}\\\nonumber
&&\leq C_\star\Big(e + \|\tau(0, \cdot)\|_{\dot{C}^\alpha} +
\|v(0, \cdot)\|_{\dot{C}^{1 + \alpha}}\Big)\big(e + \|v(0,
\cdot)\|_{\dot{C}^{1 + \alpha}} + B(t)\big)^{C\epsilon}.\end{aligned}$$ From the above inequalities and , we have $$\begin{aligned}
\label{d8}
A(t) + B(t) \leq C_\star\Big(1 + \|\tau(0, \cdot)\|_{\dot{C}^\alpha}
+ \|v(0, \cdot)\|_{\dot{C}^{1 + \alpha}}\Big)^2\end{aligned}$$ by choosing $\epsilon = \frac{1}{2C}$.
Proof of Corollary \[thm-criterion-MHD\]
========================================
In fact, it is easy to see that the tensor $H\otimes H$ satisfies the following transport equation: $$\begin{aligned}
\label{k1}
\partial_t(H\otimes H) + v\cdot\nabla(H\otimes H) = \nabla v(H\otimes
H) + (H\otimes H)(\nabla v)^t.\end{aligned}$$ Hence, the tensor $H\otimes H - \frac{1}{2}I$ plays the role of $\tau$ (However, it seems not being able to directly apply Theorem \[thm-criterion-Oldroyd\] to get Corollary \[thm-criterion-MHD\]). The rest part of the proof of Corollary \[thm-criterion-MHD\] is similar as that of Theorem \[thm-criterion-Oldroyd\].
In fact, by the assumption $\|v(0, \cdot)\|_{L^2\cap\dot{C}^{1 +
\alpha}} + \|H(0, \cdot)\|_{L^2\cap\dot{C}^\alpha} < \infty$, one can easily derive that $$\begin{aligned}
\label{k2}
\|v(0, \cdot)\|_{L^2 \cap \dot{C}^{1 + \alpha} (\mathbb{R}^2) } +
\|H\otimes H(0, \cdot)\|_{ L^1 \cap \dot{C}^\alpha (\mathbb{R}^2)} <
\infty.\end{aligned}$$ Moreover, one has the following energy law $$\begin{aligned}
\label{k3}
\|(v, H)\|_{L^\infty_T(L^2)} + 2\|\nabla v\|_{L^2_T(L^2)} = \|(v,
H)(0, \cdot)\|_{L^2},\end{aligned}$$ which gives that $$\begin{aligned}
\label{k4}
\|H\otimes H\|_{L^\infty_T(L^1)} < \infty.\end{aligned}$$ Having , and in hand, and noticing the assumption and the transport equation for $H\otimes H$, one has $$\begin{aligned}
\nonumber
\|v(t, \cdot)\|_{\dot{C}^{1 + \alpha}} + \|H\otimes H(t,
\cdot)\|_{\dot{C}^\alpha} < \infty\end{aligned}$$ by an exactly same manner as in section 3. Coming back to the transport equation for $H$ in , we have $$\begin{aligned}
\nonumber
\|H\|_{\dot{C}^\alpha} < \infty\end{aligned}$$ in a standard manner. This completes the proof of Corollary \[thm-criterion-MHD\].
Acknowledgement {#acknowledgement .unnumbered}
===============
Zhen Lei was in part supported by NSFC (grant no. 10801029), a special Postdoc Science Foundation of China (grant no. 200801175) and SGST 09DZ2272900. Nader Masmoudi was in part supported by NSF under Grant DMS-0703145. Yi Zhou is partially supported by the NSFC (grant no. 10728101), the 973 project of the Ministry of Science and Technology of China, the Doctoral Program Foundation of the Ministry of Education of China, the ¡±111¡± project (B08018) and SGST 09DZ2272900.
[00]{}
H. Bahouri and J. Y. Chemin, *Equations de transport relatives $\grave{a}$ des champs de vecteurs non-lipschitziens et m$\acute{e}$chanique des fluides*, Arch. Rational Mech. Anal., **127** (1994), pp. 159–182.
J. T. Beale, T. Kato and A. Majda, *Remarks on the breakdown of smooth solutions for the 3-D Euler equations*. Comm. Math. Phys. 94 (1984), 61–66.
J. M. Bony, *Calcul symbolique et propagation des singularit$\acute{e}$s pour les $\acute{e}$quations aux d$\acute{e}$riv$\acute{e}$es partielles non lin$\acute{e}$aires*. Ann. Sci. $\acute{e}$cole Norm. Sup. (4) **14** (1981), no. 2, 209–246.
J. Y. Chemin, *Perfect incompressible fluids*. Oxford Lecture Series in Mathematics and its Applications, 14. The Clarendon Press, Oxford University Press, New York, 1998.
J. Y. Chemin and N. Masmoudi, *About lifespan of regular solutions of equations related to viscoelastic fluids*, SIAM J. Math. Anal. **33** (2001), no. 1, 84–112.
F. Colombini and N. Lerner, *Hyperbolic operators with non-Lipschitz coefficients*, Duke Math. J., **77** (1995), pp. 657–698.
P. Constantin and N. Masmoudi, *Global well posedness for a Smoluchowski equation coupled with Navier- Stokes equations in 2d*, Comm. Math. Phys. **278** (2008), no. 1, 179–191.
C. Guillop$\acute{e}$ and J. C. Saut, *Existence results for the flow of viscoelastic fluids with a differential constitutive law*, Nonlinear Anal., **15** (1990), pp. 849–869.
C. Guillop$\acute{e}$ and J. C. Saut, *Global existence and one-dimensional nonlinear stability of shearingmotions of viscoelastic fluids of Oldroyd type*, RAIRO Mod$\textit{e}$l. Math. Anal. Num$\textit{e}$r., **24** (1990), pp. 369–401.
E. Fernandez-Cara, F. Guill$\acute{e}$n and R.R. Ortega, *Existence et unicit$\acute{e}$ de solution forte locale en temps pour des fluides non newtoniens de type Oldroyd (version Ls¨CLr)*, C. R. Acad. Sci. Paris S$\acute{e}$r. I Math., **319** (1994), pp. 411¨C416.
D. Hu and T. Leli[è]{}vre. New entropy estimates for [O]{}ldroyd-[B]{} and related models. , 5(4):909–916, 2007.
H. Kozono, T. Ogawa, and Y. Taniuchi. The critical [S]{}obolev inequalities in [B]{}esov spaces and regularity criterion to some semi-linear evolution equations. , 242(2):251–278, 2002.
H. Kozono and Y. Taniuchi, *Limitting case of the Sobolev inequality in BMO with application to the Euler equations*. Comm. Math. Phys. 214 (2000) no. 1, 191–200.
Zhen Lei, *Global existence of classical solutions for some Oldroyd-B model via the incompressible limit*, Chinese Ann. Math. Ser. B **27** (2006), no. 5, 565–580.
Zhen Lei and Yi Zhou, *BKM’s Criterion and Global Weak Solutions for Magnetohydrodynamics with Zero Viscosity*, to Appear in DCDS-A.
F.-H. Lin, P. Zhang and Z. Zhang. *On the global existence of smooth solution to the 2-d fene dumbell model*, Comm. Math. Phys. **277**, 531–553 (2008)
P.-L. Lions and N. Masmoudi, *Global solutions for some Oldroyd models of non-Newtonian flows*, Chinese Ann. Math. Ser. B, **21** (2000), pp. 131¨C146.
Pierre-Louis Lions and Nader Masmoudi. Global existence of weak solutions to some micro-macro models. , 345(1):15–20, 2007.
Nader Masmoudi. Well-posedness for the [FENE]{} dumbbell model of polymeric flows. , 61(12):1685–1714, 2008.
Nader Masmoudi. Global well posedness for the maxwell-navier-stokes system in $2$d. , 2009.
Nader Masmoudi, Ping Zhang, and Zhifei Zhang. Global well-posedness for 2[D]{} polymeric fluid models and growth estimate. , 237(10-12):1663–1675, 2008.
J. G. Oldroyd, *Non-Newtonian effects in steady motion of some idealized elastico-viscous liquids*, Proc. Roy. Soc. London Ser. A, **245** (1958), pp. 278–297.
F. Planchon. An extension of the [B]{}eale-[K]{}ato-[M]{}ajda criterion for the [E]{}uler equations. , 232(2):319–326, 2003.
[^1]: Shanghai Key Laboratory for Contemporary Applied Mathematics; School of Mathematical Sciences, Fudan University, Shanghai 200433, China. [*Email: [email protected], [email protected]*]{}
[^2]: Courant Institute of Mathematical Sciences, New York University, NY 10012, USA. [*Email: [email protected]* ]{}
[^3]: Shanghai Key Laboratory for Contemporary Applied Mathematics; School of Mathematical Sciences, Fudan University, Shanghai 200433, China. [*Email: [email protected]*]{}
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abstract: 'The estimation of variance-based importance measures (called Sobol’ indices) of the input variables of a numerical model can require a large number of model evaluations. It turns to be unacceptable for high-dimensional model involving a large number of input variables (typically more than ten). Recently, Sobol and Kucherenko have proposed the Derivative-based Global Sensitivity Measures (DGSM), defined as the integral of the squared derivatives of the model output, showing that it can help to solve the problem of dimensionality in some cases. We provide a general inequality link between DGSM and total Sobol’ indices for input variables belonging to the class of Boltzmann probability measures, thus extending the previous results of Sobol and Kucherenko for uniform and normal measures. The special case of log-concave measures is also described. This link provides a DGSM-based maximal bound for the total Sobol indices. Numerical tests show the performance of the bound and its usefulness in practice.'
address: |
$^a$ Université Paris Descartes, 45 rue des saints Pères, F-75006, France\
$^b$ IMT, F-31062, France\
$^c$ EDF R&D, 6 quai Watier, F-78401, France
author:
- 'M. Lamboni$^{ab}$, B. Iooss$^c$[^1], A.-L. Popelin$^c$, F. Gamboa$^b$'
bibliography:
- 'bibi\_dgsm.bib'
title: 'Derivative-based global sensitivity measures: general links with Sobol’ indices and numerical tests'
---
Boltzmann measure; Derivative based global sensitivity measure; Global sensitivity analysis; Log-concave measure; Poincaré inequality; Sobol’ indices
Introduction
============
With the advent of computing technology and numerical methods, computer models are now widely used to make predictions on little-known physical phenomena, to solve optimization problems or to perform sensitivity studies. These complex models often include hundreds or thousands uncertain inputs, whose uncertainties can strongly impact the model outputs (De Rocquigny ${\it{et \, al.}}$ [@derdev08], Kleijnen [@kle08], Patelli ${\it{et \, al.}}$ [@patelli10]). In fact, it is well known that, in many cases, only a small number of input variables really act in the model (Saltelli ${\it{et \, al.}}$ [@salcha00]). This number is referred to the notion of the effective dimension of a function (Caflish ${\it{et \, al.}}$ [@caflish1997]), which is a useful way to deal with the curse of dimensionality in practical applications.
Global Sensitivity Analysis (GSA) methods (Sobol [@sobol93], Saltelli ${\it{et \, al.}}$ [@salcha00]) are used to quantify the influence of model input variables (and their interaction effects) on a model reponse. It is also an objective way to determine the effective dimension by using the model simulations (Kucherenko ${\it{et \, al.}}$ [@Kucherenko2011]). A first class of GSA methods, called “screening” methods, aim at dealing with a large number of input variables (from tens to hundreds). An example of screening method is the Morris’ method (Morris [@morris91]), which allows a coarse estimation of the main effects using only a few model evaluations. While taking into account the interactions between the indices, the basic form of the Morris method did not compute precise sensitivity indices associated to the interactions between inputs. The second class of GSA methods are the popular quantitative methods, mainly based on the decomposition of the model output variance, which leads to the so-called variance-based methods and Sobol’ sensitivity indices. It allows computing the main and total effects (called first order and total Sobol’ indices) of each input variable, as well as interaction effects. However, for functions with non linear and interaction effects, the estimation procedures become particularly expensive in terms of number of required model evaluations. Hence, for this kind of model, variance-based methods can only be applied to a limited number of input variables (less than tens).
Recently, Sobol and Kucherenko [@sobol09; @sobol10] have proposed the so-called Derivative-based Global Sensitivity Measures (DGSM), which can be seen as a kind of generalization of the Morris screening method. DGSM seem computationally more tractable than variance-based measures, specially for high-dimensional models. They also theoretically proved an inequality linking DGSM to total Sobol’ indices in the case of uniform or Gaussian input variables.
In this paper, we investigate this close relationship between total Sobol’ indices and DGSM, by extending this inequality to a large class of Boltzmann probability measures. We also obtain result for the class of log-concave measures. The paper is organized as follows: Section \[sec:def\] recalls some useful definitions of Sobol’ indices and DGSM. Section \[sec:link\] establishes an inequality between these indices for a large class of Boltzmann (resp. log-concave) probability measures. Section \[sec:test\] provides some numerical simulations on two test models, illustrating how DGSM can be used in practice. We conclude in Section \[sec:con\].
Global sensitivity indices definition {#sec:def}
=====================================
Variance-based sensitivity indices
----------------------------------
Let $ Y= f({\mathbf{{X}}})$ be a model output with $d$ random input variables ${\mathbf{{X}}}=(X_1,\ldots,X_d)$. If the input variables are independent (assumption A1) and ${\mathbb{E}}\left( f^2({\mathbf{{X}}}) \right)< +\infty $ (assumption A2), we have the following unique Hoeffding decomposition (Efron and Stein [@efron81]) of $ f({\mathbf{{X}}}) $: $$\begin{aligned}
\label{eq:decsa}
f({\mathbf{{X}}}) & = & f_0 + \sum_j^d f_j({X}_j) + \sum_{i<j}^d f_{ij}({X}_i,{X}_j) + \ldots + f_{1\ldots d}({X}_1,\ldots,X_d) \\
& = & \sum_{u \subset \{1,2, \ldots d\}} f_u({X}_u),\end{aligned}$$ where $f_0 = {\mathbb{E}}\left[ f({\mathbf{X}})\right] $ corresponds to the empty subset; $f_j({X}_j) = {\mathbb{E}}\left[f({\mathbf{{X}}})| {X}_j \right] -f_0 $ and $\displaystyle f_u({X}_u) = {\mathbb{E}}\left[f({\mathbf{{X}}})| {X}_u \right] - \sum_{v \subset u } f_v({X}_v) $ for any subset $u \subset \{1,2, \ldots, d \}$ .
By regrouping all the terms in equation (\[eq:decsa\]) that contain the variable ${X}_j$ ($j=1,2, \ldots, d $) in the function called $g(\cdot)$:
$$\label{eq:g}
g({X}_j,{\mathbf{{X}}}_{\sim j}) = \sum_{u \ni j}f_u({\mathbf{{X}}}_u)\;,$$
we have the following decomposition: $$\label{eq:decdgsa}
f({\mathbf{{X}}}) = f_0+ g({X}_j,{\mathbf{{X}}}_{\sim j}) + h({\mathbf{{X}}}_{\sim j}),$$ where $ {\mathbf{{X}}}_{\sim j}$ denotes the vector containing all variables except ${X}_j$ and $h(\cdot)= f(\cdot)- f_0 - g(\cdot)$. Notice that this decomposition is also unique under assumptions A1 and A2. The function $g(\cdot)$, itself, suffices to compute the total sensitivity indices. Indeed, it contains all information relating $f({\mathbf{X}})$ to ${X}_j$.
Assume that A1, A2 hold, let $\mu({\mathbf{X}})= \mu({X}_1, \ldots, {X}_d)$ be the distribution of the input variables. For any non empty subset $u \subseteq \{1,2, \ldots, d \}$, set first $$D=\int f^2({\mathbf{{x}}}) d\mu({\mathbf{{x}}}) -f_0^2 \;,$$ $$D_u=\int f^2_u({\mathbf{{x}}}_u) d\mu({\mathbf{{x}}}_u)\;,$$ $$\label{eq:dt0}
D_{u}^{tot} = \int \sum_{v \supseteq u } f_v^2({\mathbf{{x}}}_v) d\mu({\mathbf{{x}}}_v)\;.$$ Further, the first order Sobol sensitivity indices (Sobol [@sobol93]) of ${\mathbf{{X}}}_u$ is $$\label{eq:si}
S_{u} = \frac{D_u}{D} \;,$$ The total sensitivity Sobol index of ${\mathbf{{X}}}_{u}$ (Homma and Saltelli [@homsal96]) is $$\label{eq:sit}
S_{T_u} = \frac{D_{u}^{tot}}{D} \;.$$
The following proposition gives another way to compute the total sensitivity indices.
Under assumptions A1 and A2, the total sensitivity indices of variable ${X}_j$ ($j=1,2, \ldots, d $) is obtained by the following formulas: $$\begin{aligned}
\label{eq:dt}
D_{j}^{tot} & = & \int g^2({x}_j,{\mathbf{{x}}}_{\sim j}) d\mu({\mathbf{{x}}})\end{aligned}$$ and $$\begin{aligned}
\label{eq:dt2}
D_{j}^{tot} & = & \frac{1}{2}\int \left[f({\mathbf{{x}}})-f({x}'_j, {\mathbf{{x}}}_{\sim j}) \right]^2 d\mu({\mathbf{{x}}})
d\mu({x}_j')\;.\end{aligned}$$
The first formula is an obvious consequence of equation (\[eq:decdgsa\]), and it is obtained by using the orthogonality of the summands in equation (\[eq:decsa\]). Indeed, $\displaystyle D_{j}^{tot} = \int \sum_{v \supseteq j} f_v^2({\mathbf{{x}}}_v) d\mu({\mathbf{{x}}}_v) = \int \left[\sum_{v\supseteq j} f_v({\mathbf{{x}}}_v) \right]^2 d\mu({\mathbf{{x}}})= \int g^2({x}_j,{\mathbf{{x}}}_{\sim j}) d\mu({\mathbf{{x}}})$. The later formula is proved in Sobol [@sobol2001].
Derivative-based sensitivity indices
------------------------------------
Derivative-based global sensitivity method uses the second moment of model derivatives as importance measure. This method is motivated by the fact that a high value of the derivative of the model output with respect to some input variable means that a big variation of model output is expected for a variation of the variable. This method extends the Morris method (Morris [@morris91]). Indeed, it allows to capture any small variation of the model output due to input variables.
DGSM have been first proposed in Sobol and Gresham [@sobgre95]. Then, they have been largely studied in Kucherenko ${\it{et \, al.}}$ [@kucherenko09], Sobol and Kucherenko [@sobol09; @sobol10] and Patelli ${\it{et \, al.}}$ [@patelli10b]. From now on, we assume that the function $f$ is differentiable. Two kind of DGSM are defined below:
Assume that A1 holds and that $\displaystyle \frac{\partial f({\mathbf{{X}}})}{\partial {x}_j}$ is square-integrable (assumption A3). Then, for $ j=1,2, \ldots d $, we define the DGSM indices by: $$\begin{aligned}
\label{eq:dsi}
\nu_{j} & = & {\mathbb{E}}\left[\left(\frac{\partial f({\mathbf{{X}}})}{\partial {x}_j}\right)^2\right] \\ \nonumber
& = & \int \left(\frac{\partial f({\mathbf{{x}}})}{\partial {x}_j}\right)^2 d\mu({\mathbf{{x}}}) \;.\end{aligned}$$ Let $w(\cdot)$ is be a bounded measurable function. A weighted version of the last indices is: $$\label{eq:dsip}
\tau_{j} = \int \left(\frac{\partial f({\mathbf{{x}}})}{\partial {x}_j}\right)^2 w({x}_j) d\mu({\mathbf{{x}}}) .$$
\[rem:sieqdsi\] Sobol and Kucherenko [@sobol10] showed that, for a specific weighting function $\displaystyle w({x}_j) = \frac{1-3{x}_j+3 {x}_j^2}{6} $ and for a class of linear model with respect to each input variable (following a uniform distribution over $[0,1]$), we have $ \tau_j = D_{j}^{tot} $.
By bearing in mind the decomposition in equation (\[eq:decdgsa\]), we can replace in equations (\[eq:dsi\] ) and (\[eq:dsip\]) the function $f(\cdot)$ by $g(\cdot)$. In general, $g(\cdot)$ is a $d_1$ ($d_1 \leq d$) dimension function, and this can drastically reduce the number of model evaluations for the numerical computation of $\nu$ or $\tau$. Thus, we have: $$\begin{aligned}
\label{eq:dsi2}
\nu_{j} & = & \int \left(\frac{\partial g({\mathbf{{x}}})}{\partial {x}_j}\right)^2 d\mu({\mathbf{{x}}}) \;.\end{aligned}$$ $$\label{eq:dsip2}
\tau_{j} = \int \left(\frac{\partial g({\mathbf{{x}}})}{\partial {x}_j}\right)^2 w({x}_j) d\mu({\mathbf{{x}}}) ,$$
Variance-based sensitivity indices vs. derivative-based sensitivity indices {#sec:link}
===========================================================================
As DGSM estimations need much less model evaluations than total Sobol’ indices estimations (Kucherenko ${\it{et \, al.}}$ [@kucherenko09]), it would be interesting to use the DGSM, instead of total Sobol’ indices, for factors fixing setting. A formal link is therefore necessary to provide a mathematical relation between total Sobol’ indices and DGSM. Sobol and Kucherenko [@sobol09] have established an inequality linking these two indices for uniform and Gaussian random variables (maximal bound for $S_{T_j}$). In this section, we extend the inequality for Sobolev’ space model whith the marginal distribution of input variables belonging to the class of Boltzmann measure on $\mathbb{R}$ (assumption A4). A measure $\delta$ on $\mathbb{R}$ is said to be a Boltzmann measure if it is absolutely continuous with respect to the Lebesgue measure and its density $\displaystyle d\delta({x}) = \rho({x})d{x}= c \exp[-v({x})]d{x}$. Here $v(\cdot)$ is a continuous function and $c$ a normalizing constant. Many classical continuous probability measures used in practice are Boltzmann measures (see de Rocquigny ${\it{et \, al.}}$ [@derdev08] and Saltelli ${\it{et \, al.}}$ [@salcha00]).
The class of Boltzmann probability measures includes the well known class of log-concave probability measures. In this case, $v(\cdot)$ is a convex function (assumption A5). In other words, a twice differentiable probability density function $\rho({x})$ is said to be log-concave if, and only if, $$\frac{d^2}{dx^2}[\log \rho({x})] \leq 0 \;.$$ Note that the probability measure of uniform density on a finite interval is not continuous on $\mathbb{R}$. So it cannot be considered in the class of log-concave probability measure, nor in the class of Boltzmann probability measures.
The two following propositions give the formal link between Sobol’ indices and derivative-based sensitivity indices.
\[prop:link\] Under assumptions A1, A2, A3 and A4, we have: $$\label{eq:ineq}
D_{j}^{tot} \leq C(\mu_j) \nu_{j} \,$$ with $\displaystyle C(\mu_j) = 4C_1^2$ and $\displaystyle C_1=\sup_{{x}\in \mathbb{R}} \frac{\min(F_j({x}), 1-F_j({x}))}{\rho_j({x})} $ the Cheeger constant, $F_j(\cdot)$ the cumulative probability function of ${X}_j$ and $\rho_j(\cdot)$ the density of ${X}_j$.
We recall the four assumptions:
- A1: independence between inputs $X_1$, $X_2$, …, $X_d$,
- A2: $f \in L^2(\mathbb{R})$,
- A3: $\displaystyle \frac{\partial f}{\partial x_j} \in L^2(\mathbb{R})$,
- A4: the distribution of $X_j$ is a Boltzmann probability measure.
\[proof:link1\] The resulting inequality (\[eq:ineq\]) is based on a one-dimensional $L^2$-Poincaré inequality of the type $\displaystyle \|u\|_{L^2} \leq C \|\nabla u\|_{L^2}$ for $u$ a Sobolev’ space function (see for example [@fougeres05]). It is applied here to the function $g(\cdot)$ (equation (\[eq:g\]), with $\displaystyle \int g^2({x}_j,{\mathbf{{x}}}_{\sim j}) d\mu({\mathbf{{x}}}) = D_{j}^{tot}$ (equation (\[eq:dt\])) and $\displaystyle \int \left(\frac{\partial g({\mathbf{{x}}})}{\partial {x}_j}\right)^2 d\mu({\mathbf{{x}}}) = \nu_{j}$ (equation (\[eq:dsi2\])). The constant is obtained in Bobkov [@bobkov99], and Fougères [@fougeres05] for the one-dimensional Poincaré inequality. A proof of the $d$-dimensional Poincaré inequality is given in Bakry ${\it{et \, al.}}$ [@bakry08].
\[prop:link2\] Under assumptions A1, A2, A3 and A5, we have: $$\label{eq:ineq2}
D_{j}^{tot} \leq \left[\exp(v(m))\right]^2 \nu_{j} \;,$$ with $\displaystyle C_1 = \frac{\exp(v(m))}{2}$ the Cheeger constant and $m$ the median of the measure $\mu_j$ (such that $\mu(X_j \leq m) = \mu(X_j>m)$).
We recall the assumption A5: the distribution of ${X}_j$ is a log-concave probability measure.
See proof \[proof:link1\].
Table \[tab:logconc\] shows Cheeger constant for some log-concave probability distributions that are used in practice for uncertainty and sensitivity analyses. We also give their medians and the functions $v(\cdot)$. We obtain the same results for the normal distribution ${\mathcal N}(\mu, \sigma^2)$ similar to Sobol and Kucherenko [@sobol09] but we prove them in another way (in this case, $v(m) = \log(\sigma)$). For uniform distribution ${\mathcal U}[a \, b]$, Sobol and Kucherenko [@sobol09] obtained via direct integral manipulations the inequality $\displaystyle D_{j}^{tot} \leq \frac{(b-a)^2}{\pi^2} \nu_{j}$. This relation is the classical Poincaré or Writtinger inequality (Ane ${\it{et \, al.}}$ [@ane2000]).
[p[3.3cm]{} c c c]{} Distribution & $v(x)$ & $m$ & $C_1$\
Normal ${\mathcal N}(\mu, \sigma^2)$ & $\displaystyle \frac{(x-\mu)^2}{2\sigma^2}+\log(\sigma)$ & $\mu$ & $\displaystyle \frac{\sigma}{2}$\
\
Exponential ${\mathcal E}(\lambda)$, $\lambda>0$ & $\lambda x -\log(\lambda)$ & $\displaystyle \frac{\log 2}{\lambda}$ & $\displaystyle \frac{1}{\lambda}$\
\
Beta ${\mathcal B}(\alpha,\beta)$, $\alpha,\beta \geq 1$ & $\displaystyle \log\left[x^{1-\alpha}(1-x)^{1-\beta} \right] $ & No expression & —\
\
Gamma $\Gamma(\alpha,\beta)$, scale $\alpha \geq 1$, shape $\beta > 0$ & $\displaystyle \log\left( x^{1-\alpha} \Gamma (\alpha) \right) +\frac{x}{\beta} +\alpha\log\beta$& No expression & —\
\
Gumbel ${\mathcal G}(\mu,\beta)$, scale $\beta>0$ & $\displaystyle \frac{x-\mu}{\beta} + \log\beta +\exp\left(-\frac{x-\mu}{\beta}\right)$ & $\displaystyle \mu -\beta \log(\log 2)$ & $\displaystyle \frac{\beta}{\log 2}$\
\
Weibull ${\mathcal W}(k, \lambda)$, shape $k \geq 1$, scale $\lambda > 0$ &$\displaystyle \log\left(\frac{\lambda}{k}\right) + (1-k)\log\left(\frac{x}{\lambda}\right)+\left(\frac{x}{\lambda}\right)^k $ & $\displaystyle \lambda (\log 2)^{1/k}$ & $\displaystyle \frac{\lambda(\log 2)^{(1-k)/k}}{k}$\
For general log-concave measures, no analytical expressions are available for the Cheeger constant. In this latter case or in case of non log-concave but Boltzmann measure, we can estimate the Cheeger constant by numerically evaluating the expression $\sup_{{x}\in \mathbb{R}} \frac{\min(F_j({x}), 1-F_j({x}))}{\rho_j({x})}$.
Numerical tests {#sec:test}
===============
Derivative sensitivity indices estimates
----------------------------------------
A classical estimator for the DGSM is the empirical one and is given below: $$\begin{aligned}
\label{eq:dsiest}
\widehat{\nu}_{j} &=& \frac{1}{n} \sum_{i=1}^n \left(\frac{\partial f({\mathbf{{X}}}^{(i)})}{\partial {x}_j} \right)^2.\end{aligned}$$ Experimental convergence properties of this estimator are given in Sobol and Kucherenko [@sobol09].
From definition (\[eq:decdgsa\]), we know that $\displaystyle \frac{\partial f({\mathbf{{X}}}^{(i)})}{\partial {x}_j}=\frac{\partial g({\mathbf{{X}}}^{(i)})}{\partial {x}_j}$. Estimator of $D_j^{tot}$ (see equation (\[eq:dt\])) and estimator (\[eq:dsiest\]) are based on the same function $g(\cdot)$ and it seems that estimations of these two indices will require approximately the same number of model evaluations in order to converge towards their respective values.
Computation of DGSM and Sobol’ indices can be performed with Monte Carlo-like algorithm, such as Latin Hypercube Sampling, quasi-Monte Carlo and Monte Carlo Markov Chain sampling. Kucherenko ${\it{et \, al.}}$ [@kucherenko09] have shown that quasi-Monte Carlo outperforms Monte Carlo when model has a low effective dimension. Computation of DGSM needs model gradient estimation. For complex models, model gradient computation can easily be obtained by finite difference method. Patelli and Pradlwarter [@patelli10] proposed a Monte Carlo estimation of gradient in high dimension. They used an unbiased estimator for gradients and have shown that the number of Monte Carlo evaluations $n \leq d$ is sufficient for gradient computations. In the worst case, their procedure requires the same number of model evaluations than the finite difference method. The method is very efficient when the model has a low effective dimension. In the following Sections, we compare the estimates of the Sobol indices ($S_j$ and $S_{T_j}$) and the upper bound of $S_{T_j}$ (see inequality (\[eq:ineq\])). let denote $\Upsilon_j$, the total sensitivity upper bound: $$\label{eq:upperbound}
\Upsilon_j = C \frac{\nu_j}{D} \;,$$ where $D$ is the variance of the model output $f({\mathbf{{X}}})$ and $C=4 C_1^2$. The goal of our numerical tests is just to compare the differences in terms of ranking and not to study the speed of convergence of the estimates.
Test on the Morris function
---------------------------
As a first test, we consider the Morris function (Morris [@morris91]) that includes $20$ independent and uniform input variables. The Morris function is defined by the following equation: $$\label{eq:morris}
y = \beta_0 + \sum_{i=1}^{20}\beta_i w_i + \sum_{i<j}^{20}\beta_{i,j} w_iw_j + \sum_{i<j<l}^{20}\beta_{i,j,l} w_iw_jw_l + \sum_{i<j<l<s}^{20}\beta_{i,j,l,s} w_iw_jw_lw_s \;,$$ where $\displaystyle w_i=2\left(x_i-\frac{1}{2}\right)$ except for $i=3,5,7$ where $\displaystyle w_i = 2\left(1.1 \frac{x_i}{x_i+1}-\frac{1}{2}\right)$. The coefficient values are:\
$\beta_i =20$ for $i=1, 2, \ldots, 10$,\
$\beta_{i,j} =-15$ for $i,j=1, 2, \ldots, 6$, $i<j$\
$\beta_{i,j,l} =-10$ for $i,j,l=1, 2, \ldots, 5$, $i<j<l$\
and $\beta_{1,2,3,4} = 5$.\
The remaining first and second order coefficients were generated independently from the normal distribution ${\mathcal N}(0,\,1)$ and the remaining third and fourth coefficient were set to $0$.
We replace the uniform distributions associated with several input variables by different log-concave measures of the Table \[tab:logconc\] in order to show how the bounds can be used in practical sensitivity analysis. Table \[tab:morris\] shows the probability distributions associated to each input of the Morris function.
Input Probability distribution Input Probability distribution
------- -------------------------- ------- --------------------------
$X1$ ${\mathcal U}[0,1]$ $X11$ ${\mathcal U}[0,1]$
$X2$ ${\mathcal N}(0.5,0.1)$ $X12$ ${\mathcal N}(0.5,0.1)$
$X3$ ${\mathcal E}(4)$ $X13$ ${\mathcal E}(4)$
$X4$ ${\mathcal G}(0.2,0.2)$ $X14$ ${\mathcal G}(0.2,0.2)$
$X5$ ${\mathcal W}(2,0.5)$ $X15$ ${\mathcal W}(2,0.5)$
$X6$ ${\mathcal U}[0,1]$ $X16$ ${\mathcal U}[0,1]$
$X7$ ${\mathcal U}[0,1]$ $X17$ ${\mathcal U}[0,1]$
$X8$ ${\mathcal U}[0,1]$ $X18$ ${\mathcal U}[0,1]$
$X9$ ${\mathcal U}[0,1]$ $X19$ ${\mathcal U}[0,1]$
$X10$ ${\mathcal U}[0,1]$ $X20$ ${\mathcal U}[0,1]$
: Probability distributions of the input variables of the Morris function[]{data-label="tab:morris"}
We have performed some simulations that allow computing the DGSM indices and the Sobol’ indices for the $20$ independent factors. Sobol’ indices $S_j$ and $S_{T_j}$ are obtained with the principles described in Saltelli [@saltelli02b], i.e. using two initial Monte Carlo samples of size $10^4$. For more efficient convergence properties (specially for the case of small indices), the improved formulas proposed by Sobol ${\it{et \, al.}}$ [@sobtar07] for $S_i$ and by Saltelli ${\it{et \, al.}}$ [@saltelli2010b] for $S_{T_i}$ are used. The approximation errors of these Monte Carlo estimates are calculated by repeating $20$ times the indices estimation and the mean is taken as the estimate. With $d=20$ input variables, it leads to $20 \times 10^4 \times (d+2) = 4.4\times 10^6$ model evaluations. In fact, the size of the Monte Carlo samples have been fitted to achieve acceptable absolute errors (smaller than $1\%$). However, the objective here is not to compare the algorithmic performances of DGSM and Sobol’ indices in terms of computational cost, but just to look at the inputs ranking.
The total Sobol’ indices are used in this paper as a reference. It shows that only the first $10$ inputs have some influence. Model derivatives are evaluated for each input on a Monte Carlo sample of size $1\times 10^4$ by the finite-difference method (perturbation of $0.01\%$). Then, DGSM $\nu_j$ require $2.1\times 10^5$ model evaluations. $\Upsilon_j$ is then computed using equation (\[eq:upperbound\]) where the variance of the Morris function is estimated to $D=991.521$. The results are gathered in Table \[tab:resmorris\].
Input $S_j$ $sd$ $S_{T_j}$ $sd$ $\nu_j$ $C$ $\Upsilon_j$
------- ------- ------- ----------- ------- ----------- ------- --------------
X1 0.043 0.009 0.173 0.008 2043.820 0.101 0.209
X2 0.007 0.003 0.029 0.002 2856.580 0.01 0.029
X3 0.066 0.009 0.165 0.006 31653.270 0.250 7.981
X4 0.002 0.006 0.134 0.007 2025.950 0.333 0.680
X5 0.035 0.005 0.055 0.003 4203.060 0.360 1.526
X6 0.039 0.007 0.114 0.006 1337.100 0.101 0.137
X7 0.068 0.003 0.069 0.003 6605.960 0.101 0.675
X8 0.156 0.007 0.157 0.007 1826.390 0.101 0.187
X9 0.189 0.008 0.192 0.009 2249.770 0.101 0.230
X10 0.145 0.005 0.146 0.005 1730.400 0.101 0.177
X11 0.000 0.001 0.002 0.001 22.630 0.101 0.002
X12 0.000 0.000 0.000 0.000 23.940 0.01 0.000
X13 0.000 0.001 0.001 0.000 17.670 0.250 0.004
X14 0.001 0.001 0.003 0.001 42.850 0.333 0.014
X15 0.000 0.001 0.001 0.001 19.870 0.360 0.007
X16 0.000 0.001 0.002 0.001 18.860 0.101 0.002
X17 0.000 0.001 0.002 0.001 21.400 0.101 0.002
X18 0.000 0.001 0.002 0.001 19.950 0.101 0.002
X19 0.000 0.001 0.004 0.001 54.380 0.101 0.006
X20 0.000 0.001 0.004 0.001 42.250 0.101 0.004
: Sensitivity indices (Sobol’ and DGSM) for the Morris function. For the Sobol’ indices $S_j$ and $S_{T_j}$, $20$ replicates has been used to get the standard deviation ($sd$).[]{data-label="tab:resmorris"}
In Table \[tab:resmorris\], we can first observe that the total sensitivity upper bounds $\Upsilon_j$ are always greater than the total sensitivity indices as expected. For each input, we distinguish several situations that can occur:
1. First order and total Sobol’ indices are negligible (inputs $X11$ to $X20$). In this case, we observe that the bound $\Upsilon_j$ is always negligible. For all the inputs, this test shows the high efficiency of the bound: a negligible bound warrants that the input has no influence.
2. First order and total Sobol’ indices significantly differ from zero and have approximately the same value (inputs $X7$ to $X10$). This means that the input has some influence but no interactions with other inputs. In this case, the bound $\Upsilon_j$ is relevant (close to $S_{T_j}$), except for $X7$. The interpretation of the bound gives a useful information about the total influence of the input.
3. First order Sobol’ index is negligible while total Sobol’ index significantly differs from zero (inputs $X1$ to $X6$). In this case, the bound $\Upsilon_j$ largely overstimates the total Sobol’ index $S_{T_j}$ for $X3$, $X4$ and $X5$. However, for $X_4$, we have $\Upsilon_4 < 1$ and this coarse information is still usefull. For the three other inputs, the bound is relevant.
For two inputs ($X3$ and $X5$), results can be judged as strongly unsatisfactory as the bound is useless (larger than $1$ which is the maximal value for a sensitivity index). We suspect that these results come from:
- the model non linearity with respect to these inputs (see equation (\[eq:morris\])),
- the input distributions (exponential and Weibull).
The second explanation seems to be the more convincing as these types of distribution can provide larger values during Monte Carlo simulations. In this case, departures from the central part of the input domain leads to uncontrolled derivative values of the Morris function. Indeed, it can be seen that $\nu_j$ is particularly large for $X3$ and $X5$, because of high derivative values in the estimation samples. Moreover, we have no observed the same results for $X_1$, $X_2$ and $X_4$.
As a conclusion of this first test, we argue that the bound $\Upsilon_j$ is well-suited for a screening purpose. Moreover, coupling $\Upsilon_j$ interpretation with first order Sobol’ indices $S_j$ (estimated at low cost using a smoothing technique or a metamodel, see [@salcha00; @ioo11]) can bring useful information about the presence or absence of interaction. For inputs following uniform, normal and exponential distributions,the bound is extremely efficient. In these particular cases, the bound is the best one and cannot be improved.
A case study: a flood model {#sec:case}
---------------------------
To illustrate how the Cheeger constant can be used for factors prioritization, when we use the DGSM, we consider a simple application model that simulates the height of a river compared to the height of a dyke. When the height of a river is over the height of the dyke, flooding occurs. This academic model is used as a pedagogical example in Iooss [@ioo11]. The model is based on a crude simplification of the 1D hydro-dynamical equations of SaintVenant under the assumptions of uniform and constant flowrate and large rectangular sections. It consists of an equation that involves the characteristics of the river stretch: $$S = Z_v + H -H_d - C_b \quad \mbox{with} \quad H = \left(\frac{Q}{BK_s \sqrt{\frac{Z_m-Z_v}{L} }} \right)^{0.6},$$ with $S$ the maximal annual overflow (in meters) and $H$ the maximal annual height of the river (in meters).
The model has $8$ input variables, each one follows a specific probability distribution (see Table \[tab:factors\]). Among the input variables of the model, $H_d$ is a design parameter. The randomness of the other variables is due to their spatio-temporal variability, our ignorance of their true value or some inaccuracies of their estimation. We suppose that the input variables are independent.
Input Description Unit Probability distribution
------- ----------------------------- --------- ---------------------------------------------------------------
$Q$ Maximal annual flowrate m$^3$/s Truncated Gumbel ${\mathcal G}(1013, 558)$ on $[500 , 3000 ]$
$K_s$ Strickler coefficient - Truncated normal ${\mathcal N}(30, 8)$ on $[15 , +\infty [$
$Z_v$ River downstream level m Triangular ${\mathcal T}(49, 50, 51)$
$Z_m$ River upstream level m Triangular ${\mathcal T}(54, 55, 56)$
$H_d$ Dyke height m Uniform ${\mathcal U}[7, 9]$
$C_b$ Bank level m Triangular ${\mathcal T}(55, 55.5, 56)$
$L$ Length of the river stretch m Triangular ${\mathcal T}(4990, 5000, 5010)$
$B$ River width m Triangular ${\mathcal T}(295, 300, 305)$
: Input variables of the flood model and their probability distributions[]{data-label="tab:factors"}
We also consider another model output: the associated cost (in million euros) of the dyke presence, $$C_p = \indic_{S>0} + \left[0.2 + 0.8\left( 1-\exp^{-\frac{1000}{S^4}}\right) \right]\indic_{S \leq 0} + \frac{1}{20}\left(H_d \indic_{H_d>8} + 8 \indic_{H_d \leq 8} \right),$$ with $\indic_{A}(x)$ the indicator function which is equal to 1 for $x \in A$ and 0 otherwise. In this equation, the first term represents the cost due to a flooding ($S>0$) which is 1 million euros, the second term corresponds to the cost of the dyke maintenance ($S \leq 0$) and the third term is the investment cost related to the construction of the dyke. The latter cost is constant for a height of dyke less than $8$ m and is growing proportionally with respect to the dyke height otherwise.
Sobol’ indices are estimated with the same algorithms than for the Morris function, using two initial Monte Carlo samples of size $10^5$ and $20$ replicates of the estimates. It leads to $2\times 10^7$ model evaluations in order to compute first order indices $S_j$ and total indices $S_{T_j}$ (by taking the mean of the $20$ replicates). For estimating the DGSM ($\nu_j$, weighted DGSM $\tau_j$ and the total sensitivity upper bound $\Upsilon_j$), a Sobol sequence is used with $1\times 10^4$ model evaluations.
Results of global sensitivity analysis and derivative-based global sensitivity analysis for respectively the overflow $S$ and the cost $C_p$ outputs are listed in Tables \[tab:OSI\] and \[tab:CSI\]. Global sensitivity indices show small interaction among input variables for the overflow and the cost outputs. Four input variables ($Q$, $H_d$, $K_s$, $Z_v$) drive the overflow and the cost outputs. This variable classification will serve as reference for comparison issue.
Input $S_j$ $S_{T_j} $ $\nu_j$ $\tau_j$ $\Upsilon_j$
------- ------- ------------ ----------- ---------- --------------
$Q$ 0.343 0.353 1.296e-06 1.072 2.807
$K_s$ 0.130 0.139 3.286e-03 1.033 0.198
$Z_v$ 0.185 0.186 1.123e+00 1377.41 0.561
$Z_m$ 0.003 0.003 2.279e-02 33.742 0.011
$H_d$ 0.276 0.276 8.389e-01 23.77 0.340
$C_b$ 0.036 0.036 8.389e-01 1268.90 0.105
$L$ 0.000 0.000 2.147e-08 0.268 0.000
$B$ 0.000 0.000 2.386e-05 1.070 0.000
: Sensitivity indices for the overflow output of the flood model.[]{data-label="tab:OSI"}
Input $S_j$ $S_{T_j} $ $\nu_j$ $\tau_j$ $\Upsilon_j$
------- ------- ------------ ------------ ---------- --------------
$Q$ 0.346 0.460 1.3906e-06 2.013 3.011e+00
$K_s$ 0.172 0.269 8.5307e-03 1.926 5.129e-01
$Z_v$ 0.187 0.229 1.3891e+00 1715.89 6.932e-01
$Z_m$ 0.006 0.012 4.6038e-02 68.17 2.29e-02
$H_d$ 0.118 0.179 1.5366e+00 44.04 6.227e-01
$C_b$ 0.026 0.039 9.4628e-01 1428.69 1.180e-01
$L$ 0.000 0.000 4.0276e-08 0.503 2.009e-06
$B$ 0.001 0.001 4.4788e-05 2.007 5.587e-04
: Sensitivity indices for the cost ouput of the flood model.[]{data-label="tab:CSI"}
Based on derivative sensitivity indices ($ \nu_j$) or weighted derivative sensitivity indices ($\tau_j$) we have obtained another subset of the most influential variables that are $Z_v$, $C_b$, $H_d$, $Z_m$. These results mean that, for example, the maximum annual flowrate ($Q$) does not have any impact on the overflow and the cost output. If we compare these results to the global sensitivity indices, we can infer that they are obviously wrong. This is easily explained by the fact that the input variables have different unities and that the indices $\nu_j$ and $\tau_j$ have not been renormalized by the constant depending on the probability distribution of $X_j$. By looking at the total sensitivity upper bound $\Upsilon_j$, the most influential variables are the following: $Q$, $Z_v$, $H_d$, $K_s$ for the overflow output and for the cost output. It gives the same subset of the most influential variables with some slight differences for the prioritization of the most influential variables. In conclusion, we state that $\Upsilon_j$ can provide correct information on input variance-based sensitivities.
Conclusion {#sec:con}
==========
Global sensitivity analysis, that allows exploring numerically complex model and factors fixing setting, requires a large number of model evaluations. Derivative-based global sensitivity method needs a much smaller number of model evaluations (gain factor of $10$ to $100$). The reduction of the number of model evaluations becomes more significant when the model output is controlled by a small number of input variables and when the model does not include much interaction among input variables. This is often the case in practice.
In this paper, we have produced an inequality linking the total Sobol’ index and a derivative-based sensitivity measure for a large class of probability distributions (Boltzmann measures). The new sensitivity index $\Upsilon_j$, which is defined as a constant times the crude derivative-based sensitivity, is a maximal bound of the total Sobol’ index. It improves factors fixing setting by using derivative-based sensitivities instead of variance-based sensitivities.
Two numerical tests have confirmed that the bound $\Upsilon_j$ is well-suited for a screening purpose. When total Sobol’ indices cannot be estimated because of a cpu time expensive model, $\Upsilon_j$ can provide correct information on input sensitivities. Previous studies have shown that estimating DGSM with a small derivatives’ sample (with size from tens to hundreds) allows to detect non influent inputs. In subsequent works, we propose to use jointly DGSM and first order Sobol’ indices. With these information, an efficient methodology of global sensitivity analysis can be applied and brings useful information about the presence or absence of interaction (see Iooss ${\it{et \, al.}}$ [@ioopop12]).
Acknowlegments
==============
Part of this work has been backed by French National Research Agency (ANR) through COSINUS program (project COSTA BRAVA noANR-09-COSI-015). We thank Jean-Claude Fort for helpful discussions and two anonymous reviewers for their valuable comments.
[^1]: Corresponding author: [email protected], Phone: +33 1877969, Fax: +33 130878213
|
---
abstract: |
We study the small–angle double bremsstrahlung in $e^-e^\pm$ scattering for a jet kinematics where both photons move along the electron direction. This region gives the main contribution to the cross section. We present analytic expressions for all 64 amplitudes with arbitrary helicity states of the initial and final leptons and the produced photons convenient for analytic and numerical studies. The accuracy of the obtained amplitudes is given omitting only terms of the order of $ m^2/E_j^2$, $\theta_j^2$ and $\theta_j
m/E_j$. The helicity amplitudes for the cross channel $\gamma
e^\pm \to \gamma e^+e^- + e^\pm$ are given. Several limits for the helicity amplitudes of hard or soft final particles are considered.
[*PACS:*]{} 12.20.-m; 13.85.Qk; 13.88.+e; 13.40.-f
address:
- 'Joint Institute for Nuclear Research, Dubna, 141980, Russia '
- 'Institut für Theoretische Physik and NTZ, Universität Leipzig, D-04109 Leipzig, Germany, '
- 'Novosibirsk State University, Novosibirsk, 630090, Russia'
author:
- 'E.A. Kuraev'
- 'A. Schiller'
- 'V.G. Serbo'
- 'B.G. Shaikhatdenov'
date: '2 September, 1999'
title: ' Helicity amplitudes for the small–angle process $e^-e^{\pm}\to e^-\gamma\gamma + e^{\pm}$ with both photons along one direction and its cross channel '
---
,
,
,
Quantum electrodynamics, polarized leptons, polarized photons, jet kinematics
Introduction
============
Nowadays colliders and detectors are able to work with polarized particles. Therefore QED jet–like processes are required to be described in full detail including the helicities of all particles. Among other reactions, those inelastic processes whose cross sections do not fall with energy are of special interest. In particular, QED jets formed by two or three particles in reactions of the type $$\begin{aligned}
e^+ e^- \to e^+ \gamma + e^-\,, \quad e^+ e^- \to e^+ \gamma
\gamma + e^-\,, \quad e^+ e^- \to e^+e^+ e^- + e^-\end{aligned}$$ provide a realistic model for hadronic jets. The complete set of all QED jet–like tree processes up to order $\alpha^4$ in the electromagnetic coupling have been listed in our previous paper [@KSSS]. For all of them except one we have obtained compact and simple analytic expressions for all helicity amplitudes to a high accuracy [@KSS85; @KSS86; @KSSS]. The process of double bremsstrahlung along one direction and its cross channel, discussed in the present paper, completes the calculation of all helicity amplitudes to order $\alpha^4$.
Let us define by a jet kinematics in QED a high energy reaction in which the outgoing leptons and photons are produced within a small cone[^1] relative to an axis given by their parental incoming lepton or photon. At high energies with the condition ($p_1$, $p_2$ are the incoming 4-momenta, $m_j$ are the lepton masses) $$\begin{aligned}
s= 2 p_1 p_2 = 4 E_1 E_2 \gg m_j^2
\label{mass}\end{aligned}$$ the dominant contribution to the non-decreasing cross sections is given by the region of scattering angles $\theta_j$ which is much smaller than unity though could be of the order of the typical emission angles $m_j / E_j$ or larger: $$\begin{aligned}
{m_j}/{E_j} \stackrel{<}{\sim} \theta_j \ll 1 \ .
\label{theta}\end{aligned}$$ In this kinematic region all processes have the form of two–jet processes with an exchange of a single virtual photon $\gamma^*$ in the $t$–channel (see Fig. \[fig:1\]).
In the kinematic region (\[mass\]), (\[theta\]) it is possible to obtain compact and relatively simple analytic expressions for all helicity amplitudes of the jet-like QED processes to a high accuracy omitting terms of the relative order of $$\begin{aligned}
\frac{m_j^2}{E_j^2} \,, \ \ \theta_j^2 \,, \ \
\frac{m_j}{E_j} \,\theta_j
\label{accuracy}\end{aligned}$$ only.
The amplitude $M_{fi}$ corresponding to the diagram of Fig. \[fig:1\] can be presented in the form $$\begin{aligned}
\label{9}
M_{fi} = M_{\mathrm{up}}^{\mu} \, \frac{g_{\mu \nu}} {k^2} \,
M_{\mathrm{down}}^\nu\,,\end{aligned}$$ where $M_{\mathrm{up}}^\mu$ and $M_{\mathrm{down}}^\nu$ are the amplitudes of the upper and lower block of Fig. \[fig:1\], respectively, and $(-g_{\mu \nu}/2)$ is the density matrix of the virtual photon. The transition amplitude $M_{\mathrm{up}}$ describes the scattering of an incoming particle (in our case a lepton) with a virtual photon of “mass” squared $k^2$ and polarization 4–vector $e=k_\perp/\sqrt{-k_\perp^2}$ ($k_\perp$ is the 4–vector transverse to $p_1$ and $p_2$) to some QED final state in the jet kinematics of Eqs. (\[mass\]), (\[theta\]) (similar for $M_{\mathrm{down}}$).
To introduce some notations for $e^-e^+$ collisions we use the block diagram of Fig. \[fig:1\] as an example. We work in a reference frame in which the initial particles with 4–momenta $p_1$ and $p_2$ perform a head–on collision with energies $E_1$ and $E_2$ of the same order. The $z$–axis is chosen along the momentum of the first initial particle, the azimuthal angles are denoted by $\varphi_j$ (they are referred to one fixed $x$-axis). Exploiting the freedom in choosing a general phase factor, we put the initial azimuthal angle of the electron $\varphi_1$ and that of the positron $\varphi_2$ equal to zero: $$\begin{aligned}
\varphi_1=\varphi_2=0 \,.
\label{azim}\end{aligned}$$ It is convenient to introduce “the almost light–like” 4–vectors $p$ and $p'$ $$\begin{aligned}
p &=& p_1 - \frac{m^2}{s} \, p_2\,, \quad
p'=p_2 - \frac{m^2}{s}\, p_1\,, \quad p_1^2=p_2^2=m^2 \,,
\nonumber
\\
p^2 & =& {p'}^2 = \frac{m^6} {s^2} \approx 0 \,, \ \
s = 2p_1 p_2 = 2 p p' + 3 \frac{m^4}{s} \,.
\label{light}\end{aligned}$$
With accuracy (\[accuracy\]) the matrix $g_{\mu\nu}$ from Eq. (\[9\]) can be transformed to the form $$\begin{aligned}
\label{eq:2}
g_{\mu\nu} \rightarrow
\frac{2}{s} p'_{\mu}p_{\nu}\,,\end{aligned}$$ (for details, see [@AB], §4.8.4) which results in the factorized representation for the amplitude $$\begin{aligned}
M_{fi}=\frac{s}{k^2} J_{\mathrm{up}} J_{\mathrm{down}} \,.
\label{amplitude}\end{aligned}$$ The vertex factors $J_{\mathrm{up,down}}$ are given by the block amplitudes $M_{{\mathrm{up}}}^\mu$ and $M_{{\mathrm{down}}}^\nu$ $$\begin{aligned}
J_{\mathrm{up}} = \frac{\sqrt{2}}{s} \,
M_{\mathrm{up}}^\mu \, p'_\mu \,, \ \
J_{\mathrm{down}} = \frac{\sqrt{2}}{s} \,
M_{\mathrm{down}}^\nu \, p_\nu \,.
\label{J1J2}\end{aligned}$$ The quantities $J_{\mathrm{up}}$ and $J_{\mathrm{down}}$ can be calculated in the limit $s \to \infty$ assuming that the energy fractions and transverse momenta of the final particles are finite.
We study the double bremsstrahlung in one direction for small angle $e^-e^\pm$ scattering $$\begin{aligned}
e^-(p_1)+e^{\pm}(p_2)\to e^-(p_3)+\gamma(k_1)+\gamma(k_2)+e^
{\pm}(p_4) \,.
\label{DB}\end{aligned}$$ For unpolarized particles this process has been discussed in Refs. [@BG] and [@KYF74] where the differential cross sections and photon spectra were calculated in the approximation of classical currents and equivalent photons. In Refs. [@KYF74; @KF78] the matrix element squared for the unpolarized case has been found to the accuracy (\[accuracy\]). The helicity amplitudes of the process (\[DB\]) for large angle scattering have been calculated in Refs. [@KP; @Behr]. The recent interest to multiphoton emission is related to the exact calculation of Bhabha scattering needed for LEP luminosity calibration (see, for example, Refs. [@CERN]).
In Born approximation the process (\[DB\]) (with photons not only along one direction) is described by 32 Feynman diagrams. In the considered kinematics (forward scattering at high energies) only 16 scattering–type diagrams are relevant since the contribution to the cross section which does not decrease with the energy arises from those Feynman diagrams with a virtual photon exchange in the $t$–channel. Restricting to the case in which both photons are emitted along the electron ($p_1$) direction, only six diagrams are relevant. Three of them are shown in Fig. \[fig:2\].
The double bremsstrahlung helicity amplitudes in the jet kinematics
===================================================================
The Sudakov 4-vector decomposition and kinematic invariants
-----------------------------------------------------------
The amplitude for the double forward bremsstrahlung of the process (\[DB\]) can be written with accuracy (\[mass\]), (\[theta\]) in the factorized form (for definiteness the Bhabha scattering is considered, see Fig. \[fig:2\], where the momenta and helicities of all particles are indicated) $$\begin{aligned}
&&M(e^-e^+\to e^-\gamma\gamma + e^+)
=\frac{s}{k^2} J_{\mathrm{up}}({e^- \gamma^*\to e^- \gamma \gamma})
J_{\mathrm{down}}( e^+ \gamma^* \to e^+) \,,
\\
&& k=p_4-p_2=p_1-p_3-k_1-k_2 \,.
\nonumber\end{aligned}$$ The vertex factor $J_{\mathrm{down}}$ has been calculated earlier [@KSSS]: $$\begin{aligned}
J_{\mathrm{down}} (e^+_{\lambda_{\bar{e}2}}
\gamma^* \to e^+_{\lambda_{\bar{e}4}})=\sqrt{8 \pi \alpha}\,
{\mathrm{e}}^{-{\mathrm{i}} \, \lambda_{\bar{e}4} \, \varphi_4}
\,\delta_{\lambda_{\bar{e}2},\lambda_{\bar{e}4}} \,,
\label{matelemdown}\end{aligned}$$ The quantity $\lambda_{\bar{e}2}= \pm1/2$ ($\lambda_{\bar{e}4}$) denotes the helicities of the initial (final) positron, $\varphi_4$ is the azimuthal angle of the final $e^+$.
We have to calculate the upper vertex $J_{\mathrm{up}}$. The almost light-like vectors $p$ and $p'$ have components $ p=E_1(1,0,0,1)$, $p'=E_2(1,0,0,-1)$. It is convenient to use the Sudakov decomposition of any 4–vector along $p$ and $p'$ and transverse to them. For the momenta of the photons and the initial and final electron we have $$\begin{aligned}
\label{sudakov}
&& k_i=\alpha_i p'+x_i p+k_{i\perp} \,, \quad
k_{i\perp}^2=-{\mathbf {k}}_i^2 \,, \quad (i=1,2) \,,
\nonumber
\\
&& k=\alpha_k p'+\beta_k p+k_{\perp} \,, \quad
k_\perp^2=-{\mathbf {k}}^2 \,,
\\
&& p_1= \frac{m^2}{s} p' + p \,, \quad
p_3= \alpha_3 p' + x_3 p + p_{3\perp} \,, \quad
p_{3\perp}^2=-{\mathbf {p}}_3^2\,,
\nonumber\end{aligned}$$ with the two–dimensional Euclidean vectors ${\mathbf{k}}$, ${\mathbf{k}}_i$ and ${\mathbf{p}}_3=-{\mathbf{k}}-{\mathbf{k}}_1- {\mathbf{k}}_2$ perpendicular to the beam axes, e.g. $$\begin{aligned}
{\mathbf k}=(k_x,k_y)\,.
\label{2vector}\end{aligned}$$ In the kinematics of a jet moving close to the direction of the first electron the 2–vectors ${\mathbf {k}}_i$ and ${\mathbf {p}}_3$ have typical values of a few electron masses whereas ${\mathbf {k}}$ might be much smaller. The quantities $x_{1,2}$ and $x_3=1-x_1-x_2$ are the energy fractions of the emitted photons and the scattered electron $$\begin{aligned}
x_i=\frac{k_{i0}}
{E_1},\qquad x_3=\frac{p_{30}}{E_1}=\frac{E_3}{E_1} \,.\end{aligned}$$ They are supposed to be of the order of unity while $\alpha_i$, $\alpha_3$, $\alpha_k$ and $\beta_k$ are small and vanish as $1/s$. Later on we often use complex circular components of a generic 2–vector transverse to the beam axes denoted by the corresponding Greek characters, e.g. for the 2–vector (\[2vector\]) $$\begin{aligned}
\kappa= k_x + {\mathrm{i}} \, k_y \,, \quad
\kappa^*= k_x - {\mathrm{i}} \, k_y \,.
\label{circular}\end{aligned}$$
The on–shell conditions for the incoming electron and the outgoing electron and photons lead to $$\begin{aligned}
\label{alphai}
&&s\alpha_i=\frac{{\mathbf {k}}^2_i}{x_i} \,, \quad
s\alpha_3=\frac{m^2+ {\mathbf{p}}_3^2}{x_3}\,,
\\
&&s(\alpha_k+\alpha_1+\alpha_2)=-\frac{1}{x_3}
\left[ m^2(1-x_3)+{\mathbf p}_3^2 \right] \,.
\nonumber\end{aligned}$$ From the on–shell condition for the outgoing spectator (a positron in Bhabha scattering) $p_4^2=(p_2+k)^2=m^2$ we find the expression $$\begin{aligned}
\qquad s\alpha_k\beta_k-{\mathbf {k}}^2+s\beta_k+m^2\alpha_k=0
\label{sab}\end{aligned}$$ and present the momentum transfer squared (the 4–momentum of the virtual photon $k$) in the form $$\begin{aligned}
k^2=s\alpha_k\beta_k-{\mathbf {k}}^2=
-\frac{1}{1+\alpha_k}\left({\mathbf{k}}^2
+m^2\alpha_k^2\right) \,.
\label{alpha}\end{aligned}$$ Taking into account Eq. (\[alpha\]), we observe that the Sudakov parameter $\alpha_k$ is related to the invariant mass squared of the jet generated by the scattered electron and two accompanying photons $$\begin{aligned}
\label{invmass}
(p_3+k_1+k_2)^2=(p_1-k)^2=m^2-{\mathbf {k}}^2-s\alpha_k \,.\end{aligned}$$ From Eqs. (\[alphai\]), (\[sab\]) and (\[alpha\]) explicit expressions for $\alpha_k$ and $\beta_k$ can be derived.
After replacing all Sudakov parameters by the energy fractions $x_i$, $x_3$ and the 2-vectors ${\mathbf k}_i$ and ${\mathbf k}$, the kinematic invariants appearing in the upper vertex take the following form: $$\begin{aligned}
\label{eq:5}
a_i&\equiv&-(p_1-k_i)^2+m^2=
\frac{1}{x_i}(m^2x_i^2+{\mathbf k}_i^2) \,,
\nonumber
\\
b_i&\equiv&(p_3+k_i)^2-m^2=\frac{1}{x_i x_3}(m^2 x_i^2+
{\mathbf r}_i^2) \,,
\nonumber
\\
a&\equiv&-(p_1-k_1-k_2)^2+m^2=a_1+ a_2 -\frac{1}{x_1x_2}
\left(x_1{\mathbf{k}}_2 -x_2{\mathbf{k}}_1 \right)^2
\nonumber
\\
&=& \frac{1}{x_1x_2}\biggl[x_1x_2(1-x_3)m^2
+x_2(1-x_2){\mathbf k}_1^2 +x_1(1-x_1){\mathbf k}_2^2
+2x_1x_2{\mathbf k}_1 {\mathbf k}_2\biggr] \,,
\nonumber
\\
b&\equiv&(p_3+k_1+k_2)^2-m^2=b_1+b_2+\frac{1}{x_1x_2}
\left(x_1{\mathbf{k}}_2 -x_2{\mathbf{k}}_1 \right)^2
\\
&=&\frac{1}{x_1x_2x_3}\biggl[x_1x_2(1-x_3)m^2+x_2(1-x_2)
({\mathbf {k}}_1+x_1{\mathbf {k}})^2
\nonumber
\\
&& +x_1(1-x_1)({\mathbf {k}}_2+x_2 {\mathbf {k}})^2
+2x_1x_2 ({\mathbf {k}}_1+x_1 {\mathbf {k}})
({\mathbf {k}}_2+x_2 {\mathbf {k}}) \biggr] \,,
\nonumber\end{aligned}$$ where $$\begin{aligned}
{\mathbf r}_1=x_1({\mathbf k}_2+{\mathbf k})+(1-x_2){\mathbf k}_1
\,, \quad
{\mathbf r}_2=x_2({\mathbf k}_1+{\mathbf k})+(1-x_1){\mathbf k}_2
\label{r1r2}\end{aligned}$$ or $$\begin{aligned}
\label{ri}
{\mathbf r}_i=x_3{\mathbf k}_i - x_i{\mathbf p}_3\,.\end{aligned}$$ Note that $b= - {\mathbf{k}}^2 - s \alpha_k$ and $b\to a/x_3$ in the limit ${\mathbf{k}}\to 0$.
Using the gauge invariance of an amplitude containing photons we can replace the polarization vector $e_i$ of a photon $i$[^2] with external momentum $k_i$ by $e_i + \zeta_i k_i$. The arbitrary parameter $\zeta_i$ is chosen in such a way that the polarization vector does not contain a Sudakov projection onto the light–like vector $p$: $$\begin{aligned}
e_i\equiv e^{(\lambda_i)}(k_i)= \alpha_{e_i}
p' + e_\perp^{(\lambda_i)} \,.
\label{polvector}\end{aligned}$$ The transverse component $e_\perp$ which does not depend on the 4–momentum of the photon $k_i$ is chosen as $$\begin{aligned}
\label{eq:9}
e^{(\lambda_i)}_{\perp}=- \frac{\lambda_i}{\sqrt{2}}
(0, 1, {\mathrm{i}}\, \lambda_i, 0) \,, \quad
e^{(\lambda_i)*}_{\perp}=- e^{(-\lambda_i)}_{\perp} \,.\end{aligned}$$ The Sudakov parameters $\alpha_{e_i}$ are found from the conditions $e_ik_i=0$: $$\begin{aligned}
\label{eq:10}
s\alpha_{e_i}=\frac{2 {\mathbf {e}}^{(\lambda_i)}
{\mathbf {k}}_i}{x_i}=
\frac{\sqrt{2}}{x_i}(-\delta_{\lambda_i,1}\kappa_i+
\delta_{\lambda_i,-1}\kappa_i^*)\,.\end{aligned}$$ Here $\kappa_i$ and $\kappa_i^*$ are the circular components of the vector ${\mathbf{k}}_i$ \[compare Eq. (\[circular\])\].
Helicity amplitudes for the upper vertex factor $J_{\mathrm{ up}}(e^- \gamma^* \to e^- \gamma \gamma)$
------------------------------------------------------------------------------------------------------
Now all ingredients are prepared to calculate the helicity amplitudes for the upper vertex factor. We will use the following notation indicating the helicity states of the electron $\lambda_{ei}$ and of the photons $\lambda_i$ explicitly: $$\begin{aligned}
J_{\mathrm{up}}\equiv \frac{\sqrt{2} (4 \pi \alpha)^{3/2}}{s}
\, {\cal{M}}_{\lambda_{e1}\lambda_{e3}}^{\lambda_1\,\lambda_2} \,.
\label{matelemup}\end{aligned}$$ The matrix element ${\cal{M}}$ can be represented by the Feynman diagrams
shown in Fig. \[fig:3\]. The amplitudes corresponding to that figure are $$\begin{aligned}
\label{eq:12}
{\cal{M}}=(1+{\cal{P}}_{12})Q \,, \qquad
Q={\cal{M}}_1+{\cal{M}}_2+{\cal{M}}_3\,,\end{aligned}$$ where $$\begin{aligned}
\label{amplitudes}
{\cal{M}}_1&=&\frac{1}{a_1 a}\bar u_3\hat p'(\hat p_1-\hat k_1
-\hat k_2+m)\hat e_2^*(\hat p_1-\hat k_1+m)
\hat e_1^*u_1\,,
\nonumber
\\
{\cal{M}}_2&=&-\frac{1}{a_1 b_2}\bar u_3
\hat e_2^*(\hat p_1-\hat k_1-\hat k+m)\hat p'(\hat p_1-\hat k_1+m)
\hat e_1^* u_1\,,
\\
{\cal{M}}_3&=&\frac{1}{b_2 b}\bar u_3
\hat e_2^*(\hat p_1-\hat k_1-\hat k+m)
\hat e_1^*(\hat p_1-\hat k+m) \hat p 'u_1 \,.
\nonumber\end{aligned}$$ The permutation operator ${\cal{P}}_{12}$ for the photons is defined as $$\begin{aligned}
{\cal{P}}_{12}f(k_1,e_1;k_2,e_2)=
f(k_2,e_2;k_1,e_1)\,, \qquad {\cal{P}}_{12}^2 =1. \nonumber\end{aligned}$$
The quantity $Q$ is gauge invariant with respect to the virtual photon $k$ since all permutations of this photon are taken into account. Therefore $Q$ is proportional to $k_\perp$ in the limit $k_\perp \to 0$. Indeed, using the relations $$\begin{aligned}
Q= p'_\mu Q^\mu \,, \ \ \ k_\mu Q^\mu=
(\alpha_k p' +\beta_k p +k_\perp)_\mu Q^\mu =0\,,\end{aligned}$$ we immediately obtain (neglecting the small contribution $\beta_k p_\mu Q^\mu\sim 1/s$) $$\begin{aligned}
\label{eq:13}
Q= -\frac{k_{\perp\mu}}{\alpha_k} Q^\mu \,.\end{aligned}$$
Now we transform the quantities ${\cal M}_j$ to such a form that in their sum $Q$ this noticed low $k_\perp$ behaviour is present explicitly. The reason is that in that case all individual large (compared to $k_\perp$) contributions are cancelled. The first step is to use the Dirac equations $\hat p_1 u_1 =m u_1$, $\bar u_3 \hat p_3 = m \bar u_3$ and to rearrange the amplitudes ${\cal{M}}_j$ of Eq. (\[amplitudes\]) as follows: $$\begin{aligned}
\label{eq:14}
{\cal{M}}_1&=& \bar u_3 \Biggl\{
\frac{sx_3}{a_1 a} \hat e_2^*(\hat p_1-\hat k_1+m) \hat e_1^* +
\frac{1}{a_1 a} \hat p'\hat k \hat e_2^*
(\hat p_1-\hat k_1+m) \hat e_1^* \Biggr\} u_1\,,
\nonumber
\\
{\cal{M}}_2&=& \bar u_3 \Biggl\{
-\frac{s(1-x_1)}{a_1 b_2} \hat e_2^*
(\hat p_1-\hat k_1+m) \hat e_1^*
- \frac{1}{b_2} \hat e_2^* \hat p' \hat e_1^*
\nonumber
\\
&& +\frac{1}{a_1 b_2} \hat e_2^* \hat k \hat p'
(\hat p_1- \hat k_1+m) \hat e_1^* \Biggr\} u_1 \,,
\\
{\cal{M}}_3&=& \bar u_3 \Biggl\{
\frac{s}{b_2 b} \hat e_2^* (\hat p_1-\hat k_1+m) \hat e_1^*
-\frac{s}{b_2 b} \hat e_2^* \hat k \hat e_1^*
\nonumber
\\
&& -\frac{1}{b_2 b} \hat e_2^*(\hat p_3+\hat k_2+m)
\hat e_1^* \hat k \hat p' \Biggr\} u_1 \,.
\nonumber\end{aligned}$$ From these formulae we observe that the last terms in ${\cal{M}}_1, {\cal{M}}_2, {\cal{M}}_3$ are proportional to $k_{\perp}$ up to terms of the order of (\[accuracy\]): $$\begin{aligned}
\label{eq:15}
\hat p'\hat k=\hat p' (\alpha_k \hat p' +
\beta_k \hat p + \hat k_\perp) =
\hat p'\hat k_\perp= - \hat k \hat p' \,.\end{aligned}$$ As a next step we note that the sum of the first three terms in Eqs. (\[eq:14\]) is also proportional to $k_\perp$ since \[see Eqs. (\[eq:5\])\] $$\begin{aligned}
\label{eq:16}
A \equiv \frac{x_3}{a_1 a}-\frac{1-x_1}{a_1 b_2}+\frac{1}{b_2 b} \,,
\qquad A |_{k_{\perp}\to 0}=0\,.\end{aligned}$$ Finally we consider the sum of the second terms of the quantities ${\cal{M}}_2, {\cal{M}}_3$ given in Eqs. (\[eq:14\]). Using the relations (\[eq:5\]) and (\[invmass\]) one gets $$\begin{aligned}
\label{eq:17}
-\frac{\hat p'}{b_2}-\frac{s(\alpha_k \hat p'+
\hat k_{\perp})}{b_2 b}=
-\frac{s\hat k_{\perp}}{b_2 b}+
\frac{\hat p' {\mathbf {k}}^2}{b_2 b} \,.\end{aligned}$$ Therefore, from Eqs. (\[eq:15\]), (\[eq:16\]), (\[eq:17\]) it is clearly seen that the discussed property (\[eq:13\]) $$\begin{aligned}
\left({\cal{M}}_1+{\cal{M}}_2+{\cal{M}}_3 \right)
\vert_{k_{\perp}\to 0}=0\end{aligned}$$ is obviously satisfied and the quantity $Q=\sum_{j=1}^3 {\cal{M}}_j$ is a sum of terms explicitly proportional to $k_{\perp}$: $$\begin{aligned}
\label{eq:18}
Q&=&\bar u_3\biggl\{A s \,\hat e_2^*
(\hat p_1-\hat k_1+m) \hat e_1^*
+\frac{1}{a_1 a} \hat p' \hat k_{\perp} \hat e_2^*
(\hat p_1-\hat k_1+m) \hat e_1^*
\nonumber
\\
&& -\frac{{\mathbf {k}}^2} {b_2 b} \hat e_2^* \hat e_1^* \hat p'
- \frac{s}{b_2 b} \hat e_2^* \hat k_{\perp} \hat e_1^*
\\
&& +\frac{1}{a_1 b_2} \hat e_2^* \hat k_{\perp} \hat p'
(\hat p_1-\hat k_1+m) \hat e_1^* -\frac{1}{b_2 b}
\hat e_2^* ( \hat p_3+\hat k_2+m) \hat e_1^* \hat k_{\perp}
\hat p'\biggr\}u_1\,.
\nonumber\end{aligned}$$ The subsequent calculations require to determine several bilinear spinor combinations which are listed in Appendix \[appa\].
As a result, we obtain the complete expression for the amplitude of double forward bremsstrahlung in a jet kinematics $$\begin{aligned}
\label{fullamp}
M(e^- e^+ \to e^- \gamma \gamma + e^+) =
\frac {32 \pi^2 \alpha^2 }{k^2} \,
{\cal{M}}_{\lambda_{e1}\lambda_{e3}}^{\lambda_1\;\lambda_2}\,
{\mathrm{e}}^{-{\mathrm{i}} \, \lambda_{\bar{e}4} \, \varphi_4}
\,\delta_{\lambda_{\bar{e}2},\lambda_{\bar{e}4}} \,.\end{aligned}$$ All helicity amplitudes ${\cal{M}}_{\lambda_{e1}\lambda_{e3}}^{\lambda_1\;\lambda_2}$ are expressed via the complex circular components $\kappa_i$, $\kappa$ and $\rho_i$ of the vectors ${\mathbf{k}}_i$, $\mathbf{k}$ and ${\mathbf{r}}_i$ \[see Eqs. (\[circular\]) and (\[r1r2\])\] $$\begin{aligned}
\label{eq:20}
\kappa_i=k_{ix}+ {\mathrm{i}} \, k_{iy} \,, \quad
\kappa=k_x+ {\mathrm{i}} \, k_y \,, \quad
\rho_i=r_{ix}+{\mathrm{i}}\, r_{iy}\,, \quad i=1,2\,.\end{aligned}$$ The spin non–flip amplitudes (with $\lambda_{e1}=\lambda_{e3}=+1/2 \equiv +$) being proportional to the factor $$\begin{aligned}
\label{S}
S_{\lambda_{e3}}= s \sqrt{x_3} {\mathrm{e}}^{{\mathrm{i}} \,
\varphi_3 \lambda_{e3}}\,,\end{aligned}$$ are equal to[^3] $$\begin{aligned}
\label{nonflip}
{\cal{M}}^{++}_{++}&=&
2 S_+ (1+{\cal{P}}_{12})\Biggl\{
A \frac{\kappa_1^* \rho_2^*}{x_1 x_2 x_3}
-\frac{\kappa_1^*\kappa^*}{x_1 a_1 a }
+ \frac{\kappa^* \rho_2^*}{x_2 x_3 b_2 b} \Biggr\} \,,
\nonumber
\\
{\cal{M}}^{--}_{++}&=&
x_3 S_+ \left({\cal{M}}^{++}_{++}\over S_+\right)^* \,,
\nonumber
\\
{\cal{M}}^{+-}_{++}&=&-2 S_+ \left( C + {\cal{P}}_{12} D\right)\,,
\nonumber
\\
{\cal{M}}^{-+}_{++}&=&{\cal{P}}_{12} {\cal{M}}^{+-}_{++} \,,
\\
C&=& A \left(
\frac{\kappa_1^* \kappa_2}{x_1x_2}-\frac{x_1}{x_3} m^2 \right)
-\frac{\kappa_1^* \kappa}{x_1 a_1 b_2 }+
\frac{x_1 x_2 \kappa^* \kappa + x_1 \kappa^* \kappa_2
+x_2 \kappa_1^* \kappa}{x_1 x_2 b_2 b } \,,
\nonumber
\\
D&=& (1-x_1) \Biggl(
A \frac{\kappa_1 \rho_2^*}{x_1x_2x_3}
-\frac{\kappa_1\kappa^*}{x_1 a_1 a }
+\frac{\kappa\rho_2^*}{x_2 x_3 b_2 b }\Biggr) \,.
\nonumber\end{aligned}$$ Another convenient form of $C$ useful for the soft photon limit (see Section \[softphotonlimit\]) and for the later crossing (Section \[crossingsection\]) is $$\begin{aligned}
\label{alternC}
C&=& \frac{1-x_2}{1-x_1} D^* + \Delta,
\\
\Delta&=& -\frac{1}{a}+\frac{1}{b}
+\frac{x_1x_2 m^2 +\kappa_1^* \rho_2}{x_3 a_1 b_2}
+\frac{\kappa_1^* (x_2 \kappa_1 - x_1 \kappa_2)}{x_1 a_1 a}
-\frac{(x_2 \kappa_1^* -x_1 \kappa_2^*) \rho_2}{x_2 x_3 b_2 b} \,.
\nonumber\end{aligned}$$ Note that in the limit of a soft (second) photon the function $D\propto 1/x_2$ whereas $\Delta$ remains finite.
For the spin flip amplitudes (with $\lambda_{e1}=-\lambda_{e3}=+1/2$) proportional to $ m S_-$ we obtain $$\begin{aligned}
\label{flip}
{\cal{M}}^{++}_{+-}&=&2 m S_- (1+{\cal{P}}_{12})
\Biggl\{
-\frac{1-x_3}{x_2} G^* + x_1 \left[ A \left( \frac{\kappa_1^*}{x_1}
-\frac{\kappa_2^*}{x_2} \right) + \frac{\kappa^*}{a_1 b_2} \right]
\Biggr\}\,,
\nonumber
\\
{\cal{M}}^{--}_{+-}&=&0 \,,
\nonumber
\\
{\cal{M}}^{+-}_{+-}&=& 2 m S_- \left( F + {\cal{P}}_{12} G\right) \,,
\\
{\cal{M}}^{- +}_{+-}&=& {\cal{P}}_{12} {\cal{M}}^{+-}_{+-} \,,
\nonumber
\\
F& =&x_1
\Biggl( A \frac{\rho_2}{x_2 x_3} -\frac{\kappa}{a_1 a}\Biggr)\,,
\quad
G = \frac{x_2}{x_3} \Biggl(
A \frac{\kappa_1}{x_1} + \frac{\kappa}{b_2 b}\Biggr) \,.
\nonumber\end{aligned}$$
The remaining amplitudes with $\lambda_{e1}=-1/2$ can be obtained by applying the parity conservation relation $$\begin{aligned}
{\cal{M}}_{-\lambda_{e1},-\lambda_{e3}}^{-\lambda_1,\;\,-\lambda_2}
= - (-1)^{\lambda_{e1}+ \lambda_{e3}} \left(
{\cal{M}}_{\lambda_{e1},\lambda_{e3}}^{\lambda_1, \;\, \lambda_2}
\right)^* \,.
\label{parity}\end{aligned}$$ This rule follows from the transformation properties of the electron and photon wave functions under space inversion \[see §16 in Ref. [@BLP] and Eq. (\[eq:9\])\]: $$\begin{aligned}
u_i\equiv u_{{\mathbf{p}}_i}^{(\lambda_{ei})} \to
{\mathrm{i}} \, (-1)^{s- \lambda_{ei}}
u_{{\mathbf{p}}_i}^{(-\lambda_{ei})} \,, \quad
{\mathbf{pe}}^{(\lambda_i)} \to
{\mathbf{pe}}^{(-\lambda_i)}= -
{\mathbf{pe}}^{(\lambda_i)*}\,,\end{aligned}$$ where $s=1/2$ and ${\mathbf{p}}$ is an arbitrary 2–vector. These properties give an additional factor $(-1)^{n_\gamma + 2 s - \lambda_{e1} -\lambda_{e3}}$ where $n_\gamma=2$ is the number of real photons. Therefore, under replacement (\[parity\]) we have to take the complex conjugates of the original helicity amplitudes with sign plus for the spin non-flip amplitudes (\[nonflip\]) and sign minus for the spin flip ones (\[flip\]).
All helicity amplitudes in Eqs. (\[nonflip\]) and (\[flip\]) are explicitly proportional to $\kappa$ or to the function $A$. Therefore, they vanish $\propto |{\mathbf{k}}|$ in the limit $|{\mathbf{k}}| \to 0$. It is clear (although not obvious) that also $\Delta$ in Eq. (\[alternC\]) vanishes in that limit since both $C$ and $D$ in (\[nonflip\]) tend to zero.
We note the explicit Bose symmetry between the two photons in the amplitudes (\[nonflip\]) and (\[flip\]): $$\begin{aligned}
{\cal{M}}^{\lambda_1 \;\, \lambda_2}_{\lambda_{e1} \lambda_{e3}}
(\kappa_1,x_1;\kappa_2,x_2)=
{\cal{M}}^{\lambda_2 \;\, \lambda_1}_{\lambda_{e1} \lambda_{e3}}
(\kappa_2,x_2;\kappa_1,x_1) \,.\end{aligned}$$ Since the spin–flip amplitudes are proportional to the lepton mass they are negligible compared to the spin non–flip ones for not too small scattering angles $$\begin{aligned}
\frac{m}{x_{1,2,3}E_1} \ll \theta_{1,2,3} \ll 1 \,.
\label{langle}\end{aligned}$$ We observe that in our kinematics the transitions with the largest change of helicities between initial and final particles $ \vert \lambda_{e1}- \lambda_{e3}-\lambda_1-\lambda_2 \vert=3$ are forbidden, in our case the amplitude ${\cal{M}}_{+-}^{--}=0$. This is in agreement with the vanishing amplitude for single photon bremsstrahlung with the largest change of helicities $ \vert \lambda_{e1} - \lambda_{e3} - \lambda_1 \vert=2$, see later Eq. (\[jup\]).
Limiting cases of soft and hard final particles
===============================================
The soft photon limit {#softphotonlimit}
---------------------
A useful check of the results obtained is the limiting case in which one of the real photons (belonging to the upper block $J_{\mathrm{up}}$) becomes soft. For definiteness let us suppose that this is the second photon which means that $ x_2 \ll1$, $k_2 \to 0$, but ${\mathbf{k}}_2/x_2$ remains finite.
It is known that in this limit the upper block factorizes and can be presented in the form $$\begin{aligned}
J_{\mathrm{up}}= \sqrt{4 \pi \alpha} \,
J_{\mathrm{up}}(e^-_{\lambda_{e1}} \gamma^*
\to e^-_{\lambda_{e3}} \gamma_{\lambda_1}) \, A_2 \, ,\end{aligned}$$ where $$\begin{aligned}
A_2= \left(\frac{p_1}{p_1 k_2}-\frac{p_3}{p_3 k_2} \right) e_2^{*}\end{aligned}$$ is the factor of the accompanying classical emission of the second photon. This factor is easily transformed to $$\begin{aligned}
A_2= \frac{2}{x_2}
\left( \frac{{\mathbf {k}}_2}{a_2}-\frac{{\mathbf {r}}_2}{b_2}
\right) {\mathbf {e}}^{(\lambda_2)*} \,.\end{aligned}$$ The vertex factor $J_{\mathrm{up}}(e^-_{\lambda_{e1}} \gamma^*
\to e^-_{\lambda_{e3}} \gamma_{\lambda_1})$ can be found in Ref. [@KSSS]: $$\begin{aligned}
\label{jup}
&&J_{\mathrm{up}}(e^-_{\lambda_{e1}} \gamma^*
\to e^-_{\lambda_{e3}} \gamma_{\lambda_1})=
8 \pi \alpha \,\sqrt{1-x_1} \,{\mathrm e}^{ {\mathrm i} \,
\varphi_3 \lambda_{e3}}
\\
&&\times
\left\{
\sqrt{2} \, {\mathbf{Q}}_1 {\mathbf{e}}^{(\lambda_1)*} \,
\frac{ 1- x_1 \delta_{ \lambda_1, -2 \lambda_{e1}} } {x_1}
\delta_{\lambda_{e1},\lambda_{e3}}
+ m R_1 \delta_{\lambda_1, 2 \lambda_{e1}}
\delta_{\lambda_{e1},-\lambda_{e3}}
\right\} \,.
\nonumber\end{aligned}$$ Here we use the notations (${\mathbf{q}}_1= {\mathbf{k}}_1/x_1$) $$\begin{aligned}
\label{QR}
{\mathbf {Q}}_1&=&\frac{{\mathbf {k}}_1}{a_1}-
\frac{{\mathbf {r}}_1}{x_3 b_1}
= \frac{{\mathbf{q}}_1}{m^2+{\mathbf{q}}_1^2}
-\frac{{\mathbf{k}}+{\mathbf{q}}_1}{m^2+({\mathbf{k}}
+{\mathbf{q}}_1)^2} \,,
\\
R_1&=&\frac{x_1}{a_1}- \frac{x_1}{x_3 b_1}
= \frac{1}{m^2+{\mathbf{q}}_1^2}
- \frac{1}{m^2+({\mathbf{k}}+{\mathbf{q}}_1)^2} \,.
\nonumber\end{aligned}$$
Therefore, in the soft photon limit the spin non–flip and flip amplitudes are $$\begin{aligned}
\label{softphoton}
{\cal{M}}_{\lambda_{e1} \lambda_{e1}}^{\lambda_1 \;\, \lambda_2}&=&
2 S_{\lambda_{e1}} {\mathbf{Q}}_1 {\mathbf{e}}^{(\lambda_1)*}
\frac{1-x_1 \delta_{\lambda_1,- 2 \lambda_{e1}}}{x_1} A_2 \,, \\
{\cal{M}}_{\lambda_{e1},-\lambda_{e1}}
^{\lambda_1 \ \ \;\lambda_2}&=&
\sqrt{2} m \, S_{-\lambda_{e1}} \, R_1 \, \delta_{\lambda_1,2
\lambda_{e1}}A_2 \,.
\nonumber\end{aligned}$$ These expressions coincide with those from the amplitudes (\[nonflip\]), (\[flip\]) in the limit of a soft (second) photon when $$\begin{aligned}
&&a \to a_1 \,, \quad
b \to b_1 \,, \quad
\rho_1 \to \kappa_1 + x_1 \kappa \,,
\\
&& A \to -\frac{x_3}{a_1 b_2} + \frac{1}{b_1 b_2} \,, \quad
{\cal P}_{12} A \to \frac{x_3}{a_1 a_2}- \frac{1}{b_1 a_2} \,.
\nonumber\end{aligned}$$ This can be easily shown for most of the helicity amplitudes given in Eqs. (\[nonflip\]), (\[flip\]), for the amplitude ${\cal{M}}_{++}^{+-}$ we use the form (\[alternC\]) for $C$ and neglect $\Delta$ in this limit.
The soft electron limit $m\ll E_3 \ll E_1$
------------------------------------------
For this limit we assume that the final electron has an energy $E_3$ much smaller than that of the initial lepton $E_1$, but the lepton remains ultrarelativistic. This leads to the condition $x_3 \ll 1$ whereas the value of the 2–vector ${\mathbf{p}}_3$ is comparable to those of the other 2–vectors. In that limit the helicity amplitudes vanish with $\sqrt{x_3}$ due to the factor $S_{\lambda_{e3}}$ \[see Eq. (\[S\])\] and the cross section tends to zero with $x_3$. Besides of this obvious factor the remaining dominant part of the amplitude (staying finite in that limit) can be found.
The kinematic invariants $a$, $b$ and $b_i$ are simplified to $$\begin{aligned}
&& a = m^2 +({\mathbf{k}}_1+{\mathbf{k}}_2)^2 \,, \quad
x_3 b = m^2 + ({\mathbf{k}}+{\mathbf{k}}_1+ {\mathbf{k}}_2)^2
= m^2 + {\mathbf{p}}_3^2 \,,
\nonumber
\\
&&b_i = x_i b \,.\end{aligned}$$ Additionally we have $$\begin{aligned}
A \to \frac{x_3}{a_1} \left( \frac{1}{a}-\frac{1}{x_3 b}\right)
\,, \quad
{\cal P}_{12}A \to \frac{x_3}{a_2}
\left( \frac{1}{a}-\frac{1}{x_3 b}\right) \,, \quad
\rho_i \to -x_i \pi_3 \,.\end{aligned}$$ From Eqs. (\[nonflip\]) and (\[flip\]) we derive $$\begin{aligned}
&&{\cal{M}}^{++}_{++}= 2 S_+
\left( \frac{\kappa_1^*}{x_1 a_1}+\frac{\kappa_2^*}{x_2 a_2} \right)
\left( \frac{\kappa_1^*+\kappa_2^*}{a}+\frac{\pi_3^*}{x_3 b} \right) \,,
\nonumber
\\
&&{\cal{M}}^{--}_{++}=0 \,,
\nonumber
\\
&&{\cal{M}}^{+-}_{++}= -2 x_1 S_+
\left\{ \frac{\kappa_2}{x_2 a_2}
\left( \frac{\kappa_1^*+\kappa_2^*}{a}+ \frac{\pi_3^*}{x_3 b}\right)
- m^2 \frac{1}{a_1} \left(\frac{1}{a}-\frac{1}{x_3 b}\right)
\right\} \,,
\nonumber
\\
&&{\cal{M}}^{-+}_{++}={\cal{P}}_{12} {\cal{M}}^{+-}_{++} \,,
\\
&&{\cal{M}}^{++}_{+-}=
- 2 m S_- \left(\frac{1}{a}-\frac{1}{x_3 b} \right)
\left(\frac{\kappa_1^*}{x_1 a_1}+\frac{\kappa_2^*}{x_2 a_2}
\right)\,,
\nonumber
\\
&&{\cal{M}}^{+-}_{+-}= 2 m x_1 S_-
\left\{ \frac{1}{a_1}
\left( \frac{\kappa_1+\kappa_2}{a}+\frac{\pi_3}{x_3 b} \right)
+ \left(\frac{1}{a}-\frac{1}{x_3b} \right)
\frac{\kappa_2}{x_2 a_2}
\right\} \,,
\nonumber
\\
&&{\cal{M}}^{-+}_{+-}={\cal{P}}_{12} {\cal{M}}^{+-}_{+-} \,,
\nonumber
\\
&&{\cal{M}}^{--}_{+-}=0\,.
\nonumber\end{aligned}$$
Introducing the notations $$\begin{aligned}
{\mathbf{Q}}_e =
\frac{{\mathbf{k}}_1+{\mathbf{k}}_2}{a}+
\frac{{\mathbf{p}}_3}{x_3 b}
\,, \quad
R_e =\frac{1}{a}-\frac{1}{x_3 b}\,,\end{aligned}$$ the matrix elements in the soft electron limit can be written as $$\begin{aligned}
\label{softelectron}
&&{\cal{M}}_{\lambda_{e1}\lambda_{e3}}^{\lambda_1\;\, \lambda_2}=
\sqrt{x_3} s \,{\mathrm{e}}^{{\mathrm{i}} \,
\varphi_3 \lambda_{e3} }
\left( 1 + {\cal{P}}_{12} \right)
\Biggl(
2 {\mathbf{Q}}_e{\mathbf{e}}^{(\lambda_2)*} \,
\delta_{\lambda_2,2\lambda_{e3}}
+ \sqrt{2} m R_e \delta_{\lambda_2,-2\lambda_{e3}}
\Biggr)
\nonumber
\\
&&\times \frac{1}{a_1}
\Biggl(
\frac{1-x_1\delta_{\lambda_1,-\lambda_2} }{x_1}
2 {\mathbf{k}}_1{\mathbf{e}}^{(\lambda_1)*} \,
\delta_{\lambda_2,2\lambda_{e1}}
+ \sqrt{2}m x_1 \delta_{\lambda_1,-\lambda_2}
\delta_{\lambda_2,-2\lambda_{e1}}
\Biggr) \,.\end{aligned}$$
Considering additionally a soft (second) photon we arrive at the case (\[hardphoton\]) discussed in Section \[hard\].
The limits of a hard electron or a hard photon {#hard}
----------------------------------------------
In the limit of a hard electron $x_3\to 1$ both photons become soft. Using Eqs. (\[softphoton\]) for that case, only the non–flip amplitudes remain: $$\begin{aligned}
{\cal{M}}_{\lambda_{e1}\lambda_{e3}}^{\lambda_1 \; \lambda_2}
&=& s \, A_1 A_2 \,
{\mathrm{e}}^{{\mathrm{i}} \, \varphi_3 \lambda_{e3}}
\delta_{\lambda_{e1},\lambda_{e3}} \,,
\\
A_i = \frac{2}{x_i}
\left( \frac{{\mathbf {k}}_i}{a_i}-\frac{{\mathbf {r}}_i}{b_i}
\right)
{\mathbf {e}}^{(\lambda_i)*}
&=&
2 \left(\frac{{\mathbf{k}}_i {\mathbf {e}}^{(\lambda_i)*}}
{x_i^2 m^2+{\mathbf{k}}_i^2}
-\frac{ ({\mathbf{k}}_i +x_i
{\mathbf{k}}){\mathbf {e}}^{(\lambda_i)*}}
{x_i^2m^2+({\mathbf{k}}_i + x_i {\mathbf{k}})^2}\right) \,.
\nonumber\end{aligned}$$
Now we consider the case when the first photon is hard and the soft electron remains ultrarelativistic: $x_1\to 1$ leading to $x_{2,3}\ll 1$, $|{\mathbf{r}}_2| /b_2 \ll |{\mathbf{k}}_2|/a_2$ and $ A_2 \to 2{\mathbf{k}}_{2}{\mathbf{e}}^{(\lambda_2)*}/(x_2 a_2)$. The amplitudes can be derived from Eqs. (\[softphoton\]) where ${\mathbf{Q}}_1$ and $R_1$ from Eqs. (\[QR\]) have to be taken at $x_1\to 1$: $$\begin{aligned}
\label{hardphoton}
{\cal{M}}_{\lambda_{e1} \lambda_{e3}}^{\lambda_1 \; \lambda_2}&=&
\sqrt{x_3} s \, {\mathrm{e}}^{ {\mathrm{i}} \, \varphi_3
\lambda_{e3}}
\left( 2 {\mathbf{Q}}_1
{\mathbf{e}}^{(\lambda_1)*}
\delta_{\lambda_{e3},\lambda_{e1}}
+\sqrt{2} m \, R_1 \delta_{-\lambda_{e3},\lambda_{e1}}\right)
\\
&&\times
\delta_{\lambda_1,2\lambda_{e1}}\,
\frac{2}{x_2 a_2} {\mathbf{k}}_2 {\mathbf{e}}^{(\lambda_2)*}
\,.
\nonumber\end{aligned}$$ From that expression one can see that only those amplitudes remain in which the initial electron transfers its helicity to the hard photon: $\lambda_1= 2 \lambda_{e1}$, and they vanish as $\sqrt{x_3}$.
The cross channel $\gamma e^\pm \to \gamma e^+e^- + e^\pm$ {#crossingsection}
===========================================================
Crossing relations and helicity amplitudes
------------------------------------------
We consider the reaction $$\begin{aligned}
\label{crossreaction}
\gamma(\tilde k_1) + e^\pm(p_2) \to
\gamma(\tilde k_2) + e^+(p_+) + e^-(p_-) + e^\pm(p_4)\,,\end{aligned}$$ which is one of the cross channels to the double forward bremsstrahlung considered before. We denote in the cross channel the 4–momenta of the initial photon by $\tilde k_1$ with energy $\tilde \omega_1$ and helicity $\lambda_1$, the 4–momenta (Euclidean 2–vectors) of the final lepton pair by $p_\pm$ (${\mathbf{p}}_\pm$) and the final photon by $\tilde{k}_2$ ($\tilde{\mathbf{k}}_2$) with helicities $\lambda_\pm$ and $\lambda_2$, respectively.
The amplitude for reaction (\[crossreaction\]) can be obtained in the jet kinematics by considering the cross channel of the upper vertex only: $$\begin{aligned}
M(\gamma e^\pm \to \gamma e^+e^- + e^\pm)=
\frac{\tilde s}{k^2} J_{\mathrm{up}}(\gamma \gamma^* \to \gamma
e^+e^-) J_{\mathrm{down}}(e^\pm \gamma^* \to e^\pm)\,,\end{aligned}$$ with $$\begin{aligned}
J_{\mathrm{up}}(\gamma \gamma^* \to \gamma e^+e^-)=
J_{\mathrm{up}}^{\mathrm{cross}}(e^- \gamma^* \to e^-
\gamma \gamma) \,,\quad \tilde s =2 \tilde k_1 p_2\,.\end{aligned}$$ The following 4–vector replacements have to be performed: $$\begin{aligned}
k_1\to - \tilde k_1 \,, \quad
k_2\to \tilde k_2\,, \quad
p_1\to - p_+ \,, \quad
p_3\to p_- \,,
\label{replace}\end{aligned}$$ with the remaining 4–vectors $k$, $p_2$ and $p_4$ held intact.
Obviously, the corresponding energy fractions are $$\begin{aligned}
y_\pm=\frac{E_\pm}{\tilde \omega_1}\,, \quad
y_2=\frac{\tilde\omega_2}{\tilde\omega_1} \,, \quad
y_+ +y_-+y_2=1 \,.\end{aligned}$$ The Sudakov decomposition is easily performed (using $\tilde k_1$ and $\tilde
p'=p_2 - (m^2/\tilde s)\tilde k_1$ as light–like vectors instead of $p$ and $p'$, respectively) $$\begin{aligned}
p_\pm= \frac{ m^2 +{\mathbf{p}}_\pm^2}{\tilde s y_\pm} \, \tilde p'
+ y_\pm \tilde k_1 + p_{\pm\perp}
\,, \quad
\tilde k_2= \frac{ \tilde{\mathbf{k}}_2^2}{\tilde s y_2} \, \tilde p' +
y_2\tilde k_1 +\tilde k_{2\perp} \,.
\label{sudcross}\end{aligned}$$ Comparing the kinematic invariants $a_i$, $b_i$ with their corresponding crossed invariants and using the replacements (\[replace\]) and the Sudakov decomposition (\[sudcross\]) we find the substitution rules for the energy fractions and Euclidean 2–vectors (compare Ref. [@KLS74]) $$\begin{aligned}
\label{rep1}
&& x_1 \to \frac{1}{y_+} \,, \quad
x_2 \to -\frac{y_2}{y_+} \,, \quad
x_3 \to - \frac{y_-}{y_+} \,,
\\
&& \frac{{\mathbf{k}}_1}{x_1} \to {\mathbf{p}}_+ \,, \quad
\frac{{\mathbf{r}}_1}{x_1} \to -{\mathbf{p}}_- \,, \quad
\frac{{\mathbf{k}}_2}{x_2} \to {\mathbf{p}}_+ - \frac{y_+}{y_2}
\tilde{\mathbf{k}}_2 \,, \quad
\frac{{\mathbf{r}}_2}{x_2} \to -{\mathbf{p}}_- + \frac{y_-}{y_2}
\tilde{\mathbf{k}}_2 \,,
\nonumber\end{aligned}$$ with $$\begin{aligned}
{\mathbf{p}}_+ + {\mathbf{p}}_- + \tilde{\mathbf{k}}_2
+ {\mathbf{k}} = 0 \,.\end{aligned}$$ To take into account the substitution rules for the polarizations of the particles, we have to re-introduce the azimuthal angle of the initial electron $\varphi_1$. This changes only the phase of the factor $S_{\lambda_{e3}}$ given in Eq. (\[S\]). Therefore, in this section we use $$\begin{aligned}
\label{Scomplete}
S_{\lambda_{e1}\lambda_{e3}}(\varphi_1,\varphi_3)&=&
s \sqrt{x_3} \,{\mathrm{e}}^{{\mathrm{i}} \,
(\varphi_3 \lambda_{e3} -\varphi_1 \lambda_{e1})}
=4 E_1 E_2 \sqrt{x_3} \, {\mathrm{e}}^{{\mathrm{i}} \,
(\varphi_3 \lambda_{e3} -\varphi_1 \lambda_{e1})}
\,,
\\
S_{\lambda_{e3}} &=& S_{\lambda_{e1}\lambda_{e3}}(0,\varphi_3)
\nonumber\end{aligned}$$ instead of $S_{\lambda_{e3}}$. Consequently, the helicity amplitudes given in Eqs. (\[nonflip\]) and (\[flip\]) have to be used now with the factor $S_{\lambda_{e1}\lambda_{e3}}(\varphi_1,\varphi_3)$ and they depend on both lepton azimuthal angles explicitly.
Under crossing we have the substitution rules for the helicities and the lepton azimuthal angles $$\begin{aligned}
\label{rep3}
\lambda_1 \to - \lambda_1 \,, \quad
\lambda_{e1} \to - \lambda_+ \,, \quad
\lambda_{e3} \to \lambda_- \,, \quad
\lambda_2 \to \lambda_2 \,, \quad
\varphi_{1,3} \to \varphi_{+,-} \,.\end{aligned}$$ which leads to $$\begin{aligned}
\label{Schange}
S_{\lambda_{e1}\lambda_{e3}}(\varphi_1,\varphi_3) \to
\widetilde{S}_{\lambda_+ \lambda_-}& =&
- {\mathrm{i}} \, \tilde s \sqrt{y_+ y_-} \,
{\mathrm{e}}^{{\mathrm{i}} \, (\varphi_- \lambda_- +\varphi_+
\lambda_+)}\,.\end{aligned}$$ The kinematic invariants are transformed to $$\begin{aligned}
\label{crosskin}
\left\{ {a_1 \atop b_1} \right\}
\to a_\pm&=&
\pm 2 \tilde k_1 p_\pm=\pm \frac{1}{y_\pm}
\left( m^2 +{\mathbf{p}}_\pm^2 \right) \,,
\nonumber
\\
\left\{ {a_2 \atop b_2} \right\}
\to b_\pm&=& \mp 2
\tilde k_2 p_\pm= \mp \frac{1}{y_\pm y_2}
\left[
y_2^2m^2 +\left( y_2 {\mathbf{p}}_\pm -
y_\pm \tilde {\mathbf{k}}_2 \right)^2
\right] \,,
\\
\left\{ {a \atop b} \right\}
\to \tilde a_\pm&=&
\mp k^2 \mp 2 k p_\mp=
a_\pm + b_\pm \pm \frac{1}{y_2}\tilde{\mathbf{k}}_2^2 \,,
\nonumber
\\
A \to y_- B &=&
y_- \left\{ - \frac{1}{y_+ a_+ \ \tilde a_+}
+ \frac{1-y_+}{ y_+ a_+ \ y_- b_-} + \frac{1}{ y_- b_- \ \tilde a_-}
\right\} \,.
\nonumber\end{aligned}$$
We introduce the permutation operator ${\cal {P}}$ with the properties $$\begin{aligned}
{\cal{P}} f(p_+,\lambda_+;p_-,\lambda_-)=
f(p_-,\lambda_-;p_+,\lambda_+)\,,\end{aligned}$$ or in more detail $$\begin{aligned}
{\cal{P}} f(a_\pm,b_\pm,\tilde a_\pm,y_\pm,\pi_\pm,\lambda_\pm)=
f( -a_\mp,-b_\mp,-\tilde a_\mp,y_\mp,\pi_\mp,\lambda_\mp)\,.\end{aligned}$$ As usual, the circular components of ${\mathbf{p}}_\pm$ and $\tilde
{\mathbf{k}}_2$ are used $$\begin{aligned}
\pi_{\pm}= p_{x\pm} + {\mathrm{i}} \, p_{y\pm} \,, \quad
\tilde\kappa_{2}=\tilde k_{2x} + {\mathrm{i}} \, \tilde k _{2y} \,.\end{aligned}$$ The operator ${\cal{P}}$ is similar to ${\cal{P}}_{12}$ in the original channel, in fact it will interchange $e^+ \leftrightarrow e^-$ in the crossed amplitudes.
As a result, we obtain the amplitude for reaction (\[crossreaction\]) in the considered jet kinematics $$\begin{aligned}
M(\gamma e^\pm \to \gamma e^+e^- + e^\pm)=
\frac{32 \pi^2 \alpha^2}{k^2}
\widetilde{\cal{M}}_{\lambda_{+} \lambda_{-}}
^{\lambda_1 \; \lambda_2}
{\mathrm{e}}^{-{\mathrm{i}} \, \lambda_{e4} \, \varphi_4}
\,\delta_{ \lambda_{e2} ,\lambda_{e4}}\,,
\label{fullampcross}\end{aligned}$$ where $\lambda_{e2}$ ($\lambda_{e4}$) are the helicities of the electrons or positrons moving along the $-z$ axis before and after the collision with the initial photon. Using the function $C$ in Eq. (\[alternC\]) for crossing and the notations $$\begin{aligned}
Y_1= B \pi_+ + \frac{\kappa}{y_- b_- \, \tilde a_-} \,, \quad
Y_2= \pi_- - \frac{y_-}{y_2} \tilde \kappa_2\,,\end{aligned}$$ the helicity amplitudes in the cross channel are presented as follows: $$\begin{aligned}
\label{crosshel}
&&\widetilde{\cal{M}}_{-+}^{-+} =
2 \widetilde{S}_{-+} \, y_+ (1 - {\cal{P}})
\left\{ Y_1^*Y_2^*
-\frac{ \pi_+^* \kappa^*}{y_+ a_+ \, \tilde a_+} \right\} \,,
\nonumber
\\
&&\widetilde{\cal{M}}_{-+}^{+-} =
\widetilde S_{-+}{\cal{P}}\left(\widetilde{\cal{M}}_{-+}^{-+}\over
\widetilde S_{-+}\right)^*\,,
\nonumber
\\
&&\widetilde{\cal{M}}_{-+}^{--} =
-2 \widetilde{S}_{-+}
\left( \widetilde{C} +
{\cal{P}} \widetilde{D}^* \right) \,,
\nonumber
\\
&&\widetilde {\cal{M}}_{-+}^{++} =
\widetilde S_{-+}{\cal{P}}\left(\widetilde{\cal{M}}_{-+}^{--}
\over\widetilde S_{-+}\right)^*\,,
\\
&&\widetilde{D}=
- (1-y_+) \left\{
Y_1 Y_2^* -\frac{ \pi_+ \kappa^*}{y_+ a_+ \, \tilde a_+}
\right\} \,,
\nonumber
\\
&&\widetilde{C}=
-\frac{1-y_-}{1-y_+} \widetilde{D}^*
-\frac{1}{\tilde a_+} + \frac{1}{\tilde a_-}
+\frac{y_2 \left( m^2 - Y_2 \pi_+^* \right)}{y_+ a_+ \, y_- b_-}
-\frac{\pi_+^* \tilde \kappa_2}{y_+ a_+ \, \tilde a_+}
+\frac{\tilde \kappa_2^*Y_2}{y_- b_- \, \tilde a_-} \,,
\nonumber
\\
&&\widetilde{\cal{M}}_{--}^{--} =
2 m \widetilde{S}_{--} \, (1+{\cal{P}})
\left\{ B Y_2
- \frac{ \kappa }{y_+ a_+ \, \tilde a_+} \right\} \,,
\nonumber
\\
&&\widetilde{\cal{M}}_{--}^{++} =
2 m \widetilde{S}_{--} \, y_2 \, (1+{\cal{P}}) Y_1 \,,
\nonumber
\\
&&\widetilde{\cal{M}}_{--}^{-+} =
2 m \widetilde{S}_{--} \, (1+{\cal{P}})
\left\{
(1-y_2) Y_1^* + B \frac{y_-}{y_2} \tilde \kappa_2^*
+ \frac{\kappa^* }{y_+ a_+ \, b_-}
\right\} \,,
\nonumber
\\
&&\widetilde{\cal{M}}_{--}^{+-} = 0 \,.
\nonumber\end{aligned}$$ The remaining amplitudes are derived from a relation equivalent to Eq. (\[parity\]) $$\begin{aligned}
\widetilde{\cal{M}}_{-\lambda_+,-\lambda_-}^
{-\lambda_1,\: -\lambda_2} =
- (-1)^{\lambda_+ + \lambda_-} \left(
\widetilde{\cal{M}}_{\lambda_+,\lambda_-}^
{\lambda_1, \: \, \lambda_2}\right)^*
\,.
\label{crossparity}\end{aligned}$$ The helicity amplitudes (\[crosshel\]), (\[crossparity\]) are $C$–odd. Therefore, they have to be symmetric under lepton–antilepton exchange $e^- \leftrightarrow e^+$. This symmetry, expressed by the relation $$\begin{aligned}
\widetilde{\cal{M}}_{\lambda_+ \lambda_-}^
{\lambda_1 \: \lambda_2}
\left(a_\pm,b_\pm,\tilde a_\pm,y_\pm,\pi_\pm,\varphi_\pm\right)&=& \\ \nonumber
&&\widetilde{\cal{M}}_{\lambda_- \lambda_+}^
{\lambda_1 \: \lambda_2}
\left(-a_\mp,-b_\mp,-\tilde a_\mp,y_\mp,\pi_\mp,\varphi_\mp\right)\,,\end{aligned}$$ can be easily seen for all amplitudes taking into account Eq. (\[crossparity\]) and ${\cal{P}}^2= 1$.
Note an alternative form for $\widetilde{C}$ useful for deriving the hard photon limit discussed below in Section \[crosslimits\]: $$\begin{aligned}
\widetilde{C}=
\frac{y_-}{1-y_+} \widetilde{D}^*
- B \left( \frac{y_- \tilde \kappa_2 \pi_+^*}{ y_2} - m^2 \right)
- \frac{\tilde \kappa_2 \kappa^*} {y_2 b_- \tilde a_-}
- \frac{\kappa \pi_+^*}{y_+ a_+ b_-} \,.
\label{Ccrossalt}\end{aligned}$$
Since the function $B$ analogous to $A$ tends to zero for $|{\mathbf{k}}|\to 0$, the helicity amplitudes (\[crosshel\]) are again proportional to $|{\mathbf{k}}|$ in that limit as expected from crossing. For $\widetilde{C}$ this can be seen using its form (\[Ccrossalt\]).
Soft and hard particle limits {#crosslimits}
-----------------------------
In the soft photon limit ($\tilde k_2\to 0$) we use $$\begin{aligned}
{\mathbf {Q}}=
\frac{{\mathbf{p}}_+}{m^2+{\mathbf{p}}_+^2}+
\frac{{\mathbf{p}}_-}{m^2+{\mathbf{p}}_-^2}
\,, \quad
R= \frac{1}{m^2+{\mathbf{p}}_+^2}-
\frac{1}{m^2+{\mathbf{p}}_-^2}
\label{Q+-R+-}\end{aligned}$$ and have to consider (at $y_2\to 0$) $$\begin{aligned}
\tilde a_\pm \to a_\pm \,, \quad
B \to \frac{R}{b_-} \,, \quad
{\cal{P}} B
\to \frac{R}{b_+} \,.\end{aligned}$$ This limit again leads to a factorized form of the upper vertex: $$\begin{aligned}
J_{\mathrm{up}}(\gamma_{\lambda_1} \gamma^*
\to e^+_{\lambda_{+}} e^-_{\lambda_{-}}\gamma_{\lambda_2})=
\sqrt{4 \pi \alpha} \,
J_{\mathrm{up}}(\gamma_{\lambda_1} \gamma^*
\to e^+_{\lambda_{+}} e^-_{\lambda_{-}} ) \, \widetilde{A}_2\,,\end{aligned}$$ where $J_{\mathrm{up}}(\gamma_{\lambda_1} \gamma^*
\to e^+_{\lambda_{+}} e^-_{\lambda_{-}})$ can be found in Ref. [@KSSS] and $$\begin{aligned}
\label{A2cross}
\widetilde{A}_2=
\left(\frac{p_+}{p_+ \tilde k_2}-\frac{p_-}{p_- \tilde k_2} \right)
\!
e_2^{*}
= \frac{2}{y_2}
\left( \frac{y_2 {\mathbf {p}}_+ - y_+
\tilde {\mathbf {k}}_2 }{b_+}
+\frac{y_2 {\mathbf {p}}_- - y_- \tilde {\mathbf {k}}_2 }{b_-}
\right) \! {\mathbf{e}}^{(\lambda_2)*}\end{aligned}$$ is the classical emission factor of the final photon. With notations (\[Q+-R+-\]) and (\[A2cross\]) we find the non-vanishing helicity amplitudes in that limit $$\begin{aligned}
\label{softphotoncross}
\widetilde{\cal{M}}_{\lambda_+ \lambda_-}^{\lambda_1 \: \lambda_2}
&=& {\mathrm{i}} \, \tilde s \, \sqrt{y_+ y_-}
{\mathrm{e}}^
{{\mathrm{i}}\,(\varphi_+\lambda_+ +\varphi_-\lambda_-)}
\widetilde{A}_2
\\
&\times &
\Bigg\{
2 {\mathbf{Q}} {\mathbf{e}}^{(\lambda_1)} \,
\left( y_+ - \delta_{\lambda_1, -2\lambda_+} \right)
\delta_{\lambda_+,-\lambda_-}
- \sqrt{2} m R
\delta_{\lambda_1, 2\lambda_+} \delta_{\lambda_+,\lambda_-}
\Bigg\} \,.
\nonumber\end{aligned}$$
From Eq. (\[softphotoncross\]) we derive the amplitudes in the limit of a hard electron ($y_-\to 1$, $y_+,y_2 \ll 1$) $$\begin{aligned}
\widetilde{\cal{M}}_{\lambda_+ \lambda_-}^{\lambda_1 \: \lambda_2}
&=& -{\mathrm{i}} \,
\tilde s \, \sqrt{y_+} {\mathrm{e}}^{ {\mathrm{i}} \, (\varphi_+
\lambda_+ +\varphi_- \lambda_-)} \frac{2}{b_-} {\mathbf{p}}_-
{\mathbf{e}}^{(\lambda_2)*}
\\
&\times & \Bigg\{ 2
{\mathbf{Q}} {\mathbf{e}}^{(\lambda_1)} \,
\delta_{\lambda_+,-\lambda_-}
+ \sqrt{2} m R \delta_{\lambda_+,\lambda_-}
\Bigg\} \delta_{\lambda_1, 2 \lambda_-} \,.
\nonumber\end{aligned}$$ In this case the helicity of the initial photon is transferred to that of the hard electron only, i.e., $\lambda_1=2 \lambda_-$.
The soft electron limit corresponds to the condition $y_-\ll 1$ and finite transverse momentum ${\mathbf{p}}_-$. It can be obtained as cross channel from Eq. (\[softelectron\]) performing the replacements (\[rep3\]) and (\[Schange\]) for $S_{\lambda_{e1} \lambda_{e3}}$ as well as $$\begin{aligned}
&&x_1 \to \frac{1}{y_+} \,, \quad
x_2 \to - \frac{y_2}{y_+}=-\frac{1-y_+}{y_+} \,,
\nonumber
\\
&&{\mathbf{k}}_1 \to\frac{1}{y_+} {\mathbf{p}}_+ \,, \quad
{\mathbf{k}}_2 \to - \frac{1-y_+}{y_+} {\mathbf{p}}_+ + \tilde
{\mathbf{k}}_2 \,, \quad
{\mathbf{p}}_3 \to {\mathbf{p}}_- \, ,
\\
&& a_1 \to a_+\,, \quad a_2 \to b_+\,, \quad
a \to m^2 +( {\mathbf{p}}_+ + \tilde {\mathbf{k}}_2 )^2 \,, \quad
x_3 b \to m^2 + {\mathbf{p}}_-^2 \,.
\nonumber\end{aligned}$$ In that limit the helicity amplitudes vanish as $\sqrt{y_-}$.
Finally we consider the hard final photon limit $y_2 \to 1$ and $y_\pm \ll 1$. From the general helicity amplitudes (\[crosshel\]) and using the form (\[Ccrossalt\]) of $\widetilde{C}$ we find in leading order in $y_+$ and $y_-$ $$\begin{aligned}
\widetilde{\cal{M}}_{-+}^{--}
&=&
\widetilde{\cal{M}}_{-+}^{++}
=- 2 \widetilde{S}_{-+}
\left\{ B \left( m^2 - \pi_-^* \pi_+\right)
+\frac{\kappa^* \pi_+}{y_+ a_+ \, \tilde a_+}
+\frac{\pi_-^* \kappa}{y_- a_- \, \tilde a_-} \right\}\,,
\nonumber
\\
\widetilde{\cal{M}}_{--}^{++}
&=&
\widetilde{\cal{M}}_{--}^{--}
=
2 m \widetilde{S}_{--}
(1+{\cal{P}}) \left\{ B \pi_- - \frac{\kappa}{y_+ a_+ \, \tilde a_+}
\right\} \,,
\\
\widetilde{\cal{M}}_{\lambda_+ \lambda_-}^{\,+\; -}
&=&
\widetilde{\cal{M}}_{\lambda_+ \lambda_-}^{\,-\; +}
=0\,,
\nonumber\end{aligned}$$ where in this limit $b_\pm \to - a_\pm$ and $$\begin{aligned}
B&=& {\cal{P}}B=- \frac{1}{y_+ a_+ \, \tilde a_+} -
\frac{1}{y_+ a_+ \, y_- a_-} -
\frac{1}{y_- a_- \, \tilde a_-}\,,
\\
\tilde a_+ &=& - {\cal{P}} \tilde a_-= y_- a_+ + m^2 +
( {\mathbf{p}}_+ + \tilde {\mathbf{k}}_2)^2 \,,
\nonumber\end{aligned}$$ the $a_\pm$ are defined in Eqs. (\[crosskin\]). Note that the helicity of the initial photon is transferred to that of the hard final photon, the amplitudes vanish as $\sqrt{y_+ y_-}$.
Conclusions
===========
The present work completes a series of publications [@KSS85; @KSS86; @KSSS] where the helicity analysis of Born processes with non-decreasing cross sections up to fourth order in $\alpha$ has been performed in a jet kinematics. These specific kinematic conditions provide the main contribution to the total cross section at high energies. On the other hand, under these kinematic conditions Monte Carlo simulations as well as analytical calculations by known methods (compare, for example, Ref. [@Behr]) are difficult to perform. Therefore, the results obtained can be useful for monitoring and calibrating of polarized beams at high energy colliders. They might also be used for almost exact estimates of the essential background in a number of reactions with polarized particles.
The main results of our paper are summarized in Eqs. (\[fullamp\]), (\[nonflip\]), (\[flip\]) and (\[fullampcross\]), (\[crosshel\]) which give the analytical expressions for all 64 helicity amplitudes to the high accuracy (\[accuracy\]). Let us summarize some specific features of those helicity amplitudes using as example the double bremsstrahlung process:
- The obtained formulae for the helicity amplitudes are compact and convenient for numeric calculations since in their form large compensating terms are already cancelled. Indeed, in Eqs. (\[nonflip\]) and (\[flip\]) all items are either proportional to $\kappa = k_x +{\mathrm{i}} \, k_y$ or to the function A (\[eq:16\]) which vanishes at small transverse momentum of the $t$–channel exchange photon $|{\mathbf {k}}|$. Therefore, the amplitude $$\begin{aligned}
\vert M(e^-e^+\to e^-\gamma\gamma + e^+) \vert
\propto \vert {\mathbf{k}} \vert \ \
{\mathrm{ at}} \ \ \vert {\mathbf {k}} \vert \to 0 \,.\end{aligned}$$
- The helicity amplitudes ${\cal M}_{+\;-}^{\lambda_1 \lambda_2}$ and ${\cal M}_{-\;+}^{\lambda_1 \lambda_2}$, in which the electron changes its helicity, are proportional to the electron mass $m$ (see Eq. (\[flip\])). They become small compared with the non-spin flip amplitudes ${\cal M}_{+\;+}^{\lambda_1 \lambda_2}$ and ${\cal M}_{-\;-}^{\lambda_1\lambda_2}$ for large scattering angles (\[langle\]) or small energies of both photons $\omega_{1,2}\ll E_1$.
- The amplitudes ${\cal M}_{+-}^{--}$ and ${\cal M}_{-+}^{++}$ with the maximal change of helicities $|\lambda_{e1}-\lambda_{e3}-\lambda_{1}-\lambda_{2}|=3$ are equal to zero under our kinematic conditions.
- If the energy of any final particle tends to that of the initial electron, the incoming electron transfers its helicity to that particle: $\lambda_i = 2 \lambda_{e1}$ at $\omega_i
\to E_1$ or $\lambda_{e3} = \lambda_{e1}$ at $E_3 \to E_1$.
Acknowledgements {#acknowledgements .unnumbered}
=================
This work is supported in part by Volkswagen Stiftung (Az. No. I/72 302) and by Russian Foundation for Basic Research (codes 96-15-96030, 99-02-17211 and 99-02-17730).
Calculation of necessary bilinear spinor structures {#appa}
===================================================
We have to calculate all bilinear spinor structures appearing in Eq. (\[eq:18\]) in order to find the helicity amplitudes in the $s\to \infty$ limit. For example, let us consider the numerator of the second term of Eq. (\[eq:18\]). $$\begin{aligned}
N=\bar u_3 \hat p' \hat k_{\perp} \hat e_2^*
(\hat p_1-\hat k_1+m) \hat e_1^* u_1 \,.\end{aligned}$$ We transpose $\hat e_1^*$ to the left and obtain $$\begin{aligned}
N=\bar u_3 \hat p' \hat k_{\perp} \hat e_2^*
\left[ \hat e_1^* (-\hat p_1+\hat k_1 + m) + 2 p_1 e_1^* \right]
u_1 \,.\end{aligned}$$ Using the Sudakov decomposition of the 4–vectors $e_i^*$ and $k_1$ \[see Eqs. (\[sudakov\]), (\[polvector\]), (\[eq:10\])\] and taking into account $\hat p' \hat p'=0$, $ \hat p u_1 \approx \hat p_1 u_1 = m u_1$ we find $$\begin{aligned}
N=\bar u_3 \hat p' \hat k_{\perp} \hat e_{2\perp}^*
\left[ \hat e_{1\perp}^* \hat k_{1\perp}
+ m x_1 \hat e_{1\perp}^* +
\frac{2}{x_1} {\mathbf{k}}_1 {\mathbf{e}}^{(\lambda_1)*}
\right] u_1 \,.\end{aligned}$$ Similar transformations can be performed for the other terms of Eq. (\[eq:18\]).
As a result we conclude that it is sufficient to calculate the following combinations $\bar u_3 \hat p' \hat a_\perp \hat b_\perp \dots \hat c_\perp u_1$ and $\bar u_3 \hat a_\perp \hat b_\perp \dots \hat c_\perp u_1$. Here the bispinors $u$ are defined as follows (for an initial electron with helicity +1/2) $$\begin{aligned}
u_1=\left( \begin{array}{c}
\sqrt{E_1 +m }\, w_1\\
\sqrt{E_1 -m }\, w_1
\end{array} \right)
\,, \ \
u_3=\left( \begin{array}{c}
\sqrt{E_3 +m }\, w_3\\
2 \lambda_{e3}\sqrt{E_3 -m } \,w_3
\end{array} \right)\,,\end{aligned}$$ with $$\begin{aligned}
w_1=\left(\begin{array}{c}1\\
0 \end{array}\right)
\,, \ \
w_3^{(+1/2)}=\left(\begin{array}{c}{\mathrm{e}}^{-{\mathrm{i}} \,
\varphi_3/2}
\\
\frac{\theta_3}{2}\,
{\mathrm{e}}^{{\mathrm{i}}\varphi_3/2}\end{array}\right)
\,, \ \
w_3^{(-1/2)}=\left(\begin{array}{c}-\frac{\theta_3}{2}\,
{\mathrm{e}}^{-{\mathrm{i}}\varphi_3/2}\\
{\mathrm{e}}^{{\mathrm{i}}\varphi_3/2}\end{array}\right)
\,,
\nonumber
\\\end{aligned}$$ and $$\begin{aligned}
\hat p'=E_2
\left(\begin{array}{cc}1&\sigma_z\\-\sigma_z&-1\end{array}\right)
\,, \ \
\hat a_{\perp}= - a_x \gamma_x - a_y \gamma_y \,.\end{aligned}$$ The $\sigma_i$ and $\gamma_i$ are the Pauli and Dirac matrices in the standard representation. Introducing the circular components of the Euclidean 2–vector $\mathbf{a}$ $$\begin{aligned}
a_\pm= a_x \pm {\mathrm{i}} \, a_y\,, \qquad a_\mp^*=a_\pm\,,\end{aligned}$$ we obtain $$\begin{aligned}
\hat a_\perp= - \frac{1}{2} (a_+ \gamma_- + a_- \gamma_+) \,, \quad
\gamma_\pm=\left( \begin{array}{cc}
0 &\sigma_\pm\\
-\sigma_\pm & 0
\end{array} \right) \,.\end{aligned}$$
Since $\sigma_+ w_1=0$ and $\sigma_- w_1= { 0 \choose 2}$, we have $ \gamma_+ u_1= 0$ and $\gamma_+ \gamma_- u_1= -4 u_1$. From this it follows that $$\begin{aligned}
&&\hat a_\perp u_1= - \frac{1}{2} a_+ \gamma_- u_1\,, \quad
\hat a_\perp \hat b_\perp u_1= - a_- b_+ u_1\,,
\\
&&\hat a_\perp \hat b_\perp \hat c_\perp u_1= \frac{1}{2}
a_+ b_- c_+ \gamma_- u_1 \,, \ \ \
\hat a_\perp \hat b_\perp \hat c_\perp \hat d_\perp u_1=
a_- b_+ c_- d_+ u_1 \,.
\nonumber\end{aligned}$$ Introducing the shorthand notations $$\begin{aligned}
\delta_{\lambda_{e3},\pm 1/2}=\delta_{\pm} \,,\quad
\delta_{\lambda_i,\pm 1}=\delta_{i\pm}\end{aligned}$$ and using the quantity $S_{\lambda_{e3}}$ defined in Eq. (\[S\]) we arrive at the following four basic combinations $$\begin{aligned}
\bar u_3 \hat p' u_1 &=& S_+ \delta_+ \,, \quad
\bar u_3 \hat p' \hat a_\perp u_1= - a_+ S_- \delta_- \,, \quad
\nonumber
\\
s \,\bar u_3 u_1 &=&
\frac{1}{x_3} \left[
- \pi_3 S_- \delta_- + (1+x_3) m S_+ \delta_+ \right]
\,,
\\
s \, \bar u_3 \hat a_\perp u_1 &=&
- \frac{a_+}{x_3} \left[
\pi_3^* S_+ \delta_+ + (1-x_3) m S_- \delta_- \right]\,,
\nonumber\end{aligned}$$ where $$\begin{aligned}
\pi_3=p_{3x} +{\mathrm{i}} \, p_{3y}
=-\kappa -\kappa_1 - \kappa_2 \,.\end{aligned}$$ Choosing $a_\perp= e_\perp^{(\lambda) *}$ we find $a_\pm= \mp \sqrt{2} \delta_{\lambda,\pm 1}$.
Then the relevant bilinear combinations \[using the Sudakov decomposition of all 4–vectors of Eq. (\[eq:18\])\] are found to be $$\begin{aligned}
\bar u_3\hat p'u_1&=& S_+ \delta_+ \,,
\nonumber
\\
\bar u_3 \hat p'\hat e_i^* u_1&=& \sqrt{2}\delta_{i+}S_-\delta_- \,,
\nonumber
\\
\bar u_3 \hat p' \hat e_2^* \hat e_1^* u_1&=&
2 \delta_{2-}\delta_{1+} S_+\delta_+ \,,
\nonumber
\\
\bar u_3 \hat p' \hat e_{2\perp}^* \hat k_{i\perp} u_1&=&
- \sqrt{2} \kappa_i \delta_{2-} S_+\delta_+ \,,
\nonumber
\\
\bar u_3 \hat p' \hat k_{\perp} \hat e_{i\perp}^* u_1&=&
\sqrt{2} \kappa^* \delta_{i+} S_+\delta_+ \,,
\nonumber
\\
\bar u_3 \hat p' \hat k_{\perp} \hat e_{2\perp}^*
\hat e_{1\perp}^* u_1&=&
-2 \kappa \delta_{2-} \delta_{1+} S_-\delta_- \,,
\\
\bar u_3 \hat p' \hat e_{2\perp}^* \hat e_{1\perp}^*
\hat k_{i\perp} u_1&=&
-2 \kappa_i \delta_{2+} \delta_{1-} S_-\delta_- \,,
\nonumber
\\
\bar u_3 \hat p' \hat e_{2\perp}^* \hat k_{\perp}
\hat e_{1\perp}^* u_1&=&
2 \kappa^* \delta_{2+} \delta_{1+} S_-\delta_- \,,
\nonumber
\\
\bar u_3 \hat p' \hat e_{2\perp}^* \hat e_{1\perp}^*
\hat k_{\perp} u_1&=&
-2 \kappa \delta_{2+} \delta_{1-} S_-\delta_- \,,
\nonumber
\\
\bar u_3\hat p'\hat e_{2\perp}^* \hat k_{\perp}
\hat e_{1\perp}^* \hat k_{i\perp} u_1&=&
2 \kappa \kappa_i \delta_{2-} \delta_{1-} S_+\delta_+ \,,
\nonumber
\\
\bar u_3 \hat p' \hat k_{i\perp}\hat e_{2\perp}^*
\hat e_{1\perp}^* \hat k_{\perp} u_1&=&
-2 \kappa_i^* \kappa \delta_{2+} \delta_{1-} S_+\delta_+ \,,
\nonumber
\\
s \,\bar u_3 \hat e_{i\perp}^* u_1&=&
\sqrt{2} \delta_{i+} \frac{1}{x_3}
\left[ \pi_3^* S_+ \delta_+ + (1-x_3) m S_- \delta_-\right] \,,
\nonumber
\\
s \,\bar u_3 \hat e_{2\perp}^* \hat e_{1\perp}^* u_1&=&
2 \delta_{2-} \delta_{1+} \frac{1}{x_3}
\left[ - \pi_3 S_- \delta_- + (1+x_3) m S_+ \delta_+\right] \,,
\nonumber
\\
s \, \bar u_3 \hat e_{2\perp}^* \hat e_{1\perp}^*
\hat k_{i\perp} u_1&=&
-2 \delta_{2+} \delta_{1-} \frac{\kappa_i}{x_3}
\left[ \pi_3^* S_+ \delta_+ + (1-x_3) m S_- \delta_-\right] \,,
\nonumber
\\
s \,\bar u_3 \hat e_{2\perp}^* \hat k_{\perp}
\hat e_{1\perp}^* u_1&=&
2 \delta_{2+} \delta_{1+} \frac{\kappa^*}{x_3}
\left[ \pi_3^* S_+ \delta_+ +(1-x_3) m S_- \delta_- \right] \,.
\nonumber\end{aligned}$$
[99]{}
E.A. Kuraev, A. Schiller, V.G. Serbo, D.V. Serebryakova, Eur. Phys. J. C 4 (1998) 631.
E.A. Kuraev, A. Schiller, V.G. Serbo, Nucl. Phys. B 256 (1985) 189; Phys. Lett. B 134 (1984) 455.
E.A. Kuraev, A. Schiller, V.G. Serbo, Z. Phys. C 30 (1986) 237.
A.I. Akhiezer, V.B. Berestetskii, Quantum electrodynamics, Moscow: Nauka 1981.
V.N. Baier, V.M. Galitsky, Phys. Lett. 13 (1964) 355.
E.A. Kuraev et al., Yad. Fiz. 19 (1974) 331.
E.A. Kuraev, V.S. Fadin, Novosibirsk preprint INP 78-56 (1978);\
V.N. Baier et al., Phys. Rep. 78 (1981) 293.
E.A. Kuraev, A.N. Peryshkin, Yad. Fiz. 42 (1985) 1195.
F.A. Behrends et al., Nucl. Phys. B 264 (1986) 265.
S. Jadach et al., CERN Yellow Report CERN 95-03 (1995) 343; A.B. Arbuzov et al., ibid. 369; M. Cacciari et al., ibid. 389.
V.B. Berestetskii, E.M. Lifshitz, L.P. Pitaevskii, Quantum electrodynamics, Moscow: Nauka 1989.
E.A. Kuraev, L.N. Lipatov, M.I. Strikman, Zh. Eksp. Teor. Fiz. 66 (1974) 838.
[^1]: The opening angle of the cone is characterized by the maximal polar angles of the produced particles.
[^2]: As usual, $e_i$ ($e_i^*$) are used for incoming (outgoing) photons.
[^3]: To clarify the notation we stress that in Eqs. (\[nonflip\]) and (\[flip\]) the operator ${\cal{P}}_{12}$ simply changes the indices $1\leftrightarrow 2$, for instance $$\begin{aligned}
{\cal{P}}_{12}{\cal{M}}^{+-}_{+-}=2mS_-({\cal{P}}_{12}F + G),\quad
{\cal{P}}_{12}F = x_2\left[({\cal{P}}_{12}A)\frac{\rho_1}{x_1x_3}
- \frac{\kappa}{a_2a}\right],\end{aligned}$$ with $$\begin{aligned}
{\cal{P}}_{12}A = \frac{x_3}{a_2a} - \frac{1-x_2}{a_2b_1} + \frac{1}{b_1b}.\end{aligned}$$
|
---
abstract: 'Continuous, battery-free operation of sensor nodes requires ultra-low-power sensing and data-logging techniques. In this report we show that by directly coupling a sensor/transducer signal into a Fowler-Nordheim (FN) quantum tunneling based synchronized dynamical systems, one can achieve self-powered sensing at an energy budget that is currently unachievable using conventional energy harvesting methods. The proposed device uses a differential architecture to compensate for environmental variations and the device can retain sensed information for durations ranging from hours to days. With an operating energy budget less than an attojoule, we demonstrate that when integrated with a miniature piezoelectric transducer the proposed sensor-data-logger can measure cumulative “action” due to ambient mechanical acceleration without any additional external power.'
author:
- 'Darshit Mehta, Kenji Aono and Shantanu Chakrabartty[^1][^2]'
bibliography:
- 'spsBib.bib'
title: 'A Self-powered Analog Sensor-data-logging Device based on Fowler-Nordheim Dynamical Systems'
---
Introduction
============
For sensing systems that operate in resource-constrained settings, like IoT devices or biomedical implants, utilizing a battery may be impractical due to biocompatibility, size constraints or due to technical challenges involved in replacing the battery. Self-powered sensors (SPS) can obviate the need for batteries by harvesting their operational energy directly from ambient sources, such as light [@indoorLight_mathews2014gaas] or mechanical vibrations [@wang2006piezoelectric]. SPS achieve this by first buffering ambient energy using standard power-conditioning techniques before activating the basic computation/sensing and sometimes telemetry functions [@autonomous_torah2008self; @materials_mcevoy2015]. However, when the objective is to sense and compute a simple function, like the total signal energy or a cumulative “action”, an application specific but ultra-energy-efficient variant of SPS could be designed by combining the operational physics of signal transduction, rectification and non-volatile data storage. One such SPS was reported in [@huang2011asynchronous; @chakrabartty2010self] where a cumulative measure of mechanical activity was sensed, computed and directly stored on floating-gate memories [@liangBioCas]. Similar techniques could be applied to other non-volatile technologies for sensing the event of interest as an equivalent change in magnetoresistance in MRAM [@mram_aakerman2005toward], change polarization in FeRAM [@fram1998physics], or change in electrical conductance in memristor-type systems [@memristor_chua1971]. However, all of these approaches require power conditioning such as rectification or voltage-boosting to meet the activation thresholds and to initiate the non-volatile state-change. Operational limits arise due to rectification efficiency, and due to material properties that influence diode thresholds or leakage currents. Note that some energy harvesting systems report low voltage continuous operation (i.e. $<$), however they require higher activation thresholds for initial start up conditions (e.g. $>$) [@ramadass2011battery; @mercier2012energy; @ti_bq25505].
[![a) Principle of the proposed sensor-data-logging approach where the input signal leaves its trace on a pair of synchronized dynamical system through a desynchronization process. b) Equivalent circuit model of a self-powered dynamical system where the charge on a capacitor stores the dynamical state of the system and the dynamics is governed by a leakage current $I(V_{\textrm{t}})$ and ambient stimuli $x_{\textrm{t}}$. c) Band-diagram corresponding to the tunneling junction where the electrons tunnel across the triangular energy barrier and the input signal $x_{\textrm{t}}$ modulates the barrier shape. d) Cross-section of the sensor-data-logging device showing the FN tunneling junction, the floating-gate which is coupled to a read-out transistor $P$ and a buffer $B$. e) Micrograph of the fabricated devices, with inset showing a pair of dynamical systems configured in a differential architecture.[]{data-label="scheme"}](finalFigs/fig1_new.pdf "fig:"){width="0.95\columnwidth"}]{}
We propose a self-powered system, where instead of harvesting the energy to switch between static memory states, the sensing signal is used for modulating a synchronized dynamic state. In this regard, dynamical systems, both natural and artificial, have been shown to store information in their dynamic states [@informationSingleNode_appeltant2011; @dynamicalInfo_dambre2012; @memoryTraces_ganguli2008]. In this work, we show the feasibility of this approach for self-powered sensing and data-logging, but at chip-scale. This is illustrated in Fig. \[scheme\]a which shows two synchronized dynamical systems; a sensing system and a reference system. A time-varying input signal modulates the state trajectories of the sensing dynamical system leading to its desynchronization with respect to the reference dynamical system. The relative degree of desynchronization between the two systems serves as a medium for sensing and storing the cumulative effect of the input modulation. While the principle is relatively straightforward, there exists two key challenges in implementing the proposed concept at a chip-scale. First, due to self-powering requirements, the synchronized dynamical system can only be implemented using leakage processes driven by intrinsic thermal or quantum transport of electrons. The simplest of such a system can be modeled by an equivalent circuit shown in Fig. \[scheme\]b. The capacitor $C$ in the circuit models the dynamical state (denoted by the time-varying voltage $V_{\textrm{t}}$) and the time-dependent system trajectory is determined by a leakage current $I(V_{\textrm{t}})$. The capacitor $C_{\textrm{in}}$ couples the input signal $x_{\textrm{t}}$ into the dynamical system. The challenge is that an ultra-low leakage current $I(V_{\textrm{t}})$ is required to ensure that the dynamical system is operational for the duration of sensing and data-logging. For instance, a change across a on-chip capacitor over a duration of 1 day would require a leakage current of 10 attoamperes. Even if it were possible to implement such low-leakage currents, it is difficult to ensure that the magnitude of the currents match across different devices to ensure state synchronization. The second challenge with regards to data-logging is that there exists a trade-off between the non-linearity in the dynamical systems model and the duration over which the information can be retained. As shown in supplementary Fig. \[SI\_curves\]a–b, if a constant leakage element (for example reverse leakage current) is used, not only do the system trajectories rapidly converge to the final steady state, but the modulation signal does not cause a change in the sensing system trajectory with respect to the reference system trajectory. On the other hand, a resistive or a direct-tunneling leakage element will be sensitive to the changes in modulation signal but will be unable to keep the two trajectories separate for a long period of time, leading to low retention-time. In this report, we show that a differential dynamical system [@mehta2019differential] implemented using a Fowler-Nordheim (FN) quantum tunneling device [@liangTEDTimer] can address all these challenges.
Results
=======
Differential FN tunneling device acts as a long-term reliable synchronized dynamical system
-------------------------------------------------------------------------------------------
The operating physics of a FN quantum-tunneling based dynamical system is illustrated using an energy-band diagram in Fig. \[scheme\]c [@lenzlinger1969fowler]. From a practical point-of-view, this energy-band configuration can be achieved across a thermally grown gate-oxide (silicon-di-oxide), which acts as a FN tunneling barrier separating a lightly-doped n-type semiconductor substrate and an electrically insulated but conductive polysilicon island (labeled as a floating-gate). A two-dimensional electron gas and a triangular FN tunneling barrier (as shown in Fig. \[scheme\]c) is created by applying a large potential difference across the semiconductor-floating-gate interface. Thermally-excited electrons then tunnel through the triangular FN tunneling barrier onto the floating-gate ($FG$) and cannot escape due to the surrounding electrical insulation. Each electron that tunnels through the barrier and is retained, changes the potential of the floating-gate which in turn decreases the slope of the FN tunneling barrier (shown in Fig. \[scheme\]c). Fig. \[scheme\]d shows the cross-section of such a FN tunneling device, whereby the floating-gate is coupled to a programming transistor $P$ and a source follower buffer $B$. The read-out procedure and the procedure to initialize the charge on the floating-gate is described in the Methods section. In [@liangTEDTimer], we showed that the continuous-time dynamics of this device can be modeled using a first-order differential equation which results in the change in floating-gate voltage $V_{\textrm{t}}$ at time-instant $t$ as $$\label{timerEqn}
V_{\textrm{t}} = \frac{k_2}{\log(k_1 t+ k_0)} + k_3$$ where $k_0$–$k_3$ are model parameters. The parameters $k_1$ and $k_2$ depend on the area of tunneling junction, capacitance, temperature and material properties and the device structure, the parameter $k_3$ depends on the read-out mechanisms and the parameter $k_0$ depends on the initial conditions. For the proposed sensor-data-logger we employ a differential configuration as shown in Fig. \[fig2\]a. The initial voltage (equivalently charge) on each of the floating-gates is precisely programmed through a combination of tunneling and hot-electron injection (see Calibration and Initialization in Methods) [@harrison2001cmos]. One of the FN device’s (labeled as the sensor) dynamics is modulated by an input signal $x_{\textrm{t}}$, and its desynchronization is measured with respect to a reference FN device as : $$\label{calcResponse}
\begin{split}
\hat{Y}_{\textrm{t}} = V_{\textrm{t}}^R - V_{\textrm{t}}^S
\end{split}$$ Here, $V_{\textrm{t}}^S$ and $V_{\textrm{t}}^R$ refer to the sensor and reference floating-gate voltages respectively. A capacitive divider (formed by $C_{\textrm{c}}$ and $C_{\textrm{FG2}}$) followed by a source-follower is used to read-out the floating-gate potential through the output node as shown in Fig. \[fig2\]a. The floating-node formed at the capacitive divider is independently programmed to a lower value ($\approx$ ) to ensure that the probability of unwanted tunneling or injection through the transistor $FG_2$ is low. The outputs of the sensor and reference nodes, $V_{\textrm{t}}^{sensor}$ and $V_{\textrm{t}}^{ref}$ respectively, are measured using an external data acquisition system (Keithley DAQ6510) and shown in Fig. \[fig2\]b. The differential output $Y_{\textrm{t}}$ in Fig. \[fig2\]a is measured with respect to the initial value as $$\label{calcResponse2}
Y_{\textrm{t}} = (V_{\textrm{t}}^{ref} - V_{\textrm{t}}^{sensor}) - (V_0^{ref} - V_0^{sensor}) = \Delta V_{\textrm{t}}^{ref} - \Delta V_{\textrm{t}}^{sensor} \propto \hat{Y}_{\textrm{t}}
$$ For calculating $Y_{\textrm{t}}$, we use the change from their initial voltages at time-instant $t=0$ seconds to get rid of offsets introduced at the read-out stage (Fig. \[fig2\]c). For each device, less than $1\;\%$ deviation was observed across trials, demonstrating the reliability of the tunneling dynamics and the reliability of the test setup. With respect to the differential measurements, $Y_{\textrm{t}}$ should be in a perfectly synchronized system. However, due to device mismatch and due to differences in the initialization procedure, we observe a baseline drift across all trials. This manifests as variations in device parameters $k_1$–$k_3$, which were estimated by regressing equation \[timerEqn\] to the empirical data (SI Table \[table:1\]). The estimated parameters were used for compensating for drift and to determine the final sensor output (SI Fig. \[SI\_Sync\]). Post-drift corrections are shown in Fig. \[fig2\]d, which shows the maximum difference between a pair of trials to be less than . We measured the desynchronization of the differential FN device across temperatures ranging from to . The result in Fig. \[fig2\]f shows that for a voltage range $V_{\textrm{t}}$ of , the desynchronization $Y_{\textrm{t}}$ is measured to be less than .
A simple behavioral model explains the data-logging principle
-------------------------------------------------------------
In the Methods section, we have derived a tractable mathematical model for the data sensed and stored in response to an arbitrary time-varying input signal $x_{\textrm{t}}$. We found that the output of the data-logger $Y_{\textrm{T}}$ measured at time-instant $T$ can be expressed as $$\label{recorder model}
Y_{\textrm{T}} = R(T) A_x(T)$$ where $A_x(T)$ represents the total “action” due to the input signal $x_{\textrm{t}}$ accumulated up to the time instant $T$ and $R(T)$ is a “forgetting” factor that is independent of the input signal $x_{\textrm{t}}$. $R(T)$ models the data retention capability and arises due to resynchronization of the sensor and reference FN devices, after the sensor device is perturbed by $x_{\textrm{t}}$. In the Methods section, we show that the action $A_x(T)$ can be expressed in term of device parameters as $$\label{Action model}
A_x(T) =\frac{k_1}{k_2} V_0^2\exp\left(\frac{-k_2}{V_0}\right) \int_{0}^{T} \left[\left(1+\frac{C_R x_{\textrm{t}}}{V_{\textrm{t}}}\right) \exp \left(\frac{k_2 C_{\textrm{R}} x_{\textrm{t}}}{V_{\textrm{t}}(V_{\textrm{t}}+C_{\textrm{R}} x_{\textrm{t}})} \right) - 1\right]{\mathop{}\!d}t$$ and the resynchronization term $R(T)$ can be expressed as $$\label{Resync model}
R(T) = \frac{V_{\textrm{t}}^2}{V_0^2}\exp\left(\frac{k_2}{V_0} - \frac{k_2}{V_{\textrm{T}}}\right).$$ Here $V_{\textrm{t}}$ is given by equation \[timerEqn\] with $V_0$ and $V_{\textrm{T}}$ representing the device voltage at time-instant $t=0$ and $t=T$ seconds. The parameter $C_{\textrm{R}}$ in equation \[Action model\] models a capacitive divider that is formed due to the input capacitance coupling into the floating-gate. The SI section (Figs. \[modelCompare1\] and \[modelCompare2\]) shows several examples of signals $x_{\textrm{t}}$ for which the first-order action model given by equation \[recorder model\] accurately tracks a more computationally intensive ordinary differential equation (ODE) based device model. In SI Fig. \[AC analysis\], we show the “action” $A_x(T)$ corresponding to different signal types with different magnitude and energy. The results show that $A_x(T)$ is monotonic with respect to energy and hence can be used as a measure of cumulative energy.
In our controlled experiments we subjected the FN data-logging device to a square pulse of varying magnitude but with a fixed duration of 120 seconds. This duration was chosen because it is sufficiently long enough to elicit a measurable response for the purpose of device characterization. Also, the the pulse was applied at a fixed time (1,800 seconds), after which the desynchronization $Y_{\textrm{T}}$ was measured at different values of measurement time $T$. Experiments were conducted over a duration of 10,800 seconds (3 hours), with the data-logger responses measured every 30 seconds. Each data-logger was calibrated to similar initial conditions for all experiments where the sensor and the reference nodes were initialized to equal tunneling rates. A typical experiment demonstrating the recorder in operation is shown in Fig. \[recorderExps\]a which matches the model described in the Methods section. The RMSE between the model and measured data is with an $R^2$ of 0.9999. Measurement results from three repeated trials for input signals of magnitude and are shown in Fig. \[recorderExps\]b. The signal resulted in a sensor response of for the three data-logging devices. At the end of three hours, due to resynchronization, the sensor response had dropped to . For the input, responses after the modulation were in the range of , which dropped to at the end of three hours. Though the three recorders had different responses, they were consistent across trials for the same recorder. The device responses at the end of three hours for input signals of different magnitudes are shown in Fig. \[recorderExps\]c. From the figure, it is evident that the data-logging device response is similar to a rectifier as summarized by the action model in equation \[Action model\]. The action model fits the data for this wide range of input conditions with an $R^2$ of 0.9855.
Energy budget, sensing and retention limits
-------------------------------------------
The rectification property of the FN data-logging device can be useful for measuring and logging the intensity of a time-varying signal like bio-potentials or accelerometer output. The device is sensitive to input signals of any intensity since there are no threshold requirements on the input signal to activate the sensor. The caveat being, the data retention times for small signals will be shorter due to the resynchronization (modeled by $R(T)$ in equation \[Resync model\]) and operational noise in the recorder. In a perfectly matched differential system, and in the absence of any input, the device response should be exactly , because of synchronization. However, environmental factors, mismatch between the sensing and reference nodes, or stochasticity in tunneling mechanism, cause desynchronization and the recorder response deviates from the baseline. In general, the variance in the output increases with time (See Fig. \[fig2\]d for example). This increase in variance over time is a form of operational noise in the system ($\sigma_{\textrm{t}}$). A model for $\sigma_{\textrm{t}}$ could be estimated by letting the recorder operate with inputs grounded (similar to input referred noise) and measuring the deviation of output from the baseline. Another source of noise is the readout noise ($N_{\textrm{0}}$) which limits the resolution to which charge on the floating gate can be measured. Total noise ($N_{\textrm{t}}$) is the sum of these two noise sources. While noise increases with time, recorder response decreases due to resynchronization. For a signal of given action, there will be a time instance $T_{\textrm{ret}}$, beyond which the signal-to-noise ratio (SNR) goes below a chosen threshold and input signal cannot be reconstructed with a desirable degree of certainty. We chose unity as our threshold for SNR and we defined data retention as the time at which the signal falls below system noise. The supplementary Fig. \[fig\_retention\]a shows via an illustration how data retention capacity for a given noise model can be estimated. The supplementary Fig. \[fig\_retention\]b shows that data retention capacity increases exponentially with the signal action. For a action, we could expect to measure significant deviation from baseline for over 300,000 seconds ($\approx$ 4 days). The action model can be used to estimate the energy-budget requirement on the sensing signal. Since the average FN tunneling current is $10^{-17} A$, the energy budget is less than an attojoule. Note that this is the energy to trigger desynchronization.
Noise in the system can also be described by the effective number of bits (ENOB) (supplementary Fig. \[fig\_retention\]c). For an assumed action range of , 10 bits precision can be initially expected in a system with readout noise. In a perfectly matched system, ENOB would drop to 0 at $\approx{}2\times{}10^6$ seconds (total recorder lifetime), but with the added operational noise it drops down to $\approx{}3\times{}10^5$ seconds. Readings from multiple recorders can be combined to increase the effective number of bits of the system.
Self-powered sensing of action due to ambient acceleration
----------------------------------------------------------
In this section, we demonstrate the use of the proposed sensor-data-logger for battery-free sensing of ambient acceleration. We chose a piezoelectric transducer for sensing mechanical acceleration and for directly powering the sensor-data-logger device. Note that in this regard, other transducers for e.g. photodiodes, RF antennas, thermocouples could also be directly interfaced to the FN data-logging device to create other self-powered sensing modalities. A schematic of the experimental setup is shown in Fig. \[figPiezo\]a. A PVDF (polyvinylidene difluoride) cantilever \[TE’s Measurement Specialties MiniSense 100 Vibration Sensor with nominal resonant frequency – \] was mounted on a benchtop vibration table \[3B Scientific Vibration Generator - U56001\] that is externally actuated by a function generator. The table was actuated at an off-resonant frequency of for a range of actuating amplitudes. We simultaneously measured acceleration using a 3-axis accelerometer \[Adafruit LIS3DH accelerometer\] to use as the ground truth. Results are shown in Fig.\[figPiezo\]b–c. We observed significant responses for vibration signals down to an acceleration of (). For context, a refrigerator vibrates with an acceleration of around [@roundy2005effectiveness]. The expected maximum output power of the piezo sensor is on the order of tens of nanowatts of which only a fraction is used by the recorder to store the information. In the final experiment, we electrically disconnected all power to the recording system at the 1 hour (3,600s) mark, actuated the vibration table at 1.5 hours (5,400 s) and reconnected the system at 2 hours (7,200 s) to readout the output of the data-logger. We observed vibration-induced desynchronization in this set of experiments as well, with the deviation as expected based on the earlier characterization tests.
Discussion
==========
In this report, we proposed a novel method for designing an ultra-energy-efficient sensor-data-logging device, where the energy of the sensing signal is used to modulate the state trajectories of a synchronized dynamical system. We showed that a Fowler-Nordheim (FN) quantum tunneling device [@liangTEDTimer] can be used to implement the proposed sensor-data-logger on a standard silicon process.
Our modeling study summarized in supplementary Fig. \[sysChar\] shows that there are multiple parameters, both operational and design parameters, that affect the retention time (or resynchronization) of the FN device. Change in any parameter that increases (decreases) the “action” of the signal, would also lead to faster (slower) resynchonization. Thus, its net effect on the system depends on the total duration for which the input signal was applied. The initial charge on the floating-gate and time to sample are operational parameters as they can be set at run-time, as required by the specific application. Larger time intervals allow the input signal to be integrated over a longer period of time, but it does not change the sensitivity to the signal. For a signal of given action, the measured value decreases as $T$ increases due to resynchonization (Fig. \[sysChar\]a). Initializing the device to a higher voltage leads to higher sensitivity but only up to a certain limit (Fig. \[sysChar\]b). The reason is that higher sensitivity also leads to faster resynchonization determined by design parameters $k_1$ and $k_2$. $k_1$ can be tuned by varying the area of tunneling junction and capacitance sizing. We found that increasing the area or lowering the capacitance would increase the sensitivity of the system but only to a certain point (Fig. \[sysChar\]c). Beyond this point, the gains are only marginal, at the expense of a larger footprint. Moreover, the capacitance is a function of the tunneling junction area and thus the ratio of area to capacitance is bounded and depends on the permittivity of the insulator. Smaller oxide thickness would decrease $k_2$ and sharply increase the sensitivity (Fig. \[sysChar\]d). However, at these scales the effect of other processes like direct tunneling cannot be ignored. Exploring different materials could have significant impact on both $k_1$ and $k_2$, as they affect the parameters $\alpha$ and $\beta$ (equations \[currentDensity1\] and \[params1\]). When the input signal is a single pulse, the time-of-occurrence of the pulse also plays a role in the measured response, as shown in SI Fig. \[YvsMt\]. However, this effect is weaker than that of other factors.
The desychronization based approach reduces the energy budget required for data-logging and we estimate that the proposed device can operates at an energy budget lower than an attojoule while retaining the information for at least 3 hours. In standard analog sensor circuits, a quiescent current is sourced from a power source for continuous operation. In the proposed device, the quiescent current is the FN quantum tunneling current, which is sourced from the pre-charged capacitor and ambient thermodynamic fluctuations. Hence, no external power source is needed for operation. For modulating the sensor, energy is extracted from the signal being sensed. If the energy dissipated at the input signal source (due to finite source impedance) is ignored, than the energy-budget required to modulate the state of the FN device is less than . In practice, the energy from the source is spent on charging the capacitor, and for maintaining DC voltage at the source. For example, when the magnitude of the input signal is , less than is used for charging the input capacitor. We used a digital to analog converter to generate and hold the input signal, which expends considerable power ($\sim$) to maintain a steady output. However, the most efficient power transfer or sensing is achieved when source impedance is equal to input impedance of the sensor. The input impedance of the proposed device was measured to be greater than $10^{15} \Omega$. Thus, the energy required to hold voltage of for 120 sec would be less than . Many signals of interest have power levels greater than this and they should be able to provide sufficient energy for modulating the sensor, provided the system impedance is matched to the source.
Using FN quantum tunneling to implement the dynamical system has some key advantages. Its stability allowed us to create a pair of synchronized devices which is compensated for environmental variations. Its predictability was used for modeling and we were able to derive a recorder response model that matched experimental data with $98.8\;\%$ accuracy. Its dynamics follow a $1/\log(t)$ characteristic, which yields a long operational life. The non linear response leads to rectification of input signals and offers an opportunity for time stamping and reconstruction. A more rigorous and theoretic investigation into the use of dynamical systems for information reconstruction will be the topic of future research.
At its core, the proposed device consists of four capacitors and two transistors (4C-2T), and can be implemented on any standard CMOS process. The current design is a proof-of-concept and is not optimized for sensitivity or form factor. Modeling analysis in supplementary Fig. \[sysChar\] shows that both of these parameters can be improved by minimizing the capacitance, while maintaining the capacitance ratio ($C_{\textrm{R}}$). To achieve this, an optimum balance between the input capacitor, decoupling capacitor and parasitic capacitance at the poly-substrate tunneling junction needs to be obtained. Better matching of the sensor and reference nodes (tunneling junctions, capacitors and readout circuits) using advanced analog layout techniques should be able to reduce the operational noise in the recorder and thereby increase the data retention capacity. Readout and common-mode noise can be further reduced by implementing a low-noise on-chip instrumentation amplifier. Multiple units of independent recorders could be used to increase the SNR of the recordings.
Finally, the proposed recorder could be directly integrated with FET (field-effect transistor) based sensors [@siSensor_middelhoek2000celebration; @isfet_bergveld2003thirty; @ofet_torsi2013organic], which have been developed for a wide range of applications. As there are no extrinsic powering requirements, there is a potential of integrating these devices on “smart dust” platforms as well [@smart_dust_warneke2001smart; @neuralDust_seo2016wireless].
Conclusion
==========
In this report, we have described a self-powered sensor-data-logger device that records a cumulative measure of the sensor signal intensity over its entire duration. To achieve this, we designed a pair of synchronized dynamical systems whose trajectories is modulated by an external signal. The modulation leaves its trace by desynchronizing one of the synchronized pairs. The total cumulative measure or action is stored as a dynamical state which is then measured at a later instant of time. The self-powered dynamical system was designed by exploiting the physics of Fowler-Nordheim quantum tunneling in floating gate transistors. We modeled the response of our system to an arbitrary signal and verified the model experimentally. We also demonstrated the self-powered sensing capabilities of our device by logging mechanical vibration signals produced by a small piezoelectric transducer, while being disconnected from any external power source.
Methods
=======
One-time programming {#Methods_prog .unnumbered}
--------------------
For each node of each recorder, the readout voltage was programmed to around while the tunneling node was operating in the tunneling regime. This was achieved through a combination of tunneling and injection. Specifically, VDD was set to , input to and program tunneling pin was gradually increased to . Around , the tunneling node’s potential would start increasing. The coupled readout node’s potential would also increase. When the readout potential went over , electrons would start injecting into the readout floating gate, thus ensuring its potential was clamped below . After this initial programming, VDD was set to for the rest of the experiments.
Calibration {#Methods_calibration .unnumbered}
-----------
After one-time programming, input was set to , Vprog to for 1 minute and then the floating gate was allowed to discharge naturally. Readout voltages for the sensor and reference nodes were measured every 30 seconds, for 3 hours. The rate of discharge for each node was calculated; and a state where the tunneling rates would be equal was chosen as the initial synchronization point for the remainder of the experiments.
Initialization {#Methods_init .unnumbered}
--------------
Before the start of each experiment, floating gates were initialized to the initial synchronization point, estimated in the previous section. This was done by either setting the input to stable DC point through a digital to analog converter (DAC) or if the DAC value needed was beyond its output limit, then the potential would be increased by setting Vprog pin to .
Model derivation {#Methods_Tmodel .unnumbered}
----------------
FN tunneling current density $J_{\textrm{FN}}$ across a triangular barrier can be expressed as a function of the electric field $E$ across the barrier [@lenzlinger1969fowler]: $$\label{currentDensity1}
J_{\textrm{FN}}(E)=\alpha {E}^2 \exp(-\beta /E)$$ where $\alpha$ and $\beta$ are process and device specific parameters [@lenzlinger1969fowler].
Thus, for a tunneling junction with cross-sectional area $A$ and thickness $t_{\textrm{ox}}$, the tunneling current $I_{\textrm{FN}}$ for a time-varying voltage $V_{\textrm{t}}$ is given by $$\label{currentEqn}
I_{\textrm{FN}}(V_{\textrm{t}})= A\alpha (V_{\textrm{t}}/t_{\textrm{ox}})^2 \exp(-\beta t_{\textrm{ox}}/V_{\textrm{t}}).$$
Referring to the equivalent circuit in Fig. \[fig2\]a, the dynamical system model when the sensing signal $x_{\textrm{t}}$ is absent is given by $$\label{dischargeEqn}
I_{\textrm{FN}}(V_{\textrm{t}})= - C_{\textrm{total}} \frac{{\mathop{}\!d}V_{\textrm{t}}}{{\mathop{}\!d}t}$$ where $C_{\textrm{total}} = C + C_{\textrm{in}}$ is the total capacitance at the floating-gate node. The solution of the equation can be expressed as : $$\label{timerEqn1}
V_{\textrm{t}} = \frac{k_2}{\log(k_1 t+ k_0)}$$ where $$\label{params1}
k_1 = \frac{A \alpha \beta}{Ct_{\textrm{ox}}} \: \: \: k_2 = \beta t_{\textrm{ox}}$$ depend on material properties and device structure, while
$$\label{params21}
k_0 = \exp \left(\frac{k_2}{V_0}\right)$$
depends on the initial conditions.
Now, let $$\label{dVdt}
f(V_{\textrm{t}})= -\frac{I(V_{\textrm{t}})}{C_{\textrm{total}}} = -\frac{k_1}{k_2} V_{\textrm{t}}^2 \exp\left(\frac{-k_2}{V_{\textrm{t}}}\right)$$
Desynchronization between the sensor and reference nodes shown in Fig. \[fig2\]a occurs because of differences in rates of tunneling, which are caused by differences in electric potentials across the respective floating-gates $$\label{diffEqn1}
\begin{split}
\frac{{\mathop{}\!d}Y_{\textrm{t}}}{{\mathop{}\!d}t} & = \frac{I_{\textrm{FN}}(V_{\textrm{t}}^S)}{C_{\textrm{total}}} - \frac{I_{\textrm{FN}}(V_{\textrm{t}}^R)}{C_{\textrm{total}}} \\
& = f(V_{\textrm{t}}^R) - f(V_{\textrm{t}}^S)
\end{split}$$
The reference node $V_{\textrm{t}}^R$ follows the dynamics of equation \[timerEqn1\] as it is not under the action of an external field. Thus, $V_{\textrm{t}}^R = V_{\textrm{t}}$. The potential across the sensing node is given by how much it has desynchronized from the reference node ($V_{\textrm{t}}^R - Y_{\textrm{t}}$) and the effect of the external field, $x_{\textrm{t}}$, through the input capacitor $C_{\textrm{in}}$.
$$\label{VS}
V_{\textrm{t}}^S = V_{\textrm{t}} + C_{\textrm{R}}x_{\textrm{t}} - Y_{\textrm{t}}$$
where $C_{\textrm{R}}$ is the coupling ratio due to capacitive divider formed by $C_{\textrm{in}}$ and $C_{\textrm{fg}}$. $$\label{CR}
C_{\textrm{R}} = \frac{C_{\textrm{in}}}{C_{\textrm{total}}} ; C_{\textrm{total}} = C_{\textrm{in}} + C_{\textrm{FG1}} + C_{\textrm{C}}||C_{\textrm{FG2}}$$
Substituting $V_{\textrm{t}}^R$ and $V_{\textrm{t}}^R$ in equation \[diffEqn1\] $$\label{ode}
\frac{{\mathop{}\!d}Y_{\textrm{t}}}{{\mathop{}\!d}t} = f(V_{\textrm{t}}) - f(V_{\textrm{t}} + C_{\textrm{R}}x_{\textrm{t}} - Y_{\textrm{t}})$$ Above equation is the constitutive differential equation and can be solved using numerical methods for any input signal. To obtain an explicit expression for estimating the response $Y_{\textrm{t}}$, we assume that $Y_{\textrm{t}} \ll V_{\textrm{t}}$ for all $t$, and use Taylor series expansion with first order approximation. $$\label{odeB}
\begin{split}
\frac{{\mathop{}\!d}Y_{\textrm{t}}}{{\mathop{}\!d}t} & = f(V_{\textrm{t}}) - f(V_{\textrm{t}} + C_{\textrm{R}}x_{\textrm{t}}) + \frac{{\mathop{}\!d}(f(V_{\textrm{t}}))}{{\mathop{}\!d}V_{\textrm{t}}}Y_{\textrm{t}} \\
\frac{{\mathop{}\!d}Y_{\textrm{t}}}{{\mathop{}\!d}t} & - \frac{{\mathop{}\!d}(f(V_{\textrm{t}}))}{{\mathop{}\!d}V_{\textrm{t}}}Y_{\textrm{t}} = f(V_{\textrm{t}}) - f(V_{\textrm{t}} + C_{\textrm{R}}x_{\textrm{t}})
\end{split}$$
Multiplying both sides of equation \[odeB\] by $1/f(V_{\textrm{t}})$, substituting ${\mathop{}\!d}V_{\textrm{t}} = f(V_{\textrm{t}}) {\mathop{}\!d}t $ (from equations \[dischargeEqn\] and \[dVdt\]) and simplifying: $$\label{ode3}
\begin{split}
\frac{{\mathop{}\!d}Y_{\textrm{t}}}{f(V_{\textrm{t}}) {\mathop{}\!d}t} - \frac{{\mathop{}\!d}(f(V_{\textrm{t}}))}{{f(V_{\textrm{t}})}^2 {\mathop{}\!d}t}Y_{\textrm{t}} & = \frac{1}{f(V_{\textrm{t}})}(f(V_{\textrm{t}}) - f(V_{\textrm{t}} + C_{\textrm{R}}x_{\textrm{t}})) \\
\frac{{\mathop{}\!d}}{{\mathop{}\!d}t} \left(\frac{Y_{\textrm{t}}}{f(V_{\textrm{t}})}\right) & = 1- \frac{f(V_{\textrm{t}} + C_{\textrm{R}}x_{\textrm{t}})}{f(V_{\textrm{t}})}
\end{split}$$ Integrating both sides with respect to ${\mathop{}\!d}t$ between the limits 0 and $T$: $$\label{odeSol}
\begin{split}
\frac{Y_{\textrm{T}}}{f(V_{\textrm{T}})} - \frac{Y_{\textrm{0}}}{f(V_{\textrm{0}})} & = \int_0^T{\left(1- \frac{f(V_{\textrm{t}} + C_{\textrm{R}}x_{\textrm{t}})}{f(V_{\textrm{t}})}\right)} {\mathop{}\!d}t \\
\frac{Y_{\textrm{T}}}{f(V_{\textrm{T}})} & = \int_0^T{\left(1- \frac{f(V_{\textrm{t}} + C_{\textrm{R}}x_{\textrm{t}})}{f(V_{\textrm{t}})}\right)} {\mathop{}\!d}t \\
Y_{\textrm{T}} & = f(V_{\textrm{T}}) \int_0^T{\left(1- \frac{f(V_{\textrm{t}} + C_{\textrm{R}}x_{\textrm{t}})}{f(V_{\textrm{t}})}\right)} {\mathop{}\!d}t
\end{split}$$
Substituting $f(V_{\textrm{t}})$ from equation \[dVdt\] in equation \[odeSol\] $$\label{odeSolFN}
\begin{split}
Y_{\textrm{T}} & = \frac{k_1}{k_2} V_{\textrm{T}}^2 \exp\left(\frac{-k_2}{V_{\textrm{T}}}\right) \int_{0}^{T} \left[\left(1+\frac{C_{\textrm{R}} x_{\textrm{t}}}{V_{\textrm{t}}}\right) \exp \left(\frac{k_2 C_{\textrm{R}} x_{\textrm{t}}}{V_{\textrm{t}}(V_{\textrm{t}}+C_{\textrm{R}} x_{\textrm{t}})} \right) - 1\right] {\mathop{}\!d}t
\end{split}$$
Different leakage mechanisms for dynamical logging devices
----------------------------------------------------------
Three different types of dynamical systems are simulated based on different leakage element $I(V_{\textrm{t}})$ in Fig. \[scheme\]b. All systems would respond to an external signal (a square pulse), and then resynchronize to their baseline response. This is illustrated in Fig. \[SI\_curves\]a for three different leakage elements. When the leakage elements is a resistor, the dynamics follow an exponential characteristic. However, an extremely large resistance would be required to sustain the effects of the input pulse (or transient response). As an example, for a system with $C = 1~pF, R = 1~T\Omega, V_0 = 3~V$, a 1 second long, 100 mV input signal will elicit a response that can be observed for 5.5 seconds. This is illustrated in Fig. \[SI\_curves\]b. Not that when the leakage elements is a constant current (reverse biased diode leakage), the input pulse does not elicit any change in the response. For the leakage element based on FN tunneling, which follows a $1/log(t)$ dynamics, the input pulse elicits a response that shows a much longer resynchronization time, as shown in Figs. \[SI\_curves\]a–b. This feature has been modeled and experimentally verified in the main text.
Programming and synchronization {#SI_Prog}
-------------------------------
The differential sensor-data-logging system consists of two nodes: sensor and reference node. Each node contains two floating gates decoupled via a capacitor. The charge on the four gates of the system can be individually programmed using a combination of tunneling (increases charge, course) and hot electron injection (decreases charge, fine). The programming block for each gate is selected via a switch. Injection is initiated by setting $V_{\textrm{DD}}$ = , and setting the input pin to a value (via a DAC), such that $V_{\textrm{DS}}$ is above . $V_{\textrm{DS}}$ can be modulated via the gate voltage because the PMOS is in a source follower configuration. Tunneling is realized by bring $V_{\textrm{tun}}$ to a high potential. For programming the tunneling node to be in the FN tunneling regime, we used $V_{\textrm{tun}} =$ . Except for the self-powered experiments using piezo crystals, we did not have to program the tunneling node in FN tunneling regime using $V_{\textrm{prog}}$ pin. Instead, we could set the input pin to a stable voltage (analog ground) which would push the node into FN regime. The DAC voltage was calculated each run such that tunneling node’s potential at the start of each experiment was the same (as measured by the readout node). When the needed DAC voltage exceeded , we would initiate tunneling. This process allowed us to limit the number of high voltage tunneling cycles and increase the experimental life of the recorder. This process cannot be done in actual deployment because there would not be an external DC source. Hence, for self-powered piezo experiments, we carried out tunneling for each trial.
Device parameters and drift correction
--------------------------------------
Device parameters from equation \[timerEqn\] (Main text) can be experimentally obtained by allowing the floating gate to discharge via Fowler-Nordheim tunneling and fitting the model on observed data.
We obtained the following parameters:
[r| M|M|M|M ]{} Device No. & Node & $\log (k_1)$ & $k_2$ & $k_3$\
& Sensor & 38.59 & 347.20 & -4.24\
& Ref& 39.47 & 359.04 & -4.43\
& Sensor & 44.53 & 425.01 & -4.86\
& Ref & 42.06 & 389.18 & -4.57\
& Sensor & 42.14 & 381.26 & -4.35\
& Ref & 41.16 & 370.20 & -4.30\
$k_0$ depends on the initial conditions. The starting voltages for each node can be chosen such that the sensor and reference have the same rates and are thus synchronized. However, the mismatch in other parameters causes the two nodes to drift. The drift is predictable and can be corrected as shown in Fig. \[SI\_Sync\].
$k_3$ was subsumed into $V_{\textrm{t}}$ by setting $V_{\textrm{t}} \to V_{\textrm{t}} - k_3$. The modified $V_{\textrm{t}}$ is used for derivation of the explicit model in equation \[Action model\].
Model validation {#SI_model}
----------------
The assumptions made in the derivation have been validated against a general ODE solver (Figs. \[modelCompare1\] and \[modelCompare2\]). As shown in the figure, the error was less than for a response of . Same analysis was run 100 times and the relative error was always less than $1\;\%$. The action model was computationally faster to solve than the ODE solver by a factor of $10^5$. The explicit action model led to large errors when the input signal was large (Fig. \[modelCompare2\]). The error arises due to assumption, $W \ll V$, made during linearizing the equation \[ode\] to estimate the resynchronization of the response. For large $W$, higher order terms can no longer be ignored and the resynchronization will be faster. As the 1st order model ignores these terms, it always overestimates the expected action at time T. The error in the model can be empirically reduced by fitting a model between the expected response (as generated by the ODE solver) and response calculated by the action model (Fig. \[modelCompare2\]c).
AC analysis
-----------
The action induced by an AC coupled signal is monotonic with the energy of the signal. Actions due to different waveform shapes are more similar for signals with same energy (Fig. \[AC analysis\]b) compared to signals with same amplitude (Fig. \[AC analysis\]a).
![Simulation results for recorder response to AC signals. a) Action induced by signals of different shapes as a function of amplitude of the signal. System is sensitive to biphasic signals because of the rectification. b) Action induced by signals of different shapes as a function of energy of the signal.[]{data-label="AC analysis"}](finalFigs/SI_ACsignal.png){width="7in"}
Data retention model
--------------------
Retention time for a given input signal was found by a fixed point method. First, a noise model was generated using experiments without any input modulation. Standard deviation ($\sigma_{\textrm{t}}$) was calculated across all runs as a function of time. Ideally, if the dynamics were perfectly synchronized, then the $\sigma_{\textrm{t}}$ obtained would be 0. However, we find that $\sigma_{\textrm{t}}$ increases with time due to integration of noise. We fit a rational equation on this noise. $$\sigma_{\textrm{t}} = \frac{at}{t+b}$$ We chose this equation so that it stays bounded as time approaches $\infty$. Total noise in the system is given by $$N_{\textrm{t}} = \sigma_{\textrm{t}} + N_{\textrm{0}}$$ where $N_{\textrm{0}}$ is the noise associated with readout circuits and data acquisition system.
The time of retention $T_{\textrm{ret}}$ was defined as the time instance at which the expected recorder response ($Y_{\textrm{t}}$ becomes lower than the predicted noise in the system $N_{\textrm{t}}$, i.e. the signal-to-noise ratio goes below unity. Thus at time $t = T_{\textrm{ret}}$ $$Y_{T_{\textrm{ret}}} = N_{T_{\textrm{ret}}}$$
Parametric analysis
-------------------
Using modeling and simulations, we conducted parametric analysis for our system. Parameters $T$ (Fig. \[sysChar\]a) and $V_0$ ((Fig. \[sysChar\]b) are operational parameters that can be set at run time according to application requirements. $k_1$ (Fig. \[sysChar\]c) depends on the area of the tunneling junction and on the capacitance associated with the floating gate node. $k_1$ and $k_1$ (Fig. \[sysChar\]d) are also influenced by the thickness of the insulating material and other material properties like the barrier height at the interface between the conductor and the insulator.
![Simulation results for sensitivity and parametric analysis. Default parameters are $k_1 = \exp(38.5), k_2 = 346, V_0 = 7.5V $. $Y_{\textrm{T}}'$ is the baseline response at time $T$ for a single square pulse of magnitude 100 mV and duration 1 s. a) Time of sampling determines the amount of resynchronization b) Initial programming voltage affects the sensitivity of the recorder, but its effect becomes attenuated as time of sampling increases (due to resynchronizaiton) c,d) Device parameters $k_1$ and $k_2$ can be tuned via system design and material selection to optimize the recorder response. $\alpha$ and $\beta$ are material parameters [@lenzlinger1969fowler].[]{data-label="sysChar"}](finalFigs/fig4_char.pdf){width="7in"}
Temporal dependence
-------------------
Fig. \[YvsMt\] shows the weak dependence of the recorder output to the time of occurrence of events. Earlier events lead to larger desynchronization, but have more time to recover. Modelling studies show that net result is that later events lead to a larger output at readout. However, the change in expected output was less than , smaller than the errors arising due to measurement and operational desynchronization.
[^1]: This work was supported in part by NIH research grants 1R21EY028362-01 and 1R21AR075242-01.
[^2]: D. Mehta is with the Department of Biomedical Engineering and K. Aono and S. Chakrabartty are with the Department of Electrical and Systems Engineering, Washington University in St. Louis, St. Louis, MO, 63130 USA. All correspondences regarding this paper should be addressed to Email: [email protected]
|
---
abstract: 'In statistical relational learning, knowledge graph completion deals with automatically understanding the structure of large knowledge graphs—labeled directed graphs—and predicting missing relationships—labeled edges. State-of-the-art embedding models propose different trade-offs between modeling expressiveness, and time and space complexity. We reconcile both expressiveness and complexity through the use of complex-valued embeddings and explore the link between such complex-valued embeddings and unitary diagonalization. We corroborate our approach theoretically and show that *all* real square matrices—thus all possible relation/adjacency matrices—are the real part of some unitarily diagonalizable matrix. This results opens the door to a lot of other applications of square matrices factorization. Our approach based on complex embeddings is arguably simple, as it only involves a Hermitian dot product, the complex counterpart of the standard dot product between real vectors, whereas other methods resort to more and more complicated composition functions to increase their expressiveness. The proposed complex embeddings are scalable to large data sets as it remains linear in both space and time, while consistently outperforming alternative approaches on standard link prediction benchmarks.[^1]'
author:
- |
Théo Trouillon [email protected]\
Univ. Grenoble Alpes, 700 avenue Centrale, 38401 Saint Martin d’Hères, France Christopher R. Dance [email protected]\
NAVER LABS Europe, 6 chemin de Maupertuis, 38240 Meylan, France Éric Gaussier [email protected]\
Univ. Grenoble Alpes, 700 avenue Centrale, 38401 Saint Martin d’Hères, France Johannes Welbl [email protected]\
Sebastian Riedel [email protected]\
University College London, Gower St, London WC1E 6BT, United Kingdom Guillaume Bouchard [email protected]\
Bloomsbury AI, 115 Hampstead Road, London NW1 3EE, United Kingdom\
University College London, Gower St, London WC1E 6BT, United Kingdom
bibliography:
- 'nonauto\_bib.bib'
- 'complex\_bib.bib'
title: Knowledge Graph Completion via Complex Tensor Factorization
---
complex embeddings, tensor factorization, knowledge graph, matrix completion, statistical relational learning
Introduction
============
Web-scale knowledge graph provide a structured representation of world knowledge, with projects such as the Google Knowledge Vault [@Dong:2014:KnowledgeVault]. They enable a wide range of applications including recommender systems, question answering and automated personal agents. The incompleteness of these knowledge graphs—also called knowledge bases—has stimulated research into predicting missing entries, a task known as *link prediction* or *knowledge graph completion*. The need for high quality predictions required by link prediction applications made it progressively become the main problem in statistical relational learning [@Getoor2007], a research field interested in relational data representation and modeling.
Knowledge graphs were born with the advent of the Semantic Web, pushed by the World Wide Web Consortium (W3C) recommendations. Namely, the Resource Description Framework (RDF) standard, that underlies knowledge graphs’ data representation, provides for the first time a common framework across all connected information systems to share their data under the same paradigm. Being more expressive than classical relational databases, all existing relational data can be translated into RDF knowledge graphs [@sahoo2009survey].
Knowledge graphs express data as a directed graph with labeled edges (relations) between nodes (entities). Natural redundancies between the recorded relations often make it possible to fill in the missing entries of a knowledge graph. As an example, the relation could not be recorded for all entities, but it can be inferred if the relation is known. The goal of link prediction is the automatic discovery of such regularities. However, many relations are non-deterministic: the combination of the two facts and does not always imply the fact . Hence, it is natural to handle inference probabilistically, and jointly with other facts involving these relations and entities. To this end, an increasingly popular method is to state the knowledge graph completion task as a 3D binary tensor completion problem, where each tensor slice is the adjacency matrix of one relation in the knowledge graph, and compute a decomposition of this partially-observed tensor from which its missing entries can be completed. Factorization models with low-rank embeddings were popularized by the Netflix challenge [@koren_netflix]. A partially-observed matrix or tensor is decomposed into a product of embedding matrices with much smaller dimensions, resulting in fixed-dimensional vector representations for each entity and relation in the graph, that allow completion of the missing entries. For a given fact *r(s,o)* in which the subject entity $s$ is linked to the object entity $o$ through the relation $r$, a score for the fact can be recovered as a multilinear product between the embedding vectors of $s$, $r$ and $o$, or through more sophisticated composition functions [@nickel_2016_review]. Binary relations in knowledge graphs exhibit various types of patterns: hierarchies and compositions like , or , with strict/non-strict orders or preorders, and equivalence relations like . These characteristics maps to different combinations of the following properties: reflexivity/irreflexivity, symmetry/antisymmetry and transitivity. As described in @Bordes2013, a relational model should (i) be able to learn all combinations of such properties, and (ii) be linear in both time and memory in order to scale to the size of present-day knowledge graphs, and keep up with their growth.
A natural way to handle any possible set of relations is to use the classic canonical polyadic (CP) decomposition [@hitchcock-sum-1927], which yields two different embeddings for each entity and thus low prediction performances as shown in Section \[sec:expe\]. With unique entity embeddings, multilinear products scale well and can naturally handle both symmetry and (ir)-reflexivity of relations, and when combined with an appropriate loss function, dot products can even handle transitivity [@bouchard2015approximate]. However, dealing with antisymmetric—and more generally asymmetric—relations has so far almost always implied superlinear time and space complexity [@Nickel2011; @socher2013reasoning] (see Section \[sec:mat\_case\]), making models prone to overfitting and not scalable. Finding the best trade-off between expressiveness, generalization and complexity is the keystone of embedding models.
In this work, we argue that the standard dot product between embeddings can be a very effective composition function, provided that one uses the right *representation*: instead of using embeddings containing real numbers, we discuss and demonstrate the capabilities of complex embeddings. When using complex vectors, that is vectors with entries in ${\mathbb{C}}$, the dot product is often called the *Hermitian* (or sesquilinear) dot product, as it involves the conjugate-transpose of one of the two vectors. As a consequence, the dot product is not symmetric any more, and facts about one relation can receive different scores depending on the ordering of the entities involved in the fact. In summary, complex embeddings naturally represent arbitrary relations while retaining the efficiency of a dot product, that is linearity in both space and time complexity. This paper extends a previously published article [@trouillon2016]. This extended version adds proofs of existence of the proposed model in both single and multi-relational settings, as well as proofs of the non-uniqueness of the complex embeddings for a given relation. Bounds on the rank of the proposed decomposition are also demonstrated and discussed. The learning algorithm is provided in more details, and more experiments are provided, especially regarding the training time of the models.
The remainder of the paper is organized as follows. We first provide justification and intuition for using complex embeddings in the square matrix case (Section \[sec:mat\_case\]), where there is only a single type of relation between entities, and show the existence of the proposed decomposition for all possible relations. The formulation is then extended to a stacked set of square matrices in a third-order tensor to represent multiple relations (Section \[sec:tens\_case\]). The stochastic gradient descent algorithm used to learn the model is detailed in Section \[sec:algo\], where we present an equivalent reformulation of the proposed model that involves only real embeddings. This should help practitioners when implementing our method, without requiring the use of complex numbers in their software implementation. We then describe experiments on large-scale public benchmark knowledge graphs in which we empirically show that this representation leads not only to simpler and faster algorithms, but also gives a systematic accuracy improvement over current state-of-the-art alternatives (Section \[sec:expe\]). Related work is discussed in Section \[sec:rel\_work\].
Relations as the Real Parts of Low-Rank Normal Matrices {#sec:mat_case}
=======================================================
We consider in this section a simplified link prediction task with a single relation, and introduce complex embeddings for low-rank matrix factorization.
We will first discuss the desired properties of embedding models, show how this problem relates to the spectral theorems, and discuss the classes of matrices these theorems encompass in the real and in the complex case. We then propose a new matrix decomposition—to the best of our knowledge—and a proof of its existence for all real square matrices. Finally we discuss the rank of the proposed decomposition.
Modeling Relations
------------------
Let ${\mathcal{E}}$ be a set of entities, with $|{\mathcal{E}}|=n$. The truth of the single relation holding between two entities is represented by a sign value $y_{so}\in\{-1,1\}$, where 1 represents true facts and -1 false facts, $s\in{\mathcal{E}}$ is the subject entity and $o\in{\mathcal{E}}$ is the object entity. The probability for the relation holding true is given by $${P}(y_{so}=1) = \sigma(x_{so})
\enspace
\label{observation-model0}$$ where $X\in{{\mathbb{R}}}^{n\times n}$ is a latent matrix of scores indexed by the subject (rows) and object entities (columns), $Y$ is a partially-observed sign matrix indexed in identical fashion, and $\sigma$ is a suitable sigmoid function. Throughout this paper we used the logistic inverse link function $\sigma(x) = \frac{1}{1+\mathrm{e}^{-x}}$.
### Handling Both Asymmetry and Unique Entity Embeddings
In this work we pursue three objectives: finding a generic structure for $X$ that leads to $(i)$ a computationally efficient model, $(ii)$ an expressive enough approximation of common relations in real world knowledge graphs, and $(iii)$ good generalization performances in practice. Standard matrix factorization approximates $X$ by a matrix product $UV{^\top}$, where $U$ and $V$ are two functionally-independent $n\times K$ matrices, $K$ being the rank of the matrix. Within this formulation it is assumed that entities appearing as subjects are different from entities appearing as objects. In the Netflix challenge [@koren_netflix] for example, each row $u_i$ corresponds to the user $i$ and each column $v_j$ corresponds to the movie $j$. This extensively studied type of model is closely related to the singular value decomposition (SVD) and fits well with the case where the matrix $X$ is rectangular. However, in many knowledge graph completion problems, the same entity $i$ can appear as both subject *or* object and will have two different embedding vectors, $u_i$ and $v_i$, depending on whether it appears as subject or object of a relation. It seems natural to learn unique embeddings of entities, as initially proposed by @Nickel2011 and @bordes2011learning and since then used systematically in other prominent approaches [@bordes2013translating; @Yang2015; @socher2013reasoning]. In the factorization setting, using the same embeddings for left- and right-side factors boils down to a specific case of eigenvalue decomposition: *orthogonal diagonalization*.
A real square matrix ${X}\in {{{\mathbb{R}}}^{n \times n}}$ is orthogonally diagonalizable if it can be written as ${X}= EWE{{^\top}}$, where $E, W \in {{{\mathbb{R}}}^{n \times n}}$, $W$ is diagonal, and ${E}$ orthogonal so that ${E}{E}{{^\top}}= {E}{{^\top}}{E}= I$ where $I$ is the identity matrix.
The spectral theorem for symmetric matrices tells us that a matrix is orthogonally diagonalizable if and only if it is symmetric [@cauchy1829]. It is therefore often used to approximate covariance matrices, kernel functions and distance or similarity matrices.
However as previously stated, this paper is explicitly interested in problems where matrices—and thus the they represent—can also be antisymmetric, or even not have any particular symmetry pattern at all (asymmetry). In order to both use a unique embedding for entities and extend the expressiveness to asymmetric relations, researchers have generalised the notion of dot products to *scoring functions*, also known as *composition functions*, that allow more general combinations of embeddings. We briefly recall several examples of scoring functions in Table \[tab:scoring\], as well as the extension proposed in this paper.
These models propose different trade-offs between the three essential points:
- Expressiveness, which is the ability to represent symmetric, antisymmetric and more generally asymmetric relations.
- Scalability, which means keeping linear time and space complexity scoring function.
- Generalization, for which having unique entity embeddings is critical.
RESCAL [@Nickel2011] and NTN [@socher2013reasoning] are very expressive, but their scoring functions have quadratic complexity in the rank of the factorization. More recently the <span style="font-variant:small-caps;">HolE</span> model [@nickel_2016_holographic] proposes a solution that has quasi-linear complexity in time and linear space complexity. <span style="font-variant:small-caps;">DistMult</span> [@Yang2015] can be seen as a joint orthogonal diagonalization with real embeddings, hence handling only symmetric relations. Conversely, <span style="font-variant:small-caps;">TransE</span> [@bordes2013translating] handles symmetric relations to the price of strong constraints on its embeddings. The canonical-polyadic decomposition (CP) [@hitchcock-sum-1927] generalizes poorly with its different embeddings for entities as subject and as object.
We reconcile expressiveness, scalability and generalization by going back to the realm of well-studied matrix factorizations, and making use of complex linear algebra, a scarcely used tool in the machine learning community.
### Decomposition in the Complex Domain
We introduce a new decomposition of real square matrices using unitary diagonalization, the generalization of orthogonal diagonalization to complex matrices. This allows decomposition of *arbitrary* real square matrices with unique representations of rows and columns.
Let us first recall some notions of complex linear algebra as well as specific cases of diagonalization of real square matrices, before building our proposition upon these results.
A complex-valued vector $x\in{{\mathbb{C}}}^K$, with $x={\mathrm{Re}}(x) + i{\mathrm{Im}}(x)$ is composed of a real part ${\mathrm{Re}}(x)\in{{\mathbb{R}}}^K$ and an imaginary part ${\mathrm{Im}}(x)\in{{\mathbb{R}}}^K$, where $i$ denotes the square root of $-1$. The conjugate $\overline{x}$ of a complex vector inverts the sign of its imaginary part: $\overline{x}={\mathrm{Re}}(x) - i{\mathrm{Im}}(x)$.
Conjugation appears in the usual dot product for complex numbers, called the *Hermitian* product, or *sesquilinear* form, which is defined as: $$\begin{aligned}
\left< u,v \right> &:=& \bar{u}{^\top}v\notag\\
&=&\quad\enspace {\mathrm{Re}}(u){{^\top}}{\mathrm{Re}}(v) + {\mathrm{Im}}(u){{^\top}}{\mathrm{Im}}(v) \notag\\
&& +i({\mathrm{Re}}(u){{^\top}}{\mathrm{Im}}(v) - {\mathrm{Im}}(u){{^\top}}{\mathrm{Re}}(v) )\notag\,.
\label{eqn:sesquilinear-dot}\end{aligned}$$
A simple way to justify the Hermitian product for composing complex vectors is that it provides a valid topological norm in the induced vector space. For example, $\bar{x}{^\top}x=0$ implies $x=0$ while this is not the case for the bilinear form $x {^\top}x$ as there are many complex vectors $x$ for which $x{^\top}x=0$.
This yields an interesting property of the Hermitian product concerning the order of the involved vectors: $\left< u, v \right> = \overline{\left< v, u \right>}$, meaning that the real part of the product is symmetric, while the imaginary part is antisymmetric. For matrices, we shall write ${X}^* \in {{\mathbb{C}}}^{n \times m}$ for the conjugate-transpose ${X}^*= (\overline{{X}}){{^\top}}= \overline{{X}{{^\top}}}$. The conjugate transpose is also often written ${X}^\dagger$ or $X^{\mathrm{H}}$.
A complex square matrix ${X}\in {{{\mathbb{C}}}^{n \times n}}$ is unitarily diagonalizable if it can be written as ${X}= {E}{W}{E}^*$, where ${E}, {W}\in {{{\mathbb{C}}}^{n \times n}}$, ${W}$ is diagonal, and ${E}$ is unitary such that ${E}{E}^* = {E}^*{E}= I$.
A complex square matrix ${X}$ is normal if it commutes with its conjugate-transpose so that ${X}{X}^* = {X}^*{X}$.
We can now state the spectral theorem for normal matrices.
\[spectral\_thm\]
Let ${X}$ be a complex square matrix. Then ${X}$ is unitarily diagonalizable if and only if ${X}$ is normal.
It is easy to check that all real symmetric matrices are normal, and have pure real eigenvectors and eigenvalues. But the set of purely real normal matrices also includes all real antisymmetric matrices (useful to model hierarchical relations such as ), as well as all real orthogonal matrices (including permutation matrices), and many other matrices that are useful to represent binary relations, such as assignment matrices which represent bipartite graphs. However, far from all matrices expressed as ${X}= {E}{W}{E}^*$ are purely real, and Equation (\[observation-model0\]) requires the scores $X$ to be purely real.
As we only focus on *real* square matrices in this work, let us summarize all the cases where ${X}$ is real square and ${X}= {E}{W}{E}^*$ if $X$ is unitarily diagonalizable, where ${E},{W}\in {{{\mathbb{C}}}^{n \times n}}$, ${W}$ is diagonal and ${E}$ is unitary:
- ${X}$ is symmetric if and only if ${X}$ is orthogonally diagonalizable and ${E}$ and ${W}$ are purely real.
- ${X}$ is normal and non-symmetric if and only if ${X}$ is unitarily diagonalizable and ${E}$ and ${W}$ are *not* both purely real.
- ${X}$ is not normal if and only if ${X}$ is not unitarily diagonalizable.
We generalize all three cases by showing that, for any ${X}\in {{{\mathbb{R}}}^{n \times n}}$, there exists a unitary diagonalization in the complex domain, of which the real part equals ${X}$:
$${X}= {\mathrm{Re}}({E}{W}{E}^* )\,.
\label{eqn:proj_eigen_dec}$$
In other words, the unitary diagonalization is projected onto the real subspace.
\[main\_thm\]
Suppose ${X}\in {{{\mathbb{R}}}^{n \times n}}$ is a real square matrix. Then there exists a normal matrix ${Z}\in {{{\mathbb{C}}}^{n \times n}}$ such that $ {\mathrm{Re}}({Z}) = {X}$.
Let ${Z}:= {X}+ i{X}{{^\top}}$. Then $$\begin{aligned}
{Z}^* = {X}{{^\top}}- i{X}= -i(i{X}{{^\top}}+ {X}) = -i{Z}\,,\end{aligned}$$ so that $$\begin{aligned}
{Z}{Z}^* = {Z}(-i{Z}) = (-i{Z}){Z}= {Z}^*{Z}\,.\end{aligned}$$ Therefore $Z$ is normal.
Note that there also exists a normal matrix ${Z}= {X}{{^\top}}+ i{X}$ such that $ {\mathrm{Im}}({Z}) = {X}$.
Following Theorem \[spectral\_thm\] and Theorem \[main\_thm\], any real square matrix can be written as the real part of a complex diagonal matrix through a unitary change of basis.
\[cor\_real\_diag\]
Suppose ${X}\in {{{\mathbb{R}}}^{n \times n}}$ is a real square matrix. Then there exist ${E},{W}\in {{{\mathbb{C}}}^{n \times n}}$, where ${E}$ is unitary, and ${W}$ is diagonal, such that ${X}= {\mathrm{Re}}({E}{W}{E}^* )$.
From Theorem \[main\_thm\], we can write ${X}={\mathrm{Re}}({Z}) $, where ${Z}$ is a normal matrix, and from Theorem \[spectral\_thm\], ${Z}$ is unitarily diagonalizable.
Applied to the knowledge graph completion setting, the rows of $E$ here are vectorial representations of the entities corresponding to rows and columns of the relation score matrix ${X}$. The score for the relation holding true between entities $s$ and $o$ is hence $$x_{so} = {\mathrm{Re}}(e_s{{^\top}}W \bar{e}_o)$$ where $e_s, e_o\in{{\mathbb{C}}}^n$ and $W \in{{\mathbb{C}}}^{n \times n}$ is diagonal. For a given entity, its subject embedding vector is the complex conjugate of its object embedding vector.
To illustrate this difference of expressiveness with respect to real-valued embeddings, let us consider two complex embeddings $e_s, e_o \in {{\mathbb{C}}}$ of dimension 1, with arbitrary values: $e_s = 1 - 2i$, and $e_o = -3 + i$; as well as their real-valued, twice-bigger counterparts: $e'_s = { \left(\begin{smallmatrix}1\\-2\end{smallmatrix}\right)}\in {{\mathbb{R}}}^2$ and $e'_o = { \left(\begin{smallmatrix}-3\\1\end{smallmatrix}\right)}\in {{\mathbb{R}}}^2$. In the real-valued case, that corresponds to the <span style="font-variant:small-caps;">DistMult</span> model [@Yang2015], the score is $x_{so} = e{^{\prime\top}}_s W' e'_o$. Figure \[fig:complex\_vs\_real\_decomp\] represents the heatmaps of the scores $x_{so}$ and $x_{os}$, as a function of $W \in {{\mathbb{C}}}$ in the complex-valued case, and as a function of $W' \in {{\mathbb{R}}}^2$ diagonal in the real-valued case. In the real-valued case, that is symmetric in the subject and object entities, the scores $x_{so}$ and $x_{os}$ are equal for any value of $W' \in {{\mathbb{R}}}^2$ diagonal. Whereas in the complex-valued case, the variation of $W \in {{\mathbb{C}}}$ allows to score $x_{so}$ and $x_{os}$ with any desired pair of values.
![Left: Scores $x_{so}= {\mathrm{Re}}(e_s{{^\top}}W \bar{e}_o)$ (top) and $x_{os}= {\mathrm{Re}}(e_o{{^\top}}W \overline{e}_s)$ (bottom) for the proposed complex-valued decomposition, plotted as a function of $W \in {{\mathbb{C}}}$, for fixed entity embeddings $e_s = 1 - 2i$, and $e_o = -3 + i$. Right: Scores $x_{so}= e{^{\prime\top}}_s W' e'_o$ (top) and $x_{os}= e{^{\prime\top}}_o W' e'_s$ (bottom) for the corresponding real-valued decomposition with the same number of free real-valued parameters (*i.e.* in twice the dimension), plotted as a function of $W' \in {{\mathbb{R}}}^2$ diagonal, for fixed entity embeddings $e'_s = \usebox{\vecs}$ and $e'_o = \usebox{\veco}$. By varying $W \in {{\mathbb{C}}}$, the proposed complex-valued decomposition can attribute any pair of scores to $x_{so}$ and $x_{os}$, whereas $x_{so} = x_{os}$ for all $W' \in {{\mathbb{R}}}^2$ with the real-valued decomposition.[]{data-label="fig:complex_vs_real_decomp"}](score_map_complex2.png "fig:"){width="0.49\linewidth"} ![Left: Scores $x_{so}= {\mathrm{Re}}(e_s{{^\top}}W \bar{e}_o)$ (top) and $x_{os}= {\mathrm{Re}}(e_o{{^\top}}W \overline{e}_s)$ (bottom) for the proposed complex-valued decomposition, plotted as a function of $W \in {{\mathbb{C}}}$, for fixed entity embeddings $e_s = 1 - 2i$, and $e_o = -3 + i$. Right: Scores $x_{so}= e{^{\prime\top}}_s W' e'_o$ (top) and $x_{os}= e{^{\prime\top}}_o W' e'_s$ (bottom) for the corresponding real-valued decomposition with the same number of free real-valued parameters (*i.e.* in twice the dimension), plotted as a function of $W' \in {{\mathbb{R}}}^2$ diagonal, for fixed entity embeddings $e'_s = \usebox{\vecs}$ and $e'_o = \usebox{\veco}$. By varying $W \in {{\mathbb{C}}}$, the proposed complex-valued decomposition can attribute any pair of scores to $x_{so}$ and $x_{os}$, whereas $x_{so} = x_{os}$ for all $W' \in {{\mathbb{R}}}^2$ with the real-valued decomposition.[]{data-label="fig:complex_vs_real_decomp"}](score_map_real2.png "fig:"){width="0.49\linewidth"}
This decomposition however is non-unique, a simple example of this non-uniqueness is obtained by adding a purely imaginary constant to the eigenvalues. Let $X \in {{{\mathbb{R}}}^{n \times n}}$, and $X = {\mathrm{Re}}(EWE^*)$ where $E$ is unitary, $W$ is diagonal. Then for any real constant $c \in {{\mathbb{R}}}$ we have: $$\begin{aligned}
X &=& {\mathrm{Re}}(E(W + icI)E^*)\notag\\
&=& {\mathrm{Re}}(EWE^* + ic EIE^*)\notag\\
&=& {\mathrm{Re}}(EWE^* + icI)\notag\\
&=& {\mathrm{Re}}(EWE^*)\,.\notag\\\end{aligned}$$
In general, there are many other possible couples of matrices $E$ and $W$ that preserve the real part of the decomposition. In practice however this is no synonym of low generalization abilities, as many effective matrix and tensor decomposition methods used in machine learning lead to non-unique solutions [@paatero1994positive; @Nickel2011]. In this case also, the learned representations prove useful as shown in the experimental section.
Low-Rank Decomposition
----------------------
Addressing knowledge graph completion with data-driven approaches assumes that there is a sufficient regularity in the observed data to generalize to unobserved facts. When formulated as a matrix completion problem, as it is the case in this section, one way of implementing this hypothesis is to make the assumption that the matrix has a low rank or approximately low rank. We first discuss the rank of the proposed decomposition, and then introduce the sign-rank and extend the bound developed on the rank to the sign-rank.
### Rank Upper Bound
First, we recall one definition of the rank of a matrix [@horn2012matrix].
The rank of an $m$-by-$n$ complex matrix ${\mathrm{rank}}({X})={\mathrm{rank}}({X}{{^\top}})=k$, if ${X}$ has exactly $k$ linearly independent columns.
Also note that if ${X}$ is diagonalizable so that ${X}= {E}{W}{E}^{-1}$ with ${\mathrm{rank}}({X})=k$, then ${W}$ has $k$ non-zero diagonal entries for some diagonal ${W}$ and some invertible matrix ${E}$. From this it is easy to derive a known additive property of the rank: $$\label{eq:rank_add}
{\mathrm{rank}}(B+C) \leq {\mathrm{rank}}(B) + {\mathrm{rank}}(C)$$ where $B,C \in {{\mathbb{C}}}^{m \times n}$.
We now show that any rank $k$ real square matrix can be reconstructed from a $2k$-dimensional unitary diagonalization.
\[cor\_real\_rank\] Suppose ${X}\in {{{\mathbb{R}}}^{n \times n}}$ and $rank({X}) = k$. Then there exist ${E}\in {{\mathbb{C}}}^{n \times 2k}$ such that the columns of ${E}$ form an orthonormal basis of ${{\mathbb{C}}}^{2k}$, ${W}\in {{\mathbb{C}}}^{2k \times 2k}$ is diagonal, and ${X}= {\mathrm{Re}}({E}{W}{E}^* )$.
Consider the complex square matrix ${Z}:= {X}+ i{X}{{^\top}}$. We have ${\mathrm{rank}}(i{X}{{^\top}}) = {\mathrm{rank}}({X}{{^\top}}) = {\mathrm{rank}}({X}) = k$.
From Equation (\[eq:rank\_add\]), ${\mathrm{rank}}({Z}) \leq {\mathrm{rank}}({X}) + {\mathrm{rank}}(i{X}{{^\top}}) = 2k$.
The proof of Theorem \[main\_thm\] shows that ${Z}$ is normal. Thus ${Z}= {E}{W}{E}^* $ with ${E}\in {{\mathbb{C}}}^{n \times 2k}$, ${W}\in {{\mathbb{C}}}^{2k \times 2k}$ where the columns of ${E}$ form an orthonormal basis of ${{\mathbb{C}}}^{2k}$, and ${W}$ is diagonal.
Since ${E}$ is not necessarily square, we replace the unitary requirement of Corollary \[cor\_real\_diag\] by the requirement that its columns form an orthonormal basis of its smallest dimension, $2k$. Also, given that such decomposition always exists in dimension $n$ (Theorem \[main\_thm\]), this upper bound is not relevant when ${\mathrm{rank}}({X}) \geq \frac{n}{2}$.
### Sign-Rank Upper Bound
Since we encode the truth values of each fact with $\pm 1$, we deal with square *sign matrices*: $Y \in \{-1,1\}^{n \times n}$. Sign matrices have an alternative rank definition, the *sign-rank*.
The sign-rank ${\mathrm{rank}_{\pm}}(Y)$ of an $m$-by-$n$ sign matrix $Y,$ is the rank of the $m$-by-$n$ real matrix of least rank that has the same sign-pattern as $Y,$ so that $${\mathrm{rank}_{\pm}}(Y) := \min_{{X}\in {{\mathbb{R}}}^{m \times n}} \{{\mathrm{rank}}({X})\,|\, {\mathrm{sign}}({X}) = Y \}\,,$$ where ${\mathrm{sign}}({X})_{ij} = {\mathrm{sign}}(x_{ij})$.
We define the *sign function* of $c \in {{\mathbb{R}}}$ as $${\mathrm{sign}}(c) = \left\{
\begin{array}{ll}
1 &\mbox{if } c \geq 0\\
-1 & \mbox{otherwise}\notag
\end{array}
\right.$$ where the value $c=0$ is here arbitrarily assigned to $1$ to allow zero entries in $X$, conversely to the stricter usual definition of the sign-rank.
To make generalization possible, we hypothesize that the true matrix $Y$ has a low sign-rank, and thus can be reconstructed by the sign of a low-rank score matrix $X$. The low sign-rank assumption is theoretically justified by the fact that the sign-rank is a natural complexity measure of sign matrices and is linked to learnability [@alon2016sign] and empirically confirmed by the wide success of factorization models [@nickel_2016_review].
Using Corollary \[cor\_real\_rank\], we can now show that any square sign matrix of sign-rank $k$ can be reconstructed from a rank $2k$ unitary diagonalization.
\[cor\_sign\_rank\] Suppose $Y \in \{-1,1\}^{n \times n}$, ${\mathrm{rank}_{\pm}}(Y)=k$. Then there exists ${E}\in {{\mathbb{C}}}^{n \times 2k}$, ${W}\in {{\mathbb{C}}}^{2k \times 2k}$ where the columns of ${E}$ form an orthonormal basis of ${{\mathbb{C}}}^{2k}$, and ${W}$ is diagonal, such that $Y = {\mathrm{sign}}({\mathrm{Re}}({E}{W}{E}^* ))$.
By definition, if ${\mathrm{rank}_{\pm}}(Y)=k$, there exists a real square matrix ${X}$ such that ${\mathrm{rank}}({X})=k$ and ${\mathrm{sign}}({X})=Y$. From Corollary \[cor\_real\_rank\], ${X}= {\mathrm{Re}}({E}{W}{E}^* )$ where ${E}\in {{\mathbb{C}}}^{n \times 2k}$, ${W}\in {{\mathbb{C}}}^{2k \times 2k}$ where the columns of ${E}$ form an orthonormal basis of ${{\mathbb{C}}}^{2k}$, and ${W}$ is diagonal.
Previous attempts to approximate the sign-rank in relational learning did not use complex numbers. They showed the existence of compact factorizations under conditions on the sign matrix [@nickel2014reducing], or only in specific cases [@bouchard2015approximate]. In contrast, our results show that if a square sign matrix has sign-rank $k$, then it can be exactly decomposed through a $2k$-dimensional unitary diagonalization.
Although we can only show the existence of a complex decomposition of rank $2k$ for a matrix with sign-rank $k$, the sign rank of $Y$ is often *much* lower than the rank of $Y$, as we do not know any matrix $Y \in \{-1,1\}^{n \times n}$ for which ${\mathrm{rank}_{\pm}}(Y) > \sqrt{n}$ [@alon2016sign]. For example, the $n \times n$ identity matrix has rank $n$, but its sign-rank is only 3! By swapping the columns $2j$ and $2j-1$ for $j$ in $1,\ldots,\frac{n}{2}$, the identity matrix corresponds to the relation `marriedTo`, a relation known to be hard to factorize over the reals [@nickel2014reducing], since the rank is invariant by row/column permutations. Yet our model can express it at most in rank 6, for any $n$.
Hence, by enforcing a low-rank $K \ll n$ on $E W E^*$, individual relation scores $x_{so}= {\mathrm{Re}}(e_s{^\top}W \bar{e}_o)$ between entities $s$ and $o$ can be efficiently predicted, as $e_s, e_o\in{{\mathbb{C}}}^K$ and $W \in{{\mathbb{C}}}^{K \times K}$ is diagonal.
Finding the $K$ that matches the sign-rank of $Y$ corresponds to finding the smallest $K$ that brings the 0–1 loss on $X$ to $0$, as link prediction can be seen as binary classification of the facts. In practice, and as classically done in machine learning to avoid this NP-hard problem, we use a continuous surrogate of the 0–1 loss, in this case the logistic loss as described in Section \[sec:algo\], and validate models on different values of $K$, as described in Section \[sec:expe\].
### Rank Bound Discussion
Corollaries \[cor\_real\_rank\] and \[cor\_sign\_rank\] use the aforementioned subadditive property of the rank to derive the $2k$ upper bound. Let us give an example for which this bound is strictly greater than $k$.
Consider the following $2$-by-$2$ sign matrix: $$Y=
\begin{bmatrix}
-1 & -1\\
\phantom{-}1 & \phantom{-}1\\
\end{bmatrix}\,.$$
Not only is this matrix not normal, but one can also easily check that there is no *real* normal $2$-by-$2$ matrix that has the same sign-pattern as $Y$. Clearly, $Y$ is a rank $1$ matrix since its columns are linearly dependent, hence its sign-rank is also $1$. From Corollary \[cor\_sign\_rank\], we know that there is a normal matrix whose real part has the same sign-pattern as $Y$, and whose rank is at most $2$.
However, there is no rank $1$ unitary diagonalization of which the real part equals $Y$. Otherwise we could find a 2-by-2 complex matrix ${Z}$ such that ${\mathrm{Re}}(z_{11}) < 0$ and ${\mathrm{Re}}(z_{22}) > 0$, where $z_{11} = e_1 w \bar{e}_1 = w |e_1|^2$, $z_{22} = e_2 w \bar{e}_2
= w |e_2|^2$, $e \in {{\mathbb{C}}}^2, w \in {{\mathbb{C}}}$. This is obviously unsatisfiable. This example generalizes to any $n$-by-$n$ square sign matrix that only has $-1$ on its first row and is hence rank 1, the same argument holds considering ${\mathrm{Re}}(z_{11}) < 0$ and ${\mathrm{Re}}(z_{nn}) > 0$.
This example shows that the upper bound on the rank of the unitary diagonalization showed in Corollaries \[cor\_real\_rank\] and \[cor\_sign\_rank\] can be strictly greater than $k$, the rank or sign-rank, of the decomposed matrix. However, there might be other examples for which the addition of an imaginary part could—additionally to making the matrix normal—create some linear dependence between the rows/columns and thus decrease the rank of the matrix, up to a factor of 2.\
**We summarize this section in three points:**
1. The proposed factorization encompasses all possible score matrices $X$ for a single binary relation.
2. By construction, the factorization is well suited to represent both symmetric and antisymmetric relations.
3. Relation patterns can be efficiently approximated with a low-rank factorization using complex-valued embeddings.
Extension to Multi-Relational Data {#sec:tens_case}
==================================
Let us now extend the previous discussion to models with multiple relations. Let ${\mathcal{R}}$ be the set of relations, with $|{\mathcal{R}}|=m$. We shall now write ${\mathbf{X}}\in {{\mathbb{R}}}^{m \times n \times n}$ for the score tensor, ${X}_r \in {{\mathbb{R}}}^{n \times n}$ for the score matrix of the relation $r \in {\mathcal{R}}$, and ${\mathbf{Y}}\in \{-1,1\}^{m \times n \times n}$ for the partially-observed sign tensor.
Given one relation $r \in {\mathcal{R}}$ and two entities $s,o$ $ \in {\mathcal{E}}$, the probability that the fact *r(s,o)* is true given by: $${P}(y_{rso}=1) = \sigma(x_{rso})=\sigma(\phi(r,s,o;\Theta))
\label{observation-model}$$ where $\phi$ is the scoring function of the model considered and $\Theta$ denotes the model parameters. We denote the set of all possible facts (or triples) for a knowledge graph by ${\mathcal{T}}= {\mathcal{R}}\times {\mathcal{E}}\times {\mathcal{E}}$. While the tensor ${\mathbf{X}}$ as a whole is unknown, we assume that we observe a set of true and false triples ${\Omega}= \{((r,s,o), y_{rso}) \,|\, (r,s,o) \in {\mathcal{T}}_{{\Omega}}\}$ where $y_{rso} \in \{-1,1\}$ and ${\mathcal{T}}_{{\Omega}}\subseteq{\mathcal{T}}$ is the set of observed triples. The goal is to find the probabilities of entries $y_{r's'o'}$ for a set of targeted unobserved triples $\{(r',s',o')\in {\mathcal{T}}\setminus {\mathcal{T}}_{{\Omega}}\}$.
Depending on the scoring function $\phi(r,s,o;\Theta)$ used to model the score tensor ${\mathbf{X}}$, we obtain different models. Examples of scoring functions are given in Table \[tab:scoring\].
Complex Factorization Extension to Tensors
------------------------------------------
The single-relation model is extended by jointly factorizing all the square matrices of scores into a $\mathrm{3^{rd}}$-order tensor ${\mathbf{X}}\in {{\mathbb{R}}}^{m \times n \times n}$, with a different diagonal matrix $W_r\in {{\mathbb{C}}}^{K \times K}$ for each relation $r$, and by sharing the entity embeddings $E \in {{\mathbb{C}}}^{n \times K}$ across all relations: $$\begin{aligned}
\phi(r,s,o;\Theta) &=& {\mathrm{Re}}({e}_s{{^\top}}W_{r} \bar{e}_o)\notag\\
&=& {\mathrm{Re}}(\sum_{k=1}^K w_{rk} {e}_{sk} \bar{e}_{ok})\notag\\
\label{eqn:complex-dot}
&=& {\mathrm{Re}}(\left<w_{r}, {e}_s, \bar{e}_o\right>)
\label{eqn:complex-dot1}\end{aligned}$$ where $K$ is the rank hyperparameter, $e_s, e_o \in {{\mathbb{C}}}^K$ are the rows in $E$ corresponding to the entities $s$ and $o$, $w_r = \mathrm{diag}(W_r) \in {{\mathbb{C}}}^K$ is a complex vector, and $\left<a,b,c\right> := \sum_k a_kb_kc_k$ is the component-wise multilinear dot product[^2]. For this scoring function, the set of parameters $\Theta$ is $\{e_i,w_r \in {{\mathbb{C}}}^K,
i \in {\mathcal{E}}, r \in {\mathcal{R}}\}$. This resembles the real part of a complex matrix decomposition as in the single-relation case discussed above. However, we now have a different vector of eigenvalues for every relation. Expanding the real part of this product gives: $$\begin{aligned}
{\mathrm{Re}}(\left<w_{r}, {e}_s, \bar{e}_o\right>) &=& \quad \left<{\mathrm{Re}}(w_r),{\mathrm{Re}}(e_s), {\mathrm{Re}}(e_o)\right>\notag\\
&&+ \left<{\mathrm{Re}}(w_r),{\mathrm{Im}}(e_s), {\mathrm{Im}}(e_o)\right> \notag\\
&&+ \left<{\mathrm{Im}}(w_r), {\mathrm{Re}}(e_s),{\mathrm{Im}}(e_o)\right> \notag\\
&&- \left<{\mathrm{Im}}(w_r),{\mathrm{Im}}(e_s),{\mathrm{Re}}(e_o)\right>\,.
\label{eqn:full_model}\end{aligned}$$
These equations provide two interesting views of the model:
- *Changing the representation*: Equation (\[eqn:complex-dot1\]) would correspond to <span style="font-variant:small-caps;">DistMult</span> with real embeddings (see Table \[tab:scoring\]), but handles asymmetry thanks to the complex conjugate of the object-entity embedding.
- *Changing the scoring function*: Equation (\[eqn:full\_model\]) only involves real vectors corresponding to the real and imaginary parts of the embeddings and relations.
By separating the real and imaginary parts of the relation embedding ${w}_r$ as shown in Equation (\[eqn:full\_model\]), it is apparent that these parts naturally act as weights on each latent dimension: ${\mathrm{Re}}(w_r)$ over the real part of $\left< e_o, e_s \right>$ which is symmetric, and ${\mathrm{Im}}(w)$ over the imaginary part of $\left< e_o, e_s \right>$ which is antisymmetric.
Indeed, the decomposition of each score matrix ${X}_r$ for each $r \in {\mathcal{R}}$ can be written as the sum of a symmetric matrix and an antisymmetric matrix. To see this, let us rewrite the decomposition of each score matrix ${X}_r$ in matrix notation. We write the real part of matrices with primes ${E}' = {\mathrm{Re}}({E})$ and imaginary parts with double primes ${E}'' = {\mathrm{Im}}({E})$: $$\begin{aligned}
{X}_r &=& {\mathrm{Re}}( {E}{W}_r {E}^* )\notag\\
&=& {\mathrm{Re}}( ({E}'+ i {E}'') ({W}'_r+ i {W}''_r) ({E}'- i {E}''){^\top})\notag\\
&=& ({E}' {W}_r' {E}'^{^\top}+ {E}'' {W}_r' {E}''^{^\top})
+ ({E}' {W}_r'' {E}''^{^\top}- {E}'' {W}_r'' {E}'^{^\top})\,.
\label{eq:as_symm_antisym_sum}\end{aligned}$$ It is trivial to check that the matrix ${E}' {W}_r' {E}'^{^\top}+ {E}'' {W}_r' {E}''^{^\top}$ is symmetric and that the matrix ${E}' {W}_r'' {E}''^{^\top}- {E}'' {W}_r'' {E}'^{^\top}$ is antisymmetric. Hence this model is well suited to model jointly symmetric and antisymmetric relations between pairs of entities, while still using the same entity representations for subjects and objects. When learning, it simply needs to collapse ${W}''_r = {\mathrm{Im}}({W}_r)$ to zero for symmetric relations $r \in {\mathcal{R}}$, and ${W}'_r = {\mathrm{Re}}({W}_r)$ to zero for antisymmetric relations $r \in {\mathcal{R}}$, as ${X}_r$ is indeed symmetric when ${W}_r$ is purely real, and antisymmetric when ${W}_r$ is purely imaginary.
From a geometrical point of view, each relation embedding $w_r$ is an anisotropic scaling of the basis defined by the entity embeddings $E$, followed by a projection onto the real subspace.
Existence of the Tensor Factorization
-------------------------------------
Let us first discuss the existence of the multi-relational model where the rank of the decomposition $K \leq n$, which relates to simultaneous unitary decomposition.
A family of matrices $X_1,\ldots,X_m \in {{{\mathbb{C}}}^{n \times n}}$ is simultaneously unitarily diagonalizable, if there is a single unitary matrix $E \in {{{\mathbb{C}}}^{n \times n}}$, such that $X_i = EW_iE^*$ for all $i$ in $1,\ldots,m$, where $W_i \in {{{\mathbb{C}}}^{n \times n}}$ are diagonal.
A family of normal matrices $X_1,\ldots,X_m \in {{{\mathbb{C}}}^{n \times n}}$ is a commuting family of normal matrices, if $X_i X_j^* = X_i^* X_j$, for all $i,j$ in $1,\ldots,m$.
\[thm:commuting\] Suppose $\mathcal{F}$ is the family of matrices $X_1,$ $\ldots$ $,X_m \in {{{\mathbb{C}}}^{n \times n}}$. Then $\mathcal{F}$ is a commuting family of normal matrices if and only if $\mathcal{F}$ is simultaneously unitarily diagonalizable.
To apply Theorem \[thm:commuting\] to the proposed factorization, we would have to make the hypothesis that the relation score matrices ${X}_r$ are a commuting family, which is too strong a hypothesis. Actually, the model is slightly different since we take only the real part of the tensor factorization. In the single-relation case, taking only the real part of the decomposition rids us of the normality requirement of Theorem \[spectral\_thm\] for the decomposition to exist, as shown in Theorem \[main\_thm\].
In the multiple-relation case, it is an open question whether taking the real part of the simultaneous unitary diagonalization will enable us to decompose families of arbitrary real square matrices—that is with a single unitary matrix $E$ that has *at most* $n$ columns. Though it seems unlikely, we could not find a counter-example yet.
However, by letting the rank of the tensor factorization $K$ to be greater than $n$, we can show that the proposed tensor decomposition exists for families of arbitrary real square matrices, by simply concatenating the decomposition of Theorem \[main\_thm\] of each real square matrix $X_i$.
\[thm:main\_thm\_tensors\] Suppose $X_1,\ldots,X_m \in {{{\mathbb{R}}}^{n \times n}}$. Then there exists $E \in {{\mathbb{C}}}^{n \times nm}$ and $W_i \in {{\mathbb{C}}}^{nm \times nm}$ are diagonal, such that $X_i = {\mathrm{Re}}(E W_i E^*)$ for all $i$ in $1,\ldots,m$.
From Theorem \[main\_thm\] we have $X_i = {\mathrm{Re}}(E_i W_i E_i^*)$, where $W_i \in{{\mathbb{C}}}^{n \times n}$ is diagonal, and each $E_i \in{{\mathbb{C}}}^{n \times n}$ is unitary for all $i$ in $1,\ldots,m$.
Let $E = \left[ E_1 \ldots E_m \right]$, and $$\begin{aligned}
\Lambda_i &= \begin{bmatrix}
\mathbf{0}^{((i-1)n) \times ((i-1)n)} & &\\
& W_i&& \\
& & \mathbf{0}^{((m-i)n) \times ((m-i)n)}\notag
\end{bmatrix}\end{aligned}$$ where $\mathbf{0}^{l \times l}$ the zero $l \times l$ matrix. Therefore $X_i = {\mathrm{Re}}(E \Lambda_i E^*)$ for all $i$ in $1,\ldots,m$.
By construction, the rank of the decomposition is at most $nm$. When $m \leq n$, this bound actually matches the general upper bound on the rank of the canonical polyadic (CP) decomposition [@hitchcock-sum-1927; @kruskal1989rank]. Since $m$ corresponds to the number of relations and $n$ to the number of entities, $m$ is always smaller than $n$ in real world knowledge graphs, hence the bound holds in practice.
Though when it comes to relational learning, we might expect the actual rank to be much lower than $nm$ for two reasons. The first one, as discussed above, is that we are dealing with sign tensors, hence the rank of the matrices ${X}_r$ need only match the sign-rank of the partially-observed matrices $Y_r$. The second one is that the matrices are related to each other, as they all represent the same entities in different relations, and thus benefit from sharing latent dimensions. As opposed to the construction exposed in the proof of Theorem \[thm:main\_thm\_tensors\], where other relations dimensions are canceled out. In practice, the rank needed to generalize well is indeed much lower than $nm$ as we show experimentally in Figure \[fig:mrr\_vs\_rank\].
Also, note that with the construction of the proof of Theorem \[thm:main\_thm\_tensors\], the matrix $E = \left[ E_1 \ldots E_m \right]$ is not unitary any more. However the unitary constraints in the matrix case serve only the proof of existence, which is just one solution among the infinite ones of same rank. In practice, imposing orthonormality is essentially a numerical commodity for the decomposition of dense matrices, through iterative methods for example [@saad1992numerical]. When it comes to matrix and tensor completion, and thus generalisation, imposing such constraints is more of a numerical hassle than anything else, especially for gradient methods. As there is no apparent link between orthonormality and generalisation properties, we did not impose these constraints when learning this model in the following experiments.
Algorithm {#sec:algo}
=========
Algorithm \[SGDC\] describes stochastic gradient descent (SGD) to learn the proposed multi-relational model with the AdaGrad learning-rate updates [@duchi2011adaptive]. We refer to the proposed model as <span style="font-variant:small-caps;">ComplEx</span>, for Complex Embeddings. We expose a version of the algorithm that uses only real-valued vectors, in order to facilitate its implementation. To do so, we use separate real-valued representations of the real and imaginary parts of the embeddings.
These real and imaginary part vectors are initialized with vectors having a zero-mean normal distribution with unit variance. If the training set $\Omega$ contains only positive triples, negatives are generated for each batch using the *local closed-world assumption* as in @bordes2013translating. That is, for each triple, we randomly change either the subject or the object, to form a negative example. In this case the parameter $\eta>0$ sets the number of negative triples to generate for each positive triple. Collision with positive triples in $\Omega$ is not checked, as it occurs rarely in real world knowledge graphs as they are largely sparse, and may also be computationally expensive.
Squared gradients are accumulated to compute AdaGrad learning rates, then gradients are updated. Every $s$ iterations, the parameters $\Theta$ are evaluated over the evaluation set $\Omega_v$ ([*evaluate\_AP\_or\_MRR*]{}$(\Omega_v;\Theta)$ function in Algorithm \[SGDC\]). If the data set contains both positive and negative examples, average precision (AP) is used to evaluate the model. If the data set contains only positives, then mean reciprocal rank (MRR) is used as average precision cannot be computed without true negatives. The optimization process is stopped when the measure considered decreases compared to the last evaluation (early stopping).
Bern($p$) is the Bernoulli distribution, the [*one\_random\_sample*]{}$({\mathcal{E}})$ function sample uniformly one entity in the set of all entities ${\mathcal{E}}$, and the [*sample\_batch\_of\_size\_b*]{}$(\Omega,b)$ function sample $b$ true and false triples uniformly at random from the training set $\Omega$.
For a given embedding size ${K}$, let us rewrite Equation (\[eqn:full\_model\]), by denoting the real part of embeddings with primes and the imaginary part with double primes: $e'_i = {\mathrm{Re}}(e_i)$, $e''_i = {\mathrm{Im}}(e_i)$, $w'_r = {\mathrm{Re}}(w_r)$, $w''_r = {\mathrm{Im}}(w_r)$. The set of parameters is $\Theta=\{e'_i,e''_i,w'_r,w''_r \in {{\mathbb{R}}}^K,
i \in {\mathcal{E}}, r \in {\mathcal{R}}\}$, and the scoring function involves only real vectors: $$\begin{aligned}
{999}
\phi(r,s,o;\Theta) = \quad &\left<w'_r,e'_s, e'_o\right> &&+ \left<w'_r,e''_s, e''_o\right> \notag\\
+ &\left<w''_r, e'_s, e''_o\right> &&- \left<w''_r,e''_s, e'_o\right>\end{aligned}$$ where each entity and each relation has two real embeddings.
Gradients are now easy to write: $$\begin{aligned}
{999}
\label{gradients}
{\nabla}_{e'_s} \phi(r,s,o;\Theta) &= (w'_r &&\odot e'_o) &&+ (w''_r &&\odot e''_o),\\
{\nabla}_{e''_s} \phi(r,s,o;\Theta) &= (w'_r &&\odot e''_o) &&- (w''_r &&\odot e'_o),\\
{\nabla}_{e'_o} \phi(r,s,o;\Theta) &= (w'_r &&\odot e'_s) &&- (w''_r &&\odot e''_s),\\
{\nabla}_{e''_o} \phi(r,s,o;\Theta) &= (w'_r &&\odot e''_s) &&+ (w''_r &&\odot e'_s),\\
{\nabla}_{w'_r} \phi(r,s,o;\Theta) &= (e'_s &&\odot e'_o) &&+ (e''_s &&\odot e''_o),\\
{\nabla}_{w''_r} \phi(r,s,o;\Theta) &= (e'_s &&\odot e''_o) &&- (e''_s &&\odot e'_o),\end{aligned}$$ where $\odot$ is the element-wise (Hadamard) product.
We optimized the negative log-likelihood of the logistic model described in Equation (\[observation-model\]) with $L^2$ regularization on the parameters $\Theta$: $$\begin{aligned}
\gamma(\Omega;\Theta) &=&
\sum_{((r,s,o),y) \in \Omega} \log( 1 + \exp(-y \phi(r,s,o;\Theta))) + \lambda ||\Theta||^2_2\
\label{eq:objective}\end{aligned}$$ where $\lambda \in {{\mathbb{R}}}_+$ is the regularization parameter.
To handle regularization, note that using separate representations for the real and imaginary parts does not change anything as the squared $L^2$-norm of a complex vector $v=v'+iv''$ is the sum of the squared modulus of each entry: $$\begin{aligned}
||v||^2_2 &=& \sum_j \sqrt{v_j^{\prime2} + v_j^{\prime\prime2}}^2\notag\\
&=& \sum_j v_j^{\prime2} + \sum_j v_j^{\prime\prime2}\notag\\
&=& ||v'||^2_2 + ||v''||^2_2\notag\,,\end{aligned}$$ which is actually the sum of the $L^2$-norms of the vectors of the real and imaginary parts.
We can finally write the gradient of $\gamma$ with respect to a *real* embedding $v$ for one triple $(r,s,o)$ and its truth value $y$: $$\begin{aligned}
{\nabla}_v \gamma(\{((r,s,o),y)\};\Theta) &=& -y \sigma(-y \phi(r,s,o;\Theta)) {\nabla}_v \phi(r,s,o;\Theta) + 2\lambda v\,.\end{aligned}$$
Training set $\Omega$, validation set $\Omega_v$, learning rate $\alpha\in \mathbb{R}_{++}$, rank $K\in \mathbb{Z}_{++}$, $L^2$ regularization factor $\lambda\in \mathbb{R}_{+}$, negative ratio $\eta\in \mathbb{Z}_{++}$, batch size $b\in \mathbb{Z}_{++}$, maximum iteration $m\in \mathbb{Z}_{++}$, validate every $s\in \mathbb{Z}_{++}$ iterations, AdaGrad regularizer $\epsilon = 10^{-8}$. Embeddings $e', e'', w', w''$. $e'_i\sim \mathcal{N}(\mathbf{0}^k, I^{k \times k})$ , $e''_i \sim \mathcal{N}(\mathbf{0}^k, I^{k \times k})$ for each $i \in \mathcal{E}$ $w'_i \sim \mathcal{N}(\mathbf{0}^k, I^{k \times k})$, $w''_i \sim \mathcal{N}(\mathbf{0}^k, I^{k \times k})$ for each $i \in \mathcal{R}$
$g_{e'_i} \gets \mathbf{0}^k$ , $g_{e''_i} \gets \mathbf{0}^k $ for each $i \in \mathcal{E}$ $g_{w'_i} \gets \mathbf{0}^k$ , $g_{w''_i} \gets \mathbf{0}^k $ for each $i \in \mathcal{R}$ $previous\_score \gets 0$ $\Omega_b \gets$ [*sample\_batch\_of\_size\_b*]{}$(\Omega,b)$ // Negative sampling: $\Omega_n \gets \{ \emptyset \}$ $e \gets$ [*one\_random\_sample*]{}$({\mathcal{E}})$ $\Omega_n \gets \Omega_n \cup \{((r,e,o),-1)\}$ $\Omega_n \gets \Omega_n \cup \{((r,s,e),-1)\}$ $\Omega_b \gets \Omega_b \cup \Omega_n$ // AdaGrad updates: $g_v \gets g_v + ({\nabla}_v \gamma(\{((r,s,o),y)\};\Theta))^2$ // Gradient updates: $v \gets v - \frac{\alpha}{g_v + \epsilon} {\nabla}_v \gamma(\{((r,s,o),y)\};\Theta)$ // Early stopping $current\_score \gets$ [*evaluate\_AP\_or\_MRR*]{}$(\Omega_v;\Theta)$ **break** $previous\_score \gets current\_score$ **return** $\Theta$
Experiments {#sec:expe}
===========
We evaluated the method proposed in this paper on both synthetic and real data sets. The synthetic data set contains both symmetric and antisymmetric relations, whereas the real data sets are standard link prediction benchmarks based on real knowledge graphs.
We compared <span style="font-variant:small-caps;">ComplEx</span> to state-of-the-art models, namely <span style="font-variant:small-caps;"><span style="font-variant:small-caps;">TransE</span></span> [@bordes2013translating], <span style="font-variant:small-caps;">DistMult</span> [@Yang2015], RESCAL [@Nickel2011] and also to the canonical polyadic decomposition (CP) [@hitchcock-sum-1927], to emphasize empirically the importance of learning unique embeddings for entities. For experimental fairness, we reimplemented these models within the same framework as the <span style="font-variant:small-caps;">ComplEx</span> model, using a Theano-based SGD implementation[^3] [@theano].
For the <span style="font-variant:small-caps;">TransE</span> model, results were obtained with its original max-margin loss, as it turned out to yield better results for this model only. To use this max-margin loss on data sets with observed negatives (Sections \[sec:synth\_task\] and \[sec:kinships\_umls\]), positive triples were replicated when necessary to match the number of negative triples, as described in @garcia2016combining. All other models are trained with the negative log-likelihood of the logistic model (Equation (\[eq:objective\])). In all the following experiments we used a maximum number of iterations $m=1000$, a batch size $b= \frac{|\Omega|}{100}$, and validated the models for early stopping every $s=50$ iterations.
Synthetic Task {#sec:synth_task}
--------------
To assess our claim that <span style="font-variant:small-caps;">ComplEx</span> can accurately model jointly symmetry and antisymmetry, we randomly generated a knowledge graph of two relations and 30 entities. One relation is entirely symmetric, while the other is completely antisymmetric. This data set corresponds to a $2 \times 30 \times 30$ tensor. Figure \[fig:symmetry\_example\] shows a part of this randomly generated tensor, with a symmetric slice and an antisymmetric slice, decomposed into training, validation and test sets. To ensure that all test values are predictable, the upper triangular parts of the matrices are always kept in the training set, and the diagonals are unobserved. We conducted a 5-fold cross-validation on the lower-triangular matrices, using the upper-triangular parts plus 3 folds for training, one fold for validation and one fold for testing. Each training set contains 1392 observed triples, whereas validation and test sets contain 174 triples each.
Figure \[fig:exp\_sym\_antisym\] shows the best cross-validated average precision (area under the precision-recall curve) for different factorization models of ranks ranging up to 50. The regularization parameter $\lambda$ is validated in $\{$0.1, 0.03, 0.01, 0.003,0.001, 0.0003, 0.00001, 0.0$\}$ and the learning rate $\alpha$ was initialized to 0.5.
As expected, <span style="font-variant:small-caps;">DistMult</span> [@Yang2015] is not able to model antisymmetry and only predicts the symmetric relations correctly. Although <span style="font-variant:small-caps;">TransE</span> [@bordes2013translating] is not a symmetric model, it performs poorly in practice, particularly on the antisymmetric relation. RESCAL [@Nickel2011], with its large number of parameters, quickly overfits as the rank grows. Canonical Polyadic (CP) decomposition [@hitchcock-sum-1927] fails on both relations as it has to push symmetric and antisymmetric patterns through the entity embeddings. Surprisingly, only <span style="font-variant:small-caps;">ComplEx</span> succeeds even on such simple data.
![Parts of the training, validation and test sets of the generated experiment with one symmetric and one antisymmetric relation. Red pixels are positive triples, blue are negatives, and green missing ones. Top: Plots of the symmetric slice (relation) for the 10 first entities. Bottom: Plots of the antisymmetric slice for the 10 first entities.[]{data-label="fig:symmetry_example"}](exp_symmetric_train.png "fig:"){width="0.28\linewidth"} ![Parts of the training, validation and test sets of the generated experiment with one symmetric and one antisymmetric relation. Red pixels are positive triples, blue are negatives, and green missing ones. Top: Plots of the symmetric slice (relation) for the 10 first entities. Bottom: Plots of the antisymmetric slice for the 10 first entities.[]{data-label="fig:symmetry_example"}](exp_symmetric_valid.png "fig:"){width="0.28\linewidth"} ![Parts of the training, validation and test sets of the generated experiment with one symmetric and one antisymmetric relation. Red pixels are positive triples, blue are negatives, and green missing ones. Top: Plots of the symmetric slice (relation) for the 10 first entities. Bottom: Plots of the antisymmetric slice for the 10 first entities.[]{data-label="fig:symmetry_example"}](exp_symmetric_test.png "fig:"){width="0.28\linewidth"}
![Parts of the training, validation and test sets of the generated experiment with one symmetric and one antisymmetric relation. Red pixels are positive triples, blue are negatives, and green missing ones. Top: Plots of the symmetric slice (relation) for the 10 first entities. Bottom: Plots of the antisymmetric slice for the 10 first entities.[]{data-label="fig:symmetry_example"}](exp_antisymmetric_train.png "fig:"){width="0.28\linewidth"} ![Parts of the training, validation and test sets of the generated experiment with one symmetric and one antisymmetric relation. Red pixels are positive triples, blue are negatives, and green missing ones. Top: Plots of the symmetric slice (relation) for the 10 first entities. Bottom: Plots of the antisymmetric slice for the 10 first entities.[]{data-label="fig:symmetry_example"}](exp_antisymmetric_valid.png "fig:"){width="0.28\linewidth"} ![Parts of the training, validation and test sets of the generated experiment with one symmetric and one antisymmetric relation. Red pixels are positive triples, blue are negatives, and green missing ones. Top: Plots of the symmetric slice (relation) for the 10 first entities. Bottom: Plots of the antisymmetric slice for the 10 first entities.[]{data-label="fig:symmetry_example"}](exp_antisymmetric_test.png "fig:"){width="0.28\linewidth"}
{width="49.00000%"} {width="49.00000%"} {width="65.00000%"}
Real Fully-Observed Data Sets: Kinships and UMLS {#sec:kinships_umls}
------------------------------------------------
We then compare all models on two fully observed data sets, that contain both positive and negative triples, also called the *closed-world assumption*. The Kinships data set [@denham1973detection] describes the 26 different kinship relations of the Alyawarra tribe in Australia, among 104 individuals. The unified medical language system (UMLS) data set [@mccray2003upper] represents 135 medical concepts and diseases, linked by 49 relations describing their interactions. Metadata for the two data sets is summarized in Table \[tab:kinship\_and\_umls\_meta\].
Data set $|{\mathcal{E}}|$ $|{\mathcal{R}}|$ Total number of triples
---------- ------------------- ------------------- -------------------------
Kinships 104 26 281,216
UMLS 135 49 893,025
: Number of entities $|{\mathcal{E}}|$, relations $|{\mathcal{R}}|$, and observed triples (all are observed) for the Kinships and UMLS data sets.[]{data-label="tab:kinship_and_umls_meta"}
We performed a 10-fold cross-validation, keeping 8 for training, one for validation and one for testing. Figure \[fig:exp\_kinships\_umls\] shows the best cross-validated average precision for ranks ranging up to 50, and error bars show the standard deviation over the 10 runs. The regularization parameter $\lambda$ is validated in $\{$0.1, 0.03, 0.01, 0.003, 0.001, 0.0003, 0.00001, 0.0$\}$ and the learning rate $\alpha$ was initialized to 0.5.
![Average precision (AP) for each factorization rank ranging from 5 to 50 for different state-of-the-art models on the Kinships data set (top) and on the UMLS data set (bottom).[]{data-label="fig:exp_kinships_umls"}](kinship_ap_vs_rank.png "fig:"){width="75.00000%"} ![Average precision (AP) for each factorization rank ranging from 5 to 50 for different state-of-the-art models on the Kinships data set (top) and on the UMLS data set (bottom).[]{data-label="fig:exp_kinships_umls"}](umls_ap_vs_rank.png "fig:"){width="75.00000%"}
On both data sets <span style="font-variant:small-caps;">ComplEx</span>, RESCAL and CP are very close, with a slight advantage for <span style="font-variant:small-caps;">ComplEx</span> on Kinships, and for RESCAL on UMLS. <span style="font-variant:small-caps;">DistMult</span> performs poorly here as many relations are antisymmetric both in UMLS (causal links, anatomical hierarchies) and Kinships (being father, uncle or grand-father).
The fact that CP, RESCAL and <span style="font-variant:small-caps;">ComplEx</span> work so well on these data sets illustrates the importance of having an expressive enough model, as <span style="font-variant:small-caps;">DistMult</span> fails because of ; the power of the multilinear product—that is the tensor factorization approach—as <span style="font-variant:small-caps;">TransE</span> can be seen as a sum of bilinear products [@garcia2016combining]; but not yet the importance of having unique entity embeddings, as CP works well. We believe having separate subject and object-entity embeddings works well under the closed-world assumption, because of the amount of training data compared to the number of embeddings to learn. Though when only a fractions of the positive training examples are observed (as it is most often the case), we will see in the next experiments that enforcing unique entity embeddings is key to good generalization.
Real Sparse Data Sets: FB15K and WN18
-------------------------------------
Finally, we evaluated <span style="font-variant:small-caps;">ComplEx</span> on the FB15K and WN18 data sets, as they are well established benchmarks for the link prediction task. FB15K is a subset of Freebase [@Bollacker2008], a curated knowledge graph of general facts, whereas WN18 is a subset of WordNet [@fellbaum1998wordnet], a database featuring lexical relations between words. We used the same training, validation and test set splits as in @bordes2013translating. Table \[tab:fb15k\_wn18\_meta\] summarizes the metadata of the two data sets.
---------- ------------------- ------------------- ---------- ------------ --------
Data set $|{\mathcal{E}}|$ $|{\mathcal{R}}|$ Training Validation Test
WN18 40,943 18 141,442 5,000 5,000
FB15K 14,951 1,345 483,142 50,000 59,071
---------- ------------------- ------------------- ---------- ------------ --------
: Number of entities $|{\mathcal{E}}|$, relations $|{\mathcal{R}}|$, and observed triples in each split for the FB15K and WN18 data sets.[]{data-label="tab:fb15k_wn18_meta"}
### Experimental Setup
As both data sets contain only positive triples, we generated negative samples using the local closed-world assumption, as described in Section \[sec:algo\]. For evaluation, we measure the quality of the ranking of each test triple among all possible subject and object substitutions : $r(s',o)$ and $r(s,o')$, for each $s', o'$ in ${\mathcal{E}}$, as used in previous studies [@bordes2013translating; @nickel_2016_holographic]. Mean Reciprocal Rank (MRR) and Hits at $N$ are standard evaluation measures for these data sets and come in two flavours: raw and filtered. The filtered metrics are computed *after* removing all the other positive observed triples that appear in either training, validation or test set from the ranking, whereas the raw metrics do not remove these.
Since ranking measures are used, previous studies generally preferred a max-margin ranking loss for the task [@bordes2013translating; @nickel_2016_holographic]. We chose to use the negative log-likelihood of the logistic model—as described in the previous section—as it is a continuous surrogate of the sign-rank, and has been shown to learn compact representations for several important relations, especially for transitive relations [@bouchard2015approximate]. As previously stated, we tried both losses in preliminary work, and indeed training the models with the log-likelihood yielded better results than with the max-margin ranking loss, especially on FB15K—except with <span style="font-variant:small-caps;">TransE</span>.
We report both filtered and raw MRR, and filtered Hits at 1, 3 and 10 in Table \[tab:fb15k\_wn18\_res\] for the evaluated models. The <span style="font-variant:small-caps;">HolE</span> model has recently been shown to be equivalent to <span style="font-variant:small-caps;">ComplEx</span> [@hayashi2017equivalence], we record the original results for <span style="font-variant:small-caps;">HolE</span> as reported in @nickel_2016_holographic and briefly discuss the discrepancy of results obtained with <span style="font-variant:small-caps;">ComplEx</span>. Reported results are given for the best set of hyper-parameters evaluated on the validation set for each model, after a distributed grid-search on the following values: ${K}\in \{$10, 20, 50, 100, 150, 200$\}$, $\lambda \in \{$0.1, 0.03, 0.01, 0.003, 0.001, 0.0003, 0.0$\}$, $\alpha \in \{$1.0, 0.5, 0.2, 0.1, 0.05, 0.02, 0.01$\}$, $\eta \in \{$1, 2, 5, 10$\}$ with $\lambda$ the $L^2$ regularization parameter, $\alpha$ the initial learning rate, and $\eta$ the number of negatives generated per positive training triple. We also tried varying the batch size but this had no impact and we settled with 100 batches per epoch. With the best hyper-parameters, training the <span style="font-variant:small-caps;">ComplEx</span> model on a single GPU (NVIDIA Tesla P40) takes 45 minutes on WN18 ($K=150, \eta=1$), and three hours on FB15K ($K=200, \eta=10$).
### Results {#sec:fb15k_res}
WN18 describes lexical and semantic hierarchies between concepts and contains many antisymmetric relations such as hypernymy, hyponymy, and being part of. Indeed, the <span style="font-variant:small-caps;">DistMult</span> and <span style="font-variant:small-caps;">TransE</span> models are outperformed here by <span style="font-variant:small-caps;">ComplEx</span> and <span style="font-variant:small-caps;">HolE</span>, which are on a par with respective filtered MRR scores of 0.941 and 0.938, which is expected as both models are equivalent.
Table \[tab:wn18\_detailed\_res\] shows the filtered MRR for the reimplemented models and each relation of WN18, confirming the advantage of <span style="font-variant:small-caps;">ComplEx</span> on antisymmetric relations while losing nothing on the others. 2D projections of the relation embeddings (Figures \[fig:pca12\] & \[fig:pca34\]) visually corroborate the results.
Relation name <span style="font-variant:small-caps;">ComplEx</span> RESCAL <span style="font-variant:small-caps;">DistMult</span> <span style="font-variant:small-caps;">TransE</span> CP
------------------------------- ------------------------------------------------------- -------- -------------------------------------------------------- ------------------------------------------------------ -------
hypernym **0.953** 0.935 0.791 0.446 0.109
hyponym **0.946** 0.932 0.710 0.361 0.009
member\_meronym **0.921** 0.851 0.704 0.418 0.019
member\_holonym **0.946** 0.861 0.740 0.465 0.134
instance\_hypernym **0.965** 0.833 0.943 0.961 0.233
instance\_hyponym **0.945** 0.849 0.940 0.745 0.040
has\_part **0.933** 0.879 0.753 0.426 0.035
part\_of **0.940** 0.888 0.867 0.455 0.094
member\_of\_domain\_topic **0.924** 0.865 0.914 0.861 0.007
synset\_domain\_topic\_of **0.930** 0.855 0.919 0.917 0.153
member\_of\_domain\_usage **0.917** 0.629 **0.917** 0.875 0.001
synset\_domain\_usage\_of **1.000** 0.541 **1.000** **1.000** 0.134
member\_of\_domain\_region **0.865** 0.632 0.635 **0.865** 0.001
synset\_domain\_region\_of 0.919 0.655 0.888 **0.986** 0.149
derivationally\_related\_form **0.946** 0.928 0.940 0.384 0.100
similar\_to **1.000** 0.001 **1.000** 0.244 0.000
verb\_group **0.936** 0.857 0.897 0.323 0.035
also\_see 0.603 0.302 **0.607** 0.279 0.020
: Filtered Mean Reciprocal Rank (MRR) for the models tested on each relation of the WordNet data set (WN18).[]{data-label="tab:wn18_detailed_res"}
On FB15K, the gap is much more pronounced and the <span style="font-variant:small-caps;">ComplEx</span> model largely outperforms <span style="font-variant:small-caps;">HolE</span>, with a filtered MRR of 0.692 and 59.9% of Hits at 1, compared to 0.524 and 40.2% for <span style="font-variant:small-caps;">HolE</span>. This difference of scores between the two models, though they have been proved to be equivalent [@hayashi2017equivalence], is due to the use of the aforementioned max-margin loss in the original <span style="font-variant:small-caps;">HolE</span> publication [@nickel_2016_holographic] that performs worse than the log-likelihood on this dataset, and to the generation of more than one negative sample per positive in these experiments. This has been confirmed and discussed in details by @trouillon2017comparison. The fact that <span style="font-variant:small-caps;">DistMult</span> yields fairly high scores (0.654 filtered MRR) is also due to the task itself and the evaluation measures used. As the dataset only involves true facts, the test set never includes the opposite facts $r(o,s)$ of each test fact $r(s,o)$ for *antisymmetric* relations—as the opposite fact is always false. Thus highly scoring the opposite fact barely impacts the rankings for antisymmetric relations. This is not the case in the fully observed experiments (Section \[sec:kinships\_umls\]), as the opposite fact is known to be false—for antisymmetric relations—and largely impacts the average precision of the <span style="font-variant:small-caps;">DistMult</span> model (Figure \[fig:exp\_kinships\_umls\]).
RESCAL, that represents each relation with a $K \times K$ matrix, performs well on WN18 as there are few relations and hence not so many parameters. On FB15K though, it probably overfits due to the large number of relations and thus the large number of parameters to learn, and performs worse than a less expressive model like <span style="font-variant:small-caps;">DistMult</span>. On both data sets, <span style="font-variant:small-caps;">TransE</span> and CP are largely left behind. This illustrates again the power of the multilinear product in the first case, and the importance of learning unique entity embeddings in the second. CP performs especially poorly on WN18 due to the small number of , which magnifies this subject/object difference.
Figure \[fig:mrr\_vs\_rank\] shows that the filtered MRR of the <span style="font-variant:small-caps;">ComplEx</span> model quickly converges on both data sets, showing that the low-rank hypothesis is reasonable in practice. The little gain of performances for ranks comprised between $50$ and $200$ also shows that <span style="font-variant:small-caps;">ComplEx</span> does not perform better because it has twice as many parameters for the same rank—the real and imaginary parts—compared to other linear space complexity models but indeed thanks to its better expressiveness.
![Best filtered MRR for <span style="font-variant:small-caps;">ComplEx</span> on the FB15K and WN18 data sets for different ranks. Increasing the rank gives little performance gain for ranks of $50-200$.[]{data-label="fig:mrr_vs_rank"}](mrr_vs_rank.png){width="0.70\linewidth"}
Best ranks were generally 150 or 200, in both cases scores were always very close for all models, suggesting there was no need to grid-search on higher ranks. The number of negative samples per positive sample also had a large influence on the filtered MRR on FB15K (up to +0.08 improvement from 1 to 10 negatives), but not much on WN18. On both data sets regularization was important (up to +0.05 on filtered MRR between $\lambda=0$ and optimal one). We found the initial learning rate to be very important on FB15K, while not so much on WN18. We think this may also explain the large gap of improvement <span style="font-variant:small-caps;">ComplEx</span> provides on this data set compared to previously published results—as <span style="font-variant:small-caps;">DistMult</span> results are also better than those previously reported [@Yang2015]—along with the use of the log-likelihood objective. It seems that in general AdaGrad is relatively insensitive to the initial learning rate, perhaps causing some overconfidence in its ability to tune the step size online and consequently leading to less efforts when selecting the initial step size.
Training time
-------------
As defended in , having a linear time and space complexity becomes critical when the dataset grows. To illustrate this, we report in the evolution of the filtered MRR on the validation set as a function of time, for the best set of validated hyper-parameters for each model. The convergence criterion used is the decrease of the validation filtered MRR—computed every 50 iterations—with a maximum number of iterations of 1000 (see ). All models have a linear complexity except for RESCAL that has a quadratic one in the rank of the decomposition, as it learns one matrix embedding for each relation $r \in {\mathcal{R}}$. Timings are measured on a single NVIDIA Tesla P40 GPU.
On WN18, all models reach convergence in a reasonable time, between 15 minutes and 1 hour and 20 minutes. The difference between RESCAL and the other models is not sharp there, first because its optimal embedding size ($K=50$) is lower compared to the other models. Secondly, there are only $|{\mathcal{R}}| = 18$ relations in WN18, hence the memory footprint of RESCAL is pretty similar to the other models—because it represents only relations with matrices and not entities. On FB15K, the difference is much more pronounced, as RESCAL optimal rank is similar to the other models; and with $|{\mathcal{R}}| = 1345$ relations, RESCAL has a much higher memory footprint, which implies more processor cache misses due to the uniformly-random nature of the SGD sampling.
RESCAL took more than four days to train on FB15K, whereas other models took between 40 minutes and 3 hours. While a few days might seem manageable, this could not be the case on larger data sets, as FB15K is but a small subset of Freebase that contains $|{\mathcal{R}}| = 35000$ relations [@Bollacker2008]. This experimentally supports our claim that linear complexity is required for scalability.
![Evolution of the filtered MRR on the validation set as a function of time, on WN18 (top) and FB15K (bottom) for each model for its best set of hyper-parameters. The best rank $K$ is reported in legend. Final black marker indicates that the maximum number of iterations (1000) has been reached (RESCAL on WN18, <span style="font-variant:small-caps;">TransE</span> on FB15K).[]{data-label="fig:training times"}](WN18_timings.png "fig:"){width="75.00000%"} ![Evolution of the filtered MRR on the validation set as a function of time, on WN18 (top) and FB15K (bottom) for each model for its best set of hyper-parameters. The best rank $K$ is reported in legend. Final black marker indicates that the maximum number of iterations (1000) has been reached (RESCAL on WN18, <span style="font-variant:small-caps;">TransE</span> on FB15K).[]{data-label="fig:training times"}](FB15K_timings.png "fig:"){width="75.00000%"}
### Influence of Negative Samples
We further investigated the influence of the number of negatives generated per positive training sample. In the previous experiment, due to computational limitations, the number of negatives per training sample, $\eta$, was validated over the set $\{1, 2, 5, 10\}$. On WN18 it proved to be of no help to have more than one generated negative per positive. Here we explore in which proportions increasing the number of generated negatives leads to better results on FB15K. To do so, we fixed the best validated $\lambda, {K}, \alpha$ obtained from the previous experiment. We then let $\eta$ vary in $\{1, 2, 5, 10, 20, 50, 100, 200\}$.
![Influence of the number of negative triples generated per positive training example on the filtered test MRR and on training time to convergence on FB15K for the <span style="font-variant:small-caps;">ComplEx</span> model with ${K}=200$, $\lambda=0.01$ and $\alpha=0.5$. Times are given relative to the training time with one negative triple generated per positive training sample ($=1$ on time scale).[]{data-label="fig:neg_ratio"}](neg_ratio_plot.png){width="0.70\linewidth"}
Figure \[fig:neg\_ratio\] shows the influence of the number of generated negatives per positive training triple on the performance of <span style="font-variant:small-caps;">ComplEx</span> on FB15K. Generating more negatives clearly improves the results up to 100 negative triples, with a filtered MRR of 0.737 and 64.8% of Hits@1, before decreasing again with 200 negatives, probably due to the too large class imbalance. The model also converges with fewer epochs, which compensates partially for the additional training time per epoch, up to 50 negatives. It then grows linearly as the number of negatives increases.
### WN18 Embeddings Visualization {#app:wn18_pca}
{width="80.00000%"} {width="80.00000%"}
{width="80.00000%"} {width="80.00000%"}
We used principal component analysis (PCA) to visualize embeddings of the relations of the WordNet data set (WN18). We plotted the four first components of the best <span style="font-variant:small-caps;">DistMult</span> and <span style="font-variant:small-caps;">ComplEx</span> model’s embeddings in Figures \[fig:pca12\] & \[fig:pca34\]. For the <span style="font-variant:small-caps;">ComplEx</span> model, we simply concatenated the real and imaginary parts of each embedding.
Most of WN18 relations describe hierarchies, and are thus antisymmetric. Each of these hierarchic relations has its inverse relation in the data set. For example: `hypernym` / `hyponym`, `part_of` / `has_part`, `synset_domain_topic_of` / `member_of_domain_topic`. Since <span style="font-variant:small-caps;">DistMult</span> is unable to model antisymmetry, it will correctly represent the nature of each pair of opposite relations, but not the direction of the relations. Loosely speaking, in the `hypernym` / `hyponym` pair the nature is sharing semantics, and the direction is that one entity generalizes the semantics of the other. This makes <span style="font-variant:small-caps;">DistMult</span> representing the opposite relations with very close embeddings. It is especially striking for the third and fourth principal component (Figure \[fig:pca34\]). Conversely, <span style="font-variant:small-caps;">ComplEx</span> manages to oppose spatially the opposite relations.
Related Work {#sec:rel_work}
============
We first discuss related work about complex-valued matrix and tensor decompositions, and then review other approaches for knowledge graph completion.
Complex Numbers
---------------
When factorization methods are applied, the representation of the decomposition is generally chosen in accordance with the data, despite the fact that most real square matrices only have eigenvalues in the complex domain. Indeed in the machine learning community, the data is usually real-valued, and thus eigendecomposition is used for symmetric matrices, or other decompositions such as (real-valued) singular value decomposition [@beltrami1873sulle], non-negative matrix factorization [@paatero1994positive], or canonical polyadic decomposition when it comes to tensors [@hitchcock-sum-1927].
Conversely, in signal processing, data is often complex-valued [@stoica2005spectral] and the complex-valued counterparts of these decompositions are then used. Joint diagonalization is also a much more common tool than in machine learning for decomposing sets of (complex) dense square matrices [@belouchrani1997blind; @de2001independent].
Some works on recommender systems use complex numbers as an encoding facility, to merge two real-valued relations, similarity and liking, into one single complex-valued matrix which is then decomposed with complex embeddings [@kunegis2012online; @xie2015link]. Still, unlike our work, it is not real data that is decomposed in the complex domain.
In deep learning, @danihelka2016associative proposed an LSTM extended with an associative memory based on complex-valued vectors for memorization tasks, and @hu2016initial a complex-valued neural network for speech synthesis. In both cases again, the data is first encoded in complex vectors that are then fed into the network.
Conversely to these contributions, this work suggests that processing real-valued data with complex-valued representation, through a projection onto the real-valued subspace, can be a very simple way of increasing the expressiveness of the model considered.
Knowledge Graph Completion
--------------------------
Many knowledge graphs have recently arisen, pushed by the W3C recommendation to use the resource description framework (RDF) [@cyganiak2014rdf] for data representation. Examples of such knowledge graphs include DBPedia [@dbpedia], Freebase [@Bollacker2008] and the Google Knowledge Vault [@Dong:2014:KnowledgeVault]. Motivating applications of knowledge graph completion include question answering [@bordes2014open] and more generally probabilistic querying of knowledge bases [@huang2009query; @krompass2014querying].
First approaches to relational learning relied upon probabilistic graphical models [@Getoor2007], such as bayesian networks [@Friedman1999] and markov logic networks [@richardson2006markov; @raedt2016statistical].
With the first embedding models, asymmetry of relations was quickly seen as a problem and asymmetric extensions of tensors were studied, mostly by either considering independent embeddings [@franz2009triplerank] or considering relations as matrices instead of vectors in the RESCAL model [@Nickel2011], or both [@Sutskever2009]. Direct extensions were based on uni-,bi- and trigram latent factors for triple data [@garcia2016combining], as well as a low-rank relation matrix [@Jenatton2012]. @bordes2014semantic propose a two-layer model where subject and object embeddings are first separately combined with the relation embedding, then each intermediate representation is combined into the final score.
Pairwise interaction models were also considered to improve prediction performances. For example, the Universal Schema approach [@riedel_2013_univschema] factorizes a 2D unfolding of the tensor (a matrix of entity pairs vs. relations) while @Welbl2016 extend this also to other pairs. @riedel_2013_univschema also consider augmenting the knowledge graph facts by exctracting them from textual data, as does @Toutanova2015. Injecting prior knowledge in the form of Horn clauses in the objective loss of the Universal Schema model has also been considered [@Rocktaschel2015]. @Chang2014 enhance the RESCAL model to take into account information about the entity types. For recommender systems (thus with different subject/object sets of entities), @baruch2014ternary proposed a non-commutative extension of the CP decomposition model. More recently, Gaifman models that learn neighborhood embeddings of local structures in the knowledge graph showed competitive performances [@niepert2016discriminative].
In the Neural Tensor Network (NTN) model, @socher2013reasoning combine linear transformations and multiple bilinear forms of subject and object embeddings to jointly feed them into a nonlinear neural layer. Its non-linearity and multiple ways of including interactions between embeddings gives it an advantage in expressiveness over models with simpler scoring function like <span style="font-variant:small-caps;">DistMult</span> or RESCAL. As a downside, its very large number of parameters can make the NTN model harder to train and overfit more easily.
The original multilinear <span style="font-variant:small-caps;">DistMult</span> model is symmetric in subject and object for every relation [@Yang2015] and achieves good performance on FB15K and WN18 data sets. However it is likely due to the absence of true negatives in these data sets, as discussed in Section \[sec:fb15k\_res\].
The <span style="font-variant:small-caps;">TransE</span> model from @bordes2013translating also embeds entities and relations in the same space and imposes a geometrical structural bias into the model: the subject entity vector should be close to the object entity vector once translated by the relation vector.
A recent novel way to handle antisymmetry is via the Holographic Embeddings (<span style="font-variant:small-caps;">HolE</span>) model by @nickel_2016_holographic. In <span style="font-variant:small-caps;">HolE</span> the circular correlation is used for combining entity embeddings, measuring the covariance between embeddings at different dimension shifts. This model has been shown to be equivalent to the <span style="font-variant:small-caps;">ComplEx</span> model [@hayashi2017equivalence; @trouillon2017comparison].
Discussion and Future Work
==========================
Though the decomposition proposed in this paper is clearly not unique, it is able to learn meaningful representations. Still, characterizing all possible unitary diagonalizations that preserve the real part is an interesting open question. Especially in an approximation setting with a constrained rank, in order to characterize the decompositions that minimize a given reconstruction error. That might allow the creation of an iterative algorithm similar to eigendecomposition iterative methods [@saad1992numerical] for computing such a decomposition for any given real square matrix.
The proposed decomposition could also find applications in many other asymmetric square matrices decompositions applications, such as spectral graph theory for directed graphs [@cvetkovic1997eigenspaces], but also factorization of asymmetric measures matrices such as asymmetric distance matrices [@mao2004modeling] and asymmetric similarity matrices [@pirasteh2015exploiting].
From an optimization point of view, the objective function (Equation (\[eq:objective\])) is clearly non-convex, and we could indeed not be reaching a globally optimal decomposition using stochastic gradient descent. Recent results show that there are no spurious local minima in the completion problem of positive semi-definite matrix [@ge2016matrix; @bhojanapalli2016global]. Studying the extensibility of these results to our decomposition is another possible line of future work. The first step would be generalizing these results to symmetric real-valued matrix completion, then generalization to normal matrices should be straightforward. The two last steps would be extending to matrices that are expressed as real part of normal matrices, and finally to the joint decomposition of such matrices as a tensor. We indeed noticed a remarkable stability of the scores across different random initialization of <span style="font-variant:small-caps;">ComplEx</span> for the same hyper-parameters, which suggests the possibility of such theoretical property.
Practically, an obvious extension is to merge our approach with known extensions to tensor factorization models in order to further improve predictive performance. For example, the use of pairwise embeddings [@riedel_2013_univschema; @Welbl2016] together with complex numbers might lead to improved results in many situations that involve non-compositionality. Adding bigram embeddings to the objective could also improve the results as shown on other models [@garcia2016combining]. Another direction would be to develop a more intelligent negative sampling procedure, to generate more informative negatives with respect to the positive sample from which they have been sampled. This would reduce the number of negatives required to reach good performance, thus accelerating training time. Extension to relations between more than two entities, $n$-tuples, is not straightforward, as <span style="font-variant:small-caps;">ComplEx</span>’s expressiveness comes from the complex conjugation of the object-entity, that breaks the symmetry between the subject and object embeddings in the scoring function. This stems from the Hermitian product, which seems to have no standard multilinear extension in the linear algebra literature, this question hence remains largely open.
Conclusion
==========
We described a new matrix and tensor decomposition with complex-valued latent factors called <span style="font-variant:small-caps;">ComplEx</span>. The decomposition exists for all real square matrices, expressed as the real part of normal matrices. The result extends to sets of real square matrices—tensors—and answers to the requirements of the knowledge graph completion task : handling a large variety of different relations including antisymmetric and asymmetric ones, while being scalable. Experiments confirm its theoretical versatility, as it substantially improves over the state-of-the-art on real knowledge graphs. It shows that real world relations can be efficiently approximated as the real part of low-rank normal matrices. The generality of the theoretical results and the effectiveness of the experimental ones motivate for the application to other real square matrices factorization problems. More generally, we hope that this paper will stimulate the use of complex linear algebra in the machine learning community, even and especially for processing real-valued data.
0.2in
[^1]: Code is available at: <https://github.com/ttrouill/complex>
[^2]: This is not the Hermitian extension of the multilinear dot product as there appears to be no standard definition of the Hermitian multilinear product in the linear algebra literature.
[^3]: https://github.com/lmjohns3/downhill
|
---
author:
- 'Leindert A. Boogaard'
- Jarle Brinchmann
- Nicolas Bouché
- Mieke Paalvast
- Roland Bacon
- 'Rychard J. Bouwens'
- Thierry Contini
- 'Madusha L.P. Gunawardhana'
- Hanae Inami
- 'Raffaella A. Marino'
- 'Michael V. Maseda'
- Peter Mitchell
- Themiya Nanayakkara
- Johan Richard
- Joop Schaye
- Corentin Schreiber
- Sandro Tacchella
- Lutz Wisotzki
- Johannes Zabl
bibliography:
- 'Msc1.bib'
date: 'Accepted July 20, 2018'
subtitle: 'XI. Constraining the low-mass end of the stellar mass - star formation rate relation at $z<1$[^1]'
title: The MUSE Hubble Ultra Deep Field Survey
---
Introduction {#sec:introduction}
============
How galaxies grow is one of the fundamental questions in astronomy. The picture that has emerged is that a galaxy builds up its stellar mass mainly through star formation, which is triggered by gas accretion from the cosmic web [e.g. @DekelA_09a; @VandeVoort2012], while mergers with other galaxies play only a minor role [except for massive systems; @Bundy2009].
In the past decade, star-forming galaxies have been found to form a reasonably tight quasi-linear relation between stellar mass ($M_{*}$) and star formation rate (SFR) [@Brinchmann2004; @Noeske2007a; @Elbaz2007; @Daddi2007; @Salim2007] over a wide range of masses and out to high redshifts [@Pannella2009; @Santini2009; @Oliver2010; @PengY_10a; @Rodighiero2010; @Karim2011; @Bouwens2012; @Whitaker2012; @StarkD_13a; @Whitaker2014; @Ilbert2015; @Lee2015; @Renzini2015; @Schreiber2015; @Shivaei2015; @Salmon2015; @Tasca2015; @Gavazzi2015; @Kurczynski2016; @Tomczak2016; @Santini2017; @Bisigello2017], which is often referred to as the ‘main sequence of star-forming galaxies’ or the ‘star formation sequence’. In contrast, galaxies that are undergoing a starburst or have already quenched their star formation respectively lie above and below the relation. This main sequence is close to a similar scaling relation for halos [@BirnboimY_07a; @NeisteinE_08a; @GenelS_08a; @FakhouriO_08a; @Correa2015a; @Correa2015b] where the growth rate increases super-linearly [^2] with halo mass, and this has been interpreted as supporting the picture where galaxy growth is driven by gas accretion from the cosmic web [e.g. @BoucheN_10a; @Lilly2013; @Rodriguez-PueblaA_16a; @Tacchella2016a].
This interpretation is supported by hydrodynamical simulations of galaxy formation [@Schaye2010; @Haas2013a; @Haas2013b; @Torrey2014; @Hopkins2014; @Crain2015; @Hopkins2016], where a global equilibrium relation is found between the inflow and outflow of gas and star formation in galaxies. In this picture the star formation acts as a self-regulating process, where the inflow of gas, through cooling and accretion, is balanced by the feedback from massive stars and black holes [e.g. @Schaye2010]. Furthermore, semi-analytical models [e.g. @DuttonA_10a; @Mitchell2014; @Cattaneo2011; @Cattaneo2017] and relatively simple analytic theoretical models which connect the gas supply (from the cosmological accretion) to the gas consumption can also reproduce the main features of the main sequence rather well [e.g. @BoucheN_10a; @Dave2012; @Lilly2013; @DekelA_13a; @DekelA_14a; @MitraS_15a; @Rodriguez-PueblaA_16a; @Rodriguez-PueblaA_17a] [^3].
The parameters of the $M_{*}$-SFR relation (i.e. slope, normalisation, and scatter) are thus important, as they provide us with insight into the relative contributions of different processes operating at different mass scales, in particular when comparing the values of the parameters to their counterparts in dark matter halo scaling relations. The normalisation of the star formation sequence is governed by the change in cosmological gas accretion rates and gas depletion timescales. The slope can be sensitive to the effect of various feedback processes acting on the accreted gas, which prevent (or enhance) star formation. The intrinsic scatter around the equilibrium relation is predominantly determined by the stochasticity in the gas accretion process [e.g. @ForbesJ_14a; @MitraS_17a], but can also be driven by dynamical processes that rearrange the gas inside galaxies [@Tacchella2016a]. The $M_{*}$-SFR relation is observed to be reasonably tight, with an intrinsic scatter of only $\approx 0.3$ dex [@Noeske2007a; @Salmi2012; @Whitaker2012; @Guo2013; @Speagle2014; @Schreiber2015; @Kurczynski2016 though we caution against a blind comparison as different observables probe star formation on different timescales]. Yet, it has proven to be challenging to place firm constraints on the intrinsic scatter as one needs to deconvolve the scatter due to measurement uncertainty [e.g. @Speagle2014; @Kurczynski2016; @Santini2017].
Observationally, the slope has been difficult to measure, particularly at the low-mass end, as most studies have been sensitive to galaxies with stellar masses above $\log M_*[\si{\solarmass}]\sim 10$ and often lack dynamical range in mass. In addition, while it is well known that there is significant evolution in the normalisation of the sequence with redshift, most studies have measured the slope in bins of redshift. For a flux limited sample this could introduce a bias in the slope because overlapping populations at different normalisations are not sampled equally in mass within a single redshift bin. The slope may also be mass dependent and indeed recent studies have observed that the relation turns over around a mass of $M_{*} \sim \SI{e10}{\solarmass}$ [@Whitaker2012; @Whitaker2014; @Lee2015; @Schreiber2015; @Tomczak2016] and shows a steeper slope below the turnover mass. In the low-mass regime, a (nearly) linear slope has generally been expected [e.g. @Schreiber2015; @Tomczak2016], motivated also by the fact that there is very little evolution in the faint-end slope of the blue stellar mass function with redshift [@Peng2014]. [@Leja2015] showed that the sequence cannot have a slope $a < 0.9$ at all masses because this would lead to a too high number density between $10 < \log M_{*}[\si{\solarmass}] < 11$ at $z = 1$.
In addition to the observational challenges, careful modelling is required to get reliable constraints on the parameters (slope, normalisation, scatter) of the star formation sequence. It is important to properly take selection effects into account as well as the uncertainties on both the stellar masses and star formation rates (and, if spectroscopy is lacking, also on the photometric redshifts). The latter in particular, due to the fact that there is intrinsic scatter in the relation that needs to be deconvolved from the measurement errors. Common statistical techniques do not take these complications into account self-consistently, which leads to biases in the results.
Putting the existing observations in perspective, it is clear that a large dynamical range in mass is necessary to measure the slope of the star formation sequence in the low-mass regime. Deep field studies, that can blindly detect large numbers of galaxies down to masses much below $\SI{e10}{\solarmass}$, are invaluable in this regard [e.g. @Kurczynski2016]. Yet, such studies are challenged by having to measure all observables, distances as well as stellar masses and star formation rates, from the same photometry. This can lead to undesirable correlations between different observables. At the same time the measurements suffer from the uncertainties associated with photometric redshifts. Spectroscopic follow up is crucial in this regard, but can suffer from biases due to photometric preselection.
With the advent of the *Multi Unit Spectroscopic Explorer* (MUSE; @Bacon2010) on the VLT it is now possible to address these concerns. With the deep MUSE data obtained over the *Hubble* Ultra Deep Field [HUDF; @Bacon2017] and *Hubble* Deep Field South [HDFS; @Bacon2015], we can ‘blindly’ detect star-forming galaxies in emission lines down to very low levels ($\sim \SI{e-3}{\solarmass \per \year}$) and obtain a precise spectroscopic redshift estimate at the same time [@Inami2017]. These data provides a unique view into the low-mass regime of the star formation sequence.
In this paper we present a characterisation of the low-mass end of the $M_{*}$-SFR relation, using deep MUSE observations of the HUDF and HDFS. We characterise the properties of the $M_{*}$-SFR relation down stellar masses of $M_{*} \sim \SI{e8}{\solarmass}$ and SFR $\sim \SI{e-3}{\solarmass \per \year}$, out to $z < 1$, and trace the SFR in individual galaxies with masses as low as $M_{*} \la \SI{e7}{\solarmass}$ at $z\sim0.2$. We model the relation using a self-consistent Bayesian framework and describe it with a Gaussian distribution around a plane in (log mass, log SFR, log redshift)-space. This allows us to simultaneously constrain the slope and evolution of the star formation sequence as well as the amount of intrinsic scatter, while taking into account heteroscedastic errors (i.e. a different uncertainty for each data point).
The structure of the paper is as follows. In [§\[sec:observations-methods\]]{} we first introduce the MUSE data set and outline the selection criteria used to construct our sample of star-forming galaxies. We then go into the methods used to determine a robust stellar mass and a SFR from the observed emission lines. Before looking at the results, we discuss the consistency of our SFRs in [§\[sec:cons-sfr-indic\]]{}. We then introduce the framework of our Bayesian analysis used to characterise the $M_{*}$-SFR relation ([§\[sec:modelling\]]{}) and present the results in [§\[sec:results\]]{}. We discuss the robustness of the derived parameters in [§\[sec:selection-function-completeness\]]{}. Finally, we discuss our results in the context of the literature and models, and the physical implications ([§\[sec:discussion\]]{}). We summarise with our conclusions in [§\[sec:conclusions\]]{}. Throughout this paper we assume a [@Chabrier2003] stellar initial mass function and a flat $\Lambda$CDM cosmology with $H_0 = \SI{70}{km.s^{-1}.Mpc^{-1}}$, $\Omega_m = 0.3$ and $\Omega_{\Lambda} = 0.7$.
![\[fig:redshift-sfct\] Redshift distribution of our galaxies plotted against their (dust-corrected) SFR (1$\sigma$ error bars are in grey). The colour denotes the stellar mass. The solid line depicts the lowest uncorrected SFR from [$\mathrm{H}\beta\ \lambda 4861$]{} we can detect in the HUDF at each redshift (which is effectively determined by the requirement that S/N$({\ensuremath{\mathrm{H}\gamma\ \lambda 4340}}) > 3$; see [§\[sec:star-formation-rates\]]{}).](./Images/fig1.pdf){width="\columnwidth"}
Observations and methods {#sec:observations-methods}
========================
\[sec:data\] To study the properties of the galaxy population down to low masses and star formation rates, deep spectroscopic observations are required for a large number of sources. We exploit the unique observations taken with the MUSE instrument over the *Hubble* Ultra Deep Field [@Bacon2017] and the *Hubble* Deep Field South [@Bacon2015] to investigate the star formation rates in low-mass galaxies at ${0.11}< z < {0.91}$. We provide a brief presentation of the observations and data reduction in the next section, but refer to the corresponding papers for details.
The MUSE instrument is an integral-field spectrograph situated at UT4 of the Very Large Telescope. It has a field-of-view of $\SI{1}{\arcmin} \times \SI{1}{\arcmin}$ when operating in wide-field-mode, which is fed into 24 different integral-field units. These sample the field-of-view at resolution. The spectrograph covers the spectrum across 4650Å - 9300 Å with a spectral resolution of $R \equiv \lambda / \Delta \lambda \simeq 3000$. The result of a MUSE observation is a data cube of the observed field, with two spatial and one spectral axes, i.e. an image with spectroscopic information at every pixel.
Observations, data reduction, and spectral line fitting
-------------------------------------------------------
The HUDF [@Beckwith2006] was observed with MUSE in a layered strategy. The deepest region consists of a single $1'\times 1'$ pointing with a total integration depth of 31 hours. This deep region lies embedded in a larger $3'\times 3'$ mosaic consisting of 9 individual MUSE pointings, each of which is 10 hours deep. The average full width at half maximum (FWHM) seeing measured in the data cubes is at . For the purpose of this work we use all galaxies from the region, including the deep () region, which we refer to collectively as the (MUSE) HUDF.
Because of its similar depth, we also include the MUSE observation of the HDFS [@Williams2000] which was observed as part of the commissioning activities. These observations consist of a single deep field ($1' \times 1'$) with a total integration time of 27 hours and a median seeing of .
The full data acquisition and reduction of the HUDF is detailed in [@Bacon2017] (for a description of the MUSE data reduction pipeline see Weilbacher et al., in prep.). The data reduction of the HUDF is essentially based on the reduction of the HDFS, which is detailed in [@Bacon2015], with several improvements. We use HUDF version 0.42 and HDFS version 1.0, which reach a $3\sigma$-emission line depth for a point source () of 1.5 and $3.1 \times 10^{-19}$ ( and ) and (HDFS), measured between the OH skylines at .
Sources in the HUDF were identified using both a blind and a targeted approach. The latter uses the sources from the UVUDF catalogue [@Rafelski2015] as prior information to extract objects from the MUSE cube. A blind search of the full cube was also conducted, using a tool specifically developed for MUSE cubes called ([@Bacon2017]; [Mary et al., (in prep.)]{}). A similar approach was already followed for the HDFS. Here sources were identified based on the [@Casertano2000] catalogue and blind emission line searches of the data cube were done with the automatic detection tools [^4] and [@Herenz2017] as well as through visual inspection, and cross-correlated with the corresponding photometric catalogue, as described in [@Bacon2015].
The process of determining redshifts and constructing a full catalogue from the extracted sources is described in [@Inami2017] for the HUDF (and a similar approach was followed for the HDFS). In short, redshifts were determined semi-automatically by cross-matching template spectra with the identified sources and subsequently inspected and confirmed by at least two independent investigators. For emission line galaxies an additional constraint comes from the requirement that the emission line flux is coherent in a narrow band image around the line in the MUSE cube. The typical error on the MUSE spectroscopic redshifts is $\sigma_{z} = 0.00012(1+z)$ [@Inami2017], which we will take into account in the modelling (conservatively taking $\sigma_{\log(1+z)} = 0.0005$ for all galaxies; [§\[sec:modelling\]]{})
For all detected sources one dimensional spectra are extracted using a straight sum extraction over an aperture around each source (based on the MUSE point spread function convolved with the [@Rafelski2015] segmentation map, see @Bacon2017). From the extracted 1D spectra emission line fluxes are fitted in velocity space, using an updated version of the code described in [@Tremonti2004] and [@Brinchmann2004; @Brinchmann2008]. assumes a Gaussian line profile for all lines, with the same intrinsic width and velocity. The result is a measurement of the flux and equivalent width of all emission lines present, with the uncertainties obtained from propagating the original pipeline errors. We define the signal-to-noise (S/N) in a particular spectral line as the line flux over the line flux error. We also determine the strength of the 4000 Å break, $D_{n}(4000)$, measured over $3850-3950$Å and $4000-4100$Å [@Kauffmann2003]. We note that the stellar absorption underlying the emission lines is taken into account by .
Sample selection {#sec:sample-selection}
----------------
From the HUDF and HDFS catalogues we construct our sample of star-forming galaxies using the following constraints:
1. We use [$\mathrm{H}\beta\ \lambda 4861$]{} or [$\mathrm{H}\alpha\ \lambda 6563$]{} to derive the SFR (see [§\[sec:star-formation-rates\]]{}) and in either case we always need [$\mathrm{H}\beta\ \lambda 4861$]{} (to directly probe the SFR or to correct for dust extinction in [$\mathrm{H}\alpha\ \lambda 6563$]{}). As a result, we are limited to the range of redshifts where [$\mathrm{H}\beta\ \lambda 4861$]{} falls within the MUSE spectral range. Subsequently, we only take objects into account that have a redshift $z < (9300 / 4861) - 1 = 0.913$.
2. In order to derive a robust SFR and dust correction, we only allow objects with a signal-to-noise ratio $>3$ in the relevant pair of Balmer lines. This means S/N $>3$ in either [$\mathrm{H}\beta\ \lambda 4861$]{} and [$\mathrm{H}\gamma\ \lambda 4340$]{} (for [$\mathrm{H}\beta\ \lambda 4861$]{} derived SFRs) or [$\mathrm{H}\alpha\ \lambda 6563$]{} and [$\mathrm{H}\beta\ \lambda 4861$]{} (for [$\mathrm{H}\alpha\ \lambda 6563$]{} derived SFRs).
Included in the above criteria are some galaxies that are not actively star-forming and lie on the ‘red-sequence’. Since these galaxies are not expected to lie on the $M_{*}$-SFR relation, we exclude them from the analysis based on their spectral features:
3. We remove [12]{} galaxies with a strong 4000 Å break by only allowing galaxies with a $D_{n}(4000) < 1.5$.
4. We omit galaxies with a rest-frame equivalent width in either [$\mathrm{H}\alpha\ \lambda 6563$]{} or [$\mathrm{H}\beta\ \lambda 4861$]{} of < 2Å [^5]. This removed an additional [7]{} and [5]{} objects, respectively.
In addition, [three]{} sources were removed from the sample due to severe artefacts in their emission lines (see [§\[sec:cons-sfr-indic\]]{}). All sources selected based on the MUSE data are detected in the *HST* imaging. However, [four]{} sources were removed because there photometry was severely blended, prohibiting a mass estimate.
5. We remove potential AGN from our sample in the HUDF by cross-matching our sources with the *Chandra* Deep Field South 7Ms X-ray catalogue [@Luo2016]. We also confirm the location of the sources in the star-forming region of different emission line diagnostic diagrams.
A total of [16]{} galaxies with $z<0.913$ from the MUSE catalogue are detected in X-rays. Five of these sources (including one AGN) show passive spectra without emission lines and did not pass the previous criteria. Cross-matching our star-forming sample (after applying criteria 1 through 4) left [11]{} galaxies that were detected in X-rays. Five of these sources (ID\#855, 861, 863, 895, and 902) are in the [$\mathrm{H}\alpha$]{}-subsample and six (ID\#867, 869, 874, 875, 884, and 905) are in the [$\mathrm{H}\beta$]{}-subsample. All of these sources were classified as ‘Galaxy’ in the [@Luo2016] catalogue (according to their 6 criteria based on X-ray luminosity, spectral index, flux-ratios and previous spectroscopic identification), except for ID\# 875 which was classified as an AGN and which we subsequently removed from the sample. [@Luo2016] caution however that sources classified as ‘Galaxy’ may still host low-luminosity or heavily obscured AGN.
We plot all sources from our [$\mathrm{H}\alpha\ \lambda 6563$]{}-subsample for which we have a measurement of [$\mathrm{[N\,\textsc{ii}]}\ \lambda 6584$]{} in the BPT-diagram [@Baldwin1981] in [Fig. \[fig:BPT\]]{}. We include sources for which we have a low S/N (<3) measurement of [$\mathrm{[N\,\textsc{ii}]}\ \lambda 6584$]{} as open circles. While we can only put a subsample of our sources on this diagram, all are in the star-forming region, including the [5]{} galaxies which have an X-ray detection. None of the X-ray sources classified as ‘Galaxy’ show spectral signatures of AGN activity. In [Fig. \[fig:Hb-diag\]]{} we show a similar consistency check for the [$\mathrm{H}\beta\ \lambda 4861$]{}-subsample. Because we lack access to the BPT diagram at these redshift, we instead use the diagnostics from both [@Lamareille2004] and [@Juneau2011]. Reassuringly, our sample is overall consistent with star-forming galaxies and none of the galaxies show line-ratios clearly powered by AGN activity (including, perhaps surprisingly, the single X-ray classified AGN). There is only one source which is above the discriminating line in both plots (ID\#1114), however, it is consistent within errors with being dominated by star formation and not detected in X-rays. Furthermore, its high [[\[O<span style="font-variant:small-caps;">iii</span>\]]{.nodecor}]{} flux can very well be driven by star formation and indeed it is part of the sample of high-[[\[O<span style="font-variant:small-caps;">iii</span>\]]{.nodecor}]{}/[[\[O<span style="font-variant:small-caps;">ii</span>\]]{.nodecor}]{} galaxies identified by [@Paalvast2018]. Hence, except for X-ray detected AGN ID\#875, we do not remove any additional sources from the sample. Finally, we note that none of the methods to identify AGN are individually foolproof. Therefore, we check the impact of potential misclassification of AGN and confirm that excluding (1) the sources that are above the pure star-forming line in either of the diagnostic diagrams or (2) all galaxies that are detected in X-rays (even when consistent with star formation) does not significantly affect the results.
![\[fig:BPT\] BPT-diagram [@Baldwin1981] of the sources in our [$\mathrm{H}\alpha\ \lambda 6563$]{}-subsample for which we measure [$\mathrm{[N\,\textsc{ii}]}\ \lambda 6584$]{}. All galaxies fall in the star-forming region of the diagram. The filled and open circles have S/N([$\mathrm{[N\,\textsc{ii}]}\ \lambda 6584$]{}) $>3$ and $<3$, respectively, and the [5]{} sources encircled in red are detected in X-rays [@Luo2016]. The solid and dashed curve show the AGN boundary and maximum starburst line from [@Kauffmann2003] and [@Kewley2001], respectively.](./Images/fig2.pdf){width="\columnwidth"}
{width="\columnwidth"} {width="\columnwidth"}
The final sample then consists of [179]{} star-forming galaxies, [147]{} from the HUDF, all with the highest redshift confidence [@Inami2017], and [32]{} from the HDFS, between ${0.11}< z < {0.91}$ with a mean redshift of [0.53]{} (see [Fig. \[fig:redshift-sfct\]]{}).
Stellar masses {#sec:stellar-masses}
--------------
The stellar masses of the galaxies were estimated using the Stellar Population Synthesis (SPS) code FAST [@Kriek2009]. The SPS-templates were $\chi^{2}$-fitted to the broad-band photometry of the different fields for a range of parameters. For the HUDF, we rely on the deep *HST* photometry from the UVUDF catalogue [@Rafelski2015] (containing WFC3/UVIS F225W, F275W and F336W; ACS/WFC F435W, F606W, F775W, and F850LP and WFC/IR F105W, F125W, F140W and F160W) while for the HDFS we take the WFPC2 photometry from [@Casertano2000] (F330W, F450W, F606W, and F814W). The SPS-templates were constructed from the [@Conroy2010] (FSPS) models using a discrete range of metallicities ($Z/Z_{\odot}=[0.04, 0.20, 0.50, 1.0, 1.58]$). We assumed a [@Chabrier2003] initial mass function with an exponentially declining star formation history (SFR $\propto \exp(-t/\tau)$ with $8.5 < \log(\tau/\mathrm{yr}) < 10$ in steps of 0.2 dex). The redshifts were fixed to the accurate spectroscopic values determined from the MUSE spectra. Ages were allowed to vary between $8 < \log \mathrm{Age/yr} < 10.2$ in steps of 0.2 dex. We parameterised the dust attenuation curve according to the [@Calzetti2000] dust law with the dust extinction in the visual taken to be within $0 < A_{V} < 3$ ($\Delta A_{V} = 0.1$ magnitudes). For all the parameters error estimates were obtained through Monte Carlo methods, by re-running the fitting 500 times while varying the input photometry within their photometric errors (see @Kriek2009 for details).
Stellar masses were determined for all [179]{} objects in the final sample. The distribution of masses is shown in [Fig. \[fig:mass-histogram\]]{}. With these deep MUSE observations we are mainly probing low-mass () galaxies and we can still detect star formation from emission lines in galaxies with mass $\sim$. The mass estimates with their upper and lower confidence intervals are shown for the individual objects in [Fig. \[fig:m\*-sfr\]]{}. The mean and standard deviation of the average errors on the mass estimates are $0.19 \pm 0.06$ dex for the HUDF and $0.22\pm0.12$ dex for the HDFS.
![\[fig:mass-histogram\] Histograms of the stellar mass distributions of the MUSE detected galaxies in the HUDF and the HDFS. The deep 30h observations allow us to detect and subsequently infer a stellar mass and SFR for galaxies down to $\sim \SI{e7}{\solarmass}$.](./Images/fig4.pdf){width="\columnwidth"}
Star formation rates {#sec:star-formation-rates}
--------------------
The star formation rates are inferred from the flux in the [$\mathrm{H}\alpha\ \lambda 6563$]{} or [$\mathrm{H}\beta\ \lambda 4861$]{}recombination lines emitted by [[H<span style="font-variant:small-caps;">ii</span>]{.nodecor}]{} regions, which primarily trace recent () massive star formation. Before we can infer a SFR we need to correct the measured flux in the emission lines for the attenuation by dust along the line of sight. We do this assuming a dust law according to [@Charlot2000] (i.e. $\tau \propto \lambda^{-1.3}$, appropriate for birth clouds) and using the intrinsic ratio of the Balmer recombination lines ($j_{{\ensuremath{\mathrm{H}\alpha}}}/j_{{\ensuremath{\mathrm{H}\beta}}} = 2.86$ and $j_{{\ensuremath{\mathrm{H}\beta}}}/j_{{\ensuremath{\mathrm{H}\gamma}}} = 2.14$; [@Hummer1987], for an electron temperature and density of $T = \SI{10000}{\kelvin}$ and $n_{e} = \SI{e3}{cm^{-3}}$). Hence, to derive an SFR([$\mathrm{H}\alpha\ \lambda 6563$]{}) we also require a measurement of [$\mathrm{H}\beta\ \lambda 4861$]{} and likewise for SFR([$\mathrm{H}\beta\ \lambda 4861$]{}) we also require [$\mathrm{H}\gamma\ \lambda 4340$]{}. After the dust correction we can convert the intrinsic flux to a luminosity using the measured redshift, given the assumed $\Lambda$CDM cosmology.
To determine the SFR we follow the treatment by [@Moustakas2006], which is essentially based on the relations from [@Kennicutt1998]. Out of the SFR indicators that MUSE has access to, the [$\mathrm{H}\alpha\ \lambda 6563$]{} line presents the least systematic uncertainties, but it is only available at low redshifts ($z \la 0.42$ for MUSE at ; [47]{} galaxies). We convert the [@Kennicutt1998] relation from a Salpeter to a Chabrier IMF ($0.1 < M [\si{\solarmass}] < 100$) by multiplying by a factor 0.62 (which is derived by computing the difference in total mass in both IMFs, while assuming the same number of massive () stars): $$\begin{aligned}
\label{eq:sfr-ha}
\mathrm{SFR}({\ensuremath{\mathrm{H}\alpha\ \lambda 6563}}) = 4.9 \times 10^{-42}\frac{L({\ensuremath{\mathrm{H}\alpha\ \lambda 6563}})}{\si{erg s^{-1}}}\si{\solarmass \per \year},\end{aligned}$$ where $L({\ensuremath{\mathrm{H}\alpha\ \lambda 6563}})$ is the dust-corrected luminosity. We note that this calibration assumes case B recombination and solar metallicity.
Because [$\mathrm{H}\alpha\ \lambda 6563$]{} moves out of the optical regime at $z>0.42$, the [$\mathrm{H}\beta\ \lambda 4861$]{}luminosity is the primary tracer of SFR for the majority of our sample ([132]{} galaxies). Given the intrinsic flux ratio between [$\mathrm{H}\alpha\ \lambda 6563$]{} and [$\mathrm{H}\beta\ \lambda 4861$]{}, we can convert equation [Eq. (\[eq:sfr-ha\])]{} into a SFR for $L({\ensuremath{\mathrm{H}\beta\ \lambda 4861}})$: $$\begin{aligned}
\label{eq:sfr-hb}
\mathrm{SFR}({\ensuremath{\mathrm{H}\beta\ \lambda 4861}}) = 1.4 \times 10^{-41} \frac{L({\ensuremath{\mathrm{H}\beta\ \lambda 4861}})}{\si{erg s^{-1}}}\si{\solarmass \per \year},\end{aligned}$$ where $L({\ensuremath{\mathrm{H}\beta\ \lambda 4861}})$ is corrected for dust. We note that the [$\mathrm{H}\beta\ \lambda 4861$]{} derived SFR inherits all the uncertainties from SFR$({\ensuremath{\mathrm{H}\alpha\ \lambda 6563}})$, including variations in dust reddening [@Moustakas2006].
We also investigate the SFR using the [$\mathrm{[O\,\textsc{ii}]}\ \lambda 3727$]{} nebular emission line. Here we use the calibration for the [$\mathrm{H}\alpha\ \lambda 6563$]{} SFR ([Eq. (\[eq:sfr-ha\])]{}), where we assume an intrinsic flux ratio between [$\mathrm{[O\,\textsc{ii}]}\ \lambda 3727$]{} and [$\mathrm{H}\alpha\ \lambda 6563$]{} of unity [@Moustakas2006]. Since [$\mathrm{[O\,\textsc{ii}]}\ \lambda 3727$]{} is closest to [$\mathrm{H}\beta\ \lambda 4861$]{}, we use the [$\mathrm{H}\beta\ \lambda 4861$]{}/[$\mathrm{H}\gamma\ \lambda 4340$]{} ratio to determine the dust correction, scaled to the appropriate wavelength. The consequence of this is that the addition of the [$\mathrm{[O\,\textsc{ii}]}\ \lambda 3727$]{} line as a tracer of SFR will not add any new objects to the sample. Instead, it can be used as a useful comparison, which will be discussed in [§\[sec:cons-sfr-indic\]]{}.
To estimate the uncertainty in the SFR estimates (and dust corrections), we use Monte Carlo methods to derive a confidence interval on the SFR of every individual galaxy. We create a posterior distribution on the SFR by doing 1000 draws from a Gaussian distribution centred on the measured flux, with the variance set by the measurement error squared. The median posterior SFR can then be determined, as well as the $\pm 1 \sigma$ confidence intervals, by taking the $50^{\text{th}}$, $16^{\text{th}}$ and $84^{\text{th}}$ percentile from the derived posterior distribution.
{width="\textwidth"} {width="\textwidth"}
![\[fig:tau-comparison\] Optical depths at the wavelength of [$\mathrm{H}\beta\ \lambda 4861$]{} as derived from both the [$\mathrm{H}\beta\ \lambda 4861$]{}/[$\mathrm{H}\gamma\ \lambda 4340$]{} and the [$\mathrm{H}\alpha\ \lambda 6563$]{}/[$\mathrm{H}\beta\ \lambda 4861$]{}-ratio, coloured by [$\mathrm{H}\gamma\ \lambda 4340$]{} signal-to-noise (S/N). The dashed line is the one-to-one relation. Overall the optical depths agree reasonably well, unless the [$\mathrm{H}\gamma$]{} S/N is low. Most galaxies actually show little dust ($\tau$ close to zero). The shaded area shows the regions of (unphysical) negative optical depth for each axis. We set the optical depth to zero for galaxies with negative $\tau$ as this is often consistent with the error bars and the offset is due to noise in the spectra. We note that some of the high-S/N outliers actually have discrepant Balmer ratios. If the inferred optical depth is very different, this will affect the comparison of the dust-corrected SFR from [$\mathrm{H}\beta\ \lambda 4861$]{} and [$\mathrm{H}\alpha\ \lambda 6563$]{} (see [Fig. \[fig:sfr-consistency\]]{}).](./Images/fig6.pdf){width="\columnwidth"}
Consistency of SFR indicators {#sec:cons-sfr-indic}
=============================
Before turning to the results, we first consider the consistency of the derived SFRs, by comparing the SFR estimates from different tracers for the same galaxies. In the remainder of the paper we only use the dust-corrected Balmer lines as tracers of star formation.
For a significant fraction of our galaxies ($\approx 40 \%$) we find that the Balmer line ratios are below their case B values (as stated in [§\[sec:star-formation-rates\]]{}), indicative of a negative dust correction. While this might seem surprising, this is not uncommon and similar ratios have been seen in spectra from, e.g. the SDSS [@Groves2012], MOSDEF [@Reddy2015], KBSS [@Strom2017] and ZFIRE [@Nanayakkara2017]. While ‘unphysical’, these ratios are not entirely unexpected and can have several causes.
First, these deviations can be caused by noisy spectra. Most galaxies in our sample are not very dusty and hence have a ratio close to case B. In $> 50 \%$ of the cases with deviant ratios, the case B ratio is indeed within the $1\sigma$ error bars. We conservatively apply no dust correction for all these galaxies. The mean dust correction for all galaxies in our sample is $\tau({\ensuremath{\mathrm{H}\beta}}/{\ensuremath{\mathrm{H}\gamma}}) \approx 0.6$ (setting galaxies with a negative dust correction to zero) or $\tau({\ensuremath{\mathrm{H}\beta}}/{\ensuremath{\mathrm{H}\gamma}}) \approx 1$ (including only galaxies with a positive dust correction).
Secondly, there could be a problem with the measurement. Three objects that were significantly offset from the rest of the sample showed particular problems in their emission lines. In one object (ID\#971) [$\mathrm{H}\gamma\ \lambda 4340$]{} was severely affected by an emission line from a nearby source ([$\mathrm{[O\,\textsc{iii}]}\ \lambda 4959$]{} from ID\#874 at $z=0.458$, another galaxy in our sample, coincidentally almost exactly at the observed wavelength of [$\mathrm{H}\gamma$]{}). For five other objects there was a clear problem with the fit to the [$\mathrm{H}\beta\ \lambda 4861$]{} (ID\#894, \#896, \#1027) or [$\mathrm{H}\alpha\ \lambda 6563$]{} (ID\#2, \#1426) emission lines. We subsequently removed the first four sources from the analysis; for the latter two we disregarded the [$\mathrm{H}\alpha\ \lambda 6563$]{} SFR and use the [$\mathrm{H}\beta\ \lambda 4861$]{} SFR.
A third, intriguing option is that theses objects are real. Indeed, there remains a small number of galaxies which have high-S/N spectra, but still show Balmer ratio’s below their case B values. [^6]. Similar objects have also been observed in the other surveys already referenced, such as SDSS (Jarle Brinchmann, private communication, see also @Groves2012). While these are very interesting objects on their own, a detailed analysis of these sources is beyond the scope of this paper. To be conservative and consistent, we apply no dust correction for these sources.
For some objects in the sample we measure multiple emission lines, which allows us to infer a SFR from different tracers. In any case a pair of Balmer lines (either [$\mathrm{H}\alpha$]{}/[$\mathrm{H}\beta$]{} or [$\mathrm{H}\beta$]{}/[$\mathrm{H}\gamma$]{}) is available ([§\[sec:sample-selection\]]{}), to allow for a dust correction. The majority of our sample lies at $z > 0.42$ for which [$\mathrm{H}\alpha\ \lambda 6563$]{} is not available, but (dust-corrected) [$\mathrm{[O\,\textsc{ii}]}\ \lambda 3727$]{} is available as an SFR indicator. In [Fig. \[fig:sfr-consistency\]]{} we show a comparison for all galaxies that allowed both [$\mathrm{H}\alpha\ \lambda 6563$]{} and [$\mathrm{H}\beta\ \lambda 4861$]{} (only some galaxies at $z < 0.42$) and [$\mathrm{H}\beta\ \lambda 4861$]{} and [$\mathrm{[O\,\textsc{ii}]}\ \lambda 3727$]{} derived SFRs (all redshifts). We note that [$\mathrm{H}\beta\ \lambda 4861$]{} and [$\mathrm{[O\,\textsc{ii}]}\ \lambda 3727$]{}derived SFRs are corrected for dust using the same [$\mathrm{H}\beta$]{}/[$\mathrm{H}\gamma$]{}-ratio.
In the right panels of [Fig. \[fig:sfr-consistency\]]{} we see that the [$\mathrm{H}\beta\ \lambda 4861$]{} and [$\mathrm{[O\,\textsc{ii}]}\ \lambda 3727$]{} derived SFRs agree remarkably well (standard deviation $\sigma \leq 0.28$ dex), considering that we have not taken into account the metallicity dependence of the [$\mathrm{[O\,\textsc{ii}]}\ \lambda 3727$]{} luminosity in the SFR conversion factor [e.g. @Kewley2004]. A few points scatter quite a bit, most of which have large error bars. At lower SFRs we do see that [$\mathrm{[O\,\textsc{ii}]}\ \lambda 3727$]{} predicts a lower SFR than [$\mathrm{H}\beta\ \lambda 4861$]{}, which is probably because at low SFR we are also probing low-mass and low-metallicity galaxies. Stars with a lower metallicity have a higher UV flux, which causes the ionisation equilibrium for oxygen to shift from [[\[O<span style="font-variant:small-caps;">ii</span>\]]{.nodecor}]{} to [[\[O<span style="font-variant:small-caps;">iii</span>\]]{.nodecor}]{} which diminishes the observed [$\mathrm{[O\,\textsc{ii}]}\ \lambda 3727$]{} flux. Because of the opposite effect [$\mathrm{[O\,\textsc{ii}]}\ \lambda 3727$]{}occasionally predicts a higher SFR than [$\mathrm{H}\beta\ \lambda 4861$]{} at the high-SFR end.
For a limited number of objects all three Balmer lines are in the spectral range of MUSE ($0.09 < z < 0.42$). We compare the [$\mathrm{H}\alpha\ \lambda 6563$]{} and [$\mathrm{H}\beta\ \lambda 4861$]{} derived SFRs in the left panel of [Fig. \[fig:sfr-consistency\]]{}, where we find reasonable agreement (in the HUDF, where we have most sources, they have a factor of $\sim 2$ scatter). Most of the scatter is found at low SFR, where (on average) the S/N is also the lowest. In the HDFS one object (at low S/N) is a strong outlier, but removing this source yields a similar scatter to the HUDF. Intuitively the SFRs from [$\mathrm{H}\alpha$]{} and [$\mathrm{H}\beta$]{} should agree very well, which warrants some deeper investigation into the outliers at low SFR.
The main uncertainty in the SFR estimate is the amount of dust attenuation. In [Fig. \[fig:tau-comparison\]]{} we compare the inferred optical depth from the [$\mathrm{H}\beta$]{}/[$\mathrm{H}\gamma$]{}-ratio ($\tau[{\ensuremath{\mathrm{H}\beta}}/{\ensuremath{\mathrm{H}\gamma}}]$) to the optical depth determined from the [$\mathrm{H}\alpha$]{}/[$\mathrm{H}\beta$]{} ratio ($\tau[{\ensuremath{\mathrm{H}\alpha}}/{\ensuremath{\mathrm{H}\beta}}]$). We note though that [Fig. \[fig:tau-comparison\]]{} shows the measured optical depth, while we set negative $\tau$ to zero before computing the SFR. Indeed, while many sources agree well, we see that the amount of dust correction estimated from the Balmer lines is not consistent for several objects, leading to a different SFR estimate from [$\mathrm{H}\alpha\ \lambda 6563$]{} and [$\mathrm{H}\beta\ \lambda 4861$]{}.
This tension is in part caused by the nature of the experiment, which requires that all three Balmer lines are in the spectral range of MUSE simultaneously. Necessarily then, [$\mathrm{H}\alpha\ \lambda 6563$]{} will be at the long wavelength end of the spectrograph where skylines are more prevalent, occasionally adding uncertainty to its measurement. For the low-SFR sources, however, [$\mathrm{H}\gamma\ \lambda 4340$]{} might not be very bright, adding uncertainty to the dust correction of SFR$({{\ensuremath{\mathrm{H}\beta\ \lambda 4861}}})$ for these sources (as seen at lower SFR in the left panels of [Fig. \[fig:sfr-consistency\]]{}). Indeed, most of the outliers have a low S/N in [$\mathrm{H}\gamma\ \lambda 4340$]{} (as stated earlier, for the objects with a negative dust correction from [$\mathrm{H}\beta$]{}/[$\mathrm{H}\gamma$]{}, we leave the often lower S/N measurement of [$\mathrm{H}\gamma\ \lambda 4340$]{} out of the analysis by setting $\tau({\ensuremath{\mathrm{H}\beta}}/{\ensuremath{\mathrm{H}\gamma}})$ to zero). On the other hand, the converse is not quite true: for a large number of sources with a low S/N in [$\mathrm{H}\gamma\ \lambda 4340$]{} we do have a consistent SFR estimate. For all objects we use the highest S/N lines available to infer a dust-corrected SFR, i.e. for objects which have a measurement of all three Balmer line we use the [$\mathrm{H}\alpha\ \lambda 6563$]{}, [$\mathrm{H}\beta\ \lambda 4861$]{} pair to infer a dust-corrected SFR, which generally has the highest S/N.
In summary, we have dust-corrected SFR measurement from the [$\mathrm{H}\alpha\ \lambda 6563$]{} and [$\mathrm{H}\beta\ \lambda 4861$]{}spectral lines for all galaxies at $z < 0.42$ and the [$\mathrm{H}\beta\ \lambda 4861$]{}, [$\mathrm{H}\gamma\ \lambda 4340$]{}-pair at higher redshifts. Comparing [$\mathrm{H}\alpha\ \lambda 6563$]{} and [$\mathrm{H}\beta\ \lambda 4861$]{} SFRs, we conclude that the dust correction is the largest uncertainty on the derived SFR. We always use the highest S/N line-pair available to compute a dust-corrected SFR. Comparing the [$\mathrm{H}\beta\ \lambda 4861$]{} SFRs with [$\mathrm{[O\,\textsc{ii}]}\ \lambda 3727$]{} at all redshifts, we see a very consistent picture (they have $\leq 0.3$ dex scatter in both fields). Naturally, some variations between [$\mathrm{H}\beta\ \lambda 4861$]{} and [$\mathrm{[O\,\textsc{ii}]}\ \lambda 3727$]{} SFRs are expected given the metallicity dependent nature of [$\mathrm{[O\,\textsc{ii}]}\ \lambda 3727$]{}.
Bayesian model {#sec:modelling}
==============
Definition {#sec:definition}
----------
The star formation sequence is commonly described by a power-law relation between stellar mass ($M_{*}$) and star formation rate (SFR), which evolves with redshift ($z$): $$\begin{aligned}
\label{eq:powerlaw}
\mathrm{SFR} \propto M_{*}^{a} (1+z)^{c},\end{aligned}$$ where $a$ and $c$ are the power law exponents. It has been suggested that the slope ($a$) becomes shallower in the high-mass regime ($M_{*} > \SI{e10}{\solarmass}$). In this work we will focus on the low-mass regime, for which we assume the slope is constant with mass. We will revisit this assumption in [§\[sec:low-mass-sample\]]{}. Given the lack of homogeneous studies with redshift it is still unclear whether the low-mass slope of the relation evolves with redshift. Here, we assume that the low-mass slope is independent of redshift over the range that we probe in this study. Likewise, given the large uncertainties in (the evolution of) the intrinsic scatter, we limit the number of free parameters in the model and assume that the intrinsic scatter does not depend on any of the other model parameters.
Following this description, we model the star formation sequence by a plane in ($\log M_{*}$, $\log(1+z)$, $\log{\mathrm{SFR}}$)-space: $$\begin{aligned}
\label{eq:linear}
\log{\mathrm{SFR}[\si{\solarmass\per\year}]} = a \log{\left(\frac{M_{*}}{M_{0}}\right)} +b + c\log{\left(\frac{1+z}{1+z_{0}}\right)},\end{aligned}$$ where $b$ is now a normalisation constant. We take $M_{0} = \SI{e8.5}{\solarmass}$ and $z_{0} = 0.55$ (close to the medians of the data) without the loss of generality. Galaxies scatter around this relation with an amount of intrinsic scatter in the vertical (i.e. $\log{\mathrm{SFR}}$) direction, which we denote by $\sigma_{\mathrm{intr}}$. In the lack of an obvious alternative, we take the intrinsic scatter to be Gaussian in our model.
In a statistical sense we can then say that our observations ($\log M_{*}$, $\log(1+z)$, $\log \text{SFR}$) are drawn from a Gaussian distribution around the plane defined by [Eq. (\[eq:linear\])]{}. To recover this distribution, we need to take a careful approach, taking into account the heteroscedastic errors of the measurements.
We adopt a Bayesian approach to determine the posterior distribution of the model parameters ($a, c, b, \sigma_{\mathrm{intr}}$) (see [@Andreon2010] for a lucid description of the Bayesian methodology in an astronomical context). Different approaches to construct the likelihood have been presented in the literature (see e.g. @Kelly2007 or @Hogg2010). We choose to adopt a parameterisation of the likelihood following [@Robotham2015] (hereafter ).
First, we state that our knowledge about galaxy $i$ (determined by the observations) is encompassed by the probability density function of a multivariate Gaussian distribution, $\mathcal{N}(\mathbf{x}_{i}, C_{i})$, with a mean value of: $$\begin{aligned}
\label{eq:mean}
\mathbf{x}_{i} = (\log M_{*,i}, \log(1+z_{i}), \log \mathrm{SFR}_{i})\end{aligned}$$ and a diagonal covariance matrix: $$\begin{aligned}
\label{eq:cov}
C_{i} =
\begin{pmatrix}
\sigma_{\log M_{*},i}^{2} & 0 & 0 \\
0 & \sigma_{\log (1+z),i}^{2} & 0 \\
0 & 0 & \sigma_{\log \mathrm{SFR}, i}^{2}
\end{pmatrix}\end{aligned}$$ containing the variance in each parameter. This is justified as both stellar mass and star formation rate are measured independently from different data. The covariance with redshift is negligible as the error on the spectroscopic redshift is very small.
Secondly, we parameterise the model given by [Eq. (\[eq:linear\])]{} (which is a plane in three dimensions) in terms of its normal vector $\mathbf{n}$, to avoid optimisation problems . The galaxies scatter around this plane with an amount of intrinsic Gaussian scatter, perpendicular to the plane, which we denote by $\sigma_{\perp}$. We note that perpendicular scatter $\sigma_{\perp}$ is distinct from the (commonly reported) vertical scatter $\sigma_{\mathrm{intr}}$ which lies in the $\log \mathrm{SFR}$ direction. After the analysis, we can simply transform the parameters ($\mathbf{n}, \sigma_{\perp}^{2}$) back into familiar parameters $(a,c,b,\sigma_{\mathrm{intr}}^{2})$ (using , Eq. 9).
Given the above definitions, we can express our log-likelihood [^7] as the sum over $N$ data points (see also ): $$\begin{aligned}
\label{eq:likelihood}
\ln \mathcal{L} = -\frac{1}{2}\sum_{i=1}^{N} \left[\ln \left(\sigma_{\perp}^{2} + \frac{\mathbf{n}^{\top} C_{i}\mathbf{n}}{\mathbf{n}^{\top}\mathbf{n}}\right)
+ \frac{(\mathbf{n}^{\top}[\mathbf{x}_{i} - \mathbf{n}])^2}{\sigma_{\perp}^{2} \mathbf{n}^{\top}\mathbf{n} + \mathbf{n}^{\top}C_{i}\mathbf{n}} \right],\end{aligned}$$ where all the parameters have been defined earlier.
Lastly, we have to define our priors on each component of $\mathbf{n}$ and on $\sigma_{\perp}^{2}$. As we want to impose limited prior knowledge, we express our priors as uniform distributions, with large bounds compared to the typical values of the parameters (we confirm that the results are robust, irrespective of the exact choice of bounds). $$\begin{aligned}
\label{eq:prior}
\mathbf{n} &\sim \mathcal{U}^{3}(-1000,1000) \\
\sigma_{\perp}^2 &\sim \mathcal{U}(0,1000), \notag\end{aligned}$$ where $\mathcal{U}^{n}$ is the $n$-dimensional multivariate uniform distribution and we take into account the fact that variance is always positive.
Execution {#sec:execution}
---------
With the likelihood and priors in hand we determine the posterior using *Markov chain Monte Carlo* (MCMC) methods. We use the Python implementation called [@Foreman-Mackey2013], which utilises the affine-invariant ensemble sampler for MCMC from [@Goodman2010]. samples the parameter space in parallel by setting off a predefined number of ‘walkers’, which we take to be 250.
Following [@Foreman-Mackey2013], we first initialise the walkers randomly in a large volume of parameter space. We then restart the walkers in a small Gaussian ball around the median of the posterior distribution (i.e. around the ‘best solution’). We (generously) burn in for a quarter of the total amount of iterations for each walker which we take to be 20000 for the main run ([§\[sec:global-sample\]]{}; roughly four hundred times the autocorrelation time). We note that for all subsequent runs described below we follow the same procedure, with similar results.
We take several steps to check whether the algorithm has properly converged. As an indication, one can look at both the mean acceptance fraction of the samples as well as the autocorrelation time [@Foreman-Mackey2013]. For the main run the acceptance fraction that resulted from the modelling (0.45) was well within range advocated by [@Foreman-Mackey2013] (0.2 - 0.5). The autocorrelation time was also relatively short and we let the walkers sample the posterior well over the autocorrelation time. Furthermore, we confirmed that the walkers properly explored the parameter space.
Combining the results from all walkers then gives the posterior distribution over which we can marginalise to find the posterior probability distributions for the model parameters. We will discuss the results of the modelling in [§\[sec:results\]]{}.
Model and data limitations {#sec:model-data-lims}
--------------------------
The unique aspect of the likelihood in [Eq. (\[eq:likelihood\])]{} is that it captures both the heteroscedastic errors on the observables as well as the intrinsic scatter around the plane. Furthermore, it can simultaneously describe both the slope of the sequence as well as the evolution with redshift.
It is important to determine how well we can recover the ‘true’ parameters with the observed data at hand. Our MUSE observations are constrained by the fact that we can only detect galaxies in a certain redshift range and cannot detect galaxies below the flux limit of the instrument (see [Fig. \[fig:redshift-sfct\]]{}). As the flux limit varies with redshift, this could introduce a bias in our inferred parameters. The reason behind this is that the lack of low-SFR galaxies at higher redshift will bias the posterior towards shallower slopes, with a steeper redshift evolution (see [Fig. \[fig:sfc\]]{} for an illustration). In order to correct for such a bias, we analyse a series of simulated observations. We briefly outline the procedure here, which is described in detail in .
In order to characterise the bias in the inferred parameters, we simulate galaxies from a mock star formation sequence for a range of values in each parameter, which we call $\mathbf{x}_{\mathrm{true, k}}$ (see ). After applying the redshift-dependent flux limit to the mock data, we model the remaining galaxies as described in [§\[sec:modelling\]]{} and recover the parameters, $\mathbf{x}_{\mathrm{out, k}}$. We then fit the transformation between the true and recovered parameters with an affine transformation ($\mathbf{x}_{\mathrm{out, k}} = A \mathbf{x}_{\mathrm{true, k}} + \mathbf{b}$) as outlined in [§\[sec:line-transf\]]{}. The inverse of the best-fit transformation ([Eq. (\[eq:inv-trans\])]{}) can then be used to correct the posterior density distribution as measured from the MUSE data. In the following, we provide both the uncorrected (directly fitted) and the corrected values for reference.
Star formation sequence {#sec:results}
=======================
{width="\textwidth"}
{width="70.00000%"}
![\[fig:m\*-z-sfr\] The best-fit star formation sequence for the [179]{} star-forming galaxies observed with MUSE. The symbols indicate the dust-corrected tracer used to infer the SFR. The solid line shows best-fit relation, as presented in [Eq. (\[eq:best-fit-corr\])]{}, and the dashed lines show the $1\sigma$ intrinsic scatter. We subtract the evolution from the y-axis and scale to the average redshift of the sample; $z=0.55$. After accounting for evolution, the galaxies clearly follow the star formation sequence, down to the lowest masses and SFRs. The slightly larger fraction of galaxies that scatter into the high-mass, low-SFR regime may be a result of the flattening of the relation above $M_{*} = \SI{e10}{\solarmass}$.](./Images/fig9.pdf){width="\columnwidth"}
Global sample {#sec:global-sample}
-------------
With a reliable SFR estimate in hand, we can turn to the star formation sequence between ${0.11}< z < {0.91}$ as observed by MUSE. [Fig. \[fig:m\*-sfr\]]{} shows a plot of stellar mass ($M_{*}$) versus star formation rate (SFR) for all the galaxies in the sample. The figure is based on two dust-corrected SFR indicators: the [$\mathrm{H}\beta\ \lambda 4861$]{} and [$\mathrm{H}\alpha\ \lambda 6563$]{} luminosities (Eqs. and ). The vertical grey lines indicate the errors in ($\log M_{*}$, $\log\mathrm{SFR}$) for each of the individual galaxies. The mean average error on the SFR is $\approx 0.2$ dex in both the HUDF and the HDFS.
We are able to detect star formation in galaxies down to star formation rates as low as . The galaxies appear to follow the $M_{*}$-SFR trend closely over the complete mass range, down to the lowest masses we can probe here $\sim \SI{e7}{\solarmass}$. At the high-mass end it appears we are starting to witness a flattening off of the trend, although we are primarily sensitive to the intermediate and low-mass galaxies.
We model the $M_{*}$-SFR relation with the Bayesian MCMC methodology described in detail in [§\[sec:modelling\]]{}. We show the resulting posterior probability density distribution for the parameters in [Fig. \[fig:triangle\]]{}. By marginalising over the various parameters, we recover the posterior probability distributions for the individual parameters of interest ($a,c,b,\sigma_{\text{intr}}$). These are plotted as histograms above the various axes in [Fig. \[fig:triangle\]]{}. By taking the median and the $16^{\text{th}}$ and $84^{\text{th}}$ percentile from the posterior distributions we derive the median posterior value and a $1 \sigma$ confidence interval for the parameters of interest.
The (uncorrected) best-fit (i.e. median posterior) parameters of the distribution (with their confidence intervals) that describe the star formation sequence are: $$\begin{aligned}
\label{eq:best-fit}
\log \mathrm{SFR}[\si{\solarmass\per\year}] =\; & {\ensuremath{0.79^{+0.05}_{-0.05}}}\log\left(\frac{M_*}{M_{0}}\right) {\ensuremath{-0.77^{+0.04}_{-0.04}}}\nonumber\\
&+ {\ensuremath{2.78^{+0.78}_{-0.78}}}\log\left(\frac{1+z}{1+z_{0}}\right) \pm {\ensuremath{0.46^{+0.04}_{-0.03}}}, $$ analogous to [Eq. (\[eq:linear\])]{}. The final term represents the intrinsic scatter ([$\sigma_{\text{intr}}={\ensuremath{0.46^{+0.04}_{-0.03}}}$]{}) in the vertical ($\log \text{SFR}$) direction. We note that while it is a perfectly valid option for the parameterisation of the likelihood, the posterior distribution does not favour models with zero intrinsic scatter.
[Fig. \[fig:triangle\]]{} shows that some correlations exist between the different parameters of the model, which is expected. The strongest correlation exists between slope and redshift evolution as a less steep slope requires more evolution in the normalisation to be compatible with the data. The complete covariance matrix between the different parameters is: $$\begin{aligned}
\label{eq:postcov}
\Sigma(a, c, b,\sigma_{\mathrm{intr}}) &=
\begin{pmatrix}
{
0.003 & -0.019 & -0.001 & 0.000 \\
-0.019 & 0.620 & 0.011 & 0.000 \\
-0.001 & 0.011 & 0.002 & -0.000 \\
0.000 & 0.000 & -0.000 & 0.001}\end{pmatrix}.\end{aligned}$$
We correct the posterior for observational bias, by applying [Eq. (\[eq:inv-trans\])]{}, which is indicated by the red contours in [Fig. \[fig:triangle\]]{}. This yields a steeper slope, with a significantly shallower redshift evolution: $$\begin{aligned}
\label{eq:best-fit-corr}
\log \mathrm{SFR}[\si{\solarmass\per\year}] =\; & {\ensuremath{0.83^{+0.07}_{-0.06}}}\log\left(\frac{M_*}{M_{0}}\right) {\ensuremath{-0.83^{+0.05}_{-0.05}}}\nonumber\\
&+ {\ensuremath{1.74^{+0.66}_{-0.68}}}\log\left(\frac{1+z}{1+z_{0}}\right) \pm {\ensuremath{0.44^{+0.05}_{-0.04}}}, $$ At the same time, the transformation has little effect on the intrinsic scatter. The covariance in the corrected posterior is essentially the same as the uncorrected one, with a slight increase in covariance with intrinsic scatter. $$\begin{aligned}
\label{eq:postcov-corr}
\Sigma(a, c, b,\sigma_{\mathrm{intr}}) &=
\begin{pmatrix}
{
0.004 & -0.016 & -0.002 & 0.002 \\
-0.016 & 0.459 & 0.010 & -0.003 \\
-0.002 & 0.010 & 0.002 & -0.001 \\
0.002 & -0.003 & -0.001 & 0.002}\end{pmatrix}.\end{aligned}$$
We compare the generative distribution (i.e. [Eq. (\[eq:best-fit\])]{}) with the data in [Fig. \[fig:m\*-z-sfr\]]{}. As the plane is three dimensional, we show a projection where we have subtracted the evolution with redshift from the y-axis. Overall, the distribution appears to describe the data very well and the scatter in the observations has tightened with respect to [Fig. \[fig:m\*-sfr\]]{}. For a more familiar representation we also show the resulting star formation sequence in the right panel of [Fig. \[fig:m\*-sfr\]]{}, for a number of different redshifts.
Low-mass sample $(\log M_{*} [M_{\odot}] < 9.5)$ {#sec:low-mass-sample}
------------------------------------------------
We are primarily interested in the low-mass end of the star formation sequence. Our deep MUSE sample spans a significant mass range, between $\log M_{*}[\si{\solarmass}] = 6.5 - 11$. As several studies have suggested different characteristics for the star formation sequence above and below a turnover mass of $M_{*} \sim 10^{10} \si{\solarmass}$ [e.g. @Whitaker2014; @Lee2015; @Schreiber2015], we repeat the above analysis excluding galaxies above a certain mass threshold. To be on the conservative side, we choose this mass threshold to lie at $M_{*} = 10^{9.5} \si{\solarmass}$. This excludes ${31}/{179}\approx 17.5\%$ of the sample. We include this threshold as a dashed vertical line in [Fig. \[fig:m\*-sfr\]]{}. We then repeat the modelling identically to what has been described in the previous sections.
The bias-corrected star formation sequence for galaxies that have a stellar mass below $M_{*} < 10^{9.5} \si{\solarmass}$ is: $$\begin{aligned}
\label{eq:best-fit-lowmass}
\log \mathrm{SFR}[\si{\solarmass\per\year}] =\; &{\ensuremath{0.83^{+0.10}_{-0.09}}}\log\left(\frac{M_*}{M_{0}}\right) {\ensuremath{-0.79^{+0.05}_{-0.05}}}\nonumber\\
&+ {\ensuremath{2.22^{+0.75}_{-0.76}}}\log\left(\frac{1+z}{1+z_{0}}\right) \pm {\ensuremath{0.47^{+0.06}_{-0.05}}}. $$ The result is essentially the same, with the main difference being a steeper redshift evolution. All parameters are within errors consistent with the relation for our complete sample (also for the uncorrected values, see ). This reflects the fact that we are primarily sensitive to the low-mass end of the galaxy sequence. As this fit utilises only a part of the data we will refer primarily to the fit based on all the data, [Eq. (\[eq:best-fit-corr\])]{}, as the main result in the remainder of the paper. We report the (un)corrected values for all the fits in .
The effect of redshift bins (2D) {#sec:2D-relations}
--------------------------------
Most previous studies have not modelled the redshift evolution of the star formation sequence directly, but have instead divided the data into redshift bins and adopted a non-evolving relation: $\log \mathrm{SFR} = a \log M_{*} + b$. To facilitate the comparison with the literature, we adapt our model to fit the relation in the $(\log M_{*}, \log \mathrm{SFR})$-plane, without taking the redshift evolution into account. This is easily done, by taking a two-dimensional version of our likelihood, disregarding the second, $\log(1+z)$-component in Eq. (\[eq:mean\])–(\[eq:prior\]) — the rest of the modelling is be identical. We note that we still take both heteroscedastic errors as well as intrinsic scatter into account (see [§\[sec:definition\]]{}), however, we do not apply the bias correction.
We model both the entire redshift range, as well as the $0.1 < z < 0.5$ and $0.5 < z < 1.0$ range separately (similar to other studies). The results are collected in . For the full sample the slope is significantly steeper than when we take into account the redshift evolution, when comparing to our uncorrected fits: $$\label{eq:best-fit-twod}
\log \mathrm{SFR}[\si{\solarmass\per\year}]= {\ensuremath{0.89^{+0.05}_{-0.05}}}\log\left(\frac{M_*}{M_{0}}\right) {\ensuremath{-0.82^{+0.04}_{-0.04}}}.$$
This is also the case for the smaller samples in both redshift bins, although the results are consistent with [Eq. (\[eq:best-fit\])]{} within the error bars (which are larger due to lower number statistics). The resulting relations are $$\begin{aligned}
\label{eq:best-fit-twod-0-0p5}
\log \mathrm{SFR}[\si{\solarmass\per\year}] = {\ensuremath{0.86^{+0.09}_{-0.08}}}\log\left(\frac{M_*}{M_{0}}\right) {\ensuremath{-0.92^{+0.07}_{-0.07}}},\end{aligned}$$ for $0.1 < z \leq 0.5$ and $$\begin{aligned}
\label{eq:best-fit-twod-0p5-1}
\log \mathrm{SFR}[\si{\solarmass\per\year}] = {\ensuremath{0.84^{+0.07}_{-0.06}}}\log\left(\frac{M_*}{M_{0}}\right) {\ensuremath{-0.73^{+0.06}_{-0.06}}}\end{aligned}$$ for $0.5 < z < 1.0$.
Given the significant evolution we found in the star formation sequence with redshift, this result is expected. While incidently these slopes are similar to our corrected fits, we caution that this does not imply that not modelling the redshift evolution can circumvent biases introduced by flux-limited observations.
[lccccc]{} Sample & Size & $a$ & $b$ & $c$ & $\sigma_{\mathrm{intr}}$\
3D &\
Full & [179]{}& [$0.79^{+0.05}_{-0.05}$]{}& [$-0.77^{+0.04}_{-0.04}$]{}& [$2.78^{+0.78}_{-0.78}$]{}& [$0.46^{+0.04}_{-0.03}$]{}\
$\log M_{*}[\si{\solarmass}] < 9.5$ & [148]{}& [$0.79^{+0.08}_{-0.07}$]{}& [$-0.73^{+0.04}_{-0.04}$]{}& [$3.39^{+0.91}_{-0.90}$]{}& [$0.49^{+0.04}_{-0.04}$]{}\
\
Full & [179]{}& [$0.83^{+0.07}_{-0.06}$]{}& [$-0.83^{+0.05}_{-0.05}$]{}& [$1.74^{+0.66}_{-0.68}$]{}& [$0.44^{+0.05}_{-0.04}$]{}\
$\log M_{*}[\si{\solarmass}] < 9.5$ & [148]{}& [$0.83^{+0.10}_{-0.09}$]{}& [$-0.79^{+0.05}_{-0.05}$]{}& [$2.22^{+0.75}_{-0.76}$]{}& [$0.47^{+0.06}_{-0.05}$]{}\
2D &\
Full & [179]{}& [$0.89^{+0.05}_{-0.05}$]{}& [$-0.82^{+0.04}_{-0.04}$]{}& & [$0.49^{+0.04}_{-0.04}$]{}\
$ 0.1 < z \leq 0.5$ & [72]{}& [$0.86^{+0.09}_{-0.08}$]{}& [$-0.92^{+0.07}_{-0.07}$]{}& & [$0.57^{+0.07}_{-0.06}$]{}\
$ 0.5 < z < 1.0$ & [107]{}& [$0.84^{+0.07}_{-0.06}$]{}& [$-0.73^{+0.06}_{-0.06}$]{}& & [$0.46^{+0.05}_{-0.05}$]{}\
Discussion {#sec:discussion}
==========
We have modelled the star formation sequence down to at ${0.11}< z < {0.91}$ using a Bayesian framework ([§\[sec:modelling\]]{}) that takes into account both the heteroscedastic errors on the observations as well as the intrinsic scatter in the relation. One major advantage of our framework is that we simultaneously model both the slope and the evolution in the $M_{*}$-SFR relation, while most previous studies have modelled these separately by dividing their sample into different redshift bins. As demonstrated in [§\[sec:2D-relations\]]{}, these results are not necessarily consistent, which can be attributed to evolution taking place within a single redshift bin. Another important difference is that we use the Balmer lines to trace the (dust-corrected) star formation, while most other recent studies have relied on SFRs derived from UV+IR/SED-fitting, using different dust corrections [@Whitaker2014; @Lee2015; @Schreiber2015; @Kurczynski2016].
As described in [§\[sec:global-sample\]]{}, we have found that the star formation sequence (shown in [Fig. \[fig:m\*-sfr\]]{} and [Fig. \[fig:m\*-z-sfr\]]{}) is well described by [Eq. (\[eq:best-fit-corr\])]{} (see also ). We now compare our results to other literature measurements and discuss each aspect of the star formation sequence separately, i.e. the redshift evolution, intrinsic scatter and the slope. We focus particularly on the slope, for which we find the strongest constraints, and continue with a discussion of the physical implications of our results.
Comparison with the literature {#sec:low-mass-end}
------------------------------
### Evolution with redshift {#sec:evolution-with-redshift}
We find that the normalisation in the star formation sequence increases with redshift as $(1+z)^{c}$ with [$c={\ensuremath{1.74^{+0.66}_{-0.68}}}$]{} ([$2.22^{+0.75}_{-0.76}$]{} for $M_{*} < \SI{e9.5}{\solarmass}$). The fact that the normalisation of the star formation sequence increases with redshift is well known and attributed to the change in cosmic gas accretion rates and gas depletion timescales. Most studies have probed the higher mass regime and report values in the range of $\mathrm{sSFR} \equiv \mathrm{SFR}/M_{*} \propto (1+z)^{2.5-3.5}$ at $0 < z < 3$ [e.g. @Oliver2010; @Karim2011; @Ilbert2015; @Schreiber2015; @Tasca2015]. Looking specifically at the low-mass regime, [@Whitaker2014] reports $\text{sSFR} \propto (1+z)^{1.9}$, similar to our result. Their more massive end indeed shows stronger evolution $\text{sSFR} \propto (1+z)^{2.2 - 3.5}$. [@Lee2015] on the other hand, find much steeper evolution, with $\mathrm{sSFR} \propto (1+z)^{4.12 \pm 0.1}$. We note that our parameterisation assumes a power-law type of evolution of the star formation sequence with redshift. We have decided to stick to this very common first-order approximation. Still, one should keep in mind that a more complex evolution with redshift is possible, both non-linear in time as well as a different evolution in different mass regimes. We do not find strong constraints on the redshift evolution due to our relatively small redshift range from $z=0.1$ to $z=0.91$. Still, the results from [§\[sec:2D-relations\]]{} show that it is important to take the redshift evolution into account, in order to get a robust constraint on the slope.
### Intrinsic scatter {#sec:intrinsic-scatter}
Constraining the intrinsic scatter in the star formation sequence has proven to be challenging as one has to separate the intrinsic scatter from the measurement error [e.g. @Noeske2007a; @Salim2007; @Salmi2012; @Whitaker2012; @Guo2013; @Speagle2014; @Schreiber2015]. This challenge in particular motivates our adopted model, which directly constrains the amount of intrinsic scatter in the relationship, even in the presence of measurement errors. Meanwhile, our measurements are not affected by binning, e.g. we do not boost the scatter artificially because of evolution of the star formation sequence within a single bin.
In our best fit model we find [$\sigma_{\text{intr}}={\ensuremath{0.44^{+0.05}_{-0.04}}}$]{} dex, which is larger than the value of $\sim 0.2 - 0.4$ dex that is commonly found [e.g. @Speagle2014; @Schreiber2015]. [@Kurczynski2016] determined an intrinsic scatter of $\sigma_{\rm intr} = 0.427 \pm 0.011$ in their lowest redshift bin ($0.5 < z < 1.0$) in the HUDF, similar to our result, but found significantly smaller scatter at higher redshifts. They determined the intrinsic scatter by decomposing the total scatter ($\sigma_{\rm Tot} = 0.525$) using the covariance matrix between $M_{*}$ and SFR determined from their SED fitting.
There are several effects that could potentially affect the scatter. Measurement outliers are not a cause of concern for the intrinsic scatter as they are taken into account by the likelihood approach. However, if galaxies are included in the sample that are not on the $M_{*}$-SFR relation, such as red-sequence galaxies or starbursts, then these might artificially increase the scatter. We argue that the former is unlikely as our selection criteria based on the 4000 Å break and the [$\mathrm{H}\alpha\ \lambda 6563$]{} or [$\mathrm{H}\beta\ \lambda 4861$]{} equivalent width effectively remove all red-sequence galaxies from the sample. On the other hand, our sample does include a small number of galaxies that are offset from the relation towards high SFRs. We verified however that removing all galaxies with a $\mathrm{sSFR} > \SI{10}{Gyr^{-1}}$ from the sample does not significantly increase or decrease the scatter.
Hypothetically, if the error bars on the SFR are underestimated, this will artificially boost the intrinsic scatter in the relationship. To determine the influence of the size of the error bars we redid the modelling while folding in an additional error on the SFR of 0.2 dex in quadrature (effectively doubling the average error bars); this decreased the scatter by 20% to $\sim 0.4$ dex. The sample size does not seem to affect the measurement and splitting our sample did not yield significantly larger scatter (see [§\[sec:2D-relations\]]{}).
Assuming our measured scatter is real, it might be that previous studies have underestimated the amount of intrinsic scatter. One potential danger might lie in the derivation of both stellar mass and SFR from the same photometry. Especially in SED modelling this might introduce correlations between $M_{*}$ and SFR as both are regularised through the same star formation history in the model spectrum which could artificially decrease the scatter.
More physically, the difference could also in part be due to the fact that the Balmer lines trace the SFR on shorter timescales (stars with ages and masses ) than the UV does (ages of and masses ; e.g. @Kennicutt1998 [@Kennicutt2012]). Simulations have indeed found that SFRs averaged over timescales decreasing from $10^{8}$ to $10^{6}$ yr could be significantly larger [@Hopkins2014; @Sparre2015], particularly if star formation histories are bursty [e.g. @Dominguez2015; @Sparre2017].
Furthermore, as the recent star formation histories of low-mass galaxies are more diverse, it can be expected that there is more scatter in the star formation sequence at low stellar masses. This indeed has been predicted by simulations [e.g. @Hopkins2014; @Sparre2017] as well as semi-analytical models [e.g. @MitraS_17a]. Observing such a trend requires a large and highly complete sample of galaxies over an extended mass range and hence evidence has been inconclusive. Using a large sample of galaxies from the SDSS, [@Salim2007] reported a decrease in the scatter of $-0.11~\mathrm{dex}^{-1}$ from -, but such a trend with mass has not been confirmed by studies at higher masses [@Whitaker2012; @Guo2013; @Schreiber2015; @Kurczynski2016]. Recently though, [@Santini2017] have found indications of decreasing scatter with mass in the Frontier Fields, albeit at higher redshifts ($z>1.3$).
A large and complete sample of galaxies, covering the ($\log M_{*}$, log SFR, $\log(1+z)$)-space, with independent stellar mass and SFR estimates, is required to get a firm handle on the intrinsic scatter in the star formation sequence.
### Slope {#sec:slope}
{width="\textwidth"}
We find a best-fit (median posterior) slope of the star formation sequence of [$a={\ensuremath{0.83^{+0.07}_{-0.06}}}$]{} (log SFR $\propto a \log M_{*}$). This slope is determined from galaxies that are more than an order of magnitude lower in mass than most earlier studies at $z>0$, i.e. at to , whereas most previous studies [e.g. @Speagle2014; @Lee2015; @Schreiber2015] have been primarily sensitive to a higher mass range from to . For reference, we plot the polynomial fit from [@Whitaker2014] (down to their mass completeness limit, based on stacking) in [Fig. \[fig:m\*-sfr\]]{}.
Recent studies have typically observed a shallower slope at the high-mass end, i.e. above [e.g. @Whitaker2014]. [@Gavazzi2015] find a turnover mass of $M_{*} \sim \SI{e9.7}{\solarmass}$ at $z=0.55$ (after converting their result to a Chabrier IMF), increasing with redshift. As discussed in [§\[sec:low-mass-sample\]]{}, excluding galaxies above $M_{*} > \SI{e9.5}{\solarmass}$ has no significant effect on the slope. Only $15/{179}\approx 8.5\%$ of galaxies in our sample have $M_{*} > \SI{e10}{\solarmass}$ and thus our result is not very sensitive to this turn-over. In light of this, we limit the following discussion to studies which specifically probe the mass range below the turnover of the star formation sequence.
Our best-fit slope of [$0.83^{+0.07}_{-0.06}$]{} is compared to the values found by other recent studies in [Fig. \[fig:params\]]{} where we focus on studies with similar redshift ranges (i.e. $0<z<1$) and which extend well below $M_{*} <\SI{e10}{\solarmass}$. The slope in this regime is notably steeper than the consensus relation from [@Speagle2014] who reported $a = 0.6 - 0.7$ at our redshifts, due to the fact that this compilation is for a mass range of $\log M_{*}[\si{\solarmass}] = 9.7 - 11.1$, where the slope is significantly shallower. Our slope is shallower than the low-mass power-law slope from [@Whitaker2014] ($a = 0.94 \pm 0.03$ for $M_{*} < \SI{e10.2}{\solarmass}$) from the 3D-HST catalogues in CANDELS, but is consistent with the global slope of $a = 0.88 \pm 0.06$ reported by [@Lee2015] in a large sample of star-forming galaxies in COSMOS. [@Kurczynski2016] have also presented a characterisation of the star formation sequence in the HUDF, based on the CANDELS/GOODS-S [@Santini2015] and UVUDF [@Rafelski2015] catalogues. In their lowest redshift bin ($0.5 < z < 1.0$), which goes down to $M_{*} \sim 10^{7.5} \si{\solarmass}$ they find a slope of $a=0.919 \pm 0.017$, which is also steeper (marginally consistent) compared to what find. We note that they determined both masses and SFRs from the SED modelling, taking into account the correlations between the parameters, as their study was focused particularly on measuring the intrinsic scatter, see [§\[sec:intrinsic-scatter\]]{}. In the same field [@Bisigello2017] find a slope of $0.9 \pm 0.01$ ($0.5 \leq z < 1.0$), after selecting galaxies with $\log \mathrm{sSFR[\si{\per\giga\year}]} < - 9.8$.
The Sloan Digital Sky Survey (SDSS; @York2000 [@Abazajian2009]) serves as a natural reference for Balmer line-derived SFRs in the local universe and since [@Brinchmann2004] different studies have derived the star formation sequence slope [e.g, @Salim2007; @Elbaz2007]. The most recent of these is [@Renzini2015], who measure the slope of the ridge line in the $M_{*}-N\times \mathrm{SFR}$-plane (where $N$ is the number of galaxies in every $M_{*}$-SFR bin) and find $a=0.76\pm0.01$, which is significantly flatter than our results.
Taken at face value, our slope of [$a={\ensuremath{0.83^{+0.07}_{-0.06}}}$]{} is inconsistent with a linear slope ($a=1$). A value (close to) unity may have been expected on the basis of simulations (see next section), which is also evident from the fact that several parameterisations of the star formation sequence asymptote to a linear relation at low mass [e.g. @Schreiber2015; @Tomczak2016]. An independent motivation for a near-linear value comes from the fact that there is very little evolution in the faint slope of the stellar mass function of star-forming galaxies up to $z=2$ (see, e.g. [@Tomczak2014; @Davidzon2017] for recent results). To first order, this may implies self-similar mass growth for low-mass galaxies (i.e. constant sSFR which implies a linear slope for the star formation sequence), unless balanced by mergers [@Peng2014]. [@Leja2015] investigated the link between the slope of the star formation sequence and the stellar mass function. While they do not provide precise constraints on the low-mass slope at low redshift (due to the challenge of disentangling growth through star formation and mergers), their results indicate that a sub-linear low-mass slope is still consistent with the stellar mass functions at $z<1$.
### Evolution of the low-mass slope {#sec:evolution-low-mass}
Combining results from the local universe out to redshift $z \sim 6$, [@Speagle2014] found evidence for an evolving slope at the high-mass end ($M_{*} > \SI{e9.7}{\solarmass}$), where the slope gets shallower with redshift [cf. @Abramson2016 Fig. 5]. Given the turnover in the star formation sequence at high mass, it is important to disentangle to what extent the evolution in the slope is due to different studies being sensitive to distinct mass regimes. Our data are too sparse in redshift space to simultaneously constrain the evolution of the slope (and hence we have adopted a single power-law slope for the sequence).
In light of the potential redshift evolution of the slope, we plot the slope as a function of redshift in [Fig. \[fig:params\]]{}, compared to literature results which probe the mass range $M* < \SI{e10}{\solarmass}$ at $z < 1.5$. [Fig. \[fig:params\]]{} provides evidence for evolution of the low-mass slope with redshift. However, we caution against a too strong interpretation of such a trend as the literature suffers from studies probing distinct mass ranges (sometimes including the turn-over regime). What further complicates a fair comparison is that different tracers of star formation probe different timescales and additionally use varying dust corrections, which are not necessarily consistent [e.g. @Davies2016]. A consistent analysis of the low-mass galaxy population out to higher redshifts is important to quantify potential evolution in the low-mass slope.
The MS slope — a quantitative comparison to models {#sec:model}
--------------------------------------------------
The galaxy main sequence (MS) is a natural outcome of hydrodynamical models (e.g. Fig. 1b in @Bouche2005; @Dave2008 [@GenelS_14a; @Torrey2014; @Kannan2014; @Hopkins2014; @Sparre2015; @Furlong2015]) and in semi-analytical models [e.g. @Somerville2008; @DuttonA_10a; @Cattaneo2011; @Mitchell2014; @Henriques2015; @Hirschmann2016; @Cattaneo2017]. These models have reported a slope (and scatter) that, in general, is broadly consistent with observations, but the quantitative details regarding the slope and/or the evolution of the main sequence often do not match observations.
Since the pioneering work of @Daddi2007 and @Elbaz2007, it has been noted that the redshift evolution of the main sequence normalisation, in particular around $z=2$, is a challenge for models [e.g. @Dave2008; @Damen2009; @BoucheN_10a; @DuttonA_10a; @DekelA_14a; @Torrey2014; @GenelS_14a; @Mitchell2014; @Furlong2015; @Sparre2015; @Abramson2016; @Santini2017]. Here, we focus on a quantitative comparison of the slope of the main sequence ($\mathrm{SFR} \propto M_{*}^a$) with various models, given that our study yields the tightest constraint on this parameter (compared to the other parameters in the model).
The Illustris simulations [@Vogelsberger2014; @GenelS_14a; @Sparre2015] produce a main-sequence with a slope $a$ that is slightly sub-linear with $a\lessapprox1.0$. In particular, @GenelS_14a noted that sSFR goes as $\simeq-0.1$ with stellar mass and using the results from @Sparre2015, we find that the main sequence in Illustris goes as $\mathrm{SFR} \propto M_{*}^{\approx0.95}$. The EAGLE simulations [@Schaye2015; @Crain2015] also allow an investigation of the main sequence and @Furlong2015 [their Fig. 5], showed that the sSFR is constant with $M_{*}$ from to at redshifts $z=0.1$, 1.0 and 2.0, with a relatively steep decline above . Quantitatively, below , the slope of the main sequence $a$ in @Furlong2015 is $a\approx1.04$. The MS slope for the Illustris and EAGLE simulations are shown in [Fig. \[fig:params\]]{} as the open circles and triangle symbols, respectively. In the FIRE simulations [@Hopkins2014], [@Sparre2017] focused on studying the scatter in the main sequence for different tracers of SFR and shows a slope of $a\approx0.98$ when using the FUV (their Fig. 2).
The MS slope has also been a challenge for semi-analytical models because different (regular) feedback prescriptions do not alter the MS slope as shown in @DuttonA_10a and discussed in @Mitchell2014 (however, it can alter the slope in hydrodynamical simulations, e.g. @Haas2013a [@Haas2013b; @Crain2015]). @Mitchell2014 performed a detailed comparison between predictions from the GALFORM semi-analytical models with observations and their fiducial model produces a MS slope of $a\approx0.85$ (shown in [Fig. \[fig:params\]]{} as the down-pointing triangles). Recently, the semi-analytical model of @Cattaneo2017 using the GALICS2 code was set to reproduce the local luminosity function and the local MS slope simultaneously. Their MS slope is $a\approx0.8$ (open square in [Fig. \[fig:params\]]{}), but we caution their use of an extreme feedback model, where the mass loading $\eta$ is $\eta\propto V^{-6}$, where $V$ is the halo virial velocity. Such a steep scaling between galaxy mass and wind loading is not supported by the data [e.g. @SchroetterI_16a].
@BoucheN_10a used a simple toy model for galaxy (self-)regulation with which they showed that variations in feedback prescriptions or in the laws of star formation have no impact on the MS slope. They argued that while ejective feedback alone is not sufficient to bring the theoretical slope of the main-sequence in agreement with observations, preventive feedback can easily do so as several studies have shown [@Dave2012; @Lu2015; @MitraS_15a; @MitraS_17a]. However, while the MS slope of @BoucheN_10a is sub-linear with $a\approx0.8$, a quantitative analysis reveals that the slope varies rapidly with stellar mass, likely due to the limitations of the model. Indeed, the MS slope of @BoucheN_10a goes from 0.7 at $M_{*}\sim\SI{e9.5}{\solarmass}$ to 0.9 at $M_{*}\sim\SI{e10.5}{\solarmass}$. The range of values is indicated by the light grey box in [Fig. \[fig:params\]]{}.
@MitraS_15a expanded the self-regulation model of @BoucheN_10a [@Dave2012 and others] with physically motivated parameters and attempted to determine these parameters using a Bayesian MCMC approach on a set of observed scaling relations at $0<z<2$. Their fiducial model yields a MS with a slope that is quasi-linear with $a\sim0.95$ in our mass regime, i.e. below $\SI{e10}{\solarmass}$. Their MS slope is shown as the dark grey band in [Fig. \[fig:params\]]{}.
Generally speaking, in the low-mass regime below $\SI{e10}{\solarmass}$, hydrodynamical simulations have steeper MS slopes with $a\approx1.0$ whereas our estimate ([$a={\ensuremath{0.83^{+0.07}_{-0.06}}}$]{}) at $z<1$ and recent observations covering that mass range indicate $a<1.0$ (see [Fig. \[fig:params\]]{}). The reason that models tend to predict a steeper main sequence slope lies in the underlying feature in hydrodynamical simulations and semi-analytical models, where the growth rate for dark matter halos $\dot M_{\rm h}$ scales with mass as $\dot M_{\rm h}\propto M_{\rm h}^{1.15}$ [@BirnboimY_07a; @GenelS_08a; @DekelA_09a; @FakhouriO_08a; @NeisteinE_08a], in combination with rapid gas cooling.
Implications of a shallow slope
-------------------------------
As noted originally by @Noeske2007b and discussed in @Mitchell2014 and @Abramson2016, a MS with a sub-linear slope, $\mathrm{SFR} \propto M_{*}^{a}$ with $a < 1$, implies downsizing where lower-mass galaxies have longer $e$-folding time and a later onset of star formation. This downsizing effect would be amplified if the MS slope is substantially flatter above as some studies have indicated [@Whitaker2014; @Schreiber2015; @Lee2015; @Tomczak2016]. This turnover has generally been attributed to either a morphological transition, such as bulge growth [@Abramson2014; @Lee2015; @Whitaker2015], or a reduced star formation efficiency [@Schreiber2016].
Our result, that the slope of the main sequence is sub-linear in the low-mass regime, implies that there are processes at work which either: (1) affect the conversion of the accreted gas into stars through increased (supernova) feedback or a decrease in the SF efficiency; or (2) prevent the accretion of gas onto low-mass galaxies. These two processes might conspire with the fact that the gravitational potential is shallower in low-mass galaxies [@MitraS_15a].
In hydrodynamical simulations low-mass galaxies (up to halo masses of ) obtain their gas primarily through ‘cold’-accretion [@Keres2005; @vandeVoort2011], where the gas is never heated to the virial temperature, while ‘hot’ accretion, where gas is first shock heated to the virial temperature and then cools and accretes, is dominant for more massive galaxies. A candidate process is feedback from gravitational heating, due to the formation of virial shocks [e.g. @Faucher-Giguere2011], which becomes more effective at higher masses, however, can still play a role down to halo masses of . The heating of gas through winds (from either supernovae or black hole feedback) can also prevent the gas from flowing into the galaxy [@Oppenheimer2010; @Faucher-Giguere2011; @vandeVoort2011], in particular in low-mass galaxies. However, [@Schaye2010] pointed out that this type of feedback mainly has a regulatory effect on the gas infall.
As noted by @DuttonA_10a, @BoucheN_10a, and @Mitchell2014, in semi-analytical models, the MS slope is rather insensitive to the ejective (regular) feedback mechanisms [^8], such as the heating of gas through winds and/or the star formation efficiency [@Kennicutt1998] because they act primarily on the gas content. Hence, the SFR and stellar mass are affected in a similar way, leaving the slope unchanged, unless the ejective feedback prescription is strongly mass dependent with $\eta\propto V^{-6}$, as in @Cattaneo2017. In addition, @Mitchell2014 showed that the slope is also insensitive to the gas re-incorporation prescription [see also @MitraS_15a].
Preventive processes [@BlanchardA_92a; @Gnedin2000a; @Mo2005; @Lu2007; @Okamoto2008] that tend to be mass dependent can more easily impact the MS slope, the Tully-Fischer relation, and the luminosity function as argued by @BoucheN_10a. A preventive process which can prevent the inflow of gas specifically in low-mass halos is photoionisation heating [@Quinn1996]. While it has been argued that this process is primarily effective in dwarf galaxies and becomes ineffective above halo masses of a few times [e.g. @Okamoto2008], @CantalupoS_10a suggest that photoionisation may still play a role for more massive halos if there is significant star formation.
Summary and conclusions {#sec:conclusions}
=======================
We have exploited the unique capabilities of the MUSE instrument to investigate the star formation sequence for low-mass galaxies at intermediate redshift (${0.11}< z < {0.91}$). From the large number of sources detected with MUSE in the HUDF and HDFS we have constructed a sample of [179]{}star-forming galaxies down to $M_{*} \sim \SI{e8}{\solarmass}$, with a number of objects at even lower masses ([Fig. \[fig:mass-histogram\]]{}). The accurate spectroscopic redshifts from MUSE are combined with the deep photometry available over the HUDF and HDFS to determine a robust mass estimate for the galaxies in our sample through stellar population synthesis modelling.
With MUSE we can detect star-forming galaxies down to SFR $\sim \SI{e-3}{\solarmass \per \year}$ ([Fig. \[fig:m\*-sfr\]]{}). We show that we can determine robust, dust-corrected SFR estimates from [$\mathrm{H}\alpha\ \lambda 6563$]{} and [$\mathrm{H}\beta\ \lambda 4861$]{}recombination lines, by comparing the SFRs from different tracers ([Fig. \[fig:sfr-consistency\]]{}). A dust-corrected star formation rate is inferred from the [$\mathrm{H}\alpha\ \lambda 6563$]{} and [$\mathrm{H}\beta\ \lambda 4861$]{} emission lines observed with S/N > 3 in the MUSE spectra.
We characterise the star formation sequence by a Gaussian distribution around a plane ([Eq. (\[eq:linear\])]{}). This methodology is chosen to maximally exploit the data set taking into account heteroscedastic errors. We constrain the slope, normalisation, intrinsic scatter, and evolution with redshift from the posterior probability distribution via MCMC methods ([Fig. \[fig:triangle\]]{}).
We analyse the robustness of our model and the influence of the MUSE detection limit on the derived properties of the star formation sequence, by determining how well we can recover the parameters from a sample of simulated relations (detailed in ). Using the results, we correct our inferred parameters for observational biases.
We report a best-fit description of the low-mass end of the galaxy star formation sequence of $\log \mathrm{SFR} = {\ensuremath{0.83^{+0.07}_{-0.06}}}\log M_{*} {\ensuremath{-0.83^{+0.05}_{-0.05}}}+ {\ensuremath{1.74^{+0.66}_{-0.68}}}\log(1+z)$ between ${0.11}< z < {0.91}$, shown in [Fig. \[fig:m\*-z-sfr\]]{}. The full description of our parameters, including errors and normalisation, is found in [Eq. (\[eq:best-fit-corr\])]{}.
The intrinsic scatter around the sequence is found to be [$\sigma_{\text{intr}}={\ensuremath{0.44^{+0.05}_{-0.04}}}$]{} dex (in log SFR). This is notably higher than the average value reported in literature ($\sim 0.3$ dex), which could be attributed to a combination of the Balmer lines probing star formation on shorter timescales and the star formation histories of low-mass galaxies being more diverse.
Excluding massive galaxies (with $M_{*} > \SI{e9.5}{\solarmass}$) has no significant effect on the best-fit parameters, indicating we are primarily sensitive to low-mass galaxies. Notably though, we find that the slope steepens when splitting our sample into one or multiple redshift bins, with the values going up to $\log \mathrm{SFR} [\si{\solarmass\per\year}] = {\ensuremath{0.89^{+0.05}_{-0.05}}}\log
M_{*}[\si{\solarmass}]$. This shows the importance of taking into account the evolution with redshift when deriving the properties of the star formation sequence.
The slope of the star formation sequence is an important observable as it provides information on the processes that regulate star formation in galaxies. Our slope is shallower than most simulations and (semi-)analytical models predict, which find a (super-)linear slope essentially due to the growth rate of dark matter halos. Feedback processes operating specifically in the low-mass regime, which affect the accretion of gas onto galaxies and/or subsequent star formation, are required to reconcile these differences. Models suggest that supernova feedback or a decreased star formation efficiency do not affect the slope of the star formation sequence. Instead, processes that prevent the accretion of gas onto low-mass galaxies are thought to play an important role in determining the slope of the star formation sequence in the low-mass regime.
We would like to thank the referee for providing a constructive report that helped improve the quality of the paper. LB would like to thank the participants of the Lorentz Center Workshop on *A Decade of the Star-Forming Main Sequence* for beneficial discussions. We gratefully acknowledge the developers of <span style="font-variant:small-caps;">IPython</span>, <span style="font-variant:small-caps;">Numpy</span>, <span style="font-variant:small-caps;">Matplotlib</span>, and <span style="font-variant:small-caps;">Astropy</span> [@Perez2007; @VanDerWalt2011; @Hunter2007; @Robitaille2013] and <span style="font-variant:small-caps;">Topcat</span> [@Taylor2005] for their development of the software used at various stages during this work. JB acknowledges support from Funda[ç]{}[ã]{}o para a Ci[ê]{}ncia e a Tecnologia (FCT) through national funds (UID/FIS/04434/2013) and Investigador FCT contract IF/01654/2014/CP1215/CT0003., and from FEDER through COMPETE2020 (POCI-01-0145-FEDER-007672). JS acknowledges support from the Netherlands Organisation for Scientific Research (NWO) through VICI grant 639.043.409. NB and TC acknowledge funding by the ANR FOGHAR (ANR-13-BS05-0010-02), the OCEVU Labex (ANR-11-LABX-0060), and the A\*MIDEX project (ANR-11-IDEX-0001-02) funded by the “Investissements d’avenir” French government programme. RB acknowledges support from the ERC advanced grant 339659-MUSICOS.
Simulations {#sec:simulations}
===========
Selection function and completeness {#sec:selection-function-completeness}
-----------------------------------
We have selected galaxies based on the signal-to-noise of their emission lines, without any photometric preselection. This means the selection function is essentially determined by the emission line sensitivity. In general, one might expect galaxies with higher S/N in their emission lines to have a higher SFR at a fixed mass, or similarly, for galaxies with the same S/N to have a higher SFR at higher redshift, which potentially introduces biases in our results. Additionally, we can only observe galaxies that have Balmer lines in the spectral range of MUSE ($z < 0.91$).
To investigate the influence of these selections, we determine how well we can recover the true parameters of the star formation sequence from a set of mock samples of galaxies, after applying the flux limit from our MUSE observations.
We determine the influence of the selection function on the inferred parameters by simulating mock data for a range of ‘true’ parameters. The range of values for each mock parameter is listed in , which combine to form a grid of $N={1260}$ points. The extent of grid is chosen such that it encompasses a wide range of possible parameters and we find that the results are consistent if we enlarge the grid even further (note that, if the grid is taken too large, non-linearities may arise at the extreme values which potentially bias the linear transformation approach of [§\[sec:line-transf\]]{}). We denote each set of parameters as $\mathbf{x}_{\mathrm{true}, k} = \left(\hat a, \hat c, \hat b, \hat
\sigma_{\mathrm{intr}}\right)^{T}$ with $k = 1, ..., N$.
We generate realistic mock data for each set of parameters through the following procedure: We sample [100]{} galaxies from a uniform distribution in both mass ($7.0 < \log M_{*}[\si{\solarmass}] < 10.5$) and redshift ($0.1 < z < 1$). Given the mass and redshift, we compute the SFR (via [Eq. (\[eq:linear\])]{}), i.e. assuming a mock main sequence distribution with slope $\hat a$ and evolution $\hat c$. We choose our normalisation ($\hat b$) such that a galaxy at $z=0$ has a SFR of , similar to our results and, e.g. the Milky Way [@Chomiuk2011], i.e. we take a zero-point of $b_{0} = -10$. We then sample up to $b_{\mathrm{offset}} = \pm 0.4$ dex above and below this zero-point. We provide each galaxy with a random offset from the main sequence (perpendicular to the $(\log M_{*}, \log \mathrm{SFR}$)-relation) drawn from $\mathcal{N}(0, \hat \sigma_{\mathrm{intr}})$. Finally, we apply a random measurement error for each galaxy in both log stellar mass and log SFR of 0.3 dex (i.e. drawn from $\mathcal{N}(0,0.3)$) and in log redshift of dex ($\sim \mathcal{N}(0,\num{5e-4})$), similar to the observations.
We then apply the same flux limit as our shallowest MUSE observations, namely in the with , and mark all ‘observed’ galaxies as those that fall above our detection threshold (we do not take an additional factor for dust into account as our galaxies are not very dusty on average). We then fit the observed galaxies above the flux limit. Repeating this process [30]{} times for each individual set of parameters $\mathbf{x}_{\mathrm{true}, }$ and marginalising over the combined posterior distribution, we determine the corresponding recovered parameters $\mathbf{x}_{\mathrm{out}, k} = \left( a, c, b,
\sigma_{\mathrm{intr}}\right)^{T}$.
As an example, we show one the experiment for a particular set of parameters in [Fig. \[fig:sfc\]]{}. It is clear that the recovered parameters are biased towards a shallower slope and a steeper redshift evolution. The magnitude of this bias depends on all the parameters and becomes more severe for steeper slopes and shallower redshift evolutions.
To check our methods, we also fit all simulated galaxies (without discarding any data). Reassuringly, we recover our input parameters to within the errors, even when simulating only 100 galaxies. Since our actual sample size is [179]{}galaxies, we are in principle able to recover the true parameters of the relation, even in the case of intrinsic scatter and heteroscedastic errors. One feature that does draw attention is that the redshift evolution is marginally steeper than the input relation (but admittedly poorly constrained and still consistent within the error). This can be explained due to an intricacy of the model, which assumes that the intrinsic scatter about the relation is along the normal vector to the plane ($\sigma_{\perp}$ in [§\[sec:definition\]]{}), i.e. also in the $\log(1+z)$-direction. If the data are truncated and there is a non-zero slope $(|c| > 0)$ in redshift space, this may introduce an artificial bias in the corresponding slope (and scatter) as the truncation boundaries are not parallel to the normal vector. Given the fact that our data (and mock sample) are limited in redshift space by the spectral range of MUSE, this means that we may have slight artificial bias towards a steeper redshift evolution. For interpreting the intrinsic scatter this is not a problem as we can project the scatter along the (physical) $\log \mathrm{SFR}$-axis (which is our $\sigma_{\rm intr}$).
With our simulations in hand however, we are now in place to apply a correction for both biases identified above.
[lccc]{} & min & max & step\
$\hat a$ & 0.7 & 1.1 & 0.05\
$\hat c$ & 1.5 & 4.5 & 0.5\
$b_{\mathrm{offset}}$ & -0.4 & 0.4 & 0.2\
$\hat \sigma_{\mathrm{intr}}$ & 0.3 & 0.6 & 0.1\
\
{width="\textwidth"}
Transformation {#sec:line-transf}
--------------
The simulations show a reasonably well behaved transformation between the true and recovered slope. We therefore model the mock data with an affine transformation, to be able to transform between the measured and true parameters.
We try to find the best transformation matrix $A$ and vector $\mathbf{b}$ between the measured and true parameters. For each set of input ($\mathbf{x}_{\mathrm{true}, k}$) and output ($\mathbf{x}_{\mathrm{out}, k}$) parameters we have: $$\begin{aligned}
\mathbf{x}_{\mathrm{out}, k} &\approx A \mathbf{x}_{\mathrm{true}, k} + \mathbf{b}\end{aligned}$$
We minimise the function $$\begin{aligned}
S(A, \mathbf{b}) &= \sum_{k=1}^{N} ||\mathbf{x}_{\mathrm{out}, k} - A \mathbf{x}_{\mathrm{true}, k} - \mathbf{b}||_2^2\end{aligned}$$ with respect to each component of $A$ and $\mathbf{b}$ in order to find the best-fit transformation $A$ and $\mathbf{b}$ [@Spath2004]. We note that we do not take the errors on each point $\mathbf{x}_{\mathrm{out}}, k$ into account as their magnitudes are all comparable (essentially adding a constant to the equation).
With the best-fit $A$ and $\mathbf{b}$ in hand, we can then invert the equation to obtain the relation between the observed and the recovered ‘true’ parameters, which denote as $\mathbf{x}'_{\mathrm{true}}$: $$\begin{aligned}
\label{eq:inv-trans}
\mathbf{x}'_{\mathrm{true}} &\approx A^{-1} \left(\mathbf{x}_{\mathrm{out}} - \mathbf{b}\right)\\
\begin{pmatrix}
a' \\ c' \\ b' \\ \sigma'_{\mathrm{intr}}
\end{pmatrix} &= \begin{pmatrix}
{
1.336 & 0.014 & -0.150 & 0.171 \\
0.638 & 0.863 & 0.574 & -2.621 \\
-0.178 & -0.008 & 1.175 & -0.185 \\
0.285 & 0.009 & -0.044 & 1.091}\end{pmatrix}
\left(\begin{pmatrix}
a \\ c \\ b \\ \sigma_{\mathrm{intr}}
\end{pmatrix} - \begin{pmatrix}
{
0.293 \\
0.061 \\
-0.194 \\
0.236}\end{pmatrix}\right)\end{aligned}$$
For our simulated data, we show the distribution of the difference between the recovered parameters ($\mathbf{x}'_{\mathrm{true}}$) and the true parameters ($\mathbf{x}_{\mathrm{true}}$) in [Fig. \[fig:corner-iosim\]]{}. We recover the input parameters very well, with no mean offset between the recovered and the true parameter. This shows that the transformation (i.e. $A$ and $\mathbf{b}$) are very well determined. Furthermore, the scatter in the differences is much smaller than the average uncertainty on each parameter obtained from the observations (of order $\sim 1\%$). As an illustration, we show the inverse transformation applied to the simulation by the red lines in [Fig. \[fig:sfc\]]{}, which are now in good agreement with the true values (dashed lines).
In summary, the transformation obtained from the best-fit $A$ and $\mathbf{b}$ is a very accurate description of the bias induced by the flux limit in our simulated data. We use the inverse of this transformation, [Eq. (\[eq:inv-trans\])]{}, in [§\[sec:results\]]{} to correct our inferred posterior density distribution from modelling the MUSE data.
{width="70.00000%"}
[^1]: Based on observations made with ESO telescopes at the La Silla Paranal Observatory under programme IDs ID 060.A-9100(C), 094.A-2089(B), 095.A-0010(A), 096.A-0045(A), and 096.A-0045(B).
[^2]: There is a tension between the shallow slope of the observed main sequence with the super-linear slope expected in models, which is set by the index of the initial dark matter power spectrum [@BirnboimY_07a; @NeisteinE_08a; @Correa2015a; @Correa2015b].
[^3]: For an alternative interpretation, cf. [@Gladders2013; @Kelson2014; @Abramson2016].
[^4]: <http://mpdaf.readthedocs.io/en/latest/muselet.html>
[^5]: Following the convention that emission-line equivalent widths (EQW) are negative, this translates to excluding EQW > $-2$Å.
[^6]: It is important to point out that this is not caused by stellar absorption in the continuum as this is taken into account when modelling the emission lines with .
[^7]: Throughout this paper we consistently use ‘$\log$’ for the base-10 logarithm and ‘$\ln$’ for the base-$e$ logarithm, with one exception: we stick to standard terminology and call $\ln{\mathcal{L}}$ the ‘log-likelihood’.
[^8]: with mass loading $\eta\propto V^{-1}$ or $\eta\propto V^{-2}$ for momentum or energy-driven winds, respectively.
|
---
abstract: 'In this paper, we describe our submissions to the WMT17 Multimodal Translation Task. For Task 1 (multimodal translation), our best scoring system is a purely textual neural translation of the source image caption to the target language. The main feature of the system is the use of additional data that was acquired by selecting similar sentences from parallel corpora and by data synthesis with back-translation. For Task 2 (cross-lingual image captioning), our best submitted system generates an English caption which is then translated by the best system used in Task 1. We also present negative results, which are based on ideas that we believe have potential of making improvements, but did not prove to be useful in our particular setup.'
author:
- Jindřich Helcl
- |
Jindřich Libovický\
Charles University, Faculty of Mathematics and Physics\
Institute of Formal and Applied Linguistics\
Malostransk' e n' aměst' i 25, 118 00 Prague, Czech Republic\
[{helcl, libovicky}@ufal.mff.cuni.cz]{}
bibliography:
- 'emnlp2017.bib'
title: '[CUNI]{} System for the [WMT17]{} Multimodal Traslation Task'
---
Introduction
============
Recent advances in deep learning allowed inferring distributed vector representations of both textual and visual data. In models combining text and vision modalities, this representation can be used as a shared data type. Unlike the classical natural language processing tasks where everything happens within one language or across languages, multimodality tackles how the language entities relate to the extra-lingual reality. One of these tasks is multimodal translation whose goal is using cross-lingual information in automatic image captioning.
In this system-description paper, we describe our submission to the WMT17 Multimodal Translation Task. In particular, we discuss the effect of mining additional training data and usability of advanced attention strategies. We report our results on both the 2016 and 2017 test sets and discuss efficiency of tested approaches.
The rest of the paper is organized as follows. Section \[sec:task\] introduces the tasks we handle in this paper and the datasets that were provided to the task. Section \[sec:related\] summarizes the state-of-the-art methods applied to the task. In Section \[sec:models\], we describe our models and the results we have achieved. Section \[sec:negative\] presents the negative results and Section \[sec:conclusion\] concludes the paper.
Task and Dataset Description {#sec:task}
============================
The challenge of the WMT Multimodal Translation Task is to exploit cross-lingual information in automatic image caption generation. The state-of-the-art models in both machine translation and automatic image caption generation use similar architectures for generating the target sentence. The simplicity with which we can combine the learned representations of various inputs in a single deep learning model inevitably leads to a question whether combining the modalities can lead to some interesting results. In the shared task, this is explored in two subtasks with different roles of visual and textual modalities.
In the multimodal translation task (Task 1), the input of the model is an image and its caption in English. The system then should output a German or French translation of the caption. The system output is evaluated using the METEOR [@denkowski2011meteor] and BLEU [@papineni2002bleu] scores computed against a single reference sentence. The question this task tries to answer is whether and how is it possible to use visual information to disambiguate the translation.
In the cross-lingual captioning task (Task 2), the input to the model at test-time is the image alone. However, additionally to the image, the model is supplied with the English (source) caption during training. The evaluation method differs from Task 1 in using five reference captions instead of a single one. In Task 2, German is the only target language. The motivation of Task 2 is to explore ways of easily creating an image captioning system in a new language once we have an existing system for another language, assuming that the information transfer is less complex across languages than between visual and textual modalities.
Data
----
en de fr
------------------ ------- ------- -------
Train. sentences
Train. tokens 378k 361k 410k
Avg. \# tokens 13.0 12.4 14.1
\# tokens range 4–40 2–44 4–55
Val. sentences
Val. tokens 13k 13k 14k
Avg. \# tokens 13.1 12.7 14.2
\# tokens range 4–30 3–33 5–36
OOV rate 1.28% 3.09% 1.20%
: Multi30k statistics on training and validation data – total number of tokens, average number of tokens per sentence, and the sizes of the shortest and the longest sentence.[]{data-label="tab:data"}
The participants were provided with the Multi30k dataset [@elliott2016multi30k] – a multilingual extension of Flickr30k dataset [@plummer2017flickr30k] – for both training and evaluation of their models.
The data consists of 31,014 images. In Flickr30k, each image is described with five independently acquired captions in English. Images in the Multi30k dataset are enriched with five crowd-sourced German captions. Additionally, a single German translation of one of the English captions was added for each image.
The dataset is split into training, validation, and test sets of 29,000, 1,014, and 1,000 instances respectively. The statistics on the training and validation part are tabulated in Table \[tab:data\].
For the 2017 round of the competition, an additional French translation was included for Task 1 and new test sets have been developed. Two test sets were provided for Task 1: The first one consists of 1,000 instances and is similar to the test set used in the previous round of the competition (and to the training and validation data). The second one consists of images, captions, and their translations taken from the MSCOCO image captioning dataset [@tsung2014mscoco]. A new single test set containing 1,071 images with five reference captions was added for Task 2.
The style and structure of the reference sentences in the Flickr- and MSCOCO-based test sets differs. Most of the sentences in the Multi30k dataset have a similar structure with a relatively simple subject, an active verb in present tense, simple object, and location information (e.g., “*Two dogs are running on a beach.*”). Contrastingly, the captions in the MSCOCO dataset are less formal and capture the annotator’s uncertainty about the image content (e.g., “*I don’t know, it looks like a lemon.*”).
Related Work {#sec:related}
============
Several promising neural architectures for multimodal translation task have been introduced since the first competition in 2016.
In our last year’s submission [@libovicky2016cuni], we employed a neural system that combined multiple inputs – the image, the source caption and an SMT-generated caption. We used the attention mechanism over the textual sequences and concatenated the context vectors in each decoder step.
The overall results of the WMT16 multimodal translation task did not prove the visual features to be particularly useful [@specia2016shared; @caglayan2016multimodality].
To our knowledge, @huang2016attention were the first who showed an improvement over a textual-only neural system with model utilizing distributed features explicit object recognition. @calixto2017incorporating improved state of the art using a model initializing the decoder state with the image vector, while maintaining the rest of the neural architecture unchanged. Promising results were also shown by @delbrouck2017multimodal who made a small improvement using bilinear pooling.
@elliot2017imagination brought further improvements by introducing the “imagination” component to the neural network architecture. Given the source sentence, the network is trained to output the target sentence jointly with predicting the image vector. The model uses the visual information only as a regularization and thus is able to use additional parallel data without accompanying images.
Experiments {#sec:models}
===========
All models are based on the encoder-decoder architecture with attention mechanism [@bahdanau2015neural] as implemented in Neural Monkey [@NeuralMonkey:2017].[^1] The decoder uses conditional GRUs [@firat2016cgru] with 500 hidden units and word embeddings with dimension of 300. The target sentences are decoded using beam search with beam size 10, and with exponentially weighted length penalty [@google2016bridging] with $\alpha$ parameter empirically estimated as 1.5 for German and 1.0 for French. Because of the low OOV rate (see Table \[tab:data\]), we used vocabularies of maximum 30,000 tokens and we did not use sub-word units. The textual encoder is a bidirectional GRU network with 500 units in each direction and word embeddings with dimension of 300. We use the last convolutional layer VGG-16 network [@simonyan2014vgg] of dimensionality $14\times14\times512$ for image processing. The model is optimized using the Adam optimizer [@kingma2015adam] with learning rate $10^{-4}$ with early stopping based on validation BLEU score.
Task 1: Multimodal Translation
------------------------------
We tested the following architectures with different datasets (see Section \[ssec:additional\] for details):
- purely textual (disregarding the visual modality);
- multimodal with context vector concatenation in the decoder [@libovicky2016cuni];
- multimodal with hierarchical attention combination [@libovicky2017attention] – context vectors are computed independently for each modality and then they are combined together using another attention mechanism as depicted in Figure \[fig:hier\].
Task 2: Cross-lingual Captioning
--------------------------------
We conducted two sets of experiments for this subtask. In both of them, we used an attentive image captioning model [@xu2015show] for the cross-lingual captioning with the same decoder as for the first subtask.
The first idea we experimented with was using a multilingual decoder provided with the image and a language identifier. Based on the identifier, the decoder generates the caption either in English or in German. We speculated that the information transfer from the visual to the language modality is the most difficult part of the task and might be similar for both English and German.
The second approach we tried has two steps. First, we trained an English image captioning system, for which we can use larger datasets. Second, we translated the generated captions with the multimodal translation system from the first subtask.
Acquiring Additional Data {#ssec:additional}
-------------------------
In order to improve the textual translation, we acquired additional data. We used the following technique to select in-domain sentences from both parallel and monolingual data.
We trained a neural character-level language model on the German sentences available in the training part of the Multi30k dataset. We used a GRU network with 512 hidden units and character embedding size of 128.
Using the language model, we selected 30,000 best-scoring German sentences from the SDEWAC corpus [@faass2013sdewac] which were both semantically and structurally similar to the sentences in the Multi30k dataset.
We tried to use the language model to select sentence pairs also from parallel data. By scoring the German part of several parallel corpora (EU Bookshop [@tiedemann2014billions], News Commentary [@tiedemann2012parallel] and CommonCrawl [@smith2013dirt]), we were only able to retrieve a few hundreds of in-domain sentences. For that reason we also included sentences with lower scores which we filtered using the following rules: sentences must have between 2 and 30 tokens, must be in the present tense, must not contain non-standard punctuation, numbers of multiple digits, acronyms, or named entities, and must have at most 15 % OOV rate w.r.t. Multi30k training vocabulary. We extracted additional 3,000 in-domain parallel sentences using these rules. Examples of the additional data are given in Table \[tab:example\].
By applying the same approach on the French versions of the corpora, we were pable to extract only few additional in-domain sentences. We thus trained the English-to-French models in the constrained setup only.
Following @calixto2017incorporating, we back-translated [@sennrich2016backtranslation] the German captions from the German side of the Multi30k dataset (i.e. 5+1 captions for each image), and sentences retrieved from the SDEWAC corpus. We included these back-translated sentence pairs as additional training data for the textual and multimodal systems for Task 1. The back-translation system used the same architecture as the textual systems and was trained on the Multi30k dataset only. The additional parallel data and data from the SDEWAC corpus (denoted as additional in Table \[tab:task1\]) were used only for the text-only systems because they were not accompanied by images.
For Task 2, we also used the MSCOCO [@tsung2014mscoco] dataset which consists of 300,000 images with 5 English captions for each of them.
------------------------------------------------------------------------
**SDEWAC Corpus** (with back-translation)
------------------------------------------------------------------------
zwei Männer unterhalten sich[@xt@ .80em[$\cdot$]{}@]{}
[@xt@ .80em[$\cdot$]{}@]{}*two men are talking to each other .*
ein kleines Mädchen sitzt auf einer Schaukel .[@xt@ .80em[$\cdot$]{}@]{}
[@xt@ .80em[$\cdot$]{}@]{}*a little girl is sitting on a swing .*
eine Katze braucht Unterhaltung .[@xt@ .80em[$\cdot$]{}@]{}
[@xt@ .80em[$\cdot$]{}@]{}*a cat is having a discussion .*
dieser Knabe streichelt das Schlagzeug .[@xt@ .80em[$\cdot$]{}@]{}
[@xt@ .80em[$\cdot$]{}@]{}*this professional is petting the drums .*
------------------------------------------------------------------------
**Parallel Corpora**
------------------------------------------------------------------------
[Menschen bei der Arbeit[@xt@ .80em[$\cdot$]{}@]{}*People at work*\
]{} [Männer und Frauen[@xt@ .80em[$\cdot$]{}@]{}*Men and women*\
]{} [Sicherheit bei der Arbeit[@xt@ .80em[$\cdot$]{}@]{}*Safety at work*\
]{}
Personen in der Öffentlichkeit[@xt@ .80em[$\cdot$]{}@]{}
[@xt@ .80em[$\cdot$]{}@]{}*Members of the public*
Results
-------
-------------------------------- --- ----------------------------- ----------------------------- ----------------------------- --------------------- -----------------------------
2016 Flickr MSCOCO Flickr MSCOCO
Baseline C — 19.3 / 41.9 18.7 / 37.6 44.3 / 63.1 35.1 / 55.8
Textual C 34.6 / 51.7 28.5 / 49.2 23.2 / 43.8 [**50.3**]{} / 67.0 [**43.0**]{} / [**62.5**]{}
Textual (+ Task2) U 36.6 / 53.0 28.5 / 45.7 24.1 / 40.7 — —
Textual (+ additional) U [**36.8**]{} / [**53.1**]{} [**31.1**]{} / [**51.0**]{} [**26.6**]{} / [**46.0**]{} — —
Multimodal [(concat. attn)]{} C 32.3 / 50.0 23.6 / 41.8 20.0 / 37.1 40.3 / 56.3 32.8 / 52.1
Multimodal [(hier. attn.)]{} C 31.9 / 49.4 25.8 / 47.1 22.4 / 42.7 49.9 / [**67.2**]{} 42.9 / [**62.5**]{}
Multimodal [(concat. attn.)]{} U [**36.0**]{} / [**52.1**]{} 26.3 / 43.9 23.3 / 39.8 — —
Multimodal [(hier. attn.)]{} U 34.4 / 51.7 [**29.5**]{} / [**50.2**]{} [**25.7**]{} / [**45.6**]{} — —
Task 1 winner (LIUM-CVC) C — 33.4 / 54.0 28.7 / 48.9 55.9 / 72.1 45.9 / 65.9
-------------------------------- --- ----------------------------- ----------------------------- ----------------------------- --------------------- -----------------------------
In Task 1, our best performing system was the text-only system trained with additional data. These were acquired both by the data selection method described above and by back-translation. Results of all setups for Task 1 are given in Table \[tab:task1\].
Surprisingly, including the data for Task 2 to the training set decreased the METEOR score on both of the 2017 test sets. This might have been caused by domain mismatch. However, in case of the additional parallel and SDEWAC data, this problem was likely outweighed by the advantage of having more training data.
In case of multimodal systems, adding approximately the same amount of data increased the performance more than in case of the text-only system. This suggests, that with sufficient amount of data (which is a rather unrealistic assumption), the multimodal system would eventually outperform the textual one.
The hierarchical attention combination brought major improvements over the concatenation approach on the 2017 test sets. On the 2016 test set, the concatenation approach yielded better results, which can be considered a somewhat strange result, given the similarity of the Flickr test sets.
The baseline system was Nematus [@sennrich2017nematus] trained on the textual part of Multi30k only. However, due to its low score, we suspect the model was trained with suboptimal parameters because it is in principle a model identical to our constrained textual submission.
In Task 2, none of the submitted systems outperformed the baseline which was a captioning system [@xu2015show] trained directly on the German captions in the Multi30k dataset. The results of our systems on Task 2 are shown in Table \[tab:task2\].
Task 2
----------------------------- --- ----------------------------
Baseline C [**9.1**]{} / [**23.4**]{}
Bilingual captioning C 2.3 / 17.6
en captioning + translation C 4.2 / 22.1
en captioning + translation U 6.5 / 20.6
other participant C [**9.1**]{} / 19.8
: Results of Task 2 in BLEU / METEOR points.[]{data-label="tab:task2"}
Flickr30k
-------------------------- ---------------------
@xu2015show [**19.1**]{} / 18.5
ours: Flickr30k 15.3 / [**18.7**]{}
ours: Flickr30k + MSCOCO 17.9 / 16.6
: Results of the English image captioning systems on Flickr30k test set in BLEU / METEOR points[]{data-label="tab:captioning"}
For the English captioning, we trained two models. First one was trained on the Flickr30k data only. In the second one, we included also the MSCOCO dataset. Although the captioning system trained on more data achieved better performance on the English side (Table \[tab:captioning\]), it led to extremely low performance while plugged into our multimodal translation systems (Table \[tab:task2\], rows labeled “en captioning + translation”). We hypothesize this is caused by the different styles of the sentences in the training datasets.
Our hypothesis about sharing information between the languages in a single decoder was not confirmed in this setup and the experiments led to relatively poor results.
Interestingly, our systems for Task 2 scored poorly in the BLEU score and relatively well in the METEOR score. We can attribute this to the fact that unlike BLEU which puts more emphasis on precision, METEOR considers strongly also recall.
Negative Results {#sec:negative}
================
In addition to our submitted systems, we tried a number of techniques without success. We describe these techniques since we believe it might be relevant for future developments in the field, despite the current negative result.
Beam Rescoring
--------------
Similarly to @lala2017unraveling, our oracle experiments on the validation data showed that rescoring of the decoded beam of width 100 has the potential of improvement of up to 3 METEOR points. In the oracle experiment, we always chose a sentence with the highest sentence-level BLEU score. Motivated by this observation, we conducted several experiments with beam rescoring.
We trained a classifier predicting whether a given sentence is a suitable caption for a given image. The classifier had one hidden layer with 300 units and had two inputs: the last layer of the VGG-16 network processing the image, and the last state of a bidirectional GRU network processing the text. We used the same hyper-parameters for the bidirectional GRU network as we did for the textual encoders in other experiments. Training data were taken from both parts of the Multi30k dataset with negative examples randomly sampled from the dataset, so the classes were represented equally. The classifier achieved validation accuracy of 87% for German and 74% for French. During the rescoring of the 100 hypotheses in the beam, we selected the one which had the highest predicted probability of being the image’s caption.
In other experiments, we tried to train a regression predicting the score of a given output sentence. Unlike the previous experiment, we built the training data from scored hypotheses from output beams obtained by translating the training part of the Multi30k dataset. We tested two architectures: the first one concatenates the terminal states of bidirectional GRU networks encoding the source and hypothesis sentences and an image vector; the second performs an attentive average pooling over hidden states of the RNNs and the image CNN using the other encoders terminal states as queries and concatenates the context vectors. The regression was estimating either the sentence-level BLEU score [@chen2014sbleu] or the chrF3 score [@popovic2015chrf].
Contrary to our expectations, all the rescoring techniques decreased the performance by 2 METEOR points.
Reinforcement Learning
----------------------
Another technique we tried without any success was self-critical sequence training [@rennie2016self]. This modification of the REINFORCE algorithm [@williams1992simple] for sequence-to-sequence learning uses the reward of the training-time decoded sentence as the baseline. The systems were pre-trained with the word-level cross-entropy objective and we hoped to fine-tune the systems using the REINFORCE towards sentence-level BLEU score and GLEU score [@google2016bridging].
It appeared to be difficult to find the right moment when the optimization criterion should be switched and to find an optimal mixing factor of the cross-entropy loss and REINFORCE loss. We hypothesize that a more complex objective mixing strategy (like MIXER [@ranzato2015mixer]) could lead to better results than simple objective weighting.
Conclusions {#sec:conclusion}
===========
In our submission to the 2017 Multimodal Task, we tested the advanced attention combination strategies [@libovicky2017attention] in a more challenging context and achieved competitive results compared to other submissions. We explored ways of acquiring additional data for the task and tested two promising techniques that did not bring any improvement to the system performance.
Acknowledgments {#acknowledgments .unnumbered}
===============
This research has been funded by the Czech Science Foundation grant no. P103/12/G084, the EU grant no. H2020-ICT-2014-1-645452 (QT21), and Charles University grant no. 52315/2014 and SVV project no. 260 453.
This work has been using language resources developed and/or stored and/or distributed by the LINDAT-Clarin project of the Ministry of Education of the Czech Republic (project LM2010013).
[^1]: https://github.com/ufal/neuralmonkey
|
---
abstract: 'Real-time 3D reconstruction from RGB-D sensor data plays an important role in many robotic applications, such as object modeling and mapping. The popular method of fusing depth information into a truncated signed distance function (TSDF) and applying the marching cubes algorithm for mesh extraction has severe issues with thin structures: not only does it lead to loss of accuracy, but it can generate completely wrong surfaces. To address this, we propose the directional TSDF—a novel representation that stores opposite surfaces separate from each other. The marching cubes algorithm is modified accordingly to retrieve a coherent mesh representation. We further increase the accuracy by using surface gradient-based ray casting for fusing new measurements. We show that our method outperforms state-of-the-art TSDF reconstruction algorithms in mesh accuracy.'
author:
- 'Malte Splietker and Sven Behnke[^1][^2]'
bibliography:
- 'references.bib'
title: '**Directional TSDF: Modeling Surface Orientation for Coherent Meshes** '
---
@inhibit[true]{}
Introduction
============
3D models are a useful way to describe objects or whole environments, which can be used in a variety of robotic applications like scene understanding, manipulation, and navigation. Since the publication of KinectFusion [@Newcombe2011] in 2011, TSDF fusion has turned into a de facto standard for fast registration and reconstruction using low-cost RGB-D sensors. TSDF fusion divides the modeled volume into a discretized grid of voxels and fuses distance information into it. Even though alternative approaches based on surfel predictions [@Stueckler2014; @Whelan2016] or direct meshing [@Greene2017; @Piazza2018] have emerged, TDSF fusion still remains the most popular choice.
Since the publication of KinectFusion, much work has been done to improve reconstruction speed and quality. There are, however, some fundamental limitations within the method itself. Firstly, it cannot represent anything thinner than the voxel size. While loss of fine details would be acceptable, the extracted surfaces can become completely wrong as illustrated in [Fig. \[fig:iconic\]]{}. Secondly, the state-of-the-art method of iteratively integrating new measurements is erroneous for steep observation angles and different observation directions as it overwrites and, in this way, destroys the representation.
![Directional TSDF (proposed) solves the problems of reconstructing thin objects, which state-of-the-art TSDF fusion methods have issues with (here MeshHashing [@Dong2018]). The bottom row shows the TSDFs (left: directional, right: undirected) in cross section, where the blue rectangle indicates the ground truth. The green and red colors denote areas that are in front or behind the surface, respectively. Color gradients indicate the signed distances to the object and the surface is extracted at the transition between the colors. []{data-label="fig:iconic"}](images/iconic.pdf){width="\linewidth"}
[.33]{} {width=".8\textwidth"}
[.33]{} {width=".8\textwidth"}
[.33]{} {width=".8\textwidth"}
A common way to mitigate these issues is to decrease the voxel size, which increases details and makes the problem less noticeable. However, for embedded hardware or large-scale mapping tasks a coarser voxel resolution might be required due to computation and memory restrictions. Moreover, the effect is not only affected by the voxel resolution, but also by the often depth-dependent truncation distance which is required to deal with measurement noise and usually spans multiple voxels. The problem lies in the representation itself, as the TSDF implicitly encodes surfaces as zero crossings and the density of these transitions is bounded by the voxel resolution and the truncation distance. The direction from which surfaces have been observed is only encoded indirectly by the gradient normal. This is especially problematic during fusion, because information from different directions (at corners or on opposite sides of a wall) might be contradictory within the truncation range. This leaves the TSDF in an inconsistent state with false information. Furthermore, the state-of-the-art method for data integration, voxel projection, where each voxel is projected into the camera image and associated with the nearest pixel, has disadvantages. It causes serious aliasing, incorrectly handles steep surfaces and neglects large amounts of input data, especially for larger voxel sizes. To address these issues, we introduce the directional TSDF—a novel data structure that encodes the surface orientations by dividing the modeled volume into six directions according to the positive and negative coordinate axes. This representation can handle observations of thin structures from different, opposing directions, without introducing aliasing. Data integration is done in a ray-casting fashion along the surface normals for every input point. The advantages are that all data is utilized and that the resulting representation is more accurate with respect to steep angle observations. To extract surfaces from the directional TSDF, a modified marching cubes algorithm is proposed, which can also model opposite faces while remaining computationally inexpensive.
In summary, the contributions of this paper are a novel representation which is better suited for mapping scenes from different viewing directions. We also present an improved data integration scheme which considers the actual surface gradient for determining the correct voxels for fusion. Finally, a mesh extraction method for this new representation is proposed. We thoroughly evaluate our methods on standard data sets.
Related Work
============
Surface reconstruction from range data has been an active research topic for a long time. It gained in popularity through the availability of affordable depth cameras and parallel computing hardware. Zollhöfer [et al.]{} [@Zollhoefer2018] give a comprehensive overview on modern 3D reconstruction from RGB-D data. The two main streams or research are TSDF fusion [@Newcombe2011; @Dong2018] and surfel extraction [@Stueckler2014; @Whelan2016]. TSDF-based methods make up the majority, due to their simplicity and mesh output. Surfels, however, maintain the surface and observation direction in form of a normal per surfel. Thus they can distinguish observations from different sides. Another interesting approach is presented by Schöps et al. [@Schoeps2018], who triangulate surfels to create a mesh representation.
An important step in data keeping was the switch from statically allocated voxel arrays to hash tables, allocating only required areas as proposed by Niesner [et al.]{} [@Niesner2013]. This enables scanning of large areas with limited memory and is considered state-of-the-art [@Dong2018; @Klingensmith2015; @Oleynikova2017]. We base our work on Dong et al. [@Dong2018], who further improve the data structure by tightly coupling voxel and meshing data. Signed distance data is stored on the corners of mesh cubes, which makes interpolation superfluous. Also the allocation, storage and access of vertex information is coupled to the structure, which decreases memory consumption and computation time.
As stated earlier, a major drawback of the TSDF representation is the voxel resolution, because the maximum object resolution is proportional to the voxel size. While decreasing the voxel size is one option, it is also wasteful in many areas with little detail. Steinbrücker [et al.]{} [@Steinbruecker2014] address this by dynamically adjusting the voxel resolution at the cost of additional octree nesting depth and a very complex surface extraction algorithm. This does, however, not solve the problem completely as depth-dependent noise needs to be considered in choosing the truncation range. The undirected TSDF of [Fig. \[fig:iconic\]]{} shows, how the truncation range from the opposite direction pushes the zero crossing away from the ground truth. Henry [et al.]{} [@Henry2013] dynamically create new TSDF volumes whenever the angle of the surface changes too much. This is similar to our approach, but relies on managing a huge number of separate volumes. Also the volume separation decision relies on larger surfaces; therefore it does not deal with small details. Moreover, their approach lacks a mesh extraction method and renderings are created by ray casting.
The de facto standard method for integrating measurements, voxel projection [@Newcombe2011; @Dong2018; @Steinbruecker2014; @Henry2013], suffers from aliasing effects especially for large voxel sizes and steep observation angles [@Klingensmith2015]. Curless [et al.]{} [@Curless1996] perform voxel projection onto an intermediate mesh generated from the input, thereby using the interpolated values of multiple input points to update a voxel. Many approaches have tried to overcome the issues of voxel projection TSDF fusion. Commonly data integration is weighted according to the quality of measurements. Stotko and Golla [@Stotko2015] evaluated different weighting options for fusion. This helps to compensate distance- and angle-dependent noise. To reduce the effects of integrating false information from surfaces with high observation angles, the point-to-plane distance metric can be applied [@Bylow2013]. As an alternative to voxel projection, ray casting [@Klingensmith2015] shoots a ray from the camera through every observed point and all voxels within the truncation range are updated. A sped up version using grouped ray casting was presented by Oleynikova et al. [@Oleynikova2017]. While for larger voxels the computational overhead is higher, the advantage is that there are no aliasing effects and that all information is utilized. Fossel [et al.]{} [@Fossel2015] address the issue that—especially for wide-angle sensors like LIDARs—the line-of-sight ray casting direction does not always comply with the surface direction. They estimate the surface gradient and choose the truncation range along the normal in a 2D SLAM system.
In contrast to the related works, we are proposing an improved representation based on the TSDF that utilizes the idea from Henry [@Henry2013] to represent surfaces with different orientations separate from each other. The implementation is based on the work of Dong [et al.]{} [@Dong2018], which also serves as a baseline for state-of-the-art methods using voxel projection and the marching cubes algorithm. Also we apply the gradient-based ray casting concept from Fossel et al. [@Fossel2015]. The key features of our method are:
- the directional TSDF representation that divides the modeled volume into six directions, thereby separately representing surfaces with different orientations,
- a gradient-based ray casting fusion for improved results,
- a thread-safe parallelization of ray casting fusion, and
- a modified marching cubes algorithm for mesh extraction from this representation.
Directional TSDF
================
A Signed Distance Function (SDF) denotes a function that for every 3D point yields the shortest distance to any surface. The sign denotes, whether the point is in front or behind the surface (inside an object). Let $\Omega \subset \mathbb{R}^3$ be a subset of space, e.g. a number of objects. In surface reconstruction, the points of interest lie on the boundary $\partial\Omega$. For a distance function $d$ and any point $\mathbf{p} \in \mathbb{R}^3$, the SDF $\Phi$ defines the signed distance to the surface: $$\Phi:\mathbb{R}^3 \longrightarrow \mathbb{R},~\Phi(\mathbf{p}) = \left\{
\begin{array}{ll}
-d(\mathbf{p}, \partial\Omega) & \text{if } \mathbf{p} \in \Omega, \\
\phantom{-}d(\mathbf{p}, \partial\Omega) & \text{if } \mathbf{p} \in \Omega^c.
\end{array}
\right.$$ That is, points that lie inside of the object have a negative value and the surface lies exactly at the zero crossing between positive and negative values. Consequently, most regions of the SDF are superfluous for determining the surface. The Truncated Signed Distance Function (TSDF) cuts of all values above a *truncation threshold* $\tau$, so everything outside the truncation range can be omitted. Typically TSDFs are estimated by a discretized grid of voxels and interpolation between the grid points. The truncation range is required to cope with aliasing effects and sensor noise and typically spans multiple voxels. While the method has proven to work in many scenarios, it has limitations, especially regarding thin objects, because the voxel resolution and truncation distance dictate the minimum thickness of objects. This effect occurs when observing small structures from opposite sides, as shown on the right hand side of [Fig. \[fig:iconic\]]{} where the object “blows up” as the truncation range pushes the contour further out. The cross section of the TSDF shows how far the estimated contour (transition between red and green) is away from the ground truth (blue rectangle).
The problem obviously lies in the representation itself, since a zero transition cannot be represented by fewer than two voxels (one with for each positive and negative value). We propose a new representation, called Directional TSDF (DTSDF), which stores the signed distance information in different volumes according to the surface gradient $$\Phi^d: \mathbb{R}^3 \longrightarrow \mathbb{R}^6,~\Phi^d(p) = (\Phi_D(p))_{D\in\textrm{Directions}}.$$ The $\textrm{Directions}=\{X^+, X^-, Y^+, Y^-, Z^+, Z^-\}$ are defined by the positive and negative coordinate axes $\mathbf{v} = \{(1, 0, 0)^\intercal, (-1, 0, 0)^\intercal, \cdots \}$. This way, each voxel can encode up to six surfaces, which is beneficial for modeling orthogonal or opposite surfaces and arbitrarily thin objects. Each direction spans a sector of applicable surface normals to determine which information belongs where. While the number of sectors can be arbitrarily high, six is an obvious choice due to the cuboid voxel shape. Fewer sectors are problematic, as the covered angle increases, though in principle four sectors spanning a tetrahedron would be sufficient.
Given a measurement point’s surface normal $\mathbf{n}$ and a direction vector $\mathbf{v}_{D}$, the direction-correspondence weight is defined as $$w_{D}(\mathbf{n}) = \left< \mathbf{n}, \mathbf{v}_{D} \right>.
\label{}$$ A measurement is integrated into all directions, whose weight is above a threshold of $\sin(\pi/8)$. This allows for a smooth transition between the sectors, which is important for the creation of coherent meshes. Note, that each measurement point is integrated into at most three directions.
Data Structure
--------------
Given the directional correspondence computation, it is easy to spot that not all voxels need to store information for all directions. In order to save memory, the classical voxel hashing data structure [@Dong2018; @Niesner2013] is extended to hold varying numbers of voxel arrays, each of which represents a certain direction. [Fig. \[fig:datastructure\]]{} shows the connections: The block coordinates from the world are hashed. Then a conflict-resolving hash table maps to a dynamically allocated block. For every block the up to six voxel arrays are allocated as needed. For clearer visualization, the modeled volumes in all example images are depicted in 2D, but all arguments easily extend to 3D.
![Data structure for dynamically allocating blocks and per-direction voxel arrays.[]{data-label="fig:datastructure"}](images/datastructure.pdf){width="\linewidth"}
Gradient-Directed Ray Casting Fusion
------------------------------------
Another novelty of our work is the surface gradient based ray casting fusion which, to our knowledge, has not been applied in 3D TSDF fusion before. The default method for fusing new measurements into the TSDF is voxel projection (VP), where voxels in view range and inside the camera frustum are projected onto the camera plane and are then associated with a single pixel in the depth image. [Fig. \[fig:fusion\_modes\_vp\]]{} depicts an example of the drawbacks pointed out before, where the projected SDF point (black) is far away from the projected measurement (red dot), thus gets a high SDF value, even though it is very close to a surface. Handling misassociations like those can be partially mended by using the point-to-plane metric for updating the SDF value [@Bylow2013]. [Fig. \[fig:fusion\_modes\_rc\]]{} shows ray casting fusion, where the algorithm casts a ray through every depth pixel and all intersected voxels within truncation range around the surface point are updated [@Klingensmith2015; @Oleynikova2017]. While improving the association problem and better utilizing the available data, steep observation angles remain problematic [@Fossel2015]. Our experiments have shown that this method, especially using the standard TSDF, worsens corners when rays shoot through them from different directions. Instead we extend the idea from Fossel [et al.]{} [@Fossel2015] to our method and use the surface normals as shown in [Fig. \[fig:fusion\_modes\_rcn\]]{}. Starting from the projected surface point, a ray is cast along the normal (in both directions) and all intersecting voxels in truncation range are updated. To circumvent expensive voxel-ray intersection computations, the algorithm applies voxel traversal as proposed in [@Amanatides1987].
[.325]{} ![Fusion mode comparison. A measured surface point (blue dot) of the ground truth surface (black curve) is used to update voxels (yellow squares) along the fusion ray (red).[]{data-label="fig:fusion_modes"}](images/fusion_voxel_projection.pdf "fig:"){width="\textwidth"}
[.325]{} ![Fusion mode comparison. A measured surface point (blue dot) of the ground truth surface (black curve) is used to update voxels (yellow squares) along the fusion ray (red).[]{data-label="fig:fusion_modes"}](images/fusion_raycasting.pdf "fig:"){width="\textwidth"}
[.325]{} ![Fusion mode comparison. A measured surface point (blue dot) of the ground truth surface (black curve) is used to update voxels (yellow squares) along the fusion ray (red).[]{data-label="fig:fusion_modes"}](images/fusion_raycasting_normal.pdf "fig:"){width="\textwidth"}
As the SDF corners of traversed voxels stray left and right from the ray, we furthermore apply the point-to-plane metric to increase the accuracy of the SDF. For a given surface point $\mathbf{p}$ and corresponding normal $\mathbf{n}_{\mathbf{p}}$ the distance function is $$d_{\mathrm{p2pl}}(\mathbf{x}) = (\mathbf{p} - \mathbf{x})^\intercal \mathbf{n}_{\mathbf{p}}.$$ The normals are computed using simple neighborhood estimation on the depth image. Due to noise and discretization in the depth image the estimated normals can be inaccurate, so a bilateral filter is applied. The depth image remains unfiltered to preserve reconstruction detail. While voxel projection updates every voxel exactly once, ray casting requires multiple updates per iteration. Details on the thread-safe fusion implementation are explained in [Sec. \[sec:thread\_safe\_fusion\]]{}.
For the SDF update a combined weighting scheme is applied. The noise of RGB-D cameras depends on the measured distance, which is accounted for in $w_{\mathrm{depth}}$. A high surface to view direction angle also increases inaccuracy, so it is down-weighted by $w_{\mathrm{angle}}$. The factor $w_{D}$, defined above, works the same way as $w_{\mathrm{angle}}$, but down-weights measurements that do not comply with the current fusion direction $D$. The combined weight is $$\begin{aligned}
w &= w_{\mathrm{depth}} \cdot w_{\mathrm{angle}} \cdot w_{\mathrm{D}}.\end{aligned}$$ Stotko [@Stotko2015] gives a detailed overview on weighting factors.
Directional Marching Cubes
==========================
A straight-forward and efficient method for mesh extraction from the TSDF representation is the marching cubes (MC) algorithm [@Lorensen1987]. It can be easily parallelized. The world is again divided into a regular grid of *mesh units*, which in this implementation is identical to the voxel grid. For every mesh unit, the SDF values at the corners are checked and for all edges which contain a zero transition, an appropriate set of triangles is generated. Since there are only 256 configurations for positive and negative corners, called *MC index*, a lookup table is used for efficiency.
For the directional TSDF, finding those zero transitions becomes more complicated as it is not immediately clear how to combine the information collected for different directions. Also there can be opposite faces within the same mesh unit, which the original algorithm cannot handle. Every edge must now be able to hold up to one zero transition per direction. Therefore, the data structure from [@Dong2018] is extended as depicted in [Fig. \[fig:mc\_datastructure\]]{}. The data structure makes SDF interpolation superfluous as the relevant SDF values can be directly fetched.
![Every mesh unit is responsible for storing vertices (red dots) on the thick edges $e_x, e_y, e_z$. If triangles have vertices on other edges, these are stored in adjacent voxels (yellow dots).[]{data-label="fig:mc_datastructure"}](images/mc_datastructure.pdf){width="\linewidth"}
Every mesh unit handles three edges, each of which can have up to two vertices for opposite surfaces. Triangles may have vertices on other edges, which are stored in adjacent mesh units. The outline of directional marching cubes is described in Algorithm \[alg:directional\_mc\]. The steps are explained throughout this section.
get MC index $\mathrm{mc}_D$, SDF weights $w_D$ Directional MC index filtering Inter-directional filtering Compute surface offsets for each edge Determine combined MC indices (up to 2) Allocate Vertices and Edges
Filtering
---------
It is necessary to filter the MC indices, as due to the nature of the representation some degree of false information occurs. Especially at the edge of the TSDF or at spots where different geometry collides, deviation and overhangs appear. Filtering is done in two stages, intra-directional and inter-directional.
![Filtering an implausible MC index by surface normal. The normal $\mathbf{n}$ exceeds the validity range (green semicircle) for direction $X^-$. The white and black corners of the mesh unit indicate whether the value is in front of or behind the surface, respectively. []{data-label="fig:filter_direction"}](images/filter_direction.pdf)
Firstly, all surfaces that would contradict the respective directions are discarded. This is performed using a pre-computed lookup table and comparison in cases where the orientation of the surface decides. [Fig. \[fig:filter\_direction\]]{} shows an example, where the normal of the potential surface is outside the valid range for direction $X^-$, so it is discarded. In this case the simple table lookup is insufficient, because there are configurations with the same MC index, where the normal is inside the valid range. The SDF gradient is utilized to identify these outliers.
In the second step, the different directions are weighted against each other in order to decide, whether a hypothesized surface is correct. The blue mesh unit in [Fig. \[fig:inter\_directional\_combining\]]{} shows a typical example, how the overhanging edge of direction $Y^+$ is identified as a false positive by $X^+$. To reduce false cancellations, the decision is made using a voting scheme. The credibility of a direction’s information is accounted for by the SDF weight and surface gradient w.r.t. this direction. For SDF weight $w^{\mathrm{sdf}}_D$, voxel center gradient $\nabla \Phi_D$, direction vector $\mathbf{v}_D$ and vote $a_D$, $$\label{eq:voting}
a = \sum_{D} w^{\mathrm{sdf}}_D \left<\nabla \Phi_D , \mathbf{v}_D \right> a_D,\quad a_D \in \{-1, 1\}$$ yields the consensus. If $a$ is smaller than 0, the MC index is set to 0 and no surface is extracted.
![MC index combining (orange) and inter-directional filtering (blue) is applied to TSDFs of directions $X^+$ and $Y^-$ to retrieve the combined surface on the right side.[]{data-label="fig:inter_directional_combining"}](images/inter_directional_combining.pdf){width="\linewidth"}
Surface Offset Estimation
-------------------------
From the remaining directions, the combined vertex positions are computed similar to what is described in [@Dong2018]. For every edge the MC indices of all directions are checked for potential zero transitions. The offset is added to a weighted average with the same combined weight used in . Since there can be opposite surfaces sharing the same edge, two offsets are stored per edge and the data structure is updated accordingly (c.f. [Fig. \[fig:mc\_datastructure\]]{}).
Combined MC Index
-----------------
After filtering false positives there might still be a mesh unit with multiple surface hypotheses from different directions. There are multiple ways for combining these, but intersection has shown good results. The orange encircled voxels in [Fig. \[fig:inter\_directional\_combining\]]{} are combined, such that a connection between the surfaces of the two directions is established. This combination is computationally cheap, as it can be done entirely on level of MC indices. Algorithm \[alg:mc\_index\_combining\] shows how the index-wise intersection is performed. The MC index is split into it’s up to four unconnected components. For each of these components the compatibility to the already combined indices is checked. The intersection of the indices is equivalent to the binary *and* operation.
MCIndex\[6\] combined\[2\] combined = (0, 0) combined\[0\] &= component **break** combined\[1\] &= component combined
Before extracting the final mesh, regularization between MC indices of neighboring voxels is performed for all modified blocks. This helps to reduce slits and overhangs induced by previous steps and close the surface. [Fig. \[fig:regularization\]]{} shows a mesh before and after the regularization. The procedure works solely on the MC indices by minimizing irregularities across voxel borders.
[.49]{} ![Voxel neighborhood MC index regularization.[]{data-label="fig:regularization"}](images/regularization_before.png "fig:"){width="\textwidth"}
[.49]{} ![Voxel neighborhood MC index regularization.[]{data-label="fig:regularization"}](images/regularization_after.png "fig:"){width="\textwidth"}
Thread-Safe Ray Casting Fusion {#sec:thread_safe_fusion}
==============================
The integration of new measurements into the TSDF is done by a weighted cumulative moving average. Let $D_t$ and $W_t$ be a voxel’s signed distance and weight values as time $t$. $d_t$ and $w_t$ are signed distance and weight update factors which are to be integrated. Then the update is $$\begin{aligned}
\label{eq:wcma1}
D_t &= \frac{W_{t-1} D_{t-1} + w_t d_t}{W_{t-1} + w_{t}} = \frac{\sum_{i=1}^t w_i d_i}{\sum_{i=1}^t w_i},\\
W_t &= W_{t-1} + w_t.
\label{eq:wcma2}\end{aligned}$$ In contrast to voxel-projection fusion, ray casting leads to multiple SDF updates per voxel within the same iteration. While being mathematically sound, it is problematic for a parallel implementation, as and cannot be performed atomically by most hardware. implies, that instead of applying an incremental update step for every pixel affecting a voxel, the following equivalent and thread-safe operation can be performed. Let $S_d, S_w$ be per-voxel summation values for signed distance and weight, respectively. They are initialized with zero at the beginning of each update iteration. $$\begin{aligned}
S_d & \overset{atomic}{+=} w_i d_i, &S_w \overset{atomic}{+=} w_i\end{aligned}$$ In the second step all modified voxels are iterated and the final update is computed as follows: $$\begin{aligned}
D_t &= \frac{W_{t-1} D_{t-1} + S_d}{W_{t-1} + S_w},\qquad W_t = W_{t-1} + S_w.\end{aligned}$$
Evaluation
==========
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MeshHashing [@Dong2018] serves as a baseline for recent TSDF RGB-D reconstruction algorithms with voxel hashing and marching cubes. Here, it is sometimes denoted as state-of-the-art (SOTA). Our proposed method is referred to as DTSDF. Registration is currently not implemented, so we compare the reconstruction quality and computation time.
For comparability, the well-known datasets by Zhou [et al.]{} [@Zhou2013] and the Stanford Computer Graphics Laboratory [@StanfordScanrep] are used. The Zhou dataset already provides trajectory and scans. For the Stanford dataset the 3D models are scaled, such that the longest side equals one meter. The camera performs an even circular motion with a two meter radius around the object, acquiring 1000 depth images with ground truth poses. The renderer uses the standard Kinect model $(f, c_x, c_y) = (525, 319.5, 239.5)$ with resolution $640\times480$ pixels. All experiments were run on a notebook with an Intel i7-4710HQ CPU (2.50GHz) and a GTX960M.
The truncation distance is fixed to four times the voxel size, as suggested by Oleynikova et al. [@Oleynikova2017]. Smaller factors tend to create holes in some spots, especially for voxel projection. No other parameters were changed.
[Fig. \[fig:qualitative\_comparison\]]{} shows a qualitative comparison between SOTA, DTSDF and the ground truth at voxel size. While at the given voxel resolution the amount of detail is limited, our method maintains a better surface (tail) and preserves geometry thinner than the voxel size (ridge, tail tip). The SOTA tends to enlarge the overall object. To quantify these findings we conducted a number of experiments at different voxel resolutions and measured the RMSE against the ground truth model. [Tab. \[tab:rmse\_voxel\_size\]]{} and [Tab. \[tab:rmse\_voxel\_size\_zhou\]]{} show the findings for different resolutions. Most notably, our method outperforms the SOTA in almost all cases by a good margin. Note, that we chose larger voxel sizes for the Zhou dataset due to the larger overall size of the scenes.
--------- ------- ------ ------ ------- ------- ------- -------
dataset mode 5 10 20 30 40 50
SOTA 1.55 3.67 11.83 25.27 41.02 58.36
DTSDF 1.14 1.74 3.58 6.53 14.56 21.39
SOTA 2.34 7.11 19.94 38.70 55.17 67.86
DTSDF 1.56 2.87 6.81 11.90 19.00 30.74
SOTA 1.90 3.82 9.01 17.70 27.97 37.98
DTSDF 0.98 1.23 2.18 4.17 8.59 18.37
SOTA 1.73 4.44 12.09 22.40 35.93 52.67
DTSDF 1.23 2.15 5.35 8.99 11.70 18.70
--------- ------- ------ ------ ------- ------- ------- -------
: RMSE (in ) of state-of-the-art and proposed method under different voxel sizes on the Stanford dataset.[]{data-label="tab:rmse_voxel_size"}
--------- ------- ------- ------- ------- --------
dataset mode 25 50 75 100
SOTA 12.80 27.28 57.69 102.46
DTSDF 6.86 13.38 19.60 26.53
SOTA 12.78 32.40 59.35 85.96
DTSDF 13.19 16.69 33.00 41.68
SOTA 14.96 33.91 57.85 85.52
DTSDF 12.41 25.11 41.95 51.86
SOTA 23.17 26.25 30.66 39.82
DTSDF 14.08 27.89 36.62 45.90
SOTA 7.23 13.84 26.22 39.71
DTSDF 6.89 11.47 18.85 31.20
SOTA 10.41 15.19 27.65 43.00
DTSDF 7.86 9.80 14.88 24.93
--------- ------- ------- ------- ------- --------
: RMSE (in ) of state-of-the-art and proposed method under different voxel sizes on the Zhou dataset.[]{data-label="tab:rmse_voxel_size_zhou"}
[Fig. \[fig:heatmap\_comparison\]]{} shows a distance error heatmap visualization which matches this observation. The models reconstructed by the SOTA have hotspots, wherever there is thin geometry (ears, tail, and ridge). But other areas benefit from the DTSDF, as well.
To examine the impact of the surface gradient ray casting fusion we conducted another experiment. [Tab. \[tab:rmse\_mode\]]{} shows the RMSE at voxel resolution for different algorithm modes, which are encoded as follows: *Def* and *Dir* (TSDF or DTSDF), voxel projection (VP), point-to-plane (P2PL), ray casting (RC) and ray casting along normals (RCN).
[l|XXXXXX]{} & & & & & &\
Armadillo & 3.670 & 4.915 & 4.839 & 1.931 & 2.478 & **1.741**\
Asian Dragon & 7.106 & 10.148 & 9.211 & 3.016 & 3.545 & **2.865**\
Bunny & 3.816 & 2.792 & 2.958 & 1.625 & 1.674 & **1.229**\
Dragon & 4.438 & 6.570 & 6.169 & 2.534 & 2.930 & **2.146**
Using ray casting on the default TSDF in most cases actually worsens the results, except for the bunny. This matches our earlier observation, because standard ray casting causes overshooting at corners while ray casting along normals inflates the thin parts to the maximum extent of the truncation distance. The DTSDF outperforms the SOTA even using voxel projection, but adding ray casting along normals further widens this margin. Standard ray casting, however, leads to the same problems. Only the bunny model benefits from it, because of its round shape and lack of thin, sharp edges.
The runtime of the algorithm can be split into two main components: data integration and meshing. [Tab. \[tab:total\_time\]]{} shows the total update time and proportion used for meshing on the Asian Dragon dataset. As expected, the amount of time spent on meshing increases with smaller voxel sizes. The data integration, however, remains fast even for higher resolutions: [Fig. \[fig:mean\_time\]]{} breaks down the integration step for SOTA and our proposed method. Preprocessing and allocating blocks and voxel arrays make up a significant constant factor. Fusion and recycling times are affected by the number of blocks in the view frustum. The ray casting has an additional (constant) proportion determined by the number of depth pixels, but also increases with the number of voxels due to the update accumulation step (c.f. [Sec. \[sec:thread\_safe\_fusion\]]{}).
![Mean data integration time for dataset Asian Dragon and different voxel resolutions. The left bars show the SOTA, the right side corresponds to DTSDF.[]{data-label="fig:mean_time"}](images/plots/mean_time.png){width="\linewidth"}
-- ---------- ------- ------ ------ ------ ------ ------
mode 5 10 20 30 40 50
SOTA 112.4 30.9 10.7 7.5 6.4 5.2
Proposed 199.0 74.0 31.7 19.9 15.1 13.0
SOTA 96.8 90.3 73.5 61.5 48.0 45.5
Proposed 96.8 92.8 84.5 75.9 68.8 63.5
-- ---------- ------- ------ ------ ------ ------ ------
: Mean total update time (in ms) and percentage that is used for meshing for dataset Asian Dragon.[]{data-label="tab:total_time"}
Dong [et al.]{} [@Dong2018] provide an accurate memory analysis for their MeshHashing implementation. The memory requirements for our algorithm differ from the original algorithm only by the number of voxel arrays allocated per block. [Fig. \[fig:mean\_num\_voxel\_arrays\]]{} shows the mean number of voxel arrays per block for different voxel resolutions and models. With smaller voxels the number of allocations decreases, as the number of integrations from opposite directions decreases within individual blocks. For the same reason the bunny, which is round and thicker in volume, requires fewer voxel arrays.
![Mean number of voxel arrays per block for different datasets and voxel resolutions.[]{data-label="fig:mean_num_voxel_arrays"}](images/plots/mean_num_voxel_arrays.png){width="\linewidth"}
Conclusions
===========
The proposed Directional TSDF representation and the matching modified marching cubes algorithm overcome limitations of the state-of-the-art method at the price of memory consumption and computation time. Less resolution is required for dealing with corners and thin objects which makes our method interesting for large-scale applications. Its robustness against different observation angles makes DTSDF also attractive for frame-to-model registration.
[^1]: This research has been supported by MBZIRC 2017 price money.
[^2]: All authors are with the Autonomous Intelligent Systems Group, University of Bonn, Germany. [[email protected]]{}
|
---
abstract: 'We propose a new method for calculating optical defect levels and thermodynamic charge-transition levels of point defects in semiconductors, which includes quasiparticle corrections to the Kohn-Sham eigenvalues of density-functional theory. Its applicability is demonstrated for anion vacancies at the (110) surfaces of III-V semiconductors. We find the (+/0) charge-transition level to be 0.49 eV above the surface valence-band maximum for GaAs(110) and 0.82 eV for InP(110). The results show a clear improvement over the local-density approximation and agree closely with an experimental analysis.'
author:
- Magnus Hedström
- Arno Schindlmayr
- Günther Schwarz
- Matthias Scheffler
title: 'Quasiparticle Corrections to the Electronic Properties of Anion Vacancies at GaAs(110) and InP(110)'
---
The electrical and optical properties of semiconductors depend sensitively on the electronic structure in the gap region and can hence be modified dramatically by the presence of native defects and impurities that introduce unwanted additional states inside the fundamental band gap. Most importantly, such electrically active defects can trap charge carriers (electrons or holes), counteracting the effect of intentional doping. If their concentration is sufficiently high, this process can lead to a full compensation of implanted acceptors and donors and thus eventually to Fermi-level pinning. Besides, electron-hole recombination at deep defects drastically reduces the lifetime of minority carriers, and transitions involving defect states inside the band gap may dominate optical absorption. This is especially relevant at surfaces and interfaces, where the crystal termination and the contact with other phases naturally give rise to a high number of structural defects that have been linked to the formation of Schottky barriers [@spicer80]. The central quantities are the optical defect levels inside the band gap and the charge-transition levels. The former can, in principle, be probed by direct (filled states) or inverse (empty states) photoemission. The Franck-Condon principle is well justified, as the rearrangement of the atoms happens on a much slower time scale than the electron emission or absorption, but the coupling to the atomic lattice may be visible in the line widths and shapes. The optical defect levels contain the full *electronic* relaxation in response to the created hole or the injected electron, however. The charge-transition levels, on the other hand, are thermodynamic quantities and specify the values of the Fermi energy where the charge state of the defect changes. Therefore, they are affected noticeably by the atomic relaxation taking place upon the addition or removal of an electron.
Despite considerable efforts, a reliable determination of the optical defect levels and the related charge-transition levels of deep defects still poses a very difficult challenge. Experimental investigations are thwarted by the fact that many traditional spectroscopic techniques are not applicable due to the low concentration of native point defects, while capacitance methods like deep-level transient spectroscopy [@lang74] are very sensitive but provide no elemental or structural information to identify the type of defect. For surfaces, at least, Ebert *et al.* [@ebert00] now demonstrated how the electronic structure of individual defects can be deduced using a combination of scanning tunneling microscopy (STM) and photoelectron spectroscopy, providing the first reliable experimental analysis of a charge-transition level for the P vacancy at InP(110). Theoretical approaches, on the other hand, must include accurate exchange-correlation contributions, the coupling between electronic and lattice degrees of freedom, and, in general, require the treatment of open systems in which the number of particles is not constant [@scherz93]. Previous studies that employed density-functional theory (DFT) in the local-density approximation (LDA) [@hohenberg64] indeed suffered from fundamental limitations [@ebert00; @zhang96; @kim96; @qian02]. For example, Ebert *et al.* [@ebert00] noted “that the systematic error for the calculated energies of the charge transfer levels is too large to identify the symmetry of the vacancy on the position of the defect level only.” In order to overcome this problem we here propose a new computational approach, broadly applicable to defects in the bulk as well as at surfaces, that combines DFT with many-body perturbation theory. As an example, we examine the optical defect levels and the thermodynamic charge-transition levels of anion vacancies at GaAs(110) and InP(110). The results are in close agreement with the experimental analysis [@ebert00].
The geometry of anion vacancies at the (110) surfaces of III-V semiconductors is now well understood thanks to a combination of experimental and theoretical studies. For $p$-doped materials STM images of the filled states under negative bias feature a localized hole at the position of the missing anion surrounded by a voltage-dependent depression [@lengel94]. The latter is due to a downward local band bending, indicating a positive charge of the vacancy. The charge state is, in fact, established as +1 [@chao96], which is also predicted by electronic-structure calculations [@ebert00; @zhang96; @kim96; @qian02]. STM images acquired under positive bias probe the empty $p_z$-like orbitals of the cation sublattice and show an enhancement of the cations surrounding the vacancy, initially wrongly interpreted as an upward relaxation of those atoms [@lengel94] but now understood as arising from the local depression of the electron density. DFT-LDA calculations actually show that an *inward* relaxation of the two Ga atoms enclosing the As vacancy $V_\mathrm{As}^+$ at GaAs(110) is consistent with the observations [@zhang96; @kim96]. The symmetry of the positively charged anion vacancies was initially a matter of controversy [@zhang96; @kim96]. In STM images they appear symmetric, but in a combined experimental and theoretical study of $V_\mathrm{P}^+$ at InP(110) Ebert *et al.* [@ebert00] explained the observed features as resulting from the thermal flip motion between two degenerate asymmetric configurations. This interpretation later received further confirmation [@qian03]. In $n$-doped materials, on the other hand, the vacancy is in a charge state of $-1$ [@domke98], and DFT-LDA calculations predict a symmetric relaxation for this configuration as well as the neutral vacancy [@zhang96; @kim96].
The anion vacancies at GaAs(110) or InP(110) give rise to three nondegenerate electronic states, labeled 1$a'$, 1$a''$, and 2$a'$. While the 1$a'$ state is located several eV below the valence-band maximum and always filled with two electrons, and the 2$a'$ state is too high in energy to become populated, the 1$a''$ state lies inside the band gap. It is this state, therefore, that is relevant for the discussion of charge-transition levels. Depending on the level of doping it may be occupied by zero, one, or two electrons, which corresponds to the positive, neutral, and negative charge state, respectively.
The formation energy of a surface vacancy with charge state $q$, relative to that of the neutral defect, is given by $$E^\mathrm{form}(q/0) = E^\mathrm{vac}(q,Q_q) - E^\mathrm{vac}(0,Q_0) + q \epsilon_\mathrm{F}\;,$$ where $E^\mathrm{vac}(q,Q)$ denotes the total energy of a surface featuring a single vacancy with the actual electron population $q \in \{+,0,-\}$ and geometry optimized for charge state $Q \in \{Q_+,Q_0,Q_-\}$. The final term accounts for the transfer of the charge $q$ between the defect level and the electron reservoir, i.e., the Fermi energy $\epsilon_\mathrm{F}$. The charge-transition levels $\epsilon^{q/q'}$ are defined as the values of $\epsilon_\mathrm{F}$ where the charge state of the vacancy changes, i.e., where $E^\mathrm{form}(q/0) = E^\mathrm{form}(q'/0)$, and conventionally given relative to the surface valence-band maximum. For the systems considered here the interesting transitions are $\epsilon^{+/0}$ and $\epsilon^{0/-}$. For example, the former is $$\label{Eq:ctlevel}
\epsilon^{+/0} = E^\mathrm{vac}(0,Q_0) - E^\mathrm{vac}(+,Q_+)\;.$$ All previous DFT-LDA calculations for surface point defects evaluated this energy difference directly [@ebert00; @zhang96; @kim96; @qian02], but this approach leads to systematic errors that arise because the total energies $E^\mathrm{vac}(0,Q_0)$ and $E^\mathrm{vac}(+,Q_+)$ refer to systems with different electron numbers. As is well known, the exact exchange-correlation potential in DFT exhibits a discontinuity upon addition or removal of an electron [@perdew83], which is not contained in the LDA or other jellium-based functionals. Besides, aspects like the self-interaction are treated inappropriately. As a consequence, the band gaps of semiconductors and the energies of localized defect states are not given correctly: for the P vacancy at InP(110) the experimentally determined $\epsilon^{+/0}$ level of 0.75$\pm$0.1 eV [@ebert00] contrasts noticeably with the calculated values 0.52 eV [@ebert00] and 0.388 eV [@qian02]. The variation between the two theoretical results can be traced to differences in the pseudopotentials and parallels the variation of the corresponding band gaps.
To arrive at a more accurate quantitative method that corrects the above-mentioned severe shortcomings of the LDA we rewrite Eq. (\[Eq:ctlevel\]) as $$\begin{aligned}
\label{Eq:ctlevel1}
\epsilon^{+/0} &=& \left[ E^\mathrm{vac}(+,Q_0) - E^\mathrm{vac}(+,Q_+) \right]\\ &&+ \left[ E^\mathrm{vac}(0,Q_0) - E^\mathrm{vac}(+,Q_0) \right]\nonumber\end{aligned}$$ by adding and subtracting the total energy $E^\mathrm{vac}(+,Q_0)$ of a system with the geometry of the relaxed neutral vacancy but a charge state $q = +1$. In this way the charge-transition level is decomposed into two separate contributions. The first describes the structural relaxation energy for the positive charge state and two different geometries: that of the neutral and that of the positive charge state. It is always positive. As the electron number remains constant, the problem of the discontinuity does not arise, and DFT-LDA is perfectly applicable. The second term equals the ionization energy of the neutral defect, where the removed electron is transfered to the reservoir, i.e., the Fermi energy. Determining the ionization potential from the quasiparticle band structure requires a correction of the Kohn-Sham eigenvalues, for which we employ many-body perturbation theory. Specifically, we use the $G_0W_0$ approximation for the electronic self-energy [@hedin65]. This approach is known to yield reliable band gaps for III-V semiconductors [@godby87] and their surfaces [@zhu89].
In the same spirit, $\epsilon^{0/-}$ can be written as $$\begin{aligned}
\label{Eq:ctlevel2}
\epsilon^{0/-} &=& \left[ E^\mathrm{vac}(-,Q_-) - E^\mathrm{vac}(-,Q_0) \right]\\ &&+ \left[ E^\mathrm{vac}(-,Q_0) - E^\mathrm{vac}(0,Q_0) \right]\nonumber\;.\end{aligned}$$ The first term in this case describes the energy difference of the vacancy with $q = -1$ between its own equilibrium geometry and that of the neutral charge state. It is always negative. The second term equals the electron affinity of the neutral charge state. In principle, the self-energy of the neutral charge state yields both the ionization potential and the electron affinity, which correspond to the energy of the highest occupied and the lowest unoccupied quasiparticle state, respectively. From a computational point of view, however, this procedure is inconvenient, because the neutral defect has an odd number of electrons and requires a spin-polarized calculation. Instead, we follow an equivalent approach and extract the energy levels from two separate calculations for non-spin-polarized systems with an even number of electrons. In practice, we thus determine $E^\mathrm{vac}(0,Q_0) - E^\mathrm{vac}(+,Q_0)$ as the electron affinity of the positive charge state and $E^\mathrm{vac}(-,Q_0) - E^\mathrm{vac}(0,Q_0)$ as the ionization potential of the negative charge state, both in the $Q_0$ geometry.
In the following we apply the expressions derived above to anion vacancies at the (110) surfaces of GaAs and InP. To determine the defect geometries we use DFT together with norm-conserving pseudopotentials [@bockstedte97] and the LDA exchange-correlation functional [@ceperley80]. The surfaces are simulated using a supercell with a (2$\times$4) periodicity in the \[001\] and \[110\] directions, consisting of six atomic layers separated by a vacuum buffer equivalent to four layers. A single vacancy is created at one side of the slab, while the dangling bonds at the other are passivated by pseudoatoms with noninteger nuclear charges of 0.75 and 1.25 for anion and cation termination, respectively. This mimics the continuation of the substrate by a III-V bulk layer. We use the theoretical lattice constants 5.55 Å for GaAs and 5.81 Å for InP to prevent errors resulting from a nonequilibrium unit-cell volume during the surface relaxation. The integration in reciprocal space is carried out with a mesh corresponding to eight $\mathbf{k}$-points in the two-dimensional Brillouin zone of the (1$\times$1) unit cell of the defect-free surface. In the case of charged defects we apply a uniform compensating background in order to ensure overall charge neutrality. For GaAs we thus obtain the relaxation energies $E^\mathrm{vac}(+,Q_0) - E^\mathrm{vac}(+,Q_+) = 0.30$ eV and $E^\mathrm{vac}(-,Q_-) - E^\mathrm{vac}(-,Q_0) = -0.13$ eV. The corresponding values for InP are 0.20 eV and $-0.17$ eV.
Before presenting our quasiparticle results, we first calculate the charge-transition levels strictly within the LDA by invoking the Slater-Janak transition-state approach [@slater72], where the ionization potential and the electron affinity equal the eigenvalue of the 1$a''$ level determined self-consistently with the noninteger occupancy 0.5 and 1.5, respectively. The transition state corrects, at least partially, the erroneous self-interaction of the LDA but not the discontinuity problem. The resulting energy contributions are displayed in Fig. \[Fig:variation\] relative to the surface valence-band maximum. The occurrence of slightly negative values in some cases implies that the defect level actually falls below the valence-band maximum. This is an artefact of the constrained nonequilibrium geometry: if the atomic structure is allowed to relax, then the defect level always lies inside the band gap. For all systems studied here we find that the transition-state approach yields the same results, with a deviation of less than 0.01 eV, as a straightforward evaluation of Eq. (\[Eq:ctlevel\]) and the corresponding formula for $\epsilon^{0/-}$ within the LDA. Quantitatively, our calculated value of 0.47 eV for $\epsilon^{+/0}$ in InP exhibits the same systematic underestimation of the experimentally derived charge-transition level as earlier studies at this level of approximation [@ebert00; @qian02].
![Position of the optical 1$a''$ defect level of the anion vacancies at GaAs(110) and InP(110) as a function of the occupation number. The geometry is identical in all calculations for the same material and optimized for the neutral charge state. The Slater-Janak transition-state approach (filled circles) yields the ionization potentials and electron affinities in the LDA. More accurate results are obtained by calculating the quasiparticle corrections within the $G_0W_0$ approximation (filled squares). In either case the energy zero is set to the respective surface valence-band maximum.[]{data-label="Fig:variation"}](eig.eps){width="\columnwidth"}
In order to determine the electronic contribution to the charge-transition levels more accurately we employ many-body perturbation theory. The energies derived within this framework correspond directly to the values measured in direct or inverse photoemission. We follow the usual approach to calculate the quasiparticle energies $$\epsilon_{1a''} = \epsilon_{1a''}^\mathrm{KS} + \langle \varphi_{1a''}^\mathrm{KS} | \Sigma(\epsilon_{1a''}) - V_\mathrm{xc} | \varphi_{1a''}^\mathrm{KS} \rangle$$ as a first-order correction of the Kohn-Sham eigenvalues $\epsilon_{1a''}^\mathrm{KS}$. Here $\Sigma$ is the complex, nonlocal, and frequency-dependent self-energy, which we evaluate in the $G_0W_0$ approximation using the Green function $G_0$ of the underlying Kohn-Sham system. Our numerical implementation is based on the space-time method [@rieger99]. The local exchange-correlation potential $V_\mathrm{xc}$ must be subtracted from the self-energy to avoid double counting. A more detailed discussion of our computational method can be found in Ref. [@hedstrom02]. The quasiparticle corrections, but not the Kohn-Sham eigenvalues, are obtained from a smaller (2$\times$2) surface cell, which reduces the computational effort considerably. Although we investigate charged systems, the relative self-energy shifts are insensitive to the size of the supercell, because they only include exchange-correlation effects and no electrostatic Hartree contribution. Keeping the atomic positions fixed at the optimized geometry for the neutral vacancy, we performed separate $G_0W_0$ calculations for the positive and negative charge states. The calculated 1$a''$ single-particle energies $\epsilon_{1a''}(+,Q_0)$ and $\epsilon_{1a''}(-,Q_0)$ are shown in Fig. \[Fig:variation\] with and without the self-energy correction.
$\epsilon^{+/0}$ $\epsilon^{0/-}$
----------------------- ------------------ ------------------
LDA (this work) 0.24 (0.07) 0.15
LDA (Ref. [@zhang96]) 0.4
LDA (Ref. [@kim96]) 0.24
$G_0W_0$ (this work) 0.49 (0.32) 0.60
: The charge-transition levels associated with the As vacancy at GaAs(110), given in eV. Values in brackets refer to the constrained symmetric relaxation of the positively charged vacancy. The quasiparticle band gap of 1.55 eV in this work, calculated at the theoretical lattice constant 5.55 Å, is close to the experimental value 1.52 eV.[]{data-label="Table:GaAs"}
In Table \[Table:GaAs\] we summarize the results for the As vacancy at GaAs(110). Values in brackets refer to the constrained symmetric relaxation of $V_\mathrm{As}^+$ and are included for the purpose of comparison with earlier studies. In contrast to Refs. [@zhang96; @kim96], which found a stable neutral charge state within a narrow energy window, our own calculation at the level of the LDA indicates $\epsilon^{+/0} > \epsilon^{0/-}$ and hence a direct transition from the positive to the negative charge state, but the small energetic separation is within the uncertainty of the calculation. With the $G_0W_0$ approximation we find a reversed ordering, which implies the existence of a stable neutral charge state, and a slightly increased splitting of the charge-transition levels.
$\epsilon^{+/0}$ $\epsilon^{0/-}$
------------------------- ------------------ ------------------
LDA (this work) 0.47 0.54
LDA (Ref. [@ebert00]) 0.52
LDA (Ref. [@qian02]) 0.388 0.576
$G_0W_0$ (this work) 0.82 1.09
Expt. (Ref. [@ebert00]) 0.75$\pm$0.1
: The charge-transition levels associated with the P vacancy at InP(110), given in eV. The quasiparticle band gap of 1.52 eV in this work, calculated at the theoretical lattice constant 5.81 Å, is close to the experimental value 1.42 eV.[]{data-label="Table:InP"}
The charge-transition levels for the P vacancy at InP(110) are listed in Table \[Table:InP\]. Our LDA results are similar to those reported previously [@ebert00; @qian02] and well below the experimentally deduced value of 0.75$\pm$0.1 eV [@ebert00]. The $G_0W_0$ approximation, on the other hand, yields a value for $\epsilon^{+/0}$ that lies within the experimental error bar.
In conclusion, we have developed a general computational scheme for the optical defect levels and thermodynamic charge-transition levels of point defects in semiconductors. The method is broadly applicable to the bulk as well as to surfaces. It relies on a separation of structural and electronic energy contributions that can be accurately evaluated within DFT and many-body perturbation theory, respectively. In this way the discontinuity of the exchange-correlation potential as well as other shortcomings of the LDA are treated appropriately. Our calculated (+/0) charge-transition level for the P vacancy at InP(110) is in close agreement with the experimental analysis, confirming the accuracy of this method.
We thank Jörg Neugebauer and Philipp Ebert for helpful discussions. This work was funded in part by the EU through the Nanophase Research Training Network (HPRN-CT-2000-00167) and the Nanoquanta Network of Excellence (NMP-4-CT-2004-500198).
[21]{} W. E. Spicer *et al.*, Phys. Rev. Lett. **44**, 420 (1980). D. V. Lang, J. Appl. Phys. **45**, 3014 (1974). P. Ebert *et al.*, Phys. Rev. Lett. **84**, 5816 (2000). U. Scherz and M. Scheffler, in *Semiconductors and Semimetals*, edited by E. R. Weber (Academic, New York, 1993), Vol. 38, p. 1. P. Hohenberg and W. Kohn, Phys. Rev. **136**, B864 (1964); W. Kohn and L. J. Sham, *ibid.* **140**, A1133 (1965). S. B. Zhang and A. Zunger, Phys. Rev. Lett. **77**, 119 (1996). H. Kim and J. R. Chelikowsky, Phys. Rev. Lett. **77**, 1063 (1996); Surf. Sci. **409**, 435 (1998). M. C. Qian, M. Göthelid, B. Johansson, and S. Mirbt, Phys. Rev. B. **66**, 155326 (2002). G. Lengel *et al.*, Phys. Rev. Lett. **72**, 836 (1994). K.-J. Chao, A. R. Smith, and C.-K. Shih, Phys. Rev. B **53**, 6935 (1996). M. C. Qian, M. Göthelid, B. Johansson, and S. Mirbt, Phys. Rev. B. **67**, 035308 (2003). C. Domke, P. Ebert, and K. Urban, Phys. Rev. B **57**, 4482 (1998). J. P. Perdew and M. Levy, Phys. Rev. Lett. **51**, 1884 (1983); L. J. Sham and M. Schlüter, *ibid.* **51**, 1888 (1983). L. Hedin, Phys. Rev. **139**, A796 (1965). R. W. Godby, M. Schlüter, and L. J. Sham, Phys. Rev. B **35**, R4170 (1987); X. Zhu and S. G. Louie, *ibid.* **43**, 14142 (1991); M. Rohlfing, P. Krüger, and J. Pollmann, *ibid.* **48**, 17791 (1993); B. Arnaud and M. Alouani, *ibid.* **63**, 085208 (2001). X. Zhu, S. B. Zhang, S. G. Louie, and M. L. Cohen, Phys. Rev. Lett. **63**, 2112 (1989); O. Pulci, G. Onida, R. Del Sole, and L. Reining, *ibid.* **81**, 5374 (1998). M. Bockstedte, A. Kley, J. Neugebauer, and M. Scheffler, Comput. Phys. Commun. **107**, 187 (1997); M. Fuchs and M. Scheffler, *ibid.* **119**, 67 (1999). D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. **45**, 566 (1980); J. P. Perdew and A. Zunger, Phys. Rev. B **23**, 5048 (1981). J. C. Slater, Adv. Quantum Chem. **6**, 1 (1972); J. F. Janak, Phys. Rev. B **18**, 7165 (1978). M. M. Rieger *et al.*, Comput. Phys. Commun. **117**, 211 (1999); L. Steinbeck *et al.*, *ibid.* **125**, 105 (2000). M. Hedström, A. Schindlmayr, and M. Scheffler, Phys. Status Solidi B **234**, 346 (2002).
|
---
abstract: 'We use the multiconfiguration Dirac-Hartree-Fock (MCDHF) method combined with the relativistic configuration interaction (RCI) approach (GRASP2K) to provide a consistent set of transition energies and radiative transition data for the lower $n =3$ states in all Cl-like ions of astrophysical importance, from to . We also provide excitation energies calculated for using the many-body perturbation theory (MBPT, implemented within FAC). The comparison of the present MCDHF results with MBPT and with the available experimental energies indicates that the theoretical excitation energies are highly accurate, with uncertainties of only a few hundred cm$^{-1}$. Detailed comparisons for and highlight discrepancies in the experimental energies found in the literature. Several new identifications are proposed.'
author:
- 'K. Wang'
- 'P. Jönsson'
- 'G. Del Zanna'
- 'M. Godefroid'
- 'Z. B. Chen'
- 'C. Y. Chen'
- 'J. Yan'
bibliography:
- 'ref.bib'
title: 'Large-scale multiconfiguration Dirac-Hartree-Fock calculations for astrophysics: Cl-like ions from Cr VIII to Zn XIV'
---
Introduction {#sec:intro}
============
Cl-like ions produce several strong transitions that have been widely used in astrophysics for a range of diagnostic applications, to e.g. measure electron temperatures, densities, chemical composition and even strong magnetic fields [@Zanna.2012.V541.p90]. For a review of some of the applications, see [@delzanna_mason:2018]. Providing accurate atomic data for these ions is of paramount importance.
Using the multiconfiguration Dirac-Hartree-Fock (MCDHF) and the relativistic configuration interaction (RCI) methods [@FroeseFischer.2016.V49.p182004; @Grant.2007.V.p], we provide here a consistent set of transition energies and radiative transition data with high accuracy in all Cl-like ions of astrophysical importance, from ($Z=24$) to ($Z=30$). Energy levels, wavelengths, oscillator strengths, line strengths, transition rates, and lifetimes for the main $n=3$ levels of the $3s^2 3p^5$, $3s 3p^6$, $3s^2 3p^4 3d$, $3s 3p^5 3d$, $3s^2 3p^3 3d^2$, $3s 3p^4 3d^2$, and $3s 3p^3 3d^3$ configurations are provided. To assess the accuracy of the MCDHF transition energies, we have also performed calculations and provided excitation energies for using the many-body perturbation theory (MBPT) [@Lindgren.1974.V7.p2441]. This work extends and complements our long-term theoretical efforts [@Wang.2014.V215.p26; @Wang.2015.V218.p16; @Wang.2016.V223.p3; @Wang.2016.V226.p14; @Wang.2017.V119.p189301; @Wang.2017.V194.p108; @Wang.2017.V187.p375; @Wang.2017.V229.p37; @Wang.2018.V235.p27; @Wang.2018.V239.p30; @Wang.2018.V234.p40; @Wang.2018.V208.p134; @Wang.2019.V236.p106586; @Chen.2017.V113.p258; @Chen.2018.V206.p213; @Guo.2015.V48.p144020; @Guo.2016.V93.p12513; @Si.2016.V227.p16; @Si.2017.V189.p249; @Si.2018.V239.p3; @Zhao.2018.V119.p314] to provide atomic data for L- and M-shells systems with high accuracy. For a review, see [@Jonsson.2017.V5.p16].
In Section 2 we briefly describe the calculations, while in Section 3 we present comparisons between theoretical and experimental energies for the low-lying levels of the main ions. We discuss in some detail the energies of the two main ions, and , revising some previous identifications. We then present our transition rates.
Theory and Calculations
=======================
The MCDHF method in the GRASP2K code [@Jonsson.2013.V184.p2197; @Jonsson.2007.V177.p597] and the MBPT method in the FAC code [@Gu.2008.V86.p675; @Gu.2007.V169.p154] are described by @FroeseFischer.2016.V49.p182004 and by @Lindgren.1974.V7.p2441, respectively. These two methods are also introduced in our recent papers [@Wang.2018.V235.p27; @Wang.2018.V239.p30]. For this reason, in the sections below, only the computational procedures are described.
MCDHF {#Sec:MCDHF}
-----
In our MCDHF calculations, the multireference (MR) sets for even and odd parities include
- $3s 3p^6$, $3s^2 3p^4 3d$, $3s 3p^4 3d^2$, $3p^6 3d$, $3s^2 3p^2 3d^3$, $3p^4 3d^3$, $3s 3p^2 3d^4$, $3s 3p^5 4p$, $3s^2 3p^4 4s$;
- $3s^2 3p^5$, $3s 3p^5 3d$, $3s^2 3p^3 3d^2$, $3p^5 3d^2$, $3s 3p^3 3d^3$, $3s^2 3p 3d^4$, $3s^2 3p^4 4p$, $3s 3p^5 4s$.
Initial MCDHF calculations for the MR sets for even and odd parities are performed to determine simultaneously all the orbitals $1s$, $2s$, $2p$, $3s$, $3p$, $3d$, $4s$, and $4p$ of the MR sets. Then, by allowing single and double substitutions from the $3s$, $3p$, $3d$, $4s$, $4p$ electrons of the MR sets to orbitals with $n\leq7, l\leq5$, and single excitations of the $2s$ and $2p$ electrons to orbitals with $n\leq6, l\leq5$, configuration state function (CSF) expansions are obtained. We consider the $1s$ shell as inactive, keeping its two electrons in all CSFs of the expansions. In order to monitor and obtain the convergence of computed properties, the orbital set is increased systematically layer by layer. At each stage only the outer orbitals are optimized, while the inner ones are fixed. Both valence-valence (VV) electron correlation effects and core-valence (CV) electron correlation effects are required for obtaining accurate results, though VV electron correlation effects are more important. Among CV electron correlation effects arising from subshells $2s$ and $2p$, those from $2p$ are the most important. We also checked that the contributions involving orbitals with principal quantum number $n=8$ for VV electron correlation are negligible and that opening the $1s$ shell for CV electron correlation can be omitted.
Once the orbitals optimized, the QED corrections and Breit interaction are included in the RCI calculations. The numbers of CSFs in the final even and odd expansions, are, respectively, around 26 and 23 millions, for the two parities. We use the $jj$-$LSJ$ transformation approach [@Gaigalas.2017.V5.p6; @Gaigalas.2004.V157.p239] to transform the $jj$-coupled CSFs into $LSJ$-coupled CSFs, and obtain the $LSJ$ labels used by experimentalists.
MBPT
----
The MBPT method [@Lindgren.1974.V7.p2441] is implemented in the FAC code by @Gu.2008.V86.p675 [@Gu.2007.V169.p154]. This method was used in our recent papers [@Wang.2014.V215.p26; @Wang.2015.V218.p16; @Wang.2016.V223.p3; @Wang.2016.V226.p14; @Wang.2017.V229.p37; @Wang.2018.V239.p30] to provide high accuracy atomic data for L- and M- shell ions. In the MBPT method, the Hilbert space of the full Hamiltonian is divided into two parts, i.e., a model space $M$ and a orthogonal space $N$. We included in $M$ all the CSFs of the MR set as defined above for the MCDHF calculations. All the possible CSFs generated by allowing single and double substitutions from the electrons of the MR sets are included in the $N$ space. With the maximum $l$-value of 20, the maximum $n$-values considered for the single and double excitations are, respectively, 125 and 65. The configuration interaction effects in the model space $M$ are considered non-perturbatively, i.e. are included in the self-consistent field calculations. The interaction effects between the two spaces $M$ and $N$ are considered through the second-order perturbation theory.
EVALUATION OF DATA
==================
Energy Levels and Lifetimes
---------------------------
\[Sec:en\]
Given its importance, we focus first on . The energies of the $3s^2 3p^5$, $3s 3p^6$ levels were well established, whilst not all of the $3s^2 3p^4 3d$ levels (producing the strongest lines) were identified in the earlier studies in the 1960’s and 1970’s. For a review on the accuracy of the experimental energies see [@DelZanna.2004.V422.p731], where several values were improved, and alternative/new identifications were proposed. [@DelZanna.2004.V422.p731] used the [superstructure]{} code and applied, whenever possible, Term Energy Corrections (TEC) to improve the theoretical energies. In a later scattering calculation for the $n=4$ levels, @Zanna.2012.V541.p90 revised a few of the identifications using [autostructure]{} [@badnell:2011], which were included in the CHIANTI database [@Dere.1997.V125.p149; @Dere.2019.V241.p22].
Several structure calculations have been published over the years, and some comparisons are shown in Table \[tab.lev.fe\]. In this Table, excitation energies for the lowest 197 states of the $3s^2 3p^5$, $3s 3p^6$, $3s^2 3p^4 3d$, $3s 3p^5 3d$, $3s^2 3p^3 3d^2$, and $3s 3p^4 3d^2$ configurations in from the present MCDHF and MBPT calculations are given. Observed energies from the NIST database and computed values from different sources are also included: CHIANTI, the MR-MP calculations by @Ishikawa.2010.V43.p74022, and the [grasp]{} calculations by @Aggarwal.2005.V439.p1215. We note that @Ishikawa.2010.V43.p74022 and @Aggarwal.2005.V439.p1215 provide results only for the lowest 90 states. The [grasp]{} calculations by @Aggarwal.2005.V439.p1215 focus on the estimation of cross-sections for electron impact excitation for the lowest 90 levels of the $3s^2 3p^5$, $3s3p ^6$, $3s^2 3p^4 3d$, $3s3p^5 3d$, and $3s^2 3p^3 3d^2$ configurations, and restricted electron correlation effects to the $n=3$ valence shells. In their calculations, valence-valence electron correlation effects among the levels of the $3s^2 3p^5$, $3s 3p^6$, $3s^2 3p^4 3d$, $3s 3p^5 3d$, $3p^6 3d$, $3s 3p^4 3d^2$, $3s^2 3p^3 3d^2$, and $3s^2 3p^2 3d^3$ configurations are included. Because of this limitation, results from @Aggarwal.2005.V439.p1215 are generally higher than our MCDHF values by one to several tens of thousand cm$^{-1}$. As can be seen in Table \[tab.lev.fe\], good agreement is obtained between the MCDHF, MBPT and MR-MP calculations for the lowest 90 states. For higher-lying levels (above $\#90$, mostly for the $3s^2 3p^3 3d^2$ levels) the only theoretical results available for comparison are the [autostructure]{} calculations by @Zanna.2012.V541.p90. The latter focused on the estimation of cross-sections for electron impact excitation for the higher levels and were limited to the $n=4$ configurations, restricting electron correlation to the valence shells. Because of this limitation, the [autostructure]{} excitation energies differ from our MCDHF results by $-$1 500 cm$^{-1}$ – 22 000 cm$^{-1}$ for the higher-lying levels. On the other hand, good agreement is obtained among the present accurate MCDHF and MBPT calculations.
Based on our MCDHF and MBPT excitation energies, we confirm most of experimental values from the NIST ASD and the CHIANTI database, except for a few cases, which we now discuss. For the level $\#11/3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F_{5/2}$, our energy (426602 cm$^{-1}$) is very close to the value suggested by [@jupen_etal:1993], 426707 cm$^{-1}$, allowing us to confirm their identification.
For the state $\#12/3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P_{3/2}$, the NIST energy is clearly in error. Our energy, 427951 cm$^{-1}$, is very close to the experimental value suggested by [@jupen_etal:1993], 428002 cm$^{-1}$. We therefore confirm their identification.
For the level $\#16/3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P_{3/2}$, our MCDHF value is 439619 cm$^{-1}$, relatively close to the value suggested by [@DelZanna.2004.V422.p731] (438168 cm$^{-1}$), but not to the value later proposed by @Zanna.2012.V541.p90, and included in the CHIANTI database, version 9. Our value is not close to any experimental energies previously suggested. [@jupen_etal:1993] suggested a value of 442439 cm$^{-1}$, definitely too far from our value. Considering the good agreement between our excitation energies and observed energies for the other $^{4}P$ levels, i.e., $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P_{1/2,5/2}$, we believe that the observed energy should be close (within 200 cm$^{-1}$) to our predicted one for the level $\#16/3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P_{3/2}$, and that the latest tentative identification in the CHIANTI database should be discarded.
Our energy for the level $\#19/3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D_{5/2}$ is in good agreement with the experimental value (444127 cm$^{-1}$) suggested by [@jupen_etal:1993].
For the state $\#22/3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F_{5/2}$ the NIST energy is clearly in error, while our energy supports the identification in CHIANTI, due to [@DelZanna.2004.V422.p731]. For the state $\#23/3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}F_{5/2}$, the experimental value in the CHIANTI database, suggested by [@DelZanna.2004.V422.p731] (and also [@jupen_etal:1993]), shows good agreement ($\Delta E_{\rm{CHIANTI}}$ = -41 cm$^{-1}$) with our MCDHF value, while the NIST values differs from our value by 5 388 cm$^{-1}$. Our energy for level $\#26/3s^{2}\,3p^{4}(^{1}S)~^{1}S\,3d~^{2}D_{5/2}$ (516334 cm$^{-1}$) is very close to the value suggested by [@jupen_etal:1993], 516222 cm$^{-1}$. For the state $\#35/3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F_{9/2}^{\circ}$, the experimental value from the NIST database (based on a suggestion rather than a firm observation) shows good agreement ($\Delta E_{\rm{NIST}}$ = $-$180 cm$^{-1}$) with our MCDHF value, while the CHIANTI value, based on a tentative alternative suggestion, differs from our value by $-$5 052 cm$^{-1}$.
The above comparisons clearly demonstrate the importance of the present calculations to assess the correctness of level identifications.
With regard to the lifetimes in , our MCDHF values, $\tau_{\rm MCDHF}^l$ in the length form and $\tau_{\rm MCDHF}^v$ in the velocity form, show good agreement with an average deviation of 1.5 %. Large deviations with the values for the lower levels calculated by @Zanna.2012.V541.p90 are found. This is not surprising as those calculations were aimed at the $n = 4$ configurations.
After , the most important ion (considering its abundance in astrophysical plasma) is . Compared to , the status of the experimental energies for this ion is very poor, as described in detail by a recent review of laboratory and astrophysical observations in [@delzanna_badnell:2016_ni_12]. In that paper, new scattering calculations were used to provide several tentative identifications for the unknown energy levels. Their [autostructure]{} excitation energies are generally relatively close (less than 1 000 cm$^{-1}$) to our MCDHF values, but we revise here many of the suggested experimental energies, see Table \[tab.ni\_12\]. As described in [@delzanna_badnell:2016_ni_12], the energies of many of the $3d$ levels are known only relative to the $\#5/3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D_{7/2}$ metastable state. The identification provided by [@delzanna_badnell:2016_ni_12] (a decay to the ground state at 220.247 Å) is close to the approximate value given by NIST, but is over 3 000 cm$^{-1}$ higher than our MCDHF energy. Considering the overall accuracy of our calculations, we suggest here an alternative identification. The decay of this level could indeed coincide with an transition at 221.822 Å, providing an energy of 450812 cm$^{-1}$, only 129 cm$^{-1}$ lower than our MCDHF value. This identification was not considered in [@delzanna_badnell:2016_ni_12] because the high-resolution solar spectrum of [@Behring.1976.V203.p521] did not suggest any line blending. However, we have estimated the intensity of the transition relative to those of other nearby lines using the atomic data from [@delzanna_storey:12_fe_13], and found that only about half the observed intensity is due to . The other half would be due to . We still regard this as a tentative new identification, as the decay from the $^{4}D_{5/2}$ level should be also observed. If the decay of the $^{4}D_{7/2}$ is at 221.822 Å, the decay from the $^{4}D_{5/2}$ level should be at 222.35 Å, but no line has been reported around this wavelength. Assuming that all these blends occur, we can infer the energies of several levels ($\#8$, $\#10$, $\#16$, $\#20$, $\#21$, $\#24$), shown in Table \[tab.ni\_12\]. We can see that all the inferred energies are very close to our ab initio MCDHF values, thus giving us confidence in the new identifications.
We also tentatively assign the energy of level $\#11$ ($^{4}F_{5/2}$) from a laboratory wavelength of 201.47 Å, and that of level $\#17$ ($^{4}P_{3/2}$) from a laboratory wavelength of 195.51 Å. For the levels $\#19$, $\#22$, and $\#30$ we confirm the identifications proposed by [@delzanna_badnell:2016_ni_12]. The identification of level $\#31$, the $^{2}D_{3/2}$, has been troublesome, as described in a section of the [@delzanna_badnell:2016_ni_12] paper. On the basis of the high-resolution laboratory spectrum of [@ryabtsev:1979] and the calculated intensities, [@delzanna_badnell:2016_ni_12] suggested that one of the two lines observed at 152.697 and 152.929 Å should be the decay from the $^{2}D_{3/2}$ level. The preference was given to the first one, as there is a line in the [@Behring.1976.V203.p521] solar spectrum at 152.703 Å, although too strong. Here we prefer the second option, as it gives an energy closer to our MCDHF value.
Finally, a summary of excitation energies (in cm$^{-1}$) and radiative lifetimes (in s) for the lowest 187 (196, 197, 193, 192, 194, 194) states of the $3s^2 3p^5$, $3s 3p^6$, $3s^2 3p^4 3d$, $3s 3p^5 3d$, $3s^2 3p^3 3d^2$, $3s 3p^4 3d^2$, and $3s 3p^3 3d^3$ configurations in Cr VIII (Mn IX, Fe X, Co XI, Ni XII, Cu XIII, Zn XIV) from the present MCDHF calculations are provided in Table \[tab.lev.all\]. All the states lie under the first $3p^6 3d$ state. Observed values compiled in the NIST ASD are also included in Table \[tab.lev.all\]. The differences for observed values from the present MCDHF results are about 2 500 cm$^{-1}$ – 3 200 cm$^{-1}$ in Ni XII and Cu XIII, and about 2 000 cm$^{-1}$ – 2 500 cm$^{-1}$ in Zn XIV. Whereas good agreement is obtained for the same levels in lower-$Z$ ions (Mn IX, Fe X, and Co XI). The differences (in cm$^{-1}$) between the MCDHF and NIST excitation energies for the $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P_{1/2}$, $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P_{3/2}$ and $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F_{7/2}$ levels, as a function of the nuclear charge $Z$ are shown in Figure \[figure.lev.nistlargedifferences\] as an example. For $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P_{1/2}$ and $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P_{3/2}$ four anomalies appear in Cu XIII and Zn XIV. Since in the present MCDHF calculations the same computational strategies are used for each ion, the accuracy of our calculated excitation energies should be consistent and systematic for the same level along the sequence. Therefore, the large differences indicate that several other identifications need to be revised.
Transition rates
----------------
Wavelengths $\lambda_{ij}$, transition rates $A_{ji}$, and branching fractions (${\rm BF}_{ji} = A_{ji}/ \sum \limits_{k=1}^{j-1} A_{jk}$) involving all levels considered in the present MCDHF calculations, as reported in Table \[tab.lev.all\], along with line strength $S_{ji}$ and weighted oscillator strengths $gf_{ji}$, are provided in Table \[tab.trans.all\]. E1 and E2 transition data in both length ($l$) and velocity ($v$) forms are given. For E1 and E2 transitions, we provide (last column) the uncertainty estimations of line strengths $S$ adopting the NIST ASD [@Kramida.2018.V.p] terminology (A$^{+}$ $\leq$ 2 %, A $\leq$ 3 %, B$^{+}$ $\leq$ 7 %, B $\leq$ 10 %, C$^{+}$ $\leq$ 18 %, C $\leq$ 25 %, D$^{+}$ $\leq$ 40 %, D $\leq$ 50 %, and E $>$ 50 % ) and using the method proposed by [@Kramida.2014.V212.p11]. For each E1 transition, the deviation $\delta S$ of line strengths $S_l$ in the length form and $S_v$ in the velocity form is defined as $\delta S$ = $|S_{v} - S_{l}|$/max($S_{v}$, $S_{l}$). In various ranges of $S$, the averaged uncertainties $\delta S_{av}$ of $\delta S$ for E1 transitions in Fe X are assessed to 1.1 % for $S \geq 10^{0}$; 1.3 % for $10^{0} > S \geq 10^{-1}$; 2.2 % for $10^{-1} > S \geq 10^{-2}$; 3.3 % for $10^{-2} > S \geq 10^{-3}$; 7.3 % for $10^{-3} > S \geq 10^{-4}$, 14 % for $10^{-4} > S \geq 10^{-5}$, and 29 % for $10^{-5} > S \geq 10^{-6}$. Then, the largest of $\delta S_{av}$ and $\delta S_{ij}$ is considered to be the uncertainty of each particular transition.
In Table \[tab.trans.all\], about 8.4 % E1 transitions in Fe X have the uncertainty of A+, 18.2 % of A, 30.8 % of B+, 18.1 % of B, 15.9 % of C+, 2.4 % of C, 4.3 % have the uncertainty of D+, and 1.2 % of D, while only 0.7 % have the uncertainty of E.
Using the same classification method, the uncertainties of the line strength $S$ for E2 transitions in Fe X, as well as those for E1 and E2 transitions in Cr VIII, Mn IX, Co XI, Ni XII, Cu XIII, and Zn XIV, are also listed in Table \[tab.trans.all\].
Summary
-------
The calculations of excitation energies, lifetimes, and radiative transition data for the $n=3$ states of Cl-like ions from Cr VIII to Cu XIV were performed using the MCDHF/RCI methods. Our detailed discussion of the energies of and have highlighted several discrepancies in the experimental energies in NIST and in the literature. The above comparisons clearly show the importance of the present ab initio calculations to assess the correctness of level and line identifications. Further studies are required to assess the identifications of the other ions, and further experimental work is encouraged to confirm our suggestions, especially for .
Scientific software packages {#scientific-software-packages .unnumbered}
============================
Scientific software packages including are used in the present work.
We acknowledge the support from the National Key Research and Development Program of China under Grant No. 2017YFA0402300, the Science Challenge Project of China Academy of Engineering Physics (CAEP) under Grant No. TZ2016005, the National Natural Science Foundation of China (Grant No. 11703004, No. 11674066, No. 11504421, and No. 11474034), the Natural Science Foundation of Hebei Province, China (A2019201300 and A2017201165), and the Natural Science Foundation of Educational Department of Hebei Province, China (BJ2018058). This work is also supported by the Fonds de la Recherche Scientifique - (FNRS) and the Fonds Wetenschappelijk Onderzoek - Vlaanderen (FWO) under EOS Project n$^{\rm o}$ O022818F, and by the Swedish research council under contracts 2015-04842 and 2016-04185. GDZ acknowledges support form STFC (UK) via the consolidated grant to the solar/atomic astrophysics group, DAMTP, University of Cambridge. KW expresses his gratitude to the support from the visiting researcher program at the Fudan University.
[clrrrrrrrrcc]{} 1 & $3s^{2}\,3p^{5}~^{2}P_{3/2}^{\circ}$ & 0 & 0 & 0 & 0 & 0 & 0 & 0 & & &\
2 & $3s^{2}\,3p^{5}~^{2}P_{1/2}^{\circ}$ & 15725 & 51 & -42 & -41.9 & -42 & -42 & -76 & 1.43E-02& 1.43E-02& 1.44E-02\
3 & $3s~^{2}S\,3p^{6}~^{2}S_{1/2}$ & 289634 & 522 & -398 & -385 & -398 & -702 & -4054 & 2.30E-10& 2.22E-10& 1.82E-10\
4 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D_{7/2}$ & 388485 & -148 & -919 & 224 & 223 & 329 & 731 & 1.73E-02& 1.73E-02& 1.77E-02\
5 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D_{5/2}$ & 388558 & -77 & -1094 & 151 & 156 & 274 & 713 & 1.66E-07& 2.05E-07& 3.00E-07\
6 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D_{3/2}$ & 389791 & -84 & -1266 & 259 & 228 & 289 & 731 & 1.67E-07& 1.63E-07& 2.26E-07\
7 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D_{1/2}$ & 391341 & -65 & 213 & 214 & 213 & 276 & 685 & 1.18E-07& 1.04E-07& 1.40E-07\
8 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F_{9/2}$ & 417470 & 155 & 183 & 183 & 182 & 293 & 3998 & 9.00E-02& 8.99E-02& 8.91E-02\
9 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P_{1/2}$ & 419812 & 401 & -1951 & & & 92 & 3862 & 4.50E-09& 4.54E-09& 5.43E-09\
10 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F_{7/2}$ & 422638 & 205 & 147 & 157 & 147 & 280 & 3977 & 5.93E-02& 5.93E-02& 6.06E-02\
11 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F_{5/2}$ & 426602 & 260 & 174 & 161 & 174 & 212 & 3788 & 2.13E-08& 2.18E-08& 3.36E-08\
12 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P_{3/2}$ & 427951 & 397 & -2209 & 3977 & & 137 & 3810 & 2.84E-09& 2.89E-09& 3.12E-09\
13 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F_{3/2}$ & 428253 & 288 & -525 & 45 & & 179 & 3838 & 1.17E-08& 1.15E-08& 1.05E-07\
14 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D_{3/2}$ & 434583 & 216 & -1785 & 31 & & 238 & 3730 & 2.76E-09& 2.90E-09& 3.53E-09\
15 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P_{1/2}$ & 434688 & 120 & -3685 & 112 & & 374 & 4173 & 2.81E-09& 2.74E-09& 3.56E-09\
16 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P_{3/2}$ & 439619 & 122 & -3434 & & -11253 & 362 & 4104 & 1.60E-07& 1.69E-07& 3.04E-08\
17 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F_{7/2}$ & 440785 & 520 & 54 & 55 & 54 & 96 & 6021 & 1.48E-02& 1.48E-02& 1.43E-02\
18 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P_{5/2}$ & 441779 & 131 & -1247 & 74 & -1247 & 331 & 3919 & 8.24E-09& 8.52E-09& 2.01E-08\
19 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D_{5/2}$ & 444033 & 146 & -1273 & & -1273 & 344 & 4090 & 1.28E-08& 1.31E-08& 6.58E-09\
20 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}G_{9/2}$ & 450676 & 401 & 2155 & 75 & 78 & 183 & 5963 & 1.36E-02& 1.36E-02& 1.34E-02\
21 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}G_{7/2}$ & 451066 & 548 & 17 & 18 & 17 & 116 & 6100 & 1.40E-02& 1.40E-02& 1.81E-02\
22 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F_{5/2}$ & 454034 & 665 & -1539 & -1304 & 2 & 25 & 6149 & 4.14E-08& 4.26E-08& 1.00E-07\
23 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}F_{5/2}$ & 482087 & 594 & -41 & -5388 & -41 & 151 & 8044 & 5.69E-09& 5.72E-09& 7.94E-09\
24 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}F_{7/2}$ & 485978 & 545 & 4 & 5 & 4 & 184 & 8059 & 4.97E-03& 4.98E-03& 5.03E-03\
25 & $3s^{2}\,3p^{4}(^{1}S)~^{1}S\,3d~^{2}D_{3/2}$ & 511914 & 642 & 3496 & -114 & -114 & 61 & 6518 & 9.22E-10& 9.33E-10& 1.17E-09\
26 & $3s^{2}\,3p^{4}(^{1}S)~^{1}S\,3d~^{2}D_{5/2}$ & 516334 & 588 & -112 & & -112 & 40 & 6213 & 6.41E-09& 6.43E-09& 3.85E-08\
27 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}S_{1/2}$ & 540963 & -1051 & 934 & 916 & 934 & 1353 & 10226 & 6.28E-12& 6.23E-12& 6.12E-12\
28 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P_{3/2}$ & 563592 & -180 & 616 & 606 & 616 & 1388 & 15020 & 6.74E-12& 6.82E-12& 6.36E-12\
29 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P_{1/2}$ & 569337 & -111 & 545 & 648 & 545 & 1317 & 14937 & 6.91E-12& 6.98E-12& 6.57E-12\
30 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D_{5/2}$ & 572177 & -745 & 787 & 777 & 787 & 1522 & 14128 & 5.62E-12& 5.69E-12& 5.38E-12\
31 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D_{3/2}$ & 585622 & -567 & 632 & 622 & 632 & 1341 & 13906 & 5.71E-12& 5.75E-12& 5.51E-12\
32 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}P_{1/2}^{\circ}$ & 666576 & 240 & -8912 & & & -391 & 16561 & 3.34E-10& 3.29E-10& 3.94E-10\
33 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}P_{3/2}^{\circ}$ & 669144 & 280 & -8898 & & & -422 & 16550 & 3.31E-10& 3.28E-10& 3.89E-10\
34 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}P_{5/2}^{\circ}$ & 673703 & 247 & -8896 & & & -341 & 16600 & 3.28E-10& 3.26E-10& 3.83E-10\
35 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F_{9/2}^{\circ}$ & 696841 & 830 & -5051 & -180 & -5052 & -414 & 22114 & 1.84E-10& 1.77E-10& 2.06E-10\
36 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F_{7/2}^{\circ}$ & 699761 & 943 & -5108 & -269 & & -490 & 21992 & 1.82E-10& 1.75E-10& 2.03E-10\
37 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F_{5/2}^{\circ}$ & 702836 & 991 & -5384 & -251 & & -443 & 21913 & 1.80E-10& 1.73E-10& 2.01E-10\
38 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F_{3/2}^{\circ}$ & 705654 & 1020 & -5588 & -224 & & -530 & 21828 & 1.77E-10& 1.72E-10& 2.00E-10\
39 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D_{7/2}^{\circ}$ & 727687 & 807 & -7047 & & & -117 & 24058 & 1.89E-10& 1.84E-10& 2.07E-10\
40 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D_{1/2}^{\circ}$ & 729234 & 943 & -7521 & & & -293 & 23893 & 1.86E-10& 1.81E-10& 2.03E-10\
41 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D_{5/2}^{\circ}$ & 729480 & 843 & -7292 & & & -147 & 23998 & 1.87E-10& 1.82E-10& 2.04E-10\
42 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D_{3/2}^{\circ}$ & 729748 & 885 & -7450 & & & -219 & 23906 & 1.86E-10& 1.82E-10& 2.04E-10\
43 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}F_{7/2}^{\circ}$ & 737663 & 1219 & -7738 & & -295 & -467 & 22465 & 4.10E-10& 4.05E-10& 5.06E-10\
44 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}D_{5/2}^{\circ}$ & 741800 & 1356 & -7128 & & & -389 & 24814 & 3.48E-10& 3.40E-10& 4.10E-10\
45 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}D_{3/2}^{\circ}$ & 746345 & 1461 & -6279 & & & -424 & 27413 & 2.26E-10& 2.20E-10& 2.43E-10\
46 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}F_{5/2}^{\circ}$ & 749936 & 1365 & -7552 & & & -417 & 25512 & 2.91E-10& 2.86E-10& 3.17E-10\
47 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}P_{1/2}^{\circ}$ & 760034 & 812 & -6468 & & & -331 & 24127 & 9.32E-11& 9.15E-11& 1.02E-10\
48 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}P_{3/2}^{\circ}$ & 767744 & 869 & -6436 & & & -326 & 24625 & 1.06E-10& 1.04E-10& 1.12E-10\
49 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{1/2}^{\circ}$ & 781888 & 210 & -2042 & & & 366 & 16144 & 7.04E-09& 6.81E-09& 8.72E-09\
50 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{3/2}^{\circ}$ & 782022 & 194 & -1983 & & & 391 & 16163 & 8.28E-09& 8.05E-09& 1.02E-08\
51 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{5/2}^{\circ}$ & 782285 & 170 & -1895 & & & 480 & 16186 & 1.04E-08& 1.02E-08& 1.25E-08\
52 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{7/2}^{\circ}$ & 782722 & 144 & -1792 & & & 488 & 16199 & 1.50E-08& 1.49E-08& 1.84E-08\
53 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{9/2}^{\circ}$ & 783337 & 102 & -1663 & & & 474 & 16231 & 2.62E-08& 2.62E-08& 3.23E-08\
54 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{11/2}^{\circ}$ & 784108 & 65 & -1509 & & & 484 & 16261 & 1.28E-07& 1.37E-07& 1.61E-07\
55 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}F_{5/2}^{\circ}$ & 786777 & 1778 & -3496 & & & -191 & 27880 & 8.47E-11& 8.35E-11& 8.57E-11\
56 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}F_{7/2}^{\circ}$ & 790569 & 1701 & -3462 & & & -190 & 28357 & 8.23E-11& 8.12E-11& 8.36E-11\
57 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{6}P_{3/2}^{\circ}$ & 805678 & -65 & 1206 & & & 469 & 20238 & 8.24E-09& 8.31E-09& 9.43E-09\
58 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{6}P_{5/2}^{\circ}$ & 805899 & -180 & 1367 & & & 610 & 20368 & 6.10E-09& 6.01E-09& 5.72E-09\
59 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{6}P_{7/2}^{\circ}$ & 806136 & -182 & 1567 & & & 660 & 20438 & 4.67E-09& 4.61E-09& 4.85E-09\
60 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}D_{3/2}^{\circ}$ & 813954 & 1606 & -4990 & & & 43 & 28576 & 7.71E-11& 7.56E-11& 7.06E-11\
61 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}D_{5/2}^{\circ}$ & 815339 & 1559 & -4742 & & & 97 & 28881 & 7.87E-11& 7.71E-11& 7.16E-11\
62 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D_{7/2}^{\circ}$ & 826314 & 583 & 3488 & & & 637 & 25291 & 2.99E-10& 2.90E-10& 1.93E-10\
63 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P_{1/2}^{\circ}$ & 826893 & 493 & 5379 & & & 340 & 27412 & 2.09E-09& 2.09E-09& 1.73E-10\
64 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D_{5/2}^{\circ}$ & 826991 & 656 & 3369 & & & 632 & 25448 & 2.76E-10& 2.69E-10& 1.90E-10\
65 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D_{3/2}^{\circ}$ & 827311 & 697 & 3308 & & & 951 & 25721 & 2.72E-10& 2.64E-10& 1.97E-10\
66 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D_{1/2}^{\circ}$ & 829217 & 771 & 4934 & & & 423 & 26196 & 2.37E-10& 2.30E-10& 9.35E-10\
67 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P_{3/2}^{\circ}$ & 830347 & 454 & 4339 & & & -50 & 26998 & 4.58E-09& 4.58E-09& 6.62E-10\
68 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G_{5/2}^{\circ}$ & 830949 & 772 & 2473 & & & 1609 & 24409 & 2.10E-09& 2.01E-09& 1.35E-09\
69 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G_{7/2}^{\circ}$ & 832158 & 750 & 2542 & & & 410 & 24298 & 3.07E-09& 2.91E-09& 2.81E-09\
70 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F_{3/2}^{\circ}$ & 832646 & 653 & 4945 & & & 200 & 26290 & 9.02E-10& 8.77E-10& 1.93E-09\
71 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P_{5/2}^{\circ}$ & 833659 & 471 & 4970 & & & -611 & 27252 & 7.83E-09& 7.72E-09& 2.12E-09\
72 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G_{9/2}^{\circ}$ & 834280 & 746 & 2771 & & & 425 & 24195 & 4.49E-09& 4.10E-09& 4.16E-09\
73 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G_{11/2}^{\circ}$ & 836443 & 746 & 3156 & & & 316 & 24183 & 5.56E-09& 4.75E-09& 5.16E-09\
74 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F_{5/2}^{\circ}$ & 836583 & 645 & 4400 & & & 232 & 25854 & 2.49E-09& 2.39E-09& 4.90E-09\
75 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F_{7/2}^{\circ}$ & 839581 & 623 & 2664 & & & 246 & 25292 & 2.62E-09& 2.51E-09& 2.15E-09\
76 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F_{9/2}^{\circ}$ & 842504 & 579 & 2914 & & & 287 & 25540 & 2.24E-09& 2.12E-09& 1.85E-09\
77 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H_{7/2}^{\circ}$ & 851459 & 1017 & 543 & & & 214 & 23652 & 1.05E-09& 1.02E-09& 3.36E-09\
78 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H_{9/2}^{\circ}$ & 852725 & 941 & 219 & & & 224 & 23341 & 1.34E-08& 1.30E-08& 1.90E-08\
79 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H_{11/2}^{\circ}$ & 854217 & 882 & 225 & & & 248 & 23286 & 2.88E-07& 2.07E-07& 1.93E-07\
80 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F_{7/2}^{\circ}$ & 855492 & 947 & 3953 & & & 461 & 28475 & 2.05E-10& 1.98E-10& 1.84E-10\
81 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H_{13/2}^{\circ}$ & 857044 & 835 & 473 & & & & 23016 & 3.15E-02& 3.17E-02& 2.73E-02\
82 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}P_{3/2}^{\circ}$ & 858043 & 849 & 5253 & & & 539 & 33145 & 1.17E-11& 1.15E-11& 5.03E-11\
83 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}P_{1/2}^{\circ}$ & 858212 & 304 & 9800 & & & & 42270 & 7.03E-12& 6.95E-12& 5.99E-12\
84 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F_{5/2}^{\circ}$ & 859441 & 1052 & 3944 & & & 574 & 29848 & 1.20E-10& 1.16E-10& 1.43E-10\
85 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D_{3/2}^{\circ}$ & 859712 & 675 & 8869 & & & & 41440 & 1.39E-11& 1.37E-11& 6.16E-12\
86 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H_{9/2}^{\circ}$ & 861524 & 1322 & 2642 & & & & 27162 & 1.24E-09& 1.18E-09& 1.25E-09\
87 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H_{11/2}^{\circ}$ & 863939 & 1223 & 1761 & & & & 26909 & 2.69E-09& 2.45E-09& 2.64E-09\
88 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D_{5/2}^{\circ}$ & 866545 & 1243 & 2725 & & & & 31798 & 6.62E-11& 6.43E-11& 2.03E-10\
89 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G_{5/2}^{\circ}$ & 871212 & 1095 & 1623 & & & & 28974 & 3.21E-10& 3.07E-10& 7.27E-11\
90 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G_{7/2}^{\circ}$ & 871984 & 1055 & -1678 & & & & 27675 & 2.84E-10& 2.72E-10& 6.36E-10\
91 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G_{9/2}^{\circ}$ & 875209 & 1047 & -1226 & & & & & 2.96E-10& 2.84E-10& 7.44E-10\
92 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G_{7/2}^{\circ}$ & 877873 & 1448 & 5573 & & & & & 2.07E-10& 2.00E-10& 1.66E-10\
93 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G_{9/2}^{\circ}$ & 878794 & 1400 & 5192 & & & & & 2.12E-10& 2.04E-10& 1.77E-10\
94 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G_{11/2}^{\circ}$ & 880998 & 977 & -1510 & & & & & 7.95E-10& 7.39E-10& 1.48E-09\
95 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F_{5/2}^{\circ}$ & 886966 & 1047 & 11636 & & & & & 2.24E-11& 2.19E-11& 1.93E-11\
96 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F_{7/2}^{\circ}$ & 887429 & 1042 & 11603 & & & & & 2.29E-11& 2.23E-11& 1.94E-11\
97 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F_{3/2}^{\circ}$ & 887911 & 1022 & 11498 & & & & & 2.20E-11& 2.15E-11& 2.04E-11\
98 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F_{9/2}^{\circ}$ & 889448 & 955 & 11406 & & & & & 2.35E-11& 2.29E-11& 2.03E-11\
99 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P_{3/2}^{\circ}$ & 896401 & 963 & 7577 & & & & & 3.28E-11& 3.23E-11& 3.96E-11\
100& $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}I_{11/2}^{\circ}$ & 898096 & 1711 & 2506 & & & & & 2.96E-09& 2.90E-09& 4.08E-09\
101& $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D_{1/2}^{\circ}$ & 899230 & 902 & 3388 & & & & & 3.46E-11& 3.42E-11& 2.84E-11\
102& $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}I_{13/2}^{\circ}$ & 899511 & 1693 & 2385 & & & & & 4.10E-02& 4.09E-02& 3.88E-02\
103& $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}S_{1/2}^{\circ}$ & 899995 & 902 & 4934 & & & & & 7.16E-11& 7.08E-11& 1.50E-10\
104& $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D_{3/2}^{\circ}$ & 900719 & 864 & 3763 & & & & & 4.63E-11& 4.58E-11& 4.57E-11\
105& $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D_{5/2}^{\circ}$ & 902600 & 952 & 3270 & & & & & 6.83E-11& 6.72E-11& 4.72E-11\
106& $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D_{3/2}^{\circ}$ & 905440 & 1084 & 6101 & & & & & 5.61E-11& 5.49E-11& 7.83E-11\
107& $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F_{5/2}^{\circ}$ & 905887 & 1308 & 4795 & & & & & 6.75E-11& 6.55E-11& 6.75E-11\
108& $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D_{7/2}^{\circ}$ & 906097 & 952 & 3103 & & & & & 5.30E-11& 5.22E-11& 4.52E-11\
109& $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F_{5/2}^{\circ}$ & 906596 & 1082 & 6711 & & & & & 5.10E-11& 5.02E-11& 4.24E-11\
110& $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}S)~^{4}S_{3/2}^{\circ}$ & 907868 & 395 & 4927 & & & & & 2.29E-10& 2.22E-10& 5.20E-11\
111& $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P_{1/2}^{\circ}$ & 910038 & 969 & 7453 & & & & & 3.53E-11& 3.47E-11& 4.93E-11\
112& $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F_{3/2}^{\circ}$ & 910577 & 802 & 5822 & & & & & 1.25E-10& 1.22E-10& 4.82E-11\
113& $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F_{7/2}^{\circ}$ & 911936 & 1028 & 3753 & & & & & 1.15E-10& 1.13E-10& 1.15E-10\
114& $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F_{5/2}^{\circ}$ & 912058 & 759 & 5247 & & & & & 7.50E-11& 7.37E-11& 3.87E-11\
115& $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G_{7/2}^{\circ}$ & 914356 & 1069 & 2139 & & & & & 6.67E-11& 6.56E-11& 1.07E-10\
116& $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F_{9/2}^{\circ}$ & 914876 & 801 & 2928 & & & & & 1.76E-10& 1.72E-10& 6.58E-11\
117& $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F_{7/2}^{\circ}$ & 916903 & 850 & 4217 & & & & & 9.83E-11& 9.69E-11& 3.64E-11\
118& $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G_{9/2}^{\circ}$ & 923103 & 871 & 522 & & & & & 2.88E-10& 2.83E-10& 1.43E-10\
119& $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D_{1/2}^{\circ}$ & 923999 & 671 & 5047 & & & & & 5.88E-11& 5.81E-11& 4.59E-11\
120& $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D_{3/2}^{\circ}$ & 924946 & 621 & 5146 & & & & & 4.79E-11& 4.76E-11& 4.38E-11\
121& $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P_{5/2}^{\circ}$ & 925514 & 521 & 4893 & & & & & 3.33E-11& 3.32E-11& 3.92E-11\
122& $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G_{11/2}^{\circ}$ & 929829 & 1996 & 9263 & & & & & 1.76E-11& 1.71E-11& 1.51E-11\
123& $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D_{5/2}^{\circ}$ & 929938 & 715 & 4849 & & & & & 2.79E-11& 2.77E-11& 3.87E-11\
124& $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P_{3/2}^{\circ}$ & 930904 & 548 & 4394 & & & & & 1.81E-11& 1.81E-11& 3.94E-11\
125& $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G_{9/2}^{\circ}$ & 931081 & 2028 & 10202 & & & & & 1.78E-11& 1.73E-11& 1.37E-11\
126& $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P_{1/2}^{\circ}$ & 931777 & 453 & 5323 & & & & & 1.29E-11& 1.29E-11& 3.61E-11\
127& $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D_{7/2}^{\circ}$ & 931912 & 907 & 5605 & & & & & 3.39E-11& 3.34E-11& 3.14E-11\
128& $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D_{5/2}^{\circ}$ & 933561 & 1357 & 7549 & & & & & 2.37E-11& 2.34E-11& 1.99E-11\
129& $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G_{7/2}^{\circ}$ & 933689 & 1695 & 9816 & & & & & 1.83E-11& 1.79E-11& 1.33E-11\
130& $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G_{5/2}^{\circ}$ & 934845 & 1452 & 9526 & & & & & 1.85E-11& 1.81E-11& 1.32E-11\
131& $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P_{1/2}^{\circ}$ & 939630 & 174 & 15879 & & & & & 1.22E-11& 1.21E-11& 8.24E-12\
132& $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P_{3/2}^{\circ}$ & 939850 & 104 & 17206 & & & & & 1.07E-11& 1.06E-11& 6.14E-12\
133& $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}G_{9/2}^{\circ}$ & 940286 & 2619 & 12688 & & & & & 1.97E-11& 1.91E-11& 1.58E-11\
134& $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P_{5/2}^{\circ}$ & 941075 & -33 & 17418 & & & & & 8.91E-12& 8.89E-12& 6.06E-12\
135& $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}F_{7/2}^{\circ}$ & 941344 & 1582 & 8286 & & & & & 2.37E-11& 2.33E-11& 1.94E-11\
136& $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}P_{1/2}^{\circ}$ & 942944 & 255 & 14126 & & & & & 1.36E-11& 1.35E-11& 6.20E-12\
137& $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}G_{7/2}^{\circ}$ & 946250 & 2000 & 11666 & & & & & 1.88E-11& 1.83E-11& 1.42E-11\
138& $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}P_{3/2}^{\circ}$ & 949108 & 381 & 13907 & & & & & 1.13E-11& 1.11E-11& 8.16E-12\
139& $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}H_{11/2}^{\circ}$ & 951567 & 1548 & 2012 & & & & & 2.53E-10& 2.45E-10& 6.66E-11\
140& $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}H_{9/2}^{\circ}$ & 955114 & 1907 & 1838 & & & & & 4.35E-11& 4.26E-11& 6.03E-11\
141& $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H_{9/2}^{\circ}$ & 960471 & 2067 & 15130 & & & & & 2.14E-11& 2.08E-11& 1.28E-11\
142& $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D_{7/2}^{\circ}$ & 960845 & 551 & 12147 & & & & & 7.38E-12& 7.37E-12& 6.02E-12\
143& $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}D_{5/2}^{\circ}$ & 961097 & 721 & 9558 & & & & & 2.66E-11& 2.65E-11& 3.67E-11\
144& $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}S)~^{2}D_{3/2}^{\circ}$ & 963980 & 771 & 8436 & & & & & 5.19E-11& 5.13E-11& 1.08E-10\
145& $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D_{5/2}^{\circ}$ & 964023 & 619 & 11820 & & & & & 8.30E-12& 8.29E-12& 6.49E-12\
146& $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D_{3/2}^{\circ}$ & 964961 & 713 & 11432 & & & & & 7.76E-12& 7.75E-12& 6.12E-12\
147& $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D_{1/2}^{\circ}$ & 965415 & 834 & 11101 & & & & & 7.49E-12& 7.49E-12& 6.10E-12\
148& $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H_{11/2}^{\circ}$ & 965992 & 2487 & 14920 & & & & & 1.42E-11& 1.38E-11& 1.15E-11\
149& $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D_{7/2}^{\circ}$ & 969935 & 732 & 12519 & & & & & 7.41E-12& 7.39E-12& 5.07E-12\
150& $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D_{5/2}^{\circ}$ & 971008 & 829 & 12459 & & & & & 6.76E-12& 6.75E-12& 5.09E-12\
151& $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F_{7/2}^{\circ}$ & 972582 & 778 & 16388 & & & & & 8.05E-12& 8.00E-12& 8.14E-12\
152& $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D_{3/2}^{\circ}$ & 973327 & 830 & 12206 & & & & & 6.56E-12& 6.56E-12& 5.00E-12\
153& $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F_{9/2}^{\circ}$ & 974575 & -148 & 18029 & & & & & 5.71E-12& 5.70E-12& 4.43E-12\
154& $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D_{1/2}^{\circ}$ & 975431 & 857 & 11797 & & & & & 6.40E-12& 6.40E-12& 4.92E-12\
155& $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}F_{5/2}^{\circ}$ & 976884 & 907 & 14716 & & & & & 8.85E-12& 8.78E-12& 6.84E-12\
156& $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F_{7/2}^{\circ}$ & 979945 & 7 & 13135 & & & & & 5.83E-12& 5.82E-12& 1.05E-11\
157& $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D_{5/2}^{\circ}$ & 982631 & 1591 & 9149 & & & & & 8.05E-12& 8.03E-12& 6.63E-12\
158& $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D_{3/2}^{\circ}$ & 983812 & 1355 & 9541 & & & & & 7.54E-12& 7.52E-12& 6.58E-12\
159& $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}G_{7/2}^{\circ}$ & 984424 & 1914 & 13147 & & & & & 1.76E-11& 1.72E-11& 4.53E-12\
160& $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F_{5/2}^{\circ}$ & 984497 & 111 & 16525 & & & & & 5.87E-12& 5.85E-12& 4.53E-12\
161& $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}P_{1/2}^{\circ}$ & 986925 & 1038 & 13231 & & & & & 6.54E-12& 6.52E-12& 6.17E-12\
162& $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F_{3/2}^{\circ}$ & 987039 & 363 & 14067 & & & & & 6.10E-12& 6.08E-12& 1.48E-11\
163& $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}P_{3/2}^{\circ}$ & 990290 & 894 & 12517 & & & & & 7.90E-12& 7.87E-12& 4.58E-12\
164& $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}G_{9/2}^{\circ}$ & 990339 & 1889 & 7780 & & & & & 1.84E-11& 1.80E-11& 1.44E-11\
165& $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}S_{1/2}^{\circ}$ & 992939 & 1113 & 8382 & & & & & 7.07E-12& 7.05E-12& 2.93E-11\
166& $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}S)~^{2}P_{3/2}^{\circ}$ & 993750 & 322 & 12332 & & & & & 1.21E-11& 1.20E-11& 6.95E-12\
167& $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}S)~^{2}P_{1/2}^{\circ}$ & 994309 & 531 & 12344 & & & & & 5.82E-11& 5.81E-11& 5.30E-12\
168& $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F_{5/2}^{\circ}$ & 995652 & 1188 & 13926 & & & & & 1.24E-11& 1.22E-11& 1.34E-11\
169& $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F_{7/2}^{\circ}$ & 997817 & 886 & 16594 & & & & & 8.08E-12& 8.02E-12& 6.37E-12\
170& $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D_{3/2}^{\circ}$ & 999296 & 248 & 16212 & & & & & 6.97E-12& 6.94E-12& 5.48E-12\
171& $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F_{5/2}^{\circ}$ & 999710 & 824 & 18879 & & & & & 7.89E-12& 7.81E-12& 5.87E-12\
172& $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P_{5/2}^{\circ}$ & 1002190 & 503 & 19054 & & & & & 6.97E-12& 6.93E-12& 4.66E-12\
173& $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P_{1/2}^{\circ}$ & 1003410 & 250 & 18080 & & & & & 6.11E-12& 6.09E-12& 4.55E-12\
174& $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P_{3/2}^{\circ}$ & 1003799 & 307 & 15124 & & & & & 8.24E-12& 8.22E-12& 4.98E-12\
175& $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D_{5/2}^{\circ}$ & 1003934 & 239 & 19680 & & & & & 7.37E-12& 7.32E-12& 4.86E-12\
176& $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}S_{3/2}^{\circ}$ & 1010445 & 963 & 12472 & & & & & 6.19E-12& 6.19E-12& 4.72E-12\
177& $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F_{7/2}^{\circ}$ & 1011432 & 1970 & 10602 & & & & & 1.82E-11& 1.77E-11& 1.27E-11\
178& $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}S)~^{2}D_{5/2}^{\circ}$ & 1018349 & 962 & 13306 & & & & & 6.42E-12& 6.40E-12& 5.97E-12\
179& $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}D_{3/2}^{\circ}$ & 1019218 & 923 & 14041 & & & & & 6.48E-12& 6.45E-12& 5.87E-12\
180& $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G_{7/2}^{\circ}$ & 1024704 & 579 & 17211 & & & & & 5.88E-12& 5.86E-12& 4.50E-12\
181& $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G_{9/2}^{\circ}$ & 1025042 & 483 & 17636 & & & & & 5.96E-12& 5.95E-12& 4.60E-12\
182& $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}S_{5/2}$ & 1032912 & 85 & 8576 & & & & & 1.12E-09& 1.10E-09& 2.12E-10\
183& $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D_{1/2}$ & 1039597 & 632 & 5826 & & & & & 6.78E-10& 6.75E-10& 1.60E-10\
184& $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D_{3/2}$ & 1039706 & 587 & 5714 & & & & & 6.71E-10& 6.67E-10& 1.60E-10\
185& $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D_{5/2}$ & 1039972 & 562 & 5559 & & & & & 6.99E-10& 6.92E-10& 1.61E-10\
186& $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D_{7/2}$ & 1040390 & 543 & 5464 & & & & & 7.09E-10& 7.01E-10& 1.63E-10\
187& $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D_{9/2}$ & 1041013 & 539 & 5587 & & & & & 7.30E-10& 7.21E-10& 1.66E-10\
188& $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}F_{5/2}^{\circ}$ & 1042662 & 433 & 22070 & & & & & 5.88E-12& 5.85E-12& 4.59E-12\
189& $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}F_{7/2}^{\circ}$ & 1043700 & 434 & 22134 & & & & & 6.01E-12& 5.98E-12& 4.49E-12\
190& $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}D_{3/2}^{\circ}$ & 1050888 & 1806 & 14469 & & & & & 7.36E-12& 7.26E-12& 5.32E-12\
191& $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F_{11/2}$ & 1052298 & 824 & 872 & & & & & 4.47E-10& 4.38E-10& 1.42E-10\
192& $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F_{9/2}$ & 1052884 & 859 & 1298 & & & & & 4.52E-10& 4.43E-10& 1.43E-10\
193& $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}D_{5/2}^{\circ}$ & 1053166 & 1763 & 14892 & & & & & 7.35E-12& 7.25E-12& 5.05E-12\
194& $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F_{7/2}$ & 1053290 & 879 & 1412 & & & & & 4.53E-10& 4.45E-10& 1.41E-10\
195& $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F_{1/2}$ & 1053392 & 934 & 1107 & & & & & 4.45E-10& 4.38E-10& 1.35E-10\
196& $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F_{3/2}$ & 1053433 & 2219 & 18443 & & & & & 4.40E-10& 4.33E-10& 1.18E-10\
197& $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F_{5/2}$ & 1053439 & 899 & 1368 & & & & & 4.51E-10& 4.44E-10& 1.38E-10\
[clrrrrrrrr]{} 1 & $3s^{2}\,3p^{5}~^{2}P_{3/2}^{\circ}$ & 0 & 0 & 0 & 0 &\
2 & $3s^{2}\,3p^{5}~^{2}P_{1/2}^{\circ}$ & 23678 & 23629 & 49 & 23629 & 49\
3 & $3s~^{2}S\,3p^{6}~^{2}S_{1/2}$ & 339773 & 338615 & 1158 & 338615 & 1158\
4 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D_{5/2}$ & 450451 & & & &\
5 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D_{7/2}$ & 450941 & 454000 & -3059 & 450812 TN & -129\
6 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D_{3/2}$ & 452175 & & & &\
7 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D_{1/2}$ & 454809 & & & &\
8 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F_{9/2}$ & 482542 & 485570 & -3028 & 482378 TN & 164\
9 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P_{1/2}$ & 486142 & & & &\
10 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F_{7/2}$ & 489790 & 492750 & -2960 & 489562 TN & 228\
11 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F_{5/2}$ & 496283 & & & 496352 TN & -69\
12 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P_{3/2}$ & 497035 & & & &\
13 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F_{3/2}$ & 497836 & & & &\
14 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P_{1/2}$ & 502810 & & & &\
15 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D_{3/2}$ & 503389 & & & &\
16 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F_{7/2}$ & 510045 & 513290 & -3245 & 510098 TN & -54\
17 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P_{3/2}$ & 511469 & & & 511482 TN & -13\
18 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P_{5/2}$ & 513156 & & & &\
19 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D_{5/2}$ & 517246 & & & 517550 D & -304\
20 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}G_{7/2}$ & 524204 & 526960 & -2756 & 523777 TN & -427\
21 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}G_{9/2}$ & 524407 & 527230 & -2823 & 524042 TN & -365\
22 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F_{5/2}$ & 528717 & & & 528370 D & 347\
23 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}F_{5/2}$ & 558198 & & & &\
24 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}F_{7/2}$ & 564358 & 567200 & -2842 & 564007 TN & 351\
25 & $3s^{2}\,3p^{4}(^{1}S)~^{1}S\,3d~^{2}D_{3/2}$ & 592695 & & & &\
26 & $3s^{2}\,3p^{4}(^{1}S)~^{1}S\,3d~^{2}D_{5/2}$ & 600433 & & & &\
27 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}S_{1/2}$ & 622807 & 622840 & -33 & 622836 & -29\
28 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P_{3/2}$ & 648787 & 648670 & 117 & 648630 & 157\
29 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D_{5/2}$ & 657278 & 657230 & 48 & 657240 & -38\
30 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P_{1/2}$ & 657577 & 657290 & 287 & 657360 D & 217\
31 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D_{3/2}$ & 677714 & 676420 & 1294 & 677527 TN & 187\
[crlrrrrrl]{} 24 & 1 & $3s^{2}\,3p^{5}~^{2}P_{3/2}^{\circ}$ & 0 & 0 & 0 & & & 0.95\
24 & 2 & $3s^{2}\,3p^{5}~^{2}P_{1/2}^{\circ}$ & 9927 & 9892 & 35 & 5.69E-02& 5.69E-02 & 0.95\
24 & 3 & $3s~^{2}S\,3p^{6}~^{2}S_{1/2}$ & 243659 & 242065 & 1594 & 3.26E-10& 3.07E-10 & 0.71 + 0.25 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}S$\
24 & 4 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D_{7/2}$ & 328241 & & & 2.98E-02& 2.98E-02 & 0.95\
24 & 5 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D_{5/2}$ & 328555 & & & 3.28E-07& 4.12E-07 & 0.94\
24 & 6 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D_{3/2}$ & 329365 & & & 3.76E-07& 3.84E-07 & 0.94\
24 & 7 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D_{1/2}$ & 330225 & & & 3.27E-07& 2.94E-07 & 0.94\
24 & 8 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F_{9/2}$ & 354043 & & & 2.30E-01& 2.29E-01 & 0.93 + 0.03 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}G$\
24 & 9 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P_{1/2}$ & 355764 & & & 5.69E-09& 6.02E-09 & 0.50 + 0.43 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P$\
24 & 10 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F_{7/2}$ & 357398 & & & 1.73E-01& 1.73E-01 & 0.92\
24 & 11 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F_{5/2}$ & 359822 & & & 5.31E-08& 5.37E-08 & 0.95\
24 & 12 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F_{3/2}$ & 361102 & & & 3.38E-08& 3.18E-08 & 0.91\
24 & 13 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P_{3/2}$ & 361328 & & & 3.60E-09& 3.76E-09 & 0.48 + 0.38 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P$ + 0.04 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P$\
24 & 14 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P_{1/2}$ & 368486 & & & 6.72E-09& 6.38E-09 & 0.94\
24 & 15 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D_{3/2}$ & 368491 & & & 4.29E-09& 4.29E-09 & 0.47 + 0.28 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D$ + 0.12 $3s^{2}\,3p^{4}(^{1}S)~^{1}S\,3d~^{2}D$\
24 & 16 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P_{3/2}$ & 371162 & & & 1.83E-07& 1.65E-07 & 0.83 + 0.04 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$ + 0.03 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P$\
24 & 17 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P_{5/2}$ & 373093 & & & 1.36E-08& 1.31E-08 & 0.66 + 0.13 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$ + 0.09 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D$\
24 & 18 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F_{7/2}$ & 374017 & & & 3.63E-02& 3.63E-02 & 0.53 + 0.29 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}G$ + 0.11 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}F$\
24 & 19 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D_{5/2}$ & 374346 & & & 1.25E-08& 1.19E-08 & 0.40 + 0.28 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P$ + 0.19 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D$\
24 & 20 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}G_{9/2}$ & 380465 & & & 4.23E-02& 4.23E-02 & 0.93 + 0.03 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F$\
24 & 21 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}G_{7/2}$ & 381009 & & & 4.06E-02& 4.05E-02 & 0.65 + 0.23 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F$ + 0.08 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}F$\
24 & 22 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F_{5/2}$ & 382658 & & & 1.06E-07& 1.06E-07 & 0.73 + 0.20 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}F$\
24 & 23 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}F_{5/2}$ & 408510 & & & 1.19E-08& 1.17E-08 & 0.75 + 0.18 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F$\
24 & 24 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}F_{7/2}$ & 410804 & & & 1.44E-02& 1.45E-02 & 0.77 + 0.18 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F$\
24 & 25 & $3s^{2}\,3p^{4}(^{1}S)~^{1}S\,3d~^{2}D_{3/2}$ & 434065 & & & 2.13E-09& 2.07E-09 & 0.68 + 0.23 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$\
24 & 26 & $3s^{2}\,3p^{4}(^{1}S)~^{1}S\,3d~^{2}D_{5/2}$ & 436440 & & & 2.22E-08& 2.47E-08 & 0.72 + 0.19 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$\
24 & 27 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}S_{1/2}$ & 462431 & 461540 & 891 & 7.44E-12& 7.27E-12 & 0.68 + 0.24 $3s~^{2}S\,3p^{6}~^{2}S$ + 0.03 $3s~^{2}S\,3p^{4}(^{3}P)~^{2}P\,3d^{2}(^{3}P)~^{2}S$\
24 & 28 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P_{3/2}$ & 480629 & 479310 & 1319 & 7.87E-12& 7.79E-12 & 0.50 + 0.41 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P$ + 0.02 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D$\
24 & 29 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P_{1/2}$ & 484202 & 482910 & 1292 & 8.01E-12& 7.93E-12 & 0.51 + 0.43 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P$\
24 & 30 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D_{5/2}$ & 489060 & 487780 & 1280 & 6.51E-12& 6.48E-12 & 0.65 + 0.20 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$ + 0.09 $3s^{2}\,3p^{4}(^{1}S)~^{1}S\,3d~^{2}D$\
24 & 31 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D_{3/2}$ & 497347 & 496170 & 1177 & 6.60E-12& 6.58E-12 & 0.61 + 0.17 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$ + 0.12 $3s^{2}\,3p^{4}(^{1}S)~^{1}S\,3d~^{2}D$\
24 & 32 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}P_{1/2}^{\circ}$ & 561641 & & & 4.79E-10& 4.80E-10 & 0.81 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
24 & 33 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}P_{3/2}^{\circ}$ & 563278 & & & 4.76E-10& 4.79E-10 & 0.80 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
24 & 34 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}P_{5/2}^{\circ}$ & 566104 & & & 4.72E-10& 4.77E-10 & 0.80 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
24 & 35 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F_{9/2}^{\circ}$ & 587238 & & & 2.62E-10& 2.51E-10 & 0.84 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
24 & 36 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F_{7/2}^{\circ}$ & 589367 & & & 2.59E-10& 2.50E-10 & 0.82 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
24 & 37 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F_{5/2}^{\circ}$ & 591395 & & & 2.57E-10& 2.48E-10 & 0.82 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
24 & 38 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F_{3/2}^{\circ}$ & 593123 & & & 2.55E-10& 2.47E-10 & 0.82 + 0.11 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
24 & 39 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D_{7/2}^{\circ}$ & 612907 & & & 2.64E-10& 2.60E-10 & 0.78 + 0.07 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
24 & 40 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D_{5/2}^{\circ}$ & 613956 & & & 2.64E-10& 2.59E-10 & 0.77 + 0.07 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
24 & 41 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D_{1/2}^{\circ}$ & 614167 & & & 2.64E-10& 2.59E-10 & 0.79 + 0.07 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
24 & 42 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D_{3/2}^{\circ}$ & 614244 & & & 2.64E-10& 2.59E-10 & 0.78 + 0.07 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
24 & 43 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}F_{7/2}^{\circ}$ & 622378 & & & 6.18E-10& 6.18E-10 & 0.72 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
24 & 44 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}D_{5/2}^{\circ}$ & 625434 & & & 5.09E-10& 5.00E-10 & 0.42 + 0.32 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}F^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D^{\circ}$\
24 & 45 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}D_{3/2}^{\circ}$ & 628881 & & & 3.47E-10& 3.40E-10 & 0.73 + 0.07 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}D^{\circ}$\
24 & 46 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}F_{5/2}^{\circ}$ & 630533 & & & 4.35E-10& 4.29E-10 & 0.42 + 0.33 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}D^{\circ}$ + 0.05 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
24 & 47 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}P_{1/2}^{\circ}$ & 642258 & & & 1.35E-10& 1.33E-10 & 0.40 + 0.22 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}P^{\circ}$ + 0.17 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P^{\circ}$\
24 & 48 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}P_{3/2}^{\circ}$ & 646769 & & & 1.49E-10& 1.47E-10 & 0.40 + 0.19 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}P^{\circ}$ + 0.17 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P^{\circ}$\
24 & 49 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{1/2}^{\circ}$ & 658995 & & & 1.71E-08& 1.66E-08 & 0.96\
24 & 50 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{3/2}^{\circ}$ & 659069 & & & 1.99E-08& 1.93E-08 & 0.96\
24 & 51 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{5/2}^{\circ}$ & 659218 & & & 2.53E-08& 2.50E-08 & 0.96\
24 & 52 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{7/2}^{\circ}$ & 659454 & & & 3.66E-08& 3.66E-08 & 0.96\
24 & 53 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{9/2}^{\circ}$ & 659779 & & & 6.55E-08& 6.67E-08 & 0.96\
24 & 54 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{11/2}^{\circ}$ & 660184 & & & 2.98E-07& 3.23E-07 & 0.96\
24 & 55 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}F_{5/2}^{\circ}$ & 664053 & & & 1.20E-10& 1.18E-10 & 0.56 + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
24 & 56 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}F_{7/2}^{\circ}$ & 666318 & & & 1.17E-10& 1.15E-10 & 0.56 + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
24 & 57 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{6}P_{3/2}^{\circ}$ & 678735 & & & 2.04E-08& 2.07E-08 & 0.96\
24 & 58 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{6}P_{5/2}^{\circ}$ & 678804 & & & 1.47E-08& 1.45E-08 & 0.95\
24 & 59 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{6}P_{7/2}^{\circ}$ & 678959 & & & 1.20E-08& 1.19E-08 & 0.95\
24 & 60 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}D_{3/2}^{\circ}$ & 687228 & & & 1.11E-10& 1.09E-10 & 0.55 + 0.20 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$\
24 & 61 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}D_{5/2}^{\circ}$ & 687892 & & & 1.12E-10& 1.10E-10 & 0.56 + 0.20 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$\
24 & 62 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D_{7/2}^{\circ}$ & 696602 & & & 3.87E-10& 3.78E-10 & 0.57 + 0.21 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
24 & 63 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D_{5/2}^{\circ}$ & 697208 & & & 3.63E-10& 3.55E-10 & 0.57 + 0.21 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
24 & 64 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D_{3/2}^{\circ}$ & 697658 & & & 3.55E-10& 3.46E-10 & 0.55 + 0.21 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.05 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
24 & 65 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P_{1/2}^{\circ}$ & 697946 & & & 1.01E-09& 9.92E-10 & 0.09 + 0.28 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.23 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
24 & 66 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D_{1/2}^{\circ}$ & 698619 & & & 4.41E-10& 4.31E-10 & 0.41 + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
24 & 67 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P_{3/2}^{\circ}$ & 700132 & & & 5.54E-09& 5.53E-09 & 0.41 + 0.32 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$\
24 & 68 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G_{5/2}^{\circ}$ & 701756 & & & 4.97E-09& 4.70E-09 & 0.48 + 0.18 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$\
24 & 69 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P_{5/2}^{\circ}$ & 702261 & & & 1.21E-08& 1.18E-08 & 0.38 + 0.29 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$\
24 & 70 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G_{7/2}^{\circ}$ & 702439 & & & 5.13E-09& 4.78E-09 & 0.58 + 0.24 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$\
24 & 71 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F_{3/2}^{\circ}$ & 703204 & & & 2.20E-09& 2.15E-09 & 0.22 + 0.37 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$ + 0.18 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
24 & 72 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G_{9/2}^{\circ}$ & 703547 & & & 6.26E-09& 5.64E-09 & 0.61 + 0.28 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.02 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$\
24 & 73 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G_{11/2}^{\circ}$ & 704785 & & & 6.96E-09& 5.96E-09 & 0.63 + 0.32 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$\
24 & 74 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F_{5/2}^{\circ}$ & 705555 & & & 3.58E-09& 3.48E-09 & 0.34 + 0.19 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
24 & 75 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F_{7/2}^{\circ}$ & 707545 & & & 3.64E-09& 3.52E-09 & 0.39 + 0.20 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
24 & 76 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F_{9/2}^{\circ}$ & 709489 & & & 3.26E-09& 3.14E-09 & 0.44 + 0.22 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
24 & 77 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H_{7/2}^{\circ}$ & 720039 & & & 1.69E-09& 1.64E-09 & 0.79 + 0.04 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F^{\circ}$ + 0.03 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{2}F^{\circ}$\
24 & 78 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H_{9/2}^{\circ}$ & 720631 & & & 3.50E-08& 3.40E-08 & 0.90 + 0.04 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
24 & 79 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H_{11/2}^{\circ}$ & 721403 & & & 4.33E-07& 3.23E-07 & 0.91 + 0.04 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
24 & 80 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F_{7/2}^{\circ}$ & 722076 & & & 2.62E-10& 2.53E-10 & 0.32 + 0.23 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H^{\circ}$\
24 & 81 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H_{13/2}^{\circ}$ & 722827 & & & 5.67E-02& 5.64E-02 & 0.97\
24 & 82 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F_{5/2}^{\circ}$ & 725024 & & & 1.69E-10& 1.64E-10 & 0.34 + 0.24 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
24 & 83 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D_{3/2}^{\circ}$ & 725289 & & & 4.14E-11& 4.09E-11 & 0.22 + 0.21 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}D^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}D^{\circ}$\
24 & 84 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}P_{1/2}^{\circ}$ & 726577 & & & 8.43E-12& 8.41E-12 & 0.53 + 0.27 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}P^{\circ}$ + 0.05 $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}P^{\circ}$\
24 & 85 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}P_{3/2}^{\circ}$ & 727059 & & & 9.45E-12& 9.44E-12 & 0.47 + 0.21 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}P^{\circ}$ + 0.05 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$\
24 & 86 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H_{9/2}^{\circ}$ & 727749 & & & 1.76E-09& 1.66E-09 & 0.43 + 0.36 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}H^{\circ}$\
24 & 87 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H_{11/2}^{\circ}$ & 729335 & & & 3.03E-09& 2.75E-09 & 0.30 + 0.41 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
24 & 88 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D_{5/2}^{\circ}$ & 730183 & & & 8.73E-11& 8.49E-11 & 0.23 + 0.24 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}D^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D^{\circ}$\
24 & 89 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G_{5/2}^{\circ}$ & 735559 & & & 5.64E-10& 5.39E-10 & 0.61 + 0.29 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$\
24 & 90 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G_{7/2}^{\circ}$ & 736243 & & & 4.99E-10& 4.78E-10 & 0.59 + 0.25 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.03 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H^{\circ}$\
24 & 91 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G_{9/2}^{\circ}$ & 738060 & & & 5.18E-10& 4.96E-10 & 0.54 + 0.20 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.05 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G^{\circ}$\
24 & 92 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G_{7/2}^{\circ}$ & 740600 & & & 2.92E-10& 2.81E-10 & 0.72 + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G^{\circ}$\
24 & 93 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G_{9/2}^{\circ}$ & 740955 & & & 3.00E-10& 2.89E-10 & 0.74 + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G^{\circ}$\
24 & 94 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G_{11/2}^{\circ}$ & 741225 & & & 1.01E-09& 9.44E-10 & 0.54 + 0.18 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H^{\circ}$\
24 & 95 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F_{5/2}^{\circ}$ & 749890 & & & 2.86E-11& 2.79E-11 & 0.43 + 0.35 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.07 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F^{\circ}$\
24 & 96 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F_{7/2}^{\circ}$ & 750046 & & & 2.89E-11& 2.82E-11 & 0.43 + 0.34 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.07 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F^{\circ}$\
24 & 97 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F_{3/2}^{\circ}$ & 750368 & & & 2.83E-11& 2.76E-11 & 0.44 + 0.37 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.07 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F^{\circ}$\
24 & 98 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F_{9/2}^{\circ}$ & 750974 & & & 2.93E-11& 2.85E-11 & 0.44 + 0.35 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.07 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F^{\circ}$\
24 & 99 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{2}P_{3/2}^{\circ}$ & 757804 & & & 4.06E-11& 4.03E-11 & 0.20 + 0.24 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}P^{\circ}$ + 0.17 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P^{\circ}$\
24 & 100 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}I_{11/2}^{\circ}$ & 758244 & & & 8.18E-09& 8.05E-09 & 0.94\
24 & 101 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}I_{13/2}^{\circ}$ & 758854 & & & 8.61E-02& 8.59E-02 & 0.97\
24 & 102 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}S_{1/2}^{\circ}$ & 760236 & & & 9.31E-11& 9.29E-11 & 0.30 + 0.23 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}S^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}P^{\circ}$\
24 & 103 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D_{1/2}^{\circ}$ & 761407 & & & 4.29E-11& 4.25E-11 & 0.43 + 0.24 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
24 & 104 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D_{3/2}^{\circ}$ & 761985 & & & 6.02E-11& 5.96E-11 & 0.29 + 0.18 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
24 & 105 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D_{5/2}^{\circ}$ & 762803 & & & 8.86E-11& 8.71E-11 & 0.11 + 0.15 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
24 & 106 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}D_{3/2}^{\circ}$ & 764460 & & & 1.02E-10& 9.95E-11 & 0.05 + 0.27 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}S)~^{4}S^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}S^{\circ}$\
24 & 107 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F_{5/2}^{\circ}$ & 765031 & & & 8.50E-11& 8.25E-11 & 0.25 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D^{\circ}$\
24 & 108 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}S)~^{4}S_{3/2}^{\circ}$ & 765072 & & & 1.26E-10& 1.24E-10 & 0.31 + 0.15 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}S^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D^{\circ}$\
24 & 109 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D_{7/2}^{\circ}$ & 765443 & & & 6.59E-11& 6.49E-11 & 0.24 + 0.25 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
24 & 110 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D_{5/2}^{\circ}$ & 765586 & & & 6.30E-11& 6.22E-11 & 0.20 + 0.17 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F^{\circ}$ + 0.15 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
24 & 111 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P_{1/2}^{\circ}$ & 766753 & & & 5.35E-11& 5.30E-11 & 0.14 + 0.22 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}S^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}S^{\circ}$\
24 & 112 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F_{7/2}^{\circ}$ & 769236 & & & 8.50E-11& 8.33E-11 & 0.29 + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
24 & 113 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F_{3/2}^{\circ}$ & 769734 & & & 1.83E-10& 1.79E-10 & 0.40 + 0.36 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
24 & 114 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F_{5/2}^{\circ}$ & 770760 & & & 1.28E-10& 1.25E-10 & 0.36 + 0.36 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
24 & 115 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F_{7/2}^{\circ}$ & 771934 & & & 1.14E-10& 1.12E-10 & 0.20 + 0.18 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G^{\circ}$\
24 & 116 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F_{9/2}^{\circ}$ & 772740 & & & 2.50E-10& 2.44E-10 & 0.33 + 0.25 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G^{\circ}$\
24 & 117 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G_{7/2}^{\circ}$ & 774144 & & & 3.31E-10& 3.28E-10 & 0.21 + 0.24 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}G^{\circ}$\
24 & 118 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G_{9/2}^{\circ}$ & 778054 & & & 5.19E-10& 5.14E-10 & 0.21 + 0.25 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.20 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}G^{\circ}$\
24 & 119 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D_{1/2}^{\circ}$ & 781751 & & & 6.92E-11& 6.87E-11 & 0.36 + 0.29 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.15 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
24 & 120 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D_{3/2}^{\circ}$ & 782328 & & & 5.90E-11& 5.89E-11 & 0.27 + 0.25 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
24 & 121 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P_{5/2}^{\circ}$ & 782615 & & & 4.47E-11& 4.49E-11 & 0.48 + 0.11 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
24 & 122 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D_{5/2}^{\circ}$ & 785884 & & & 4.60E-11& 4.61E-11 & 0.19 + 0.33 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.19 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
24 & 123 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G_{11/2}^{\circ}$ & 786207 & & & 2.22E-11& 2.16E-11 & 0.38 + 0.26 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G^{\circ}$ + 0.25 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
24 & 124 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P_{3/2}^{\circ}$ & 786432 & & & 3.22E-11& 3.24E-11 & 0.68 + 0.07 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
24 & 125 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D_{7/2}^{\circ}$ & 786690 & & & 4.65E-11& 4.63E-11 & 0.34 + 0.25 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
24 & 126 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G_{9/2}^{\circ}$ & 787234 & & & 2.23E-11& 2.18E-11 & 0.35 + 0.25 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G^{\circ}$ + 0.24 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
24 & 127 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P_{1/2}^{\circ}$ & 787273 & & & 2.55E-11& 2.57E-11 & 0.78 + 0.05 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.05 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}P^{\circ}$\
24 & 128 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G_{7/2}^{\circ}$ & 788612 & & & 2.24E-11& 2.19E-11 & 0.37 + 0.27 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G^{\circ}$ + 0.26 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
24 & 129 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G_{5/2}^{\circ}$ & 789011 & & & 2.47E-11& 2.42E-11 & 0.22 + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G^{\circ}$ + 0.17 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
24 & 130 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}F_{5/2}^{\circ}$ & 789780 & & & 2.60E-11& 2.56E-11 & 0.17 + 0.16 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
24 & 131 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}G_{9/2}^{\circ}$ & 793895 & & & 2.56E-11& 2.49E-11 & 0.81 + 0.05 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}G^{\circ}$ + 0.03 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$\
24 & 132 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P_{1/2}^{\circ}$ & 794451 & & & 1.06E-11& 1.06E-11 & 0.40 + 0.27 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
24 & 133 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P_{3/2}^{\circ}$ & 794460 & & & 9.95E-12& 9.97E-12 & 0.27 + 0.38 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
24 & 134 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}F_{7/2}^{\circ}$ & 794481 & & & 2.90E-11& 2.86E-11 & 0.18 + 0.23 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}G^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
24 & 135 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P_{5/2}^{\circ}$ & 795136 & & & 9.22E-12& 9.23E-12 & 0.27 + 0.36 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
24 & 136 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}G_{7/2}^{\circ}$ & 798011 & & & 2.48E-11& 2.42E-11 & 0.64 + 0.07 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}G^{\circ}$\
24 & 137 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}P_{1/2}^{\circ}$ & 798764 & & & 1.44E-11& 1.43E-11 & 0.38 + 0.22 $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}P^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P^{\circ}$\
24 & 138 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}H_{11/2}^{\circ}$ & 803074 & & & 4.12E-10& 4.01E-10 & 0.65 + 0.23 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H^{\circ}$ + 0.03 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$\
24 & 139 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}P_{3/2}^{\circ}$ & 803118 & & & 1.37E-11& 1.35E-11 & 0.34 + 0.23 $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}P^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P^{\circ}$\
24 & 140 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}H_{9/2}^{\circ}$ & 805959 & & & 9.87E-11& 9.69E-11 & 0.80 + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H^{\circ}$\
24 & 141 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}S)~^{2}D_{5/2}^{\circ}$ & 811908 & & & 6.21E-11& 6.17E-11 & 0.37 + 0.18 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}D^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D^{\circ}$\
24 & 142 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H_{9/2}^{\circ}$ & 811985 & & & 2.26E-11& 2.20E-11 & 0.48 + 0.44 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H^{\circ}$\
24 & 143 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D_{7/2}^{\circ}$ & 812004 & & & 9.11E-12& 9.09E-12 & 0.49 + 0.20 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,4p~^{4}D^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
24 & 144 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}S)~^{2}D_{3/2}^{\circ}$ & 813332 & & & 8.55E-11& 8.39E-11 & 0.44 + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}D^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D^{\circ}$\
24 & 145 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D_{5/2}^{\circ}$ & 813616 & & & 1.02E-11& 1.01E-11 & 0.38 + 0.21 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,4p~^{4}D^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
24 & 146 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D_{3/2}^{\circ}$ & 814964 & & & 9.55E-12& 9.53E-12 & 0.44 + 0.17 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,4p~^{4}D^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
24 & 147 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H_{11/2}^{\circ}$ & 815374 & & & 1.83E-11& 1.78E-11 & 0.46 + 0.33 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}H^{\circ}$\
24 & 148 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D_{1/2}^{\circ}$ & 815605 & & & 8.97E-12& 8.97E-12 & 0.48 + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.14 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,4p~^{4}D^{\circ}$\
24 & 149 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D_{7/2}^{\circ}$ & 819006 & & & 8.74E-12& 8.77E-12 & 0.28 + 0.19 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,4p~^{4}D^{\circ}$ + 0.15 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
24 & 150 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}D_{5/2}^{\circ}$ & 819895 & & & 1.10E-11& 1.10E-11 & 0.03 + 0.21 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,4p~^{2}D^{\circ}$ + 0.17 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
24 & 151 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D_{3/2}^{\circ}$ & 822162 & & & 8.59E-12& 8.62E-12 & 0.11 + 0.26 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D^{\circ}$\
24 & 152 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D_{1/2}^{\circ}$ & 823059 & & & 8.31E-12& 8.33E-12 & 0.12 + 0.26 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D^{\circ}$\
24 & 153 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F_{7/2}^{\circ}$ & 823721 & & & 1.08E-11& 1.07E-11 & 0.07 + 0.15 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F^{\circ}$ + 0.13 $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}F^{\circ}$\
24 & 154 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F_{5/2}^{\circ}$ & 826795 & & & 1.05E-11& 1.04E-11 & 0.13 + 0.13 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.12 $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}F^{\circ}$\
24 & 155 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F_{9/2}^{\circ}$ & 827521 & & & 6.83E-12& 6.87E-12 & 0.35 + 0.27 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.17 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
24 & 156 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F_{7/2}^{\circ}$ & 831519 & & & 1.17E-11& 1.18E-11 & 0.16 + 0.23 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,4p~^{4}D^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
24 & 157 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}G_{7/2}^{\circ}$ & 832359 & & & 2.08E-11& 2.04E-11 & 0.53 + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G^{\circ}$\
24 & 158 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F_{5/2}^{\circ}$ & 833974 & & & 7.51E-12& 7.55E-12 & 0.29 + 0.26 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
24 & 159 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F_{3/2}^{\circ}$ & 835654 & & & 1.03E-11& 1.03E-11 & 0.17 + 0.18 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.17 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,4p~^{4}D^{\circ}$\
24 & 160 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}G_{9/2}^{\circ}$ & 835959 & & & 2.39E-11& 2.34E-11 & 0.55 + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G^{\circ}$\
24 & 161 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}S)~^{2}P_{1/2}^{\circ}$ & 838982 & & & 9.31E-11& 9.44E-11 & 0.54 + 0.20 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}P^{\circ}$ + 0.05 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,4p~^{2}P^{\circ}$\
24 & 162 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}S)~^{2}P_{3/2}^{\circ}$ & 839158 & & & 5.63E-11& 5.59E-11 & 0.47 + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}P^{\circ}$ + 0.06 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,4p~^{2}P^{\circ}$\
24 & 163 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D_{5/2}^{\circ}$ & 839427 & & & 2.58E-11& 2.57E-11 & 0.15 + 0.21 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,4p~^{2}D^{\circ}$ + 0.12 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,4p~^{4}D^{\circ}$\
24 & 164 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{2}P_{1/2}^{\circ}$ & 840879 & & & 1.53E-11& 1.55E-11 & 0.23 + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}P^{\circ}$\
24 & 165 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}P_{3/2}^{\circ}$ & 841661 & & & 1.48E-11& 1.49E-11 & 0.12 + 0.18 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,4p~^{2}D^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{2}P^{\circ}$\
24 & 166 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F_{5/2}^{\circ}$ & 842102 & & & 1.39E-11& 1.37E-11 & 0.30 + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F^{\circ}$\
24 & 167 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F_{7/2}^{\circ}$ & 842242 & & & 1.06E-11& 1.05E-11 & 0.22 + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$ + 0.17 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,4p~^{2}F^{\circ}$\
24 & 168 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D_{3/2}^{\circ}$ & 843495 & & & 1.85E-11& 1.87E-11 & 0.11 + 0.24 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,4p~^{2}D^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$\
24 & 169 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D_{3/2}^{\circ}$ & 846507 & & & 9.13E-12& 9.16E-12 & 0.13 + 0.20 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}D^{\circ}$\
24 & 170 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F_{5/2}^{\circ}$ & 846712 & & & 1.33E-11& 1.33E-11 & 0.23 + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$ + 0.08 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,4p~^{2}F^{\circ}$\
24 & 171 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}S_{1/2}^{\circ}$ & 847864 & & & 1.67E-11& 1.68E-11 & 0.19 + 0.33 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,4p~^{2}S^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
24 & 172 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D_{5/2}^{\circ}$ & 848557 & & & 9.76E-12& 9.76E-12 & 0.08 + 0.17 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$\
24 & 173 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P_{5/2}^{\circ}$ & 850100 & & & 7.81E-12& 7.86E-12 & 0.39 + 0.30 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
24 & 174 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P_{3/2}^{\circ}$ & 850139 & & & 8.36E-12& 8.44E-12 & 0.39 + 0.31 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
24 & 175 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P_{1/2}^{\circ}$ & 850829 & & & 9.38E-12& 9.47E-12 & 0.34 + 0.26 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.08 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,4p~^{2}S^{\circ}$\
24 & 176 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F_{7/2}^{\circ}$ & 853699 & & & 2.05E-11& 2.02E-11 & 0.39 + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}F^{\circ}$\
24 & 177 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}S_{3/2}^{\circ}$ & 861261 & & & 1.55E-11& 1.60E-11 & 0.43 + 0.26 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}S)~^{4}S^{\circ}$ + 0.19 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,4p~^{4}S^{\circ}$\
24 & 178 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G_{9/2}^{\circ}$ & 868955 & & & 7.08E-12& 7.13E-12 & 0.43 + 0.35 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G^{\circ}$\
24 & 179 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G_{7/2}^{\circ}$ & 869104 & & & 7.02E-12& 7.07E-12 & 0.42 + 0.36 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G^{\circ}$\
24 & 180 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}S_{5/2}$ & 869605 & & & 1.63E-09& 1.58E-09 & 0.90 + 0.08 $3s^{2}\,3p^{2}(^{3}P)~^{3}P\,3d^{3}(^{4}P)~^{6}S$\
24 & 181 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}D_{3/2}^{\circ}$ & 871550 & & & 2.07E-11& 2.07E-11 & 0.10 + 0.31 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,4p~^{2}D^{\circ}$ + 0.15 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}S)~^{2}D^{\circ}$\
24 & 182 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}D_{5/2}^{\circ}$ & 872037 & & & 2.14E-11& 2.17E-11 & 0.10 + 0.39 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,4p~^{2}D^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}S)~^{2}D^{\circ}$\
24 & 183 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D_{1/2}$ & 876216 & & & 1.00E-09& 9.73E-10 & 0.59 + 0.30 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}D$ + 0.06 $3s^{2}\,3p^{2}(^{3}P)~^{3}P\,3d^{3}(^{4}F)~^{6}D$\
24 & 184 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D_{3/2}$ & 876271 & & & 1.00E-09& 9.74E-10 & 0.58 + 0.30 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}D$ + 0.06 $3s^{2}\,3p^{2}(^{3}P)~^{3}P\,3d^{3}(^{4}F)~^{6}D$\
24 & 185 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D_{5/2}$ & 876397 & & & 1.01E-09& 9.79E-10 & 0.55 + 0.31 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}D$ + 0.06 $3s^{2}\,3p^{2}(^{3}P)~^{3}P\,3d^{3}(^{4}F)~^{6}D$\
24 & 186 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D_{7/2}$ & 876602 & & & 1.02E-09& 9.90E-10 & 0.52 + 0.33 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}D$ + 0.06 $3s^{2}\,3p^{2}(^{3}P)~^{3}P\,3d^{3}(^{4}F)~^{6}D$\
24 & 187 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D_{9/2}$ & 876903 & & & 1.05E-09& 1.01E-09 & 0.49 + 0.36 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}D$ + 0.06 $3s^{2}\,3p^{2}(^{3}P)~^{3}P\,3d^{3}(^{4}F)~^{6}D$\
25 & 1 & $3s^{2}\,3p^{5}~^{2}P_{3/2}^{\circ}$ & 0 & 0 & 0 & & & 0.96\
25 & 2 & $3s^{2}\,3p^{5}~^{2}P_{1/2}^{\circ}$ & 12582 & 12546 & 36 & 2.79E-02& 2.79E-02 & 0.96\
25 & 3 & $3s~^{2}S\,3p^{6}~^{2}S_{1/2}$ & 265836 & 265408 & 428 & 2.72E-10& 2.63E-10 & 0.71 + 0.25 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}S$\
25 & 4 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D_{7/2}$ & 357786 & & & 2.26E-02& 2.26E-02 & 0.94\
25 & 5 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D_{5/2}$ & 358013 & & & 2.31E-07& 2.92E-07 & 0.93\
25 & 6 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D_{3/2}$ & 359021 & & & 2.47E-07& 2.47E-07 & 0.93\
25 & 7 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D_{1/2}$ & 360186 & & & 1.92E-07& 1.69E-07 & 0.94\
25 & 8 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F_{9/2}$ & 385248 & & & 1.40E-01& 1.40E-01 & 0.92 + 0.04 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}G$\
25 & 9 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P_{1/2}$ & 387221 & & & 4.97E-09& 5.06E-09 & 0.50 + 0.42 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P$ + 0.02 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D$\
25 & 10 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F_{7/2}$ & 389459 & & & 9.88E-02& 9.87E-02 & 0.91 + 0.02 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}G$ + 0.02 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F$\
25 & 11 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F_{5/2}$ & 392563 & & & 3.34E-08& 3.45E-08 & 0.94\
25 & 12 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F_{3/2}$ & 393962 & & & 5.10E-09& 5.20E-09 & 0.47 + 0.24 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P$ + 0.19 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P$\
25 & 13 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P_{3/2}$ & 394108 & & & 5.66E-09& 5.74E-09 & 0.24 + 0.46 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F$ + 0.19 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P$\
25 & 14 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D_{3/2}$ & 400946 & & & 3.50E-09& 3.69E-09 & 0.46 + 0.27 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D$ + 0.11 $3s^{2}\,3p^{4}(^{1}S)~^{1}S\,3d~^{2}D$\
25 & 15 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P_{1/2}$ & 401058 & & & 4.30E-09& 4.17E-09 & 0.94\
25 & 16 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P_{3/2}$ & 404712 & & & 1.89E-07& 1.93E-07 & 0.80 + 0.04 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P$ + 0.04 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$\
25 & 17 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P_{5/2}$ & 406810 & & & 1.03E-08& 1.06E-08 & 0.60 + 0.16 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$ + 0.11 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D$\
25 & 18 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F_{7/2}$ & 406834 & & & 2.34E-02& 2.33E-02 & 0.52 + 0.30 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}G$ + 0.10 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}F$\
25 & 19 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D_{5/2}$ & 408473 & & & 1.34E-08& 1.36E-08 & 0.36 + 0.34 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P$ + 0.17 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D$\
25 & 20 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}G_{9/2}$ & 414854 & & & 2.37E-02& 2.36E-02 & 0.92 + 0.04 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F$\
25 & 21 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}G_{7/2}$ & 415368 & & & 2.36E-02& 2.36E-02 & 0.63 + 0.24 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F$ + 0.09 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}F$\
25 & 22 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F_{5/2}$ & 417655 & & & 6.63E-08& 6.85E-08 & 0.72 + 0.20 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}F$ + 0.02 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$\
25 & 23 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}F_{5/2}$ & 444709 & & & 8.18E-09& 8.27E-09 & 0.75 + 0.18 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F$\
25 & 24 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}F_{7/2}$ & 447724 & & & 8.34E-03& 8.37E-03 & 0.77 + 0.18 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F$\
25 & 25 & $3s^{2}\,3p^{4}(^{1}S)~^{1}S\,3d~^{2}D_{3/2}$ & 472345 & & & 1.40E-09& 1.41E-09 & 0.68 + 0.24 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$\
25 & 26 & $3s^{2}\,3p^{4}(^{1}S)~^{1}S\,3d~^{2}D_{5/2}$ & 475614 & & & 1.10E-08& 1.12E-08 & 0.73 + 0.18 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$\
25 & 27 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}S_{1/2}$ & 500867 & 501710 & -843 & 6.86E-12& 6.81E-12 & 0.69 + 0.24 $3s~^{2}S\,3p^{6}~^{2}S$ + 0.03 $3s~^{2}S\,3p^{4}(^{3}P)~^{2}P\,3d^{2}(^{3}P)~^{2}S$\
25 & 28 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P_{3/2}$ & 521492 & 521840 & -348 & 7.31E-12& 7.41E-12 & 0.50 + 0.40 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P$ + 0.03 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D$\
25 & 29 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P_{1/2}$ & 526049 & 526380 & -331 & 7.47E-12& 7.56E-12 & 0.51 + 0.43 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P$\
25 & 30 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D_{5/2}$ & 530004 & 530560 & -556 & 6.08E-12& 6.18E-12 & 0.65 + 0.20 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$ + 0.09 $3s^{2}\,3p^{4}(^{1}S)~^{1}S\,3d~^{2}D$\
25 & 31 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D_{3/2}$ & 540681 & 541160 & -479 & 6.17E-12& 6.24E-12 & 0.61 + 0.17 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$ + 0.12 $3s^{2}\,3p^{4}(^{1}S)~^{1}S\,3d~^{2}D$\
25 & 32 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}P_{1/2}^{\circ}$ & 613989 & & & 3.93E-10& 3.84E-10 & 0.81 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
25 & 33 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}P_{3/2}^{\circ}$ & 616058 & & & 3.91E-10& 3.83E-10 & 0.81 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
25 & 34 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}P_{5/2}^{\circ}$ & 619684 & & & 3.87E-10& 3.81E-10 & 0.80 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
25 & 35 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F_{9/2}^{\circ}$ & 641922 & & & 2.16E-10& 2.05E-10 & 0.84 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
25 & 36 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F_{7/2}^{\circ}$ & 644435 & & & 2.14E-10& 2.03E-10 & 0.82 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
25 & 37 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F_{5/2}^{\circ}$ & 646947 & & & 2.11E-10& 2.02E-10 & 0.81 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
25 & 38 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F_{3/2}^{\circ}$ & 649167 & & & 2.09E-10& 2.00E-10 & 0.82 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
25 & 39 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D_{7/2}^{\circ}$ & 670178 & & & 2.20E-10& 2.12E-10 & 0.78 + 0.07 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
25 & 40 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D_{5/2}^{\circ}$ & 671546 & & & 2.19E-10& 2.11E-10 & 0.77 + 0.07 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
25 & 41 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D_{1/2}^{\circ}$ & 671577 & & & 2.18E-10& 2.10E-10 & 0.79 + 0.07 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
25 & 42 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D_{3/2}^{\circ}$ & 671827 & & & 2.18E-10& 2.11E-10 & 0.78 + 0.07 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
25 & 43 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}F_{7/2}^{\circ}$ & 679981 & & & 4.95E-10& 4.84E-10 & 0.72 + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
25 & 44 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}D_{5/2}^{\circ}$ & 683565 & & & 4.11E-10& 3.97E-10 & 0.45 + 0.30 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}F^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D^{\circ}$\
25 & 45 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}D_{3/2}^{\circ}$ & 687549 & & & 2.75E-10& 2.65E-10 & 0.72 + 0.07 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}D^{\circ}$\
25 & 46 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}F_{5/2}^{\circ}$ & 690037 & & & 3.50E-10& 3.40E-10 & 0.45 + 0.30 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}D^{\circ}$ + 0.05 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
25 & 47 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}P_{1/2}^{\circ}$ & 701130 & & & 1.10E-10& 1.08E-10 & 0.41 + 0.23 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}P^{\circ}$ + 0.17 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P^{\circ}$\
25 & 48 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}P_{3/2}^{\circ}$ & 707080 & & & 1.23E-10& 1.20E-10 & 0.39 + 0.19 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}P^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P^{\circ}$\
25 & 49 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{1/2}^{\circ}$ & 720570 & & & 1.08E-08& 1.04E-08 & 0.96\
25 & 50 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{3/2}^{\circ}$ & 720674 & & & 1.27E-08& 1.22E-08 & 0.96\
25 & 51 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{5/2}^{\circ}$ & 720879 & & & 1.60E-08& 1.57E-08 & 0.96\
25 & 52 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{7/2}^{\circ}$ & 721208 & & & 2.31E-08& 2.29E-08 & 0.96\
25 & 53 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{9/2}^{\circ}$ & 721667 & & & 4.08E-08& 4.11E-08 & 0.96\
25 & 54 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{11/2}^{\circ}$ & 722239 & & & 1.94E-07& 2.11E-07 & 0.96\
25 & 55 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}F_{5/2}^{\circ}$ & 725356 & & & 9.88E-11& 9.60E-11 & 0.57 + 0.18 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
25 & 56 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}F_{7/2}^{\circ}$ & 728322 & & & 9.63E-11& 9.37E-11 & 0.56 + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
25 & 57 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{6}P_{3/2}^{\circ}$ & 742346 & & & 1.28E-08& 1.29E-08 & 0.96\
25 & 58 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{6}P_{5/2}^{\circ}$ & 742486 & & & 9.31E-09& 9.12E-09 & 0.95\
25 & 59 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{6}P_{7/2}^{\circ}$ & 742687 & & & 7.33E-09& 7.21E-09 & 0.94\
25 & 60 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}D_{3/2}^{\circ}$ & 750584 & & & 9.07E-11& 8.79E-11 & 0.56 + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$\
25 & 61 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}D_{5/2}^{\circ}$ & 751570 & & & 9.20E-11& 8.91E-11 & 0.57 + 0.20 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$\
25 & 62 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D_{7/2}^{\circ}$ & 761622 & & & 3.35E-10& 3.24E-10 & 0.56 + 0.21 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
25 & 63 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D_{5/2}^{\circ}$ & 762285 & & & 3.11E-10& 3.01E-10 & 0.55 + 0.21 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
25 & 64 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P_{1/2}^{\circ}$ & 762657 & & & 1.90E-09& 1.87E-09 & 0.12 + 0.37 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.29 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
25 & 65 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D_{3/2}^{\circ}$ & 762725 & & & 3.05E-10& 2.95E-10 & 0.52 + 0.20 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.05 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
25 & 66 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D_{1/2}^{\circ}$ & 764011 & & & 2.92E-10& 2.82E-10 & 0.53 + 0.20 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.05 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
25 & 67 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P_{3/2}^{\circ}$ & 765419 & & & 4.47E-09& 4.43E-09 & 0.31 + 0.39 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$\
25 & 68 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G_{5/2}^{\circ}$ & 766570 & & & 3.23E-09& 3.06E-09 & 0.51 + 0.18 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$\
25 & 69 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G_{7/2}^{\circ}$ & 767494 & & & 4.02E-09& 3.77E-09 & 0.57 + 0.22 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.05 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$\
25 & 70 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F_{3/2}^{\circ}$ & 768021 & & & 1.44E-09& 1.39E-09 & 0.17 + 0.33 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$ + 0.20 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
25 & 71 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P_{5/2}^{\circ}$ & 768062 & & & 1.06E-08& 1.04E-08 & 0.41 + 0.31 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$\
25 & 72 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G_{9/2}^{\circ}$ & 769053 & & & 5.38E-09& 4.88E-09 & 0.61 + 0.27 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.02 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$\
25 & 73 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G_{11/2}^{\circ}$ & 770720 & & & 6.31E-09& 5.40E-09 & 0.63 + 0.32 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$\
25 & 74 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F_{5/2}^{\circ}$ & 771158 & & & 3.01E-09& 2.89E-09 & 0.13 + 0.32 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$ + 0.18 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
25 & 75 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F_{7/2}^{\circ}$ & 773644 & & & 3.12E-09& 2.99E-09 & 0.38 + 0.20 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
25 & 76 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F_{9/2}^{\circ}$ & 776054 & & & 2.73E-09& 2.60E-09 & 0.44 + 0.22 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
25 & 77 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H_{7/2}^{\circ}$ & 786023 & & & 1.32E-09& 1.27E-09 & 0.77 + 0.04 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
25 & 78 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H_{9/2}^{\circ}$ & 786907 & & & 2.12E-08& 2.06E-08 & 0.88 + 0.05 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
25 & 79 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H_{11/2}^{\circ}$ & 788003 & & & 3.48E-07& 2.52E-07 & 0.90 + 0.05 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
25 & 80 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F_{7/2}^{\circ}$ & 788941 & & & 2.28E-10& 2.19E-10 & 0.31 + 0.22 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H^{\circ}$\
25 & 81 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H_{13/2}^{\circ}$ & 790039 & & & 4.15E-02& 4.14E-02 & 0.97\
25 & 82 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D_{3/2}^{\circ}$ & 791909 & & & 1.96E-11& 1.93E-11 & 0.03 + 0.18 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}P^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}D^{\circ}$\
25 & 83 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F_{5/2}^{\circ}$ & 792396 & & & 1.40E-10& 1.35E-10 & 0.32 + 0.23 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
25 & 84 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}P_{1/2}^{\circ}$ & 792521 & & & 7.64E-12& 7.58E-12 & 0.53 + 0.27 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}P^{\circ}$ + 0.05 $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}P^{\circ}$\
25 & 85 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}P_{3/2}^{\circ}$ & 793409 & & & 1.07E-11& 1.07E-11 & 0.36 + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}P^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$\
25 & 86 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H_{9/2}^{\circ}$ & 794857 & & & 1.47E-09& 1.39E-09 & 0.43 + 0.35 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}H^{\circ}$\
25 & 87 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H_{11/2}^{\circ}$ & 796810 & & & 2.82E-09& 2.57E-09 & 0.29 + 0.41 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
25 & 88 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D_{5/2}^{\circ}$ & 798438 & & & 7.52E-11& 7.27E-11 & 0.22 + 0.24 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}D^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D^{\circ}$\
25 & 89 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G_{5/2}^{\circ}$ & 803531 & & & 4.36E-10& 4.16E-10 & 0.59 + 0.30 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$\
25 & 90 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G_{7/2}^{\circ}$ & 804296 & & & 3.80E-10& 3.63E-10 & 0.57 + 0.25 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H^{\circ}$\
25 & 91 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G_{9/2}^{\circ}$ & 806747 & & & 3.93E-10& 3.77E-10 & 0.51 + 0.19 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G^{\circ}$\
25 & 92 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G_{7/2}^{\circ}$ & 809421 & & & 2.44E-10& 2.34E-10 & 0.70 + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.03 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G^{\circ}$\
25 & 93 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G_{9/2}^{\circ}$ & 810005 & & & 2.51E-10& 2.42E-10 & 0.72 + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
25 & 94 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G_{11/2}^{\circ}$ & 811071 & & & 9.04E-10& 8.44E-10 & 0.53 + 0.17 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H^{\circ}$\
25 & 95 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F_{5/2}^{\circ}$ & 818679 & & & 2.50E-11& 2.44E-11 & 0.43 + 0.34 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.07 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F^{\circ}$\
25 & 96 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F_{7/2}^{\circ}$ & 818962 & & & 2.54E-11& 2.47E-11 & 0.43 + 0.33 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.07 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F^{\circ}$\
25 & 97 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F_{3/2}^{\circ}$ & 819362 & & & 2.47E-11& 2.40E-11 & 0.45 + 0.36 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.07 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F^{\circ}$\
25 & 98 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F_{9/2}^{\circ}$ & 820362 & & & 2.59E-11& 2.52E-11 & 0.44 + 0.35 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.07 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F^{\circ}$\
25 & 99 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P_{3/2}^{\circ}$ & 827356 & & & 3.61E-11& 3.56E-11 & 0.18 + 0.23 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}P^{\circ}$ + 0.20 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{2}P^{\circ}$\
25 & 100 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}I_{11/2}^{\circ}$ & 828428 & & & 4.88E-09& 4.80E-09 & 0.93\
25 & 101 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}I_{13/2}^{\circ}$ & 829370 & & & 5.86E-02& 5.85E-02 & 0.97\
25 & 102 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}S_{1/2}^{\circ}$ & 830240 & & & 7.86E-11& 7.77E-11 & 0.26 + 0.20 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}S^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}P^{\circ}$\
25 & 103 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D_{1/2}^{\circ}$ & 830697 & & & 3.90E-11& 3.84E-11 & 0.37 + 0.20 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
25 & 104 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D_{3/2}^{\circ}$ & 831604 & & & 5.17E-11& 5.09E-11 & 0.30 + 0.19 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
25 & 105 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D_{5/2}^{\circ}$ & 832908 & & & 7.62E-11& 7.45E-11 & 0.17 + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
25 & 106 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D_{3/2}^{\circ}$ & 835209 & & & 6.62E-11& 6.44E-11 & 0.09 + 0.13 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
25 & 107 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F_{5/2}^{\circ}$ & 835642 & & & 7.47E-11& 7.21E-11 & 0.25 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}F^{\circ}$\
25 & 108 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D_{7/2}^{\circ}$ & 835955 & & & 5.83E-11& 5.72E-11 & 0.26 + 0.25 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
25 & 109 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}D_{5/2}^{\circ}$ & 836251 & & & 5.68E-11& 5.57E-11 & 0.04 + 0.17 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
25 & 110 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}S)~^{4}S_{3/2}^{\circ}$ & 836705 & & & 2.33E-10& 2.27E-10 & 0.52 + 0.26 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}S^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}P^{\circ}$\
25 & 111 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}P_{1/2}^{\circ}$ & 838485 & & & 4.35E-11& 4.27E-11 & 0.00 + 0.21 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}S^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{2}P^{\circ}$\
25 & 112 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F_{3/2}^{\circ}$ & 840346 & & & 1.50E-10& 1.46E-10 & 0.39 + 0.33 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
25 & 113 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F_{7/2}^{\circ}$ & 840797 & & & 9.35E-11& 9.11E-11 & 0.27 + 0.15 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$\
25 & 114 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F_{5/2}^{\circ}$ & 841611 & & & 9.75E-11& 9.54E-11 & 0.34 + 0.34 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
25 & 115 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F_{7/2}^{\circ}$ & 843211 & & & 8.33E-11& 8.18E-11 & 0.10 + 0.16 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$\
25 & 116 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F_{9/2}^{\circ}$ & 843996 & & & 2.10E-10& 2.05E-10 & 0.31 + 0.24 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G^{\circ}$\
25 & 117 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G_{7/2}^{\circ}$ & 845640 & & & 1.79E-10& 1.77E-10 & 0.18 + 0.20 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
25 & 118 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G_{9/2}^{\circ}$ & 850632 & & & 3.93E-10& 3.87E-10 & 0.20 + 0.24 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.19 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}G^{\circ}$\
25 & 119 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D_{1/2}^{\circ}$ & 853175 & & & 6.46E-11& 6.37E-11 & 0.37 + 0.27 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
25 & 120 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D_{3/2}^{\circ}$ & 853932 & & & 5.44E-11& 5.40E-11 & 0.27 + 0.23 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.15 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
25 & 121 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P_{5/2}^{\circ}$ & 854358 & & & 3.98E-11& 3.97E-11 & 0.48 + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
25 & 122 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D_{5/2}^{\circ}$ & 858246 & & & 3.68E-11& 3.65E-11 & 0.17 + 0.30 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.17 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
25 & 123 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G_{11/2}^{\circ}$ & 858364 & & & 1.95E-11& 1.90E-11 & 0.38 + 0.26 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G^{\circ}$ + 0.24 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
25 & 124 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P_{3/2}^{\circ}$ & 858955 & & & 2.52E-11& 2.52E-11 & 0.66 + 0.08 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
25 & 125 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D_{7/2}^{\circ}$ & 859514 & & & 4.08E-11& 4.03E-11 & 0.32 + 0.24 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
25 & 126 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G_{9/2}^{\circ}$ & 859529 & & & 1.97E-11& 1.92E-11 & 0.34 + 0.24 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G^{\circ}$ + 0.23 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
25 & 127 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P_{1/2}^{\circ}$ & 859868 & & & 1.87E-11& 1.87E-11 & 0.76 + 0.06 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}P^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$\
25 & 128 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G_{7/2}^{\circ}$ & 861419 & & & 1.98E-11& 1.93E-11 & 0.37 + 0.27 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G^{\circ}$ + 0.26 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
25 & 129 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}F_{5/2}^{\circ}$ & 861505 & & & 2.50E-11& 2.46E-11 & 0.15 + 0.09 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
25 & 130 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G_{5/2}^{\circ}$ & 862495 & & & 2.10E-11& 2.05E-11 & 0.30 + 0.20 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$ + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
25 & 131 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}G_{9/2}^{\circ}$ & 867370 & & & 2.22E-11& 2.15E-11 & 0.79 + 0.04 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}G^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$\
25 & 132 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P_{1/2}^{\circ}$ & 867516 & & & 1.11E-11& 1.10E-11 & 0.42 + 0.28 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
25 & 133 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P_{3/2}^{\circ}$ & 867719 & & & 1.00E-11& 9.99E-12 & 0.40 + 0.29 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
25 & 134 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}F_{7/2}^{\circ}$ & 868068 & & & 2.62E-11& 2.57E-11 & 0.16 + 0.24 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}G^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
25 & 135 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P_{5/2}^{\circ}$ & 868757 & & & 8.96E-12& 8.93E-12 & 0.30 + 0.38 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
25 & 136 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}P_{1/2}^{\circ}$ & 870987 & & & 1.36E-11& 1.34E-11 & 0.40 + 0.20 $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}P^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P^{\circ}$\
25 & 137 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}G_{7/2}^{\circ}$ & 872339 & & & 2.14E-11& 2.09E-11 & 0.63 + 0.07 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.05 $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}F^{\circ}$\
25 & 138 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}P_{3/2}^{\circ}$ & 876293 & & & 1.23E-11& 1.21E-11 & 0.34 + 0.23 $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}P^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P^{\circ}$\
25 & 139 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}H_{11/2}^{\circ}$ & 877496 & & & 3.19E-10& 3.10E-10 & 0.63 + 0.25 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$\
25 & 140 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}H_{9/2}^{\circ}$ & 880754 & & & 6.60E-11& 6.48E-11 & 0.80 + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H^{\circ}$ + 0.05 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H^{\circ}$\
25 & 141 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H_{9/2}^{\circ}$ & 886529 & & & 2.12E-11& 2.07E-11 & 0.50 + 0.41 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H^{\circ}$\
25 & 142 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}S)~^{2}D_{5/2}^{\circ}$ & 887015 & & & 4.09E-11& 4.06E-11 & 0.32 + 0.18 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}D^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D^{\circ}$\
25 & 143 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D_{7/2}^{\circ}$ & 887785 & & & 8.01E-12& 8.00E-12 & 0.52 + 0.20 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
25 & 144 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}S)~^{2}D_{3/2}^{\circ}$ & 889218 & & & 7.59E-11& 7.47E-11 & 0.41 + 0.16 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}D^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D^{\circ}$\
25 & 145 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D_{5/2}^{\circ}$ & 890229 & & & 8.59E-12& 8.57E-12 & 0.42 + 0.20 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
25 & 146 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H_{11/2}^{\circ}$ & 890896 & & & 1.59E-11& 1.55E-11 & 0.45 + 0.32 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H^{\circ}$ + 0.15 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}H^{\circ}$\
25 & 147 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D_{3/2}^{\circ}$ & 891098 & & & 8.21E-12& 8.19E-12 & 0.45 + 0.23 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
25 & 148 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D_{1/2}^{\circ}$ & 891510 & & & 8.14E-12& 8.12E-12 & 0.48 + 0.23 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
25 & 149 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D_{7/2}^{\circ}$ & 896249 & & & 7.63E-12& 7.63E-12 & 0.22 + 0.16 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
25 & 150 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D_{5/2}^{\circ}$ & 897167 & & & 7.11E-12& 7.12E-12 & 0.22 + 0.21 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D^{\circ}$ + 0.20 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
25 & 151 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F_{7/2}^{\circ}$ & 898593 & & & 8.98E-12& 8.90E-12 & 0.06 + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F^{\circ}$ + 0.10 $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}F^{\circ}$\
25 & 152 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D_{3/2}^{\circ}$ & 899069 & & & 6.97E-12& 6.98E-12 & 0.05 + 0.23 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.23 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D^{\circ}$\
25 & 153 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D_{1/2}^{\circ}$ & 900738 & & & 6.85E-12& 6.86E-12 & 0.07 + 0.25 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.25 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D^{\circ}$\
25 & 154 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F_{9/2}^{\circ}$ & 901469 & & & 6.20E-12& 6.20E-12 & 0.35 + 0.26 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.17 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
25 & 155 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F_{5/2}^{\circ}$ & 902254 & & & 9.61E-12& 9.50E-12 & 0.13 + 0.15 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}F^{\circ}$\
25 & 156 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F_{7/2}^{\circ}$ & 905821 & & & 6.33E-12& 6.33E-12 & 0.32 + 0.26 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.15 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
25 & 157 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D_{5/2}^{\circ}$ & 906822 & & & 8.60E-12& 8.56E-12 & 0.35 + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D^{\circ}$\
25 & 158 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D_{3/2}^{\circ}$ & 908224 & & & 8.49E-12& 8.45E-12 & 0.27 + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$\
25 & 159 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}G_{7/2}^{\circ}$ & 908711 & & & 1.95E-11& 1.91E-11 & 0.55 + 0.15 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G^{\circ}$\
25 & 160 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F_{5/2}^{\circ}$ & 909407 & & & 6.32E-12& 6.31E-12 & 0.31 + 0.28 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.15 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
25 & 161 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F_{3/2}^{\circ}$ & 911458 & & & 6.42E-12& 6.42E-12 & 0.29 + 0.27 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.15 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
25 & 162 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}S_{1/2}^{\circ}$ & 911617 & & & 7.01E-12& 7.00E-12 & 0.23 + 0.22 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}P^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{2}P^{\circ}$\
25 & 163 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}G_{9/2}^{\circ}$ & 913338 & & & 2.07E-11& 2.03E-11 & 0.55 + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G^{\circ}$\
25 & 164 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}P_{3/2}^{\circ}$ & 914187 & & & 9.58E-12& 9.56E-12 & 0.38 + 0.23 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}S)~^{2}P^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{2}P^{\circ}$\
25 & 165 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}S)~^{2}P_{1/2}^{\circ}$ & 916291 & & & 1.12E-11& 1.12E-11 & 0.30 + 0.20 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}S^{\circ}$ + 0.18 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}P^{\circ}$\
25 & 166 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}S)~^{2}P_{3/2}^{\circ}$ & 917529 & & & 1.32E-11& 1.32E-11 & 0.24 + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$\
25 & 167 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P_{1/2}^{\circ}$ & 917552 & & & 1.73E-11& 1.73E-11 & 0.17 + 0.26 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}S)~^{2}P^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}P^{\circ}$\
25 & 168 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F_{5/2}^{\circ}$ & 919601 & & & 1.33E-11& 1.32E-11 & 0.42 + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F^{\circ}$\
25 & 169 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F_{7/2}^{\circ}$ & 921248 & & & 8.89E-12& 8.82E-12 & 0.26 + 0.18 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$ + 0.15 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
25 & 170 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D_{3/2}^{\circ}$ & 923259 & & & 7.48E-12& 7.46E-12 & 0.19 + 0.11 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$\
25 & 171 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F_{5/2}^{\circ}$ & 923665 & & & 8.71E-12& 8.63E-12 & 0.12 + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{2}F^{\circ}$\
25 & 172 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P_{5/2}^{\circ}$ & 925715 & & & 7.56E-12& 7.53E-12 & 0.27 + 0.24 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
25 & 173 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P_{3/2}^{\circ}$ & 926347 & & & 7.94E-12& 7.93E-12 & 0.19 + 0.24 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}S)~^{2}P^{\circ}$\
25 & 174 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P_{1/2}^{\circ}$ & 926595 & & & 6.45E-12& 6.44E-12 & 0.44 + 0.35 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
25 & 175 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D_{5/2}^{\circ}$ & 927147 & & & 7.89E-12& 7.86E-12 & 0.08 + 0.14 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$\
25 & 176 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}S_{3/2}^{\circ}$ & 931723 & & & 6.46E-12& 6.46E-12 & 0.49 + 0.23 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}S)~^{4}S^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
25 & 177 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F_{7/2}^{\circ}$ & 932821 & & & 2.02E-11& 1.97E-11 & 0.43 + 0.21 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}F^{\circ}$\
25 & 178 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D_{5/2}^{\circ}$ & 939629 & & & 6.83E-12& 6.80E-12 & 0.16 + 0.28 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}S)~^{2}D^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}D^{\circ}$\
25 & 179 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}D_{3/2}^{\circ}$ & 940272 & & & 6.86E-12& 6.82E-12 & 0.13 + 0.31 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}S)~^{2}D^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}D^{\circ}$\
25 & 180 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G_{7/2}^{\circ}$ & 947253 & & & 6.35E-12& 6.36E-12 & 0.42 + 0.35 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G^{\circ}$\
25 & 181 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G_{9/2}^{\circ}$ & 947346 & & & 6.45E-12& 6.46E-12 & 0.44 + 0.34 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G^{\circ}$\
25 & 182 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}S_{5/2}$ & 950467 & & & 1.35E-09& 1.35E-09 & 0.90 + 0.07 $3s^{2}\,3p^{2}(^{3}P)~^{3}P\,3d^{3}(^{4}P)~^{6}S$\
25 & 183 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D_{1/2}$ & 957170 & & & 8.33E-10& 8.35E-10 & 0.60 + 0.29 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}D$ + 0.06 $3s^{2}\,3p^{2}(^{3}P)~^{3}P\,3d^{3}(^{4}F)~^{6}D$\
25 & 184 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D_{3/2}$ & 957258 & & & 8.35E-10& 8.35E-10 & 0.58 + 0.29 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}D$ + 0.06 $3s^{2}\,3p^{2}(^{3}P)~^{3}P\,3d^{3}(^{4}F)~^{6}D$\
25 & 185 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D_{5/2}$ & 957448 & & & 8.41E-10& 8.39E-10 & 0.55 + 0.30 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}D$ + 0.06 $3s^{2}\,3p^{2}(^{3}P)~^{3}P\,3d^{3}(^{4}F)~^{6}D$\
25 & 186 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D_{7/2}$ & 957749 & & & 8.53E-10& 8.49E-10 & 0.51 + 0.32 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}D$ + 0.06 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F$\
25 & 187 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D_{9/2}$ & 958189 & & & 8.75E-10& 8.72E-10 & 0.48 + 0.36 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}D$ + 0.06 $3s^{2}\,3p^{2}(^{3}P)~^{3}P\,3d^{3}(^{4}F)~^{6}D$\
25 & 188 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}F_{5/2}^{\circ}$ & 963396 & & & 6.32E-12& 6.31E-12 & 0.31 + 0.26 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
25 & 189 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}F_{7/2}^{\circ}$ & 964148 & & & 6.45E-12& 6.44E-12 & 0.29 + 0.24 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{2}F^{\circ}$\
25 & 190 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F_{11/2}$ & 968723 & & & 5.31E-10& 5.22E-10 & 0.89 + 0.07 $3s^{2}\,3p^{2}(^{3}P)~^{3}P\,3d^{3}(^{4}F)~^{6}F$\
25 & 191 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F_{9/2}$ & 969187 & & & 5.35E-10& 5.26E-10 & 0.83 + 0.06 $3s^{2}\,3p^{2}(^{3}P)~^{3}P\,3d^{3}(^{4}F)~^{6}F$ + 0.03 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}D$\
25 & 192 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F_{7/2}$ & 969508 & & & 5.36E-10& 5.28E-10 & 0.83 + 0.06 $3s^{2}\,3p^{2}(^{3}P)~^{3}P\,3d^{3}(^{4}F)~^{6}F$ + 0.03 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}D$\
25 & 193 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F_{5/2}$ & 969645 & & & 5.34E-10& 5.27E-10 & 0.85 + 0.07 $3s^{2}\,3p^{2}(^{3}P)~^{3}P\,3d^{3}(^{4}F)~^{6}F$ + 0.03 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D$\
25 & 194 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F_{1/2}$ & 969657 & & & 5.29E-10& 5.21E-10 & 0.90 + 0.07 $3s^{2}\,3p^{2}(^{3}P)~^{3}P\,3d^{3}(^{4}F)~^{6}F$\
25 & 195 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F_{3/2}$ & 969668 & & & 5.31E-10& 5.24E-10 & 0.88 + 0.07 $3s^{2}\,3p^{2}(^{3}P)~^{3}P\,3d^{3}(^{4}F)~^{6}F$\
25 & 196 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}D_{3/2}^{\circ}$ & 970762 & & & 8.07E-12& 7.96E-12 & 0.27 + 0.19 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D^{\circ}$\
26 & 1 & $3s^{2}\,3p^{5}~^{2}P_{3/2}^{\circ}$ & 0 & 0 & 0 & & & 0.96\
26 & 2 & $3s^{2}\,3p^{5}~^{2}P_{1/2}^{\circ}$ & 15725 & 15683.1 & 41.9 & 1.43E-02& 1.43E-02 & 0.96\
26 & 3 & $3s~^{2}S\,3p^{6}~^{2}S_{1/2}$ & 289634 & 289249 & 385 & 2.30E-10& 2.22E-10 & 0.72 + 0.25 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}S$\
26 & 4 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D_{7/2}$ & 388485 & 388709 & -224 & 1.73E-02& 1.73E-02 & 0.94\
26 & 5 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D_{5/2}$ & 388558 & 388709 & -151 & 1.66E-07& 2.05E-07 & 0.92\
26 & 6 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D_{3/2}$ & 389791 & 390050 & -259 & 1.67E-07& 1.63E-07 & 0.92\
26 & 7 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D_{1/2}$ & 391341 & 391555 & -214 & 1.18E-07& 1.04E-07 & 0.93\
26 & 8 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F_{9/2}$ & 417470 & 417653 & -183 & 9.00E-02& 8.99E-02 & 0.92 + 0.05 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}G$\
26 & 9 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P_{1/2}$ & 419812 & & & 4.50E-09& 4.54E-09 & 0.50 + 0.42 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P$ + 0.03 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D$\
26 & 10 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F_{7/2}$ & 422638 & 422795 & -157 & 5.93E-02& 5.93E-02 & 0.88 + 0.04 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}G$ + 0.03 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F$\
26 & 11 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F_{5/2}$ & 426602 & 426763 & -161 & 2.13E-08& 2.18E-08 & 0.94\
26 & 12 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P_{3/2}$ & 427951 & 431928 & -3977 & 2.84E-09& 2.89E-09 & 0.38 + 0.29 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P$ + 0.18 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F$\
26 & 13 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F_{3/2}$ & 428253 & 428298 & -45 & 1.17E-08& 1.15E-08 & 0.72 + 0.09 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P$ + 0.07 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P$\
26 & 14 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D_{3/2}$ & 434583 & 434614 & -31 & 2.76E-09& 2.90E-09 & 0.44 + 0.27 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D$ + 0.10 $3s^{2}\,3p^{4}(^{1}S)~^{1}S\,3d~^{2}D$\
26 & 15 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P_{1/2}$ & 434688 & 434800 & -112 & 2.81E-09& 2.74E-09 & 0.94\
26 & 16 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P_{3/2}$ & 439619 & & & 1.60E-07& 1.69E-07 & 0.76 + 0.06 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P$ + 0.06 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P$\
26 & 17 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F_{7/2}$ & 440785 & 440840 & -55 & 1.48E-02& 1.48E-02 & 0.50 + 0.30 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}G$ + 0.10 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}F$\
26 & 18 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P_{5/2}$ & 441779 & 441853 & -74 & 8.24E-09& 8.52E-09 & 0.57 + 0.16 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$ + 0.12 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D$\
26 & 19 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D_{5/2}$ & 444033 & & & 1.28E-08& 1.31E-08 & 0.35 + 0.36 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P$ + 0.16 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D$\
26 & 20 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}G_{9/2}$ & 450676 & 450751 & -75 & 1.36E-02& 1.36E-02 & 0.91 + 0.05 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F$\
26 & 21 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}G_{7/2}$ & 451066 & 451084 & -18 & 1.40E-02& 1.40E-02 & 0.62 + 0.24 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F$ + 0.09 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}F$\
26 & 22 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F_{5/2}$ & 454034 & 452730 & 1304 & 4.14E-08& 4.26E-08 & 0.71 + 0.21 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}F$ + 0.03 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$\
26 & 23 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}F_{5/2}$ & 482087 & 476699 & 5388 & 5.69E-09& 5.72E-09 & 0.74 + 0.18 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F$ + 0.03 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$\
26 & 24 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}F_{7/2}$ & 485978 & 485983 & -5 & 4.97E-03& 4.98E-03 & 0.76 + 0.18 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F$\
26 & 25 & $3s^{2}\,3p^{4}(^{1}S)~^{1}S\,3d~^{2}D_{3/2}$ & 511914 & 511800 & 114 & 9.22E-10& 9.33E-10 & 0.67 + 0.24 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$\
26 & 26 & $3s^{2}\,3p^{4}(^{1}S)~^{1}S\,3d~^{2}D_{5/2}$ & 516334 & & & 6.41E-09& 6.43E-09 & 0.73 + 0.17 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$ + 0.02 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F$\
26 & 27 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}S_{1/2}$ & 540963 & 541879 & -916 & 6.28E-12& 6.23E-12 & 0.69 + 0.23 $3s~^{2}S\,3p^{6}~^{2}S$ + 0.03 $3s~^{2}S\,3p^{4}(^{3}P)~^{2}P\,3d^{2}(^{3}P)~^{2}S$\
26 & 28 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P_{3/2}$ & 563592 & 564198 & -606 & 6.74E-12& 6.82E-12 & 0.50 + 0.39 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P$ + 0.03 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D$\
26 & 29 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P_{1/2}$ & 569337 & 569985 & -648 & 6.91E-12& 6.98E-12 & 0.51 + 0.44 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P$\
26 & 30 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D_{5/2}$ & 572177 & 572954 & -777 & 5.62E-12& 5.69E-12 & 0.65 + 0.20 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$ + 0.09 $3s^{2}\,3p^{4}(^{1}S)~^{1}S\,3d~^{2}D$\
26 & 31 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D_{3/2}$ & 585622 & 586244 & -622 & 5.71E-12& 5.75E-12 & 0.60 + 0.17 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$ + 0.12 $3s^{2}\,3p^{4}(^{1}S)~^{1}S\,3d~^{2}D$\
26 & 32 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}P_{1/2}^{\circ}$ & 666576 & & & 3.34E-10& 3.29E-10 & 0.81 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
26 & 33 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}P_{3/2}^{\circ}$ & 669144 & & & 3.31E-10& 3.28E-10 & 0.81 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
26 & 34 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}P_{5/2}^{\circ}$ & 673703 & & & 3.28E-10& 3.26E-10 & 0.80 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
26 & 35 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F_{9/2}^{\circ}$ & 696841 & 696661 & 180 & 1.84E-10& 1.77E-10 & 0.84 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
26 & 36 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F_{7/2}^{\circ}$ & 699761 & 699492 & 269 & 1.82E-10& 1.75E-10 & 0.82 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
26 & 37 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F_{5/2}^{\circ}$ & 702836 & 702585 & 251 & 1.80E-10& 1.73E-10 & 0.81 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.02 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D^{\circ}$\
26 & 38 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F_{3/2}^{\circ}$ & 705654 & 705430 & 224 & 1.77E-10& 1.72E-10 & 0.82 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
26 & 39 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D_{7/2}^{\circ}$ & 727687 & & & 1.89E-10& 1.84E-10 & 0.77 + 0.07 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
26 & 40 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D_{1/2}^{\circ}$ & 729234 & & & 1.86E-10& 1.81E-10 & 0.79 + 0.07 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
26 & 41 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D_{5/2}^{\circ}$ & 729480 & & & 1.87E-10& 1.82E-10 & 0.76 + 0.07 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
26 & 42 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D_{3/2}^{\circ}$ & 729748 & & & 1.86E-10& 1.82E-10 & 0.77 + 0.07 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
26 & 43 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}F_{7/2}^{\circ}$ & 737663 & & & 4.10E-10& 4.05E-10 & 0.71 + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
26 & 44 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}D_{5/2}^{\circ}$ & 741800 & & & 3.48E-10& 3.40E-10 & 0.47 + 0.28 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}F^{\circ}$ + 0.05 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D^{\circ}$\
26 & 45 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}D_{3/2}^{\circ}$ & 746345 & & & 2.26E-10& 2.20E-10 & 0.71 + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}D^{\circ}$\
26 & 46 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}F_{5/2}^{\circ}$ & 749936 & & & 2.91E-10& 2.86E-10 & 0.47 + 0.28 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}D^{\circ}$ + 0.05 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
26 & 47 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}P_{1/2}^{\circ}$ & 760034 & & & 9.32E-11& 9.15E-11 & 0.41 + 0.23 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}P^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P^{\circ}$\
26 & 48 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}P_{3/2}^{\circ}$ & 767744 & & & 1.06E-10& 1.04E-10 & 0.39 + 0.19 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}P^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P^{\circ}$\
26 & 49 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{1/2}^{\circ}$ & 781888 & & & 7.04E-09& 6.81E-09 & 0.96\
26 & 50 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{3/2}^{\circ}$ & 782022 & & & 8.28E-09& 8.05E-09 & 0.96\
26 & 51 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{5/2}^{\circ}$ & 782285 & & & 1.04E-08& 1.02E-08 & 0.96\
26 & 52 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{7/2}^{\circ}$ & 782722 & & & 1.50E-08& 1.49E-08 & 0.96\
26 & 53 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{9/2}^{\circ}$ & 783337 & & & 2.62E-08& 2.62E-08 & 0.96\
26 & 54 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{11/2}^{\circ}$ & 784108 & & & 1.28E-07& 1.37E-07 & 0.96\
26 & 55 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}F_{5/2}^{\circ}$ & 786777 & & & 8.47E-11& 8.35E-11 & 0.58 + 0.18 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
26 & 56 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}F_{7/2}^{\circ}$ & 790569 & & & 8.23E-11& 8.12E-11 & 0.56 + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
26 & 57 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{6}P_{3/2}^{\circ}$ & 805678 & & & 8.24E-09& 8.31E-09 & 0.95\
26 & 58 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{6}P_{5/2}^{\circ}$ & 805899 & & & 6.10E-09& 6.01E-09 & 0.95\
26 & 59 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{6}P_{7/2}^{\circ}$ & 806136 & & & 4.67E-09& 4.61E-09 & 0.94\
26 & 60 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}D_{3/2}^{\circ}$ & 813954 & & & 7.71E-11& 7.56E-11 & 0.56 + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$\
26 & 61 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}D_{5/2}^{\circ}$ & 815339 & & & 7.87E-11& 7.71E-11 & 0.57 + 0.20 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$\
26 & 62 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D_{7/2}^{\circ}$ & 826314 & & & 2.99E-10& 2.90E-10 & 0.55 + 0.21 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
26 & 63 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P_{1/2}^{\circ}$ & 826893 & & & 2.09E-09& 2.09E-09 & 0.13 + 0.38 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.30 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
26 & 64 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D_{5/2}^{\circ}$ & 826991 & & & 2.76E-10& 2.69E-10 & 0.52 + 0.20 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.05 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
26 & 65 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D_{3/2}^{\circ}$ & 827311 & & & 2.72E-10& 2.64E-10 & 0.47 + 0.18 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$\
26 & 66 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D_{1/2}^{\circ}$ & 829217 & & & 2.37E-10& 2.30E-10 & 0.56 + 0.21 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.05 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
26 & 67 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P_{3/2}^{\circ}$ & 830347 & & & 4.58E-09& 4.58E-09 & 0.29 + 0.36 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$\
26 & 68 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G_{5/2}^{\circ}$ & 830949 & & & 2.10E-09& 2.01E-09 & 0.50 + 0.17 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$\
26 & 69 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G_{7/2}^{\circ}$ & 832158 & & & 3.07E-09& 2.91E-09 & 0.56 + 0.21 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.05 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$\
26 & 70 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F_{3/2}^{\circ}$ & 832646 & & & 9.02E-10& 8.77E-10 & 0.14 + 0.27 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$ + 0.17 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
26 & 71 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P_{5/2}^{\circ}$ & 833659 & & & 7.83E-09& 7.72E-09 & 0.39 + 0.30 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$\
26 & 72 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G_{9/2}^{\circ}$ & 834280 & & & 4.49E-09& 4.10E-09 & 0.61 + 0.26 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.03 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$\
26 & 73 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G_{11/2}^{\circ}$ & 836443 & & & 5.56E-09& 4.75E-09 & 0.63 + 0.32 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$\
26 & 74 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F_{5/2}^{\circ}$ & 836583 & & & 2.49E-09& 2.39E-09 & 0.12 + 0.29 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
26 & 75 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F_{7/2}^{\circ}$ & 839581 & & & 2.62E-09& 2.51E-09 & 0.38 + 0.19 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
26 & 76 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F_{9/2}^{\circ}$ & 842504 & & & 2.24E-09& 2.12E-09 & 0.44 + 0.22 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
26 & 77 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H_{7/2}^{\circ}$ & 851459 & & & 1.05E-09& 1.02E-09 & 0.74 + 0.05 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
26 & 78 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H_{9/2}^{\circ}$ & 852725 & & & 1.34E-08& 1.30E-08 & 0.86 + 0.07 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
26 & 79 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H_{11/2}^{\circ}$ & 854217 & & & 2.88E-07& 2.07E-07 & 0.87 + 0.07 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
26 & 80 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F_{7/2}^{\circ}$ & 855492 & & & 2.05E-10& 1.98E-10 & 0.30 + 0.22 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.15 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H^{\circ}$\
26 & 81 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H_{13/2}^{\circ}$ & 857044 & & & 3.15E-02& 3.17E-02 & 0.97\
26 & 82 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}P_{3/2}^{\circ}$ & 858043 & & & 1.17E-11& 1.15E-11 & 0.31 + 0.13 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}P^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}D^{\circ}$\
26 & 83 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}P_{1/2}^{\circ}$ & 858212 & & & 7.03E-12& 6.95E-12 & 0.53 + 0.27 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}P^{\circ}$ + 0.05 $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}P^{\circ}$\
26 & 84 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F_{5/2}^{\circ}$ & 859441 & & & 1.20E-10& 1.16E-10 & 0.31 + 0.22 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
26 & 85 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D_{3/2}^{\circ}$ & 859712 & & & 1.39E-11& 1.37E-11 & 0.17 + 0.23 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}P^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}D^{\circ}$\
26 & 86 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H_{9/2}^{\circ}$ & 861524 & & & 1.24E-09& 1.18E-09 & 0.42 + 0.34 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}H^{\circ}$\
26 & 87 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H_{11/2}^{\circ}$ & 863939 & & & 2.69E-09& 2.45E-09 & 0.28 + 0.40 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
26 & 88 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D_{5/2}^{\circ}$ & 866545 & & & 6.62E-11& 6.43E-11 & 0.22 + 0.24 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}D^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D^{\circ}$\
26 & 89 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G_{5/2}^{\circ}$ & 871212 & & & 3.21E-10& 3.07E-10 & 0.56 + 0.30 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.03 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
26 & 90 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G_{7/2}^{\circ}$ & 871984 & & & 2.84E-10& 2.72E-10 & 0.54 + 0.24 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G^{\circ}$\
26 & 91 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G_{9/2}^{\circ}$ & 875209 & & & 2.96E-10& 2.84E-10 & 0.46 + 0.17 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G^{\circ}$\
26 & 92 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G_{7/2}^{\circ}$ & 877873 & & & 2.07E-10& 2.00E-10 & 0.68 + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.03 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
26 & 93 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G_{9/2}^{\circ}$ & 878794 & & & 2.12E-10& 2.04E-10 & 0.69 + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
26 & 94 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G_{11/2}^{\circ}$ & 880998 & & & 7.95E-10& 7.39E-10 & 0.51 + 0.16 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H^{\circ}$\
26 & 95 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F_{5/2}^{\circ}$ & 886966 & & & 2.24E-11& 2.19E-11 & 0.43 + 0.33 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.07 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F^{\circ}$\
26 & 96 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F_{7/2}^{\circ}$ & 887429 & & & 2.29E-11& 2.23E-11 & 0.42 + 0.32 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.07 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F^{\circ}$\
26 & 97 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F_{3/2}^{\circ}$ & 887911 & & & 2.20E-11& 2.15E-11 & 0.45 + 0.36 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.07 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F^{\circ}$\
26 & 98 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F_{9/2}^{\circ}$ & 889448 & & & 2.35E-11& 2.29E-11 & 0.43 + 0.34 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.07 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F^{\circ}$\
26 & 99 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P_{3/2}^{\circ}$ & 896401 & & & 3.28E-11& 3.23E-11 & 0.18 + 0.22 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}P^{\circ}$ + 0.20 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{2}P^{\circ}$\
26 & 100 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}I_{11/2}^{\circ}$ & 898096 & & & 2.96E-09& 2.90E-09 & 0.92 + 0.02 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}H^{\circ}$\
26 & 101 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D_{1/2}^{\circ}$ & 899230 & & & 3.46E-11& 3.42E-11 & 0.41 + 0.19 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.15 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
26 & 102 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}I_{13/2}^{\circ}$ & 899511 & & & 4.10E-02& 4.09E-02 & 0.97\
26 & 103 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}S_{1/2}^{\circ}$ & 899995 & & & 7.16E-11& 7.08E-11 & 0.31 + 0.23 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}S^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{2}P^{\circ}$\
26 & 104 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D_{3/2}^{\circ}$ & 900719 & & & 4.63E-11& 4.58E-11 & 0.30 + 0.18 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
26 & 105 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D_{5/2}^{\circ}$ & 902600 & & & 6.83E-11& 6.72E-11 & 0.18 + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
26 & 106 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D_{3/2}^{\circ}$ & 905440 & & & 5.61E-11& 5.49E-11 & 0.09 + 0.13 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
26 & 107 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F_{5/2}^{\circ}$ & 905887 & & & 6.75E-11& 6.55E-11 & 0.27 + 0.11 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}F^{\circ}$\
26 & 108 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D_{7/2}^{\circ}$ & 906097 & & & 5.30E-11& 5.22E-11 & 0.25 + 0.25 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
26 & 109 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F_{5/2}^{\circ}$ & 906596 & & & 5.10E-11& 5.02E-11 & 0.03 + 0.15 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
26 & 110 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}S)~^{4}S_{3/2}^{\circ}$ & 907868 & & & 2.29E-10& 2.22E-10 & 0.53 + 0.27 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}S^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}P^{\circ}$\
26 & 111 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P_{1/2}^{\circ}$ & 910038 & & & 3.53E-11& 3.47E-11 & 0.17 + 0.20 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}S^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{2}P^{\circ}$\
26 & 112 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F_{3/2}^{\circ}$ & 910577 & & & 1.25E-10& 1.22E-10 & 0.38 + 0.29 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
26 & 113 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F_{7/2}^{\circ}$ & 911936 & & & 1.15E-10& 1.13E-10 & 0.21 + 0.21 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.18 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$\
26 & 114 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F_{5/2}^{\circ}$ & 912058 & & & 7.50E-11& 7.37E-11 & 0.33 + 0.30 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
26 & 115 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G_{7/2}^{\circ}$ & 914356 & & & 6.67E-11& 6.56E-11 & 0.11 + 0.19 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}G^{\circ}$\
26 & 116 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F_{9/2}^{\circ}$ & 914876 & & & 1.76E-10& 1.72E-10 & 0.30 + 0.23 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G^{\circ}$\
26 & 117 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F_{7/2}^{\circ}$ & 916903 & & & 9.83E-11& 9.69E-11 & 0.17 + 0.15 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$\
26 & 118 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G_{9/2}^{\circ}$ & 923103 & & & 2.88E-10& 2.83E-10 & 0.19 + 0.23 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.20 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
26 & 119 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D_{1/2}^{\circ}$ & 923999 & & & 5.88E-11& 5.81E-11 & 0.38 + 0.25 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
26 & 120 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D_{3/2}^{\circ}$ & 924946 & & & 4.79E-11& 4.76E-11 & 0.27 + 0.21 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
26 & 121 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P_{5/2}^{\circ}$ & 925514 & & & 3.33E-11& 3.32E-11 & 0.49 + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
26 & 122 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G_{11/2}^{\circ}$ & 929829 & & & 1.76E-11& 1.71E-11 & 0.38 + 0.25 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G^{\circ}$ + 0.24 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
26 & 123 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D_{5/2}^{\circ}$ & 929938 & & & 2.79E-11& 2.77E-11 & 0.13 + 0.26 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
26 & 124 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P_{3/2}^{\circ}$ & 930904 & & & 1.81E-11& 1.81E-11 & 0.62 + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
26 & 125 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G_{9/2}^{\circ}$ & 931081 & & & 1.78E-11& 1.73E-11 & 0.32 + 0.23 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G^{\circ}$ + 0.22 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
26 & 126 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P_{1/2}^{\circ}$ & 931777 & & & 1.29E-11& 1.29E-11 & 0.72 + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.07 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}P^{\circ}$\
26 & 127 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D_{7/2}^{\circ}$ & 931912 & & & 3.39E-11& 3.34E-11 & 0.28 + 0.23 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
26 & 128 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D_{5/2}^{\circ}$ & 933561 & & & 2.37E-11& 2.34E-11 & 0.00 + 0.12 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
26 & 129 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G_{7/2}^{\circ}$ & 933689 & & & 1.83E-11& 1.79E-11 & 0.34 + 0.25 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G^{\circ}$ + 0.24 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
26 & 130 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G_{5/2}^{\circ}$ & 934845 & & & 1.85E-11& 1.81E-11 & 0.32 + 0.22 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$ + 0.21 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
26 & 131 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P_{1/2}^{\circ}$ & 939630 & & & 1.22E-11& 1.21E-11 & 0.41 + 0.24 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
26 & 132 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P_{3/2}^{\circ}$ & 939850 & & & 1.07E-11& 1.06E-11 & 0.42 + 0.28 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
26 & 133 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}G_{9/2}^{\circ}$ & 940286 & & & 1.97E-11& 1.91E-11 & 0.77 + 0.06 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}G^{\circ}$\
26 & 134 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P_{5/2}^{\circ}$ & 941075 & & & 8.91E-12& 8.89E-12 & 0.31 + 0.39 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
26 & 135 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}F_{7/2}^{\circ}$ & 941344 & & & 2.37E-11& 2.33E-11 & 0.13 + 0.28 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}G^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
26 & 136 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}P_{1/2}^{\circ}$ & 942944 & & & 1.36E-11& 1.35E-11 & 0.39 + 0.17 $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}P^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P^{\circ}$\
26 & 137 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}G_{7/2}^{\circ}$ & 946250 & & & 1.88E-11& 1.83E-11 & 0.60 + 0.07 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.05 $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}F^{\circ}$\
26 & 138 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}P_{3/2}^{\circ}$ & 949108 & & & 1.13E-11& 1.11E-11 & 0.35 + 0.22 $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}P^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}P^{\circ}$\
26 & 139 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}H_{11/2}^{\circ}$ & 951567 & & & 2.53E-10& 2.45E-10 & 0.59 + 0.26 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$\
26 & 140 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}H_{9/2}^{\circ}$ & 955114 & & & 4.35E-11& 4.26E-11 & 0.79 + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H^{\circ}$ + 0.03 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H^{\circ}$\
26 & 141 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H_{9/2}^{\circ}$ & 960471 & & & 2.14E-11& 2.08E-11 & 0.53 + 0.38 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H^{\circ}$\
26 & 142 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D_{7/2}^{\circ}$ & 960845 & & & 7.38E-12& 7.37E-12 & 0.50 + 0.20 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
26 & 143 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}D_{5/2}^{\circ}$ & 961097 & & & 2.66E-11& 2.65E-11 & 0.17 + 0.26 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}S)~^{2}D^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D^{\circ}$\
26 & 144 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}S)~^{2}D_{3/2}^{\circ}$ & 963980 & & & 5.19E-11& 5.13E-11 & 0.38 + 0.15 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}D^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D^{\circ}$\
26 & 145 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D_{5/2}^{\circ}$ & 964023 & & & 8.30E-12& 8.29E-12 & 0.34 + 0.20 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
26 & 146 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D_{3/2}^{\circ}$ & 964961 & & & 7.76E-12& 7.75E-12 & 0.40 + 0.23 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
26 & 147 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D_{1/2}^{\circ}$ & 965415 & & & 7.49E-12& 7.49E-12 & 0.46 + 0.24 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
26 & 148 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H_{11/2}^{\circ}$ & 965992 & & & 1.42E-11& 1.38E-11 & 0.44 + 0.30 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H^{\circ}$ + 0.18 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}H^{\circ}$\
26 & 149 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D_{7/2}^{\circ}$ & 969935 & & & 7.41E-12& 7.39E-12 & 0.20 + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D^{\circ}$\
26 & 150 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D_{5/2}^{\circ}$ & 971008 & & & 6.76E-12& 6.75E-12 & 0.21 + 0.19 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
26 & 151 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F_{7/2}^{\circ}$ & 972582 & & & 8.05E-12& 8.00E-12 & 0.05 + 0.11 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
26 & 152 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D_{3/2}^{\circ}$ & 973327 & & & 6.56E-12& 6.56E-12 & 0.04 + 0.23 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.22 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D^{\circ}$\
26 & 153 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F_{9/2}^{\circ}$ & 974575 & & & 5.71E-12& 5.70E-12 & 0.35 + 0.26 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.17 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
26 & 154 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D_{1/2}^{\circ}$ & 975431 & & & 6.40E-12& 6.40E-12 & 0.06 + 0.25 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.25 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D^{\circ}$\
26 & 155 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}F_{5/2}^{\circ}$ & 976884 & & & 8.85E-12& 8.78E-12 & 0.14 + 0.15 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F^{\circ}$\
26 & 156 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F_{7/2}^{\circ}$ & 979945 & & & 5.83E-12& 5.82E-12 & 0.31 + 0.25 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.15 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
26 & 157 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D_{5/2}^{\circ}$ & 982631 & & & 8.05E-12& 8.03E-12 & 0.34 + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D^{\circ}$\
26 & 158 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D_{3/2}^{\circ}$ & 983812 & & & 7.54E-12& 7.52E-12 & 0.23 + 0.11 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}D^{\circ}$\
26 & 159 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}G_{7/2}^{\circ}$ & 984424 & & & 1.76E-11& 1.72E-11 & 0.55 + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G^{\circ}$\
26 & 160 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F_{5/2}^{\circ}$ & 984497 & & & 5.87E-12& 5.85E-12 & 0.31 + 0.27 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
26 & 161 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}P_{1/2}^{\circ}$ & 986925 & & & 6.54E-12& 6.52E-12 & 0.21 + 0.23 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}S^{\circ}$ + 0.17 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{2}P^{\circ}$\
26 & 162 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F_{3/2}^{\circ}$ & 987039 & & & 6.10E-12& 6.08E-12 & 0.26 + 0.24 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
26 & 163 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}P_{3/2}^{\circ}$ & 990290 & & & 7.90E-12& 7.87E-12 & 0.36 + 0.15 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{2}P^{\circ}$ + 0.15 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}S)~^{2}P^{\circ}$\
26 & 164 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}G_{9/2}^{\circ}$ & 990339 & & & 1.84E-11& 1.80E-11 & 0.54 + 0.20 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}G^{\circ}$\
26 & 165 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}S_{1/2}^{\circ}$ & 992939 & & & 7.07E-12& 7.05E-12 & 0.27 + 0.22 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}S^{\circ}$ + 0.15 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}P^{\circ}$\
26 & 166 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}S)~^{2}P_{3/2}^{\circ}$ & 993750 & & & 1.21E-11& 1.20E-11 & 0.26 + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P^{\circ}$\
26 & 167 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}S)~^{2}P_{1/2}^{\circ}$ & 994309 & & & 5.82E-11& 5.81E-11 & 0.46 + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}P^{\circ}$\
26 & 168 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F_{5/2}^{\circ}$ & 995652 & & & 1.24E-11& 1.22E-11 & 0.42 + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
26 & 169 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F_{7/2}^{\circ}$ & 997817 & & & 8.08E-12& 8.02E-12 & 0.26 + 0.15 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
26 & 170 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D_{3/2}^{\circ}$ & 999296 & & & 6.97E-12& 6.94E-12 & 0.04 + 0.15 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.15 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$\
26 & 171 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F_{5/2}^{\circ}$ & 999710 & & & 7.89E-12& 7.81E-12 & 0.10 + 0.13 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F^{\circ}$\
26 & 172 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P_{5/2}^{\circ}$ & 1002190 & & & 6.97E-12& 6.93E-12 & 0.28 + 0.24 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
26 & 173 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P_{1/2}^{\circ}$ & 1003410 & & & 6.11E-12& 6.09E-12 & 0.43 + 0.32 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
26 & 174 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P_{3/2}^{\circ}$ & 1003799 & & & 8.24E-12& 8.22E-12 & 0.16 + 0.21 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}S)~^{2}P^{\circ}$\
26 & 175 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D_{5/2}^{\circ}$ & 1003934 & & & 7.37E-12& 7.32E-12 & 0.08 + 0.15 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
26 & 176 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}S_{3/2}^{\circ}$ & 1010445 & & & 6.19E-12& 6.19E-12 & 0.51 + 0.26 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}S)~^{4}S^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
26 & 177 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F_{7/2}^{\circ}$ & 1011432 & & & 1.82E-11& 1.77E-11 & 0.42 + 0.23 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}F^{\circ}$\
26 & 178 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}S)~^{2}D_{5/2}^{\circ}$ & 1018349 & & & 6.42E-12& 6.40E-12 & 0.30 + 0.15 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}D^{\circ}$\
26 & 179 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}D_{3/2}^{\circ}$ & 1019218 & & & 6.48E-12& 6.45E-12 & 0.14 + 0.33 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}S)~^{2}D^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}D^{\circ}$\
26 & 180 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G_{7/2}^{\circ}$ & 1024704 & & & 5.88E-12& 5.86E-12 & 0.41 + 0.34 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G^{\circ}$\
26 & 181 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G_{9/2}^{\circ}$ & 1025042 & & & 5.96E-12& 5.95E-12 & 0.44 + 0.33 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G^{\circ}$\
26 & 182 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}S_{5/2}$ & 1032912 & & & 1.12E-09& 1.10E-09 & 0.90 + 0.07 $3s^{2}\,3p^{2}(^{3}P)~^{3}P\,3d^{3}(^{4}P)~^{6}S$\
26 & 183 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D_{1/2}$ & 1039597 & & & 6.78E-10& 6.75E-10 & 0.60 + 0.27 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}D$ + 0.06 $3s^{2}\,3p^{2}(^{3}P)~^{3}P\,3d^{3}(^{4}F)~^{6}D$\
26 & 184 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D_{3/2}$ & 1039706 & & & 6.71E-10& 6.67E-10 & 0.58 + 0.28 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}D$ + 0.06 $3s^{2}\,3p^{2}(^{3}P)~^{3}P\,3d^{3}(^{4}F)~^{6}D$\
26 & 185 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D_{5/2}$ & 1039972 & & & 6.99E-10& 6.92E-10 & 0.55 + 0.29 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}D$ + 0.06 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F$\
26 & 186 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D_{7/2}$ & 1040390 & & & 7.09E-10& 7.01E-10 & 0.51 + 0.31 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}D$ + 0.08 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F$\
26 & 187 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D_{9/2}$ & 1041013 & & & 7.30E-10& 7.21E-10 & 0.47 + 0.36 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}D$ + 0.07 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F$\
26 & 188 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}F_{5/2}^{\circ}$ & 1042662 & & & 5.88E-12& 5.85E-12 & 0.30 + 0.27 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
26 & 189 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}F_{7/2}^{\circ}$ & 1043700 & & & 6.01E-12& 5.98E-12 & 0.28 + 0.25 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
26 & 190 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}D_{3/2}^{\circ}$ & 1050888 & & & 7.36E-12& 7.26E-12 & 0.27 + 0.21 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D^{\circ}$\
26 & 191 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F_{11/2}$ & 1052298 & & & 4.47E-10& 4.38E-10 & 0.89 + 0.07 $3s^{2}\,3p^{2}(^{3}P)~^{3}P\,3d^{3}(^{4}F)~^{6}F$ + 0.02 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}G$\
26 & 192 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F_{9/2}$ & 1052884 & & & 4.52E-10& 4.43E-10 & 0.81 + 0.06 $3s^{2}\,3p^{2}(^{3}P)~^{3}P\,3d^{3}(^{4}F)~^{6}F$ + 0.04 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}D$\
26 & 193 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}D_{5/2}^{\circ}$ & 1053166 & & & 7.35E-12& 7.25E-12 & 0.31 + 0.18 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D^{\circ}$ + 0.17 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D^{\circ}$\
26 & 194 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F_{7/2}$ & 1053290 & & & 4.53E-10& 4.45E-10 & 0.80 + 0.06 $3s^{2}\,3p^{2}(^{3}P)~^{3}P\,3d^{3}(^{4}F)~^{6}F$ + 0.05 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}D$\
26 & 195 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F_{1/2}$ & 1053392 & & & 4.45E-10& 4.38E-10 & 0.90 + 0.07 $3s^{2}\,3p^{2}(^{3}P)~^{3}P\,3d^{3}(^{4}F)~^{6}F$\
26 & 196 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F_{3/2}$ & 1053433 & & & 4.40E-10& 4.33E-10 & 0.86 + 0.07 $3s^{2}\,3p^{2}(^{3}P)~^{3}P\,3d^{3}(^{4}F)~^{6}F$\
26 & 197 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F_{5/2}$ & 1053439 & & & 4.51E-10& 4.44E-10 & 0.83 + 0.06 $3s^{2}\,3p^{2}(^{3}P)~^{3}P\,3d^{3}(^{4}F)~^{6}F$ + 0.03 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D$\
27 & 1 & $3s^{2}\,3p^{5}~^{2}P_{3/2}^{\circ}$ & 0 & 0 & 0 & & & 0.96\
27 & 2 & $3s^{2}\,3p^{5}~^{2}P_{1/2}^{\circ}$ & 19405 & 19345 & 60 & 7.62E-03& 7.62E-03 & 0.96\
27 & 3 & $3s~^{2}S\,3p^{6}~^{2}S_{1/2}$ & 314310 & 313630 & 680 & 1.97E-10& 1.94E-10 & 0.72 + 0.25 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}S$\
27 & 4 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D_{5/2}$ & 419410 & & & 1.23E-07& 1.47E-07 & 0.91\
27 & 5 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D_{7/2}$ & 419571 & & & 1.35E-02& 1.35E-02 & 0.93 + 0.02 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F$\
27 & 6 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D_{3/2}$ & 420884 & & & 1.14E-07& 1.14E-07 & 0.91\
27 & 7 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D_{1/2}$ & 422921 & & & 7.41E-08& 6.79E-08 & 0.92 + 0.02 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P$\
27 & 8 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F_{9/2}$ & 449939 & & & 6.02E-02& 6.01E-02 & 0.91 + 0.06 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}G$\
27 & 9 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P_{1/2}$ & 452803 & & & 4.18E-09& 4.23E-09 & 0.49 + 0.41 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P$ + 0.04 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D$\
27 & 10 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F_{7/2}$ & 456136 & & & 3.76E-02& 3.76E-02 & 0.84 + 0.05 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F$ + 0.05 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}G$\
27 & 11 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F_{5/2}$ & 461202 & & & 1.37E-08& 1.41E-08 & 0.93\
27 & 12 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P_{3/2}$ & 462366 & & & 2.44E-09& 2.50E-09 & 0.35 + 0.26 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P$ + 0.20 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F$\
27 & 13 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F_{3/2}$ & 462871 & & & 1.05E-08& 1.05E-08 & 0.65 + 0.10 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P$ + 0.08 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P$\
27 & 14 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P_{1/2}$ & 468635 & & & 1.89E-09& 1.87E-09 & 0.94\
27 & 15 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D_{3/2}$ & 468719 & & & 2.16E-09& 2.26E-09 & 0.41 + 0.25 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D$ + 0.10 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P$\
27 & 16 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F_{7/2}$ & 475190 & & & 9.24E-03& 9.24E-03 & 0.48 + 0.29 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}G$ + 0.09 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}F$\
27 & 17 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P_{3/2}$ & 475212 & & & 1.14E-07& 1.19E-07 & 0.72 + 0.08 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P$ + 0.07 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P$\
27 & 18 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P_{5/2}$ & 477259 & & & 6.78E-09& 6.98E-09 & 0.56 + 0.15 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$ + 0.12 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D$\
27 & 19 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D_{5/2}$ & 480307 & & & 1.15E-08& 1.17E-08 & 0.35 + 0.37 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P$ + 0.16 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D$\
27 & 20 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}G_{9/2}$ & 487209 & & & 7.95E-03& 7.95E-03 & 0.90 + 0.06 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F$\
27 & 21 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}G_{7/2}$ & 487366 & & & 8.40E-03& 8.39E-03 & 0.61 + 0.25 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F$ + 0.10 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}F$\
27 & 22 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F_{5/2}$ & 491077 & & & 2.60E-08& 2.68E-08 & 0.69 + 0.21 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}F$ + 0.04 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$\
27 & 23 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}F_{5/2}$ & 519938 & & & 4.04E-09& 4.10E-09 & 0.74 + 0.18 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F$ + 0.03 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$\
27 & 24 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}F_{7/2}$ & 524874 & & & 3.01E-03& 3.02E-03 & 0.76 + 0.18 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F$\
27 & 25 & $3s^{2}\,3p^{4}(^{1}S)~^{1}S\,3d~^{2}D_{3/2}$ & 552048 & & & 6.20E-10& 6.32E-10 & 0.66 + 0.24 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$\
27 & 26 & $3s^{2}\,3p^{4}(^{1}S)~^{1}S\,3d~^{2}D_{5/2}$ & 557936 & & & 3.95E-09& 4.00E-09 & 0.72 + 0.16 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$ + 0.03 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F$\
27 & 27 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}S_{1/2}$ & 581659 & 582510 & -851 & 5.75E-12& 5.77E-12 & 0.69 + 0.23 $3s~^{2}S\,3p^{6}~^{2}S$ + 0.03 $3s~^{2}S\,3p^{4}(^{3}P)~^{2}P\,3d^{2}(^{3}P)~^{2}S$\
27 & 28 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P_{3/2}$ & 606071 & 606420 & -349 & 6.23E-12& 6.35E-12 & 0.49 + 0.39 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P$ + 0.04 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D$\
27 & 29 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P_{1/2}$ & 613220 & 613480 & -260 & 6.41E-12& 6.53E-12 & 0.51 + 0.44 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P$\
27 & 30 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D_{5/2}$ & 614654 & 615140 & -486 & 5.21E-12& 5.31E-12 & 0.66 + 0.21 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$ + 0.08 $3s^{2}\,3p^{4}(^{1}S)~^{1}S\,3d~^{2}D$\
27 & 31 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D_{3/2}$ & 631325 & 631680 & -355 & 5.31E-12& 5.37E-12 & 0.60 + 0.16 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$ + 0.13 $3s^{2}\,3p^{4}(^{1}S)~^{1}S\,3d~^{2}D$\
27 & 32 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}P_{1/2}^{\circ}$ & 720012 & & & 2.85E-10& 2.74E-10 & 0.82 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
27 & 33 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}P_{3/2}^{\circ}$ & 723164 & & & 2.83E-10& 2.73E-10 & 0.81 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
27 & 34 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}P_{5/2}^{\circ}$ & 728815 & & & 2.79E-10& 2.70E-10 & 0.80 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
27 & 35 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F_{9/2}^{\circ}$ & 752717 & & & 1.59E-10& 1.50E-10 & 0.85 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
27 & 36 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F_{7/2}^{\circ}$ & 756020 & & & 1.57E-10& 1.49E-10 & 0.81 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.02 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D^{\circ}$\
27 & 37 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F_{5/2}^{\circ}$ & 759717 & & & 1.54E-10& 1.47E-10 & 0.80 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.03 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D^{\circ}$\
27 & 38 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F_{3/2}^{\circ}$ & 763243 & & & 1.52E-10& 1.45E-10 & 0.81 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
27 & 39 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D_{7/2}^{\circ}$ & 786251 & & & 1.63E-10& 1.57E-10 & 0.76 + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
27 & 40 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D_{1/2}^{\circ}$ & 787884 & & & 1.59E-10& 1.53E-10 & 0.78 + 0.07 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
27 & 41 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D_{5/2}^{\circ}$ & 788547 & & & 1.60E-10& 1.54E-10 & 0.75 + 0.07 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
27 & 42 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D_{3/2}^{\circ}$ & 788786 & & & 1.60E-10& 1.54E-10 & 0.77 + 0.07 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
27 & 43 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}F_{7/2}^{\circ}$ & 796271 & & & 3.39E-10& 3.25E-10 & 0.70 + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
27 & 44 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}D_{5/2}^{\circ}$ & 801030 & & & 2.97E-10& 2.84E-10 & 0.49 + 0.27 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}F^{\circ}$ + 0.05 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D^{\circ}$\
27 & 45 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}D_{3/2}^{\circ}$ & 806158 & & & 1.87E-10& 1.80E-10 & 0.70 + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D^{\circ}$ + 0.04 $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}P^{\circ}$\
27 & 46 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}F_{5/2}^{\circ}$ & 811099 & & & 2.42E-10& 2.33E-10 & 0.49 + 0.26 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}D^{\circ}$ + 0.05 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
27 & 47 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}P_{1/2}^{\circ}$ & 819821 & & & 7.97E-11& 7.75E-11 & 0.40 + 0.24 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}P^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P^{\circ}$\
27 & 48 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}P_{3/2}^{\circ}$ & 829678 & & & 9.15E-11& 8.85E-11 & 0.38 + 0.19 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}P^{\circ}$ + 0.15 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P^{\circ}$\
27 & 49 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{1/2}^{\circ}$ & 844168 & & & 4.69E-09& 4.51E-09 & 0.96\
27 & 50 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{3/2}^{\circ}$ & 844330 & & & 5.55E-09& 5.37E-09 & 0.96\
27 & 51 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{5/2}^{\circ}$ & 844672 & & & 6.83E-09& 6.66E-09 & 0.96\
27 & 52 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{7/2}^{\circ}$ & 845249 & & & 9.95E-09& 9.80E-09 & 0.96\
27 & 53 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{9/2}^{\circ}$ & 846081 & & & 1.72E-08& 1.71E-08 & 0.96\
27 & 54 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{11/2}^{\circ}$ & 847108 & & & 8.89E-08& 9.40E-08 & 0.96\
27 & 55 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}F_{5/2}^{\circ}$ & 849224 & & & 7.34E-11& 7.10E-11 & 0.58 + 0.18 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
27 & 56 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}F_{7/2}^{\circ}$ & 854044 & & & 7.11E-11& 6.89E-11 & 0.57 + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
27 & 57 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{6}P_{3/2}^{\circ}$ & 870108 & & & 5.42E-09& 5.43E-09 & 0.95\
27 & 58 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{6}P_{5/2}^{\circ}$ & 870449 & & & 4.08E-09& 4.01E-09 & 0.94\
27 & 59 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{6}P_{7/2}^{\circ}$ & 870717 & & & 3.04E-09& 2.99E-09 & 0.93\
27 & 60 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}D_{3/2}^{\circ}$ & 878378 & & & 6.62E-11& 6.41E-11 & 0.56 + 0.18 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$\
27 & 61 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}D_{5/2}^{\circ}$ & 880335 & & & 6.80E-11& 6.56E-11 & 0.58 + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$\
27 & 62 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P_{1/2}^{\circ}$ & 892029 & & & 1.84E-09& 1.85E-09 & 0.39 + 0.31 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$\
27 & 63 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D_{7/2}^{\circ}$ & 892064 & & & 2.72E-10& 2.67E-10 & 0.53 + 0.21 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
27 & 64 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D_{3/2}^{\circ}$ & 892589 & & & 2.53E-10& 2.48E-10 & 0.39 + 0.15 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$\
27 & 65 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D_{5/2}^{\circ}$ & 892629 & & & 2.55E-10& 2.50E-10 & 0.47 + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.05 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$\
27 & 66 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D_{1/2}^{\circ}$ & 895556 & & & 2.01E-10& 1.97E-10 & 0.57 + 0.22 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.05 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
27 & 67 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P_{3/2}^{\circ}$ & 896204 & & & 2.93E-09& 2.91E-09 & 0.24 + 0.31 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$\
27 & 68 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G_{5/2}^{\circ}$ & 896231 & & & 1.27E-09& 1.25E-09 & 0.48 + 0.15 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D^{\circ}$\
27 & 69 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G_{7/2}^{\circ}$ & 897738 & & & 2.34E-09& 2.31E-09 & 0.55 + 0.20 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.05 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$\
27 & 70 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F_{3/2}^{\circ}$ & 898592 & & & 6.12E-10& 6.03E-10 & 0.10 + 0.18 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D^{\circ}$\
27 & 71 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P_{5/2}^{\circ}$ & 900361 & & & 5.45E-09& 5.50E-09 & 0.12 + 0.36 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.28 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
27 & 72 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G_{9/2}^{\circ}$ & 900589 & & & 3.79E-09& 3.75E-09 & 0.61 + 0.25 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.03 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$\
27 & 73 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F_{5/2}^{\circ}$ & 903224 & & & 2.09E-09& 2.07E-09 & 0.11 + 0.26 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
27 & 74 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G_{11/2}^{\circ}$ & 903368 & & & 4.99E-09& 4.81E-09 & 0.63 + 0.33 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$\
27 & 75 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F_{7/2}^{\circ}$ & 906713 & & & 2.20E-09& 2.18E-09 & 0.37 + 0.19 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
27 & 76 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F_{9/2}^{\circ}$ & 910202 & & & 1.81E-09& 1.78E-09 & 0.44 + 0.22 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
27 & 77 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H_{7/2}^{\circ}$ & 917605 & & & 8.69E-10& 8.51E-10 & 0.72 + 0.05 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
27 & 78 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H_{9/2}^{\circ}$ & 919404 & & & 9.42E-09& 9.34E-09 & 0.83 + 0.08 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$ + 0.02 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$\
27 & 79 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H_{11/2}^{\circ}$ & 921402 & & & 2.58E-07& 2.18E-07 & 0.85 + 0.08 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
27 & 80 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F_{7/2}^{\circ}$ & 923122 & & & 1.85E-10& 1.80E-10 & 0.28 + 0.21 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H^{\circ}$\
27 & 81 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}P_{1/2}^{\circ}$ & 924816 & & & 6.57E-12& 6.57E-12 & 0.52 + 0.28 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}P^{\circ}$ + 0.06 $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}P^{\circ}$\
27 & 82 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}P_{3/2}^{\circ}$ & 924998 & & & 8.78E-12& 8.78E-12 & 0.40 + 0.17 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}P^{\circ}$ + 0.05 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}D^{\circ}$\
27 & 83 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H_{13/2}^{\circ}$ & 925260 & & & 2.38E-02& 2.41E-02 & 0.97\
27 & 84 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D_{3/2}^{\circ}$ & 927091 & & & 1.78E-11& 1.78E-11 & 0.21 + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}D^{\circ}$ + 0.15 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}P^{\circ}$\
27 & 85 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F_{5/2}^{\circ}$ & 927489 & & & 1.05E-10& 1.02E-10 & 0.30 + 0.22 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
27 & 86 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H_{9/2}^{\circ}$ & 929133 & & & 1.07E-09& 1.06E-09 & 0.41 + 0.33 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}H^{\circ}$\
27 & 87 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H_{11/2}^{\circ}$ & 932133 & & & 2.60E-09& 2.51E-09 & 0.26 + 0.39 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
27 & 88 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D_{5/2}^{\circ}$ & 935794 & & & 5.95E-11& 5.82E-11 & 0.21 + 0.23 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}D^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D^{\circ}$\
27 & 89 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G_{5/2}^{\circ}$ & 940060 & & & 2.27E-10& 2.24E-10 & 0.51 + 0.29 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.05 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
27 & 90 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G_{7/2}^{\circ}$ & 940708 & & & 2.13E-10& 2.10E-10 & 0.51 + 0.24 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G^{\circ}$\
27 & 91 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G_{9/2}^{\circ}$ & 944877 & & & 2.25E-10& 2.21E-10 & 0.41 + 0.16 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G^{\circ}$\
27 & 92 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G_{7/2}^{\circ}$ & 947422 & & & 1.78E-10& 1.75E-10 & 0.64 + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
27 & 93 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G_{9/2}^{\circ}$ & 948821 & & & 1.77E-10& 1.73E-10 & 0.65 + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
27 & 94 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G_{11/2}^{\circ}$ & 952498 & & & 7.02E-10& 6.89E-10 & 0.49 + 0.16 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H^{\circ}$\
27 & 95 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F_{5/2}^{\circ}$ & 956140 & & & 2.05E-11& 2.01E-11 & 0.42 + 0.32 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.07 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F^{\circ}$\
27 & 96 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F_{7/2}^{\circ}$ & 956877 & & & 2.10E-11& 2.07E-11 & 0.42 + 0.31 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.06 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F^{\circ}$\
27 & 97 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F_{3/2}^{\circ}$ & 957402 & & & 1.99E-11& 1.96E-11 & 0.45 + 0.35 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.07 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F^{\circ}$\
27 & 98 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F_{9/2}^{\circ}$ & 959724 & & & 2.18E-11& 2.14E-11 & 0.43 + 0.33 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.06 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F^{\circ}$\
27 & 99 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P_{3/2}^{\circ}$ & 966434 & & & 3.00E-11& 2.98E-11 & 0.17 + 0.20 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}P^{\circ}$ + 0.19 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{2}P^{\circ}$\
27 & 100 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D_{1/2}^{\circ}$ & 968736 & & & 2.87E-11& 2.84E-11 & 0.43 + 0.19 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
27 & 101 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}I_{11/2}^{\circ}$ & 968863 & & & 1.83E-09& 1.81E-09 & 0.90 + 0.03 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}H^{\circ}$ + 0.02 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H^{\circ}$\
27 & 102 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}S_{1/2}^{\circ}$ & 970661 & & & 7.87E-11& 7.86E-11 & 0.33 + 0.25 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}S^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{2}P^{\circ}$\
27 & 103 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D_{3/2}^{\circ}$ & 970802 & & & 4.22E-11& 4.19E-11 & 0.29 + 0.17 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
27 & 104 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}I_{13/2}^{\circ}$ & 970925 & & & 2.84E-02& 2.84E-02 & 0.97\
27 & 105 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D_{5/2}^{\circ}$ & 973399 & & & 6.25E-11& 6.17E-11 & 0.19 + 0.15 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
27 & 106 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D_{3/2}^{\circ}$ & 976694 & & & 4.77E-11& 4.69E-11 & 0.08 + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
27 & 107 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F_{5/2}^{\circ}$ & 977205 & & & 6.20E-11& 6.06E-11 & 0.28 + 0.11 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}F^{\circ}$\
27 & 108 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D_{7/2}^{\circ}$ & 977355 & & & 4.88E-11& 4.81E-11 & 0.25 + 0.24 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
27 & 109 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D_{5/2}^{\circ}$ & 978100 & & & 4.53E-11& 4.48E-11 & 0.04 + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
27 & 110 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}S)~^{4}S_{3/2}^{\circ}$ & 980412 & & & 2.07E-10& 2.08E-10 & 0.52 + 0.27 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}S^{\circ}$ + 0.03 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P^{\circ}$\
27 & 111 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F_{3/2}^{\circ}$ & 982012 & & & 1.07E-10& 1.06E-10 & 0.34 + 0.23 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D^{\circ}$\
27 & 112 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P_{1/2}^{\circ}$ & 982886 & & & 2.86E-11& 2.83E-11 & 0.18 + 0.19 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}S^{\circ}$ + 0.17 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{2}P^{\circ}$\
27 & 113 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F_{5/2}^{\circ}$ & 983647 & & & 5.90E-11& 5.84E-11 & 0.31 + 0.27 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
27 & 114 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F_{7/2}^{\circ}$ & 984061 & & & 1.28E-10& 1.26E-10 & 0.22 + 0.20 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
27 & 115 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F_{9/2}^{\circ}$ & 986936 & & & 1.48E-10& 1.46E-10 & 0.29 + 0.22 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G^{\circ}$\
27 & 116 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F_{7/2}^{\circ}$ & 986968 & & & 6.88E-11& 6.80E-11 & 0.13 + 0.21 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}G^{\circ}$\
27 & 117 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G_{7/2}^{\circ}$ & 989546 & & & 5.55E-11& 5.52E-11 & 0.11 + 0.15 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$\
27 & 118 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D_{1/2}^{\circ}$ & 995837 & & & 5.33E-11& 5.30E-11 & 0.39 + 0.23 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.18 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
27 & 119 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D_{3/2}^{\circ}$ & 997019 & & & 4.15E-11& 4.15E-11 & 0.26 + 0.18 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.18 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
27 & 120 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G_{9/2}^{\circ}$ & 997046 & & & 1.92E-10& 1.91E-10 & 0.18 + 0.22 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.22 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
27 & 121 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P_{5/2}^{\circ}$ & 997806 & & & 2.77E-11& 2.77E-11 & 0.49 + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
27 & 122 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G_{11/2}^{\circ}$ & 1002346 & & & 1.61E-11& 1.59E-11 & 0.37 + 0.25 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G^{\circ}$ + 0.24 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
27 & 123 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}F_{5/2}^{\circ}$ & 1002546 & & & 2.19E-11& 2.18E-11 & 0.16 + 0.21 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
27 & 124 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G_{9/2}^{\circ}$ & 1003569 & & & 1.65E-11& 1.64E-11 & 0.29 + 0.21 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G^{\circ}$ + 0.20 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
27 & 125 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P_{3/2}^{\circ}$ & 1003868 & & & 1.31E-11& 1.31E-11 & 0.57 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
27 & 126 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P_{1/2}^{\circ}$ & 1004525 & & & 9.40E-12& 9.40E-12 & 0.64 + 0.13 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.09 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}P^{\circ}$\
27 & 127 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D_{7/2}^{\circ}$ & 1005515 & & & 2.61E-11& 2.59E-11 & 0.23 + 0.19 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$\
27 & 128 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D_{5/2}^{\circ}$ & 1006988 & & & 2.06E-11& 2.05E-11 & 0.13 + 0.08 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
27 & 129 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G_{7/2}^{\circ}$ & 1007191 & & & 1.83E-11& 1.81E-11 & 0.27 + 0.22 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G^{\circ}$ + 0.20 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
27 & 130 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G_{5/2}^{\circ}$ & 1008356 & & & 1.69E-11& 1.68E-11 & 0.31 + 0.21 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$ + 0.20 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
27 & 131 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P_{1/2}^{\circ}$ & 1012949 & & & 1.32E-11& 1.32E-11 & 0.17 + 0.36 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}P^{\circ}$\
27 & 132 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P_{3/2}^{\circ}$ & 1013315 & & & 1.20E-11& 1.21E-11 & 0.43 + 0.25 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
27 & 133 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}G_{9/2}^{\circ}$ & 1014424 & & & 1.79E-11& 1.77E-11 & 0.74 + 0.07 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
27 & 134 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P_{5/2}^{\circ}$ & 1014665 & & & 9.19E-12& 9.20E-12 & 0.39 + 0.30 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
27 & 135 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}F_{7/2}^{\circ}$ & 1016008 & & & 2.16E-11& 2.14E-11 & 0.10 + 0.33 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}G^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
27 & 136 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}P_{1/2}^{\circ}$ & 1016463 & & & 1.52E-11& 1.51E-11 & 0.35 + 0.12 $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}P^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
27 & 137 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}G_{7/2}^{\circ}$ & 1021432 & & & 1.65E-11& 1.63E-11 & 0.54 + 0.07 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.07 $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}F^{\circ}$\
27 & 138 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}P_{3/2}^{\circ}$ & 1023140 & & & 1.04E-11& 1.03E-11 & 0.35 + 0.22 $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}P^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}P^{\circ}$\
27 & 139 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}H_{11/2}^{\circ}$ & 1027142 & & & 2.05E-10& 2.03E-10 & 0.56 + 0.28 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H^{\circ}$ + 0.05 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$\
27 & 140 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}H_{9/2}^{\circ}$ & 1030665 & & & 2.71E-11& 2.69E-11 & 0.73 + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H^{\circ}$ + 0.02 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}G^{\circ}$\
27 & 141 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D_{7/2}^{\circ}$ & 1034872 & & & 6.85E-12& 6.88E-12 & 0.48 + 0.18 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
27 & 142 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H_{9/2}^{\circ}$ & 1035717 & & & 2.51E-11& 2.48E-11 & 0.55 + 0.30 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H^{\circ}$ + 0.05 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}H^{\circ}$\
27 & 143 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}D_{5/2}^{\circ}$ & 1036288 & & & 1.66E-11& 1.66E-11 & 0.14 + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}S)~^{2}D^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
27 & 144 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D_{5/2}^{\circ}$ & 1038959 & & & 8.48E-12& 8.50E-12 & 0.24 + 0.17 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.17 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
27 & 145 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D_{3/2}^{\circ}$ & 1039767 & & & 7.41E-12& 7.43E-12 & 0.40 + 0.22 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
27 & 146 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}S)~^{2}D_{3/2}^{\circ}$ & 1040197 & & & 3.60E-11& 3.60E-11 & 0.37 + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}D^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D^{\circ}$\
27 & 147 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D_{1/2}^{\circ}$ & 1040382 & & & 6.92E-12& 6.95E-12 & 0.44 + 0.25 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
27 & 148 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H_{11/2}^{\circ}$ & 1042412 & & & 1.29E-11& 1.28E-11 & 0.43 + 0.28 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H^{\circ}$ + 0.20 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}H^{\circ}$\
27 & 149 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F_{7/2}^{\circ}$ & 1044561 & & & 7.08E-12& 7.11E-12 & 0.05 + 0.19 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.18 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
27 & 150 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D_{5/2}^{\circ}$ & 1045657 & & & 6.47E-12& 6.52E-12 & 0.19 + 0.17 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.17 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D^{\circ}$\
27 & 151 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D_{7/2}^{\circ}$ & 1047677 & & & 7.41E-12& 7.41E-12 & 0.11 + 0.11 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
27 & 152 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D_{3/2}^{\circ}$ & 1048439 & & & 6.19E-12& 6.25E-12 & 0.03 + 0.22 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.21 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D^{\circ}$\
27 & 153 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F_{9/2}^{\circ}$ & 1048640 & & & 5.31E-12& 5.33E-12 & 0.36 + 0.25 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.17 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
27 & 154 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D_{1/2}^{\circ}$ & 1051038 & & & 5.98E-12& 6.03E-12 & 0.06 + 0.25 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.24 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D^{\circ}$\
27 & 155 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F_{5/2}^{\circ}$ & 1052646 & & & 8.20E-12& 8.18E-12 & 0.13 + 0.15 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}F^{\circ}$\
27 & 156 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F_{7/2}^{\circ}$ & 1055116 & & & 5.43E-12& 5.45E-12 & 0.31 + 0.24 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
27 & 157 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D_{5/2}^{\circ}$ & 1059308 & & & 7.43E-12& 7.47E-12 & 0.33 + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D^{\circ}$\
27 & 158 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D_{3/2}^{\circ}$ & 1060273 & & & 6.76E-12& 6.79E-12 & 0.20 + 0.11 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
27 & 159 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F_{5/2}^{\circ}$ & 1060853 & & & 5.53E-12& 5.55E-12 & 0.29 + 0.26 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
27 & 160 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}G_{7/2}^{\circ}$ & 1061383 & & & 1.60E-11& 1.59E-11 & 0.55 + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G^{\circ}$\
27 & 161 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}P_{1/2}^{\circ}$ & 1062949 & & & 6.09E-12& 6.16E-12 & 0.20 + 0.23 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}S^{\circ}$ + 0.18 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{2}P^{\circ}$\
27 & 162 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F_{3/2}^{\circ}$ & 1063736 & & & 5.77E-12& 5.80E-12 & 0.23 + 0.21 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
27 & 163 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}P_{3/2}^{\circ}$ & 1067193 & & & 6.82E-12& 6.88E-12 & 0.30 + 0.16 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{2}P^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}S)~^{2}P^{\circ}$\
27 & 164 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}G_{9/2}^{\circ}$ & 1068851 & & & 1.66E-11& 1.65E-11 & 0.53 + 0.20 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}G^{\circ}$\
27 & 165 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}S_{1/2}^{\circ}$ & 1070083 & & & 6.09E-12& 6.15E-12 & 0.27 + 0.22 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}S^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}P^{\circ}$\
27 & 166 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}S)~^{2}P_{3/2}^{\circ}$ & 1071229 & & & 1.02E-11& 1.02E-11 & 0.25 + 0.24 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}D^{\circ}$\
27 & 167 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F_{5/2}^{\circ}$ & 1072803 & & & 1.16E-11& 1.15E-11 & 0.41 + 0.12 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
27 & 168 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}S)~^{2}P_{1/2}^{\circ}$ & 1072964 & & & 1.13E-10& 1.14E-10 & 0.54 + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}P^{\circ}$\
27 & 169 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F_{7/2}^{\circ}$ & 1075595 & & & 7.39E-12& 7.41E-12 & 0.25 + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
27 & 170 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P_{3/2}^{\circ}$ & 1076338 & & & 6.47E-12& 6.52E-12 & 0.18 + 0.20 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$\
27 & 171 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F_{5/2}^{\circ}$ & 1076836 & & & 7.15E-12& 7.15E-12 & 0.09 + 0.11 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F^{\circ}$\
27 & 172 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P_{5/2}^{\circ}$ & 1079776 & & & 6.47E-12& 6.51E-12 & 0.29 + 0.24 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
27 & 173 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P_{1/2}^{\circ}$ & 1081470 & & & 5.82E-12& 5.87E-12 & 0.41 + 0.29 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.05 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}S^{\circ}$\
27 & 174 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D_{5/2}^{\circ}$ & 1081984 & & & 6.90E-12& 6.93E-12 & 0.09 + 0.15 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
27 & 175 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D_{3/2}^{\circ}$ & 1082852 & & & 8.80E-12& 8.88E-12 & 0.01 + 0.17 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.15 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}S)~^{2}P^{\circ}$\
27 & 176 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}S_{3/2}^{\circ}$ & 1089838 & & & 5.91E-12& 6.00E-12 & 0.50 + 0.27 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}S)~^{4}S^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
27 & 177 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F_{7/2}^{\circ}$ & 1091609 & & & 1.63E-11& 1.63E-11 & 0.40 + 0.24 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}F^{\circ}$\
27 & 178 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}S)~^{2}D_{5/2}^{\circ}$ & 1097645 & & & 6.05E-12& 6.10E-12 & 0.31 + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}D^{\circ}$\
27 & 179 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}D_{3/2}^{\circ}$ & 1098837 & & & 6.13E-12& 6.16E-12 & 0.14 + 0.34 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}S)~^{2}D^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}D^{\circ}$\
27 & 180 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G_{7/2}^{\circ}$ & 1103182 & & & 5.51E-12& 5.54E-12 & 0.41 + 0.33 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G^{\circ}$\
27 & 181 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G_{9/2}^{\circ}$ & 1103923 & & & 5.57E-12& 5.60E-12 & 0.44 + 0.32 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G^{\circ}$\
27 & 182 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}S_{5/2}$ & 1116322 & & & 9.50E-10& 9.40E-10 & 0.90 + 0.07 $3s^{2}\,3p^{2}(^{3}P)~^{3}P\,3d^{3}(^{4}P)~^{6}S$\
27 & 183 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D_{1/2}$ & 1122804 & & & 5.89E-10& 5.89E-10 & 0.62 + 0.26 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}D$ + 0.06 $3s^{2}\,3p^{2}(^{3}P)~^{3}P\,3d^{3}(^{4}F)~^{6}D$\
27 & 184 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}F_{5/2}^{\circ}$ & 1122853 & & & 5.48E-12& 5.52E-12 & 0.30 + 0.27 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
27 & 185 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D_{3/2}$ & 1122982 & & & 5.89E-10& 5.88E-10 & 0.59 + 0.27 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}D$ + 0.06 $3s^{2}\,3p^{2}(^{3}P)~^{3}P\,3d^{3}(^{4}F)~^{6}D$\
27 & 186 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D_{5/2}$ & 1123346 & & & 5.93E-10& 5.90E-10 & 0.54 + 0.28 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}D$ + 0.08 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F$\
27 & 187 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D_{7/2}$ & 1123907 & & & 6.02E-10& 5.97E-10 & 0.49 + 0.30 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}D$ + 0.11 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F$\
27 & 188 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}F_{7/2}^{\circ}$ & 1124253 & & & 5.64E-12& 5.68E-12 & 0.27 + 0.26 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
27 & 189 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D_{9/2}$ & 1124755 & & & 6.22E-10& 6.17E-10 & 0.46 + 0.35 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}D$ + 0.09 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F$\
27 & 190 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}D_{3/2}^{\circ}$ & 1131893 & & & 6.79E-12& 6.81E-12 & 0.26 + 0.22 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D^{\circ}$\
27 & 191 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}D_{5/2}^{\circ}$ & 1134908 & & & 6.78E-12& 6.79E-12 & 0.31 + 0.18 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D^{\circ}$ + 0.17 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D^{\circ}$\
27 & 192 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F_{11/2}$ & 1136827 & & & 3.84E-10& 3.78E-10 & 0.88 + 0.06 $3s^{2}\,3p^{2}(^{3}P)~^{3}P\,3d^{3}(^{4}F)~^{6}F$ + 0.03 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}G$\
27 & 193 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F_{9/2}$ & 1137570 & & & 3.89E-10& 3.84E-10 & 0.78 + 0.06 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}D$ + 0.06 $3s^{2}\,3p^{2}(^{3}P)~^{3}P\,3d^{3}(^{4}F)~^{6}F$\
28 & 1 & $3s^{2}\,3p^{5}~^{2}P_{3/2}^{\circ}$ & 0 & 0 & 0 & & & 0.96\
28 & 2 & $3s^{2}\,3p^{5}~^{2}P_{1/2}^{\circ}$ & 23678 & 23629 & 49 & 4.19E-03& 4.19E-03 & 0.96\
28 & 3 & $3s~^{2}S\,3p^{6}~^{2}S_{1/2}$ & 339773 & 338615 & 1158 & 1.71E-10& 1.66E-10 & 0.72 + 0.25 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}S$\
28 & 4 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D_{5/2}$ & 450451 & & & 9.35E-08& 1.11E-07 & 0.90\
28 & 5 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D_{7/2}$ & 450941 & 454000 & -3059 & 1.07E-02& 1.07E-02 & 0.93 + 0.03 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F$\
28 & 6 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D_{3/2}$ & 452175 & & & 8.00E-08& 7.86E-08 & 0.89 + 0.02 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P$\
28 & 7 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D_{1/2}$ & 454809 & & & 4.79E-08& 4.39E-08 & 0.90 + 0.03 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P$ + 0.02 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P$\
28 & 8 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F_{9/2}$ & 482542 & 485570 & -3028 & 4.18E-02& 4.18E-02 & 0.90 + 0.07 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}G$\
28 & 9 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P_{1/2}$ & 486142 & & & 3.97E-09& 3.99E-09 & 0.48 + 0.40 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P$ + 0.06 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D$\
28 & 10 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F_{7/2}$ & 489790 & 492750 & -2960 & 2.53E-02& 2.53E-02 & 0.80 + 0.08 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F$ + 0.07 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}G$\
28 & 11 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F_{5/2}$ & 496283 & & & 8.99E-09& 9.20E-09 & 0.93\
28 & 12 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P_{3/2}$ & 497035 & & & 2.42E-09& 2.48E-09 & 0.26 + 0.30 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F$ + 0.19 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P$\
28 & 13 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F_{3/2}$ & 497836 & & & 5.91E-09& 5.93E-09 & 0.45 + 0.17 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P$ + 0.13 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P$\
28 & 14 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P_{1/2}$ & 502810 & & & 1.30E-09& 1.28E-09 & 0.93\
28 & 15 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D_{3/2}$ & 503389 & & & 1.71E-09& 1.77E-09 & 0.36 + 0.22 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D$ + 0.19 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F$\
28 & 16 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F_{7/2}$ & 510045 & 513290 & -3245 & 5.71E-03& 5.71E-03 & 0.46 + 0.28 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}G$ + 0.13 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F$\
28 & 17 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P_{3/2}$ & 511469 & & & 7.22E-08& 7.40E-08 & 0.67 + 0.10 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P$ + 0.09 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P$\
28 & 18 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P_{5/2}$ & 513156 & & & 5.63E-09& 5.74E-09 & 0.56 + 0.14 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$ + 0.12 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D$\
28 & 19 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D_{5/2}$ & 517246 & & & 9.71E-09& 9.79E-09 & 0.36 + 0.36 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P$ + 0.15 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D$\
28 & 20 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}G_{7/2}$ & 524204 & 526960 & -2756 & 5.12E-03& 5.11E-03 & 0.61 + 0.25 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F$ + 0.10 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}F$\
28 & 21 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}G_{9/2}$ & 524407 & 527230 & -2823 & 4.76E-03& 4.75E-03 & 0.89 + 0.07 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F$\
28 & 22 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F_{5/2}$ & 528717 & & & 1.66E-08& 1.69E-08 & 0.68 + 0.22 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}F$ + 0.04 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$\
28 & 23 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}F_{5/2}$ & 558198 & & & 2.94E-09& 2.97E-09 & 0.73 + 0.18 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F$ + 0.04 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$\
28 & 24 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}F_{7/2}$ & 564358 & 567200 & -2842 & 1.88E-03& 1.88E-03 & 0.76 + 0.18 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F$\
28 & 25 & $3s^{2}\,3p^{4}(^{1}S)~^{1}S\,3d~^{2}D_{3/2}$ & 592695 & & & 4.25E-10& 4.30E-10 & 0.65 + 0.25 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$ + 0.02 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F$\
28 & 26 & $3s^{2}\,3p^{4}(^{1}S)~^{1}S\,3d~^{2}D_{5/2}$ & 600433 & & & 2.51E-09& 2.54E-09 & 0.72 + 0.15 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$ + 0.03 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F$\
28 & 27 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}S_{1/2}$ & 622807 & 622840 & -33 & 5.30E-12& 5.28E-12 & 0.69 + 0.23 $3s~^{2}S\,3p^{6}~^{2}S$ + 0.03 $3s~^{2}S\,3p^{4}(^{3}P)~^{2}P\,3d^{2}(^{3}P)~^{2}S$\
28 & 28 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P_{3/2}$ & 648787 & 648670 & 117 & 5.80E-12& 5.88E-12 & 0.49 + 0.38 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P$ + 0.04 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D$\
28 & 29 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D_{5/2}$ & 657278 & 657230 & 48 & 4.87E-12& 4.93E-12 & 0.66 + 0.21 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$ + 0.08 $3s^{2}\,3p^{4}(^{1}S)~^{1}S\,3d~^{2}D$\
28 & 30 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P_{1/2}$ & 657577 & 657290 & 287 & 5.99E-12& 6.06E-12 & 0.51 + 0.44 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P$\
28 & 31 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D_{3/2}$ & 677714 & 676420 & 1294 & 4.98E-12& 5.00E-12 & 0.59 + 0.16 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$ + 0.13 $3s^{2}\,3p^{4}(^{1}S)~^{1}S\,3d~^{2}D$\
28 & 32 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}P_{1/2}^{\circ}$ & 773571 & & & 2.50E-10& 2.43E-10 & 0.82 + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
28 & 33 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}P_{3/2}^{\circ}$ & 777377 & & & 2.48E-10& 2.41E-10 & 0.81 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
28 & 34 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}P_{5/2}^{\circ}$ & 784275 & & & 2.44E-10& 2.39E-10 & 0.80 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
28 & 35 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F_{9/2}^{\circ}$ & 808802 & & & 1.40E-10& 1.34E-10 & 0.85 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
28 & 36 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F_{7/2}^{\circ}$ & 812466 & & & 1.38E-10& 1.33E-10 & 0.80 + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.03 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D^{\circ}$\
28 & 37 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F_{5/2}^{\circ}$ & 816861 & & & 1.35E-10& 1.31E-10 & 0.79 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.03 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D^{\circ}$\
28 & 38 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F_{3/2}^{\circ}$ & 821226 & & & 1.33E-10& 1.28E-10 & 0.80 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.03 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D^{\circ}$\
28 & 39 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D_{7/2}^{\circ}$ & 845010 & & & 1.46E-10& 1.41E-10 & 0.74 + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
28 & 40 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D_{1/2}^{\circ}$ & 846703 & & & 1.40E-10& 1.36E-10 & 0.77 + 0.07 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
28 & 41 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D_{5/2}^{\circ}$ & 847957 & & & 1.41E-10& 1.37E-10 & 0.73 + 0.06 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
28 & 42 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D_{3/2}^{\circ}$ & 848173 & & & 1.41E-10& 1.37E-10 & 0.76 + 0.07 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
28 & 43 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}F_{7/2}^{\circ}$ & 855065 & & & 2.86E-10& 2.78E-10 & 0.68 + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
28 & 44 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}D_{5/2}^{\circ}$ & 860423 & & & 2.62E-10& 2.54E-10 & 0.50 + 0.25 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}F^{\circ}$ + 0.05 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D^{\circ}$\
28 & 45 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}D_{3/2}^{\circ}$ & 866094 & & & 1.59E-10& 1.55E-10 & 0.68 + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D^{\circ}$ + 0.05 $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}P^{\circ}$\
28 & 46 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}F_{5/2}^{\circ}$ & 872788 & & & 2.07E-10& 2.01E-10 & 0.50 + 0.25 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}D^{\circ}$ + 0.05 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
28 & 47 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}P_{1/2}^{\circ}$ & 879669 & & & 7.00E-11& 6.87E-11 & 0.40 + 0.24 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}P^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P^{\circ}$\
28 & 48 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}P_{3/2}^{\circ}$ & 892076 & & & 8.15E-11& 7.96E-11 & 0.37 + 0.18 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}P^{\circ}$ + 0.15 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P^{\circ}$\
28 & 49 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{1/2}^{\circ}$ & 905894 & & & 3.18E-09& 3.09E-09 & 0.96\
28 & 50 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{3/2}^{\circ}$ & 906076 & & & 3.79E-09& 3.70E-09 & 0.95\
28 & 51 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{5/2}^{\circ}$ & 906492 & & & 4.67E-09& 4.59E-09 & 0.95\
28 & 52 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{7/2}^{\circ}$ & 907227 & & & 6.80E-09& 6.76E-09 & 0.95\
28 & 53 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{9/2}^{\circ}$ & 908313 & & & 1.16E-08& 1.15E-08 & 0.95\
28 & 54 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{11/2}^{\circ}$ & 909650 & & & 6.24E-08& 6.62E-08 & 0.96\
28 & 55 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}F_{5/2}^{\circ}$ & 911783 & & & 6.56E-11& 6.41E-11 & 0.58 + 0.17 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
28 & 56 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}F_{7/2}^{\circ}$ & 917807 & & & 6.32E-11& 6.18E-11 & 0.56 + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
28 & 57 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{6}P_{3/2}^{\circ}$ & 933895 & & & 3.67E-09& 3.70E-09 & 0.94\
28 & 58 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{6}P_{5/2}^{\circ}$ & 934363 & & & 2.83E-09& 2.81E-09 & 0.94\
28 & 59 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{6}P_{7/2}^{\circ}$ & 934651 & & & 2.06E-09& 2.05E-09 & 0.91 + 0.02 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D^{\circ}$ + 0.02 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
28 & 60 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}D_{3/2}^{\circ}$ & 942786 & & & 5.86E-11& 5.73E-11 & 0.56 + 0.18 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$\
28 & 61 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}D_{5/2}^{\circ}$ & 945446 & & & 6.04E-11& 5.89E-11 & 0.58 + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$\
28 & 62 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P_{1/2}^{\circ}$ & 956413 & & & 1.54E-09& 1.56E-09 & 0.39 + 0.31 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$\
28 & 63 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D_{3/2}^{\circ}$ & 956877 & & & 2.40E-10& 2.37E-10 & 0.29 + 0.15 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
28 & 64 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D_{7/2}^{\circ}$ & 957284 & & & 2.52E-10& 2.50E-10 & 0.52 + 0.20 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
28 & 65 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D_{5/2}^{\circ}$ & 957551 & & & 2.49E-10& 2.47E-10 & 0.39 + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$\
28 & 66 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G_{5/2}^{\circ}$ & 960986 & & & 7.35E-10& 7.29E-10 & 0.44 + 0.13 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D^{\circ}$\
28 & 67 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D_{1/2}^{\circ}$ & 961465 & & & 1.74E-10& 1.72E-10 & 0.57 + 0.22 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.05 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
28 & 68 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P_{3/2}^{\circ}$ & 961484 & & & 1.37E-09& 1.37E-09 & 0.09 + 0.26 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.21 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
28 & 69 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G_{7/2}^{\circ}$ & 962703 & & & 1.75E-09& 1.75E-09 & 0.55 + 0.19 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.05 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$\
28 & 70 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D_{3/2}^{\circ}$ & 964374 & & & 4.76E-10& 4.74E-10 & 0.07 + 0.19 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D^{\circ}$ + 0.15 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
28 & 71 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G_{9/2}^{\circ}$ & 966429 & & & 3.10E-09& 3.12E-09 & 0.60 + 0.25 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.03 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$\
28 & 72 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P_{5/2}^{\circ}$ & 966494 & & & 3.61E-09& 3.68E-09 & 0.10 + 0.31 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.25 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
28 & 73 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F_{5/2}^{\circ}$ & 969616 & & & 1.78E-09& 1.78E-09 & 0.10 + 0.22 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
28 & 74 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G_{11/2}^{\circ}$ & 969887 & & & 4.27E-09& 4.20E-09 & 0.63 + 0.33 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$\
28 & 75 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F_{7/2}^{\circ}$ & 973506 & & & 1.78E-09& 1.78E-09 & 0.36 + 0.19 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
28 & 76 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F_{9/2}^{\circ}$ & 977570 & & & 1.42E-09& 1.41E-09 & 0.43 + 0.21 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
28 & 77 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H_{7/2}^{\circ}$ & 982999 & & & 7.46E-10& 7.39E-10 & 0.69 + 0.06 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
28 & 78 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H_{9/2}^{\circ}$ & 985481 & & & 7.49E-09& 7.49E-09 & 0.79 + 0.09 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$ + 0.02 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$\
28 & 79 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H_{11/2}^{\circ}$ & 988054 & & & 2.39E-07& 2.04E-07 & 0.82 + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
28 & 80 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F_{7/2}^{\circ}$ & 990379 & & & 1.69E-10& 1.66E-10 & 0.27 + 0.21 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.17 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H^{\circ}$\
28 & 81 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}P_{1/2}^{\circ}$ & 991466 & & & 6.18E-12& 6.22E-12 & 0.52 + 0.28 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}P^{\circ}$ + 0.06 $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}P^{\circ}$\
28 & 82 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}P_{3/2}^{\circ}$ & 991893 & & & 8.04E-12& 8.10E-12 & 0.41 + 0.17 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}P^{\circ}$ + 0.05 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}D^{\circ}$\
28 & 83 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H_{13/2}^{\circ}$ & 993199 & & & 1.80E-02& 1.82E-02 & 0.97\
28 & 84 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D_{3/2}^{\circ}$ & 994242 & & & 1.73E-11& 1.74E-11 & 0.06 + 0.21 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$ + 0.20 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}D^{\circ}$\
28 & 85 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F_{5/2}^{\circ}$ & 995077 & & & 9.27E-11& 9.17E-11 & 0.29 + 0.21 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
28 & 86 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H_{9/2}^{\circ}$ & 996179 & & & 9.27E-10& 9.29E-10 & 0.40 + 0.32 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}H^{\circ}$\
28 & 87 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H_{11/2}^{\circ}$ & 999916 & & & 2.51E-09& 2.46E-09 & 0.25 + 0.38 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
28 & 88 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D_{5/2}^{\circ}$ & 1004705 & & & 5.53E-11& 5.46E-11 & 0.20 + 0.23 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}D^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
28 & 89 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G_{5/2}^{\circ}$ & 1008570 & & & 1.51E-10& 1.50E-10 & 0.44 + 0.27 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
28 & 90 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G_{7/2}^{\circ}$ & 1008894 & & & 1.60E-10& 1.59E-10 & 0.47 + 0.23 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G^{\circ}$\
28 & 91 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G_{9/2}^{\circ}$ & 1014174 & & & 1.74E-10& 1.73E-10 & 0.36 + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$\
28 & 92 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G_{7/2}^{\circ}$ & 1016507 & & & 1.54E-10& 1.52E-10 & 0.61 + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.05 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
28 & 93 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G_{9/2}^{\circ}$ & 1018573 & & & 1.44E-10& 1.43E-10 & 0.59 + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$\
28 & 94 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G_{11/2}^{\circ}$ & 1024070 & & & 6.02E-10& 5.99E-10 & 0.48 + 0.15 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H^{\circ}$\
28 & 95 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F_{5/2}^{\circ}$ & 1024785 & & & 1.89E-11& 1.88E-11 & 0.41 + 0.30 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.06 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F^{\circ}$\
28 & 96 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F_{7/2}^{\circ}$ & 1025870 & & & 1.96E-11& 1.94E-11 & 0.40 + 0.30 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.06 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F^{\circ}$\
28 & 97 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F_{3/2}^{\circ}$ & 1026424 & & & 1.83E-11& 1.81E-11 & 0.45 + 0.34 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.07 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F^{\circ}$\
28 & 98 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F_{9/2}^{\circ}$ & 1029756 & & & 2.05E-11& 2.03E-11 & 0.41 + 0.31 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.06 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F^{\circ}$\
28 & 99 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{2}P_{3/2}^{\circ}$ & 1035915 & & & 2.78E-11& 2.78E-11 & 0.18 + 0.19 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}P^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P^{\circ}$\
28 & 100 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D_{1/2}^{\circ}$ & 1037526 & & & 2.51E-11& 2.51E-11 & 0.43 + 0.18 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.17 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
28 & 101 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}I_{11/2}^{\circ}$ & 1038912 & & & 1.16E-09& 1.15E-09 & 0.88 + 0.04 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}H^{\circ}$ + 0.03 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H^{\circ}$\
28 & 102 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D_{3/2}^{\circ}$ & 1040305 & & & 3.98E-11& 3.98E-11 & 0.27 + 0.16 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
28 & 103 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}S_{1/2}^{\circ}$ & 1040856 & & & 8.02E-11& 8.08E-11 & 0.33 + 0.26 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}S^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{2}P^{\circ}$\
28 & 104 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}I_{13/2}^{\circ}$ & 1041858 & & & 1.98E-02& 1.98E-02 & 0.97\
28 & 105 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D_{5/2}^{\circ}$ & 1043735 & & & 5.96E-11& 5.93E-11 & 0.04 + 0.19 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
28 & 106 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F_{3/2}^{\circ}$ & 1047364 & & & 4.06E-11& 4.03E-11 & 0.09 + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D^{\circ}$\
28 & 107 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F_{5/2}^{\circ}$ & 1048083 & & & 5.78E-11& 5.70E-11 & 0.29 + 0.11 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}F^{\circ}$\
28 & 108 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D_{7/2}^{\circ}$ & 1048148 & & & 4.64E-11& 4.60E-11 & 0.25 + 0.23 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
28 & 109 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D_{5/2}^{\circ}$ & 1049175 & & & 4.03E-11& 4.01E-11 & 0.05 + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
28 & 110 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}S)~^{4}S_{3/2}^{\circ}$ & 1052156 & & & 1.50E-10& 1.51E-10 & 0.43 + 0.24 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}S^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$\
28 & 111 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F_{3/2}^{\circ}$ & 1053173 & & & 9.91E-11& 9.86E-11 & 0.24 + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D^{\circ}$\
28 & 112 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F_{5/2}^{\circ}$ & 1054771 & & & 4.72E-11& 4.71E-11 & 0.28 + 0.23 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
28 & 113 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P_{1/2}^{\circ}$ & 1055512 & & & 2.32E-11& 2.31E-11 & 0.20 + 0.17 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{2}P^{\circ}$ + 0.17 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}S^{\circ}$\
28 & 114 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F_{7/2}^{\circ}$ & 1055634 & & & 1.13E-10& 1.12E-10 & 0.19 + 0.18 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$\
28 & 115 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F_{9/2}^{\circ}$ & 1058542 & & & 1.22E-10& 1.21E-10 & 0.28 + 0.21 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G^{\circ}$\
28 & 116 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F_{7/2}^{\circ}$ & 1059401 & & & 9.97E-11& 9.94E-11 & 0.12 + 0.23 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}G^{\circ}$\
28 & 117 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D_{7/2}^{\circ}$ & 1062075 & & & 3.44E-11& 3.45E-11 & 0.16 + 0.15 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
28 & 118 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D_{1/2}^{\circ}$ & 1066990 & & & 4.76E-11& 4.76E-11 & 0.40 + 0.21 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
28 & 119 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D_{3/2}^{\circ}$ & 1068428 & & & 3.60E-11& 3.62E-11 & 0.26 + 0.20 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.17 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
28 & 120 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P_{5/2}^{\circ}$ & 1069527 & & & 2.36E-11& 2.37E-11 & 0.48 + 0.11 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
28 & 121 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G_{9/2}^{\circ}$ & 1070797 & & & 1.03E-10& 1.03E-10 & 0.18 + 0.22 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.21 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
28 & 122 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G_{11/2}^{\circ}$ & 1074216 & & & 1.49E-11& 1.48E-11 & 0.36 + 0.24 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G^{\circ}$ + 0.24 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
28 & 123 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}F_{5/2}^{\circ}$ & 1074547 & & & 1.86E-11& 1.86E-11 & 0.18 + 0.18 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
28 & 124 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G_{9/2}^{\circ}$ & 1075347 & & & 1.61E-11& 1.61E-11 & 0.23 + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G^{\circ}$ + 0.18 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
28 & 125 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P_{3/2}^{\circ}$ & 1076190 & & & 1.03E-11& 1.04E-11 & 0.50 + 0.11 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
28 & 126 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P_{1/2}^{\circ}$ & 1076538 & & & 7.59E-12& 7.64E-12 & 0.57 + 0.18 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.10 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}P^{\circ}$\
28 & 127 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G_{7/2}^{\circ}$ & 1078489 & & & 1.92E-11& 1.92E-11 & 0.20 + 0.15 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
28 & 128 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D_{5/2}^{\circ}$ & 1080105 & & & 1.69E-11& 1.70E-11 & 0.12 + 0.13 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
28 & 129 & $3s~^{2}S\,3p^{3}(^{2}D)~^{3}D\,3d^{3}(^{2}H)~^{4}G_{7/2}^{\circ}$ & 1080487 & & & 2.01E-11& 2.01E-11 & 0.00 + 0.16 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.15 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
28 & 130 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G_{5/2}^{\circ}$ & 1081445 & & & 1.58E-11& 1.58E-11 & 0.24 + 0.17 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$ + 0.15 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
28 & 131 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P_{1/2}^{\circ}$ & 1085743 & & & 1.33E-11& 1.33E-11 & 0.12 + 0.29 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}P^{\circ}$\
28 & 132 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P_{3/2}^{\circ}$ & 1086535 & & & 1.39E-11& 1.40E-11 & 0.42 + 0.22 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
28 & 133 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P_{5/2}^{\circ}$ & 1088052 & & & 9.74E-12& 9.82E-12 & 0.38 + 0.29 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
28 & 134 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}G_{9/2}^{\circ}$ & 1088086 & & & 1.66E-11& 1.65E-11 & 0.70 + 0.08 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.05 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
28 & 135 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}P_{1/2}^{\circ}$ & 1090258 & & & 1.85E-11& 1.85E-11 & 0.29 + 0.18 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
28 & 136 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}F_{7/2}^{\circ}$ & 1090378 & & & 1.97E-11& 1.98E-11 & 0.07 + 0.41 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}G^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
28 & 137 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}G_{7/2}^{\circ}$ & 1096353 & & & 1.43E-11& 1.43E-11 & 0.44 + 0.09 $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}F^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
28 & 138 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}P_{3/2}^{\circ}$ & 1096998 & & & 9.70E-12& 9.64E-12 & 0.34 + 0.21 $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}P^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}P^{\circ}$\
28 & 139 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}H_{11/2}^{\circ}$ & 1102302 & & & 1.63E-10& 1.62E-10 & 0.51 + 0.30 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H^{\circ}$ + 0.05 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$\
28 & 140 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}H_{9/2}^{\circ}$ & 1105506 & & & 1.86E-11& 1.86E-11 & 0.63 + 0.24 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H^{\circ}$ + 0.03 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}G^{\circ}$\
28 & 141 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D_{7/2}^{\circ}$ & 1108422 & & & 6.48E-12& 6.54E-12 & 0.47 + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.15 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
28 & 142 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D_{5/2}^{\circ}$ & 1110711 & & & 1.24E-11& 1.25E-11 & 0.20 + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}S)~^{2}D^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}D^{\circ}$\
28 & 143 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H_{9/2}^{\circ}$ & 1110718 & & & 3.33E-11& 3.33E-11 & 0.53 + 0.22 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}H^{\circ}$\
28 & 144 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D_{5/2}^{\circ}$ & 1113389 & & & 8.25E-12& 8.32E-12 & 0.20 + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.15 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
28 & 145 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D_{3/2}^{\circ}$ & 1114327 & & & 6.25E-12& 6.31E-12 & 0.38 + 0.25 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.17 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
28 & 146 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D_{1/2}^{\circ}$ & 1114979 & & & 6.46E-12& 6.53E-12 & 0.42 + 0.25 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
28 & 147 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}S)~^{2}D_{3/2}^{\circ}$ & 1115454 & & & 5.52E-11& 5.56E-11 & 0.38 + 0.15 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}D^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D^{\circ}$\
28 & 148 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F_{7/2}^{\circ}$ & 1118655 & & & 6.73E-12& 6.80E-12 & 0.05 + 0.21 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.19 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
28 & 149 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H_{11/2}^{\circ}$ & 1118689 & & & 1.20E-11& 1.19E-11 & 0.41 + 0.26 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H^{\circ}$ + 0.23 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}H^{\circ}$\
28 & 150 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D_{5/2}^{\circ}$ & 1119632 & & & 6.31E-12& 6.39E-12 & 0.17 + 0.20 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.15 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D^{\circ}$\
28 & 151 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F_{9/2}^{\circ}$ & 1122027 & & & 4.98E-12& 5.03E-12 & 0.36 + 0.24 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.17 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
28 & 152 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D_{7/2}^{\circ}$ & 1122339 & & & 6.98E-12& 7.02E-12 & 0.10 + 0.10 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
28 & 153 & $3s~^{2}S\,3p^{3}(^{2}P)~^{1}P\,3d^{3}(^{4}F)~^{4}D_{3/2}^{\circ}$ & 1122872 & & & 5.90E-12& 5.99E-12 & 0.00 + 0.21 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.19 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D^{\circ}$\
28 & 154 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D_{1/2}^{\circ}$ & 1126015 & & & 5.61E-12& 5.69E-12 & 0.06 + 0.24 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.23 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D^{\circ}$\
28 & 155 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F_{5/2}^{\circ}$ & 1128040 & & & 7.63E-12& 7.66E-12 & 0.13 + 0.15 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}F^{\circ}$\
28 & 156 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F_{7/2}^{\circ}$ & 1129740 & & & 5.11E-12& 5.17E-12 & 0.30 + 0.23 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
28 & 157 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D_{5/2}^{\circ}$ & 1135535 & & & 6.79E-12& 6.87E-12 & 0.30 + 0.15 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D^{\circ}$\
28 & 158 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D_{3/2}^{\circ}$ & 1136224 & & & 6.17E-12& 6.23E-12 & 0.17 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
28 & 159 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F_{5/2}^{\circ}$ & 1137025 & & & 5.34E-12& 5.39E-12 & 0.26 + 0.23 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
28 & 160 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}G_{7/2}^{\circ}$ & 1137839 & & & 1.47E-11& 1.47E-11 & 0.55 + 0.13 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G^{\circ}$\
28 & 161 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}P_{1/2}^{\circ}$ & 1138327 & & & 5.70E-12& 5.80E-12 & 0.19 + 0.22 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}S^{\circ}$ + 0.17 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{2}P^{\circ}$\
28 & 162 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F_{3/2}^{\circ}$ & 1139894 & & & 5.43E-12& 5.49E-12 & 0.19 + 0.18 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
28 & 163 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D_{3/2}^{\circ}$ & 1143608 & & & 6.29E-12& 6.37E-12 & 0.03 + 0.23 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}P^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{2}P^{\circ}$\
28 & 164 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}S_{1/2}^{\circ}$ & 1146423 & & & 5.66E-12& 5.75E-12 & 0.25 + 0.20 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}S^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
28 & 165 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}G_{9/2}^{\circ}$ & 1147163 & & & 1.51E-11& 1.51E-11 & 0.52 + 0.20 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}G^{\circ}$\
28 & 166 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D_{3/2}^{\circ}$ & 1148111 & & & 8.41E-12& 8.47E-12 & 0.29 + 0.21 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}S)~^{2}P^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}D^{\circ}$\
28 & 167 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F_{5/2}^{\circ}$ & 1149261 & & & 1.10E-11& 1.11E-11 & 0.39 + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
28 & 168 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}S)~^{2}P_{1/2}^{\circ}$ & 1151057 & & & 8.26E-11& 8.36E-11 & 0.54 + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}P^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P^{\circ}$\
28 & 169 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P_{3/2}^{\circ}$ & 1152737 & & & 6.09E-12& 6.18E-12 & 0.23 + 0.20 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}S^{\circ}$\
28 & 170 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F_{7/2}^{\circ}$ & 1153058 & & & 6.82E-12& 6.89E-12 & 0.24 + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
28 & 171 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F_{5/2}^{\circ}$ & 1153479 & & & 6.52E-12& 6.56E-12 & 0.07 + 0.11 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$\
28 & 172 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P_{5/2}^{\circ}$ & 1156893 & & & 6.02E-12& 6.10E-12 & 0.29 + 0.25 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
28 & 173 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P_{1/2}^{\circ}$ & 1159268 & & & 5.59E-12& 5.67E-12 & 0.38 + 0.26 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}S^{\circ}$\
28 & 174 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}D_{5/2}^{\circ}$ & 1159657 & & & 6.48E-12& 6.54E-12 & 0.12 + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D^{\circ}$\
28 & 175 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}S)~^{2}P_{3/2}^{\circ}$ & 1161801 & & & 8.90E-12& 9.03E-12 & 0.17 + 0.15 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P^{\circ}$\
28 & 176 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}S_{3/2}^{\circ}$ & 1168605 & & & 5.72E-12& 5.84E-12 & 0.47 + 0.27 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}S)~^{4}S^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
28 & 177 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F_{7/2}^{\circ}$ & 1171543 & & & 1.44E-11& 1.45E-11 & 0.38 + 0.25 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}F^{\circ}$\
28 & 178 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}S)~^{2}D_{5/2}^{\circ}$ & 1176435 & & & 5.74E-12& 5.82E-12 & 0.30 + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}D^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D^{\circ}$\
28 & 179 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}D_{3/2}^{\circ}$ & 1178131 & & & 5.86E-12& 5.92E-12 & 0.13 + 0.34 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}S)~^{2}D^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}D^{\circ}$\
28 & 180 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G_{7/2}^{\circ}$ & 1181215 & & & 5.26E-12& 5.32E-12 & 0.40 + 0.32 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G^{\circ}$\
28 & 181 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G_{9/2}^{\circ}$ & 1182444 & & & 5.25E-12& 5.32E-12 & 0.45 + 0.31 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G^{\circ}$\
28 & 182 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}S_{5/2}$ & 1200732 & & & 8.07E-10& 7.83E-10 & 0.90 + 0.07 $3s^{2}\,3p^{2}(^{3}P)~^{3}P\,3d^{3}(^{4}P)~^{6}S$\
28 & 183 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}F_{5/2}^{\circ}$ & 1202566 & & & 5.13E-12& 5.20E-12 & 0.29 + 0.27 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
28 & 184 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}F_{7/2}^{\circ}$ & 1204376 & & & 5.32E-12& 5.39E-12 & 0.26 + 0.26 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
28 & 185 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D_{1/2}$ & 1206858 & & & 5.02E-10& 4.93E-10 & 0.62 + 0.25 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}D$ + 0.06 $3s^{2}\,3p^{2}(^{3}P)~^{3}P\,3d^{3}(^{4}F)~^{6}D$\
28 & 186 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D_{3/2}$ & 1207098 & & & 5.02E-10& 4.91E-10 & 0.59 + 0.25 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}D$ + 0.05 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F$\
28 & 187 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D_{5/2}$ & 1207583 & & & 5.05E-10& 4.93E-10 & 0.53 + 0.26 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}D$ + 0.10 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F$\
28 & 188 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D_{7/2}$ & 1208307 & & & 5.12E-10& 4.99E-10 & 0.48 + 0.29 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}D$ + 0.14 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F$\
28 & 189 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D_{9/2}$ & 1209430 & & & 5.31E-10& 5.17E-10 & 0.44 + 0.35 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}D$ + 0.12 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F$\
28 & 190 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}D_{3/2}^{\circ}$ & 1212288 & & & 6.30E-12& 6.37E-12 & 0.25 + 0.22 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D^{\circ}$ + 0.15 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D^{\circ}$\
28 & 191 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}D_{5/2}^{\circ}$ & 1216239 & & & 6.30E-12& 6.36E-12 & 0.31 + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D^{\circ}$\
28 & 192 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F_{11/2}$ & 1222325 & & & 3.30E-10& 3.20E-10 & 0.87 + 0.06 $3s^{2}\,3p^{2}(^{3}P)~^{3}P\,3d^{3}(^{4}F)~^{6}F$ + 0.04 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}G$\
29 & 1 & $3s^{2}\,3p^{5}~^{2}P_{3/2}^{\circ}$ & 0 & 0 & 0 & & & 0.96\
29 & 2 & $3s^{2}\,3p^{5}~^{2}P_{1/2}^{\circ}$ & 28599 & 28560 & 39 & 2.38E-03& 2.38E-03 & 0.96\
29 & 3 & $3s~^{2}S\,3p^{6}~^{2}S_{1/2}$ & 367983 & & & 1.48E-10& 1.41E-10 & 0.72 + 0.25 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}S$\
29 & 4 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D_{5/2}$ & 483611 & & & 7.17E-08& 8.56E-08 & 0.89 + 0.02 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F$\
29 & 5 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D_{7/2}$ & 484537 & & & 8.38E-03& 8.38E-03 & 0.92 + 0.04 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F$\
29 & 6 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D_{3/2}$ & 485578 & & & 5.64E-08& 5.47E-08 & 0.87 + 0.03 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P$ + 0.02 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$\
29 & 7 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D_{1/2}$ & 488926 & & & 3.13E-08& 2.86E-08 & 0.88 + 0.04 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P$ + 0.03 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P$\
29 & 8 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F_{9/2}$ & 517229 & & & 3.01E-02& 3.01E-02 & 0.88 + 0.09 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}G$\
29 & 9 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P_{1/2}$ & 521843 & & & 3.80E-09& 3.78E-09 & 0.46 + 0.39 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P$ + 0.09 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D$\
29 & 10 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F_{7/2}$ & 525478 & & & 1.80E-02& 1.80E-02 & 0.74 + 0.11 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F$ + 0.10 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}G$\
29 & 11 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F_{3/2}$ & 533713 & & & 2.50E-09& 2.55E-09 & 0.36 + 0.17 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P$ + 0.13 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P$\
29 & 12 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F_{5/2}$ & 533818 & & & 5.90E-09& 6.02E-09 & 0.92\
29 & 13 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P_{3/2}$ & 535200 & & & 3.45E-09& 3.45E-09 & 0.24 + 0.26 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F$ + 0.18 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P$\
29 & 14 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P_{1/2}$ & 539178 & & & 9.05E-10& 8.86E-10 & 0.92\
29 & 15 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D_{3/2}$ & 540688 & & & 1.37E-09& 1.41E-09 & 0.31 + 0.30 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F$ + 0.18 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D$\
29 & 16 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F_{7/2}$ & 547435 & & & 3.49E-03& 3.49E-03 & 0.42 + 0.25 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}G$ + 0.18 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F$\
29 & 17 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P_{3/2}$ & 550433 & & & 4.47E-08& 4.45E-08 & 0.62 + 0.12 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P$ + 0.12 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P$\
29 & 18 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P_{5/2}$ & 551424 & & & 4.62E-09& 4.67E-09 & 0.56 + 0.12 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$ + 0.12 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D$\
29 & 19 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D_{5/2}$ & 556863 & & & 7.81E-09& 7.78E-09 & 0.36 + 0.34 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P$ + 0.15 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D$\
29 & 20 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}G_{7/2}$ & 563580 & & & 3.16E-03& 3.15E-03 & 0.60 + 0.25 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F$ + 0.11 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}F$\
29 & 21 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}G_{9/2}$ & 564294 & & & 2.90E-03& 2.90E-03 & 0.88 + 0.09 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F$\
29 & 22 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F_{5/2}$ & 568949 & & & 1.06E-08& 1.07E-08 & 0.66 + 0.23 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}F$ + 0.05 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$\
29 & 23 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}F_{5/2}$ & 598882 & & & 2.17E-09& 2.18E-09 & 0.72 + 0.17 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F$ + 0.06 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$\
29 & 24 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}F_{7/2}$ & 606457 & & & 1.20E-03& 1.20E-03 & 0.76 + 0.18 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F$\
29 & 25 & $3s^{2}\,3p^{4}(^{1}S)~^{1}S\,3d~^{2}D_{3/2}$ & 635883 & & & 2.93E-10& 2.95E-10 & 0.64 + 0.25 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$ + 0.03 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F$\
29 & 26 & $3s^{2}\,3p^{4}(^{1}S)~^{1}S\,3d~^{2}D_{5/2}$ & 645927 & & & 1.61E-09& 1.62E-09 & 0.72 + 0.14 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$ + 0.04 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F$\
29 & 27 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}S_{1/2}$ & 666411 & 663840 & 2571 & 4.87E-12& 4.82E-12 & 0.69 + 0.23 $3s~^{2}S\,3p^{6}~^{2}S$ + 0.03 $3s~^{2}S\,3p^{4}(^{3}P)~^{2}P\,3d^{2}(^{3}P)~^{2}S$\
29 & 28 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P_{3/2}$ & 693761 & 690990 & 2771 & 5.39E-12& 5.42E-12 & 0.49 + 0.37 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P$ + 0.05 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D$\
29 & 29 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D_{5/2}$ & 702068 & 699480 & 2588 & 4.53E-12& 4.56E-12 & 0.66 + 0.21 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$ + 0.08 $3s^{2}\,3p^{4}(^{1}S)~^{1}S\,3d~^{2}D$\
29 & 30 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P_{1/2}$ & 704445 & 701500 & 2945 & 5.58E-12& 5.60E-12 & 0.51 + 0.43 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P$\
29 & 31 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D_{3/2}$ & 726875 & 724240 & 2635 & 4.65E-12& 4.64E-12 & 0.58 + 0.15 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$ + 0.14 $3s^{2}\,3p^{4}(^{1}S)~^{1}S\,3d~^{2}D$\
29 & 32 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}P_{1/2}^{\circ}$ & 828012 & & & 2.25E-10& 2.27E-10 & 0.82 + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
29 & 33 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}P_{3/2}^{\circ}$ & 832544 & & & 2.23E-10& 2.25E-10 & 0.81 + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
29 & 34 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}P_{5/2}^{\circ}$ & 840856 & & & 2.20E-10& 2.23E-10 & 0.79 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
29 & 35 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F_{9/2}^{\circ}$ & 865875 & & & 1.27E-10& 1.25E-10 & 0.85 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
29 & 36 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F_{7/2}^{\circ}$ & 869844 & & & 1.25E-10& 1.24E-10 & 0.80 + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.03 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D^{\circ}$\
29 & 37 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F_{5/2}^{\circ}$ & 875005 & & & 1.22E-10& 1.21E-10 & 0.78 + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.04 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D^{\circ}$\
29 & 38 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F_{3/2}^{\circ}$ & 880348 & & & 1.19E-10& 1.18E-10 & 0.79 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.04 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D^{\circ}$\
29 & 39 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D_{7/2}^{\circ}$ & 904776 & & & 1.34E-10& 1.34E-10 & 0.71 + 0.08 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}F^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
29 & 40 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D_{1/2}^{\circ}$ & 906481 & & & 1.26E-10& 1.27E-10 & 0.76 + 0.08 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
29 & 41 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D_{5/2}^{\circ}$ & 908539 & & & 1.27E-10& 1.28E-10 & 0.70 + 0.06 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
29 & 42 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D_{3/2}^{\circ}$ & 908755 & & & 1.27E-10& 1.27E-10 & 0.74 + 0.07 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
29 & 43 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}F_{7/2}^{\circ}$ & 914882 & & & 2.44E-10& 2.46E-10 & 0.65 + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$ + 0.06 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D^{\circ}$\
29 & 44 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}D_{5/2}^{\circ}$ & 920826 & & & 2.38E-10& 2.38E-10 & 0.50 + 0.24 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}F^{\circ}$ + 0.05 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D^{\circ}$\
29 & 45 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}D_{3/2}^{\circ}$ & 927000 & & & 1.39E-10& 1.39E-10 & 0.66 + 0.06 $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}P^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D^{\circ}$\
29 & 46 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}F_{5/2}^{\circ}$ & 935848 & & & 1.80E-10& 1.81E-10 & 0.51 + 0.23 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}D^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
29 & 47 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}P_{1/2}^{\circ}$ & 940501 & & & 6.28E-11& 6.30E-11 & 0.39 + 0.25 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}P^{\circ}$ + 0.15 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P^{\circ}$\
29 & 48 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}P_{3/2}^{\circ}$ & 955900 & & & 7.44E-11& 7.43E-11 & 0.36 + 0.18 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}P^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P^{\circ}$\
29 & 49 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{1/2}^{\circ}$ & 968376 & & & 2.21E-09& 2.18E-09 & 0.95\
29 & 50 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{3/2}^{\circ}$ & 968564 & & & 2.66E-09& 2.64E-09 & 0.95\
29 & 51 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{5/2}^{\circ}$ & 969056 & & & 3.30E-09& 3.30E-09 & 0.94\
29 & 52 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{7/2}^{\circ}$ & 969976 & & & 4.80E-09& 4.82E-09 & 0.94\
29 & 53 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{9/2}^{\circ}$ & 971371 & & & 8.00E-09& 8.04E-09 & 0.95\
29 & 54 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{11/2}^{\circ}$ & 973071 & & & 4.49E-08& 4.74E-08 & 0.95 + 0.02 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
29 & 55 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}F_{5/2}^{\circ}$ & 975437 & & & 6.04E-11& 6.05E-11 & 0.57 + 0.17 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
29 & 56 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}F_{7/2}^{\circ}$ & 982865 & & & 5.78E-11& 5.80E-11 & 0.56 + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
29 & 57 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{6}P_{3/2}^{\circ}$ & 998400 & & & 2.57E-09& 2.60E-09 & 0.93\
29 & 58 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{6}P_{5/2}^{\circ}$ & 999061 & & & 2.04E-09& 2.05E-09 & 0.93\
29 & 59 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{6}P_{7/2}^{\circ}$ & 999321 & & & 1.45E-09& 1.46E-09 & 0.89 + 0.03 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D^{\circ}$ + 0.03 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
29 & 60 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}D_{3/2}^{\circ}$ & 1008208 & & & 5.34E-11& 5.34E-11 & 0.55 + 0.17 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$\
29 & 61 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}D_{5/2}^{\circ}$ & 1011763 & & & 5.52E-11& 5.51E-11 & 0.57 + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$\
29 & 62 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P_{1/2}^{\circ}$ & 1021406 & & & 1.26E-09& 1.29E-09 & 0.38 + 0.31 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$\
29 & 63 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D_{3/2}^{\circ}$ & 1021482 & & & 2.22E-10& 2.22E-10 & 0.21 + 0.18 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
29 & 64 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D_{5/2}^{\circ}$ & 1023004 & & & 2.61E-10& 2.62E-10 & 0.29 + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
29 & 65 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D_{7/2}^{\circ}$ & 1023319 & & & 2.40E-10& 2.41E-10 & 0.49 + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{6}P^{\circ}$\
29 & 66 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G_{5/2}^{\circ}$ & 1026650 & & & 4.50E-10& 4.52E-10 & 0.38 + 0.21 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$\
29 & 67 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P_{3/2}^{\circ}$ & 1027679 & & & 8.26E-10& 8.33E-10 & 0.08 + 0.23 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.18 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
29 & 68 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D_{1/2}^{\circ}$ & 1028290 & & & 1.53E-10& 1.54E-10 & 0.57 + 0.23 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.05 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
29 & 69 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G_{7/2}^{\circ}$ & 1028405 & & & 1.31E-09& 1.32E-09 & 0.54 + 0.18 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$\
29 & 70 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D_{3/2}^{\circ}$ & 1031242 & & & 4.02E-10& 4.04E-10 & 0.07 + 0.19 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D^{\circ}$ + 0.18 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
29 & 71 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G_{9/2}^{\circ}$ & 1033180 & & & 2.52E-09& 2.56E-09 & 0.60 + 0.25 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.03 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$\
29 & 72 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P_{5/2}^{\circ}$ & 1033423 & & & 2.39E-09& 2.45E-09 & 0.09 + 0.27 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.21 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
29 & 73 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F_{5/2}^{\circ}$ & 1037155 & & & 1.55E-09& 1.57E-09 & 0.08 + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
29 & 74 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G_{11/2}^{\circ}$ & 1037384 & & & 3.57E-09& 3.55E-09 & 0.62 + 0.34 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$\
29 & 75 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F_{7/2}^{\circ}$ & 1041308 & & & 1.37E-09& 1.39E-09 & 0.35 + 0.19 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
29 & 76 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F_{9/2}^{\circ}$ & 1045854 & & & 1.06E-09& 1.06E-09 & 0.40 + 0.21 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
29 & 77 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H_{7/2}^{\circ}$ & 1048958 & & & 6.71E-10& 6.73E-10 & 0.66 + 0.07 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
29 & 78 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H_{9/2}^{\circ}$ & 1052399 & & & 8.04E-09& 8.08E-09 & 0.72 + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$\
29 & 79 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H_{11/2}^{\circ}$ & 1055531 & & & 2.42E-07& 2.06E-07 & 0.79 + 0.12 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$ + 0.02 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H^{\circ}$\
29 & 80 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F_{7/2}^{\circ}$ & 1058629 & & & 1.57E-10& 1.56E-10 & 0.26 + 0.20 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.17 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H^{\circ}$\
29 & 81 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}P_{1/2}^{\circ}$ & 1059109 & & & 5.88E-12& 5.96E-12 & 0.52 + 0.28 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}P^{\circ}$ + 0.06 $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}P^{\circ}$\
29 & 82 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}P_{3/2}^{\circ}$ & 1059849 & & & 7.68E-12& 7.77E-12 & 0.41 + 0.17 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}P^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}D^{\circ}$\
29 & 83 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H_{13/2}^{\circ}$ & 1062245 & & & 1.35E-02& 1.37E-02 & 0.98\
29 & 84 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D_{3/2}^{\circ}$ & 1062306 & & & 1.60E-11& 1.62E-11 & 0.05 + 0.21 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$ + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}D^{\circ}$\
29 & 85 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F_{5/2}^{\circ}$ & 1063495 & & & 8.35E-11& 8.37E-11 & 0.28 + 0.21 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
29 & 86 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H_{9/2}^{\circ}$ & 1064029 & & & 8.11E-10& 8.20E-10 & 0.39 + 0.31 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}H^{\circ}$\
29 & 87 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H_{11/2}^{\circ}$ & 1068683 & & & 2.43E-09& 2.41E-09 & 0.24 + 0.38 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H^{\circ}$\
29 & 88 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D_{5/2}^{\circ}$ & 1074523 & & & 5.39E-11& 5.38E-11 & 0.18 + 0.21 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}D^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
29 & 89 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G_{7/2}^{\circ}$ & 1077898 & & & 1.22E-10& 1.23E-10 & 0.43 + 0.21 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G^{\circ}$\
29 & 90 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G_{5/2}^{\circ}$ & 1078160 & & & 9.96E-11& 9.98E-11 & 0.35 + 0.23 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
29 & 91 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G_{9/2}^{\circ}$ & 1084474 & & & 1.39E-10& 1.39E-10 & 0.30 + 0.25 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$\
29 & 92 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G_{7/2}^{\circ}$ & 1086501 & & & 1.32E-10& 1.32E-10 & 0.57 + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
29 & 93 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G_{9/2}^{\circ}$ & 1089475 & & & 1.14E-10& 1.14E-10 & 0.53 + 0.11 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$\
29 & 94 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F_{5/2}^{\circ}$ & 1094216 & & & 1.78E-11& 1.78E-11 & 0.40 + 0.28 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.06 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F^{\circ}$\
29 & 95 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F_{7/2}^{\circ}$ & 1095741 & & & 1.86E-11& 1.86E-11 & 0.39 + 0.28 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
29 & 96 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F_{3/2}^{\circ}$ & 1096282 & & & 1.70E-11& 1.70E-11 & 0.44 + 0.33 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.07 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F^{\circ}$\
29 & 97 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G_{11/2}^{\circ}$ & 1097147 & & & 5.09E-10& 5.10E-10 & 0.46 + 0.15 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H^{\circ}$\
29 & 98 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F_{9/2}^{\circ}$ & 1100930 & & & 1.98E-11& 1.97E-11 & 0.39 + 0.30 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
29 & 99 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{2}P_{3/2}^{\circ}$ & 1106161 & & & 2.62E-11& 2.64E-11 & 0.17 + 0.17 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}P^{\circ}$ + 0.15 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P^{\circ}$\
29 & 100 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D_{1/2}^{\circ}$ & 1107002 & & & 2.24E-11& 2.25E-11 & 0.43 + 0.18 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.17 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
29 & 101 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}I_{11/2}^{\circ}$ & 1109676 & & & 7.44E-10& 7.47E-10 & 0.84 + 0.05 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}H^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H^{\circ}$\
29 & 102 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D_{3/2}^{\circ}$ & 1110643 & & & 3.86E-11& 3.89E-11 & 0.24 + 0.15 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
29 & 103 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}S_{1/2}^{\circ}$ & 1111951 & & & 8.20E-11& 8.32E-11 & 0.34 + 0.26 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}S^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{2}P^{\circ}$\
29 & 104 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}I_{13/2}^{\circ}$ & 1113787 & & & 1.38E-02& 1.38E-02 & 0.97\
29 & 105 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D_{5/2}^{\circ}$ & 1115050 & & & 5.90E-11& 5.93E-11 & 0.16 + 0.17 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
29 & 106 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F_{3/2}^{\circ}$ & 1118820 & & & 3.44E-11& 3.45E-11 & 0.10 + 0.15 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
29 & 107 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D_{7/2}^{\circ}$ & 1119891 & & & 4.53E-11& 4.54E-11 & 0.24 + 0.21 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
29 & 108 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F_{5/2}^{\circ}$ & 1119900 & & & 5.41E-11& 5.39E-11 & 0.29 + 0.11 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}F^{\circ}$\
29 & 109 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D_{5/2}^{\circ}$ & 1121188 & & & 3.57E-11& 3.59E-11 & 0.05 + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}D^{\circ}$\
29 & 110 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}S)~^{4}S_{3/2}^{\circ}$ & 1124613 & & & 9.82E-11& 9.89E-11 & 0.26 + 0.17 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$ + 0.15 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}S^{\circ}$\
29 & 111 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P_{3/2}^{\circ}$ & 1125710 & & & 1.06E-10& 1.07E-10 & 0.03 + 0.23 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}S)~^{4}S^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}S^{\circ}$\
29 & 112 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F_{5/2}^{\circ}$ & 1126909 & & & 3.92E-11& 3.94E-11 & 0.24 + 0.19 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
29 & 113 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F_{7/2}^{\circ}$ & 1128195 & & & 9.19E-11& 9.22E-11 & 0.16 + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$\
29 & 114 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P_{1/2}^{\circ}$ & 1129219 & & & 1.91E-11& 1.92E-11 & 0.21 + 0.17 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{2}P^{\circ}$ + 0.15 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}S^{\circ}$\
29 & 115 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F_{9/2}^{\circ}$ & 1131124 & & & 9.81E-11& 9.86E-11 & 0.28 + 0.20 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G^{\circ}$\
29 & 116 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G_{7/2}^{\circ}$ & 1132901 & & & 1.79E-10& 1.80E-10 & 0.12 + 0.23 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}G^{\circ}$\
29 & 117 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D_{7/2}^{\circ}$ & 1136009 & & & 2.50E-11& 2.52E-11 & 0.21 + 0.17 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.15 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
29 & 118 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D_{1/2}^{\circ}$ & 1138918 & & & 4.24E-11& 4.27E-11 & 0.40 + 0.20 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
29 & 119 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D_{3/2}^{\circ}$ & 1140639 & & & 3.15E-11& 3.19E-11 & 0.25 + 0.21 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.18 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
29 & 120 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P_{5/2}^{\circ}$ & 1142229 & & & 2.05E-11& 2.08E-11 & 0.48 + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
29 & 121 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G_{9/2}^{\circ}$ & 1145353 & & & 3.33E-11& 3.34E-11 & 0.18 + 0.20 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$\
29 & 122 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G_{11/2}^{\circ}$ & 1146946 & & & 1.40E-11& 1.40E-11 & 0.35 + 0.24 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$ + 0.24 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
29 & 123 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}F_{5/2}^{\circ}$ & 1147494 & & & 1.65E-11& 1.67E-11 & 0.20 + 0.15 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
29 & 124 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G_{9/2}^{\circ}$ & 1148325 & & & 2.12E-11& 2.13E-11 & 0.03 + 0.17 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}G^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}G^{\circ}$\
29 & 125 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P_{1/2}^{\circ}$ & 1149343 & & & 6.56E-12& 6.64E-12 & 0.51 + 0.22 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.11 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}P^{\circ}$\
29 & 126 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P_{3/2}^{\circ}$ & 1149366 & & & 8.67E-12& 8.78E-12 & 0.43 + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$\
29 & 127 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G_{7/2}^{\circ}$ & 1152153 & & & 1.57E-11& 1.58E-11 & 0.28 + 0.19 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$ + 0.17 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
29 & 128 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G_{5/2}^{\circ}$ & 1154150 & & & 1.43E-11& 1.44E-11 & 0.22 + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G^{\circ}$ + 0.18 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
29 & 129 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}F_{7/2}^{\circ}$ & 1155257 & & & 2.16E-11& 2.17E-11 & 0.11 + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
29 & 130 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D_{5/2}^{\circ}$ & 1155666 & & & 1.43E-11& 1.45E-11 & 0.14 + 0.14 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
29 & 131 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P_{1/2}^{\circ}$ & 1159522 & & & 1.33E-11& 1.34E-11 & 0.09 + 0.24 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.22 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}P^{\circ}$\
29 & 132 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P_{3/2}^{\circ}$ & 1161013 & & & 1.63E-11& 1.66E-11 & 0.40 + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
29 & 133 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P_{5/2}^{\circ}$ & 1162805 & & & 1.08E-11& 1.10E-11 & 0.36 + 0.28 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
29 & 134 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}G_{9/2}^{\circ}$ & 1162838 & & & 1.57E-11& 1.58E-11 & 0.67 + 0.10 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
29 & 135 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}P_{1/2}^{\circ}$ & 1165776 & & & 2.34E-11& 2.36E-11 & 0.25 + 0.22 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.15 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
29 & 136 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}G_{7/2}^{\circ}$ & 1165805 & & & 1.82E-11& 1.84E-11 & 0.50 + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.05 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
29 & 137 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}P_{3/2}^{\circ}$ & 1172025 & & & 9.13E-12& 9.16E-12 & 0.34 + 0.21 $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}P^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}P^{\circ}$\
29 & 138 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F_{7/2}^{\circ}$ & 1172501 & & & 1.21E-11& 1.22E-11 & 0.04 + 0.32 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}G^{\circ}$ + 0.11 $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}F^{\circ}$\
29 & 139 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}H_{11/2}^{\circ}$ & 1178629 & & & 1.29E-10& 1.29E-10 & 0.47 + 0.33 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H^{\circ}$\
29 & 140 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}H_{9/2}^{\circ}$ & 1181056 & & & 1.42E-11& 1.43E-11 & 0.51 + 0.31 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H^{\circ}$\
29 & 141 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D_{7/2}^{\circ}$ & 1183070 & & & 6.34E-12& 6.43E-12 & 0.46 + 0.16 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
29 & 142 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D_{5/2}^{\circ}$ & 1186059 & & & 1.04E-11& 1.05E-11 & 0.06 + 0.22 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}D^{\circ}$\
29 & 143 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H_{9/2}^{\circ}$ & 1187055 & & & 4.89E-11& 4.92E-11 & 0.49 + 0.25 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}H^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H^{\circ}$\
29 & 144 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D_{5/2}^{\circ}$ & 1188615 & & & 7.55E-12& 7.66E-12 & 0.23 + 0.13 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}D^{\circ}$\
29 & 145 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D_{3/2}^{\circ}$ & 1189838 & & & 6.00E-12& 6.09E-12 & 0.30 + 0.25 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.20 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
29 & 146 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D_{1/2}^{\circ}$ & 1190698 & & & 6.12E-12& 6.22E-12 & 0.41 + 0.26 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
29 & 147 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}S)~^{2}D_{3/2}^{\circ}$ & 1191663 & & & 3.52E-11& 3.58E-11 & 0.34 + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}D^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D^{\circ}$\
29 & 148 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D_{7/2}^{\circ}$ & 1193795 & & & 6.49E-12& 6.60E-12 & 0.09 + 0.23 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.20 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
29 & 149 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D_{5/2}^{\circ}$ & 1194607 & & & 6.40E-12& 6.52E-12 & 0.23 + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D^{\circ}$\
29 & 150 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F_{9/2}^{\circ}$ & 1196169 & & & 4.74E-12& 4.81E-12 & 0.35 + 0.23 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.17 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
29 & 151 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H_{11/2}^{\circ}$ & 1196317 & & & 1.13E-11& 1.13E-11 & 0.39 + 0.26 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}H^{\circ}$ + 0.23 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H^{\circ}$\
29 & 152 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F_{7/2}^{\circ}$ & 1198031 & & & 6.69E-12& 6.76E-12 & 0.06 + 0.11 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D^{\circ}$\
29 & 153 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D_{3/2}^{\circ}$ & 1198225 & & & 5.78E-12& 5.89E-12 & 0.04 + 0.19 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
29 & 154 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D_{1/2}^{\circ}$ & 1201878 & & & 5.32E-12& 5.42E-12 & 0.06 + 0.24 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.23 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D^{\circ}$\
29 & 155 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F_{5/2}^{\circ}$ & 1204449 & & & 7.16E-12& 7.25E-12 & 0.13 + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}F^{\circ}$\
29 & 156 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F_{7/2}^{\circ}$ & 1205241 & & & 4.88E-12& 4.95E-12 & 0.28 + 0.21 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
29 & 157 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D_{5/2}^{\circ}$ & 1212896 & & & 6.21E-12& 6.31E-12 & 0.24 + 0.13 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D^{\circ}$\
29 & 158 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D_{3/2}^{\circ}$ & 1213186 & & & 5.72E-12& 5.81E-12 & 0.15 + 0.11 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
29 & 159 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F_{5/2}^{\circ}$ & 1214536 & & & 5.32E-12& 5.40E-12 & 0.20 + 0.18 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
29 & 160 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}P_{1/2}^{\circ}$ & 1214700 & & & 5.42E-12& 5.54E-12 & 0.18 + 0.22 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}S^{\circ}$ + 0.17 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{2}P^{\circ}$\
29 & 161 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}G_{7/2}^{\circ}$ & 1215329 & & & 1.37E-11& 1.38E-11 & 0.54 + 0.13 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G^{\circ}$\
29 & 162 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F_{3/2}^{\circ}$ & 1216807 & & & 5.13E-12& 5.22E-12 & 0.15 + 0.15 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}P^{\circ}$\
29 & 163 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F_{3/2}^{\circ}$ & 1221357 & & & 6.03E-12& 6.14E-12 & 0.06 + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}P^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D^{\circ}$\
29 & 164 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}S_{1/2}^{\circ}$ & 1223590 & & & 5.35E-12& 5.46E-12 & 0.23 + 0.18 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}S^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
29 & 165 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D_{3/2}^{\circ}$ & 1225996 & & & 6.97E-12& 7.07E-12 & 0.34 + 0.16 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}S)~^{2}P^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}D^{\circ}$\
29 & 166 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F_{5/2}^{\circ}$ & 1226570 & & & 1.05E-11& 1.07E-11 & 0.37 + 0.15 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
29 & 167 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}G_{9/2}^{\circ}$ & 1226860 & & & 1.39E-11& 1.40E-11 & 0.51 + 0.21 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}G^{\circ}$\
29 & 168 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P_{3/2}^{\circ}$ & 1230159 & & & 5.97E-12& 6.09E-12 & 0.23 + 0.21 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}S^{\circ}$\
29 & 169 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}S)~^{2}P_{1/2}^{\circ}$ & 1230322 & & & 6.14E-11& 6.26E-11 & 0.54 + 0.15 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}P^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P^{\circ}$\
29 & 170 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F_{5/2}^{\circ}$ & 1231135 & & & 6.03E-12& 6.11E-12 & 0.05 + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
29 & 171 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}F_{7/2}^{\circ}$ & 1231757 & & & 6.37E-12& 6.47E-12 & 0.15 + 0.23 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{2}F^{\circ}$\
29 & 172 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P_{5/2}^{\circ}$ & 1235108 & & & 5.69E-12& 5.79E-12 & 0.29 + 0.24 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
29 & 173 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P_{1/2}^{\circ}$ & 1238433 & & & 5.45E-12& 5.56E-12 & 0.35 + 0.23 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}S^{\circ}$\
29 & 174 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}D_{5/2}^{\circ}$ & 1238470 & & & 6.13E-12& 6.22E-12 & 0.13 + 0.17 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D^{\circ}$\
29 & 175 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}S)~^{2}P_{3/2}^{\circ}$ & 1242150 & & & 8.43E-12& 8.61E-12 & 0.18 + 0.15 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{2}P^{\circ}$\
29 & 176 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}S_{3/2}^{\circ}$ & 1248749 & & & 5.76E-12& 5.91E-12 & 0.44 + 0.27 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}S)~^{4}S^{\circ}$ + 0.05 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}P^{\circ}$\
29 & 177 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F_{7/2}^{\circ}$ & 1252774 & & & 1.22E-11& 1.24E-11 & 0.34 + 0.26 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}F^{\circ}$\
29 & 178 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}S)~^{2}D_{5/2}^{\circ}$ & 1256495 & & & 5.54E-12& 5.66E-12 & 0.30 + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}D^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D^{\circ}$\
29 & 179 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}D_{3/2}^{\circ}$ & 1258916 & & & 5.72E-12& 5.83E-12 & 0.12 + 0.34 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}S)~^{2}D^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}D^{\circ}$\
29 & 180 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G_{7/2}^{\circ}$ & 1260308 & & & 5.19E-12& 5.27E-12 & 0.39 + 0.29 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G^{\circ}$\
29 & 181 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G_{9/2}^{\circ}$ & 1262085 & & & 5.01E-12& 5.09E-12 & 0.45 + 0.30 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G^{\circ}$\
29 & 182 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}F_{5/2}^{\circ}$ & 1283358 & & & 4.85E-12& 4.94E-12 & 0.28 + 0.27 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
29 & 183 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F_{7/2}^{\circ}$ & 1285617 & & & 5.08E-12& 5.17E-12 & 0.26 + 0.26 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
29 & 184 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}S_{5/2}$ & 1288182 & & & 6.80E-10& 6.48E-10 & 0.89 + 0.07 $3s^{2}\,3p^{2}(^{3}P)~^{3}P\,3d^{3}(^{4}P)~^{6}S$\
29 & 185 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}D_{3/2}^{\circ}$ & 1293681 & & & 5.93E-12& 6.03E-12 & 0.24 + 0.23 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D^{\circ}$ + 0.15 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D^{\circ}$\
29 & 186 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D_{1/2}$ & 1293758 & & & 4.25E-10& 4.10E-10 & 0.63 + 0.24 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}D$ + 0.05 $3s^{2}\,3p^{2}(^{3}P)~^{3}P\,3d^{3}(^{4}F)~^{6}D$\
29 & 187 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D_{3/2}$ & 1294076 & & & 4.25E-10& 4.08E-10 & 0.59 + 0.24 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}D$ + 0.07 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F$\
29 & 188 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D_{5/2}$ & 1294710 & & & 4.27E-10& 4.10E-10 & 0.52 + 0.24 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}D$ + 0.12 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F$\
29 & 189 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D_{7/2}$ & 1295618 & & & 4.33E-10& 4.14E-10 & 0.47 + 0.27 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}D$ + 0.16 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F$\
29 & 190 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D_{9/2}$ & 1297072 & & & 4.51E-10& 4.30E-10 & 0.42 + 0.34 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}D$ + 0.15 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F$\
29 & 191 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}D_{5/2}^{\circ}$ & 1298773 & & & 5.93E-12& 6.02E-12 & 0.31 + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D^{\circ}$\
29 & 192 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}P_{1/2}^{\circ}$ & 1304995 & & & 3.54E-12& 3.58E-12 & 0.35 + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}P^{\circ}$ + 0.13 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}P^{\circ}$\
29 & 193 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}P_{3/2}^{\circ}$ & 1308685 & & & 3.74E-12& 3.79E-12 & 0.30 + 0.16 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}P^{\circ}$ + 0.11 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}P^{\circ}$\
29 & 194 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F_{11/2}$ & 1310833 & & & 2.82E-10& 2.69E-10 & 0.86 + 0.06 $3s^{2}\,3p^{2}(^{3}P)~^{3}P\,3d^{3}(^{4}F)~^{6}F$ + 0.05 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}G$\
30 & 1 & $3s^{2}\,3p^{5}~^{2}P_{3/2}^{\circ}$ & 0 & 0 & 0 & & & 0.96\
30 & 2 & $3s^{2}\,3p^{5}~^{2}P_{1/2}^{\circ}$ & 34252 & 34210 & 42 & 1.39E-03& 1.39E-03 & 0.96\
30 & 3 & $3s~^{2}S\,3p^{6}~^{2}S_{1/2}$ & 393679 & & & 1.33E-10& 1.26E-10 & 0.72 + 0.25 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}S$\
30 & 4 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D_{5/2}$ & 513704 & & & 5.73E-08& 6.78E-08 & 0.87 + 0.03 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F$ + 0.02 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$\
30 & 5 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D_{7/2}$ & 515195 & & & 6.91E-03& 6.91E-03 & 0.90 + 0.05 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F$\
30 & 6 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D_{3/2}$ & 515895 & & & 4.15E-08& 4.02E-08 & 0.85 + 0.03 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P$ + 0.03 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$\
30 & 7 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D_{1/2}$ & 520073 & & & 2.15E-08& 1.98E-08 & 0.85 + 0.06 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P$ + 0.04 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P$\
30 & 8 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F_{9/2}$ & 548852 & & & 2.24E-02& 2.24E-02 & 0.87 + 0.10 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}G$\
30 & 9 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P_{1/2}$ & 554830 & & & 3.80E-09& 3.80E-09 & 0.45 + 0.38 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P$ + 0.11 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D$\
30 & 10 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F_{7/2}$ & 557977 & & & 1.37E-02& 1.37E-02 & 0.67 + 0.15 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F$ + 0.13 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}G$\
30 & 11 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D_{3/2}$ & 567124 & & & 2.31E-09& 2.37E-09 & 0.07 + 0.34 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F$ + 0.15 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P$\
30 & 12 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F_{5/2}$ & 568681 & & & 3.99E-09& 4.06E-09 & 0.91 + 0.02 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}D$ + 0.02 $3s^{2}\,3p^{4}(^{1}S)~^{1}S\,3d~^{2}D$\
30 & 13 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P_{3/2}$ & 569934 & & & 2.63E-09& 2.65E-09 & 0.26 + 0.20 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P$ + 0.17 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F$\
30 & 14 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P_{1/2}$ & 572616 & & & 6.53E-10& 6.39E-10 & 0.92 + 0.02 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P$\
30 & 15 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F_{3/2}$ & 575529 & & & 1.13E-09& 1.16E-09 & 0.40 + 0.26 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$ + 0.15 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D$\
30 & 16 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F_{7/2}$ & 582360 & & & 2.13E-03& 2.13E-03 & 0.39 + 0.24 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F$ + 0.23 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}G$\
30 & 17 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P_{5/2}$ & 586915 & & & 3.82E-09& 3.86E-09 & 0.56 + 0.12 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D$ + 0.11 $3s^{2}\,3p^{4}(^{1}S)~^{1}S\,3d~^{2}D$\
30 & 18 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P_{3/2}$ & 587045 & & & 2.94E-08& 2.92E-08 & 0.57 + 0.14 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P$ + 0.13 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P$\
30 & 19 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D_{5/2}$ & 594075 & & & 6.17E-09& 6.15E-09 & 0.37 + 0.33 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}P$ + 0.16 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D$\
30 & 20 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}G_{7/2}$ & 600401 & & & 1.98E-03& 1.98E-03 & 0.60 + 0.25 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F$ + 0.11 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}F$\
30 & 21 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}G_{9/2}$ & 601798 & & & 1.79E-03& 1.79E-03 & 0.87 + 0.10 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F$\
30 & 22 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F_{5/2}$ & 606674 & & & 6.99E-09& 7.07E-09 & 0.63 + 0.24 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}F$ + 0.05 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$\
30 & 23 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}F_{5/2}$ & 636928 & & & 1.65E-09& 1.66E-09 & 0.70 + 0.17 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F$ + 0.07 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$\
30 & 24 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}F_{7/2}$ & 646116 & & & 7.75E-04& 7.76E-04 & 0.75 + 0.18 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F$\
30 & 25 & $3s^{2}\,3p^{4}(^{1}S)~^{1}S\,3d~^{2}D_{3/2}$ & 676568 & & & 2.08E-10& 2.10E-10 & 0.62 + 0.25 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$ + 0.04 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{4}F$\
30 & 26 & $3s^{2}\,3p^{4}(^{1}S)~^{1}S\,3d~^{2}D_{5/2}$ & 689456 & & & 1.04E-09& 1.05E-09 & 0.71 + 0.13 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$ + 0.05 $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}F$\
30 & 27 & $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}S_{1/2}$ & 707311 & 705310 & 2001 & 4.55E-12& 4.50E-12 & 0.69 + 0.23 $3s~^{2}S\,3p^{6}~^{2}S$ + 0.02 $3s~^{2}S\,3p^{4}(^{3}P)~^{2}P\,3d^{2}(^{3}P)~^{2}S$\
30 & 28 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P_{3/2}$ & 735959 & 733660 & 2299 & 5.10E-12& 5.12E-12 & 0.48 + 0.36 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P$ + 0.06 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$\
30 & 29 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D_{5/2}$ & 743971 & 741880 & 2091 & 4.29E-12& 4.31E-12 & 0.66 + 0.21 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$ + 0.07 $3s^{2}\,3p^{4}(^{1}S)~^{1}S\,3d~^{2}D$\
30 & 30 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}P_{1/2}$ & 748807 & 746370 & 2437 & 5.28E-12& 5.31E-12 & 0.50 + 0.43 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}P$\
30 & 31 & $3s^{2}\,3p^{4}(^{3}P)~^{3}P\,3d~^{2}D_{3/2}$ & 773815 & 771700 & 2115 & 4.41E-12& 4.40E-12 & 0.58 + 0.14 $3s^{2}\,3p^{4}(^{1}D)~^{1}D\,3d~^{2}D$ + 0.14 $3s^{2}\,3p^{4}(^{1}S)~^{1}S\,3d~^{2}D$\
30 & 32 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}P_{1/2}^{\circ}$ & 883033 & & & 1.99E-10& 1.98E-10 & 0.82 + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
30 & 33 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}P_{3/2}^{\circ}$ & 888377 & & & 1.97E-10& 1.96E-10 & 0.81 + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
30 & 34 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}P_{5/2}^{\circ}$ & 898267 & & & 1.93E-10& 1.94E-10 & 0.79 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
30 & 35 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F_{9/2}^{\circ}$ & 923637 & & & 1.12E-10& 1.10E-10 & 0.85 + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
30 & 36 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F_{7/2}^{\circ}$ & 927859 & & & 1.11E-10& 1.09E-10 & 0.79 + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.04 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D^{\circ}$\
30 & 37 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F_{5/2}^{\circ}$ & 933870 & & & 1.07E-10& 1.06E-10 & 0.76 + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.05 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D^{\circ}$\
30 & 38 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F_{3/2}^{\circ}$ & 940337 & & & 1.05E-10& 1.03E-10 & 0.78 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.05 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D^{\circ}$\
30 & 39 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D_{7/2}^{\circ}$ & 965174 & & & 1.22E-10& 1.21E-10 & 0.67 + 0.12 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}F^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
30 & 40 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D_{1/2}^{\circ}$ & 966865 & & & 1.12E-10& 1.11E-10 & 0.75 + 0.08 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
30 & 41 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D_{5/2}^{\circ}$ & 969955 & & & 1.13E-10& 1.12E-10 & 0.66 + 0.06 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.06 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}D^{\circ}$\
30 & 42 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D_{3/2}^{\circ}$ & 970242 & & & 1.13E-10& 1.12E-10 & 0.72 + 0.07 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
30 & 43 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}F_{7/2}^{\circ}$ & 975418 & & & 2.02E-10& 2.02E-10 & 0.60 + 0.09 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}D^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
30 & 44 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}D_{5/2}^{\circ}$ & 981924 & & & 2.12E-10& 2.11E-10 & 0.49 + 0.24 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}F^{\circ}$ + 0.05 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D^{\circ}$\
30 & 45 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}D_{3/2}^{\circ}$ & 988531 & & & 1.20E-10& 1.19E-10 & 0.64 + 0.07 $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}P^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D^{\circ}$\
30 & 46 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}F_{5/2}^{\circ}$ & 999998 & & & 1.54E-10& 1.53E-10 & 0.51 + 0.21 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}D^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
30 & 47 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}P_{1/2}^{\circ}$ & 1001924 & & & 5.57E-11& 5.56E-11 & 0.38 + 0.25 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}P^{\circ}$ + 0.15 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P^{\circ}$\
30 & 48 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}P_{3/2}^{\circ}$ & 1020803 & & & 6.70E-11& 6.65E-11 & 0.34 + 0.17 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}P^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P^{\circ}$\
30 & 49 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{1/2}^{\circ}$ & 1030986 & & & 1.52E-09& 1.50E-09 & 0.95\
30 & 50 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{3/2}^{\circ}$ & 1031153 & & & 1.84E-09& 1.83E-09 & 0.94\
30 & 51 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{5/2}^{\circ}$ & 1031714 & & & 2.33E-09& 2.32E-09 & 0.93\
30 & 52 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{7/2}^{\circ}$ & 1032839 & & & 3.35E-09& 3.35E-09 & 0.93\
30 & 53 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{9/2}^{\circ}$ & 1034599 & & & 5.48E-09& 5.50E-09 & 0.94\
30 & 54 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{6}F_{11/2}^{\circ}$ & 1036720 & & & 3.20E-08& 3.37E-08 & 0.94 + 0.03 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
30 & 55 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}F_{5/2}^{\circ}$ & 1039816 & & & 5.50E-11& 5.48E-11 & 0.57 + 0.17 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
30 & 56 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}F_{7/2}^{\circ}$ & 1048856 & & & 5.24E-11& 5.22E-11 & 0.55 + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
30 & 57 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{6}P_{3/2}^{\circ}$ & 1063027 & & & 1.78E-09& 1.80E-09 & 0.92\
30 & 58 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{6}P_{5/2}^{\circ}$ & 1063926 & & & 1.47E-09& 1.47E-09 & 0.92\
30 & 59 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{6}P_{7/2}^{\circ}$ & 1064096 & & & 1.02E-09& 1.02E-09 & 0.87 + 0.04 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D^{\circ}$ + 0.03 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
30 & 60 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}D_{3/2}^{\circ}$ & 1074158 & & & 4.89E-11& 4.87E-11 & 0.52 + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$\
30 & 61 & $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}D_{5/2}^{\circ}$ & 1078875 & & & 5.00E-11& 4.97E-11 & 0.57 + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$\
30 & 62 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F_{3/2}^{\circ}$ & 1085774 & & & 1.83E-10& 1.82E-10 & 0.20 + 0.16 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
30 & 63 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P_{1/2}^{\circ}$ & 1086298 & & & 1.00E-09& 1.02E-09 & 0.38 + 0.30 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$\
30 & 64 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D_{5/2}^{\circ}$ & 1088233 & & & 2.67E-10& 2.68E-10 & 0.19 + 0.18 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$\
30 & 65 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D_{7/2}^{\circ}$ & 1089463 & & & 2.27E-10& 2.29E-10 & 0.47 + 0.18 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{6}P^{\circ}$\
30 & 66 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G_{5/2}^{\circ}$ & 1092578 & & & 3.04E-10& 3.05E-10 & 0.32 + 0.28 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
30 & 67 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G_{7/2}^{\circ}$ & 1094135 & & & 9.39E-10& 9.45E-10 & 0.53 + 0.17 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$\
30 & 68 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D_{3/2}^{\circ}$ & 1094139 & & & 5.85E-10& 5.91E-10 & 0.21 + 0.20 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
30 & 69 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D_{1/2}^{\circ}$ & 1095382 & & & 1.33E-10& 1.34E-10 & 0.57 + 0.23 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.05 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
30 & 70 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P_{3/2}^{\circ}$ & 1098468 & & & 3.43E-10& 3.46E-10 & 0.06 + 0.20 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.18 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D^{\circ}$\
30 & 71 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G_{9/2}^{\circ}$ & 1100141 & & & 2.00E-09& 2.04E-09 & 0.60 + 0.25 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.03 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
30 & 72 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P_{5/2}^{\circ}$ & 1100497 & & & 1.59E-09& 1.62E-09 & 0.08 + 0.23 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.18 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
30 & 73 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G_{11/2}^{\circ}$ & 1105152 & & & 2.89E-09& 2.88E-09 & 0.61 + 0.35 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$\
30 & 74 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F_{5/2}^{\circ}$ & 1105161 & & & 1.32E-09& 1.34E-09 & 0.07 + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$\
30 & 75 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F_{7/2}^{\circ}$ & 1109344 & & & 9.24E-10& 9.33E-10 & 0.33 + 0.18 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
30 & 76 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F_{9/2}^{\circ}$ & 1114028 & & & 7.55E-10& 7.59E-10 & 0.31 + 0.18 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H^{\circ}$\
30 & 77 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H_{7/2}^{\circ}$ & 1114813 & & & 6.51E-10& 6.53E-10 & 0.62 + 0.08 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$\
30 & 78 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H_{9/2}^{\circ}$ & 1119712 & & & 1.68E-08& 1.68E-08 & 0.59 + 0.15 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
30 & 79 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H_{11/2}^{\circ}$ & 1123084 & & & 2.64E-07& 2.19E-07 & 0.76 + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$ + 0.03 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H^{\circ}$\
30 & 80 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F_{7/2}^{\circ}$ & 1127221 & & & 1.43E-10& 1.42E-10 & 0.25 + 0.19 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.17 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H^{\circ}$\
30 & 81 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}P_{1/2}^{\circ}$ & 1127406 & & & 5.55E-12& 5.62E-12 & 0.51 + 0.29 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}P^{\circ}$ + 0.06 $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}P^{\circ}$\
30 & 82 & $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}P_{3/2}^{\circ}$ & 1128437 & & & 8.12E-12& 8.22E-12 & 0.36 + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}P^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}D^{\circ}$\
30 & 83 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}D_{3/2}^{\circ}$ & 1130686 & & & 1.21E-11& 1.22E-11 & 0.17 + 0.19 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}P^{\circ}$ + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$\
30 & 84 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H_{13/2}^{\circ}$ & 1131676 & & & 9.68E-03& 9.74E-03 & 0.98\
30 & 85 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H_{9/2}^{\circ}$ & 1131962 & & & 7.00E-10& 7.09E-10 & 0.38 + 0.30 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}H^{\circ}$\
30 & 86 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F_{5/2}^{\circ}$ & 1132037 & & & 7.40E-11& 7.41E-11 & 0.26 + 0.20 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
30 & 87 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H_{11/2}^{\circ}$ & 1137730 & & & 2.29E-09& 2.28E-09 & 0.37 + 0.23 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H^{\circ}$\
30 & 88 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D_{5/2}^{\circ}$ & 1144516 & & & 5.29E-11& 5.29E-11 & 0.04 + 0.20 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$ + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}D^{\circ}$\
30 & 89 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G_{7/2}^{\circ}$ & 1146968 & & & 9.40E-11& 9.40E-11 & 0.39 + 0.20 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G^{\circ}$\
30 & 90 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G_{5/2}^{\circ}$ & 1148127 & & & 6.71E-11& 6.71E-11 & 0.26 + 0.19 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
30 & 91 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G_{9/2}^{\circ}$ & 1155027 & & & 1.13E-10& 1.13E-10 & 0.25 + 0.30 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$\
30 & 92 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G_{7/2}^{\circ}$ & 1156669 & & & 1.09E-10& 1.09E-10 & 0.53 + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
30 & 93 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G_{9/2}^{\circ}$ & 1160820 & & & 8.61E-11& 8.62E-11 & 0.47 + 0.12 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}H^{\circ}$\
30 & 94 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F_{5/2}^{\circ}$ & 1163827 & & & 1.67E-11& 1.67E-11 & 0.38 + 0.27 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.06 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F^{\circ}$\
30 & 95 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F_{7/2}^{\circ}$ & 1165866 & & & 1.77E-11& 1.77E-11 & 0.36 + 0.26 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
30 & 96 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F_{3/2}^{\circ}$ & 1166352 & & & 1.57E-11& 1.57E-11 & 0.44 + 0.31 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.07 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}F^{\circ}$\
30 & 97 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G_{11/2}^{\circ}$ & 1171011 & & & 4.20E-10& 4.22E-10 & 0.44 + 0.15 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H^{\circ}$\
30 & 98 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F_{9/2}^{\circ}$ & 1172636 & & & 1.91E-11& 1.90E-11 & 0.37 + 0.28 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
30 & 99 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D_{1/2}^{\circ}$ & 1176376 & & & 1.97E-11& 1.99E-11 & 0.42 + 0.18 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.15 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
30 & 100 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{2}P_{3/2}^{\circ}$ & 1176478 & & & 2.47E-11& 2.48E-11 & 0.16 + 0.15 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}P^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P^{\circ}$\
30 & 101 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}I_{11/2}^{\circ}$ & 1180385 & & & 4.77E-10& 4.79E-10 & 0.80 + 0.06 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}H^{\circ}$ + 0.05 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H^{\circ}$\
30 & 102 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D_{3/2}^{\circ}$ & 1181053 & & & 3.72E-11& 3.75E-11 & 0.20 + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
30 & 103 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}S_{1/2}^{\circ}$ & 1183176 & & & 8.31E-11& 8.45E-11 & 0.33 + 0.27 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}S^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{2}P^{\circ}$\
30 & 104 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}I_{13/2}^{\circ}$ & 1186006 & & & 9.43E-03& 9.42E-03 & 0.98\
30 & 105 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D_{5/2}^{\circ}$ & 1186545 & & & 5.92E-11& 5.95E-11 & 0.17 + 0.15 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
30 & 106 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P_{3/2}^{\circ}$ & 1190327 & & & 2.84E-11& 2.85E-11 & 0.02 + 0.17 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$\
30 & 107 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D_{7/2}^{\circ}$ & 1191795 & & & 4.39E-11& 4.40E-11 & 0.23 + 0.18 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
30 & 108 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F_{5/2}^{\circ}$ & 1191934 & & & 4.76E-11& 4.75E-11 & 0.29 + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}F^{\circ}$\
30 & 109 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D_{5/2}^{\circ}$ & 1193337 & & & 3.08E-11& 3.10E-11 & 0.05 + 0.16 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}D^{\circ}$\
30 & 110 & $3s~^{2}S\,3p^{3}(^{2}D)~^{3}D\,3d^{3}(^{2}D)~^{4}S_{3/2}^{\circ}$ & 1197219 & & & 7.43E-11& 7.46E-11 & 0.00 + 0.18 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$ + 0.18 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}S)~^{4}S^{\circ}$\
30 & 111 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}S)~^{4}S_{3/2}^{\circ}$ & 1198766 & & & 9.96E-11& 1.01E-10 & 0.29 + 0.16 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}S^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D^{\circ}$\
30 & 112 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F_{5/2}^{\circ}$ & 1199349 & & & 3.33E-11& 3.36E-11 & 0.21 + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
30 & 113 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G_{7/2}^{\circ}$ & 1201072 & & & 7.37E-11& 7.39E-11 & 0.15 + 0.13 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
30 & 114 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P_{1/2}^{\circ}$ & 1203303 & & & 1.56E-11& 1.57E-11 & 0.23 + 0.17 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{2}P^{\circ}$ + 0.15 $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}P^{\circ}$\
30 & 115 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F_{9/2}^{\circ}$ & 1203863 & & & 7.50E-11& 7.55E-11 & 0.28 + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}F^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}G^{\circ}$\
30 & 116 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F_{7/2}^{\circ}$ & 1206655 & & & 2.09E-10& 2.10E-10 & 0.17 + 0.21 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}G^{\circ}$\
30 & 117 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D_{7/2}^{\circ}$ & 1210546 & & & 2.04E-11& 2.06E-11 & 0.24 + 0.19 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F^{\circ}$ + 0.17 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
30 & 118 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D_{1/2}^{\circ}$ & 1210870 & & & 3.61E-11& 3.63E-11 & 0.40 + 0.21 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
30 & 119 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D_{3/2}^{\circ}$ & 1212876 & & & 2.73E-11& 2.76E-11 & 0.24 + 0.22 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
30 & 120 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P_{5/2}^{\circ}$ & 1215151 & & & 1.76E-11& 1.79E-11 & 0.47 + 0.13 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
30 & 121 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G_{9/2}^{\circ}$ & 1218635 & & & 1.61E-11& 1.62E-11 & 0.25 + 0.15 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G^{\circ}$\
30 & 122 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G_{11/2}^{\circ}$ & 1219910 & & & 1.30E-11& 1.31E-11 & 0.34 + 0.25 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$ + 0.23 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
30 & 123 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}F_{5/2}^{\circ}$ & 1220718 & & & 1.48E-11& 1.49E-11 & 0.20 + 0.12 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
30 & 124 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P_{1/2}^{\circ}$ & 1222253 & & & 5.78E-12& 5.84E-12 & 0.46 + 0.25 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.11 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{4}P^{\circ}$\
30 & 125 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P_{3/2}^{\circ}$ & 1222657 & & & 7.46E-12& 7.55E-12 & 0.37 + 0.13 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$\
30 & 126 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G_{9/2}^{\circ}$ & 1223111 & & & 4.48E-11& 4.51E-11 & 0.09 + 0.24 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.23 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}G^{\circ}$\
30 & 127 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G_{7/2}^{\circ}$ & 1225886 & & & 1.38E-11& 1.39E-11 & 0.30 + 0.21 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$ + 0.19 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
30 & 128 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G_{5/2}^{\circ}$ & 1228333 & & & 1.29E-11& 1.30E-11 & 0.27 + 0.24 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G^{\circ}$ + 0.23 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
30 & 129 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D_{5/2}^{\circ}$ & 1230365 & & & 1.19E-11& 1.21E-11 & 0.19 + 0.09 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$\
30 & 130 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}F_{7/2}^{\circ}$ & 1230821 & & & 2.07E-11& 2.08E-11 & 0.13 + 0.15 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
30 & 131 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}P_{1/2}^{\circ}$ & 1233667 & & & 1.34E-11& 1.35E-11 & 0.25 + 0.22 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
30 & 132 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P_{3/2}^{\circ}$ & 1235996 & & & 1.87E-11& 1.90E-11 & 0.38 + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
30 & 133 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}G_{9/2}^{\circ}$ & 1238064 & & & 1.51E-11& 1.51E-11 & 0.63 + 0.10 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}G)~^{4}G^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}G^{\circ}$\
30 & 134 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P_{5/2}^{\circ}$ & 1238285 & & & 1.22E-11& 1.24E-11 & 0.32 + 0.26 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$\
30 & 135 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}G_{7/2}^{\circ}$ & 1241497 & & & 1.65E-11& 1.67E-11 & 0.56 + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.04 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
30 & 136 & $3p^{5}~^{2}P\,3d^{2}(^{1}D)~^{2}P_{1/2}^{\circ}$ & 1242252 & & & 2.83E-11& 2.85E-11 & 0.00 + 0.24 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.23 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}P^{\circ}$\
30 & 137 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}P_{3/2}^{\circ}$ & 1247555 & & & 8.45E-12& 8.47E-12 & 0.33 + 0.20 $3s~^{2}S\,3p^{5}~^{1}P\,3d~^{2}P^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}P^{\circ}$\
30 & 138 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F_{7/2}^{\circ}$ & 1249051 & & & 9.45E-12& 9.52E-12 & 0.03 + 0.21 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}G^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
30 & 139 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}H_{11/2}^{\circ}$ & 1255358 & & & 9.87E-11& 9.90E-11 & 0.42 + 0.35 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H^{\circ}$\
30 & 140 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}H_{9/2}^{\circ}$ & 1256623 & & & 1.18E-11& 1.19E-11 & 0.40 + 0.36 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H^{\circ}$\
30 & 141 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D_{7/2}^{\circ}$ & 1258292 & & & 6.39E-12& 6.48E-12 & 0.46 + 0.16 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
30 & 142 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}D_{5/2}^{\circ}$ & 1261588 & & & 9.23E-12& 9.37E-12 & 0.11 + 0.23 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$\
30 & 143 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D_{5/2}^{\circ}$ & 1263739 & & & 6.38E-12& 6.47E-12 & 0.27 + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}D^{\circ}$\
30 & 144 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H_{9/2}^{\circ}$ & 1263980 & & & 5.86E-11& 5.90E-11 & 0.43 + 0.33 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}H^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H^{\circ}$\
30 & 145 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D_{3/2}^{\circ}$ & 1265487 & & & 5.88E-12& 5.96E-12 & 0.23 + 0.22 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.21 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
30 & 146 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D_{1/2}^{\circ}$ & 1266770 & & & 5.74E-12& 5.83E-12 & 0.39 + 0.26 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D^{\circ}$\
30 & 147 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}D_{3/2}^{\circ}$ & 1267978 & & & 2.04E-11& 2.08E-11 & 0.11 + 0.29 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}S)~^{2}D^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
30 & 148 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}D_{7/2}^{\circ}$ & 1269202 & & & 6.15E-12& 6.26E-12 & 0.11 + 0.25 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.21 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
30 & 149 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D_{5/2}^{\circ}$ & 1269869 & & & 6.56E-12& 6.68E-12 & 0.25 + 0.11 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}S)~^{2}D^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$\
30 & 150 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F_{9/2}^{\circ}$ & 1270518 & & & 4.51E-12& 4.58E-12 & 0.35 + 0.21 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
30 & 151 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{4}D_{3/2}^{\circ}$ & 1273724 & & & 5.66E-12& 5.78E-12 & 0.20 + 0.17 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.15 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D^{\circ}$\
30 & 152 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F_{7/2}^{\circ}$ & 1274087 & & & 6.37E-12& 6.43E-12 & 0.07 + 0.11 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}F^{\circ}$\
30 & 153 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}H_{11/2}^{\circ}$ & 1274740 & & & 1.06E-11& 1.06E-11 & 0.20 + 0.36 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}H^{\circ}$ + 0.30 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}H^{\circ}$\
30 & 154 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}D_{1/2}^{\circ}$ & 1277823 & & & 4.95E-12& 5.05E-12 & 0.06 + 0.22 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}D^{\circ}$ + 0.22 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}D)~^{4}D^{\circ}$\
30 & 155 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F_{7/2}^{\circ}$ & 1281036 & & & 4.63E-12& 4.70E-12 & 0.26 + 0.20 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
30 & 156 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F_{5/2}^{\circ}$ & 1281113 & & & 6.63E-12& 6.71E-12 & 0.13 + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}F^{\circ}$\
30 & 157 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D_{3/2}^{\circ}$ & 1290375 & & & 5.31E-12& 5.39E-12 & 0.09 + 0.13 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
30 & 158 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D_{5/2}^{\circ}$ & 1290535 & & & 5.73E-12& 5.83E-12 & 0.20 + 0.13 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
30 & 159 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}S_{1/2}^{\circ}$ & 1291196 & & & 5.05E-12& 5.17E-12 & 0.22 + 0.17 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}P^{\circ}$ + 0.16 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{2}P^{\circ}$\
30 & 160 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F_{5/2}^{\circ}$ & 1292756 & & & 5.14E-12& 5.21E-12 & 0.18 + 0.15 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D^{\circ}$\
30 & 161 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}G_{7/2}^{\circ}$ & 1293112 & & & 1.26E-11& 1.27E-11 & 0.53 + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
30 & 162 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{4}F_{3/2}^{\circ}$ & 1293693 & & & 4.73E-12& 4.82E-12 & 0.14 + 0.13 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}P^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
30 & 163 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}D_{3/2}^{\circ}$ & 1299720 & & & 5.72E-12& 5.82E-12 & 0.15 + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}P^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{4}F^{\circ}$\
30 & 164 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P_{1/2}^{\circ}$ & 1300727 & & & 4.98E-12& 5.08E-12 & 0.14 + 0.21 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}S^{\circ}$ + 0.17 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}S^{\circ}$\
30 & 165 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F_{5/2}^{\circ}$ & 1303923 & & & 9.85E-12& 9.96E-12 & 0.34 + 0.17 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
30 & 166 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D_{3/2}^{\circ}$ & 1304008 & & & 5.72E-12& 5.80E-12 & 0.38 + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}D^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}S)~^{2}P^{\circ}$\
30 & 167 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}G_{9/2}^{\circ}$ & 1307251 & & & 1.26E-11& 1.28E-11 & 0.49 + 0.21 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.07 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}G^{\circ}$\
30 & 168 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P_{3/2}^{\circ}$ & 1308018 & & & 5.92E-12& 6.05E-12 & 0.21 + 0.20 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}S^{\circ}$\
30 & 169 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F_{5/2}^{\circ}$ & 1309122 & & & 5.55E-12& 5.62E-12 & 0.03 + 0.12 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}F^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F^{\circ}$\
30 & 170 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}S)~^{2}P_{1/2}^{\circ}$ & 1309937 & & & 4.98E-11& 5.07E-11 & 0.53 + 0.16 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}P^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}P^{\circ}$\
30 & 171 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}F_{7/2}^{\circ}$ & 1311003 & & & 5.87E-12& 5.96E-12 & 0.15 + 0.22 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}F)~^{2}F^{\circ}$\
30 & 172 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P_{5/2}^{\circ}$ & 1313779 & & & 5.31E-12& 5.40E-12 & 0.28 + 0.24 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
30 & 173 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}D_{5/2}^{\circ}$ & 1317785 & & & 5.72E-12& 5.80E-12 & 0.17 + 0.14 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}D^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D^{\circ}$\
30 & 174 & $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P_{1/2}^{\circ}$ & 1318274 & & & 5.21E-12& 5.31E-12 & 0.32 + 0.20 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{4}P^{\circ}$ + 0.09 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}S^{\circ}$\
30 & 175 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}S)~^{2}P_{3/2}^{\circ}$ & 1322966 & & & 7.16E-12& 7.31E-12 & 0.16 + 0.15 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{4}P^{\circ}$ + 0.08 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{3}P)~^{2}P^{\circ}$\
30 & 176 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{4}S_{3/2}^{\circ}$ & 1329307 & & & 5.92E-12& 6.08E-12 & 0.37 + 0.25 $3s^{2}\,3p^{3}(^{4}S)~^{4}S\,3d^{2}(^{1}S)~^{4}S^{\circ}$ + 0.06 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}P^{\circ}$\
30 & 177 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}F_{7/2}^{\circ}$ & 1334371 & & & 9.01E-12& 9.14E-12 & 0.25 + 0.25 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F^{\circ}$ + 0.11 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$\
30 & 178 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}S)~^{2}D_{5/2}^{\circ}$ & 1336757 & & & 5.27E-12& 5.37E-12 & 0.29 + 0.12 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D^{\circ}$ + 0.12 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}D^{\circ}$\
30 & 179 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G_{7/2}^{\circ}$ & 1340063 & & & 5.48E-12& 5.56E-12 & 0.35 + 0.22 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.15 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}G)~^{2}F^{\circ}$\
30 & 180 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}S)~^{2}D_{3/2}^{\circ}$ & 1340208 & & & 5.56E-12& 5.66E-12 & 0.33 + 0.11 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}D^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}D^{\circ}$\
30 & 181 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}F)~^{2}G_{9/2}^{\circ}$ & 1342258 & & & 4.75E-12& 4.82E-12 & 0.45 + 0.29 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}G^{\circ}$ + 0.10 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}G^{\circ}$\
30 & 182 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}F_{5/2}^{\circ}$ & 1364559 & & & 4.53E-12& 4.61E-12 & 0.27 + 0.27 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
30 & 183 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}P)~^{2}F_{7/2}^{\circ}$ & 1367304 & & & 4.78E-12& 4.87E-12 & 0.25 + 0.25 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}F^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}D)~^{2}F^{\circ}$\
30 & 184 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}S_{5/2}$ & 1373575 & & & 5.95E-10& 5.66E-10 & 0.88 + 0.07 $3s^{2}\,3p^{2}(^{3}P)~^{3}P\,3d^{3}(^{4}P)~^{6}S$\
30 & 185 & $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D_{3/2}^{\circ}$ & 1375218 & & & 5.49E-12& 5.59E-12 & 0.23 + 0.23 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}D^{\circ}$ + 0.14 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D^{\circ}$\
30 & 186 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D_{1/2}$ & 1378396 & & & 3.74E-10& 3.60E-10 & 0.64 + 0.23 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}D$ + 0.05 $3s^{2}\,3p^{2}(^{3}P)~^{3}P\,3d^{3}(^{4}F)~^{6}D$\
30 & 187 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D_{3/2}$ & 1378809 & & & 3.72E-10& 3.58E-10 & 0.59 + 0.23 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}D$ + 0.08 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F$\
30 & 188 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D_{5/2}$ & 1379633 & & & 3.75E-10& 3.59E-10 & 0.51 + 0.23 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}D$ + 0.15 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F$\
30 & 189 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D_{7/2}$ & 1380738 & & & 3.79E-10& 3.62E-10 & 0.45 + 0.25 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}D$ + 0.20 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F$\
30 & 190 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}D_{5/2}^{\circ}$ & 1381742 & & & 5.49E-12& 5.58E-12 & 0.31 + 0.20 $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{1}G)~^{2}D^{\circ}$ + 0.15 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{3}P)~^{2}D^{\circ}$\
30 & 191 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}D_{9/2}$ & 1382583 & & & 3.96E-10& 3.77E-10 & 0.40 + 0.32 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}P)~^{6}D$ + 0.18 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F$\
30 & 192 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}P_{1/2}^{\circ}$ & 1386661 & & & 3.22E-12& 3.25E-12 & 0.36 + 0.14 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}P^{\circ}$ + 0.13 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}P^{\circ}$\
30 & 193 & $3s^{2}\,3p^{3}(^{2}D)~^{2}D\,3d^{2}(^{3}F)~^{2}P_{3/2}^{\circ}$ & 1391125 & & & 3.43E-12& 3.47E-12 & 0.29 + 0.15 $3s^{2}\,3p^{3}(^{2}P)~^{2}P\,3d^{2}(^{1}D)~^{2}P^{\circ}$ + 0.11 $3s~^{2}S\,3p^{5}~^{3}P\,3d~^{2}P^{\circ}$\
30 & 194 & $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}F_{11/2}$ & 1397265 & & & 2.49E-10& 2.38E-10 & 0.85 + 0.06 $3s~^{2}S\,3p^{4}(^{3}P)~^{4}P\,3d^{2}(^{3}F)~^{6}G$ + 0.06 $3s^{2}\,3p^{2}(^{3}P)~^{3}P\,3d^{3}(^{4}F)~^{6}F$\
[lllllllllllll]{} 26 & 1 & 2 & 6.3594E+03& M1& 9.998E-01& 6.984E+01& 8.469E-07& 1.332E+00& & & &\
26 & 1 & 2 & 6.3594E+03& E2& 2.156E-04& 1.506E-02& 1.826E-10& 2.798E-01& 1.455E-02& 1.764E-10 &2.702E-01& D+\
26 & 1 & 3 & 3.4526E+02& E1& 6.951E-01& 3.023E+09& 1.080E-01& 1.228E-01& 3.118E+09& 1.114E-01 &1.267E-01& B+\
26 & 1 & 4 & 2.5741E+02& M2& 1.000E+00& 5.777E+01& 4.591E-09& 3.503E+01& & & &\
26 & 1 & 5 & 2.5736E+02& E1& 1.000E+00& 6.011E+06& 3.581E-04& 3.034E-04& 4.880E+06& 2.908E-04 &2.463E-04& C\
26 & 1 & 6 & 2.5655E+02& E1& 9.980E-01& 5.986E+06& 2.363E-04& 1.995E-04& 6.120E+06& 2.416E-04 &2.040E-04& B\
26 & 1 & 7 & 2.5553E+02& E1& 4.240E-01& 3.598E+06& 7.045E-05& 5.926E-05& 4.229E+06& 8.279E-05 &6.965E-05& C+\
26 & 1 & 9 & 2.3820E+02& E1& 1.573E-01& 3.496E+07& 5.948E-04& 4.664E-04& 3.656E+07& 6.220E-04 &4.877E-04& B\
26 & 1 & 10& 2.3661E+02& M2& 2.060E-01& 3.473E+00& 2.332E-10& 1.382E+00& & & &\
26 & 1 & 11& 2.3441E+02& E1& 1.000E+00& 4.702E+07& 2.324E-03& 1.793E-03& 4.594E+07& 2.271E-03 &1.752E-03& B+\
26 & 1 & 12& 2.3367E+02& E1& 6.148E-01& 2.168E+08& 7.097E-03& 5.460E-03& 2.133E+08& 6.986E-03 &5.374E-03& B+\
26 & 1 & 13& 2.3351E+02& E1& 9.868E-01& 8.461E+07& 2.767E-03& 2.127E-03& 8.602E+07& 2.813E-03 &2.162E-03& B+\
26 & 1 & 14& 2.3011E+02& E1& 6.951E-01& 2.521E+08& 8.006E-03& 6.065E-03& 2.404E+08& 7.635E-03 &5.784E-03& B+\
26 & 1 & 15& 2.3005E+02& E1& 8.702E-01& 3.093E+08& 4.907E-03& 3.717E-03& 3.161E+08& 5.016E-03 &3.799E-03& B+\
26 & 1 & 16& 2.2747E+02& E1& 2.348E-01& 1.464E+06& 4.542E-05& 3.402E-05& 1.360E+06& 4.221E-05 &3.161E-05& C+\
26 & 1 & 17& 2.2687E+02& M2& 1.850E-01& 1.252E+01& 7.730E-10& 4.038E+00& & & &\
26 & 1 & 18& 2.2636E+02& E1& 1.000E+00& 1.214E+08& 5.595E-03& 4.169E-03& 1.174E+08& 5.411E-03 &4.032E-03& B+\
26 & 1 & 19& 2.2521E+02& E1& 1.000E+00& 7.796E+07& 3.557E-03& 2.637E-03& 7.608E+07& 3.471E-03 &2.574E-03& B+\
26 & 1 & 21& 2.2170E+02& M2& 1.395E-01& 9.964E+00& 5.874E-10& 2.863E+00& & & &\
26 & 1 & 22& 2.2025E+02& E1& 1.000E+00& 2.417E+07& 1.055E-03& 7.648E-04& 2.346E+07& 1.024E-03 &7.423E-04& B\
26 & 1 & 23& 2.0743E+02& E1& 1.000E+00& 1.759E+08& 6.808E-03& 4.649E-03& 1.749E+08& 6.771E-03 &4.624E-03& B+\
26 & 1 & 24& 2.0577E+02& M2& 6.584E-03& 1.326E+00& 6.733E-11& 2.624E-01& & & &\
26 & 1 & 25& 1.9535E+02& E1& 6.124E-01& 6.640E+08& 1.519E-02& 9.772E-03& 6.589E+08& 1.508E-02 &9.697E-03& B+\
26 & 1 & 26& 1.9367E+02& E1& 1.000E+00& 1.560E+08& 5.263E-03& 3.356E-03& 1.556E+08& 5.250E-03 &3.347E-03& B+\
26 & 1 & 27& 1.8486E+02& E1& 7.680E-01& 1.223E+11& 1.253E+00& 7.628E-01& 1.232E+11& 1.262E+00 &7.681E-01& A+\
26 & 1 & 28& 1.7743E+02& E1& 9.745E-01& 1.445E+11& 2.729E+00& 1.594E+00& 1.429E+11& 2.698E+00 &1.576E+00& A+\
26 & 1 & 29& 1.7564E+02& E1& 2.482E-01& 3.592E+10& 3.323E-01& 1.921E-01& 3.544E+10& 3.278E-01 &1.896E-01& A+\
26 & 1 & 30& 1.7477E+02& E1& 1.000E+00& 1.780E+11& 4.891E+00& 2.814E+00& 1.756E+11& 4.826E+00 &2.777E+00& A+\
26 & 1 & 31& 1.7076E+02& E1& 4.309E-02& 7.541E+09& 1.319E-01& 7.413E-02& 7.443E+09& 1.301E-01 &7.316E-02& A\
|
---
abstract: 'Recently, algorithms for calculation of 3-loop propagator diagrams in HQET and on-shell QCD with a heavy quark have been constructed and implemented. These algorithms (based on integration by parts recurrence relations) reduce an arbitrary diagram to a combination of a finite number of basis integrals. Here I discuss various ways to calculate non-trivial bases integrals, either exactly or as expansions in $\varepsilon$. Some integrals of these two classes are related to each other by inversion, which provides a useful cross-check.'
address: 'Budker Institute of Nuclear Physics, Novosibirsk'
author:
- 'A.G. Grozin'
title: Multiloop calculations in HQET
---
I presented a review talk about multiloop calculations in HQET at this conference in Pisa in 1995 [@BG:95]. Methods of calculation of two-loop propagator diagrams in HQET [@BG:91] and on-shell massive QCD [@B:92; @FT:92], based on integration by parts [@CT:81], were discussed there. Recently, three-loop HQET [@G:00] and on-shell [@MR:00] algorithms have been constructed. Here I discuss this substantial progress.
Three-loop massless diagrams {#QCD}
============================
First, I briefly remind you the classic method of calculation of 3-loop massless propagator diagrams. There are 3 generic topologies of such diagrams. They can be reduces, using integration by parts, to 6 basis integrals [@CT:81]. This algorithm is implemented in the package Mincer [@Mincer] (first written in SCHOONSCHIP [@Sch] and later rewritten in FORM [@Form]), and in the package Slicer [@Sli] written in REDUCE [@H:99; @G:97].
Four basis integrals are trivial. One is a two-loop diagram with a non-integer power of the middle line. It can be found as a particular case of a more general expression [@K:96; @BGK:97] for the the two-loop diagram with three non-integer powers via a hypergeometric ${}_3F_2$ function of the unit argument, with indices tending to integers at $\varepsilon\to0$. There is a rather straightforward algorithm for expanding such functions in $\varepsilon$, with coefficients expressed via multiple $\zeta$-values. I have implemented it in REDUCE in the summer of 2000, some results produced by this program are published in [@G:01]. It is clearly presented as Algorithm A in [@MUW:02]; this paper also contains other, more complicated, algorithms. The algorithms of [@MUW:02] are implemented in the C++ library nestedsums [@W:02] based on the computer-algebra library GiNaC [@BFK:00]. This implementation is very convenient; unfortunately, it requires one to install an outdated version of GiNaC. The Algorithm A seems to be also implemented in FORM [@V:99], but I could not understand how to use it. Using my REDUCE procedure or nestedsums [@W:02], one can quickly find as many terms of expansion of this basis integral in $\varepsilon$ as needed, in terms of multiple $\zeta$-values. They can be expressed, up to weight 9, via a minimum set of independent $\zeta$-values, using the results of [@B:96; @V:99]. The two-loop diagram with a non-integer power of the middle line can also be expressed [@K:85] via an ${}_3F_2$ function of the argument $-1$. Expanding this expression in $\varepsilon$ (say, using nestedsums [@W:02]), we encounter more general Euler–Zagier sums, which were also considered in [@B:96]. Reducing them to the minimal basis, we obtain, of course, the same $\varepsilon$-expansion of our basis integral.
Using this expansion and integration-by-parts relations, it is easy to recover the well-known result for the 3-loop ladder diagram, which is finite $\varepsilon=0$: $20\zeta(5)+\mathcal{O}(\varepsilon)$. The last and most difficult basis diagram is non-planar. It is also finite at $\varepsilon=0$. Using gluing of its external vertices [@CT:81], one can easily understand that it has the same value $20\zeta(5)$ at $\varepsilon=0$. There is no easy way to find further terms of its $\varepsilon$- expansion.
(92,50) (46,25)[(0,0)[]{}]{} (13.5,26)[(0,0)\[b\][$n_1$]{}]{} (28.5,26)[(0,0)\[b\][$n_2$]{}]{} (63.5,26)[(0,0)\[b\][$n_1$]{}]{} (78.5,26)[(0,0)\[b\][$n_2$]{}]{} (8.5,37.5)[(0,0)\[r\][$n_3$]{}]{} (33,37.5)[(0,0)\[l\][$n_4$]{}]{} (58.5,39)[(0,0)\[r\][$d-n_1-n_3$]{}]{} (57,36)[(0,0)\[r\][${}-n_5-n_7$]{}]{} (83,39)[(0,0)\[l\][$d-n_2-n_4$]{}]{} (84,36)[(0,0)\[l\][${}-n_5-n_8$]{}]{} (22,33.75)[(0,0)\[l\][$n_5$]{}]{} (72,33.75)[(0,0)\[l\][$n_5$]{}]{} (17.7,39)[(0,0)\[r\][$n_7$]{}]{} (25,39)[(0,0)\[l\][$n_8$]{}]{} (67.7,39)[(0,0)\[r\][$n_7$]{}]{} (75,39)[(0,0)\[l\][$n_8$]{}]{} (21,46)[(0,0)\[b\][$n_6$]{}]{} (71,46)[(0,0)\[b\][$d-n_6-n_7-n_8$]{}]{} (46,30)[(0,0)[$=$]{}]{} (11,1)[(0,0)\[b\][$n_1$]{}]{} (21,1)[(0,0)\[b\][$n_3$]{}]{} (31,1)[(0,0)\[b\][$n_2$]{}]{} (61,1)[(0,0)\[b\][$n_1$]{}]{} (71,1)[(0,0)\[b\][$n_3$]{}]{} (81,1)[(0,0)\[b\][$n_2$]{}]{} (9.5,10)[(0,0)\[r\][$n_4$]{}]{} (32,10)[(0,0)\[l\][$n_5$]{}]{} (59.5,12)[(0,0)\[r\][$d-n_1-n_4$]{}]{} (57.5,9)[(0,0)\[r\][${}-n_6$]{}]{} (82,12)[(0,0)\[l\][$d-n_2-n_5$]{}]{} (83,9)[(0,0)\[l\][${}-n_7$]{}]{} (17,10)[(0,0)\[l\][$n_6$]{}]{} (25,10)[(0,0)\[r\][$n_7$]{}]{} (67,10)[(0,0)\[l\][$n_6$]{}]{} (75,10)[(0,0)\[r\][$n_7$]{}]{} (21,16)[(0,0)\[b\][$n_8$]{}]{} (71,16)[(0,0)\[b\][$d-n_3-n_6-n_7-n_8$]{}]{} (46,5)[(0,0)[$=$]{}]{}
Three-loop HQET diagrams {#HQET}
========================
There are 10 generic topologies of 3-loop HQET propagator diagrams. They can be reduced, using integration by parts, to 8 basis integrals [@G:00]. This algorithm is implemented in the REDUCE package Grinder [@G:00], available at http://www-ttp.physik.uni-karlsruhe.de/Progdata/ttp00/ttp00-01/. Five basis integrals are trivial. Two can be expressed via ${}_3F_2$ hypergeometric functions of the unit argument [@BB:94; @G:00]. Their expansions in $\varepsilon$ can be obtained in the same way as in the massless case, the results are presented in [@G:01]. The last and most difficult basis integral was found in [@G:01] up to the finite term in $\varepsilon$, using direct integration in the coordinate space. More terms of its $\varepsilon$-expansion were recently obtained in [@CM:02] using inversion, as explained in the next Section.
Three-loop on-shell diagrams {#OS}
============================
Calculations of on-shell diagrams with massive quarks in QCD are necessary for obtaining coefficients in the HQET Lagrangian and $1/m$ HQET expansions of QCD operators by matching. There are 2 generic topologies of 2-loop on-shell propagator diagrams with a single non-zero mass. They can be reduced, using integration by parts, to 3 basis integrals. This algorithm is implemented in the REDUCE package RECURSOR [@B:92] and the FORM package SHELL2 [@FT:92]. Two basis integrals are trivial, and the third one is expressed via two ${}_3F_2$ hypergeometric functions of the unit argument. However, some of their indices tend to half-integers at $\varepsilon\to0$, and the algorithm of expansion in $\varepsilon$ discussed in Sect. \[QCD\] is not applicable. This approach was used for QCD/HQET matching of heavy-light quark currents [@BG:95a] and chromomagnetic interaction [@CG:97].
The case when there is another non-zero mass was systematically studied in [@DG:99]. There are 4 basis integrals, 2 of them trivial, and 2 are expressed via ${}_3F_2$ hypergeometric functions of the mass ratio squared. Finite parts at $\varepsilon\to0$ are expressed via dilogarithms. More terms of expansions of the general results [@DG:99] in $\varepsilon$ were recently obtained [@AMR:02]. The REDUCE package [@DG:99] is available at http://wwwthep.physik.uni-mainz.de/Publications/progdata/mzth9838/ Mm.red.
There are 11 generic topologies of 3-loop on-shell propagator diagrams with a single non-zero mass (10 of them are the same as in HQET, and one involves a heavy-quark loop). They can be reduced, using integration by parts, to 18 basis integrals [@MR:00]. This algorithm is implemented as the FORM package SHELL3 [@MR:00]. The basis integrals are mostly known from QED [@LR:96].
Some on-shell diagrams are related to HQET ones by inversion of Euclidean integration momenta. One- and two-loop relations were presented in [@BG:95]. Three-loop relations are shown in Fig. \[Fig\]. The second of them was used in [@CM:02] to relate the the convergent ladder HQET diagram at $\varepsilon=0$ to the known on-shell ladder diagram.
[99]{}
D.J. Broadhurst and A.G. Grozin, in *New computing techniques in physics research IV*, ed. B. Denby and D. Perret-Gallix, World Scientific (1996) 217.
D.J. Broadhurst and A.G. Grozin, Phys. Lett. B267 (1991) 105.
N. Gray, D.J. Broadhurst, W. Grafe, K. Schilcher, Z. Phys. C48 (1990) 673;\
D.J. Broadhurst, N. Gray, K. Schilcher, Z. Phys. C52 (1991) 111;\
D.J. Broadhurst, Z. Phys. C54 (1992) 599.
J. Fleischer and O.V. Tarasov, Phys. Lett. B283 (1992) 129; Comput. Phys. Commun. 71 (1992) 193.
F.V. Tkachov, Phys. Lett. B100 (1981) 65;\
K.G. Chetyrkin and F.V. Tkachov, Nucl. Phys. B192 (1981) 159.
A.G. Grozin, J. High Energy Physics 03 (2000) 013; hep-ph/0002266.
K. Melnikov and T. van Ritbergen, Phys. Lett. B482 (2000) 99; Nucl. Phys. B591 (2000) 515.
S.G. Gorishny, S.A. Larin, F.V. Tkachev, Preprint INR P-0330, Moscow (1984);\
S.G. Gorishny, S.A. Larin, L.R. Surguladze, F.V. Tkachev, Comput. Phys. Commun. 55 (1989) 381;\
S.A. Larin, F.V. Tkachev, J.A.M. Vermaseren, Preprint NIKHEF-H/91-18, Amsterdam (1991).
M. Veltman, SCHOONSCHIP, CERN (1967);\
H. Strubbe, Comput. Phys. Commun. 8 (1974) 1.
J.A.M. Vermaseren, Symbolic manipulations with FORM, Amsterdam (1991); math-ph/0010025 (2002).
D.J. Broadhurst, Preprint OUT-4102-41 (1992), see in: D.J. Broadhurst, A.L. Kataev, O.V. Tarasov, Phys. Lett. B298 (1993) 445, and in: D.J. Broadhurst, hep-th/9909185.
A.C. Hearn, REDUCE User’s Manual, Version 3.7 (1999).
A.G. Grozin, Using REDUCE in High Energy Physics, Cambridge University Press (1997).
A.V. Kotikov, Phys. Lett. B375 (1996) 240.
D.J. Broadhurst, J.A. Gracey, D. Kreimer, Z. Phys. C75 (1997) 559.
A.G. Grozin, in *QCD: theory and experiment*, AIP conference proceedings 602 (2001) 271.
S. Moch, P. Uwer, S. Weinzierl, J. Math. Phys. 43 (2002) 3363.
S. Weinzierl, Comput. Phys. Commun. 145 (2002) 357.
C. Bauer, A. Frink, R. Kreckel, Preprint MZ-TH-00-17 (2000), cs.sc/0004015.
J.A.M. Vermaseren, Int. J. Mod. Phys. A14 (1999) 2037.
D.I. Kazakov, Theor. Math. Phys. 62 (1985) 84.
D.J. Broadhurst, Preprint OUT-4102-62 (1996), hep-th/9604128;\
J.M. Borwein, D.M. Bradley, D.J. Broadhurst, Electronic J. Combinatorics 4 No. 2 (1997) \#R5, hep-th/9611004;\
J.M. Borwein, D.M. Bradley, D.J. Broadhurst, P. Lisonek, Electronic J. Combinatorics 5 (1998) \#R38, math.NT/9812020.
M. Beneke and V.M. Braun, Nucl. Phys. B426 (1994) 301.
A. Czarnecki and K. Melnikov, Phys. Rev. D66 (2002) 011502.
D.J. Broadhurst and A.G. Grozin, Phys. Rev. D52 (1995) 4082.
A. Czarnecki and A.G. Grozin, Phys. Lett. B405 (1997) 142.
A.I. Davydychev and A.G. Grozin, Phys. Rev. D59 (1999) 054023.
M. Argeri, P. Mastrolia, E. Remiddi, Nucl. Phys. B631 (2002) 388.
S. Laporta and E. Remiddi, Phys. Lett. B379 (1996) 283;\
K. Melnikov and T. van Ritbergen, Phys. Rev. Lett. 84 (2000) 1673.
|
---
abstract: 'We give the first explicit computations of rational homotopy groups of spaces of “long knots” in Euclidean spaces. We define a spectral sequence which converges to these rational homotopy groups whose $E^1$ term is defined in terms of braid Lie algebras. For odd $k$ we establish a vanishing line for this spectral sequence, show the Euler characteristic of the rows of this $E^1$ term is zero, and make calculations of $E^2$ in a finite range.'
address:
- |
Department of Mathematics and Computer Science\
Saint Louis University\
St. Louis, MO 63103
- |
Department of Mathematics\
Brown University\
Providence, RI 02906
author:
- 'Kevin P. Scannell'
- 'Dev P. Sinha'
bibliography:
- 'data.bib'
date: 'October 27, 2000'
title: 'A one-dimensional embedding complex'
---
[^1]
Introduction
============
In this paper we introduce a spectral sequence which converges to the rational homotopy groups of ${Emb(I, {\mathbb R}^k \times I)}$, for $k \geq 3$, which is the space of embeddings of an interval in ${{\mathbb R}}^k \times I$ with fixed endpoints and tangent vectors at those endpoints (essentially, the space of long knots in ${{\mathbb R}}^{k+1}$). Our starting point is the work of [@Si00], which defines such spectral sequences in terms of the topology of configuration spaces. The paper [@Si00] in turn builds upon work of Goodwillie and his collaborators [@GK00; @GW99a; @We99], who have built a powerful theory for studying spaces of embeddings in general.
The rational homotopy groups of configuration spaces, which comprise the $E^1$ term, are Lie algebras which are well-known (we call them “braid Lie algebras”). Just as the study of cohomology of embedding spaces gives rise to the study of graph cohomology, which has been studied extensively [@BN95; @Ko94; @To00; @Va92], our complexes of braid Lie algebras are interesting new objects in quantum algebra. Similar complexes were described by Kontsevich in his plenary talk [@Ko00].
We start by reviewing the computation of the rational homotopy groups of ordered configurations of points in Euclidean space as a graded Lie algebra under Whitehead product, as well as some basics of free Lie algebras, which appear as subalgebras of these homotopy groups. At that point, we will have the necessary algebraic background to define the chain complexes which are the rows of the $E^1$ term of our spectral sequence. It turns out that through $E^2$, our spectral sequence for the homotopy groups of ${Emb(I, {\mathbb R}^k \times I)}$ depends, up to regrading, only on the parity of $k$. We focus on odd $k$. We prove some fundamental facts about these complexes, such as the vanishing of their Euler characteristic. We proceed to describe algorithms for computing the homology of these chain complexes, and in the final section present the results of these computations in low dimensions. In some cases, the classes which arise in $E^2$ must survive, implying the existence of non-trivial spherical families of embeddings. Non-zero higher differentials are also possible. We end with a brief discussion of the case of $k$ even, which pertains to the theory of finite-type knot invariants.
The second author would like to thank Tom Goodwillie for many helpful discussions and Ira Gessel for help in simplifying the proof of Theorem \[chi\].
The Rational Homotopy Groups of Configuration Spaces {#compFn}
====================================================
We remind the reader of the computations of the rational homotopy groups of configuration spaces [@FN62] and their Lie algebra structure under Whitehead product. Throughout this paper, $\pi_\ast(X)$ will denote the homotopy groups of $X$ tensored with the rational numbers. Let $F(M, n)$ denote the space of ordered configurations of $n$ distinct points in a manifold $M$. We consider the projection $\rho: F(M,n) \to F(M, n-1)$ defined by forgetting the last point in the configuration, which is in fact a fiber bundle whose fiber is $M \setminus \{ (n-1) \; {\rm points} \}$. Let $\iota$ denote the inclusion of the fiber. When $M =
{{\mathbb R}}^{k+1}$, the fibers are homotopy equivalent to $\bigvee_{n-1} S^k$, and the projection map admits a section, by adding a point (say in a fixed direction at a large distance) to a configuration of $n-1$ points. This section leads to a splitting of the long exact sequence of a fibration into split short exact sequences $$0 \to \pi_i(\bigvee_{n-1} S^k) \to \pi_i ({F({\mathbb R}^k \times I, n)}) \to
\pi_i(F({{\mathbb R}}^{k+1}, n-1)) \to 0.$$ By induction, we find that additively $$\pi_i({F({\mathbb R}^k \times I, n)}) \cong \bigoplus_{j=1}^{n-1} \pi_i(\bigvee_j S^k).$$
We now compute the structure of the rational homotopy groups $\pi_\ast({F({\mathbb R}^k \times I, n)})$ as a Lie algebra under the Whitehead product.
\[defB\] Let ${{\mathcal B}}^o_{n}$ (respectively ${{\mathcal B}}^e_n$) be the Lie algebra (super Lie algebra for ${{\mathcal B}}^e_n$) generated over ${{\mathbb Q}}$ by classes $x_{ij}$ for $1 \leq i, j \leq n$ with relations
1. $x_{ij} = x_{ji}$ (respectively, $-x_{ji}$ for ${{\mathcal B}}^e_n$). \[rel1\]
2. $x_{ii} = 0$ \[rel2\]
3. $[x_{ij}, x_{\ell m}] = 0 \; \; \text{if} \;\; \{ i, j \} \cap
\{\ell, m\} = \emptyset$ \[rel3\]
4. $[x_{ij}, x_{j\ell}] = [x_{j\ell}, x_{\ell i}] = [x_{\ell i}, x_{ij}]$. \[rel4\]
We call ${{\mathcal B}}^o_n$ and ${{\mathcal B}}^e_n$ [*braid Lie algebras*]{}.
\[piconfig\] There is a Lie algebra isomorphism between $\pi_\ast({F({\mathbb R}^k \times I, n)})$ and ${{\mathcal B}}^o_n$ if $k$ is odd or ${{\mathcal B}}^e_n$ if $k$ is even.
We first define classes which generate $\pi_\ast({F({\mathbb R}^k \times I, n)})$ as a Lie algebra. Pick a basepoint in ${F({\mathbb R}^k \times I, n)}$, say with $z_i = (2i, 0, \ldots, 0)$ for definiteness. There are $\binom{n}{2}$ generators of $\pi_k({F({\mathbb R}^k \times I, n)})$, corresponding to distinct pairs $\{i, j\} \subseteq \{1, \ldots, n\}$, which we now realize geometrically. We define $b_{ij} \in \pi_k({F({\mathbb R}^k \times I, n)})$ as the class represented by the composite of two maps. First, we collapse $S^k$ onto $S^k \vee I$ by sending the “southern hemisphere” of $S^k$ to $I$ through the height function. Next, choose a path $\gamma_{ij}$ from $z_i$ to the point $(2j - 1, 0, \ldots, 0)$ in the complement of the other configuration points, and let $\iota_j$ denote the map which sends $S^k$ to the unit sphere about the point $z_j$. To define $b_{ij}$ we compose the collapse map above with the map $S^k \vee I$ to ${F({\mathbb R}^k \times I, n)}$ which sends $t\in I$ to ${F({\mathbb R}^k \times I, n)}$ as $(z_1, \ldots, z_{i-1}, \gamma_{ij}(t), z_{i+1}, \ldots, z_n )$ and $S^k$ to ${F({\mathbb R}^k \times I, n)}$ as $v \mapsto (z_1, \ldots, z_{i-1}, \iota_j(v), z_{i+1}, \ldots, z_n )$.
To see inductively that these classes are generators of $\pi_k({F({\mathbb R}^k \times I, n)})$, we simply note that $b_{in}$ is equal to the image under $\iota_\ast$ of the generator of $\pi_k (\bigvee_{n-1} S^k)$ defined by the inclusion of the $i$th wedge factor.
It is simple to check that these $b_{ij}$ satisfy the relations for $x_{ij}$ in the definition of ${{\mathcal B}}^o_n$. Note that from the usual graded commutativity of the Whitehead product, brackets in $b_{ij}$ anti-commute when $k$ is odd and commute when $k$ is even. Note that $b_{ij} = (-1)^{k+1} b_{ji}$ and $b_{ii} = 0$ so that relations (\[rel1\]) and (\[rel2\]) are satisfied.
We next verify that the $b_{ij}$ satisfy relation (\[rel3\]). Recall that if $\{f\}$ and $\{g\}$ are elements of $\pi_k(X)$ then $[\{f\}, \{g\}] = 0$ if and only if $f \vee g \colon S^k \vee S^k \to X$ extends to $S^k \times S^k$. If $\{ i, j \} \cap \{\ell, m \} = \emptyset$, the map $b_{ij} \vee b_{\ell m}$ may be so extended by sending $$v \times w \mapsto
(p_1, \ldots, p_{i-1}, \iota_{ij}(v), p_{i+1}, \ldots, p_{\ell-1},
\iota_{\ell m}(w), p_{\ell+1}, \ldots),$$ where $\iota_{ij}$ is the composite of the collapse map of $S^k$ onto $S^k \vee I$ with $\iota_j \vee \gamma_{ij}$. Informally we say that $z_i$ can travel around $z_j$ and $z_\ell$ can travel around $z_m$ without having their paths (the images of $S^k$ under the projection onto the $i$th and $\ell$th coordinates) intersect.
Next, we verify that the $b_{ij}$ satisfy relation (\[rel4\]). Equivalently, we claim that $[b_{j \ell}, b_{ij} + b_{i\ell}] = 0$. Informally, we say that $b_{ij} + b_{i\ell}$ is represented by a map in which $z_i$ travels around $z_j$ and $z_\ell$ but no other points in the configuration, and this may happen simultaneously as $z_j$ travels around $z_\ell$, giving an extension of $(b_{ij} + b_{i\ell}) \vee b_{j\ell}$ to $S^k \times S^k$ similar to the one given for $b_{ij} \vee b_{\ell m}$.
We claim that relations (\[rel1\]) through (\[rel4\]) are a complete set of relations for $\pi_\ast({F({\mathbb R}^k \times I, n)})$. This follows from the fact that these relations may be used to reduce to an additive basis of Lie algebra monomials of the form $[ \cdots [ b_{im}, b_{jm}] \cdots b_{\ell m} \cdots ]$, where $i, j, \ell < m \leq n$. We exhibit this claim algorithmically when discussing the computations in §\[S:Algs\]; see in particular Algorithm 5.2.
The fiber sequence above leads us to identify some subalgebras of $\pi_\ast({F({\mathbb R}^k \times I, n)})$ which are free Lie algebras. Tensored with the rationals, the homotopy groups of wedges of spheres, $\pi_{\ast+1}(\bigvee_j S^k)$ for $k>1$, are well known [@Hi55; @Wh78] to form free Lie algebra under Whitehead product, with $j$ generators in degree $\ast + 1= k $. Since the inclusion map $\iota : \bigvee_{n-1} S^k \to {F({\mathbb R}^k \times I, n)}$ is injective on homotopy, and by naturality of Whitehead products, the image of these homotopy groups under $\iota_\ast$ in $\pi_\ast({F({\mathbb R}^k \times I, n)})$ is a free Lie algebra which is generated by the classes $b_{in}$.
In the development of the spectral sequence we will in fact need the rational homotopy groups of $FT(k,n) = F({{\mathbb R}}^k \times I, n) \times (S^k)^n$. We call $FT(k,n)$ the space of tangential configurations, thinking of the points in $S^k$ as unit tangent vectors at points of a configuration. Recall that the homotopy groups of a product of spaces is a direct sum of their homotopy groups, and all Whitehead products between these summands are zero. Let $\Lambda^o$ (respectively $\Lambda^e$) be the free Lie algebra (respectively super Lie algebra) on one generator. Let ${{\mathcal{BT}}}^o_n$ denote ${{\mathcal B}}^o_n \oplus n\Lambda^o$ and similarly for ${{\mathcal{BT}}}^e_n$.
\[C:comp\] There is a Lie algebra isomorphism between $\pi_{\ast+1}(FT(k,n))$ and ${{\mathcal{BT}}}^o_n$ if $k$ is odd or ${{\mathcal{BT}}}^e_n$ if $k$ is even.
These isomorphisms respect the gradings involved. We may grade ${{\mathcal{BT}}}^o_n$ according to the number of generators appearing in a bracket. The $d$th graded summand of ${{\mathcal{BT}}}^o_n$ coincides with $\pi_{d(k-1) + 1}(FT(k,n))$.
Free Lie algebras
=================
Let ${{\mathcal L}}(A)$ denote the free Lie algebra over ${{\mathbb Q}}$ on a set $A$ of symbols. For our explicit computations, we must choose an additive basis for ${{\mathcal L}}(A)$. Natural labels for elements of free Lie algebras can be obtained from rooted, planar binary trees (hereafter, referred to as simply a [*trees*]{}) with leaves labeled by elements of $A$. Such a tree prescribes a bracketing of the elements which label the leaves. The number of leaves is the [*degree*]{} of the tree. Trees with a root but no branches (degree one) are identified with the set of symbols $A$. When the context is clear, we will identify trees with the free Lie algebra elements they produce. The obvious product of two trees $x$ and $y$ (a tree with a new root, left subtree $x$, and right subtree $y$) corresponds to the product in the Lie algebra and will therefore also be denoted $[x,y]$.
A set $\mathcal{H}$ of trees is called a [*Hall set*]{} for ${{\mathcal L}}(A)$ [@Re93 §4.1] if the following conditions hold:
1. $\mathcal{H}$ has a total order $\leq$
2. $A \subset \mathcal{H}$
3. If $h = [h_1,h_2] \in \mathcal{H}$ then $h_2 \in \mathcal{H}$ and $h < h_2$.
4. For any tree $h = [h_1,h_2]$ of degree at least two, we have $h \in \mathcal{H}$ if and only if $h_1, h_2 \in \mathcal{H}$, $h_1 < h_2$, and either $h_1 \in A$ or $h_1 = [x,y]$ with $h_2 \leq y$. \[C:final\]
It is straightforward to show that a Hall set forms an additive basis of ${{\mathcal L}}(A)$ [@Re93 Thm. 4.9] (cf. Algorithm 5.1 below). The basis elements comprising a fixed Hall set will be called [*Hall trees*]{}. It is easy to see that (many) Hall sets exist [@Re93 Prop. 4.1]; for completeness, we give a quick description of an algorithm for creating one.
[**Algorithm 3.1**]{} (Generating a Hall set).
1. 2. 3. 4.
[**Example.**]{} The output of Algorithm 3.1 with $A = \{a,b\}$ and $d = 5$ is the following list: $$\begin{aligned}
&a
\hspace{3ex}
[a,[[[a,b],b],b]]
\hspace{3ex}
[a,[a,[a,[a,b]]]]
\hspace{3ex}
[a,[a,[[a,b],b]]]
\hspace{3ex}
[a,[[a,b],b]] \\
&[a,[a,[a,b]]]
\hspace{3ex}
[a,[a,b]]
\hspace{3ex}
[[a,[a,b]],[a,b]]
\hspace{3ex}
[a,b]\\
&[[a,b],[[a,b],b]]
\hspace{3ex}
[[a,b],b]
\hspace{3ex}
[[[a,b],b],b]
\hspace{3ex}
[[[[a,b],b],b],b]
\hspace{3ex}
b\end{aligned}$$
The following result will be used in computing the Euler characteristic of the chain complexes which appear as the rows of the $E^1$ term of our spectral sequence.
[@Re93 Cor. 4.14] \[L:count\] The number of Hall trees for $\mathcal{L}(A)$ of degree $d$ equals $$\frac{1}{d} \sum_{j | d} \mu(j) |A|^{d/j}$$ where $\mu$ is the Möbius function.
The spectral sequence
=====================
In this section we present an explicit realization of the spectral sequence introduced in [@Si00 §4] which converges to the rational homotopy groups of ${Emb(I, {\mathbb R}^k \times I)}$. The spectral sequence arises from models of ${Emb(I, {\mathbb R}^k \times I)}$ which are reminiscent of cosimplicial spaces, but whose combinatorics are based on Stasheff polytopes instead of simplices. The entries are (Fulton-MacPherson compactified versions of) the ordered tangential configuration spaces $FT(k,n) = F({{\mathbb R}}^k \times I, n) \times (S^k)^n$
We can now describe the $E^1$ term of this spectral sequence. We first describe an unreduced version (which will be denoted throughout by the addition of a tilde $\tilde{E}^1$), followed by the reduced version (denoted simply $E^1$). Recall that one way to obtain the $E^1$ term of the homotopy spectral sequence of a cosimplicial space is by first passing to homotopy groups of the entries, which if all entries are simply connected defines a cosimplicial abelian group. The $E^1$ term is then the chain complex associated to this cosimplicial abelian group, which is bi-graded because the homotopy groups themselves are graded. We show in [@Si00] that even though our models are based on Stasheff polyhedra, applying homotopy groups to these models gives rise to cosimplicial abelian groups. Hence the $\tilde{E}^1$ term of our spectral sequence is the chain complex of the cosimplicial abelian group: $$\pi_\ast( FT(k, 0)) = pt. \overset{\Rightarrow}{\leftarrow} \pi_\ast(FT(k,1))
\overset{\Rrightarrow}{\Leftarrow} \pi_\ast(FT(k, 2)) \cdots$$
Here the coface maps $d^i_\ast$ are induced by maps $d^i$ on configuration spaces (or rather their Fulton-MacPherson compactifications) which are “doubling” the $i$th point in a tangential configuration in the direction of the unit tangent vector determined by the $i$th factor of $S^k$, or if $i$ = $0$ or $n$ by adding a point to the configuration at $(\vec{0}, 0)$ or $(\vec{0}, 1) \in {{\mathbb R}}^k \times I$. The codegeneracy maps $s^i$ are defined by forgetting a point in the configuration.
\[T:dpsmain\] There is a second-quadrant spectral sequence whose $E^1$ term is given by $\tilde{E}^1_{-p, q} = \pi_q(FT(k, p))$ and $d^1$ given by $\Sigma_i (-1)^i d^i_\ast$ which converges to $\pi_\ast({Emb(I, {\mathbb R}^k \times I)})$.
We now make the coface and codegeneracy maps algebraically explicit. Recall from Corollary \[C:comp\] that the rational homotopy groups of $FT(k,n)$ are isomorphic to the Lie algebra ${{\mathcal{BT}}}^o_n$ (or ${{\mathcal{BT}}}^e_n$) generated by classes $x_{ij}$ and $y_i$ for $1 \leq i,j \leq n$ and with relations defined as in Definition \[defB\] and so that $[y_i, x_{j \ell}] = 0$ for all $i, j, \ell$ and $[y_i, y_j] = 0$ for $i \neq j$.
Define $\sigma^\ell(i)$ to be $i$ if $i<\ell$ and $i+1$ if $i> \ell$. For $0 \leq \ell \leq n+1$ define $\partial^\ell \colon {{\mathcal{BT}}}^o_n \to {{\mathcal{BT}}}^o_{n+1}$ (respectively from ${{\mathcal{BT}}}^e_n$ to ${{\mathcal{BT}}}^e_{n+1}$) to be the Lie algebra homomorphism defined on generators as follows. $$\partial^\ell (x_{ij}) =
\begin{cases}
x_{\sigma^\ell(i) \sigma^\ell(j)} &\text{if $i,j \neq \ell$} \\
x_{i\sigma^\ell(j)} + x_{i+1,\sigma^\ell(j)} &\text{if $i = \ell$}
\end{cases}$$
$$\partial^\ell (y_i) =
\begin{cases}
y_{\sigma^\ell(i)} &\text{if $i \neq \ell$} \\
x_{i,i+1} + y_i + y_{i+1} &\text{if $i < j = \ell$}
\end{cases}$$
For $1 \leq \ell \leq n$ define $\phi^\ell \colon {{\mathcal{BT}}}^o_n \to {{\mathcal{BT}}}^o_{n-1}$ (respectively from ${{\mathcal{BT}}}^e_n$ to ${{\mathcal{BT}}}^e_{n-1}$) to be the Lie algebra homomorphism defined on generators as follows. $$\phi^\ell(x_{ij}) =
\begin{cases}
x_{\sigma^\ell(i) \sigma^\ell(j)} &\text{if $i,j \neq \ell$} \\
0 &\text{if $i$ or $j$ = $\ell$}
\end{cases}$$ $$\phi^\ell(y_{i}) =
\begin{cases}
y_{\sigma^\ell(i)} &\text{if $i \neq \ell$} \\
0 &\text{if $i$ = $\ell$}
\end{cases}$$
The following proposition is immediate from the definitions of the classes $b_{ij}$ and the maps $d^\ell$ and $s^\ell$.
Under the isomorphisms of Corollary \[C:comp\] the homomorphisms $d^\ell_\ast$ and $s^\ell_\ast$ coincide with $\partial^\ell$ and $\phi^\ell$ respectively.
Making Theorem \[T:dpsmain\] algebraically explicit using Corollary \[C:comp\] and the previous proposition leads us to the following spectral sequence whose $E^1$ term is defined in terms of braid Lie algebras.
\[algvers\] There is a second-quadrant spectral sequence which converges to $\pi_\ast({Emb(I, {\mathbb R}^k \times I)})$ such that $\tilde{E}^1_{-n, d(k-1)+1}$ is isomorphic to the $d$th graded summand of ${{\mathcal{BT}}}^o_n$ (respectively ${{\mathcal{BT}}}^e_n$), $E^1_{-n,q} = 0$ when $q-1$ is not a multiple of $k-1$, and $d^1$ is given by $\Sigma_i (-1)^i \partial^i$.
For the rest of the paper, we focus on the case in which $k$ is odd.
A useful reduction when studying cosimplicial abelian groups is the replacement of the $n$th group by the intersection of the kernels of the codegeneracy maps. Such a reduction does not change the homology of the associated chain complex. First note that the codegeneracy maps $\phi^\ell \colon {{\mathcal{BT}}}^o_n \to
{{\mathcal{BT}}}^o_{n-1}$ respect the direct sum decomposition ${{\mathcal{BT}}}^o_n = {{\mathcal B}}^o_n \oplus n\Lambda^o$. Restricted to the $\Lambda^o$ factors, the intersection of the kernel of the $\phi^\ell$ is zero unless $n$ is equal to one, in which case it is all of $\Lambda^o$. Restricted to the ${{\mathcal B}}^o_n$ factor, the kernel of the codegeneracy map $\phi^n \colon {{\mathcal B}}^o_n \to
{{\mathcal B}}^o_{n-1}$ is the subalgebra generated by the classes $x_{in}$, which is in fact a free Lie algebra (see the remarks following the proof of Theorem \[piconfig\]). We identify the kernel of all of the $\phi^\ell$ as a submodule of this free Lie algebra.
For $n>1$, let $M_{d,n}$ be the submodule of of the degree $d$ summand of ${{\mathcal{BT}}}^o_n$ generated by brackets of the classes $x_{in}$ such that each $i$ from $1$ to $n-1$ appears as an index. Let $M_{1,1} = {{\mathcal{BT}}}^o_1$.
\[reducedss\] There is a spectral sequence which converges to $\pi_\ast({Emb(I, {\mathbb R}^k \times I)})$ whose $E^1$ term is given by $E^1_{-n, d(k-1)+1} = M_{d,n}$ and whose $d^1$ is the restriction to this submodule of the $d^1$ of Corollary \[algvers\].
Note that $M_{d,n} = 0$ for $d<n-1$, which leads to the following vanishing theorem.
\[vanish\] In the spectral sequence of Theorem \[reducedss\], $E^1_{-p, q} = 0$ if $q<p(k-1) + 2 - k$.
It is interesting to note that while the modules $M_{d, n}$ may be defined purely in terms of the free Lie algebra (on $n-1$ generators), the boundary maps between them require extending the free Lie algebra to a braid Lie algebra. From the algebraic definition of $d^1$ it is not obvious that its restriction to $M_{d,n}$ maps to $M_{d,n+1}$.
Since computing the $E^2$ term amounts to computing the cohomology of the complexes $M_{d,\ast}$, as a warmup we will compute the rank of $M_{d,n}$, which we denote $R(d,n)$, and will show that $\chi (M_{d,\ast}) = 0$. Recall that the number of Hall trees of degree $d$ with $n$ symbols is equal by Lemma \[L:count\] to $\frac{1}{d} {\sum}_{j|d} \mu(j) n^{d/j}$. We may produce a basis of $M_{d,n}$ by first considering all brackets of degree $d$ and throwing away ones in which fewer than $n$ elements appear. We find that $$R(d,n) = \frac{1}{d} {\sum}_{i=0}^n (-1)^i \binom{n}{i}
{\sum}_{j|d} \mu(j) i^{d/j}.$$
We pause to define $S(d,n) = {\sum}_{i=0}^n (-1)^i \binom{n}{i} i^d$, which are essentially Stirling numbers. There is a combinatorial interpretation of $S(d,n)$ as the number of surjections from a $d$ element set onto an $n$ element set (to verify this, count all set maps and subtract the non-surjections). Note as well that the $S(d,n)$ have a generating function, as $${\sum}_{m=0}^\infty S(m,n) \frac{x^m}{m!} = (e^x -1)^m.$$
Reordering the summations of $R(d,n)$ we find the following:
\[P:count\] $R(d,n) = \frac{1}{d} {\sum}_{j|d} \mu(j) S(d/j, n)$.
We may give $R(d,n)$ a combinatorial interpretation in line with this equality as the number of surjections of a $d$ element set to an $n$ element set which are not invariant under any cyclic permutation of the $d$ element set, modulo cyclic permutations of the $d$ element set. It would be interesting to find a bijection between such equivalence classes of surjections and a basis of $M_{d,n}$. Such a combinatorial interpretation would be particularly interesting for $M_{n,n}$ which, along with its $\Sigma_n$ action by permuting the letters, is known as $Lie(n)$ and arises in the calculus of functors approach to homotopy theory [@AM99].
\[chi\] The Euler characteristic of $M_{d,\ast}$ is zero for $d > 2$.
The Euler characteristic of the complex $M_{d,\ast}$ is by definition ${\sum}_{\ell = 1}^d (-1)^\ell R(d,\ell)$, which after applying Proposition \[P:count\], reversing the order of summation, and ignoring zero terms, is equal to $$\frac{1}{d} {\sum}_{j|d} \mu(j) {\sum}_{\ell = 1}^{d/j} (-1)^\ell S(d/j, \ell).$$ We claim that ${\sum}_{\ell = 1}^m (-1)^\ell S(m, \ell) = (-1)^m$, which can be verified by computing the coefficient of $x^m/m!$ of ${\sum}_{p = 0}^m (-1)^p (e^x - 1)^p$. Hence the Euler characteristic is equal to $\frac{1}{d} {\sum}_{j|d} \mu(j) (-1)^{d/j}$, which is zero if $d > 2$.
Algorithms {#S:Algs}
==========
In this section we provide a detailed description of the methods used to compute the boundary operators in the complexes described above. These algorithms can be performed by hand for the complexes of small degree $d$, but are best implemented on the modern electronic computer otherwise.
Because the product of two Hall trees is not necessarily a Hall tree, one must have an algorithm which takes an arbitrary tree representing a free Lie algebra element, and expresses it as a linear combination of Hall trees. The proof that this algorithm terminates and produces the desired result is contained in the proof of Theorem 4.9 in [@Re93].
[**Algorithm 5.1**]{} (Hallification).
1. 2. 3. 4. 5.
The following algorithm uses the relations for ${{\mathcal B}}^o_{n}$ from Definition \[defB\] and the Jacobi identity to express elements of ${{\mathcal B}}^o_{n}$ in a standardized form. It will be used in the computation of the boundary operator $\partial^n$ in Algorithm 5.3 below.
We say that a bracket in the classes $x_{ij}$ for $1 \leq i, j \leq n$ is [*pure*]{} if either all $x_{ij}$ which appear are of the form $x_{in}$ or none are of this form.
[**Algorithm 5.2**]{} (Standard basis for ${{\mathcal B}}^o_n$).
1. 2. 3. 4. 5. 6. 7.
A simple induction argument shows that this argument terminates and produces the desired result. Namely, we associate to a bracket $t$ the pair $(a,b)$ where $a$ is the number of generators $x_{ij}$ with $j < n$ appearing in $t$, and $b$ is the degree of the smallest impure sub-bracket found in step (2). We order such pairs lexicographically, with the minimum $(0,0)$ being achieved by pure brackets. At every step, this algorithm produces brackets whose associated pairs are less than that of the original. Steps (5) and (6), corresponding to $b=2$, clearly reduce $a$. Step (7) leaves $a$ unchanged but reduces $b$, since the sub-bracket $[x,y]$ of $s$ which is initially pure becomes impure in all terms which occur after applying the Jacobi identity. Finally note that since $b$ in such an associated pair is bounded by the degree of the bracket, there are only finitely many pairs less than a given one, so the algorithm must terminate after a finite number of recursive steps.
Note that the terms in the linear combination output by Algorithm 5.2 which do not involve any $x_{in}$ can be run recursively through the algorithm as elements of ${{\mathcal B}}^o_{n-1}$, yielding the standard form claimed in the proof of Theorem \[piconfig\].
The final algorithm is the heart of the calculation; it computes $\partial^\ell$ for $\ell = 0, \ldots, n$, exploiting the fact that these maps are Lie algebra homomorphisms. Observe that $\partial^{n+1}$ is simply the natural inclusion of ${{\mathcal{BT}}}^o_n$ into ${{\mathcal{BT}}}^o_{n+1}$ and therefore requires no detailed description.
[**Algorithm 5.3**]{} (Boundary operator).
1. 2. 3. 4. 5. 6.
An example of this algorithm is worked out by hand in the next section.
Results {#S:calcs}
=======
In this section we present some results of the computations described in the previous section. We will choose the gradings to correspond to the case $k=3$, i.e. embeddings in ${{\mathbb R}}^4$.
First we note that in the degree one case, we have $E^1_{-1,3} = {{\mathbb Q}}$, generated by $y_1$, $E^1_{-2, 3} = {{\mathbb Q}}$ generated by $x_{12}$, and $d^1$ is an isomorphism. In degree two, the only non-zero entry is $E^1_{-3,5} = {{\mathbb Q}}$, generated by $[x_{13}, x_{23}]$, implying $E^2_{-3,5} = {{\mathbb Q}}$.
We proceed by working out the first non-trivial boundary operator $d^1 : E^1_{-3,7} \to E^1_{-4,7}$ by hand. These spaces are by definition $M_{3,3}$ and $M_{3,4}$. Bases are obtained by creating, with Algorithm 3.1, Hall bases for the free Lie algebra generated by $\{x_{13}, x_{23}\}$ (resp. $\{x_{14}, x_{24}, x_{34}\}$) and selecting the elements which have degree $3$ and such that all possible values of $i$ appear. It turns out that each space is two-dimensional; the first is generated by $[x_{13},[x_{13},x_{23}]]$ and $[[x_{13},x_{23}],x_{23}]$ and the second by $[x_{14},[x_{24},x_{34}]]$ and $[[x_{14},x_{34}],x_{24}]$. Algorithm 5.3 is straightforward for $\ell \neq 3$; in these cases we have: $$\begin{aligned}
&\partial^0[x_{13},[x_{13},x_{23}]] = [x_{24},[x_{24},x_{34}]] \\
&\partial^0[[x_{13},x_{23}],x_{23}] = [[x_{24},x_{34}],x_{34}],\end{aligned}$$ while $$\begin{aligned}
\partial^1[x_{13},[x_{13},x_{23}]] &= [x_{14}+x_{24},[x_{14}+x_{24},x_{34}]] \\
&= [x_{14}+x_{24},[x_{14},x_{34}]+[x_{24},x_{34}]] \\
&= [x_{14},[x_{14},x_{34}]]+[x_{14},[x_{24},x_{34}]]+[x_{24},[x_{14},x_{34}]]+[x_{24},[x_{24},x_{34}]] \\
&= [x_{14},[x_{14},x_{34}]]+[x_{14},[x_{24},x_{34}]]-[[x_{14},x_{34}],x_{24}]+[x_{24},[x_{24},x_{34}]] \\\end{aligned}$$ and $$\begin{aligned}
\partial^1[[x_{13},x_{23}],x_{23}] &= [[x_{14}+x_{24},x_{34}],x_{34}] \\
&= [[x_{14},x_{34}]+[x_{24},x_{34}],x_{34}] \\
&= [[x_{14},x_{34}],x_{34}]+[[x_{24},x_{34}],x_{34}], \\\end{aligned}$$ where the last line in the first case comes from an application of Algorithm 5.1 for the free Lie algebra over $\{x_{14},x_{24},x_{34}\}$. Similarly we have for $\ell = 2$ $$\begin{aligned}
\partial^2[x_{13},[x_{13},x_{23}]] &= [x_{14},[x_{14},x_{24}]]+
[x_{14},[x_{14},x_{34}]] \\
\partial^2[[x_{13},x_{23}],x_{23}] &= [[x_{14},x_{24}],x_{24}]+
[[x_{14},x_{24}],x_{34}]+
[[x_{14},x_{34}],x_{24}]+
[[x_{14},x_{34}],x_{34}] \\
&= [[x_{14},x_{24}],x_{24}]+
[[x_{14},x_{34}],x_{34}]+
2\ast[[x_{14},x_{34}],x_{24}]+
[x_{14},[x_{24},x_{34}]], \end{aligned}$$ where again the last line comes from Algorithm 5.1. Finally, as noted above, $\partial^4$ is the natural inclusion: $$\begin{aligned}
&\partial^4[x_{13},[x_{13},x_{23}]] = [x_{13},[x_{13},x_{23}]] \\
&\partial^4[[x_{13},x_{23}],x_{23}] = [[x_{13},x_{23}],x_{23}].\end{aligned}$$ The case $\ell = 3$ is much more computationally taxing, as it requires the use of Algorithm 5.2: $$\begin{aligned}
\partial^3[x_{13},[x_{13},x_{23}]] &= [x_{13}+x_{14},[x_{13}+x_{14},x_{23}+x_{24}]] \\
&= [x_{13},[x_{13},x_{23}]]+
[x_{14},[x_{14},x_{24}]]+
[x_{24},[x_{34},x_{14}]]+
[x_{14},[x_{34},x_{24}]] \hspace{4ex} \text{ (Alg. 5.2)} \\
&= [x_{13},[x_{13},x_{23}]]+
[x_{14},[x_{14},x_{24}]]+
[[x_{14},x_{34}],x_{24}]-
[x_{14},[x_{24},x_{34}]] \hspace{4ex} \text{ (Alg. 5.1)} \\
\partial^3[[x_{13},x_{23}],x_{23}] &= [[x_{13}+x_{14},x_{23}+x_{24}],x_{23}+x_{24}] \\
&= [[x_{13},x_{23}],x_{23}]+
[[x_{14},x_{24}],x_{24}]+
[[x_{14},x_{34}],x_{24}]+
[[x_{24},x_{34}],x_{14}] \hspace{4ex} \text{ (Alg. 5.2)} \\
&= [[x_{13},x_{23}],x_{23}]+
[[x_{14},x_{24}],x_{24}]+
[[x_{14},x_{34}],x_{24}]-
[x_{14},[x_{24},x_{34}]] \hspace{4ex} \text{ (Alg. 5.1)} \\\end{aligned}$$ Since $d^1 = \Sigma_i (-1)^i \partial^i$, we have from the above calculations that $$\begin{aligned}
d^1[x_{13},[x_{13},x_{23}]] &= 0 \\
d^1[[x_{13},x_{23}],x_{23}] &= 2\ast[x_{14},[x_{24},x_{34}]] +
[[x_{14},x_{34}],x_{24}]. \end{aligned}$$ and so the matrix for the boundary operator with respect to our chosen bases is given by $\left(
\begin{smallmatrix}
0 & 2 \\
0 & 1
\end{smallmatrix}
\right)$. We conclude that the boundary operator has rank one, and so $E^2_{-3,7} \cong E^2_{-4,7} \cong {{\mathbb Q}}$. Further (computer) calculations yield the $E^1$ and $E^2$ terms for $k$ odd given in Tables 1 and 2.
${{\mathbb Q}}^{120}$ ${{\mathbb Q}}^{300}$ ${{\mathbb Q}}^{260}$ ${{\mathbb Q}}^{89}$ ${{\mathbb Q}}^9$ 13
----------------------- ----------------------- ----------------------- ---------------------- ------------------- ----------------- ----------------- ----
12
${{\mathbb Q}}^{24}$ ${{\mathbb Q}}^{48}$ ${{\mathbb Q}}^{30}$ ${{\mathbb Q}}^6$ 11
10
${{\mathbb Q}}^6$ ${{\mathbb Q}}^9$ ${{\mathbb Q}}^3$ 9
8
${{\mathbb Q}}^2$ ${{\mathbb Q}}^2$ 7
6
${{\mathbb Q}}$ 5
4
${{\mathbb Q}}$ ${{\mathbb Q}}$ 3
-7 -6 -5 -4 -3 -2 -1
: $E^1$ term for $k=3$
${{\mathbb Q}}$ ${{\mathbb Q}}$ 13
----------------- ------------------- ----------------- ----------------- ----------------- ---- ----
12
${{\mathbb Q}}^2$ ${{\mathbb Q}}$ ${{\mathbb Q}}$ 11
10
9
8
${{\mathbb Q}}$ ${{\mathbb Q}}$ 7
6
${{\mathbb Q}}$ 5
4
3
-7 -6 -5 -4 -3 -2
: $E^2$ term for $k=3$
These low-dimensional computations do not reveal any regular behavior. Note that, as allowed because the Euler characteristic of the rows is zero, some rows vanish while most do not. Note as well that there is no additional vanishing along the edge of the vanishing line of Corollary \[vanish\].
All of the classes in Table 2 survive to $E^\infty$ except perhaps those in bidegrees $(-6, 13)$ and $(-3, 11)$ which could support a $d^3$ differential.
There are non-trivial classes in $\pi_n(Emb(I, {{\mathbb R}}^3 \times I))$ for $n = 2,3,4,5,6$.
It would be interesting to find explicit spherical families of embeddings which represent these classes. One expects the evaluation map $$\Delta^n \times {Emb(I, {\mathbb R}^k \times I)}\to F({{\mathbb R}}^k \times I, n) \times (S^k)^n$$ to play a central role in relating these homotopy groups to those of $F({{\mathbb R}}^k \times I, n) \times (S^k)^n$ which appear in our spectral sequence.
We conclude with a brief description of some of the methods used to verify the computer calculations (beyond merely computing examples by hand and comparing with the computer output, which was done extensively). Algorithm 3.1 was checked by an independent function which verified that the generated trees were Hall, and checked the number of elements in the resulting Hall set against the dimension count given by Lemma \[L:count\]. It was verified in the course of computing the $E^2$ term in Table 2 that $d^2 = 0$ for each of the chain complexes comprising $E^1$. A similar mathematical fact which was not hard-coded into the application is that the image of $M_{d,n}$ under $d^1$ lands in $M_{d,n+1}$ despite the fact that this is not the case for the individual homomorphisms $\partial^\ell$. The ranks of the boundary operators were verified using the linear algebra capabilities of a symbolic mathematics package (Maple). Finally, a nice check of the system as a whole was provided by varying the algorithm for generating Hall sets (noting the choices made in Algorithm 3.1) and verifying that the ranks of all boundary operators remained unchanged.
Further Work
============
In further work [@SS01] we will investigate the case of $k$ even, which includes the case of classical knots. Though the spectral sequences of [@Si00] do not necessarily converge, one can use those methods to produce knot invariants, which we show are of finite type. In particular, an optimistic view of rational homotopy theory predicts that the module of classes along the vanishing line of our spectral sequence (which for $k = 2$ is the anti-diagonal) is isomorphic to the module of primitives in the Hopf algebra of finite type invariants [@BN95]. To prove such a conjecture would involve relating the combinatorics of braid Lie algebras to those of Feynman diagrams, which could give a satisfactory explanation in terms of algebraic topology of the appearance of Feynman diagrams in the study of knots.
[^1]: The first named author was partially supported by NSF grant DMS-0072515
|
---
abstract: 'Least-squares fits are popular in many data analysis applications, and so we review some theoretical results in regard to the optimality of this fit method. It is well-known that common variants of the least-squares fit applied to Poisson-distributed data produce biased estimates, but it is not well-known that the bias can be overcome by iterating an appropriately weighted least-squares fit. We prove that the iterated fit converges to the maximum-likelihood estimate. Using toy experiments, we show that the iterated weighted least-squares method converges faster than the equivalent maximum-likelihood method when the statistical model is a linear function of the parameters and it does not require problem-specific starting values. Both can be a practical advantage. The equivalence of both methods also holds for binomially distributed data. We further show that the unbinned maximum-likelihood method can be derived as a limiting case of the iterated least-squares fit when the bin width goes to zero, which demonstrates the deep connection between the two methods.'
address:
- 'Max Planck Institute for Nuclear Physics, Heidelberg, Germany'
- 'Rostock University, Rostock, Germany'
author:
- Hans Dembinski
- Michael Schmelling
- Roland Waldi
title: 'Application of the Iterated Weighted Least-Squares Fit to counting experiments'
---
Introduction
============
In this paper, we review some theoretical results on least-squares methods, in particular, when they yield optimal estimates. We show how they can be applied to counting experiments without sacrificing optimality. The insights discussed here are known in the statistics community [@nw72; @cfy76], but less so in the high-energy physics community. Standard text books on statistical methods and papers, see [*e.g.*]{} [@james2006statistical; @baker1984clarification], correctly warn about biased results when standard variants of the least-squares fit are applied to counting experiments with small numbers of events, but do not show that these can be overcome. The results presented here are of practical relevance for fits of linear models, where the iterated weighted least-squares method discussed in this paper converges faster than the standard maximum-likelihood method and does not require starting values near the optimum.
The least-squares fit is a popular tool of statistical inference. It can be applied in situations with $k$ measurements $\{y_i|i=1, \dots, k\}$, described by a model with $m$ parameters ${\boldsymbol{p}} = (p_j|j=1,\dots, m)$ that predicts the expectation values $\operatorname*{E}[y_i] = \mu_i({\boldsymbol{p}})$ for the measurements. The measurements differ from the expectation values by unknown residuals $\epsilon_i$ = $y_i - \mu_i({\boldsymbol{p}})$. The solution ${\hat{{\boldsymbol{p}}}}$ that minimizes the sum ${Q}({\boldsymbol{p}})$ of squared residuals, $${Q}({\boldsymbol{p}}) = \sum_{i=1}^{k} \big(y_i - \mu_i({\boldsymbol{p}})\big)^2,
\label{eq:ls}$$ is taken as the best fit of the model to the data.
More generally, the measurements and the model predictions can be regarded as $k$-dimensional vectors ${\boldsymbol{y}} = (y_i|i = 1, \dots, k)$ and ${\boldsymbol{\mu}} = (\mu_i|i = 1, \dots, k)$, for which one wants to minimize a distance measure. In [Eq.(\[eq:ls\])]{}, we minimize the squared Euclidean distance. A generalization is the bilinear form $${Q}({\boldsymbol{p}}) = ({\boldsymbol{y}} - {\boldsymbol{\mu}})^T {\boldsymbol{W}} ({\boldsymbol{y}} - {\boldsymbol{\mu}}),
\label{eq:vls}$$ where ${\boldsymbol{W}}$ is a positive-definite symmetric matrix of weights. This variant is called *weighted least squares* (WLS). [Eq.(\[eq:ls\])]{} is recovered with ${\boldsymbol{W}} = {\boldsymbol{1}}$. An important special case is when the weight matrix is equal to the inverse of the true covariance matrix ${\boldsymbol{C}}$ of the measurements, ${\boldsymbol{W}} = {\boldsymbol{C}}^{-1}$ with ${\boldsymbol{C}} = \operatorname*{E}[{\boldsymbol{y}} {\boldsymbol{y}}^T] - \operatorname*{E}[{\boldsymbol{y}}] \operatorname*{E}[{\boldsymbol{y}}]^T$. For uncorrelated measurements, [Eq.(\[eq:vls\])]{} simplifies to the familiar form $${Q}({\boldsymbol{p}}) = \sum_{i=1}^{k} \big(y_i - \mu_i({\boldsymbol{p}})\big)^2 \big/ \sigma_i^2,
\label{eq:wls}$$ with variances $\sigma_i^2 = \operatorname*{E}[y_i^2] - \operatorname*{E}[y_i]^2$.
Aitken [@aitken1934least] showed in a generalization to the Gauss-Markov theorem [@james2006statistical p. 152] that minimizing $Q({\boldsymbol{p}})$ with ${\boldsymbol{W}} \propto {\boldsymbol{C}}^{-1}$ produces an optimal, in the sense as detailed below, solution for linear models ${\boldsymbol{\mu}}({\boldsymbol{p}}) = {\boldsymbol{X}} {\boldsymbol{p}}$, where ${\boldsymbol{X}}$ is a constant $k\times m$ matrix. The theorem applies when the covariance matrix ${\boldsymbol{C}}$ is finite and non-singular. Then, ${Q}({\boldsymbol{p}})$ has a unique minimum at $${\hat{{\boldsymbol{p}}}}= ({\boldsymbol{X}}^T \, {\boldsymbol{C}}^{-1} \,{\boldsymbol{X}})^{-1} \, {\boldsymbol{X}}^T \, {\boldsymbol{C}}^{-1} \, {\boldsymbol{y}}.
\label{eq:wls_linear_solution}$$ The best fit parameters ${\hat{{\boldsymbol{p}}}}$ in this case are a linear function of the measurements ${\boldsymbol{y}}$ with the covariance matrix $${\boldsymbol{C}}_{p} = ({\boldsymbol{X}}^T \, {\boldsymbol{C}}^{-1} \,{\boldsymbol{X}})^{-1} .
\label{eq:wls_covmat}$$ If the measurements are unbiased, $\operatorname*{E}[{\boldsymbol{y}}] = {\boldsymbol{\mu}}$, this solution is the *best linear unbiased estimator* (BLUE). Like all linear estimators, [Eq.(\[eq:wls\_linear\_solution\])]{} is unbiased if the input is unbiased. In addition, it has minimal variance of all linear estimators. This is true for any shape of the data distribution and any sample size. These excellent properties may be compromised in practical applications, since the covariance matrix ${\boldsymbol{C}}$ is often only approximately known. The least-squares approach is often regarded as a special case of the more general *maximum-likelihood* (ML) approach. The ML principle states that the best fit of a model should maximize the likelihood $L$, which is proportional to the joint probability of all measurements under the model. In practice, it is more convenient to work with ${\ln\!{L}}$ rather than $L$, so that the product of probabilities turns into a sum of log-probabilities, $${\ln\!{L}}({\boldsymbol{p}}) = \ln \prod_{i=1}^k P_i(y_i; {\boldsymbol{p}}) = \sum_{i=1}^k \ln P_i(y_i; {\boldsymbol{p}}).
\label{eq:ml}$$ Here $P_i(y_i; {\boldsymbol{p}})$ is the value of the probability density at $y_i$ for continuous outcomes or the actual probability for discrete outcomes. The ML method needs a fully specified probability distribution for each measurement, while the WLS method uses only the first two moments.
The parameter vector ${\hat{{\boldsymbol{p}}}}$ that maximizes [Eq.(\[eq:ml\])]{} is called the *maximum-likelihood estimate* (MLE). MLEs have optimal asymptotic properties; asymptotic here means in the limit of infinite samples. They are *consistent* (asymptotically unbiased) and *efficient* (asymptotically attaining minimal variance) [@james2006statistical]. In many practical cases of inference, in particular when data are Poisson-distributed, this method is known to produce good estimates also for finite samples. These properties make the ML fit the recommended tool for the practicioner [@baker1984clarification; @james2006statistical].
The WLS fit can be derived as a special case of a ML fit, if one considers normally distributed measurements $y_i$ with expectations $\mu_i$ and variances $\sigma_i^2$, where each measurement has the probability density function (PDF) $$P_i(y_i;\mu_i,\sigma_i) = \frac{1}{\sqrt{2\pi \sigma_i^2}} \exp\left(-\frac{(y_i-\mu_i)^2}{2\sigma_i^2}\right).$$ For fixed $\sigma_i^2$, we obtain [Eq.(\[eq:wls\])]{} from [Eq.(\[eq:ml\])]{}, $$\begin{aligned}
{\ln\!{L}}({\boldsymbol{p}}) = - \sum_{i=1}^k \frac{(y_i - \mu_i({\boldsymbol{p}}))^2}{2 \sigma_i^2} + c \equiv -\frac{1}{2} {Q}({\boldsymbol{p}}) + c,
\label{eq:ml_wls_normal}\end{aligned}$$ where the constant term $c$ depends only on the fixed variances $\sigma_i^2$. Constant terms do not affect the location ${\hat{{\boldsymbol{p}}}}$ of the maximum of ${\ln\!{L}}$ and the minimum of ${Q}$. We will often drop them from equations.
This derivation shows that for Gaussian PDFs a ML and a WLS fit give identical results when the same fixed variances are used, even if they are not the true variances. This does not hold in general, but is relevant in this context. When data are Poisson-distributed and have small counts, common implementations of the WLS fit are biased as we will show in the following section. The bias does not originate from the skewed shape of the Poisson distribution however, but rather from the fact that the weights are either biased or not fixed.
To these standard methods, we add the *iterated weighted least-squares* (IWLS) fit [@nw72]. It yields maximum-likelihood estimates when data are Poisson or binomially distributed with only the probabilities as free parameters [@cfy76]. This extends the strict equivalence between ML and WLS fits to a larger class of problems, an extension which is highly relevant in practice, since counts in histograms are Poisson distributed, and counted fractions are binomially distributed with the denominator considered fixed. The iterations are used to successively update estimates of the variances $\sigma_i^2$, which are kept constant during minimization.
When IWLS and ML fits are equivalent, which one is recommended? We conducted toy experiments where IWLS and ML fits are carried out numerically, as is common in practice. We found similar convergence rates for both methods when the model is non-linear, and a significantly faster convergence for the IWLS fit if the model is linear. This makes the IWLS fit a useful addition to the toolbox.
We have seen how the WLS fit can be derived from the ML fit under certain conditions. Inversely, we will show that the unbinned ML fit can be derived as a limiting case from the IWLS fit under weak conditions. The derivation shows that the two approaches are deeply connected.
Least-Squares Variants In Use
=============================
Standard variants of the WLS fit used in practice produce biased estimates when the fit is applied to Poisson-distributed data with small counts. The bias is often attributed to the breakdown of the normal approximation to the Poisson distribution, but it is actually related to how the unknown true variances $\sigma_i^2$ in [Eq.(\[eq:wls\])]{} are replaced by estimates.
We demonstrate this along a simple example. We fit the single parameter $\mu$ of the Poisson-distribution $$P(n;\mu) = e^{-\mu} \, \mu^{n} / n!,
\label{eq:poisson}$$ to $k$ counts $\{n_i|i=1,\dots,k\}$ sampled from it. The maximum-likelihood estimate for $\mu$ can be computed analytically by maximizing [Eq.(\[eq:ml\])]{}. We solve $\partial{\ln\!{L}}/\partial\mu \equiv \partial_\mu {\ln\!{L}}= 0$ for $\mu$ and obtain the arithmetic average $$\hat\mu = \frac{1}{k} \sum_{i=1}^k n_i,
\label{eq:poisson_mle}$$ which is unbiased and has minimal variance. We will now apply variants of the WLS fit to the same problem, which differ in how they substitute the unknown true variance.
[**Variance computed for each sample.**]{} For a single isolated sample, the unbiased estimate of $\mu$ is $\hat\mu_i = n_i$, with variance $\operatorname*{Var}[n_i] = \mu \simeq \hat\mu_i = n_i$. This is the origin of the well-known $\sqrt{n}$-estimate for the standard deviation of a count $n$. With this variance estimate, we get $${Q}(\mu) = \sum_{i=1}^k (n_i - \mu)^2/n_i.$$ This form is called Neyman’s $\chi^2$ in the statistics literature [@baker1984clarification]. Replacing the true variance $\mu$ by its sample estimate $n_i$ is an application of the *bootstrap principle* discussed by Efron and Tibshirani [@EfroTibs93]. To obtain the minimum, we solve $\partial_\mu {Q}= 0$ for $\mu$ and obtain the harmonic average $$\frac{1}{\hat\mu} = \frac{1}{k} \sum_{i=1}^{k} \frac{1}{n_i}.$$ The solution is biased and breaks down for samples with $n_i=0$. The variance estimates here are constant (they do not vary with $\hat\mu$), but differ from sample to sample. This treatment ignores the fact that the true variance is the same for all samples in this setup.
[**Variance computed from model.**]{} Another choice is to directly insert $\operatorname*{Var}[n_i] = \mu$ in the formula, $${Q}(\mu) = \sum_{i=1}^k (n_i - \mu)^2/\mu.
\label{eq:mupar}$$ This form, called Pearson’s $\chi^2$ [@baker1984clarification], is a conceptual improvement, because $\mu$ is the exact but unknown value of the variance. However, the variance $\mu$ now varies together with the expectation value $\mu$. Solving $\partial_\mu {Q}= 0$ for $\mu$ yields the quadratic average $$\hat\mu = \sqrt{\frac{1}{k} \sum_{i=1}^k n_i^2}.$$ This estimate is also biased, but can handle samples with $n_i=0$. The bias may come at a surprise, since we used the exact value for the variance after all. The failure here can be traced back to the fact that the variance estimates $\sigma_i^2 = \mu$ are not fixed during the minimization. A small positive bias on $\mu$ in [Eq.(\[eq:mupar\])]{} leads to a second order increase in the numerator, which is overcompensated by a first order increase of the denominator. In other words, the fit tends to increase the variance even at the cost of a small bias in the expectation when given this freedom, because overall it yields a reduction of ${Q}$.
[**Constant variance.**]{} Finally, we simply use $\sigma_i^2 = c$, where $c$ is an arbitrary constant, $${Q}(\mu) = \sum_{i=1}^k (n_i - \mu)^2/c.$$ We solve $\partial_\mu {Q}= 0$ for $\mu$ and obtain the optimal maximum-likelihood estimate [Eq.(\[eq:poisson\_mle\])]{} as the solution; the constant $c$ drops out.
This seems counter-intuitive, since we used a constant for all samples instead of a value close or equal to the true variance. However, this case satisfies all conditions of the Gauss-Markov theorem. The expectation values are trivial linear functions of the parameter $\mu_i = \mu$. The variances $\sigma_i^2$ are all equal and only need to be known up to a global scaling factor, hence any constant $c$ will do.
We learned that keeping the variance estimates constant during minimization is important, but the estimates should in general be as close to the true variances as possible. An iterated fit can satisfy both requirements.
Iterated Weighted Least-Squares
===============================
The iterated (re)weighted least-squares methods (IWLS or IRLS) are well known in statistical regression [@nw72], and can be applied to fits with $k$ measurements $\{y_i|i=1, \dots, k\}$ described by a model with $m$ parameters ${\boldsymbol{p}} = (p_j|j=1,\dots, m)$, which predicts the expectations $\operatorname*{E}[y_i] = \mu_i({\boldsymbol{p}})$ and variances $\operatorname*{Var}[y_i] = \sigma^2_i({\boldsymbol{p}})$ of each measurement. We will discuss the special application where the $y_i$ are entries of a histogram. One then minimizes the sum of squared residuals $${Q}({\boldsymbol{p}}) = \sum_{i=1}^{k} (y_i - \mu_i({\boldsymbol{p}})\big)^2/\sigma_i^2({\hat{{\boldsymbol{p}}}}),
\label{eq:iwls}$$ where the $\sigma_i^2$ are constant within one iteration of the fit and computed from the model using the parameter estimate ${\hat{{\boldsymbol{p}}}}$ that minimized ${Q}({\boldsymbol{p}})$ in the previous iteration. A convenient choice for the first iteration is $\sigma_i^2 = 1$. One iterates until ${\hat{{\boldsymbol{p}}}}$ converges.
In particle physics, we often work with samples drawn from two monoparametric distributions of the exponential family:
- Poisson distribution. Example: fitting a distribution function to a histogram of counts.
- Binomial distribution with fixed number of trials. Example: fitting an efficiency function to two histograms with generated and accepted events.
Charles, Frome, and Yu [@cfy76] derived that the IWLS fit gives the exact same result as the ML fit for a family of distributions. We demonstrate this in the appendix for the special distributions discussed here.
The Hessian matrices of second derivatives are also equal up to a constant factor, $\partial_{p_l} \partial_{p_m} {Q}= -2 \partial_{p_l} \partial_{p_m} {\ln\!{L}}$. The inverse of the Hessian is an estimate of the covariance matrix of the solution, an important uncertainty estimate in practical applications.
We emphasize that the equivalence does not depend on the size of the data sample or on the functional form of the model that predicts the expectation values for the measurements. In particular, when the IWLS fit is applied to histograms, it is not biased by small counts per bin or even empty bins.
Including systematic uncertainties
----------------------------------
A formal discussion of how systematic uncertainties can be handled with the IWLS fit is outside of the scope of this paper, but we note that it can include systematic uncertainties. Barlow [@2017arXiv170103701B] discusses how correlated systematic uncertainties can be handled in a least-squares fit. One minimizes [Eq.(\[eq:vls\])]{} in each iteration with a matrix $${\boldsymbol{C}} = {\boldsymbol{C}}'({\hat{{\boldsymbol{p}}}}) + {\boldsymbol{C}}_\text{sys}({\hat{{\boldsymbol{p}}}}),$$ where ${\boldsymbol{C}}'$ is the current estimate of the stochastic covariance computed from the previous solution, and ${\boldsymbol{C}}_\text{sys}$ is a current covariance matrix that represents the systematic uncertainties of the measurements. The matrix ${\boldsymbol{C}}_\text{sys}$ may be a function of the parameter vector. Like the covariance matrix ${\boldsymbol{C}}'$, it is kept constant during each iteration, and updated between iterations using the current value of ${\hat{{\boldsymbol{p}}}}$. This approach has been successfully applied in a combination of measurements from the CDF and D0 experiments [@Aaltonen:2013wca].
IWLS or ML fit?
---------------
When the IWLS and the ML fits are equivalent, which one should be used in practice? The two methods produce the same results in analytical problems, but can have different performance in numerical problems. In practice, the extrema of the log-likelihood function ${\ln\!{L}}({\boldsymbol{p}})$ and the weighted least-squares function ${Q}({\boldsymbol{p}})$ are usually found with a local optimizer, like the [`MIGRAD`]{} algorithm in the [`MINUIT`]{} package [@1975CoPhC..10..343J; @iminuit]. Computing the functions is sometimes expensive, when the fitted data sets are large and the model has many parameters. Numerical methods are therefore judged based on the number of function evaluations required to converge to the optimum within some tolerance. Another criterion is robustness, the ability to converge to the right optimum from a point in the neighborhood of the solution.
To address these points, we conducted toy experiments with Poisson-distributed counts $n_i$ and find that the ML method requires less function evaluations than the IWLS methods in general. However, the rate of convergence of the IWLS method can be greatly accelerated, when the model that computes the count expectation $E[n_i] = \mu_i({\boldsymbol{p}})$ is linear in the parameters, $\mu_i = {\boldsymbol{X}}_i \, {\boldsymbol{p}}$, where ${\boldsymbol{X}}_i$ is a vector of constants. The maximum of ${\ln\!{L}}({\boldsymbol{p}})$ usually cannot be found analytically in this case, but the minimum of ${Q}({\boldsymbol{p}})$ is given by [Eq.(\[eq:wls\_linear\_solution\])]{} in each iteration of the IWLS fit. When the computing time is dominated by the evaluation of ${Q}({\boldsymbol{p}})$ or ${\ln\!{L}}({\boldsymbol{p}})$, solving the IWLS fit is faster than the ML fit. The IWLS fit also does not require a problem-specific starting point for the optimization in this case. We call this special variant the L-IWLS fit. All three methods are able to handle fits that have bounded parameters, which are common in particle physics. In our toy experiments, the parameters are bounded to be non-negative. Details are given in the next section.
Whether the ML or the IWLS fits are more robust in the above sense is more difficult to say. No general proofs can be given for either method. Our toy studies suggest the following order of increasing robustness: IWLS, L-IWLS, ML. In some toy experiments, the IWLS methods require many more iterations than average, producing a long tail in the distribution of iteration counts. Such tails are not observed for the ML fit. It is likely, however, than a more sophisticated implementation of the IWLS fit than ours could improve the robustness of this method.
Performance in toy experiments
------------------------------
![Example of a toy data set (points) to test the performance of the ML, IWLS, and L-IWLS fits (see text). *Left:* The curve represent the fit result of the three methods for this data set (which are identical). *Right:* Intermediate parameter states during the optimization for the ML, IWLS, and L-IWLS fits (see text), in each iteration of the respective algorithms until the stopping criterion is reached. The L-IWLS fit often converges quadratically. The IWLS fit is slowed down by the artifical dampening that we introduced to avoid oscillations.[]{data-label="fig:poisson_example_toy"}](poisson_example_toy.pdf){width="\columnwidth"}
We compare the performances of ML, IWLS, and L-IWLS fits in a series of 1000 toy experiments with Poisson-distributed samples. We use a linear model for the expectation with two parameters, $\mu(x, {\boldsymbol{p}}) = (p_0 + p_1\, x^2)$, with $x\in [0, 1]$ as an independent variable. For the true parameters ${\boldsymbol{p}}_\text{truth} = (1, 10)$, we simulate 10 pairs $(x_i, n_i)$. The $x_i$ are evenly spaced over the interval $[0, 1]$, $\mu_i$ is calculated for each $x_i$ based on the true parameters, and finally a random sample $n_i$ is drawn for each $\mu_i$ from the Poisson distribution. The model is then fitted to each toy data set using the following three methods. One of the toy experiments is shown in [Fig.(\[fig:poisson\_example\_toy\])]{}.
- ML fit: Starting from [Eq.(\[eq:ml\_poisson\])]{} we use the [`MIGRAD`]{} algorithm from the [`MINUIT`]{} package to find the minimum. We pass the exact analytical gradient to [`MIGRAD`]{} for this problem, replacing the numerical approximation that [`MINUIT`]{} uses otherwise. We restrict the parameter range to $p_k \ge 0$ and add an epsilon to $\mu$ whenever it appears in a denominator to avoid division by zero.
- IWLS fit: We use Newton’s method to update ${\boldsymbol{p}}$, $${\boldsymbol{p}}_{n+1} = {\boldsymbol{p}}_{n} - {\boldsymbol{H}}^{-1} \, {\boldsymbol{\partial}}_{{\boldsymbol{p}}} {Q},$$ with the exact analytical gradient ${\boldsymbol{\partial}}_{{\boldsymbol{p}}} {Q}$ and Hessian ${\boldsymbol{H}}$ for this problem. Since the model is linear and the function ${Q}$ quadratic, Newton’s method yields the exact solution for the given gradient and Hessian matrix, but without taking the boundary condition $p_k \ge 0$ into account. We resolve this in an ad hoc way, by setting negative parameter values are set to zero.
Since the covariance matrix is fixed in each Newton step, each step fulfills the requirements of the IWLS method. We update the covariance matrix after each step for the computation of the next step. To check for convergence, we use the [`MINUIT`]{} criterion, which is based on the estimated distance-to-minimum and deviations in the diagonal elements of the inverted Hessian [@1975CoPhC..10..343J].
This approach works very well for most toy experiments, but in some rare cases ($< 1\,\%$) the solution starts to oscillate indefinitely between two states. We resolve this again in an ad hoc way by averaging the updated parameter vector with the previous one, ${\boldsymbol{p}}_{n+1} := ({\boldsymbol{p}}_{n+1} + {\boldsymbol{p}}_{n})/2$ after each Newton step. This slows down the convergence rate drastically, but avoids the oscillations.
- L-IWLS fit: We solve [Eq.(\[eq:wls\_linear\_solution\])]{} with the [`NNLS`]{} [@doi:10.1137/1.9781611971217] algorithm as implemented in [`SciPy`]{} [@scipy], and iterate. It solves [Eq.(\[eq:wls\_linear\_solution\])]{} under the boundary condition $p_k \ge 0$. To check for convergence, we again use the [`MINUIT`]{} criterion.
![Application of the maximum-likelihood (ML), the iterated weighted least-squares (IWLS) fit and its specialization for linear models (L-IWLS) to 1000 toy experiments with Poisson-distributed samples (see text). [*Top row:*]{} Histograms of the two fitted parameters of the model $\hat p_0$ and $\hat p_1$ are shown, overlayed for all three fit methods. The histograms are nearly identical. [*Bottom row:*]{} Normalized histograms of the number of evalutions of the model function for the the fit methods in double-logarithmic scale.[]{data-label="fig:poisson_example"}](poisson_example.pdf){width="\columnwidth"}
We note that our application of the general IWLS fit to a problem with a linear model is artifical. We only do this here to compare all three fitting methods on the same problem. For the IWLS and ML fits, we use the optimistic starting point ${\boldsymbol{p}}_\text{truth} = (1, 10)$. The ML and IWLS fits therefore run under ideal conditions compared to the L-IWLS fit, which does not require a specific starting point. In practice, one will usually start with a less ideal starting point, which slows down the convergence of ML and IWLS fits compared to the L-IWLS fit.
In case of the ML and IWLS fits, we increase the call counter for each evaluation of ${Q}$ or ${\ln\!{L}}$ and each evaluation of their gradients for all values of $z$ by one. In case of the L-IWLS fit, we count one application of the NNLS algorithm as one call, since it requires essentially one computation of the gradient.
The results are shown in [Fig.(\[fig:poisson\_example\])]{}. As expected, the results are equal within the numerical accuracies of the numerical algorithms, which stop when [`MINUIT`]{}’s standard convergence criterion is reached. This criterion roughly gives a precision of about $10^{-3}$ in the parameter relative to its uncertainty.
The average number of calls required to converge is different: 19.3 for ML, 23.6 for IWLS, and 4.8 for L-IWLS. The L-IWLS fit is the fastest to converge, requiring only a quarter of the function evaluations of the ML fit. The IWLS fit is the slowest, it requires about 20% more calls on average than the ML fit. This is mainly due to artificial dampening. In cases where the dampening is not needed, the IWLS fit converges as rapidly as the L-IWLS fit. Since we chose a linear model for this performance study in order to compare all three methods, a Newton’s step computes the exact solution to the fitting problem for the current covariance matrix estimate.
An investigation shows that the convergence issues of the IWLS fit appear when a parameter of the model is very close to zero. If this is not the case and no dampening is applied, the IWLS and L-IWLS fits produce identical results. [`MINUIT`]{} was designed to handle such cases well and shows a much more stable convergence rate. This suggests that the issues of the IWLS fit can be overcome as well with a more sophisticated implementation, but this comes at the cost of a slower convergence in favorable cases. The overall performance of the IWLS fit will probably not surpass that of the more straight-forward ML fit.
In conclusion, we recommend the L-IWLS fit for linear models and the ML fit for non-linear models.
Unbinned maximum-likelihood from IWLS
=====================================
In the introduction, we reviewed how the WLS fit can be derived as a special case of the ML fit, when measurements are normally distributed with known variance. Alternatively, the WLS fit can also be derived from geometric principles without relying on the ML principle. We will now show that the unbinned ML fit can be derived as a limiting case of the IWLS fit.
For the unbinned ML fit of a known probability density $f(x; {\boldsymbol{p}})$ of a continuous stochastic variable $x$ with parameters ${\boldsymbol{p}}$, one maximizes the sum of logarithms of the model density evaluated at the measurements $\{x_i| i = 1 \dots k \}$, $${\ln\!{L}}({\boldsymbol{p}}) = \sum_{i=1}^k \ln f(x_i; {\boldsymbol{p}}).
\label{eq:ml_unbinned}$$ The maximum is found by solving the system of equations $\partial_{p_j} {\ln\!{L}}({\boldsymbol{p}}) = 0$. The density $f(x; {\boldsymbol{p}})$ must be at least once differentiable in ${\boldsymbol{p}}$.
To derive these equations as a limit of the IWLS fit, we assume that $f(x;{\boldsymbol{p}})$ is finite everywhere in $x$, so that the probability density is not concentrated in discrete points.
We start by considering a histogram of $k$ samples $x_i$. Since the samples are independently drawn from a PDF, the histogram counts $n_l$ are uncorrelated and Poisson-distributed. Following the IWLS approach, we minimize the function $${Q}({\boldsymbol{p}}) = \sum_{l} \frac{(n_l - k P_l)^2}{k \hat P_l}$$ and iterate, where $P_l({\boldsymbol{p}}) = \int_{x_l}^{x_l+\Delta x} f(x; {\boldsymbol{p}}) \,\text{d}x$ is the expected fraction of the samples in bin $l$, and $\hat P_l = P_l({\hat{{\boldsymbol{p}}}})$ is the value based on the fitted parameters ${\hat{{\boldsymbol{p}}}}$ from the previous iteration. Expansion of the squares yields three terms, $${Q}({\boldsymbol{p}}) = \sum_l \frac{n_l^2}{k \hat P_l} - 2 \sum_l \frac{n_l\,P_l}{\hat P_l} + k \sum_l \frac{P_l^2}{\hat P_l}.$$ The first term is proportional to $1/\Delta x$, but not a function of ${\boldsymbol{p}}$. Therefore it does not contribute to the minimum obtained by solving the equations $\partial_{p_j} {Q}({\boldsymbol{p}}) = 0$. We drop it in the following and consider only the second and third term, which both are functions of ${\boldsymbol{p}}$.
We investigate the limit $\Delta x \rightarrow 0$. Since $f(x)$ is finite everywhere, we have ultimately either zero or one count in each bin. With $P_l \rightarrow f(x_l; {\boldsymbol{p}}) \, \Delta x$, the second term has a finite limit $$\sum_l \frac{n_l \, P_l}{\hat P_l}
\xrightarrow{\Delta x \rightarrow 0}
\sum_{i=1}^k \frac{f(x_i; {\boldsymbol{p}}) \, \Delta x}{f(x_i; {\hat{{\boldsymbol{p}}}}) \, \Delta x}
= \sum_{i=1}^k \frac{f(x_i; {\boldsymbol{p}})}{f(x_i; {\hat{{\boldsymbol{p}}}})},$$ where only bins around the measurements $x_i$ with one entry contribute ($n_l=1$), and the bin widths cancel. The third term also has a finite limit, $$\sum_l \frac{P_l^2}{\hat P_l} \xrightarrow{\Delta x \rightarrow 0} \sum_l \frac{f^2(x_l; {\boldsymbol{p}})\,(\Delta x)^2}{f(x_l; {\hat{{\boldsymbol{p}}}})\,\Delta x} = \int \frac{f^2(x; {\boldsymbol{p}})}{f(x; {\hat{{\boldsymbol{p}}}})}\,\text{d}x.$$ One $\Delta x$ cancels in the ratio and in the limit $\Delta x\to0$ the remaining sum is the very definition of a Riemann integral.
We now consider the derivatives $\partial_{p_j}{Q}({\boldsymbol{p}})$ in the limit of many iterations. We assume that the iterations converge, so that the previous solution ${\hat{{\boldsymbol{p}}}}$ approaches the next solution ${\boldsymbol{p}}$. We get $$\begin{aligned}
\partial_{p_j} {Q}({\boldsymbol{p}}) &= -2 \sum_{i=1}^k \frac{\partial_{p_j} f(x_i; {\boldsymbol{p}})}{f(x_i; {\hat{{\boldsymbol{p}}}})} + k \int \frac{2 f(x; {\boldsymbol{p}}) \, \partial_{p_j} f(x; {\boldsymbol{p}})}{f(x; {\hat{{\boldsymbol{p}}}})} \, \text{d}x \\
&\xrightarrow{{\hat{{\boldsymbol{p}}}}\rightarrow {\boldsymbol{p}}}
- 2 \sum_{i=1}^k \frac{\partial_{p_j} f(x_i; {\boldsymbol{p}})}{f(x_i; {\boldsymbol{p}})}
+ 2 k \partial_{p_j} \int f(x; {\boldsymbol{p}}) \, \text{d}x.
\end{aligned}
\label{eq:lsq_limit}$$ The last term vanishes in the limit, because $\int f(x;{\boldsymbol{p}})\,\text{d}x = 1$ is constant.
We finally obtain the equivalence $$\partial_{p_j} {Q}({\boldsymbol{p}})
\xrightarrow{\Delta x
\rightarrow 0,\,{\hat{{\boldsymbol{p}}}}\rightarrow {\boldsymbol{p}}}
- 2\sum_{i=1}^k \frac{\partial_{p_j} f(x_i; {\boldsymbol{p}})}{f(x_i; {\boldsymbol{p}})} \\
= -2 \sum_{i=1}^k \partial_{p_j} \ln f(x_i; {\boldsymbol{p}})
= -2 \partial_{p_j} {\ln\!{L}}({\boldsymbol{p}}).$$ The derivatives are equal up to a constant factor, which means that the solutions of $\partial_{p_j} {Q}({\boldsymbol{p}}) = 0$ and $\partial_{p_j} {\ln\!{L}}({\boldsymbol{p}}) = 0$ are equal. In other words, the IWLS solution in the limit of infinitesimal bins is found by minimizing the negative log-likelihood of the probability density. The latter is effectively a shortcut to the solution, which does not require iterations.
We showed the equivalence for the case when measurements consist of a single variable $x_i$ per event for simplicity, but it also holds for the general case of a set of n $n$-dimensional vectors $\{{\boldsymbol{x}}_{i} | i=1 \dots k\}$ with ${\boldsymbol{x}}_i = (x_{ji} | j=1\dots n)$ and a corresponding $n$-dimensional probability density $f({\boldsymbol{x}}; {\boldsymbol{p}})$. In this case, one would repeat the derivation starting from an $n$-dimensional histogram.
The derivation provides some insights.
- The absolute values of ${Q}({\hat{{\boldsymbol{p}}}})$ and $-2{\ln\!{L}}({\hat{{\boldsymbol{p}}}})$ at the solution ${\hat{{\boldsymbol{p}}}}$ are not equal. They differ by an (infinite) additive constant.
- The derivatives $\partial_{p_j} {Q}({\boldsymbol{p}})$ and $-2 \partial_{p_j} {\ln\!{L}}({\boldsymbol{p}})$ differ in general when ${\boldsymbol{p}}$ is not the solution ${\hat{{\boldsymbol{p}}}}$, because the second term in [Eq.(\[eq:lsq\_limit\])]{} does not vanish for ${\boldsymbol{p}} \ne {\hat{{\boldsymbol{p}}}}$.
In practice, the second point means that the [`MINUIT`]{} package produces the same error estimates for the solution ${\hat{{\boldsymbol{p}}}}$ if the [`HESSE`]{} algorithm is used, but not if the [`MINOS`]{} algorithm is used. The [`HESSE`]{} algorithm numerically computes and inverts the Hessian matrix of second derivatives at the minimum, which gives identical results for ${Q}$ and $-2{\ln\!{L}}$. The [`MINOS`]{} algorithm scans the neighborhood of the minimum, which for ${Q}$ and $-2{\ln\!{L}}$ usually has a different shape.
Notes on goodness-of-fit tests {#sec:gof}
==============================
For a goodness-of-fit (GoF) test, one computes a test statistic for a probabilistic model and a set of measurements. The test statistic is designed to have a known probability distribution when the measurements are truly distributed according to the probabilistic model. If the value for a particular model is very improbable, the model may be rejected.
It is well-known that the minimum value ${Q}({\hat{{\boldsymbol{p}}}})$ is $\chi^2$-distributed with expectation $(k-m)$, if the measurements are normally distributed, where $k$ and $m$ are the number of measurements and number of fitted parameters, respectively [@james2006statistical]. This GoF property is so useful and frequently applied, that the function ${Q}({\boldsymbol{p}})$ is often simply called *chi-square*.
In general, ${Q}({\hat{{\boldsymbol{p}}}})$ is not $\chi^2$-distributed for measurements that are not normally distributed around the model expectations. Approximately, it holds for Poisson and binomially distributed measurements when counts are not close to zero, and fractions are neither too close to zero or one. Stronger statements can be made about the expectation value of ${Q}({\hat{{\boldsymbol{p}}}})$. For linear models with $m$ parameters and $k$ unbiased measurements with known covariance matrix ${\boldsymbol{C}}$, the expectation of ${Q}({\hat{{\boldsymbol{p}}}})$ is guaranteed to be $$\operatorname*{E}[{Q}({\hat{{\boldsymbol{p}}}})] = k - m,
\label{eq:liwls_gof}$$ regardless of the sample size and the distribution of the measurements, as shown in the appendix. Therefore, the well-known quality criterion that the *reduced* $\chi^2$ should be close to unity, ${Q}({\hat{{\boldsymbol{p}}}})/(k-m) \simeq 1$, is often useful even if measurements are not normally distributed.
We saw previously that $-2{\ln\!{L}}({\hat{{\boldsymbol{p}}}})$ differs from ${Q}({\hat{{\boldsymbol{p}}}})$ by an infinite additive constant, which is a hint that it cannot straight-forwardly replace the latter as a GoF statistic. When used with unbinned data, ${\ln\!{L}}({\hat{{\boldsymbol{p}}}})$ is ill-suited as a GoF test statistic. Heinrich [@2003sppp.conf...52H] presented striking examples when ${\ln\!{L}}({\hat{{\boldsymbol{p}}}})$ carries no information of how well the model fits the measurements. Cousins [@cousins2013; @cousins2016] gave an intuitive explanation for this fact. The IWLS fit provides a maximum-likelihood estimate for measurements that follow a Poisson or binomial distribution and a GoF test statistic as a side result, which in general is not exactly $\chi^2$-distributed, but its distribution can often be obtained from a Monte Carlo simulation.
Conclusions
===========
An iterated weighted least-squares fit applied to measurements, which are Poisson- or binomially distributed around model expectations, provides the exact same solution as a maximum-likelihood fit. This holds for any model and any sample size. When the two fit methods are equivalent, the maximum-likelihood fit is still recommended, except when the model is linear. In this case, the minimum of the weighted least-squares problem can be found analytically in each iteration, which usually needs less computations overall than numerically maximizing the likelihood and requires no problem-specific starting point. The iterated weighted least-squares fit provides a goodness-of-fit statistic in addition, while the maximum-likelihood fit usually does not. Of course, a goodness-of-fit statistic can always be separately computed after the optimization, but in case of the maximum-likelihood it requires implementing two functions in a computer program instead of one.
Whether the two fit methods give equivalent results depends only on the probability distribution of the measurements around the model expectations. Here we presented proofs of the equivalence for Poisson and binomial distributions. In the statistics literature [@nw72; @cfy76], more general proofs are given that hold also for some other distributions.
Acknowledgments
===============
We thank Bob Cousins for a critical reading of the manuscript and for valuable pointers to the primary statistical literature, and the anonymous reviewer for suggestions to clarify additional points.
Equivalence of ML and ILWS for Poisson-distributed data
=======================================================
A common task is to fit a model to a histogram with $k$ bins, each with a count $n_i$. Especially in multi-dimensional histograms some bins may have few or even zero entries. This poses a problem for a conventional weighted least-squares fit, but not for a ML fit or an IWLS fit.
A ML fit of a model with $m$ parameters ${\boldsymbol{p}} = (p_j|j=1,\dots, m)$ to a sample of $k$ Poisson-distributed numbers $\{n_i|i=1,\dots, k\}$ with expectation values $\operatorname*{E}[n_i] = \mu_i({\boldsymbol{p}})$ is performed by maximizing the log-likelihood $${\ln\!{L}}({\boldsymbol{p}}) = \sum_{i=1}^k n_i \ln \mu_i - \sum_{i=1}^k \mu_i,
\label{eq:ml_poisson}$$ which is obtained by taking the logarithm of the product of Poisson probabilities (\[eq:poisson\]) of the data under the model, and dropping terms that do not depend on ${\boldsymbol{p}}$.
To find the maximum, we set the $m$ first derivatives $$\frac{\partial {\ln\!{L}}}{\partial p_j}
= \sum_{i=1}^k \frac{n_i}{\mu_i}
\frac{\partial \mu_i}{\partial p_j}
- \sum_{i=1}^k
\frac{\partial \mu_i}{\partial p_j}$$ for $j = 1$ to $m$ to 0. We get a system of equations $$\sum_{i=1}^k \frac{n_i - \mu_i}{\mu_i}
\, \frac{\partial \mu_i}{\partial p_j} = 0.
\label{eq:ml_poisson_diff}$$
We now approach the same problem as an IWLS fit. The sum of weighted squared residuals is $${Q}= \sum_{i=1}^k \frac{(n_i - \mu_i)^2}{\hat \mu_i},
\label{eq:wls_poisson}$$ where $\hat \mu_i$ is the expected variance computed from the model, using the parameter estimate ${\hat{{\boldsymbol{p}}}}$ from the previous iteration. To find the minimum, we again set the $m$ first derivatives to 0 and obtain $$\frac{\partial {Q}}{\partial p_j} =
-2\sum_{i=1}^k \frac{n_i - \mu_i}{\hat\mu_i}
\, \frac{\partial \mu_i}{\partial p_j} = 0.
\label{eq:wls_poisson_diff}$$
[Eq.(\[eq:wls\_poisson\_diff\])]{} and [Eq.(\[eq:ml\_poisson\_diff\])]{} yield identical solutions in the limit $\hat \mu_i \rightarrow \mu_i$, and so do their solutions. The limit is approached by iterating the fit, so that we actually obtain the maximum-likelihood estimate from the IWLS fit. Remarkably, this does not depend on the size of the counts $n_i$ per bin. The equivalence holds even when many bins with zero entries are present. To obtain this result, the $\hat \mu_i$ must be constant. If $\hat \mu_i$ was replaced by $\mu_i$ in [Eq.(\[eq:wls\_poisson\])]{}, extra non-vanishing terms would appear in [Eq.(\[eq:wls\_poisson\_diff\])]{}.
As already mentioned, when the expectations are linear functions, the unique analytical solution to [Eq.(\[eq:wls\_poisson\_diff\])]{} is given by [Eq.(\[eq:wls\_linear\_solution\])]{}, with ${\boldsymbol{C}}^{-1} = (\delta_{ij}/\hat \mu_i|i,j = 1,\dots,k)$ and $y_i = n_i$. An analytical solution of [Eq.(\[eq:ml\_poisson\_diff\])]{} is not known to the authors. The IWLS fit converges faster than the ML fit in this case.
Equivalence of ML and IWLS for binomially distributed data
==========================================================
Another common task is to obtain an efficiency function of a selection or trigger as a function of an observable. One collects a histogram of generated events with bin contents $N_i$, and a corresponding histogram of accepted events with bin contents $n_i$. The $N_i$ are considered as constants here, while the $n_i$ are drawn from the binomial distribution. The goal is to obtain a model function that best describes the efficiencies $\epsilon_i$ that best describe the drawn samples $n_i$. A single least-squares fit will give biased results when many $n_i$ are close to either 0 or $N_i$, but not a ML or an IWLS fit.
A ML fit of a model with $m$ parameters ${\boldsymbol{p}} = (p_j|j=1,\dots, m)$ for a sample of $k$ binomially distributed numbers $\{n_i|i=1,\dots, k\}$ with expectations $\operatorname*{E}[n_i] = \mu_i({\boldsymbol{p}}) = \epsilon_i({\boldsymbol{p}}) \, N_i$ is performed by maximizing the log-likelihood $${\ln\!{L}}({\boldsymbol{p}}) = \sum_{i=1}^k n_i \ln \mu_i
+ \sum_{i=1}^k (N_i-n_i) \ln(N_i - \mu_i),
\label{eq:ml_binom}$$ which is obtained by taking the logarithm of the product of binomial probabilities to observe $n_i$ when $\mu_i = \epsilon_i\,N_i$ are expected, $$P(n_i;\mu_i,N_i) = \binom{N_i}{n_i} \, \epsilon_i^{n_i} \, (1-\epsilon_i)^{N_i-n_i} = \binom{N_i}{n_i} \frac{\mu_i^{n_i} (N_i-\mu_i)^{N_i-n_i}} {N_i^{N_i}},$$ and dropping terms that do not depend on ${\boldsymbol{p}}$. A binomial distribution has two parameters $(\mu_i,N_i)$, but it is a monoparametric distribution in this context since the $N_i$ are known and only the $\mu_i$ are free parameters.
Again we set the $m$ first derivatives $$\frac{\partial {\ln\!{L}}}{\partial p_j}
= \sum_{i=1}^k \frac{n_i }{ \mu_i}\,\frac{\partial \mu_i}{\partial p_j}
- \sum_{i=1}^k \frac{N_i - n_i}{N_i - \mu_i}\,
\frac{\partial \mu_i}{\partial p_j}$$ to zero for $j = 1$ to $m$. The minimum is obtained by solving $$\sum_{i=1}^k \frac{n_i - \mu_i}{\mu_i (1 - \mu_i/N_i)} \,
\frac{\partial \mu_i}{\partial p_j} = 0.
\label{eq:ml_binom_diff}$$
For the IWLS fit, we need to minimize the sum $${Q}({\boldsymbol{p}}) = \sum_{i=1}^k \frac{(n_i - \mu_i)^2}
{\hat \mu_i (1-\hat\mu_i/N_i)}.
\label{eq:wls_binom}$$ where the variances for the binomial distribution with expectation $\mu_i$ are $\sigma_i^2 = \operatorname*{Var}[n_i] = N_i \epsilon_i (1-\epsilon_i) = \mu_i (1-\mu_i/N_i)$. Again, we replaced $\mu_i$ in the variance by the constant estimate $\hat \mu_i$ from the previous iteration. Setting the $m$ first derivatives to 0, we obtain $$\frac{\partial {Q}}{\partial p_j} =
-2\sum_{i=1}^k \frac{n_i - \mu_i}{\hat\mu_i(1-\hat\mu_i/N_i)}
\, \frac{\partial \mu_i}{\partial p_j} = 0
\label{eq:wls_binom_diff}$$
Like in the previous case, [Eq.(\[eq:ml\_binom\_diff\])]{} and [Eq.(\[eq:wls\_binom\_diff\])]{} yield identical solutions in the limit $\hat \mu_i \rightarrow \mu_i$, which is approached by iterating the minimization. Again, we obtain the maximum-likelihood estimate with the IWLS fit.
Like in the previous case, the L-IWLS fit for a linear model converges faster than the ML fit, while the IWLS fit converges more slowly than the ML fit in general.
Expectation of ${Q}({\hat{{\boldsymbol{p}}}})$ for linear models
================================================================
We compute the expectation of ${Q}$ in [Eq.(\[eq:vls\])]{}, evaluated at the solution ${\hat{{\boldsymbol{p}}}}$ from [Eq.(\[eq:wls\_linear\_solution\])]{} for linear models with $\operatorname*{E}[{\boldsymbol{y}}] = {\boldsymbol{X}} {\boldsymbol{p}}$, where ${\boldsymbol{X}}$ is a fixed $k \times m$ matrix, and where the measurements ${\boldsymbol{y}}$ have a known finite covariance matrix ${\boldsymbol{C}}$. Similar proofs are found in the literature [@KS]. The covariance matrix of ${\hat{{\boldsymbol{p}}}}$ is obtained by error propagation with the matrix ${\boldsymbol{M}} = ({\boldsymbol{X}}^T C^{-1} {\boldsymbol{X}})^{-1} {\boldsymbol{X}}^T {\boldsymbol{C}}^{-1}$ and ${\hat{{\boldsymbol{p}}}}= M {\boldsymbol{y}}$ as $${\boldsymbol{C}}_p = {\boldsymbol{M}} {\boldsymbol{C}} {\boldsymbol{M}}^T = ({\boldsymbol{X}}^T {\boldsymbol{C}}^{-1} {\boldsymbol{X}})^{-1},
\label{eq:cp}$$\
where we used that ${\boldsymbol{C}}^{-1}$ and $({\boldsymbol{X}}^T {\boldsymbol{C}}^{-1} {\boldsymbol{X}})^{-1}$ are symmetric matrices.
The expectation is a linear operator. Since the solution ${\hat{{\boldsymbol{p}}}}= {\boldsymbol{M}} {\boldsymbol{y}}$ is a linear function of the measurement, we have $$\operatorname*{E}[{\hat{{\boldsymbol{p}}}}] = {\boldsymbol{M}} \operatorname*{E}[{\boldsymbol{y}}] = {\boldsymbol{M}} {\boldsymbol{X}} {\boldsymbol{p}} = {\boldsymbol{p}},$$ in other words, ${\hat{{\boldsymbol{p}}}}$ is an unbiased estimate of ${\boldsymbol{p}}$.
We expand ${Q}$ evaluated at ${\hat{{\boldsymbol{p}}}}$, $${Q}({\hat{{\boldsymbol{p}}}}) = {\boldsymbol{y}}^T {\boldsymbol{C}}^{-1} {\boldsymbol{y}}
- {\boldsymbol{y}}^T {\boldsymbol{C}}^{-1} {\boldsymbol{X}} {\hat{{\boldsymbol{p}}}}- {\hat{{\boldsymbol{p}}}}^T {\boldsymbol{X}}^T {\boldsymbol{C}}^{-1} {\boldsymbol{y}}
+ {\hat{{\boldsymbol{p}}}}^T {\boldsymbol{X}}^T {\boldsymbol{C}}^{-1} {\boldsymbol{X}} {\hat{{\boldsymbol{p}}}},$$ which simplifies with ${\boldsymbol{C}}_p^{-1} {\hat{{\boldsymbol{p}}}}= {\boldsymbol{X}}^T {\boldsymbol{C}}^{-1} {\boldsymbol{y}}$ and [Eq.(\[eq:cp\])]{} to $${Q}({\hat{{\boldsymbol{p}}}}) = {\boldsymbol{y}}^T {\boldsymbol{C}}^{-1} {\boldsymbol{y}} - {\hat{{\boldsymbol{p}}}}^T {\boldsymbol{C}}_p^{-1} {\hat{{\boldsymbol{p}}}}.$$ The scalar result of a bilinear form is trivially equal to the trace of this bilinear form, and a cyclic permutation inside the trace then yields $${Q}({\hat{{\boldsymbol{p}}}}) = \operatorname*{Tr}({\boldsymbol{C}}^{-1} {\boldsymbol{y}} {\boldsymbol{y}}^T) - \operatorname*{Tr}({\boldsymbol{C}}_p^{-1} {\hat{{\boldsymbol{p}}}}{\hat{{\boldsymbol{p}}}}^T).$$ We compute the expectation on both sides and get, using linearity of trace and expectation, $$\operatorname*{E}[{Q}({\hat{{\boldsymbol{p}}}})] = \operatorname*{Tr}({\boldsymbol{C}}^{-1} \operatorname*{E}[{\boldsymbol{y}} {\boldsymbol{y}}^T]) - \operatorname*{Tr}({\boldsymbol{C}}_p^{-1} \operatorname*{E}[{\hat{{\boldsymbol{p}}}}{\hat{{\boldsymbol{p}}}}^T]).$$ The definition of the covariance matrix ${\boldsymbol{C}} = \operatorname*{E}[{\boldsymbol{y}} {\boldsymbol{y}}^T] - \operatorname*{E}[{\boldsymbol{y}}] \operatorname*{E}[{\boldsymbol{y}}]^T$ is inserted, and vice versa for ${\boldsymbol{C}}_p$. We get $$\operatorname*{E}[{Q}({\hat{{\boldsymbol{p}}}})] = \operatorname*{Tr}({\boldsymbol{C}}^{-1} {\boldsymbol{C}} + {\boldsymbol{C}}^{-1} \operatorname*{E}[{\boldsymbol{y}}] \operatorname*{E}[{\boldsymbol{y}}]^T)
- \operatorname*{Tr}({\boldsymbol{C}}_p^{-1} {\boldsymbol{C}}_p + {\boldsymbol{C}}_p^{-1} \operatorname*{E}[{\hat{{\boldsymbol{p}}}}] \operatorname*{E}[{\hat{{\boldsymbol{p}}}}^T]).$$ The trace of a matrix multiplied with its inverse is equal to the number of diagonal elements, which is $k$ in case of ${\boldsymbol{C}}$ and $m$ in case of ${\boldsymbol{C}}_p$. We use this, $\operatorname*{E}[{\boldsymbol{y}}] = {\boldsymbol{X}} {\boldsymbol{p}}$, $\operatorname*{E}[{\hat{{\boldsymbol{p}}}}] = {\boldsymbol{p}}$, and again the linearity of the trace, to get $$\operatorname*{E}[{Q}({\hat{{\boldsymbol{p}}}})] = k + \operatorname*{Tr}({\boldsymbol{C}}^{-1} {\boldsymbol{X}} {\boldsymbol{p}} {\boldsymbol{p}}^T {\boldsymbol{X}}^T)
- \big(m + \operatorname*{Tr}({\boldsymbol{C}}_p^{-1} {\boldsymbol{p}} {\boldsymbol{p}}^T)\big).$$ The remaining traces are identical and cancel, $$\operatorname*{Tr}({\boldsymbol{C}}^{-1} {\boldsymbol{X}} {\boldsymbol{p}} {\boldsymbol{p}}^T {\boldsymbol{X}}^T)
= \operatorname*{Tr}({\boldsymbol{X}}^T {\boldsymbol{C}}^{-1} {\boldsymbol{X}} {\boldsymbol{p}} {\boldsymbol{p}}^T)
= \operatorname*{Tr}({\boldsymbol{C}}_p^{-1} {\boldsymbol{p}} {\boldsymbol{p}}^T),$$ and so we finally obtain the result $$\operatorname*{E}[{Q}({\hat{{\boldsymbol{p}}}})] = k - m,$$ which is independent of the PDFs that describe the scatter of the measurements ${\boldsymbol{y}}$ around the expectation values $\operatorname*{E}[{\boldsymbol{y}}]$.
[10]{}
J. A. Nelder, R. W. M. Wedderburn, , J. R. Statist. Soc. [**A135**]{}, 370–384 (1972).
A. Charles, E.L. Frome, P. L. Yu, , J. Am. Stat. Assoc. [**71**]{}, 169–171 (1976).
F. James, , World Scientific Publishing Company, 2006.
S. Baker and R. D. Cousins, , Nuclear Instruments and Methods in Physics Research [**221**]{}, 437 (1984).
A. C. Aitken, , Proc. R. Soc. Edinburgh [**55**]{}, 42 (1935).
B. Efron and R. J. Tibshirani, , Monographs on Statistics and Applied Probability No. 57, Chapman & Hall/CRC, Boca Raton, Florida, USA, 1993.
R. J. [Barlow]{}, , arXiv.1701.03701 (2017).
T.A. Aaltonen [*et al.*]{} (CDF, D0 collaborations), , Phys. Rev. [**D89**]{}, 072001 (2014).
F. [James]{} and M. [Roos]{}, , Computer Physics Communications [**10**]{}, 343 (1975).
iminuit team, , <https://github.com/iminuit/iminuit> (2013), accessed: 2018-03-05.
C. Lawson and R. Hanson, , Society for Industrial and Applied Mathematics, 1995, <https://doi.org/10.1137/1.9781611971217>.
E. Jones [*et al.*]{}, , <http://www.scipy.org> (2001), accessed: 2018-03-05.
J. [Heinrich]{}, , in [*Statistical Problems in Particle Physics, Astrophysics, and Cosmology*]{}, edited by L. [Lyons]{}, R. [Mount]{}, and R. [Reitmeyer]{}, p. 52, 2003, arXiv:physics/0310167.
R.D. [Cousins]{}, , <http://www.physics.ucla.edu/~cousins/stats/cousins_saturated.pdf> (2013), accessed: 2018-07-19.
R.D. [Cousins]{}, , <http://cousins.web.cern.ch/cousins/ongoodness6march2016.pdf> (2016), accessed: 2018-07-19.
see, e.g., M. G. Kendall, A. Stuart, [*The Advanced Theory of Statistics*]{}, vol 2, Sect. 19.9, 3rd edition, Charles Griffin & Co., 1961.
|
---
author:
- |
[^1]\
Institute for Theoretical Physics, KIT, 76128 Karlsruhe, Germany\
E-mail:
- |
Francisco Campanario\
Theory Division, IFIC, University of Valencia-CSIC, E-46980 Paterna, Valencia, Spain\
E-mail:
- |
Sebastian Sapeta\
CERN PH-TH, CH-1211, Geneva 23, Switzerland\
E-mail:
- |
Dieter Zeppenfeld\
Institute for Theoretical Physics, KIT, 76128 Karlsruhe, Germany\
E-mail:
bibliography:
- 'LSWZAC.bib'
title: Anomalous couplings in WZ production beyond NLO QCD
---
Introduction
============
The production of two heavy vector bosons is an interesting process, as it allows to study the interactions between the gauge bosons. Leptonic decays of the vector bosons can be used to discriminate backgrounds and get precise measurements of the kinematics. Searches for new physics include new resonances decaying to vector boson pairs and changes to their couplings due to new particles in loops. Those can appear in tails of distributions or as modifications in angular distributions sensitive to spin. To measure diboson production and search for deviations from the Standard Model (SM) high center of mass energies and high luminosities are required, such that the most recent LHC upgrades promise to enhance those analyses.
In this contribution, we study WZ production as a representative diboson production process. We calculate NLO QCD corrections to WZ and WZj production with VBFNLO and combine them to get accuracy using the LoopSim approach [@Rubin:2010xp]. This improves the prediction especially for high-$\pt$ vector bosons. Recently a calculation of the inclusive cross section of WZ production at NNLO QCD was reported in Ref. [@Grazzini:2016swo], but there are no differential distributions available yet. [^2]
We study Anomalous Couplings (AC) using an Effective Field Theory (EFT) approach. This allows to parametrize deviations from the Standard Model (SM) interactions, especially in triple and quartic gauge couplings in a general way without choosing a specific model. Effects of AC are most visible in the region of high invariant-mass diboson pairs with high transverse momentum of the final state particles. In particular in the high-$\pt$ region, the LoopSim approach should approximate the full NNLO corrections closely.
WZ production has been measured at the LHC in several decay channels. The fully leptonic decay, as studied in Ref. [@Aad:2012twa; @Aad:2014pha; @Aad:2016ett; @Aaboud:2016yus; @Khachatryan:2016tgp], has little background, but at the same time the smallest cross section. Semi-leptonic final states, [@Chatrchyan:2012bd; @Aad:2013pdy], have a larger cross sections but suffer from backgrounds due to top production and single-vector-boson + jets. Measuring the cross section allows to set limits on anomalous triple gauge couplings, as many of the analyses do. For details on diboson measurements see also the contributions by S.L. Barnes and N. Woods in these proceedings.
Calculational setup
===================
For the simulation, we use [@Arnold:2008rz; @Baglio:2014uba; @Arnold:2011wj] in combination with LoopSim [@Rubin:2010xp]. The setup is comparable to previous work with LoopSim on WZ [@Campanario:2012fk], WW [@Campanario:2013wta] and ZZ production [@Campanario:2015nha]. LoopSim generates a merged sample of WZ@NLO and WZj@NLO [@Campanario:2010hp] to give us WZ@$\nNLO$. For the study of AC, we are interested in phase space regions with high transverse momentum and high invariant mass of electroweak particles. These can get large contributions from additional QCD radiation of $\mathcal{O}( \als \ln^2 \ptj / m_Z )$, which are included in LoopSim. Compared to the full NNLO calculation, LoopSim misses the finite 2-loop contributions. Those are expected to be on the level of a few percent, since they are suppressed by $\als^2 $ and are not enhanced, like some real emission contributions. Due to the missing contributions the prediction of the total cross section, the low-$\pt$ phase space region and the scale dependence is only NLO.
Anomalous Couplings
-------------------
To parametrize AC, we use the set of dimension-6 operators based on the HISZ basis [@Hagiwara:1993ck] as updated in Ref. [@Hankele:2006ma]. They extend the SM via $\mathcal{L} = \mathcal{L_{\text{SM}}} + \sum_i \frac{f_i}{\Lambda^2}\mathcal{O}_i$. Assuming C and P invariance, only three operators contribute to the WWZ vertex: $$\begin{aligned}
\ow &= \owval, \\
\ob &= \obval, \\
\owww &= \owwwval .
\label{eq:wwz-operators}\end{aligned}$$ We focus on the $\ow$ operator for this work. It is a suitable representative for AC as it leads to a term with non-SM Lorentz structure in the WWZ vertex. To get meaningful limits on AC, one has to consider a complete set of operators, which affect a given observable, including correlations between operators.
Diboson production processes are competitive in limiting dimension 6 operators and are therefore included in global fits of AC limits, like Refs. [@Corbett:2012ja; @Masso:2012eq; @Butter:2016cvz]. Our values used for the couplings are in the allowed region of Ref. [@Corbett:2012ja] and represent typical values current measurements are sensitive to.
Validity of EFT approach {#sec:acinterfer}
------------------------
Effective Field Theory (EFT) for triple gauge couplings assumes an expansion in $\frac{f}{\Lambda^2}$. This depends on both the coupling of new physics as well as its energy scale. The expansion is only valid if the scale of the considered observable is smaller than the new physics scale entering $\Lambda$.
For high invariant masses, the EFT operators violate unitarity of the S-matrix. This is a sign that the EFT approach is no longer valid and should be replaced by a UV complete model. To still get physical predications, one can apply a unitarization procedure. We use the form factor $$\begin{aligned}
FF = \left( 1+\frac{m_{\text{WZ}}}{\Lambda_{\text{FF}}^2} \right)^{\text{-n}}, \quad
\text{with } \Lambda_{\text{FF}} = \SI{2}{TeV} \text{ and n} = 1 .
\label{eq:ffdef}\end{aligned}$$ The value for the form factor scale is determined with the form factor tool [@VBFNLOFF:2014], such that the operator does not lead to unitarity violation in $2\rightarrow 2$ scattering. The exponent n is chosen to cancel the leading divergence of the EFT operator. Using the form factor tool the scale $\Lambda_{\text{FF}}$ is fixed, such that this description leaves no free parameter in the unitarization scheme. For most plots, we will focus on phase space regions significantly below $\Lambda_\text{FF}$ where neither the form factor nor most unitarization procedures would have a visible effect.
Even below the scale of unitarity violation, there is an ambiguity in how to make predictions using an EFT. In the calculation of the squared matrix element $\left|\mathcal{M}\right|^2$, there are interference terms between the SM and AC as well as purely AC terms. If one considers an amplitude with contributions from AC operators of dimension 6 and dimension 8, the terms are: $$\begin{aligned}
\mathcal{M} &= \MSM + \underbrace{\MACs}_{\nicefrac{1}{\Lambda^2}} +
\underbrace{\MACe}_{\nicefrac{1}{\Lambda^4}} +\mathcal{O}(\Lambda^{-6}) \\
\left|\mathcal{M}\right|^2 &=
\underbrace{\left|\MSM\right|^2}_{\nicefrac{1}{\Lambda^0}}
+ \underbrace{2 \text{Re} \MSM^* {\MACs}}_{\nicefrac{1}{\Lambda^2}}
+ \underbrace{\left| \MACs\right|^2 }_{\nicefrac{1}{\Lambda^4}}
+ \underbrace{2 \text{Re} \MSM^* {\MACe}}_{\nicefrac{1}{\Lambda^4}}
+ \underbrace{\left|\MACe\right|^2}_{\nicefrac{1}{\Lambda^8}}
+\mathcal{O}(\Lambda^{-6})\end{aligned}$$
We include both the $ \MSM^* \MACs$ and the ${\left|\MACs\right|}^2$ term in our calculation. By naive power counting, one would assume that the latter should be considered simultaneously with dim-8 operators, that contribute via $\MSM^* \MACe$. This is in general not the case because the SM amplitude is suppressed by the weak coupling, such that $\left| \MACs \right|^2$ can naturally be larger than $\MSM^* \MACe$.
Besides this size argument, there are also practical reasons to include the $\left| \MAC \right|^2$ term. Without it, one can generate (unphysical) negative cross sections when negative interference exceeds the SM contribution.
To be independent of this ambiguity, one can restrict oneself to phase space regions where the squared term is not relevant. Based on the sign dependence of the interference, we will study in which phase space regions both terms contribute in \[sec:acnnlo\].
Numerical results
=================
We consider the LHC at run 2 with pp collisions at . The jets are clustered using the anti-$k_t$ algorithm [@Cacciari:2008gp] with a cone radius of $R=0.4$. To simulate typical detector acceptance, we impose a minimal set of inclusive cuts $$\label{eq:basiccuts}
\begin{aligned}
% \pt_l &> \SI{15}{GeV} & \pt_{j} &>\SI{30}{GeV} & \slashed E_{\text{T}} &> \SI{30}{GeV} \\
\pt_l &> 15 \text{GeV} & \pt_{j} &> 30 \text{GeV} & \slashed E_{\text{T}} &> 30 \text{GeV} \\
|y_j| &< 4.5 & |\eta_l| &< 2.5 & R_{lj} &> 0.4 \\
120 \text{GeV} &> m_{ll} > 60 \text{GeV} ,
\end{aligned}$$ where the $m_{ll}$ cut is applied only to same-flavor leptons with opposite sign coming from the $Z$ boson.
We consider decays $W\rightarrow e \nu_e, Z\rightarrow \mu \mu$. Adding the other leptonic final states increases the number of expected events by a factor of $4$.[^3]
For the renormalization scale $\mur$ and factorization scale $\muf$, we use $$\mu_0 = \HT = \HTval .$$
The theoretical uncertainty is estimated using a simultaneous variation of the two scales by a factor of 2: $\mur=\muf=\{0.5,2\}\mu_0$. We use the default values set in VBFNLO 3.0 for electroweak constants and NNPDF23 [@Ball:2012cx] for the parton distribution functions.
WZ production at QCD
--------------------
Results for WZ production at QCD using Loopsim were first presented in Ref. [@Campanario:2012fk]. As shown there, the corrections due to on top of NLO QCD can be sizeable. A typical electroweak observable, like the $\pt$ of the hardest lepton ($\ptlmax$), is enhanced by about 25% at . Changes are substantially larger for observables sensitive to extra radiation. For $\htjets$, the correction is a factor of 5 at . These distributions are shown in \[fig:loopsim\_large\_kfac\]. In both cases, the corrections are outside of the scale variation.
{width=".45\textwidth"} {width=".45\textwidth"}
AC at QCD {#sec:acnnlo}
---------
For AC, we include both the $\MSM^* \MAC$ interference term as well as the $\left|\MAC\right|^2$ term, as discussed in \[sec:acinterfer\]. The interference term is sensitive to the sign of the AC, while the squared term is not. Which of the two terms dominated can thus be seen in distributions by looking for dependence on the coupling sign.
{width="48.00000%"} {width="48.00000%"}\
To show the typical effect of AC, we consider $\ptlmax$ at NLO QCD in \[fig:motivation\_highptl\]. For $\ptlmax$ and the chosen AC values, at around , the destructive interference is maximal, while above , the $\left|\MAC\right|^2$ term dominates. The interference region gives access to the sign.
The corrections at $\nNLO$ shown in \[fig:loopsim\_large\_kfac\] are of comparable size to those due to AC. Therefore, an analysis based on a prediction at NLO QCD for the SM might mistake a deviation for a detection of AC, while a prediction at higher order in QCD might match the measurement. shows a comparison of SM $\nNLO$ and AC NLO that are of similar size. With and LoopSim, predictions for AC can also be made at $\nNLO$ accuracy. The $\nNLO/\text{NLO}$ K-factor depends on the specific value of the anomalous coupling, such that extrapolating the SM K-factor to AC predictions gives inaccurate results. This can be seen in \[fig:ptlmax-nnlo-kfac\].
{width="45.00000%"} {width="45.00000%"}\
![The ratio $\left( \rm{d}\sigma_{\nNLO}/\rm{d}\sigma_{\text{NLO}} \right)_{
\text{SM},\text{AC} } $ for $\ptlmax$ is shown. This K-factor depends significantly on the value of the anomalous coupling, ranging from 1.1 to 1.3 for the bin at . Therefore, AC can not be accurately described by rescaling an AC prediction at NLO with a SM $\nNLO$/NLO K-factor. []{data-label="fig:ptlmax-nnlo-kfac"}](plots/loopsim-wz-ac/SMNLO-ratioonly_WpZ_ptlmax_rebin2.pdf){width="45.00000%"}
Dynamical jet veto
------------------
In Ref. [@Campanario:2014lza], we suggested a dynamical jet veto to improve the sensitivity to AC in WZj production. We use this veto and study its effect on LoopSim corrections. AC effects typically grow with the invariant mass or momentum transfer at triple gauge vertices. To enhance their signal, the focus is on high-invariant mass boson pairs and high transverse momentum bosons/leptons in the final state. When a high-$\pt$ vector boson is required, this occurs about half of the time due to recoil against a jet (instead of the second vector boson). To reduce those events, one introduces a jet veto. A traditional fixed-$\pt$ jet veto rejects all events with additional jets above a certain threshold. This introduces logarithms of the veto scale, which need to be resummed. Also this veto cuts away relevant phase space, since in very-high invariant-mass regions (for example $m_{WZ} = \SI{1}{TeV}$) most events will have additional jet radiation at , which is soft compared to the EW system and does not reduce the AC sensitivity. Thus, a fixed veto at will remove relevant signal in that phase space region.
The dynamical veto is based on $$\xjet = \xjetval, \text{ where }
\et = E \frac{\left|\vec p_{\rm{t}}\right|}{\left| \vec p \right|} .
\label{eq:xjetdef}$$ The definition of $\et$ here differs from the one chosen in Ref. [@Campanario:2014lza], where $m_T = \sqrt{m^2 + \pt^2}$ was used instead. At small $\pt$, the latter is dominated by the mass and thus leads to small $\xjet$ values for inclusive samples, while $\et$ generates a broader distribution also in the low-$\pt$ region and leads to better discrimination between SM and AC contributions. In the high-$\pt$ region or for massless particles, these definitions become identical: $m_{\rm{T}} = \et = \pt$.
![ $\xjet$ as defined in \[eq:xjetdef\] for inclusive and boosted ($\ptz > \SI{200}{GeV})$ cuts. As visible especially in the boosted case, there are two relevant phase space regions around small $\xjet$ and for $\xjet \approx 0.5$. The latter is not sensitive to AC, such that the AC analysis should focus on the region below $\xjet = 0.2~\text{to}~0.3$. []{data-label="fig:xjet-nnlo-ac"}](plots/loopsim-wz-ac/SMNLO_WpZ_x_j_ET_rebin2.pdf "fig:"){width="45.00000%"} ![ $\xjet$ as defined in \[eq:xjetdef\] for inclusive and boosted ($\ptz > \SI{200}{GeV})$ cuts. As visible especially in the boosted case, there are two relevant phase space regions around small $\xjet$ and for $\xjet \approx 0.5$. The latter is not sensitive to AC, such that the AC analysis should focus on the region below $\xjet = 0.2~\text{to}~0.3$. []{data-label="fig:xjet-nnlo-ac"}](plots/loopsim-wz-ac/SMNLO_WpZ_x_j_ET_boosted_rebin2.pdf "fig:"){width="45.00000%"}
In \[fig:xjet-nnlo-ac\], we show the $\xjet$ distribution at NLO and QCD for the SM as well as for two exemplary AC values at QCD. The ratio to the SM NLO prediction shows a non-flat effect of the corrections, enhancing large $\xjet$ values. AC contribute at small $\xjet$ values. Vetoing $\xjet > 0.2$ cuts away both the region not sensitive to AC as well as the region with the largest higher-order corrections.
The veto only induces modest logarithms proportional to $\ln(\xjet) = \ln(0.2)$ and offers an alternative to a fixed-$\pt$ veto without the need for resummation. is a good testing ground for jet vetos, as all terms with potentially large logarithms are included. Jet veto studies at NLO are not sufficient as they miss the two-jet final state, which is dominant in some phase space regions.
![ The effect of anomalous couplings in combination with a jet veto is shown for the cuts $\xjet < 0.2$ and $\xjet < 0.4$. Scale variation bands are given at NLO and . An estimate of the statistical uncertainty is given as a grey band, for which an integrated luminosity of is assumed as well as a factor $4$ for all lepton flavor combinations in the decays. []{data-label="fig:ptl-xjet-veto-ac"}](plots/loopsim-wz-ac/SMNLO-ratios_WpZ_ptlmax_vetoET_02_lowpt_rebin2.pdf "fig:"){width="45.00000%"} ![ The effect of anomalous couplings in combination with a jet veto is shown for the cuts $\xjet < 0.2$ and $\xjet < 0.4$. Scale variation bands are given at NLO and . An estimate of the statistical uncertainty is given as a grey band, for which an integrated luminosity of is assumed as well as a factor $4$ for all lepton flavor combinations in the decays. []{data-label="fig:ptl-xjet-veto-ac"}](plots/loopsim-wz-ac/SMNLO-ratios_WpZ_ptlmax_vetoET_04_lowpt_rebin2.pdf "fig:"){width="45.00000%"}
shows the $\pt$ of the hardest lepton when an additional cut based on $\xjet$ is introduced. The veto is designed to cut away jet-dominated events while allowing harder radiation than a traditional fixed jet veto would, especially in the tails of distributions. The veto reduces the corrections. Instead of an increase of 25%, we see a decrease of the cross section by up to 10%. By cutting away the hard jet events, it also increases the sensitivity to AC.
In \[fig:ptl-xjet-veto-ac\] scale variation bands are given at NLO and . Those bands are not a reliable uncertainty estimate, since with jet vetoes the scale dependence is artificially reduced. For an extended discussion and possible better estimates see e.g. Ref. [@Stewart:2011cf]. Besides the scale variation bands, we also show a band for the assumed statistical error. This is a rough estimate based on of data in bins. For extrapolation from one combination of lepton families in the decays of $W$ and $Z$ bosons to all 4 possibilities, we assume a factor $4$. In final states such as $\mu^+ \nu_\mu \mu^+ \mu^-$, there are two different combinations possible for the reconstruction of the $Z$ boson. We neglect corrections due to imperfect reconstruction and to identical particle effects in that case.
![The K-factor for $\xjet$ of $\nNLO$ compared to NLO is shown for the SM and two values of the $\ow$ coupling. The change of $\nNLO$ effects visible in \[fig:xjet-nnlo-ac\] can be reduced to their dependence on $\xjet$. Different AC values show similar $\xjet$ K-factors and one could thus approximate $\nNLO$ effects by correcting with the shown K-factor. The fluctuations visible in this plot are due to Monte Carlo statistics. []{data-label="fig:xj-nnlo-kfac"}](plots/loopsim-wz-ac/SMNLO-ratioonly_WpZ_x_j_ET_rebin2.pdf){width=".5\textwidth"}
Considering $\nNLO$ corrections to different observables, we find that SM and AC receive corrections with different shapes, such that a description of AC that is based on LO or NLO and rescaled with SM K-factors is not sufficient. In \[fig:xj-nnlo-kfac\], the corrections to $\xjet$ are shown for the SM and different AC values. They are identical (within the simulation uncertainty), such that potentially a K-factor binned in $\xjet$ could describe AC beyond NLO QCD.
Conclusions {#sec:conclusions}
===========
We presented a calculation of WZ production at the LHC at $\nNLO$ QCD including Anomalous Couplings (AC) using VBFNLO in combination with LoopSim.
To enhance the sensitivity to AC, we use a jet veto based on $\xjet$, as defined in \[eq:xjetdef\]. This reduces the large non-AC contribution where a high-$\pt$ jet recoils against a vector boson. The $\xjet$ cut improves the sensitivity to anomalous couplings without introducing large logarithms, as is expected for the traditional fixed-$\pt$ jet veto. Furthermore, the $\xjet$ veto scales with the hardness of the event, such that we include more phase space in the tails of the distribution. Thereby the $\xjet$ cut preserves the region sensitive to AC also at high $\ptz$.
While currently most limits on AC are based on the high-$\pt$ tails of distributions, we suggest to also study the interference region. Since high precision NNLO calculations for vector boson pair production are now becoming available also for distributions [@Grazzini:2016ctr], the theoretical uncertainties in the interference regions will soon be small enough for a meaningful analysis. This double analysis has the advantage of being sensitive to new physics from both strong coupling (large deviations in the high energy tail) as well as to intermediate coupling physics at lower energy scales, where also electroweak corrections are still expected to be modest.
The dominant kinematical effects of corrections are already present at NLO QCD. Thus, exploratory analyses for AC measurements can be performed with NLO programs. For full data analysis, however, the higher precision of NNLO calculations will ultimately be necessary.
FC has been partially supported by the Spanish Government and ERDF funds from the European Commission (Grants No. FPA2014-53631-C2-1-P , FPA2014-57816-P, and SEV-2014-0398) RR is supported by the Graduiertenkolleg “GRK 1694: Elementarteilchenphysik bei höchster Energie und höchster Präzision.”
[^1]: KA-TP-32-2016
[^2]: On NNLO QCD calculations of diboson production see the contribution by S. Kallweit, “NNLO di-boson production.”
[^3]: The factor 4 is not exact, because of small corrections due to the not-ideal reconstruction in states with identical flavors and the Pauli interference effect.
|
---
abstract: 'While the existence of a spin-liquid ground state of the spin-1/2 kagome Heisenberg antiferromagnet (KHAF) is well established, the discussion of the effect of an interlayer coupling (ILC) by controlled theoretical approaches is still lacking. Here we study this problem by using the coupled-cluster method to high orders of approximation. We consider a stacked KHAF with a perpendicular ILC $J_\perp$, where we study ferro- as well as antiferromagnetic $J_\perp$. We find that the spin-liquid ground state (GS) persists until relatively large strengths of the ILC. Only if the strength of the ILC exceeds about 15% of the intralayer coupling the spin-liquid phase gives way for $q=0$ magnetic long-range order, where the transition between both phases is continuous and the critical strength of the ILC, $|J^c_\perp|$, is almost independent of the sign of $J_\perp$. Thus, by contrast to the quantum GS selection of the strictly two-dimensional KHAF at large spin $s$, the ILC leads first to a selection of the $q=0$ GS. Only at larger $|J_\perp|$ the ILC drives a first-order transition to the $\sqrt{3}\times\sqrt{3}$ long-range ordered GS. As a result, the stacked spin-1/2 KHAF exhibits a rich GS phase diagram with two continuous and two discontinuous transitions driven by the ILC.'
author:
- |
O. Götze and J. Richter\
\
title: 'The route to magnetic order in the spin-$1/2$ kagome Heisenberg antiferromagnet: The role of interlayer coupling'
---
[*Introduction.–*]{} The search for exotic quantum spin liquid (QSL) states and fractionalized quasiparticles in frustrated magnets attracts currently much attention both from the theoretical and experimental side. One of the most promising, fascinating, and, at same time, challenging problems is the investigation of the ground state (GS) of the quantum antiferromagnet on the kagome lattice. Over the last 25 years a plethora of theoretical approaches has been applied to understand the GS properties of the spin-$1/2$ kagome antiferromagnet (KAFM), see, e.g., Refs. [@singh1992; @Waldtmann1998; @Capponi2004; @Singh2007; @Evenbly2010; @Yan2011; @lauchli2011; @iqbal2011; @nakano2011; @goetze2011; @schollwoeck2012; @becca2013; @ioannis2013; @bruce2014; @Ioannis2014; @Vishwanath2015; @Becca2015; @Oitmaa2016]. Clearly, the GS of the $s=1/2$ Heisenberg KAFM does not exhibit GS magnetic long-range order (LRO). However, there is a long-standing debate on the nature of the quantum GS. Recent large-scale numerical studies [@Yan2011; @lauchli2011; @schollwoeck2012] provide arguments for a gapped $\mathbb{Z}_2$ topological QSL for spin $s=1/2$. However, the gap state is not fully proven, and also a gapless spin liquid is suggested, see, e.g., Refs. [@iqbal2011; @becca2013; @Becca2015].
A natural question is that for the stability of the QSL phase against modifications of the paradigmatic pure $s=1/2$ KAFM. Several recent investigations have been focused on $s>1/2$ [@goetze2011; @cepas2011; @Lauchli_s1_2014; @Weichselbaum_s1_2014; @satoshi_s1_2014; @Weichselbaum_s1_2014a; @Kumar2015; @Liu2016], anisotropic models [@cepas2008; @Mila2009; @zhito_XXZ_2014; @XXZ_s12_2014; @wir_XXZ_2015; @XXZ_s12_2015; @Becca2015; @fradkin2015; @Chernyshev2015; @Jaubert2015; @Liu2016] as well as KAFMs with further-neighbor couplings [@XXZ_s12_2015; @Domenge2005; @Janson2008; @Bishop2010; @tay2011; @Li2012; @Balents2012; @Thomale2014; @Trebst2014; @Gong2015; @Gong2015a; @Schollwoeck2015; @Bieri2015; @Laeuchli2015; @Thomale2015]. It has been found that such modifications of the pure KAFM may play a crucial role either to modify the QSL state or even to establish GS magnetic LRO of $\sqrt{3} \times \sqrt{3}$ or of $q=0$ symmetry. At that the input from experiments plays an important role to trigger the theoretical hunt for exotic quantum states [@herbertsmithite2007; @herbertsmithite2007a; @herbertsmithite2009; @herbertsmithite2010; @herbertsmithite2012; @kapellasite2010; @kapellasite2014; @Bernu2013; @Jeschke2013; @Thomale2015; @edwartsite2013; @barlowite2014; @barlowite2014a; @tanaka2009; @Tanaka2014; @Tanaka2015; @Bieri2015a]. Prominent examples for $s=1/2$ kagome compounds are herbertsmithite [@herbertsmithite2007; @herbertsmithite2007a; @herbertsmithite2009; @herbertsmithite2010; @herbertsmithite2012] and kapellasite [@kapellasite2010; @kapellasite2014]. Both compounds do not show magnetic order down to very low temperatures [@herbertsmithite2007; @herbertsmithite2007a; @herbertsmithite2009; @herbertsmithite2010; @herbertsmithite2012; @kapellasite2010; @kapellasite2014]. However, the underlying magnetic model is quite different. Herbertsmithite is likely the best realization of a spin-$1/2$ Heisenberg KAFM with only nearest-neighbor (NN) exchange couplings. On the other hand, the model for kapellasite contains noticeable further-neighbor couplings $J_d$ along the diagonals of the hexagons [@Janson2008; @Bernu2013; @Jeschke2013; @Thomale2015]. Except the kagome compounds without magnetic order there are several kagome magnets which exhibit a phase transition to a long-range ordered state at a critical temperature $T_c$. Examples are edwardsite [@edwartsite2013], barlowite [@barlowite2014; @barlowite2014a] or the family of kagome compounds Cs$_2$Cu$_3$MF$_{12}$ (M=Zr, Hf, Sn) [@tanaka2009; @Tanaka2014; @Tanaka2015]. For an overview on the relation between extended models and kagome compounds we refer the interested reader to Ref. [@Thomale2015].
Bearing in mind the huge number of theoretical studies of purely two-dimensional (2D) kagome models, see, e.g., Refs. [@singh1992; @Waldtmann1998; @Capponi2004; @Singh2007; @Evenbly2010; @Yan2011; @lauchli2011; @iqbal2011; @nakano2011; @goetze2011; @schollwoeck2012; @becca2013; @ioannis2013; @bruce2014; @Ioannis2014; @Vishwanath2015; @Oitmaa2016; @cepas2011; @Lauchli_s1_2014; @Weichselbaum_s1_2014; @satoshi_s1_2014; @Weichselbaum_s1_2014a; @Kumar2015; @cepas2008; @Mila2009; @zhito_XXZ_2014; @XXZ_s12_2014; @wir_XXZ_2015; @XXZ_s12_2015; @Becca2015; @fradkin2015; @Chernyshev2015; @Jaubert2015; @Domenge2005; @Janson2008; @Bishop2010; @tay2011; @Li2012; @Balents2012; @Thomale2014; @Trebst2014; @Gong2015; @Gong2015a; @Schollwoeck2015; @Bieri2015; @Laeuchli2015; @Thomale2015; @Liu2016], the investigation of the role of interlayer coupling (ILC) $J_{il}$ so far has been widely ignored. The reason for that might be related to the fact that most of the controlled approaches with satisfactory accuracy, such as large-scale exact diagonalization, density matrix renormalization group (DMRG), or entanglement renormalization techniques are designed for low-dimensional quantum systems. Thus, for example, for the three-dimensional (3D) counterpart of the KAFM, the quantum pyrochlore Heisenberg antiferromagnet (HAFM), precise GS data are missing so far. To the best of our knowledge the stacked kagome spin-$1/2$ HAFM was studied only in an early paper by using a rotational invariant Green’s function approach [@RGM2004]. Certainly, one can expect that in kagome compounds an ILC is present. The geometry and the strength of $J_{il}$ may differ from compound to compound. Unquestionably, an ILC is crucial to establish magnetic LRO at finite temperatures, at least if the spin anisotropy is negligible.
In the present paper we study the spin-$1/2$ HAFM on the stacked kagome lattice described by $$\begin{aligned}
\label{ham}
H=\sum_n\Bigg(\sum_{\langle ij \rangle}{\bf s}_{i,n} \cdot {\bf s}_{j,n}
\Bigg)
+ J_\perp \sum_{i,n} {\bf s}_{i,n} \cdot {\bf s}_{i,n+1},\end{aligned}$$ where $n$ labels the kagome layers and $J_\perp$ is a perpendicular (i.e. non-frustrated) ILC. The expression in brackets represents the kagome HAFM model of the layer $n$ with NN intralayer couplings $J=1$. For $J_\perp$ we consider antiferromagnetic (AFM) as well as ferromagnetic (FM) couplings.
The questions we want to address in the present paper are as follows: Is the perpendicular ILC $J_{\perp}$ able to establish magnetic LRO for kagome $s=1/2$ layers with AFM isotropic NN interactions, at all? As we will demonstrate below the answer is ’yes’. Then, as consequent questions arise: Does the magnetically disordered GS survive a (small) finite (non-frustrated) ILC? Which GS magnetic LRO (i.e. $\sqrt{3} \times \sqrt{3}$ or $q=0$) is selected? Is the sign of $J_{\perp}$ relevant? If for $|J_{\perp}|>0$ GS magnetic LRO is present, we may expect that for the 3D system at hand a finite critical temperature $T_c$ exists. From previous studies of coupled low-dimensional Heisenberg spin systems [@Irkhin1997; @Troyer2005] we know that $T_c$ may grow as a logarithmic function of $J_\perp$ slightly beyond the quantum phase transition to GS magnetic LRO.
In order to address the above asked questions concerning the role of the ILC we use the coupled cluster method (CCM)[@bishop98a; @bishop04] to high orders of approximation. The CCM is a very general [*ab initio*]{} many-body technique that has been successfully applied to strongly frustrated quantum magnets [@Bishop2010; @goetze2011; @wir_XXZ_2015; @Li2012; @Schm:2006; @darradi08; @Zinke2008; @farnell09; @richter2010; @farnell11; @archi2014; @bishop2014; @gapj1j2_2015; @jiang2015; @Li2015]. The precision of the method has been demonstrated for kagome spin systems in Refs. [@goetze2011] and [@wir_XXZ_2015]. Thus, the CCM GS energy for the $s=1/2$ isotropic Heisenberg KAFM is close to best available DMRG results [@Yan2011; @schollwoeck2012]. By contrast to exact diagonalization, DMRG, or entanglement renormalization techniques the CCM can be applied straightforwardly to 3D systems [@Schm:2006; @bishop00].
[*Coupled cluster method (CCM).–*]{} We illustrate here only some basic relevant features of the CCM. At that we follow Refs. [@goetze2011] and [@wir_XXZ_2015], where the CCM was applied to the 2D KAFM. For more general information on the CCM, see, Refs. [@roger90; @bishop91a; @zeng98; @bishop00; @bishop04]. Note first that the CCM yields results directly for number of sites $N\to\infty$. As a starting point of the CCM calculation we choose a normalized reference state $|\Phi\rangle$. From a quasi-classical point of view that is for the system at hand the stacked coplanar $\sqrt{3}\times\sqrt{3}$ or $q=0$ state (see, e.g., Refs. [@zhito_XXZ_2014; @wir_XXZ_2015; @Harris1992; @sachdev1992; @chub92; @henley1995]). We perform a rotation of the local axes of each of the spins such that all spins in the reference state align along the negative $z$ axis. Within the framework of the local spin coordinates we define a complete set of multispin creation operators $C_I^+ \equiv (C^{-}_{I})^{\dagger}$ related to this reference state: $|{\Phi}{\rangle}= |\downarrow\downarrow\downarrow\cdots\rangle ; \mbox{ }
C_I^+
= { s}_{n}^+ \, , \, { s}_{n}^+{ s}_{m}^+ \, , \, { s}_{n}^+{ s}_{m}^+{
s}_{k}^+ \, , \, \ldots \; ,
$. Here the spin operators are defined in the local rotated coordinate frames. The indices $n,m,k,\ldots$ denote arbitrary lattice sites. The ket and bra GS eigenvectors $|\Psi{\rangle}$ and ${\langle}\tilde{\Psi}|$ of the spin system are given by $|\Psi{\rangle}=e^S|\Phi{\rangle}\; , \mbox{ } S=\sum_{I\neq 0}a_IC_I^+ \; ; \;
$ ${\langle}\tilde{ \Psi}|={\langle}\Phi |\tilde{S}e^{-S} \; , \mbox{ } \tilde{S}=1+
\sum_{I\neq 0}\tilde{a}_IC_I^{-} .$ The coefficients $a_I$ and $\tilde{a}_I$ in the CCM correlation operators, $S$ and $\tilde{S}$, can be determined by the ket-state and bra-state equations $\langle\Phi|C_I^-e^{-S}He^S|\Phi\rangle = 0 \; ; \; \langle\Phi|{\tilde S}e^{-S}[H, C_I^+]e^S|\Phi\rangle = 0 \; ; \; \forall
I\neq 0.$ Each equation belongs to a certain configuration index $I$, i.e., it corresponds to a certain configuration of lattice sites $n,m,k,\dots\;$. From the Schrödinger equation, $H|\Psi{\rangle}=E_0|\Psi{\rangle}$, we get for the GS energy $E_0={\langle}\Phi|e^{-S}He^S|\Phi{\rangle}$. The magnetic order parameter (sublattice magnetization) is given by $ M = -\frac{1}{N} \sum_{i=1}^N {\langle}\tilde\Psi|{ s}_i^z|\Psi{\rangle}$, where ${s}_i^z$ is expressed in the transformed coordinate system. For the solution of the ket-state and bra-state equations we use the well established LSUB$m$ approximation scheme, in order to truncate the expansions of $S$ and $\tilde S$, cf., e.g., Refs. [@roger90; @bishop91a; @Bishop2010; @goetze2011; @wir_XXZ_2015; @Li2012; @Schm:2006; @darradi08; @Zinke2008; @farnell09; @richter2010; @farnell11; @bishop2014; @gapj1j2_2015; @jiang2015; @Li2015]. In the LSUB$m$ scheme no more than $m$ spin flips spanning a range of no more than $m$ contiguous lattice sites are included. Using an efficient parallelized CCM code [@cccm] we can solve the CCM equations up to LSUB8 for $s=1/2$. Following Refs. [@goetze2011; @wir_XXZ_2015] we extrapolate the ‘raw’ LSUB$m$ data to the limit $m \to \infty$. Here we use two schemes, namely an extrapolation using $m=4,5,\ldots,8$ (scheme I) and separetely an extrapolation using $m=4,6,8$ (scheme II). The former one corresponds to that used for the 2D KAFM [@goetze2011; @wir_XXZ_2015], whereas scheme II (i.e., omitting the odd LSUB$m$ approximation levels) is more appropriate for magnets with collinear AFM correlations [@bishop04; @Schm:2006; @darradi08; @Zinke2008; @richter2010; @farnell11; @archi2014; @gapj1j2_2015]. By comparing the results of both schemes we can get an idea on the precision of the extrapolated data.
For the GS energy the ansatz $e_0(m)=E_0(m)/N = e_0(m\to\infty) + a_1/m^2 + a_2/m^4$ provides accurate data for the extrapolated energy $ e_0(m\to\infty)$, whereas for the magnetic order parameter $M$ the ansatz $M(m)=M(m\to\infty) +b_1(1/m)^{x}+b_2(1/m)^{x+1}$ is appropriate. The choice of the leading exponent $x$ is a subtle issue, since $x$ might be different in semi-classical GS phases with well-pronounced magnetic LRO and near a quantum critical point, see [@Bishop2010; @goetze2011; @wir_XXZ_2015; @Li2012; @darradi08; @Zinke2008; @Schm:2006; @richter2010; @farnell11; @gapj1j2_2015; @Li2015]. For the kagome problem at hand we start from a magnetically disordered phase at $J_\perp=0$ and search for quantum phase transitions to GSs with magnetic LRO. For that the extrapolation of $M$ with $x=1/2$ is the best choice as it has been demonstrated in many previous CCM investigations [@Bishop2010; @goetze2011; @wir_XXZ_2015; @Li2012; @darradi08; @richter2010; @farnell11; @gapj1j2_2015; @Li2015]. Thus, the CCM treatment of the celebrated spin-half $J_1$-$J_2$ model on the square lattice using $x=1/2$ [@darradi08; @gapj1j2_2015] yields quantum critical points, which are in very good agreement with best available numerical results obtained by DMRG with explicit implementation of SU(2) spin rotation symmetry [@Gong2014].
[*Results and Discussion.–*]{} We start with a brief discussion of the GS energy per spin $e_0=E_0/N$, shown in the insets of Figs. \[fig1\](a) and (b) for the $\sqrt{3}\times\sqrt{3}$ and $q=0$ reference states, respectively. We see that $e_0$ converges quickly as the level $m$ of the LSUB$m$ approximation increases. Hence, the extrapolation with leading order $1/m^2$ can be considered as very accurate, as it has been demonstrated in many cases, where data from other precise methods are available to compare with, see, e.g. Refs. [@bishop04; @goetze2011; @wir_XXZ_2015]. Moreover, the results of both extrapolation schemes are almost indistinguishable. The shape of the curves and the magnitude of the energies is very similar for both states. From Ref. [@goetze2011] we know that at $J_\perp=0$ the $q=0$ state has slightly lower energy. The extrapolated GS energy behaves smoothly as changing the sign of $J_\perp$. The magnetic order parameter $M$ for the $\sqrt{3}\times\sqrt{3}$ and $q=0$ states is shown in the main panel of Figs. \[fig1\](a) and (b). Of course, $M$ is zero for $J_\perp=0$ [@goetze2011; @wir_XXZ_2015]. As a main result, we find that the ILC is able to establish magnetic LRO for kagome $s=1/2$ layers with AFM NN Heisenberg interactions. The critical ILCs, where magnetic LRO sets in, are (i) $J_\perp=-0.100$, $J_\perp=+0.102$ ($\sqrt{3}\times\sqrt{3}$ state) and $J_\perp=-0.154$, $J_\perp=+0.151$ ($q=0$ state) for scheme I, and (ii) $J_\perp=-0.104$, $J_\perp=+0.110$ ($\sqrt{3}\times\sqrt{3}$ state) and $J_\perp=-0.135$, $J_\perp=+0.130$ ($q=0$ state) for scheme II. Thus, there is a reasonable agreement of the critical ILCs obtained by both extrapolation schemes. We notice that the amount of the critical $|J_\perp|$ is of comparable size as the spin gap estimated, e.g., in Refs. [@Capponi2004] and [@schollwoeck2012]. We may also compare with the square-lattice $J_1$-$J_2$ HAFM in the limit of strong frustration, i.e., at $J_2/J_1 \sim 0.5$. The critical ILC $J_{\perp}$ found by various approaches [@Schm:2006; @Holt2011; @Fan2014] is $J_{\perp} \approx
0.12 - 0.2 J_1$, i.e., its size is comparable to that reported here for the kagome system. The behavior of $M$ near the critical $J_{\perp}$ indicates a typical second-order transition, where the slope of $M$ is quite steep. On the FM side ($J_\perp < 0$) there is a monotonic increase of $M$ with increasing $|J_\perp|$, and, both schemes I and II lead to very similar $M(J_\perp)$ curves. By contrast, on the AFM side ($J_\perp > 0$) there is a noticeable difference between both schemes. That can be attributed to emerging collinear AFM correlations along the AFM $J_\perp$ bonds that may lead to a different scaling of odd and even LSUB$m$ data [@ccm_odd_even]. We mention, that the maximum value of $M$ remains small even at $J_\perp \sim
1$. Note that for AFM $J_\perp$ we have calculated data up to $J_\perp = 100$. For the extrapolation scheme II relevant in the limit of large $J_\perp$ we do not find indications for a breakdown of LRO at a finite $J_\perp$, rather there is a monotonic decrease of $M$ with increasing $J_\perp$ reaching adiabatically $M=0$ at infinite $J_\perp$, cf. also Ref. [@Zinke2008].
Next we discuss the question which magnetic LRO is selected by quantum fluctuations. As it has been very recently demonstrated [@zhito_XXZ_2014; @wir_XXZ_2015] the mechanism of quantum selection of the GS LRO in the KAFM is very subtle and it is related to topologically nontrivial, looplike high-order spin-flip processes [@zhito_XXZ_2014]. As a result, the energy difference between competing states is very small, e.g., about $10^{-4}J$ for the $XXZ$-KAFM [@zhito_XXZ_2014; @wir_XXZ_2015]. Hence, it is crucial to have a theory at hand that provides very accurate results for the GS energy and is able to take into account such high-order spin-flip processes. These criteria are fulfilled by the CCM, if high orders of approximation are considered. Thus, the quantum selection of the $\sqrt{3}\times\sqrt{3}$ GS vs. the $q=0$ GS obtained by non-linear spin-wave theory is also obtained by CCM for $s>1/2$ [@goetze2011]. Very recently, a direct comparison of CCM and non-linear spin-wave data for energy differences (which are also of the order of few $10^{-3}$) for the $XXZ$ KAFM for large $s$ has been given, see Fig. 3 in Ref. [@wir_XXZ_2015], which provides evidence that both independent approaches agree very well. Thus, we may conclude that our results for the quantum selection are trustworthy. We show our results for the energy difference $\delta e = e_0^{\sqrt{3} \times \sqrt{3}}- e_0^{q=0} $ as a function of $J_\perp$ in Fig. \[fig2\]. We mention first that both extrapolation schemes I and II yield consistent results for $\delta e $. At low values of $|J_\perp|$ the $q=0$ reference state yields lower energy, i.e. $\delta e > 0$. That is in accordance with Refs. [@goetze2011] and [@wir_XXZ_2015], where the case $J_\perp=0$ was considered. On both sides $\delta e$ is still positive at those values of $J_\perp$, where the sublattice magnetizations $M_{\sqrt{3} \times \sqrt{3}}$ or $M_{q=0}$ become larger than zero. Hence, our results provide evidence that there is a magnetic disorder-to-order transition to ${q=0}$ LRO at $J_\perp \sim -(0.14 \ldots 0.15)$ and $J_\perp \sim +(0.13 \ldots 0.15)$, respectively, where this transition is likely continuous. Note that the quantum selection of the $q=0$ GS LRO is contrary to the semi-classical large-$s$ order-by-disorder selection of the $\sqrt{3} \times \sqrt{3}$ LRO found for the 2D spin-$s$ KAFM. Further increasing the strength of $J_\perp$ leads to a second transition from ${q=0}$ to $\sqrt{3} \times \sqrt{3}$ LRO on the FM side at $J_\perp=-0.435$ (scheme I) and $J_\perp=-0.467$ (scheme II). At the AFM side we find $J_\perp=0.310$ (scheme I) and $J_\perp=0.252$ (scheme II). By contrast to the first transition this second transition is a discontinuous one between two ordered GS phases with different symmetries. On the FM side we may understand the realization of $\sqrt{3}
\times \sqrt{3}$ LRO in terms of the large-$s$ order-by-disorder GS selection of the $\sqrt{3}
\times \sqrt{3}$ state. Increasing the strength of the FM ILC leads to an effective composite spin with higher spin quantum number. However, this kind of mechanism does not work for AFM $J_\perp$, and to clarify the mechanism responsible for changing the GS selection remains an open question.
Collecting our results we obtain a sketch of the GS phase diagram of the stacked spin-1/2 Heisenberg KAFM as shown in Fig. \[fig3\]. The system exhibits four transitions, two continuous ones between a QSL state and a magnetically ordered state with $q=0$ symmetry at $J_\perp \sim -(0.14 \ldots 0.15)$ and $J_\perp \sim +(0.13 \ldots 0.15)$, and two discontinuous ones between states with magnetic LRO of $q=0$ and $\sqrt{3}
\times \sqrt{3}$ symmetry at $J_\perp \sim -(0.44 \ldots 0.47)$ and $J_\perp \sim +(0.25 \ldots 0.31)$. We further argue, that this kind of phase diagram is specific for the extreme quantum case $s=1/2$. From Ref. [@goetze2011] we know that already for $s=1$ (and also for $s>1$) the $\sqrt{3}\times\sqrt{3}$ reference state has the lower energy. It seems to be very unlikely that this preference of the $\sqrt{3}\times\sqrt{3}$ state is changed by $J_\perp$.
[*Concluding remarks.*]{} Let us discuss the relation of our findings to the previous results based on a rotational invariant Green’s function method (RGM) [@RGM2004], where the existence of a non-magnetic GS for arbitrary values of $J_\perp$ was reported. To evaluate this discrepancy we have to assess the accuracy of the current CCM approach and of the RGM approach. First we mention that the CCM is a systematic approach taking into account all spin-flip processes up to a well-defined order. On the other hand, the decoupling of the equation of motion used in the RGM contains uncontrolled elements of approximation. Meanwhile, there is ample of experience in applying the RGM on frustrated quantum magnets, see, e.g., Ref. [@haertel2013] and references therein. In its minimal version (used in Ref. [@RGM2004]), where as many vertex parameters are used as independent conditions for them can be formulated, the accuracy of the description of GS properties seems to be limited [@Barabanov1994; @Ihle2001; @Schm:2006]. In particular, the rotational invariant decoupling strongly overestimates the region of QSL phases. Thus, for the square-lattice $s=1/2$ $J_1$-$J_2$ HAFM the minimal version of the RGM predicts a QSL phase in an extremely wide region $0.1 \lesssim J_2/J_1 \lesssim 1.7$, cf., e.g., Refs. [@Barabanov1994; @Ihle2001], instead of $0.44 \lesssim J_2/J_1 \lesssim 0.6$, obtained by recent DMRG calculations [@Gong2014] and also by the CCM [@darradi08; @gapj1j2_2015]. Another indication is the fairly poor GS energy of $e_0=-0.4296$ [@RGM2004; @bern], that is more than 2% above the best available DMRG energy $e_0=-0.4386$. (Note that the CCM energy obtained in Ref. [@goetze2011] is $e_0=-0.4372$.) Thus we have evidence that the CCM description of the GS properties is much more reliable than the RGM in its minimal version.
Let us finally discuss the relevance of our results for experiments on kagome compounds. In real kagome compounds typically the interlayer coupling is more sophisticated than that we consider in our paper. Thus, there is only an indirect relation of our results to those compounds, which concerns the general question for the crossover from a purely 2D to a quasi-2D and finally to a three-dimensional system. However, there is at least one example with stacked (unshifted) kagome layers, namely barlowite. As it has been pointed out very recently, through isoelectronic substitution in barlowite this kagome system fits to our model system [@barlow2016].
A main finding of our paper is that the QSL phase can be observed even if there is a sizeable ILC. Therefore, in accordance with the experimental observation the ILC of about 5% of the intralayer coupling as reported for herbertsmithite [@janson_diss] and the ILC of about 6-7% predicted for the modified barlowite system [@barlow2016] is not sufficient to destroy the QSL phase. On the other hand, if the ILC is large enough (about 15% of the intralayer coupling in our model system) magnetic LRO can be established, where the $q=0$ symmetry is favorable if $J_\perp$ is of moderate strength. Thus, the observed $q=0$ magnetic order found in Cs$_2$Cu$_3$SnF$_{12}$ and ascribed in Refs. [@Tanaka2014] and [@Tanaka2015] to anisotropy terms could also be attributed to the ILC without further anisotropy terms. The facilitation of the $\sqrt{3}\times\sqrt{3}$ magnetic long-range order found in the present paper for larger values of $|J_\perp|$ is related to a very small energy gain. In real compounds even very small additional terms in the relevant spin Hamiltonian such as further distance exchange couplings may therefore be more relevant.
Acknowledgments
===============
We thank H. Rosner, H. Tanaka, O. Janson and O. Derzhko for fruitful discussions.
[99]{}
R. R. P. Singh and D. A. Huse, Phys. Rev. Lett. [**68**]{}, 1766 (1992).
C. Waldtmann H.-U. Everts, B. Bernu, C. Lhuillier, P. Sindzingre, P. Lecheminant, and L. Pierre, Eur. Phys. J. B [**2**]{}, 501 (1998).
S. Capponi, A. Läuchli, and M. Mambrini, Phys. Rev. B [**70**]{}, 104424 (2004).
R. R. P. Singh and D. A. Huse, Phys. Rev. B [**76**]{}, 180407(R) (2007).
G. Evenbly and G. Vidal, Phys. Rev. Lett. [**104**]{}, 187203 (2010).
S. Yan, D. A. Huse, and S. R. White, Science [**332**]{}, 1173 (2011). H. Nakano and T. Sakai, J. Phys. Soc. Jpn. [**80**]{}, 053704 (2011).
A. M. Läuchli, J. Sudan, and E. S. S[ø]{}rensen, Phys. Rev. B [**83**]{}, 212401 (2011).
Y. Iqbal, F. Becca, and D. Poilblanc, Phys. Rev. B [**84**]{}, 020407(R) (2011). O. Götze, D.J.J. Farnell, R.F. Bishop, P.H.Y. Li, and J. Richter, Phys. Rev. B [**84**]{}, 224428 (2011).
S. Depenbrock, I. P. McCulloch, and U. Schollwöck, Phys. Rev. Lett. [**109**]{}, 067201 (2012).
Y. Iqbal, F. Becca, S. Sorella, D. Poilblanc, Phys. Rev. B [**87**]{}, 060405(R) (2013). I. Rousochatzakis, R. Moessner, J. van den Brink, Phys. Rev. B [**88**]{}, 195109 (2013).
Z. Y. Xie, J. Chen, J. F. Yu, X. Kong, B. Normand, T. Xiang, Phys. Rev. X [**4**]{}, 011025 (2014). I. Rousochatzakis, Y. Wan, O. Tchernyshyov, and F. Mila, Phys. Rev. B [**90**]{}, 100406(R) (2014).
W.-J. Hu, S.S. Gong, F. Becca, and D.N. Sheng, Phys. Rev. B [**92**]{}, 2015(R) (2015).
M. P. Zaletel and A. Vishwanath, Phys. Rev. Lett. [**114**]{}, 077201 (2015).
J. Oitmaa and R. R. P. Singh, Phys. Rev. B [**93**]{}, 014424 (2016) O. Cépas and A. Ralko, Phys. Rev. B [**84**]{}, 020413 (2011).
T. Liu, W. Li, A. Weichselbaum, J. von Delft, and G. Su, Phys. Rev. B 91, [**060403**]{} (2015).
H. J. Changlani and A. M. Läuchli, Phys. Rev. B [**91**]{}, 100407 (2015).
S. Nishimoto and M. Nakamura, Phys. Rev. B [**92**]{}, 140412(R) (2015).
W. Li, A. Weichselbaum, J. von Delft, and H.-H. Tu, Phys. Rev. B [**91**]{}, 224414 (2015). P. Ghosh, A.K. Verma, and B. Kumar, Phys. Rev. B [**93**]{}, 014427 (2016).
Tao Liu, Wei Li, Gang Su, arXiv:1603.01935v1.
O. Cepas, C. M. Fong, P. W. Leung, C. Lhuillier, Phys. Rev. B [**78**]{}, 140405(R) (2008). I. Rousochatzakis, S. R. Manmana, A. M. Läuchli, B. Normand, and F. Mila, Phys. Rev. B [**79**]{}, 214415 (2009).
A. L. Chernyshev and M. E. Zhitomirsky, Phys. Rev. Lett. [**113**]{}, 237202 (2014).
Y.C. He and Y. Chen, Phys. Rev. Lett. [**114**]{}, 037201 (2015)
O. Götze and J. Richter, Phys. Rev. B [**91**]{}, 104402 (2015).
W. Zhu, S. S. Gong, and D. N. Sheng, Phys. Rev. B [**92**]{}, 014424 (2015).
K. Kumar, K. Sun, and E. Fradkin, Phys. Rev. B [**92**]{}, 094433 (2015). A. L. Chernyshev and M. E. Zhitomirsky, Phys. Rev. B [**92**]{}, 144415 (2015).
K. Essafi, O. Benton, and L.D.C. Jaubert, Nat. Commun. [**7**]{}, 10297 (2016) J.-C. Domenge, P. Sindzingre, C. Lhuillier, and L. Pierre, Phys. Rev. B [**72**]{}, 024433 (2005). O. Janson, J. Richter, and H. Rosner, Phys. Rev. Lett. [**101**]{}, 106403 (2008).
R.F. Bishop, P.H.Y. Li, D.J.J. Farnell, and C.E. Campbell, Phys. Rev. B [**82**]{}, 104406 (2010).
T. Tay and O. I. Motrunich, Phys. Rev. B [**84**]{}, 020404(R) (2011). P.H.Y. Li, R.F. Bishop, C.E. Campbell, D.J.J. Farnell, Phys. Rev. B [**86**]{}, 214403 (2012).
H.-C. Jiang, Z. Wang, and L. Balents, Nature Physics [**8**]{}, 902 (2012). B. Bauer, L. Cincio, B.P. Keller, M. Dolfi, G. Vidal, S. Trebst, and A.W.W. Ludwig, Nature Communications [**5**]{}, 5137 (2014).
R. Suttner, C. Platt, J. Reuther, and R. Thomale, Phys. Rev. B [**89**]{}, 020408(R) (2014).
W.J. Hu, W. Zhu, Y. Zhang, S.S. Gong, F. Becca, and D.N. Sheng Phys. Rev. B [**91**]{}, 041124(R) (2015) S.S. Gong, W. Zhu, L. Balents, and D. N. Sheng Phys. Rev. B [**91**]{}, 075112 (2015).
F. Kolley, S. Depenbrock, I. P. McCulloch, U. Schollwöck, and V. Alba, Phys. Rev. B [**91**]{}, 104418 (2015).
S. Bieri, L. Messio, B. Bernu, and C. Lhuillier, Phys. Rev. B [**92**]{}, 060407(R) (2015). A. Wietek, A.Sterdyniak, and A. M. Laeuchli, Phys. Rev. B [**92**]{}, 125122 (2015).
Y. Iqbal, H. O. Jeschke, J. Reuther, R. Valenti, I.I. Mazin, M. Greiter, and R. Thomale, Phys. Rev. B [**92**]{}, 220404 (2015). P. Mendels, F. Bert, M. A. de Vries, A. Olariu, A. Harrison, F. Duc, J. C. Trombe, J. S. Lord, A. Amato, and C. Baines, Phys. Rev. Lett. [**98**]{}, 077204 (2007).
J. S. Helton, K. Matan, M. P. Shores, E. A. Nytko, B. M. Bartlett, Y. Yoshida, Y. Takano, A. Suslov, Y. Qiu, J.-H. Chung, D. G. Nocera, and Y. S. Lee, Phys. Rev. Lett. [**98**]{}, 107204 (2007). M. A. de Vries, J. R. Stewart, P. P. Deen, J. Piatek, G. N. Nilsen, H. M. Ronnow, and A. Harrison, Phys. Rev. Lett. [**103**]{}, 237201 (2009). D. Wulferding, P. Lemmens, P. Scheib, J. Röder, P. Mendels, S. Chu, T. Han, and Y. S. Lee, Phys. Rev. B [**82**]{}, 144412 (2010). T. H. Han, J. S. Helton, S. Chu, D. G. Nocera, J. A. Rodriguez-Rivera, C. Broholm, and Y. S. Lee, Nature (London) [**492**]{}, 406 (2012).
B. Fak, E. Kermarrec, L. Messio, B. Bernu, C. Lhuillier, F. Bert, P. Mendels, B. Koteswararao, F. Bouquet, J. Ollivier, A. D. Hillier, A. Amato, R. H. Colman, and A. S. Wills, Phys. Rev. Lett. [**109**]{}, 037208 (2012). E. Kermarrec, A. Zorko, F. Bert, R. H. Colman, B. Koteswararao, F. Bouquet, P. Bonville, A. Hillier, A. Amato, J. van Tol, A. Ozarowski, A. S. Wills, and P. Mendels, Phys. Rev. B [**90**]{}, 205103 (2014). B. Bernu, C. Lhuillier, E. Kermarrec, F. Bert, P. Mendels, R. H. Colman, and A. S. Wills, Phys. Rev. B [**87**]{}, 155107 (2013). H. O. Jeschke, F. Salvat-Pujol, and R. Valenti, Phys. Rev. B [**88**]{}, 075106 (2013).
H. Ishikawa, Y. Okamoto, and Z. Hiroi, J. Phys. Soc. Jpn. [**82**]{}, 063710 (2013).
T.-H. Han, J. Singleton, J. A. Schlueter, Phys. Rev. Lett. [**113**]{}, 227203 (2014). H. O. Jeschke, F. Salvat-Pujol, E. Gati, N. H. Hoang, B. Wolf, M. Lang, J. A. Schlueter, and R. Valenti, Phys. Rev. B [**92**]{}, 094417 (2015). T. Ono, K. Morita, M. Yano, H. Tanaka, K. Fujii, H. Uekusa, Y. Narumi, and K. Kindo, Phys. Rev. B [**79**]{}, 174407 (2009). T. Ono, K. Matan, Y. Nambu, T. J. Sato, K. Katayama, S. Hirata, and H. Tanaka, J. Phys. Soc. Jpn. [**83**]{}, 043701 (2014).
K. Katayama, N. Kurita, and H. Tanaka, Phys. Rev. B [**91**]{}, 214429 (2015). D. Boldrin, B. Fak, M. Enderle, S. Bieri, J. Ollivier, S. Rols, P. Manuel, and A. S. Wills, Phys. Rev. B [**91**]{}, 220408(R) (2015).
D. Schmalfu[ß]{}, J. Richter and D. Ihle, Phys. Rev. B [**70**]{} 184412 (2004)
V. Yu. Irkhin and A. A. Katanin, Phys. Rev. B [**55**]{}, 12318 (1997).
C. Yasuda, S. Todo, K. Hukushima, F. Alet, M. Keller, M. Troyer, and H. Takayama, Phys. Rev. Lett. [**94**]{}, 217201 (2005).
R.F. Bishop, in [*Microscopic Quantum Many-Body Theories and Their Applications*]{}, edited by J. Navarro and A. Polls, Lecture Notes in Physics [**510**]{} (Springer, Berlin, 1998), p.1.
D.J.J. Farnell and R.F. Bishop, in [*Quantum Magnetism*]{}, Lecture Notes in Physics [**645**]{}, 307 (2004)
D. Schmalfu[ß]{}, R. Darradi, J. Richter, J. Schulenburg, and D. Ihle, Phys. Rev. Lett. [**97**]{}, 157201 (2006).
R. Darradi, O. Derzhko, R. Zinke, J. Schulenburg, S.E. Krüger and J. Richter, Phys. Rev. B [**78**]{}, 214415 (2008).
R. Zinke, J. Schulenburg, and J. Richter, Eur. Phys. J. B [**61**]{}, 147 (2008).
D.J.J. Farnell, R. Zinke, J. Schulenburg, and J. Richter, J. Phys.: Cond. Matter [**21**]{}, 406002 (2009).
J. Richter, R. Darradi, J. Schulenburg, D.J.J. Farnell, and H. Rosner, Phys. Rev. B [**81**]{}, 174429 (2010).
D.J.J. Farnell, R.F. Bishop, P.H.Y. Li, J. Richter, and C.E. Campbell, Phys. Rev. B [**84**]{}, 012403 (2011).
D.J.J. Farnell, O. Götze, J. Richter, R.F. Bishop, and Phys. Rev. B [**89**]{}, 184407 (2014).
R. F. Bishop, P. H. Y. Li, and C. E. Campbell, Phys. Rev. B [**89**]{}, 214413 (2014).
J.-J. Jiang, Y.-J Liu, F. Tang, C.-H. Yang, and Y.-B. Sheng, Physica B: Cond. Mat. [**463**]{}, 30 (2015). J. Richter, R. Zinke, D.J.J. Farnell, Eur. Phys. J. B [**88**]{}, 2 (2015).
P. H. Y. Li, R. F. Bishop, and C. E. Campbell, Phys. Rev. B [**91**]{}, 014426 (2015). R.F. Bishop, D.J.J. Farnell, S.E. Krüger, J.B. Parkinson, J. Richter, and C. Zeng, J. Phys.: Condens. Matter [ **12**]{}, 6887 (2000).
M. Roger and J.H. Hetherington, Phys. Rev. B [**41**]{}, 200 (1990).
R.F. Bishop, J.B. Parkinson, and Y. Xian, [ Phys. Rev. B]{} [**43**]{}, R13782 (1991); [ Phys. Rev. B]{} [**44**]{}, 9425 (1991).
C. Zeng, D.J.J. Farnell, and R.F. Bishop, J. Stat. Phys. [**90**]{}, 327 (1998).
A. B. Harris, C. Kallin and A. J. Berlinsky, Phys. Rev. B [**45**]{}, 2899 (1992). S. Sachdev, Phys. Rev. B [**45**]{}, 12377 (1992).
A. Chubukov, Phys. Rev. Lett. [**69**]{}, 832 (1992).
C. L. Henley and E. P. Chan, J. Magn. Magn. Mater. [**140-144**]{}, 1693 (1995).
For the numerical calculation we use the program package ‘The crystallographic CCM’ (D.J.J. Farnell and J. Schulenburg).
S.-S. Gong, W. Zhu, D.N. Sheng, O.I. Motrunich, M.P.A. Fisher, Phys. Rev. Lett. [**113**]{}, 027201 (2014).
M. Holt, O. P. Sushkov, D. Stanek, and G. S. Uhrig, Phys. Rev. B [**83**]{}, 144528 (2011).
Z. Fan and Q.-L. Jie, Phys. Rev. B [**89**]{}, 054418 (2014). D. J. J. Farnell and R. F. Bishop, Int. J. Mod. Phys. B [**22**]{}, 3369 (2008).
A.F. Barabanov and V.M. Berezovskii, J. Phys. Soc. Jpn. [**63**]{}, 3974 (1994); Phys. Lett. A [**186**]{}, 175 (1994); Zh. Eksp. Teor. Fiz. [**106**]{}, 1156 (1994) (JETP [**79**]{}, 627 (1994).
L. Siurakshina, D. Ihle, and R. Hayn, Phys. Rev. B [**64**]{}, 104406 (2001).
M. Härtel et al., Phys. Rev. B [**87**]{}, 054412 (2013).
B.H. Bernhard, B. Canals, and C. Lacroix, Phys. Rev. B [**[66]{}**]{}, 104424 (2002).
O. Janson, [*DFT based microscopic magnetic modeling for low-dimensional spin systems*]{}, Ph.D. thesis, Technische Universit[ä]{}t Dresden (2012)
D. Guterding, R. Valenti, and H. O. Jeschke, arXiv:1605.08162.
|
---
abstract: 'Studying unidentified $\gamma$-ray sources is important as they may hide new discoveries. We conducted a multiwavelength analysis of 13 unidentified Fermi-LAT sources in the 3FGL catalogue that have no known counterparts (Unidentified Gamma-ray Sources, UnIDs). The sample was selected for sources that have a single radio and X-ray candidate counterpart in their uncertainty ellipses. The purpose of this study is to find a possible blazar signature and to model the Spectral Energy Distribution (SED) of the selected sources using an empirical log parabolic model. The results show that the synchrotron emission of all sources peaks in the infrared (IR) band and that the high-energy emission peaks in MeV to GeV bands. The SEDs of sources in our sample are all blazar like. In addition, the peak position of the sample reveals that 6 sources (46.2%) are Low Synchrotron Peaked (LSP) blazars, 4 (30.8%) of them are High Synchrotron Peaked (HSP) blazars, while 3 of them (23.0%) are Intermediate Synchrotron Peaked (ISP) blazars.'
---
Introduction
============
The Large Area Telescope (LAT) on board the *Fermi* Gamma-ray Space Telescope detects high-energy photons between 20MeV and 300GeV ([@Atwood2009 Atwood et al. 2009]). In the 3FGL catalog among 3033 sources detected, 33% are still unassociated to any counterparts ([@Acero2015 Acero et al. 2015]) while in the latest catalog (4FGL) more than 38% remain unassociated ([@Collabo2019 The Fermi-LAT collaboration 2019]). Among 2023 sources that have been already identified in 3FGL, 1100 are Active Galactic Nuclei (AGN) of blazar type ( 57%). Blazars are a subclass of AGN with their relativistic jets pointing at Earth. The two main classes of blazars according to their optical properties are: (1) Flat Spectrum Radio Quasars (FSRQs) characterised by the strong emission lines and (2) BL Lacs characterised by weak or no emission lines at all. In addition, blazars are classified according to the position of their peak frequency ([@Massaro2004 Massaro et al.2004]): Low Synchrotron-Peaked (LSP) objects with $\rm{\nu_{p}^{synch}\leq 10^{14}\,Hz}$, Intermediate Synchrotron-Peaked (ISP) objects with $\rm{10^{14}\,Hz\leq \nu_{p}^{synch}\leq 10^{15}\,Hz}$, and High Synchrotron-Peaked (HSP) objects with $\rm{\nu_{p}^{synch}\geq 10^{15}\,Hz}$. In this research, we determined the type of emissions from a sample of 13 blazar candidates and classified them according to the shape of their SEDs.
Sample selection {#sect.2}
================
The 3FGL catalogue includes 1010 unidentified sources ([@Acero2015 Acero et al. 2015]). The release of the 3FGL catalogue enabled the *Swift*-XRT survey of the *Fermi*-LAT unassociated sources with the purpose of performing their follow-up observations in an attempt to find their potential X-ray counterparts. Two common features that all blazars share are that they are radio emitters (radio-loud objects) and that they are highly variable. [@Yang2015 Yang & Fan (2005)] showed that there is a strong correlation between radio and $\rm{\gamma}$-ray emission in blazars when they are in a their flaring high state. However, the most recent studies are continuously debating about the correlation between radio to $\rm{\gamma}$-ray bands ([@Massaro2017 Massaro et al. 2017]; [@Bruni2018 Bruni et al. 2018]). The selected sample takes advantage of the fact that most of the studied blazars exhibit strong radio-X-ray emission ([@Padovani2007 Padovani et al. 2007]). In addition, most of the known $\gamma$-ray bright AGN are above a Galactic latitude of $|b|> 10$ degrees, due to possible source confusion as well as higher diffuse background in the Galactic plane ([@Ackermann2015 Ackermann et al., 2015]). A sample of unassociated sources was selected based on the following criteria:
1. Only a single XRT detection lies in the uncertainty ellipse of the 3$\sigma$ of the LAT unassociated source.
2. Only a single radio source lies in the uncertainty box of *Fermi*-LAT uncertainty region which is coincident with the XRT detection.
3. Only sources with $|b| >10$ degrees in order to limit for any source confusion in the Galactic plane.
Applying all cuts to the population of unassociated sources listed in the 3FGL and observed by *Swift*, we isolated a sample of 13 unassociated sources assumed to be potential blazar candidates.
Data analysis {#sect.3}
=============
The *Swift*-XRT/UVOT data available were considered. We used standard extraction methods to produce *Swift*/XRT and *Swift*/UVOT spectra.
The *Fermi*-LAT spectral data point were extracted using Fermi Science tools (v10r0p5 released on June 24, 2015). The analysis takes into account the source region and the region of interest (ROI). We selected events within the energy range $\rm{100\,MeV-300\,GeV}$, a maximum zenith angle of 90 degrees, a ROI of 10 degrees.
A complete multiwavelength SED was obtained by supplementing the XRT/UVOT and *Fermi*-LAT data with other data from across the whole electromagnetic spectrum. These data were obtained using SEDbuilder tool[^1].
In order to identify the type of emission from the source, we apply an empirical model on the data. The SED shows that the log-parabolic model fits the data well for blazars, in different energy bands ([@Massaro2004 Massaro et al. 2004]; [@Tramacere2009 Tramacere et al. 2009]). The model used includes two log parabolas, one fitting the low-energy spectrum (synchrotron emission) and another one fitting the high-energy spectrum. In addition to these two log-parabolas, three other components are added: absorption, extinction, and a blackbody.
Results and discussions {#sect.4}
=======================
The SEDs resulted from the fitting and reported herein show the two typical bump known as the main signature of blazars. The low-energy bump (peaking between IR and optical bands) is interpreted as synchrotron emission from highly accelerated relativistic electrons and the high-energy bump (peaking at X-ray and $\rm{\gamma}$-ray energy bands) is related to high-energy emission, which could be either leptonic (Inverse Compton) or hadronic in nature. Based on the position of the $\nu{}^\mathrm{sync}_\mathrm{peak}$ ([@Abdo2010 Abdo et al., 2010]), we found that 6 ($\sim 46.2\%$) sources of our sample are LSP objects, 3 ($23.0\%$) sources are ISP objects and 4 (30.8%) of them are HSP objects. Figure 1 shows an example of the SED for 3FGL J1220.1$-$3715 where data are fitted by the model. The lower panel represents residuals.
![A sample of modeled SED of our sample with a log parabolic model. The model includes two absorbed logarithmic parabolas shown in blue, a blackbody in red dashed, and the total unabsorbed model shown in black. The extracted data are black colored while 3FGL data are in cyan and non-simultaneous archival data are in magenta.[]{data-label="1"}](./12201_new.pdf){width="0.6\columnwidth"}
For most of the cases, our classification is in agreement with [@Parkinson2016 Parkinson et al. (2016)], [@Lefaucheur2017 Lefaucheur & Pita (2017)] and [@Salvetti2017 Salvetti et al. (2017)], who classified the sources of our sample as AGN and blazars. All three works use Machine Learning techniques. We finally used the WISE IR colour-colour plot taking into account sources that have WISE matching sources. Only 9 sources have matching counterparts in WISE catalogue. As a results we found that only 7 sources lies in the WISE blazar region which emphasize their candidacy to be blazars.
2010, *ApJ*, 716, 30
2015, *ApJS*, 218, 23
2015 *ApJ*, 810, 14
2009, *ApJ*, 697, 1071
2018, *ApJ*, 854, L23
2000, *in Manset N., Veillet C., Crabtree D., 653 eds, Astronomical Society of the Pacific Conference Series Vol. 216, 654 Astronomical Data Analysis Software and Systems*, IX. p. 591
2017, *A&A*, 602, A86
2004, *A&A*, 413, 489
2017, *ApJ*, 834, 113
2007, *ApJ*, 662, 182
2016, *ApJ*, 820, 8
2017, *MNRAS*, 470, 1291
2019, *arXiv e-prints, p. arXiv:* 1902.10045
2009, *A&A*, 706 501, 879
2005, *Chinese J. Astron. Astrophys.,* 5, 229
[^1]: <http://www.asdc.asi.it/>
|
---
abstract: 'The properties of Josephson devices are strongly affected by geometrical effects. A loop-shaped superconducting electrode tightly couples a long Josephson tunnel junction with the surrounding electromagnetic field. Due to the fluxoid conservation, any change of the magnetic flux linked to the loop results in a variation of the shielding current circulating around the loop, which, in turn, affects the critical current of the Josephson junction. This method allows the realization of a novel family of robust superconducting devices (not based on the quantum interference) which can function as a general-purpose magnetic sensors. The best performance is accomplished without compromising the noise performance by employing an in-line-type junction few times longer than its Josephson penetration length. The linear (rather than periodic) response to magnetic flux changes over a wide range is just one of its several advantages compared to the most sensitive magnetic detectors currently available, namely the Superconducting Quantum Interference Devices (SQUID). We will also comment on the drawbacks of the proposed system and speculate on its noise properties.'
author:
- Roberto Monaco
title: |
Magnetic Sensors based on Long Josephson Tunnel Junctions\
An Alternative to dc-SQUIDs
---
[[email protected]]{}
Introduction
============
A Josephson quantum interferometer, which in its conventional form is realized by a superconducting loop interrupted by two Josephson junctions, lies at the core of the most sensitive magnetic detectors currently available, namely the Superconducting Quantum Interference Devices (SQUIDs)[@handbook]. Its working principle is that a variation, $\Delta \Phi_e$, of the external flux linked to the loop, smaller than one half of the magnetic flux quantum, $\Phi_0\equiv h/2e$, produces a measurable modulation, $\Delta I_c$, of the junctions maximum zero-voltage current (critical current), $I_c$. Best operational conditions for the [*bare*]{} interferometer (uncoupled loop and unshunted junctions) require[@tesche] that the shielding parameter, $\beta_{L} \equiv 2 L_{int} I_{c} /\Phi_0$, is approximately equal to unity and the best performance is achieved with an optimal value of the interferometer inductance $L_{int} =O( 100\,$pH) resulting in an average current responsivity $\left\langle I_\Phi \right\rangle \equiv \Delta I_c / \Delta \Phi_e \sim 10-20\mu A/\Phi_0$, but only in a flux range as small as $\Phi_0/2$. It took more than 30 years to turn the Josephson interferometers into the nowadays SQUIDs that can measure, at their best, magnetic fields as low as several attotestlas or magnetic fluxes as small as $1\,\mu \Phi_0$.
![3D sketch (not to scale) of the window-type in-line Josephson tunnel junction whose top electrode (in gray) is shaped as a rectangular loop. The base electrode is in black and the area of the tunneling insulating layer in between is hatched.[]{data-label="geometry"}](fig1.jpg){width="8.0cm" height="5.0cm"}
A constant flux-to-current transfer coefficient $I_\Phi \equiv dI_c / d \Phi_e \approx 10\mu A/\Phi_0$ in a flux range of hundreds of flux quanta has been recently reported[@PRB12] using a Long Josephson Tunnel Junction (LJTJ) built on top of a superconducting loop, despite that the samples were not optimized for sensor applications. In its simplest configuration, a DOubly-Connected-Electrode Long Josephson Tunnel Junction (DOCELJTJ), consists of a LJTJ for which at least one electrode is shaped as a loop. This device is sketched in Fig.\[geometry\] in which the junction bottom electrode is in black, while the top electrode - in the shape of a rectangular thin-film loop - is in gray and the tunneling area in between has a wavy hatched pattern. As it is generally accomplished with bare Josephson interferometers[@ketchen], also for the DOCELJTJs the intrinsic flux sensitivity can be enhanced by several orders of magnitude, by efficiently coupling its loop to the secondary coil of a multi-turn input coil superconducting flux transformer. However, we will limit the interest to the bare DOCELJTJ and point out at some specific expedients that can be adopted to increase its flux signal-to-noise-ratio. A theoretical analysis of this system, corroborated by experiments, has been recently reported[@PRB12] in which the static sine-Gordon equation for a one-dimensional in-line LJTJ has been coupled to the quantization[@london] of the fluxoid in the doubly connected electrode; however, no mention was given in Ref.[@PRB12] about the the prospects of exploiting the DOCELJTJ as a magnetic sensor. In the next section we will review the working principles of this novel magnetic sensor and demonstrate that, with a proper design, it is competitive with the optimized Josephson interferometers. We will also comment on the drawbacks of the proposed system and speculate on its noise properties. Later on we will discuss its many advantages and point out at a few of its disadvantages, as well. Further discussion on noise limitation will be given in Section 3. In Section 4 the performance of a prototype device will be presented and commented upon. Finally, the conclusions will be drawn in the Section 5.
The method
==========
Denoting with $L_{loop}$ the loop inductance and taking as positive the currents flowing clockwise, the DOCELJTJ working principle is the following. The internal magnetic flux, $\Phi_i$, within the loop is the sum of the externally applied flux, $\Phi_e$, and the self-flux, $\Phi_s$, produced by the shielding current, $I_{cir}$, which circulates around the loop to restore the initial flux: $\Phi_i=\Phi_e+\Phi_s=n \Phi_0$. The last equality is a direct consequence of the London’s fluxoid quantization law[@london], where $n=0, \pm 1, \pm 2, ...$ is the number of flux quanta trapped in the loop at the time of its superconducting transition. Due to the fluxoid conservation[@mercereau], any change in $\Phi_e$ corresponds to a variation of the circulating current, $I_{cir}\equiv \Phi_s/L_{loop}=(n\Phi_0- \Phi_e)/ L_{loop}$, which, in turn, alters the induced radial magnetic field, $H_{\rho} \propto I_{cir}$, at the loop surface, i.e., a superconducting loop acts as a flux-to-field transformer. If a magnetic field sensor is placed above (or below) part of the loop, it will thus detect the changes of the external magnetic flux, $\Phi_e$, linked to the loop. This remarkable property was first exploited by Pannetier [*et al.*]{}[@pannetier] who used high-sensitivity giant magnetoresistive sensors. However, within the context of superconducting thin films, the most natural (and sensitive) magnetic sensors are the Josephson junctions; in this case, any change in the external flux, $\Phi_e$, is measured by the variation, $\Delta I_c$, of the junction critical current. In essence, a DOCELJTJ is a flux-to-(critical) current transducer[@PRB09] which achieves its best performance with a one-dimensional Josephson tunnel junction whose width $W$ is smaller and whose length ${\rm{L}}$ is larger than the Josephson penetration length, $\lambda _J\equiv \sqrt{\Phi_0/ 2\pi \mu _{0}d_{e} J_{c}}$, setting the length unit of the LJTJ; $\mu _{0}$ is the vacuum magnetic permeability, $d_e $ the junction magnetic thickness[@wei] and $J_c$ the junction critical current density. To bias the LJTJ, a dc current $I$ is fed into the loop at an arbitrary point $O$ and is inductively split in the two loop arms before crossing the LJTJ barrier; $I$ is taken out via the junction bottom electrode. With the bias current injected and extracted at the junction extremities we have the so called *in-line* geometrical configuration treated in the pioneering works[@ferrelOS; @stuehmbasa; @radparvar85] on LJTJs soon after the discovery of the Josephson effect[@joseph]. The junction critical current can be measured by standard time-of-fligth techniques[@fulton; @castellano] based on AD conversion and peak detection with a resolution better than $1$ part in $10^3$, and it can be determined with at least one order of magnitude better accuracy, by measuring the switching current distributions[@russo].
Upon the assumption that the junction width $W$ is larger than the magnetic thickness, $d_e$, of the Josephson sandwich, the radial magnetic field generated by the circulating current is $H_{\rho} = \Lambda_t I_{cir}/W$; $\Lambda_t$ is the inductance per unit length of the junction top electrode normalized to the self-inductance per unit length, $\mathcal{L}_J=\mu_0 d_e/W_{}$, of the tunnel junction seen as a two-conductor transmission line[@scott76; @vanDuzer]. Of course, if the loop is formed by the bottom, rather than the top electrode, the normalized inductance per unit length of the junction bottom electrode, $\Lambda_b=1-\Lambda_t$, must replace $\Lambda_t$ in the above expression. Ideal symmetric junctions have $\Lambda_t=\Lambda_b=1/2$. For applications it is better to realize the loop with the top electrode which, typically, with respect to the bottom electrode, has a smaller width and so a larger inductance per unit length.
![Theoretical magnetic diffraction patterns $I_c(\Phi_e)$ for a very long noise-free in-line Josephson tunnel junction with different values of the balance parameter $0 \leq \alpha \Lambda_t \leq 1$. The critical current is normalized to $I_0 = J_c W_{} \lambda_J$ and the critical magnetic flux is $\Phi_c = 2 L_{loop} I_0/\Lambda_t$.[]{data-label="mdpp"}](fig2.jpg){width="8.0cm" height="6cm"}
As reported by several authors[@stuehmbasa; @vaglio], the largest supercurrent carried by an in-line LJTJ is $4I_0$, where $I_0 \equiv J_c W \lambda_J$ is a characteristic junction current (generally from a fraction of a milliampere to a few milliamperes) and depends on the junction normalized length, $\ell\equiv {\rm{L}}/\lambda_J$, as $I_0(\ell)= I_0 \tanh \ell/2$. For $L>> \lambda_J$, strictly speaking in the limit $L \to \infty$, the threshold curves, $I_c(\Phi_e),$ in the Meissner regime and in the absence of noise, have been computed in Ref.[@PRB12] and are shown in Fig. \[mdpp\] for different values of the product $\alpha \Lambda_t$. Here $\alpha$ ($1-\alpha$) is a geometrical design parameter corresponding to the fraction of the bias current $I$ diverted into the right (left) arm of the loop; moving the current entry point $O$ along the loop, then the paths of least impedance change and $\alpha$ can take any value between $0$ and $1$. Since $\alpha$ and $\Lambda_t$ both belong to the interval $[0,1]$, so does their product. In Fig.2 the critical current $I_c$ is normalized to $I_0$ and the critical magnetic flux, $\Phi_c\equiv 2L_{loop}I_0/\Lambda_t$, is the theoretical flux value that fully suppresses the critical current. We have numerically computed the gradual crossover of $\Phi_c$ from short ($\ell\simeq 1$) to long ($\ell>2\pi$) junctions and it was found to be carefully described by the empirical relationship: $\Phi_c(\ell)= \Phi_c \coth \ell/\pi$.
At variance with a Josephson interferometer, a DOCELJTJ has a linear, rather than periodic, response to external flux changes; the wide range of linearity of the threshold curves is very attractive for the realization of high-dynamics sensors also in very noisy environments and makes the use of a flux-locked loop superfluous. The device sensitivity to flux changes is measured by the absolute slope $I_\Phi\equiv |dI_c/d\Phi_e|$:
$$\label{gain}
I_\Phi=\begin{cases} \frac{1}{\alpha L_{loop}}& \textrm{for} \quad -\Phi_c \leq \Phi_e \leq \Phi_{max}\\
\frac{\Lambda_t}{(1-\alpha \Lambda_t)L_{loop}} & \textrm{for} \quad \Phi_{max} \leq \Phi_e \leq \Phi_c
\end{cases}$$
where $\Phi_{max}\equiv (2\alpha \Lambda_t -1)\Phi_c$. All-Niobium proof-of-principle DOCELJTJs were realized with $8\lambda_J$-long in-line junctions having the base electrode shaped as either a rectangular or annular closed path[@PRB12]. Their measurements accurately reproduced the theoretical predictions in Eqs.(\[gain\]) in a wide flux range and for different $\alpha$ values. An intrinsic flux sensitivity larger than that of an optimized (bare) Josephson interferometer was attained with loop inductances less than $100\,$pH. It was also found that the device response can be tuned by an external magnetic field applied in the junction plane. In passing, we note that the product, $I_\Phi \Phi_c$, of the device sensitivity and dynamic range is independent on the loop inductance. Interestingly, the barrier electrical and geometrical parameters, such as the Josephson current density, $J_c$, and the junction width, $W_{}$, do not appear explicitly in Eqs.(\[gain\]); the only requirement is that ${\rm{L}}>>\lambda_J$. For a finite length junction, $I_\Phi$ results to be proportional to the product $\tanh \ell/2 \times \tanh \ell/\pi$ suggesting the use of very long junctions; however, for $\ell= 2\pi$, that product is already as large as $0.96$. In practical cases, it suffices to have $L \simeq 4-5 \lambda_J$ (and $W\leq \lambda_J/2$). By further decreasing the junction length, one would experience a progressive reduction of both the flux sensitivity and the linearity range. Incidentally, an enhanced stability against thermal fluctuations has been numerically reported[@pankratov] for the zero-voltage metastable states of (overlap-type) LJTJs with $L \simeq 5 \lambda_j$. We like to stress that, the DOCELJTJ loop not being interrupted by any Josephson element, the shielding parameter $\beta_L$ looses its importance which makes the junction critical current and the loop inductance independent design parameters.
![3D sketch (not to scale) of a double loop DoCELJTJ. The base electrode is in black, the top electrode in gray and the tunneling insulating layer in between is white wavy hatched.[]{data-label="fig3"}](fig3.png){width="8cm"}
The threshold curves of Fig. \[mdpp\] are symmetric with respect to the inversion of both the supercurrent and the external flux, that is, $I_c^-(\Phi_e) = -I_c^+(-\Phi_e).$ Further, in the range $[-\Phi_{max},\Phi_{max}]$, the slope of the positive and negative critical currents, respectively, $I_{c}^+$ and $I_{c}^-$, is the same, i.e., any change in $\Phi_e$ modulates them in a concord fashion. This implies that the offset current, $I_c^{off} \equiv I_{c}^+ + I_{c}^-$, changes twice as fast and, $ I_{c}^+$ and $I_{c}^-$ being independent, its root mean square noise is $\sqrt{2}$ times larger than that of a single critical current; this enhances the signal-to-noise ratio by a factor $\sqrt{2}$. Notably, since the change with temperature of $I_{c}^+$ is numerically the same but opposite to that of $I_{c}^-$, one more advantage of the offset current, as compared to the single critical current, is its substantially reduced dependence on temperature.
Next, we consider the case when both electrodes are doubly connected as it is sketched in Fig. \[fig3\]. The theoretical analysis of a double loop device, not yet available, should account for two fluxoid quantizations (one in each loop) and the mutual magnetic interaction between the loops. Nevertheless, this double loop configuration is expected to be two times more sensitive to external flux changes: in fact, the shielding currents in the two loops circulate in opposite directions, but also on opposite sides with respect to the barrier, so that their respective radial magnetic fields add each other in the barrier plane. In passing, we note that flux-focusing washer loops[@jaycox] or fractional-turn loops[@Zimmerman] made by many loops in parallel can also be usefully employed in the embodiment of a DOCELJTJ. Last but not least, if a parallel array of $N$ junctions is distributed along the loop perimeter, the resulting flux sensitivity will be $N$ times larger than that for just one junction and, at the same time, the signal-to-noise ratio is expected to increase by a factor $\sqrt{N}$.
Noise properties
================
The ultimate performance of any device depends on its noise. To estimate the value of the minimum detectable change of the external flux, it is important to know the spectral density, $S_\Phi$, of the flux noise generated under working conditions. Different sources contribute to the noise: we will only consider the intrinsic ones, since the extra noise induced by any eventual input circuitry has been already fully investigated in the context of the dc SQUIDs[@kleiner; @koelle].
Any thermal fluctuation in the loop energy is felt as a magnetic flux noise $ \left\langle \Phi_n^2 \right\rangle=k_B T L_{loop}$, where $k_B$ is Boltzmann’s constant and $T$ is the bath temperature; as for SQUIDs, ultra low noise applications demand small operating temperatures and loop inductances. However, the periodicity of any a Josephson interferometer would be completely wiped out by the noise, if $\left\langle \Phi_n^2 \right\rangle \geq \Phi_0^2/4$; the constraint $L_{int}< \Phi_0^2/4k_B T$ needed to observe quantum interference[@tesche] imposes $L_{int}< 15\,$nH, at liquid $He$ temperature and $L_{int}< 1\,$nH at liquid $N_2$ temperature. In the case of a DOCELJTJ, for a given operating temperature, is the requested measurement accuracy that determines the upper limit for the loop inductance.
The vast majority of dc SQUIDs are used at frequencies, $f$, below $1$kHz; if we focus on quasi-static or, at most, radio-frequency applications and assume thermal equilibrium with temperature $T >> hf/k_B$, we can disregard the shot noise due to the interaction of the current through the barrier and the electromagnetic field in the junction cavity[@likharev]. Furthermore, since the LJTJ is not shunted and operates in the zero-voltage state, we do not have to consider the flicker or $1/f$ noise; in addition, the Johnson noise generated in the large internal sub-gap resistance can be neglected in high-quality junctions operating well below their critical temperatures. At last, the only noise source intrinsic to the LJTJ is represented by the thermally induced escape from the zero-voltage state[@fulton] that manifests as a (critical) current noise with spectral density, $S_I$. The relative intensity of the thermal fluctuations is given by the dimensionless parameter $\Gamma\equiv k_B T/E_J$, where $E_J$ is the energy of the LJTJ. By adding the magnetostatic energy stored in and between the junction electrodes to the Josephson energy associated with the Cooper-pair tunneling current, it was found[@mc] that $E_J$ never exceeds $8E_0$, where $E_0\equiv \Phi_0 I_0/2\pi$ is the well-known fluxon rest energy; $E_0$ depends on the junction’s electrical and geometrical parameters and represents its characteristic energy unit. Then, in our case, $\Gamma = \pi k_B T/4I_0 \Phi_0$, so that large $I_0$ values can be chosen to reduce the effects of the fluctuation; this fact is of paramount importance for the development of high-temperature sensors (with $I_0=0.5\,$mA, $\Gamma =O(10^{-3})$ at $T=77\,$K.) In Ref.[@castellano], for bias currents near the critical current, the activation energy for (not very) long junctions was found to vary approximately as $(1-I/I_c)^{3/2}$, just as it does for short junctions[@fulton], but its magnitude is scaled by a factor that depends on the junction normalized length and on the applied magnetic field and goes to zero at the critical field. Smaller activation energies result in a broader probability density for the escape from the zero-voltage state[@dino], i.e., in a larger flux noise spectral density $S_\Phi=S_I/I_\Phi$.
The need for a cryogenic environment is the obvious drawback common to all superconducting devices. In our specific case, the main limitation is given by the smallest achievable size of the loop. Using the all-Niobium fabrication processes, high quality Josephson barriers can be attained[@highjc] with critical current densities as high as $10$kA/cm$^2$ corresponding to $\lambda_J \simeq 5 \mu$m; this is about the radius of the smallest useful ring for flux sensing applications. The high $J_c$ values required to reduce the Josephson depth, $\lambda_J$, at the same time, increase the characteristic current, $I_0$, so providing a broader linearity range, $\Phi_c$, and a smaller temperature parameter, $\Gamma$. The use a LJTJ is a drawback when an external shunt resistor would be used, as in the dc-SQUID, because here the involved resistance to be shunted to have a non hysteretic response is intrinsically lower than the one of the parallel connection of two (very) small junctions.
The prototype
=============
![Exploded diagram (not to scale) of a double loop device integrating two rectangular washer-type loops. The base electrode is in black, the top electrode in gray and the tunneling insulating layer in between is white wavy hatched.[]{data-label="fig4"}](fig4.png){width="7cm"}
Fig. \[fig4\] depicts the exploded view of a symmetric double loop DOCELJTJ realized with two rectangular flux-focusing washers[@jaycox] deformed in such a way that each acts as a ground plane for the other. The outer dimensions of each washer are $800\times 1000\, \mu m^2$, while the loop dimensions are $100\times 200\, \mu m^2$. Details on the samples fabrication and the experimental setup can be found in Ref.[@PRB12]. Here we will only point out the relevant electrical and geometrical parameters. We used high quality $Nb/Al-Al_{ox}/Nb$ LJTJs fabricated on silicon substrates using the trilayer technique in which the Josephson junction is realized in a window opened in a $200\,$nm thick $SiO_2$ insulator layer. The LJTJ had width $W_{}=1.5\,\mu$m and length ${\rm{L}}=100\,\mu$m and Josephson current density was $J_c\simeq 3.1\,$kA/cm$^2$ (at $T=4.2\,$K). In the junction area the bottom and top electrodes were, respectively, $10$ and $6\,\mu m$ wide and $100$ and $350\,nm$ thick. From the analysis of the experimental magnetic diffraction pattern in presence of a transverse magnetic field it is possible to derive peculiar system quantities such as the maximum critical current, $I_c^{max}=1.8\,$mA, the characteristic current, $I_0= 0.45\,$mA, and the symmetry parameter, $\alpha \Lambda_t= 0.55$. The magnetic current responsivity was found to be close to $2\mu A/nT$. The wires inside the cryoprobe were not filtered, so that the measurements were affected by a root mean square noise current $\left\langle I_n^2\right\rangle^{1/2}\sim 1\,\mu$A with an integration time of $0.1\,$s; therefore, the change in the static magnetic flux density that the bare DOCELJTJ could detect with a signal-to-noise ratio of one was $500pT$ in a $\pm 1\mu T$ range. We consider this as an encouraging figure of merit for a prototype sensor tested in an environment which is not meant for extremely low noise operations. The effective capture area of such a device is rather difficult to evaluate (as well as its self inductance); however, even assuming an underestimated value of $0.1\, mm^2$, we end up with a flux sensitivity better than $25m\Phi_0$ in a range of $\pm 100\Phi_0$. It is straightforward that a wider washer with a smaller loop area will drastically enhance the performance of the bare sensor. Furthermore, coupling the device to the secondary loop of a flux transformer would provide further orders of magnitudes improvements.
Conclusions
===========
Long Josephson tunnel junctions were traditionally used to investigate the physics of non-linear phenomena[@barone]. In the last decade they have been employed to shed light on other fundamental concepts in physics such as the symmetry principles and how they are broken[@PRL06; @PRB08; @gordeeva]. In this paper we have discussed how one or more long Josephson tunnel junctions can be integrated with a superconducting loop to provide very large sensitivity to magnetic flux. The sensor principle is to capture the flux related to the field to be measured by the aim of a superconducting loop, perpendicular to the applied field. A supercurrent runs in the loop to avoid magnetic field penetration and to keep the superconductor in the Meissner state. If this loop is narrow, the circulating current density will become relatively high and will locally create a high magnetic field and a high density of field lines. A LJTJ placed above or below part of the loop will thus detect a local field through the changes of its critical current. The accuracy of the critical current measurements depends on the switching probability (or escape rate) caused by thermal noise. We note that the LJTJ is sensitive to parallel fields, whereas the device (superconducting loop) is only sensitive to perpendicular fields. The physical property of superconductors which makes the operation of these devices possible is the quantization of the fluxoid associated with a closed loop of superconductor. We stress that the detection of magnetic flux with LJTJs is not based on the Josephson interference, but it only relies on the fluxoid conservation in the loop. This magnetometer combines ease of use, low noise, high dynamic performance and stability against thermal drifts. At the same time, it retains the advantages of high speed and low power inherent in Josephson devices. Its linear response is particularly advantageous also for the measurement of absolute flux and for noise thermometry. In addition, it is fully compatible with any present low- and high-T$_c$ thin film technology developed for the fabrication of Josephson junctions[@hilow]. Further, the demand on the external electronics is reduced, although a DOCELJTJ can benefit, in all respects, of the accessory circuitries (shunt resistors, input coil, modulation and feedback coil, flux transformer, tunable resonators and so on). In this light, the proposed device could conveniently replace the Josephson interferometer in the many dc SQUID sensors developed over the years to cover a wide range of applications (magnetometer, gradiometer, galvanometer, amplifier, etc.). Ultimately, the performances of any DOCELJTJ based device are intimately related to its intrinsic noise. Unfortunately, the noise properties of long Josephson junctions have not found an adequate interest in the Josephson community, and, in particular, the thermal effects in in-line LJTJs still remain an unexplored land. Further investigations have been planned to remedy this lack.
[50]{} , edited by J. Clarke and A. I. Braginski (Wiley-VCH Verlag GmbH & Co. KgaA, Weinheim, Germany, 2006), Vol 2; R. L. Fagaly, [*Rev. Sci. Instrum.*]{} [**77**]{}, 101101 (2006).
C.D. Tesche and J. Clarke, [*J. Low Temp. Phys.*]{} [**29**]{}, 301 (1977).
R. Monaco, J. Mygind, and V.P. Koshelets, [*Phys. Rev. B*]{} [**85**]{}, 094514 (2012).
M.B. Ketchen and J.M. Jaycox , [*App. Phys.Letts.*]{} [**40**]{}, 736 (1982).
F. London, [*Phys. Rev.*]{} [**74**]{}, 562 (1948).
J. E. Mercereau and L. T. Crane, [*Phys. Rev. Lett.*]{} [**12**]{}, 191 (1964).
M. Pannetier, C. Fermon, G. Le Goff, J. Simola, E. Kerr, [*Science*]{}, [**304**]{}, 1648 (2004); M. Pannetier, C. Fermon, G. Le Goff, J. Simola, E. Kerr, M. Welling, and R.J. Wijngaarden, [*IEEE Trans. on Appl. Supercond.*]{} [**15**]{}, 892 (2005).
R. Monaco, J. Mygind, R.J. Rivers, and V.P. Koshelets, [*Phys. Rev. B*]{} [**80**]{}, 180501(R) (2009); John R. Kirtley and Francesco Tafuri, [*Physics*]{} [**2**]{}, 92 (2009).
M. Weihnacht, [*Phys. Status Solidi*]{} [**32**]{}, K169 (1969).
R.A. Ferrel, and R.E. Prange, [*Phys. Rev. Letts.*]{}[**10**]{}, 479 (1963); C.S. Owen and D.J. Scalapino, [*Phys. Rev.*]{}[**164**]{}, 538 (1967).
D.L. Stuehm, and C.W. Wilmsem, [*J. Appl. Phys.*]{}[**45**]{}, 429 (1974); S. Basavaiah and R. F. Broom, [*IEEE Trans. Magn.*]{}[**MAG-11**]{}, 759 (1975).
M. Radparvar and J. E. Nordman, [*IEEE Trans. on Magn.*]{} [**MAG-21**]{}, 888 (1985).
B. D. Josephson, [*Rev. Mod. Phys.*]{} [**36**]{}, 216 (1964).
T. A. Fulton and L. N. Dunkleberger, [*Phys. Rev. B*]{} [**9**]{}, 4760 (1974).
M.G. Castellano, G. Torrioli, C. Cosmelli, A. Costantini, F. Chiarello, P. Carelli, G. Rotoli, M. Cirillo, R.L. Kautz, [*Phys. Rev. B*]{} [**54**]{} 15417 (1996).
R. Russo, C. Granata, P. Walke, A. Vettoliere, E. Esposito, M. Russo, [*J. Nanopart. Res.*]{} [**13**]{}, 5661 (2011).
A.C. Scott, F.Y.F. Chu and S.A. Reible, [*J. Appl. Phys.*]{} [**47**]{}, 3272 (1976).
Theodore Van Duzer, Charles W. Turner, [*Principles of Superconductive Devices and Circuits*]{}(Prentice Hall- New Jersey, 2nd Edition, 1998).
A. Barone, W.J. Johnson and R. Vaglio, [*J. Appl. Phys.*]{} [**46**]{}, 3628 (1975).
K. G. Fedorov and A. L. Pankratov, [*Phys. Rev. B*]{} [**76**]{}, 024504 (2007).
J.M. Jaycox and M.B. Ketchen, [*IEEE Trans. Magn.*]{} [**17**]{}, 403 (1981).
J.E. Zimmerman, [*Jou. Appl. Phys.*]{} [**42**]{}, 4483 (1971).
R. Kleiner, D. Koelle, F. Ludwig, and J. Clarke, [*Proceedings of the IEEE*]{} [**92**]{}, 1534 (2004).
D. Koelle, R. Kleiner, F. Ludwig, E. Dantsker and J. Clarke, [*Rev. Mod. Phys.*]{} [**71**]{}, 631 (1999).
K.K. Likharev, [*Dynamics of Josephson Junctions and Circuits*]{}(Gordon & Breach Science Publishers, London, 1984).
A.C. Scott and W.J. Johnson, [*Appl. Phys. Letts.*]{}, [**14**]{}, 316 (1969); D.W. McLaughlin and A.C. Scott, [*Phys. Rev. A*]{}[**18**]{}, 1652 (1978).
B. Ruggiero, C. Granata, E. Esposito, M. Russo and P. Silvestrini, [*Appl. Phys. Lett.*]{} [**75**]{}, 121 (1999).
R. E. Miller, W. H. Mallison, A. W. Kleinsasser, K. A. Delin, and E. M. Macedo, [*App. Phys.Letts.*]{} [**63**]{}, 1423 (1993); H. Sugiyama, A. Fujimaki, and H. Hayakawa, [*IEEE Trans. Appl. Superc.*]{} [**5**]{}, 2739 (1995).
A. Barone and G. Paternò, [*Physics and Applications of the Josephson Effect*]{}(Wiley, New York, 1982).
R. Monaco, J. Mygind, M. Aaroe, R.J. Rivers and V.P. Koshelets, [*Phys. Rev. Lett.*]{} [**96**]{}, 180604 (2006).
R. Monaco, M. Aaroe, J. Mygind, R.J. Rivers, and V.P. Koshelets, [*Phys. Rev. B*]{} [**77**]{}, 054509 (2008).
A.V. Gordeeva and A.L. Pankratov, [*Phys. Rev. B*]{} [**81**]{}, 212504 (2010).
M. Gurvitch, M. A. Washington, and H. A. Huggins, [*App. Phys. Letts.*]{} [**42**]{}, 472 (1983); H. Hilgenkamp and J. Mannhart, [*Rev. Mod. Phys.*]{} [**74**]{}, 485 (2002).
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abstract: |
In this note we study the existence of the Lelong-Demailly number of a negative plurisubharmonic current with respect to a positive plurisubharmonic function on an open subset of $\C^n$. Then we establish some estimates of the Lelong-Demailly numbers of positive or negative plurisubharmonic currents.\
**Sur les Nombres de Lelong-Demailly des courants plurisousharmoniques.**\
<span style="font-variant:small-caps;">Résumé.</span> Dans cette note, on étudie l’existence du nombre de Lelong-Demailly d’un courant négatif plurisousharmonique relativement à une fonction positive plurisousharmonique sur un ouvert de $\C^n$ puis on donne quelques estimations des nombres de Lelong-Demailly des courants positifs ou négatifs plurisousharmoniques.
address: |
Department of Mathematics\
Faculty of sciences of Gabès\
University of Gabès\
6072 Gabès Tunisia.
author:
- Noureddine Ghiloufi
title: 'On the Lelong-Demailly numbers of plurisubharmonic currents'
---
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0.5cm
Version française abrégée {#version-française-abrégée .unnumbered}
=========================
L’existence des nombres de Lelong des courants positifs a été résolu par P. Lelong dans les années 1950 pour le cas des courants *fermés*, puis ce résultat a été étendu par Skoda au cas des courants *positifs plurisousharmoniques*. En revanche, il existe des courants *négatifs plurisousharmoniques* qui n’admettent pas de nombres de Lelong, on peut voir par exemple que $\log(|z_2|^2)[z_1=0]$ est un exemple de courant négatif plurisousharmonique de bidimension $(1,1)$ sur la boule unité de $\C^2$ qui n’admet pas de nombre de Lelong en 0.\
Le principal objectif de cette note est de traiter le problème de l’existence des nombres de Lelong généralisés introduits par Demailly [@De], d’un courant négatif plurisousharmonique $T$ de bidimension $(p,p)$ sur un ouvert $\Omega$ de $\C^n$; pour cela on note ${\mathrm{PSH}}(T,\Omega)$ l’ensemble des fonctions $\varphi$ positives plurisousharmoniques semi-exhaustives dont le logarithme $\log\varphi$ est plurisousharmonique sur $\Omega$, et telles que le produit extérieur $T{\wedge}(dd^c\varphi)^p$ soit bien défini. Le nombre de Lelong-Demailly de $T$ relativement à un poids $\varphi$ tel que $\varphi\in{\mathrm{PSH}}(T,\Omega)$ est $\nu(T,\varphi):= \lim_{r\to0^+} \nu(T,\varphi,r)$, où $\nu(T,\varphi,.)$ est la fonction définie par $$\nu(T,\varphi,r):=\frac1{r^p}\int_{\{\varphi<r\}}T{\wedge}(dd^c\varphi)^p.$$ Le résultat principal de cette note est:\
**Théorème \[theo1\].** *Soient $T$ un courant négatif plurisousharmonique de bidimension $(p,p)$ sur $\Omega$ et $\varphi\in{\mathrm{PSH}}(T,\Omega)$. Si la fonction $t\longmapsto \frac{\nu(dd^cT,\varphi,t)}{t}$ est intégrable au voisinage de $0$, alors le nombre de Lelong-Demailly $\nu(T,\varphi)$ du courant $T$ relativement à $\varphi$ existe.*
Introduction
============
The existence of Lelong numbers of positive currents was proved by P. Lelong in the 1950’s in the case of *closed* currents, then this result was extended by H. Skoda to the case of *positive plurisubharmonic* currents. However, there are *negative* plurisubharmonic currents which do not admit Lelong numbers, for example $-(-\log(|z_2|^2))^\epsilon[z_1=0]$ is a negative plurisubharmonic current of bidimension $(1,1)$ on the unit ball of $\C^2$, which admits no Lelong number at 0 for all $0<\epsilon\leq1$.\
The main purpose of the first part of this note is to study the existence of generalized Lelong numbers, introduced by Demailly [@De], in the case of negative plurisubharmonic currents of bidimension $(p,p)$ on an open set $\Omega$ of $\C^n$. In the second part, we give some proprieties of the Lelong-Demailly numbers of positive or negative plurisubharmonic currents. In particular, we prove that the Lelong-Demailly numbers do not depend to the system of coordinates. To this aim, we consider a non-negative plurisubharmonic function $\varphi$ on $\Omega$ such that $\log\varphi$ is plurisubharmonic on $\{\varphi> 0\}$ and for every $r>0,\ r_2>r_1>0$, we set $$\begin{array}{l}
{\displaystyle}B_\varphi(r):=\{z\in\Omega;\ \varphi(z)<r\},\\
{\displaystyle}B_\varphi(r_1,r_2):=\{z\in\Omega;\ r_1\leq\varphi(z)<r_2\}=B_\varphi(r_2)\smallsetminus B_\varphi(r_1) \\
{\displaystyle}\beta_\varphi:=dd^c\varphi=\frac{i}{\pi}\partial\overline{\partial}\varphi,\quad \alpha_\varphi:=dd^c\log\varphi \hbox{ on } \{\varphi>0\}.
\end{array}$$ We assume that $\varphi$ is semi-exhaustive, i.e. there exists $R=R(\varphi)>0$ such that $B_\varphi(R)$ is relatively compact in $\Omega$. If $S$ is a positive (or negative) current of bidimension $(p,p)$ on the set $\Omega$ then we denote ${\mathrm{PSH}}(S,\Omega)$ the set of non-negative semi-exhaustive plurisubharmonic functions $\varphi$ such that $\log\varphi$ is also plurisubharmonic on $\Omega$ and the exterior product $S{\wedge}(dd^c\varphi)^p$ is well defined. Finally, we denote by $\mathcal I_S(\varphi):=\{k>0;\ \varphi^k\in {\mathrm{PSH}}(S,\Omega)\}$ for every $\varphi\in {\mathrm{PSH}}(S,\Omega)$; in particular, if $\varphi$ is $\mathcal C^2$ then $\mathcal I_S(\varphi)\supset[1,+\infty[$ for every current $S$.\
The Lelong-Demailly number of $S$ relatively to the weight $\varphi$ is $\nu(S,\varphi):= \lim_{r\to0^+} \nu(S,\varphi,r)$ where $\nu(S,\varphi,.)$ is the function defined on $]0,R(\varphi)[$ by $$\nu(S,\varphi,r):=\frac1{r^p}\int_{B_\varphi(r)}S{\wedge}\beta_\varphi^p.$$ The classical Lelong number is given by the choice $\varphi(z)=\varphi_0(z):=|z|^2$.
\[exple1\] Let $S_\epsilon(z_1,z_2)=(|z_2|^{2\epsilon}-1)[z_1=0]$ where $\epsilon>0$. Then $S_\epsilon$ is a negative plurisubharmonic current of bidimension $(1,1)$ on the unit ball $\mathbb B$ of $\C^2$, $\varphi_0\in {\mathrm{PSH}}(S_\epsilon,\mathbb B)$ and $\mathcal I_{S_\epsilon}(\varphi_0)=\mathcal I_{dd^cS_\epsilon}(\varphi_0)={}]0,+\infty[.$
A simple computation proves that $$S_\epsilon{\wedge}dd^c(\varphi_0^k)=\frac{k^2}\pi (|z_2|^{2\epsilon}-1)|z_2|^{2(k-1)}idz_2{\wedge}d\overline{z}_2{\wedge}[z_1=0]$$ and $dd^cS_\epsilon=\frac{\epsilon^2}\pi|z_2|^{2(\epsilon-1)}idz_2{\wedge}d\overline{z}_2{\wedge}[z_1=0]$, hence $S_\epsilon{\wedge}dd^c(\varphi_0^k)$ is well defined for all $k>0$. Therefore, $\mathcal I_{S_\epsilon}(\varphi_0)=\mathcal I_{dd^cS_\epsilon}(\varphi_0)={}]0,+\infty[$. One can check easily that $$\nu(S_\epsilon,\varphi_0^k,r)=2k^2\left(\frac{r^{\epsilon/k}}{\epsilon+k}-\frac1k\right)\quad \hbox{and}\quad \nu(dd^cS_\epsilon,\varphi_0,r)=2\epsilon r^{\epsilon},$$ hence $\nu(S_\epsilon,\varphi_0^k)=-2k$.
Main result
===========
In the following, we will use a Lelong-Jensen formula proved by Demailly [@De]. In [@To], Toujani used an analogue of this formula to prove the existence of the directional Lelong-Demailly numbers of positive plurisubharmonic currents.
\[lem1\](See [@De] or [@To]) Let $S$ be a positive or negative plurisubharmonic current of bidimension $(p,p)$ on $\Omega$ and $\varphi\in {\mathrm{PSH}}(S,\Omega)$. Then, for all $0<r_1<r_2<R(\varphi)$, $$\label{eq 1.1}
\begin{array}{lcl}
\nu(S,\varphi,r_2)-\nu(S,\varphi,r_1) & = &{\displaystyle}\frac1{r_2^p}\int_{B_\varphi(r_2)} S{\wedge}\beta_\varphi^p -\frac1{r_1^p} \int_{B_\varphi(r_1)}S{\wedge}\beta_\varphi^p\\
& = & {\displaystyle}\int_{B_\varphi(r_1,r_2)}S{\wedge}\alpha_\varphi^p\\
& & {\displaystyle}+\int_{r_1}^{r_2}\left(\frac1{t^p} -\frac1{r_2^p} \right)dt\int_{B_\varphi(t)}dd^cS{\wedge}\beta_\varphi^{p-1}\\
& &{\displaystyle}+\left(\frac1{r_1^p}-\frac1{r_2^p}\right)\int_0^{r_1}dt\int_{B_\varphi(t)} dd^cS{\wedge}\beta_\varphi^{p-1}.
\end{array}$$
According to Lemma \[lem1\], if $S$ is positive plurisubharmonic then $\nu(S,\varphi,.)$ is a non-negative increasing function on $]0,R(\varphi)[$, so $\nu(S,\varphi):=\lim_{r\to0^+} \nu(S,\varphi,r)$ exists.\
It is well known that if $S$ is a positive plurisubharmonic current, then for every $\varphi\in {\mathrm{PSH}}(S,\Omega)$ the function $t\longmapsto\frac{\nu(dd^cS,\varphi,t)}t$ is integrable in the neighborhood of 0. Throughout this note, for every negative plurisubharmonic current $T$ on $\Omega$, we say that a function $\varphi\in {\mathrm{PSH}}(T,\Omega)$ satisfies condition $(C)$ if the function $t\longmapsto \frac{\nu(dd^cT,\varphi,t)}{t}$ is integrable on a neighborhood of 0. In particular, if $\varphi$ satisfies condition $(C)$, we must have $\nu(dd^cT,\varphi)=0$.\
Now we state the main result concerning the case of negative plurisubharmonic currents.
\[theo1\] Let $T$ be a negative plurisubharmonic current of bidimension $(p,p)$ on $\Omega$ and $\varphi\in {\mathrm{PSH}}(T,\Omega)$ satisfying condition $(C)$. Then, the Lelong-Demailly number $\nu(T,\varphi)$ of the current $T$ relatively to $\varphi$ exists.
For every $0<r<R(\varphi)$, we set $$f(r)=\frac1{r^p}\int_{B_\varphi(r)}T{\wedge}\beta_\varphi^p+\frac1{r^p}\int_0^r dt\int_{B_\varphi(t)}dd^cT{\wedge}\beta_\varphi^{p-1} -\int_0^r\frac{dt}{t^p}\int_{B_\varphi(t)}dd^cT{\wedge}\beta_\varphi^{p-1}.$$ Thanks to condition $(C)$, the function $f$ is well defined and non-positive on $]0,R(\varphi)[$. Indeed, $$f(r) =\nu(T,\varphi,r)+\int_0^r\left(\frac{t^p}{r^p}-1\right)\frac{\nu(dd^cT,\varphi,t)}{t}dt\leq0$$ because the function $\nu(dd^cT,\varphi,.)$ is non-negative on $]0,R(\varphi)[$.\
For $0<r_1<r_2<R(\varphi)$ we set $A(r_1,r_2):=f(r_2)-f(r_1)$. The current $T$ is negative plurisubharmonic, thus lemma \[lem1\] gives $$\label{eq 1.2}
\begin{array}{lcl}
A(r_1,r_2) & = &{\displaystyle}\nu(T,\varphi,r_2)- \nu(T,\varphi,r_1)+ \frac1{r_2^p}\int_0^{r_2}t^{p-1} \nu(dd^cT,\varphi,t)dt\\
& & {\displaystyle}-\frac1{r_1^p}\int_0^{r_1} t^{p-1}\nu(dd^cT,\varphi,t)dt -\int_{r_1}^{r_2}\frac{\nu(dd^cT,\varphi,t)}{t}dt\\
&=&{\displaystyle}\int_{B_\varphi(r_1,r_2)}T{\wedge}\alpha_\varphi^p\leq 0.
\end{array}$$ Therefore, $f$ is a non-positive decreasing function on $]0,R(\varphi)[$, and this implies the existence of the limit $\varrho:=\lim_{r\to0^+} f(r)$. The hypothesis of integrability of $\nu(dd^cT,\varphi,t)/t$ and the fact that $(t^p/r^p-1)$ is uniformly bounded give $${\displaystyle}\lim_{r\to0^+}\int_0^r\left(\frac{t^p}{r^p}-1\right)\frac{\nu(dd^cT,\varphi,t)}{t}dt=0.$$ Therefore, $\varrho=\lim_{r\to0^+}f(r)=\lim_{r\to0^+}\nu(T,\varphi,r)=\nu(T,\varphi)$.
If $T$ is a positive (resp. negative) plurisubharmonic current of bidimension $(p,p)$ on $\Omega$ and $\varphi\in{\mathrm{PSH}}(T,\Omega)$ (resp. satisfying condition $(C)$) then the extension $\widetilde{T{\wedge}\alpha_\varphi^p}$ of $T{\wedge}\alpha_\varphi^p$ by 0 over the compact set $\{\varphi=0\}$ exists and we have $$\int_{B_\varphi(r)}\widetilde{T{\wedge}\alpha_\varphi^p}=f(r)-\nu(T,\varphi).$$
Indeed, it suffice to use the lemma \[lem1\] to prove that $\int_{B_\varphi(\epsilon,r)}T{\wedge}\alpha_\varphi^p$ is uniformly bounded and then tend $\epsilon$ to 0 to prove the equality of the remark.\
A natural question arises: does the existence of the Lelong-Demailly number of the negative plurisubharmonic current $T$ implies the existence of $\widetilde{T{\wedge}\alpha_\varphi^p}$? so condition $(C)$ will be necessary in Theorem \[theo1\].\
In the following proposition, we give some properties of Lelong-Demailly numbers in both cases of negative or positive plurisubharmonic currents.
\[pro1\] Let $T$ be a positive or negative plurisubharmonic current of bidimension $(p,p)$ on $\Omega$ and $\varphi\in {\mathrm{PSH}}(T,\Omega)$. Then, for every $k\in\mathcal I_T(\varphi)$ and every $r\in{}]0,R(\varphi)[$, we have $$\label{eq 1.3}
\nu(T,\varphi^k,r^k)=k^p\left[\nu(T,\varphi,r)+\int_0^r\frac{\nu(dd^cT,\varphi,t)}t \left(\frac{t^p}{r^p} -\frac{t^{kp}}{r^{kp}}\right)dt\right].$$ In particular, if $T$ is negative, then $\nu(T,\varphi)$ exists if and only if for all $k\in\mathcal I_T(\varphi)$, $\nu(T,\varphi^k)$ exists.\
Furthermore, in both cases $($with the assumption that $\nu(T,\varphi)$ exists if $T$ is negative$)$ we have $$\nu(T,\varphi^k)=k^p\nu(T,\varphi).$$
The previous equality is due to Demailly in the case of closed positive currents.
Let $\epsilon>0$ be sufficiently small. By replacing $\varphi$ with $\varphi_\epsilon=\varphi+\epsilon$ and $\psi:=\varphi^k$ with $\psi_\epsilon=\varphi_\epsilon^k$, when $T$ is negative (respectively positive), Equality (\[eq 1.2\]) (resp. Equality (\[eq 1.1\])) gives for $0<r_1< \epsilon<r<R(\varphi)$ $$\label{eq 1.4}
\begin{array}{lcl}
{\displaystyle}\nu(T,\varphi_\epsilon,r) & = & {\displaystyle}\int_{B_{\varphi_\epsilon}(\epsilon,r)}T{\wedge}\alpha_{\varphi_\epsilon}^p -\frac1{r^p}\int_\epsilon^rt^{p-1}\nu(dd^cT,\varphi_\epsilon,t)dt \\
& &{\displaystyle}\hfill +\int_\epsilon^r\frac{\nu(dd^cT,\varphi_\epsilon,t)}t dt
\end{array}$$ and $$\label{eq 1.5}
\begin{array}{lcl}
\nu(T,\psi_\epsilon,r^k) & =& {\displaystyle}\int_{B_{\psi_\epsilon}(\epsilon^k,r^k)}T{\wedge}\alpha_{\psi_\epsilon}^p -\frac1{r^{kp}}\int_{\epsilon^k}^{r^k}t^{p-1}\nu(dd^cT,\psi_\epsilon,t)dt\\
& &{\displaystyle}\hfill+\int_{\epsilon^k}^{r^k}\frac{\nu(dd^cT,\psi_\epsilon,t)}t dt\\
& = & {\displaystyle}\int_{B_{\psi_\epsilon}(\epsilon^k,r^k)}T{\wedge}\alpha_{\psi_\epsilon}^p - \frac k{r^{kp}}\int_\epsilon^r s^{kp-1}\nu(dd^cT,\psi_\epsilon,s^k)ds\\
& &{\displaystyle}\hfill+k\int_\epsilon^r\frac{\nu(dd^cT,\psi_\epsilon,s^k)}s ds\\
& =& {\displaystyle}k^p\int_{B_{\varphi_\epsilon}(\epsilon,r)}T{\wedge}\alpha_{\varphi_\epsilon}^p -\frac{k^p}{r^{kp}}\int_\epsilon^r s^{kp-1}\nu(dd^cT,\varphi_\epsilon,s)ds \\
& &{\displaystyle}\hfill +k^p\int_\epsilon^r\frac{\nu(dd^cT,\varphi_\epsilon,s)}s ds\\
& = &{\displaystyle}k^p\left(\nu(T,\varphi_\epsilon,r)+\frac1{r^p}\int_\epsilon^rt^{p-1} \nu(dd^cT,\varphi_\epsilon,t)dt \right)\\
& & \hfill{\displaystyle}-\frac{k^p}{r^{kp}}\int_\epsilon^r s^{kp-1}\nu(dd^cT,\varphi_\epsilon,s)ds\\
& = &{\displaystyle}k^p\left(\nu(T,\varphi_\epsilon,r)+\int_\epsilon^r \frac{\nu(dd^cT,\varphi_\epsilon,t)}t \left(\frac{t^p}{r^p} -\frac{t^{kp}}{r^{kp}}\right)dt \right).
\end{array}$$ Here we have used successively Equality (\[eq 1.4\]) with $\psi_\epsilon$ instead of $\varphi_\epsilon$ in the first equality, then the change of variable $t=s^k$, next the fact that $\nu(dd^cT,\psi_\epsilon,s^k)=k^{p-1}\nu(dd^cT,\varphi_\epsilon,s)$ (equality proved by Demailly in the case of closed positive currents), and finally Equality (\[eq 1.4\]).\
When $\epsilon\to0$, Equality (\[eq 1.5\]) implies Equality (\[eq 1.3\]).\
Thanks to Equality (\[eq 1.3\]), for every $r\in{}]0,R(\varphi)[$, we have $$\nu(T,\varphi,r)-\frac1{k^p}\nu(T,\varphi^k,r^k)=-\int_0^r\nu(dd^cT,\varphi,t) \left(\frac{t^{p-1}}{r^p} -\frac{t^{kp-1}}{r^{kp}}\right)dt.$$ The current $dd^cT$ is positive and closed, hence $\nu(dd^cT,\varphi,.)$ is a non-negative increasing function and $\nu(dd^cT,\varphi)=0$. We consider two disjoint cases:
- *First case $k\geq 1$.* We have $$\begin{array}{lcl}
0 & \leq &{\displaystyle}\int_0^r\nu(dd^cT,\varphi,t) \left(\frac{t^{p-1}}{r^p} -\frac{t^{kp-1}}{r^{kp}}\right)dt\\
& \leq &{\displaystyle}\nu(dd^cT,\varphi,r)\int_0^r \left(\frac{t^{p-1}}{r^p} -\frac{t^{kp-1}}{r^{kp}}\right)dt\\
& = &{\displaystyle}\nu(dd^cT,\varphi,r)\frac{k-1}{kp}.
\end{array}$$ So $$-\frac{k-1}{kp}\nu(dd^cT,\varphi,r)\leq\nu(T,\varphi,r)-\frac1{k^p}\nu(T,\varphi^k,r^k)\leq 0$$
- *Second case $k< 1$.* A similar calculation shows that we have $$0\leq\nu(T,\varphi,r)-\frac1{k^p}\nu(T,\varphi^k,r^k)\leq -\frac{k-1}{kp}\nu(dd^cT,\varphi,r).$$
In the two cases, both terms (of the right-hand and the left-hand) have the same limit 0 when $r\to0^+$. This completes the proof of the proposition.
Applications
============
The following corollaries are immediate consequences of Theorem \[theo1\] and/or Proposition \[pro1\].
Let $T$ be a negative plurisubharmonic current of bidimension $(p,p)$ on $\Omega$ and $\varphi\in {\mathrm{PSH}}(T,\Omega)$. If there exists $k\in\mathcal I_T(\varphi)$ such that the function $t\longmapsto \frac{\nu(dd^cT,\varphi,t^{\frac1k})}{t}$ is integrable on a neighborhood of $0$, then the Lelong-Demailly number $\nu(T,\varphi)$ of $T$ relatively to $\varphi$ exists.
We can prove this corollary in two different ways:
- *First way*. Thanks to Proposition \[pro1\], to prove that $\nu(T,\varphi)$ exists, it suffices to show that $\nu(T,\varphi^k)$ exists; for this we observe that $$\frac{\nu(dd^cT,\varphi^k,t)}t=\frac{\nu(dd^cT,\varphi^k,(t^{\frac1k})^k)}t =k^{p-1}\frac{\nu(dd^cT,\varphi,t^{\frac1k})}t$$ is integrable on a neighborhood of 0 (hypothesis). Hence, thanks to Theorem \[theo1\], $\nu(T,\varphi^k)$ exists.
- *Second way*. We remark that the integrability of $\frac{\nu(dd^cT,\varphi,t^{\frac1k})}t$ is equivalent to the integrability of $\frac{\nu(dd^cT,\varphi,t)}t$. Hence, thanks to Theorem \[theo1\], $\nu(T,\varphi)$ exists. In fact if we take $t=s^k$ we obtain $$\int_0^{r_0}\frac{\nu(dd^cT,\varphi,t^{\frac1k})}t dt=k\int_0^{r_0^k}\frac{\nu(dd^cT,\varphi,s)}s ds.$$
Let $T$ be a negative plurisubharmonic current of bidimension $(p,p)$ on $\Omega$ and $\varphi\in {\mathrm{PSH}}(T,\Omega)$ such that $\mathcal I_T(\varphi)$ is non bounded $($this is always the case if $\varphi$ is $\mathcal C^2)$. Then for every $r\in{}]0,R(\varphi)[$ one has $$\nu(T,\varphi,r)\leq -\int_0^r\nu(dd^cT,\varphi,t) \frac{t^{p-1}}{r^p}dt.$$ In particular, $$\nu(T,\varphi,r)\leq -\frac{1-s^{-p}}p\nu(dd^cT,s\varphi,r)\quad\forall s\geq1.$$
Thanks to Proposition \[pro1\], for every $k\in\mathcal I_T(\varphi)$, we have $$u(k):=\frac1{k^p}\nu(T,\varphi^k,r^k)=\nu(T,\varphi,r)+\int_0^r\frac{\nu(dd^cT,\varphi,t)}t \left(\frac{t^p}{r^p} -\frac{t^{kp}}{r^{kp}}\right)dt.$$ The function $u$ is non-positive and increasing on $\mathcal I_T(\varphi)$, so $$\lim_{k\to+\infty, k\in\mathcal I_T(\varphi)}u(k)=\nu(T,\varphi,r)+\int_0^r\nu(dd^cT,\varphi,t) \frac{t^{p-1}}{r^p}dt\leq 0.$$ If $s=1$ the inequality is clear. For $s>1$ we have $$\begin{array}{lcl}
{\displaystyle}\nu(T,\varphi,r)\leq -\int_0^r\nu(dd^cT,\varphi,t) \frac{t^{p-1}}{r^p}dt
& \leq & {\displaystyle}-\int_{r/s}^r\nu(dd^cT,\varphi,t) \frac{t^{p-1}}{r^p}dt \\
& \leq &{\displaystyle}-\frac{1-s^{-p}}p\nu(dd^cT,\varphi,\frac rs).
\end{array}$$
\[theo2\] Let $T$ be a positive or negative plurisubharmonic current of bidimension $(p,p)$ on $\Omega$ and $\varphi,\ \psi\in {\mathrm{PSH}}(T,\Omega)$ such that $$\liminf_{\varphi(z)\to0}\frac{\log \psi(z)}{\log \varphi(z)}\geq \ell$$ where $\mathcal I_T(\varphi)$ contains a neighborhood $\mathscr V(\ell)$ of $\ell$. One has
- $\nu(T,\psi)\geq\ell^p\nu(T,\varphi)$ if $T$ is positive.
- $\nu(T,\psi)\leq\ell^p\nu(T,\varphi)$ if $T$ is negative $($with the assumption that $\psi$ satisfies condition $(C))$.
In particular,if $\log \psi(z)\sim\ell\log \varphi(z)$ for $\varphi(z)\in\mathscr V(0)$ then $\nu(T,\psi)=\ell^p\nu(T,\varphi).$
This type of theorem is called a “comparison theorem”; such a statement has been proved by Demailly in the case of positive closed currents. Furthermore, one can deduce the invariance of the Lelong-Demailly numbers under changes of coordinate systems.
One can replace $\psi$ by $\psi^k$ with $1<k\in\mathcal I_T(\psi)$, in order to assume that $$\liminf_{\varphi(z)\to0}\frac{\log \psi(z)}{\log \varphi(z)}> \ell.$$ As a consequence $$\lim_{\varphi(z)\to0}\frac{\psi(z)}{\varphi(z)^\ell}=0.$$ For $\epsilon>0$ small sufficiently, we set $\Psi_\epsilon:=\psi+\epsilon\varphi^\ell \underset{\varphi(z)\in\mathscr V(0)}\sim\epsilon\varphi^\ell$. Thanks to the dominated convergence theorem, $$\lim_{\epsilon\to0}\frac1{r^p}\int_{\Psi_\epsilon<r}T{\wedge}\beta_{\Psi_\epsilon}^p =\frac1{r^p}\int_{\psi<r}T{\wedge}\beta_{\psi}^p.$$
- *First assume $T\geq 0$.* As the function $\nu(T,\Psi_\epsilon,.)$ is increasing then $$\begin{array}{lcl}
\nu(T,\Psi_\epsilon,r)&=&{\displaystyle}\frac1{r^p}\int_{\Psi_\epsilon<r}T{\wedge}\beta_{\Psi_\epsilon}^p\\
& \geq &{\displaystyle}\lim_{\rho\to0}\frac1{\rho^p}\int_{\Psi_\epsilon<\rho}T{\wedge}\beta_{\Psi_\epsilon}^p =\lim_{\rho\to0}\frac1{\rho^p}\int_{\epsilon\varphi^\ell<\rho}T{\wedge}\beta_{\Psi_\epsilon}^p \\
& \geq &{\displaystyle}\lim_{\rho\to0}\frac1{\rho^p}\int_{\epsilon\varphi^\ell<\rho}T{\wedge}\beta_{\epsilon\varphi^\ell}^p =\nu(T,\varphi^\ell).
\end{array}$$
because $\Psi_\epsilon\sim\epsilon\varphi^\ell$ and $T{\wedge}(dd^c(\psi+\epsilon\varphi^\ell))^p\geq T{\wedge}(\epsilon dd^c(\varphi^\ell))^p$. If $\epsilon\to0$, we obtain $\nu(T,\psi,r)\geq\nu(T,\varphi^\ell)$. Hence, if $r\to0$, $\nu(T,\psi)\geq\nu(T,\varphi^\ell)$.
- *Now, assume $T\leq0$.* Thanks to Theorem \[theo1\], Equality (\[eq 1.2\]) gives $$\begin{array}{lcl}
{\displaystyle}\nu(T,\Psi_\epsilon,r)+\int_0^r\frac{\nu(dd^cT,\Psi_\epsilon,t)}t\left(\frac{t^p}{r^p}-1\right)dt & \leq &{\displaystyle}\nu(T,\Psi_\epsilon) \\
& = &{\displaystyle}\lim_{\rho\to0}\frac1{\rho^p}\int_{\Psi_\epsilon<\rho}T{\wedge}\beta_{\Psi_\epsilon}^p
\end{array}$$ and a similar calculation proves that $$\nu(T,\Psi_\epsilon,r)+\int_0^r\frac{\nu(dd^cT,\Psi_\epsilon,t)}t\left(\frac{t^p}{r^p}-1\right)dt \leq \nu(T,\varphi^\ell).$$ So if $\epsilon\to0$ we obtain $$\nu(T,\psi,r)+\int_0^r\frac{\nu(dd^cT,\psi,t)}t\left(\frac{t^p}{r^p}-1\right)dt \leq \nu(T,\varphi^\ell).$$ When $r\to0$, the last inequality gives $\nu(T,\psi)\leq \nu(T,\varphi^\ell).$
In the particular case $\log \psi(z)\sim\ell\log \varphi(z)$, for every $\epsilon>0$, there exists $\eta>0$ so that if $|\varphi(z)|<\eta$, then $\varphi(z)^{\ell(1+\epsilon)}\leq \psi(z)\leq \varphi(z)^{\ell(1-\epsilon)}$ and for $\epsilon$ small enough, $\ell(1+\epsilon),\ \ell(1-\epsilon)\in\mathscr V(\ell)\subset\mathcal I_T(\varphi)$. Therefore we can apply the previous inequalities to obtain $\ell^p(1+\epsilon)^p\nu(T,\varphi)\leq\nu(T,\psi)\leq \ell^p(1-\epsilon)^p\nu(T,\varphi)$ (resp. $\ell^p(1-\epsilon)^p\nu(T,\varphi)\leq\nu(T,\psi)\leq \ell^p(1+\epsilon)^p\nu(T,\varphi)$ in the case $T\geq0$ (resp. $T\leq0$). Hence, when $\epsilon\to0$, we obtain $\nu(T,\psi)=\ell^p\nu(T,\varphi)$.
Acknowledgements {#acknowledgements .unnumbered}
================
I thank Professor Jean-Pierre Demailly for several suggestions which enabled me to clarify and improve the original form of this note. I thank also Professor Khalifa Dabbek for useful discussions concerning this work.
[X-XX1]{} **J.-P. Demailly**, Sur les nombres de Lelong associés à l’image directe d’un courant positif fermé, Annales de l’institut Fourier, tome 32, $n^\circ 2$ p37-66 (1982). **M. Toujani**, Nombre de Lelong directionnel d’un courant positif plurisousharmonique, C. R. Acad. Sci. Paris, Ser. I 343 p705-710 (2006).
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abstract: 'We calculate the indirect charge carrier mediated Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between magnetic impurities for two selected graphene nanoflakes containing four hexagonal rings in their structure, differing by their geometry. We describe the electronic structure of either charge-neutral or doped nanoflakes using the tight-binding approximation with the Hubbard term, which is treated within the molecular-field approximation. We find pronounced differences in the RKKY coupling energies, dependent on the placement of the pair of magnetic moments in the nanostructure and on the edge form. For an odd total number of electrons in the structure, we predict in some circumstances the existence of ferromagnetic coupling with leading first-order perturbational contribution, while for an even number of charge carriers the usual, second-order mechanism dominates. Therefore, doping of the nanoflake with a single charge carrier is found to be able to change the coupling from an antiferromagnetic to a ferromagnetic one for some geometries.'
author:
- 'Karol Sza[ł]{}owski'
title: The RKKY Coupling between Impurity Spins in Graphene Nanoflakes
---
Introduction
============
Two-dimensional graphene, being one of the most promising contemporary materials [@Geim1; @Geim2] offers unique physical properties due to its electronic structure.[@Novoselov1; @CastroNeto1; @Abergel1] Since its discovery it has attracted concerted theoretical and experimental efforts aimed at understanding its rich physics and making basis for future electronics. One of the directions is focused on integrating the charge and spin degrees of freedom, to create novel spintronic devices.[@Rocha1; @Yazev2; @Zhang1] This encourages studies of magnetic properties of graphene.
Recently, increasing interest has been focused on the graphene nanostructures.[@Snook1; @Raeder1; @Fernandez1] Such nanostructures can be engineered to merge the properties of graphene with ultrasmall sizes, and can serve as possible building blocks of novel devices (see for example Refs. ), especially in context of transport properties, showing giant magnetoresistance.[@Krompiewski1] Significant efforts are made to understand magnetism emerging in finite-size graphene structures without magnetic impurities introduced (an extensive review of that particular topic is presented in the Ref. ). However, the possibility of introducing magnetic impurity atoms to the graphene lattice is also taken into account.[@Santos1; @Wu1; @Kang1]
Among the topics attracting the interests of researchers, the problem of indirect magnetic coupling between magnetic impurity spins in graphene, mediated by charge carriers, is worth mentioning. This kind of interaction, known as Ruderman-Kittel-Kasuya-Yosida (RKKY) coupling,[@Ruderman1; @Kasuya1; @Yosida1] can be expected to show unique properties in graphene, different from the behaviour in two-dimensional metals,[@BealMonod1] owing to the peculiar, linear dispersion relation for the charge carriers in the vicinity of Dirac points. Various calculations of RKKY interaction in graphene sheets are present in the literature, exploiting the bipartite nature of the graphene crystalline lattice in various (mainly perturbational) approaches.[@Vozmediano1; @Dugaev1; @Cheianov1; @Saremi1; @Bunder1; @Brey1; @Sherafati1; @Sherafati2; @Kogan1] Moreover, tight-binding calculations in real space were performed for such a system.[@Annica1; @Annica2] Recently, also the problem of two Kondo impurities attracted some attention. [@Uchoa1; @Hu1]
However, the main efforts have so far focused on the calculations of RKKY coupling properties in infinite graphene planes. On the other hand, the coupling in ultrasmall nanoflakes (or nanodisks, [@Ezawa1] containing just a few hexagonal rings) can be expected to deviate from the predictions for infinite system, as the dominance of the edge in a nanoflake substantially modifies the electronic structure and severely breaks the translational symmetry. Especially, the peculiar features of the electronic state at the zigzag edge of graphene (or even graphite) are known both from theoretical predictions[@Fujita1; @Nakada1] and experimental observations for various systems (e.g. Ref. ), especially quantum-dot systems.[@Ritter1] The significance of the edge structure has been already found for example in the transport properties of nanoflakes.[@Krompiewski1] Moreover, in ultrasmall molecule-like structures, the existence of discrete electronic states significantly separated in energy allows us to expect a range of novel phenomena. Therefore, a sound motivation appears for studies of RKKY coupling between magnetic moments in nanosized graphene structures, which is the aim of the present work.
To achieve this goal, we use the real-space tight-binding approximation (TBA) supplemented with the Hubbard term for finite, ultrasmall graphene nanoflakes. We perform the exact diagonalization of the single-particle Hamiltonian resulting from the mean field approximation (MFA). Not limiting the calculations to charge-neutral structures, we take into account the possibility of varying the charge concentration electron by electron (charge doping). One of our goals is to search for the configuration for which the coupling changes its character from ferro- to antiferromagnetic as a result of adding or removing a single electron from the system, in close vicinity to the equilibrium electron concentration. In principle, such a possibility might be opened by placing the nanostructure between two tunneling electrodes, providing a single-electron control of mutual orientation of impurity spins (let us mention that the idea of doping-controlled coupling between magnetic moments in bilayer graphene has been very recently raised in the Ref. ). The phenomenon of Coulomb blockade in some graphene nanostructures with two electrodes was studied in Ref. , while another route to modifying the charge concentration is the absorption of gas molecules (as shown in Ref. ).
Theory
======
The analysis of RKKY interaction in graphene-based structures consisting of $N$ carbon atoms bases on the reliable description of the electronic properties. The most commonly used approach to that problem is the TBA [@Wallace1; @Reich1; @CastroNeto1]. The main advantage of the TBA method is the ability to include the full, realistic band structure of graphene with finite bandwidth, not being restricted to the linear part of the dispersion relation close to Dirac points. Therefore, the use of certain cut-off schemes is not necessary to obtain convergent results and the issue of the possible unphysical modifications of the RKKY range function is absent. Moreover, the TBA facilitates the non-perturbational treatment of the problem of magnetic impurities. In our study we will adopt this approach, taking into account the electronic hopping for nearest-neighbours only.
The total Hamiltonian for the ultrasmall graphene structure with two magnetic impurities can be written as $$\mathcal{H}=\mathcal{H}_{0}+\mathcal{H}_{C}+\mathcal{H}_{imp},$$ and contains the contribution from the TBA hopping term $\mathcal{H}_{0}$, Hubbard term $\mathcal{H}_{C}$ as well as impurity potential $\mathcal{H}_{imp}$.
The TBA term has the following form: $$\begin{aligned}
\mathcal{H}_{0}=&-t\,\sum_{\left\langle i,j\right\rangle,\sigma}^{}{\left(c^{\dagger}_{i,\sigma}\,c_{j,\sigma}+c^{\dagger}_{j,\sigma}\,c_{i,\sigma}\right)}.\end{aligned}$$ Here, $c^{\dagger}_{i,\sigma}$ and $c_{i,\sigma}$ denote creation and annihilation operators for $p^{z}$ electrons at sites $i$ and $j$ in the nanoflake, with spin $\sigma=\,\uparrow\,,\,\downarrow$. The summation over nearest-neighbour sites is denoted by $\left\langle i,j\right\rangle$. The parameter $t$ (usually taken as 2.8 eV) is the hopping integral between nearest neighbours. In the further numerical results, all the energies are normalized to $t$.
In order to incorporate the electron-electron correlations induced by the Coulombic interactions, we include the Hubbard term in the Hamiltonian: $$\label{eq:hubbard1}
\mathcal{H}_{C}=U\sum_{i}^{}{n_{i,\uparrow}\,n_{i,\downarrow}},$$ with the electron number operators $n_{i,\sigma}=c^{\dagger}_{i,\sigma}\,c_{i,\sigma}$.
The Hubbard term, capturing the on-site coulombic repulsion only, neglects the long-range part of the interaction. The usefulness of this model is a subject of debate (see the recent review in Ref. ). However, it is of noticeably wide use in the theory of carbon (nano)structures; see Refs. and the recent work on carbon nanotubes in Ref. Along the lines of the discussion in Ref. , $U$ should be treated as an effective parameter describing the influence of Coulombic interactions. Its value might result from a competition between on-site repulsion and the interaction between electrons located on different sites (especially nearest neighbour). Also, some recent results evidence suppression of the influence of the long-range interactions [@Reed1].
Contrary to the choice of hopping integrals in the TBA term, the situation with the value of on-site Coulomb repulsion parameter $U$ appears less clear. The values accepted vary between $U=2$ eV [@Jung1] and even $U\simeq 10$ eV.[@Wehling1] In our study, if we include the Hubbard term, we accept a moderate value of $U/t=1$, close to the choice of Yazyev [@Yazyev1] and Potasz *et al.* [@Potasz1].
The interaction of the $z$-component of the on-site impurity spin $S^{z}_{k}$ (located in at lattice site $k$) with an electron spin $s_{i}^{z}=\left(n_{i,\uparrow}-n_{i,\downarrow}\right)/2$ at the same site is described by the Anderson-Kondo Hamiltonian. For two impurities, located at the sites $a$ and $b$, we have: $$\label{eq:Himp1}
\mathcal{H}_{imp}=\frac{1}{2}J\left(S^{z}_{a} s^{z}_{a}+S^{z}_{b}s^{z}_{b}\right),$$ where $J$ is a spin-dependent impurity potential (contact potential).
Let us note that we select the Ising form of the interaction Hamiltonian \[\[eq:Himp1\]\] just for simplicity of the calculations. It leads to the final interaction between magnetic impurities described by $\mathcal{H}^{RKKY}=J^{RKKY}\,S^{z}_{a}S^{z}_{b}$. The usage of the Heisenberg exchange Hamiltonian, of the form $H_{imp}=\frac{1}{2}J\left(\mathbf{S}_{a}\mathbf{s}_{a}+\mathbf{S}_{b}\mathbf{s}_{b}\right)$, would just yield $\mathcal{H}^{RKKY}=J^{RKKY} \mathbf{S}_{a}\mathbf{S}_{b}$, without any modification to the indirect exchange integral itself, which is the only subject of our interest in the present work.
In order to evaluate the RKKY coupling between the impurities at $T=0$, the ground state energy must be found first in the presence of $n=N+\Delta n$ electrons in the structure, for both parallel and antiparallel orientation of the impurity spins. In undoped graphene each carbon atom donates one $p^{z}$ electron, so that $\Delta n=0$ (i.e. half-filling) characterizes the state of charge neutrality. In general, however, the number of electrons present in the system can vary hypotetically between $0$ (empty energetic spectrum) and $2N$ (completely filled spectrum). We limit our further considerations to $\left|\Delta n\right|\leq 6$, in order not to lose the accuracy of electronic spectrum reproduction by means of the TBA method. The RKKY coupling energy between the impurity spins can be related to the difference of total energies by $$\label{eq:rkky}
E\left(S^{z}_a=\uparrow,S^{z}_b=\downarrow\right)-E\left(S^{z}_a=\uparrow,S^{z}_b=\uparrow\right)=2S^2 J^{RKKY},$$ where the positive value of $J^{RKKY}$ corresponds to ferromagnetic coupling (F) and the negative value to antiferromagnetic coupling (AF). In our case (when electronic structure description involves NN hopping only) the coupling values calculated for $\pm\Delta n$ are identical (i.e., electron and hole doping lead to the same results) due to electron-hole transformation symmetry of the Hamiltonian on the finite bipartite lattice.
Let us note that the indirect charge carrier mediated interaction resulting from our calculations can be not necessarily the usual form of RKKY coupling, which is proportional to the square of the contact potential $J$, as resulting from the second order perturbation calculus. [@Kogan1] However, we will use in general the term “RKKY interaction” to name the indirect charge carrier mediated coupling between impurity spins, even with different characteristic features, resulting from the simultaneous presence of other mechanisms.
In order to deal with the Hubbard term in the Hamiltonian for quite a large system, we adopt the mean field approximation (MFA), which consists in replacement of the form $n_{i,\uparrow}\,n_{i,\downarrow}\simeq n_{i,\uparrow}\,\left\langle n_{i,\downarrow}\right\rangle+n_{i,\downarrow}\,\left\langle n_{i,\uparrow}\right\rangle-\left\langle n_{i,\uparrow}\right\rangle\,\left\langle n_{i,\downarrow}\right\rangle$. This approach has been shown recently to compare successfully with some exact diagonalization method for graphene nanostructures [@Feldner1] especially for $U/t<2$ and has been applied to studies of edge magnetic polarization in graphene sheets [@Fujita1; @Jung1; @Yazyev2; @Feldner2] or RKKY in infinite graphene.[@Annica2] The approximation leads to the effective Hamiltonian defined in single-particle space. The pair of coupled effective single-particle Hamiltonians, $\mathcal{H}^{MFA}_{\uparrow}$ and $\mathcal{H}^{MFA}_{\downarrow}$, for spin-up and spin-down electrons, is obtained (the Hamiltonians depend on the electronic densities $\left\langle n_{i,\sigma}\right\rangle$ and can be treated self-consistently).
The Hamiltonian matrix for MFA Hamiltonians can be found in the orthonormal basis of single-electron atomic orbitals $c^{\dagger}_{i,\sigma}\left|0\right\rangle$. Then, $N$ single-particle eigenstates indexed by $\mu$ can be found for each electron spin orientation $\sigma$ in the form of linear combination of atomic orbitals $\left|\psi_{\mu,\sigma}\right\rangle=\left(1/\sqrt{N}\right)\sum_{i=1}^{N}{\gamma^{\mu}_{i,\sigma} c^{\dagger}_{i,\sigma}\left|0\right\rangle}$. Here, the coefficients $\gamma^{\mu}_{i,\sigma}$ for $\mu=1,\dots,n$ set up an eigenvector of the Hamiltonian matrix corresponding to the eigenvalue $\epsilon^{\mu}_{\sigma}$.
Let us sort the eigenvalues in an ascending order, so that $\epsilon^{1}_{\sigma}$ is the lowest one, etc. Then, the total electronic density at site $i$ in the presence of $n_{\sigma}$ electrons with spin orientation $\sigma$ in the system in the ground state at the temperature $T=0$ can be calculated as $\left\langle n_{i,\sigma}\right\rangle =\sum_{\mu=1}^{ n^{\sigma}}{\left|\gamma^{\mu}_{i,\sigma}\right|^2}$. The corresponding ground-state energy is
$$\label{eq:energy1}
E\left(S^{z}_a,S^{z}_b\right)=\sum_{\sigma=\uparrow,\downarrow}^{}{\sum_{\mu=1}^{n^{\sigma}}{\epsilon^{\mu}_{\sigma}}}.$$
The values of $n^{\uparrow}$ and $n^{\downarrow}$ (which add up to the given value of $n$) should be selected so to minimize the total energy.
After the calculation, the obtained values of electron densities are substituted back into the MFA Hamiltonians and the numerical procedure is repeated iteratively until the satisfactory convergence of the eigenvalues and eigenvectors is achieved. Then the obtained self-consistent numeric value of total energy can be used to calculate RKKY coupling according to the formula given in Eq. \[eq:rkky\].
The numerical results and discussion
====================================
For our calculations, we selected two graphene nanoflakes, consisting of four hexagonal rings with an even number of carbon atoms. The first one is similar to a pyrene molecule and contains $N=16$ carbon atoms, and its edge has a zigzag-like character. The second one, triphenylene-like, composed of $N=18$ carbon atoms, possesses the armchair-type edge. Both structures are schematically depicted in Fig. \[fig:N16density\](c) and Fig. \[fig:N18density\](c).
To study the influence of broken translational symmetry on the RKKY coupling in nanostructures, we calculated $J^{RKKY}$ between two impurities being either nearest or second neighbours, focusing in each case on two different locations of impurity pairs in a given nanostructure. We denote these cases as 1a,1b and 2a,2b, respectively. The considered pairs of magnetic impurities are presented schematically for both graphene nanoflakes in Fig. \[fig:N16density\](c) and Fig. \[fig:N18density\](c). Let us mention that nearest neighbours correspond to two ions in two different sublattices while second neighbours constitute a pair of ions in the same sublattice (referring to the subdivision of the bipartite graphene lattice into A and B sublattices for the finite system). In our calculations, we set $S=1/2$. In the presentation of further results we focus mainly on the influence of charge doping on the characteristics of indirect coupling.
Pyrene-like nanoflake
---------------------
In order to gain some insight into the electronic structure of the pyrene-like nanoflake, we plot the predicted energy levels resulting from diagonalization of the Hamiltonian in the absence of magnetic impurities and for $U/t=0$ in Fig. \[fig:N16density\](a). The energy states are numbered by $\mu$ and sorted according to ascending energy and each of the states is doubly (spin) degenerate. No additional degeneracy is observed. In the case of charge neutrality the HOMO-LUMO (highest occupied molecular orbital-lowest unoccupied molecular orbital) gap amounts to 0.89 eV (which is in concert with the value resulting from the calculations in Ref. , Fig. 2). In order to visualize the electronic densities assigned to the distinct states, we present Fig. \[fig:N16density\](b). There, the values of $\left|\gamma^{\mu}_{i,\sigma}\right|^2$ (probabilities of finding the electron at the given lattice site $i$ for a given state $\mu$) are plotted on the nanoflake scheme, for selected orbitals which are HOMO orbitals for $|\Delta n|\leq 6$. Let us observe that the subsequent states are characterized by significant variability of the corresponding partial charges distribution. If the selected state is occupied only by a single electron, the distribution of partial charge for this orbital reflects also the spin density.
In Fig. \[fig:N161st\] we present the values of RKKY exchange integrals calculated for two positions of magnetic impurities, 1a and 1b, as marked in Fig. \[fig:N16density\](c). In order to show qualitatively the influence of increasing contact potential $J$, we prepared the plot for two selected values of $J/t=0.2$ and $0.4$. Moreover, to enable the observation of the Hubbard term effect, we took into consideration the case of $U/t=0$ as well as $U/t=1$. The indirect exchange values are calculated for various deviations in electron number $\Delta n$ from charge neutrality, for $|\Delta n|\leq 6$. As can be observed for both impurity pair positions, the coupling is antiferromagnetic for charge-neutral structure (expected for a bipartite lattice half-filled with the electrons, especially for undoped infinite graphene; e.g. [@Annica1]). The interaction is relatively weak and its non-linear rise with increasing $J$ values is also observable. The situation is quite similar for $|\Delta n|=2$, the only difference being the stronger coupling. However, it is quite striking that changing the electronic concentration by one charge carrier from $\Delta n=0$ results in switching of the coupling sign from antiferromagnetic to ferromagnetic, which takes place for $|\Delta n|=3$ as well. In these cases, it can be observed that the change of $J^{RKKY}$ with increasing $J$ is noticeably slower than for even $\Delta n$ values.
In order to study more systematically the dependence of exchange coupling on contact potential $J$, we performed the appropriate calculations for nearest-neighbour ions, the results of which are presented in the Fig. \[fig:N161stlog\] in double logarithmic scale, which allows us to linearize power-law dependences. The case of 1a impurity pair for $U/t=0$ is illustrated in the Fig. \[fig:N161stlog\](a). It can be verified that two kinds of dependences of $J^{RKKY}$ on $J$ can be distinguished. For selected even total numbers of electrons in the nanoflake ($|\Delta n|=0,2$) the usual dependence $J^{RKKY}\propto J^{2}$ is obeyed, which (according to Fig. \[fig:N161st\]) coincides with antiferromagnetic sign of interaction. On the contrary, for odd numbers of electrons ($|\Delta n|=1,3$) we deal with ferromagnetic interaction with $J^{RKKY}\propto |J|$. The latter behaviour tends to convert into to the quadratic dependence on $J$ (and antiferromagnetic exchange) provided that the potential $J$ is strong enough. For clarity, the data for $|\Delta n|=4,5,6$ were omitted, as very close to the results for $\Delta n=0$. Let us also observe that qualitatively the same situation is met in presence of the Hubbard term, for $U/t=1$ \[Fig. \[fig:N161stlog\](b)\]. It is worth special emphasis that the usual derivation of RKKY exchange integral, within the framework of second-order perturbation calculus, yields always the coupling proportional to the square of the contact potential, which has been recently recapitulated for bipartite graphene lattice. [@Kogan1] Therefore, it is of special interest to identify the source of the unusual linear behaviour. Quite similarly, such a behaviour can be observed for the pair of magnetic impurities in location 1b for example when $|\Delta n|=3$ \[Fig. \[fig:N161stlog\](c)\] (there, the data for $|\Delta n|=0,1,2$ are identical, the same being true for the case of $|\Delta n|=4,6$). Thus, it can be deduced that for selected odd total electron numbers we deal with $J^{RKKY}\propto |J|$.
To investigate more deeply the selected case of two impurities in position 1a and $\Delta n=-1$ and identify the origin of the mentioned unusual behaviour, we plot the energy difference between the AF and F orientations of impurity spins (note that $J^{RKKY}\propto \Delta E^{AF-F}$) as a function of the contact potential $J$ in Fig. \[fig:N16deltaE2\](a). For $\Delta n=-1$, the ground state is characterized by unequal number of spin-up and spin-down electrons. In the plot we resolve the contributions coming from the highest energy orbital occupied by a single electron as well as the total contribution originating from the lower energy orbitals occupied by pairs of electrons of opposite spins. It is noticeable that the latter contribution favours the AF state and is proportional to $J^2$ for not too strong contact potentials (as expected on the basis of second-order perturbation calculus). On the contrary, the orbital occupied by a single electron gives rise to the energy difference which is proportional to $J$ and tends to prefer the F state of the impurities. In the limit of low $J$, the situation is ruled by the singly coupled state so that the total indirect exchange is ferromagnetic and linear in contact potential, while the increase of $J$ leads first to compensation of both contributions and then to domination of the ordinary perturbational $J^2$-proportional behaviour. The linearity of energy difference in $J$ can be explained basing on the fact that for a single electron occupying a given orbital, the leading correction to energy of the orbital \[coming from the term Eq. (\[eq:Himp1\]) in the Hamiltonian\] is of the first- order perturbational kind. If the orbital is occupied by a pair of electrons with opposite spins, first-order corrections for them are of opposite values and cancel each other, while the second-order corrections give rise to the ordinary RKKY interaction. However, when there is no second electron, the uncompensated first-order contribution dominates. Under such circumstances, for F polarization of impurity spins, the first-order correction to the electronic orbital energy due to the presence of two impurities at sites $a$ and $b$ amounts to $\Delta E^{F}=-\frac{1}{2}|J|S \left(|\gamma^{\mu}_{a}|^2+|\gamma^{\mu}_{b}|^2\right)$ (corresponding to the direction of electron spin which minimizes the total energy). Let us assume without loss of generality that $|\gamma^{\mu}_{a}|^2\leq |\gamma^{\mu}_{b}|^2$. If the polarization of impurity spins is AF, then the corresponding correction is $\Delta E^{AF}=-\frac{1}{2}|J|S\left(|\gamma^{\mu}_{b}|^2-|\gamma^{\mu}_{a}|^2\right)$. The resulting energy difference $\Delta E ^{AF-F}=|J|S|\gamma^{\mu}_{a}|^2>0$ clearly makes a ferromagnetic contribution to the indirect coupling. Its magnitude is proportional to the smaller of the electronic densities on the impurity pair sites; thus it is maximized for equal electronic densities on both sites. Such a contribution to indirect coupling yields some resemblance to the double exchange mechanism. [@Nolting] However, let us still use the term RKKY interaction to characterize the indirect charge carrier mediated coupling which results from our calculations.
Let us observe, that in some cases we deal with the situation when the electronic density for a given orbital $\mu$ vanishes at least at one of the sites at which the impurity spins are localized. Such an orbital does not indicate any energy difference between AF and F orientation of impurity spins and thus does not give any contribution to the RKKY exchange integral. This is, for example, the case for $\Delta n=-5$ and impurities in the 1a position; see the corresponding electronic densities for the orbital $\mu=6$ in Fig. \[fig:N16density\](b), which is the highest energy orbital occupied by a single electron. Such a situation prevents the indirect interaction from switching to the F sign (like for $|\Delta n|=1,3$), even though the total number of electrons in the system is odd. Quite a similar situation can be observed for impurities in the 1b position, since the electronic density for the orbital $\mu=8$ also vanishes at one of the impurity sites. Therefore, the coupling for $\Delta n=0$ and $\Delta n=-1$ is exactly the same. We note that the value of $J^{RKKY}$ is also unchanged for $\Delta n=-2$, which corresponds to the case when the highest energy orbital $\mu=7$ is occupied by two electrons with opposite spins. However, in that case we can observe that the unperturbed electronic densities are almost equal at both 1b impurity sites. Therefore, the AF state of impurities changes the orbital energy particularly weakly \[as can be seen in Fig. \[fig:N16deltaE\](b) for $\mu=7$\] and the energy changes for spin-up and spin-down electron for F impurity polarization cancel each other. Therefore, a doubly occupied state $\mu=7$ also gives a particularly weak contribution to exchange integral. This explains the robustness of weak AF RKKY coupling for 1b pair location with respect to deviations from charge neutrality. On the other hand, the same feature of the $\mu=7$ orbital when it is singly occupied gives rise to a particularly strong ferromagnetic contribution to the coupling (seen clearly for $\Delta n=-3$, when $J^{RKKY}\propto |J|$). As mentioned before, this strong ferromagnetic contribution can be explained as a first-order perturbational effect.
To generalize, the possibility of ferromagnetic coupling for nearest-neighbour impurities is open provided that the number of electrons in the system is odd. Under such a condition, the nanoflake gains nonzero magnetic moment, which originates from the HOMO orbital. An additional condition is that the electronic density for the highest energy occupied orbital cannot vanish at both impurity locations. The coupling is particularly enhanced when the electronic densities at the impurity sites are high and close to each other. Then it appears straightforward to explain, that it is energetically favourable to have both impurity spins parallel, as the energy of such a configuration is significantly lowered by the first-order term. Let us notice that in general, for a given weak contact potential $J$, the ferromagnetic couplings, if present, are much stronger than the corresponding antiferromagnetic interactions, which can be seen in Fig. \[fig:N161stlog\].
Let us observe that the presence of a Hubbard term with $U/t=1$ influences remarkably the interaction, especially for the case of charge neutrality, where it leads to strong enhancement of the AF coupling, and eventually it is able to suppress totally any ferromagnetic behaviour for odd electron number, provided that the contact potential is strong enough. This tendency is observable for both nearest-neighbour impurities in the 1a and 1b positions.
For second-neighbour impurities (results plotted in Fig. \[fig:N162nd\]), it is observed that the interaction for both 2a and 2b impurity locations \[see Fig. \[fig:N16density\](c)\] is almost always ferromagnetic, regardless of the number of electrons, with an exception of $|\Delta n|=4$. Let us mention that the ferromagnetic coupling is expected for an infinite charge-neutral graphene when considering the impurities belonging to the same sublattice, which is the case for second neighbours (e.g. Ref. ). However, for the nanoflakes, pronounced differences emerge depending on the location of the impurity pair. In the case of the 2a location, the coupling is much enhanced for an odd number of electrons (which can be attributed to the same mechanism as that described for F coupling between nearest neighbours; see the linear dependence of $J^{RKKY}$ on $J$ in Fig. \[fig:N162nd\] for $|\Delta n|=1,3$). For even values of $\Delta n$ the interaction energy is much weaker. The presence of the Hubbard term with $U/t=1$ tends to build up the coupling. The interaction between impurities in the 2b position exhibits traces of vanishing electronic density of the orbital $\mu=8$ \[see Fig. \[fig:N16density\](b)\], in analogy to the situation met for the 1b pair. Here, the interaction is strongly damped for $|\Delta n|\leq 2$, while for $|\Delta n|=3$ we observe enhanced F coupling owing to the mechanism mentioned earlier. What is quite interesting, $|\Delta n|=4$ converts the coupling into an antiferromagnetic one.
In order to comment on the influence of the Hubbard term, we can conclude that its dominant result is to enhance the magnitude of the coupling. The Hubbard term associated energy is lowered when the electronic densities for opposite-spin electrons tend to become unequal, hence acting in hand with the impurity potential in creating an imbalance in spin-up and spin-down electronic densities. This could qualitatively justify why the presence of the Hubbard term increases the magnitude of RKKY coupling, as observed for infinite graphene by Black-Schaffer. [@Annica2] Let us exemplify this for the particular case of the 2b impurity pair. The energy differences between the AF and F states of on-site impurities are presented in Figs. \[fig:N16deltaE2\](b) and \[fig:N16deltaE2\](c), in the absence and in the presence of Hubbard term, respectively. The total energy difference is plotted together with the separated contribution of the orbitals $\mu=1,2,6$, each occupied by a pair of opposite-spin electrons. The orbitals $\mu=1,2$ are characterized by especially high electronic densities on the impurity sites. Here, all the terms are proportional to $J^2$, i.e., present an ordinary perturbational behaviour for low $J$. As is visible, the energy differences are much more pronounced for all orbitals in the presence of $U/t=1$.
Triphenylene-like nanoflake
---------------------------
The electronic structure of the triphenylene-like nanoflake (obtained from diagonalization of the Hamiltonian in the absence of magnetic impurities and for $U/t=0$) is plotted in Fig. \[fig:N18density\](a). What can be observed is that some states acquire additional two-fold degeneracy, in addition to ordinary spin-degeneracy. For charge-neutral nanoflake we have the HOMO-LUMO gap of 1.37 eV. The electronic densities assigned to the distinct states are presented in Fig. \[fig:N16density\](b) for selected orbitals which constitute HOMO orbitals when $|\Delta n|\leq 6$. There, the values of $\left|\gamma^{\mu}_{i,\sigma}\right|^2$ (partial charges present at the lattice sites for a given state) are plotted on the nanoflake scheme. If the selected state is occupied only by a single electron, the distribution of partial charge for this orbital reflects also the spin density.
Figures \[fig:N181st\](a) and \[fig:N181st\](b) show the values of the RKKY coupling between the impurities in two nearest-neighbour positions 1a and 1b \[see Fig. \[fig:N18density\](c)\]. For an undoped case, an antiferromagnetic interaction is always predicted, again in agreement with expectations for the bipartite lattice. The interesting situation arises for the impurity pair 1b, where the doping with $|\Delta n|=1$ switches the interaction sign to the ferromagnetic one. This sign remains robust against further doping, up to $|\Delta n|\leq 3$, and also the magnitude of coupling varies only weakly. Moreover, the linear dependence of F interaction on $J$ is visible. This phenomenon, present for odd $\Delta n$ values, sparks off from the mechanism described earlier, involving orbitals occupied with a single electron. Here, it is visible in Fig. \[fig:N18density\](b) that the degenerate orbital $\mu=8,9$ is characterized by a large electronic density on both 1a impurity sites. Actually, breaking the symmetry of the nanoflake by introducing the magnetic impurities causes that it is most energetically favourable to have large electronic density at both sites of the 1a pair. What is interesting is that the linear dependence of $J^{RKKY}$ on $J$ survives also for $|\Delta n|=2$. Here, it can be attributed to the fact that for the F polarization of impurity spins, the ground state of the system involves both $\mu=8$ and $\mu=9$ orbitals, each occupied by a single electron, instead of just one doubly occupied orbital. Let us note that the electronic densities for degenerate orbitals $\mu=8,9$ fulfill the condition $|\gamma^{8}_{i,\sigma}|^2+|\gamma^{9}_{i,\sigma}|^2=1/9$, while the orbitals are orthogonal. Therefore, if for one of the orbitals the maximum electronic density at 1a pair sites is achieved, then the second orbital is characterized by vanishing electron density at the same sites. As a consequence, in the F state of the impurities, only one of the orbitals gives contribution to the indirect coupling. The situation is thus analogous to the one expected for the total odd number of electrons. The described occasion of having F coupling for $\Delta n$ even is intimately connected with an additional degeneracy of some electronic orbitals.
For the 1b pair, the coupling switches its sign from AF to F each time when $\Delta n$ changes from odd to even. The coupling magnitudes are relatively weak. Ferromagnetic couplings are still linearly dependent on $J$. This time, the degeneracy of the orbital $\mu=8,9$ does not play a role at $\Delta n=2$ and the coupling is AF.
Let us observe that for the magnetic impurities in the 1a position, the presence of the Hubbard term with $U/t=1$ tends to enhance the coupling magnitude, conserving its sign. On the other hand, for the 1b pair, the Hubbard term strongly pushes the interaction toward antiferromagnetic for low $\Delta n$.
The RKKY exchange integrals calculated for second-neighbour magnetic impurities \[in position 2a, 2b; see Fig. \[fig:N18density\](c)\] are presented in Fig. \[fig:N182nd\]. For 2a placement of the impurities, the interaction is almost always ferromagnetic, but its magnitude is significantly enhanced for odd values of $\Delta n$. For the 2b pair of impurities, the coupling is particularly strong (and ferromagnetic) for $|\Delta n|=1,2,3$, which can be attributed to the same mechanism as described previously in the case of the 1b pair.
Concluding remarks
==================
We have performed a tight-binding based study of RKKY interaction in two graphene nanoflakes, different in their geometries but both containing four hexagonal rings. We focused on the influence of possible charge doping on the magnitude and sign of an indirect interaction. We also incorporated a Hubbard term in the Hamiltonian to estimate the effect of prototypic Coulomb interaction. In general, we found a pronounced dependence of the RKKY coupling integrals on the location of the pair of on-site magnetic moments in the nanostructure.
In relation to the odd number of charge carriers in the nanoflakes, we observed a specific contribution to indirect interaction, originating from the highest-energy electronic orbital occupied by a single electron and giving rise to nonzero spin distributed over the structure. This contribution is proportional to the magnitude of the contact potential $J$ and results from an uncompensated first-order correction to the orbital energy due to the presence of a pair of ferromagnetically polarized impurities. This mechanism, which may be regarded as somewhat similar to double exchange, always produces ferromagnetic contribution to indirect exchange and for sufficiently low $J$ it can dominate over the typical contribution proportional to $J^2$ originating from second-order perturbational correction to the energy (being the clue of the ordinary RKKY interaction). Such a ferromagnetic contribution can either change the coupling sign from AF to F for nearest-neighbour impurities or enhance the ferromagnetic coupling between second neighbours, as shown for two types of nanoflakes. The details of the mechanism depend also on the presence or absence of degeneracies in energy spectrum of the nanoflake.
The mechanism leading to the indirect ferromagnetic coupling linearly proportional to $|J|$ might be potentially useful, since it allows switching from AF coupling characteristic of nearest-neighbour impurities for charge neutrality to F coupling by adding or removing a single electron from the nanoflake. Especially advantageous conditions for such a situation occur for a triphenylene-like nanoflake for impurities in the 1a position. What is more, adding one or two further charge carriers does not alter the coupling sign in these particular situations.
The influence of the Hubbard term is mainly to enhance the magnitude of the coupling, as noted in Ref. for infinite graphene. Unequal electronic densities for opposite-spin electrons lower the energy coming from the Hubbard term, and the presence of the local spin-dependent impurity potential acts in the same direction. Therefore, the Hubbard term tends to increase the interaction energy in general. On the other hand, the presence of additional long-range repulsion between the electrons residing on different sites should qualitatively tend to compensate the effect of on-site repulsion. This effect could be captured in treating the $U$ parameter not necessarily as a true on-site energy, but rather in the spirit of the effective parameter (as mentioned before, in connection with Ref. )). Sometimes other changes are detected, which can be attributed to the fact that the presence of the Hubbard term itself causes some redistribution of the charge densities over lattice sites with respect to the case when the Coulombic correlations are neglected. These changes also modify the indirect coupling. The application of the MFA approximation to the Hubbard model appears justified for the ground state. [@Nolting] However, development of the more accurate approaches (albeit consuming less resources than exact diagonalization) would be valuable with a view to the studies of temperature properties of the nanoflakes.
The present calculations are performed for $T=0$, exploiting ground-state properties of the system. However, in the ultrasmall, molecule-like structures, the separation of the discrete electronic energy levels is usually of the order of tenths of $t$, where $t=2.8$ eV \[see Fig. \[fig:N16density\](c) and \[fig:N18density\](c)\]. Therefore, it appears that no significant redistribution of the charge carriers between the states can happen due to thermal excitations up to the temperatures of interest (i.e., room temperature). As a consequence, the results for coupling energies appear valid also for nonzero temperatures. Perhaps some finite temperature modifications can be expected in the presence of degenerate spectrum, as the impurity-induced energy splitting between the otherwise degenerate states is considerably small (note the interesting thermodynamics of the undoped metallic-like nanoflakes with degenerate zero-energy states, discussed by Ezawa ).
In general, we find an immense influence of the electronic structure of the nanostructures on the properties of RKKY interaction, being dependent on the wavefunctions of single orbitals, which are very strongly shaped by the nanoflake geometry. This distinguishes our case from the case of an infinite system, where propagating states are involved, and might open the door for designing the appropriate structures to guarantee the expected indirect coupling features.
The author is deeply grateful to T. Balcerzak for critical reading of the manuscript.
The numerical calculations have been performed partly with the Wolfram Mathematica 8.0.1 software[@Wolfram1] and partly using the *LAPACK* package[@laug].
This work has been supported by Polish Ministry of Science and Higher Education by a special purpose grant to fund the research and development activities and tasks associated with them, serving the development of young scientists and doctoral students.
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---
author:
- 'Christian Forster, Luca Carlone, Frank Dellaert, Davide Scaramuzza[^1]'
bibliography:
- 'main.bib'
title: 'On-Manifold Preintegration for Real-Time Visual-Inertial Odometry'
---
This paper has been accepted for publication in *IEEE Transactions on Robotics*.\
DOI: [10.1109/TRO.2016.2597321](http://dx.doi.org/10.1109/TRO.2016.2597321)\
IEEE Explore: <http://ieeexplore.ieee.org/document/7557075/>\
Please cite the paper as:\
Christian Forster, Luca Carlone, Frank Dellaert, Davide Scaramuzza,\
“On-Manifold Preintegration for Real-Time Visual-Inertial Odometry”,\
in *IEEE Transactions on Robotics*, 2016.\
Supplementary Material {#supplementary-material .unnumbered}
======================
- Video of the experiments: [\\videolink](\videolink)
- Source-code for preintegrated IMU and structureless vision factors <https://bitbucket.org/gtborg/gtsam>.
Appendix {#appendix .unnumbered}
========
[Christian Forster]{} (1986, Swiss) obtained his Ph.D. in Computer Science (2016) at the University of Zurich under the supervision of Davide Scaramuzza. Previously, he received a B.Sc. degree in Mechanical Engineering (2009) and a M.Sc. degree in Robotics, Systems and Control (2012) at ETH Zurich, Switzerland. In 2011, he was a visiting researcher at CSIR (South Africa), and in 2014 at Georgia Tech in the group of Frank Dellaert. He is broadly interested in developing real-time computer vision algorithms that enable robots to perceive the three dimensional environment.
[Luca Carlone]{} is a research scientist in the Laboratory for Information and Decision Systems at the Massachusetts Institute of Technology. Before joining MIT, he was a postdoctoral fellow at Georgia Tech (2013-2015), a visiting researcher at the University of California Santa Barbara (2011), and a visiting researcher at the University of Zaragoza (2010). He got his Ph.D. from Politecnico di Torino, Italy, in 2012. His research interests include nonlinear estimation, optimization, and control applied to robotics, with special focus on localization, mapping, and decision making for navigation in single and multi robot systems.
[Frank Dellaert]{} is Professor in the School of Interactive Computing at Georgia Tech. He graduated from Carnegie Mellon in 2001 with a Ph.D. in Computer Science, after obtaining degrees from Case Western Reserve University and K.U. Leuven before that. Professor Dellaert’s research focuses on large-scale inference for autonomous robot systems, on land, air, and in water. He pioneered the use of several probabilistic methods in both computer vision and robotics. With Dieter Fox and Sebastian Thrun, he has introduced the Monte Carlo localization method for estimating and tracking the pose of robots, which is now a standard and popular tool in mobile robotics. Most recently, he has investigated 3D reconstruction in large-scale environments by taking a graph-theoretic view, and pioneered the use of sparse factor graphs as a representation in robot navigation and mapping. He and his group released an open-source large-scale optimization toolbox, GTSAM.
[Davide Scaramuzza]{} (1980, Italian) is Professor of Robotics at the University of Zurich, where he does research at the intersection of robotics, computer vision, and neuroscience. He did his PhD in robotics and computer Vision at ETH Zurich and a postdoc at the University of Pennsylvania. From 2009 to 2012, he led the European project sFly, which introduced the world’s first autonomous navigation of micro drones in GPS-denied environments using visual-inertial sensors as the only sensor modality. For his research contributions, he was awarded an SNSF-ERC Starting Grant, the IEEE Robotics and Automation Early Career Award, and a Google Research Award. He coauthored the book Introduction to Autonomous Mobile Robots (published by MIT Press).
[^1]: C.Forster is with the Robotics and Perception Group, University of Zurich, Switzerland. E-mail: [[email protected]]{}
L.Carlone is with the Laboratory for Information & Decision Systems, Massachusetts Institute of Technology, USA. E-mail: [[email protected]]{}
F.Dellaert is with the College of Computing, Georgia Institute of Technology, USA. E-mail: [[email protected]]{}
D.Scaramuzza is with the Robotics and Perception Group, University of Zurich, Switzerland. E-mail: [[email protected]]{}
This research was partially funded by the Swiss National Foundation (project number 200021-143607, “Swarm of Flying Cameras”), the National Center of Competence in Research Robotics (NCCR), the UZH Forschungskredit, the NSF Award 11115678, and the USDA NIFA Award GEOW-2014-09160.
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author:
- 'J. C. Guella[^1] , V. A. Menegatto &A. P. Peron'
title: '**Strictly positive definite kernels on a product of circles**'
---
[**Key words and phrases:**]{} strictly positive definite kernels, isotropy, product of circles, Schoenberg theorem, Skolem-Mahler-Lech theorem.\
[**2010 Math. Subj. Class.:**]{} 33C50; 33C55; 42A16; 42A32; 42A82; 42B05; 43A35.
Introduction
============
Positive definite functions and kernels have a long history in mathematics, entering as an important tool in harmonic analysis and other areas as well. In the spherical setting, they can be traced back to the remarkable paper of I. J. Schoenberg published in 1942 ([@schoen]), where a characterization for the continuous, isotropic and positive definite kernels on a single sphere was obtained. This characterization is far-reaching, having applications in approximation theory, spatial statistics, geomathematics, discrete geometry, etc. We mention [@cheney; @dai; @gneiting; @musin] and references therein for some applications of positive definite functions and kernels on spheres.
In this paper, we will be concerned with positive definite kernels on a product of circles. As so, we will recast the basic concepts and results from Schoenberg’s work that applies to circles, up to the point we can state what the main contribution in the present paper is.
We will write $S^1$ to denote the unit circle in $\mathbb{R}^{2}$. Continuity of a kernel $K$ on $S^1$ will be attached to the usual geodesic distance on $S^1$ and that will be extended to the product $S^1 \times S^1$ in the usual way. The [*isotropy or radiality*]{} of a kernel $K$ on $S^1$ refers to the existence of a function $K_r$ on $[-1,1]$ so that $$K(x,y)=K_r(x \cdot y), \quad x,y \in S^1,$$ in which $\cdot$ is the usual inner product of $\mathbb{R}^2$. For a kernel $K$ on $S^1 \times S^1$, isotropy corresponds to the property $$K((x,z),(y,w))=K_r(x \cdot y, z\cdot w), \quad x,y,z,w \in S^1,$$ in which the function $K_r$ has now domain $[-1,1]^2$. In both cases, we will call $K_r$ the [*isotropic part*]{} of the kernel $K$. Finally, the [*positive definiteness*]{} of a real kernel $K$ on an infinite set $X$ refers to the validity of the inequality $$\sum_{\mu,\nu=1}^n c_\mu c_\nu K(x_\mu, x_\nu) \geq 0,$$ whenever $n$ is a positive integer, $x_1, x_2, \ldots, x_n$ are distinct points on $X$ and the $c_\mu$ are real scalars. The [*strict positive definiteness*]{} of $K$ demands both, its positive definiteness and that the inequalities above be strict whenever at least one of the $c_\mu$ is nonzero. We will apply these definitions to the cases in which either $X=S^1$ or $X=S^1 \times S^1$.
According to a result of Schoenberg in [@schoen], a real, continuous and isotropic kernel $K$ on $S^1$ is positive definite if, and only if, the isotropic part $K_r$ of $K$ has the form $$K_r(t)=\sum_{k=0}^\infty a_k P_k(t), \quad t \in [-1,1],$$ in which all the $a_k$ are nonnegative, $P_k$ is the Tchebyshev polynomial (of first kind) of degree $k$ (see [@szego]), and $\sum_{k=0}^\infty a_k P_k(1)<\infty$. In the nineties, many attempts were made in order to deduce a similar characterization for the strict case ([@ron; @sun]) but that only appeared in [@mene] (see also [@barbosa]): a kernel having Schoenberg’s representation described above is strictly positive definite on $S^1$ if, and only if, the set $\{k:a_{|k|}>0\}$ intersects every full arithmetic progression in $\mathbb{Z}$. Both results described above extends to the complex setting, that is, to the case in which $S^1$ is replaced with the unit circle in $\mathbb{C}$, the positive definite kernel is allowed to assume complex values and the scalars $c_\mu$ are now complex numbers. This extension is also discussed in [@mene].
Moving to $S^1 \times S^1$, a theorem proved in [@jean] includes a characterization for the positive definiteness of a real, continuous and isotropic kernel $K$ on $S^1 \times S^1$ as those having an isotropic part in the form $$\label{repre}
K_r(t,s)=\sum_{k,l=0}^\infty a_{k,l} P_k(t)P_l(s), \quad t,s \in [-1,1],$$ where all the coefficients $a_{k,l}$ are nonnegative and $\sum_{k,l=0}^\infty a_{k,l} P_k(1)P_l(1)<\infty$. The results in this paper will converge to the following characterization for strict positive definiteness on $S^1 \times S^1$.
\[mainintro\] Let $K$ be a real, continuous, isotropic and positive definite kernel on $S^1 \times S^1$. It is strictly positive definite if, and only if, the set $\{(k,l): a_{|k|,|l|} >0\}$ extracted from the series representation for the isotropic part $K_r$ of $K$ intersects all the translations of every lattice of $\mathbb{Z}^2$.
The paper proceeds as follows. In Section 2, we present a technical result that leads to an alternative formulation for the concept of strict positive definiteness and use it to ratify the necessity part of Theorem \[mainintro\]. In Section 3, after presenting a series of three technical results, we finally state and prove a result that not only includes Theorem \[mainintro\] but also provides an alternative characterization for the strict positive definiteness on $S^1 \times S^1$. In Section 4, we provide a different path one could follow in order to simplify the proof of Theorem \[repre\]. In Section 5, we indicate how the main results of the paper can be extended to the complex setting.
Strict positive definiteness on $S^1 \times S^1$
================================================
This section is divided in two parts: in the first one we explore a little bit deeper the concept of strict positive definiteness on $S^1 \times S^1$. The outcome Proposition \[prop-equiv cAc=0\] implies an obvious equivalence for that concept. In the second part, we use this alternative description in order to obtain a proof for the necessity half of Theorem \[mainintro\].
For a function $K_r$ representable as in (\[repre\]), we will write $$J_K:=\{(k,l)\in\mathbb{Z}_+^2: a_{k,l}>0\}.$$ It is an easy matter to verify that the strict positive definiteness of the kernel $K$ with isotropic part given by (\[repre\]) depends upon $J_K$ only and not on the actual values of the Fourier coefficients $a_{k,l}$.For distinct points $(x_1,w_1),(x_2,w_2), \ldots,(x_n,w_n)$ on $S^1 \times S^1$, we will write $A=(A_{\mu\nu})$, in which $$A_{\mu\nu}=K_r(x_\mu \cdot x_\nu, w_\mu \cdot w_\nu), \quad \mu,\nu=1,2,\ldots,n.$$ We will also represent the points above in polar form: $$x_\mu=(\cos \t_\mu, \sin \t_\mu), \quad w_\mu=(\cos \p_\mu, \sin \p_\mu), \quad \t_\mu,\p_\mu \in [0,2\pi),\quad \mu=1,2,\ldots,n.$$
\[prop-equiv cAc=0\] Let $K$ be a nonzero, real, continuous, isotropic and positive definite kernel on $S^1 \times S^1$. For distinct points $(x_1,w_1),(x_2,w_2), \ldots,(x_n,w_n)$ on $S^1 \times S^1$ and a column vector $c=(c_\mu)$ in $\mathbb{R}^n$, the following statements are equivalent:\
$(i)$ $c^{t}Ac=0$;\
$(ii)$ The double equality $$\sum_{\mu=1}^{n}c_{\mu}e^{ik\t_{\mu}}e^{il\p_{\mu}}=\sum_{\mu=1}^{n}c_{\mu}e^{ik\t_{\mu}}e^{-il\p_{\mu}}=0$$ holds for all $(k,l)\in J_K$.
The normalization one decides to adopt for the Tchebyshev polynomials is of no importance in this paper. So, we will write $$P_k(x_\mu \cdot x_\nu)=\frac{2}{k} \cos k(\t_\mu -\t_\nu), \quad k>0, \quad x_\mu \in S^1, \quad \mu=1,2,\ldots,n,$$ while $P_0(x_\mu \cdot x_\nu)=1$, $\mu=1,2,\ldots,n$. Now, computing each expression $$\sum_{\mu,\nu=1}^{n} c_\mu c_\nu P_k(x_\mu \cdot x_\nu)P_l(w_\mu \cdot w_\nu)$$ in the cases when both $k$ and $l$ are nonzero and when at least one of them is zero, it is not hard to see that the equality $c^{t}Ac=0$ corresponds to $$\left|\sum_{\mu=1}^{n}c_{\mu}e^{ik\t_{\mu}}e^{il\p_{\mu}}\right|^2+
\left|\sum_{\mu=1}^{n}c_{\mu}e^{ik\t_{\mu}}e^{-il\p_{\mu}}\right|^2=0, \quad (k,l)\in J_K.$$ But that is equivalent to the statement in $(ii)$.
From now on, we will deal with subgroups of $\mathbb{Z}^2$ and their translations. The subgroups of $\mathbb{Z}^2$ can be classified as follows.
\[class\] A nontrivial subgroup of $\mathbb{Z}^2$ belongs to one of the following categories:\
$(i)$ $(0,b)\mathbb{Z}:=\{(0,pb): p \in \mathbb{Z}\}$, $b>0$;\
$(ii)$ $(a,b)\mathbb{Z}:=\{(pa, pb): p \in \mathbb{Z}\}$, $a>0$;\
$(iii)$ $(a,b)\mathbb{Z}+(0,d)\mathbb{Z}:=\{(pa,pb+qd): p,q \in \mathbb{Z}\}$, $a,d>0$.
We advise the reader that there are different ways to describe the subgroups of $\mathbb{Z}^2$ (for instance, the one presented in [@pinkus] is slightly different and quite elegant). The subgroups that fit into Lemma \[class\]-$(iii)$ will be called [*lattices*]{}. The set of lattices of $\mathbb{Z}^2$ encompasses all the subgroups of rank 2. If $ad=1$, then a lattice becomes the whole $\mathbb{Z}^2$, otherwise it is a proper subgroup of $\mathbb{Z}^2$. The lattices having the form $$(a\mathbb{Z},b\mathbb{Z}):=\{(pa, qb): q,p\in \mathbb{Z}\},\quad a,b>0,$$ will be called [*rectangular lattices*]{} of $\mathbb{Z}^2$. By [*translates of subgroups*]{} of $\mathbb{Z}^2$, we will mean sets of the form $(j,j')+S$, in which $(j,j')$ is a fixed element of $\mathbb{Z}^2$ and $S$ is a subgroup of $\mathbb{Z}^2$.
The next result is very close to the necessity part of Theorem \[mainintro\].
\[juru\] Let $K$ be a real, continuous, isotropic and positive definite kernel on $S^1 \times S^1$.If $K$ is strictly positive definite, then the set $\{(k,l): (|k|,|l|)\in J_K\}$ intersects all the translations of each rectangular lattice of $\mathbb{Z}^2$.
Assume $K$ is strictly positive definite and write $S=(a\mathbb{Z}, b\mathbb{Z})$ with $a,b>0$. We will show that $$\{(k,l): (|k|,|l|)\in J_K\}$$ intersects $(j,j')+S$, whenever $j\in \{0,1,\ldots, a-1\}$ and $j'\in \{0,1,\ldots, b-1\}$. There is nothing to prove if $a=b=1$. In the other cases, we will assume that $$\{(k,l): (|k|,|l|)\in J_K\} \cap (j+a\mathbb{Z}, j'+b\mathbb{Z})=\emptyset,$$ and will reach a contradiction. In the case in which $a=1$ and $b\geq 2$, the assumption on $J_K$ implies that $l-j',-l-j' \not \in b\mathbb{Z}$, whenever $(k,l) \in J_K$. In particular, $$\sum_{\mu=1}^b\left(e^{i2\pi \mu /b}\right)^{l-j'}=\sum_{\mu=1}^b \left(e^{i2\pi \mu /b}\right)^{-l-j'}=0,$$ and, consequently, $$\sum_{\mu=1}^b \left[\mbox{Re\,}\left(e^{i2\pi \mu j'/b}\right)\right] \left(e^{i2\pi \mu/b}\right)^l=0, \quad (k,l) \in J_K.$$ The real scalars $c_\mu:=\mbox{Re\,}(e^{i2\pi \mu j'/b})$, $\mu=1,2,\ldots,b$, are not all zero and the points $$(x_\mu,w_\mu)=(1, e^{i2\pi \mu/b}), \quad \mu=1,2,\ldots,b,$$ are distinct in $S^1\times S^1$. Thus, under the light of Proposition \[prop-equiv cAc=0\], we have a contradiction with the strict positive definiteness of $K$. The case in which $a\geq 2$ and $b=1$ is similar. To conclude the proof, we now assume $a,b\geq 2$ and adapt the procedure employed in the first case. If $(k,l) \not \in (j+a\mathbb{Z}, j'+b\mathbb{Z})$, then either $k-j \not \in a\mathbb{Z}$ or $l-j' \not \in b\mathbb{Z}$. Hence, we may conclude that $$\sum_{\mu=1}^a \left(e^{i2\pi \mu /a}\right)^{k-j} \sum_{\nu=1}^b\left(e^{i2\pi \nu /b}\right)^{l-j'}=0,$$ that is, $$\sum_{\mu=1}^a \sum_{\nu=1}^b e^{-i2\pi \mu j /a}e^{-i2\pi \nu j'/b} \left(e^{i2\pi \mu /a}\right)^k \left(e^{i2\pi \nu /b}\right)^l=0.$$ Repeating the argument with the assumption $(-k,-l) \not \in (j+a\mathbb{Z}, j'+b\mathbb{Z}),$ we conclude that $$\sum_{\mu=1}^a \sum_{\nu=1}^b e^{i2\pi \mu j /a}e^{i2\pi \nu j'/b} \left(e^{i2\pi \mu /a}\right)^k \left(e^{i2\pi \nu /b}\right)^l=0.$$ Thus, since $(k,l)$ is arbitrary, $$\sum_{\mu=1}^a \sum_{\nu=1}^b \left[\mbox{Re\,}\left(e^{i2\pi \mu j /a}e^{i2\pi \nu j'/b}\right)\right] \left(e^{i2\pi \mu /a}\right)^k \left(e^{i2\pi \nu /b}\right)^l=0, \quad (k,l) \in J_K.$$ By an analogous procedure, now taking into account that $(-k,l),(k,-l) \not \in (j+a\mathbb{Z}, j'+b\mathbb{Z})$, the conclusion is $$\sum_{\mu=1}^a \sum_{\nu=1}^b \left[\mbox{Re\,}\left(e^{i2\pi \mu j /a}e^{i2\pi \nu j'/b}\right)\right] \left(e^{i2\pi \mu /a}\right)^k \left(e^{-i2\pi \nu /b}\right)^l=0, \quad (k,l) \in J_K.$$ Therefore, since the numbers $\mbox{Re\,}\left[e^{i2\pi \mu j /a}e^{i2\pi \nu j'/b}\right]$ are not all zero and the $ab$ points $$(x_\mu,w_\nu)=(e^{i2\pi \mu/a}, e^{i2\pi \nu/b}), \quad \mu=1,2,\ldots,a,\quad \nu=1,2,\ldots, b,$$ are distinct in $S^1 \times S^1$, we have reached a contradiction once again.
In order to ratify the necessity part of Theorem \[mainintro\], the following lemma becomes handy.
\[tech\] The lattice $L=(a,b)\mathbb{Z}+(0,d)\mathbb{Z}$, $a,d>0$, can be decomposed in the form $$L=\bigcup_{(j,j')\in A}\left[(j,j')+(ad\mathbb{Z}, ad\mathbb{Z})\right],$$ in which $A=L\cap \{(\alpha,\beta) \in \mathbb{Z}^2: 0\leq \alpha,\beta < ad\}$.
For $p,q \in \mathbb{Z}$, we can certainly write $$(pa,pb+qd)=(j+\alpha ad, j'+ \beta ad),$$ in which $j,j'\in \{0,1,\ldots,ad-1\}$. Since $$(j,j')=(p-\alpha d)(a,b)+(\alpha b +q-\beta a)(0,d),$$ it is clear that $(j,j') \in L$. These arguments show that $L$ is a subset of the union quoted in the statement of the lemma. As for the reverse inclusion, first observe that if $\alpha,\beta \in \mathbb{Z}$, we have that $$(\alpha ad,\beta ad)=d\alpha (a, b) +(a\beta-b\alpha)(0,d) \in L.$$ Since $L$ is a subgroup of $\mathbb{Z}^2$, $(j,j')+(\alpha ad,\beta ad) \in L$ whenever $(j,j')\in A$.
The theorem below is now evident.
\[main2\] Let $K$ be a real, continuous, isotropic and positive definite kernel on $S^1 \times S^1$. If $K$ is strictly positive definite, then the set $\{(k,l): (|k|,|l|)\in J_K\}$ intersects all the translations of each lattice in $\mathbb{Z}^2$.
Sufficiency via M. Laurent’s theorem
====================================
In this section, we will prove that the necessary conditions for the strict positive definiteness of a continuous, isotropic and positive definite kernel on $S^1 \times S^1$ presented in Theorem \[juru\] are also sufficient.
We begin recalling an elementary bi-dimensional version of the Skolem-Mahler-Lech Theorem due to M. Laurent ([@laurent; @laurent1]). The original Skolem-Mahler-Lech Theorem is discussed in details in [@everest]. This very same bi-dimensional version was used in [@pinkus] in order to characterize certain strictly positive definite kernels on complex Hilbert spaces.
\[laurent\] Let $\{(x_1,w_1), (x_2,w_2), \ldots, (x_n,w_n)\}$ be a subset of $\mathbb{C}^2$. For $n$ complex numbers $c_1,c_2, \ldots, c_n$, define a double sequence $\{b_{k,l}:k,l \in \mathbb{Z}\}$ through the formula $$b_{k,l}:=\sum_{\mu=1}^{n}c_\mu x_\mu^k w_\mu^l, \quad k,l\in \mathbb{Z}.$$ Then, the set $\{(k,l): b_{k,l}=0\}$ is the union of a finite number of translates of subgroups of $\mathbb{Z}^2$.
The technical lemma below adds to Theorem \[laurent\] when the points are distinct and belong to $\Omega_2 \times \Omega_2$, in which $\Omega_2$ is the unit circle in $\mathbb{C}$.
\[notallzero\] Let $(x_1,w_1), (x_2,w_2), \ldots, (x_n,w_n)$ be distinct points in $\Omega_2 \times \Omega_2$. For complex number $c_1,c_2, \ldots, c_n$, define $$b_{k,l}=\sum_{\mu=1}^{n}c_\mu x_\mu^k w_\mu^l, \quad k,l \in \mathbb{Z}.$$ If $\{(k,l): b_{k,l}=0\}=\mathbb{Z}^2$, then all the $c_\mu$ are zero.
We will write the components of the points in polar form $x_\mu=e^{i\t_\mu}$, $w_\mu=e^{i\p_\mu}$, $\mu=1,2,\ldots,n$, and will assume, as we can, that the $n$ points $(\t_1,\p_1), (\t_2, \p_2), \ldots, (\t_n, \p_n)$ are distinct in $[0,2\pi)^2$. Choose $\alpha,\beta \in \mathbb{Z}$ in such a way that all the elements in the set $$\left\{\alpha\frac{\t_\mu-\t_\nu}{2\pi}+\beta \frac{\p_\mu-\p_\nu}{2\pi}: \mu,\nu=1,2,\ldots,n; \mu \neq \nu\right\}$$ are nonzero. Next, pick $\gamma\in \mathbb{Z}_+$ arbitrarily large so that $$\left\{\frac{\alpha}{\gamma}\frac{\t_\mu-\t_\nu}{2\pi}+\frac{\beta}{\gamma} \frac{\p_\mu-\p_\nu}{2\pi}: \mu,\nu=1,2,\ldots,n; \mu \neq \nu\right\}\subset (-1,1)\setminus \{0\}.$$ For each pair $(\mu,\nu)$, $\mu \neq \nu$, for which $$\frac{\alpha}{\gamma} \frac{\t_\mu-\t_\nu}{2\pi}+\frac{\beta}{\gamma} \frac{\p_\mu-\p_\nu}{2\pi} \in \mathbb{Q},$$ let $p_{\mu\nu}$ be a positive integer $>\gamma$ satisfying $$p_{\mu\nu}\left(\frac{\alpha}{\gamma} \frac{\t_\mu-\t_\nu}{2\pi}+\frac{\beta}{\gamma} \frac{\p_\mu-\p_\nu}{2\pi} \right)\in \mathbb{Z}.$$ Finally, select an integer $q$ so that $q$ is greater then all the $p_{\mu\nu}$ and each set $\{q,p_{\mu\nu}\}$ is coprime. If $\{(k,l): b_{k,l}=0\}=\mathbb{Z}^2$, then we may infer that $$\sum_{\mu=1}^n c_\mu e^{i\t_\mu k}e^{i\p_\mu l}=0, \quad (k,l)=(0,0),(\alpha q,\beta q),(2\alpha q,2\beta q),\ldots, ((n-1)\alpha q,(n-1)\beta q).$$ The matrix of the system above has $\mu\nu$-entries given by $$\left[e^{i(\alpha \t_\mu +\beta \p_\mu)q}\right]^{\nu},\quad \mu,\nu=1,2,\ldots,n,$$ and, consequently, it is a Vandermonde matrix. So, the proof of the lemma will be complete as long as we show that the $n$ points $e^{i(\alpha \t_\mu +\beta \p_\mu)q}$, $\mu=1,2,\ldots, n$, are distinct. But, for $\mu \neq \nu$, $$e^{i(\alpha \t_\mu +\beta \p_\mu)q}= e^{i(\alpha \t_\nu +\beta \p_\nu)q}$$ if, and only if, $$q \left(\alpha\frac{\t_\mu -\t_\nu}{2\pi} +\beta\frac{\p_\mu -\p_\nu}{2\pi}\right)\in \mathbb{Z}.$$ If all the numbers $$\left(\alpha\frac{\t_\mu -\t_\nu}{2\pi} +\beta\frac{\p_\mu -\p_\nu}{2\pi}\right),\quad \mu \neq \nu,$$ are irrational, we are done. Otherwise, there would be integers $j$ and $j'$ such that $$\frac{j}{\gamma q}=\frac{ j'}{p_{\mu\nu}}\in (-1,1)\setminus \{0\}.$$ for some pair $(\mu,\nu)$, $\mu \neq \nu$. Since $\{q,p_{\mu\nu}\}$ is coprime, then $p_{\mu\nu}$ would divide $\gamma$, contradicting our choice of $p_{\mu\nu}$.
The next result reveals that if a proper subset $A$ of $\mathbb{Z}^2$ is a finite union of translates of subgroups of $\mathbb{Z}^2$, then there exists a rectangular lattice $H$ of $\mathbb{Z}^2$ and $(j,j') \in \mathbb{Z}^2$ so that $[(j,j')+H]\cap A=\emptyset$.
\[elattice\] Let $A$ be a proper subset of $\mathbb{Z}^2$. If $A$ is a finite union of translates of subgroups of $\mathbb{Z}^2$ and $(j,j')\in \mathbb{Z}^2\setminus A$, then there exists a rectangular lattice $H$ of $\mathbb{Z}^2$ such that $(j,j')+H \subset \mathbb{Z}^2\setminus A$.
If $A$ is a finite union of translates of subgroups of $\mathbb{Z}^2$, we can write $$A=F \cup [(j_1,j_1') +G_1] \cup [(j_2,j_2') +G_2] \cup \cdots \cup [(j_r,j_r') +G_r]$$ in which $F$ is a finite (possibly empty) subset of $\mathbb{Z}^2$, $(j_1,j_1'), (j_2,j_2'), \ldots, (j_r,j_r')\in \mathbb{Z}^2$ and $G_1, G_2, \ldots, G_r$ are nontrivial subgroups of $\mathbb{Z}^2$. It suffices to prove the lemma in the case in which $F=\emptyset$. Indeed, if a solution $(j,j')+H$ is available for that case, we can pick a convenient subgroup $H_1$ of $H$ so that $(j,j')+H_1$ avoids all the elements of $F$. So, assume that $F=\emptyset$ and fix $(j,j') \in \mathbb{Z}^2 \setminus A$. We can assume all the $G_i$ have rank 2. Indeed, if $G_i$ has rank 1 for some $i$, we can pick $(\alpha,\beta) \in \mathbb{Z}^2$ such that $$(\alpha , \beta)\mathbb{Z} \cap [(j_i-j,j_i'-j')+G_i]=\emptyset.$$ Hence, $$[(j,j')+(\alpha , \beta)\mathbb{Z}] \cap [(j_i,j_i')+G_i]=\emptyset,$$ and, therefore, $$(j,j') \not \in (j_i,j_i')+(\alpha , \beta)\mathbb{Z} +G_i.$$ In particular, $(\alpha , \beta)\mathbb{Z} +G_i$ is a subgroup of rank 2 and we can replace $(j_i,j_i)+G_i$ with $(j_i,j_i')+(\alpha , \beta)\mathbb{Z} +G_i$ in the union decomposition for $A$ keeping $(j,j')$ in $\mathbb{Z}^2\setminus A$. If all the $G_i$ have rank 2, the proof of the lemma proceeds as follows. Let $m_i$ be the index of $G_i$ in $\mathbb{Z}^2$, $i=1,2,\ldots,r$, and pick a common multiple $m$ of all the $m_i$. The subgroup $(m\mathbb{Z},m\mathbb{Z})$ is a rectangular lattice and, by the definition of index of a subgroup, it follows that $$(m\mathbb{Z},m\mathbb{Z}) \subset G_i \quad i=1,2,\ldots,r.$$ In particular, $$[(j,j')+(m\mathbb{Z},m\mathbb{Z})]\cap G_i=\emptyset, \quad i=1,2,\ldots,r,$$ and, consequently, $(j,j')+(m\mathbb{Z},m\mathbb{Z}) \subset \mathbb{Z}^2\setminus A$.
Next, we prove a refinement of the sufficiency part of Theorem \[mainintro\].
\[sufic\] Let $K$ be a real, continuous, isotropic and positive definite kernel on $S^1 \times S^1$. If $\{(k,l): (|k|,|l|)\in J_K\}$ intersects all the translations of each rectangular lattice of $\mathbb{Z}^2$, then $K$ is strictly positive definite.
Let us assume that $\{(k,l): (|k|,|l|)\in J_K\}$ intersects all the translations of each rectangular lattice of $\mathbb{Z}^2$. For a fixed $n\geq 2$, $n$ distinct points $(\t_1,\p_1), (\t_2, \p_2), \ldots, (\t_n, \p_n)$ in $[0,2\pi)^2$ and $c_1, c_2, \ldots, c_n$ real numbers, not all zero, we intend to show that either $\sum_{\mu=1}^n c_\mu e^{i\t_\mu k}e^{-i\p_\mu l} \neq 0$ or $\sum_{\mu=1}^n c_\mu e^{i\t_\mu k}e^{i\p_\mu l}\neq 0$, for some $(k,l) \in J_K$. A help of Proposition \[prop-equiv cAc=0\] will lead to the strict positive definiteness of $K$. In order to achieve the conclusion mentioned above, define $$b_{k,l}=\sum_{\mu=1}^n c_\mu e^{i\t_\mu k}e^{i\p_\mu l}, \quad k,l\in \mathbb{Z}.$$ On one hand, Lemma \[notallzero\] and the fact that at least one $c_\mu$ is nonzero imply that $\{(k,l): b_{k,l}=0\}\neq \mathbb{Z}^2$.Theorem \[laurent\] asserts that $\{(k,l): b_{k,l}=0\}$ is the union of a finite number of translations of subgroups of $\mathbb{Z}^2$ while Lemma \[elattice\] guarantees the existence of a rectangular lattice of $\mathbb{Z}^2$, a translation of which belongs to $\mathbb{Z}^2 \setminus \{(k,l): b_{k,l}=0\}$. Thus, due to our assumption on $\{(k,l) : (|k|,|l|) \in J_K\}$, we immediately have that $$\{(k,l) : (|k|,|l|) \in J_K\}\not \subset \{(k,l): b_{k,l}=0\}.$$ Therefore, there must exist at least one pair $(k,l)$ in $\{(k,l): (|k|,|l|)\in J_K\}$ for which $$\sum_{\mu=1}^n c_\mu e^{i\t_\mu k}e^{i\p_\mu l}\neq 0.$$ Since the $c_\mu$ are real, the result follows.
The following characterizations are now evident.
Let $K$ be a real, continuous, isotropic and positive definite kernel on $S^1 \times S^1$. The following assertions are equivalent:\
$(i)$ $K$ is strictly positive definite;\
$(ii)$ $\{(k,l): (|k|,|l|)\in J_K\}$ intersects all the translations of each lattice in $\mathbb{Z}^2$;\
$(iii)$ $\{(k,l): (|k|,|l|)\in J_K\}$ intersects all the translations of each rectangular lattice in $\mathbb{Z}^2$.
To conclude the section, we present an alternative guise for the previous characterizations. That will require the following technical, but elementary lemma.
Let $K$ be a subset of $\mathbb{Z}^2$ that intersects all the translations of each lattice in $\mathbb{Z}^2$. Then, every such intersection is an infinite set.
Let $L=(j,j')+(a,b)\mathbb{Z}+(0,d)\mathbb{Z}$, $a,d>0$, and assume that $K\cap L$ is finite. Write $$(j+p_1a,j'+p_1b+q_1d), (j+p_2a,j'+p_2b+q_2d), \ldots, (j+p_na,j'+p_nb+q_nd),$$ to denote the elements in the intersection and define $$p:=\max\{|p_1|,|p_2|,\ldots,|p_n|\} \quad \mbox{and} \quad q:=\max\{|q_1|,|q_2|,\ldots,|q_n|\}.$$ We will reach a contradiction, analyzing four different cases.\
[*Case1.*]{} $p=q=0$: The intersection contains just one element, $(j,j')$. We now look at the translation $$L':=(j+2a,j'+2b)+(3a,3b)\mathbb{Z} +(0,d)\mathbb{Z}\subset L$$ of the sublattice $(3a,3b)\mathbb{Z} +(0,d)\mathbb{Z}$ of $(a,b)\mathbb{Z}+(0,d)\mathbb{Z}$. If $(j,j')\in L'$, then $$\left\{
\begin{array}{c}
j+2a+3ar = j \\
j'+2b+3br+ds = j'
\end{array}
\right.$$ for some $r,s\in\mathbb{Z}$. But, since $a(3r+2)\neq0$, $r\in\mathbb{Z}$, this is impossible. In particular, $K\cap L'=\emptyset$, a contradiction to our basic assumption.\
[*Case 2.*]{} [*$p=0$ and $q>0$*]{}: Here we consider the sublattice $(a,b)\mathbb{Z} +(0,{2(2q+1)}d)\mathbb{Z}$ of $(a,b)\mathbb{Z}+(0,d)\mathbb{Z}$ and look at its translation $$L'':=(j,j'+2qd) + (a,b)\mathbb{Z} +(0,2(2q+1)d)\mathbb{Z} \subset L.$$ If $(j+ra,j'+2qd+rb+2s(2q+1)d)=(j,j'+q_\mu d)$ for some $\mu \in \{1,2,\ldots,n\}$ and $r,s \in \mathbb{Z}$, then $$\left\{
\begin{array}{c}
ra=0\\
2qd+rb+2s(2q+1)d=q_\mu d
\end{array}
\right.$$ and, consequently, $2q+2s(2q+1)=q_\mu$. However, due to the definition of $q$, no integer $s$ can satisfy the previous equality. Thus, $L''\cap K=\emptyset$, another contradiction.\
[*Case 3.*]{} [*$p>0$ and $q=0$*]{}: Its is similar to the previous case.\
[*Case 4.*]{} $p,q>0$: Here we consider the sublattice $(2(2p+1)a,2(2p+1)b)\mathbb{Z} +(0,qd)\mathbb{Z}$ of $(a,b)\mathbb{Z}+(0,d)\mathbb{Z}$ and its translation $$L''':= (j+2pa,j'+2pb)+(2(2p+1)a,2(2p+1)b)\mathbb{Z} +(0,qd)\mathbb{Z} \subset L.$$ If $$(j+2pa+2r(2p+1)a, j'+2pb+2r(2p+1)b + sqd)= (j+p_\mu a,j'+p_\mu b+q_\mu d)$$ for some $\mu \in \{1,2,\ldots,n\}$ and $r,s \in \mathbb{Z}$, we will have that $2p+2r(2p+1)=p_\mu$. As in Case 2, we can deduce that $L'''\cap K=\emptyset$, a contradiction to our initial assumption on $K$.
It is now clear that the following additional result holds.
Let $K$ be a real, continuous, isotropic and positive definite kernel on $S^1 \times S^1$. The following assertions are equivalent:\
$(i)$ $K$ is strictly positive definite;\
$(ii)$ $\{(k,l): (|k|,|l|)\in J_K\}$ intersects all the translations of each lattice in $\mathbb{Z}^2$ infinitely many times;\
$(iii)$ $\{(k,l): (|k|,|l|)\in J_K\}$ intersects all the translations of each rectangular lattice in $\mathbb{Z}^2$ infinitely many times.
Appendix: a direct proof for Theorem \[main2\]
==============================================
Here, we include a self-contained direct proof for Theorem \[main2\]. A few passages incorporates new arguments.
Let $L=(j,j')+(a,b)\mathbb{Z}+(0,d)\mathbb{Z}$, $a,d\geq1$, be a lattice of $\mathbb{Z}^2$. If $d=1$, then we can assume that $a>1$ because, otherwise, $L=\mathbb{Z}^2$ and the the conclusion of the Theorem holds trivially. We will suppose that $K$ is strictly positive definite and that $\{(k,l): (|k|,|l|)\in J_K\} \cap L=\emptyset$ and will reach a contradiction. For each $(k,l)\in J_K$, we consider all possible cases attached to our assumption. Since $(k,l) \not \in L$, either $k-j \not \in a\mathbb{Z}$ or $$k-j \in a\mathbb{Z} \quad \mbox{and } \quad l -j' -(k-j)b/a \not \in d\mathbb{Z}.$$ In the first possibility, we have that $a\geq 2$ and we can conclude that $$\sum_{\mu=1}^a e^{-i2\pi \mu j/a} e^{i 2 \pi \mu k/a}=\sum_{\mu=1}^{a} (e^{i2\pi \mu/a})^{k-j}=0.$$ In the other, $d\geq 2$ and the conclusion is $$\sum_{\nu=1}^d \left[e^{i2\pi \nu /d} \right]^{l-j'-(k-j)b/a}=0.$$ The final outcome is $$\sum_{\mu=1}^a \left[e^{i2\pi \mu /a} \right]^{k-j} \sum_{\nu=1}^d \left[e^{i2\pi \nu /d} \right]^{l-j'-(k-j)b/a}=0, \quad (k,l) \in J_K.$$ A similar argument now using the fact that $(-k,-l) \not \in L$, leads to $$\sum_{\mu=1}^a \left[e^{i2\pi \mu /a} \right]^{-k-j} \sum_{\nu=1}^d \left[e^{i2\pi \nu /d} \right]^{-l-j'+(k+j)b/a}=0, \quad (k,l) \in J_K.$$ These two pieces of information can be put together to ensure the equality $$\sum_{\mu=1}^a \sum_{\nu=1}^d \left[\mbox{Re\,}( e^{-i2\pi \mu j/a} e^{-i2\pi \nu j'/d} e^{i 2 \pi \nu j b/ad})\right] e^{i2\pi k(\mu-\nu b/d)/a} e^{i2\pi l \nu /d}= 0, \quad (k,l) \in J_K,$$ with $a\geq 2$ or $d\geq 2$. Repeating the procedure above in the cases $(-k,l) \not \in L$ and $(k,-l) \not \in L$, we can conclude that $$\sum_{\mu=1}^a \sum_{\nu=1}^d \left[\mbox{Re\,}( e^{-i2\pi \mu j/a} e^{-i2\pi \nu j'/d} e^{i 2 \pi \nu j b/ad})\right] e^{i2\pi k(\mu-\nu b/d)/a} e^{-i2\pi l \nu /d}=
0, \quad (k,l) \in J_K,$$ under the same setting for $a$ and $d$. The angles $$\theta_{\mu\nu}= \dfrac{2\pi\left(\mu-\nu b/d\right)}{a}, \quad \p_\nu=\dfrac{2\pi \nu}{d},\quad \mu=1,2,\ldots,a, \quad \nu=1,2,\ldots,d,$$ define $ad$ distinct points $$x_{\mu\nu}=(\cos\theta_{\mu\nu},\sin\theta_{\mu\nu}), \quad w_\nu=(\cos\p_\nu,\sin\phi_\nu),\quad \mu=1,2,\ldots,a, \quad \nu=1,2,\ldots,d.$$ in $S^1 \times S^1$. Indeed, if $x_{\mu_1\nu}=x_{\mu_2\nu}$, for a fixed $\nu \in \{1,2,\ldots, d\}$ and $\mu_1, \mu_2 \in \{1,2,\ldots,a\}$, it is promptly seen that $\t_{\mu_1\nu}=\t_{\mu_2\nu}$, that is, $\mu_1=\mu_2$. On the other hand, at least one of the numbers $$r_{\mu\nu}:=\mbox{Re\,}( e^{-i2\pi \mu j/a} e^{-i2\pi \nu j'/d} e^{i 2 \pi \nu j b/ad}), \quad \mu=1,2,\ldots, a,\quad \nu=1,2,\ldots, d,$$ is not zero. Indeed, this is obvious if $j=0$. If $j\neq 0$ and $d>1$, then one of the three numbers $r_{11}$, $r_{2,1}$ and $r_{1d}$ must be nonzero. Otherwise, all three numbers $$4\left(\frac{j}{a}+\frac{j'}{d}-\frac{j b}{ad}\right),\quad 4\left(\frac{2j}{a}+\frac{j'}{d}-\frac{jb}{ad}\right), \quad \mbox{and} \quad 4\left(\frac{j}{a}+j'-\frac{j b}{a}\right)$$ belong to $1+2\mathbb{Z}$. In particular, $$4 j \in 2a\mathbb{Z},\quad \mbox{and}\quad 4(j+j'a-jb) \in a(1+2\mathbb{Z})$$ what generates a contradiction (even=odd).If $j\neq 0$ and $d=1$, then $a\geq 2$ and there is just one value for $\nu$, namely $\nu=1$. In this case, the numbers $r_{\mu\nu}$ become $$r_{\mu}:= \mbox{Re\,}[ e^{i 2\pi j(b-\mu)/a}],\quad \mu=1,2,\ldots, a.$$ If $a=2$, it is easily seen that $r_1\neq 0$. If $a>2$, a calculation similar to another one made above yields that one of the three numbers $r_1$, $r_2$ and $r_a$ is nonzero. Recalling Proposition \[prop-equiv cAc=0\] once again, we now have reached a contradiction with the strict positive definiteness of $K$.
Another appendix: the complex circle
====================================
All the major results demonstrated in this paper can be adapted to hold for positive definiteness on the complex circle $\Omega_2$. In that case, we replace $S^1$ with $\Omega_2$, we allow the kernels to assume complex values and the scalars $c_\mu$ in the definition of positive definiteness can be complex numbers (the quadratic form in the definition of positive definiteness is Hermitian). We will sketch what these results are and refer the interested reader to [@jean; @mene] where the necessary adaptations for the proofs can be prospected from.
Let $K: \Omega_2 \times \Omega_2 \to \mathbb{C}$ be a continuous kernel and assume that $$K((x,z),(y,w))=K_r(x\cdot y, z\cdot w),\quad x,y,z,w\in \Omega_2,$$ for some function $K_r : \Omega_2 \times \Omega_2 \to \mathbb{C}$, in which $\cdot$ is now the usual inner product of $\mathbb{C}$.It is positive definite if, and only if, the function $K_r$ is of the form $$K_r(z,w)=\sum_{k,l\in\mathbb{Z}}a_{k,l}z^k w^l, \quad z,w \in \Omega_2,$$ in which $a_{k,l} \geq 0$, $k,l \in \mathbb{Z}$ and $\sum_{k,l\in\mathbb{Z}}a_{k,l}<\infty$. Taking the above representation for granted, we can define $I_K:=\{(k,l): a_{k,l}>0\}$. For distinct points $(x_1,w_1),(x_2,w_2), \ldots,(x_n,w_n)$ on $\Omega_2 \times \Omega_2$ and a column vector $c$ in $\mathbb{C}^n$, the quadratic form $\overline{c}^{t}Ac=0$ corresponds to $$\sum_{\mu=1}^{n}c_{\mu}e^{ik\t_{\mu}}e^{il\p_{\mu}}=0, \quad (k,l)\in I_K,$$ in which $\t_\mu$ and $\p_\mu$ are the arguments in the polar representation of $x_\mu$ and $w_\mu$ respectively. In particular, this reveals that the proofs we have developed in Sections 2 and 3 simplify in the present complex setting.
We close the paper stating what the main characterization for strict positive definiteness on $\Omega_2 \times \Omega_2$ becomes.
Let $K$ be a continuous and positive definite kernel on $\Omega_2 \times \Omega_2$ as described above. It is strictly positive definite if, and only if, $I_K$ intersects all the translations of each rectangular lattice of $\mathbb{Z}^2$.
[**Acknowledgement.**]{} The arguments presented in the proof of Lemma \[elattice\] are originally due to H. Borges and E. Tengan. We thank them for providing such elegant algebraic details.
[1]{}
Barbosa, V. S.; Menegatto, V. A., Strictly positive definite kernels on two-point compact homogeneous spaces. ArXiv:1505.00591. Cheney, E. W., Approximation using positive definite functions. Approximation theory VIII, Vol. 1 (College Station, TX, 1995), 145-168, Ser. Approx. Decompos., 6, World Sci. Publ., River Edge, NJ, 1995. Dai, Feng; Xu, Yuan, Approximation theory and harmonic analysis on spheres and balls. Springer Monographs in Mathematics. Springer, New York, 2013. Everest, G.; van der Poorten, A.; Shparlinski, I.; Ward, T., Recurrence sequences. Mathematical Surveys and Monographs, 104. American Mathematical Society, Providence, RI, 2003. Gneiting, T., Strictly and non-strictly positive definite functions on spheres. [*Bernoulli*]{} 19 (2013), no. 4, 1327-1349. Guella, J. C.; Menegatto, V. A.; Peron, A. P., An extension of a theorem of Schoenberg to products of spheres. ArXiv:1503.08174. Laurent, M., Équations exponentielles-polynômes et suites récurrentes linéaires. II. [*J. Number Theory*]{} 31 (1989), no. 1, 24-53. Laurent, M., Équations diophantiennes exponentielles. [*Invent. Math.*]{} 78 (1984), no. 2, 299-327. Menegatto, V. A.; Oliveira, C. P.; Peron, A. P., Strictly positive definite kernels on subsets of the complex plane. [*Comput. Math. Appl.*]{} 51 (2006), no. 8, 1233-1250. Musin, O. R., Positive definite functions in distance geometry. European Congress of Mathematics, 115-134, Eur. Math. Soc., Zürich, 2010. Pinkus, A., Strictly Hermitian positive definite functions. [*J. Anal. Math.*]{} 94 (2004), 293-318. Ron, A.; Sun, Xingping, Strictly positive definite functions on spheres in Euclidean spaces. [*Math. Comp.*]{} 65 (1996), no. 216, 1513-1530. Schoenberg, I. J., Positive definite functions on spheres. [*Duke Math. J.*]{} 9, (1942), 96-108. Sun, Xingping, Strictly positive definite functions on the unit circle. [*Math. Comp.*]{} 74 (2005), no. 250, 709-721. Szegö, G., Orthogonal polynomials. Fourth edition. American Mathematical Society, Colloquium Publications, Vol. XXIII. American Mathematical Society, Providence, R.I., 1975.
J. C. Guella, V. A. Menegatto and A. P. Peron\
Departamento de Matemática,\
ICMC-USP - São Carlos, Caixa Postal 668,\
13560-970 São Carlos SP, Brasil\
e-mails: [email protected]; [email protected]; [email protected]
[^1]: All authors partially supported by FAPESP, under grants $\#$ 2012/22161-3, $\#$2014/00277-5 and 2014/25796-5 respectively.
|
---
address:
- University of Arizona
- Yale University
- Northwestern University
- 'Université Paris-7'
author:
- Paul Bressler
- Mikhail Kapranov
- Boris Tsygan
- Eric Vasserot
title: 'Riemann-Roch for real varieties'
---
6.5in
\[thm\][Corollary]{} \[thm\][Proposition]{} \[thm\][Lemma]{} \[thm\][Claim]{} \[thm\][Conjecture]{}
\[thm\][Remark]{} \[thm\][Example]{}
\[thm\][Definition]{}
**Introduction**
[**(0.1)**]{} Let $\Sigma$ be an oriented real analytic manifold of dimension $d$ and $X$ be a complex envelope of $\Sigma$, i.e., a complex manifold of the same dimension containing $\Sigma$ as a totally real submanifold. Then, (real) geometric objects on $\Sigma$ can be viewed as (complex) geometric objects on $X$ involving cohomology classes of degree $d$. For example, a $C^\infty$-function $f$ on $\Sigma$ can be considered as a section of ${\mathcal B}_\Sigma$, the sheaf of hyperfunctions on $\Sigma$ which, according to Sato, can be defined as $${\mathcal B}_\Sigma = \underline{H}^d_\Sigma({{\cal O}}_X), \leqno (0.1.1)$$ where ${{\cal O}}_X$ is the sheaf of holomorphic functions. So $f$ can be viewed as a class in $d$th local cohomology.
More generally, the equality (0.1.1) suggests that various results of holomorphic geometry on $X$ should have consequences for the purely real geometry on $\Sigma$, consequences that involve raising the cohomological degree by $d$. The goal of this paper is to investigate the consequences of one such result, the Grothendieck-Riemann-Roch theorem (GRR).
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[**(0.2)**]{} Let $p: X\to B$ be a smooth proper morphism of complex algebraic manifolds. We denote the fibers of $p$ by $X_b=p^{-1}(b)$ and assume them to be of dimension $d$. If $\mathcal{E}$ is an algebraic vector bundle on $X$, the GRR theorem says that $$ch_m(Rp_*({\mathcal{E}})) = \int_{X/B} \biggl[ ch({\mathcal E})\cdot {\operatorname{Td}}({\mathcal T}_{X/B})\biggr]_{2m+2d}
\quad \in \quad H^{2m}(B,
\mathbb{C}). \leqno (0.2.1)$$ Here $\int_{X/B}: H^{2m+2d}(X, {\mathbb C})\to H^{2m}(B, {\mathbb C})$ is the cohomological direct image (integration over the fibers of $p$).
In the case $m=1$ the class on the left comes from the class, in the Picard group of $B$, of the determinantal line bundle $\det (Rp_*{\mathcal E})$ whose fiber, at a generic point $b\in B$, is $$\det\, H^\bullet(X_b, {\mathcal E}) \quad =\quad \bigotimes_i \left( \Lambda^{max} \, H^i(X_b, {\mathcal E})
\right) ^{\otimes \, (-1)^i}. \leqno (0.2.2)$$ Deligne \[D1\] posed the problem of describing $\det(Rp_*{\mathcal E})$ in a functorial way as a refinement of GRR for $m=1$. This problem makes sense already for the case $B=pt$ when we have to describe the 1-dimensional vector space (0.2.2) as a functor of $\mathcal E$. Deligne solved this problem for a family of curves and further results have been obtained in \[E\].
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[**(0.3)**]{} To understand the real counterpart of (0.2.1), assume first that $B=pt$, so $X=X_{pt}$ and let $\Sigma\subset X$ be as in (0.1). Denote by $E$ the restriction of $\mathcal{E}$ to $\Sigma$ and by $C^\infty_\Sigma(E)$ the sheaf of its $C^\infty$ sections. Then, similarly to (0.1.1) we have the embedding $$C^\infty_\Sigma(E)\subset \underline{H}^d_\Sigma({\mathcal E}).
\leqno (0.3.1)$$ Assume further that $d=1$, so $X$ is an algebraic curve, and that $\Sigma$ is a small circle in $X$ cutting it into two pieces: $X_+$ (a small disk) and $X_-$. Let $\mathcal{E}_\pm =
{\mathcal E}|_{X_\pm}$. We are then in the situation of the Krichever correspondence \[PS\]. Namely, the space $\Gamma(E)$ of $L^2$-sections has a canonical polarization in the sense of Pressley and Segal \[PS\] and therefore possesses a determinantal gerbe ${\operatorname{Det}}\, \Gamma(E)$. The latter is a category with every Hom-set made into a $\mathbb{C}^*$-torsor (a 1-dimensional vector space with zero deleted). The extensions ${\mathcal E}_\pm$ of $E$ to $X_\pm$ define two objects $[\mathcal {E}_\pm]$ of this gerbe, and $$\det\, H^\bullet(X,{ \mathcal E}) = {\operatorname{Hom}}_{{\operatorname{Det}}\, \Gamma(E)} ([{\mathcal E}_+], [{\mathcal E}_-]).
\leqno (0.3.2)$$ The real counterpart of the problem of describing the $\mathbb{C}^*$-torsor $\det\, H^\bullet(X,
{\mathcal E})$ is then the problem of describing the gerbe ${\operatorname{Det}}\, \Gamma(E)$. If we now have a family $p: X\to B$ as before (with $d=1$), equipped with a subfamily of circles $q: \Sigma\to B$,$\Sigma\subset X$, then we have an $\mathcal{O}^*_B$-gerbe ${\operatorname{Det}}\, q_*(E)$ which, according to the the classification of gerbes \[Bre\], has a class in $H^2(B, \mathcal{O}^*_B)$. The latter group maps naturally to $H^3(B, \mathbb{Z})$ and in fact can be identified with the Deligne cohomology group $H^3(B, {\mathbb Z}_{ D}(1))$, see \[Bry\]. The Real Riemann-Roch for a circle fibration describes the above class (modulo 2-torsion) as $$[{\operatorname{Det}}\, q_*(E)] = \int_{\Sigma/B} ch_2(E)\quad \in\quad H^3(B, {\mathbb Z}_{\mathcal D}(2))\otimes
{\mathbb Z}\left[ {1\over 2}\right].
\leqno (0.3.3)$$ Here $\int_{\Sigma/B}: H^4(\Sigma, {\mathbb Z}_{ D}(2))\to H^3(B, {\mathbb Z}_{ D}(1))$ is the direct image in Deligne cohomology. Note the absense of the characteristic classes of ${\mathcal T}_{\Sigma/B}$ (they are 2-torsion for a real rank one bundle). If one is interested in the image of the determinantal class in $H^3(B, \mathbb{Z})$, then one can understand the RHS of the above formula in the purely topological sense.
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Both sides of (0.3.3) do not involve anything other than $q:\Sigma\to B$ and a vector bundle $E$ on $\Sigma$ (equipped with CR-structures coming from the embeddings into $X,\mathcal{E}$). One has a similar result for any $C^\infty$ circle fibration (no CR structure) and any $C^\infty$ complex bundle $E$ on $\Sigma$. In this case we get a gerbe with lien $C_B^{\infty *}$, the sheaf of invertible complex valued $C^\infty$-functions on $B$ and its class lies in $H^2(B, C_B^{\infty *}) = H^3(B, \mathbb{Z})$. It is this, purely $C^\infty$ setting, that we adopt and generalize in the present paper.
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[**(0.4)**]{} Let now $\Sigma$ be a compact oriented $C^\infty$-manifold of arbitrary dimension $d$ and $E$ a $C^\infty$ complex vector bundle on $\Sigma$. One expects that the space $\Gamma(E)$ should have some kind of $d$-fold polarization, giving rise to a “determinantal $d$-gerbe”, ${\operatorname{Det}}\, \Gamma(E)$. This structure is rather clear when $\Sigma$ is a 2-torus but in general the theory of higher gerbes is not fully developed. In any case one expects that a $C^\infty$ family of such gerbes over a base $B$ gives a class in $H^{d+1}(B, C_B^{\infty *}) = H^{d+2}(B, \mathbb{Z})$. In this paper we consider a $C^\infty$ family $q:\Sigma\to B$ of relative dimension $d$ and a $C^\infty$ bundle $E$ on $\Sigma$. We then define by means of the Chern-Weil approach, what should be the characteristic class of the would-be $d$-gerbe ${\operatorname{Det}}(q_*(E))$: $$C_1(q_*(E)) \quad\in\quad H^{d+2}(B, \mathbb{C}).\leqno (0.4.1)$$ We denote it by $C_1$ since it is a kind of $d$-fold delooping of the usual first Chern (determinantal) class. We then show the compatibility of this class with the gerbe approach whenever the latter can be carried out rigorously. Our main result is the Real Riemann-Roch theorem (RRR): $$C_1(q_*E) = \int_{\Sigma/B}\biggl[ ch(E)\cdot {\operatorname{Td}}({\mathcal T}_{\Sigma/B})\biggr]_{2d+2}
\quad\in\quad
H^{d+2}(B, {\mathbb C}).
\leqno (0.4.2)$$ Here ${\mathcal T}_{\Sigma/B}$ is the complexified relative tangent bundle and $\int_{\Sigma/B}$, the integration along the fibers of $q$, lowers the degree by $d$.
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Note that the above theorem is a statement of purely real geometry and is quite different from the “Riemann-Roch theorem for differentiable manifolds” proved by Atiyah and Hirzebruch \[AH\]. The latter expresses properties of a Dirac operator on a real manifold $\Sigma$, while our RRR deals with the $\overline{\partial}$-operator on a complex envelope $X$ of $\Sigma$. The $d=1$ case above can be deduced from a result of Lott \[Lo\] on “higher" index forms for Dirac operators (because the polarization in the circle case can be described in terms of the signs of eigenvalues of the Dirac operator). In general, however, our results proceed in a different direction.
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[**(0.5)**]{} Our definition of $C_1(q_*E)$ uses the description of the cyclic homology of differential operators \[BG\] \[W\] which provides a construction of a natural Lie algebra cohomology class $\gamma$ of the Atiyah algebra, i.e., of the Lie algebra of infinitesimal automorphisms of a pair $(\Sigma, E)$ where $\Sigma$ is a compact oriented $d$-dimensional $C^\infty$-manifold and $E$ is a vector bundle on $\Sigma$. The intuition with higher gerbes suggests that this class comes in fact from a group cohomology class of the infinite-dimensional group of all the automorphisms of $(\Sigma, E)$, see (3.7.7) and, moreover, that there are similar classes coming from the higher Chern classes (3.7.8). This provides a new point of view on the rather classical subject of “cocycles on gauge groups and Lie algebras" i.e., on groups of diffeomorphisms of manifolds and automorphisms of vector bundles as well as their Lie algebra analogs.
There have been two spurs of interest in this subject. The first one was the study of the cohomology of the Lie algebras of vector fields following the work of Gelfand-Fuks, see \[F\] for a systematic account. In particular, Bott \[Bo\] produced a series of cohomology classes of the Lie algebra of vector fields on a compact manifold and integrated them to group cohomology classes of the group of diffeomorphisms. Later, group cocycles have been studied with connections with various anomalies in physics, see \[RSF\].
From our point of view, the approach of \[RSF\] can be seen as producing “integrals of products of Chern classes" in families over a base $B$, (cf. \[D1\] \[E\]), in other words, as producing the ingredients for the right hand side of a group-theoretical RRR. This is the same approach that leads to the construction of the Morita-Miller characteristic classes for surface fibrations \[Mo\]. The anomalies themselves, however, should be seen as the classes whose existence is conjectured in (3.7.7-8) and whose description through integrals of products of Chern classes constitutes the RRR.
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[**(0.6)**]{} As far as the proof of the RRR goes, we use two types of techniques. The first is that of differential graded Lie algebroids (which can be seen as infinitesimal analogs of higher groupoids appearing in the heuristic discussion above). The second technique is that of “formal geometry" of Gelfand and Kazhdan, i.e., reduction of global problems in geometry of manifolds and vector bundles to problems related to cohomology of Lie algebras of formal vector fields and currents. The first work relating Riemann-Roch to Lie algebra cohomology was \[FT\] and this approach was further developed in \[BNT\]. To prove the RRR we use results of \[NT\] and \[BNT\] on the Lie algebra cohomology of formal Atiyah algebras.
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[**(0.7)**]{} We are grateful to K. C. H. Mackenzie for pointing out several inaccuracies in an earlier version. The second author would like to acknowledge support from NSF, Université Paris-7 and Max-Planck Institut fuer Mathematik.
**1. Background on Lie algebroids, groupoids and gerbes.**
[**(1.1) Conventions.**]{} All manifolds will be understood to be $C^\infty$ unless otherwise specified. For a manifold $\Sigma$ we denote by $C^\infty_\Sigma$ the sheaf of $\mathbb C$-valued $C^\infty$-functions. By a vector bundle over $\Sigma$ we mean a locally trivial, $C^\infty$ complex vector bundle, possibly infinite-dimensional. For such a bundle $E$ we denote by $C^\infty(E)=C^\infty_\Sigma(E)$ the sheaf of smooth sections, which is a locally free sheaf of $C^\infty_\Sigma$-modules. By ${\mathcal T}_\Sigma$ we denote the [*complexified*]{} tangent bundle of $\Sigma$, so its sections are derivations of $C^\infty_\Sigma$. We denote by ${{\cal D}}_\Sigma$ the sheaf of differential operators acting on $C^\infty_\Sigma$ and by ${{\cal D}}_{\Sigma, E}$ the sheaf of differential operators acting from sections of $E$ to sections of $E$. The notations ${{\cal D}}(\Sigma)$ and ${{\cal D}}(\Sigma, E)$ will be used for the spaces of global sections of ${{\cal D}}_\Sigma$ and ${{\cal D}}_{\Sigma, E}$.
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[**(1.2) Lie algebroids.**]{} Recall \[Mac\] that a Lie algebroid on $\Sigma$ consists of a vector bundle $\mathcal G$, a morphism of vector bundles $\alpha:{\mathcal G}
\to{\mathcal T}_\Sigma$ (the anchor map) and a Lie algebra structure in $C^\infty({\mathcal G})$ satisfying the properties:
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(1.2.1) $\alpha$ takes the Lie bracket on sections of $\mathcal G$ to the standard Lie bracket on vector fields.
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(1.2.2) For any smooth function $f$ on $\Sigma$ and sections $x,y$ of $\mathcal G$ we have $$[fx,y] - f\cdot [x,y] = \operatorname{Lie}_{\alpha(y)}(f)\cdot x.$$
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A Lie algebroid is called transitive, if $\alpha$ is surjective.
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[**(1.2.3) Examples.**]{} (a) When $\Sigma=pt$, a Lie algebroid is the same as a Lie algebra.
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\(b) ${\mathcal T}_X$ with the standard Lie bracket and $\alpha={\operatorname{id}}$ is a Lie algebroid.
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\(c) If $\alpha=0$, then the bracket in $\mathcal G$ is $C^\infty_\Sigma$-linear. In this case we say that $\mathcal G$ is a bundle of Lie algebras: every fiber of $\mathcal G$ is a Lie algebra.
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For a fixed $\Sigma$ we will speak about morphims of Lie algebroids on $\Sigma$, understanding morphisms of Lie algebroids in the sense of \[Mac\] which are identical on $\Sigma$. Thus a morphism ${{\cal G}}\to {{\cal G}}'$ is a morphism of vector bundles commuting with brackets and the anchor maps.
Note that for any transitive Lie algebroid $\mathcal G$ the kernel ${\operatorname{Ker}}(\alpha)\subset {\mathcal G}$ is a bundle of Lie algebras, i.e., a Lie algebroid with trivial anchor map, and the maps in the short exact sequence $$0\to {\operatorname{Ker}}(\alpha)\to {\mathcal G}\buildrel \alpha\over\to {\mathcal T}_X \to 0
\leqno (1.2.4)$$ are morphisms of Lie algebroids.
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[**(1.3) The de Rham complex of a Lie algebroid.**]{} Let $\mathcal G$ be a Lie algebroid on $\Sigma$. We denote $${\operatorname{DR}}^i({\mathcal G}) = {\operatorname{Hom}}(\Lambda^i{\mathcal G}, C^\infty_\Sigma).$$ The differential $d: {\operatorname{DR}}^i({\mathcal G}) \to {\operatorname{DR}}^{i+1}({\mathcal G})$ is defined by the standard formula of Cartan: for an antisymmetric $i$-linear function $l: {\mathcal G}^i\to C^\infty_\Sigma $ we set $$dl(x_1, ..., x_{i+1}) = \sum_{j=1}^{i+1} (-1)^j l(x_1, ...,\widehat{x_j}, ... , x_{i+1}) +
\sum_{j<k} (-1)^{j+k} l([x_j, x_k], x_1, ..., \widehat{x_j}, ..., \widehat{x_k}, ...,
x_{i+1}). \leqno (1.3.1).$$ We get a complex ${\operatorname{DR}}^\bullet({\mathcal G})$ called the de Rham complex of $\mathcal G$. A morphism of Lie algebroids $\phi: {\mathcal G}\to
{\mathcal H}$ gives rise to the morphism of de Rham complexes $\phi^*: {\operatorname{DR}}^\bullet ({{\cal H}})\to{\operatorname{DR}}^\bullet ({{\cal G}})$.
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[**(1.3.2) Examples.**]{} (a) if $\Sigma=pt$, so ${{\cal G}}$ is a Lie algebra, then ${\operatorname{DR}}^\bullet({{\cal G}}) = C^\bullet({{\cal G}})$ is the cochain complex of ${{\cal G}}$ with trivial coefficients.
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\(b) If ${{\cal G}}= {{\cal T}}_\Sigma$, then ${\operatorname{DR}}^\bullet({{\cal G}}) = \Omega^\bullet_\Sigma$ is the $C^\infty$ de Rham complex of $\Sigma$.
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[**(1.4) The enveloping algebra of a Lie algebroid.**]{} Let ${{\cal G}}$ be a Lie algebroid on $\Sigma$, as before. The enveloping algebra $U({{\cal G}})$ is the sheaf of associative algebras on $\Sigma$ defined by generators $x\in{{\cal G}}$ (local sections) and $f\in C^\infty_\Sigma$ (local functions) subject to the relations: $$xy-yx = [x,y]. \leqno (1.4.1)$$ $$f\cdot x - x\cdot f = {\operatorname{Lie}}_{\alpha(x)}(f).\leqno (1.4.2)$$
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[**(1.4.3) Examples.**]{} (a) If $\Sigma=pt$, so ${{\cal G}}$ is a Lie algebra, then $U({{\cal G}})$ is the usual enveloping algebra of ${{\cal G}}$.
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\(b) If ${{\cal G}}= {{\cal T}}_\Sigma$, then $U({{\cal G}}) = {{\cal D}}_\Sigma$ is the sheaf of differential operators $C^\infty_\Sigma\to C^\infty_\Sigma$.
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\(c) If ${{\cal G}}$ is any Lie algebroid, then the anchor map $\alpha$ induces a morphism $$U(\alpha): U({{\cal G}}) \to U({{\cal T}}_\Sigma) = {{\cal D}}_\Sigma$$ of sheaves of associative algebras. In particular, $C^\infty_\Sigma$ is a left $U({{\cal G}})$-module.
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The sheaf $U({{\cal G}})$ has an increasing ring filtration $\{U^m({{\cal G}})\}$ with $U^m({{\cal G}})$ generated by products involving at most $m$ sections of ${{\cal G}}$. The following is then standard.
(1.4.4) Proposition. The associated graded sheaf of algebras $\operatorname{gr} \, U({{\cal G}})$ is identified with the symmetric algebra $S^\bullet({{\cal G}})$.
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[**(1.5) The Koszul resolution.**]{} Let ${{\cal G}}$ be a Lie algebroid on $\Sigma$. We have then the complex $$...\to U({{\cal G}})\otimes \Lambda^2{{\cal G}}\to U({{\cal G}})\otimes{{\cal G}}\to U({{\cal G}}) \to C^\infty_\Sigma\to 0.
\leqno (1.5.1)$$ with the differential defined by: $$d(u\otimes (\gamma_1\wedge ...\wedge \gamma_n)) = \sum_{j=1}^n (-1)^j (u\gamma_j) \otimes \bigl(\gamma_1\wedge ... \wedge
\widehat{\gamma_j}\wedge ... \wedge \gamma_n\bigr) +$$ $$+ \sum_{j<k} (-1)^{j+k} u\otimes \bigl([\gamma_i, \gamma_j] \wedge ... \wedge \widehat{\gamma_i} \wedge ...
\wedge \widehat{\gamma_j} \wedge ... \wedge \gamma_n\bigr).$$
(1.5.2) Proposition. The complex (1.5.1) is exact and thus provides a locally free resolution of $C^\infty_\Sigma$ as a $U({{\cal G}})$-module.
(1.5.3) Corollary. We have $${\operatorname{DR}}^\bullet ({{\cal G}}) \simeq \underline{R{\operatorname{Hom}}}_{U({{\cal G}})} (C^\infty_\Sigma,
C^\infty_\Sigma).$$
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[**(1.6) The Atiyah algebra.**]{} Let $G$ be a Lie group, ${{\frak g}}$ be its Lie algebra, and $\rho: P\to\Sigma$ a principal $G$-bundle on $\Sigma$. The Atiyah algebra ${{\cal A}}_P$ is the sheaf of Lie algebras on $\Sigma$ whose sections are $G$-invariant vector fields on $P$: $${{\cal A}}_P= (\rho_* {{\cal T}}_P)^G. \leqno (1.6.1)$$ The map $\alpha=d\rho$ makes ${{\cal A}}_P$ into a transitive Lie algebroid of the form $$0\to{\operatorname{Ad}}(P) \to {{\cal A}}_P\buildrel\alpha\over \longrightarrow {{\cal T}}_\Sigma\to 0.\leqno (1.6.2)$$ Here ${\operatorname{Ad}}(P)$ is the bundle of Lie algebras on $\Sigma$ associated to $P$ via the adjoint representation.
If $\Sigma = \bigcup U_i$ is a covering in which $P$ is trivialized: $P|_{U_i} = U_i\times G$, and $g_{ij}: U_i\cap U_j\to G$ are the transition functions, then ${{\cal A}}_P$ is glued out of ${{\cal A}}_P|_{U_i} = {{\cal T}}_{U_i}\times {{\frak g}}$ via the transition functions $$(v,x) \mapsto (v, i_v(dg_{ij}\cdot g_{ij}^{-1}) + {\operatorname{Ad}}_{g_{ij}}(x)).\leqno (1.6.3)$$
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[**(1.6.4) Example.**]{} Let $G= GL_r({{\Bbb C}})$, so ${{\frak g}}= {\frak {gl}}_r({{\Bbb C}})$. A principal $G$-bundle $P$ corresponds then to a rank $r$ vector bundle $E$ on $\Sigma$. In this case ${{\cal A}}_P$ will also be denoted ${{\cal A}}_E$ and has a well known alternative description. It consists of differential operators $L: E\to E$ such that:
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\(a) $L$ has order $\leq 1$.
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\(b) The first order symbol of $L$ (which is a priori a section of ${{\cal T}}_\Sigma\otimes {\operatorname{End}}(E)$) lies in the subsheaf ${{\cal T}}_\Sigma = {{\cal T}}_\Sigma\otimes 1$.
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[**(1.7) Modules over Lie algebroids.**]{} We follow \[Mac\], see also \[Kal\] §3 for a more algebraic language. Let ${{\cal G}}$ be a Lie algebroid on $\Sigma$. A ${{\cal G}}$-module is a vector bundle ${{\cal M}}$ on $\Sigma$ equipped with a Lie algebra action $(x, m)\mapsto xm$ of ${{\cal G}}$ in the sections which satisfies the two twisted linearity properties: $$x(f\cdot m) - f\cdot (xm) = ({\operatorname{Lie}}_{\alpha(x)} f)\cdot m, \quad
f\in C^\infty_\Sigma, x\in{{\cal G}}, m\in{{\cal M}}.\leqno (1.7.1)$$
(1.7.2) For any local section $x$ of ${{\cal G}}$ the operator $m\mapsto xm$ on sections of ${{\cal M}}$ belongs to the Atiyah algebra ${{\cal A}}_{{\cal M}}$, and for different $m$ these operators define a morphism of Lie algebroids ${{\cal G}}\to{{\cal A}}_{{\cal M}}$.
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[**(1.7.3) Examples.**]{} (a) For any ${{\cal G}}$ the trivial bundle (whose sheaf of sections is) $C^\infty_\Sigma$ is a ${{\cal G}}$-module with the ${{\cal G}}$ action given via the anchor map and the Lie derivations of functions.
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\(b) Unless the anchor of ${{\cal G}}$ is trivial, the bracket in ${{\cal G}}$ does not make ${{\cal G}}$ into a ${{\cal G}}$-module, as the map from ${{\cal G}}$ to the Atiyah algebroid of ${{\cal G}}$ as a vector bundle is not linear over functions.
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\(c) An ideal in ${{\cal G}}$ is a sub-Lie algebroid ${{\cal G}}'$ such that $[{{\cal G}}, {{\cal G}}']\subset {{\cal G}}'$. In this case ${{\cal G}}'$ is a ${{\cal G}}$-module provided its anchor is trivial.
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Any ${{\cal G}}$-module has a structure of a sheaf of modules over the sheaf of rings $U({{\cal G}})$.
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[**(1.8) Cohomology of Lie algebroids.**]{} Let ${{\cal M}}$ be a ${{\cal G}}$-module. The de Rham complex ${\operatorname{DR}}^\bullet({{\cal G}}, {{\cal M}})$ with coefficients in ${{\cal M}}$ is defined by $${\operatorname{DR}}^i({{\cal G}}, {{\cal M}}) = \underline {{\operatorname{Hom}}}(\Lambda^i{\mathcal G}, {{\cal M}}). \leqno (1.8.1)$$ with the differential of $l: {{\cal G}}^i\to {{\cal M}}$ defined by the modification of (1.3.1): $$dl(x_1, ..., x_{i+1}) = \sum_{j=1}^{i+1} (-1)^j x_j (l(x_1, ...,\widehat{x_j}, ... , x_{i+1})) +
\sum_{j<k} (-1)^{j+k} l([x_j, x_k], x_1, ..., \widehat{x_j}, ..., \widehat{x_k}, ...,
x_{i+1}). \leqno (1.8.2).$$ Its cohomology sheaves will be denoted $\underline{H}_{{\operatorname{Lie}}}^i({{\cal G}}, {{\cal M}})$ and the corresponding cohomology groups of the complex of global smooth sections of ${\operatorname{DR}}^\bullet({{\cal G}}, {{\cal M}})$ by simply $H^i_{{\operatorname{Lie}}}({{\cal G}}, {{\cal M}})$. See \[Mac\], §7.1. As before, it is easy to see that $${\operatorname{DR}}^\bullet({{\cal G}}, {{\cal M}}) \simeq \underline{R{\operatorname{Hom}}}_{U({{\cal G}})} (C^\infty_\Sigma, {{\cal M}}).\leqno (1.8.3)$$ Therefore $$\underline{H^i}_{{\operatorname{Lie}}}({{\cal G}}, {{\cal M}}) = \underline{{\operatorname{Ext}}}^i_{U({{\cal G}})}(C^\infty_\Sigma, {{\cal M}}),\quad
H^i_{{\operatorname{Lie}}}({{\cal G}},{{\cal M}}) = {\operatorname{Ext}}^i_{U({{\cal G}})}(C^\infty_\Sigma, {{\cal M}}).\leqno (1.8.4)$$
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[**(1.8.5) Example.**]{} The trivial bundle $C^\infty_\Sigma$ is always a ${{\cal G}}$-module and for ${{\cal G}}= {{\cal T}}_\Sigma$ we have $H^i_{{\operatorname{Lie}}}({{\cal T}}_\Sigma, C^\infty_\Sigma) = H^i(\Sigma, {{\Bbb C}})$ (topological cohomology).
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[**(1.9) The Hochschild-Serre spectral sequence and the transgression.**]{} Let $$0\to{{\cal G}}'\to{{\cal G}}\to{{\cal G}}''\to 0\leqno (1.9.1)$$ be an extension of Lie algebroids on $\Sigma$, so ${{\cal G}}'$ is an ideal in ${{\cal G}}$. Note that ${{\cal G}}'$ is then a bundle of Lie algebras. Let ${{\cal M}}$ be a ${{\cal G}}$-module. Then for every point $x\in\Sigma$ the fiber ${{\cal M}}_x$ is a module over the Lie algebra ${{\cal G}}'_x$. Assume that for any $i\geq 0$ the Lie algebra cohomology spaces $H^i_{{\operatorname{Lie}}}({{\cal G}}'_x, {{\cal M}}_x)$ have dimension independent on $x\in\Sigma$. Then the sheaves $\underline{H}^i_{{\operatorname{Lie}}} ({{\cal G}}', {{\cal M}})$ are vector bundles on $\Sigma$ and these vector bundles have natural structures of ${{\cal G}}''$-modules. In this case we have (a Lie algebroid generalization of) the Hochshild-Serre spectral sequence with $$E_2^{pq} = H^p_{{\operatorname{Lie}}}({{\cal G}}'', \underline{H}^q_{{\operatorname{Lie}}}({{\cal G}}', {{\cal M}}))\Rightarrow
H^{p+q}_{{\operatorname{Lie}}}({{\cal G}}, {{\cal M}}).\leqno (1.9.2)$$ The construction is parallel to the classical (Lie algebra) case as in \[F\]. One uses the short exact sequence (1.9.1) to produce, in a standard way, a filtration on ${\operatorname{DR}}^\bullet({{\cal G}}, {{\cal M}})$. See \[Mac\], §7.4 for the treatment of the case ${{\cal G}}'' = {{\cal T}}_\Sigma$ which is the only case we will use in this paper.
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[**(1.9.3) Example.**]{} Similarly to the classical case, one can use (1.9.2) (or elementary considerations) to identify $H^2_{{\operatorname{Lie}}}({{\cal G}}, {{\cal M}})$ with the set of isomorphism classes of central extensions of Lie algebroids $$0\to {{\cal M}}\to\widetilde{{{\cal G}}}\to{{\cal G}}\to 0.$$ Central extensions of this type with ${{\cal G}}= {{\cal T}}_\Sigma$, ${{\cal M}}= C^\infty_\Sigma$, and the ${{\cal G}}$-action on ${{\cal M}}$ being the standard one (by Lie derivations), were called in \[Kal\] Picard Lie algebroids. The set of their isomorphism classes is thus identified with $H^2_{{\operatorname{Lie}}}({{\cal T}}_\Sigma, C^\infty_\Sigma)$ which is the same as the topological (de Rham) cohomology $H^2(\Sigma, {{\Bbb C}})$.
.1cm Fix $n>0$ and assume that $$H^j({{\cal G}}', {{\cal M}}) = 0, \quad 0<j<n, \leqno (1.9.4)$$ In this case $E_2^{0,n}=E_{n+1}^{0,n}$ as well as $E_2^{0, n+1} = E_{n+1}^{0, n+1}$. We obtain therefore the [*transgression map*]{} $$d_{n+1}: E_{n+1}^{0,n} = E_2^{0,n} =
H^n_{{\operatorname{Lie}}}({{\cal G}}', {{\cal M}})^{{{\cal G}}''} \to H^{n+1}_{{\operatorname{Lie}}}({{\cal G}}'', {{\cal M}}^{{{\cal G}}'})= E_2^{n+1, 0} = E_{n+1}^{n+1, 0}.
\leqno (1.9.5)$$ We will use this map later in the paper. Without the assumption (1.9.4) we have that $E_{n+1}^{0,n}$ is a subspace of $E_2^{0,n} = H^n_{{\operatorname{Lie}}}({{\cal G}}', {{\cal M}})^{{{\cal G}}''}$ namely the intersection of the kernels of $d_2, ..., d_n$. For convenience we will call elements of this space [*transgressive*]{} elements of $E_2^{0,n}$. Similarly, $E_{n+1} ^{n+1, 0}$ is a quotient space of $E_2^{n+1, 0} = H^{n+1}_{{\operatorname{Lie}}}({{\cal G}}'', {{\cal M}}^{{{\cal G}}'})$ by the union of images of $d_2, ..., d_n$.
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[**(1.9.6) Example.**]{} Suppose that $n=2$ and $\Sigma = pt$, so (1.9.1) is a central extension of Lie algebras and ${{\cal M}}$ is a ${{\cal G}}$-module in the usual sense. Let $\gamma\in E_2^{0,2} = H^2_{{\operatorname{Lie}}}({{\cal G}}', {{\cal M}})^{{{\cal G}}''}$ be a ${{\cal G}}''$-invariant class in $H^2$ and $$0\to{{\cal M}}\to \widetilde{{{\cal G}}}' \to {{\cal G}}' \to 0$$ be a central extension representing $\gamma$. The class $\gamma$ is transgressive, (i.e., annihilated by $d_2$) if and only if $\widetilde{{{\cal G}}'}$ can be made into a ${{\cal G}}$-equivariant central extension (as opposed to the fact that the class of the extension remains unchanged under the ${{\cal G}}$-action or, what is the same, under ${{\cal G}}''$-action). Given such an equivariant extension, one obtains a crossed module of Lie algebras (i.e., a dg-Lie algebra situated in degrees (-1) and 0) $$\widetilde{{{\cal G}}}''\buildrel\partial\over\longrightarrow {{\cal G}},$$ with ${\operatorname{Ker}}(\partial) = {{\cal M}}$ and ${\operatorname{Coker}}(\partial) = {{\cal G}}''$. As well known (see, e.g., \[L\], Example E.10.3), such a crossed module represents an element in $H^3({{\cal G}}'', {{\cal M}})$, and this element is the lifting of $d_3(\beta)$. Different choices of equivariant structure on $\widetilde{{{\cal G}}'}$ correspond to the ambiguity of the values of $d_3$ modulo the image of $d_2$. One can generalize this picture easily to the case of an arbitrary $\Sigma$.
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[**(1.10) Reminder on gerbes.**]{} We follow the same conventions as in \[KV2\] and use \[Bre\] as the background reference.
If $B$ is a topological space and and ${{\cal F}}$ is a sheaf of abelian groups on $B$, then we can speak of ${{\cal F}}$-gerbes (= gerbes with lien ${{\cal F}}$). Recall that such a gerbe ${\frak{ G}}$ consists of the following data:
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\(1) A category ${\frak{ G}}(U)$ given for all open $U\subset B$, the restriction functors $r_{UV}: {\frak{G}}(U)\to {\frak {G}}(V)$ given for any morphism $V\subset U$ and natural isomorphisms of functors $s_{UVW}: r_{VW}\circ r_{UV}\Rightarrow
r_{UW}$ given for each $W\subset V\subset U$ and satisfying the transitivity conditions.
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\(2) The structure of ${{{\cal F}}}|_U$-torsor (possibly empty) on each sheaf ${\underline{\operatorname{Hom}}}_{{\frak {G}}(U)}
(x,y)$ compatible with the $r_{UV}$ and such that the composition of morphisms is bi-additive.
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These data have to satisfy the local uniqueness and gluing properties for which we refer to \[Bre\].
By a sheaf of ${{\cal F}}$-groupoids we will mean a sheaf of categories ${\frak{ C}}$ on $B$ (so both ${ {\operatorname{Ob}}}\, {\frak {C}}$ and ${\operatorname{Mor}}\, {\frak {C}}$ are sheaves of sets) in which each sheaf ${\underline{\operatorname{Hom}}}_{{\frak {C}}(U)}(x,y)$ is either empty or is made into a sheaf of ${{{\cal F}}}|_U$-torsors so that the composition is biadditive. A sheaf ${\frak {C}}$ of ${{\cal F}}$-groupoids is called locally connected if locally on $B$ all the ${{\operatorname{Ob}}} \, {\frak {C}}(U)$ and ${\operatorname{Hom}}_{{\frak {C}}(U)}(x,y)$ are nonempty.
Each sheaf of ${{\cal F}}$-groupoids can be seen as a fibered category over $B$, in fact it is a pre-stack, see, e.g., \[LM\]. Recall (see, e.g., [*loc. cit.*]{} Lemma 2.2) that for any pre-stack ${\frak {C}}$ there is an associated stack ${\frak {C}}^{^\sim}$. If ${\frak {C}}$ is a locally connected sheaf of ${{\cal F}}$-groupoids, then ${\frak {C}}^{^\sim}$ is an ${{\cal F}}$-gerbe.
As well known (see, e.g., \[Bre\]), the set formed by ${{{\cal F}}}$-gerbes up to equivalence is identified with $H^2(B, {{{\cal F}}})$. The identification of the set of isomorphism classes of Picard Lie algebroids in Example 1.9.3 can be seen as an infinitesimal analog of this fact. Given an ${{\cal F}}$-gerbe ${\frak{ G}}$, we denote by $[{\frak {G}}]\in H^2(B, {{{\cal F}}})$ its class. Given a sheaf ${\frak{ C}}$ of ${{\cal F}}$-groupoids, we denote by $[{\frak {C}}]$ the class of the corresponding gerbe.
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Let $B$ be a $C^\infty$-manifold. We will be particularly interested in $C^{\infty *}_B$-gerbes on $B$. Recall that we have the exponential sequence of sheaves on $B$: $$0\to \underline {{{\Bbb Z}}}_B \to C^\infty_B\buildrel e^{2\pi i x} \over\longrightarrow
C^{\infty *}_B\to 0.\leqno (1.10.1)$$ The corresponding coboundary map $$\delta_n: H^n(B, C^{\infty *}_B\to H^{n+1}(B, {{\Bbb Z}})\leqno (1.10.2)$$ is an isomoprhism for $n\geq 1$ since $C^\infty_B$ is a soft sheaf. Thus $[{\frak {G}}]$ give rise to a class in $H^3(B, {{\Bbb Z}})$.
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Let ${\frak {G}}$ be a $C^{\infty *}_B$-gerbe. Recall \[Bry\], that a [*connective structure*]{} $\Delta$ on ${\frak {G}}$ is a set of data that associates to each open $U\subset B$ and each object $x\in {\operatorname{Ob}}\, {\frak{G}}(U)$ a sheaf $\Delta(x)$ of $\Omega^1_U$-torsors (whose sections can be thought of as “formal connections" in $x$) and for any local (iso)morphism $g: x\to y$ over $U$ an identification of torsors $g_*: \Delta(x)\to\Delta(y)$, satisfying the compatibility property plus the following gauge condition: if $x=y$ so $g\in C^{\infty *}(U)$ is an invertible function, then $g_*(\nabla) = \nabla - g^{-1} d(g)$.
A [*curving*]{} of a connective structure $\Delta$ is a rule $K$ associating to any $x$ as above and any global object $\nabla\in\Delta(x)$ a 2-form $K(\nabla)\in\Omega^2(U)$ satisfying the compatibility with pullbacks, invariance under isomorphisms as well as the gauge condition: $K(\nabla + \alpha) = K(\nabla) + d\alpha$, $\alpha\in\Omega^1(U)$. In this situation Brylinski defined the 3-curvature of the connective structure and curving, which is a closed 3-form $S = S_{\Delta, K}\in\Omega^3(B)$.
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[**(1.10.3) Example.**]{} let $G$ be a Lie group and $$1\to {{\Bbb C}}^*\to \widetilde{G}\to G\to 1$$ be a central extension of Lie groups. Let $\rho: P\to B$ be a principal $G$-bundle. We have then the $C^{\infty *}_B$-gerbe ${\operatorname{Lift}}_G^{\widetilde{G}}(P)$ whose objects over $U\subset B$ are liftings of $P|_U$ to a principal $\widetilde{G}$-bundle over $U$, compare \[Bl\]. Let $\nabla_P$ be a connection on $P$. Then ${\operatorname{Lift}}_G^{\widetilde G}(P)$ has a connective structure $\Delta$ which to every lifting $\widetilde {P}$ of $P$ to a $\widetilde{G}$-bundle associates the space of all connections on $\widetilde{P}$ extending $\nabla_P$. Further, let $R_\nabla\in\Omega^2(B)\otimes {\operatorname{Ad}}(P)$ be the curvature of $\nabla$. A choice of a lifting of $R_\nabla$ to a form $\widetilde{R}_\nabla\in\Omega^2(B)\otimes
{\operatorname{Ad}}(\widetilde{P})$ gives a curving $K$ on $\Delta$. This curving associates to any section $\widetilde{\nabla}$ of $\Delta(\widetilde{P})$, i.e., to a connection on $\widetilde{P}$ extending $\nabla$, the 2-form $R_{\widetilde {\nabla}} -
\widetilde{R}_\nabla$, where $R_{\widetilde{\nabla}}$ is the curvature of $\widetilde{\nabla}$.
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We will need the following result (\[Bry\], Thm. 5.3.12).
(1.10.4) Theorem. If ${\frak{G}}$ is a $C^{\infty *}_B$-gerbe with a connective structure $\Delta$ and a curving $K$, then the class of $S_{\Delta, K}$ in $H^3(B, {{\Bbb C}})$ is integral and is equal to the image of $\delta_2 [{\frak {G}}]$ under the natural map from $H^3(B, {{\Bbb Z}})$ to $H^3(B, {{\Bbb C}})$.
**2. Background on homology of differential operators.**
[**(2.1) Conventions.**]{} Let $A$ be an associative algebra over ${{\Bbb C}}$. We denote by ${\operatorname{Hoch}}_\bullet(A)$ the Hochschild complex of $A$ with coefficients in $A$: $$...\rightarrow A\otimes A\otimes A \to A\otimes A \to A,\leqno (2.1.1)$$ $$d(a_0\otimes ...\otimes a_p) = \sum_{i=0}^{p-1} (-1)^i a_0\otimes ... \otimes a_i a_{i+1}\otimes ... \otimes a_p
+
(-1)^p a_p a_0 \otimes a_1\otimes ... \otimes a_{p-1}.$$ By $HH_\bullet (A)$ wew denote the homology of ${\operatorname{Hoch}}_\bullet(A)$. As well known, $$HH_\bullet(A) = \operatorname{Tor}^{A\otimes A^{op}}_\bullet(A, A).\leqno (2.1.2)$$ Put $$\tau(a_0\otimes ... \otimes a_p) = (-1)^p a_1\otimes ... \otimes a_p\otimes a_0.\leqno (2.1.3)$$ The cyclic complex of $A$ is defined as the total compplex $$CC_\bullet(A) = \operatorname{Tot}_\bullet \biggl\{ \cdots{\operatorname{Hoch}}_\bullet(A)\buildrel 1-\tau\over\rightarrow
{\operatorname{Hoch}}_\bullet(A)\buildrel N\over\rightarrow {\operatorname{Hoch}}_\bullet(A)\buildrel 1-\tau\over\rightarrow
{\operatorname{Hoch}}_\bullet(A)\biggr\}.
\leqno (2.1.4)$$ Here $N= 1+\tau+\tau^2+...+\tau^n$ on ${\operatorname{Hoch}}_n(A)$.
The cyclic homology $HC_\bullet(A)$ is the homology of the complex $CC_\bullet(A)$. We recall the fundamental result relating the cyclic homology with the Lie algebra homology of the algebra of matrices, see \[L\].
(2.1.5) Theorem. $$H_\bullet^{Lie} ({\frak gl}(A)) = S^\bullet(HC_{\bullet-1}(A)).$$
(2.1.6) Corollary. If $HC_j(A)=0$ for $j=0, ..., p-1$, then $H_j^{Lie}({\frak gl}(A))=0$ for $j=0, ..., p$, and $H_{p+1}^{Lie}({\frak gl}(A)) = HC_p(A)$.
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[**(2.2) Homology of differential operators: algebro-geometric version.**]{} Let $X$ be a smooth affine algebraic variety over ${{\Bbb C}}$ of dimension $d$ and ${{\cal E}}$ be an algebraic vector bundle on $X$. Then the Hochschild-Kostant-Rosenberg theorem (together with Morita invariance of $HH_\bullet$) gives an identification: $$HH_m({\operatorname{End}}({{\cal E}})) = \Omega^m(X), \leqno (2.2.1)$$ where on the right we have the space of global regular $m$-forms on $X$. Further, $$HC_m({\operatorname{End}}({{\cal E}})) = \Omega^m(X) d\Omega^{m-1}(X)
\oplus H^{m-2}(X, {{\Bbb C}}) \oplus H^{m-4}(X, {{\Bbb C}}) \oplus ..
\leqno (2.2.2)$$ where on the right we have the usual topological (de Rham) cohomology, see \[L\] Th. 3.4.12. Let ${{\cal D}}({{\cal E}})$ be the ring of global differential operators from ${{\cal E}}$ to ${{\cal E}}$. Then the results of \[BG\] \[W\] imply: $$HH_m({{\cal D}}({{\cal E}}))= H^{2d-m}(X, {{\Bbb C}}).\leqno (2.2.3)$$ Further, $$HC_m({{\cal D}}({{\cal E}})) = \bigoplus_{i\geq 0} H^{2d-m+2i}(X, {{\Bbb C}}). \leqno (2.2.4)$$ We recall that the approach of [*loc. cit*]{} is to use the filtration by the degree of differential operators and realize the $E_1$-term of the corresponding spectral sequence for $HH$ as the complex of form on the cotangent bundle with the differential adjoint to the de Rham differential by means of the symplectic form. The spectral sequence is then seen to degenerate at $E_2$.
Let us note the particular case when $X={{\Bbb A}}^d$ and $E= {{\cal O}}_{{{\Bbb A}}^d}$ is the trivial bundle of rank 1. Then ${{\cal D}}({{\cal E}}) = W_d$ is the Weyl algebra with generators $x_i, \partial_i$, $i=1, ..., d$, and relations $$[x_i, x_j] = [\partial_i, \partial_j] = 0, \quad [\partial_i, x_j] = \delta_{ij}\cdot 1.$$ The above results imply that
$$HH_i(W_d) = 0 \quad \text {if} \quad i\neq 2d, \quad \quad HH_{2d}(W_d) = {{\Bbb C}}. \leqno (2.2.5)$$ and $$HC_{i}(W_d) = {{\Bbb C}}, \, i-2d\in 2{{\Bbb Z}}_+, \quad HC_i(W_d)=0, \, i-2d\notin 2{{\Bbb Z}}_+ .\leqno (2.2.6)$$
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[**(2.3) The $C^\infty$ version.**]{} Let $\Sigma$ be an oriented $C^\infty$-manifold of dimension $d$ and $E$ be a smooth complex vector bundle on $\Sigma$. We have then the algebras ${\operatorname{End}}(E)$, ${{\cal D}}(E)$ of smooth endomorphisms and differential operators on $E$. Following \[W\] we present the analogs of the results cited in (2.2) for these algebras. These rings have natural Fréchét topologies. As pointed out in loc. cit., to get reasonable results, all tensor products occuring in the Hochschild and cyclic complexes of the above algebras should be taken taken in the category of topological vector spaces, i.e., be completed. In plain terms, this means that the ${\operatorname{End}}(E)^{\otimes p}$ should be understood as the ring of endomorphisms of the vector bundle $E^{\boxtimes p}$ on the $p$-fold Cartesian product $\Sigma^p$ and similarly for differential operators. Under these conventions, we have: $$HH_m({{\cal D}}(E))= H^{2d-m}(\Sigma, {{\Bbb C}}), \leqno (2.3.1)$$ $$HC_m({{\cal D}}(E)) = \bigoplus_{i\geq 0} H^{2d-m+2i}(\Sigma, {{\Bbb C}}), \leqno (2.3.2)$$ where on the right we have the topological cohomology.
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[**(2.3.3) Remark.**]{} The Lie algebra cochain complexes of ${{\cal D}}(E)$ and of ${\frak {gl}}_N {{\cal D}}(E) = {{\cal D}}(E\otimes {{\Bbb C}}^r)$ involve exterior products of these algebras over ${{\Bbb C}}$. If we understand these products in the completed sense as above (compare also with Fuks \[F\]), then the analog of (2.1.5) holds, and we have the following.
(2.3.4) Corollary. Let $\Sigma$ be a compact, oriented $C^\infty$ manifold of dimension $d$. Then, for $N\gg 0$ we have: $$H_i^{{\operatorname{Lie}}} {\frak{gl}}_N {{\cal D}}(E)=0, \quad 0<i<d+1;$$ $$H_{d+1}^{{\operatorname{Lie}}} {\frak{gl}}_N{{\cal D}}(E) = {{\Bbb C}}.$$
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[**(2.4) The formal series version.**]{} Let $$\widehat{W}_d = W_d \otimes_{{{\Bbb C}}[x_1, ..., x_d]} {{\Bbb C}}[[x_1. ..., x_d]]$$ be the algebra of differential operators whose coefficients are formal power series. Similarly to the above, we consider the Hochschild and cyclic complexes of $\widehat{W}_d$ using the adic topology on ${{\Bbb C}}[[x_1, ..., x_d]]$ and taking completions. Thus $\widehat{W}_d^{\otimes p}$ is understood as the ring of differential operators whose coefficients are power series in $p$ groups of $d$ variables. With this understanding, we have the analog of (2.2.5): $$HH_{2d}(\widehat{W}_d) = {{\Bbb C}}, \quad HH_i(\widehat{W}_d)=0, \, i\neq 2d.\leqno (2.4.1)$$ For the proof, see \[FT\]. One can also apply the spectral sequence argument of \[BG\] and \[W\] and then use the Poincare lemma on the (contangent bundle to the) formal disk.
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Our next step is to consider such formal completions simultaneously at all points of a given $C^\infty$-manifold $\Sigma$. So let $\Sigma, E$ be as above. Let $\widehat{{\operatorname{Hoch}}}_p ({{\cal D}}(E))$ be the completion of ${{\cal D}}(E^{\boxtimes(p+1)})$ (differential operators in the bundle $E^{\boxtimes (p+1)}$ on $\Sigma^{p+1}$) along the diagonal $\Sigma\subset\Sigma^{p+1}$. This is a sheaf of $\Sigma$.
Then the Hochschild differential extends to $\widehat{{\operatorname{Hoch}}}_\bullet({{\cal D}}(E))$, making it into a complex ,and we denote by its homology. Similarly, we define the completed cyclic complex $\widehat{CC}_\bullet({{\cal D}}(E))$ by the procedure identical to (2.1.4) and denote its homology by $\widehat{HC}_\bullet({{\cal D}}(E))$. Thus, $\widehat{HH}_\bullet ({{\cal D}}(E))$ and $\widehat{CC}_\bullet({{\cal D}}(E))$ are sheaves on $\Sigma$.
(2.4.2) Proposition. We have $\widehat{HH}_p({{\cal D}}(E)) =\underline{{{\Bbb C}}}_\Sigma$ (constant sheaf) for $p=2d$ and it is equal to 0 for $p\neq 2d$.
[*Proof:*]{} It is clearly enough to consider the case when $\Sigma$ is an open ball in ${{\Bbb R}}^d$ and $E$ is trivial and to prove that in this case the complex of global sections of $\widehat{HH}_\bullet({{\cal D}}(E))$ is exact everywhere except degree $2d$ where the cohomology is isomoprhic to ${{\Bbb C}}$. This will imply that the only sheaf of cohomology of $\widehat{HH}_\bullet({{\cal D}}(E))$ in this case (and thus in the general case) is $\underline{{{\Bbb C}}}_\Sigma$. So we make this assumption in the rest of the proof.
We start with the case of $\widehat{Hoch}_\bullet(C^\infty_\Sigma)$ defined, as before, using functions on the completion of $\Sigma^{\bullet +1}$ along the diagonal. Such functions form a flat module over $C^\infty(\Sigma\times\Sigma)$ so by the interperation of $HH$ as $\operatorname{Tor}$, see (2.1.2) we see that $$\widehat{HH}_\bullet(C^\infty_\Sigma) = \Omega^i_\Sigma,$$ and the same will hold if we replace $C^\infty_\Sigma$ by a matrix algebra (i.e., take $E$ of higher rank).
Next, we replace $C^\infty_\Sigma$ by the sheaf of commutative algebras $${{\cal A}}= S^\bullet({{\cal T}}_\Sigma)$$ (polynomial functions on the contangent bundle) and define $\widehat{Hoch}_\bullet({{\cal A}})$ using the completions of sheaves of sections of ${{\cal A}}^{\boxtimes (p+1)}$ on $\Sigma^{p+1}$ along the diagonals. The same argument with flatness will apply, so we conclude that $$\widehat{HH}_\bullet({{\cal A}}) = p_* (\Omega^\bullet_{T^*\Sigma}),\leqno
(2.4.2.1)$$ where $p: T^*\Sigma\to \Sigma$ is the projection. Again, a similar statement will hold for matrices.
Finally, we use the approach of \[BG\] \[W\] and consider the spectral sequence for $\widehat{HH}_\bullet({{\cal D}}(E))$ associated to the filtration by degree of operators. We get the $E_1$-term to be (2.4.2.1) with the differential being the adjoint of the de Rham differential on $T^*\Sigma$. Since we assumed $\Sigma$ to eb a ball, we conclude that the $E_2$-term reduces to one space ${{\Bbb C}}$.
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Further, we need a relative version of the above statements. Let $$q: \Sigma\to B$$ be a submersion (smooth fibration) of $C^\infty$-manifolds, whose fibers are of dimension $d$ and are oriented. Let $E$ be $C^\infty$-bundle on $\Sigma$, as above. We have then the subring $${{\cal D}}_{\Sigma/B}(E) \subset {{\cal D}}(E),\leqno (2.4.3)$$ consistsing of differential operators that are $q^{-1}C^\infty_B$-linear, i.e., act along the fibers only.
Let $\Sigma^{p+1}_B\subset \Sigma^{p+1}$ be the $(p+1)$-fold fiber product of $\Sigma$ over $B$. We denote by $E^{\boxtimes (p+1)}_B$ the restriction of $E^{\boxtimes (p+1)}$ to $\Sigma^{p+1}_B$.
Let $\widehat{{\operatorname{Hoch}}}_p({{\cal D}}_{\Sigma/B}(E))$ denote the completion of ${{\cal D}}_{\Sigma^{p+1}_B/B}(E^{\boxtimes (p+1)}_B)$ along the diagonal. Then the Hochschild differential extends to $\widehat{{\operatorname{Hoch}}}_p({{\cal D}}_{\Sigma/B}(E))$. We also define the completed cyclic complex $\widehat{CC}_\bullet({{\cal D}}_{\Sigma/B}(E))$ by implementing (2.1.4).
(2.4.4) Theorem. (a) The complex $\widehat{{\operatorname{Hoch}}}_p({{\cal D}}_{\Sigma/B}(E))$ is acyclic in degrees other than $2d$, and its $2d$th cohomology sheaf is isomorphic to $q^{-1} C^\infty_B$. In other words, we have a quasiisomorphism in the derived category of sheaves of $q^{-1}C^\infty_B$-modules on $\Sigma$: $$\mu_{{\cal D}}: \widehat{{\operatorname{Hoch}}}_p({{\cal D}}_{\Sigma/B}(E)) \to q^{-1} C^\infty_B [2d].$$ (b) We have $H^i(\widehat{CC}_\bullet({{\cal D}}_{\Sigma/B}(E))=0$ unless $i=-2d+k$, $k\in {{\Bbb Z}}_+$, and $$H^{-2d+k}(\widehat{CC}_\bullet ({{\cal D}}_{\Sigma/B}(E)) = q^{-1} C^\infty_B.$$
[*Proof:*]{} Similar to (2.4.2).
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As a corollary of Theorem 2.4.4 we have a morphism (no longer an isomorphism) in the derived category $$\nu_{{\cal D}}: \widehat{CC}_\bullet({{\cal D}}_{\Sigma/B}(E))\to q^{-1} C^\infty_B[2d]. \leqno (2.4.5)$$
**3. Characteristic classes from Lie algebra cohomology.**
[**(3.1) The finite-dimensional case.**]{} Let $G$ be a Lie group with Lie algebra $\frak g$. We denote by $C^\bullet({\frak g})$ the cochain complex of $\frak g$ with trivial coefficients ${{\Bbb C}}$ and by $H^n({\frak g})$ its $n$th cohomology space.
Let $\gamma\in H^n({\frak g})$ be a cohomology class. We want to associate (under certain conditions) to $\gamma$ a characteristic class of principal $G$-bundles. In other words, we want to produce, for each $C^\infty$-manifold $B$ and each smooth principal $G$-bundle $P$ on $B$, a topological (de Rham) cohomology class $$c_\gamma(P)\in H^{n+1}(B) = H^{n+1}(B, {{\Bbb C}}) \leqno (3.1.1)$$ (note the shift of degree by $1$).
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Indeed, let a principal $G$-bundle $\rho: P\to B$ be given and ${{\cal A}}_P$ be its Atiyah algebra. We have then the extension of Lie algebroids (1.6.2) on $B$ and the corresponding Hochschild-Serre spectral sequence (1.9.2) which in our case has the form: $$E_2^{pq} = H^p_{{\operatorname{Lie}}}({{\cal T}}_B, \underline{H}^q_{{\operatorname{Lie}}}({\operatorname{Ad}}(P), C^\infty_B))\Rightarrow H^{p+q}_{{\operatorname{Lie}}}({{\cal A}}_P, C^\infty_B).
\leqno (3.1.2)$$ This sequence was considered in \[McK\], Thm. 7.4.19. Note that $\underline{H}^q_{{\operatorname{Lie}}}({\operatorname{Ad}}(P), C^\infty_B)$ is the cohomology of the cochain complex of ${\operatorname{Ad}}(P)$ as a Lie algebra over $C^\infty_B$, i.e., of the complex of bundles formed by the duals of the fiberwise exterior products of fibers of ${\operatorname{Ad}}(P)$. We will also use the notation $C^\bullet({\operatorname{Ad}}(P)_{/B})$ for this complex.
(3.1.3) Lemma. For any $q\geq 0$ the bundle $H^q_{{\operatorname{Lie}}}({\operatorname{Ad}}(P), C^\infty_B) =
H^q({\operatorname{Ad}}(P)_{/B})$ on $B$ formed by the Lie algebra cohomology spaces of the fibers of ${\operatorname{Ad}}(P)$ is canonically identified with the trivial bundle with fiber $H^q({{\frak g}})$.
[*Proof:*]{} This follows from the fact the the adjoint action of $G$ on ${{\frak g}}$ induces the trivial action on $H^q({{\frak g}})$.
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Therefore $$E_2^{pq} = H^p(B)\otimes H^{q}({{\frak g}}). \leqno (3.1.4)$$ In particular, our class $\gamma\in H^n({{\frak g}})$ gives an element $1\otimes\gamma\in E_2^{0n}$.
Assume now that we have $n>0$ such that the Lie algebra ${{\frak g}}$ satisfies the sphericity condition: $$H^i({{\frak g}}) = 0, \quad 0<i<n. \leqno (3.1.5)$$ Then we are in the situation of (1.9.4), so we have the transgression map (1.9.5) which in our case has the form $$d_{n+1}: H^n({{\frak g}})\to H^{n+1}(B), \leqno (3.1.6)$$ and we define $$c_\gamma(P) = d_{n+1}(1\otimes\gamma). \leqno (3.1.7)$$ Without the assumption (3.1.5) we have that $c_\gamma(P)$ is defined only if $1\otimes\gamma$ is transgressive (i.e., annihilated by $d_2, ..., d_n$ and takes value not in $H^{n+1}(B)$ but in the quotient of $H^{n+1}(B)$ by the images of $d_2, ..., d_n$.
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[**(3.1.8) Examples.**]{}
\(a) Let $n=1$. Then the condition (3.1.5) is trivially satisfied. A class $\gamma$ is just a trace functional $\gamma: {{\frak g}}\to{{\Bbb C}}$. The class $c_\gamma(P)\in H^2(B)$ can be obtained by choosing a connection $\nabla$ in $P$ with curvature $R\in \Omega^2_B\otimes{{\frak g}}$ and taking the class of the closed 2-form $\gamma(R)\in\Omega^2_B$. Alternatively, one can use $\gamma$ to produce a trace functional $\gamma_P: {\operatorname{Ad}}(P)\to C^\infty_B$ and then use $\gamma_P$ to push forward the extension (1.6.2) to a central extension of Lie algebroids $$0\to C^\infty_B\to {{\cal G}}\to {{\cal T}}_B\to 0.$$ As well known (1.7) the set of isomorphism classes of such central extensions is identified with $H^2_{{\operatorname{Lie}}}({{\cal T}}_B, C^\infty_B) = H^2(B, {{\Bbb C}})$.
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\(b) Let $n=2$, so $\gamma$ is represented by a central extension $$0\to {{\Bbb C}}\to \widetilde{{{\frak g}}} \to {{\frak g}}\to 0.\leqno (3.1.9)$$ A sufficient condition for $\gamma$ to be basic for any $P$ is that $\widetilde{{{\frak g}}}$ can be made into a $G$-equivariant central extension, compare (1.9.6). Suppose that such an equivariant structure has been chosen. Then the class $c_\gamma(P)\in H^3(B, {{\Bbb C}})$ can be constructed as follows. We have the representation $\widetilde{{\operatorname{Ad}}}$ of $G$ on $\widetilde{{{\frak g}}}$, and therefore an extension of associated vector bundles on $B$: $$0\to C^\infty_B \to\widetilde{{\operatorname{Ad}}}(P) \to {\operatorname{Ad}}(P)\to 0.$$ Choose a connection $\nabla$ in $P$. Then we have associated linear connections $\nabla_{{\operatorname{Ad}}}$ in ${\operatorname{Ad}}(P)$ and $\nabla_{\widetilde{{\operatorname{Ad}}}}$ in $\widetilde{{\operatorname{Ad}}}(P)$. We also have the curvature $R_\nabla\in \Omega^2(B)\otimes{\operatorname{Ad}}(P)$. Choose a lifting $\widetilde{R_\nabla}$ of $R_\nabla$ to $\Omega^2(B)\otimes \widetilde{{\operatorname{Ad}}}(P)$, and take $$S = \nabla_{\widetilde{{\operatorname{Ad}}}}(\widetilde{R}_\nabla)\in \Omega^3(B)\otimes\widetilde{{\operatorname{Ad}}}(P).$$ By the Bianchi identity $\nabla(R_\nabla)=0$ and so $S$ lies in the tensor product of $\Omega^3(B)$ and the subbundle $C^\infty_B\subset \widetilde{{\operatorname{Ad}}}(P))$, i.e., it is a scalar differential form $S\in\Omega^3(B)$. Further, it is clear that $S$ is a closed 3-form. The class $c_\gamma(P)$ is then the class of the form $S$. A different choice of an equivariant structure on $\widetilde{{{\frak g}}}$ leads to change of the class of $S$ by an element from the image of $d_2$.
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\(c) Let $G= GL_N({{\Bbb C}})$, so ${{\frak g}}= {\frak {gl}}_N({{\Bbb C}})$. Then $H^\bullet({{\frak g}})$ is the exterior algebra on generators $\gamma_1, ..., \gamma_N$ with $\gamma_i\in H^{2i-1}({{\frak g}})$. A principal $G$-bundle $P$ on $B$ is the same as a rank $N$ vector bundle $E$. In this case each $1\otimes\gamma_i$ is transgressive, and $c_{\gamma_i}(P)$ is the image of $c_i(E)\in H^{2i}(B)$ under the natural projection $H^{2i}(B)\to E_{n+1}^{0, n+1}$. Here $c_i(E)$ is the usual $i$th Chern class of $E$.
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[**(3.2) Other interpretations.**]{} Here we collect, for future use, some more or less straightforward reformulations of the construction of $c_\gamma(P)$.
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[*(a) The Chern-Weil picture.*]{} If we choose a connection $\nabla$ in $P$, then the sequence (1.6.2) splits (such splitting is in fact the definition of a connection following Atiyah). So we can identify $$\Omega^\bullet(P)^G = {\operatorname{DR}}^\bullet({{\cal A}}_P) = \Omega^\bullet_P\otimes C^\bullet({\operatorname{Ad}}(P)_{/B}). \leqno (3.2.1)$$ Let $R$ be the curvature of $\nabla$. Then the differential in the RHS of (3.2.1) has the form $\partial+\nabla +i_R$, where $\partial$ is the differential in $C^\bullet({{\frak g}})$ and $$i_R: \Omega^\bullet_B\otimes C^\bullet({{\frak g}}) \to \Omega^{\bullet +2}_B\otimes
C^{\bullet -1}({{\frak g}})\leqno (3.2.2)$$ is the contraction with $R$. This leads to a definition of $c_\gamma(P)$ in terms of differential forms. Namely, we have an injective and a surjective morphisms of complexes: $$\Omega^\bullet_B = \Omega^\bullet_B\otimes C^0({{\frak g}})\buildrel\phi\over \hookrightarrow
\Omega^\bullet_B\otimes C^\bullet({{\frak g}}) \buildrel \psi\over\longrightarrow \Omega^0_B\otimes
C^\bullet({{\frak g}}).\leqno (3.2.3)$$ Here $\psi$ is identified with the projection to ${\operatorname{gr}}^0_F$, where $F$ is the filtration from (3.1.2). If our class $\gamma$ is basic, then it lifts uniquely to a class in $H^n( {\operatorname{Coker}}(\phi))$, so $c_\gamma(P)$ is the image of that lifted class under the coboundary map corresponding to the short exact sequence $$0\to \Omega^\bullet_B\buildrel\phi\over\hookrightarrow \Omega^\bullet_B\otimes C^\bullet({{\frak g}})
\to {\operatorname{Coker}}(\phi)\to 0. \leqno (3.2.4)$$
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[*(b) The differential graded picture.*]{} Let ${\mathfrak A}$ denote the cone of the map $i : {\operatorname{Ad}}(P)\to{{\cal A}}_P$ viewed as a differential graded Lie algebroid. Thus ${{\cal A}}_P$ is put in degree 0, and ${\operatorname{Ad}}(P)$ in degree $(-1)$. The anchor map $\alpha$ induces the quasi-isomorphism of Lie algebroids ${\mathfrak A} @>>> {\mathcal T}_B$, hence the map of respective universal enveloping (differential graded) algebras ${ U}({\mathfrak A}) @>>> { U}({\mathcal T}_B)=
{\mathcal D}_B$ (the latter concentrated in degree zero) which is a quasi-isomorphism. Define the map $${\operatorname{DR}}^{\bullet}({\mathcal A}_P)/{\operatorname{DR}}^{\bullet}({{{\cal T}}_B}) @>{\delta}>> {\operatorname{DR}}^{\bullet+1}({\mathfrak A})
\leqno (3.2.5)$$ as follows. For $X\in {\operatorname{Ad}}(P)$, denote by ${\underline X}$ the element $(X,0)$ in the cone ${\mathfrak A}$ of $i$; for $Y \in {\mathcal A}_P$, denote the element $(0,Y)$ simply by $Y$. Given a $p$-cochain $\omega$ from ${\operatorname{DR}}^{\bullet}({\mathcal A}_P$, define the cochain $\delta \omega$ by $$\delta \omega ({\underline X}_1, \ldots, {\underline X}_q, Y_1, \ldots, Y_r)=\omega ({\underline X}_1, Y_1, \ldots, Y_r)
\leqno (3.2.6)$$ for $q=1$ and zero for $q\neq 1$.
It is easy to see that the sequence $${\operatorname{DR}}^{\bullet}({\mathcal A}_P{\operatorname{DR}}^{\bullet}({{{\cal T}}_B}) @>{\delta}>>
{\operatorname{DR}}^{\bullet+1}({\mathfrak A}) \leftarrow {\operatorname{DR}}^{\bullet+1}({{\cal T}}_B)=\Omega^{\bullet+1}_B
\leqno (3.2.7)$$ represents the boundary map $$H^\bullet({\operatorname{DR}}^\bullet({{\cal A}}_P)/{\operatorname{DR}}^\bullet({{\cal T}}_B))\to H^{\bullet+1}({\operatorname{DR}}^\bullet({{\cal T}}_B)) = H^{\bullet+1}(B). \leqno
(3.2.8)$$ A basic class $\gamma$ as above defines an $n$-dimensional cohomology class $\tilde\gamma$ of ${\operatorname{DR}}^{\bullet}({\mathcal A}_P)/{\operatorname{DR}}^{\bullet}({{{\cal T}}_B})$, and $c_\gamma(P)$ is the image of $\tilde\gamma$ under (3.2.7).
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[*(c) The ${{\cal D}}$-module picture.*]{} Consider the short exact sequence $$0\to C^{\geq 1}({\operatorname{Ad}}(P)_{/B}) \to C^\bullet({\operatorname{Ad}}(P)_{/B}) \to C^\infty_B\to 0
\leqno (3.2.9)$$ coming from the fact that $C^\infty_B = C^0({\operatorname{Ad}}(P)_{/B})$ is the 0th term of the relative cochain complex. If $\mathfrak A$ is as in (b), then all three complexes in (3.2.9) are graded $U({\mathfrak A})$-modules in the following way. Elements $Y=(0,Y), Y\in {{\cal A}}$, act via the adjoint action. Elements $\underline{X} = (X,0), X\in{\operatorname{Ad}}(P)$, act by exterior multiplication. The action of $U({\mathfrak A})$ on $C^\infty_B$ is via the quasiisomorphism with ${{\cal D}}_B$.
Note that (3.2.9) splits as a short exact sequence of complexes of vector bundles but not of $U({\mathfrak A})$-modules. We will use the corresponding connecting morphism $$\delta: C^\infty_B\to C^{\geq 1}({\operatorname{Ad}}(P)_{/B})[1]
\leqno (3.2.10)$$ in $D(U({\mathfrak A}))$, the derived category of differential graded $U({\mathfrak A})$-modules.
As $\mathfrak A$ is quasiisomorphic to ${{\cal T}}_B$, the DG algebra $U({\mathfrak A})$ is quasiisomorphic to ${{\cal D}}_B$, and the category $D(U({\mathfrak A}))$ is equivalent to $D({{\cal D}}_B)$. Now recall (1.5.3) that $$H^m(B) = {\operatorname{Hom}}_{D({{\cal D}}_B)}(C^\infty_B, C^\infty_B[m]). \leqno (3.2.11)$$ On the other hand, suppose that ${{\frak g}}$ is such that $H^i({{\frak g}})=0$ for $0<i<n$, see (3.1.11). Then $H^i({\operatorname{Ad}}(P)_{/B}) = H^i({{\frak g}})\otimes C^\infty_B = 0$ for $0<i<n$ as well. In other words, the complex $C^{\geq 1}({\operatorname{Ad}}(P)_{/B})$ is acyclic in degrees $<n$ and therefore each class $\xi$ in its $n$th cohomology (which is isomorphic to $H^n({{\frak g}})\otimes C^\infty_B$) defines a morphism in the derived category of complexes of vector bundles $$\tilde\xi: C^{\geq 1}({\operatorname{Ad}}(P)_{/B})\to C^\infty_B[n]. \leqno (3.2.12)$$ Further, “constant” class $\xi$, i.e., a class of the form $\gamma\otimes 1$, $\gamma\in H^n({{\frak g}})$, defines in fact a morphism in the category $D(U({\mathfrak A}))\sim D({{\cal D}}_B)$. Composing $\widetilde{\gamma\otimes 1}$ with $\delta$, we get a morphism $$C^\infty_B\to C^\infty_B[n+1], \leqno (3.2.13)$$ i.e., a class in $H^{n+1}(B)$.
(3.2.14) Proposition. The class in $H^{n+1}(B)$ corresponding to (3.2.13) is equal to $c_\gamma(P)$.
[*Proof:*]{} This follows directly from the definitions (in fact, we could take (3.2.13) as the definition of $c_\gamma(P)$). Indeed, the morphism in the derived category from the cohomology of a quotient complex such as $C^\infty_B$ to the homology of a subcomplex such as $C^{\geq 1}({\operatorname{Ad}}(P)_{/B})$ acyclic up to degree $n$, is precisely the differential $d_{n+1}$ in the corresponding spectral sequence.
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[**(3.3) Infinite-dimensional groups.**]{} Slightly reformulating the approach of K.-T. Chen \[C\], we introduce the following definition.
(3.3.1) Definition. A differentiable space is an ind-object in the category of $C^\infty$-manifolds.
For background on ind-objects, see \[D2\]. Thus a differentiable space $M$ is a formal limit $``\lim\limits_{\longrightarrow}{}''{}_{\alpha\in A} M_\alpha$ of (finite-dimensional) $C^\infty$-manifolds. In particular, $M$ defines a functor $$S\mapsto M(S) = C^\infty(S, M) = \lim_{\longrightarrow} C^\infty(S, M_\alpha)
\leqno (3.3.2)$$ on such manifolds and can in fact be identified with this functor. In practice, however, we will identify $M$ with the set $M(pt) = \lim\limits_{\longrightarrow} M_\alpha$ with (3.2.2) providing an additional structure on this set (description of what it means for an element of this set to vary in a smooth family). For a differential space $M$ we define (compare \[C\]) the space of $p$-forms (in particular, of $C^\infty$-functions) on $M$ by $$\Omega^p(M) = \lim_{\longleftarrow} \Omega^p(M_\alpha).\leqno (3.3.3)$$ For a point $m\in M(pt)$ the tangent space $T_mM$ is defined by $$T_mM = \lim_{\longrightarrow} \, T_sS,\leqno (3.3.4)$$ where the limit is taken over $C^\infty$-maps $(S,s)\to (M,m)$.
A differentiable group $G$ is a group object in the category of differentiable spaces. For such a group the space ${{\frak g}}= T_eG$ is a Lie algebra in the standard way.
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[**(3.3.5) Examples.**]{} [*(a) Groups of diffeomorphisms.*]{} Let $\Sigma_0$ be a compact oriented $C^\infty$-manifold of dimension $d$. Then we have a differentiable group $G = \operatorname{Diffeo}(\Sigma_0)$ of orientation preserving diffeomorphisms. The corresponding functor (3.3.2) is as follows. A smooth map $S\to \operatorname{Diffeo}(\Sigma_0)$ is a diffeomorphism of $S\times\Sigma_0$ preserving the projection to $S$. The Lie algebra of this group is $\operatorname{Vect}(\Sigma_0)$, the algebra of $C^\infty$ vector fields.
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[*(b) Gauge groups.*]{} Let $\Sigma_0$ be as before and $E_0$ be a $C^\infty$ complex vector bundle on $\Sigma_0$ Then we have the differentiable group ${\operatorname{Aut}}(E_0)$ of $C^\infty$-automorphisms of $E_0$ (the differentiable structure defined similarly to (a)). Its Lie algebra is ${\operatorname{End}}(E_0)$.
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[*(c) Atiyah groups.*]{} Let $\Sigma_0, E_0$ be as before. The Atiyah group $AT(\Sigma_0, E_0)$ consists of pairs $(\phi,f)$, where $\phi$ is an orientation preserving diffeomorphism of $\Sigma_0$, and $f: \phi^*E_0\to E_0$ is an isomorphism of vector bundles. Thus we have an extension of differentiable groups: $$1\to{\operatorname{Aut}}(E_0) \to AT(\Sigma_0, E_0) \to \operatorname{Diffeo}(\Sigma_0)\to 1.$$ The Lie algebra of $AT(\Sigma_0, E_0)$ is ${{\cal A}}_{E_0}(\Sigma_0)$, the algebra of global $C^\infty$-sections of the Atiyah Lie algebroid.
In fact, the Atiyah group is a particular case of constructions in \[Mac\], §1.4.4-7, namely the group of bisections of the frame groupoid of $E_0$.
More generally, one can replace the vector bundle in Examples (b), (c) by a principal bundle with an arbitrary structure Lie group. In this paper we will be interested in the vector bundle case and will concentrate on the Example (c) as the most general.
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Let us now describe a class of principal bundles with structure groups as in (c). Suppose $q: \Sigma\to B$ is a smooth fibration with compact oriented fibers of dimension $d$. Suppose that $B$ is connected. Then all the fibers $\Sigma_b = q^{-1}(b), b\in B$, are diffeomorphic to each other. Let $\Sigma_0$ be one such fiber. Futher, let $E$ be a smooth ${{\Bbb C}}$-vector bundle on $\Sigma$ and $E_b= E|_{\Sigma_b}$. Then for different $b$ the pairs $(\Sigma_b, E_b)$ are isomorphic, in particular, isomorphic to $(\Sigma_0, E_0)$. Let $G= AT(\Sigma_0, E_0)$. We have then the principal $G$-bundle $$\rho: P=P(\Sigma/B, E)\to B
\leqno (3.4.5)$$ whose fiber $P_b=\rho^{-1}(b), b\in B$, consists of isomoprhisms of pairs $(\Sigma_0, E_0)\to (\Sigma_b, E_b)$.
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For any differentiable $G$-bundle $P$ over a finite-dimensional base $B$ the Atiyah algebra ${{\cal A}}_P$ can be defined by (1.6.3). In the example where $G=AT(\Sigma_0, E_0)$ and $P= P(\Sigma/B, E)$, this gives $${{\cal A}}_{P(\Sigma/B, E)} = q_* {{\cal A}}_E \leqno (3.4.6)$$ (the sheaf-theoretic direct image of the Atiyah algebra of $E$).
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[**(3.5) The first Chern class.**]{} Let $q: \Sigma\to B$ and $E$ be as before, so that we have a principal bundle $P=P(\Sigma/B, E)\to B$ with structure group $G=AT(\Sigma_0, E_0)$. AS the corresponding Lie algebra ${{\frak g}}= {{\cal A}}_{E_0}(\Sigma_0)$ consists of global sections of the Atiyah Lie algebroid of $\Sigma_0$, we have the embeddings $${{\frak g}}\hookrightarrow {{\cal D}}(E_0) \hookrightarrow {\mathfrak{gl}} ({{\cal D}}(E_0)).\leqno (3.5.1)$$ By Corollary 2.3.4, ${\mathfrak gl} ({{\cal D}}(E_0))$ has a unique continuous (in the Frechet topology) cohomology class $c$ in degree $d+1$. We denote by $\gamma$ the restriction of $c$ to ${{\frak g}}$.
(3.5.2) Proposition-Definition. The class $1\otimes\gamma$ is transgressive, so $d_{d+2}(1\otimes\gamma)$ is defined. Moreover, the Lie algebroid ${{\cal A}}_P$ naturally embeds into a bigger Lie algebroid $$0\to q_* ({\mathfrak gl}({{\cal D}}_{\Sigma/B}(E)))\to {{\cal A}}_{\Sigma/B, E} \buildrel \alpha\over
\rightarrow {{\cal T}}_B\to 0. \leqno (3.5.3)$$ Here the fibers of ${\operatorname{Ker}}(\alpha)$ are Lie algebras isomorphic to ${ \mathfrak {gl}}({{\cal D}}_{\Sigma_0, E_0}))$ via an isomoprhism defined uniquely up to an inner automorphism and thus satisfy the sphericity condition (3.1.5) with $n=d+1$. Therefore $d_{d+2}(1\otimes\gamma)$ admits a canonical lifting to a class $$C_1(q_*E) := c_\gamma(P) \in H^{d+2}(B, {{\Bbb C}}).$$
This class will be the main object of study in the rest of the paper.
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To prove Proposition 3.5.2, it is enough to describe the construction of ${{\cal A}}_{\Sigma/B, E}$, as the Hochschild-Serre spectral sequence for it maps into the analogous sequence for ${{\cal A}}_P$.
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[**(3.5.4) Construction of ${{\cal A}}_{\Sigma/B, E}$.**]{} We start with the Atiyah Lie algebroid on $\Sigma$: $$0\to {\operatorname{End}}(E)\buildrel i\over\rightarrow{{\cal A}}_E\buildrel\alpha\over
\rightarrow {{\cal T}}_\Sigma\to 0.\leqno (3.5.5)$$
Let ${ U}({\mathcal A})_{E/B}$ denote the centralizer of $q^{-1} C^\infty_B$ in ${ U}({ A_E})$. Let $F_\bullet{ U}({\mathcal A}_E)$ denote the filtration defined by $F_{-1}{ U}({\mathcal A}_E) = 0$, $F_0{ U}({\mathcal A}_E) =
{ U}({\mathcal A})_{E/B}$ and $[F_i{ U}({\mathcal A}_E), q^{-1} C^\infty_B]\subseteq
F_{i-1}{ U}({\mathcal A}_E)$. Then, $F_1{ U}({\mathcal A}_E)$ is a Lie algebra under the commutator, ${ U}({\mathcal A})_{E/B}$ is a Lie ideal in $F_1{ U}({\mathcal A}_E)$, and there is an exact sequence $$0 \to { U}({\mathcal A})_{E/B} \to
F_1{ U}({\mathcal A}_E) \to q^{-1}{\mathcal T}_B \to 0
\leqno (3.5.6)$$ exhibiting $F_1{ U}({\mathcal A}_E)$ as a transitive $q^{-1} C^\infty_B$-algebroid.
The inclusion ${ {{\cal A}}}_{E} \to
{\mathcal D}_{\Sigma} (E)$ induces the surjective map ${ U}({\mathcal A}_{ E})\to {\mathcal D}_{\Sigma} (E)$ with kernel being the ideal generated by the relation which identifies $1\in C^\infty_\Sigma \subset{ U}({\mathcal A}_{E})$ with $1\in{\underline{\operatorname{End}}}_{C^\infty_\Sigma}({ E})\subset{\mathcal A}_{ E}$. The exact sequence (3.5.6) reduces to the exact sequence $$0 \to {\mathcal D}_{\Sigma/B,{ E}}\to
F_1{\mathcal D}_{\Sigma,{ E}} \to q^{-1}{\mathcal T}_B \to 0.
\leqno (3.5.7)$$
Replacing ${ E}$ by its tensor product by the trivial bundle of rank $r$ in the above example, (3.5.7) can be rewritten as $$0 \to {\mathfrak gl}_r({\mathcal D}_{\Sigma/B, { E}}) \to
F_1{\mathfrak gl}_r({\mathcal D}_{\Sigma, { E}}) \to q^{-1}{\mathcal T}_B \to 0.
\leqno (3.5.8)$$ Taking the limit over inclusions ${\mathfrak gl}_r \to {\mathfrak gl}_{r+1}$ we obtain a $q^{-1}C^\infty_B$-algebroid $$0\to {\mathfrak gl}({\mathcal D}_{\Sigma/B, { E}}) \to
{\mathcal A}_{q, { E}} \to q^{-1}{\mathcal T}_B \to 0. \leqno (3.5.9)$$ Put $${\mathfrak A}_{q, {E}}={\operatorname{Cone}}\biggl\{{\mathfrak gl}({\mathcal D}_{\Sigma/B, { E}}) \to
{\mathcal A}_{q, { E}}\biggr\}. \leqno (3.5.10)$$ There are quasi-isomorphisms $${\mathfrak A}_{q, {E}} \to q^{-1}C^\infty_B, \quad
{ U}_{q^{-1} C^\infty_B}({\mathfrak A}_{q, { E}})\to q^{-1}{\mathcal D}_B.$$
Taking the direct image of (3.5.9) under $q$ and pulling back by the canonical map ${\mathcal T}_B \to q_* q^{-1}{\mathcal T}_B$ we obtain the transitive (because ${ R}^1\pi_*{\mathfrak gl}_r({\mathcal D}_{\Sigma/B, { E}}) = 0$) Lie algebroid on $B$: $$0 \to {\mathcal G} \to
{\mathcal A}_{\Sigma/B, {E}} \to {\mathcal T}_B \to 0, \leqno (3.5.11)$$ where ${\mathcal G} = q_*{\mathfrak gl}({\mathcal D}_{\Sigma/B, { E}})$, as we wanted. Let ${\mathfrak A}_{\Sigma/B, { E}}$ denote the differential graded Lie algebroid on $B$ equal to the cone of the inclusion ${\mathcal G} \to {\mathcal A}_{\Sigma/B, E}$. Let $$\delta_{\Sigma/B} : C^\infty_B \to C_+({\mathcal G})[1]
\leqno (3.5.12)$$ denote the connecting homomorphism; it is a morphism in the derived category of differential graded modules over the universal enveloping (differential graded) algebra ${ U}({\mathfrak A}_{\Sigma/B})$.
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[**(3.6) Smooth cohomology and characteristic classes.**]{} A more traditional way of getting characteristic classes of principal $G$-bundle is by using group cohomology classes of $G$. Let us present a framework which we will then compare with the Lie algebra framework above.
Let $S$ be a topological space and ${{\cal F}}$ be a sheaf of abelian groups on $S$. We denote by $\Phi^\bullet({{\cal F}})$ the standard Godement resolution of ${{\cal F}}$ by flabby sheaves. Thus $\Phi^0({{\cal F}}) = DS({{\cal F}})$ is the sheaf of (possibly discontinuous) sections of the (etale space associated to) ${{\cal F}}$, and $\Phi^{n+1}({{\cal F}}) = DS(\Phi^n({{\cal F}}))$. In this and the next sections we write $R\Gamma(S, {{\cal F}})$ for the complex of global sections $\Gamma(S, \Phi^\bullet({{\cal F}}))$.
Let $G$ be a differentiable group and $B_\bullet G$ be its classifying space. Thus $B_\bullet G= (B_nG)_{n\geq 0}$ is a simplicial object in the category of differentiable spaces with $B_nG=G^n$, and the face and degeneracy maps given by the standard formulas. We define the smooth cohomology of $G$ with coefficients in ${{\Bbb C}}^*$ to be $$H^n_{sm}(G, {{\Bbb C}}^*) = {{\Bbb H}}^n(B_\bullet G, C^{\infty *}). \leqno (3.6.1)$$ Here the hypercohomology on the right is defined as the cohomology of the double complex whose rows are the complexes $R\Gamma(B_nG, C^{\infty *}_{B_nG})$ and the differential between the neighboring slices coming from the simplicial structure on $B_\bullet G$. This is a version of the Segal cohomology theory for topological groups (\[F\], p. 305). In particular, we have a spectral sequence $$H^i(B_n G, {{\Bbb C}}^*) \Rightarrow H^{i+n}_{sm}(G, {{\Bbb C}}^*).\leqno (3.6.2))$$ We will use some other natural (complexes of) sheaves on $B_\bullet G$ to get natural cohomology theories for $G$. For example, the Deligne cohomology $$H^n_{sm}(G, {{\Bbb Z}}_D(p)) = {{\Bbb H}}^n(B_\bullet G, {{\Bbb Z}}_D(p)), \leqno (3.6.3)$$ where for any differentiable space $M$ we set $${{\Bbb Z}}_D(p) = \biggl\{ \underline{{{\Bbb Z}}}_M\to \Omega^0_M\to\Omega^1_m\to ... \to \Omega^{p-1}_M\biggr\},
\leqno (3.6.4)$$ with $\underline{{{\Bbb Z}}}_M$ put in degree 0, compare \[Bry\].
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Let $B$ be a $C^\infty$-manifold and ${{{{\cal U}}}} = \{U_i\}_{i\in I}$ be an open covering of $B$. We denote by $N_\bullet{{{{\cal U}}}}$ the simplicial nerve of ${{{{\cal U}}}}$, i.e., the simplicial manifold with $$N_n{{{{\cal U}}}} = \coprod_{i_0, ..., i_n} U_{i_0} \cap ...\cap U_{i_n}.\leqno (3.6.5)$$ As well known, $N_\bullet{{{{\cal U}}}}$ is homotopy equivalent to $B$, so for any sheaf ${{\cal F}}$ on $B$ we have $${{\Bbb H}}^i(N_\bullet {{{\cal U}}}, {{\cal F}}_\bullet) = H^i(B, {{\cal F}}), \leqno (3.6.6)$$ where ${{\cal F}}_\bullet$ is the natural sheaf on $N_\bullet{{{{\cal U}}}}$ whose $n$th component is the sheaf on (3.6.6) formed by the restrictions of ${{\cal F}}$.
Let $\rho: P\to B$ be a principal $G$-bundle and suppose that $P$ is trivial on each $U_i$. Then a collection of trivializations (i.e., sections) $\tau = (\tau_i: U_i\to P)$ gives a morphism of simplicial differentiable spaces $$u_\tau: N_\bullet{{{\cal U}}}\to B_\bullet G.\leqno (3.6.7)$$ Given a class $\beta\in H^n_{sm}(G, {{\Bbb C}}^*)$, we define the characteristic class $${\frak {c}}_\beta(P) = u_\phi^* (\beta)\in H^n(B, C^{\infty *}_B). \leqno (3.6.8)$$ Similarly one can define characteristic classes corresponding to group cohomology classes with values in the Deligne cohomology.
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[**(3.7) Integrality and integrability.**]{} Let $G$ be as in (3.6), and ${{\frak g}}$ be the Lie algebra of $G$. We construct the “derivative" map $$\partial: H^n_{sm}(G, {{\Bbb C}}^*) \to H^n_{Lie}({{\frak g}}, {{\Bbb C}}).\leqno (3.7.1)$$ To do this, we first remark that for any topological space $S$, any sheaf of abelain groups $
{{\cal F}}$ on $S$ and any point $s_0\in S$ we have a natural morphism of complexes $$\epsilon_{s_0}: R\Gamma(S, {{\cal F}}) \to {{\cal F}}_{s_0},$$ where ${{\cal F}}_{s_0}$ is the stalk of ${{\cal F}}$ at $s_0$. To construct $\epsilon_{s_0}$, we first project $R\Gamma(S, {{\cal F}}) = \Gamma(S, \Phi^\bullet({{\cal F}}))$ to its 0th term $\Gamma(S, \Phi^0({{\cal F}}))$ which, by definition, is the space of all sections $\phi = (s\mapsto \phi_s)$ of the etale space of ${{\cal F}}$. Thus any such $\phi$ is a rule which to any point $s\in S$ associates an element of ${{\cal F}}_s$. We define $\epsilon_{s_0}$ by further mapping any $\phi$ as above to $\phi_{s_0}\in{{\cal F}}_{s_0}$.
We now specialize to $S= B_mG=G^m$, to $s_0 = e_m :=(1, ..., 1)$ and to ${{\cal F}}= C^{\infty *}_{S}$. We get a morphism from the double complex $$\{R\Gamma(B_mG, C^{\infty *}_{B_mG})\}_{m\geq 0}\leqno (3.7.2)$$ to the complex of stalks $${{\Bbb C}}^* \to C^{\infty *}_{G, e_1} \to C^{\infty *}_{G\times G, e_2} \to ...\leqno (3.7.3)$$ An $n$-cocycle in (3.7.2) gives thus a germ of a smooth function $$\xi = \xi(g_1, ..., g_n): G^n\to {{\Bbb C}}^*$$ satisfying the group cocycle equation (on a neighborhood of $e_{n+1}$ in $G^{n+1}$). Similarly to \[F\], p. 293, one associates to $\xi$ a Lie algebra cocycle $\partial(\xi)\in C^{n}({{\frak g}})$ by $$\partial(\xi)(x_1, ..., x_n) = {d\over dt} \log \xi(\exp(t x_1), ..., \exp (t x_n))\biggr|_{t=0}. \leqno (3.7.4)$$
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A Lie algebra cohomology class $\gamma\in H^n({{\frak g}}, {{\Bbb C}})$ will be called integrable, if it lies in the image of the map $\partial$ from (3.7.1). Consider the exponential exact sequence (1.10.1) of sheaves on $B$: and ts coboundary map $\delta_n$ from (1.10.2). The intuition with determinantal $d$-gerbes (0.4) suggests the following.
(3.7.5) Conjecture. (a) The class $\gamma\in H^{d+1}({{\cal A}}_{E_0}(\Sigma_0))$ constructed in (3.5) is integrable and comes from a natural class $\beta\in H^{d+1}_{sm}(AT(\Sigma_0, E_0), {{\Bbb C}}^*)$ (the “higher determinantal class"). (b) Further, for any $q: \Sigma\to B$ and $E$ as above, the class $C_1(q_*E) = c_\gamma(P) \in H^{d+2}(B, {{\Bbb C}})$ is integral and is the image of the following class in the integral cohomology: $$\delta_{d+1}({\frak{c}}_\beta(P))\in H^{d+2}(B, {{\Bbb Z}}).$$
This conjecture holds for $d=1$ (i.e., for the case of a circle fibration). We will verify this in Section 5. In general, the property (b) seems to follow from (a) in virtue of some compatibility result between group cohomology classes with coefficients in ${{\Bbb C}}^*$ and Lie algebra cohomology classes with coefficients in ${{\Bbb C}}$. Here we present a $d=1$ version of such a result.
Let $G$ be a differentiable group with Lie algebra ${{\frak g}}$. Let $\beta\in H^2_{sm}(G, {{\Bbb C}}^*)$ and $\gamma = \partial(\beta)\in H^2_{{\operatorname{Lie}}}({{\frak g}}, {{\Bbb C}})$ be the derivative of $\beta$. Suppose $\beta$ is represented by an extension of differentiable groups $$1\to {{\Bbb C}}^* \to \widetilde{G}\to G\to 1,\leqno (3.7.6)$$ whose Lie algebra is the extension (3.1.9) representing $\gamma$. Let $\rho: P\to B$ be a principal $G$-bundle over a $C^\infty$-manifold $B$. Then we have the characteristic class $c_\gamma(P)\in H^3(B, {{\Bbb C}})$ (lifting to $H^3$ well defined because $\widetilde{{{\frak g}}}$ is a ${G}$-module via the adjoint representation of $\widetilde{G}$, see Example 3.1.8(b)). On the other hand, $\beta$ gives rise to a class ${\frak{c}}_\beta(P)\in H^2(B, C_B^{\infty *})$, see (3.6.8).
(3.7.7) Proposition. In the above situation $c_\gamma(P)\in H^3(B, {{\Bbb C}})$ is the image of $\delta_2({\frak {c}}_\beta(P))\in H^3(B, {{\Bbb Z}})$ under the natural homomorphism from the integral to the complex cohomology.
[*Proof:*]{} This follows from the result of Brylinski (1.10.4) using Example 3.1.8(b) and an obvious generalization of (1.10.3) to differentiable groups.
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We further conjecture the existence of the natural “deloopings" of the higher Chern classes as well, i.e., the existence of classes $$\beta_m \in H^{d+2m}_{sm}(AT(\Sigma_0, E_0), {{\Bbb Z}}_D(m)), \quad m\geq 1, \leqno (3.7.8)$$ which then give characteristic classes in families: $$C_m(q_*E) \in H^{d+2m}(B, {{\Bbb Z}}_D(m)).\leqno (3.7.9)$$
**4. The Real Riemann-Roch.**
Here is the main result of the present paper.
(4.1) Theorem. Let $q:\Sigma\to B$ be a $C^\infty$ fibration with compact oriented fibers of dimension $d$. Let $E$ be a complex $C^\infty$ vector bundle on $\Sigma$. Then: $$C_1(q_*E) = \int_{\Sigma/B}\biggl[ ch(E)\cdot {\operatorname{Td}}({\mathcal T}_{\Sigma/B})\biggr]_{2d+2}
\quad\in\quad
H^{d+2}(B, {\mathbb C}).)$$
The proof consists of several steps.
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[**(4.2) A ${{\cal D}}$-module interpretation of $C_1$ using ${{\cal A}}_{\Sigma/B, E}$.**]{} We use the notations of (3.5.3-12) and introduce the following abbreviations: $${{\cal G}}= q_*({\mathfrak gl} ({{\cal D}}_{\Sigma/B}(E))).\leqno (4.2.1)$$ This is a bundle of infinite-dimensional Lie algebras on $B$. $${\mathfrak A} = {\mathfrak A}_{q,E}.\leqno (4.2.2)$$ This is a DG Lie algebroid on $\Sigma$ quasiisomorphic to ${{\cal T}}_\Sigma$. $$U{\mathfrak A} = U_{q^{-1} C^\infty_B} ({\mathfrak A}_{q,E}). \leqno (4.2.3)$$ This is a sheaf of DG-algebras on $\Sigma$ quasiisomorphic to $q^{-1}{{\cal D}}_B$.
Now, $U{\mathfrak A}$ acts on $C_+({\mathfrak gl}({{\cal D}}_{\Sigma/B, E}))_B$, the reduced relative Lie cochain complex. Further, it acts on the relative Hochschild and cyclic complexes. In the same spirit as in (3.2)(c), elements $Y=(0,Y), Y\in{{\cal A}}_{q,E}$, act via the adjoint action. Elements of the form $\underline{X}=(X,0)$ act via the “insertion operators” $$\iota_X(a_0\otimes ...\otimes a_p) = \sum_{i=0}^p (-1)^i
a_0\otimes ... \otimes a_i\otimes X\otimes a_{i+1}\otimes ... \otimes a_p.\leqno (4.2.4)$$ Denoting by $b,B$ the standard operators on Hoschshild cochains, see \[L\], we have $$[b, \iota _X] = {\operatorname{ad}}(X), \quad [b, \iota_X]=0.$$ Therefore $U{{{\mathfrak A}}}$ acts on both the Hochschild and the cyclic complexes. This action extends to the completions described in (2.4). Further, the morphisms $\mu_{{\cal D}}, \nu_{{\cal D}}$ from (2.4.4-5) are in fact morphisms in $D(U{{\mathfrak A}})$. Indeed, there is a spectral sequence $$E_2^{pq} = {\operatorname{Ext}}^p_{q^{-1}{{\cal D}}_B}\bigl( \underline H^q(\widehat{{\operatorname{Hoch}}}^\bullet(
{{\cal D}}_{\Sigma/B}(E))), C^\infty_B\bigr) \Rightarrow
{\operatorname{Ext}}^{p+q}_{U{{\mathfrak A}}} (\bigr( \widehat{{\operatorname{Hoch}}}^\bullet({{\cal D}}_{\Sigma/B}(E)), C^\infty_B),
\leqno (4.2.5)$$ and similarly for the cyclic complex. The map $\mu_{{\cal D}}$ defines an element of $E_2^{0d}$, and $E_2^{pq}=0$ for $q<d$, so $\mu_{{\cal D}}$ gives rise to a well defined class in ${\operatorname{Ext}}^d$ on the RHS of (4.2.5). Similarly for $\nu_{{\cal D}}$.
Let $$\alpha: C_+({\mathfrak gl}({{\cal D}}_{\Sigma/B}(E)))_B\to q^{-1} C^\infty_B[2d]
\leqno (4.2.6)$$ denote the composition $$C_+({\mathfrak gl}({{\cal D}}_{\Sigma/B}(E)))_B\buildrel\beta\over\longrightarrow
CC_\bullet({{\cal D}}_{\Sigma/B}(E))_B[1] \to \widehat{CC}_\bullet({{\cal D}}_{\Sigma/B}(E))_B[1]
\buildrel \nu_{{\cal D}}[1]\over\longrightarrow q^{-1} C^\infty_B[2d+1].\leqno (4.2.7)$$ Here the first morphism $\beta$ is the standard map from the Lie algebra chain complex to the cyclic complex, see \[L\], (10.2.3).
It is checked directly that $\beta$ commutes with the operators $\iota_X$, so it is $U{{\mathfrak A}}$-invariant. Therefore, all maps in (4.2.7) and the map $\alpha$ are morphisms in $D(U{{\mathfrak A}})$.
Let us now thake the $C^\infty$ direct image and define the morphism $$\int_{\Sigma/B}\alpha: C_+({{\cal G}})_B\to C^\infty_B[d] \leqno (4.2.8)$$ as the composition $$C_+({{\cal G}})_B \to q_* C_+{\mathfrak gl}({{\cal D}}_{\Sigma/B}(E))_B \buildrel\sim\over
\longrightarrow Rq_*C_+ {\mathfrak gl} ({{\cal D}}_{\Sigma/B}(E))_B \buildrel\alpha\over\longrightarrow$$ $$\to Rq_*q^{-1} C^\infty_B[2d+1]\buildrel \int_{\Sigma/B}\over\longrightarrow C^\infty_B[d+1].
\leqno (4.2.9)$$ Here the last map is the integration over the relative (topological) fundamental class of $\Sigma/B$.
Consider the composition $$C^\infty_B\buildrel\delta_{\Sigma/B}\over\longrightarrow
C_+({{\cal G}})_B[1]\buildrel \int_{\Sigma/B}\alpha\over\longrightarrow C^\infty_B[d+2],
\leqno (4.2.10)$$ where $\delta_{\Sigma/B}$ is as in (3.5.12). As both maps in (4.2.10) are morphisms in $D({{\cal D}}_B)$, the composition (denote it $C$) is an element $$C\in {\operatorname{Ext}}^{d+2}_{{{\cal D}}_B}(C^\infty_B, C^\infty_B) = H^{d+2}(B, {{\Bbb C}}).$$
(4.2.11) Proposition. We have $C= C_1(q_*E)$.
[*Proof:*]{} This follows from the interpretation of $C_1(q_*E) = c_\gamma(P(\Sigma/B, E))$ given in (3.2) (b), (c) and from the compatibility of the Atiyah algebroid of $P(\Sigma/B)$ with ${{\cal A}}_{\Sigma/B, E}$.
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[**(4.3) A local RRR in the total space.**]{} Proposition 4.2.11 reduces the RRR to the following “local" statement taking place in the total space $\Sigma$.
(4.3.1) Theorem. Let $\xi$ be the morphism in $D(q^{-1} {{\cal D}}_B)$ defined as the composition $$q^{-1}{{\cal O}}_B\to C_+{\mathfrak gl} ({{\cal D}}_{\Sigma/B, E}))[1] \to q^{-1}{{\cal O}}_B[2d+2].$$ Then the class in $${\operatorname{Ext}}^{2d+2}_{q^{-1}{{\cal D}}_B}(q^{-1} {{\cal O}}_B, q^{-1} {{\cal O}}_B) = H^{2d+2}(\Sigma, {{\Bbb C}}),$$ corresponding to $\xi$, is equal to $$\biggl[ch(E)\cdot {\operatorname{Td}}({{\cal T}}_{\Sigma/B})\biggr]_{2d+2}.$$
We now concentrate on the proof of Theorem 4.3.1. First, we remind the definition of periodic cyclic homology \[L\]. Let $A$ be an associative algebra. The “negative" cyclic complex of $A$ is defined, similarly to (2.1.4), as $$CC_\bullet^-(A) = \operatorname{Tot} \biggl\{ {\operatorname{Hoch}}_\bullet(A) \buildrel N\over\longrightarrow
{\operatorname{Hoch}}_\bullet(A)\buildrel 1-\tau\over\longrightarrow {\operatorname{Hoch}}_\bullet(A) \to ...\biggr\}
\leqno (4.3.2)$$ Here, the grading of the copies of ${\operatorname{Hoch}}_\bullet(A)$ in the horizontal direction goes in increasing integers 0,1,2 etc. So $CC_\bullet^-(A)$ s a module over the formal Taylor series ring ${{\Bbb C}}[[u]]$ where $u$ has degree $(-2)$. The original cyclic complex is a module over the polynomial ring ${{\Bbb C}}[u^{-1}]$. Finally, the periodic cyclic complex $CC_\bullet^{per}(A)$ is obtained by merging together $CC_\bullet(A)$ and $CC_\bullet^{-}(A)$ into one double complex which is repeated 2-periodically both in the positive and negative horizontal directions. In other words, $$CC_\bullet^{per}(A) = CC_\bullet^-(A)\otimes _{{{\Bbb C}}[[u]]} {{\Bbb C}}((u)).\leqno (4.3.3)$$ We extend these construction to other situations (see §2) where the tensor products are understood in the sense of various completions. In particular, the morphism $\nu_D$ from (2.4.5) extends to morphisms $$\nu_D^-: CC^-_\bullet({{\cal D}}_{\Sigma/B, E})\to q^{-1}C^\infty_B[2d] [[u]], \quad
\nu_D^{per}: CC_\bullet^{per}({{\cal D}}_{\Sigma/B, E}) \to q^{-1}C^\infty_B[2d]((u)).$$ These morphisms include into the commutative diagram $$\begin{CD}
CC^-_\bullet ({\mathcal D}_{\Sigma/B , { E}}) @>>> CC^{per}_\bullet ({\mathcal D}_{\Sigma/B, { E}})
@>>> CC({\mathcal D}_{X/S, {\mathcal E}})[2] \\
@V{\nu_{\mathcal D}^-}VV @V{\nu_{\mathcal D}^{per}}VV
@VV{\nu_{\mathcal D}}V \\
{\mathcal O}_S[2d][[u]] @>>> {\mathcal O}_S[2d]((u))
@>{Res_{u=0}}>> {\mathcal O}_S[2d+2]
\end{CD} \leqno (4.3.4)$$ We now want to reduce Theorem 4.3.1 to the following statement.
(4.3.5) Theorem. The composition $$C^\infty_B \buildrel +1 \over\rightarrow CC_\bullet^{per}({{\cal D}}_{\Sigma/B, E}) \buildrel
\nu_{{\cal D}}^{per}\over\longrightarrow C^\infty_B[2d]((u))$$ is equal to $$\sum_{i=0}^\infty u^i \cdot \bigl[{\operatorname{ch}}(E){\operatorname{Td}}({{\cal T}}_{\Sigma/B})\bigr]_{2(d-i)}.$$
Indeed, suppose we know Theorem 4.3.5. To prove Theorem 4.3.1, it would then be sufficient to prove that the composition $$q^{-1} C^\infty_B\to C_+({\frak{gl}}({{\cal D}}_{\Sigma/B}))[1] \to CC_\bullet ({{\cal D}}_{\Sigma/B, E})[2]
\leqno (4.3.6)$$ is equal to the composition $$q^{-1}C^\infty_B \buildrel +1\over\rightarrow CC_\bullet^{per} ({{\cal D}}_{\Sigma/B, E}) \to
CC_\bullet ({{\cal D}}_{\Sigma/B, E})[2],\leqno (4.3.7)$$ as the latter one is related to Chern and Todd via (4.3.5). In order to perform the comparison, let $K$ be the cone of the inclusion $C_+({\frak{gl}}({{\cal D}}_{\Sigma/B, E})\to C_\bullet({\frak{gl}}({{\cal D}}_{\Sigma/B, E}))$, so that we have a quasi-isomorphism $K\to q^{-1}C^\infty_B$ as well as an isomorphism of distinguished triangles $$\begin{CD}
C_\bullet({\mathfrak gl}({\mathcal D}_{\Sigma/B, { E}})) @>>> K @>>> C_+({\mathfrak
gl}({\mathcal D}_{\Sigma/B, { E}}))[1] \\
@VVV @VVV @VVV \\
C({\mathfrak gl}({\mathcal D}_{\Sigma/B, {\mathcal E}})) @>>> q^{-1}C^\infty_B
@>>> C_+({\mathfrak gl}({\mathcal D}_{\Sigma/B, E}))[1]
\end{CD}$$ (with the top row a short exact sequence of complexes). Notice now that there is a morphism of distinguished triangles $$\begin{CD}
C_\bullet ({\mathfrak gl}({\mathcal D}_{\Sigma/B, {E}})) @>>> K @>>> C_+({\mathfrak
gl}({\mathcal D}_{\Sigma/B, {E}}))[1] \\
@VVV @VVV @VVV \\
CC_\bullet^-({\mathcal D}_{\Sigma/B, { E}}) @>>> CC_\bullet^{per}({\mathcal D}_{\Sigma/B, { E}}) @>>>
CC_\bullet ({\mathcal D}_{\Sigma/B, { E}})[2]
\end{CD}$$ It remains to notice further that the diagram $$q^{-1}C^\infty_B@<<< K @>>> CC_\bullet ^{per}({\mathcal D}_{\Sigma/B, { E}})$$ represents the morphism $C^\infty_B @>{1}>> CC^{per}_\bullet ({\mathcal
D}_{\Sigma/B, { E}})$ in the derived category and the proof is finished.
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[**(4.4) Proof of Theorem 4.3.5.**]{} This statement can be deduced from the results of \[NT\] on the cohomology of the Lie algebras of formal vector fields and formal matrix functions. We recall the setting of \[NT\] which extends that of the Chern-Weil definition of characteristic classes. Recall that the latter provides a map $$S^\bullet[[{\mathfrak{h}}_0]]^{H_0} \to H^{2\bullet}(\Sigma, {{\Bbb C}}),\leqno (4.4.1)$$ where $H_0 = GL_d({{\Bbb C}})\times GL_r({{\Bbb C}})$ with $r= {\operatorname{rk}}(E)$, while $\mathfrak{h}_0$ is the Lie algebra of $H_0$, i.e., ${\mathfrak{gl}}_d({{\Bbb C}}) \oplus {\mathfrak{gl}}_r({{\Bbb C}})$. To be precise, the elementary symmetric functions of the two copies of $\mathfrak{gl}$ are mapped to the Chern classes of ${{\cal T}}_{\Sigma/B}$ and $E$.
In \[NT\], this construction was generalized in the following way. Let $k=\dim(B)$, and ${\widehat {\mathfrak g}}$ be the Lie algebra of formal differential operators of the form $$\sum_{i=1}^k P_i(y_1, \ldots, y_k){\frac{\partial}{\partial{y_i}}}+\sum_{j=1}^d Q_j(x_1, \ldots, x_d, y_1, \ldots, y_k){\frac{\partial}{\partial{x_i}}}+
R(x_1, \ldots, x_d, y_1, \ldots, y_k)\leqno (4.4.2)$$ where $P_i$, $Q_j$ are formal power series, $R(x)$ is an $r\times r$ matrix whose entries are power series. Thus ${\widehat {\mathfrak g}}$ is the formal version of the relative Atiyah algebra. Consider the Lie subalgebra $\mathfrak{h}$ of fields such that all $P_i$ and $Q_j$ are of degree one and all entries of $R$ are of degree zero. We can identify this subalgebra with $${\mathfrak h}={\mathfrak {gl}}_d ({\mathbb C}) \oplus {\mathfrak {gl}}_k ({\mathbb C})\oplus {\mathfrak {gl}}_r({\mathbb C})$$ Let $${H} ={\operatorname{GL}}_d ({\mathbb C}) \times {\operatorname{GL}}_k ({\mathbb C})\times {\operatorname{GL}}_r ({\mathbb C})$$ be the corresponding Lie group. Thus $({\widehat{\mathfrak g}}, H)$ form a Harish-Chandra pair. Following the ideas of “formal geometry" (or “localization") of Gelfand and Kazhdan, one sees that every $({\widehat{\mathfrak g}}, H)$-module $L$ induces a sheaf ${\mathcal L}$ on $\Sigma$. Similarly, a complex $L^\bullet$ of modules gives rise to a complex of sheaves ${{\cal L}}^\bullet$. A complex $L^\bullet$ of modules is called [*homotopy constant*]{}, i.e. the action of ${\widehat {\mathfrak g}}$ extends to an action of the differential graded Lie algebra $({\widehat {\mathfrak g}}[\epsilon], {\frac{\partial}{\partial{\epsilon}}})$. Here $\epsilon$ is a formal variable of degree $-1$ and square zero. In this case, there is a generalization of the Chern-Weil map constructed in \[NT\] : $${\operatorname{CW}}:{\mathbb H}^{\bullet}({\mathfrak h}_0[\epsilon], {\mathfrak h}_0; L^{\bullet}) \to {\mathbb H}^{\bullet}(\Sigma,{\mathcal L}^{\bullet}),
\leqno (4.4.3)$$ which gives (4.4.1) when $L={\mathbb C}$ with the trivial action. Consider the following $({\widehat{\mathfrak g}}, H)$-modules: $${\mathcal D}=\biggl\{\sum P_{\alpha}(x_1, \ldots, x_d, y_1, \ldots, y_k){\partial_x ^{\alpha}}\biggr\},$$ where $P_{\alpha}$ are $r\times r$ matrices whose entries are power series, and $$\Omega ^{\bullet} =\biggl\{ \sum _I P_I(x_1, \ldots, x_d, y_1, \ldots, y_k) d^Ix\biggr\},$$ which is the space of differential forms whose coefficients are formal power series. The latter is a complex with the (fiberwise) De Rham differential. Moreover, $\Omega ^{\bullet}$ is homotopy constant ($\epsilon {\widehat {\mathfrak g}}$ acts by exterior multiplication). The Hochschild, cyclic, etc. complexes of ${\mathcal D}$ inherit the $({\widehat{\mathfrak g}}, H)$-module structure; moreover, they also become homotopy constant (the $\epsilon X \in \epsilon {\widehat {\mathfrak g}}$ acts by operators $\iota _X$ from (4.2.4)). One constructs (\[BNT\], pt. II, Lemma 3.2.4) a class $$\nu \in {\mathbb H}^0 ({\mathfrak h}_0[\epsilon], {\mathfrak h}_0; {\underline {\operatorname{Hom}}}
({\operatorname{CC}}^{\operatorname{per}}_{-{\bullet}}({\mathcal D}),\Omega^{2d+\bullet})), \leqno (4.4.3)$$ such that ${\operatorname{CW}}(\nu)$ coincides with $$\nu_{\mathcal D} \in {\mathbb H}^0 (X; {\underline {\operatorname{Hom}}}({\operatorname{CC}}^{\operatorname{per}}_{-{\bullet}}({\mathcal D}_{\Sigma/B}),
\Omega^{2d+\bullet}_{\Sigma/B})).$$ To be precise, the cited lemma concerns the Weyl algebra of power series in both coordinates and derivations with the Moyal product (clearly, differential operators of finite order form a subalgebra). Second, the construction there is for the relative cohomology of the pair $({{\frak g}}, {\mathfrak h})$ but it extends to the case of the pair $({{\frak g}}[\epsilon], {\mathfrak h})$ of which $({\mathfrak h}_0 [\epsilon], {\mathfrak h}_0)$ is a sub-pair.
The cochain $\nu$ is actually independent of $y$. There is the canonical class $1$ in ${\operatorname {HC_0}}^{\operatorname {per}}({\mathcal D})$; it is ${\mathfrak h}_0$-invariant, and it is shown in \[NT\] how to extend it to a class in ${\mathbb H}^0 ({\mathfrak h}_0[\epsilon], {\mathfrak h}_0; {\operatorname{CC}}^{\operatorname{per}}_{-{\bullet}}({\mathcal D}))$. On the other hand, $${\mathbb H}^0 ({\mathfrak h}_0[\epsilon], {\mathfrak h}_0; \Omega^{\bullet})$$ can be naturally identified with $${\mathbb H}^0 ({\mathfrak h}_0[\epsilon], {\mathfrak h}_0; {\mathbb C})$$ It remains to show that $$\nu(1) = \sum [{\operatorname{ch}}\cdot{\operatorname{Td}}]_{2(d+i)}\cdot u^i$$ where ${\operatorname{ch}}$ is the corresponding invariant power series in $H^{\bullet}({\mathfrak {gl}}_r [\epsilon],
{\mathfrak {gl}}_r; {\mathbb C})$ and ${\operatorname{Td}}$ is the corresponding invariant power series in $H^{\bullet}({\mathfrak {gl}}_d [\epsilon],
{\mathfrak {gl}}_d; {\mathbb C})$. This was carried out in \[BNT\], Lemma 5.3.2.
**5. Comparison with the gerbe picture**
[**(5.1) $L^2$-sections of a vector bundle on a circle.**]{} Let $\Sigma$ be an oriented $C^\infty$-manifold diffeomorphic to the circle $S^1$ with the standard orientation, and let $E$ be a complex $C^\infty$-vector bundle on $\Sigma$. Choose a smooth Riemannian metric $g$ on $\Sigma$ and a smooth Hermitian metric $h$ on $E$. Let $\Gamma(\Sigma, E)$ be the space of $C^\infty$-sections of $E$. The choise of $g,h$ defines a positive definite scalar product on this space and we denote by $L^2_{g,h}(\Sigma, E)$ the Hilbert space obtained by completion with respect to this scalar product.
(5.1.1) Lemma. For a different choice $g', h'$ of metrics on $\Sigma, E$ we have a canonical identification of topological vector spaces $$L^2_{g,h}(\Sigma, E)\to L^2_{g', h'}(\Sigma, E).$$
[*Proof:*]{} The Hilbert norms on $\Gamma(\Sigma, E)$ associated to $(g,h)$ and $(g', h')$ are equivalent, since $\Sigma$ is compact.
.1cm
So we will denote the completion simply by $L^2(\Sigma, E)$.
.2cm
Consider now the case when $\Sigma=S^1$ is the standard circle and $E={{\Bbb C}}^r$ is the trivial bundle of rank $r$. In this case $L^2(\Sigma, E) = L^2(S^1)^{\oplus r}$. Let us denote this Hilbert space by $H$. It comes with a polarization in the sense of Pressley and Segal \[PS\]. In other words, $H$ is decomposed as $H_+\oplus H_-$ where $H_+, H_-$ are infinite-dimensional ortogonal closed subspaces defined as follows.
$H_+$ consists of vector-functions extending holomorphically into the unit disk $D_+ = \{ |z| <1\}$. The space $H_-$ consists of vector functions extending holomorphically into the opposite annulus $D_- = \{|z|>1\}$ and vanishing at $\infty$.
The decomposition $H= H_+\oplus H_-$ yields the groups $GL_{res}(H)\subset GL(H)$, see \[PS\] (6.2.1), as well as the Sato Grassmannian $Gr(H)$ on which $GL_{res}(H)$ acts transitively. We recall that $Gr(H)$ consists of closed subspaces $W\subset H$ whose projection to $H_+$ is a Fredholm operator and the projection to $H_-$ is a Hilbert-Schmidt operator, see \[PS\] (7.1.1).
Given arbitrary $\Sigma, E$ as before, we can choose an orientation preserving diffeomorphism $\phi:
S^1\to\Sigma$ and a trivialization $\psi: \phi^*E\to {{\Bbb C}}^r$. This gives an identification $$u_{\phi, \psi}: L^2(\Sigma, E)\to H= L^2(S^1)^{\oplus r}.$$ In particular, we get a distinguished set of subspaces in $L^2(\Sigma, E)$, namely $$Gr_{\phi, \psi}(\Sigma, E) = u_{\phi, \psi}^{-1}(Gr(H)),$$ and a distinguished subgroup of its automorphisms, namely $$GL_{res}^{\phi, \psi}( L^2(\Sigma, E)) = u_{\phi, \psi}^{-1} GL_{res}(H) u_{\phi, \psi}.$$
(5.1.2) Lemma. The subgroup $GL_{res}^{\phi, \psi}(L^2(\Sigma, E))$ and the set $Gr_{\phi, \psi}(L^2(\Sigma, E))$ are independent on the choice of $\phi$ and $\psi$.
[*Proof:*]{} Any two choices of $\phi, \psi$ differ by an element of the Atiyah group $AT(S^1, {{\Bbb C}}^r)$, see Example 3.3.5(c). This group being a semidirect product of $\operatorname{Diffeo}(S^1)$ and $GL_r C^\infty (S^1)$, our statement follows from the known fact that both of these groups are subgroups of $GL_{res}(H)$, see \[PS\].
So we will drop $\phi, \psi$ from the notation, writing $Gr(L^2(\Sigma, E))$ and $GL_{res}(L^2(\Sigma, E))$.
Recall further that $Gr(H)\times Gr(H)$ is equipped with a line bundle $\Delta$ (the relative determinantal bundle) which has the following additional structures:
.2cm
\(a) Equivariance with respect to $GL_{res}(H)$.
.1cm
\(b) A multiplicative structure, i.e., an identification $$p_{12}^*\Delta\otimes p_{23}^*\Delta \to p_{13}^*\Delta\leqno (5.1.3)$$ of vector bundles on $Gr(H)\times Gr(H)\times Gr(H)$, which is equivariant under $GL_{res}(H)$ and satisfies the associatiivity, unit and inversion properties.
.2cm
It follows from the above, that we have a canonically defined line bundle (still denoted $\Delta$) on $Gr(L^2(\Sigma, E))$ equivariant under $GL_{res}(L^2(\Sigma, E))$ and equipped with a multiplicative structure. For $W, W'\in Gr(L^2(\Sigma, E))$ we denote by $\Delta_{W, W'}$ the fiber of $\Delta$ at $(W, W')$.
As well known, the multiplicative bundle $\Delta$ gives rise to a category (${{\Bbb C}}^*$-gerbe) ${{\cal D}}et\, L^2(\Sigma, E)$ whose objects for the set $Gr(L^2(\Sigma, E))$, while $${\operatorname{Hom}}_{{{\cal D}}et\, L^2(\Sigma, E)} (W, W') = \Delta_{W, W'}-\{0\}.$$ The composition of morphisms comes from the identification $$\Delta_{W, W'}\otimes \Delta_{W', W''}\to \Delta_{W, W''}$$ given by (5.1.3).
.3cm
[**(5.2) $L^2$-direct image in a circle fibration.**]{} Let now $q:\Sigma\to B$ be a fibration in oriented circles and $E$ be a vector bundle on $\Sigma$. We have then a bundle of Hilbert spaces $q_*^{L^2}(E)$ whose fiber at $b\in B$ is $L^2(\Sigma_b, E_b)$. Further, by (5.2) this bundle has a $GL_{res}(H)$-structure, where $H= L^2(S^1)^{\oplus r}$. Therefore we have the associated bundle of Sato Grassmannians $Gr(q_*^{L^2}(E))$ on $B$ and the (fiberwise) multiplicative line bundle $\Delta$ on $$Gr(q_*^{L^2}(E))\times_B Gr(q_*^{L^2}(E)).$$ We define a sheaf of $C_B^{\infty*}$-groupoids on $B$ whose local objects are local sections of $Gr(q_*^{L^2}(E))$ and for any two such sections defined on $U\subset B$ $${\underline{\operatorname{Hom}}}(s_1, s_2) = (s_1, s_2)^*\Delta - 0_U,$$ where $0_U$ stands for the zero section of the induced line bundle. This sheaf of groupoids is locally connected and so gives rise to a $C_B^{\infty *}$-gerbe which we denote ${{\cal D}}et (q_* E)$. So we have the class $$\bigl[ {{\cal D}}et(q_* e) \bigr] \in H^2(B, C_B^{\infty *}).$$ Alternatively, consider the Atiyah group $G= AT(S^1, {{\Bbb C}}^r)$, see Example 3.3.5(c). By the above, $G\subset GL_{res}(H)$. The determinantal ${{\Bbb C}}^*$-gerbe ${{\cal D}}et(H)$ (over a point) with $G$-action gives a central extension $\widetilde{G}$ of $G$ by ${{\Bbb C}}^*$. A circle fibration $q:\Sigma\to B$ gives a principal $G$-bundlle $P(\Sigma/B)$, as in (3.4.5), and the following is clear.
(5.2.1) Proposition. The gerbe ${{\cal D}}et(q_*E)$ is equivalent to ${\operatorname{Lift}}_G^{\widetilde{G}}(P(\Sigma/B, E))$, see Example 1.10.3.
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Consider the exponential sequence (1.10.1) of sheaves on $B$ and the corresponding coboundary map $\delta_2$, see (1.10.2). Then we have the class $$\delta_2 \bigl[ {{\cal D}}et(q_*E)\bigr] \in H^3(B, {{\Bbb Z}}).$$
(5.2.2) Theorem. The image of $\delta \bigl[ {{\cal D}}et(q_*E)\bigr]$ in $H^3(B, {{\Bbb C}})$ coincides with negative of the class $C_1(q_*E)$ defined in (3.5.2).
To prove Theorem 5.2.2, we apply Proposition 3.7.7 to $G= AT(S^1, {{\Bbb C}}^r)$ and $\beta$ being the class of the central extension $\widetilde{G}$. Then ${{\frak g}}= {{\cal A}}_{{{\Bbb C}}^r}(S^1)$ is the Atiyah algebra of the trivial bundle on $S^1$ and $\gamma$ is the class of the “trace" central extension induced from the Lie algebra ${\frak{gl}}_{res}(H)$ of $GL_{res}(H)$. We have the embeddings $${{\frak g}}\subset {\frak{gl}}_r({{\cal D}}(S^1)) \subset {\frak{gl}}_{res}(H),$$ and the trace central extension is represented by an explicit cocycle $\Psi$ of ${\frak{gl}}_{res}(H)$ (going back to \[T\]). Let $z$ be the standard complex coordinate on $S^1$ such that $|z|=1$. Then the formula for the restriction of $\Psi$ to ${\frak{gl}}_r({{\cal D}}(S^1))$ was given in \[KP\], see also \[KR\], formula (1.5.2): $$\Psi(f(z) \partial_z^m, g(z) \partial_z^n) = {m! n!\over (m+n+1)!} {\operatorname{Res}}_{z=0} dz \cdot {\operatorname{Tr}}(f^{(n+1}(z) g^{(m)}(z)),
\leqno (5.2.3)$$ where $f^{(n)}$ means the $n$th derivative in $z$. Our statement now reduces to the following.
(5.2.4) Lemma. The second Lie cohomology class of ${\frak{gl}}_r {{\cal D}}(S^1)$ given by the cocycle $\Psi$ is equal to the negative of the class corresponding to the fundamental class of $S^1$ via the identification (2.3.4).
[*Proof:*]{} As the space of (continuous) Lie algebra homology in question is 1-dimensional, it is enough to evaluate the cocycle $\Psi$ on the Lie algebra 2-homology class $\sigma$ from (2.3.4) and to show that this value is precisely equal to 1. For this it is enough to consider $r=1$. Let ${{\cal D}}= {{\cal D}}(S^1)$ for simplicity.
We need to recall the explicit form of the identification (2.3.1) for the case $n=1$ (first Hochschild homology maps to the second Lie algebra homology). In other words, we need to recall the definition of the map. $$\epsilon: HH_1({{\cal D}})\to H_2^{{\operatorname{Lie}}}({\frak{gl}}({{\cal D}}))\to {{\Bbb C}}.$$ As explained in \[BG\] and \[W\], this map is defined via the order filtration $F$ on the ring ${{\cal D}}$ and uses the corresponding spectral sequence. This means we need to start with a Hochschild 1-cycle $\sigma = \sum P_i\otimes Q_i\in {{\cal D}}\otimes{{\cal D}}$ and form its highest symbol cycle $${\operatorname{Smbl}}(\sigma) = \sum {\operatorname{Smbl}}(P_i)\otimes {\operatorname{Smbl}}(Q_i)\in {\operatorname{gr}}({{\cal D}})\otimes{\operatorname{gr}}({{\cal D}}),$$ which gives an element in ${\operatorname{Hoch}}_1({\operatorname{gr}}({{\cal D}}))$. As ${\operatorname{gr}}({{\cal D}})$ is the ring of polynomial functions on $T^*S^1$, Hochschild-Kostant-Rosenberg gives $HH_1({\operatorname{gr}}({{\cal D}})) = \Omega^1({T^*S^1})$, the space of 1-forms on $T^*S^1$ polynomial along the fibers. So the class of ${\operatorname{Smbl}}(\sigma)$ is a 1-form $\omega = \omega(\sigma)$ in $T^*S^1$. This is an element of the $E_1$-term of the spectral sequence for the Hochschild homology of the filtered ring ${{\cal D}}$.
Further, one denotes by $*$ the symplectic Hodge operator in forms on $T^*S^1$. The results of [*loc. cit.*]{} imply the differential in the $E_1$-term is $*d*$ where $d$ is the de Rham differential on $T^*S^1$ while higher differentials vanish. This means that under our assumptions $*\omega(\sigma)$ is a closed 1-form and $$\epsilon(\sigma) = \int_{S^1} *\omega(\sigma).$$ To show Lemma 5.3.5 we need to exhibit just one $\sigma$ as above such that $$0\neq \epsilon (\sigma) = \Psi(\sigma) := \sum \Psi(P_i, Q_i) .$$
We take $$\sigma = z^2\otimes z^{-1} \partial_z - 2 z\otimes\partial_z.$$ Then one sees that $\sigma$ is a Hochschild 1-cycle and $\Psi(\sigma) = 1$. On the other hand, let $\theta$ be the real coordinate on $S^1$ so that $z=\exp(2\pi i \theta)$. Then the real coordinates on $T^*S^1$ are $\theta, \xi$ with $\xi = {\operatorname{Smbl}}(\partial/\partial \theta)$, so the Poisson bracket $\{\theta, \xi\}$ is equal to 1. In terms of the coordinate $z$ it means that $\xi = {\operatorname{Smbl}}(z \partial/\partial z)$ and $\{z, \xi\} = z$. Therefore $${\operatorname{Smbl}}(\sigma) = z^2\otimes z^{-2}\xi - 2 z\otimes z^{-1}\xi$$ and hence $$\omega (\sigma) = z^2 d(z^{-2}\xi) = 2 z d(z^{-1}\xi) = -dz - z^{-1}\xi,$$ see \[L\] p. 11. The symplectic (volume) form on $T^*S^1$ is $(dz/z)\wedge d\xi$, so the symplectic Hodge operator is given by $$*d\xi = dz/z, \quad *dz/z = d\xi, \quad *^2=1.$$ Therefore $$*\omega(\sigma) = -dz/z -\xi d\xi, \quad \int_{S^1} *\omega(\sigma) = -1$$ and we are done.
**References**
\[AH\] M. F. Atiyah, F. Hirzebruch, [*Riemann-Roch theorems for differentiable manifolds*]{}, Bull. Amer. Math. Soc. [**65**]{} (1959), p. 276–281.
.1cm
\[Bl\] S. Bloch, $K_2$ [*and algebraic cycles*]{}, Ann. of Math. (2) [**99**]{} (1974), 349–379.
.1cm
\[Bo\] R. Bott, [*On the characteristic classes of groups of diffeomorphisms*]{}, L’Enseignment Math. [**23**]{} (1977), 209-220.
.1cm
\[BNT\] P. Bressler, R. Nest, B. Tsygan, [*Riemann-Roch theorems via deformation quantization. I, II*]{}, Adv. Math. [**167**]{} (2002), no. 1, p. 1–25, 26–73.
.1cm
\[Bry\] L. Breen, [*On the classification of 2-gerbes and 2-stacks,*]{} Astérisque, [**225**]{} (1994), 160 pp.
.1cm
\[Bry\] J. L. Brylinski, [*Loop Spaces, Characteristic Classes and Geometric Quantization*]{}, Birkhauser, Boston, 1993.
.1cm
\[BG\] J.-L. Brylinski, E. Getzler, [*The homology of algebras of pseudodifferential symbols and the noncommutative residue*]{}, $K$-Theory [**1**]{} (1987), no. 4, 385–403.
.1cm
\[C\] K. T. Chen, [*Iterated path integrals*]{}, Bull. Amer. Math. Soc. [**83**]{} (1977), no. 5, 831–879.
.1cm
\[D1\] P. Deligne, [*Le determinant de la cohomologie*]{}, Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), 93–177, Contemp. Math., 67, Amer. Math. Soc., Providence, RI, 1987.
.1cm
\[D2\] P. Deligne, [*Le groupe fondamental de la droite projective moins trois points*]{}, Galois groups over $Q$ (Berkeley, CA, 1987), 79–297, Math. Sci. Res. Inst. Publ., [**16**]{}, Springer, New York, 1989.
.1cm
\[E\] R. Elkik, [*Fibrés d’intersections et intégrales de classes de Chern,*]{} Ann. Sci. École Norm. Sup. (4) [**22**]{} (1989), no. 2, 195–226.
.1cm
\[FT\] B. Feigin, B. Tsygan, [*Riemann-Roch theorem and Lie algebra cohomology, I*]{}, Proceedings of the Winter School on Geometry and Physics (Srní, 1988). Rend. Circ. Mat. Palermo (2) Suppl. [**21**]{} (1989), 15–52.
.1cm
\[F\] D. B. Fuks, [*Cohomology of Infinite-Dimensional Lie Algebras*]{}, Consultants Bureau, New York and London, 1986.
.1cm
\[HS\] V. Hinich, V. Schechtman, [*Deformation theory and Lie algebra homology, I,II*]{}, Algebra Colloq. [**4**]{} (1997), no. 2, 213–240, 291–316.
.1cm
\[KP\] V. Kac, D. Peterson, [*Spin and wedge representations of infinite-dimensional Lie algebras and groups*]{}, Proc. Natl. Acad. Sci. USA, [**78**]{} (1981), 3308-3312.
.1cm
\[KR\] V. Kac, A. Radul, [*Quasifinite highest weight modules over the Lie algebra of differential operators on the circle,*]{} Comm. Math. Phys. [**157**]{} (1993), no. 3, 429–457.
.1cm
\[Kal\] R. Kallstrom, [*Smooth Modules over Lie Algebroids I*]{}, preprint math.AG/9808108.
.1cm
\[KV1\] M. Kapranov, E. Vasserot, [*Vertex algebras and the formal loop space*]{}, Publ. Math. Inst. Hautes Etudes Sci. [**100**]{} (2004), 209–269.
.1cm
\[KV2\] M. Kapranov, E. Vasserot, [*Formal loops II: a local Riemann-Roch theorem for determinantal gerbes*]{}, preprint math.AG/0509646.
.1cm
\[LM\] G. Laumon, L. Moret-Bailly, [*Champs Algébriques*]{}, Springer-Verlag, Berlin, 2000.
.1cm
\[L\] J. L. Loday, [*Cyclic Homology*]{}, Second edition, Springer-Verlag, Berlin, 1998.
.1cm
\[Lo\] J. Lott, [*Higher-degree analogs of the determinant line bundle*]{}, Comm. Math. Phys. [**230**]{} (2002), no. 1, 41–69. .1cm
\[Mac\] K. C. H. McKenzie, [*General Theory of Lie Groupoids and Lie Algebroids*]{}, London Mathematical Society Lecture Note Series, [**213**]{}. Cambridge University Press, Cambridge, 2005.
.1cm
\[Mo\] S. Morita, [*Geometry of Characteristic Classes*]{}, Translations of Mathematical Monographs, [**199**]{}. Iwanami Series in Modern Mathematics. American Mathematical Society, Providence, RI, 2001
.1cm
\[NT\] R. Nest, B. Tsygan, [*Algebraic index theorem for families*]{}, Adv. Math. [**113**]{} (1995), no. 2, 151–205.
.1cm
\[PS\] A. Pressley, G. B. Segal, [*Loop Groups*]{}, Cambridge University Press, 1986.
.1cm
\[RSF\] A. G. Reiman, M. A. Semenov-Tyan-Shanskii, L. D. Faddeev, [*Quantum anomalies and cocycles on gauge groups,*]{} Funkt. Anal. Appl. [**18**]{} (1984), No. 4, 64-72.
.1cm
\[T\] J. Tate, [*Residues of differentials on curves*]{}, Ann. Sci. École Norm. Sup. (4) [**1** ]{} (1968) 149–159.
.1cm
\[W\] M. Wodzicki, [*Cyclic homology of differential operators*]{}, Duke Math. J. [**54**]{} (1987), no. 2, 641–647.
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abstract: 'The randomization of a complete first order theory $T$ is the complete continuous theory $T^R$ with two sorts, a sort for random elements of models of $T$, and a sort for events in an underlying probability space. We give necessary and sufficient conditions for an element to be definable over a set of parameters in a model of $T^R$.'
address:
- 'University of Wisconsin-Madison, Department of Mathematics, Madison, WI 53706-1388'
- 'University of Illinois at Chicago, Department of Mathematics, Statistics, and Computer Science, Science and Engineering Offices (M/C 249), 851 S. Morgan St., Chicago, IL 60607-7045, USA'
author:
- 'Uri Andrews, Isaac Goldbring, and H. Jerome Keisler'
title: Definable Closure in Randomizations
---
Introduction
============
A randomization of a first order structure ${\mathcal{M}}$, as introduced by Keisler \[Kei1\] and formalized as a metric structure by Ben Yaacov and Keisler \[BK\], is a continuous structure ${\mathcal{N}}$ with two sorts, a sort for random elements of ${\mathcal{M}}$, and a sort for events in an underlying atomless probability space. Given a complete first order theory $T$, the theory $T^R$ of randomizations of models of $T$ forms a complete theory in continuous logic, which is called the randomization of $T$. In a model ${\mathcal{N}}$ of $T^R$, for each $n$-tuple $\vec{ a}$ of random elements and each first order formula $\varphi(\vec v)$, the set of points in the underlying probability space where $\varphi(\vec{ a})$ is true is an event denoted by $\l\varphi(\vec{ a})\rr$.
In a first order structure ${\mathcal{M}}$, an element $b$ is *definable over* a set $A$ of elements of ${\mathcal{M}}$ (called parameters) if there is a tuple $\vec a$ in $A$ and a formula $\varphi(u,\vec a)$ such that $${\mathcal{M}}\models (\forall u)(\varphi(u,\vec a)\leftrightarrow u=b).$$ In a general metric structure ${\mathcal{N}}$, an element $ b$ is said to be *definable over* a set of parameters $ A$ if there is a sequence of tuples $\vec{ a}_n$ in $ A$ and continuous formulas $\Phi_n(x,\vec{ a}_n)$ whose truth values converge uniformly to the distance from $x$ to $ b$. In this paper we give necessary and sufficient conditions for definability in a model of the randomization theory $T^R$. These conditions can be stated in terms of sequences of first order formulas. The results in this paper will be applied in a forthcoming paper about independence relations in randomizations.
In Theorem \[t-definableB\], we show that an event ${\mathsf{E}}$ is definable over a set $ A$ of parameters if and only if it is the limit of a sequence of events of the form $\l\varphi_n(\vec{ a}_n)\rr$, where each $\varphi_n$ is a first order formula and each $\vec{ a}_n$ is a tuple from $ A$.
In Theorem \[t-separable\], we show that a random element $ b$ is definable over a set $ A$ of parameters if and only if $ b$ is the limit of a sequence of random elements $ b_n$ such that for each $n$, $$\l(\forall u)(\varphi_n(u,\vec{ a}_n)\leftrightarrow u= b_n)\rr$$ has probability one for some first order formula $\varphi_n(u,\vec v)$ and a tuple $\vec{ a}_n$ from $ A$. In Section 4 we give some consequences in the special case that the underlying first order theory $T$ is $\aleph_0$-categorical.
Continuous model theory in its current form is developed in the papers \[BBHU\] and \[BU\]. The papers \[Go1\], \[Go2\], \[Go3\] deal with definability questions in metric structures. Randomizations of models are treated in \[AK\], \[Be\], \[BK\], \[EG\], \[GL\], \[Ke1\], and \[Ke2\].
Preliminaries
=============
We refer to \[BBHU\] and \[BU\] for background in continuous model theory, and follow the notation of \[BK\]. We assume familiarity with the basic notions about continuous model theory as developed in \[BBHU\], including the notions of a theory, structure, pre-structure, model of a theory, elementary extension, isomorphism, and $\kappa$-saturated structure. In particular, the universe of a pre-structure is a pseudo-metric space, the universe of a structure is a complete metric space, and every pre-structure has a unique completion. In continuous logic, formulas have truth values in the unit interval $[0,1]$ with $0$ meaning true, the connectives are continuous functions from $[0,1]^n$ into $[0,1]$, and the quantifiers are $\sup$ and $\inf$. A *tuple* is a finite sequence, and $A^{<{\mathbb{N}}}$ is the set of all tuples of elements of $A$.
The theory $T^R$
----------------
We assume throughout that $L$ is a finite or countable first order signature, and that $T$ is a complete theory for $L$ whose models have at least two elements.
The *randomization signature* $L^R$ is the two-sorted continuous signature with sorts ${\mathbb{K}}$ (for random elements) and $\BB$ (for events), an $n$-ary function symbol $\l\varphi(\cdot)\rr$ of sort ${\mathbb{K}}^n\to\BB$ for each first order formula $\varphi$ of $L$ with $n$ free variables, a $[0,1]$-valued unary predicate symbol $\mu$ of sort $\BB$ for probability, and the Boolean operations $\top,\bot,\sqcap, \sqcup,\neg$ of sort $\BB$. The signature $L^R$ also has distance predicates $d_\BB$ of sort $\BB$ and $d_{\mathbb{K}}$ of sort ${\mathbb{K}}$. In $L^R$, we use ${{\mathsf{B}}},{{\mathsf{C}}},\ldots$ for variables or parameters of sort $\BB$. ${{\mathsf{B}}}\doteq{{\mathsf{C}}}$ means $d_\BB({{\mathsf{B}}},{{\mathsf{C}}})=0$, and ${{\mathsf{B}}}\sqsubseteq{{\mathsf{C}}}$ means ${{\mathsf{B}}}\doteq{{\mathsf{B}}}\sqcap{{\mathsf{C}}}$.
A pre-structure for $T^R$ will be a pair ${\mathcal{P}}=({\mathcal{K}},{\mathcal{B}})$ where ${\mathcal{K}}$ is the part of sort ${\mathbb{K}}$ and ${\mathcal{B}}$ is the part of sort $\BB$. The *reduction* of ${\mathcal{P}}$ is the pre-structure ${\mathcal{N}}=({\widehat}{{\mathcal{K}}},{\widehat}{{\mathcal{B}}})$ obtained from ${\mathcal{P}}$ by identifying elements at distance zero, and the associated mapping from ${\mathcal{P}}$ onto ${\mathcal{N}}$ is called the *reduction map*. The *completion* of ${\mathcal{P}}$ is the structure obtained by completing the metrics in the reduction of ${\mathcal{P}}$. A pre-structure ${\mathcal{P}}$ is called *pre-complete* if the reduction of ${\mathcal{P}}$ is already the completion of ${\mathcal{P}}$.
In \[BK\], the randomization theory $T^R$ is defined by listing a set of axioms. We will not repeat these axioms here, because it is simpler to give the following model-theoretic characterization of $T^R$.
\[d-nice\] Given a model ${\mathcal{M}}$ of $T$, a *nice randomization of* ${\mathcal{M}}$ is a pre-complete structure $({\mathcal{K}},{\mathcal{B}})$ for $L^R$ equipped with an atomless probability space $(\Omega,{\mathcal{B}},\mu )$ such that:
1. ${\mathcal{B}}$ is a $\sigma$-algebra with $\top,\bot,\sqcap, \sqcup,\neg$ interpreted by $\Omega,\emptyset,\cap,\cup,\setminus$.
2. ${\mathcal{K}}$ is a set of functions $a\colon\Omega\to M$.
3. For each formula $\psi(\vec{x})$ of $L$ and tuple $\vec{a}$ in ${\mathcal{K}}$, we have $$\l\psi(\vec{a})\rr=\{\omega\in\Omega:{\mathcal{M}}\models\psi(\vec{a}(\omega))\}\in{\mathcal{B}}.$$
4. ${\mathcal{B}}$ is equal to the set of all events $ \l\psi(\vec{a})\rr$ where $\psi(\vec{v})$ is a formula of $L$ and $\vec{a}$ is a tuple in ${\mathcal{K}}$.
5. For each formula $\theta(u, \vec{v})$ of $L$ and tuple $\vec{b}$ in ${\mathcal{K}}$, there exists $a\in{\mathcal{K}}$ such that $$\l \theta(a,\vec{b})\rr=\l(\exists u\,\theta)(\vec{b})\rr.$$
6. On ${\mathcal{K}}$, the distance predicate $d_{\mathbb{K}}$ defines the pseudo-metric $$d_{\mathbb{K}}(a,b)= \mu \l a\neq b\rr .$$
7. On ${\mathcal{B}}$, the distance predicate $d_\BB$ defines the pseudo-metric $$d_\BB({{\mathsf{B}}},{{\mathsf{C}}})=\mu ( {{\mathsf{B}}}\triangle {{\mathsf{C}}}).$$
For each first order theory $T$, the *randomization theory* $T^R$ is the set of sentences that are true in all nice randomizations of models of $T$.
It follows that for each first order sentence $\varphi$, if $T\models\varphi$ then $T^R\models \l\varphi\rr\doteq \top$. The following basic facts are from \[BK\], Theorem 2.1 and Proposition 2.2, Example 3.4 (ii), Proposition 2.7, and Theorem 2.9.
\[f-complete\] For every complete first order theory $T$, the randomization theory $T^R$ is complete.
\[f-T\^R\] Every model ${\mathcal{M}}$ of $T$ has nice randomizations.
\[f-perfectwitnesses\] (Fullness) Every pre-complete model ${\mathcal{P}}=({\mathcal{K}},{\mathcal{B}})$ of $T^R$ has perfect witnesses, i.e.,
1. For each first order formula $\theta(u,\vec v)$ and each $\vec{b }$ in ${\mathcal{K}}^n$ there exists $a \in{\mathcal{K}}$ such that $$\l\theta(a,\vec b)\rr \doteq
\l(\exists u\,\theta)(\vec{b })\rr;$$
2. For each ${{\mathsf{B}}}\in{\mathcal{B}}$ there exist $a ,b \in{\mathcal{K}}$ such that ${{\mathsf{B}}}\doteq\l a=b \rr$.
\[c-two\] Every model ${\mathcal{N}}$ of $T^R$ has a pair of elements $ c, d$ such that $\l c\ne d\rr=\top$.
Every model of $T$ has at least two elements, so $T\models(\exists u)(\exists v)u\ne v$. The result follows by applying Fullness twice.
\[f-qe\] (Strong quantifier elimination) Every formula $\Phi$ in the continuous language $L^R$ is $T^R$-equivalent to a formula with the same free variables and no quantifiers of sort ${\mathbb{K}}$ or $\BB$.
\[l-glue\] Let ${\mathcal{P}}=({\mathcal{K}},{\mathcal{B}})$ be a pre-complete model of $T^R$ and let $a ,b \in{\mathcal{K}}$ and ${{\mathsf{B}}}\in{\mathcal{B}}$. Then there is an element $c \in{\mathcal{K}}$ that agrees with $a $ on ${{\mathsf{B}}}$ and agrees with $b $ on $\neg{{\mathsf{B}}}$, that is, ${{\mathsf{B}}}\sqsubseteq\l c =a \rr$ and $(\neg{{\mathsf{B}}})\sqsubseteq\l c =b \rr$.
In Lemma \[l-glue\], we will call $c$ a *characteristic function of ${\mathsf{B}}$ with respect to $a,b$*.
Note that the distance between any two characteristic functions of an event ${\mathsf{B}}$ with respect to elements $a,b$ is zero. In particular, in a model of $T^R$, the characteristic function is unique.
By Fact \[f-perfectwitnesses\] (2), there exist $d,e\in{\mathcal{K}}$ such that ${{\mathsf{B}}}\doteq\l d=e\rr$. The first order sentence $$(\forall u)(\forall v)(\forall x)(\forall y)(\exists z)[(x=y\rightarrow z=u)\wedge( x\neq y\rightarrow z=v)]$$ is logically valid, so we must have $$\l (\exists z)[(d=e\rightarrow z=a)\wedge (d\ne e\rightarrow z=b)]\rr\doteq \top.$$ By Fact \[f-perfectwitnesses\] (1) there exists $c\in{\mathcal{K}}$ such that $$\l d=e\rightarrow c=a\rr\doteq\top,\quad \l d\ne e\rightarrow c=b\rr\doteq \top,$$ so $\l d=e\rr\sqsubseteq \l c=a\rr$ and $\l d\ne e\rr\sqsubseteq\l c=b\rr$.
We will need the following result, which is a consequence of Theorem 3.11 of \[Be\]. Since the setting in \[Be\] is quite different from the present paper, we give a direct proof here.
\[p-representation\] Every model of $T^R$ is isomorphic to the reduction of a nice randomization of a model of $T$.
Let ${\mathcal{N}}=({\widehat}{{\mathcal{K}}},{\widehat}{{\mathcal{B}}})$ be a model of $T^R$ of cardinality $\kappa$. Let $\Omega$ be the Stone space of the Boolean algebra ${\widehat}{{\mathcal{B}}}=({\widehat}{{\mathcal{B}}},\top,\bot,\sqcap, \sqcup,\neg)$. Thus $\Omega$ is a compact topological space, the points of $\Omega$ are ultrafilters, we may identify ${\widehat}{{\mathcal{B}}}$ with the Boolean algebra of clopen sets of $\Omega$, and $\mu^{\mathcal{N}}$ is a finitely additive probability measure on ${\widehat}{{\mathcal{B}}}$.
We next show that $\mu$ is $\sigma$-additive on ${\widehat}{{\mathcal{B}}}$. To do this, we assume that ${{\mathsf{A}}}_0\supseteq{{\mathsf{A}}}_1\supseteq\cdots$ in ${\widehat}{{\mathcal{B}}}$ and ${{\mathsf{C}}} =\bigcap_n{{\mathsf{A}}}_n\in{\widehat}{{\mathcal{B}}}$, and prove that $\mu({\mathsf{C}})=\lim_{n\to\infty}\mu({{\mathsf{A}}}_n)$. Indeed, the family $\{{\mathsf{C}}\cup(\Omega\setminus{{\mathsf{A}}}_n)\colon n\in{\mathbb{N}}\}$ is an open covering of $\Omega$, so by the topological compactness of $\Omega$, we have $\Omega=\bigcup_{k=0}^n ({\mathsf{C}}\cup(\Omega\setminus{{\mathsf{A}}}_k))$ for some $n\in{\mathbb{N}}$. Then ${\mathsf{C}}={{\mathsf{A}}}_n$, so $\mu({\mathsf{C}})=\mu({{\mathsf{A}}}_n)=\lim_{n\to\infty}\mu({{\mathsf{A}}}_n)$.
By the Caratheodory theorem, there is a complete probability space $(\Omega,{\mathcal{B}},\mu)$ such that ${\mathcal{B}}\supseteq{\widehat}{{\mathcal{B}}}$, $\mu$ agrees with $\mu^{\mathcal{N}}$ on ${\widehat}{{\mathcal{B}}}$, and for each ${{\mathsf{B}}}\in{\mathcal{B}}$ and $m>0$ there is a countable sequence ${{\mathsf{A}}}_{m0}\subseteq {{\mathsf{A}}}_{m1}\subseteq\cdots$ in ${\widehat}{{\mathcal{B}}}$ such that $$\label{eq-rep}
B\subseteq \bigcup_n {{\mathsf{A}}}_{mn} \mbox{ and } \mu\left(\bigcup_n {{\mathsf{A}}}_{mn}\right)\le\mu({{\mathsf{B}}})+1/m.$$ Note that since the probability space $(\Omega,{\mathcal{B}},\mu)$ is complete, every subset of $\Omega$ that contains a set in ${\mathcal{B}}$ of measure one also belongs to ${\mathcal{B}}$ and has measure one.
We claim that for each ${\mathsf{B}}\in{\mathcal{B}}$ there is a unique event $f({\mathsf{B}})\in{\widehat}{{\mathcal{B}}}$ such that $\mu(f({\mathsf{B}})\triangle{{\mathsf{B}}})=0$. The uniqueness of $f({\mathsf{B}})$ follows from the fact that the distance function $d_\BB({\mathsf{C}},{\mathsf{D}})=\mu({\mathsf{C}}\triangle{\mathsf{D}})$ is a metric on ${\widehat}{{\mathcal{B}}}$. To show the existence of $f({\mathsf{B}})$, for each $m>0$ let ${{\mathsf{A}}}_{m0}\subseteq {{\mathsf{A}}}_{m1}\subseteq\cdots$ be as in (\[eq-rep\]). Note that $({{\mathsf{A}}}_{m0},{{\mathsf{A}}}_{m1},\ldots)$ is a Cauchy sequence of events in the model ${\mathcal{N}}$, so there is an event ${{\mathsf{C}}}_m\in{\widehat}{{\mathcal{B}}}$ such that ${{\mathsf{C}}}_m=\lim_{n\to\infty} {{\mathsf{A}}}_{mn}$. Hence $\lim_{n\to\infty}\mu({{\mathsf{A}}}_{mn}\triangle {{\mathsf{C}}}_m)=0$, so $\mu((\bigcup_n {{\mathsf{A}}}_{mn})\triangle {{\mathsf{C}}}_m)=0$. Then $({{\mathsf{C}}}_1,{{\mathsf{C}}}_2,\ldots)$ is a Cauchy sequence, so there is an event $f({\mathsf{B}})=\lim_{m\to\infty} {{\mathsf{C}}}_m$ in ${\widehat}{{\mathcal{B}}}$ with $\mu(f({\mathsf{B}})\triangle{{\mathsf{B}}})=0$.
We make some observations about the mapping $f\colon{\mathcal{B}}\to{\widehat}{{\mathcal{B}}}$. If ${\mathsf{B}}, {\mathsf{C}}\in{\mathcal{B}}$ and $d_\BB({\mathsf{B}},{\mathsf{C}})=0$, then $f({\mathsf{B}})=f({\mathsf{C}})$. For each ${\mathsf{B}}, {\mathsf{C}}\in{\mathcal{B}}$, we have $$f({\mathsf{B}}\cup{\mathsf{C}})=f({\mathsf{B}})\cup f({\mathsf{C}}),\qquad f({\mathsf{B}}\cap{\mathsf{C}})=f({\mathsf{B}})\cap f({\mathsf{C}}),$$ $$\Omega\setminus f({\mathsf{B}})=f(\Omega\setminus{\mathsf{B}}), \qquad \mu({\mathsf{B}})=\mu(f({\mathsf{B}})).$$ Moreover, the mapping $f$ sends ${\mathcal{B}}$ onto ${\widehat}{{\mathcal{B}}}$, because if ${\mathsf{C}}\in{\widehat}{{\mathcal{B}}}$ then ${\mathsf{C}}\in{\mathcal{B}}$ and $f({\mathsf{C}})={\mathsf{C}}$. Therefore the mapping ${\widehat}f$ that sends the equivalence class of each ${\mathsf{B}}\in{\mathcal{B}}$ under $d_\BB$ to $f({\mathsf{B}})$ is well defined and is an isomorphism from the reduction of the pre-structure $({\mathcal{B}},\sqcup,\sqcap,\neg.\top,\bot,\mu)$ onto the measured algebra $({\widehat}{{\mathcal{B}}},\sqcup,\sqcap,\neg.\top,\bot,\mu)$.
A model ${\mathcal{M}}$ of $T$ is *$\kappa^+$-universal* if every model of $T$ of cardinality $\le\kappa$ is elementarily embeddable in ${\mathcal{M}}$. By Theorem 5.1.12 in \[CK\], every $\kappa$-saturated model of $T$ is $\kappa^+$-universal, so $\kappa^+$-universal models of $T$ exist. We now assume that ${\mathcal{M}}$ is a $\kappa^+$-universal model of $T$, and prove that ${\mathcal{N}}$ is isomorphic to the reduction of a nice randomization of ${\mathcal{M}}$ with the underlying probability space $(\Omega,{\mathcal{B}},\mu)$.
In the following paragraphs, we will use boldface letters ${\boldsymbol{b}},{\boldsymbol{d}},\ldots$ for elements of ${\widehat}{{\mathcal{K}}}$. Let $L_{{\widehat}{{\mathcal{K}}}}$ be the first order signature formed by adding a constant symbol for each element ${\boldsymbol{b}}\in{\widehat}{{\mathcal{K}}}$. For each $\omega\in\Omega$, the set of $L_{{\widehat}{{\mathcal{K}}}}$-sentences $$U(\omega)=\{\psi(\vec{{\boldsymbol{b}}})\colon \omega\in\l\psi(\vec{{\boldsymbol{b}}})\rr\}$$ is consistent with $T$ and has cardinality $\le\kappa$. By the Compactness and Löwenheim-Skolem theorems, each $U(\omega)$ has a model $({\mathcal{M}}_\omega,{{\boldsymbol{b}}}_\omega)_{{\boldsymbol{b}}\in{\widehat}{{\mathcal{K}}}}$ of cardinality $\le\kappa$. Since ${\mathcal{M}}$ is $\kappa^+$-universal, for each $\omega\in\Omega$ we may choose an elementary embedding $h_\omega\colon{\mathcal{M}}_\omega\prec{\mathcal{M}}$. Then $({\mathcal{M}},h_\omega({{\boldsymbol{b}}}_\omega))_{{\boldsymbol{b}}\in{\widehat}{{\mathcal{K}}}}\models U(\omega)$ for every $\omega\in\Omega$. It follows that for each formula $\psi(\vec v)$ of $L$ and each tuple $\vec {{\boldsymbol{b}}}\in{\widehat}{{\mathcal{K}}}^{<{\mathbb{N}}}$, $$\l\psi(\vec{{\boldsymbol{b}}})\rr=\{\omega\in\Omega\colon{\mathcal{M}}_\omega\models\psi(\vec {{\boldsymbol{b}}}_\omega)\}=
\{\omega\in\Omega\colon {\mathcal{M}}\models\psi(h_\omega(\vec {{\boldsymbol{b}}}_\omega))\}\in{\widehat}{{\mathcal{B}}}.$$ For each formula $\psi(\vec v)$ of $L$ and tuple $\vec c$ of functions in $M^\Omega$, define $$\l\psi(\vec c)\rr:=\{\omega\in\Omega\colon {\mathcal{M}}\models\psi(\vec c(\omega))\}.$$
Let ${\mathcal{K}}$ be the set of all functions $a\colon\Omega\to M$ such that for some element ${\boldsymbol{b}}\in{\widehat}{{\mathcal{K}}}$, we have $$\mu(\{\omega\in\Omega\colon a(\omega)=h_\omega({{\boldsymbol{b}}}_\omega)\})=1.$$ We claim that for each $a\in{\mathcal{K}}$ there is a unique element $f(a)\in{\widehat}{{\mathcal{K}}}$ such that $$\mu(\{\omega\in\Omega\colon a(\omega)=h_\omega(f(a)_\omega)\})=1.$$ The existence of $f(a)$ is guaranteed by the definition of ${\mathcal{K}}$. To prove uniqueness, suppose ${\boldsymbol{b}}, {\boldsymbol{d}}\in{\widehat}{{\mathcal{K}}}$ and $$\mu(\{\omega\in\Omega\colon a(\omega)=h_\omega({{\boldsymbol{b}}}_\omega)\})=\mu(\{\omega\in\Omega\colon a(\omega)=h_\omega({{\boldsymbol{d}}}_\omega)\})=1.$$ Then $$\mu(\{\omega\in\Omega\colon h_\omega({{\boldsymbol{b}}}_\omega)=h_\omega({{\boldsymbol{d}}}_\omega) \})=1,$$ so $$\mu(\l{\boldsymbol{b}}={\boldsymbol{d}}\rr)=\mu(\{\omega\in\Omega\colon {{\boldsymbol{b}}}_\omega={{\boldsymbol{d}}}_\omega\})=1,$$ and hence $d_{\mathbb{K}}({\boldsymbol{b}},{\boldsymbol{d}})=0$. Since $d_{\mathbb{K}}$ is a metric on ${\widehat}{{\mathcal{K}}}$, it follows that ${\boldsymbol{b}}={\boldsymbol{d}}$.
We now make some observations about the mapping $f\colon{\mathcal{K}}\to{\widehat}{{\mathcal{K}}}$. This mapping sends ${\mathcal{K}}$ onto ${\widehat}{{\mathcal{K}}}$, because for each ${\boldsymbol{b}}\in{\widehat}{{\mathcal{K}}}$, we have $f(a)={\boldsymbol{b}}$ where $a$ is the element of ${\mathcal{K}}$ such that $a(\omega)=h_\omega({{\boldsymbol{b}}}_\omega)$ for all $\omega\in\Omega$. Suppose $\vec c\in{{\mathcal{K}}}^{<{\mathbb{N}}}$ and $\vec {{\boldsymbol{d}}}=f(\vec c)$. We have $\vec {{\boldsymbol{d}}}\in{\widehat}{{\mathcal{K}}}^{<{\mathbb{N}}}$ and $$\l\psi(\vec {{\boldsymbol{d}}})\rr=\{\omega\in\Omega\colon {\mathcal{M}}\models\psi(h_\omega(\vec {{\boldsymbol{d}}}_\omega))\}\doteq
\{\omega\in\Omega\colon {\mathcal{M}}\models\psi(\vec c(\omega))\}=\l\psi(\vec c)\rr.$$ Since the probability space $(\Omega,{\mathcal{B}},\mu)$ is complete, $\l\psi(\vec {{\boldsymbol{d}}})\rr\in{\widehat}{{\mathcal{B}}}\subseteq{\mathcal{B}}$, and $\l\psi(\vec {{\boldsymbol{d}}})\rr\doteq\l\psi(\vec c)\rr$, we have $\l\psi(\vec c)\rr\in{\mathcal{B}}$ and $\l\psi(\vec {{\boldsymbol{d}}})\rr=f(\l\psi(\vec c)\rr)$. Therefore, if $a,c\in{\mathcal{K}}$ and $d_{\mathbb{K}}(a,c)=0$, then $d_{\mathbb{K}}(f(a),f(c))=0$, and hence $f(a)=f(c)$. This shows that ${\mathcal{P}}=({\mathcal{K}},{\mathcal{B}})$ is a well-defined pre-complete structure for $L^R$, and that the mapping ${\widehat}f$ that sends the equivalence class of each ${{\mathsf{B}}}\in{\mathcal{B}}$ to $f({\mathsf{B}})$, and the equivalence class of each $a\in{\mathcal{K}}$ to $f(a)$, is an isomorphism from the reduction of ${\mathcal{P}}$ to ${\mathcal{N}}$.
It remains to show that ${\mathcal{P}}$ is a nice randomization of ${\mathcal{M}}$. It is clear that ${\mathcal{P}}$ satisfies conditions (1)-(3) in Definition \[d-nice\].
Proof of (4): We have already shown that $\l\psi(\vec c)\rr\in{\mathcal{B}}$ for each formula $\psi(\vec v)$ of $L$ and each tuple $\vec c$ in ${\mathcal{K}}$. For the other direction, let ${\mathsf{B}}\in{\mathcal{B}}$. By Corollary \[c-two\], there exist $a,e\in{\mathcal{K}}$ such that $\l a\ne e\rr\doteq\Omega$. We may choose a function $b\in M^\Omega$ such that $b(\omega)=e(\omega)$ whenever $a(\omega)\ne e(\omega)$, and $b(\omega)\ne a(\omega)$ for all $\omega\in\Omega$. Then $b\in{\mathcal{K}}$ and $\l a\ne b\rr=\Omega$. By Lemma \[l-glue\], there exists $c\in{\mathcal{K}}$ which is a characteristic function of ${\mathsf{B}}$ with respect to $a,b$. Then $\l c=a\rr\doteq{\mathsf{B}}$. Let $d\in M^\Omega$ be the function such that $d(\omega)=a(\omega)$ for $\omega\in{\mathsf{B}}$, and $d(\omega)=b(\omega)$ for $\omega\in\neg{\mathsf{B}}$. Then $\mu(\l c=d\rr)=1$, so $d\in{\mathcal{K}}$, and $\l a=d\rr={\mathsf{B}}$. Thus (4) holds with $\psi$ being the sentence $a=d$.
Proof of (5): Consider a formula $\theta(u,\vec v)$ of $L$ and a tuple $\vec b$ in ${\mathcal{K}}$. By Fullness, there exists $c\in{\mathcal{K}}$ such that $$\l\theta(c,\vec b)\rr\doteq\l(\exists u)\theta(u,\vec b)\rr.$$ We may choose a function $a\in M^\Omega$ such that for all $\omega\in\Omega$, $${\mathcal{M}}\models [\theta(c(\omega),\vec b(\omega))\leftrightarrow (\exists u)\theta(u,\vec b)] \mbox{ implies } a(\omega)=c(\omega),$$ and $${\mathcal{M}}\models [(\exists u)\theta(u,\vec b(\omega))\rightarrow\theta(a(\omega),\vec b(\omega))].$$ Then $\mu(\l a=c\rr)=1$, so $a\in{\mathcal{K}}$ and $$\l\theta(a,\vec b)\rr=\l(\exists u)\theta(u,\vec b)\rr,$$ as required.
Proof of (6) and (7): By Fact \[f-T\^R\], the properties $$(\forall x)(\forall y) d_{\mathbb{K}}(x,y)=\mu(\l x\ne y\rr),\quad (\forall {\mathsf{U}})(\forall {\mathsf{V}})d_\BB({\mathsf{U}},{\mathsf{V}})=\mu({\mathsf{U}}\triangle{\mathsf{V}})$$ hold in some model of $T^R$. By Fact \[f-complete\], these properties hold in all models of $T^R$, and thus in ${\mathcal{N}}$. Therefore (6) and (7) hold for ${\mathcal{P}}$.
Types and Definability
----------------------
For a first order structure ${\mathcal{M}}$ and a set $A$ of elements of ${\mathcal{M}}$, ${\mathcal{M}}_A$ denotes the structure formed by adding a new constant symbol to ${\mathcal{M}}$ for each $a\in A$. The *type realized by* a tuple $\vec b$ over the parameter set $A$ in ${\mathcal{M}}$ is the set $\tp^{\mathcal{M}}(\vec b/A)$ of formulas $\varphi(\vec u,{\vec a})$ with $\vec a\in A^{<{\mathbb{N}}}$ satisfied by $\vec b$ in ${\mathcal{M}}_A$. We call $\tp^{{\mathcal{M}}}(\vec b/A)$ an *$n$-type* if $n=|\vec b|$.
In the following, let ${\mathcal{N}}$ be a continuous structure and let $ A$ be a set of elements of ${\mathcal{N}}$. ${\mathcal{N}}_{ A}$ denotes the structure formed by adding a new constant symbol to ${\mathcal{N}}$ for each $ a\in A$. The *type* $\tp^{\mathcal{N}}(\vec{ b}/ A)$ *realized* by $\vec { b}$ over the parameter set $ A$ in ${\mathcal{N}}$ is the function $p$ from formulas to $[0,1]$ such that for each formula $\Phi(\vec{x},\vec { a})$ with $\vec{ a}\in { A}^{<{\mathbb{N}}}$, we have $\Phi(\vec{x},\vec { a})^p=\Phi(\vec { b},\vec { a})^{\mathcal{N}}$.
We now recall the notions of definable element and algebraic element from \[BBHU\]. An element ${ b}$ is *definable over* $ A$ in ${\mathcal{N}}$, in symbols $ b\in\dcl^{\mathcal{N}}( A)$, if there is a sequence of formulas $\langle\Phi_k(x,\vec{ a}_k)\rangle$ with $\vec{ a}_k\in{ A}^{<{\mathbb{N}}}$ such that the sequence of functions $\langle\Phi_k(x,\vec{ a}_k)^{\mathcal{N}}\rangle$ converges uniformly in $x$ to the distance function $d(x, b)^{\mathcal{N}}$ of the corresponding sort. $ b$ is *algebraic over $ A$* in ${\mathcal{N}}$, in symbols $ b\in\acl^{\mathcal{N}}( A)$, if there is a compact set $C$ and a sequence of formulas $\langle\Phi_k(x,\vec { a}_k)\rangle$ with $\vec{ a}_k\in{ A}^{<{\mathbb{N}}}$ such that $b\in C$ and the sequence of functions $\langle\Phi_k(x,\vec{ a}_k)^{\mathcal{N}}\rangle$ converges uniformly in $x$ to the distance function $d(x,C)^{\mathcal{N}}$ of the corresponding sort.
If the structure ${\mathcal{N}}$ is clear from the context, we will sometimes drop the superscript and write $\tp, \dcl, \acl$ instead of $\tp^{\mathcal{N}}, \dcl^{\mathcal{N}}, \acl^{\mathcal{N}}$.
\[f-definable\] (\[BBHU\], Exercises 10.7 and 10.10) For each element $ b$ of ${\mathcal{N}}$, the following are equivalent, where $p=\tp^{\mathcal{N}}( b/ A)$:
1. $ b$ is definable over $ A$ in ${\mathcal{N}}$;
2. in each model ${\mathcal{N}}'\succ{\mathcal{N}}$, $ b$ is the a unique element that realizes $p$ over $ A$;
3. $ b$ is definable over some countable subset of $ A$ in ${\mathcal{N}}$.
\[f-algebraic\] (\[BBHU\], Exercise 10.8 and 10.11) For each element $ b$ of ${\mathcal{N}}$, the following are equivalent, where $p=\tp^{\mathcal{N}}( b/ A)$:
1. $ b$ is algebraic over $ A$ in ${\mathcal{N}}$;
2. in each model ${\mathcal{N}}'\succ{\mathcal{N}}$, the set of elements $ b$ that realize $p$ over $ A$ in ${\mathcal{N}}'$ is compact.
3. $ b$ is algebraic over some countable subset of $ A$ in ${\mathcal{N}}$.
\[f-definableclosure\] (Definable Closure, Exercises 10.10 and 10.11 in \[BBHU\])
1. If $ A\subseteq{\mathcal{N}}$ then $\dcl( A)=\dcl(\dcl( A))$ and $\acl( A)=\acl(\acl( A))$.
2. If $ A$ is a dense subset of $ B$ and $ B\subseteq{\mathcal{N}}$, then $\dcl(A)=\dcl( B)$ and $\acl(A)=\acl( B)$.
It follows that for any $ A\subseteq{\mathcal{N}}$, $\dcl( A)$ and $\acl( A)$ are closed with respect to the metric in ${\mathcal{N}}$.
We now turn to the case where ${\mathcal{N}}$ is a model of $T^R$. In that case, a set of elements of ${\mathcal{N}}$ may contain elements of both sorts ${\mathbb{K}}, \BB$. But as we will now explain, we need only consider definability over sets of parameters of sort ${\mathbb{K}}$.
\[r-sortK-definability\] Let ${\mathcal{N}}=({\widehat}{{\mathcal{K}}},{\widehat}{{\mathcal{B}}})$ be a model of $T^R$. Since every model of $T$ has at least two elements, ${\mathcal{N}}$ has a pair of elements $ a, b$ of sort ${\mathbb{K}}$ such that ${\mathcal{N}}\models\l a= b\rr=\bot$. For each event ${{\mathsf{D}}}\in{\widehat}{{\mathcal{B}}}$, let $1_{{\mathsf{D}}}$ be the characteristic function of ${{\mathsf{D}}}$ with respect to $ a, b$. Then in the model ${\mathcal{N}}$, ${{\mathsf{D}}}$ is definable over $\{ a, b,1_{{\mathsf{D}}}\}$, and $1_{{\mathsf{D}}}$ is definable over $\{ a, b,{{\mathsf{D}}}\}$.
By Fact \[f-definable\].
In view of Remark \[r-sortK-definability\] and Fact \[f-definableclosure\], if $C$ is a set of parameters in ${\mathcal{N}}$ of both sorts, and there are elements $a,b\in C$ such that ${\mathcal{N}}\models\l a= b\rr=\bot$, then an element of either sort is definable over $C$ if and only if it is definable over the set of parameters of sort ${\mathbb{K}}$ obtained by replacing each element of $C$ of sort $\BB$ by its characteristic function with respect to $ a, b$. For this reason, in a model ${\mathcal{N}}$ of $T^R$ we will only consider definability over sets of parameters of sort ${\mathbb{K}}$. We write $\dcl_\BB( A)$ for the set of elements of sort $\BB$ that are definable over $ A$ in ${\mathcal{N}}$, and write $\dcl( A)$ for the set of elements of sort ${\mathbb{K}}$ that are definable over $ A$ in ${\mathcal{N}}$. Similarly for $\acl_\BB( A)$ and $\acl( A)$.
Conventions and Notation
------------------------
We will assume hereafter that ${\mathcal{N}}=({\widehat}{{\mathcal{K}}},{\widehat}{{\mathcal{B}}})$ is a model of $T^R$, ${\mathcal{P}}=({\mathcal{K}},{\mathcal{B}})$ is a nice randomization of a model ${\mathcal{M}}\models T$ with probability space $(\Omega,{\mathcal{B}},\mu)$, and ${\mathcal{N}}$ is the reduction of ${\mathcal{P}}$. The existence of ${\mathcal{P}}$ is guaranteed by Proposition \[p-representation\].
We will use boldfaced letters ${\boldsymbol{a}},{\boldsymbol{b}},\ldots$ for elements of ${\widehat}{{\mathcal{K}}}$. For each element ${\boldsymbol{a}}\in{\widehat}{{\mathcal{K}}}$, we will choose once and for all an element $a\in{\mathcal{K}}$ such that the image of $a$ under the reduction map is ${\boldsymbol{a}}$. It follows that for each first order formula $\varphi(\vec v)$, $\l\varphi(\vec{{\boldsymbol{a}}})\rr$ is the image of $\l\varphi(\vec a)\rr$ under the reduction map. For any countable set $ A\subseteq{\widehat}{{\mathcal{K}}}$ and each $\omega\in\Omega$, we define $$A(\omega)=\{a(\omega)\colon {\boldsymbol{a}}\in A\}.$$ When $ A\subseteq{\widehat}{{\mathcal{K}}}$, $\cl( A)$ denotes the closure of $ A$ in the metric $d_{\mathbb{K}}$. When $ B\subseteq{\widehat}{{\mathcal{B}}}$, $\cl( B)$ denotes the closure of $ B$ in the metric $d_\BB$, and $\sigma( B)$ denotes the smallest $\sigma$-subalgebra of ${\widehat}{{\mathcal{B}}}$ containing $ B$.
Randomizations of Arbitrary Theories {#s-arb}
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Definability in Sort $\BB$
--------------------------
We characterize the set of elements of ${\widehat}{{\mathcal{B}}}$ that are definable in ${\mathcal{N}}$ over a set of parameters $ A\subseteq{\widehat}{{\mathcal{K}}}$.
For each $A\subseteq {\widehat}{{\mathcal{K}}}$, we say that an event ${{\mathsf{E}}}$ is *first order definable* over $A$, in symbols ${{\mathsf{E}}}\in\fo_\BB(A)$, if ${\mathsf{E}}=\l\varphi(\vec{{\boldsymbol{a}}})\rr$ for some first order formula $\varphi(\vec v)$ and tuple $\vec{{\boldsymbol{a}}}$ in ${A}^{<{\mathbb{N}}}$.
\[t-definableB\] For each $ A\subseteq {\widehat}{{\mathcal{K}}}$, $\dcl_\BB( A)=\cl(\fo_\BB( A))=\sigma(\fo_\BB( A))$.
By quantifier elimination (Fact \[f-qe\]), in any elementary extension ${\mathcal{N}}'\succ{\mathcal{N}}$, two events have the same type over $ A$ if and only if they have the same type over $\fo_\BB( A)$. Then by Fact \[f-definable\], $\dcl_\BB( A)=\dcl_\BB(\fo_\BB( A))$. Moreover, $\dcl_\BB(\fo_\BB( A))$ is equal to the definable closure of $\fo_\BB( A)$ in the pure measured algebra $({\widehat}{{\mathcal{B}}},\mu)$. By Observation 16.7 in \[BBHU\], the definable closure of $\fo_\BB( A)$ in $({\widehat}{{\mathcal{B}}},\mu)$ is equal to $\sigma(\fo_\BB( A))$, so $\dcl_\BB( A)=\sigma(\fo_\BB( A))$. Since $\fo_\BB( A)$ is a Boolean subalgebra of ${\widehat}{{\mathcal{B}}}$, $\cl(\fo_\BB( A))$ is a Boolean subalgebra of ${\widehat}{{\mathcal{B}}}$. By metric completeness, $\cl(\fo_\BB( A))$ is a $\sigma$-algebra and $\sigma(\fo_\BB( A))$ is closed, so $\cl(\fo_\BB( A))=\sigma(\fo_\BB( A))$.
\[c-event-noparameters\] The only events that are definable without parameters in ${\mathcal{N}}$ are $\top$ and $\bot$.
For every first order sentence $\varphi$, either $T\models\varphi$ and $T^R\models\l\varphi\rr=\top$, or $T\models\neg\varphi$ and $T^R\models\l\varphi\rr=\bot$. So $\fo_\BB(\emptyset)=\{\top,\bot\}$.
First Order and Pointwise Definability
--------------------------------------
To prepare the way for a characterization of the definable elements of sort ${\mathbb{K}}$, we introduce two auxiliary notions, one that is stronger than definability in sort ${\mathbb{K}}$ and one that is weaker than definability in sort ${\mathbb{K}}$. We will work in the nice randomization ${\mathcal{P}}=({\mathcal{K}},{\mathcal{B}})$ of ${\mathcal{M}}$, and let $A$ be a subset of ${\widehat}{{\mathcal{K}}}$ and ${\boldsymbol{b}}$ be an element of ${\widehat}{{\mathcal{K}}}$.
A first order formula $\varphi(u,\vec v)$ is *functional* if $$T\models(\forall \vec v)(\exists ^{\le 1} u)\varphi(u,\vec v).$$
We say that ${\boldsymbol{b}}$ is *first order definable on ${{\mathsf{E}}}$ over $A$* if there is a functional formula $\varphi(u,\vec v)$ and a tuple $\vec{{\boldsymbol{a}}}\in {A}^{<{\mathbb{N}}}$ such that ${\mathsf{E}}=\l \varphi({\boldsymbol{b}},\vec{{\boldsymbol{a}}})\rr$.
We say that ${\boldsymbol{b}}$ is *first order definable over $ A$*, in symbols ${\boldsymbol{b}}\in\fo( A)$, if ${\boldsymbol{b}}$ is first order definable on $\top$ over $A$.
\[r-definableover\] ${\boldsymbol{b}}$ is first order definable over $ A$ if and only if there is a first order formula $\varphi(u,\vec v)$ and a tuple $\vec{{\boldsymbol{a}}}$ from $ A$ such that $$\mu(\l(\forall u)(\varphi(u,\vec{{\boldsymbol{a}}})\leftrightarrow u={\boldsymbol{b}})\rr)=1.$$
First order definability has finite character, that is, ${\boldsymbol{b}}$ is first order definable over $ A$ if and only if ${\boldsymbol{b}}$ is first order definable over some finite subset of $ A$.
If ${\boldsymbol{b}}$ is first order definable on ${{\mathsf{E}}}$ over $A$, then ${\mathsf{E}}$ is first order definable over $A\cup\{{\boldsymbol{b}}\}$.
If ${\boldsymbol{b}}$ is first order definable on ${{\mathsf{D}}}$ over $A$, and ${\mathsf{E}}$ is first order definable over $A\cup\{{\boldsymbol{b}}\}$, then ${\boldsymbol{b}}$ is first order definable on ${{\mathsf{D}}}\sqcap{\mathsf{E}}$ over $A$.
\[l-firstorderdefinable\] If ${\boldsymbol{b}}$ is first order definable over $ A$ then ${\boldsymbol{b}}$ is definable over $ A$ in ${\mathcal{N}}$. Thus $\fo( A)\subseteq\dcl( A)$.
Let ${\mathcal{N}}'\succ{\mathcal{N}}$ and suppose that $\tp^{{\mathcal{N}}'}({\boldsymbol{b}})=\tp^{{\mathcal{N}}'}({\boldsymbol{d}})$. Then $$\l \varphi({\boldsymbol{b}},\vec{{\boldsymbol{a}}})\rr=\l \varphi({\boldsymbol{d}},\vec{{\boldsymbol{a}}})\rr=\top.$$ Since $\varphi$ is functional, $$\l(\forall t)(\forall u) (\varphi(t,\vec{{\boldsymbol{a}}})\wedge\varphi(u,\vec{{\boldsymbol{a}}})\rightarrow t=u)\rr=\top.$$ Then $\l {\boldsymbol{b}} = {\boldsymbol{d}}\rr=\top$, so ${\boldsymbol{b}}={\boldsymbol{d}}$, and by Fact \[f-definable\], ${\boldsymbol{b}}\in\dcl( A)$.
When $A$ is countable, we define $$\l b\in\dcl^{{\mathcal{M}}}(A)\rr:=\{\omega\in\Omega\colon b(\omega)\in \dcl^{\mathcal{M}}(A(\omega))\}.$$
\[l-pointwisemeasurable\] If $A$ is countable, then $$\l b\in\dcl^{{\mathcal{M}}}(A)\rr= \bigcup\{\l\theta(b,\vec a)\rr\colon\theta(u,\vec v) \mbox{ functional, } \vec{{\boldsymbol{a}}}\in A^{<{\mathbb{N}}}\},$$ and $\l b\in\dcl^{{\mathcal{M}}}(A)\rr\in{\mathcal{B}}$.
Note that for every first order formula $\theta(u,\vec v)$, the formula $$\theta(u,\vec v)\wedge(\exists^{\le 1}u)\,\theta(u,\vec v)$$ is functional. Therefore $\omega\in\l b\in\dcl^{{\mathcal{M}}}(A)\rr$ if and only if $b(\omega)\in \dcl^{\mathcal{M}}(A(\omega))$, and this holds if and only if there is a functional formula $\theta(u,\vec v)$ and a tuple $\vec {{\boldsymbol{a}}}\in A^{<{\mathbb{N}}}$ such that ${\mathcal{M}}\models \theta(b(\omega),\vec a(\omega)).$ Since $A$ and $L$ are countable, $\l b\in\dcl^{{\mathcal{M}}}(A)\rr$ is the union of countably many events in ${\mathcal{B}}$, and thus belongs to ${\mathcal{B}}$.
When $A$ is countable, we say that ${\boldsymbol{b}}$ is *pointwise definable over $A$* if $$\mu(\l b\in\dcl^{{\mathcal{M}}}(A)\rr)=1.$$
\[c-pointwisedefinable\] If $A$ is countable, then ${\boldsymbol{b}}$ is pointwise definable over $A$ if and only if there is a function $f$ on $\Omega$ such that:
1. For each $\omega\in \Omega$, $f(\omega)$ is a pair $\<\theta_\omega(u,\vec v),\vec a_\omega\>$ where $\theta_\omega(u,\vec v)$ is functional and $\vec a_\omega\in A^{|\vec v|}$;
2. $f$ is $\sigma(\fo_\BB(A))$-measurable (i.e., the inverse image of each point belongs to $\sigma(\fo_\BB(A))$);
3. ${\mathcal{M}}\models \theta_\omega(b(\omega),\vec a_\omega(\omega))$ for almost every $\omega\in\Omega$.
If $\omega\in\l b\in\dcl^{{\mathcal{M}}}(A)\rr$, let $f(\omega)$ be the first pair $\<\theta_\omega,\vec a_\omega\>$ such that $\theta_\omega(u,\vec v)$ is functional, $\vec a_\omega\in A^{|\vec v|}$, and ${\mathcal{M}}\models \theta_\omega(b(\omega),\vec a_\omega(\omega))$. Otherwise let $f(\omega)=\<\bot,\emptyset\>$. The result then follows from Lemma \[l-pointwisemeasurable\].
\[l-pointwisedefinable\] If ${\boldsymbol{b}}$ is definable over $ A$ in ${\mathcal{N}}$, then ${\boldsymbol{b}}$ is pointwise definable over some countable subset of $A$.
By Fact \[f-definable\] (3), we may assume that $A$ is countable. By Lemma \[l-pointwisemeasurable\], the measure $r:= \mu(\l b\in\dcl^{{\mathcal{M}}}(A)\rr)$ exists. Suppose ${\boldsymbol{b}}$ is not pointwise definable over $A$. Then $r<1$. For each finite collection $\chi_1(u,\vec v),\ldots,\chi_n(u,\vec v)$ of first order formulas, each tuple $\vec{{\boldsymbol{a}}}\in A^{<{\mathbb{N}}}$, and each $\omega\in\Omega\setminus \l b\in\dcl^{{\mathcal{M}}}(A)\rr$, the sentence $$(\exists u)[u\ne b(\omega)\wedge\bigwedge_{i=1}^n [\chi_i(b(\omega),\vec a(\omega))\leftrightarrow \chi_i(u,\vec a(\omega))]$$ holds in ${\mathcal{M}}$, because $b(\omega)$ is not definable over $A(\omega)$. Therefore in ${\mathcal{P}}$ we have $$\mu\l (\exists u)[u\ne b\wedge\bigwedge_{i=1}^n [\chi_i(b,\vec a)\leftrightarrow \chi_i(u,\vec a)]\rr\ge 1-r.$$ By condition \[d-nice\] (5), there is an element ${\boldsymbol{d}}\in{\widehat}{{\mathcal{K}}}$ such that $$\mu\l d\ne b\wedge\bigwedge_{i=1}^n [\chi_i(b,\vec a)\leftrightarrow \chi_i(d,\vec a)]\rr\ge 1-r.$$ It follows that $ \mu(\l d\ne b\rr)\ge 1-r $, and $\l \chi_i(b,\vec a)\rr\doteq\l \chi_i(d,\vec a)\rr$ for each $i\le n$. By compactness, in some elementary extension of ${\mathcal{N}}$ there is an element ${\boldsymbol{d}}$ such that $\mu\l{\boldsymbol{d}}\ne{\boldsymbol{b}}\rr\ge 1-r$, and $\l\chi({\boldsymbol{b}},\vec{{\boldsymbol{a}}})\rr=\l\chi({\boldsymbol{d}},\vec{{\boldsymbol{a}}})\rr$ for each first order formula $\chi(u,\vec v)$. Then ${\boldsymbol{d}}\ne{\boldsymbol{b}}$, and by quantifier elimination, $\tp({\boldsymbol{d}}/A)=\tp({\boldsymbol{b}}/A)$. Hence by Fact \[f-definable\] (2), ${\boldsymbol{b}}\notin\dcl( A)$.
The following example shows that the converse of Lemma \[l-pointwisedefinable\] fails badly.
Let ${\mathcal{M}}$ be a finite structure with a constant symbol for every element. Then every element of ${\mathcal{K}}$ is pointwise definable without parameters, but the only elements of ${\widehat}{{\mathcal{K}}}$ that are definable without parameters are the equivalence classes of constant functions $b\colon\Omega\to{\mathcal{M}}$.
Definability in Sort ${\mathbb{K}}$
-----------------------------------
We will now give necessary and sufficient conditions for an element of ${\boldsymbol{b}}\in {\widehat}{{\mathcal{K}}}$ to be definable over a parameter set $ A\subseteq{\widehat}K$ in ${\mathcal{N}}$.
\[t-dcl\] ${\boldsymbol{b}}$ is definable over $ A$ if and only if there exist pairwise disjoint events $\{{\mathsf{E}}_n\colon n\in{\mathbb{N}}\}$ such that $\sum_{n\in{\mathbb{N}}}\mu( {\mathsf{E}}_n)=1$, and for each $n$, ${{\mathsf{E}}}_n$ is definable over $ A$, and ${\boldsymbol{b}}$ is first order definable on ${\mathsf{E}}_n$ over $A$.
$(\Rightarrow)$: Suppose ${\boldsymbol{b}}\in \dcl(A)$. By Lemma \[l-pointwisedefinable\], ${\boldsymbol{b}}$ is pointwise definable over some countable subset $A_0$ of $A$. The set of all events ${\mathsf{C}}$ such that ${\boldsymbol{b}}$ is first order definable on ${\mathsf{C}}$ over $A_0$ is countable, and may be arranged in a list $\{{\mathsf{C}}_n\colon n\in{\mathbb{N}}\}$. Let ${\mathsf{E}}_0={\mathsf{C}}_0$, and $${\mathsf{E}}_{n+1}={\mathsf{C}}_{n+1}\sqcap\neg({\mathsf{C}}_0\sqcup\cdots\sqcup{\mathsf{C}}_n).$$ The events ${\mathsf{E}}_n$ are pairwise disjoint, and for each $n$ we have $${\mathsf{E}}_0\sqcup\cdots\sqcup{\mathsf{E}}_n={\mathsf{C}}_0\sqcup\cdots\sqcup{\mathsf{C}}_n.$$ By Remarks \[r-definableover\], for each $n$, ${\boldsymbol{b}}$ is first order definable on ${\mathsf{E}}_n$ over $A$. By Lemma \[l-pointwisemeasurable\] and pointwise definability, $$\sum_{n\in{\mathbb{N}}}\mu({\mathsf{E}}_n)=\lim_{n\to\infty}\mu({\mathsf{C}}_0\sqcup\cdots\sqcup{\mathsf{C}}_n)=\mu(\l\dcl^{{\mathcal{M}}}(A_0)\rr)=1.$$ By Remarks \[r-definableover\], ${{\mathsf{E}}}_n$ is definable over $ A\cup\{{\boldsymbol{b}}\}$, and since ${\boldsymbol{b}}$ is definable over $ A$, ${{\mathsf{E}}}_n$ is definable over $ A$ by Fact \[f-definableclosure\].
$(\Leftarrow)$: Let ${\mathsf{E}}_n$ be as in the theorem. For each $n$, we have ${\mathsf{E}}_n=\l\theta_n({\boldsymbol{b}},\vec{{\boldsymbol{a}}}_n)\rr$ for some functional formula $\theta_n$ and tuple $\vec {{\boldsymbol{a}}}_n\in A^{<{\mathbb{N}}}$. Since ${\mathsf{E}}_n$ is definable over $A$, by Theorem \[t-definableB\] there is a sequence of formulas $\psi_k(\vec v)$ and tuples $\vec{{\boldsymbol{a}}_k}\in A^{<{\mathbb{N}}}$ such that $$\lim_{k\to\infty}d_\BB(\l\psi_k(\vec{{\boldsymbol{a}}}_k)\rr,\l\theta_n({\boldsymbol{b}},\vec{{\boldsymbol{a}}})\rr)=0.$$ Suppose ${\boldsymbol{d}}$ has the same type over $ A$ as ${\boldsymbol{b}}$ in some elementary extension ${{\mathcal{N}}}'$ of ${{\mathcal{N}}}$. Then $$\lim_{k\to\infty}d_\BB(\l\psi_k(\vec{{\boldsymbol{a}}}_k)\rr,\l\theta_n({\boldsymbol{d}},\vec{{\boldsymbol{a}}})\rr)=0.$$ Hence $$\l\theta_n({\boldsymbol{d}},\vec{{\boldsymbol{a}}}_n)\rr = \l\theta_n({\boldsymbol{b}},\vec{{\boldsymbol{a}}}_n)\rr={{\mathsf{E}}}_n$$ in ${{\mathcal{N}}}'$. Since $\theta_n(u,\vec v)$ is functional, we have $\l\theta_n({\boldsymbol{b}},\vec{{\boldsymbol{a}}})\rr\sqsubseteq\l{\boldsymbol{d}}={\boldsymbol{b}}\rr$ for each $n$. Then $$\mu(\l {\boldsymbol{d}}={\boldsymbol{b}}\rr)\ge\sum_{n\in{\mathbb{N}}}\mu({{\mathsf{E}}}_n)=1,$$ so ${\boldsymbol{d}} ={\boldsymbol{b}}$. Then by Fact \[f-definable\], ${\boldsymbol{b}}\in\dcl( A)$.
\[c-definableK-noparameters\] An element ${\boldsymbol{b}} \in{\widehat}{{\mathcal{K}}}$ is definable without parameters if and only if ${\boldsymbol{b}}$ is first order definable without parameters. Thus $\dcl(\emptyset)=\fo(\emptyset)$.
$(\Rightarrow)$: Suppose ${\boldsymbol{b}}\in\dcl(\emptyset)$. By Theorem \[t-dcl\], there is an event ${\mathsf{E}}$ such that $\mu({\mathsf{E}})>0$, ${{\mathsf{E}}}$ is definable without parameters, and ${\boldsymbol{b}}$ is first order definable on ${\mathsf{E}}$ without parameters. By Corollary \[c-event-noparameters\] we have ${{\mathsf{E}}}=\top$, so ${\boldsymbol{b}}$ is first order definable without parameters.
$(\Leftarrow)$: By Lemma \[l-firstorderdefinable\].
\[c-definable-finite\] If $\fo_\BB( A)$ is finite, then $\dcl_\BB( A)=\fo_\BB( A)$ and $\dcl( A)=\fo( A)$.
$\dcl_\BB( A)=\fo_\BB( A)$ follows from Theorem \[t-definableB\]. Lemma \[l-firstorderdefinable\] gives $\dcl( A)\supseteq\fo( A)$. For the other inclusion, suppose ${\boldsymbol{b}}\in\dcl( A)$. By Theorem \[t-dcl\], there is a finite partition ${\mathsf{E}}_0,\ldots,{\mathsf{E}}_k$ of $\top$, a tuple $\vec {{\boldsymbol{a}}}\in A^{<{\mathbb{N}}}$, and first order formulas $\psi_i(\vec v)$ such that ${\mathsf{E}}_i=\l\psi_i(\vec{{\boldsymbol{a}}})\rr$ and ${\boldsymbol{b}}$ is first order definable on ${\mathsf{E}}_i$. Then there are functional formulas $\varphi_i(u,\vec v)$ such that ${\mathsf{E}}_i\doteq\l\varphi_i({\boldsymbol{b}},\vec{{\boldsymbol{a}}})\rr$. We may take the formulas $\psi_i(\vec v)$ to be pairwise inconsistent and such that $T\models\bigvee_{i=0}^n\psi(\vec v)$. Then $\bigwedge_{i=0}^n (\psi_i(\vec v)\rightarrow\varphi_i(u,\vec v))$ is a functional formula such that $$\l\bigwedge_{i=0}^n (\psi_i(\vec{{\boldsymbol{a}}})\rightarrow\varphi_i({\boldsymbol{b}},\vec {{\boldsymbol{a}}}))\rr=\top,$$ so ${\boldsymbol{b}}$ is first order definable over $ A$.
\[c-dcl2\] ${\boldsymbol{b}}$ is definable over $ A$ if and only if:
1. ${\boldsymbol{b}}$ is pointwise definable over some countable subset of $A$;
2. for each functional formula $\varphi(u,\vec v)$ and tuple $\vec {{\boldsymbol{a}}}\in A^{<{\mathbb{N}}}$, $\l \varphi ({\boldsymbol{b}}, \vec{{\boldsymbol{a}}})\rr$ is definable over $ A$.
$(\Rightarrow)$: Suppose ${\boldsymbol{b}} \in\dcl( A)$. Then (1) holds by Lemma \[l-pointwisedefinable\]. $\l \varphi ({\boldsymbol{b}}, \vec{{\boldsymbol{a}}})\rr$ is obviously definable over $ A\cup\{{\boldsymbol{b}}\}$, so $\l \varphi ({\boldsymbol{b}}, \vec{{\boldsymbol{a}}})\rr$ is definable over $ A$ by Fact \[f-definableclosure\], and thus (2) holds.
$(\Leftarrow)$: Assume conditions (1) and (2). By (1) and Lemma \[l-pointwisemeasurable\], there is a sequence of functional formulas $\theta_n(u,\vec v)$ and tuples $\vec{{\boldsymbol{a}}}_n\in A^{<{\mathbb{N}}}$ such that $$\l b\in\dcl^{{\mathcal{M}}}(A)\rr= \bigcup_{n\in{\mathbb{N}}}\l\theta_n(b,\vec a_n)\rr\doteq\Omega.$$ Let ${\mathsf{E}}_n=\l\theta_n({\boldsymbol{b}},\vec{{\boldsymbol{a}}}_n)\rr$, so ${\boldsymbol{b}}$ is first order definable on ${\mathsf{E}}_n$ over $A$. By Remark \[r-definableover\], we may take the ${\mathsf{E}}_n$ to be pairwise disjoint, and thus $\sum_{n\in{\mathbb{N}}}\mu({\mathsf{E}}_n)=1$. By (2), ${{\mathsf{E}}}_n$ is definable over $ A$ for each $n$. Then by Theorem \[t-dcl\], ${\boldsymbol{b}} \in\dcl( A)$.
\[c-dcl3\] ${\boldsymbol{b}}$ is definable over $ A$ if and only if:
1. $b$ is pointwise definable over some countable subset of $A$;
2. $\fo_\BB( A\cup\{{\boldsymbol{b}}\})\subseteq\dcl_\BB( A)$.
\[t-separable\] ${\boldsymbol{b}}$ is definable over $ A$ if and only if ${\boldsymbol{b}}=\lim_{m\to\infty} {\boldsymbol{b}}_m$, where each ${\boldsymbol{b}}_m$ is first-order definable over $ A$. Thus $\dcl( A)=\cl(\fo( A))$.
$(\Rightarrow)$: Suppose that ${\boldsymbol{b}}\in \dcl( A)$. If $A$ is empty, then ${\boldsymbol{b}}$ is already first order definable from $ A$ by Corollary \[c-definableK-noparameters\]. Assume $A$ is not empty and let ${\boldsymbol{c}}\in A$. Let $\{{\mathsf{E}}_n\colon n\in{\mathbb{N}}\}$ be as in Theorem \[t-dcl\], and fix an $\varepsilon>0$. Then for some $n$, $\sum_{k=0}^n\mu( {\mathsf{E}}_k)>1-\varepsilon$. For each $k$, ${{\mathsf{E}}}_k$ is definable over $ A$, so by Theorem \[t-definableB\], there is an event ${\mathsf{D}}_k\in\fo_\BB(A)$ such that $\mu({\mathsf{D}}_k\triangle{\mathsf{E}}_k)<\varepsilon/n$. Since the events ${\mathsf{E}}_k$ are pairwise disjoint, we may also take the events ${\mathsf{D}}_k$ to be pairwise disjoint. We have ${\mathsf{E}}_k=\l\theta_k({\boldsymbol{b}},\vec{{\boldsymbol{a}}}_k)\rr$ for some functional $\theta_k(u,\vec v)$, so we may assume that ${\mathsf{D}}_k$ has the additional properties that ${\mathsf{D}}_k\sqsubseteq\l(\exists ! u)\theta_k(u,\vec{{\boldsymbol{a}}}_k)\rr$, and that ${\mathsf{D}}_k=\l\psi_k(\vec{{\boldsymbol{a}}}_k)\rr$ for some formula $\psi_k(\vec v)$. Then there is a unique element ${\boldsymbol{d}}\in{\widehat}{{\mathcal{K}}}$ such that $$\begin{cases}
{\mathcal{M}}\models\theta_k(d(\omega),\vec a_k(\omega)) & \mbox{ if } k \le n \mbox{ and } \omega\in\l\psi_k(\vec{a}_k)\rr,\\
d(\omega) = c(\omega) & \mbox{ if } \omega\in\Omega\setminus\bigcup_{k=0}^n \l\psi_k(\vec{a}_k)\rr.
\end{cases}$$ Then ${\boldsymbol{d}}$ is first order definable over $ A$, and $d_{\mathbb{K}}({\boldsymbol{b}},{\boldsymbol{d}})<\varepsilon$.
$(\Leftarrow)$: This follows because first order definability implies definability (Lemma \[l-firstorderdefinable\]) and the set $\dcl( A)$ is metrically closed (Fact \[f-definableclosure\] (2)).
The following result was proved in \[Be\] by an indirect argument using Lascar types. We give a simple direct proof here.
For any model ${\mathcal{N}}=({\widehat}{{\mathcal{K}}},{\widehat}{{\mathcal{B}}})$ of $T^R$ and set $ A\subseteq{\widehat}{{\mathcal{K}}}$, $\acl_\BB( A)=\dcl_\BB( A)$ and $\acl( A)=\dcl( A)$.
By Facts \[f-definable\] and \[f-algebraic\], we may assume ${\mathcal{N}}$ is $\aleph_1$-saturated and $ A$ is countable. Suppose an event ${{\mathsf{E}}}\in{\widehat}{{\mathcal{B}}}$ is not definable over $ A$. By Fact \[f-definable\] and $\aleph_1$-saturation there exists ${\mathsf{D}}\in{\widehat}{{\mathcal{B}}}$ such that $\tp({\mathsf{D}}/ A)=\tp({\mathsf{E}}/ A)$ but $d_\BB({\mathsf{D}},{\mathsf{E}})>0$. By $\aleph_1$-saturation again, there is a countable sequence of events $\<{{\mathsf{F}}}_n\colon n\in{\mathbb{N}}\>$ in ${\widehat}{{\mathcal{B}}}$ such that $$\mu({\mathsf{C}}\cap{{\mathsf{F}}}_n)=\mu({\mathsf{C}}\setminus{{\mathsf{F}}}_n)=\mu({\mathsf{C}})/2$$ for each $n$ and each event ${\mathsf{C}}$ in the Boolean algebra generated by $$\fo_\BB(A)\cup\{{\mathsf{D}},{\mathsf{E}}\}\cup\{{{\mathsf{F}}}_k\colon k<n\}.$$ For each $n$, let $${{\mathsf{D}}}_n=({\mathsf{D}}\cap{{\mathsf{F}}}_n)\cup({\mathsf{E}}\setminus{{\mathsf{F}}}_n).$$ Then for each ${\mathsf{C}}\in\fo_\BB(A)$ and $n\in{\mathbb{N}}$, we have $$\mu({{\mathsf{D}}}_n\cap{\mathsf{C}})=\mu({\mathsf{D}}\cap {\mathsf{C}})/2 + \mu({\mathsf{E}}\cap {\mathsf{C}})/2 =\mu({\mathsf{E}}\cap {\mathsf{C}}).$$ By quantifier elimination, $\tp({\mathsf{D}}_n/ A)=\tp({\mathsf{E}}/ A)$ for each $n\in{\mathbb{N}}$. Moreover, whenever $k< n$ we have $${{\mathsf{D}}}_n\setminus{{\mathsf{D}}}_k=(({\mathsf{D}}\setminus{{\mathsf{D}}}_k)\cap{{\mathsf{F}}}_n)\cup(({\mathsf{E}}\setminus{{\mathsf{D}}}_k)\setminus{{\mathsf{F}}}_n),$$ so $$\mu({{\mathsf{D}}}_n\setminus{{\mathsf{D}}}_k)=\mu({\mathsf{D}}\setminus{{\mathsf{D}}}_k)/2+\mu({\mathsf{E}}\setminus{{\mathsf{D}}}_k)/2.$$ Note that whenever $\tp({\mathsf{D}}'/A)=\tp({\mathsf{D}}''/A)$, we have $\mu({\mathsf{D}}')=\mu({\mathsf{D}}'')$, and hence $$\mu({\mathsf{D}}'\setminus{\mathsf{D}}'')=\mu({\mathsf{D}}''\setminus{\mathsf{D}}')=d_\BB({\mathsf{D}}',{\mathsf{D}}'')/2.$$ Therefore $$d_\BB({{\mathsf{D}}}_n,{{\mathsf{D}}}_k)=d_\BB({\mathsf{D}},{{\mathsf{D}}}_k)/2 + d_\BB({\mathsf{E}},{{\mathsf{D}}}_k)/2\ge d_\BB({\mathsf{D}},{\mathsf{E}})/2.$$ It follows that the set of realizations of $\tp({\mathsf{E}}/A)$ is not compact, and ${\mathsf{E}}$ is not algebraic over $ A$. This shows that $\acl_\BB( A)=\dcl_\BB( A)$.
Now suppose ${\boldsymbol{b}}\in \acl( A)\setminus \dcl( A)$. There is an element ${\boldsymbol{c}}\in{\widehat}{{\mathcal{K}}}$ such that $\tp({\boldsymbol{b}}/ A)=\tp({\boldsymbol{c}}/ A)$ but $d_{\mathbb{K}}({\boldsymbol{b}},{\boldsymbol{c}})>0$. For each first order formula $\psi(u,\vec v)$ and $\vec{{\boldsymbol{a}}}\in A^{<{\mathbb{N}}}$, $\l\psi({\boldsymbol{b}},\vec{{\boldsymbol{a}}})\rr\in \acl_\BB(\{{\boldsymbol{b}}\}\cup A)\subseteq\acl_\BB(\acl(A))$. By Fact \[f-definableclosure\], $\l\psi({\boldsymbol{b}},\vec{{\boldsymbol{a}}})\rr\in \acl_\BB(A)$. By the preceding paragraph, $\l\psi({\boldsymbol{b}},\vec{{\boldsymbol{a}}})\rr\in \dcl_\BB( A)$. Since $\tp({\boldsymbol{b}}/ A)=\tp({\boldsymbol{c}}/ A)$, we have $\tp(\l\psi({\boldsymbol{b}},\vec{{\boldsymbol{a}}})\rr/A)=\tp(\l\psi({\boldsymbol{c}},\vec{{\boldsymbol{a}}})\rr/A)$. By Fact \[f-definable\], it follows that $\l\psi({\boldsymbol{b}},\vec{{\boldsymbol{a}}})\rr=\l\psi({\boldsymbol{c}},\vec{{\boldsymbol{a}}})\rr$ for every first order formula $\psi(u,\vec v)$. Then $\tp(b(\omega)/A(\omega))=\tp(c(\omega)/A(\omega))$ for $\mu$-almost all $\omega$. By $\aleph_1$-saturation, there are countably many independent events ${\mathsf{D}}_n\in{\widehat}{{\mathcal{B}}}$ such that ${\mathsf{D}}_n\sqsubseteq\l{\boldsymbol{b}} \ne{\boldsymbol{c}}\rr$ and $\mu({\mathsf{D}}_n)=d_{\mathbb{K}}({\boldsymbol{b}},{\boldsymbol{c}})/2$. Let ${\boldsymbol{c}}_n$ agree with ${\boldsymbol{c}}$ on ${\mathsf{D}}_n$ and agree with ${\boldsymbol{b}}$ elsewhere. We have $\tp({\boldsymbol{c}}_n/A)=\tp({\boldsymbol{b}}/A)$ for every $n\in{\mathbb{N}}$, and $d_{\mathbb{K}}({\boldsymbol{c}}_n,{\boldsymbol{c}}_k)=d_{\mathbb{K}}({\boldsymbol{b}},{\boldsymbol{c}})/2$ whenever $k< n$. Thus the set of realizations of $\tp({\boldsymbol{b}}/A)$ is not compact, contradicting the fact that ${\boldsymbol{b}}\in \acl( A)$.
A Special Case: $\aleph_0$-categorical theories
===============================================
Definability and $\aleph_0$-Categoricity
----------------------------------------
We use our preceding results to characterize $\aleph_0$-categorical theories in terms of definability in randomizations.
\[t-categorical\] The following are equivalent:
1. $T$ is $\aleph_0$-categorical;
2. $\fo_\BB( A)$ is finite for every finite $ A$;
3. $\dcl_\BB( A)$ is finite for every finite $ A$;
4. $\fo_\BB(A)=\dcl_\BB(A)$ for every finite $A$;
5. $\fo( A)$ is finite for every finite $ A$;
6. $\dcl( A)$ is finite for every finite $ A$.
7. $\fo(A)=\dcl(A)$ for every finite $A$;
By the Ryll-Nardzewski Theorem (see \[CK\], Theorem 2.3.13), (1) is equivalent to
\(0) For each $n$ there are only finitely many formulas in $n$ variables up to $T$-equivalence.
Assume (0) and let $A\subseteq{\widehat}{{\mathcal{K}}}$ be finite. Then (2) holds. Moreover, there are only finitely many functional formulas in $|A|+1$ variables, so (5) holds. Then by Corollary \[c-definable-finite\], (3), (4), (6), and (7) hold.
Now assume that (0) fails.
*Proof that (2) and (3) fail*: For some $n$ there are infinitely many formulas in $n$ variables that are not $T$-equivalent. Hence there is an $n$-type $p$ in $T$ without parameters that is not isolated. So there are formulas $\varphi_1(\vec v), \varphi_2(\vec v),\ldots$ in $p$ such that for each $k>0$, $T\models \varphi_{k+1}\rightarrow\varphi_k$ but the formula $\theta_k=\varphi_k\wedge\neg\varphi_{k+1}$ is consistent with $T$. The formulas $\theta_k$ are consistent but pairwise inconsistent. By Fullness, for each $k>0$ there exists an $n$-tuple $\vec {{\boldsymbol{b}}}_k\in{\widehat}{{\mathcal{K}}}^n$ such that $\l\theta_k(\vec{{\boldsymbol{b}}}_k)\rr=\top$. Since the measured algebra $({\widehat}{{\mathcal{B}}},\mu)$ is atomless, there are pairwise disjoint events ${\mathsf{E}}_1,{\mathsf{E}}_2,\ldots$ in ${\widehat}{{\mathcal{B}}}$ such that $\mu({\mathsf{E}}_k)=2^{-k}$ for each $k>0$. By applying Lemma \[l-glue\] $k$ times, we see that for each $k>0$ there is an $n$-tuple $\vec{{\boldsymbol{a}}}_k\in{\widehat}{{\mathcal{K}}}^n$ that agrees with $\vec{{\boldsymbol{b}}}_i$ on ${\mathsf{E}}_i$ whenever $0<i\le k$. Whenever $0<k\le j$, we have $\mu(\l \vec {{\boldsymbol{a}}}_k=\vec{{\boldsymbol{a}}}_j\rr)\ge 1-2^{-k}$. So $\<\vec {{\boldsymbol{a}}}_1,\vec{{\boldsymbol{a}}}_2,\ldots\>$ is a Cauchy sequence, and by metric completeness the limit $\vec {{\boldsymbol{a}}}=\lim_{k\to\infty}\vec{{\boldsymbol{a}}}_k$ exists in ${\widehat}{{\mathcal{K}}}^n$. Let $A=\range(\vec{{\boldsymbol{a}}})$. For each $k>0$ we have ${\mathsf{E}}_k=\l\vec{{\boldsymbol{a}}}=\vec{{\boldsymbol{b}}}_k\rr=\l\theta_k(\vec{{\boldsymbol{a}}})\rr$, so ${\mathsf{E}}_k\in\fo_\BB(A)$. Thus $\fo_\BB(A)$ is infinite, so (2) fails and (3) fails.
*Proof that (4) fails*: Let ${\mathsf{E}}_k$ be as in the preceding paragraph. The set $\fo_\BB(A)$ is countable. But the closure $\cl(\fo_\BB(A))$ is uncountable, because for each set $S\subseteq{\mathbb{N}}\setminus\{0\},$ the supremum $\bigsqcup_{k\in S}{\mathsf{E}}_k$ belongs to $\cl(\fo_\BB(A))$. Thus by Theorem \[t-definableB\], $$\dcl_\BB(A)=\cl(\fo_\BB(A))\ne\fo_\BB(A),$$ and (4) fails.
*Proof that (5), (6), and (7) fail*: By Corollary \[c-two\], there exist ${\boldsymbol{c}}, {\boldsymbol{d}}\in{\mathcal{K}}$ such that $\l {\boldsymbol{c}}\ne{\boldsymbol{d}}\rr=\top$. Let $C$ be the finite set $C=A\cup\{{\boldsymbol{c}},{\boldsymbol{d}}\}$. By Remark \[r-sortK-definability\], for any event ${\mathsf{D}}\in\fo_\BB( A)$, the characteristic function $1_{{\mathsf{D}}}$ of ${\mathsf{D}}$ with respect to ${\boldsymbol{c}},{\boldsymbol{d}}$ is definable over $C$. Moreover, we always have $d_{\mathbb{K}}(1_{{\mathsf{D}}},1_{{\mathsf{E}}})=d_\BB({\mathsf{D}},{\mathsf{E}})$. It follows that $\fo(C)$ is infinite, so (5) and (6) fail. To show that (7) fails, we take an event ${\mathsf{D}}\in \dcl_\BB(A)\setminus\fo_\BB(A)$. By Theorem \[t-definableB\] we have ${\mathsf{D}}\in\cl(\fo_\BB(A))$. It follows that $1_{{\mathsf{D}}}\in\cl(\fo(C))$, so by Theorem \[t-separable\], $1_{{\mathsf{D}}}\in\dcl(C)$. Hence $\dcl(C)$ is uncountable. But $\fo(C)$ is countable, so (7) fails.
By the Ryll-Nardzewski Theorem, if $T$ is $\aleph_0$-categorical then for each $n$, $T$ has finitely many $n$-types; so each type $p$ in the variables $(u,\vec v)$ has an *isolating formula*, that is, a formula $\varphi(u,\vec v)$ such that $T\models \varphi(u,\vec v)\leftrightarrow \bigwedge p$.
We now characterize the definable closure of a finite set $ A\subseteq {\widehat}{{\mathcal{K}}}$ in the case that $T$ is $\aleph_0$-categorical. Hereafter, when $A$ is a finite subset of ${\widehat}{{\mathcal{K}}}$, $\vec{{\boldsymbol{a}}}$ will denote a finite tuple whose range is $A$.
\[c-cat-definable1\] Suppose that $T$ is $\aleph_0$-categorical, ${\boldsymbol{b}}\in{\widehat}{{\mathcal{K}}}$, and $A$ is a finite subset of ${\widehat}{{\mathcal{K}}}$. Then ${\boldsymbol{b}}\in \dcl( A)$ if and only if:
1. ${\boldsymbol{b}}$ is pointwise definable over $A$;
2. for every isolating formula $\varphi(u,\vec v)$, if $\mu(\l\varphi({\boldsymbol{b}},\vec{{\boldsymbol{a}}})\rr)>0$ then $$\l\varphi({\boldsymbol{b}},\vec{{\boldsymbol{a}}})\rr=\l(\exists u)\varphi(u,\vec{{\boldsymbol{a}}})\rr.$$
$(\Rightarrow)$: Suppose ${\boldsymbol{b}}\in \dcl({ A})$. (1) holds by Lemma \[l-pointwisedefinable\]. Suppose $\varphi(u,\vec v)$ is isolating and $\mu(\l\varphi({\boldsymbol{b}},\vec{{\boldsymbol{a}}})\rr)>0$. We have $\l\varphi({\boldsymbol{b}},\vec{{\boldsymbol{a}}})\rr\in\fo_\BB(\{{\boldsymbol{b}}\}\cup{ A})$, so by Corollary \[c-dcl3\], $\l\varphi({\boldsymbol{b}},\vec{{\boldsymbol{a}}})\rr\in\dcl_\BB(A)$. By Theorem \[t-categorical\], $\l\varphi({\boldsymbol{b}},\vec{{\boldsymbol{a}}})\rr\in\fo_\BB(A)$. We note that $(\exists u)\varphi(u,\vec v)$ is an isolating formula, so $\l(\exists u)\varphi(u,\vec{{\boldsymbol{a}}})\rr$ is an atom of $\fo_\BB(A)$. Therefore (2) holds.
$(\Leftarrow)$: Assume (1) and (2). By (2), for every isolating formula $\varphi(u,\vec v)$ such that $\mu(\l\varphi({\boldsymbol{b}},\vec{{\boldsymbol{a}}})\rr)>0$, we have $$\l\varphi({\boldsymbol{b}},\vec{{\boldsymbol{a}}})\rr\in\fo_\BB(A).$$ Every formula $\theta(u,\vec v)$ is $T$-equivalent to a finite disjunction of isolating formulas in the variables $(u,\vec v)$. It follows that $\fo_\BB(A\cup\{{\boldsymbol{b}}\})\subseteq\fo_\BB(A)$. Therefore by Corollary \[c-dcl3\], ${\boldsymbol{b}}\in\dcl(A)$.
\[c-cat-definable2\] Suppose that $T$ is $\aleph_0$-categorical, ${\boldsymbol{b}}\in{\widehat}{{\mathcal{K}}}$, and $A$ is a finite subset of ${\widehat}{{\mathcal{K}}}$. Then ${\boldsymbol{b}}\in \dcl({ A})$ if and only if for every isolating formula $\psi(\vec v)$ there is a functional formula $\varphi(u,\vec v)$ such that $\l\psi(\vec{{\boldsymbol{a}}})\rr\sqsubseteq\l\varphi({\boldsymbol{b}},\vec {{\boldsymbol{a}}})\rr.$
$(\Rightarrow)$: Suppose ${\boldsymbol{b}}\in \dcl({ A})$. By Theorem \[t-categorical\], ${\boldsymbol{b}}$ is first order definable over $\vec{{\boldsymbol{a}}}$, so there is a functional formula $\varphi(u,\vec v)$ such that $\l\varphi({\boldsymbol{b}},\vec{{\boldsymbol{a}}})\rr=\top$. Then for every isolating $\psi(\vec v)$ we have $\l\psi(\vec{{\boldsymbol{a}}})\rr\sqsubseteq\l\varphi({\boldsymbol{b}},\vec{{\boldsymbol{a}}})\rr.$
$(\Leftarrow)$: There is a finite set $\{\psi_0(\vec v),\ldots,\psi_k(\vec v)\}$ that contains exactly one isolating formula for each $|\vec{{\boldsymbol{a}}}|$-type of $T$. By hypothesis, for each $i\le k$ there is a functional formula $\varphi_i(u,\vec v)$ such that $\l\psi_i(\vec{{\boldsymbol{a}}})\rr\sqsubseteq\l\varphi_i({\boldsymbol{b}},\vec{{\boldsymbol{a}}})\rr.$ Since the formulas $\psi_i(\vec v)$ are pairwise inconsistent, the formula $\bigvee_{i=0}^k (\psi_i(\vec v)\wedge\varphi_i(u,\vec v))$ is functional, and $$\l \bigvee_{i=0}^k (\psi_i(\vec{{\boldsymbol{a}}})\wedge\varphi_i({\boldsymbol{b}},\vec{{\boldsymbol{a}}}))\rr=\top.$$ Hence ${\boldsymbol{b}}$ is first order definable over $\vec{{\boldsymbol{a}}}$, so by Lemma \[l-firstorderdefinable\] we have ${\boldsymbol{b}}\in \dcl({ A})$.
The Theory $\DLO^R$
-------------------
We will use Corollary \[c-cat-definable2\] to give a more natural characterization of the definable closure of a finite parameter set in a model of $\DLO^R$, where $\DLO$ is the theory of dense linear order without endpoints. Note that in $\DLO$, every type in $(v_1,\ldots,v_n)$ has an isolating formula of the form $\bigwedge_{i=1}^{n-1} u_i\alpha_i u_{i+1}$ where $\{u_1,\ldots u_n\}=\{v_1,\ldots,v_n\}$ and each $\alpha_i\in\{<,=\}$. (This formula linearly orders the equality-equivalence classes).
\[c-DLO-definable\] Let $T=\DLO$, ${\boldsymbol{b}}\in{\widehat}{{\mathcal{K}}}$, and $A$ be a finite subset of ${\widehat}{{\mathcal{K}}}$. Then ${\boldsymbol{b}}\in \dcl({ A})$ if and only if for every isolating formula $\psi(v_1,\ldots,v_n)$ there is an $i\in\{1,\ldots,n\}$ such that $\l\psi(\vec {{\boldsymbol{a}}})\rr\sqsubseteq\l {\boldsymbol{b}}= {\boldsymbol{a}}_i\rr.$
For any ${\mathcal{M}}\models\DLO$ and parameter set $A$, we have $\dcl^{{\mathcal{M}}}(A)=A$. Therefore for every isolating formula $\psi(v_1,\ldots,v_n)$ and functional formula $\varphi(u,v_1,\ldots,v_n)$ there exists $i\in\{1,\ldots,n\}$ such that $$\DLO\models(\psi(v_1,\ldots,v_n)\wedge\varphi(u,v_1,\ldots,v_n))\rightarrow u=v_i.$$ The result now follows from Corollary \[c-cat-definable2\].
In the theory $\DLO$, we define $\min(u,v)$ and $\max(u,v)$ in the usual way. For ${\boldsymbol{a}},{\boldsymbol{b}}\in{\widehat}{{\mathcal{K}}}$, we let $\min({\boldsymbol{a}},{\boldsymbol{b}})$ be the unique element ${\boldsymbol{e}}\in{\widehat}{{\mathcal{K}}}$ such that $$\l e=\min(a,b)\rr=\top,$$ and similarly for $\max$. For finite subsets $A$ of ${\widehat}{{\mathcal{K}}}$, $\min(A)$ and $\max(A)$ are defined by repeating the two-variable functions $\min$ and $\max$ in the natural way.
We next show that in $\DLO^R$, the definable closure of a finite set can be characterized as the closure under a “choosing function” of four variables.
In the theory $\DLO$, let $\ell$ be the function of four variables defined by the condition $$\ell(u,v,x,y)=x \mbox{ if } u < v, \mbox{ and } \ell(u,v,x,y) = y \mbox{ if not } u < v.$$ For ${\boldsymbol{a}},{\boldsymbol{b}},{\boldsymbol{c}},{\boldsymbol{d}}\in{\mathcal{K}}$, let $\ell({\boldsymbol{a}},{\boldsymbol{b}},{\boldsymbol{c}},{\boldsymbol{d}})$ be the unique element ${\boldsymbol{e}}\in{\widehat}{{\mathcal{K}}}$ such that $\l e=\ell(a,b,c,d)\rr=\top$. Given a set $ A\subseteq{\widehat}{{\mathcal{K}}}$, let $\lcl( A)$ be the closure of $ A$ under the function $\ell$.
Note that in $\DLO$, the function $\ell$ is definable without parameters. In both $\DLO$ and $\DLO^R$, $\min(u,v)=\ell(u,v,u,v)$, and $\max(u,v)=\ell(u,v,v,u)$.
\[p-DLO\] Let $T=\DLO$. Then for every finite subset $A$ of ${\widehat}{{\mathcal{K}}}$, $\dcl( A)=\lcl( A)$.
It is clear that $\lcl( A)\subseteq\dcl( A)$.
We prove the other inclusion. If $A$ is empty, the result is trivial, so we assume $A$ is non-empty. Let ${\boldsymbol{0}}=\min(A), {\boldsymbol{1}}=\max(A)$. We have ${\boldsymbol{0}}, {\boldsymbol{1}}\in\lcl( A)$. Let $\Omega_0=\l 0<1\rr$. Note that $\Omega\setminus\Omega_0=\l 0=1\rr$. If $\mu(\Omega_0)=0$, then $ A$ is a singleton, and we trivially have $\lcl( A)=\dcl( A)= A$. We may therefore assume that $\mu(\Omega_0)>0$. To simplify notation we will instead assume that $\Omega_0=\Omega$; the argument in the general case is similar.
In the following, all characteristic functions are understood to be with respect to ${\boldsymbol{0}}, {\boldsymbol{1}}$. Note that $\ell({\boldsymbol{a}},{\boldsymbol{b}},{\boldsymbol{0}},{\boldsymbol{1}})$ is the characteristic function of the event $\l {\boldsymbol{a}} <{\boldsymbol{b}}\rr$. If ${\boldsymbol{d}}$ is the characteristic function of an event ${\mathsf{D}}$ and ${\boldsymbol{e}}$ is the characteristic function of an event ${\mathsf{E}}$, then $\ell({\boldsymbol{d}},{\boldsymbol{1}},{\boldsymbol{1}},{\boldsymbol{0}})$ is the characteristic function of $\neg{\mathsf{D}}$, $\min({\boldsymbol{d}},{\boldsymbol{e}})$ is the characteristic function of ${\mathsf{D}}\sqcap{\mathsf{E}}$, and $\max({\boldsymbol{d}},{\boldsymbol{e}})$ is the characteristic function of ${\mathsf{D}}\sqcup{\mathsf{E}}$. It follows that for every quantifier-free first order formula $\varphi(\vec v)$ of $\DLO$ with $|\vec v|=|\vec{{\boldsymbol{a}}}|$, the characteristic function of the event $\l\varphi(\vec{{\boldsymbol{a}}})\rr$ belongs to $\lcl(A)$. Since $\DLO$ admits quantifier elimination, the characteristic function of every event that is first order definable over $A$ belongs to $\lcl( A)$. Hence by Theorem \[t-categorical\], the characteristic function of every event in $\dcl_\BB(A)$ belongs to $\lcl(A)$. Moreover, for every ${\boldsymbol{c}}\in A$ and event ${\mathsf{D}}\in\dcl_\BB(A)$ with characteristic function ${\boldsymbol{d}}$, ${\boldsymbol{c}}\upharpoonright{{\mathsf{D}}}:=\ell({\boldsymbol{d}},{\boldsymbol{1}},{\boldsymbol{0}},{\boldsymbol{c}})$ is the element that agrees with ${\boldsymbol{c}}$ on ${\mathsf{D}}$ and agrees with ${\boldsymbol{0}}$ on the complement of ${\mathsf{D}}$, so ${\boldsymbol{c}}\upharpoonright{{\mathsf{D}}}$ belongs to $\lcl( A)$. Let $\{{\mathsf{D}}_1,\ldots,{\mathsf{D}}_n\}$ be the set of atoms of $\dcl_\BB( A)$ (which is finite because $\DLO$ is $\aleph_0$-categorical). By Corollary \[c-DLO-definable\], every element of $\dcl( A)$ has the form $$\max({\boldsymbol{c}}_1\upharpoonright{\mathsf{D}}_1,\ldots,{\boldsymbol{c}}_n\upharpoonright{\mathsf{D}}_n)$$ for some ${\boldsymbol{c}}_1,\ldots,{\boldsymbol{c}}_n\in A$. Therefore $\dcl( A)\subseteq\lcl( A)$.
In this example we show that the exchange property fails for $\DLO^R$, even though it holds for $\DLO$. Thus the exchange property is not preserved under randomizations. Let $T=\DLO$. By Fullness, there exist elements ${\boldsymbol{a}}, {\boldsymbol{b}}\in{\widehat}{{\mathcal{K}}}$ such that $\max({\boldsymbol{a}},{\boldsymbol{b}})\notin\{{\boldsymbol{a}},{\boldsymbol{b}}\}$. Let ${\boldsymbol{c}}=\max({\boldsymbol{a}},{\boldsymbol{b}}), {\boldsymbol{d}}=\min({\boldsymbol{a}},{\boldsymbol{b}})$. It is easy to check that $$\dcl(\{{\boldsymbol{a}},{\boldsymbol{b}}\})=\{{\boldsymbol{a}},{\boldsymbol{b}},{\boldsymbol{c}},{\boldsymbol{d}}\}, \quad \dcl(\{{\boldsymbol{a}},{\boldsymbol{c}}\})=\{{\boldsymbol{a}},{\boldsymbol{c}}\},\quad \dcl(\{{\boldsymbol{a}}\})=\{{\boldsymbol{a}}\}.$$ Thus ${\boldsymbol{c}}\in\dcl(\{{\boldsymbol{a}},{\boldsymbol{b}}\})\setminus\dcl(\{{\boldsymbol{a}}\})$ but ${\boldsymbol{b}}\notin\dcl(\{{\boldsymbol{a}},{\boldsymbol{c}}\})$.
References {#references .unnumbered}
==========
\[AK\] Uri Andrews and H. Jerome Keisler. Randomizations of Theories with Countably Many Countable Models. To appear. Available online at www.math.wisc.edu/$\sim$Keisler.
\[Be\] Itaï Ben Yaacov. On Theories of Random Variables. To appear, Israel J. Math. ArXiv:0901.1584v3 (2001).
\[BBHU\] Itaï Ben Yaacov, Alexander Berenstein, C. Ward Henson and Alexander Usvyatsov. Model Theory for Metric Structures. To appear, Lecture Notes of the London Math. Society.
\[BK\] Itaï Ben Yaacov and H. Jerome Keisler. Randomizations of Models as Metric Structures. Confluentes Mathematici 1 (2009), pp. 197-223.
\[BU\] Itaï Ben Yaacov and Alexander Usvyatsov. Continuous first order logic and local stability. Transactions of the American Mathematical Society 362 (2010), no. 10, 5213-5259.
\[CK\] C.C.Chang and H. Jerome Keisler. Model Theory. Dover 2012.
\[EG\] Clifton Early and Isaac Goldbring. Thorn-Forking in Continuous Logic. Journal of Symbolic Logic 77 (2012), 63-93.
\[Go1\] Isaac Goldbring. Definable Functions in Urysohn’s Metric Space. To appear, Illinois Journal of Mathematics.
\[Go2\] Isaac Goldbring. An Approximate Herbrand’s Theorem and Definable Functions in Metric Structures. Math. Logic Quarterly 50 (2012), 208-216.
\[Go3\] Isaac Goldbring. Definable Operators on Hilbert Spaces. Notre Dame Journal of Formal Logic 53 (2012), 193-201.
\[GL\] Isaac Goldbring and Vinicius Lopes. Pseudofinite and Pseudocompact Metric Structures. To appear, Notre Dame Journal of Formal Logic. Available online at www.homepages.math.uic.edu/$\sim$isaac.
\[Ke1\] H. Jerome Keisler. Randomizing a Model. Advances in Math 143 (1999), 124-158.
\[Ke2\] H. Jerome Keisler. Separable Randomizations of Models. To appear. Available online at www.math.wisc.edu/$\sim$Keisler.
|
---
abstract: 'In the framework of simple spin-boson Hamiltonian we study an interplay between dynamic and spectral roots to stochastic-like behavior. The Hamiltonian describes an initial vibrational state coupled to discrete dense spectrum reservoir. The reservoir states are formed by three sequences with rationally independent periodicities $1\, ;\, 1 \pm \delta $ typical for vibrational states in many nanosize systems (e.g., large molecules containing $C H_2$ fragment chains, or carbon nanotubes). We show that quantum evolution of the system is determined by a dimensionless parameter $\delta \, \Gamma $, where $\Gamma $ is characteristic number of the reservoir states relevant for the initial vibrational level dynamics. When $\delta \, \Gamma > 1$ spectral chaos destroys recurrence cycles and the system state evolution is stochastic-like. In the opposite limit $\delta \, \Gamma < 1$ dynamics is regular up to the critical recurrence cycle $k_c$ and for larger $k > k_c$ dynamic mixing leads to quasi-stochastic time evolution. Our semi-quantitative analytic results are confirmed by numerical solution of the equation of motion. We anticipate that both kinds of stochastic-like behavior (namely, due to spectral mixing and recurrence cycle dynamic mixing) can be observed by femtosecond spectroscopy methods in nanosystems in the spectral window $10^{11} \, -\, 10^{13} \, s^{-1} $.'
author:
- 'V.A.Benderskii'
- 'E. I. Kats'
title: 'Dynamic and spectral mixing in nanosystems.'
---
Quantum dynamics for an initial vibrational level $\epsilon _s^0$ coupled to a set of discrete dense levels $\epsilon _n^0$ can be described in a framework of so-called spin-boson Hamiltonian [@CL83], [@LC87]
$$\label{n1}
H = \epsilon _s^0 b_s^+b_s + \sum _{n}\epsilon _n^0 b_n^+b_n + \sum _n (b_n^+b_s + b_n b_s^+)
\, ,$$
where $b_s^+(b_s)$ is initial level creation (annihilation) operator (i.e., excitation of the initial vibrational level from the system ground state, which is assumed so deep that its influence on the system dynamics can be neglected), and $b_n^+(b_n)$ are similar operators for the discrete reservoir levels. $C_n$ stands for the coupling matrix elements. For this Hamiltonian time dependent wave function can be presented in a series over full orthogonal set of wave functions of the unperturbed (uncoupled) initial and reservoir states with time dependent coefficients (amplitudes) $a_s(t)$
$$\label{n2}
a_s(t) = \sum _{n}\left \{\left (\frac{d F}{dE}\right )^{-1} \exp (- i E t)\right \}_{E = \epsilon _n}
\, ,$$
where $F(E) = 0$ is a secular equation to find the eigenstates $\epsilon _n$ of the Hamiltonian (\[n1\]).
Common wisdom [@GR93], [@HA01], [@ME04], [@PW07] claims that stochastic-like dynamics is a feature typical for random matrix Hamiltonians, with random eigenstate spectra. However for a system with discrete dense spectrum (e.g., vibrational states in nano-particles or medium size molecules) there is another dynamic root to quasi-stochastic behavior. Indeed for such a system with discrete spectrum, dynamic evolution is represented by periodically repeating steps (recurrence cycles). When time is going on, the initial vibrational level population oscillates faster and faster and corresponding response signals become broader. Eventually at a certain critical cycle number $k_c$, the cycles are overlapped in time. Then for any finite accuracy of time or frequency measurements, time evolution looks as irregular, quasi-stochastic, indistinguishable from truly chaotic behavior. It was demonstrated recently [@BF07], [@BG09], [@BK09] for the simplest version of the spin-boson Hamiltonian (\[n1\]) with $\epsilon _n^0 \equiv n$, and $C_n \equiv C$ (so-called Zwanzig approximation [@ZW60]). Here (and in what follows) we utilize a characteristic reservoir spacing as energy unit, and $n=0$ is the reservoir level coinciding with $\epsilon _s$. It is worth noting that this dynamic mixing occurs in a single Hamiltonian system (not for an ensemble of equivalent systems).
Since there are mentioned above two roots to stochastic-like time evolution, two natural questions arise, namely, (i) how the both mechanisms are interrelated, and (ii) whether they are independent ones. To answer these questions is a purpose of our presentation. We study quantum dynamics for a version of spin-boson Hamiltonian (\[n1\]), where the both ingredients yielding to stochastic time evolution (spectral and dynamic) may be presented simultaneously and can be tuned by model Hamiltonian parameters. We assume that the reservoir discrete bare spectrum in the Hamiltonian (\[n1\]) is formed by three equidistant sequences with different periods $$\label{n3}
\epsilon _n^0 = \pm 3 n \, ;\, \epsilon _1^0 = \pm (1-\delta )(3n + 1)\, ;\, \epsilon _2^0(n) = \pm (1+ \delta )(3n +2)
\, ,$$ where spectral shift dimensionless parameter $\delta \leq 1/2$. It is worth noting that such kind of triplet structures are quite frequently observed in the spectra of many molecular systems, forming notorious Snyder sequences [@SN60] (see also more recent publications on high resolution spectroscopy of nanotubes [@IS03], phospholipid molecules [@RO04], [@TH10] and fullerenes [@CH06]). When the total number of levels $N \to \infty $ and the sequence periods $1\, ,\, 1 \pm \delta $ are rationally independent, the spectrum of the reservoir becomes mixed. In own turn, according to the ergodic properties [@SI76], [@GR06] chaotic behavior of an ensemble of systems (random eigen-value distribution) holds if its individual system is mixing. That is why we term our case as exhibiting of stochastic-like behavior (cf. with definition [@LU04]).
In more practical terms mixing phenomenon can be related to reservoir level ordering. Indeed, level ordering and interlevel spacings depend on a cycle number. In a zero cycle the spectrum is formed by triplets in the following sequence $$\label{n4a}
\epsilon _2(n-1) < \epsilon _0(n) < \epsilon _1(n) < \epsilon _2(n)
\, .$$ When $n$ increases the quantities $p_1^{(0)} = \epsilon _0(n) - \epsilon _2(n-1)$ and $p_2^{(0)} = \epsilon _1(n) - \epsilon _0(n)$ become smaller and fill with a step $3\delta $ intervals $[0\, , \, 1 \pm \delta ]$. Level splitting between neighboring triplets $p_3^{(0)} = \epsilon _1(n) - \epsilon _0(n)$ increases with a step $6\delta $ approaching to 3. In the next cycle $k=1$ the level ordering is different $$\label{n4b}
\epsilon _1(n) < \epsilon _0(n) < \epsilon _2(n-1) < \epsilon _1(n+1)
\, ,$$ and the quantities $p_1^{(1)}(n) = \epsilon _2(n-1) - \epsilon _0(n)$ and $p_2^{(1)}(n) = \epsilon _0(n) - \epsilon _1(n)$ are increased with the same as in the $k=0$ cycle, step $3\delta $ up to a limit value 3/2. One can check that in the next even cycles the splittings $p_1^{(2k)}$ and $p_2^{(2k)}$ decrease, whereas $p_3^{(2k)}$ increases. In the odd cycles the opposite variation of the splittings hold ($p_1$ and $p_2$ increase, and $p_3$ decreases). Eventually levels from different triplets fill in the $k \to \infty $ limit almost uniformly and densely the interval $[0\, ,\, 6\delta ]$ with odd-even alternations with the recurrence cycle number. By rather lengthy and boring but straitforward calculations we find that for cycle number $k \geq 1$, the variables $p_1$ and $p_2$ fill uniformly the intervals $$\label{n5}
\left [a_k^{(1)}\, ,\, \frac{3}{2} - a_{k+1}^{(1)}\right ] \, ; \, \left [a_k^{(2)}\, ,\, \frac{3}{2} - a_{k+1}^{(2)}\right ]
\, ,$$ with the different order of $p_1$ and $p_2$ in even and odd cycles. The $p_3$ fills the interval $$\label{n6}
\left [3 - a_k^{(3)}\, ,\, a_{k+1}^{(3)}\right ]
\, .$$ In own turn the above intervals limits $a_k^{(j)}$ with $j = 1 , 2 , 3$ satisfy the following relations $$\label{n7a}
a_{k}^{(1 , 2)} = \left \{\frac{1}{6\delta }(3k + 2) \pm \frac{1}{3} \right \} \, ;\, a_{k}^{(3)} = \left \{\frac{1}{6\delta }(3k+2) - \frac{1}{2}\right \}
\, ,$$ where $\{X\}$ stands for the fractional part of $X$. At $k \gg 1$ the (\[n7a\]) can be generated approximately by so-called fractional part recurrence relation $$\label{n7}
a_{k+1}^{(j)} \simeq \left \{\frac{1}{2\delta }\, a_k^{(j)}\right \}
\, .$$ As it is known [@ZA85] the fractional part transformation is the mixing one (i.e., stochastic) with characteristic correlation time (in our notations) $\propto \ln (1/\delta )$. Thus our model reservoir spectrum is the reservoir with spectral mixing. This phenomenon can be formulated as the mixing of $p_j$ parameters within the interval $[0\, ,\, 3]$. The interlevel spacing distribution density is approximately (over the parameter $1/N$) a constant $\simeq 5/9$ in the interval $[0\, ,\, 3/2]$ and another constant $\simeq 1/9$ in the interval $[3/2\, ,\, 3]$. Without going to more subtle mathematical details of random sequences we calculate numerically the distribution function $\rho (\epsilon )$ for our model spectrum. When the total number $N$ of the reservoir levels increases the distribution approaches to that with two uniformly distributed parts (see the fig. 1, where we show also the widely used in the literature Wigner distribution (see e.g., [@ME04], [@PW07], and also [@WI67]) which holds for orthogonal Gaussian random matrices [@ME04]).
![Interlevel spacing distribution density $\rho (\epsilon )$ for the reservoir (\[n3\]) with 3 rationally independent periods ($\delta = 0.049$). Empty circles ($C^2=0$), stars ($C^2=1$), and filled circles (both distribution densities coincide) are density 5/9 in the interval $[0\, , \, 3/2]$ and another constant 1/9 in the interval $[3/2\, ,\, 3]$. Wigner distribution is shown for comparison by dashed line.[]{data-label="fig:1"}](fig1.ps){width="50.00000%"}
Let us come back to the secular equation to find the eigenstates $$\begin{aligned}
\label{n8}
F(\epsilon ) = \epsilon - \sum_{n=-N}^{n = + N}\left (\frac{C_{3n}^2}{\epsilon -\epsilon _{3n}^0}
+ \frac{C^2_{3n+1}}{\epsilon - \epsilon _{3n+1}^0}
+ \frac{C_{3n+2}^2}{\epsilon -\epsilon _{3n+2}^0}\right ) = 0
\, .\end{aligned}$$ For the ease of algebra (if needed this approximation can be relaxed) we assume following Zwanzig [@ZW60] approximation $$C_{3n} = C_{3n+1} = C_{3n+2} \equiv C$$ and in this case the series entering (\[n8\]) can be written explicitly in terms of trigonometric functions $$\label{n9}
F(\epsilon ) = \epsilon - \frac{\Gamma }{3}\left [\cot \left(\frac{\pi \epsilon }{3}\right ) + \cot \left (\frac{\pi \epsilon }{3(1-\delta )}- \frac{\pi }{3}\right ) + \cot \left (\frac{\pi \epsilon }{3(1+\delta )} - \frac{2\pi }{3} \right ) \right ] = 0
\, .$$ Here $\Gamma = \pi C^2$ characterizes the window of the reservoir states, where at $\delta = 0$ the coupling to the initial vibrational level $\epsilon _s =0$ contributes essentially to system time evolution (the reservoir levels with the quantum numbers $\leq \Gamma $ govern the system dynamics).
When $\delta = 0$ our model is reduced to a single equidistant reservoir Zwanzig model, and in this case dynamic stochastic-like behavior (due to recurrence cycle mixing) occurs [@BF07], [@BK09] for the cycle number $k > k_c^0 = \pi \Gamma $. If $\delta \neq 0$ analysis is more involved. One can check by tedious but direct calculations that the trigonometric equation (\[n9\]) has one and only one root in each interval between the values of the bare energies $\epsilon _0^0(n)\, ,\, \epsilon _1^0(n)$, and $\epsilon _2^0(n)$ given by the expressions (\[n3\]). Armed with this knowledge the formal solution (\[n2\]) for the initial vibrational level amplitude $a_s(t)$ can be represented as a finite series over the partial recurrence cycle amplitudes $a_s^{(k)}(\tau _k)$ (where $k$ is confined within the interval $0 \leq k \leq [t/(2\pi )]$ and $[y]$ means an integer part of $y$) calculated in the local cycle time $\tau _k = t - 2\pi k$. In own turn $a_s^{(k)}(\tau _k)$ can be presented in terms of generalized Lommel and Laguerre polynomials [@BE53]. To avoid this lengthy and cumbersome mathematics in this publication we restrict ourselves to the only small $\delta \ll 1$ limit, when the secular equation is reduced to $$\label{n10}
\epsilon - \Gamma \cot (\pi \epsilon )[1 + R(\epsilon \, ,\, \delta )] = 0
\, ,$$ with $$\begin{aligned}
\label{n11}
R(\epsilon \, ,\, \delta ) = \left (1 + q\frac{\sin (\pi \epsilon /3)}{\sin (\pi \epsilon )}\right )^{-1}\left [q\left (\frac{\sin (\pi \epsilon /3)}{\sin (\pi \epsilon )} - \frac{\cos (\pi \epsilon /3)}{3\cos (\pi \epsilon )}\right )
- 2 \delta \frac{\sin (\pi \epsilon /3)}{\cos (\pi \epsilon )}(\sqrt {3} +q) \right ]
\, ,\end{aligned}$$ where $$\label{n12}
q = 1 + 4 \sin(\pi \delta \epsilon )\cos (\pi \delta \epsilon + \pi /3)
\, .$$ All qualitative features of the quantum time evolution for our model can be seen in this limit $\delta \ll 1$. Since $F(\epsilon )$ depends on $\pi \epsilon /3$ in each recurrence cycle $k$ its amplitude $a_s^{(k)}$ has not only the main Loschmidt echo signal but also two sattelites shifted with respect to the main signal by $2\pi /3$ and $4\pi /3$ (see fig. 2 where we plot initial vibrational level population time evolution). Intensities of the sattelites are proportional to $\delta $, and when the cycle number $k$ increases the main echo signal and its sattelites acquire a fine internal structure. The number of components in the fine structure of the echo signals increases with $k$ and starting with a certain threshold $k_c$ start to overlap. For $\delta \ll 1$ $$\label{n13}
k_c^{-1} = (\pi \Gamma )^{-1} + 3 \delta
\, .$$ At $k> k_c$ the triplets formed by the main echo signal and sattelite triplets are strongly mixed and time evolution becomes chaotic. In the fig. 3 we computed $k_c(\delta )$. It is clear that the numerical results are in a good agreement with our approximated formula (\[n13\]).
![Initial vibrational level population dynamics ($C^2=1$). From the top to the bottom: $\delta = 0 \, ;\, \delta = 0.019 \, ;\, \delta = 0.049 \, ;\, \delta = 0.079 \, ; \, \delta = 0.12 $. Critical cycle numbers $k_c$ are indicated by arrows.[]{data-label="fig:2"}](fig2.ps){width="50.00000%"}
This formula (\[n13\]) is our main result in this paper. It shows how the spectral source of the chaotic behavior ($\delta $ in the (\[n13\])) interplays with the dynamic source of stochastic-like time evolution ($\Gamma $ in the (\[n13\])). Namely, $k_c(\delta )$ is determined by the condition that the triplets from the cycle $k_c$ are mixed with those from the next cycle. The spectral chaos contribution ($\delta \neq 0$) reduces considerably the number of cycles with regular dynamics. For $\delta = 0.12 $ (still $\delta \ll 1$) the intensities of sattelites and the main echo component are comparable and the regular triplet structure is completely broken after 2 initial cycles. We illustrate this in the fig. 2, where we show numerically calculated $|a_s(t)|^2$ (i.e., the initial vibrational level population) for $\delta = 0\, ,\, 0.019\, ,\, 0.049\, ,\, 0.079\, ,\, 0.12 $ (a bit bizarre numerical values of $\delta $ are chosen to get better approximation for rationally independent sequences of the reservoir levels).
![Critical cycle number dependence on $\delta $: $C^2=1$ - open circles; $C^2 = 2$ - stars; $C^2 =4$ - squares.[]{data-label="fig:3"}](fig3.ps){width="50.00000%"}
Thus we arrive at the general conclusion. Dynamic, stochastic-like time evolution is determined by the local spectrum characteristics (mean interlevel spacing) and occurs only for sufficiently large recurrence cycle number. On the contrary spectral truly chaotic dynamics is governed by the global spectrum structure (e.g., in our spin-boson model Hamiltonian (\[n1\]) by the reservoir triplet structure (\[n3\])). It is instructive also to show contributions of the reservoir states into the initial vibrational level wave function amplitude $a_s(t)$. According to the expression (\[n2\]) we computed the quantity $A_n \equiv (d F/dE)^{-1}|_{E = \epsilon _n}$ and plotted the results in the fig. 4. These quantities $A_n$ also manifest themselves simultaneous effects of dynamic and spectral mixing. For the pure Zwanzig model (no spectral mixing, i.e., $\delta =0$), the coefficients $A_n$ decrease monotonously with $\epsilon _n$. However, upon increasing $\delta $ (spectral mixing), $A_n$ distribution becomes more and more irregular.
![Amplitudes $A_n \equiv \left (\partial F/\partial E \right )^{-1}_{E=\epsilon _n}$ (\[n2\]) of the reservoir states contributing into the initial vibrational level evolution. From the top to the bottom: $\delta =0 \, ;\, \delta = 0.049 \, ;\, \delta = 0.12 $, and $C^2=1$.[]{data-label="fig:4"}](fig4.ps){width="50.00000%"}
Our motivation in this work is not a pure curiosity. As a matter of fact quantum dynamics of various systems (ranging from relatively small molecules up to large photochromic molecules and their protein complexes or molecules confined near interfaces [@BE02] (see also [@FE03]) is an active area of experimental researches. The femtosecond spectroscopy data (which allow to study time evolution of one initially prepared by optical pumping state) manifest variety of possible regimes including not only weakly damped more or less regular oscillations but also very irregular long time behavior with a number of peaks corresponding to a partial recovering of the initial state population. This generic feature is omnipresent in the systems with complex and irregular vibrational relaxation. The triplet model investigated in this paper reflects the spirit of minimalist approaches, in that it is simple yet based on a physical principle. Understanding all its limitations, we nevertheless hope that our crude theory captures the essential elements of vibrational relaxation in nano-systems.
A.O.Caldeira, A.J.Leggett, Ann. Phys., [**149**]{}, 587 (1983).
A.J.Leggett, S.Chakravarty, A.T.Dorsey, M.P.A.Fisher, A.Garg, M.Zweger, Rev. Mod. Phys., [**59**]{}, 1 (1987).
P.Grigolini, Quantum Mechanical Irreversibility, World Scientific, Singapore (1993).
F.Haake, Quantum signature of chaos, 2d Edition, Springer, Berlin (2001).
M.L.Mehta, Random matrices, 3d Edition, Academic Press, New York (2004).
T.Papenbrock, H.A.Weldenmuller, Rev. Mod. Phys., [ bf 79]{}, 997 (2007).
V.A.Benderskii, L.A.Falkovsky, E.I.Kats, JETP Lett., [**86**]{}, 311 (2007).
V.A.Benderskii, L.N.Gak, E.I.Kats, JETP, [**108**]{}, 160 (2009), and JETP, [**109**]{}, 505 (2009).
V.A.Benderskii, E.I.Kats, Eur. Phys. J., D, [**54**]{}, 597 (2009).
R.Zwanzig, Lectures in Theor. Phys., [**3**]{}, 106 (1960).
R.C.Snyder, J.Mol. Spectr., [**4**]{}, 411 (1960).
T.Ishioka, et al., Spectrochimica Acta A, [**59**]{}, 671 (2003).
K.R.Rodriguez, et al., J. Chem. Phys., [**121**]{}, 8671 (2004).
O.P.Charkin, et al., J. Inorg. Chem., [**51**]{}, Suppl.1, 1 (2006).
J.M.Thomas, Angew. Chem., [**43**]{}, 2606 (2010).
Ya.G.Sinai, Introduction to ergodic theory, Princeton University Press, Princeton (1976).
R.M.Gray, Probability, random processes and ergodic properties, Springer, Berlin (2006).
S.Luzatto, Arxiv, Math. (2004).
G.M.Zaslavskii, Chaos in dynamical systems, Harwood, New York (1985).
E.P.Wigner, SIAM Rev., [**9**]{}, 1 (1967).
H.Bateman, A.Erdelyi, Higher Transcendental Functions, vol.2, McGraw Hill, New York (1953).
A.V.Benderskii, K.B.Eisental, J. Phys. Chem., A, [**106**]{}, 7482 (2002).
C.J.Fecko, J.D.Eaves, J.J.Loparo, A.Tokmakoff, P.L.Geissler, Science, [**301**]{}, 1698 (2003).
Figure captions
Fig. 1
Interlevel spacing distribution density $\rho (\epsilon )$ for the reservoir (\[n3\]) with 3 rationally independent periods ($\delta = 0.049$). Empty circles ($C^2=0$), stars ($C^2=1$), and filled circles (both distribution densities coincide) are numerical computation performed for 166 levels (10 cycles). Solid line - our theoretical prediction: $\rho (\epsilon ) \simeq 5/9$ in the interval $[0\, , \, 3/2]$ and $\rho (\epsilon ) \simeq 1/9$ in the interval $[3/2\, ,\, 3]$. Wigner distribution is shown for comparison by dashed line.
Fig. 2
Initial vibrational level population dynamics ($C^2=1$). From the top to the bottom: $\delta = 0 \, ;\, \delta = 0.019 \, ;\, \delta = 0.049 \, ;\, \delta = 0.079 \, ; \, \delta = 0.12 $. Critical cycle numbers $k_c$ are indicated by arrows.
Fig. 3
Critical cycle number dependence on $\delta $: $C^2=1$ - open circles; $C^2 = 2$ - stars; $C^2 =4$ - squares.
Fig. 4
Amplitudes $A_n \equiv \left (\partial F/\partial E \right )^{-1}_{E=\epsilon _n}$ (\[n2\]) of the reservoir states contributing into the initial vibrational level evolution. From the top to the bottom: $\delta =0 \, ;\, \delta = 0.049 \, ;\, \delta = 0.12 $, and $C^2=1$.
|
---
abstract: 'The electromagnetic component waves, comprising together with their generating oscillatory massless charge a material particle, will be Doppler shifted when the charge hence particle is in motion, with a velocity $v$, as a mere mechanical consequence of the source motion. We illustrate here that two such component waves generated in opposite directions and propagating at speed $c$ between walls in a one-dimensional box, superpose into a traveling beat wave of wavelength ${\mit\Lambda}_d$$=(\frac{v}{c}){\mit\Lambda}$ and phase velocity $c^2/v+v$ which resembles directly L. de Broglie’s hypothetic phase wave. This phase wave in terms of transporting the particle mass at the speed $v$ and angular frequency ${\mit\Omega}_d=2\pi v /{\mit\Lambda}_d$, with ${\mit\Lambda}_d$ and ${\mit\Omega}_d$ obeying the de Broglie relations, represents a de Broglie wave. The standing-wave function of the de Broglie (phase) wave and its variables for particle dynamics in small geometries are equivalent to the eigen-state solutions to Schrödinger equation of an identical system.'
author:
- |
J X Zheng-Johansson$^1$ and P-I Johansson$^2$\
\
\
[August, 2006]{}
title: '[**[Developing de Broglie Wave[^1] ]{}**]{}'
---
Introduction
============
As it stood at the turn of the 20th century, M. Planck’s quantum theory suggested that energy ($\eng$) is associated with a certain periodic process of frequency ($\nu$), $\eng=h \nu$; and A. Einstein’s mass-energy relation suggested the total energy of a particle ($\eng$) is connected to its mass ($m$), $\eng=mc^2$. Planck and Einstein together implied that mass was associated with a periodic process $mc^2=h \nu$, and accordingly a larger $\nu$ with a moving mass. Incited by such a connection but also a clash with this from Einstein’s relativity theory which suggested a moving mass is associated with a slowing-down clock and thus a smaller $\nu$, L. de Broglie put forward in 1923 [@deBroglie1923] a hypothesis that a matter particle (moving at velocity $v$) consists of an internal periodic process describable as a packet of phase waves of frequencies dispersed about $\nu$, having a phase velocity $W=\frac{\nu}{k}
%=\frac{\nu'}{k'}
=c^2/v$, with $c$ the speed of light, and a group velocity of the phase-wave packet equal to $v$. Despite the hypothetic phase wave appeared supernatural and is today not held a standard physics notion, the de Broglie wave has proven in modern physics to depict accurately the matter particles, and the de Broglie relations proven their fundamental relations.
So inevitably the puzzles with the de Broglie wave persist, involving the hypothetic phase waves or not, and are unanswered prior to our recent unification work[@Unif1]: What is waving with a de Broglie wave and more generally Schrödinger’s wave function? If de Broglie’s phase wave is indeed a reality, what is then transmitted at a speed ($W$) being $\frac{c}{v}$ times the speed of light $c$? How is the de Broglie (phase) wave related to the particle’s charge, which if accelerated generates according to Maxwell’s theory electromagnetic (EM) waves of speed $c$, and how is it in turn related to the EM waves, which are commonplace emitted or absorbed by a particle which changes its internal state? In [@Unif1] we showed that a physical model able to yield all of the essential properties of a de Broglie particle, in terms of solutions in a unified framework of the three basic mechanics, is provided by a single harmonic oscillating, massless charge $+e$ or $-e$ (termed a [*vaculeon*]{}) and the resulting electromagnetic waves. The solutions for a basic material particle generally in motion, with the charge quantity (accompanied with a spin) and energy of the charge as the sole inputs, predict accurately the inertial mass, total wave function, total energy equal to the mass times $c^2$, total momentum, kinetic energy and linear momentum of the particle, and that the particle is a de Broglie wave, it obeys Newton’s laws of motion, the de Broglie relations, the Schrödinger equation in small geometries, Newton’s law of gravitation, and the Galileo-Lorentz-Einstein transformation at high velocities. In this paper we give a self-contained illustration of the process by which the electromagnetic component waves of such a particle in motion superpose into a de Broglie (phase) wave.
Particle; component waves; dynamic variables
============================================
A free massless vaculeon charge ($q$) endowed with a kinetic energy $\Eng_q$ at its creation, being not dissipatable except in a pair annihilation, will tend to move about in the vacuum, and yet at larger displacement restored, fully if $\Eng_q$ below a threshold, toward equilibrium by the potential field of the surrounding dielectric vacuum being now polarized under the charge’s own field [@Unif1]. As a result the charge oscillates in the vacuum, at a frequency $\W_q$; once in addition uni-directionally driven, it will also be traveling at a velocity $v$ here along $X$-axis in a one-dimensional box of length $L$, firstly in $+X$-direction. Let axis $X'$ be attached to the moving charge, $X'= X-\vel T$; let $v$ be low so that $(v/c)^2\rightarrow 0$, with $c$ the velocity of light; accordingly $T'=T$. The charge will according to Maxwell’s theory generate electromagnetic waves to both $+X$- and $-X$- directions, having the standard plane wave solution, given in dimensionless displacements (of the medium or fields): $$\begin{aligned}
\label{eq-ux1p}
\varphi^{\dagsup} (X',T)
= C_1 \sin[K^{\dagsup} X'-\W^{\dagsup} T +\alpha_0], \quad ({\rm a})
\cr
\varphi^{\ddagsup}(X',T)=
- C_1 \sin[K^{\ddagsup} X'+\W^{\ddagsup} T -\alpha_0], \quad({\rm b})\end{aligned}$$ where $\left[{K^{\dagsup} \atop K^{\ddagsup}}\right]
=\lim_{(v/c)^2\rightarrow 0}
\left[{k^{\dagsup} \atop k^{\ddagsup}}\right]
=K\pm K_d$, with $\left[{k^{\dagsup} \atop k^{\ddagsup}}\right]= \frac{K}{1\mp v/c}$ wavevectors Doppler-shifted due to the source motion from their zero-$v$ value, $K$; ${\mit\Lambda}=2\pi/K$. On defining $k_d= \sqrt{(k^{\dagsup}-K) (K-k^{\ddagsup}) }=\lf(\frac{v}{c}\rt)k$, with $k=\g K$ and $\g=1/\sqrt{1-(v/c)^2}$, its classic-velocity limit gives, $$\begin{aligned}
\label{eq-Kd}
K_d=\lim_{(v/c)^2\rightarrow 0} k_d
=\lf(\frac{v}{c}\rt)K;\end{aligned}$$ $\left[{\W^{\dagsup}\atop \W^{\ddagsup}} \right]
=\left[{K^{\dagsup}c \atop K^{\ddagsup}c }\right]=\W \pm K_d c$, with $\W=cK$; $\W=\W_q$ for the classical electromagnetic radiation; and $\a_0$ is the initial phase. Assuming $\Eng_q$ is large and radiated in $J(>>1)$ wave periods if without re-fuel, the wavetrain of $\varphi^j$ of a length $L_\varphi=J L$ will wind about the box $L$ in $J>>1$ loops.
The electromagnetic wave $\varphi^j$ of angular frequency $ \w^j=k^jc$, $j=\dagger $ or $ \ddagger$, has according to M. Planck a wave energy $\eng^j$ $= \hbar \w^j $, with $2\pi\hbar=h$ the Planck constant. The waves are here the components of a particle; the geometric mean of their wave energies, $\sqrt{\eng^{\dagsup}\eng^{\ddagsup}}$ $
=\hbar\sqrt{\w^{\dagsup}\w^{\ddagsup}}=\g \hbar \W$ gives thereby the total energy of the particle. $\eng_v=\g \hbar \W -\hbar \W =\frac{\hbar}{2} (\frac{v}{c})^2\W [1+\frac{3}{4}(\frac{v}{c})^2+\ldots]$ gives further the particle’s kinetic energy and in a similar fashion its linear momentum $p_v$ (see [@Unif1]); and $$\begin{aligned}
\label{eq-Engv}
&\Eng_v=\lim_{(v/c)^2 \rightarrow 0}\eng_v=\frac{1}{2}\hbar \left(\frac{v}{c}\right)^2 \W,
\\
\label{eq-Pv}
& P_v=\lim_{(v/c)^2 \rightarrow 0}p_v=\sqrt{2m_0 \Eng_v}
= \hbar \left(\frac{v}{c}\right)K.\end{aligned}$$ The above continues to indeed imply as L. de Broglie noted that a moving mass has a larger $\g \W/2\pi$ ($=\nu$), and thus a clash with the time-dilation of Einstein’s moving clock. This conflict however vanishes when the underlying physics becomes clear-cut\[2,2006b\].
Propagating total wave of particle {#SecI.V}
===================================
\[Sec-NdbTrw\]
A tagged wave front of say $\varphi^{\dagsup} (X',T)$ generated by the vaculeon charge, of $v>0$, to its right at location $X'$ at time $T$, will after a round-trip of distance $2L$ in time $\delta T=2L/c$ return from left, and propagate again to the right to $X'$ at time $T^*=T+\delta T$. Here it gains a total extra phase $\alpha' =K2L +2\pi$ due to $2L$ (with $\frac{K^{\dagsup} +K^{\ddagsup} }{2} =K$) and the twice reflections at the massive walls, and becomes $$\varphi^{\dagsup}_r(X', T^*)
= C_1 \sin[K^{\dagsup} X'-\W^{\dagsup} T +\alpha_0+\alpha']. \eqno(\ref{eq-ux1p}{\rm a})'$$ $\varphi_r^{\dagsup} $ meets with $\varphi^{\dagsup} (X',T^*)$ just generated to the right, an identical wave except for an $\a'$, and superpose with it to a maximum if assuming $K2L=N2\pi$, $N=0,1, \ldots$, returning the same $\varphi^{\dagsup} $ (assuming normalized). Meanwhile, $\varphi^{\dagsup}_r(X', T^*)$ meets $\varphi^{\ddagsup} (X'_1,T^*)$ just
![ (a) The time development of electromagnetic waves with wave speed $c$ and wavelength ${\mit\Lambda}$, $\varphi^{\dagsup}$ generated to the right of (\[eq-ux1p\][a]{})$'$ and $\varphi^{\ddagsup}$ to the left of (\[eq-ux1p\][b]{}) by a charge ($\ominus$) traveling at velocity $\vel$ in $+X$ direction in a one-dimensional box of side $L $. (b) $\varphi^{\dagsup}$ and $\varphi^{\ddagsup}$ superpose to a beat, or de Broglie phase wave $\widetilde{\psi}$ of (\[eq-Ad1\]) traveling at phase velocity $W\simeq \frac{c^2}{\vel}$, of wavelength ${\mit\Lambda}_d$. For the plot: ${\mit\Lambda}=0.067 {\mit\Lambda}_d $, and $\a_0=-\frac{\pi}{2}$; $T'=T-\frac{\Taum}{4}$; $\vel=(\frac{{\mit\Lambda}}{{\mit\Lambda}_d}) c \ll c$. []{data-label="figI.4-dBwav-trv"}](fig1-dBwtrvA.eps){width="85.00000%"}
![ (a) The time development of electromagnetic waves with wave speed $c$ and wavelength ${\mit\Lambda}$, $\varphi^{\dagsup}$ generated to the right of (\[eq-ux1p\][a]{})$'$ and $\varphi^{\ddagsup}$ to the left of (\[eq-ux1p\][b]{}) by a charge ($\ominus$) traveling at velocity $\vel$ in $+X$ direction in a one-dimensional box of side $L $. (b) $\varphi^{\dagsup}$ and $\varphi^{\ddagsup}$ superpose to a beat, or de Broglie phase wave $\widetilde{\psi}$ of (\[eq-Ad1\]) traveling at phase velocity $W\simeq \frac{c^2}{\vel}$, of wavelength ${\mit\Lambda}_d$. For the plot: ${\mit\Lambda}=0.067 {\mit\Lambda}_d $, and $\a_0=-\frac{\pi}{2}$; $T'=T-\frac{\Taum}{4}$; $\vel=(\frac{{\mit\Lambda}}{{\mit\Lambda}_d}) c \ll c$. []{data-label="figI.4-dBwav-trv"}](fig1-dBwtrvB.eps){width="85.00000%"}
generated by the charge to the left (Figure \[figI.4-dBwav-trv\]a), and superpose with it to ${\widetilde \psi}=\varphi^{\dagsup}_r +\varphi^{\ddagsup}$. Using the trigonometric identity (TI), denoting ${\widetilde \psi}(X',T)={\widetilde \psi}(X',T^*)$, this is $
{\widetilde \psi}(X',T)$ $ =2C_1\cos (KX' -K_d cT )$ $\times \sin(K_d X'-$ $\W T+\a_0 )
$. With $X'=X-vT$, we have on the $X$-axis: $$\begin{aligned}
\label{eq-Ad1}\label{eq-ada}
&&\widetilde{\psi}(X,T)
=\widetilde{\Phimit}(X,T) \widetilde{\Psimit}(X,T),
\qquad \qquad \qquad \qquad
\\
\label{eq-Ad1a}
&&\widetilde{\Phimit} (X,T)= 2 C_1 \cos(KX-2K_dcT), \qquad\qquad
\\
\label{eq-Ad1b}
&&\widetilde{\Psimit} (X,T)=\sin[\Kd X- (\W+\W_d)T+\a_0], \qquad\end{aligned}$$ where $Kv=K_d c$, and $$\begin{aligned}
\label{eq-nud4}
\W_d = K_d \vel =\lf(\frac{v}{c}\rt)^2\W.\end{aligned}$$ $\widetilde{\psi}$ expressed by (\[eq-ada\]) is a [*traveling beat wave*]{}, as plotted versus $X$ in Figure \[figI.4-dBwav-trv\]b for consecutive time points during $\Taum/2$, or Figure \[fig-wvpac\]a during $\Taum_d/2$. $\widetilde{\psi}$ is due to all the component waves of the charge of the particle while moving in one direction, and thus represents the (propagating) total wave of the particle, to be identified as a [*de Broglie phase wave*]{} below.
$\widetilde{\psi}$ in (\[eq-ada\]) has one product component $\widetilde{\Phimit}$ oscillating rapidly on the $X$-axis with the wavelength ${\mit\Lambda}=2\pi/K$, and propagating at the speed of light $c$ at which the total wave energy is transported. The other, $\widetilde{\Psimit}$, envelops about $\widetilde{\Phimit}$, modulating it into a slow varying beat $\widetilde{\psi}$. $\widetilde{\Psimit}$, thus $\widetilde{\psi}$, has a wavevector, wavelength and angular frequency: $$\begin{aligned}
\label{eq-Wbeat1}
K_{\beat }=K_d, \quad {\mit\Lambda}_{\beat}=\frac{2\pi}{K_{\beat}}=\frac{2\pi}{K_{d}}={\mit\Lambda}_d, \quad
\W_{\beat }= \W +\W_d;
\\ \label{eq-lamd}
{\rm where} \quad {\mit\Lambda}_d=
\lf(\frac{c}{\vel}\rt){\mit\Lambda}.\qquad\qquad \qquad\end{aligned}$$ As follows (\[eq-Wbeat1\]), the beat $\widetilde{\psi}$ travels at the [*phase velocity*]{} $$\refstepcounter{equation}
\label{eq-V1}
W= \frac{ \W_{\beat } }{K_{\beat } }
= \frac{\W}{\Kd} +\vel = \left(\frac{c}{\vel}\right)c+\vel.
\eqno(\ref{eq-V1})$$
De Broglie wave
===============
The beat wave, of the wave variables ${\mit\Lambda}_d$ and $ K_d =2\pi/{\mit\Lambda}_d$, transports with it the mass of the particle at the velocity $v$, and in this context represents thereby a periodic process of (the center of mass of) the particle, of a wavelength and wavevector equal to ${\mit\Lambda}_d$ and $K_d$ of the beat wave. $K_d$ and $v$ define for the particle dynamics an angular frequency, $K_d v =\W_d$, as expressed by (\[eq-nud4\]). Combining (\[eq-lamd\]) and (\[eq-Pv\]), and (\[eq-nud4\]) and (\[eq-Engv\]) yield just the [*de Broglie relations*]{}: \[eq-px3\] \[eq-e3\]\[eq-e3b\] $$\displaylines{
\hfill\qquad \Pfp = \frac{h}{{\mit\Lambda}_d}; \qquad\quad\hfill
(\ref{eq-px3}) \hfill\hfill
\qquad\quad
\Efp = \frac{1}{2} \hbar \W_d. \qquad \hfill\quad(\ref{eq-e3}) \nonumber
}$$ Accordingly $K_d$, ${\mit\Lambda}_d$, and $\W_d$ represent the de Broglie wavevector, wavelength and angular frequency. The beat wave $\widetilde{\psi}$ of a phase velocity $W$ resembles thereby the [*de Broglie phase wave*]{} and it in the context of transporting the particle mass represents the [*de Broglie wave*]{} of the particle.
Virtual source. Reflected total particle wave
=============================================
At an earlier time $T_1=T-\Delta T$, at a distance $L$ advancing its present location $X$, with $\Delta T =L/v$, the actual charge was traveling to the left, let axis $X''(=X+\vel T)$ be fixed to it. This past-time charge, said being virtual, generated similarly at location $X''$ at time $T_1$ one component wave ${\varphi^{\dagsup}}^{\imc}(X'',T_1^*)$ to the right, which after traversing $2L$ returned from left to $X''$ at time $T_1^* =T_1+\delta T$ as ${\varphi^{\dagsup}}^{\imc}_r(X'',T_1^*)
=C_1 \sin(K_{\mbox{-}}^{\dagsup} X''-\W_{\mbox{-}}^{\dagsup}T_1^* +\a_0 +\a')$, where $K^{\dagsup}_{\mbox{-}}= K- \Kd$, $ K^{\ddagsup}_{\mbox{-}}= K+ \Kd$, and $\W^{j}_{\mbox{-}}=K^j_{\mbox{-}}c$ are the Doppler shifted wavevectors and angular frequencies; $\a' =(2N+1)\pi$ as earlier. Here at $X''$ and $T_1^*$, ${\varphi^{\dagsup}}^{\imc}_r$ meets the wave the virtual charge just generated to the left, ${\varphi^{\ddagsup}}^{\imc}(X'',T_1^*)
=-C_1 \sin(K_{\mbox{-}}^{\ddagsup} X''+\W_{\mbox{-}}^{\dagsup}T_1^* -\a_0)$, and superpose with it to ${\widetilde \psi}^{\imc } (X,T_1^*)= {\varphi^{\dagsup}}^{\imc}_r +{\varphi^{\ddagsup}}^{\imc}$ $=2C_1\cos (KX''+K_d cT_1) \sin[-K_d X'' - 2\W T_1 -\a_0]$.
Having $J>>1$ and being nondamping, $ {\widetilde \psi}^{\imc }$ will be looping continuously, up to the present time $T$. Its present form ${\widetilde \psi}^{\imc }(X'',T)$ is then as if just produced by the virtual charge at time $T$ but at a location of a distance $L$ advancing the actual charge; it accordingly has a phase advance $\beta =\frac{(K^{\dagsup}-K^{\dagsup}_{\mbox{-}})}{2} L=K_d L $ relative to $\widetilde{\psi}$ (the phase advance in time yields no newer feature). Including this $\beta$, using TI and with some algebra, ${\widetilde \psi}^{\imc }(X'',T)$ writes on axis $X$ as $$\begin{aligned}
\label{eq-psivir-W}
&{\widetilde \psi}^{\imc } (X,T)
=\widetilde{\Phimit}^{\imc}(X,T) \widetilde{\Psimit}^{\imc}(X,T),
\\
&
\widetilde{\Phimit}^{\imc}(X,T) = 2C_1 \cos[(KX+2K_dcT ], \end{aligned}$$ $$\refstepcounter{equation} \label{eq-psivir-WPsi}
\widetilde{\Psimit}^{\imc} (X,T) = - \sin[\Kd X + (\W +\W_d)T +\a_0+\beta]. \eqno(\ref{eq-psivir-WPsi})$$ ${\widetilde \psi}^{\imc }$ of the virtual or reflected charge is seen to be similarly a traveling beat or de Broglie phase wave to the left of a phase velocity $-W$, and of $K_\beat$, ${\mit\Lambda}_\beat$ and $\W_\beat$ as of (\[eq-Wbeat1\]).
![ (a) The beat waves $\widetilde{\psi}$ traveling at a phase velocity $W$ to the right as in Figure \[figI.4-dBwav-trv\]b and $\widetilde{\psi}^{\imc}$ at $-W$ to the left, of a wavelength ${\mit\Lambda}_d$, due to the right- and left- traveling actual and virtual sources respectively. (b) $\widetilde{\psi}$ and $\widetilde{\psi}^{\imc}$ superpose to a standing beat or de Broglie phase wave $\psi$ of wavelength ${\mit\Lambda}_d$, angular frequency $\sim\W$. Along with the $\psi$ process, the particle’s center of mass ($\ominus$) is transported at the velocity $v$, of a period $\frac{2\pi}{\W_d}={\mit\Lambda}_d/v$. []{data-label="fig-wvpac"}](fig2-dBWStanA.eps){width="85.00000%"}
![ (a) The beat waves $\widetilde{\psi}$ traveling at a phase velocity $W$ to the right as in Figure \[figI.4-dBwav-trv\]b and $\widetilde{\psi}^{\imc}$ at $-W$ to the left, of a wavelength ${\mit\Lambda}_d$, due to the right- and left- traveling actual and virtual sources respectively. (b) $\widetilde{\psi}$ and $\widetilde{\psi}^{\imc}$ superpose to a standing beat or de Broglie phase wave $\psi$ of wavelength ${\mit\Lambda}_d$, angular frequency $\sim\W$. Along with the $\psi$ process, the particle’s center of mass ($\ominus$) is transported at the velocity $v$, of a period $\frac{2\pi}{\W_d}={\mit\Lambda}_d/v$. []{data-label="fig-wvpac"}](fig2-dBWStanB.eps){width="85.00000%"}
Standing total wave and de Broglie wave
=======================================
Now if $K_d L(=\beta)=n \pi$, i.e. $$\begin{aligned}
\label{eq-Kdn}
\label{eq-kd3}
\Kd_n = \frac{n\pi}{L}, \quad n=1,2,\ldots, \end{aligned}$$ and accordingly ${\mit\Lambda}_{dn}
=\frac{2L}{n}$, then $ {\widetilde \psi}^{\imc }$ and $ {\widetilde \psi}$ superposed onto themselves from different loops are each a maximum. Also, at ($X$, $T$), $ {\widetilde \psi}^{\imc }$ and $ {\widetilde \psi}$ meet and superpose, as $\psi
=\widetilde{\Phimit}\widetilde{\Psimit}+\widetilde{\Phimit}^{\imc}\widetilde{\Psimit}^{\imc}$. On the scale of ${\mit\Lambda}_d$, or $K_d$, the time variations in $\widetilde{\Phimit}$ and $\widetilde{\Phimit}^{\imc}$ are higher-order ones; thus for $K>>K_d$, we have to a good approximation $ \widetilde{\Phimit}(X,T)\simeq
\widetilde{\Phimit}^{\imc}(X,T)\simeq
2C_1\cos (K X)=F(X)$. Thus $\psi (X,T)= F(X) [\widetilde{\Psimit}+\widetilde{\Psimit}^{\imc}]
= C_4\cos(KX)\sin[(\W +\W_d)T] \cos (K_d X+\a_0)$; $C_4=4C_1$. The mechanical condition at the massive walls $\psi(0,T)=\psi(L,T)=0$ requires $\a_0=-\frac{\pi}{2}$. Hence finally $$\begin{aligned}
\label{eq-beatstd}
&\psi(X,T)= \Phimit(X,T)\Psimit_\Xssub(X);
\\
\label{eq-PsiA}
&\Psimit_\Xssub(X)=\sin (K_d X),
\\ \label{eq-Phi}\label{eq-PhiA}
& \Phimit(X,T)=C_4\cos(KX)\sin[(\W +\W_d)T]. \end{aligned}$$ $\psi$ of (\[eq-beatstd\]) is a [*standing beat*]{}, or [*standing de Broglie phase wave*]{}; it includes all of the component waves due to both the actual and virtual charges and hence represents the (standing) total wave of the particle.
Eigen-state wave function and variables {#SecI.5.2.b}
=======================================
\[SecI.Vb\] \[SecI.VIb\] \[Sec-NdBstnd\]
Equation (\[eq-e3\]) showed the particle’s kinetic energy is transported at the angular frequency $\frac{1}{2}\W_d$, half the value $\W_d$ for transporting the particle mass, and is a source motion effect of order $(\frac{v}{c})^2$. This is distinct from, actually exclusive of, the source motion effect, of order $v$, responsible for the earlier beat wave formation. We here include the order $(\frac{v}{c})^2$ effect only simply as a multiplication factor to $\psi$, and thus have $\psi' =\psi(X,T) e^{-i \frac{1}{2}\hbar \W_d T}$ which describes the particle’s kinetic energy transportation. Furthermore, in typical applications $K\gg \Kd$, $\W\gg \W_d$; thus on the scale of ($K_d$, $\W_d$), we can to a good approximation ignore the rapid oscillation in $\Phimit$ of (\[eq-PhiA\]), and have $$\Phimit (X,T)
\simeq C_4
\equiv{\rm Constant} \eqno(\ref{eq-PhiA})'$$ and $\psi (X,T)=C \Psimit_\Xssub(X)$. The time-dependent wave function, in energy terms, is thus $\Psimit(X,T)$ $= \psi'(X,T)=\psi e^{-i\frac{\W_d}{2}T}=C \Psimit_\Xssub(X) e^{-i\frac{\W_d}{2}T}$, or $$\begin{aligned}
\label{eq-psipac}
\Psimit(X,T)
=C \sin (\Kd X) e ^{-i \frac{1}{2} \W_d T},\end{aligned}$$ where $C=\frac{1}{\int^L_0 \psi^2 d X}
=\frac{ \sqrt{2/L} }{ C_4}$ is a normalization constant. With (\[eq-kd3\]) for $K_{dn}$ in (\[eq-px3\])–(\[eq-e3\]), for a fixed $L$ we have the permitted dynamic variables \[eq-Pvn\] \[eq-Evn\] $$\displaylines{
\hfill \qquad \Pm_{\vel n}= \frac{nh } {2L}, \hfill
\quad \ \ (\ref{eq-Pvn})
\quad \qquad \hfill\hfill
\Eng_{\vel n} =\frac{n^2 {h}^2 }{8 M L^2}, \qquad \hfill(\ref{eq-Evn}) \nonumber
}$$ where $n=1,2,\ldots.$. These dynamic variables are seen to be quantized, pronouncingly for $L$ not much greater than ${\mit\Lambda}_d$, as the direct result of the standing wave solutions. As shown for the three lowest energy levels in Figure \[figI.6-dBwav-n\]a, the permitted $\Psimit(X,T_0)\equiv \Psimit(X)$, with $T_0$ a fixed point in time, describing the envelopes (dotted lines) for $\psi(X,T_0)\equiv \psi(X)$ (solid lines) which rapid oscillation has no physical consequence to the particle dynamics, are in complete agreement with the corresponding solution to Schrödinger equation for an identical system, $\Psimit_S(X)$ (the same dotted lines).
![ (a) The total wave of particle $\psi(X)$ of (\[eq-beatstd\]) with rapid oscillation, and the de Broglie wave $\Psimit(X)$ as the envelop, for three lowest energy levels $n=1,2,3$; $\Psimit$ coincides with Schrödinger eigen-state functions $\Psimit_S$. (b) The corresponding probabilities. []{data-label="figI.6-dBwav-n"}](fig3-stan-n.eps){width="88.00000%"}
The total wave of a particle, hence its total energy $\Eng(X)$, mass, size, all extend in (real) space throughout the wave path. A portion of the particle, hence the probability in finding the particle, at position $X$ in space is proportional to $\Eng(X)$ stored in the infinitesimal volume at $X$, $ \Eng (X)=B\psi^2(X) \propto \psi^2(X)$ (Fig\[figI.6-dBwav-n\]b), with $B$ a conversion constant [@Unif1].
With (\[eq-Pvn\]) in $\Delta \Pm_{\vel} = \Pm_{\vel.n+1} -\Pm_{\vel.n} $ we have $ \Delta \Pm_{\vel} 2L = h $, which reproduces Heisenberg’s uncertainty relation. It follows from the solution that the uncertainty in finding a particle in real space results from the particle is an extensive wave over $L$, and in momentum space from the standing wave solution where waves interfering destructively are cancelled and inaccessible to an external observer.
Concluding remarks
==================
We have seen that the total wave superposed from the electromagnetic component waves generated by a traveling oscillatory vaculeon charge, which together make up our particle, has actually the requisite properties of a de Broglie wave. It exhibits in spatial coordinate the periodicity of the de Broglie wave, by the wavelength ${\mit\Lambda}_d$, facilitated by a beat or de Broglie phase wave traveling at a phase velocity $\sim c^2/v$, with the beat in the total wave resulting naturally from the source-motion resultant Doppler differentiation of the electromagnetic component waves. ${\mit\Lambda}_d$ conjoined with the particle’s center-of-mass motion leads to a periodicity of the de Broglie particle wave on time axis, the angular frequency $\W_d$. The ${\mit\Lambda}_d$ and $\W_d$ obey the de Broglie relations. The particle’s standing wave solutions in confined space agree completely with the Schrödinger solutions for an identical system.
[10]{}
De Broglie L., [*Comptes rendus*]{}, 1923, v. 177, 507-510; PhD Thesis, Univ. of Paris, 1924; [*Phil. Mag.*]{} 1924, v.47, 446.
Zheng-Johansson J.X. and Johansson P.-I., [*Unification of Classical, Quantum and Relativistic Mechanics and of the Four Forces*]{} (Foreword by Lundin R.), Nova Sci. Pub. Inc., N.Y., 2006a, 2nd printing (later 2006); [*Inference of Basic Laws of Classical, Quantum and Relativistic Mechanics from First-Principles Classical-Mechanics Solutions*]{}, (Foreword by Lundin R.), Nova Sci. Pub. Inc., N.Y., 2006b; [*Proceedings of the IV$^{th}$ International Symposium on Quantum Theory and Symmetries*]{}, ed. Dobrev V.K., Heron Press Science Series, Sofia, 2006c (one of two papers with Lundin R.); arxiv: phyiscs/0411134 (v4); phyiscs/0411245; phyiscs/0501037(v3).
[^1]: In: Progress in Physics, v.4, 32-35, 2006; recompiled, self-contained treatment of one of topics from: J.X. Zheng-Johansson and P-I. Johansson, [*Unification of Classical, Quantum and Relativistic Mechanics and of the Four Forces*]{}, Nova Sci. Pub., NY, 2nd printing, later 2006, a revised and enlarged edition of the 1st printing of April 2006.
|
---
abstract: 'In this paper we propose a global optimization-based approach to jointly matching a set of images. The estimated correspondences simultaneously maximize pairwise feature affinities and cycle consistency across multiple images. Unlike previous convex methods relying on semidefinite programming, we formulate the problem as a low-rank matrix recovery problem and show that the desired semidefiniteness of a solution can be spontaneously fulfilled. The low-rank formulation enables us to derive a fast alternating minimization algorithm in order to handle practical problems with thousands of features. Both simulation and real experiments demonstrate that the proposed algorithm can achieve a competitive performance with an order of magnitude speedup compared to the state-of-the-art algorithm. In the end, we demonstrate the applicability of the proposed method to match the images of different object instances and as a result the potential to reconstruct category-specific object models from those images.'
author:
- |
Xiaowei Zhou, Menglong Zhu, Kostas Daniilidis\
GRASP Laboratory, University of Pennsylvania\
[{xiaowz,menglong,kostas}@seas.upenn.edu]{}
bibliography:
- 'mybib-abbr.bib'
title: 'Multi-Image Matching via Fast Alternating Minimization'
---
Introduction
============
Finding feature correspondences between two images is a fundamental problem in computer vision with various applications such as structure from motion, image registration, shape analysis, to name a few. While previous efforts were mostly focused on matching a pair of images, many tasks require to find correspondences across multiple images. A typical example is nonrigid structure from motion [@bregler2000recovering; @dai2012simple], where one can hardly reconstruct a nonrigid shape from two frames. Furthermore, recent work has shown that leveraging multi-way information can dramatically improve matching results compared to pairwise matching [@pachauri2013solving; @huang2013consistent].
The most important constraint for joint matching is the cycle consistency, i.e., the composition of matches along a loop of images should be identity, as illustrated in . Given pairwise matches, one can possibly identify true or false matches by checking all cycles in the image collection. But there are many difficulties for this approach [@chen2014near]. For example, the input pairwise matches are often very noisy with many false matches and missing matches, and the features detected from different images may only have a partial overlap even if the same feature detector is applied [@mikolajczyk2005comparison]. Therefore, it is likely that very few consistent cycles can be found. Moreover, how to sample cycles is not straightforward due to the huge number of possibilities [@huang2013consistent]. Recent work on joint matching has shown that, if all feature correspondences within multiple images are denoted by a large binary matrix, the cycle consistency can be translated into the fact that such a matrix should be positive semidefinite and low-rank [@kim2012exploring; @pachauri2013solving; @huang2013consistent]. Based on this observation, convex optimization-based algorithms were proposed, which achieved the state-of-the-art performances with theoretical guarantees [@huang2013consistent; @chen2014near]. But these algorithms rely on semidefinite programming (SDP), which is not computationally efficient to handle image matching problems in practice.
![An illustration of consistent multi-image matching. []{data-label="fig:demo"}](figures/demo.png){width="0.8\linewidth"}
In this paper, we propose a novel algorithm for multi-image matching. The inputs to our algorithm are original similarities between feature descriptors such as SIFT descriptors [@lowe2004distinctive] and deep features [@weinzaepfel2013deepflow], or optimized affinities provided by existing graph matching solvers [@leordeanu2005spectral]. The outputs are feature correspondences between all pairs of images. Unlike many previous methods starting from quantized pairwise matches [@pachauri2013solving; @chen2014near], we postpone the decision until we optimize for both pairwise affinities and multi-image consistency. Instead of using SDP relaxation, we formulate the problem as a low-rank matrix recovery problem and employ the nuclear-norm relaxation for rank minimization (). We show that the positive semidefiniteness of a desired solution can be spontaneously fulfilled (). Moreover, we derive a fast alternating minimization algorithm to globally solve the problem in the low-dimensional variable space (). Besides validating our method on both simulated and real benchmark datasets, we also demonstrate the applicability of the proposed method combined with deep learning and graph matching to match images with different objects and reconstruct category-specific object models ().
Related work
============
The early work on joint matching aimed to select cycle-consistent matches and identify incorrect matches from bad cycles [@zach2010disambiguating; @nguyen2011optimization]. The assumption for this family of methods is that correct matches are dominant in the raw input. Otherwise, it is difficult to find a sufficient number of closed cycles [@huang2013consistent]. Some works proposed to use the cycle consistency as an explicit constraint for sparse feature matching [@yan2013joint; @yan2014graduated; @yan2015consistency; @yan2015multi] or pixel-wise flow computation [@zhou2015flowweb], but the resulting optimization problems are nonconvex and can hardly be solved globally. Recent results in [@kim2012exploring; @huang2013consistent; @pachauri2013solving] showed that the consistent matches could be extracted from the spectrum (top eigenvectors) of the matrix composed of all pairwise matches. The rationale behind this spectral technique is that the problem can be formulated as a quadratic integer program and relaxed into a generalized Rayleigh problem. But the relaxation assumes full feature correspondences (bijection) between images [@pachauri2013solving]. Recently, Huang and Guibas [@huang2013consistent] proposed an elegant solution based on convex relaxation and derived the theoretical conditions for exact recovery. The result is further improved in [@chen2014near] by assuming that the underlying rank of the variable matrix can be reliably estimated. In these works, the problem is formulated as SDP, which has a limited computational efficiency in real applications.
Regarding methodology, our work is inspired by the recent advances on low-rank matrix recovery which make use of convex relaxation [@candes2009exact; @candes2011robust] and explore the underlying low-rank structure to accelerate computation [@cabral2013unifying; @hastie2014matrix]. Our work is also related to some other problems that aim to find global estimates from pairwise estimates such as rotation averaging [@hartley2013rotation; @wang2013exact] and model fusion [@ye2012robust].
Preliminaries and notation
==========================
Suppose we have $n$ images and $p_i$ features from each image $i$. The objective is to find feature correspondences between all pairs of images. Before introducing the proposed method, we first give a brief introduction to pairwise matching techniques and the definition of cycle consistency.
Pairwise matching {#sec:pairwise}
-----------------
To match an image pair $(i,j)$, one can compute similarities for all pairs of feature points from two images and store them in a matrix $\bfS_{ij}\in\RR{p_i}{p_j}$.
We represent the feature correspondences for image pair $(i,j)$ by a partial permutation matrix $\bfX_{ij}\in\Bin{p_i}{p_j}$, which satisfies the doubly stochastic constraints: $$\begin{aligned}
\bfzero \leq \bfX_{ij}\bfone \leq \bfone, ~~~~, \bfzero \leq \bfX_{ij}^T\bfone \leq \bfone. \label{eq:ds-constr}\end{aligned}$$ To find $\bfX_{ij}$, we can maximize the inner product between $\bfX_{ij}$ and $\bfS_{ij}$ subject to the constraints in resulting in a linear assignment problem, which has been well studied and can be efficiently solved by the Hungarian algorithm.
In image matching, spatial rigidity is usually preferred, i.e., the relative location between two features in an image should be similar to that between their correspondences in the other image. This problem is well known as graph matching and formulated as a quadratic assignment problem (QAP). While QAP is NP-hard, many efficient algorithms have been proposed to solve it approximately, e.g., [@leordeanu2005spectral; @berg2005shape; @cho2010reweighted]. Those solvers basically relax the binary constraint on the permutation matrix, solve the optimization, and output the confidence of a candidate match being correct. We refer readers to the related literature for details. Here we aim to emphasize that the outputs of graph matching solvers are basically optimized affinity scores of candidate matches, which consider both feature similarity and spatial rigidity. We will use these scores (saved in $\bfS_{ij}$) as our input in some cases.
Cycle consistency
-----------------
Some recent work proposed to use the cycle consistency as a constraint to match a bunch of images [@pachauri2013solving; @yan2014graduated; @chen2014near]. The cycle consistency can be described by $$\begin{aligned}
\bfX_{ij}=\bfX_{iz}\bfX_{zj}, \label{eq:3-cycle}\end{aligned}$$ for any three images $(i,j,z)$ and can be extended to the case with more images.
The recent results in [@huang2013consistent; @pachauri2013solving] show that the cycle consistency can be described more concisely by introducing a virtual “universe" that is defined as the set of unique features that appear in the image collection. Each point in the universe may be observed by several images and the corresponding image points should be matched. In this way, consistent matching should satisfy $\bfX_{ij} = \bfA_i\bfA_j^T$, where $\bfA_i\in\Bin{p_i}{k}$ denotes the map from Image $i$ to the universe, $k$ is the number of points in the universe, and $k\geq p_i$ for all $i$.
Suppose the correspondences for all $m=\sum_{i=1}^{n}p_i$ features in the image collection is denoted by $\bfX\in\Bin{m}{m}$: $$\begin{aligned}
\bfX=\left(\begin{array}{cccc}
\bfX_{11} & \bfX_{12} & \cdots & \bfX_{1n}\\
\bfX_{21} & \bfX_{22} & \cdots & \bfX_{2n}\\
\vdots & \vdots & \ddots & \vdots\\
\bfX_{n1} & \cdots & \cdots & \bfX_{nn}
\end{array}\right),\label{eq:bigX}\end{aligned}$$ and all $\bfA_i$s are concatenated as rows in a matrix $\bfA\in\Bin{m}{k}$. Then, one can write $\bfX$ as $$\begin{aligned}
\bfX=\bfA\bfA^T, \label{eq:univ}\end{aligned}$$ From , it is clear to see that a desired $\bfX$ should be both positive semidefinite and low-rank: $$\begin{aligned}
\bfX \succeq 0, ~~~~ \rank{\bfX}\leq k. \label{eq:lrsdp-constr}\end{aligned}$$
Using the cycle consistency can be effectively imposed without checking all cycles of pairwise matches. Moreover, partial matching is allowed, while bijection needs to be assumed in .
Joint matching via rank minimization
====================================
Given affinity scores $\{\bfS_{ij}~|~1\leq i,j \leq n\}$, we aim to find globally consistent matches $\bfX$. Note that $\bfS_{ij}$ can be all-zero if matching is not performed for a pair $(i,j)$. Moreover, affinity scores can be computed from either feature similarities or graph matching solvers according to specific scenarios, as described in .
Formulation {#sec:formulation}
-----------
We formulate the problem as a low-rank matrix recovery problem. We maximize the inner product between $\bfX_{ij}$ and $\bfS_{ij}$ for all $i$ and $j$ as multiple linear assignment problems. At the same time, we minimize the rank of $\bfX$ to enforce the cycle consistency. We ignore the positive semidefinite constraint on $\bfX$ and will explain the reasons later.
To make the optimization tractable, we make the following relaxations: (1) $\bfX$ is treated as a real matrix $\bfX\in[0,1]^{m\times m}$ instead of a binary matrix, which is a general practice in solving matching problems. Experimentally, we found that the solution values were very close to 0 or 1 and could be stably quantized by a threshold of 0.5. This might be attributed to the existence of a linear term in the cost function [@maciel2003global]. (2) Rank of $\bfX$ is replaced by the nuclear norm $\|\bfX\|_*$ (sum of singular values), which is a tight convex relaxation proven to be very effective in various low-rank problems such as matrix completion [@candes2009exact] and robust principal component analysis [@candes2011robust].
The estimated $\bfX$ should be sparse since at most one value in each row of $\bfX_{ij}$ can be nonzero. To induce sparsity, we minimize the sum of values in $\bfX$. Combining all three terms, we obtain the following cost function: $$\begin{aligned}
\label{eq:raw}
f(\bfX) &= -\sum_{i=1}^{n}\sum_{j=1}^{n} \InProd{\bfS_{ij}}{\bfX_{ij}} + \alpha\InProd{\bfone}{\bfX} + \lambda\|\bfX\|_*, \nonumber \\
&= -\InProd{\bfS-\alpha\bfone}{\bfX} + \lambda\|\bfX\|_*,\end{aligned}$$ where $\InProd{\cdot}{\cdot}$ denotes the inner product and $\bfS\in\RR{m}{m}$ is the matrix collecting all $\bfS_{ij}$s. $\alpha$ is the weight of sparsity, which can be interpreted as a threshold to remove small scores in $\bfS_{ij}$s. In our implementation, we normalize the scores to let them lie between 0 and 1 and empirically set $\alpha=0.1$. $\lambda$ controls the weight of the nuclear norm. We will discuss $\lambda$ in and .
Besides the doubly stochastic constraints in , additional constraints shall be imposed on $\bfX$ after relaxation: $$\begin{aligned}
& \bfX_{ii} = \bfI_{p_i}, ~~~~ 1 \leq i \leq n, \label{eq:diag-constr} \\
& \bfX_{ij} = \bfX_{ji}^T, ~~ 1 \leq i,j \leq n, i\neq j, \label{eq:sym-constr} \\
& \bfzero \leq \bfX \leq \bfone, \label{eq:bound-constr}\end{aligned}$$ where constrains self-matching to be identity, constrains $\bfX$ to be symmetric, and constrains the values in $\bfX$ to lie in $[0,1]$.
Finally, we obtain the following optimization problem: $$\begin{aligned}
\label{eq:basic}
\min_{\bfX} ~ & \InProd{\bfW}{\bfX} + \lambda \|\bfX\|_*, \nonumber \\
\st ~~ & \bfX \in \mathcal{C},\end{aligned}$$ where $\bfW=\alpha\bfone-\bfS$ and $\mathcal{C}$ denotes the set of matrices satisfying the constraints given in , , and .
Upon our experimental observation, the result doesn’t degrade noticeably when removing the doubly stochastic constraints in . This might be attributed to the existence of the sparsity regularizer. Therefore, we remove in implementation to accelerate the computation.
Positive semidefiniteness {#sec:psd}
-------------------------
We ignore the positive semidefinite constraint for two reasons: (1) solving SDP is generally unscalable; (2) with the constraints in and , the solution to turns out to be nearly positive semidefinite if $\lambda$ is sufficiently large.[^1]
Suppose $\sigma_1,\cdots,\sigma_m$ are eigenvalues of $\bfX$. From , we have $X_{ii}=1$ for all $i$, and $\sum_{i=1}^{m}\sigma_i=\trace{\bfX}=m$, which implies that the sum of $\sigma_i$s is fixed. From , we have $\bfX$ is symmetric, and $\sigma_i$s are all real numbers. When we choose a large $\lambda$, $\|\bfX\|_*=\sum_{i=1}^{m}|\sigma_i|$ dominates the cost function, and a solution with all nonnegative $\sigma_i$s will give the lowest cost, because $\sum_{i=1}^{m}|\sigma_i|\geq\sum_{i=1}^{m}\sigma_i=m$ and the equality holds iff. $\sigma_i\geq 0$ for all $i$.
The boundness $\|\bfX\|_*\geq m$ also implies that the solution to will be insensitive to $\lambda$ when $\lambda$ is sufficiently large, and then minimizing the nuclear norm is equivalent to adding a positive semidefinite constraint. The effect of $\lambda$ is experimentally illustrated in .
Fast alternating minimization {#sec:optimization}
=============================
Optimization in the low-rank space {#sec:factorization}
----------------------------------
The nuclear norm minimization in is convex and the state-of-the-art methods to solve this family of problems are the proximal method [@parikh2013proximal] or ADMM [@boyd2010distributed] based on iterative singular value thresholding [@cai2010singular]. However, singular value decomposition (SVD) needs to be performed in each iteration, which is extremely expensive even for a medium-sized problem. For instance, if there are 20 images with 500 features per image to match, we have to optimize for an $10,000\times 10,000$ matrix. A single SVD for such a matrix takes hundreds of seconds on a typical PC even if a partial SVD solver [@larsen2004propack] is used. See and .
$[n,p]$ $m=np$ MatchLift Partial SVD MatchALS
------------ ----------------- ----------- ------------- ---------- --
$[5,20]$ $1.0\times10^2$ 0.005 0.016 0.001
$[10,20]$ $2.0\times10^2$ 0.009 0.016 0.003
$[20,20]$ $4.0\times10^2$ 0.034 0.033 0.009
$[20,100]$ $2.0\times10^3$ 1.472 2.023 0.283
$[20,500]$ $1.0\times10^4$ 166.8 219.3 9.804
: The CPU time (seconds) for one iteration of MatchALS, MatchLift [@chen2014near] and partial SVD [@larsen2004propack]. $n$, $p$, and $m$ denote the number of images, the number of points per image, and the dimension of $\bfX$, respectively. We set $k=2p$ for MatchALS. []{data-label="tab:time-simu"}
Fortunately, recent results on low-rank optimization have shown that one can solve the problem more efficiently via a change of variables $\bfX=\bfA\bfB^T$ [@cabral2013unifying; @hastie2014matrix], where $\bfA,\bfB\in\RR{m}{k}$ are new variables with a smaller dimension $k<m$. More importantly, the change of variables will not introduce additional local minima if $k$ is larger than the rank of the original solution. This result was originally derived for low-rank SDP [@burer2005local; @kulis2007fast] but also applies here since a nuclear-norm minimization problem can be rewritten as SDP [@recht2010guaranteed].
Inspired by these works, we propose the following low-rank factorization-based formulation in order to leverage the underlying low dimensionality of our problem: $$\begin{aligned}
\label{eq:als}
\min_{\bfA,\bfB} ~ & \InProd{\bfW}{\bfA\bfB^T} + \lambda \|\bfA\bfB^T\|_*, \nonumber \\
\st ~~ & \bfA\bfB^T \in \mathcal{C}.\end{aligned}$$ Moreover, with the following equation [@recht2010guaranteed], $$\begin{aligned}
\label{nuclear_fro}
\|\bfX\|_* = \min_{\bfA,\bfB:\bfA\bfB^T=\bfX} ~ \frac{1}{2}\left(\|\bfA\|_F^2+\|\bfB\|_F^2\right),\end{aligned}$$ we finally obtain the following formulation: $$\begin{aligned}
\label{eq:als}
\min_{\bfA,\bfB} ~ & \InProd{\bfW}{\bfA\bfB^T} + \frac{\lambda}{2} \|\bfA\|_F^2 + \frac{\lambda}{2} \|\bfB\|_F^2, \nonumber \\
\st ~~ & \bfA\bfB^T \in \mathcal{C}.\end{aligned}$$ The selection of matrix dimension $k$ is critical to the success of change of variables, while it directly affects the computational complexity. We will first provide the algorithm, analyze its complexity and then discuss the selection of $k$.
Algorithms
----------
The problem in is not straightforward to solve due to the constraint on the product of variables. Instead, we rewrite the problem as $$\begin{aligned}
\label{eq:als1}
\min_{\bfX,\bfA,\bfB} ~ & \InProd{\bfW}{\bfX} + \frac{\lambda}{2} \|\bfA\|_F^2 + \frac{\lambda}{2} \|\bfB\|_F^2, \nonumber \\
\st ~~ & \bfX = \bfA\bfB^T, ~~ \bfX \in \mathcal{C},\end{aligned}$$ and apply the ADMM [@boyd2010distributed] to solve .
The augmented Lagrangian of reads: $$\begin{aligned}
\mathcal{L}_{\mu}\left(\bfX,\bfA,\bfB,\bfY\right) = & \InProd{\bfW}{\bfX} + \frac{\lambda}{2} \|\bfA\|_F^2 + \frac{\lambda}{2} \|\bfB\|_F^2 \\
+ & \InProd{\bfY}{\bfX-\bfA\bfB^T} + \frac{\mu}{2}\|\bfX-\bfA\bfB^T\|_F^2 \nonumber\end{aligned}$$ where $\bfY$ is the dual variable and $\mu$ is a parameter controlling the step size in optimization. We keep the constraint $\bfX\in\mathcal{C}$ since it can be easily handled as we will show later. Then, the ADMM alternately updates each primal variable by minimizing $\mathcal{L}_{\mu}$ and updates the dual variable via gradient ascent while fixing all other variables. The overall algorithm is summarized in .
\[alg:MatchALS\]
randomly initialize $\bfA$ and $\bfB$, $\bfY=\bfzero$ $\bfW=\alpha\bfone-\bfS$ quantize $\bfX$ with a threshold equal to 0.5.
Minimizing $\mathcal{L}_{\mu}$ over $\bfA$ turns out to be a regularized least squares problem with a closed-form solution given in Step 4 in . The update of $\bfB$ can be solved similarly. The update of $\bfX$ requires to solve: $$\begin{aligned}
\min_{\bfX\in\mathcal{C}} \| \bfX - \bfA\bfB^T + \frac{1}{\mu}\left(\bfW+\bfY\right) \|_F^2,\end{aligned}$$ and the solution turns out to be a projection to $\mathcal{C}$. Since the constraints in $\mathcal{C}$ are all linear, the projection can be solved conveniently. We denote the solution by ${\mathcal{P}_{\mathcal{C}}\left(\cdot\right)}$ and leave the details to the supplementary material.
Computational complexity {#sec:complexity}
------------------------
The time complexity of an iteration in is dominated by matrix multiplication that requires $O(m^2k)$ flops[^2]. We compare it to the state-of-the-art algorithm MatchLift [@chen2014near], which is based on SDP. The time complexity of an iteration in MatchLift is dominated by the eigenvalue decomposition that requires $O(m^3)$ flops. As $m$ is much larger than $k$, MatchALS has a lower complexity compared to MatchLift. Moreover, matrix multiplication is parallelizable and has been inherently multithreaded in Matlab, while the parallelization of eigenvalue decomposition is an open problem. Both MatchALS and MatchLift are based on ADMM and require similar numbers of iterations to converge upon our observation.
The CPU time for some problem sizes is shown in . The algorithms are implemented in Matlab and tested on a PC with an Intel i7 3.4GHz CPU and 8G RAM. We also compare the time cost of partial SVD using PROPACK [@larsen2004propack], a toolbox widely used to solve large-scale matrix completion problems [@lin2010augmented]. In partial SVD, only $r$ leading singular vectors are computed, which is much faster than full SVD when $r/m$ is extremely small. But it is not efficient for a relatively large $r$. In our problem, $r$ should be larger than the true rank and we test partial SVD with $r=p$ in .
Selection of $k$ and rank reduction {#sec:rank}
-----------------------------------
From the previous subsections we see that $k$ determines the complexity of MatchALS and $k$ should be larger than the rank of true solution, i.e. the size of universe. While some spectral techniques have been proposed in previous work for rank estimation [@chen2014near], we found that the estimation was inaccurate when the input was noisy and incomplete. Fortunately, our solution doesn’t depend on $k$ if $k$ is larger than the underlying true rank (demonstrated later in ). A heuristic choice is to set $k=2\hat{r}$, where $\hat{r}$ is a rough estimate of the size of universe.
In real applications, there are likely to be many isolated features in each image which don’t have any correspondence in other images. However, the constraint in implies that every image feature must be matched to a point in the universe. To see this, recall that we hope $\bfX=\bfA\bfA^T$ in . If diagonal values of $\bfX$ are all ones, every row of $\bfA$ has a unit norm, which indicates a match to the universe. Therefore, the size of universe is dramatically increased by those isolated features, and consequently a very large $k$ needs to be selected, which severely increases the computation. To address this issue, we loose the constraint in to be $$\begin{aligned}
\trace{\bfX} &= m', \nonumber \\
\mbox{off-diagnal values}\{\bfX_{ii}\} &= \bfzero, ~ 1\leq i \leq n, \label{eq:diag-constr-new}\end{aligned}$$ where $m'\leq m$ is a predefined constant. When $m'=m$, is reduced to . When $m'<m$, we allow some rows and columns in $\bfX$ to be null, which is most likely to happen for the rows and columns corresponding to the isolated features, since “switching" them off will not lose many affinity scores but be able to reduce the nuclear norm immediately. By using such a “rank reduction" strategy, the algorithm can automatically prune the isolated features and reduce the size of universe, which enables us to select a smaller $k$ for better computational efficiency. We set $m'=m$ in simulation since there is no isolated feature and $m'=0.7m$ in real experiments.
Experiments {#sec:experiments}
===========
Simulation
----------
We evaluate the performance of the proposed method using synthesized data. Given a permutation matrix $\bfX$ and the ground truth $\bfX^*$, we measure the error rate by intersection over union: $$\begin{aligned}
1 - \frac{\left|\tau(\bfX)\cap\tau(\bfX^*)\right|}{\left|\tau(\bfX)\cup\tau(\bfX^*)\right|},\end{aligned}$$ where $\tau$ denotes the matches defined by a permutation matrix and $|\cdot|$ means the size of a set.
### Matching errors
Spectral
![The 2D plot of matching errors under various problem settings for the spectral method [@pachauri2013solving], MatchLift [@chen2014near] and the proposed MatchALS. In the left column, the number of images $n$ and the input error rate $\rho_e$ are varying, while the observation ratio $\rho_o=0.6$. In the right column, $\rho_o$ and $\rho_e$ are varying, while $n=20$. Lower intensity indicates smaller error and overall a larger dark region indicates a better performance.[]{data-label="fig:simu-error"}](figures/result-avg-err-sp-1.pdf "fig:"){width="0.45\linewidth"} ![The 2D plot of matching errors under various problem settings for the spectral method [@pachauri2013solving], MatchLift [@chen2014near] and the proposed MatchALS. In the left column, the number of images $n$ and the input error rate $\rho_e$ are varying, while the observation ratio $\rho_o=0.6$. In the right column, $\rho_o$ and $\rho_e$ are varying, while $n=20$. Lower intensity indicates smaller error and overall a larger dark region indicates a better performance.[]{data-label="fig:simu-error"}](figures/result-avg-err-sp-2.pdf "fig:"){width="0.45\linewidth"}\
MatchLift
![The 2D plot of matching errors under various problem settings for the spectral method [@pachauri2013solving], MatchLift [@chen2014near] and the proposed MatchALS. In the left column, the number of images $n$ and the input error rate $\rho_e$ are varying, while the observation ratio $\rho_o=0.6$. In the right column, $\rho_o$ and $\rho_e$ are varying, while $n=20$. Lower intensity indicates smaller error and overall a larger dark region indicates a better performance.[]{data-label="fig:simu-error"}](figures/result-avg-err-lift-1.pdf "fig:"){width="0.45\linewidth"} ![The 2D plot of matching errors under various problem settings for the spectral method [@pachauri2013solving], MatchLift [@chen2014near] and the proposed MatchALS. In the left column, the number of images $n$ and the input error rate $\rho_e$ are varying, while the observation ratio $\rho_o=0.6$. In the right column, $\rho_o$ and $\rho_e$ are varying, while $n=20$. Lower intensity indicates smaller error and overall a larger dark region indicates a better performance.[]{data-label="fig:simu-error"}](figures/result-avg-err-lift-2.pdf "fig:"){width="0.45\linewidth"}\
MatchALS
![The 2D plot of matching errors under various problem settings for the spectral method [@pachauri2013solving], MatchLift [@chen2014near] and the proposed MatchALS. In the left column, the number of images $n$ and the input error rate $\rho_e$ are varying, while the observation ratio $\rho_o=0.6$. In the right column, $\rho_o$ and $\rho_e$ are varying, while $n=20$. Lower intensity indicates smaller error and overall a larger dark region indicates a better performance.[]{data-label="fig:simu-error"}](figures/result-avg-err-mm-1.pdf "fig:"){width="0.45\linewidth"} ![The 2D plot of matching errors under various problem settings for the spectral method [@pachauri2013solving], MatchLift [@chen2014near] and the proposed MatchALS. In the left column, the number of images $n$ and the input error rate $\rho_e$ are varying, while the observation ratio $\rho_o=0.6$. In the right column, $\rho_o$ and $\rho_e$ are varying, while $n=20$. Lower intensity indicates smaller error and overall a larger dark region indicates a better performance.[]{data-label="fig:simu-error"}](figures/result-avg-err-mm-2.pdf "fig:"){width="0.45\linewidth"}
We follow the settings in [@chen2014near] to evaluate the performance of MatchALS and compare it to alternative methods. The size of universe is fixed as 20 points and in each image a random sample of the points are observed with a probability denoted by $\rho_o$. The number of images is denoted by $n$. Then, the ground-truth pairwise matches are established, and random corruptions are simulated by removing some true matches and adding some false matches to achieve an error rate of $\rho_e$. Finally, the corrupted permutation matrix is fed into as the input affinity scores.
We evaluate the performance of MatchALS under various $\rho_o$, $\rho_e$ and $n$. We compare MatchALS to two related methods: MatchLift [@chen2014near] and the spectral method [@pachauri2013solving]. Both of the alternative methods require to know the size of universe and we provide the true value $r^*=20$. For MatchALS parameters, we set $k=2r^*$ and $\lambda=50$.
The output error rates under various settings are shown in . When the number of images is sufficiently large, all methods can achieve nearly exact recovery even if the input error rate is larger than $50\%$, which demonstrates the power of joint matching. MatchALS and MatchLift achieve very similar performances and outperform the spectral method especially when the observation ratio is small. Compared to MatchLift, the proposed method obtains a competitive performance without exactly knowing the true rank and requires much less computation time.
### Sensitivity to parameters {#sec:param}
![The estimation error versus the input rank $\hat{r}$ and the weight of nuclear norm $\lambda$. The true rank $r^*=20$. Here we set $k=\hat{r}$ for MatchALS.[]{data-label="fig:sensitivity"}](figures/result-sensitivity-rank.pdf "fig:"){width="0.49\linewidth"} ![The estimation error versus the input rank $\hat{r}$ and the weight of nuclear norm $\lambda$. The true rank $r^*=20$. Here we set $k=\hat{r}$ for MatchALS.[]{data-label="fig:sensitivity"}](figures/result-sensitivity-lambda.pdf "fig:"){width="0.49\linewidth"}
The sensitivity of MatchALS to the parameters in is illustrated in . The figure shows that MatchALS is insensitive to the predefined dimension of factor matrices $k$ when $k$ is larger than the true rank $r^*$, as we explained in . When $k<r^*$, the problem in is no longer equivalent to the original convex problem in , and consequently the alternating minimization fails. In practice, we choose $k=2\hat{r}$ as a compromise between safety and efficiency. The right panel in illustrates that the algorithm is insensitive to $\lambda$ when $\lambda$ is sufficiently large as we explained in . In all our experiments, we set $\lambda=50$.
Real experiments
----------------
### Graffiti datasets {#sec:graffiti}
Graffiti Bikes Light\
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{width="0.3\linewidth"} {width="0.3\linewidth"} {width="0.3\linewidth"}
We evaluate the performance of our algorithm on six benchmark datasets from the Graffiti datasets[^3]. In each dataset, there are six images of a scene with various image transformations such as viewpoint change, blurring, illumination variation, etc.
We detect 1000 affine covariant features [@mikolajczyk2005comparison] with SIFT [@lowe2004distinctive] descriptors from each image using the VLFeat library [@vedaldi08vlfeat]. For each image pair $(i,j)$, we compute the inner products between feature descriptors as affinity scores and only keep the scores larger than 0.7 and collect them in $S_{ij}$. If the ratio between the first and the second largest scores in a row/column is smaller than 1.1, we set all scores in this row/column to be zero in order to remove indistinctive features. After computing all $\bfS_{ij}$, we remove the features that have candidate matches in less than two images since they have no contribution to joint matching. Finally, we input the affinity scores to to obtain the optimized joint matches.
For evaluation, we adopt the metric used in [@chen2014near]: for a testing point in an image, we calculate the distance between its estimated correspondence and the true correspondence in another image. If the distance is smaller than a threshold, we regard that a correct match is found for this testing point. Then, we plot the percentages of testing points with correct matches versus the threshold values and obtain a curve analogous to a precision-recall curve. If a testing point is not aligned with any detected point, its estimated correspondence is obtained by interpolation. In this experiment, we use all detected feature points in the first image as testing points and evaluate the matches from the first image to the other five images. True correspondences are computed from the homography matrices provided in the datasets.
The performance curves on three datasets are shown in . A curve closer to the upper-left corner indicates a better performance. The area under curve and computation time for all datasets are summarized in . All of the joint matching methods achieve obvious improvements compared to the original pairwise matching. MatchALS and MatchLift perform similarly and outperforms the spectral method, which coincides with the observation in simulation. Regarding computation time, MatchALS achieves a remarkable speedup ($\sim$30 times on average) compared to MatchLift.
-------------------------------------------------------------------------------------------
Original MatchALS Spectral MatchLift
---------- ---------- ---------------------------------------------- ---------- -----------
Graffiti 60.2% **[87.3%]{} & 75.6% & 80.6%\
Bikes & 76.8% & **[94.3%]{} & 86.7% & 92.5%\
Boat & 86.2% & **[93.9%]{} & 87.7% & 91.7%\
Light & 76.0% & 93.9% & 90.0% & **[94.0%]{}\
Bark & 71.7% & **[92.2%]{} & 91.2% & 90.0%\
UBC & 88.0% & **[97.0%]{} & 92.9% & 96.8%\
Time & - & 85.8 & 86.4 & 2518.4\
************
-------------------------------------------------------------------------------------------
: The matching scores and the average computation time (seconds) on the Graffiti datasets. The score is calculated as the area under the curve shown in .[]{data-label="tab:graffiti"}
We select three image pairs to visually demonstrate the effect of joint matching in . A match with a deviation less than five pixels from the ground truth is declared as true. Clearly, the joint matching can prune the false matches (fewer blue lines), complete some missing matches (denser yellow lines), and achieve almost correct matching for these image pairs with large disparities in viewpoints, blurring and illumination changes.
### Matching different objects {#sec:cars}
{width="0.8\linewidth"}\
{width="0.4\linewidth"} {width="0.4\linewidth"}\
Recent years have witnessed growing interest in reconstructing category-specific object models from single images, which is still an open problem [@vicente2014reconstructing; @carreira2014virtual]. Among a series of challenges, feature matching for different object instances is the foremost and previous work usually assumed that correspondences of some keypoints were given [@vicente2014reconstructing; @carreira2014virtual]. In this section, we demonstrate the applicability of joint matching to solve this problem.
We use the FG3DCar datasets [@Lin2014jointly] and try to match the images of different car models in the same category (e.g., sedan or SUV). Following the general practice in object reconstruction [@vicente2014reconstructing; @carreira2014virtual], we assume segmentation is provided such that background can be ignored, and we only match images with similar views. We select nine sedans and eight SUVs and match two sets of images separately. Note that the car models are all different from each other. See for examples.
To exact descriptive features we first detect image edges by the structured forests [@dollar2013structured] and sample a number of points on the edges with constant spacing. On average, we obtain $\sim$600 feature points for each image. Since the object appearance is changed from image to image and the features are automatically extracted, the matching is extremely difficult. Inspired by recent works [@weinzaepfel2013deepflow; @carreira2014virtual], we adopt deep features, i.e., middle-layer responses of convolutional neural nets (CNN), as descriptors for feature matching. More specifically, we use the publicly available deep learning toolbox Caffe [@jia2014caffe] and the pre-trained CNN Alexnet [@krizhevsky2012imagenet]. We feed a $192\times 192$ patch around each feature point forward through the Alexnet. The center columns of conv4 and conv5 layers are concatenated and normalized to form a 640 dimensional feature vector. To leverage the prior on object rigidity, we use pairwise graph matching solved by the Reweighted Random Walk algorithm [@cho2010reweighted] and collect the output scores of candidate matches as affinity scores. Then, we delete the points with candidate matches in less than two images and run MatchALS.
![The performance curve of car image matching. “Deep" represents deep features. “GM" denotes graph matching. “Joint" means joint matching using the proposed method.[]{data-label="fig:cars-plot"}](figures/sedan-plot.pdf){width="0.5\linewidth"}
We adopt the same metric introduced in for quantitative evaluation and use the manually-annotated landmarks provided in the datasets as ground truth. The result on the sedan images is shown in . Matching with SIFT features fails since local image patterns are different for two cars. Graph matching with deep features obtains a much better performance, which is further improved by the proposed joint matching algorithm. We obtain a very similar result on the SUV images, which is not plotted.
The results are visualized in . The corresponding parts of cars are basically matched in spite of the large differences in appearances and viewpoints. Note that the features are automatically detected and therefore not fully overlapped for two images. For a simple demonstration, we run rigid reconstruction from the estimated feature correspondences by triangulation with an orthographic camera model and the viewpoints provided in the dataset. Despite some noises and missing points, we can clearly see the 3D structures of a sedan and a SUV. We believe that more appealing reconstructions can be obtained by using sophisticated reconstruction techniques and more information such as object silhouette and surface smoothness, while they are out of the scope of this paper.
Conclusion
==========
In this paper, we proposed a practical solution to multi-image matching. We use pairwise feature similarities or graph matching scores as input and obtain accurate matches by an efficient algorithm that globally optimizes for both feature affinities and cycle consistency of matches. The experiments not only validate the effectiveness of the proposed method but also demonstrate that joint matching is a promising approach to matching images with different object instances as the first step towards reconstructing object models from crowd-sourced image collections. As future work, we would like to explore more applications and incremental algorithms for joint matching.
**Acknowledgments**: Grateful for support through the following grants: NSF-DGE-0966142, NSF-IIS-1317788, NSF-IIP-1439681, NSF-IIS-1426840, ARL RCTA W911NF-10-2-0016, and ONR N000141310778
[^1]: We use the term “nearly positive semidefinite" to refer to the property that the negative eigenvalues of a matrix, if there exist, are negligible compared to the norm of the matrix.
[^2]: The detail is given in the supplementary material
[^3]: http://www.robots.ox.ac.uk/ vgg/data/data-aff.html
|
---
abstract: 'In preparation for deep extragalactic imaging with the James Webb Space Telescope, we explore the clustering of massive halos at $z=8$ and $10$ using a large N-body simulation. We find that halos with masses $10^9$ to $10^{11}$ ${\ensuremath{h^{-1}\;M_\odot}}$, which are those expected to host galaxies detectable with JWST, are highly clustered with bias factors ranging from 5 and 30 depending strongly on mass, as well as on redshift and scale. This results in correlation lengths of 5–10[$h^{-1}\;{\rm Mpc}$]{}, similar to that of today’s galaxies. Our results are based on a simulation of 130 billion particles in a box of $250{\ensuremath{h^{-1}\;{\rm Mpc}}}$ size using our new high-accuracy [<span style="font-variant:small-caps;">Abacus</span>]{}simulation code, the corrections to cosmological initial conditions of @2016MNRAS.461.4125G, and the *Planck* 2015 cosmology. We use variations between sub-volumes to estimate the detectability of the clustering. Because of the very strong inter-halo clustering, we find that surveys of order 25[$h^{-1}\;{\rm Mpc}$]{} comoving transverse size may be able to detect the clustering of $z=8$–10 galaxies with only 500-1000 survey objects if the galaxies indeed occupy the most massive dark matter halos.'
author:
- Hao Zhang
- 'Daniel J. Eisenstein'
- 'Lehman H. Garrison'
- 'Douglas W. Ferrer'
bibliography:
- 'bibliography.bib'
title: Testing the Detection Significance on the Large Scale Structure by a JWST Deep Field Survey
---
Introduction
============
Galaxy formation is strongly influenced by large-scale structure. In dark matter halos of high mass, gas is easier to cool and can thus form stars and galaxies [@1997MNRAS.286..795K]. The halos hosting luminous galaxies at high redshift are expected to be massive, rare, and therefore highly clustered. This in turn implies that the galaxies should be highly clustered, corresponding to large bias values [@1999MNRAS.303..188K; @2017MNRAS.469.4428J]. Observations on Lyman $\alpha$ emitters [@2014PASJ...66R...1T; @2015MNRAS.453.1843S; @2017arXiv170407455O], Lyman $\alpha$ blobs , Lyman break galaxies [@2014ApJ...793...17B; @2016ApJ...821..123H; @2017arXiv170406535H] and HST deep field observations [@2006ApJ...648L...5O; @2013ApJ...768..196S; @2014ApJ...796L..27R] have supported this, finding large angular clustering and field-to-field density variations.
As massive halos are extreme fluctuations in the density field, the resulting number of these host sites and their clustering is unusually sensitive to the cosmological model. Therefore, measuring the clustering can be indicative for the halo mass function. This has been a common application of halo occupation distribution modeling, a method in which the association of galaxies to halos of a given mass leads to detailed predictions of galaxy clustering [for a review, see @2002PhR...372....1C].
However, the extreme sensitivity of the clustering on the cosmological properties also requires careful control of the initial conditions and numerical methods. In this paper, we present a large-scale high-resolution N-body simulation to investigate halo clustering at high redshifts. Our work includes several improvements that we argue will improve the reliability of the results. First, we adopt our cosmological parameters from the most recent Planck measurements . The increase in the matter density, $\Omega_m h^2$, relative to previous CMB results increases the small-scale fluctuations in LCDM and hence the abundance of halos at a given mass. Second, we utilize our new N-body cosmological code [<span style="font-variant:small-caps;">Abacus</span>]{}, which features high force accuracy. We adopt a small particle mass of $10^7{\ensuremath{h^{-1}\;M_\odot}}$ so that halos of $10^{10}{\ensuremath{h^{-1}\;M_\odot}}$, which we expect will be typical of detectable galaxies with JWST, will be well-resolved. The high speed of Abacus allows us to still run a box of 130 billion particles filling $248.8{\ensuremath{h^{-1}\;{\rm Mpc}}}$, big enough to capture most of the large-scale modes relevant to the formation of these halos. Third, we utilize the corrections to the linear-theory initial conditions highlighted by @2016MNRAS.461.4125G and also use the methods in that paper to include second-order perturbation theory terms in the initial conditions.
In order to compare with upcoming JWST deep field surveys at high redshifts, we analyze time slices of our simulation at $z=10$ and $z=8$. Since the simulation volume of around $(250{\ensuremath{h^{-1}\;{\rm Mpc}}})^2$ is much larger than the volume of a typical JWST survey, we are able to cut the simulation into many sub-volumes and use the variations between them to estimate the covariance of the clustering. Here, we choose to divide the region into 10 $\times$ 10 boxes, investigate the clustering in our simulated halo catalogs, and predict a detection significance for one box representing a single survey.
While we were preparing this publication, the work of [@2017arXiv170702312B] was published which investigated clustering at $z > 7$ using a cosmological hydrodynamical simulation called the BLUETIDES simulation [@2016MNRAS.455.2778F]. Their research obtained results that are compatible with ours.
In Section 2, we introduce the simulation used in this work. In Section 3, we describe our methodology and then present our results. In section 4, we give our conclusion and a discussion.
Cosmological Simulation
=======================
[<span style="font-variant:small-caps;">Abacus</span>]{}is a code for cosmological N-body simulation (Ferrer et al., in prep.; Metchnik & Pinto, in prep.) that is both extremely fast and highly accurate, aided by recent computational techniques and commodity hardware for high performance computing. Abacus utilizes a novel fully-disjoint split between the near-field and far-field gravitational sources, solving the former on GPU hardware and the latter with a variant of a multipole method. The result is very high speed, in excess of 20 million particle updates per second on a single 24-core workstation. Further, Abacus is built to store most of its data on a high-speed disk system, allowing us to run multi-terabyte problems on a single computer with only modest amounts of RAM.
In this paper, we use a single $5120^3$ simulation of a $(248.8{\ensuremath{h^{-1}\;{\rm Mpc}}})^3$ box. This results in a particle mass of $10^7{\ensuremath{h^{-1}\;M_\odot}}$, suitable to robustly identify halos with masses around $10^{10}{\ensuremath{h^{-1}\;M_\odot}}$. We evolve the simulation using a standard leap-frog integration with 225 time steps from $z=199$ to $z=10$ and 67 more to $z=8$. All particles have the same time step. The simulation was run on a single commodity-based 24-core dual Xeon workstation with 256 GB of RAM, 2 NVidia GeForce GTX 980 Ti GPUs, and a RAID system providing over 1.5 GB/sec of disk speed, with each time step taking about 2.2 hours.
[@PhysRevD.73.103507] showed that solutions to the discrete N-body problem do not correctly recover the continuum linear perturbation theory found in cosmological textbooks for wavenumbers near the Nyquist wavenumber. Most Fourier modes grow too slowly, although a few grow too quickly. While the effects are small for modes much larger than the inter-particle spacing, we are nevertheless concerned that the formation of extreme halos is very sensitive to small changes in perturbation amplitude.
We therefore use the initial conditions proposed by [@2016MNRAS.461.4125G], whose method seeks to cancel out these linear theory errors at a given target redshift (here chosen to be $z=49$). This method is careful to use only the longitudinal linear-theory growing mode, which differs from the wavevector in the discrete theory. It then adjusts the initial displacement amplitudes of each mode so as to compensate for the non-standard growth function that will be encountered between the initial redshift of $z=199$ and the target redshift. Finally, we include second-order effects on the initial perturbations by inverting the particle displacements and using the sum of the forces in both cases to isolate the second-order forces, which are then applied as displacements and velocities assuming the continuum limit. Such second-order corrections are known to be important for the formation of the most massive halos [@doi:10.1111/j.1365-2966.2006.11040.x; @2015PhDT.......115S].
We adopted the cosmology of $\Omega_m = 0.31415$, $\Omega_{DE} = 1 - \Omega_{m}$, $\Omega_{K} = 0$, $h = 0.6726$ from the Planck measurements . The linear power spectra was calculated by the package of “Code for Anisotropies in the Microwave Background” (CAMB) [@2000ApJ...538..473L]. We started at the initial redshift of $z=199$ and used the output time slice at $z = 10$ and $8$.
The group finding algorithm that we adopt is the Friends-of-Friends algorithm [@1982ApJ...259..449P; @1982ApJ...257..423H], which connects all pairs of particles within a certain critical distance and then identifies clumps of interconnected particles above a certain multiplicity threshold as a halo. As our goal is to establish a more robust prediction of halo abundance and clustering, we require at least 300 particles in a halo and focus on the case of 1000 particles ($10^{10}{\ensuremath{h^{-1}\;M_\odot}}$) and at redshift of $z=10$ as our fiducial case. This ensures that halos are robustly found. For example, @2016MNRAS.461.4125G found that such multiplicity yielded well-converged results with respect to particle discreteness when using the initial conditions developed in that work.
We make the halo catalogs from our simulation available at <http://nbody.rc.fas.harvard.edu/public/JWST_products/>. Further documentation of the data files are given in [@2017ApJS...Submit...G] and at <https://lgarrison.github.io/AbacusCosmos/>.
Large Scale Structure of High-Redshift Halos
============================================
Clustering Methodology
----------------------
### Two-point Clustering Statistics {#2PCFmethod}
We aim to study the clustering of halos as a function of their mass using the two-point clustering statistics: the familiar two-point correlation function (2PCF) and power spectrum. We define samples based on thresholds in halo mass and compare the results between different threshold values, as well as to the correlations of the matter field and of linear theory. It is worth noting that halos of the requisite mass are treated as containing only one galaxy. Halo occupation distribution models commonly assign additional satellite galaxies to the most massive halos, which can further increase the clustering strength, particularly at intra-halo separations but also at inter-halo separations.
For our analysis of the halo clustering, we split the simulation volume into 100 rectangular pieces, each 25 by 25 by 250$h^{-1}$ comoving Mpc. We introduce the sub-volumes so that we can use the dispersion among the sub-volumes to determine the covariance matrix of the 2PCF. However, it is also the case that these volumes correspond to roughly the scale of a substantial JWST survey, about 13$'$ wide and $\Delta z=2$ at these redshifts.
We compute the 2PCF in each sub-volume, ignoring any periodicity, using the @1993ApJ...412...64L estimator $$\label{eq:xiDR}
\xi(\vec{r}) = \frac{DD - 2DR + RR}{RR}$$ where DD, DR and RR indicate the counts in each separation bin of data-data, data-random and random-random halo pairs, respectively. The random catalog is a uniform distribution across the entire volume. In detail, we use the simplicity of the rectangular volume to accelerate the DR and RR calculations by building interpolative functions to return the volumes of spheres near to the boundaries. We confirm that the mean of the sub-volume results is very similar, save at the largest separations, to the result for the full periodic simulation volume, where the DR and RR counts are trivially computed in the infinite sampling limit.
For the correlations of the non-linear matter field and linear theory, we first obtain their power spectra and then compute the 2PCFs from the power spectra based on the inverse Fourier Transform relation described in Eq. \[eq:xi\]. $$\label{eq:xi}
\xi(r) = \int dk \frac{k^2}{2\pi^2} \frac{\sin(kr)}{kr} e^{-(k\Sigma)^2}P(k)$$
The power spectra of the halos and of the matter field has been calculated in the conventional way using Fourier transforms of a large periodic gridded representation of the density field. Shot noise is removed as presented in @2015MNRAS.453L..11B and we divide by the transfer function of the grid with aliasing [@2005ApJ...620..559J].
### Detection Significance
Based on the 2PCFs of the 100 sub-volumes, the $(j, k)$ entry of the covariance matrix $\mathbf{C}_{jk}$ is given by $$\mathbf{C}_{jk} = \frac{1}{N - 1}\sum_{i = 1}^{N} \mathbf{d}_{ij} \mathbf{d}_{ik},
\label{eq:cov}$$ where $\mathbf{d}_{ij}$ denotes the j-th separation bin of the 2PCF in the i-th sub-volume. We then compute the detection significance by $$\chi^2 = \sum_{i = 1}^{n}\sum_{j = 1}^{n}(\mathbf{d}_{obs, i} - \mathbf{d}_{mean, i})(\mathbf{{C}}^{-1})_{ij}(\mathbf{d}_{obs, j} - \mathbf{d}_{mean, j})
\label{eq:chi2}$$ where $\mathbf{d}_{mean, i} = \frac{1}{N}(\sum_{k = 1}^{N} \mathbf{d}_{k,i})$. We use $\mathbf{d}_{obs,i} = 0$ to correspond to the unclustered case, which we interpret as a non-detection.
Results
-------
### Halo Sample Overview
\[propsbyN\]
------ ---------------------------------------- ----------------------- ------------------------------------- ------------------------------------- ------------------------------------- ------------------------------------- ---------------------------- -------------------------- ---- ----
($10^9{\ensuremath{h^{-1}\;M_\odot}}$) of halos $1{\ensuremath{h^{-1}\;{\rm Mpc}}}$ $5{\ensuremath{h^{-1}\;{\rm Mpc}}}$ $1{\ensuremath{h^{-1}\;{\rm Mpc}}}$ $5{\ensuremath{h^{-1}\;{\rm Mpc}}}$ 0.1[$h\;{\rm Mpc}^{-1}$]{} 1[$h\;{\rm Mpc}^{-1}$]{}
300 3.0 296364 10.9 0.85 0.19 0.060 1996 185 85 52
450 4.5 138720 15.2 1.04 0.25 0.070 2468 257 51 27
700 7.0 57127 22.5 1.31 0.35 0.087 3190 380 27 20
1000 10.0 26864 32.4 1.61 0.45 0.106 3976 536 19 15
1500 15.0 10668 52.4 2.02 0.64 0.130 5250 863 10 8
2000 20.0 5341 76.6 2.34 0.88 0.149 6283 1246 5 4
$8.25 \times 10^{-2}$ $1.43 \times 10^{-2}$ 83.3 1.017
$7.04 \times 10^{-2}$ $1.36 \times 10^{-2}$ 77.8 0.931
------ ---------------------------------------- ----------------------- ------------------------------------- ------------------------------------- ------------------------------------- ------------------------------------- ---------------------------- -------------------------- ---- ----
\[propsbyN8\]
------ ---------------------------------------- ----------------------- ------------------------------------- ------------------------------------- ------------------------------------- ------------------------------------- ---------------------------- -------------------------- ----- -----
($10^9{\ensuremath{h^{-1}\;M_\odot}}$) of halos $1{\ensuremath{h^{-1}\;{\rm Mpc}}}$ $5{\ensuremath{h^{-1}\;{\rm Mpc}}}$ $1{\ensuremath{h^{-1}\;{\rm Mpc}}}$ $5{\ensuremath{h^{-1}\;{\rm Mpc}}}$ 0.1[$h\;{\rm Mpc}^{-1}$]{} 1[$h\;{\rm Mpc}^{-1}$]{}
300 3.0 1916736 4.8 0.53 0.11 0.043 1201 84 199 105
450 4.5 1038761 6.0 0.62 0.13 0.047 1412 106 160 117
700 7.0 515064 8.0 0.74 0.16 0.055 1703 139 119 62
1000 10.0 284623 10.3 0.86 0.19 0.061 2003 178 99 50
1500 15.0 140220 14.3 1.07 0.24 0.069 2488 243 63 38
2000 20.0 82914 18.3 1.23 0.30 0.077 2913 309 48 26
3000 30.0 38051 27.3 1.55 0.42 0.097 3720 455 27 17
4000 40.0 21267 37.6 1.83 0.52 0.114 4535 628 15 10
6000 60.0 8894 59.9 2.46 0.76 0.162 6184 972 9 6
8000 80.0 4678 84.3 3.02 1.02 0.189 8028 1392 6 3
$0.127$ $2.10 \times 10^{-2}$ 113 1.58
$0.105$ $2.03 \times 10^{-2}$ 116 1.39
------ ---------------------------------------- ----------------------- ------------------------------------- ------------------------------------- ------------------------------------- ------------------------------------- ---------------------------- -------------------------- ----- -----
We begin in Fig. \[fig1\] with the mass distribution of our halo samples at $z=10$ and $z=8$, obtained by Friends of Friends algorithm. The number of halos above a series of mass cut are available in Tables \[propsbyN\] and \[propsbyN8\]. Below we primarily use the case of low particle number cut ${\ensuremath{N_{\rm min}}}= 1000$ as an illustration.
![A histogram of comoving number density of halos in bins of halo particle multiplicity. We divide the halo counts by the logarithmic bin width to yield the comoving number density per logarithmic mass bin. Recall that each particle is $10^7$[$h^{-1}\;M_\odot$]{}. The two histograms at $z=10$ and $z=8$ are overplotted.[]{data-label="fig1"}](fig1.pdf){width="7cm"}
![A thin slice through the simulation box showing halos larger than 300 particles ($3\times 10^9 {\ensuremath{h^{-1}\;M_\odot}}$) at $z=10$. Each halo is plotted as a circle with radius proportional to the 90th percentile of the radial particle distribution (“`r90`”); the radii are inflated by a factor of 10 for plotting purposes. We imagine this slice as a side-on view of what an observer to the left of the box would see; thus, the horizontal axis is redshift and the vertical axis is angular position. The depth of the slice is 25 comoving ${\ensuremath{h^{-1}\;{\rm Mpc}}}$, or 13.2’, which is the size of one of our “sub-volumes”. The horizontal shaded region demarcates the same width.[]{data-label="fig2"}](fig2.pdf){width="\linewidth"}
Fig. \[fig2\] shows a 25${\ensuremath{h^{-1}\;{\rm Mpc}}}$ thick slice of our simulation at $z=10$, the thickness chosen to match the width of one of our sub-volumes. The shaded region shows the same width, which allows one to gauge survey-to-survey variations by eye. One can see that there will indeed be such variations, depending on the chance intersection of the survey pencil beam with clusters and voids.
### Clustering in 3D Real Space: Halos, Matter Field and Linear Theory
Following the methods presented in Section \[2PCFmethod\], we compute the 2PCF of the $z=10$ halos containing more than $N = 1000$ particles and show the result in Fig. \[fig3\] along with the 2PCF of the $z=10$ matter field and linear theory. We adopt the ${\ensuremath{N_{\rm min}}}= 1000$, ${\ensuremath{M_{\rm min}}}= 10^{10}$ ${\ensuremath{h^{-1}\;M_\odot}}$ case as our representative one. This corresponds to 270 objects in a $13' \times 13'$ region at $z=10$. To remove the steep scale dependence of the 2PCF, we choose to plot the expression $r^2\xi(r)$ in the upper panel of Fig. \[fig3\]. This choice is common in the low-redshift literature. As we can see, the matter field 2PCF is consistent with the prediction of the linear theory, while the halo 2PCF is larger by a factor of order $10^2$ to $10^3$, corresponding to clustering biases of 10–30.
To highlight the scale-dependent bias, we repeat these results in the lower panel of Fig. \[fig3\] after dividing by the linear theory correlations function. This shows a notable increase in bias at scales below 2${\ensuremath{h^{-1}\;{\rm Mpc}}}$ compared to a possible plateau at large scale. We stress that the comoving diameter of a $10^{10}{\ensuremath{h^{-1}\;M_\odot}}$ halo is about $50h^{-1}$ kpc, so this scale dependence in the bias is occuring well outside the halo scale and indeed beyond even the $300h^{-1}$ kpc scale of the initial Lagrangian volume corresponding to this mass. We further note that this scale dependence occurs even though we have omitted satellite galaxies from our analysis. Indeed, because two halos cannot be closer together than the sum of their radii, our 2PCF drop precipitously at small scales. We do not plot results interior to 0.3[$h^{-1}\;{\rm Mpc}$]{} so as to comfortably avoid this effect.
![Comparison of the matter 2PCF (green), the halo 2PCF (blue), and the 2PCF predicted by linear theory at $z=10$ (red) for halos containing more than 1000 particles in a 1% sub-field, in $r^2\xi(r)$ (upper panel) and $\xi(r)/\xi_\mathrm{ref}(r)$ (lower panel). Note that the error bars in both cases indicate the standard deviation of a 1% sub-volume in our $10 \times 10$ partitioning (not the error on the mean for the full simulation volume). The y axis is in $r^2\xi(r)$ in the upper panel, where the flat profile of the halo 2PCF indicates the $r^{-2}$ power law relationship. In the lower panel, we can see that the matter power spectra is basically consistent with the prediction of linear theory, except for the distances below the grid scale where the matter 2PCF gets larger by a factor of 4. The halo 2PCF is highly biased by a multiple of $2 \times 10^2$ to $2 \times 10^3$. []{data-label="fig3"}](fig3.pdf){width="8cm"}
We then computed the corresponding power spectra for the three cases. These are shown in Fig. \[fig4\], with the linear theory power spectrum being the reference. Again, we obtain a qualitatively similar result as in the previous plot (Fig. \[fig3\]).
![Comparison of the matter power spectrum (green), the halo power spectrum (blue), and the power spectrum predicted by linear theory at $z=10$ (red, taken as reference). The matter power spectra is very consistent with the prediction of linear theory. The halo power spectra, however, has a very high bias at the order of $10^2$ to $10^3$.[]{data-label="fig4"}](fig4.pdf){width="8cm"}
Finally we investigate the dependence of the 2PCF on the halo mass cut. In Fig. \[fig5\] and \[fig6\], we plot $r^2\xi(r)$ at $z=10$ and $z=8$, respectively, for a range of halo mass cuts (3, 4.5, 7, 10, 15, 20 $\times 10^9\ {\ensuremath{h^{-1}\;M_\odot}}$ for both cases, and additionally 30, 40, 60, 80 $\times 10^9\ {\ensuremath{h^{-1}\;M_\odot}}$ for $z=8$). In Fig. \[fig5\], we find a very strong increase in bias for the increasing mass cut. Further, the correlation functions are shallower than $r^{-2}$ at lower masses, but steeper than $r^{-2}$ at higher masses. Again, this increase is occuring even though we have not included any satellite galaxies in our catalogs and it involves scales beyond the virialized diameters of the halos.
Fig. \[fig6\] shows the same progression at $z=8$. The clustering amplitudes at fixed mass are smaller at low redshift, indicating that clustering bias is falling faster than the growth function is increasing. However, the clustering amplitudes at fixed number density are more comparable. Tables \[propsbyN\] and \[propsbyN8\] report some characteristic values of different measurements of the clustering. For the range of the particle number cutoffs mentioned above, we present the various statistics, each at two representative values (1${\ensuremath{h^{-1}\;{\rm Mpc}}}$ and 5${\ensuremath{h^{-1}\;{\rm Mpc}}}$ for the 2PCF, 0.1${\ensuremath{h\;{\rm Mpc}^{-1}}}$ and 1${\ensuremath{h\;{\rm Mpc}^{-1}}}$ for power spectrum). As comparisons, we also give the corresponding 2PCFs and power spectrum values obtained from the matter density field and linear theory instead of halos. Comparing the square root of the ratio of the 2PCF indicates bias factor ranging from 5 to 30.
![The 2PCFs for $z=10$ halos by different mass cutoff represented by the minimum number of particles in the halo finder. For example, the curve labeled 300 shows the 2PCF for all halos that contains more than 300 particles. One can see the strong trend that the higher mass samples have larger 2PCF amplitudes and higher bias, i.e., that more massive halos are more clustered than less massive halos. For the 2PCF of the highest mass sample, the shot noise in this low number density sample is substantial. For this sample, we include the standard deviation of the mean 2PCF of our entire simulation box (not as in Fig. \[fig3\] where the errors for a 1% sub-volume were plotted). The errors for the higher density samples are substantially smaller. We note that comparisons between curves are partially correlated due to both large-scale structure and the overlapping mass ranges of the halo selections. This implies that ratios between samples are more tightly constrained than the variance within a sample would suggest.[]{data-label="fig5"}](fig5.pdf){width="8cm"}
![The same as Fig. \[fig5\], but at redshift $z=8$. We extend the upper limit of the low mass cutoff up to $8 \times 10^{10}$ ${\ensuremath{h^{-1}\;M_\odot}}$ so that the sample size of the most massive halos remains around 5,000 (see Table \[propsbyN\] and \[propsbyN8\]).[]{data-label="fig6"}](fig6.pdf){width="8cm"}
### Halos Clustering in Projected 2D Sky Plane
In many imaging surveys, our knowledge of the line-of-sight position of galaxies would be limited to the precision of photometric redshifts. Our measurement of small-scale clustering will then rely on the angular distribution, with the photometric redshifts used to bound the projection effects. To approximate this situation in our simulation, we project all of the coordinates onto the sky plane by assigning the uniform value to the $x$ coordinate, which is along the redshift direction. We label the resulting correlation function as $w(r)$, the 2D 2PCF. Given the $250h^{-1}$ comoving Mpc depth of our box, this corresponds at these redshifts to a projection of about $\Delta z=2$, which is typical of photometric redshift accuracy in Lyman-break samples.
We investigate the dependence of the 2D 2PCF on the mass cut value. The result is shown in Fig. \[fig7\], which is analogous to Fig. \[fig5\] in 3D real space. This time we plot the function of $r w(r)$, equivalent to a $r^{-2}$ correlation function power-law slope. The stratified structure showing an increasing bias with higher halo mass cut is similar to its 3D counterpart shown in Fig. \[fig5\].
![The same as Fig. \[fig5\], except that this plot shows the 2D 2PCF (projected onto the sky plane) for the corresponding halo particle number cutoffs.[]{data-label="fig7"}](fig7.pdf){width="8cm"}
### Halos Clustering in 3D Redshift Space
With precise spectroscopic redshifts, one can make more accurate clustering measurements. In this case, one must contend with the redshift-space distortions caused by peculiar velocities. Fig. \[fig8\] shows the 2PCFs in real space and in redshift space for the ${\ensuremath{N_{\rm min}}}= 1000$ halos at $z=10$. The distorted 2PCF gives lower correlation on smaller scales and higher correlation on larger scales. This is expected from the effects of small-scale peculiar velocities, which tends to make nearby objects appear further apart.
![The redshift-space 2PCF at $z=10$ for ${\ensuremath{N_{\rm min}}}= 1000$ halos compared with the real-space 2PCF of the same sample. Redshift-space distortions caused by the peculiar velocities of the halo centers of mass bring about a substantial decrease of clustering at small separation and an enhancement at large separations. The error bars are the standard deviations of the mean 2PCFs for the full simulation volume in each case, indicating that the redshift space distortion effect will significantly affect our detection.[]{data-label="fig8"}](fig8.pdf){width="8cm"}
Detectability
-------------
We calculate the detectability of these 2PCF based on the covariance matrix derived from the 100 2PCFs corresponding to our 100 sub-volumes. The $(i, j)$-th entry of the covariance matrix here is defined as the correlation of $i$-th and $j$-th separation bins in the 2PCF over the 100 sub-volumes; see Eq. \[eq:cov\]. The off-diagonal entries are the correlations between two different bins, and the diagonal entries are just variations of each separation bin. We use 8 bins of radial separation, so as to limit the biases that result from inverting a noisy estimate of the covariance matrix [@2014MNRAS.439.2531P].
In Fig. \[fig9\], we plot an example of the reduced covariance matrix, defined as $\mathscr{C}_{ij} = C_{ij} / \sqrt{C_{ii} C_{jj}}$. From the plot, we see higher correlations for closer separated bins, decaying for progressively farther separated bins. If the variation only consists of Poisson shot noise, then we would expect the variation at each separation bin to be uncorrelated, which is rejected by such a strong off-diagonal covariance. This indicates that there is indeed substantial contribution from the sample variance of large-scale structure in our 100 sub-volumes. Repeating this with samples of different mass thresholds shows as expected that the correlations of sparser samples have more diagonally dominated covariances.
We then use the covariance matrix to estimate the detectability of the 2PCF, according to Eq. \[eq:chi2\]. The $\chi^2$ statistic here is equivalently the difference in $\chi^2$ between that of the measured 2PCF and a null $\xi=0$ result. The interpretation of this in terms of detection significance depends on one’s choice of model. If one had a fully unconstrained model, then one could claim a clustering detection only if the result was unusually large compared to a $\chi^2$ distribution with degrees of freedom equal to the number of bins. In our case with 8 bins, finding $\chi^2>20$ would be a 99% confident detection. However, it is more cosmologically interesting to investigate smooth models, which sharply limits the number of parameters. As an extreme, if one’s model were simply a rescaling of the observed clustering, then one would have 1 degree of freedom, and the significance would be $\sqrt{\chi^2}$ $\sigma$. More likely, one would additionally include a power-law slope or other scale-dependent parameter. The resulting interpretations are hence model dependent, but we suggest that a $\chi^2$ of 20-25 would be good goal in designing a survey large enough for a first detection of large-scale clustering.
Our results for both angular (2D) and spectroscopic redshift-space (3D) clustering are shown in Tables \[propsbyN\] and \[propsbyN8\]. Note that the $\chi^2$ values refer to a 1% subvolume, i.e., about 13$'$ square and $\Delta z\approx 2$ deep, while the number of halos refers to the number in the full box. Studying the results, we find that a first detection of the large-scale correlations could result from an angular survey of 500-1000 galaxies, if the galaxies are indeed populating only the most massive halos. Adding spectroscopy increases the detection sensitivity by removing noise from projection; however, this is most effective when the samples are denser. It is important to stress that these results utilize only scales above 300$h^{-1}$ comoving kpc, which is much larger than the virial radius of these halos. In other words, this is only sensitive to the inter-halo clustering; additional signal from intra-halo (or one-halo) clustering at small separations would boost the detection significance but might be less easily related to the halo mass distribution.
![The covariance matrix corresponding to the 8 distance bins characterizing the fluctuation of the 2PCFs in 100 sub-volumes in Fig. \[fig3\], where a halo number cutoff of ${\ensuremath{N_{\rm min}}}= 1000$ and $z=10$ has been implemented. Each entry is normalized by the formula of $\mathscr{C}_{ij} = C_{ij} / \sqrt{C_{ii}C_{jj}} $, where $\mathscr{C}_{ij}, C_{ij}$ are the normalized and raw entry of the covariance matrix, respectively. Such normalization guarantees all of the diagonal entries to be converted to 1, and all of the off-diagonal entries into the interval of $[-1, 1]$. From this plot we learn that the correlation tends to be larger for closer pair of distance bins, indicating that the 2PCFs for the 100 sub-volumes are fluctuating in a smooth and positively correlated way. Note that the matrix will get closer to a diagonal matrix if we replace ${\ensuremath{N_{\rm min}}}= 1000$ with a larger number, indicating the fact that higher ${\ensuremath{N_{\rm min}}}$ samples will be more strongly influenced by shot noise which is uncorrelated.[]{data-label="fig9"}](fig9.pdf){width="8cm"}
Conclusions
===========
In this paper, we have investigated the clustering of massive halos at $z=8$ and $10$ using a cosmological N-body simulation. We measured the 2PCFs and power spectra of the halo catalog above a range of cutoff masses and compared them with the same measures for matter field and the prediction of linear theory, finding high values of the clustering bias, typically 10–20. We also measured the angular correlation function by doing a line-of-sight projection and found consistent biases.
We then calculated the detectability of this clustering for an example JWST survey. We set its full volume to $(250 {\ensuremath{h^{-1}\;{\rm Mpc}}})^3$. We divide our full simulation into $10 \times 10$ sub-volumes with equal size and estimate the 2PCF covariance matrix for a single sub-volume. We then measured the $\chi^2$ of the mean 2PCF relative to the null clustering signal. Based on the angular correlation function at $z=10$ of a sample exceeding $10^{10}$ ${\ensuremath{h^{-1}\;M_\odot}}$, we derived an expectation of $\chi_{2D}^2 = 15$ relative to a null clustering signal from a sample of 270 galaxies. With spectroscopic information to remove false pairs from projection, this significance would increase to $\chi_{3D}^2 = 19$ for the 3D redshift-space correlation function. Hence, we find that the samples of 500–1000 galaxies could yield a detectable large-scale clustering signal ($\chi^2>20$) if indeed the detected galaxies inhabit the most massive dark matter halos. If the joint distribution of galaxy luminosity (or more precisely, detectability) and halo mass has more scatter, then the typical host halo mass will decrease as will the clustering amplitude.
These results indicate that the inter-halo clustering of $z\approx8$–10 galaxies could be detectable with achievable sample sizes and that the amplitude of the clustering signal can offer some selection between galaxy formation hypotheses. However, we remind that our results include only the effect of halo clustering. Galaxy formation may yet depend on additional effects, such as large-scale radiative feedback and reionization, that could cause additional large-scale clustering. Distinguishing such signals from those of halo clustering might be possible in the shape of the 2PCF or the signatures of higher-point correlations, but any interpretations of early clustering signals will need to include this caveat.
We next compare the clustering measurement at high redshift presented by [@2017arXiv170702312B] obtained from BLUETIDES, a hydrodynamical simulation code that incorporates physics of galaxies, with our clustering measurements from a ABACUS, a pure dark-matter N-body gravitational code. The BLUETIDES analysis gets a bias factor of $10.8 \pm 0.7$ for galaxies, which is consistent with our measurements for dark matter halos. In addition, analyzing the results of these two papers via Halo Occupation Distribution (HOD) modeling helps constraining the galaxy-dark matter halo connection (see Section 3 in [@2017arXiv170702312B] for the detailed methods). However, our simulation is purely gravitational on dark matter halos without any assumption on smaller scale physics about galaxies, which thus provides a more robust probe of clustering at high redshifts. Another unique feature of our paper is our focus on detectability of clustering from proposed deep field surveys at high redshift.
Our [<span style="font-variant:small-caps;">Abacus</span>]{}code is a robust gravitational N-body cosmology simulation code in the following senses. First, we adopted the latest cosmological parameters from Planck mission (see, for example, ). Second, we adopted an improved set of initial conditions as described in [@2016MNRAS.461.4125G], which only takes longitudinal wave mode, compensates for the non-standard growing factor across the simulated redshift range and take into account second order effects. Therefore, our [<span style="font-variant:small-caps;">Abacus</span>]{}code is capable of doing simulations which properly evolve the non-linear fluctuations.
Our investigation clearly reinforces the expectation for upcoming high-redshift surveys that there will be significant field-to-field variations in galaxy populations at $z \approx 10$. But these variations come with an opportunity, that the clustering signal can be measured with moderately scoped surveys, giving a route to constrain the mass distribution of the host halos of these early galaxies.
We make the halo catalogs from our simulation available at <http://nbody.rc.fas.harvard.edu/public/JWST_products/>, so that the simulation can be used for additional analyses of clustering and the generation of JWST mock catalogs.
Acknowledgement {#acknowledgement .unnumbered}
===============
We thank Marc Metchnik and Philip Pinto for their contributions to the Abacus simulation code. DJE, LHG, and DWF have been supported by grant AST-1313285 from the National Science Foundation, and DJE is additionally supported as a Simons Foundation Investigator.
|
---
abstract: 'We introduce a model for the adaptive evolution of a network of company ownerships. In a recent work it has been shown that the empirical global network of corporate control is marked by a central, tightly connected “core” made of a small number of large companies which control a significant part of the global economy. Here we show how a simple, adaptive “rich get richer” dynamics can account for this characteristic, which incorporates the increased buying power of more influential companies, and in turn results in even higher control. We conclude that this kind of centralized structure can emerge without it being an explicit goal of these companies, or as a result of a well-organized strategy.'
author:
- 'Sebastian M. Krause'
- 'Tiago P. Peixoto'
- Stefan Bornholdt
bibliography:
- 'bib.bib'
title: Spontaneous centralization of control in a network of company ownerships
---
Introduction
============
The worldwide network of company ownership provides crucial information for the systemic analysis of the world economy [@schweitzer_networks_2009; @farmer_complex_2012]. A complete understanding of its properties and how they are formed has a wide range of potential applications, including assessment and evasion of systemic risk [@battiston_debtrank_2012], collusion and antitrust regulation [@gulati_strategicnetworks_2000; @gilo_collusion_2006], market monitoring [@diamond_delegated_1984; @chirinko_germanbanks_2006], and strategic investment [@teece_cooperation_1992]. Recently, Vitali et al [@vitali_network_2011] inferred the network structure of global corporate control, using the Orbis 2007 marketing database [^1]. Analyzing its structure, they found a tightly connected “core” made of a small number of large companies (mostly financial institutions) which control a significant part of the global economy. A central question which arises is what is the dominant mechanism behind this centralization of control. The answer is not obvious, since the decision of firms to buy other firms can be driven by diverse goals: Banks act as financial intermediaries doing monitoring for uninformed investors [@diamond_delegated_1984; @chirinko_germanbanks_2006], managers can improve their power by buying other firms instead of paying dividends [@jensen_takeovers_1986], speculation on stock prices as well as dividend earnings can be a significant source of revenue [@modiglia_cost_1958; @porta_dividend_2000; @jensen_takeovers_1986], and companies can have strategic advantages, e.g. due to knowledge sharing [@teece_cooperation_1992; @hamel_strategic_1991; @dyer_relational_1998]. Another possible hypothesis for control centralization is that managers collude to form influential alliances: Indeed, agents (e.g. board members) often work for different firms in central positions [@battiston_boards_2004]. Although all these factors are likely to play a role, we here investigate a different hypothesis, namely that a centralized structure may arise spontaneously, as a result of a simple “richt-get-richer” dynamics [@simon_richer_1955], without any explicit underlying strategy from the part of the companies. We consider a simple adaptive feedback mechanism [@gross_adaptive_2008], which incorporates the indirect control that companies have on other companies they own, which in turn increases their buying power. The higher buying power can then be used to buy portions of more important companies, or a larger number of less important ones, which further increases their relative control, and progressively marginalizes smaller companies. We show that this simple dynamical ingredient suffices to reproduce many of the qualitative features observed in the real data [@vitali_network_2011], including the emergence of a core-periphery structure and the relative portion of control exerted by the dominating core. Although this does not preclude the possibility that companies may take advantage and further consolidate their privileged positions in the network, it does suggest that deliberate strategizing may not be the dominating factor which leads to global centralization.
Model description {#sec:model}
=================
We consider a network of $N$ companies, where a directed edge between two nodes $j\to i$ means company $j$ owns a portion of company $i$. The relative amount of $i$ which $j$ owns is given by the matrix $w_{ij}$ (i.e. the ownership shares), such that $\sum_j w_{ij} = 1$. We note that it is possible for self-loops to exist, i.e. a company can in principle buy its own shares. In the following, we describe a model with two main mechanisms: 1. The evolution of the relative control of companies, given a static network; 2. The evolution of the network topology via adaptive rewiring of the edges.
Evolution of control
--------------------
Here we assume that if $j$ owns $i$, it exerts some influence on $i$ in a manner which is proportional to $w_{ij}$. If we let $v_j$ describe the relative amount of control a company $j$ has on other companies, we can write $$\label{eq:v}
v_j = 1-\alpha + \alpha \sum_i A_{ij} w_{ij} v_i,$$ where $A_{ij}$ is the adjacency matrix, the parameter $\alpha$ determines the propagation of control and $1-\alpha$ is an intrinsic amount of independence between companies [^2]. We further assume that the control value $v_j$ directly affects other features such as profit margins, and overall market influence, such that the buying power of companies with large $v_j$ is also increased. This means that the ownership of a company $i$ is distributed among the owners $j$, proportionally to their control $v_j$, i.e. $$\label{eq:w}
w_{ij} = \frac{A_{ij} v_j}{\sum_l A_{il} v_l},$$ (see Fig. \[fig:control\]). These equations are assumed to evolve in a faster time scale, such that equilibrium is reached before the topology changes, as described in the next section.
![Illustration of the control of firms including indirect control (left) and the ownership being proportional to the control (right), as described in the text.[]{data-label="fig:control"}](graph_control.pdf "fig:"){width=".35\columnwidth"} ![Illustration of the control of firms including indirect control (left) and the ownership being proportional to the control (right), as described in the text.[]{data-label="fig:control"}](graph_weights.pdf "fig:"){width=".35\columnwidth"}
Evolution of the network topology
---------------------------------
Companies may decide to buy or sell shares of a given company at a given time. The actual mechanisms regulating these decisions are in general complicated and largely unknown, since they may involve speculation, actual market value, and other factors, which we do not attempt to model in detail here. Instead, we describe these changes probabilistically, where an edge may be deleted or inserted randomly in the network, and such moves may be accepted or rejected depending on how much it changes the control of the nodes involved. For simplicity, we force the total amount of edges in the network to be kept constant, such that a random edge deletion is always accompanied by a random edge insertion. Such “moves” may be rejected or accepted, based on the change they bring to the $v_j$ values of the companies involved. If we let $m$ be the company which buys new shares of company $l$, and $j$ which sells shares of company $i$, the probability that the move is accepted is $$\label{eq:rewire}
p = \min\left(1,e^{\beta(\tilde{w}_{lm} v_l - w_{ij} v_i)}\right),$$ where $w_{ij}$ is computed before the move and $\tilde{w}_{lm}$ afterwards, and the parameter $\beta$ determines the capacity companies have to foresee the advantage of the move, such that for $\beta=0$ all random moves are accepted, and for $\beta\to\infty$ they are only accepted if the net gain is positive (see Fig. \[fig:adaptiv\]). Note that in Eq. \[eq:rewire\] it is implied that companies with larger control will tend to buy more than companies with smaller control, which is well justified by our assumption that control is correlated with profit and wealth.
![Illustration of the adaptive process, before the rewiring (left) and afterwards (right), as described in the text.[]{data-label="fig:adaptiv"}](graph_adaptiv_0.pdf "fig:"){width=".35\columnwidth"} ![Illustration of the adaptive process, before the rewiring (left) and afterwards (right), as described in the text.[]{data-label="fig:adaptiv"}](graph_adaptiv_1.pdf "fig:"){width=".35\columnwidth"}
The overall dynamics is composed by performing many rewiring steps as described above, until an equilibrium is reached, i.e. the observed network properties do not change any longer. In order to preserve a separation of time scales between the control and rewiring dynamics, we performed a sufficiently large number of iterations of Eqs. \[eq:v\] and \[eq:w\] before each attempted edge move.
Centralization of control {#sec:concentration}
=========================
A typical outcome of the dynamics can be seen in Fig. \[fig:condensation\] for a network with $N=3\times10^4$ nodes and average degree $\left<k\right>=2$, after an equilibration time of about $6\times10^{9}$ steps. In contrast to the case with $\beta=0$, which results in a fully random graph, for a sufficiently high value of $\beta$ the distribution of firm ownerships (i.e. the out-degree of the nodes) becomes very skewed, with a bimodal form. We can divide the most powerful companies into a broad range which owns shares from $10$ to about $150$ other companies, and a separate group with $k_{\text{out}}>150$. The correlation matrix of this network shows that these high-degree nodes are connected strongly among themselves, and own a large portion of the remaining companies (see Fig. \[fig:condensation\]). This corresponds to a highly connected “core” of about 45 nodes with $\left<k_{\text{sub}}\right>\approx
39.8$, which is highlighted in red in Fig. \[fig:condensation\]c and can be seen separately in Fig. \[fig:condensation\]d. The distribution of in-degree (not shown) is bimodal as well with highest values for the inner core. With values up to $k_{\rm in}=50$, the highest in-degree (number of owners) is considerably below the highest out-degree (number of firms owned at once).
{width="\textwidth"} (a)
{width="\textwidth"} (b)
{width="100.00000%"} (c)
{width="\textwidth"} (d)
![Left: Distribution of inherited control $v_i - (1-\alpha)$ for $\alpha = 0.5$ and different values of $\beta$; Right: Relative fraction of control as a function of fraction of most powerful companies.[]{data-label="fig:condensation_v"}](pic_new_dens_control_k2_a0,5_N30000.pdf "fig:"){width=".49\columnwidth"} ![Left: Distribution of inherited control $v_i - (1-\alpha)$ for $\alpha = 0.5$ and different values of $\beta$; Right: Relative fraction of control as a function of fraction of most powerful companies.[]{data-label="fig:condensation_v"}](pic_new_control_fraction_a0,5_N30000.pdf "fig:"){width=".49\columnwidth"}
Similarly to the out-degree, the distribution of control values $v_i$ is also bimodal for larger values of $\beta$, as can be seen in Fig. \[fig:condensation\_v\], and is strongly correlated with the out-degree values. The total fraction of companies controlled by the most powerful ones is very large, as shown on the right panel of Fig. \[fig:condensation\_v\]. For instance, we see that a fraction of around $0.15\%$ of the central core controls about $57\%$ of all companies. The companies with intermediary values of control (and out-degree) also possess a significant part of the global control, e.g. around $.85\%$ of the most powerful control an additional $25\%$ of the network. It is important to emphasize the difference between these two classes of companies for two reasons: Firstly the inner core inherits control from intermediate companies without the need to gather up all the minor companies. In fact the ownership links going out from the inner core (about $10^4$) is enough to cover the direct control of only a third of all companies, while the effective control is more than a half. Secondly, the fraction of intermediary companies increases for larger networks. For a network with $N=3\times10^5$, the inner core includes a fraction of only $0.04\%$, controlling an effective $41\%$ of the total companies. Nonetheless, all the most powerful companies together account for around $1\%$ of the network and $82\%$ of the total control; values which do not change considerably with system size.
Let us compare the results presented so far with empirical data presented in [@vitali_network_2011]. For different reasons, this comparison can only be qualitative. First of all, the empirical data includes economic agents with different functions (shareholders, transnational companies and participated companies) out of different sectors (eg. financial and real economy), while we consider identical agents. Secondly, we force every company to be owned 100%, while the empirical data neglects restrained shares and diversified holdings. Thirdly, the control analysis in [@vitali_network_2011] is done somewhat differently: All the $600,508$ economic agents were considered for the topological characterization, while many companies (80% of all agents there) were neglected for the control analysis. In the empirical data, a strongly connected component of $1,318$ companies controls more than a half of all companies arranged in the out component. This concentration is compatible with the core-periphery structure presented in Fig. \[fig:condensation\], however the empirical data does not show a distinct bimodal structure. Nonetheless, there are highly connected substructures in the core, e.g. a structure with 22 highly connected financial companies ($\left<k_{\rm
sub}\right>\approx 12$) was highlighted in [@battiston_debtrank_2012]. The control concentration in the empirical data was reported as a fraction of $0.5\%$ which controls $80\%$ of the network. This is similar to the results of our model (see Fig. \[fig:condensation\_v\] on the right). There are, however, features that our model does not reproduce, the most important of which being the out-degree distribution of the network, which in [@vitali_network_2011] is very broad, and displays no discernible scales, where in our case it is either bimodal or Poisson-like. One possible explanation for this discrepancy is that we have focused on equilibrium steady-state configurations of the dynamics, whereas the real economy is surely far away from such an equilibrium. A more precise model would need to incorporate such transient dynamics in a more realistic way. Nevertheless, the general tendency of the control to be concentrated on relatively few companies is evident in such equilibrium states, and features very prominently in the empirical data as well.
Transition to centralization
----------------------------
To investigate the transition from homogeneous no centralized networks with increasing $\beta$, we measured the inverse participation ratio $I=\left[\frac{1}{TN}\sum_{ti} v_i(t)^2 \right]^{-1}$ with the time $t$ summing over a sufficiently long time window of length $T$ after equilibration. Since $\frac{1}{N}\leq I\leq 1$, we expect $I=1$ in the perfectly homogeneous case where $v_i=1$ for all nodes, and $I=\frac{1}{N}$ if only one node has $v_i > 0$, and the control is maximally concentrated. As can be seen in Fig. \[fig:condensation\_trans\], we observe a smooth transition from very homogeneous companies connected in fully random manner for $\beta=0$, to a pronounced concentration of control for increased $\beta$, for which the aforementioned core-periphery is observed. The transition becomes more abrupt when either the average degree $\left<k\right>$ is increased or the parameter $\alpha$ (which determines the fraction of inherited control) is decreased.
![Inverse participation ratio $I=\left[\frac{1}{TN}\sum_{ti}
v_i(t)^2 \right]^{-1}$ as a function of $\beta$, for a network with $N=10^4$, and for (left) $\left<k\right>=2$ and different values of $\alpha$ and (right) $\alpha=0.5$ and different values of $\left<k\right>$.[]{data-label="fig:condensation_trans"}](pic_new_concentration_over_beta_k2___N10000.pdf "fig:"){width="0.49\columnwidth"} ![Inverse participation ratio $I=\left[\frac{1}{TN}\sum_{ti}
v_i(t)^2 \right]^{-1}$ as a function of $\beta$, for a network with $N=10^4$, and for (left) $\left<k\right>=2$ and different values of $\alpha$ and (right) $\alpha=0.5$ and different values of $\left<k\right>$.[]{data-label="fig:condensation_trans"}](pic_new_concentration_over_beta_kvar_a0,5_N10000.pdf "fig:"){width="0.49\columnwidth"}
![Distribution of out degrees (left) and inherited control $v_i-(1-\alpha)$ (right) for $\beta=10$, $\left<k\right>=2$ and $N=30000$ as in Fig. \[fig:condensation\] and \[fig:condensation\_v\], but for different values of $\alpha$.[]{data-label="fig:condensation_dist"}](pic_new_dens_k_k2_b10_N30000.pdf "fig:"){width="0.49\columnwidth"} ![Distribution of out degrees (left) and inherited control $v_i-(1-\alpha)$ (right) for $\beta=10$, $\left<k\right>=2$ and $N=30000$ as in Fig. \[fig:condensation\] and \[fig:condensation\_v\], but for different values of $\alpha$.[]{data-label="fig:condensation_dist"}](pic_new_dens_control_k2_b10_N30000.pdf "fig:"){width="0.49\columnwidth"}
Centralization of control can emerge in different ways depending on the parameters $\alpha$ and $\beta$. In Fig. \[fig:condensation\_dist\], it is shown that different values of $\alpha$ for a high value of $\beta=10$ can lead to a detached controlling core ($\alpha=0.2$) or to broadly distributed control values ($\alpha=0.8$). With smaller values of $\alpha$, indirect control is suppressed and companies can gain power only by owning large numbers of marginal companies. E.g.: for $\alpha=0.2$, this leads to a highly connected core of $41$ companies having $\left<k_{\rm sub}\right>\approx 18.2$, the rest of the companies have very little influence. For larger values of $\alpha$, indirect control has a larger effect, which leads to a hierarchical network where companies with small numbers of owned firms $k_{\rm out}$ may nevertheless inherit large control values $v_i$. The case with $\alpha=0.5$ and $\beta=10$ shown in Figs. \[fig:condensation\] and \[fig:condensation\_v\] exhibits a mixture of these two scenarios. The transition to a centralized core also occurs when increasing $\beta$ and keeping $\alpha$ constant (see right panel in Fig. \[fig:condensation\_trans\]).
![Left: Graph layout of a $10\times 10$ lattice with $\alpha=0.9$. The vertex sizes and colors correspond to the $v_i$ values, and the edge thickness to the $w_{ij}$ values. Right: Distribution of inherited control $v_i-(1-\alpha)$ for static poisson graphs having $\left<k\right>=2$ and $N=30\,000$, with different values of $\alpha$ (for $\alpha=0.5$ and $\alpha=0.8$ shifted). The dashed line is a power law with exponent $-1$.[]{data-label="fig:lattice"}](lattice.pdf "fig:"){width="0.41\columnwidth"} ![Left: Graph layout of a $10\times 10$ lattice with $\alpha=0.9$. The vertex sizes and colors correspond to the $v_i$ values, and the edge thickness to the $w_{ij}$ values. Right: Distribution of inherited control $v_i-(1-\alpha)$ for static poisson graphs having $\left<k\right>=2$ and $N=30\,000$, with different values of $\alpha$ (for $\alpha=0.5$ and $\alpha=0.8$ shifted). The dashed line is a power law with exponent $-1$.[]{data-label="fig:lattice"}](pic_new_dens_control_tau0_k2_N30000.pdf "fig:"){width="0.57\columnwidth"}
One interesting aspect of the centralization of control as we have formulated is that it is not entirely dependent on the adaptive dynamics, and occurs also to some extent on graphs which are static. Simply solving Eqs. \[eq:v\] and \[eq:w\] will lead to a non-trivial distribution of control values $v_i$ which depend on the (in this case fixed) network topology and the control inheritance parameter $\alpha$. In Fig. \[fig:lattice\] is shown on the left the control values obtained for a square $2D$ lattice with periodic boundary conditions, and bidirectional edges. What is observed is a spontaneous symmetry breaking, where despite the topological equivalence shared between all nodes, a hierarchy of control is formed, which is not unique and will vary between each realization of the dynamics. A similar behavior is also observed for fully random graphs, as shown on the right of Fig. \[fig:lattice\], where the distribution of control values becomes increasingly broader for larger values of $\alpha$, asymptotically approaching a power-law $\rho(v) \sim v^{-1}$ for $\alpha\to1$. This behavior is similar to a phase transition at $\alpha=1$, where at this point Eq. \[eq:v\] no longer converges to a solution.
Conclusion
==========
We have tested the hypothesis that a rich-get-richer process using a simple, adaptive dynamics is capable of explaining the phenomenon of concentration of control observed in the empirical network of company ownership [@vitali_network_2011]. The process we proposed incorporates the indirect control that companies have on other companies they own, which increases their buying power in a feedback fashion, and allows them to gain even more control. In our model, the system spontaneously organizes into a steady-state comprised of a well-defined core-periphery structure, which reproduces many qualitative observations in the real data presented in [@vitali_network_2011], such as the relative portion of control exerted by the dominating companies. Our model shows that this kind of centralized structure can emerge without it being an explicit goal of the companies involved. Instead, it can emerge simply as a result of individual decisions based on local knowledge only, with the effect that powerful companies can increase their relative advantage even further.
It is interesting to compare our model to other agent based models featuring agents competing for centrality. The emergence of hierarchical, centralized states with interesting patterns of global order was reported for agents creating links according to game theory [@holme_centrality_2006; @lee_multiadaptive_2011; @do_patterns_2010] as well as for very simple effective rules of rewiring according to measured centrality [@koenig_centrality_2011; @bardoscia_climbing_2013]. The latter is combined with phase transitions according to the noise in the rewiring process. The stylized model of a society studied in [@bardoscia_climbing_2013] shows a hierarchical structure, if the individuals have a preference for social status. The intuitive emergence of hierarchy is associated with shrinking mobility of single agents within the hierarchy. This effect is present in our model as well and deserves further investigation.
Our results may shed light on certain antitrust regulation strategies. As we found that a simple mechanism without collusion suffices for control centralization, any regulation which is targeted to diminish such activities may prove fruitless. Instead, targeting the self-organizing features which lead to such concentration, such as e.g. limitations on the indirect control of shareholders representing other companies, may appear more promising.
[^1]: <http://www.bvdinfo.com/products/company-information/international/orbis>
[^2]: Eq. \[eq:v\] can be seen as a weighted version of the Katz centrality index [@katz_new_1953], which is one of many ways of measuring the relative centrality of nodes in a directed network, such as PageRank [@page_PageRank_1999] and HITS [@kleinberg_authoritative_1999]. It converges for $0\leq\alpha<
1$ and we enforce normalization with $\sum_i v_i=N$.
|
---
abstract: 'We study indefinite quaternion algebras over totally real fields $F$, and give an example of a cohomological construction of $p$-adic Jacquet-Langlands functoriality using completed cohomology. We also study the (tame) levels of $p$-adic automorphic forms on these quaternion algebras and give an analogue of Mazur’s ‘level lowering’ principle.'
author:
- James Newton
title: 'Completed cohomology of Shimura curves and a $p$-adic Jacquet-Langlands correspondence'
---
Introduction
============
Our primary goal in this paper is to give some examples of $p$-adic interpolation of Jacquet-Langlands functoriality by studying the completed cohomology [@Emint] of Shimura curves. The first example of a $p$-adic interpolation of Jacquet-Langlands functoriality appears in work of Chenevier [@MR2111512], who produces a rigid analytic map between the eigencurve for a definite quaternion algebra over ${\ensuremath{\mathbb{Q}}\xspace}$ (as constructed in [@Bu1]) and the Coleman-Mazur eigencurve for ${\ensuremath{\mathrm{GL}}\xspace}_2/{\ensuremath{\mathbb{Q}}\xspace}$ [@CM].
In this paper we will be concerned with the Jacquet-Langlands transfer between automorphic representations of the multiplicative group of quaternion algebras over a totally real field $F$, which are split at one infinite place (so the attached Shimura varieties are one-dimensional). These quaternion algebras will always be split at places dividing $p$. For the purposes of this introduction we will suppose that $F={\ensuremath{\mathbb{Q}}\xspace}$. Let $D$ be a quaternion algebra over ${\ensuremath{\mathbb{Q}}\xspace}$, split at the infinite place and with discriminant $N$, and let $D'$ be a quaternion algebra over ${\ensuremath{\mathbb{Q}}\xspace}$, split at the infinite place and with discriminant $Nql$ for $q,l$ distinct primes (coprime to $Np$). In this situation there is a rather geometric description of the Jacquet-Langlands transfer from automorphic representations of $(D'\otimes_{\ensuremath{\mathbb{Q}}\xspace}{\ensuremath{\mathbb{A}}\xspace}_{\ensuremath{\mathbb{Q}}\xspace})^\times$ to automorphic representations of $(D\otimes_{\ensuremath{\mathbb{Q}}\xspace}{\ensuremath{\mathbb{A}}\xspace}_{\ensuremath{\mathbb{Q}}\xspace})^\times$, arising from the study of the reduction modulo $q$ or $l$ of integral models of Shimura curves attached to $D$ and $D'$ (where we consider Shimura curves attached to $D$ with Iwahori level at $q$ and $l$). For the case $N=1$ and $F={\ensuremath{\mathbb{Q}}\xspace}$ this is one of the key results in [@Ribet100], and it was extended to general totally real fields $F$ by Rajaei [@Raj]. Using the techniques from these papers, together with Emerton’s completed cohomology [@Emint], we can then describe a $p$-adic Jacquet-Langlands correspondence in a geometric way — the result obtained is stronger than just the existence of a map between eigenvarieties, which is all that can be deduced from interpolatory techniques as in [@MR2111512].
Our techniques also allow us to prove a $p$-adic analogue of Mazur’s principle (as extended to the case of totally real fields by Jarvis [@JarMazPrin]), generalising Theorem 6.2.4 of [@Emlg] to totally real fields. This should have applications to questions of local-global compatibility at $l \ne p$ in the $p$-adic setting. Having proved a level lowering result for completed cohomology, we can deduce a level lowering result for eigenvarieties. For $F={\ensuremath{\mathbb{Q}}\xspace}$ these level lowering results have already been obtained by Paulin [@NonComp], using the results of [@Emlg] (in an indirect way, rather than by applying Emerton’s construction of the eigencurve). A related approach to overconvergent Jacquet-Langlands functoriality is pursued in [@HIS], using Stevens’s cohomological construction of the eigencurve from overconvergent modular symbols.
We briefly indicate the main results proven in what follows:
Level lowering
--------------
We prove an analogue of Mazur’s principle for $p$-adic automorphic forms (see Theorems \[mazprin\] and \[mazprinjacquet\] for precise statements). Roughly speaking, we show that if a two-dimensional $p$-adic representation $\rho$ of $\operatorname{Gal}(\overline{F}/F)$ ($F$ a totally real field) is unramified at a finite place ${\ensuremath{\mathfrak{q}}\xspace}\nmid p$ and occurs in the completed cohomology of a system of Shimura curves with tame level equal to $U_0({\ensuremath{\mathfrak{q}}\xspace})$ at the ${\ensuremath{\mathfrak{q}}\xspace}$-factor, then the system of Hecke eigenvalues attached to $\rho$ occurs in the completed cohomology of Shimura curves with tame level equal to ${\ensuremath{\mathrm{GL}}\xspace}_2({\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{q}}\xspace})$ at the ${\ensuremath{\mathfrak{q}}\xspace}$-factor.
A $p$-adic Jacquet-Langlands correspondence
-------------------------------------------
We show the existence of a short exact sequence of admissible unitary Banach representations of ${\ensuremath{\mathrm{GL}}\xspace}_2(F\otimes_{\ensuremath{\mathbb{Q}}\xspace}{\ensuremath{\mathbb{Q}}\xspace}_p)$ (Theorem \[RRES\] — in fact the exact sequence is defined integrally) which encodes a $p$-adic Jacquet-Langlands correspondence from a quaternion algebra $D'/F$, split at one infinite place, with discriminant $\mathfrak{N}{\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}$, to a quaternion algebra $D/F$, split at the same infinite place and with discriminant $\mathfrak{N}$. Applying Emerton’s eigenvariety construction to this exact sequence gives a map between eigenvarieties (Theorem \[ocjl\]), i.e. an overconvergent Jacquet-Langlands correspondence in the same spirit as the main theorem of [@MR2111512]. As a corollary of these results we also obtain level raising results (Corollary \[levelraising\]).
Preliminaries
=============
Vanishing cycles {#sec:van}
----------------
In this section we will follow the exposition of vanishing cycles given in the first chapter of [@Raj] - we give some details to fix ideas and notation. Let ${\ensuremath{\mathcal{O}}\xspace}$ be a characteristic zero Henselian discrete valuation ring, residue field $k$ of characteristic $l$, fraction field $L$. Let ${\ensuremath{\mathscr{X}}\xspace}\rightarrow S=\operatorname{Spec}({\ensuremath{\mathcal{O}}\xspace})$ be a proper, generically smooth curve with reduced special fibre, and define $\Sigma$ to be the set of singular points on ${\ensuremath{\mathscr{X}}\xspace}\otimes \overline{k}$. Moreover assume that a neighbourhood of each point $x \in \Sigma$ is étale locally isomorphic over $S$ to $\operatorname{Spec}({\ensuremath{\mathcal{O}}\xspace}[t_1,t_2]/(t_1 t_2 - a_x))$ with $a_x$ a non-zero element of ${\ensuremath{\mathcal{O}}\xspace}$ with valuation $e_x=v(a_x)>0$. Define morphisms $i$ and $j$ to be the inclusions: $$i: {\ensuremath{\mathscr{X}}\xspace}\otimes \overline{k} \rightarrow {\ensuremath{\mathscr{X}}\xspace}, j: {\ensuremath{\mathscr{X}}\xspace}\otimes\overline{L} \rightarrow {\ensuremath{\mathscr{X}}\xspace}.$$ For ${\ensuremath{\mathscr{F}}\xspace}$ a constructible torsion sheaf on ${\ensuremath{\mathscr{X}}\xspace}$, with torsion prime to $l$, the Grothendieck (or Leray) spectral sequence for the composition of the two functors $\Gamma({\ensuremath{\mathscr{X}}\xspace},-)$ and $j_*$ gives an identification $$R\Gamma({\ensuremath{\mathscr{X}}\xspace}\otimes\overline{L},j^*{\ensuremath{\mathscr{F}}\xspace})=R\Gamma({\ensuremath{\mathscr{X}}\xspace},R(j)_*j^*{\ensuremath{\mathscr{F}}\xspace}).$$ Applying the proper base change theorem we also have $$R\Gamma({\ensuremath{\mathscr{X}}\xspace},R(j)_*j^*{\ensuremath{\mathscr{F}}\xspace})\simeq R\Gamma({\ensuremath{\mathscr{X}}\xspace}\otimes\overline{k},i^*R(j)_*j^*{\ensuremath{\mathscr{F}}\xspace}).$$ The natural adjunction $id\implies R(j)_*j^*$ gives a morphism $i^*{\ensuremath{\mathscr{F}}\xspace}\rightarrow i^*R(j)_*j^*{\ensuremath{\mathscr{F}}\xspace}$ of complexes on ${\ensuremath{\mathscr{X}}\xspace}\otimes\overline{k}$. We denote by $R\Phi({\ensuremath{\mathscr{F}}\xspace})$ the mapping cone of this morphism (the complex of *vanishing cycles*), and denote by $R\Psi({\ensuremath{\mathscr{F}}\xspace})$ the complex $i^*R(j)_*j^*{\ensuremath{\mathscr{F}}\xspace}$ (the complex of *nearby cycles*). There is a distinguished triangle $$i^*{\ensuremath{\mathscr{F}}\xspace}\rightarrow R\Psi({\ensuremath{\mathscr{F}}\xspace}) \rightarrow R\Phi({\ensuremath{\mathscr{F}}\xspace}) \rightarrow,$$ whence a long exact sequence of cohomology $$\cdots\rightarrow H^i({\ensuremath{\mathscr{X}}\xspace}\otimes\overline{k},i^*{\ensuremath{\mathscr{F}}\xspace})\rightarrow H^i({\ensuremath{\mathscr{X}}\xspace}\otimes\overline{k},R\Psi({\ensuremath{\mathscr{F}}\xspace}))\rightarrow H^i({\ensuremath{\mathscr{X}}\xspace}\otimes\overline{k},R\Phi({\ensuremath{\mathscr{F}}\xspace}))\rightarrow\cdots$$ As on page 36 of [@Raj] there is a *specialisation exact sequence*, with ‘$(1)$’ denoting a Tate twist of the $\operatorname{Gal}(\overline{L}/L)$ action: $$\label{speces}
\minCDarrowwidth20pt\begin{CD}0 @>>> H^1({\ensuremath{\mathscr{X}}\xspace}\otimes\overline{k},i^*{\ensuremath{\mathscr{F}}\xspace})(1) @>>> H^1({\ensuremath{\mathscr{X}}\xspace}\otimes\overline{L},j^*{\ensuremath{\mathscr{F}}\xspace})(1) @>\beta>> \bigoplus_{x\in\Sigma} R^1\Phi({\ensuremath{\mathscr{F}}\xspace})_x(1) \\
@>\gamma>> H^2({\ensuremath{\mathscr{X}}\xspace}\otimes\overline{k},i^*{\ensuremath{\mathscr{F}}\xspace})(1) @>{sp(1)}>> H^2({\ensuremath{\mathscr{X}}\xspace}\otimes\overline{L},j^*{\ensuremath{\mathscr{F}}\xspace})(1) @>>> 0.\end{CD}$$We can extend the above results to lisse étale ${\ensuremath{\mathbb{Z}}\xspace}_p$-sheaves ($p \ne l$), by taking inverse limits of the exact sequences for the sheaves ${\ensuremath{\mathscr{F}}\xspace}/p^i{\ensuremath{\mathscr{F}}\xspace}$. Since our groups satisfy the Mittag-Leffler condition we preserve exactness. In particular we obtain the specialisation exact sequence (\[speces\]) for ${\ensuremath{\mathscr{F}}\xspace}$. From now on ${\ensuremath{\mathscr{F}}\xspace}$ will denote a ${\ensuremath{\mathbb{Z}}\xspace}_p$-sheaf, and we define
$$X({\ensuremath{\mathscr{F}}\xspace}) := \ker(\gamma) = \mathrm{im}(\beta) \subset \bigoplus_{x\in\Sigma} R^1\Phi({\ensuremath{\mathscr{F}}\xspace})_x(1),$$ so there is a short exact sequence $$\minCDarrowwidth20pt\begin{CD}0 @>>> H^1({\ensuremath{\mathscr{X}}\xspace}\otimes\overline{k},i^*{\ensuremath{\mathscr{F}}\xspace})(1) @>>> H^1({\ensuremath{\mathscr{X}}\xspace}\otimes\overline{L},j^*{\ensuremath{\mathscr{F}}\xspace})(1) @>>> X({\ensuremath{\mathscr{F}}\xspace}) @>>> 0.\end{CD}$$
As in section 1.3 of [@Raj] there is also a cospecialisation sequence $$\label{cospeces}
\minCDarrowwidth20pt\begin{CD}0 @>>> H^0(\widetilde{{\ensuremath{\mathscr{X}}\xspace}\otimes \overline{k}},R\Psi({\ensuremath{\mathscr{F}}\xspace})) @>>> H^0(\widetilde{{\ensuremath{\mathscr{X}}\xspace}\otimes \overline{k}},{\ensuremath{\mathscr{F}}\xspace}) @>{\gamma'}>> \bigoplus_{x\in\Sigma} H^1_x({\ensuremath{\mathscr{X}}\xspace}\otimes \overline{k},R\Psi({\ensuremath{\mathscr{F}}\xspace})) \\
@>{\beta'}>> H^1({\ensuremath{\mathscr{X}}\xspace}\otimes \overline{k},R\Psi({\ensuremath{\mathscr{F}}\xspace})) @>>> H^1({\ensuremath{\mathscr{X}}\xspace}\otimes \overline{k},{\ensuremath{\mathscr{F}}\xspace}) @>>> 0,\end{CD}$$where we write $\widetilde{{\ensuremath{\mathscr{X}}\xspace}\otimes \overline{k}}$ for the normalisation of ${\ensuremath{\mathscr{X}}\xspace}\otimes \overline{k}$, and (in the above, and from now on) we also denote by ${\ensuremath{\mathscr{F}}\xspace}$ the appropriate pullbacks of ${\ensuremath{\mathscr{F}}\xspace}$ and $R\Psi({\ensuremath{\mathscr{F}}\xspace})$ (e.g. $i^*{\ensuremath{\mathscr{F}}\xspace}$). The normalisation map is denoted $r:\widetilde{{\ensuremath{\mathscr{X}}\xspace}\otimes \overline{k}}\rightarrow {\ensuremath{\mathscr{X}}\xspace}\otimes \overline{k}$. We can now define $$\check{X}({\ensuremath{\mathscr{F}}\xspace}) := \mathrm{im}(\beta'),$$ and apply Corollary 1 of [@Raj] to get the following:
We have the following diagram made up of two short exact sequences: $$\xymatrix{& 0\ar[d]\\
&\check{X}({\ensuremath{\mathscr{F}}\xspace})\ar[d]\\
0\ar[r]&H^1({\ensuremath{\mathscr{X}}\xspace}\otimes \overline{k}, {\ensuremath{\mathscr{F}}\xspace}) \ar[r]\ar[d]&H^1({\ensuremath{\mathscr{X}}\xspace}\otimes\overline{L}, {\ensuremath{\mathscr{F}}\xspace})\ar[r]&X({\ensuremath{\mathscr{F}}\xspace})(-1)\ar[r]&0\\
&H^1({\ensuremath{\mathscr{X}}\xspace}\otimes \overline{k}, r_*r^*{\ensuremath{\mathscr{F}}\xspace})\ar[d]\\
&0}$$
Note that the map $\check{X}({\ensuremath{\mathscr{F}}\xspace})\rightarrow H^1({\ensuremath{\mathscr{X}}\xspace}\otimes \overline{k},{\ensuremath{\mathscr{F}}\xspace})$ is the one induced by the fact that $$\check{X}({\ensuremath{\mathscr{F}}\xspace}) \subset H^1({\ensuremath{\mathscr{X}}\xspace}\otimes \overline{k},R\Psi({\ensuremath{\mathscr{F}}\xspace}))$$ is in the image of the injective map $$H^1({\ensuremath{\mathscr{X}}\xspace}\otimes\overline{k},{\ensuremath{\mathscr{F}}\xspace})\rightarrow H^1({\ensuremath{\mathscr{X}}\xspace}\otimes\overline{k},R\Psi({\ensuremath{\mathscr{F}}\xspace})).$$ Finally, as in (1.13) of [@Raj] there is an injective map $$\lambda: X({\ensuremath{\mathscr{F}}\xspace})\rightarrow \check{X}({\ensuremath{\mathscr{F}}\xspace})$$ coming from the monodromy pairing.
Shimura curves and their bad reduction
======================================
Notation
--------
Let $F/{\ensuremath{\mathbb{Q}}\xspace}$ be a totally real number field of degree $d$. We denote the infinite places of $F$ by $\tau_1,...,\tau_d$. We will be comparing the arithmetic of two quaternion algebras over $F$. The first quaternion algebra will be denoted by $D$, and we assume that it is split at $\tau_1$ and at all places in $F$ dividing $p$, and non-split at $\tau_i$ for $i \ne 1$ together with a finite set $S$ of finite places. If $d=1$ then we furthermore assume that $D$ is split at some finite place (so we avoid working with non-compact modular curves). Fix two finite places ${\ensuremath{{\mathfrak{q}_1}}\xspace}$ and ${\ensuremath{{\mathfrak{q}_2}}\xspace}$ of $F$, which do not divide $p$ and where $D$ is split, and denote by $D'$ a quaternion algebra over $F$ which is split at $\tau_1$ and non-split at $\tau_i$ for $i \ne 1$ together with each finite place in $S \cup \{{\ensuremath{{\mathfrak{q}_1}}\xspace},{\ensuremath{{\mathfrak{q}_2}}\xspace}\}$. We fix maximal orders ${\ensuremath{\mathcal{O}}\xspace}_D$ and ${\ensuremath{\mathcal{O}}\xspace}_{D'}$ of $D$ and $D'$ respectively, and an isomorphism $D\otimes_F {\ensuremath{\mathbb{A}}\xspace}_F^{({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace})} \cong D'\otimes_F {\ensuremath{\mathbb{A}}\xspace}_F^{({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace})}$ compatible with the choice of maximal orders in $D$ and $D'$.
We write ${\ensuremath{\mathfrak{q}}\xspace}$ to denote one of the places ${\ensuremath{{\mathfrak{q}_1}}\xspace},{\ensuremath{{\mathfrak{q}_2}}\xspace}$, and denote the completion $F_{\ensuremath{\mathfrak{q}}\xspace}$ of $F$ by $L$, with ring of integers ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{q}}\xspace}$ and residue field $k_{\ensuremath{\mathfrak{q}}\xspace}$. Denote $F\cap {\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{q}}\xspace}$ by ${\ensuremath{\mathcal{O}}\xspace}_{({\ensuremath{\mathfrak{q}}\xspace})}$. We have the absolute Galois group $G_{\ensuremath{\mathfrak{q}}\xspace}=\operatorname{Gal}(\overline{L}/L)$ with its inertia subgroup $I_{\ensuremath{\mathfrak{q}}\xspace}$.
Let $G$ and $G'$ denote the reductive groups over ${\ensuremath{\mathbb{Q}}\xspace}$ arising from the unit groups of $D$ and $D'$ respectively. Note that $G$ and $G'$ are both forms of $\mathrm{Res}_{F/{\ensuremath{\mathbb{Q}}\xspace}}({\ensuremath{\mathrm{GL}}\xspace}_2/F)$. For $U$ a compact open subgroup of $G({\ensuremath{\mathbb{A}}\xspace}_f)$ and $V$ a compact open subgroup of $G'({\ensuremath{\mathbb{A}}\xspace}_f)$ we have complex (disconnected) Shimura curves
$$M(U)({\ensuremath{\mathbb{C}}\xspace}) = G({\ensuremath{\mathbb{Q}}\xspace})\backslash G({\ensuremath{\mathbb{A}}\xspace}_f)\times ({\ensuremath{\mathbb{C}}\xspace}-{\ensuremath{\mathbb{R}}\xspace})/U$$ $$M'(U)({\ensuremath{\mathbb{C}}\xspace}) = G'({\ensuremath{\mathbb{Q}}\xspace})\backslash G'({\ensuremath{\mathbb{A}}\xspace}_f)\times ({\ensuremath{\mathbb{C}}\xspace}-{\ensuremath{\mathbb{R}}\xspace})/V,$$ where $G({\ensuremath{\mathbb{Q}}\xspace})$ and $G'({\ensuremath{\mathbb{Q}}\xspace})$ act on ${\ensuremath{\mathbb{C}}\xspace}-{\ensuremath{\mathbb{R}}\xspace}$ via the $\tau_1$ factor of $G({\ensuremath{\mathbb{R}}\xspace})$ and $G'({\ensuremath{\mathbb{R}}\xspace})$ respectively.
Both these curves have canonical models over $F$, which we denote by $M(U)$ and $M'(U)$. We follow the conventions of [@carmauv] to define this canonical model — see [@BDJ] for a discussion of two different conventions used when defining canonical models of Shimura curves.
Integral models and coefficient sheaves {#subsec:intco}
---------------------------------------
Let $U \subset G({\ensuremath{\mathbb{A}}\xspace}_f)$ be a compact open subgroup (we assume $U$ is a product of local factors over the places of $F$), with $U_{\ensuremath{\mathfrak{q}}\xspace}= U_0({\ensuremath{\mathfrak{q}}\xspace})$ (the matrices in ${\ensuremath{\mathrm{GL}}\xspace}_2({\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{q}}\xspace})$ which have upper triangular image in ${\ensuremath{\mathrm{GL}}\xspace}_2(k_{\ensuremath{\mathfrak{q}}\xspace})$), and $U$ unramified at all the finite places $v$ where $D$ is non-split (i.e. $U_v={\ensuremath{\mathcal{O}}\xspace}_{D,v}^\times$). Let $\Sigma(U)$ denote the set of finite places where $U$ is ramified. In Theorem 8.9 of [@JarMazPrin], Jarvis constructs an integral model for $M(U)$ over ${\ensuremath{\mathcal{O}}\xspace}_{{\ensuremath{\mathfrak{q}}\xspace}}$ which we will denote by ${\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}(U)$. In fact this integral model only exists when $U$ is sufficiently small (a criterion for this is given in [@JarMazPrin Lemma 12.1]), but this will not cause us any difficulties (see the remark at the end of this subsection). Section 10 of [@JarMazPrin] shows that ${\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}(U)$ satisfies the conditions imposed on ${\ensuremath{\mathscr{X}}\xspace}$ in section \[sec:van\] above.
We also have integral models for places dividing the discriminant of our quaternion algebra. Let $V \subset G'({\ensuremath{\mathbb{A}}\xspace}_f)$ be a compact open subgroup with $V_{\ensuremath{\mathfrak{q}}\xspace}= {\ensuremath{\mathcal{O}}\xspace}_{D',{\ensuremath{\mathfrak{q}}\xspace}}^\times$. We let ${\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}'(V)$ denote the integral model for $M'(V)$ over ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{q}}\xspace}$ coming from Theorem 5.3 of [@VarII] (this is denoted $\mathfrak{C}$ in [@Raj]). This arises from the ${\ensuremath{\mathfrak{q}}\xspace}$-adic uniformisation of $M'(V)$ by a formal ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{q}}\xspace}$-scheme, and gives integral models for varying $V$ which are compatible under the natural projections, and again satisfy the conditions necessary to apply section \[sec:van\].
We now want to construct sheaves on ${\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}(U)$ and ${\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}'(V)$ corresponding to the possible (cohomological) weights of Hilbert modular forms. Let $k = (k_1,\ldots k_d)$ be a $d$-tuple of integers (indexed by the embeddings $\tau_i$ of $F$ into ${\ensuremath{\mathbb{R}}\xspace}$ if the reader prefers), all $\ge 2$ and of the same parity. The $d$-tuple of integers $v$ is then characterised by having non-negative entries, at least one of which is zero, with $k+2v=(w,...,w)$ a parallel vector. We will now define $p$-adic lisse étale sheaves ${\ensuremath{\mathscr{F}_{k}}\xspace}$ on ${\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}(U)$, as in [@Car; @milne; @Sai]. Denote by $Z$ the centre of the algebraic group $G$. Note that $Z \cong\mathrm{Res}_{F/{\ensuremath{\mathbb{Q}}\xspace}}({\ensuremath{\mathbb{G}}\xspace}_{m,F})$. We let $Z_s$ denote the maximal subtorus of $Z$ that is split over ${\ensuremath{\mathbb{R}}\xspace}$ but which has no subtorus split over ${\ensuremath{\mathbb{Q}}\xspace}$. It is straightforward to see that $Z_s$ is the subtorus given by the kernel of the norm map $$N_{F/{\ensuremath{\mathbb{Q}}\xspace}}:\mathrm{Res}_{F/{\ensuremath{\mathbb{Q}}\xspace}}({\ensuremath{\mathbb{G}}\xspace}_{m,F})\rightarrow {\ensuremath{\mathbb{G}}\xspace}_{m,{\ensuremath{\mathbb{Q}}\xspace}}.$$ In the notation of [@milne Ch. III] the algebraic group $G^c$ is the quotient of $G$ by $Z_s$. Take $F'\subset {\ensuremath{\mathbb{C}}\xspace}$ a number field of finite degree, Galois over ${\ensuremath{\mathbb{Q}}\xspace}$ and containing $F$ such that $F'$ splits both $D$ and $D'$. Since $F'/{\ensuremath{\mathbb{Q}}\xspace}$ is normal, $F'$ contains all the Galois conjugates of $F$ and we can identify the embeddings $\{\tau_i:F\rightarrow {\ensuremath{\mathbb{C}}\xspace}\}$ with the embedding $\{\tau_i:F\rightarrow F'\}$, via the inclusion $F' \subset {\ensuremath{\mathbb{C}}\xspace}$. Since $D\otimes_{F,\tau_i} F' \cong M_2(F')$ for each $i$, we obtain representations $\xi_i$ of $D^\times=G({\ensuremath{\mathbb{Q}}\xspace})$ on the space $W_i=F'^2$. Let $\nu:G({\ensuremath{\mathbb{Q}}\xspace})\rightarrow F^\times$ denote the reduced norm. Then $$\xi^{(k)} = \otimes_{i=1}^d (\tau_i\circ \nu)^{v_i}\otimes\operatorname{Sym}^{k_i-2}(\xi_i)$$ is a representation of $G({\ensuremath{\mathbb{Q}}\xspace})$ acting on $W_k=\otimes_{i=1}^d \operatorname{Sym}^{k_i-2}(W_i)$. In fact $\xi^{(k)}$ arises from an algebraic representation of $G$ on the ${\ensuremath{\mathbb{Q}}\xspace}$-vector space underlying $W_k$ (or rather the associated ${\ensuremath{\mathbb{Q}}\xspace}$-vector space scheme). For $z \in Z({\ensuremath{\mathbb{Q}}\xspace})=F^\times$ the action of $\xi^{(k)}(z)$ is multiplication by $N_{F/{\ensuremath{\mathbb{Q}}\xspace}}(z)^{w-2}$, so $\xi^{(k)}$ factors through $G^c$. The representation $\xi^{(k)}$ gives rise to a represpentation of $G({\ensuremath{\mathbb{Q}}\xspace}_p)$ on $W_k\otimes_{\ensuremath{\mathbb{Q}}\xspace}{\ensuremath{\mathbb{Q}}\xspace}_p = W_k \otimes_{F'} (F' \otimes_{\ensuremath{\mathbb{Q}}\xspace}{\ensuremath{\mathbb{Q}}\xspace}_p)$, so taking a prime ${\ensuremath{\mathfrak{p}}\xspace}$ in $F'$ above $p$ and projecting to the relevant factor of $F' \otimes_{\ensuremath{\mathbb{Q}}\xspace}{\ensuremath{\mathbb{Q}}\xspace}_p$ we obtain an $F'_{\ensuremath{\mathfrak{p}}\xspace}$-linear representation $W_{k,{\ensuremath{\mathfrak{p}}\xspace}}$ of $G({\ensuremath{\mathbb{Q}}\xspace}_p)$.
Now we may define a ${\ensuremath{\mathbb{Q}}\xspace}_p$-sheaf on $M(U)({\ensuremath{\mathbb{C}}\xspace})$:
$${\ensuremath{\mathscr{F}}\xspace}_{k,{\ensuremath{\mathbb{Q}}\xspace}_p,{\ensuremath{\mathbb{C}}\xspace}} = G({\ensuremath{\mathbb{Q}}\xspace})\backslash(G({\ensuremath{\mathbb{A}}\xspace}_f) \times ({\ensuremath{\mathbb{C}}\xspace}- {\ensuremath{\mathbb{R}}\xspace}) \times W_{k,{\ensuremath{\mathfrak{p}}\xspace}} / U,$$ where $G({\ensuremath{\mathbb{Q}}\xspace})$ acts on $W_{k,{\ensuremath{\mathfrak{p}}\xspace}}$ by the representation described above (via its embedding into $G({\ensuremath{\mathbb{Q}}\xspace}_p)$), and $U$ acts only on the $G({\ensuremath{\mathbb{A}}\xspace}_f)$ factor, by right multiplication. Note that this definition only works because $\xi^{(k)}$ factors through $G^c$.
Choose an ${\ensuremath{\mathcal{O}}\xspace}_{F'_{\ensuremath{\mathfrak{p}}\xspace}}$ lattice $W_{k,{\ensuremath{\mathfrak{p}}\xspace}}^0$ in $W_{k,{\ensuremath{\mathfrak{p}}\xspace}}$. After projection to $G({\ensuremath{\mathbb{Q}}\xspace}_p)$, $U$ acts on $W_{k,{\ensuremath{\mathfrak{p}}\xspace}}$, and if $U$ is sufficiently small it stabilises $W_{k,{\ensuremath{\mathfrak{p}}\xspace}}^0$. Pick a normal open subgroup $U_1 \subset U$ acting trivially on $W_{k,{\ensuremath{\mathfrak{p}}\xspace}}^0/p^n$. We now define a ${\ensuremath{\mathbb{Z}}\xspace}/p^n{\ensuremath{\mathbb{Z}}\xspace}$-sheaf on $M(U)({\ensuremath{\mathbb{C}}\xspace})$ by $${\ensuremath{\mathscr{F}}\xspace}_{k,{\ensuremath{\mathbb{Z}}\xspace}/p^n{\ensuremath{\mathbb{Z}}\xspace},{\ensuremath{\mathbb{C}}\xspace}} = (M(U_1)({\ensuremath{\mathbb{C}}\xspace})\times (W_{k,{\ensuremath{\mathfrak{p}}\xspace}}^0/p^n)) / (U/U_1).$$ Now on the integral model ${\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}(U)$ we can define an étale sheaf of ${\ensuremath{\mathbb{Z}}\xspace}/p^n{\ensuremath{\mathbb{Z}}\xspace}$-modules by $${\ensuremath{\mathscr{F}}\xspace}_{k,{\ensuremath{\mathbb{Z}}\xspace}/p^n{\ensuremath{\mathbb{Z}}\xspace}} = ({\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}(U_1)\times (W_{k,{\ensuremath{\mathfrak{p}}\xspace}}^0/p^n)) / (U/U_1).$$ By section 2.1 of [@Car] this construction is independent of the choice of lattice $W_{k,{\ensuremath{\mathfrak{p}}\xspace}}^0$. Taking inverse limits of these ${\ensuremath{\mathbb{Z}}\xspace}/p^n{\ensuremath{\mathbb{Z}}\xspace}$-sheaves we get a ${\ensuremath{\mathbb{Z}}\xspace}_p$- (in fact ${\ensuremath{\mathcal{O}}\xspace}_{F'_{\ensuremath{\mathfrak{p}}\xspace}}$-) lisse étale sheaf on ${\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}(U)$ which will be denoted by ${\ensuremath{\mathscr{F}_{k}}\xspace}$.
The same construction applies to ${\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}'(V)$, and we also denote the resulting étale sheaves on ${\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}'(V)$ by ${\ensuremath{\mathscr{F}_{k}}\xspace}$. Note that we chose $F'$ so that it split both $D$ and $D'$ - this allows $F'_{\ensuremath{\mathfrak{p}}\xspace}$ to serve as a field of coefficients for both ${\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}'(U)$ and ${\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}'(V)$.
From now on we work with a coefficient field $E$. This can be any finite extension of $F'_{\ensuremath{\mathfrak{p}}\xspace}$. Denote the ring of integers in $E$ by ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$, and fix a uniformiser $\varpi$. The sheaf ${\ensuremath{\mathscr{F}_{k}}\xspace}$ on the curves ${\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}(U)$ and ${\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}'(V)$ will now denote the sheaves constructed above with coefficients extended to ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$.
Note that in the above discussion we assumed that the level subgroup $U$ was ‘small enough’. In the applications below we will be taking direct limits of cohomology spaces as the level at $p$, $U_p$, varies over all compact open subgroups of $G({\ensuremath{\mathbb{Q}}\xspace}_p)$, so we can always work with a sufficiently small level subgroup by passing far enough through the direct limit.
Hecke algebras {#sec:hecke}
--------------
We will briefly recall the definition of Hecke operators, as in [@DT; @Raj]. Let $U$ and $U'$ be two (sufficiently small) compact open subgroups of $G({\ensuremath{\mathbb{A}}\xspace}_f)$ as in the previous section, and suppose we have $g \in G({\ensuremath{\mathbb{A}}\xspace}_f)$ satisfying $g^{-1}U'g \subset U$. Then there is a (finite) étale map $\rho_g:M(U')\rightarrow M(U)$, corresponding to right multiplication by $g$ on $G({\ensuremath{\mathbb{A}}\xspace}_f)$. Now for $v$ a finite place of $F$, with $D$ unramified at $v$, let $\eta_v$ denote the element of $G({\ensuremath{\mathbb{A}}\xspace}_f)$ which is equal to the identity at places away from $v$ and equal to $\begin{pmatrix}1 & 0 \\0 & \varpi_v \end{pmatrix}$ at the place $v$, where $\varpi_v$ is a fixed uniformiser of ${\ensuremath{\mathcal{O}}\xspace}_{F,v}$. We also use $\varpi_v$ to denote the element of $G({\ensuremath{\mathbb{A}}\xspace}_f)$ which is equal to the identity at places away from $v$ and equal to $\begin{pmatrix}\varpi_v & 0 \\0 & \varpi_v \end{pmatrix}$ at the place $v$. Then the Hecke operator $T_v$ (or $\mathsf{U}_v$ if $U_v$ is not maximal compact) is defined by means of the correspondence on $M(U)$ coming from the maps $$\rho_1,\rho_{\eta_v}:M(U\cap\eta_v U \eta_v^{-1})\rightarrow M(U),$$ whilst $S_v$ is defined by the correspondence $$\rho_1,\rho_{\varpi_v}:M(U)\rightarrow M(U).$$ We can make the same definitions for the curve $M'(V)$.
The Hecke algebra ${\ensuremath{\mathbb{T}}\xspace}_k(U)$ is defined to be the ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$-algebra of endomorphisms of $H^1(M(U)_{\overline{F}},{\ensuremath{\mathscr{F}_{k}}\xspace})_E$ generated by Hecke operators $T_v, S_v$ for finite places $v \nmid p{\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}$ such that $U$ and $D$ are both unramified at $v$. The Hecke algebra ${\ensuremath{\mathbb{T}}\xspace}'_k(V)$ is defined to be the ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$-algebra of endomorphisms of $H^1(M'(V)_{\overline{F}},{\ensuremath{\mathscr{F}_{k}}\xspace})_E$ generated by Hecke operators $T_v, S_v$ for finite places $v \nmid p$ such that $V$ and $D'$ are both unramified at $v$.
The Hecke algebras ${\ensuremath{\mathbb{T}}\xspace}_k(U)$ and ${\ensuremath{\mathbb{T}}\xspace}'_k(V)$ are semilocal, and their maximal ideals correspond to mod $p$ Galois representations arising from Hecke eigenforms occuring in $H^1(M(U)_{\overline{F}},{\ensuremath{\mathscr{F}_{k}}\xspace})_{\overline{E}}$ and $H^1(M'(V)_{\overline{F}},{\ensuremath{\mathscr{F}_{k}}\xspace})_{\overline{E}}$ respectively. Given a maximal ideal ${\ensuremath{\mathfrak{m}}\xspace}$ of ${\ensuremath{\mathbb{T}}\xspace}_k(U)$ we denote the Hecke algebra’s localisation at ${\ensuremath{\mathfrak{m}}\xspace}$ by ${\ensuremath{\mathbb{T}}\xspace}_k(U)_{{\ensuremath{\overline{\rho}}\xspace}}$, where ${\ensuremath{\overline{\rho}}\xspace}: \operatorname{Gal}(\overline{F}/F) \rightarrow {\ensuremath{\mathrm{GL}}\xspace}_2(\overline{{\ensuremath{\mathbb{F}}\xspace}}_p)$ is the mod $p$ Galois representation attached to ${\ensuremath{\mathfrak{m}}\xspace}$, and more generally for any ${\ensuremath{\mathbb{T}}\xspace}_k(U)$-module $M$, denote its localisation at ${\ensuremath{\mathfrak{m}}\xspace}$ by $M_{{\ensuremath{\overline{\rho}}\xspace}}$. We apply the same notational convention to ${\ensuremath{\mathbb{T}}\xspace}'_k(V)$ and modules for this Hecke algebra.
The Hecke algebra ${\ensuremath{\mathbb{T}}\xspace}_k(U)$ acts faithfully on the ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$-torsion free quotient\
$H^1(M(U)_{\overline{F}},{\ensuremath{\mathscr{F}_{k}}\xspace})^{\mathrm{tf}}$ of $H^1(M(U)_{\overline{F}},{\ensuremath{\mathscr{F}_{k}}\xspace})$. Moreover, since the action of the Hecke operators on $H^1(M(U)_{\overline{F}},{\ensuremath{\mathscr{F}_{k}}\xspace})^{\mathrm{tors}}$ is Eisenstein (see [@DT Lemma 4]), if ${\ensuremath{\overline{\rho}}\xspace}$ as above is irreducible then $H^1(M(U)_{\overline{F}},{\ensuremath{\mathscr{F}_{k}}\xspace})^{\mathrm{tf}}_{{\ensuremath{\overline{\rho}}\xspace}}$ can be naturally viewed as an ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$-direct summand of $H^1(M(U)_{\overline{F}},{\ensuremath{\mathscr{F}_{k}}\xspace})$, and we denote it by $H^1(M(U)_{\overline{F}},{\ensuremath{\mathscr{F}_{k}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}$. The same consideration applies to any subquotient $M$ of $H^1(M(U)_{\overline{F}},{\ensuremath{\mathscr{F}_{k}}\xspace})$, and for the Hecke algebra ${\ensuremath{\mathbb{T}}\xspace}_k'(V)$. Note that if $k=(2,...,2)$ then $H^1(M(U)_{\overline{F}},{\ensuremath{\mathscr{F}_{k}}\xspace})=H^1(M(U)_{\overline{F}},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})$ is already ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$-torsion free.
Mazur’s principle {#mazprinclass}
-----------------
In the next two subsections we summarise the results of [@JarMazPrin] and [@Raj]. First we fix a (sufficiently small) compact open subgroup $U \subset G({\ensuremath{\mathbb{A}}\xspace}_f)$ such that $U_{\ensuremath{\mathfrak{q}}\xspace}= {\ensuremath{\mathrm{GL}}\xspace}_2({\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{q}}\xspace})$. For brevity we denote the compact open subgroup $U\cap U_0({\ensuremath{\mathfrak{q}}\xspace})$ by $U({\ensuremath{\mathfrak{q}}\xspace})$.
We apply the theory of Section \[sec:van\] to the curve ${\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}(U({\ensuremath{\mathfrak{q}}\xspace}))$ and the ${\ensuremath{\mathbb{Z}}\xspace}_p$-sheaf given by ${\ensuremath{\mathscr{F}_{k}}\xspace}$, for some weight vector $k$, to obtain ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$-modules $X_{\ensuremath{\mathfrak{q}}\xspace}(U({\ensuremath{\mathfrak{q}}\xspace}),{\ensuremath{\mathscr{F}_{k}}\xspace})$ and $\check{X}_{\ensuremath{\mathfrak{q}}\xspace}(U({\ensuremath{\mathfrak{q}}\xspace}),{\ensuremath{\mathscr{F}_{k}}\xspace})$. We have short exact sequences $$\label{ccses}\minCDarrowwidth10pt\begin{CD}0 @>>> H^1({\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}(U({\ensuremath{\mathfrak{q}}\xspace}))\otimes \overline{k_{\ensuremath{\mathfrak{q}}\xspace}}, {\ensuremath{\mathscr{F}_{k}}\xspace}) @>>> H^1({\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}(U({\ensuremath{\mathfrak{q}}\xspace}))\otimes\overline{L}, {\ensuremath{\mathscr{F}_{k}}\xspace}) @>>> X_{\ensuremath{\mathfrak{q}}\xspace}(U({\ensuremath{\mathfrak{q}}\xspace}),{\ensuremath{\mathscr{F}_{k}}\xspace})(-1) @>>> 0,\end{CD}$$ $$\label{cccoses}\minCDarrowwidth10pt\begin{CD}0@>>>\check{X}_{\ensuremath{\mathfrak{q}}\xspace}(U({\ensuremath{\mathfrak{q}}\xspace}),{\ensuremath{\mathscr{F}_{k}}\xspace})@>>> H^1({\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}(U({\ensuremath{\mathfrak{q}}\xspace}))\otimes \overline{k_{\ensuremath{\mathfrak{q}}\xspace}}, {\ensuremath{\mathscr{F}_{k}}\xspace})@>>> H^1({\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}(U({\ensuremath{\mathfrak{q}}\xspace}))\otimes \overline{k_{\ensuremath{\mathfrak{q}}\xspace}}, r_*r^*{\ensuremath{\mathscr{F}_{k}}\xspace})@>>> 0.\end{CD}$$The above short exact sequences are equivariant with respect to the $G_{\ensuremath{\mathfrak{q}}\xspace}$ action, as well as the action of ${\ensuremath{\mathbb{T}}\xspace}_k(U({\ensuremath{\mathfrak{q}}\xspace}))$ and the Hecke operator $\mathsf{U}_{\ensuremath{\mathfrak{q}}\xspace}$. The sequence \[cccoses\] underlies the proof of Mazur’s principle (see [@JarMazPrin] and Theorem \[mazprin\] below).
Ribet-Rajaei exact sequence {#sec:RR}
---------------------------
We also apply the theory of Section \[sec:van\] to the curve ${\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}'(V)$ and the sheaf ${\ensuremath{\mathscr{F}_{k}}\xspace}$, where $V \subset G'({\ensuremath{\mathbb{A}}\xspace}_f)$ is a (sufficiently small) compact open subgroup which is unramified at ${\ensuremath{\mathfrak{q}}\xspace}$. We obtain ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$-modules $Y_{\ensuremath{\mathfrak{q}}\xspace}(V,{\ensuremath{\mathscr{F}_{k}}\xspace})$ and $\check{Y}_{\ensuremath{\mathfrak{q}}\xspace}(V,{\ensuremath{\mathscr{F}_{k}}\xspace}),$ where $Y$ and $\check{Y}$ correspond to $X$ and $\check{X}$ respectively. As in equation 3.2 of [@Raj] these modules satisfy the ($G_{\ensuremath{\mathfrak{q}}\xspace}$ and ${\ensuremath{\mathbb{T}}\xspace}'_k(V)$ equivariant) short exact sequences $$\minCDarrowwidth10pt\begin{CD}0@>>>H^1({\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}'(V)\otimes \overline{k_{\ensuremath{\mathfrak{q}}\xspace}}, {\ensuremath{\mathscr{F}_{k}}\xspace}) @>>>H^1({\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}'(V)\otimes\overline{L}, {\ensuremath{\mathscr{F}_{k}}\xspace})@>>>Y_{\ensuremath{\mathfrak{q}}\xspace}(V,{\ensuremath{\mathscr{F}_{k}}\xspace})(-1)@>>>0,\end{CD}$$ $$\minCDarrowwidth10pt\begin{CD}0@>>>\check{Y}_{\ensuremath{\mathfrak{q}}\xspace}(V,{\ensuremath{\mathscr{F}_{k}}\xspace})@>>>H^1({\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}'(V)\otimes \overline{k_{\ensuremath{\mathfrak{q}}\xspace}}, {\ensuremath{\mathscr{F}_{k}}\xspace})@>>>H^1({\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}'(V)\otimes \overline{k_{\ensuremath{\mathfrak{q}}\xspace}}, r_*r^*{\ensuremath{\mathscr{F}_{k}}\xspace})@>>>0.\end{CD}$$
It is remarked in section 3 of [@Raj] that in fact $H^1({\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}'(V)\otimes \overline{k_{\ensuremath{\mathfrak{q}}\xspace}}, r_*r^*{\ensuremath{\mathscr{F}_{k}}\xspace})=0$, since the irreducible components of the special fibre of ${\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}'(V)$ are rational curves, so we have $$\check{Y}_{\ensuremath{\mathfrak{q}}\xspace}(V,{\ensuremath{\mathscr{F}_{k}}\xspace})\cong H^1({\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}'(V)\otimes \overline{k_{\ensuremath{\mathfrak{q}}\xspace}}, {\ensuremath{\mathscr{F}_{k}}\xspace}).$$
We can now relate our constructions for the two quaternion algebras $D$ and $D'$. This requires us to now distinguish between the two places ${\ensuremath{{\mathfrak{q}_1}}\xspace}$ and ${\ensuremath{{\mathfrak{q}_2}}\xspace}$. We make some further assumptions on the levels $U$ and $V$. First we assume that $U_{\ensuremath{{\mathfrak{q}_1}}\xspace}= {\ensuremath{\mathcal{O}}\xspace}_{D,{\ensuremath{{\mathfrak{q}_1}}\xspace}} \cong {\ensuremath{\mathrm{GL}}\xspace}_2({\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{{\mathfrak{q}_1}}\xspace})$ and $U_{\ensuremath{{\mathfrak{q}_2}}\xspace}= {\ensuremath{\mathcal{O}}\xspace}_{D,{\ensuremath{{\mathfrak{q}_2}}\xspace}} \cong {\ensuremath{\mathrm{GL}}\xspace}_2({\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{{\mathfrak{q}_2}}\xspace})$, and similarly that $V$ is also maximal at both ${\ensuremath{{\mathfrak{q}_1}}\xspace}$ and ${\ensuremath{{\mathfrak{q}_2}}\xspace}$. We furthermore assume that the factors of $U$ away from ${\ensuremath{{\mathfrak{q}_1}}\xspace}$ and ${\ensuremath{{\mathfrak{q}_2}}\xspace}$, $U^{{\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}}$, match with $V^{{\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}}$ under the fixed isomorphism $D\otimes_F {\ensuremath{\mathbb{A}}\xspace}_{F,f}^{({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace})} \cong D'\otimes_F {\ensuremath{\mathbb{A}}\xspace}_{F,f}^{({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace})}$. We will use the notation $U({\ensuremath{{\mathfrak{q}_1}}\xspace}), U({\ensuremath{{\mathfrak{q}_2}}\xspace}), U({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace})$ to denote $U\cap U_0({\ensuremath{{\mathfrak{q}_1}}\xspace}), U\cap U_0({\ensuremath{{\mathfrak{q}_2}}\xspace}), U\cap U_0({\ensuremath{{\mathfrak{q}_1}}\xspace})\cap U_0({\ensuremath{{\mathfrak{q}_2}}\xspace})$ respectively.
We fix a Galois representation $${\ensuremath{\overline{\rho}}\xspace}: \operatorname{Gal}(\overline{F}/F) \rightarrow {\ensuremath{\mathrm{GL}}\xspace}_2(\overline{{\ensuremath{\mathbb{F}}\xspace}}_p),$$ satisfying two assumptions. Firstly, we assume that ${\ensuremath{\overline{\rho}}\xspace}$ arises from a Hecke eigenform in $H^1(M'(V)_{\overline{F}},{\ensuremath{\mathscr{F}_{k}}\xspace})$. Secondly, we assume that ${\ensuremath{\overline{\rho}}\xspace}$ is irreducible. Equivalently, the corresponding maximal ideal ${\ensuremath{\mathfrak{m}}\xspace}'$ in ${\ensuremath{\mathbb{T}}\xspace}'_k(V)$ is not Eisenstein.
There is a surjection $${\ensuremath{\mathbb{T}}\xspace}_k(U({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace})) \rightarrow {\ensuremath{\mathbb{T}}\xspace}'_k(V)$$ arising from the classical Jacquet–Langlands correspondence, identifying ${\ensuremath{\mathbb{T}}\xspace}'_k(V)$ as the ${\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}$-new quotient of ${\ensuremath{\mathbb{T}}\xspace}_k(U({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}))$. Hence there is a (non-Eisenstein) maximal ideal of ${\ensuremath{\mathbb{T}}\xspace}_k(U({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}))$ associated to ${\ensuremath{\overline{\rho}}\xspace}$ (in fact this can be established directly from the results contained on pg. $55$ of [@Raj], but for expository purposes it is easier to rely on the existence of the Jacquet–Langlands correspondence). We can therefore regard all ${\ensuremath{\mathbb{T}}\xspace}'_k(V)_{{\ensuremath{\overline{\rho}}\xspace}}$-modules as ${\ensuremath{\mathbb{T}}\xspace}_k(U({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}))_{{\ensuremath{\overline{\rho}}\xspace}}$-modules. Now [@Raj Theorem 3] implies the following:
\[RRESclass\]There are ${\ensuremath{\mathbb{T}}\xspace}_k(U({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}))_{{\ensuremath{\overline{\rho}}\xspace}}$-equivariant short exact sequences $$\minCDarrowwidth15pt\begin{CD}0@>>> Y_{\ensuremath{{\mathfrak{q}_1}}\xspace}(V,{\ensuremath{\mathscr{F}_{k}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}} @>>>X_{\ensuremath{{\mathfrak{q}_2}}\xspace}(U({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathscr{F}_{k}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}@>{i^\dagger}>>X_{\ensuremath{{\mathfrak{q}_2}}\xspace}(U({\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathscr{F}_{k}}\xspace})^2_{{\ensuremath{\overline{\rho}}\xspace}}@>>>0,\end{CD}$$ $$\minCDarrowwidth15pt\begin{CD}0@>>>\check{X}_{\ensuremath{{\mathfrak{q}_2}}\xspace}(U({\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathscr{F}_{k}}\xspace})^2_{{\ensuremath{\overline{\rho}}\xspace}}@>{i}>>\check{X}_{\ensuremath{{\mathfrak{q}_2}}\xspace}(U({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathscr{F}_{k}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}} @>>>\check{Y}_{\ensuremath{{\mathfrak{q}_1}}\xspace}(V,{\ensuremath{\mathscr{F}_{k}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}@>>>0.\end{CD}$$
In the theorem, the maps labelled $i$ and $i^\dagger$ are the natural ‘level raising’ map and its adjoint. To be more precise $i=(i_1,i_{\eta_{\ensuremath{\mathfrak{q}}\xspace}})$ where $i_g$ is the map coming from the composition of the natural map $$H^1(M(U)_{\overline{F}},{\ensuremath{\mathscr{F}_{k}}\xspace}) \rightarrow H^1(M(U({\ensuremath{\mathfrak{q}}\xspace}))_{\overline{F}},\rho_{g}^*{\ensuremath{\mathscr{F}_{k}}\xspace})$$ with the map coming from the sheaf morphism $\rho_{g}^*{\ensuremath{\mathscr{F}_{k}}\xspace}\rightarrow {\ensuremath{\mathscr{F}_{k}}\xspace}$ as in [@DT pg. 449]. The map $i^\dagger=(i_1^\dagger,i_{\eta_{\ensuremath{\mathfrak{q}}\xspace}}^\dagger)$ where $i_g^\dagger$ is the map coming from the composition of the map arising from the sheaf morphism ${\ensuremath{\mathscr{F}_{k}}\xspace}\rightarrow \rho_g^*{\ensuremath{\mathscr{F}_{k}}\xspace}$ (again see [@DT pg. 449]) with the map $$H^1(M(U({\ensuremath{\mathfrak{q}}\xspace}))_{\overline{F}},\rho_g^*{\ensuremath{\mathscr{F}_{k}}\xspace}) \rightarrow H^1(M(U)_{\overline{F}},{\ensuremath{\mathscr{F}_{k}}\xspace})$$ arising from the trace map $\rho_{g*}\rho_g^*{\ensuremath{\mathscr{F}_{k}}\xspace}\rightarrow {\ensuremath{\mathscr{F}_{k}}\xspace}$.
We think of the short exact sequences in the above Theorem as a geometric realisation of the Jacquet–Langlands correspondence between automorphic forms for $G'$ and for $G$. The proof of the above theorem relies on relating the spaces involved to the arithmetic of a third quaternion algebra $\overline{D}$ (which associated reductive algebraic group $\overline{G}/{\ensuremath{\mathbb{Q}}\xspace}$) which is non-split at all the infinite places of $F$ and at the places in $S \cup \{{\ensuremath{{\mathfrak{q}_2}}\xspace}\}$. However, we will just focus on the groups $G'$ and $G$, and view $\overline{G}$ as playing an auxiliary role.
Completed cohomology of Shimura curves
======================================
Completed cohomology {#subsec:ccoho}
--------------------
We now consider compact open subgroups $U^p$ of $G({\ensuremath{\mathbb{A}}\xspace}_f^{(p)})$, with factor at $v$ equal to ${\ensuremath{\mathcal{O}}\xspace}_{D_v}^\times$ at all places $v$ of $F$ where $D$ is non-split. Then we have the following definitions, as in [@Emint; @Emlg]:
We write $H^n_D(U^p,{\ensuremath{\mathscr{F}_{k}}\xspace}/\varpi^s)$ for the ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$ module $$\varinjlim_{U_p} H^n(M(U_p U^p)_{\overline{F}},{\ensuremath{\mathscr{F}_{k}}\xspace}/\varpi^s),$$ which has a smooth action of $G({\ensuremath{\mathbb{Q}}\xspace}_p)=\prod_{{\ensuremath{v}\xspace}| p} {\ensuremath{\mathrm{GL}}\xspace}_2(F_{\ensuremath{v}\xspace})$ and a continuous action of $G_F$.
We then define a $\varpi$-adically complete ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$-module $$\widetilde{H}^n(U^p,{\ensuremath{\mathscr{F}_{k}}\xspace}) := \varprojlim_s H^n(U^p,{\ensuremath{\mathscr{F}_{k}}\xspace}/\varpi^s),$$ which has a continuous action of $G({\ensuremath{\mathbb{Q}}\xspace}_p)$ and $G_F$.
The ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$-module $\widetilde{H}^n_D(U^p,{\ensuremath{\mathscr{F}_{k}}\xspace})$ is a $\varpi$-adically admissible $G({\ensuremath{\mathbb{Q}}\xspace}_p)$ representation over ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$ (in the sense of [@Emordone Definition 2.4.7]).
This follows from [@Emint Theorem 2.2.11], taking care to remember the ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$-integral structure.
Write $\widetilde{H}^n(U^p,{\ensuremath{\mathscr{F}_{k}}\xspace})_{{\ensuremath{E}}}$ for the admissible continuous Banach $G({\ensuremath{\mathbb{Q}}\xspace}_p)$ representation $$\widetilde{H}^n(U^p,{\ensuremath{\mathscr{F}_{k}}\xspace})\otimes_{{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}}{\ensuremath{E}}.$$
\[hhat\] There is a canonical isomorphism $$\widetilde{H}^n(U^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}) = \varprojlim_s H^n(U^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})/\varpi^sH^n(U^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}),$$ compatible with $G_F$ and $G({\ensuremath{\mathbb{Q}}\xspace}_p)$ actions.
Since $M(U_p U^p)_{\overline{F}}$ is proper of dimension $1$, this follows from Corollary 2.2.27 of [@Emint]
Note that in the notation of [@Emint] the above lemma says that $\widetilde{H}^n(U^p) = \widehat{H}^n(U^p)$.
The following lemma allows us to restrict to considering completed cohomology with trivial coefficients.
\[changeweight\]There is a canonical $G({\ensuremath{\mathbb{Q}}\xspace}_p)$, $G_F$ equivariant isomorphism $$\widetilde{H}^n(U^p,{\ensuremath{\mathscr{F}_{k}}\xspace}) \cong \widetilde{H}^n(U^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})\otimes_{{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}} W^0_{k,{\ensuremath{\mathfrak{p}}\xspace}}.$$
This follows from Theorem 2.2.17 of [@Emint].
Let $\overline{F^\times}$ be the closure of $F^\times$ in ${\ensuremath{\mathbb{A}}\xspace}_F^{f,\times}$. There is an isomorphism between $\widetilde{H}^0(U^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})$ and the space of continuous functions $$\mathscr{C}(\overline{F^\times} \backslash {\ensuremath{\mathbb{A}}\xspace}_F^{f,\times} / \det(U^p), {\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}),$$ with ${\ensuremath{\mathrm{GL}}\xspace}_2({\ensuremath{\mathbb{Q}}\xspace}_p)$ action on the right hand side induced by $g$ acting on ${\ensuremath{\mathbb{A}}\xspace}_F^{f,\times}$ by multiplication by $\det(g)$.
This follows from the fact that the product of the sign map on $({\ensuremath{\mathbb{C}}\xspace}- {\ensuremath{\mathbb{R}}\xspace})$ and the reduced norm map on $G({\ensuremath{\mathbb{A}}\xspace}_f)$ induces a map $$G({\ensuremath{\mathbb{Q}}\xspace})\backslash G({\ensuremath{\mathbb{A}}\xspace}_f)\times ({\ensuremath{\mathbb{C}}\xspace}- {\ensuremath{\mathbb{R}}\xspace})/U \rightarrow F^\times \backslash {\ensuremath{\mathbb{A}}\xspace}_{F,f}^\times\times \{\pm\} / \det(U) = F^\times \backslash {\ensuremath{\mathbb{A}}\xspace}_{F,f}^\times / \det(U)$$ with connected fibres.
The space $\widetilde{H}^2(U^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})$ is equal to $0$.
Exactly as for Proposition 4.3.6 of [@Emint]
If $v$ is a prime in $F$ such that $U^p$ and $D$ are both unramified at $v$ then there are Hecke operators $T_v, S_v$ acting on the spaces $H^1(U^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}/\varpi^s)$, defined as in Section \[sec:hecke\]. By continuity this extends to an action of the Hecke operators on the space $\widetilde{H}^1_D(U^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})$, so we can make the following definition:
The Hecke algebra ${\ensuremath{\mathbb{T}}\xspace}(U^p)$ is defined to be $\varprojlim_{U_p}{\ensuremath{\mathbb{T}}\xspace}_{(2,...,2)}(U^p U_p)$. We endow it with the projective limit topology given by taking the $\varpi$-adic topology on each finitely generated ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$-algebra ${\ensuremath{\mathbb{T}}\xspace}_{(2,...,2)}(U^p U_p)$.
The topological ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$-algebra ${\ensuremath{\mathbb{T}}\xspace}(U^p)$ acts faithfully on $\widetilde{H}^1_D(U^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})$, and is topologically generated by Hecke operators $T_v, S_v$ for primes $v \nmid p$ such that $U^p$ and $D$ are both unramified at $v$. Note that the isomorphisms of lemmas \[hhat\] and \[changeweight\] are equivariant with respect to the Hecke action, so we will often just work with the completed cohomology spaces defined with trivial coefficients.
An ideal $I$ of ${\ensuremath{\mathbb{T}}\xspace}(U^p)$ is *Eisenstein* if the map $\lambda: T(U^p) \rightarrow T(U^p)/I$ satisfies $\lambda(T_q) = \epsilon_1(q) + \epsilon_2(q)$ and $\lambda(qS_q) = \epsilon_1(q)\epsilon_2 (q)$ for all $q \notin \{p\}\cup\Sigma$, for some characters $\epsilon_1,\epsilon_2 : {\ensuremath{\mathbb{Z}}\xspace}_{p\Sigma}^\times \rightarrow T(U^p)/I$.
The Hecke algebra ${\ensuremath{\mathbb{T}}\xspace}(U^p)$ is semilocal and Noetherian, with maximal ideals corresponding to mod $p$ Galois representations arising from Hecke eigenforms in (an extension of scalars of) $H^1_D(U^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}/\varpi)$. Given a maximal ideal ${\ensuremath{\mathfrak{m}}\xspace}$ of ${\ensuremath{\mathbb{T}}\xspace}(U^p)$ we denote the Hecke algebra’s localisation at ${\ensuremath{\mathfrak{m}}\xspace}$ by ${\ensuremath{\mathbb{T}}\xspace}(U^p)_{{\ensuremath{\overline{\rho}}\xspace}}$, where ${\ensuremath{\overline{\rho}}\xspace}: \operatorname{Gal}(\overline{F}/F) \rightarrow {\ensuremath{\mathrm{GL}}\xspace}_2(\overline{{\ensuremath{\mathbb{F}}\xspace}}_p)$ is the mod $p$ Galois representation attached to ${\ensuremath{\mathfrak{m}}\xspace}$, and more generally for any ${\ensuremath{\mathbb{T}}\xspace}(U^p)$-module $M$, denote its localisation at ${\ensuremath{\mathfrak{m}}\xspace}$ by $M_{{\ensuremath{\overline{\rho}}\xspace}}$.
We now fix such a ${\ensuremath{\overline{\rho}}\xspace}$, and assume that it is irreducible. There is a deformation $\rho_{{\ensuremath{\overline{\rho}}\xspace},U^p}^m$ of ${\ensuremath{\overline{\rho}}\xspace}$ to ${\ensuremath{\mathbb{T}}\xspace}(U^p)_{{\ensuremath{\overline{\rho}}\xspace}}$ (taking a projective limit, over compact open subgroups $U_p$ of $G({\ensuremath{\mathbb{Q}}\xspace}_p)$, of the deformations to ${\ensuremath{\mathbb{T}}\xspace}_{(2,...,2)}(U^pU_p)_{{\ensuremath{\overline{\rho}}\xspace}}$). For a closed point $\mathscr{P} \in \operatorname{Spec}({\ensuremath{\mathbb{T}}\xspace}(U^p)_{{\ensuremath{\overline{\rho}}\xspace}}[\frac{1}{p}])$ we denote by $\rho(\mathscr{P})$ the attached Galois representation, defined over the characteristic $0$ field $k(\mathscr{P})$.
Given a ${\ensuremath{\mathbb{T}}\xspace}(U^p)_{{\ensuremath{\overline{\rho}}\xspace}}$-module $M$ and a closed point $\mathscr{P}$ of $\operatorname{Spec}({\ensuremath{\mathbb{T}}\xspace}(U^p)_{{\ensuremath{\overline{\rho}}\xspace}}[\frac{1}{p}])$, we denote by $M[\mathscr{P}]$ the $k(\mathscr{P})$-vector space $$\{m \in M\otimes_{{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}}k(\mathscr{P}) : Tm = 0 \hbox{ for all } T \in \mathscr{P}\}.$$ Suppose we have a homomorphism $\lambda: {\ensuremath{\mathbb{T}}\xspace}(U^p)_{{\ensuremath{\overline{\rho}}\xspace}}\rightarrow E'$, where $E'/E$ is a finite field extension. We say that the system of Hecke eigenvalues $\lambda$ *occurs in* a ${\ensuremath{\mathbb{T}}\xspace}(U^p)_{{\ensuremath{\overline{\rho}}\xspace}}$-module $M$ if $M\otimes_{{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}} E'$ contains a non-zero element where ${\ensuremath{\mathbb{T}}\xspace}(U^p)$ acts via the character $\lambda$. In other words, $\lambda$ occurs in $M$ if and only if $M[\ker(\lambda)] \ne 0$. We say that $\lambda$ is Eisenstein whenever $\ker{\lambda}$ is.
We have the following result, generalising [@Emlg Proposition 5.5.3]. Denote by $X$ the ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$-module $\operatorname{Hom}_{{\ensuremath{\mathbb{T}}\xspace}(U^p)_{{\ensuremath{\overline{\rho}}\xspace}}[G_F]}(\rho_{{\ensuremath{\overline{\rho}}\xspace},U^p}^m,\widetilde{H}^1_D(U^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}})$.
\[galfac\] The natural evaluation map $$ev:\rho_{{\ensuremath{\overline{\rho}}\xspace},U^p}^m \otimes_{{\ensuremath{\mathbb{T}}\xspace}(U^p)_{{\ensuremath{\overline{\rho}}\xspace}}} X \rightarrow \widetilde{H}^1_D(U^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}$$ is an isomorphism.
First let $\mathscr{P} \in \operatorname{Spec}({\ensuremath{\mathbb{T}}\xspace}(U^p)_{{\ensuremath{\overline{\rho}}\xspace}}[\frac{1}{p}])$ be the prime ideal of the Hecke algebra corresponding to a classical point coming from a Hecke eigenform in $$H^1(M(U^pU_p),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}\otimes_{{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}}k(\mathscr{P}).$$ Then the image of the map $$\rho(\mathscr{P}) \otimes_{k(\mathscr{P})}X[\mathscr{P}] \rightarrow \widetilde{H}^1_D(U^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}\otimes_{{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}} k(\mathscr{P})$$ induced by $ev$ contains $H^1(M(U^pU_p)_{{\ensuremath{\overline{\rho}}\xspace}},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})
[\mathscr{P}].$ From this, it is straightforward to to see that the image of the map $$ev_E: E \otimes_{{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}}\rho_{{\ensuremath{\overline{\rho}}\xspace},U^p}^m \otimes_{{\ensuremath{\mathbb{T}}\xspace}(U^p)_{{\ensuremath{\overline{\rho}}\xspace}}} X \rightarrow \widetilde{H}^1_D(U^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace},E}$$ is dense, since it will contain the dense subspace $H^1_D(U^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace},E}$. Now $X$ is a $\varpi$-adically admissible $G(Q_p)$-representation over ${\ensuremath{\mathbb{T}}\xspace}(U^p)_{{\ensuremath{\overline{\rho}}\xspace}}$, hence by [@Emlg Proposition 3.1.3] the image of $ev_E$ is closed, therefore $ev_E$ is surjective. We now note that the evaluation map $${\ensuremath{\overline{\rho}}\xspace}\otimes_{k({\ensuremath{\mathfrak{p}}\xspace})} \operatorname{Hom}_{G_F}({\ensuremath{\overline{\rho}}\xspace},H^1_D(U^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}/\varpi)_{{\ensuremath{\overline{\rho}}\xspace}}) \rightarrow H^1_D(U^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}/\varpi)_{{\ensuremath{\overline{\rho}}\xspace}}$$ is injective, by the irreducibility of ${\ensuremath{\overline{\rho}}\xspace}$. We conclude the proof of the proposition by applying the following lemma (see [@Emlg Lemma 3.1.6]).
Let $\pi_1$ and $\pi_2$ be two $\varpi$-adically admissible $G({\ensuremath{\mathbb{Q}}\xspace}_p)$ representations over a complete local Noetherian ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$-algebra $A$ (with maximal ideal [$\mathfrak{m}$]{}), which are torsion free as ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$-modules. Suppose $f: \pi_1 \rightarrow \pi_2$ is a continuous $A[G({\ensuremath{\mathbb{Q}}\xspace}_p)]$-linear morphism, satisfying
1. The induced map $\pi_1 \otimes_{{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}} E \rightarrow \pi_2 \otimes_{{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}} E$ is a surjection
2. The induced map $\pi_1/\varpi [{\ensuremath{\mathfrak{m}}\xspace}] \rightarrow \pi_2/\varpi [{\ensuremath{\mathfrak{m}}\xspace}]$ is an injection.
Then $f$ is an isomorphism.
First we show that $f$ is an injection. Denote the kernel of $f$ by $K$. It is a $\varpi$-adically admissible $G({\ensuremath{\mathbb{Q}}\xspace}_p)$ representation over $A$, so in particular $$K/\varpi =\bigcup_{i\ge 1} K/\varpi [{\ensuremath{\mathfrak{m}}\xspace}^i].$$ Our second assumption on $f$ then implies that $K/\varpi = 0$, which implies $K=0$, since $K$ is $\varpi-$adically separated. We now know that $f$ is an injection, and out first assumption on $f$ implies that it has ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$-torsion cokernel $C$. Since $\pi_1$ is ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$-torsion free there is an exact sequence $$0 \rightarrow C[\varpi] \rightarrow \pi_1/\varpi \rightarrow \pi_2/\varpi.$$ Since $\pi_1/\varpi =\bigcup_{i\ge 1} \pi_1/\varpi [{\ensuremath{\mathfrak{m}}\xspace}^i]$ we have $C[\varpi] =\bigcup_{i\ge 1} C[\varpi][{\ensuremath{\mathfrak{m}}\xspace}^i],$ but the second assumption on $f$ implies that $C[\varpi][{\ensuremath{\mathfrak{m}}\xspace}]=0$. Therefore $C[\varpi]=0$, and since $C$ is ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$-torsion we have $C=0$. Therefore $f$ is an isomorphism.
Completed vanishing cycles
--------------------------
We now assume that our tame level $U^p$ is unramified at the prime ${\ensuremath{\mathfrak{q}}\xspace}$, and let ${\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace}$ denote the tame level $U^p \cap U_0({\ensuremath{\mathfrak{q}}\xspace})$. Fix a non-Eisenstein maximal ideal ${\ensuremath{\mathfrak{m}}\xspace}$ of ${\ensuremath{\mathbb{T}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace})$. We are going to ‘$\varpi$-adically complete’ the constructions of subsection \[mazprinclass\].
For ${\ensuremath{\mathscr{F}}\xspace}$ equal to either ${\ensuremath{\mathscr{F}_{k}}\xspace}$ or $r_*r^*{\ensuremath{\mathscr{F}_{k}}\xspace}$ we define ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$-modules with smooth $G({\ensuremath{\mathbb{Q}}\xspace}_p)$ actions $$H^1_{{\ensuremath{\mathrm{red}}\xspace}}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}}\xspace}) = \varinjlim_{U_p} H^1({\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}(U_p{\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace})\otimes \overline{k_{\ensuremath{\mathfrak{q}}\xspace}}, {\ensuremath{\mathscr{F}}\xspace}),$$ $$X_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})=\varinjlim_{U_p}X_{\ensuremath{\mathfrak{q}}\xspace}(U_p{\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace}),$$ and $$\check{X}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})=\varinjlim_{U_p}\check{X}_{\ensuremath{\mathfrak{q}}\xspace}(U_p{\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace}).$$
\[smoothses\] Equation \[ccses\] induces a short exact sequence, equivariant with respect to Hecke, $G({\ensuremath{\mathbb{Q}}\xspace}_p)$ and $G_{\ensuremath{\mathfrak{q}}\xspace}$ actions: $$\minCDarrowwidth10pt\begin{CD}0@>>>H^1_{{\ensuremath{\mathrm{red}}\xspace}}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace}) @>>>H^1({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})@>>>X_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})(-1)@>>>0.\end{CD}$$ Similarly, Equation \[cccoses\] induces $$\minCDarrowwidth10pt\begin{CD}0@>>>\check{X}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})@>>>H^1_{{\ensuremath{\mathrm{red}}\xspace}}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})@>>>H^1_{{\ensuremath{\mathrm{red}}\xspace}}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},r_*r^*{\ensuremath{\mathscr{F}_{k}}\xspace})@>>>0.\end{CD}$$
This just follows from taking the direct limit of the relevant exact sequences.
\[completedstuff\] We define ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$-modules with continuous $G({\ensuremath{\mathbb{Q}}\xspace}_p)$ actions (again ${\ensuremath{\mathscr{F}}\xspace}$ equals either ${\ensuremath{\mathscr{F}_{k}}\xspace}$ or $r_*r^* {\ensuremath{\mathscr{F}_{k}}\xspace}$) $$\widetilde{H}^1_{\ensuremath{\mathrm{red}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}}\xspace}) = \varprojlim_s H^1_{{\ensuremath{\mathrm{red}}\xspace}}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}}\xspace})/\varpi^sH^1_{{\ensuremath{\mathrm{red}}\xspace}}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}}\xspace}),$$ $$\widetilde{X}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace}) = \varprojlim_s X_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})/\varpi^sX_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})$$ and $$\widetilde{\check{X}}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace}) = \varprojlim_s \check{X}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})/\varpi^s\check{X}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace}).$$
Now $\varpi$-adically completing the short exact sequences of Proposition \[smoothses\], we obtain
\[prop:es\] We have short exact sequences of $\varpi$-adically admissible $G({\ensuremath{\mathbb{Q}}\xspace}_p)$-representations over ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$, equivariant with respect to Hecke, $G({\ensuremath{\mathbb{Q}}\xspace}_p)$ and $G_{\ensuremath{\mathfrak{q}}\xspace}$ actions: $$\minCDarrowwidth10pt\begin{CD}0@>>>\widetilde{H}^1_{{\ensuremath{\mathrm{red}}\xspace}}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})_{\ensuremath{\mathfrak{m}}\xspace}@>>>\widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})_{\ensuremath{\mathfrak{m}}\xspace}@>>>\widetilde{X}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})_{\ensuremath{\mathfrak{m}}\xspace}(-1)@>>>0,\end{CD}$$ $$\minCDarrowwidth10pt\begin{CD}0@>>>\widetilde{\check{X}}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})_{\ensuremath{\mathfrak{m}}\xspace}@>>>\widetilde{H}^1_{{\ensuremath{\mathrm{red}}\xspace}}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})_{\ensuremath{\mathfrak{m}}\xspace}@>>>\widetilde{H}^1_{{\ensuremath{\mathrm{red}}\xspace}}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},r_*r^*{\ensuremath{\mathscr{F}_{k}}\xspace})_{\ensuremath{\mathfrak{m}}\xspace}@>>>0.\end{CD}$$
The existence of a short exact sequence of ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$-modules follows from the fact that $X_{\ensuremath{\mathfrak{q}}\xspace}(U_p{\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})_{\ensuremath{\mathfrak{m}}\xspace}$ and $H^1_{{\ensuremath{\mathrm{red}}\xspace}}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},r_*r^*{\ensuremath{\mathscr{F}_{k}}\xspace})_{\ensuremath{\mathfrak{m}}\xspace}$ is always ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$-torsion free. The module $X_{\ensuremath{\mathfrak{q}}\xspace}(U_p{\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})$ is torsion free even without localising at ${\ensuremath{\mathfrak{m}}\xspace}$ (see the remark following [@Raj Corollary 1]), whilst for $H^1_{{\ensuremath{\mathrm{red}}\xspace}}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},r_*r^*{\ensuremath{\mathscr{F}_{k}}\xspace})_{\ensuremath{\mathfrak{m}}\xspace}$ this follows as in [@DT Lemma 4]. The fact that we obtain a short exact sequence of $\varpi$-adically admissible ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}[G({\ensuremath{\mathbb{Q}}\xspace}_p)]$-modules follows from Proposition 2.4.4 of [@Emordone].
It is now straightforward to deduce the analogue of Lemma \[changeweight\]:
There are canonical $G({\ensuremath{\mathbb{Q}}\xspace}_p)$, $G_{\ensuremath{\mathfrak{q}}\xspace}$ and Hecke equivariant isomorphisms $$\widetilde{H}^1_{{\ensuremath{\mathrm{red}}\xspace}}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})_{\ensuremath{\mathfrak{m}}\xspace}\cong \widetilde{H}^1_{{\ensuremath{\mathrm{red}}\xspace}}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{\ensuremath{\mathfrak{m}}\xspace}\otimes_{{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}} W^0_{k,{\ensuremath{\mathfrak{p}}\xspace}},$$ $$\widetilde{\check{X}}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})_{\ensuremath{\mathfrak{m}}\xspace}\cong \widetilde{\check{X}}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{\ensuremath{\mathfrak{m}}\xspace}\otimes_{{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}} W^0_{k,{\ensuremath{\mathfrak{p}}\xspace}},$$ $$\widetilde{X}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})_{\ensuremath{\mathfrak{m}}\xspace}\cong \widetilde{X}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{\ensuremath{\mathfrak{m}}\xspace}\otimes_{{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}} W^0_{k,{\ensuremath{\mathfrak{p}}\xspace}}.$$
Mazur’s principle {#mazurs-principle}
-----------------
We can now proceed as in Theorem 6.2.4 of [@Emlg], following Jarvis’s approach to Mazur’s principle over totally real fields [@JarMazPrin]. First we fix an absolutely irreducible mod $p$ Galois representation ${\ensuremath{\overline{\rho}}\xspace}$, coming from a maximal ideal of ${\ensuremath{\mathbb{T}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace})$. We have the following proposition
\[unramred\] The injection $$\widetilde{H}^1_{{\ensuremath{\mathrm{red}}\xspace}}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}) \hookrightarrow \widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})$$ induces an isomorphism $$\widetilde{H}^1_{{\ensuremath{\mathrm{red}}\xspace}}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}} \cong (\widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}})^{I_{\ensuremath{\mathfrak{q}}\xspace}}.$$
This follows from Proposition 4 and (2.2) in [@Raj].
There is a Hecke operator $\mathsf{U}_{\ensuremath{\mathfrak{q}}\xspace}$ acting on $H^1({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})$, which induces an action on the spaces $\check{X}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})$ and $X_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})$. This extends by continuity to give an action of $\mathsf{U}_{\ensuremath{\mathfrak{q}}\xspace}$ on $\widetilde{\check{X}}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})$ and $\widetilde{X}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})$.
\[frobaction\] The Frobenius element of $\operatorname{Gal}(\overline{k}_{\ensuremath{\mathfrak{q}}\xspace}/k_{\ensuremath{\mathfrak{q}}\xspace})$ acts on $\widetilde{\check{X}}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})$ by $(\mathbf{N}{\ensuremath{\mathfrak{q}}\xspace})\mathsf{U}_{\ensuremath{\mathfrak{q}}\xspace}$ and on $\widetilde{X}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})$ by $\mathsf{U}_{\ensuremath{\mathfrak{q}}\xspace}$.
Section 6 of [@Car] tells us that Frobenius acts as required on the dense subspaces $\check{X}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})$ and $X_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})$, so by continuity we are done.
\[oldred\] There is a natural isomorphism $$\widetilde{H}^1_{{\ensuremath{\mathrm{red}}\xspace}}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},r_*r^*{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}) \cong \widetilde{H}^1_D(U^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})^{\oplus 2}.$$
This follows from Lemma 16.1 in [@JarMazPrin].
\[monodromy\] The monodromy pairing map $\lambda: X_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}) \rightarrow \check{X}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})$ induces a $G_{\ensuremath{\mathfrak{q}}\xspace}$ and ${\ensuremath{\mathbb{T}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace})$-equivariant isomorphism $$\widetilde{X}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}} \cong \widetilde{\check{X}}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}.$$
This follows from Proposition 5 in [@Raj].
Note that the isomorphism in the above lemma is not $\mathsf{U}_{\ensuremath{\mathfrak{q}}\xspace}$ equivariant.
Before giving an analogue of Mazur’s principle, we will make explicit the connection between the modules $\widetilde{X}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}$ and a space of newforms. There are natural level raising maps $i:\widetilde{H}^1_D(U^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{\oplus 2} \rightarrow \widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}$, and their adjoints $i^\dagger: \widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}} \rightarrow \widetilde{H}^1_D(U^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{\oplus 2}$, defined by taking the limit of maps at finite level defined as in the paragraph after Theorem \[RRESclass\].
\[def:new\] The space of ${\ensuremath{\mathfrak{q}}\xspace}$-newforms $$\widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{{\ensuremath{\mathfrak{q}}\xspace}\mhyphen\mathrm{new}}$$ is defined to be the kernel of the map $$i^\dagger: \widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}} \rightarrow \widetilde{H}^1_D(U^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{\oplus 2}.$$
Composing the injection $\widetilde{H}^1_\mathrm{red}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}} \hookrightarrow \widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}$ with $i^\dagger$ gives a map, which we also denote by $i^\dagger$, from $\widetilde{H}^1_\mathrm{red}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}$ to $\widetilde{H}^1_D(U^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{\oplus 2}.$ Recall that we can also define a map $\omega$ between these spaces, by composing the natural map $$\widetilde{H}^1_\mathrm{red}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}} \rightarrow \widetilde{H}^1_\mathrm{red}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},r_*r^*{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}$$ with the isomorphism of Lemma \[oldred\]. However these maps are not equal, as remarked in [@Ribet100 pg. 447]. Fortunately, they have the same kernel, as we now show (essentially by following the proof of [@Ribet100 Theorem 3.11]).
\[prop:fix\] The kernel of the map $$i^\dagger: \widetilde{H}^1_\mathrm{red}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}\rightarrow\widetilde{H}^1_D(U^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{\oplus 2}$$ is equal to the image of $\widetilde{\check{X}}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}$ in $\widetilde{H}^1_\mathrm{red}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}$.
Since all our spaces are ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$-torsion free, it suffices to show that the kernel of the map $$i^\dagger: H^1({\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}(U_p{\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace})\otimes \overline{k_{\ensuremath{\mathfrak{q}}\xspace}}, {\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}} \rightarrow H^1({\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}(U_pU^p)\otimes \overline{k_{\ensuremath{\mathfrak{q}}\xspace}}, {\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{\oplus 2}$$ is equal to the image of $\check{X}_{\ensuremath{\mathfrak{q}}\xspace}(U_p{\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}$ in $H^1({\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}(U_p{\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace})\otimes \overline{k_{\ensuremath{\mathfrak{q}}\xspace}}, {\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}$ for every $U_p$. In fact this just needs to be checked after tensoring everything with $E$. For brevity we write $U$ for $U_pU^p$ and $U({\ensuremath{\mathfrak{q}}\xspace})$ for $U_p{\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace}$. We complete the proof by relating our constructions to the Jacobians of Shimura curves. For simplicity we assume that our Shimura curves have just one connected component, but the argument is easily modified to work in the disconnected case by taking products of Jacobians of connected components. The degeneracy maps $\rho_1,\rho_{\eta_{\ensuremath{\mathfrak{q}}\xspace}}$ induce by Picard functoriality a map $\rho:\operatorname{Jac}(M(U)/F) \times \operatorname{Jac}(M(U)/F) \rightarrow \operatorname{Jac}(M(U({\ensuremath{\mathfrak{q}}\xspace}))/F)$ with finite kernel, giving the map $i$ on cohomology. The dual map $\rho^\dagger$ (induced by Albanese functoriality from the degeneracy maps) gives the map $i^\dagger$ on cohomology. Let $J^0$ denote the connected component of $0$ in the special fibre at ${\ensuremath{\mathfrak{q}}\xspace}$ of the Néron model of $\operatorname{Jac}(M(U({\ensuremath{\mathfrak{q}}\xspace}))/F)$. The map $\omega$ arises from a map $\Omega: J^0 \rightarrow \operatorname{Jac}({\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}(U)\otimes \overline{k_{\ensuremath{\mathfrak{q}}\xspace}}) \times \operatorname{Jac}({\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}(U)\otimes \overline{k_{\ensuremath{\mathfrak{q}}\xspace}})$, which identifies $\operatorname{Jac}({\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}(U)\otimes \overline{k_{\ensuremath{\mathfrak{q}}\xspace}}) \times \operatorname{Jac}({\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}(U)\otimes \overline{k_{\ensuremath{\mathfrak{q}}\xspace}})$ as the maximal Abelian variety quotient of $J^0$ (which is a semi-Abelian variety). The composition of $\Omega$ with the map induced by $\rho$ on the special fibre gives an auto-isogeny $A$ of $\operatorname{Jac}({\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}(U)\otimes \overline{k_{\ensuremath{\mathfrak{q}}\xspace}}) \times \operatorname{Jac}({\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}(U)\otimes \overline{k_{\ensuremath{\mathfrak{q}}\xspace}})$. The quasi-auto-isogeny $(\deg{\rho})^{-1}A$ gives an automorphism of $H^1({\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}(U_pU^p)\otimes \overline{k_{\ensuremath{\mathfrak{q}}\xspace}}, E)_{{\ensuremath{\overline{\rho}}\xspace}}^{\oplus 2}$. Denote the inverse of this automorphism by $\gamma$. Then the construction of the automorphism $\gamma$ implies that there is a commutative diagram $$\minCDarrowwidth10pt\begin{CD}H^1({\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}(U_p{\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace})\otimes \overline{k_{\ensuremath{\mathfrak{q}}\xspace}}, E)_{{\ensuremath{\overline{\rho}}\xspace}} @>{\omega}>>H^1({\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}(U_pU^p)\otimes \overline{k_{\ensuremath{\mathfrak{q}}\xspace}}, E)_{{\ensuremath{\overline{\rho}}\xspace}}^{\oplus 2}@.\\
@| @V{\gamma}VV \\
H^1({\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}(U_p{\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace})\otimes \overline{k_{\ensuremath{\mathfrak{q}}\xspace}}, E)_{{\ensuremath{\overline{\rho}}\xspace}} @>{i^\dagger}>>H^1({\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}(U_pU^p)\otimes \overline{k_{\ensuremath{\mathfrak{q}}\xspace}}, E)_{{\ensuremath{\overline{\rho}}\xspace}}^{\oplus 2}@.,\end{CD}$$ which identifies the kernel of $i^\dagger$ with the kernel of $\omega$.
\[prop:Xnew\] A system of Hecke eigenvalues $\lambda: {\ensuremath{\mathbb{T}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace}) \rightarrow E'$ occurs in $$\widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{{\ensuremath{\mathfrak{q}}\xspace}\mhyphen\mathrm{new}}$$ if and only if it occurs in $\widetilde{X}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}$.
Proposition \[prop:es\] and Lemma \[oldred\], together with Definition \[def:new\] and Proposition \[prop:fix\] imply that we have the following commutative diagram of $\varpi$-adically admissible $G({\ensuremath{\mathbb{Q}}\xspace}_p)$ representations over ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$: $$\minCDarrowwidth10pt\begin{CD}0@>>>\widetilde{\check{X}}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}@>>>\widetilde{H}^1_{{\ensuremath{\mathrm{red}}\xspace}}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}} @>{\omega}>>\widetilde{H}^1_D(U^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{\oplus 2}@>>>0\\
@. @V{\alpha}VV @V{\beta}VV @V{\gamma}VV\\
0@>>>\widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{{\ensuremath{\mathfrak{q}}\xspace}\mhyphen\mathrm{new}}@>>>\widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}} @>{i^\dagger}>>\widetilde{H}^1_D(U^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{\oplus 2},\end{CD}$$ where the two rows are exact and the maps $\alpha$, $\beta$ and $\gamma$ are injections. Note that both the existence of the map $\gamma$ and its injectivity follow from Proposition \[prop:fix\]. Moreover $${\ensuremath{\mathrm{coker}}\xspace}(\beta)=\widetilde{X}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}(-1) \cong \widetilde{\check{X}}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}(-1),$$ where the second isomorphism comes from the monodromy pairing (see Lemma \[monodromy\]). Applying the snake lemma we see that ${\ensuremath{\mathrm{coker}}\xspace}(\alpha)\hookrightarrow{\ensuremath{\mathrm{coker}}\xspace}(\beta)$, so (applying Lemma \[monodromy\] once more to the source of the map $\alpha$) we have an exact sequence $$\minCDarrowwidth10pt\begin{CD}0@>>>\widetilde{X}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}} @>\alpha>>\widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{{\ensuremath{\mathfrak{q}}\xspace}\mhyphen\mathrm{new}} @>>>\widetilde{X}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}(-1).\end{CD}$$ From this exact sequence it is easy to deduce that the systems of Hecke eigenvalues occurring in $\widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{{\ensuremath{\mathfrak{q}}\xspace}\mhyphen\mathrm{new}}$ and $\widetilde{X}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}$ are the same.
\[mazprin\] Let $\rho: \operatorname{Gal}(\overline{F}/F) \rightarrow {\ensuremath{\mathrm{GL}}\xspace}_2(\overline{{\ensuremath{\mathbb{Q}}\xspace}}_p)$ be a Galois representation, with irreducible reduction $\overline{\rho}: \operatorname{Gal}(\overline{F}/F) \rightarrow {\ensuremath{\mathrm{GL}}\xspace}_2(\overline{{\ensuremath{\mathbb{F}}\xspace}}_p)$. Suppose the following two conditions are verified:
1. $\rho$ is unramified at the prime ${\ensuremath{\mathfrak{q}}\xspace}$
2. $\rho(\mathrm{Frob}_{\ensuremath{\mathfrak{q}}\xspace})$ is not a scalar
3. There is a system of Hecke eigenvalues $\lambda:{\ensuremath{\mathbb{T}}\xspace}(U^p)_{{\ensuremath{\overline{\rho}}\xspace}} \rightarrow E'$ attached to $\rho$ (i.e. $\rho \cong \rho(\ker(\lambda))$), with $\lambda$ occurring in $\widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}).$
Then $\lambda$ occurs in $\widetilde{H}^1_D(U^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})$.
We take $\rho$ as in the statement of the theorem. The third assumption, combined with Proposition \[galfac\] implies that there is a prime ideal $\mathscr{P}=\ker(\lambda)$ of ${\ensuremath{\mathbb{T}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace})$, and an embedding $$\rho \hookrightarrow \widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})[\mathscr{P}].$$ Our aim is to show that $\widetilde{H}^1_D(U^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})[\mathscr{P}]\ne 0$.
Let ${\ensuremath{\mathbb{T}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace})^*$ denote the finite ring extension of ${\ensuremath{\mathbb{T}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace})$ obtained by adjoining the operator $\mathsf{U}_{\ensuremath{\mathfrak{q}}\xspace}$. Since $\rho$ is irreducible there is a prime $\mathscr{P}^*$ of ${\ensuremath{\mathbb{T}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace})^*$ such that there is an embedding $$\rho \hookrightarrow \widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})[\mathscr{P}^*].$$
Recall that by Propositions \[prop:es\] and \[unramred\] and Lemma \[oldred\] there is a $G_{\ensuremath{\mathfrak{q}}\xspace}$ and ${\ensuremath{\mathbb{T}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace})$-equivariant short exact sequence $$\minCDarrowwidth10pt\begin{CD}0@>>>\widetilde{\check{X}}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}@>>>\widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{I_{\ensuremath{\mathfrak{q}}\xspace}} @>{\omega}>>\widetilde{H}^1_D(U^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{\oplus 2}@>>>0.\end{CD}$$ Suppose for a contradiction that $\widetilde{H}^1_D(U^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}[\mathscr{P}]=0$
Let $V$ denote the $G_{\ensuremath{\mathfrak{q}}\xspace}$-representation obtained from $\rho$ by restriction. Since $\rho$ is unramified at ${\ensuremath{\mathfrak{q}}\xspace}$, there is an embedding $V \hookrightarrow \widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})^{I_{\ensuremath{\mathfrak{q}}\xspace}}_{{\ensuremath{\overline{\rho}}\xspace}}[\mathscr{P}^*]$. The above short exact sequence then implies that the embedding $$\widetilde{\check{X}}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}[\mathscr{P}]\hookrightarrow \widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{I_{\ensuremath{\mathfrak{q}}\xspace}}[\mathscr{P}]$$ is an isomorphism, and hence the embedding $$\widetilde{\check{X}}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}[\mathscr{P}^*]\hookrightarrow \widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{I_{\ensuremath{\mathfrak{q}}\xspace}}[\mathscr{P}^*]$$ is also an isomorphism.
This implies that there is an embedding $V \hookrightarrow \widetilde{\check{X}}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}[\mathscr{P}^*]$, so by Lemma \[frobaction\] $\rho(\mathrm{Frob}_{\ensuremath{\mathfrak{q}}\xspace})$ is a scalar $(\mathbf{N}{\ensuremath{\mathfrak{q}}\xspace})\alpha$ where $\alpha$ is the $\mathsf{U}_{\ensuremath{\mathfrak{q}}\xspace}$ eigenvalue coming from $\mathscr{P}^*$, contradicting our second assumption.
Mazur’s principle and the Jacquet functor
-----------------------------------------
After taking locally analytic vectors and applying Emerton’s locally analytic Jacquet functor (see [@MR2292633]) we can prove a variant of Theorem \[mazprin\]. We let $B$ denote the Borel subgroup of $G({\ensuremath{\mathbb{Q}}\xspace}_p)$ consisting of upper triangular matrices, and let $T$ denote the maximal torus contained in $B$. We then have a locally analytic Jacquet functor $J_B$ as defined in [@MR2292633], which can be applied to the space of locally analytic vectors $(V)^{an}$ of an admissible continuous Banach $E[G({\ensuremath{\mathbb{Q}}\xspace}_p)]$-representation $V$. The following lemma follows from Proposition \[galfac\], but we give a more elementary separate proof.
\[galembed\] Let $\lambda:{\ensuremath{\mathbb{T}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace})\rightarrow E$ be a system of Hecke eigenvalues such that $\lambda$ occurs in $J_B(\widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},E)^{an})$. Suppose the attached Galois representation $\rho$ is absolutely irreducible.Then there is a non-zero $\operatorname{Gal}(\overline{F}/F)$-equivariant map (necessarily an embedding, since $\rho$ is irreducible) $$\rho \hookrightarrow J_B(\widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},E)^{an}).$$
To abbreviate notation we let $M$ denote $J_B(\widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},E)^{an})$. Since $M$ is an essentially admissible $T$ representation, the system of Hecke eigenvalues $\lambda$ occurs in the $\chi$-isotypic subspace $M^\chi$ for some continuous character $\chi$ of $T$. So (again by essential admissibility) $M^{\chi,\lambda}$ is a non-zero finite dimensional $\operatorname{Gal}(\overline{F}/F)$-representation over $E$, and therefore by the Eichler-Shimura relations (as in section 10.3 of [@carmauv]) and the irreducibility of $\rho$ the desired embedding exists.
\[mazprinjacquet\] Suppose that the system of Hecke eigenvalues $\lambda:{\ensuremath{\mathbb{T}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace})\rightarrow E$ occurs in $J_B(\widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},E)^{an})$, that the attached Galois representation $$\rho : \operatorname{Gal}(\overline{F}/F) \rightarrow {\ensuremath{\mathrm{GL}}\xspace}_2(E)$$ is unramified at ${\ensuremath{\mathfrak{q}}\xspace}$, with $\rho(Frob_{\ensuremath{\mathfrak{q}}\xspace})$ not a scalar, and that the reduction $$\overline{\rho}: \operatorname{Gal}(\overline{F}/F) \rightarrow {\ensuremath{\mathrm{GL}}\xspace}_2(\overline{{\ensuremath{\mathbb{F}}\xspace}}_p)$$ is irreducible. Then $\lambda$ occurs in $J_B(\widetilde{H}^1_D(U^p,E)^{an})$.
We take $\rho$ as in the statement of the theorem. Let ${\ensuremath{\mathfrak{m}}\xspace}$ be the maximal ideal of ${\ensuremath{\mathbb{T}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace})$ attached to $\overline{\rho}$, and let $\mathscr{P}$ be the prime ideal of ${\ensuremath{\mathbb{T}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace})$ attached to $\rho$, so applying Lemma \[galembed\] gives an embedding $$\rho \hookrightarrow J_B(\widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},E)^{an}_{{\ensuremath{\overline{\rho}}\xspace}})[\mathscr{P}].$$ Our aim is to show that $J_B(\widetilde{H}^1_D(U^p,E)^{an}_{{\ensuremath{\overline{\rho}}\xspace}})[\mathscr{P}]\ne 0$. Let ${\ensuremath{\mathbb{T}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace})^*$ denote the finite ring extension of ${\ensuremath{\mathbb{T}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace})$ obtained by adjoining the operator $\mathsf{U}_{\ensuremath{\mathfrak{q}}\xspace}$. Since $\rho$ is irreducible there is a prime $\mathscr{P}^*$ of ${\ensuremath{\mathbb{T}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace})^*$ such that there is an embedding $$\rho \hookrightarrow J_B(\widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},E)^{an}_{{\ensuremath{\overline{\rho}}\xspace}})[\mathscr{P}^*].$$ By Propositions \[prop:es\] and \[unramred\] and Lemma \[oldred\] there is a short exact sequence $$\minCDarrowwidth10pt\begin{CD}0@>>>\widetilde{\check{X}}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},E)_{{\ensuremath{\overline{\rho}}\xspace}}@>>>\widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},E)_{{\ensuremath{\overline{\rho}}\xspace}}^{I_{\ensuremath{\mathfrak{q}}\xspace}} @>>>\widetilde{H}^1_D(U^p,E)_{{\ensuremath{\overline{\rho}}\xspace}}^2@>>>0.\end{CD}$$ Taking locally analytic vectors (which is exact) and applying the Jacquet functor (which is left exact), we get an exact sequence $$\minCDarrowwidth10pt\begin{CD}0@>>>J_B(\widetilde{\check{X}}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},E)_{{\ensuremath{\overline{\rho}}\xspace}}^{an})@>>>J_B((\widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},E)^{I_{\ensuremath{\mathfrak{q}}\xspace}}_{{\ensuremath{\overline{\rho}}\xspace}})^{an}) @>{\omega}>>J_B(\widetilde{H}^1_D(U^p,E)_{{\ensuremath{\overline{\rho}}\xspace}}^{an})^2.\end{CD}$$
Note that $J_B((\widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},E)^{I_{\ensuremath{\mathfrak{q}}\xspace}}_{{\ensuremath{\overline{\rho}}\xspace}})^{an})$ is equal to $J_B(\widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},E)^{an}_{{\ensuremath{\overline{\rho}}\xspace}})^{I_{\ensuremath{\mathfrak{q}}\xspace}}$, since under the isomorphism of Proposition \[galfac\] they both correspond to $$(\rho^m_{{\ensuremath{\overline{\rho}}\xspace},{\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace}})^{I_{\ensuremath{\mathfrak{q}}\xspace}}\otimes_{{\ensuremath{\mathbb{T}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}} J_B(X_E^{an}).$$ Let $V$ denote the $G_{\ensuremath{\mathfrak{q}}\xspace}$-representation obtained from $\rho$ by restriction. Since $\rho$ is unramified at ${\ensuremath{\mathfrak{q}}\xspace}$, there is an embedding $V\hookrightarrow J_B((\widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},E)^{I_{\ensuremath{\mathfrak{q}}\xspace}}_{{\ensuremath{\overline{\rho}}\xspace}})^{an})[\mathscr{P}^*]$. Suppose for a contradiction that $J_B(\widetilde{H}^1_D(U^p,E)^{an}_{{\ensuremath{\overline{\rho}}\xspace}})[\mathscr{P}]=0$. The above exact sequence then implies that the embedding $$J_B(\widetilde{\check{X}}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},E)_{{\ensuremath{\overline{\rho}}\xspace}}^{an})[\mathscr{P}]\hookrightarrow J_B((\widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},E)^{I_{\ensuremath{\mathfrak{q}}\xspace}}_{{\ensuremath{\overline{\rho}}\xspace}})^{an}) [\mathscr{P}]$$ is an isomorphism, and hence the embedding $$J_B(\widetilde{\check{X}}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},E)_{{\ensuremath{\overline{\rho}}\xspace}}^{an})[\mathscr{P}^*]\hookrightarrow J_B((\widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},E)^{I_{\ensuremath{\mathfrak{q}}\xspace}}_{{\ensuremath{\overline{\rho}}\xspace}})^{an}) [\mathscr{P}^*]$$ is also an isomorphism.
This implies that there is an embedding $V \hookrightarrow J_B(\widetilde{\check{X}}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},E)_{{\ensuremath{\overline{\rho}}\xspace}}^{an})[\mathscr{P}^*]$, so by Lemma \[oldred\] $\rho(\mathrm{Frob}_{\ensuremath{\mathfrak{q}}\xspace})$ is a scalar $(\mathbf{N}{\ensuremath{\mathfrak{q}}\xspace})\alpha$ corresponding to the $(\mathbf{N}{\ensuremath{\mathfrak{q}}\xspace})\mathsf{U}_{\ensuremath{\mathfrak{q}}\xspace}$ eigenvalue in $\mathscr{P}^*$. We then obtain a contradiction exactly as in the proof of Theorem \[mazprin\].
A $p$-adic Jacquet–Langlands correspondence {#pJL}
-------------------------------------------
We now wish to relate the arithmetic of $p$-adic automorphic forms for $G$ and $G'$. The definitions and results of subection \[subsec:ccoho\] apply to $G'$. We denote the associated completed cohomology spaces of tame level $V^p$ by $$\widetilde{H}^1_{D'}(V^p,{\ensuremath{\mathscr{F}}\xspace}_k),$$ and its Hecke algebra by ${\ensuremath{\mathbb{T}}\xspace}'(V^p)$. The integral models ${\ensuremath{\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}}\xspace}'$ and the vanishing cycle formalism also allows us to define some more topological ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$-modules:
\[Ycompletedstuff\] We define ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$-modules with smooth $G({\ensuremath{\mathbb{Q}}\xspace}_p)$ actions $$Y_{\ensuremath{\mathfrak{q}}\xspace}(V^p,{\ensuremath{\mathscr{F}_{k}}\xspace})=\varinjlim_{V_p}Y_{\ensuremath{\mathfrak{q}}\xspace}(V_pV^p,{\ensuremath{\mathscr{F}_{k}}\xspace}),$$ and $$\check{Y}_{\ensuremath{\mathfrak{q}}\xspace}(V^p,{\ensuremath{\mathscr{F}_{k}}\xspace})=\varinjlim_{V_p}\check{Y}_{\ensuremath{\mathfrak{q}}\xspace}(V_pV^p,{\ensuremath{\mathscr{F}_{k}}\xspace}).$$
We define ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$-modules with continuous $G({\ensuremath{\mathbb{Q}}\xspace}_p)$ actions $$\widetilde{Y}_{\ensuremath{\mathfrak{q}}\xspace}(V^p,{\ensuremath{\mathscr{F}_{k}}\xspace}) = \varprojlim_s Y(V^p,{\ensuremath{\mathscr{F}_{k}}\xspace})/\varpi^sY(V^p,{\ensuremath{\mathscr{F}_{k}}\xspace})$$ and $$\widetilde{\check{Y}}_{\ensuremath{\mathfrak{q}}\xspace}(V^p,{\ensuremath{\mathscr{F}_{k}}\xspace}) = \varprojlim_s \check{Y}_{\ensuremath{\mathfrak{q}}\xspace}(V^p,{\ensuremath{\mathscr{F}_{k}}\xspace})/\varpi^s\check{Y}_{\ensuremath{\mathfrak{q}}\xspace}(V^p,{\ensuremath{\mathscr{F}_{k}}\xspace}).$$
Fixing a non-Eisenstein maximal ideal ${\ensuremath{\mathfrak{m}}\xspace}'$ of ${\ensuremath{\mathbb{T}}\xspace}'(V^p)$, then in exactly the same way as we obtained Proposition \[prop:es\] we get:
\[prop:Yses\] We have a short exact sequence of $\varpi$-adically admissible $G({\ensuremath{\mathbb{Q}}\xspace}_p)$ representations over ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$, equivariant with respect to ${\ensuremath{\mathbb{T}}\xspace}'(V^p)_{{\ensuremath{\mathfrak{m}}\xspace}'}$, $G({\ensuremath{\mathbb{Q}}\xspace}_p)$ and $G_{\ensuremath{\mathfrak{q}}\xspace}$ actions: $$\minCDarrowwidth10pt\begin{CD}0@>>>\widetilde{\check{Y}}_{\ensuremath{\mathfrak{q}}\xspace}(V^p,{\ensuremath{\mathscr{F}_{k}}\xspace})_{{\ensuremath{\mathfrak{m}}\xspace}'}@>>>\widetilde{H}^1_{D'}(V^p,{\ensuremath{\mathscr{F}_{k}}\xspace})_{{\ensuremath{\mathfrak{m}}\xspace}'}@>>>\widetilde{Y}_{\ensuremath{\mathfrak{q}}\xspace}(V^p,{\ensuremath{\mathscr{F}_{k}}\xspace})_{{\ensuremath{\mathfrak{m}}\xspace}'}(-1)@>>>0.\end{CD}$$
Fix $U^p \subset G({\ensuremath{\mathbb{A}}\xspace}_f^{(p)})$ such that $U^p$ is unramified at the places ${\ensuremath{{\mathfrak{q}_1}}\xspace},{\ensuremath{{\mathfrak{q}_2}}\xspace}$, and matches with $V^p$ at all other places. We now fix an irreducible mod $p$ Galois representation ${\ensuremath{\overline{\rho}}\xspace}$, arising from a (necessarily non-Eisenstein) maximal ideal ${\ensuremath{\mathfrak{m}}\xspace}'$ of ${\ensuremath{\mathbb{T}}\xspace}'(V^p)$. There is a corresponding maximal ideal ${\ensuremath{\mathfrak{m}}\xspace}'_0$ of ${\ensuremath{\mathbb{T}}\xspace}'_{(2,...,2)}(V^pV_p)$ for $V_p$ a small enough compact open subgroup of $G'({\ensuremath{\mathbb{Q}}\xspace}_p)$, and this pulls back by the map ${\ensuremath{\mathbb{T}}\xspace}_{(2,...,2)}(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace})U_p) \rightarrow {\ensuremath{\mathbb{T}}\xspace}'_{(2,...,2)}(V^pV_p)$ to a maximal ideal ${\ensuremath{\mathfrak{m}}\xspace}_0$ of ${\ensuremath{\mathbb{T}}\xspace}_{(2,...,2)}(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace})U_p)$ for $U_p$ the compact open subgroup of $G({\ensuremath{\mathbb{Q}}\xspace}_p)$ which is identified with $V_p$ under the isomorphism $G({\ensuremath{\mathbb{Q}}\xspace}_p) \cong G'({\ensuremath{\mathbb{Q}}\xspace}_p)$. Hence there is a map ${\ensuremath{\mathbb{T}}\xspace}(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}))_{{\ensuremath{\overline{\rho}}\xspace}} \rightarrow {\ensuremath{\mathbb{T}}\xspace}'(V^p)_{{\ensuremath{\overline{\rho}}\xspace}}$.
We can give a refinement of Proposition \[galfac\] in this situation. Firstly we note that the Galois representation $\rho_{{\ensuremath{\overline{\rho}}\xspace},V^p}^m:G_F \rightarrow {\ensuremath{\mathrm{GL}}\xspace}_2({\ensuremath{\mathbb{T}}\xspace}'(V^p)_{{\ensuremath{\overline{\rho}}\xspace}}\otimes_{{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}}E)$ has an unramified sub-$G_{\ensuremath{\mathfrak{q}}\xspace}$ representation $V_{0,E}$ (which is a free ${\ensuremath{\mathbb{T}}\xspace}'(V^p)_{{\ensuremath{\overline{\rho}}\xspace}}\otimes_{{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}}E$-module of rank one), with unramified quotient $V_{1,E}\cong V_{0,E}(-1)$ (also a free ${\ensuremath{\mathbb{T}}\xspace}'(V^p)_{{\ensuremath{\overline{\rho}}\xspace}}\otimes_{{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}}E$-module of rank one). This follows from the fact that each representation $\rho:G_F\rightarrow {\ensuremath{\mathrm{GL}}\xspace}_2({\ensuremath{\mathbb{T}}\xspace}'_{(2,...,2)}(V^pV_p)_{{\ensuremath{\overline{\rho}}\xspace}}\otimes_{{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}}E)$ has a unique unramified subrepresentation of the right kind. Denote by $Y$ the ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$-module $\operatorname{Hom}_{{\ensuremath{\mathbb{T}}\xspace}'(V^p)_{{\ensuremath{\overline{\rho}}\xspace}}[G_F]}(\rho_{{\ensuremath{\overline{\rho}}\xspace},V^p}^m,\widetilde{H}^1_{D'}(V^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}})$. By proposition \[galfac\] we have an isomorphism $$ev_E: E\otimes_{{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}}\rho_{{\ensuremath{\overline{\rho}}\xspace},V^p}^m \otimes_{{\ensuremath{\mathbb{T}}\xspace}'(V^p)_{{\ensuremath{\overline{\rho}}\xspace}}} Y \cong \widetilde{H}^1_{D'}(V^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}.$$
\[galfacY\] The isomorphism $ev_E$ identifies the image of the embedding $$\widetilde{\check{Y}}_{\ensuremath{\mathfrak{q}}\xspace}(V^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace},E}\rightarrow\widetilde{H}^1_{D'}(V^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace},E}$$ with $V_{0,E} \otimes_{{\ensuremath{\mathbb{T}}\xspace}'(V^p)_{{\ensuremath{\overline{\rho}}\xspace}}} Y$. It follows that the quotient $\widetilde{Y}_{\ensuremath{\mathfrak{q}}\xspace}(V^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace},E}(-1)$ is identified with $V_{1,E} \otimes_{{\ensuremath{\mathbb{T}}\xspace}'(V^p)_{{\ensuremath{\overline{\rho}}\xspace}}} Y$.
We proceed as in the proof of proposition \[galfac\]. First let $\mathscr{P} \in \operatorname{Spec}({\ensuremath{\mathbb{T}}\xspace}'(V^p)_{{\ensuremath{\overline{\rho}}\xspace}}\otimes_{{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}}E)$ be the prime ideal of the Hecke algebra corresponding to a classical point coming from a Hecke eigenform in $H^1(M'(V^pV_p),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}\otimes_{{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}}k(\mathscr{P})$. Taking $V_p$ invariants, extending scalars to $k(\mathscr{P})$ and taking the $\mathscr{P}$ torsion parts, the isomorphism $ev_E$ induces an isomorphism $$\rho(\mathscr{P}) \otimes_{k(\mathscr{P})}\operatorname{Hom}_{k(\mathscr{P})[G_F]}(\rho(\mathscr{P}),H^1(M'(V^pV_p),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace},E}[\mathscr{P}]) \cong H^1(M'(V^pV_p),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace},E}[\mathscr{P}],$$ under which $\check{Y}_{\ensuremath{\mathfrak{q}}\xspace}(V^pV_p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace},E}[\mathscr{P}]$ is identified with $$V_{0,E}(\mathscr{P})\otimes_{k(\mathscr{P})}\operatorname{Hom}_{k(\mathscr{P})[G_F]}(\rho(\mathscr{P}),H^1(M'(V^pV_p),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace},E}[\mathscr{P}]).$$A density argument now completes the proof.
Passing to the direct limit over compact open subgroups $U_p$ of $G({\ensuremath{\mathbb{Q}}\xspace}_p)$ and $\varpi$-adically completing the short exact sequences of Theorem \[RRESclass\], we obtain
\[RRES\] We have the following short exact sequences of $\varpi$-adically admissible $G({\ensuremath{\mathbb{Q}}\xspace}_p)$-representations over ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$, equivariant with respect to ${\ensuremath{\mathbb{T}}\xspace}(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}))_{{\ensuremath{\overline{\rho}}\xspace}}$ and $G({\ensuremath{\mathbb{Q}}\xspace}_p)\cong G'({\ensuremath{\mathbb{Q}}\xspace}_p)$ actions: $$\minCDarrowwidth10pt\begin{CD}0@>>>\widetilde{Y}_{\ensuremath{{\mathfrak{q}_1}}\xspace}(V^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}} @>>>\widetilde{X}_{\ensuremath{{\mathfrak{q}_2}}\xspace}(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}@>{i^\dagger}>>\widetilde{X}_{\ensuremath{{\mathfrak{q}_2}}\xspace}(U^p({\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})^2_{{\ensuremath{\overline{\rho}}\xspace}}@>>>0,\end{CD}$$ $$\minCDarrowwidth10pt\begin{CD}0@>>>\widetilde{\check{X}}_{\ensuremath{{\mathfrak{q}_2}}\xspace}(U^p({\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})^2_{{\ensuremath{\overline{\rho}}\xspace}}@>>>\widetilde{\check{X}}_{\ensuremath{{\mathfrak{q}_2}}\xspace}(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}} @>>>\widetilde{\check{Y}}_{\ensuremath{{\mathfrak{q}_1}}\xspace}(V^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}@>>>0.\end{CD}$$
The proof is as for Proposition \[prop:es\]. We just need to check that the short exact sequences of Theorem \[RRESclass\] are compatible as $U_p$ and $V_p$ vary, but this is clear from the proof of [@Raj Theorem 3], since the descriptions of the dual graph of the special fibres of $\mathbb{M}_{\ensuremath{{\mathfrak{q}_2}}\xspace}(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace})U_p)$ and $\mathbb{M}'_{\ensuremath{{\mathfrak{q}_1}}\xspace}(V^pV_p)$ are compatible as $U_p$ and $V_p$ vary.
We regard Theorem \[RRES\] (especially the first exact sequence therein) as a geometric realisation of a $p$-adic Jacquet–Langlands correspondence. The following propositions explain why this is reasonable. We let $\widetilde{X}_{\ensuremath{{\mathfrak{q}_2}}\xspace}(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{{\ensuremath{{\mathfrak{q}_1}}\xspace}\mhyphen\mathrm{new}}$ denote the kernel of the map $$i^\dagger: \widetilde{X}_{\ensuremath{{\mathfrak{q}_2}}\xspace}(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}\rightarrow \widetilde{X}_{\ensuremath{{\mathfrak{q}_2}}\xspace}(U^p({\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})^{\oplus 2}_{{\ensuremath{\overline{\rho}}\xspace}}.$$
\[prop:Xdoublenew\] A system of Hecke eigenvalues $\lambda: {\ensuremath{\mathbb{T}}\xspace}(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace})) \rightarrow E'$ occurs in $\widetilde{H}^1_D(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{{\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}\mhyphen\mathrm{new}}$ if and only if it occurs in $\widetilde{X}_{\ensuremath{{\mathfrak{q}_2}}\xspace}(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{{\ensuremath{{\mathfrak{q}_1}}\xspace}\mhyphen\mathrm{new}}$.
We have the following commutative diagram: $$\minCDarrowwidth20pt\begin{CD}0@. 0@. \\
@VVV @VVV \\
\widetilde{X}_{\ensuremath{{\mathfrak{q}_2}}\xspace}(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{{\ensuremath{{\mathfrak{q}_1}}\xspace}\mhyphen\mathrm{new}} @>{\alpha}>>\widetilde{H}^1_D(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{{\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}\mhyphen\mathrm{new}} @.\\
@VVV @VVV \\
\widetilde{X}_{\ensuremath{{\mathfrak{q}_2}}\xspace}(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}} @>{\beta}>> \widetilde{H}^1_D(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})^{{\ensuremath{{\mathfrak{q}_2}}\xspace}\mhyphen\mathrm{new}}_{{\ensuremath{\overline{\rho}}\xspace}} @.\\
@VVV @VVV \\
\widetilde{X}_{\ensuremath{{\mathfrak{q}_2}}\xspace}(U^p({\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})^{\oplus 2}_{{\ensuremath{\overline{\rho}}\xspace}} @>{\gamma}>> (\widetilde{H}^1_D(U^p({\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{{\ensuremath{{\mathfrak{q}_2}}\xspace}\mhyphen\mathrm{new}})^{\oplus 2} @.\\
@VVV @. \\
0@. @.\end{CD}$$ where the two columns are exact, and the maps $\alpha$, $\beta$ and $\gamma$ are injections. Note that we are applying Lemma \[monodromy\]. By the snake lemma we have an exact sequence $$\minCDarrowwidth15pt\begin{CD}0@>>> {\ensuremath{\mathrm{coker}}\xspace}(\alpha) @>>> {\ensuremath{\mathrm{coker}}\xspace}{\beta} @>>> {\ensuremath{\mathrm{coker}}\xspace}{\gamma},\end{CD}$$ and the proof of Proposition \[prop:Xnew\] implies that there are injections $$\begin{CD}@.{\ensuremath{\mathrm{coker}}\xspace}{\beta} \hookrightarrow \widetilde{X}_{\ensuremath{{\mathfrak{q}_2}}\xspace}(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}(-1),\\@.{\ensuremath{\mathrm{coker}}\xspace}{\gamma} \hookrightarrow \widetilde{X}_{\ensuremath{{\mathfrak{q}_2}}\xspace}(U^p({\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})^{\oplus 2}_{{\ensuremath{\overline{\rho}}\xspace}}(-1).\end{CD}$$ This shows that there is a Hecke-equivariant injection $${\ensuremath{\mathrm{coker}}\xspace}(\alpha)\hookrightarrow\widetilde{X}_{\ensuremath{{\mathfrak{q}_2}}\xspace}(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{{\ensuremath{{\mathfrak{q}_1}}\xspace}\mhyphen\mathrm{new}}(-1).$$ Therefore, we have an exact sequence $$\minCDarrowwidth15pt\begin{CD}0@>>>\widetilde{X}_{\ensuremath{{\mathfrak{q}_2}}\xspace}(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{{\ensuremath{{\mathfrak{q}_1}}\xspace}\mhyphen\mathrm{new}}@>\alpha>>\widetilde{H}^1_D(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{{\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}\mhyphen\mathrm{new}} \\@. @>>>\widetilde{X}_{\ensuremath{{\mathfrak{q}_2}}\xspace}(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{{\ensuremath{{\mathfrak{q}_1}}\xspace}\mhyphen\mathrm{new}}(-1).\end{CD}$$ From this last exact sequence it is easy to deduce that the systems of Hecke eigenvalues occurring in $\widetilde{H}^1_D(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{{\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}\mhyphen\mathrm{new}}$ and $\widetilde{X}_{\ensuremath{{\mathfrak{q}_2}}\xspace}(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{{\ensuremath{{\mathfrak{q}_1}}\xspace}\mhyphen\mathrm{new}}$ are the same.
A system of Hecke eigenvalues $\lambda: {\ensuremath{\mathbb{T}}\xspace}'(V^p) \rightarrow E'$ occurs in\
$\widetilde{H}_{D'}^1(V^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}$ if and only if it occurs in $\widetilde{Y}_{\ensuremath{{\mathfrak{q}_1}}\xspace}(V^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}$.
This follows from Proposition \[galfacY\].
Combined with Theorem \[RRES\] this has as a consequence
The first exact sequence of Theorem \[RRES\] induces a bijection between the systems of eigenvalues for ${\ensuremath{\mathbb{T}}\xspace}'(V^p)$ appearing in $\widetilde{H}_{D'}^1(V^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}$ and the systems of eigenvalues for ${\ensuremath{\mathbb{T}}\xspace}(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}))$ appearing in $\widetilde{H}^1_D(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{{\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}\mhyphen\mathrm{new}}$.
Eigenvarieties and an overconvergent Jacquet–Langlands correspondence
=====================================================================
In this section we will apply Emerton’s construction of eigenvarieties to deduce an overconvergent Jacquet–Langlands correspondence from Theorem \[RRES\].
Locally algebraic vectors {#localg}
-------------------------
Fix a maximal ideal ${\ensuremath{\mathfrak{m}}\xspace}$ of ${\ensuremath{\mathbb{T}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace})$ which is not Eisenstein. Recall that given an admissible $G({\ensuremath{\mathbb{Q}}\xspace}_p)$-representation $V$ over ${\ensuremath{E}}$, Schneider and Teitelbaum [@STei] define an exact functor by passing to the locally ${\ensuremath{\mathbb{Q}}\xspace}_p$-analytic vectors $V^{an}$, and the $G({\ensuremath{\mathbb{Q}}\xspace}_p)$-action on $V^{an}$ differentiates to give an action of the Lie algebra ${\ensuremath{\mathfrak{g}}\xspace}$ of $G({\ensuremath{\mathbb{Q}}\xspace}_p)$.
There are natural isomorphisms (for ${\ensuremath{\mathscr{F}}\xspace}= {\ensuremath{\mathscr{F}_{k}}\xspace}$ or ${\ensuremath{r_*r^*\mathscr{F}_{k}}\xspace}$)
1. $H^0({\ensuremath{\mathfrak{g}}\xspace},\widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})^{an}_{{\ensuremath{\mathfrak{m}}\xspace},{\ensuremath{E}}})\cong H^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})_{{\ensuremath{\mathfrak{m}}\xspace},{\ensuremath{E}}},$
2. $H^1({\ensuremath{\mathfrak{g}}\xspace},\widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})^{an}_{{\ensuremath{\mathfrak{m}}\xspace},{\ensuremath{E}}})\cong 0,$
3. $H^0({\ensuremath{\mathfrak{g}}\xspace},\widetilde{H}^1_{\ensuremath{\mathrm{red}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}}\xspace})^{an}_{{\ensuremath{\mathfrak{m}}\xspace},{\ensuremath{E}}})\cong H^1_{\ensuremath{\mathrm{red}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}}\xspace})_{{\ensuremath{\mathfrak{m}}\xspace},{\ensuremath{E}}},$
4. $H^1({\ensuremath{\mathfrak{g}}\xspace},\widetilde{H}^1_{\ensuremath{\mathrm{red}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}}\xspace})^{an}_{{\ensuremath{\mathfrak{m}}\xspace},{\ensuremath{E}}})\cong 0,$
5. $H^0({\ensuremath{\mathfrak{g}}\xspace},\widetilde{X}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})^{an}_{{\ensuremath{\mathfrak{m}}\xspace},{\ensuremath{E}}})\cong X_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})_{{\ensuremath{\mathfrak{m}}\xspace},{\ensuremath{E}}},$
6. $H^0({\ensuremath{\mathfrak{g}}\xspace},\widetilde{\check{X}}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})^{an}_{{\ensuremath{\mathfrak{m}}\xspace},{\ensuremath{E}}})\cong \check{X}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})_{{\ensuremath{\mathfrak{m}}\xspace},{\ensuremath{E}}},$
The first and second natural isomorphisms comes from the spectral sequence (for completed étale cohomology) discussed in [@Emint Proposition 2.4.1], since localising at ${\ensuremath{\mathfrak{m}}\xspace}$ kills $H^i_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})$ and $\widetilde{H}^i_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})$ for $i=0,2$.
The same argument works for the third and fourth isomorphisms, since as noted in the proof of [@Emint Proposition 2.4.1] the spectral sequence of [@Emint Corollary 2.2.18] can be obtained using the Hochschild-Serre spectral sequence for étale cohomology, which applies equally well to the cohomology of the special fibre of $\mathbb{M}_{\ensuremath{\mathfrak{q}}\xspace}(U_p{\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace})$.
We now turn to the fifth isomorphism. By the first short exact sequence of Proposition \[prop:es\] and the exactness of localising at ${\ensuremath{\mathfrak{m}}\xspace}$, inverting $\varpi$ and then passing to locally analytic vectors, there is a short exact sequence $$\minCDarrowwidth20pt\begin{CD}0@>>>\widetilde{H}^1_{{\ensuremath{\mathrm{red}}\xspace}}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})^{an}_{{\ensuremath{\mathfrak{m}}\xspace},{\ensuremath{E}}} @>>>\widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})^{an}_{{\ensuremath{\mathfrak{m}}\xspace},{\ensuremath{E}}}\\@.@>>>\widetilde{X}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})(-1)^{an}_{{\ensuremath{\mathfrak{m}}\xspace},{\ensuremath{E}}}@>>>0.\end{CD}$$ Taking the long exact sequence of ${\ensuremath{\mathfrak{g}}\xspace}$-cohomology then gives an exact sequence $$\xymatrix{0\ar[r]&H^0({\ensuremath{\mathfrak{g}}\xspace},\widetilde{H}^1_{\ensuremath{\mathrm{red}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})^{an}_{{\ensuremath{\mathfrak{m}}\xspace},{\ensuremath{E}}}) \ar[r]&H^0({\ensuremath{\mathfrak{g}}\xspace},\widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})^{an}_{{\ensuremath{\mathfrak{m}}\xspace},{\ensuremath{E}}})\\\ar[r]&H^0({\ensuremath{\mathfrak{g}}\xspace},\widetilde{X}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})(-1)^{an}_{{\ensuremath{\mathfrak{m}}\xspace},{\ensuremath{E}}})\ar[r]&H^1({\ensuremath{\mathfrak{g}}\xspace},\widetilde{H}^1_{\ensuremath{\mathrm{red}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})^{an}_{{\ensuremath{\mathfrak{m}}\xspace},{\ensuremath{E}}}),}$$ so applying the third, first and fourth part of the proposition to the first, second and fourth terms in this sequence respectively we get a short exact sequence $$\minCDarrowwidth20pt\begin{CD}0@>>> H^1_{{\ensuremath{\mathrm{red}}\xspace}}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})_{{\ensuremath{\mathfrak{m}}\xspace},{\ensuremath{E}}} @>>>H^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})_{{\ensuremath{\mathfrak{m}}\xspace},{\ensuremath{E}}}\\@. @>>>H^0({\ensuremath{\mathfrak{g}}\xspace},\widetilde{X}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})(-1)^{an}_{{\ensuremath{\mathfrak{m}}\xspace},{\ensuremath{E}}})@>>>0,\end{CD}$$ so identifying this with the short exact sequence of Proposition 2.3.2 gives the desired isomorphism $$H^0({\ensuremath{\mathfrak{g}}\xspace},\widetilde{X}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})^{an}_{{\ensuremath{\mathfrak{m}}\xspace},{\ensuremath{E}}})\cong X_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})_{{\ensuremath{\mathfrak{m}}\xspace},{\ensuremath{E}}}.$$
Finally the sixth isomorphism follows from the first four in the same way as the fifth (using the second short exact sequence of Proposition \[prop:es\].
Given a continuous $G({\ensuremath{\mathbb{Q}}\xspace}_p)$-representation $V$ we denote by $V^{alg}$ the space of locally algebraic vectors in $V$. Let $E(1)$ denote a one dimensional $E$-vector space on which $G({\ensuremath{\mathbb{Q}}\xspace}_p) \times \operatorname{Gal}(\overline{L}/L)$ acts by $\prod_{v\mid p}N_{F_v/{\ensuremath{\mathbb{Q}}\xspace}_p}\circ \det \otimes \epsilon$, where $\epsilon$ is the unramified character of $\operatorname{Gal}(\overline{L}/L)$ sending $Frob_\lambda$ to $N_{F/{\ensuremath{\mathbb{Q}}\xspace}}(\lambda)$. For any $n \in {\ensuremath{\mathbb{Z}}\xspace}$ let $E(n)=E(1)^{\otimes n}$. We can now proceed as in Theorem 7.4.2 of [@Emlgc] to deduce
There are natural $G({\ensuremath{\mathbb{Q}}\xspace}_p) \times G_{\ensuremath{\mathfrak{q}}\xspace}\times {\ensuremath{\mathbb{T}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace})$-equivariant isomorphisms
1. $\bigoplus_{k,n\in {\ensuremath{\mathbb{Z}}\xspace}} H^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})_{{\ensuremath{\mathfrak{m}}\xspace},{\ensuremath{E}}}\otimes_{{\ensuremath{E}}} W_{k,{\ensuremath{\mathfrak{p}}\xspace}}^\vee \otimes_{{\ensuremath{E}}} {\ensuremath{E}}(n) \cong \widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})^{alg}_{{\ensuremath{\mathfrak{m}}\xspace},{\ensuremath{E}}}.$
2. $\bigoplus_{k,n\in {\ensuremath{\mathbb{Z}}\xspace}} H^1_{\ensuremath{\mathrm{red}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})_{{\ensuremath{\mathfrak{m}}\xspace},{\ensuremath{E}}}\otimes_{{\ensuremath{E}}} W_{k,{\ensuremath{\mathfrak{p}}\xspace}}^\vee \otimes_{{\ensuremath{E}}} {\ensuremath{E}}(n) \cong \widetilde{H}^1_{\ensuremath{\mathrm{red}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})^{alg}_{{\ensuremath{\mathfrak{m}}\xspace},{\ensuremath{E}}}.$
3. $\bigoplus_{k,n\in {\ensuremath{\mathbb{Z}}\xspace}} H^1_{\ensuremath{\mathrm{red}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{r_*r^*\mathscr{F}_{k}}\xspace})_{{\ensuremath{\mathfrak{m}}\xspace},{\ensuremath{E}}}\otimes_{{\ensuremath{E}}} W_{k,{\ensuremath{\mathfrak{p}}\xspace}}^\vee \otimes_{{\ensuremath{E}}} {\ensuremath{E}}(n) \cong \widetilde{H}^1_{\ensuremath{\mathrm{red}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},r_*r^*{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})^{alg}_{{\ensuremath{\mathfrak{m}}\xspace},{\ensuremath{E}}}.$
4. $\bigoplus_{k,n\in {\ensuremath{\mathbb{Z}}\xspace}} X_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})_{{\ensuremath{\mathfrak{m}}\xspace},{\ensuremath{E}}}\otimes_{{\ensuremath{E}}} W_{k,{\ensuremath{\mathfrak{p}}\xspace}}^\vee \otimes_{{\ensuremath{E}}} {\ensuremath{E}}(n) \cong \widetilde{X}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})^{alg}_{{\ensuremath{\mathfrak{m}}\xspace},{\ensuremath{E}}}.$
5. $\bigoplus_{k,n\in {\ensuremath{\mathbb{Z}}\xspace}} \check{X}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathscr{F}_{k}}\xspace})_{{\ensuremath{\mathfrak{m}}\xspace},{\ensuremath{E}}}\otimes_{{\ensuremath{E}}} W_{k,{\ensuremath{\mathfrak{p}}\xspace}}^\vee \otimes_{{\ensuremath{E}}} {\ensuremath{E}}(n) \cong \widetilde{\check{X}}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})^{alg}_{{\ensuremath{\mathfrak{m}}\xspace},{\ensuremath{E}}}.$
In all the above direct sums, the index $k$ runs over $d$-tuples with all entries the same parity and $\ge 2$.
The proof is as for Theorem 7.4.2 of [@Emlgc], since $$\{(\xi^{(k)})^\vee \otimes (\otimes_{i=1}^d \det \circ \xi_i)^n\}_{k,n}$$ is a complete set of isomorphism class representatives of irreducible algebraic representations of $G$ which factor through $G^c$.
Applying the same arguments to $G'$, with ${\ensuremath{\mathfrak{m}}\xspace}'$ a non-Eisenstein maximal ideal of ${\ensuremath{\mathbb{T}}\xspace}'(V^p)$ we also obtain
\[Ylocalg\] There are natural $G'({\ensuremath{\mathbb{Q}}\xspace}_p) \times G_{\ensuremath{\mathfrak{q}}\xspace}\times {\ensuremath{\mathbb{T}}\xspace}'(V^p)$-equivariant isomorphisms
1. $\bigoplus_{k,n\in {\ensuremath{\mathbb{Z}}\xspace}} H^1_{D'}(V^p,{\ensuremath{\mathscr{F}_{k}}\xspace})_{{\ensuremath{\mathfrak{m}}\xspace}',{\ensuremath{E}}}\otimes_{{\ensuremath{E}}} W_{k,{\ensuremath{\mathfrak{p}}\xspace}}^\vee \otimes_{{\ensuremath{E}}} {\ensuremath{E}}(n) \cong \widetilde{H}^1_{D'}(V^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})^{alg}_{{\ensuremath{\mathfrak{m}}\xspace}',{\ensuremath{E}}}.$
2. $\bigoplus_{k,n\in {\ensuremath{\mathbb{Z}}\xspace}} Y_{\ensuremath{\mathfrak{q}}\xspace}(V^p,{\ensuremath{\mathscr{F}_{k}}\xspace})_{{\ensuremath{\mathfrak{m}}\xspace}',{\ensuremath{E}}}\otimes_{{\ensuremath{E}}} W_{k,{\ensuremath{\mathfrak{p}}\xspace}}^\vee \otimes_{{\ensuremath{E}}} {\ensuremath{E}}(n) \cong \widetilde{Y}_{\ensuremath{\mathfrak{q}}\xspace}(V^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})^{alg}_{{\ensuremath{\mathfrak{m}}\xspace}',{\ensuremath{E}}}.$
3. $\bigoplus_{k,n\in {\ensuremath{\mathbb{Z}}\xspace}} \check{Y}_{\ensuremath{\mathfrak{q}}\xspace}(V^p,{\ensuremath{\mathscr{F}_{k}}\xspace})_{{\ensuremath{\mathfrak{m}}\xspace}',{\ensuremath{E}}}\otimes_{{\ensuremath{E}}} W_{k,{\ensuremath{\mathfrak{p}}\xspace}}^\vee \otimes_{{\ensuremath{E}}} {\ensuremath{E}}(n) \cong \widetilde{\check{Y}_{\ensuremath{\mathfrak{q}}\xspace}}(V^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})^{alg}_{{\ensuremath{\mathfrak{m}}\xspace}',{\ensuremath{E}}}.$
Cofreeness results
------------------
We now give analogues of Corollary 5.3.19 in [@Emlg].
\[prop:injsmooth\] Choose $U_p$ small enough so that $U_p$ is pro-$p$ and $U_p{\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace}$ is neat. Then for each $s > 0$,
- $H^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}/\varpi^s)_{\ensuremath{\mathfrak{m}}\xspace}$,
- $H^1_{\ensuremath{\mathrm{red}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}/\varpi^s)_{\ensuremath{\mathfrak{m}}\xspace}$,
- $H^1_{\ensuremath{\mathrm{red}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},r_*r^* {\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}/\varpi^s)_{\ensuremath{\mathfrak{m}}\xspace}$,
are injective as smooth representations of ${\ensuremath{U_p/\overline{F^{\times} \cap U_pU^p}}\xspace}$ over ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}/\varpi^s$.
In the statement of the proposition, we write ${\ensuremath{U_p/\overline{F^{\times} \cap U_pU^p}}\xspace}$ to denote the quotient of $U_p$ by the closure in $U_p$ of the factor at $p$ of $F^\times \cap U_pU^p$.
We follow the proof of Proposition 5.3.15 in [@Emlg], with modifications due to the presence of infinitely many global units ${\ensuremath{\mathcal{O}}\xspace}_F^\times$. Let $L$ be any finitely generated smooth representation of ${\ensuremath{U_p/\overline{F^{\times} \cap U_pU^p}}\xspace}=U_p/\overline{F^\times\cap U_pU^p({\ensuremath{\mathfrak{q}}\xspace})}$ over ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}/\varpi^s$, with Pontriagin dual $L^\vee := \operatorname{Hom}_{{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}/\varpi^s}(L,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}/\varpi^s)$. For brevity we denote ${\ensuremath{U_p/\overline{F^{\times} \cap U_pU^p}}\xspace}$ by $\overline{U_p}$. There is an induced local system $\mathscr{L}^\vee$ on each of the Shimura curves $M(U'_p{\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace})$ as $U'_p$ varies over the open normal subgroups of $U_p$. As in Proposition 5.3.15 of [@Emlg] there is a natural isomorphism $$H^1(M(U_p{\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace}),\mathscr{L}^\vee)_{\ensuremath{\mathfrak{m}}\xspace}\cong \operatorname{Hom}_{\overline{U_p}}(L,H^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}/\varpi^s)_{\ensuremath{\mathfrak{m}}\xspace}).$$ Now starting from a short exact sequence $0\rightarrow L_0 \rightarrow L_1 \rightarrow L_2\rightarrow 0$ of finitely generated smooth $\overline{U_p}$-representations over ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}/\varpi^s$ we obtain a short exact sequence of sheaves on $M(U_p{\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace})$: $$\minCDarrowwidth10pt\begin{CD}0 @>>> \mathscr{L}^\vee_2 @>>> \mathscr{L}^\vee_1 @>>> \mathscr{L}^\vee_0 @>>> 0.\end{CD}$$ Taking the associated long exact cohomology sequence and localising at ${\ensuremath{\mathfrak{m}}\xspace}$ gives another short exact sequence $$\minCDarrowwidth10pt\begin{CD}0 @>>> H^1(M(U_p{\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace}),\mathscr{L}^\vee_2)_{\ensuremath{\mathfrak{m}}\xspace}@>>> H^1(M(U_p{\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace}),\mathscr{L}^\vee_1)_{\ensuremath{\mathfrak{m}}\xspace}\\@.@>>> H^1(M(U_p{\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace}),\mathscr{L}^\vee_0)_{\ensuremath{\mathfrak{m}}\xspace}@>>> 0,\end{CD}$$ or equivalently $$\minCDarrowwidth10pt\begin{CD}0 @>>> \operatorname{Hom}_{\overline{U_p}}(L_2,H^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}/\varpi^s)_{\ensuremath{\mathfrak{m}}\xspace}) @>>> \operatorname{Hom}_{\overline{U_p}}(L_1,H^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}/\varpi^s)_{\ensuremath{\mathfrak{m}}\xspace}) \\ @.@>>>\operatorname{Hom}_{\overline{U_p}}(L_0,H^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}/\varpi^s)_{\ensuremath{\mathfrak{m}}\xspace}) @>>> 0.\end{CD}$$Therefore we conclude that $H^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}/\varpi^s)_{\ensuremath{\mathfrak{m}}\xspace}$ is injective as a smooth representation of $\overline{U_p}$ over ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}/\varpi^s$. The same argument applies to the other cohomology spaces (using Lemma \[oldred\] for the final case).
We say that a topological representation $V$ of a topological group $\Gamma$ over ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$ is *cofree* if there is a topological isomorphism of representations $V \cong \mathscr{C}(\Gamma,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})^r$ for some integer $r$, where $\mathscr{C}(\Gamma,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})$ denotes the space of continuous functions from $\Gamma$ to ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$ with the right regular action of $\Gamma$.
\[cor:cofree1\] If $U_p$ is small enough so that $U_p$ is pro-$p$ and $U_p{\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace}$ is neat, then $\widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{\ensuremath{\mathfrak{m}}\xspace}$, $\widetilde{H}^1_{\ensuremath{\mathrm{red}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{\ensuremath{\mathfrak{m}}\xspace}$ and $\widetilde{H}^1_{\ensuremath{\mathrm{red}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},r_*r^*{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{\ensuremath{\mathfrak{m}}\xspace}$ are all cofree representations of ${\ensuremath{U_p/\overline{F^{\times} \cap U_pU^p}}\xspace}$.
The proof of Corollary 5.3.19 in [@Emlg] goes through.
\[cor:cofree2\] If $U_p$ is small enough so that $U_p$ is pro-$p$ and $U_p{\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace}$ is neat, then $\widetilde{X}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{\ensuremath{\mathfrak{m}}\xspace}$ is a cofree representation of ${\ensuremath{U_p/\overline{F^{\times} \cap U_pU^p}}\xspace}$.
Taking the first exact sequence of Proposition \[prop:es\] and applying the functor $\operatorname{Hom}_{cts}(-,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})$ gives a short exact sequence (we can ignore the Tate twist for the purposes of this Corollary) $$\begin{aligned}
0\rightarrow \operatorname{Hom}_{cts}(\widetilde{X}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{\ensuremath{\mathfrak{m}}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}) &\rightarrow \operatorname{Hom}_{cts}(\widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{\ensuremath{\mathfrak{m}}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})\\ &\rightarrow \operatorname{Hom}_{cts}(\widetilde{H}^1_{{\ensuremath{\mathrm{red}}\xspace}}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{\ensuremath{\mathfrak{m}}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})\rightarrow 0.\end{aligned}$$ Corollary \[cor:cofree1\] implies that the second and third non-zero terms in this sequence are free ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}[[{\ensuremath{U_p/\overline{F^{\times} \cap U_pU^p}}\xspace}]]$-modules, so $\operatorname{Hom}_{cts}(\widetilde{X}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{\ensuremath{\mathfrak{m}}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})$ is projective, hence free since ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}[[{\ensuremath{U_p/\overline{F^{\times} \cap U_pU^p}}\xspace}]]$ is local. Applying the functor $\operatorname{Hom}_{cts}(-,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})$ again gives the desired result.
Similarly we obtain
\[cor:cofree3\] If $V_p$ is small enough so that $V_p$ is pro-$p$ and $V_pV^p$ is neat, then
- $\widetilde{H}^1_{D'}(V^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\mathfrak{m}}\xspace}'}$,
- $\widetilde{Y_{\ensuremath{\mathfrak{q}}\xspace}}(V^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\mathfrak{m}}\xspace}'}$,
- $\widetilde{\check{Y}_{\ensuremath{\mathfrak{q}}\xspace}}(V^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\mathfrak{m}}\xspace}'}$,
are cofree representations of ${\ensuremath{V_p/\overline{F^{\times} \cap V_pV^p}}\xspace}$.
Jacquet-Langlands maps between eigenvarieties
---------------------------------------------
We can now apply the results of section \[pJL\] to deduce some cases of *overconvergent* Jacquet-Langlands functoriality, or in other words maps between eigenvarieties interpolating the classical Jacquet-Langlands correspondence. Let $\widehat{T}$ denote the rigid analytic variety parametrising continuous characters $\chi: T \rightarrow {\ensuremath{\mathbb{C}}\xspace}_p^\times$.
\[def:evar\] Suppose $V$ is a $\varpi$-adically admissible $G({\ensuremath{\mathbb{Q}}\xspace}_p)$-representation over ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$, with a commuting ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$-linear action of a commutative Noetherian ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$-algebra $\mathbf{A}$. The essentially admissible locally analytic $T$-representation $J_B(V_E^{an})$ gives rise (by duality) to a coherent sheaf $\mathscr{M}$ on $\widehat{T}$, and the action of $\mathbf{A}$ gives rise to a coherent sheaf on $\widehat{T}$ of ${\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}$-algebras $\mathscr{A} \hookrightarrow End_{\widehat{T}}(\mathscr{M})$. Define the *Eigenvariety* $$\mathscr{E}(V,\mathbf{A})$$ to be the rigid analytic space given by taking the relative spectrum of $\mathscr{A}$ over $\widehat{T}$.
Suppose $V$ and $\mathbf{A}$ are as in Definition \[def:evar\], further suppose that $V$ is a (non-zero) cofree representation of $U_p/X$ for some compact open subgroup $U_p$ of $G({\ensuremath{\mathbb{Q}}\xspace}_p)$, $X$ some closed subgroup of $Z({\ensuremath{\mathbb{Q}}\xspace}_p) \cap U_p$. Denote by $T_0$ the intersection $T \cap U_p$. Then $\mathscr{E}(V,\mathbf{A})$ is equidimensional of dimension equal to the dimension of the rigid analytic variety $\widehat{T_0/X}$ parametrising continuous characters of $T_0/X$ which are trivial on $X$.
This is a combination of a mild generalisation of the proof of [@MR2292633 Proposition 4.2.36] and [@cclr Corollary 4.1]. We will outline the argument. The cofreeness assumption on $V$ implies that $V_E^{an}$ is isomorphic to $\mathscr{C}^{an}(U_p/X,E)^r$, where $\mathscr{C}^{an}(U_p/X,E)$ denotes the space of locally analytic functions from $U_p/X$ to $E$. We write $M$ for the strong dual of $J_B(V_E^{an})$, which is equal to the space of global sections $\mathscr{M}(\widehat{T})$. $M$ is a module for the Fréchet algebra of locally analytic functions $\mathscr{C}^{an}(\widehat{T_0/X},E)$. We then write $\widehat{T_0/X}$ as a union of admissible affinoid subdomains $\mathrm{MaxSpec}(A_n)$ and apply the argument of [@cclr Corollary 4.1] to $M\widehat{\otimes}_{\mathscr{C}^{an}(\widehat{T_0/X},E)} A_n$ which is the finite slope part of an orthonormalisable Banach $A_n$-module with respect to some compact operator.
In our applications we will have $X$ equal to the closure in $U_p$ of the $p$-factor of $F^\times \cap U_pU^p$ for some tame level $U^p$. The dimension of $\widehat{T_0/X}$ will therefore depend on the Leopoldt conjecture for $F$. To be precise, it will equal $$\dim(\widehat{T_0})- (d-1 - \delta)=d+1+\delta,$$ where $\delta$ is the defect in Leopoldt’s conjecture for $F$. Applying the above lemma to the cofreeness results of the previous section, we have the following:
\[lotsequidim\] The following eigenvarieties are all equidimensional of dimension equal to $d+1+\delta$:
- $\mathscr{E}(\widetilde{H}^1_D(U^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{\ensuremath{\mathfrak{m}}\xspace},{\ensuremath{\mathbb{T}}\xspace}(U^p)_{\ensuremath{\mathfrak{m}}\xspace})$
- $\mathscr{E}(\widetilde{X}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{\ensuremath{\mathfrak{m}}\xspace},{\ensuremath{\mathbb{T}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace})_{\ensuremath{\mathfrak{m}}\xspace})$
- $\mathscr{E}(\widetilde{H}_{D'}^1(V^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\mathfrak{m}}\xspace}'},{\ensuremath{\mathbb{T}}\xspace}'(V^p)_{{\ensuremath{\mathfrak{m}}\xspace}'})$
- $\mathscr{E}(\widetilde{Y}_{\ensuremath{\mathfrak{q}}\xspace}(V^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\mathfrak{m}}\xspace}'},{\ensuremath{\mathbb{T}}\xspace}'(V^p)_{{\ensuremath{\mathfrak{m}}\xspace}'})$
- $\mathscr{E}(\widetilde{\check{Y}}_{\ensuremath{\mathfrak{q}}\xspace}(V^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\mathfrak{m}}\xspace}'},{\ensuremath{\mathbb{T}}\xspace}'(V^p)_{{\ensuremath{\mathfrak{m}}\xspace}'})$
\[lnew\] The map $\widetilde{X}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}) \rightarrow \widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})$ induced by composing the (monodromy pairing) embedding $\widetilde{X}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}) \rightarrow \widetilde{\check{X}}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})$ with the embedding $\widetilde{\check{X}}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}) \rightarrow \widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})$ provided by Proposition \[prop:es\] induces an isomorphism of eigenvarieties $$\mathscr{E}(\widetilde{X}_{\ensuremath{\mathfrak{q}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{\ensuremath{\mathfrak{m}}\xspace},{\ensuremath{\mathbb{T}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace})_{\ensuremath{\mathfrak{m}}\xspace}) \cong \mathscr{E}(\widetilde{H}^1_D({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})^{{\ensuremath{\mathfrak{q}}\xspace}\mhyphen\mathrm{new}}_{\ensuremath{\mathfrak{m}}\xspace},{\ensuremath{\mathbb{T}}\xspace}({\ensuremath{U^p({\ensuremath{\mathfrak{q}}\xspace})}\xspace})_{\ensuremath{\mathfrak{m}}\xspace}).$$
This follows from Proposition \[prop:Xnew\].
We have a similar result for the group $G'$.
\[lemmaYH1\] The map $\widetilde{Y}_{\ensuremath{\mathfrak{q}}\xspace}(V^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}) \rightarrow \widetilde{H}^1_{D'}(V^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})$ induced by composing the (monodromy pairing) embedding $\widetilde{Y}_{\ensuremath{\mathfrak{q}}\xspace}(V^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}) \rightarrow \widetilde{\check{Y}}_{\ensuremath{\mathfrak{q}}\xspace}(V^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})$ with the embedding\
$\widetilde{\check{Y}}_{\ensuremath{\mathfrak{q}}\xspace}(V^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}) \rightarrow \widetilde{H}_{D'}^1(V^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})$ provided by Proposition \[prop:Yses\] induces an isomorphism between eigenvarieties $$\mathscr{E}(\widetilde{Y}_{\ensuremath{\mathfrak{q}}\xspace}(V^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\mathfrak{m}}\xspace}'},{\ensuremath{\mathbb{T}}\xspace}'(V^p)_{{\ensuremath{\mathfrak{m}}\xspace}'}) \cong \mathscr{E}(\widetilde{H}_{D'}^1(V^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\mathfrak{m}}\xspace}'},{\ensuremath{\mathbb{T}}\xspace}'(V^p)_{{\ensuremath{\mathfrak{m}}\xspace}'}).$$
This follows from Proposition \[galfacY\].
\[lemmaXnew\] The map $\widetilde{X}_{\ensuremath{{\mathfrak{q}_2}}\xspace}(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{\ensuremath{\mathfrak{m}}\xspace}^{{\ensuremath{{\mathfrak{q}_1}}\xspace}\mhyphen\mathrm{new}}\rightarrow \widetilde{H}^1_D(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{\ensuremath{\mathfrak{m}}\xspace}^{{\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}\mhyphen\mathrm{new}}$ induces an isomorphism between eigenvarieties $$\mathscr{E}(\widetilde{X}_{\ensuremath{{\mathfrak{q}_2}}\xspace}(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{\ensuremath{\mathfrak{m}}\xspace}^{{\ensuremath{{\mathfrak{q}_1}}\xspace}\mhyphen\mathrm{new}},{\ensuremath{\mathbb{T}}\xspace}(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}))_{\ensuremath{\mathfrak{m}}\xspace})\cong \mathscr{E}(\widetilde{H}^1_D(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{\ensuremath{\mathfrak{m}}\xspace}^{{\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}\mhyphen\mathrm{new}},{\ensuremath{\mathbb{T}}\xspace}(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}))_{\ensuremath{\mathfrak{m}}\xspace}).$$
This follows from Proposition \[prop:Xdoublenew\].
We now put ourselves in the situation of section \[pJL\], so $U^p$ and $V^p$ are isomorphic at places away from ${\ensuremath{{\mathfrak{q}_1}}\xspace}$ and ${\ensuremath{{\mathfrak{q}_2}}\xspace}$, and ${\ensuremath{\mathfrak{m}}\xspace}$ and ${\ensuremath{\mathfrak{m}}\xspace}'$ give rise to the same irreducible mod $p$ Galois representation ${\ensuremath{\overline{\rho}}\xspace}$.
*(Overconvergent Jacquet-Langlands correspondence)*\[ocjl\] The map $$\widetilde{Y}_{\ensuremath{{\mathfrak{q}_1}}\xspace}(V^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}} \rightarrow \widetilde{X}_{\ensuremath{{\mathfrak{q}_2}}\xspace}(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}$$ given by Theorem \[RRES\] induces an isomorphism between eigenvarieties $$\mathscr{E}(\widetilde{H}^1_{D'}(V^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}},{\ensuremath{\mathbb{T}}\xspace}'(V^p)_{{\ensuremath{\overline{\rho}}\xspace}}) \cong \mathscr{E}(\widetilde{H}^1_D(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{{\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}\mhyphen\mathrm{new}},{\ensuremath{\mathbb{T}}\xspace}(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}))_{{\ensuremath{\overline{\rho}}\xspace}}).$$
It is clear that Theorem \[RRES\] induces an isomorphism $\mathscr{E}(\widetilde{Y}_{\ensuremath{{\mathfrak{q}_1}}\xspace}(V^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}} ,{\ensuremath{\mathbb{T}}\xspace}'(V^p)_{{\ensuremath{\overline{\rho}}\xspace}}) \cong \mathscr{E}(\widetilde{X}_{\ensuremath{{\mathfrak{q}_2}}\xspace}(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{{\ensuremath{{\mathfrak{q}_1}}\xspace}\mhyphen\mathrm{new}},{\ensuremath{\mathbb{T}}\xspace}(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}))_{{\ensuremath{\overline{\rho}}\xspace}})$. Our theorem now follows by applying Lemmas \[lemmaYH1\] and \[lemmaXnew\].
Finally we note that level raising results in the same spirit as those in [@chicomp; @cclr] follow from the equidimensionality of the eigenvarieties described in this section. In particular we have
\[weirdlr\] The following eigenvarieties are equidimensional (of dimension $d+1+\delta$):
- $\mathscr{E}(\widetilde{H}^1_{D}(U^p({\ensuremath{\mathfrak{q}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{{\ensuremath{\mathfrak{q}}\xspace}\mhyphen\mathrm{new}},{\ensuremath{\mathbb{T}}\xspace}(U^p({\ensuremath{\mathfrak{q}}\xspace}))_{{\ensuremath{\overline{\rho}}\xspace}})$
- $\mathscr{E}(\widetilde{H}^1_D(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{{\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}\mhyphen\mathrm{new}},{\ensuremath{\mathbb{T}}\xspace}(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}))_{{\ensuremath{\overline{\rho}}\xspace}})$
The first claim follows from the second part of Corollary \[lotsequidim\] and Lemma \[lnew\]. The second claim follows from the fourth part of Corollary \[lotsequidim\] and Theorem \[ocjl\].
Recall the natural map $$i_1:\check{X}_{\ensuremath{{\mathfrak{q}_2}}\xspace}(U({\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}\rightarrow\check{X}_{\ensuremath{{\mathfrak{q}_2}}\xspace}(U({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}$$ appearing in Section \[sec:RR\], which is an injection by Theorem \[RRES\]. This induces an embedding $i_1$ from $$\mathscr{E}(\widetilde{H}^1_{D}(U^p({\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{{\ensuremath{{\mathfrak{q}_2}}\xspace}\mhyphen\mathrm{new}},{\ensuremath{\mathbb{T}}\xspace}(U^p({\ensuremath{{\mathfrak{q}_2}}\xspace}))_{{\ensuremath{\overline{\rho}}\xspace}})$$ to $$\mathscr{E}(\widetilde{H}^1_D(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{{\ensuremath{{\mathfrak{q}_2}}\xspace}\mhyphen\mathrm{new}},{\ensuremath{\mathbb{T}}\xspace}(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}))_{{\ensuremath{\overline{\rho}}\xspace}}).$$ The following corollary characterises the intersection of the image of $i_1$ (the ${\ensuremath{{\mathfrak{q}_1}}\xspace}$-old points of the eigenvariety) with $$\mathscr{E}(\widetilde{H}^1_{D}(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{{\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}\mhyphen\mathrm{new}},{\ensuremath{\mathbb{T}}\xspace}(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}))_{{\ensuremath{\overline{\rho}}\xspace}}).$$
\[levelraising\] Suppose we have a point $$x \in \mathscr{E}(\widetilde{H}^1_{D}(U^p({\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{{\ensuremath{{\mathfrak{q}_2}}\xspace}\mhyphen\mathrm{new}},{\ensuremath{\mathbb{T}}\xspace}(U^p({\ensuremath{{\mathfrak{q}_2}}\xspace}))_{{\ensuremath{\overline{\rho}}\xspace}}).$$ Then $$i_1(x) \in \mathscr{E}(\widetilde{H}^1_{D}(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{{\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}\mhyphen\mathrm{new}},{\ensuremath{\mathbb{T}}\xspace}(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}))_{{\ensuremath{\overline{\rho}}\xspace}})$$ if and only if the $T_{\ensuremath{{\mathfrak{q}_1}}\xspace}$ and $S_{\ensuremath{{\mathfrak{q}_1}}\xspace}$ eigenvalues of $x$ satisfy $$T_{\ensuremath{{\mathfrak{q}_1}}\xspace}(x)^2-(\mathbf{N}{\ensuremath{{\mathfrak{q}_1}}\xspace}+1)^2S_{\ensuremath{{\mathfrak{q}_1}}\xspace}(x)=0.$$
Note that if the conditions of this Corollary are satisfied, then Corollary \[weirdlr\] implies that $i_1(x)$ lies in an equidimensional family of newforms, with ‘full’ dimension.
A standard calculation shows that the composite $i^\dagger\circ i$ acts by the matrix $$\begin{pmatrix}
\mathbf{N}{\ensuremath{{\mathfrak{q}_1}}\xspace}+1 & T_{\ensuremath{{\mathfrak{q}_1}}\xspace}\\ S_{\ensuremath{{\mathfrak{q}_1}}\xspace}^{-1}T_{\ensuremath{{\mathfrak{q}_1}}\xspace}& \mathbf{N}{\ensuremath{{\mathfrak{q}_1}}\xspace}+1
\end{pmatrix}$$ on $(\check{X}_{\ensuremath{{\mathfrak{q}_2}}\xspace}(U({\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace},E}^{an})^{\oplus 2}$. Write $\widetilde{x}$ for a class in $\check{X}_{\ensuremath{{\mathfrak{q}_2}}\xspace}(U({\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace},E'}^{an}$ giving rise to the point $x$ (where $E'/E$ is a field over which $x$ is defined) . Now if the criterion $T_{\ensuremath{{\mathfrak{q}_1}}\xspace}(x)^2-(\mathbf{N}{\ensuremath{{\mathfrak{q}_1}}\xspace}+1)^2S_{\ensuremath{{\mathfrak{q}_1}}\xspace}(x)=0$ is satisfied there is a non-zero element $\widetilde{y}$ of $$E'\cdot \widetilde{x} \oplus E'\cdot \widetilde{x} \subset (\check{X}_{\ensuremath{{\mathfrak{q}_2}}\xspace}(U({\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace},E'}^{an})^{\oplus 2}$$ in the kernel of $i^\dagger\circ i$ — namely, $$((\mathbf{N}{\ensuremath{{\mathfrak{q}_1}}\xspace}+1)S_{\ensuremath{{\mathfrak{q}_1}}\xspace}(x)\cdot \widetilde{x},-T_{\ensuremath{{\mathfrak{q}_1}}\xspace}(x)\cdot \widetilde{x}).$$ The element $i(\widetilde{y})$ then gives rise to the point $$i_1(x)\in\mathscr{E}(\check{X}_{\ensuremath{{\mathfrak{q}_2}}\xspace}(U({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{{\ensuremath{{\mathfrak{q}_1}}\xspace}\mhyphen\mathrm{new}},{\ensuremath{\mathbb{T}}\xspace}(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}))_{{\ensuremath{\overline{\rho}}\xspace}}),$$ so we are done by Lemmas \[monodromy\] and \[lemmaXnew\]. Conversely if $$i_1(x) \in \mathscr{E}(\widetilde{H}^1_{D}(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{{\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}\mhyphen\mathrm{new}},{\ensuremath{\mathbb{T}}\xspace}(U^p({\ensuremath{{\mathfrak{q}_1}}\xspace}{\ensuremath{{\mathfrak{q}_2}}\xspace}))_{{\ensuremath{\overline{\rho}}\xspace}})$$ then the local Galois representation of $G_{\ensuremath{{\mathfrak{q}_1}}\xspace}$ attached to $x$ must be a sum of two unramified characters $\chi_1,\chi_2$ with $\chi_1(Frob_{\ensuremath{{\mathfrak{q}_1}}\xspace})/\chi_2(Frob_{\ensuremath{{\mathfrak{q}_1}}\xspace})=(\mathbf{N}{\ensuremath{{\mathfrak{q}_1}}\xspace})^{\pm 1}$, since it is unramified and $i_1(x)$ lies in a family of classical ${\ensuremath{{\mathfrak{q}_1}}\xspace}$-newforms. This implies that $T_{\ensuremath{{\mathfrak{q}_1}}\xspace}(x)^2-(\mathbf{N}{\ensuremath{{\mathfrak{q}_1}}\xspace}+1)^2S_{\ensuremath{{\mathfrak{q}_1}}\xspace}(x)=0$.
We would also like to prove a level raising result characterising the intersection of ${\ensuremath{\mathfrak{q}}\xspace}$-old and ${\ensuremath{\mathfrak{q}}\xspace}$-new forms with tame level $U^p({\ensuremath{\mathfrak{q}}\xspace})$. Corollary \[weirdlr\] would be enough to deduce such a result, provided one shows that the map $$i:(\widetilde{H}^1_{D}(U^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace},E}^{an})^{\oplus 2} \rightarrow \widetilde{H}^1_{D}(U^p({\ensuremath{\mathfrak{q}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace},E}^{an}$$ is an injection. This would obviously follow from injectivity of the map $$i:\widetilde{H}^1_{D}(U^p,{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}}^{\oplus 2} \rightarrow \widetilde{H}^1_{D}(U^p({\ensuremath{\mathfrak{q}}\xspace}),{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace})_{{\ensuremath{\overline{\rho}}\xspace}},$$ but this is equivalent to showing the injectivity of $$i:H^1(M(U_pU^p)_{\overline{F}},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}/\varpi)_{{\ensuremath{\overline{\rho}}\xspace}}^{\oplus 2} \rightarrow H^1(M(U_pU^p({\ensuremath{\mathfrak{q}}\xspace}))_{\overline{F}},{\ensuremath{\mathcal{O}}\xspace}_{\ensuremath{\mathfrak{p}}\xspace}/\varpi)_{{\ensuremath{\overline{\rho}}\xspace}}$$ for all compact open subgroups $U_p$ of $G({\ensuremath{\mathbb{Q}}\xspace}_p)$ — this version of Ihara’s lemma is not known in this generality (as far as the author is aware). The approach of [@DT] (which is extended to Shimura curves over totally real fields in [@Cheng-preprint]), via crystalline techniques, can work only when $U_p$ is maximal compact (since one needs $M(U_pU^p({\ensuremath{\mathfrak{q}}\xspace}))_{\overline{F}}$ to have good reduction at places dividing $p$).
Acknowledgements {#acknowledgements .unnumbered}
----------------
This paper is based on part of the author’s PhD thesis, written under the supervision of Kevin Buzzard, to whom I am grateful for providing such excellent guidance. This research crucially relies on work of Matthew Emerton, whom I also thank for several helpful conversations and communicating details of an earlier draft of [@Emlg]. The author was supported by an Engineering and Physical Sciences Research Council doctoral training grant during the bulk of the time spent researching the contents of this paper. The writing up process was completed whilst the author was a member of the Institute for Advanced Study, supported by National Science Foundation grant DMS-0635607. The author is currently supported by Trinity College, Cambridge. I thank all these institutions for their support.
[10]{}
K. Buzzard. On $p$-adic families of automorphic forms. In [*Modular Curves and Abelian Varieties*]{}, number 224 in Progress in Mathematics. Birkhauser, 2004.
K. Buzzard, F. Diamond, and F. Jarvis. On [S]{}erre’s conjecture for mod [$\ell$]{} [G]{}alois representations over totally real fields. , 155(1):105–161, 2010.
H. Carayol. Sur la mauvaise réduction des courbes de [S]{}himura. , 59(2):151–230, 1986.
H. Carayol. Sur les représentations [$l$]{}-adiques associées aux formes modulaires de [H]{}ilbert. , 19(3):409–468, 1986.
G. Chenevier. Une correspondance de [J]{}acquet-[L]{}anglands [$p$]{}-adique. , 126(1):161–194, 2005.
C. Cheng. On [I]{}hara’s lemma for [S]{}himura curves, 2010.
R. Coleman and B. Mazur. The eigencurve. In [*Galois representations in arithmetic algebraic geometry ([D]{}urham, 1996)*]{}, volume 254 of [*London Math. Soc. Lecture Note Ser.*]{}, pages 1–113. Cambridge Univ. Press, Cambridge, 1998.
F. Diamond and R. Taylor. Non-optimal levels of mod $l$ modular representations. , 115:435–462, 1994.
M. Emerton. Jacquet modules of locally analytic representations of [$p$]{}-adic reductive groups. [I]{}. [C]{}onstruction and first properties. , 39(5):775–839, 2006.
M. Emerton. A local-global compatibility conjecture in the [$p$]{}-adic [L]{}anglands programme for [${\rm GL}_{2/{\Bbb Q}}$]{}. , 2(2, part 2):279–393, 2006.
M. Emerton. On the interpolation of systems of eigenvalues attached to automorphic [H]{}ecke eigenforms. , 164(1):1–84, 2006.
M. Emerton. Local-global compatibility in the $p$-adic [L]{}anglands programme for $\mathrm{GL}_2/\mathbb{Q}$, 2010. Preprint at <http://www.math.northwestern.edu/~emerton/preprints.html>.
M. Emerton. Ordinary parts of admissible representations of $p$-adic reductive groups [I]{}. definition and first properties. , (331):355–402, 2010. Repr[é]{}sentations $p$-adiques de groupes $p$-adiques [III]{} : M[é]{}thodes globales et g[é]{}om[é]{}triques.
M. Harris, A. Iovita, and G. Stevens. The [J]{}acquet-[L]{}anglands correspondence via $l$-adic uniformization. n preparation.
F. Jarvis. Mazur’s principle for totally real fields of odd degree. , 116(1):39–79, 1999.
J. S. Milne. Canonical models of (mixed) [S]{}himura varieties and automorphic vector bundles. In [*Automorphic forms, [S]{}himura varieties, and [$L$]{}-functions, [V]{}ol. [I]{} ([A]{}nn [A]{}rbor, [MI]{}, 1988)*]{}, volume 10 of [*Perspect. Math.*]{}, pages 283–414. Academic Press, Boston, MA, 1990.
J. Newton. Geometric level raising for $p$-adic automorphic forms. , 147(2):335–354, 2011.
J. Newton. Level raising and completed cohomology. , 2011(11):2565–2576, 2011.
A. Paulin. Failure of the local to global principle in the eigencurve, 2011. To appear in Manuscripta Mathematica.
A. Rajaei. On the levels of mod [$l$]{} [H]{}ilbert modular forms. , 537:33–65, 2001.
K. A. Ribet. On modular representations of [${\rm Gal}(\overline{\bf Q}/{\bf Q})$]{} arising from modular forms. , 100(2):431–476, 1990.
T. Saito. Hilbert modular forms and [$p$]{}-adic [H]{}odge theory. , 145(5):1081–1113, 2009.
P. Schneider and J. Teitelbaum. Algebras of [$p$]{}-adic distributions and admissible representations. , 153(1):145–196, 2003.
Y. Varshavsky. -adic uniformization of unitary [S]{}himura varieties. [II]{}. , 49(1):75–113, 1998.
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---
abstract: 'We consider the infinite dimensional linear programming (inf-LP) approach for solving stochastic control problems. The inf-LP corresponding to problems with uncountable state and input spaces is in general computationally intractable. By focusing on linear systems with quadratic cost (LQG), we establish a connection between this approach and the well-known Riccati LMIs. In particular, we show that the semidefinite programs known for the LQG problem can be derived from the pair of primal and dual inf-LPs. Furthermore, we establish a connection between multi-objective and chance constraint criteria and the inf-LP formulation.'
address:
- 'Automatic Control Laboratory, ETH Zürich, Switzerland ([email protected])'
- 'Mechanical Engineering, University of Texas at Dallas ([email protected])'
author:
- Maryam Kamgarpour
- Tyler Summers
bibliography:
- 'references\_maryamka.bib'
title: On infinite dimensional linear programming approach to stochastic control
---
stochastic control, linear programming, semidefinite programming
Introduction
============
Optimal control of discrete time stochastic systems can be addressed via the dynamic programming (DP) [@bellman1957dp] principle of optimality. For an infinite horizon average or discounted cost problem, the optimal cost function and control policy can be computed as the fixed point of the so-called dynamic programming operator. In general, computing this fixed point is challenging and thus, several approximate approaches based on the DP principle of optimality have been developed.
An alternative approach to solving stochastic control problems is linear programming (LP) [@puterman2009markov; @hernandez1996discrete]. If the control and input spaces are uncountable, the corresponding LP is infinite dimensional (inf-LP). In the primal form of this LP, the optimization variable is the *occupation measure*, which measures infinite horizon occupancy of state and inputs in each Borel subset of the product state input space. An optimal policy may be derived from the optimal occupation measure, while the optimal value function is the optimizer of the dual of this LP.
In addition to providing an elegant alternative formulation of the optimality conditions for a stochastic control solution, in the LP approach constraints have a natural interpretation. By properly constraining the occupation measure, one can ensure probabilistic constraints on the state trajectory or can ensure bounds on multiple objectives. Such formulations of constrained stochastic control were considered in [@borkar1994ergodic; @feinberg1996constrained; @altman1999constrained; @hernandez2000constrained; @hernandez2003constrained].
The inf-LP formulation is in general computationally intractable. For problems with polynomial data, this inf-LP can be approximated via a sequence of semidefinite programs (SDPs) [@savorgnan2009discrete; @summers2013approximate]. These recent works are among the few that explore the inf-LP approach for computation of optimal value function and policies in a stochastic control problem.
The abstract inf-LP work has not attempted to establish clear connections with the well known, computationally tractable Linear Matrix Inequality (LMI) formulations of optimal control. In particular, for a stochastic linear system with quadratic cost (LQG), one can formulate the so-called Riccati LMI to find the optimal value function of the LQG problem [@boyd1994linear; @balakrishnan2003semidefinite]. Similarly, the well known LMI formulations have not attempted to show how these results can be derived from a more general approach to stochastic optimal control, namely the inf-LP approach.
In this work, we establish the connection between the inf-LP approach and the well-known Riccati LMIs for LQG problems. This inf-LP in general, includes infinitely many constraints on the occupation measure. The relaxation of these constraints to moments up to order two of the occupation measure and taking the dual of this problem results in the well-known Riccati LMI solution approaches. Since the variables in the relaxation of primal inf-LP are discounted moments of the state and input, moment constraints or certain class of chance constraints can be naturally encoded in the inf-LP formulation.
Our paper is organized as follows. In Section \[sec:problem\] we review the inf-LP approach to discrete-time infinite horizon discounted stochastic control. In Section \[sec:approach\] we apply the approach to LQG problems. In Section \[sec:numerical\] we provide numerical case studies. In Section \[sec:conclusion\] we summarize the results.
Inf-LP approach to Stochastic control {#sec:problem}
=====================================
Consider the discrete-time stochastic system $$\begin{aligned}
\label{eq:abstract_dynamics}
x_{t+1} \sim \tau(B_x|x_t,u_t),\end{aligned}$$ where $x_t \in X$, $u_t \in U$, and $\tau(.|x,u)$ is a stochastic kernel. It assigns a probability distribution to $B_x \in {\mathcal}{B}(X)$ given $x$ and $u$, where ${\mathcal}{B}(X)$ is the set of Borel subsets of $X$. The stochastic control problem is defined by $$\begin{aligned}
\label{opt:lp_abstract}
\min \limits_{\pi \in \Pi} \; & \mathbb{E}_{\nu_0}^\pi \sum_{t=0}^\infty \alpha^t c_0(x_t,u_t).\end{aligned}$$ Above, $c_0: X \times U \rightarrow {\mathbb{R}}_+$ is the running cost and $\alpha \in (0,1)$ is a discount factor, $\nu_0$ is an initial state distribution. We consider randomized policies $\pi \in \Pi$, where $\Pi$ is the set of probability measures on $U$ given $X$. That is, for each $x \in X$, $\pi(x)$ gives a probability distribution on the input space $U$. The expectation $\mathbb{E}$ is with respect to the probability measure induced by $\nu_0$, $\pi$ and $\tau$.
The solution to the stochastic control problem above can be characterized as the solution of an infinite dimensional linear program (inf-LP). To present this inf-LP, we first define the infinite dimensional optimization spaces for the primal and dual LPs. Define the weight functions $$\begin{aligned}
\label{eq:weights}
w(x,u) = {\epsilon}+c_0(x,u) , \; \tilde{w}(x) =\min_{u \in U} w(x,u), \end{aligned}$$ where ${\epsilon}> 0$ so that the weights are bounded away from zero. Let ${\mathcal}{F}(X\times U), {\mathcal}{F}(X)$ denote the space of real valued measurable functions with bounded $w, \tilde{w}$-norms, respectively. That is, for $f \in {\mathcal}{F}(X\times U), \tilde{f} \in {\mathcal}{F}(X)$: $$\begin{aligned}
\sup_{(x,u)}\frac{|f(x,u)|}{w(x,u)} < \infty, \quad \sup_{x}\frac{|\tilde{f}(x)|}{\tilde{w}(x)} < \infty.\end{aligned}$$ Let ${\mathcal}{M}(X\times U), {\mathcal}{M}(X)$ denote the space of measures with finite $w, \tilde{w}$-variations, respectively. That is, for $\mu \in {\mathcal}{M}(X\times U), \tilde{\mu} \in {\mathcal}{M}(X)$: $$\begin{aligned}
\label{eq:measure}
\int_{X\times U} w d\mu < \infty, \quad \int_{X} \tilde{w} d\tilde{\mu} < \infty.\end{aligned}$$
Define the linear map $T: {\mathcal}{M}({X\times U}) \rightarrow {\mathcal}{M}(X)$ as: $$\begin{aligned}
\label{eq:T_map}
[T\mu ](B) = \tilde{\mu}(B) - \alpha \int_{X \times U} \tau(B|x,u)\mu(dx,du), \end{aligned}$$ where $\tilde{\mu}(B) := \mu(B, U)$ and $B \in {\mathcal}{B}(X)$. Analogously, define the linear map $T^*: {\mathcal}{F}(X) \rightarrow {\mathcal}{F}(X \times U)$ as: $$\begin{aligned}
[T^*v] \;(x,u) = v(x)- \alpha \int_X \tau(dy|x,u)v(y).\end{aligned}$$ Note that the second term above $\int_X \tau(dy|x,u)v(y)$, is the expectation of the function $v$ under the stochastic kernel $\tau$. One can verify that $T$ and $T^*$ are adjoint operators: $$\begin{aligned}
<T^*v, \mu>_{X\times U}\, =\, <v, T \mu>_X,\end{aligned}$$ where the bilinear maps are given by: $$\begin{aligned}
<c, \mu>_{X\times U} &= \int_{X\times U} c(x,u)\mu(dx,du), \\
<v, \nu>_X &= \int_X v(x) \nu(dx). \end{aligned}$$
In the remainder, for simplicity, we drop the subscript of $<.\;,\;.>$ since the space is clear from the context. To formulate the inf-LP corresponding to stochastic control, we need the following standard assumptions [@hernandez1996discrete].
\[asm:lp\_assm\]
(a) \[asm:c0\_compact\] The cost $c_0$ is lower semi-continuous and inf-compact, that is, for every $x \in X$, $r \in {\mathbb{R}}$, the set $\{ u \in U \; | \; c_0(x,u) \leq r\}$ is non-empty and compact.
(b) \[asm:tau\_cont\] The stochastic kernel $\tau$ is weakly continuous.
(c) \[asm:q\_norm\] $\sup_{X\times U} \int_X \tilde{w}(y) \tau(dy | x, u)/w(x,u) < \infty$.
(d) \[asm:nu0\] $\nu_0 \in {\mathcal}{M}_+(X)$.
Let ${\mathcal}{M}_+(X \times U) \subset {\mathcal}{M}(X \times U)$ denote the cone of non-negative measures. For $\nu_0 \in {\mathcal}{M}_+(X)$, the constraint on $\mu \in {\mathcal}{M}(X \times U)$, denoted by $\nu_0 - T \mu = 0$ refers to $$\begin{aligned}
\label{const:measure}
& \nu_0(B_x) - [T\mu](B_x) =0, \quad \forall B_x \in {\mathcal}{B}(X).\end{aligned}$$
The stochastic control problem , can be equivalently formulated as the following inf-LP: $$\begin{aligned}
\label{opt:lp_primal}\tag{P-SC}
\min_{\mu \in {\mathcal}{M}_+(K)} \quad &<c_0, \mu> \\
\label{con:measure_inf}
\mbox{s.t. } \quad &\nu_0 - T\mu =0.\end{aligned}$$
We summarize the idea of the proof and refer the readers to [@hernandez1996discrete] for details. Given a policy $\pi \in \Pi$, one can define $\mu \in {\mathcal}{M}_+(X\times U)$ as $$\begin{aligned}
\label{eq:discounted_ocupancy}
\mu(B_x , B_u) = \sum_{t = 0}^\infty \alpha^t \mathbb{P}^\pi_{\nu_0}\{ (x_t,u_t) \in (B_x ,B_u)\}, \end{aligned}$$ where $B_x \in {\mathcal}{B}(X), B_u \in {\mathcal}{B}(U)$. This measure corresponds to discounted probability of $(x_t,u_t)$ being in any Borel subset of $X \times U$ and is referred to as the occupation measure. It can be verified that the occupation measure satisfies $\nu_0 - T\mu =0$. Furthermore, given any $\mu \in {\mathcal}{M}_+(X\times U)$, there exists a policy $\varphi \in \Pi$, satisfying $$\begin{aligned}
\label{eq:conditional}
\mu(B_x, B_u) = \int_{B_x} \varphi(B_u|x)\tilde{\mu}(dx),\end{aligned}$$ for all $B_x \in {\mathcal}{B}(X), B_u \in {\mathcal}{B}(U)$ \[Proposition D.8(a) in [@hernandez1996discrete]\]. It can be shown that the cost corresponding to the policy $\varphi$ is $$\begin{aligned}
\label{eq:cost_policy}
\mathbb{E}_{\nu_0}^\varphi \sum_{t=0}^\infty \alpha^t c_0(x_t,u_t) = <c_0,\mu>. \end{aligned}$$ Putting the above results together, the problem of finding the optimal policy for can be equivalently formulated as finding a measure minimizing subject to .
Whereas the inf-LP above provides the optimal occupation measure and the optimal policy for the stochastic control problem, the dual of this inf-LP can be used to find the optimal value function. Furthermore, the duality gap is zero [@hernandez1996discrete].
To define this dual inf-LP, let the constraint on $v \in {\mathcal}{F}(X)$, denoted by $c_0-T^*v \geq 0$ refer to $$\begin{aligned}
\label{const:valuefunction}
& c_0(x,u)- [T^*v](x,u) \geq 0, \; \forall (x,u) \in X\times U. \end{aligned}$$ The dual inf-LP is given by: $$\begin{aligned}
\label{opt:lp_dual}\tag{D-SC}
\max\limits_{v \in {\mathcal}{F}(X)} \quad & < v, \nu_0> \\
\label{con:constraint_u_inf}
\mbox{s.t. } \quad & c_0 -T^*v\geq 0.\end{aligned}$$ **Remark.** Constraint is the Bellman inequality. In particular, based on the Bellman principle of optimality, a function $v^*$ is the optimal value function of the stochastic control if and only if $ c_0 -T^*v= 0$. Thus, the optimizer of the above inf-LP satisfies the Bellman equality.
Inf-LP Approach to LQG Problems {#sec:approach}
===============================
Consider the linear system as a specialization of : $$\begin{aligned}
\label{eq:linear_dynamics}
x_{t+1} = Ax_t + B u_t + \omega_t, \end{aligned}$$ where $x \in {\mathbb{R}}^n$, $u \in {\mathbb{R}}^m$, $\omega \in {\mathbb{R}}^n$, $\omega_t$, are independent identically distributed Gaussian random variables and for all $t$, $E\{\omega_t\} = 0$ and $E\{\omega_t \omega_t^T\} = W$. The initial state is independent of the stochastic noise and has a distribution $\nu_0$, with mean $E\{x_0\} = m_0$ and covariance $E\{x_0 x_0^T\} = \Sigma_0$. The discounted linear quadratic Gaussian (LQG) problem is formulated as: $$\begin{aligned}
\label{eq:lqg_cost}
\min \limits_{\pi \in \Pi} \; &\mathbb{E}_{\nu_0}^\pi \sum_{t=0}^\infty \alpha^t (x^T_t Q_0 x_t + u^T_t R_0 u_t ).\end{aligned}$$ We assume the pair $(A,B)$ is controllable and the pair $(A,C)$ is observable, where $Q_0= C^TC$. Denote by ${\mathcal}{S}^n$, ${\mathcal}{S}^n_+$ and ${\mathcal}{S}^n_{++}$ the set of $n \times n$ symmetric, symmetric positive semidefinte and symmetric positive definite matrices, respectively. We assume $Q_0 \in {\mathcal}{S}^n_+$ and $R_0 \in {\mathcal}{S}^m_{++}$.
To apply the inf-LP approach to this problem, first we verify Assumption as follows. The weight functions in the LQG problem are $w(x,u) = {\epsilon}+ x^T Q_0 x + u^T R_0 u$, $\tilde{w}(x) = {\epsilon}+ x^T Q_0x$. Thus, ${\mathcal}{F}(X\times U)$ and ${\mathcal}{F}(X)$ are spaces of functions over $X\times U = {\mathbb{R}}^{n\times m}, X = {\mathbb{R}}^n$ respectively, that do not grow faster than quadratic functions. Furthermore, by definition , ${\mathcal}{M}({\mathbb{R}}^{n\times m}), {\mathcal}{M}({\mathbb{R}}^n)$ are sets of measures that have bounded variance. From $Q_0 \in {\mathcal}{S}_+, R_0 \in {\mathcal}{S}_{++}$, part (a) of Assumptions holds. The stochastic kernel $\tau$ is Gaussian, which is continuous and has finite variance, satisfying part (b). Part (c) holds since $$\begin{aligned}
& \int_X w_0(y) \tau(dy | x, u) = \epsilon + x^T A^T Q_0Ax \\
&+ 2x^TA^TQ_0Bu + u^TB^TQBu+ \text{Tr}(Q_0W) \in {\mathcal}{F}({\mathbb{R}}^{n\times m}).\end{aligned}$$ Finally, part (d) holds due to finite variance of the initial state distribution $\nu_0$.
Primal and Dual SDPs
--------------------
We consider a relaxation of the inf-LP , resulting in an equivalent restriction of , to obtain a tractable formulation of these inf-LPs. These formulations are then connected to the well-known Riccati LMIs to solve the LQG [@balakrishnan2003semidefinite].
First, consider the constraint $\nu_0-T\mu = 0$ in . It can be verified that this is equivalent to $ <v,\nu_0-T\mu> = 0, \; \forall v \in {{\mathcal}{F}}(X)$. We relax this constraint by restricting ${\mathcal}{F}$ to a subset $\hat{{\mathcal}{F}}(X)$. In particular, define $$\begin{aligned}
\label{eq:quadratic_functions}
\hat{{\mathcal}{F}}(X)= \{ v \in {\mathcal}{F}(X) \; | \; v(x) = x^T P x + q^Tx + r \},\end{aligned}$$ $P \in {\mathcal}{S}^n, q \in {\mathbb{R}}^n, r \in {\mathbb{R}}$. Since $v \in {\mathcal}{F}(X)$ are quadratic, the infinitely many constraint on the measure $\mu$ are relaxed to a set of finite constraints on its moments of order up to two. These constraints will be derived as follows.
Introduce moments of measure $\mu$ as: $$\begin{aligned}
m & = \int_{X \times U} \mu(dx, du) = \mu(X\times U) \in {\mathbb{R}}_+, \\
m_x & = \int_X x \mu(dx,U) \in {\mathbb{R}}^n, \\
Z_{xx} & = \int_X x x^T \mu(dx,U) \in {\mathcal}{S}^n_+. \end{aligned}$$ Similarly, $m_u, Z_{xu}, Z_{uu}$ are defined. For any $S \in {\mathbb{R}}^{n\times n}$, $\text{Tr}(S)$ denotes trace of the matrix $S$.
\[lem:primal\_sdp\] By constraining ${\mathcal}{F}(X)$ to ${\mathcal}{\hat{F}}(X)$, we obtain a relaxation of as $$\begin{aligned}
\label{sdp:primal}\tag{P-LQG}
& \min \quad \text{Tr}(Q_0 Z_{xx}) +\text{Tr}( R_0 Z_{uu}) \\
\nonumber
\mbox{s.t. } \quad & \Sigma_0 + m_0 m_0^T - Z_{xx} + \alpha A Z_{xx}A^T
+ A Z_{xu}B^T \\
\label{con:quad_measure}\tag{C2}
&+ B Z_{xu}^T A^T + B Z_{uu} B^T +m W ) = 0_{n\times n}\\
\label{con:mean}\tag{C1}
\quad & m_0 - m_x + \alpha(A m_x + Bm_u) = 0_{n\times 1},\\
\label{con:total}\tag{C0}
\quad & \alpha m -m + 1 =0,\\
\label{con:quad_measure2}\tag{Cp}
\quad &Z:= \left[ \begin{array}{ccc} m & m_x^T & m_u^T \\ m_x & Z_{xx} & Z_{xu} \\ m_u & Z_{xu}^T & Z_{uu}\end{array} \right] \succeq 0,\end{aligned}$$ over the variables $(m,m_x, m_u, {Z_{xx}, Z_{xu}, Z_{uu}})$.
Based on the definition of the moments, the cost functions in Problem can be expressed as: $$\begin{aligned}
< c_0, \mu> = \text{Tr}(Q_0 Z_{xx}) + \text{Tr}(R_0 Z_{uu}).\end{aligned}$$ Next, expanding $ <v,\nu_0-T\mu> = 0$ we obtain the first term as $$\begin{aligned}
<v,\nu_0 >&= <x^TPx + q^T x + r, \nu_0> \\
&= \text{Tr}(P \Sigma_0) + m_0^TPm_0 + q^T m_0 + r.\end{aligned}$$ Using definition of $T$ in , we expand $<v, -T\mu>$. The first term is: $$\begin{aligned}
&<v, \tilde{\mu}>\,= \,<x^TPx + q^T x + r, -\tilde{\mu}> = \\
&- \int_X v(x) \mu(dx,U) = - \text{Tr}(PZ_{xx}) - q^T m_x - m r.\end{aligned}$$ The second term is obtained as: $$\begin{aligned}
&\alpha \times \big(\text{Tr}(P (A Z_{xx}A^T + AZ_{xu}B^T + B Z_{xu}^T A^T + B Z_{uu}B^T) \big) \\
&+ q^T(Am_x + B m_u) + m \big(\text{Tr}(PW) + r\big).\end{aligned}$$ In the above, we used the fact that the measure $\tau(dx|y,u)$ has mean $Ay + Bu$ and covariance $W$. Each of the terms in the constraint expansion above are linear in the variables $P, q, r$. Since $<v, \nu_0-T\mu>=0$ must hold for all $v \in \hat{{\mathcal}{F}}(X)$, that is, for all $P \in S^{n}, q \in {\mathbb{R}}^n, r \in {\mathbb{R}}$, the corresponding coefficients of these variables need to equal zero. From this, we obtain the set of affine constraints, , , . Constraint holds since $Z$ is moment of a positive measure $\mu$ [@lasserre2009moments].
Similarly, we can obtain the dual SDP as follows.
\[lem:dual\_sdp\] By constraining ${\mathcal}{\hat{F}}(X)$ to ${\mathcal}{F}(X)$, we obtain a restriction of as follows: $$\begin{aligned}
\label{sdp:dual}\tag{D-LQG}
& \min \; \text{Tr}(P\Sigma_0) + \text{Tr}(Pm_0 m_0^T) + q^Tm_0 + r \\
\nonumber
& \mbox{s.t. } \quad \left[ \begin{array}{ccc} s_0 & s_1^T & s_2^T \\ s_1 & S_{11} & S_{12} \\ s_2 & S_{12}^T & S_{22} \end{array} \right] \succeq 0, \end{aligned}$$ with optimization variables, $P, q, r$ and $$\begin{aligned}
& s_0= r(\alpha-1) + \alpha \text{Tr}(PW), \\
& s_1 = \frac{1}{2} (-I + \alpha A^T)q, \; s_2 = \frac{\alpha}{2} B^Tq, \\
&\small{\left[ \begin{array}{cc}S_{11} & S_{12} \\ S_{12}^T & S_{22} \end{array} \right] = \left[ \begin{array}{cc} \alpha A^T P A-P + Q_0 & \alpha A^T P B \\ \alpha B^T P A & R_0 + \alpha B^T P B\end{array} \right]. }\end{aligned}$$ Furthermore, this SDP is the dual of .
The term $<v, \nu_0>$ in the cost function was discussed in Proof of Proposition . For $v \in \hat{{\mathcal}{F}}(X)$, Constraint becomes $$\begin{aligned}
& x^T (-P + \alpha A^T P A + Q_0)x+ 2 \alpha x^T A^T P B u \\
+& u^T (\alpha B^T P B + R_0)u + q^T(-I + \alpha A)x \\
+ & \alpha q^T B u + r(\alpha-1) + \alpha \text{Tr}(PW) \geq 0, \; \forall (x,u) \in X \times U. \end{aligned}$$ An equivalent way of writing the above constraint is: $$\begin{aligned}
& \left[ \begin{array}{c} 1 \\ x \\ a \end{array} \right]^T
\left[ \begin{array}{ccc} s_0 & s_1 & s_2 \\ s_1^T & S_{11} & S_{12} \\ s_2^T & S_{12}^T & S_{22} \end{array} \right]
\left[ \begin{array}{c} 1 \\ x \\ a \end{array} \right] \succeq 0,\end{aligned}$$ which leads to the constraint in . Using SDP duality [@vandenberghe1996semidefinite], it can also be verified that is dual of .
**Remark.** The primal and dual SDPs above are a generalization of the existing results in literature due to the additional terms arising from zero and first order moments of $\mu$. If $m_0 = 0$ from controllability of the pair $(A,B)$ we have $m_x = 0$, $m_u = 0$. Thus, removing the corresponding dual variable $q \in {\mathbb{R}}^n$, we can obtain that $r = \frac{\alpha}{1-\alpha}\text{Tr}(PW)$. This leads to the standard results in [@willems1971least; @boyd1994linear; @balakrishnan2003semidefinite]: $$\begin{aligned}
\label{sdp:dual2}\tag{D-LQG0}
& \min \; \text{Tr}(P\Sigma_0) + \frac{\alpha}{1-\alpha}\text{Tr}(PW) \\
\nonumber
& \mbox{s.t. } \quad S:= \left[ \begin{array}{cc} A^T P A-P + Q_0& \alpha A^T P B \\ \alpha B^T P & R_0 + \alpha B^T P B\end{array} \right] \succeq 0. \end{aligned}$$ In the rest of the paper, we consider $m_0=0$ and thus, we work with and its dual.
If the SDP and its dual have non-empty optimal sets, the complementary slackness condition holds [@vandenberghe1996semidefinite]: $$\begin{aligned}
\nonumber
&Z^* S^* = 0 \iff \left[ \begin{array}{cc} Z_{xx} & Z_{xu} \\ Z_{xu}^T & Z_{uu} \end{array} \right] \times \\
& \left[ \begin{array}{cc} -P + Q_0 + \alpha A^T P A & \alpha A^T P B \\ \alpha B^T P A & R_0 + \alpha B^T P B\end{array} \right] = 0, \end{aligned}$$ where we dropped $^*$ from individual terms above. Expanding above equality, we obtain that $$\begin{aligned}
\nonumber
0 = \;& -P + Q + \alpha A^T P A +\\
\label{eq:p_riccati}
& \alpha A^T P B (R + \alpha B^T P B)^{-1}\alpha B^T P A, \\
\label{eq:lqg_gain}
Z_{xu}^T Z_{xx}^{-1} = \;& (R + \alpha B^T P B)^{-1}\alpha B^T P A, \\
\label{eq:covariance_0}
Z_{uu} = \;& Z_{xu}^T Z_{xx}^{-1}Z_{xu}.\end{aligned}$$ Equation is the algebraic Riccati equation of the infinite horizon discounted LQG problem. Equation provides the optimal controller gain, $K = Z_{xu}^T Z_{xx}^{-1}$.
**Remark.** An alternative derivation of the optimal policy is provided by considering the occupation measure $\mu$ in the inf-LP. Since $ \frac{1}{m}\mu({X \times U}) = (1-\alpha) \mu({X \times U}) $ is a Gaussian measure (alternatively, by considering only the knowledge of the first and second order moments of this measure), the conditional measure $\varphi(u|x)$ can be obtained as $$\begin{aligned}
\varphi(u|x) \sim {\mathcal}{N}(m_{u|x},Z_{u|x}), \end{aligned}$$ with the conditional mean $m_{u|x} = m_u + Z^T_{xu} Z_{xx}^{-1} x$ and covariance $Z_{u|x} = \frac{1}{m} (Z_{uu} - Z^T_{xu}Z_{xx}^{-1} Z_{xu})$. By complementary slackness of , the covariance of this measure is zero and thus, the optimal policy predicted by inf-LP is deterministic and is equal to $\varphi(x) = Z^T_{xu} Z_{xx}^{-1} x$.
Constrained LQG {#sec:clqg}
---------------
One of the advantages of the inf-LP is that constraints of the form $\mathbb{E}_{\nu_0}^\pi \sum_{t=0}^\infty \alpha^t c_i(x_t,u_t) \leq \beta_i$, $i = 1, 2, \dots, N$, can readily be incorporated. In particular, from the definition of the occupation measure : $$\mathbb{E}_{\nu_0}^\pi \sum_{t=0}^\infty \alpha^t c_i(x_t,u_t) \leq \beta_i \iff <c_i, \mu> \; \leq \; \beta_i.$$ Such constraints correspond to multi-objective stochastic control, where $c_i$, for $i=1, \dots, N$, denotes a set of additional objectives and $\beta_i \in {\mathbb{R}}_+$ are desired bounds.
For the LQG problem, let $c_1(x_t,u_t) = x_t^TQ_1x_t + u_t^T R_1 u_t$. Then, constraint $<c_1, \mu> \; \leq\; \beta_1$ is equivalent to $\text{Tr}(Q_1Z_{xx}) + \text{Tr}(R_1Z_{uu}) \leq \beta_1$ in the primal SDP . From our derivation of , it can be verified that the dual SDP with the additional constraint is $$\begin{aligned}
\label{sdp:cdual}\tag{C-LQG}
& \min \; \text{Tr}(P\Sigma_0) + \frac{\alpha}{1-\alpha}\text{Tr}(PW) - {\gamma}\beta \\
\nonumber
& \mbox{s.t. } \quad \left[ \begin{array}{cc} -P + Q + \alpha A^T P A & \alpha A^T P B \\ \alpha B^T P A & R + \alpha B^T P B\end{array} \right]\succeq 0, \end{aligned}$$ where $Q = Q_0 + \beta Q_1$ and $R = R_0 + \beta R_1$ and the optimization variables are $P$ and the dual multiplier of the constraint ${\gamma}> 0$. This is consistent with alternative derivations in multi-criterion LQG in [@boyd1994linear].
Due to $Z_{xx}$ and $Z_{uu}$ corresponding to the second order discounted moments of the occupation measure, constraints $\text{Tr}(Q_1Z_{xx}) \leq \beta_1$ can also be used to pose chance constraints on the state of the form $$\begin{aligned}
\label{eq:chanc_constraint}
\sum_{t=0}^\infty \alpha^t\mathbb{P}_{\nu_0}^\pi ( |g^T x_t |< h) \geq 1-{\epsilon}.\end{aligned}$$
Given a policy $\pi$, let $\mu$ be the resulting occupation measure. Let $Q_1 = gg^T$, and $\beta_1 = {\epsilon}h^2$ $$\begin{aligned}
\text{Tr}(Q_1Z_{xx}) \leq \beta_1 \Rightarrow \sum_{t=0}^\infty \alpha^t P^\pi_{\nu_0} ( |g^T x_t |\leq h) \geq 1-{\epsilon}. \end{aligned}$$ Furthermore, for the case in which $\pi(x) = Kx$ and in the LQG setting, letting $\beta_1 = \frac{h^2}{ 2 (\text{erf}^{-1}(1-{\epsilon}/m)^2) }$,where denotes the error function, the constraint above is also sufficient for the chance constraint.
Define $X_h = \{ x \; : \; |g^T x |\geq h \}$, so that $\mu(X_h, U) = \sum_{t=0}^\infty \alpha^t \mathbb{P}_{\nu_0}^\pi \big( x_t \in X_h \big)$. Then, the chance constraint can be written as $\mu(X_h,U) \leq \epsilon$. Now $$\begin{aligned}
0 \leq h^2{\mathbf}{1}_{X_h} \leq (g^T x)^2 {\mathbf}{1}_{X_h}\end{aligned}$$ Taking integral with respect to $\mu$ we obtain that $$\begin{aligned}
h^2\mu({X_h}, U) &\leq \int_X (g^T x)^2 {\mathbf}{1}_{X_h}\mu(dx,U) \leq g^TZ_{xx}g.\end{aligned}$$ It follows that $\frac{g^TZ_{xx}g}{h^2} < {\epsilon}\Rightarrow \mu(X_h,U) < {\epsilon}$.
If $\pi$ is linear, the resulting occupation measure $\mu$ is a scaled (by factor $m = \frac{1}{1-\alpha})$ Gaussian measure. Since the cumulative distribution function of the Gaussian distribution is invertible and is given through the function, we have $$\begin{aligned}
\mu(X_h, U) < {\epsilon}& \iff 1 + \text{erf}(\frac{-h}{\sigma \sqrt{2}}) \leq \frac{\epsilon}{m} \\
& \iff \frac{h^2}{ 2 (\text{erf}^{-1}(1-{\epsilon}/m)^2) } \geq g^TZ_{xx}g. \qed\end{aligned}$$
**Remark.** The above chance constraints were also considered in infinite horizon average cost LQG problems [@schildbach2015linear]. The authors derived analogous SDPs for the average cost criterion. Their approach was through considering the steady-state second order moments of the closed loop linear system, corresponding to a linear policy. As shown here, this approach is equivalent to the relaxation of the primal infinite dimensional LP from which the occupation measure and the corresponding second order moments were derived.
Numerical Case Studies {#sec:numerical}
======================
Our goal is to use the constrained LQG formulation of the previous section to study effects of multi-objective and chance constraints on the LQG problem. To this end, we consider two linear systems, each with a nominal objective, and closed-loop infinite horizon second order moment constraints on states or inputs. In both cases, we solved SDP using the parser CVX [@grant2008cvx] with the solver SDPT3 [@toh1999sdpt3].
Our first example is a second order system to study effects of chance constraints . In our second example, we consider a model for a miniature coaxial helicopter linearized around a hover maneuver, with a nominal objective of minimizing deviations from hover. Our secondary objective is to minimize control energy.
Two-state system with state constraints
---------------------------------------
The system dynamics parameters are $$\begin{aligned}
A = \left[ \begin{array}{cc}
1 \;& 0.1 \\
0 \;& 1 \\
\end{array} \right], \;
B = \left[ \begin{array}{c}
0 \\
1 \\
\end{array} \right], \;
W = \left[ \begin{array}{cc}
0.1 & 0 \\
0 & 0.1\\
\end{array} \right].\end{aligned}$$ The primary and secondary objective parameters are $$\begin{aligned}
Q_0 = \left[ \begin{array}{cc}
1 & 0 \\
0 & 1 \\
\end{array} \right], \;
Q_1 = \left[ \begin{array}{cc}
0 & 0 \\
0 & 1 \\
\end{array} \right], \;
R_0 = 1, R_1 = 0.\end{aligned}$$ The discount factor is $\alpha = 0.99 $ and $x_0 \sim {\mathcal}{N}(m_0, \Sigma_0)$, with mean $m_0 = [-0.46, 0.58]^T$ and covariance $\Sigma_0 = I$. The second objective, $\text{Tr}(Q_1Z_{xx})$ is constrained to be less than a parameter $\beta$. As such, we require $\sum_{t=0}^\infty \alpha^t P^\pi_{\nu_0} ( [0,\,1] x_t \leq h) \geq 1-{\epsilon}$. This has the interpretation of a soft constraint for the second state to remain close to zero.
We vary $\beta$ between the value of the second objective achieved using the optimal policy for first objective without the constraint and a lower tighter value that forces the constraint to be active. Figure shows the optimal state covariance $Z^*_{xx}$. It is seen that as the constraint is tightened, the discounted occupancy ellipse changes to one with less variation in the second state, due to definition of $Q_1$, but more variation in the first state. The cost is 378 without the constraint and 981 with $\beta = 15$.
\[fig:tradeoff1\]
Miniature coaxial helicopter
----------------------------
We now consider a simplified eight-state model of a miniature two-rotor coaxial helicopter linearized around a hover maneuver based on [@kunz2013fast; @summers2013approximate]. The states of the system are the three-dimensional position and heading deviations from a desired hover pose in an inertial reference frame and the associated velocities in a body reference frame. There are four inputs: pitch, roll, thrust, and yaw, used for forward flight, sideways flight, vertical flight, and heading change, respectively. Pitch and roll are actuated with a swashplate mechanism connected to two servos. Thrust is actuated by the rotational speed of the rotor motors, and yaw is actuated by a rotational speed difference of the rotor motors. The pitch and roll angles and velocities are neglected in the model, and the pitch and roll inputs are assumed to act directly on the lateral position states.
The dynamics are discretized in time by Euler integration with sampling time $t_s$. The parameters of the the discrete-time system dynamics are $$\begin{aligned}
&A = \left[ \begin{array}{cc}
I_4 & t_s I_4 \\
0 & I_4 + t_s \text{diag} ([k_x, k_y, k_z, k_\psi]) \\
\end{array} \right], \; \\
&B = \left[ \begin{array}{c}
0_4 \\
t_s \text{diag} ([b_x, b_y, b_z, b_\psi]) \\
\end{array} \right], \;
W = \left[ \begin{array}{cc}
0_4 & 0 \\
0 & 0.1I_4 \\
\end{array} \right],\end{aligned}$$ where $[k_x,k_y,k_z,k_\psi ] = [-0.5,\, -0.5,\, 0,\, -5]$ represent fuselage drag parameters and $[b_x,b_y,b_z,b_\psi] = [2.0,2.1,11,18]$ represent inertial parameters mapping actuator influence to state derivatives. The values of the parameters are taken from [@kunz2013fast] and are based on a grey-box system identification with experimental data. We consider a scenario in which the deviations from the desired hover pose are to be minimized, subject to an infinite-horizon closed-loop constraint on the expected discounted control energy. The trade-off between trajectory optimization and energy cost minimization is a classical control tradeoff. It is encoded in our framework with the primary and secondary cost parameters $$\begin{aligned}
Q_0 = I_8 , \;
Q_1 = 0_8 , \;
R_0 = 0_4 ,
R_1 = I_4 .\end{aligned}$$ The discount factor is $\alpha = 0.99$ and the threshold is $\beta = 50$. The constraint on control energy is achieved with the constrained LQG formulation by solving . This constraint is satisfied at a price of poorer regulation compared to the unconstrained case as illustrated by the closed-loop system performance in Fig. 2 .
\[fig:tradeoff2\]
Conclusions {#sec:conclusion}
===========
We established a link between the semidefinite programs (SDPs) for solving the LQG problem and the infinite dimensional linear programming (inf-LP) approach to stochastic control. The inf-LP approach is an equivalent alternative to the Dynamic Programming principle of optimality. While the inf-LP formulation has been known since 1950’s, its computational aspects and connections with existing control theoretic results have not been fully explored. We showed that the LMI derived from the occupation measure formulation of the inf-LP corresponds to the dual of the well-known Riccati LMI. Furthermore, given second order moments of the occupation measure, we showed that multi objective and chance constraints have a natural interpretation in this framework, and these formulations coincide with alternative approaches to derive the results. We illustrated the constrained LQG problem with two numerical case studies.
Extensions of this work to continuous-time LQG, and average cost LQG problems are straightforward and will complete the picture. While a rich theory of approximate dynamic programming (ADP) exists, it would be interesting to enrich the approximation procedures, through advanced optimization techniques for solving the inf-LPs corresponding to stochastic control. There has been recent promising steps towards this objective [@sutter2014approximation; @adp_reach_nikos]. It will be interesting to further apply these techniques to large-scale and constrained stochastic control problems.
|
---
abstract: 'We used our newly developed magnetohydrodynamic (MHD) code to perform 2.5D simulations of a fast-mode MHD wave interacting with coronal holes (CH) of varying Alfvén speed which result from assuming different CH densities. We find that this interaction leads to effects like reflection, transmission, stationary fronts at the CH boundary and the formation of a density depletion that moves in the opposite direction to the incoming wave. We compare these effects with regard to the different CH densities and present a comprehensive analysis of morphology and kinematics of the associated secondary waves. We find that the density value inside the CH influences the phase speed as well as the amplitude values of density and magnetic field for all different secondary waves. Moreover, we observe a correlation between the CH density and the peak values of the stationary fronts at the CH boundary. The findings of reflection and transmission on the one hand and the formation of stationary fronts caused by the interaction of MHD waves with CHs on the other hand, strongly support the theory that large scale disturbances in the corona are fast-mode MHD waves.'
author:
- Isabell Piantschitsch
- Bojan Vršnak
- Arnold Hanslmeier
- Birgit Lemmerer
- Astrid Veronig
- 'Aaron Hernandez-Perez'
- Jaša Čalogović
bibliography:
- 'references.bib'
title: |
Numerical Simulation of coronal waves interacting with coronal holes:\
II. Dependence on Alfvén speed inside the coronal hole
---
Introduction {#sec:intro}
============
Large-scale propagating disturbances in the corona or coronal waves, as they are also called, were directly observed for the first time by the Extreme-ultraviolet Imaging Telescope (EIT; @Delaboudiniere1995) onboard the Solar and Heliospheric Observatory (SOHO; @DomingoFleck1995). They are driven by solar flares or alternatively by coronal mass ejections (CMEs) (for a comprehensive review see, [[*e.g.*]{}]{},@Vrsnak_Cliver2008) and can be observed over the entire solar surface.
Inconsistencies regarding the analysis and comparison of observations and simulations led to the development of different theories on how to interpret coronal waves [@Long2017]. Coronal waves can either be described by wave theories, which consider the disturbances as fast-mode MHD waves [@Vrsnak_Lulic2000; @Lulic_etal2013; @Warmuth2004; @Veronig2010; @Thompson1998; @Wang2000; @Wu2001; @Ofman2002; @Patsourakos2009; @Patsourakos_etal.2009; @Schmidt_Ofman2010]. Alternatively, coronal waves can be explained by so called pseudo-wave theories, which consider the observed disturbances as a result of the reconfiguration of the coronal magnetic field, caused by either continuous small-scale reconnection [@Attrill2007a; @Attrill2007b; @van_Driel-Gesztelyi_etal_2008], Joule heating [@Delanee_Hochedez2007] or stretching of magnetic field lines [@Chen_etal2002]. Effects like reflection, refraction or transmission of coronal waves at a coronal hole (CH) boundary support the wave theory whereas the existence of stationary bright fronts was one of the primary reasons for the development of the competing pseudo-wave theory. Another, alternative, approach is hybrid models which try to combine both wave and pseudo-wave theories by interpreting the outer envelope of a CME as a pseudo-wave which is followed by a freely propagating fast-mode-MHD wave [@Chen_etal2002; @Chen_etal2005; @Zhukov_Auchere2004; @Cohen_etal2009; @Chen_Wu2011; @Downs_etal2011; @Cheng_etal2012; @Liu_Nitta_etal2010]. Recent observations also include both the wave and non-wave approach into the interpretation of an individual EUV wave event [@Zong2017].
However, among these competing theories the wave interpretation is regarded as the best supported approach [@Long2017; @Warmuth2015]. Observational evidence for the wave character of these large scale propagating disturbances is given by various authors who report about waves being reflected and refracted at a CH [@Kienreich_etal2012; @Veronig_etal2008; @Long_etal2008; @Gopalswamy_etal2009] or waves being transmitted through a CH [@Olmedo2012] or EIT wave fronts pushing plasma downwards [@Veronig2011; @Harra2011], which is also consistent with the interpretation that EIT waves are fast-mode MHD waves. Recent observations also show that fast EUV waves are able to form bright stationary fronts at the boundary of a magnetic separatrix layer [@Chandra2016]. Furthermore, studies on simulating coronal waves indicate, that stationary wave fronts at a CH boundary can be produced by the interaction of a fast-mode MHD wave with obstacles like a CH [@Piantschitsch2017] or a magnetic quasi-separatrix layer [@Chen2016] and therefore confirm the above mentioned observations.
In @Piantschitsch2017 we used a newly developed MHD code to perform 2.5D simulations which showed that the interaction of an MHD wave with a low density region like a CH leads to effects like reflection and transmission of the incoming wave. Moreover, we observed stationary features at the CH boundary and the formation of a density depletion which is moving in opposite direction of the incoming wave propagation. We found reflections inside the CH which subsequently led to additional transmissive and reflective features outside the CH. We showed that the incoming wave pushes the CH boundary in the direction of wave propagation. Additionally, we compared phase speeds and positions of the incoming wave and the resulting waves after the interaction with a CH and found good agreement with observational cases where waves were being reflected and refracted at a CH [@Kienreich_etal2012] or transmitted through a CH [@Olmedo2012].
In @Piantschitsch2017 we assumed a certain initial density amplitude for the incoming wave and a fixed CH density for our simulations. In this paper we focus on the comparison of different CH densities and on how these various densities change the kinematics of the secondary waves (i.e. reflected, transmitted and traversing waves) and the stationary features at the CH boundary. These different CH densities lead subsequently to different Alfvén speeds inside the CH. We will show that there is an influence of the CH density and the Alfvén speed, respectively, on the amplitude values of the secondary waves and the peak values of the stationary features.
In Section 2 we describe the initial conditions and present the numerical method we use for our simulations. A comprehensive description of the morphology of the reflected, traversing and transmitted waves will be presented in Section 3. In Section 4 we analyze the kinematic measurements of secondary waves and stationary features and compare the cases of varying CH densities. In Section 5 and Section 6 we discuss the conclusions that can be drawn from our simulation results.
Numerical setup
===============
Algorithm and Equations
-----------------------
We use our newly developed code to perform 2.5D simulations of MHD wave propagation and its interaction with low density regions of varying density. In this code we numerically solve the standard homogeneous MHD equations (for detailed description of the equations see @Piantschitsch2017) by applying the so called Total Variation Diminishing Lax-Friedrichs (TVDLF) scheme, first described by @Toth_Odstrcil1996. This scheme is a fully explicit method and achieves second order accuracy in space and time. The simulations are performed by using a $500\times300$ resolution and a dimensionless length of the computational box equal to $1.0$ both in the $x$- and $y$-direction. Transmissive boundary conditions are used for the simulation boundaries.
Initial Conditions
------------------
We assume an idealized case with zero pressure all over the computational box and a homogeneous magnetic field in the vertical direction. The initial setup describes five different cases for the density distribution inside the CH, starting from a density value of $\rho_{CH}=0.1$ and going up to $\rho_{CH}=0.5$. The detailed initial conditions for all parameters are as follows:
$$\rho(x) =
\begin{cases}
\Delta\rho\cdot cos^2(\pi\frac{x-x_0}{\Delta x})+\rho_0 & 0.05\leq x\leq0.15 \\
0.1 \lor 0.2 \lor 0.3 \lor 0.4 \lor 0.5 & \:\:0.4\leq x\leq0.6 \\
\qquad \qquad1.0 & \:\:\qquad\text{else}
\end{cases}$$
$$v_x(x) =
\begin{cases}
2\cdot \sqrt{\frac{\rho(x)}{\rho_0}} -2.0& 0.05\leq x\leq0.15 \\
\:\:0 & \:\:\qquad\text{else}
\end{cases}$$
$$B_z(x) =
\begin{cases}
\:\:\rho(x) & 0.05\leq x\leq0.15 \\
\:\: 1.0 & \:\:\qquad\text{else}
\end{cases}$$
$$B_{x}=B_{y}=0,\qquad0\leq x\leq1$$
$$v_{y}=v_{z}=0,\qquad0\leq x\leq1$$
where $\rho_{0}=1.0$, $\triangle\rho=0.5$, $x_{0}=0.1$, $\triangle x=0.1$.
![Initial conditions for density $\rho$, plasma flow velocity $v_x$ and magnetic field in $z$-direction $B_z$ for five different densities inside the CH, starting from $\rho_{CH}=0.1$ (blue solid line), increased by steps of $0.1$ and ending with $\rho_{CH}=0.5$ (black solid line in the range $0.4\leq x\leq0.6$).[]{data-label="InitCond_1D"}](paper2_update_init_cond_1D_resize.pdf){width="48.00000%"}
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Figure \[InitCond\_1D\] shows a vertical cut through the 2D initial conditions for density, $\rho$, $z$-component of the magnetic field, $B_{z}$, and plasma flow velocity in $x$-direction, $v_{x}$. In Figure \[InitCond\_1D\]a we see an overlay of five different vertical cuts of the 2D density distribution at $y=0$ ($\rho_{CH}=0.1$), $y=0.25$ ($\rho_{CH}=0.2$), $y=0.5$ ($\rho_{CH}=0.3$), $y=0.75$ ($\rho_{CH}=0.4$) and $y=1$ ($\rho_{CH}=0.5$). In the range $0.05\leq x \leq 0.15$ we created a wave with the initial amplitude of $\rho=1.5$ (for detailed description see equation (1)). We can see that the initial density amplitude of the incoming wave has the same value in all five cases whereas the initial CH densities within the range $0.4\leq x \leq 0.6$ vary from one vertical cut to another. The background density is equal to 1.0 everywhere. Figure \[InitCond\_1D\]b and Figure \[InitCond\_1D\]c show the initial conditions for plasma flow velocity in $x$-direction, $v_{x}$, and $z$-component of the magnetic field, $B_{z}$. We can see that $B_{z}$ and $v_{x}$ are defined as functions of $\rho$ in the range $0.05\leq x \leq 0.15$. The initial amplitudes for $v_{x}$ and $B_{z}$ are the same for all cases of varying density distribution. The background magnetic field in the $z$-direction is equal to $1.0$ over the whole computational box whereas the magnetic field components in $x$- and $y$-direction are equal to zero everywhere (see Equations (3) and (4)). The background plasma flow velocity in the $x$-direction is equal to zero and the plasma flow velocities for the $y$- and $z$-directions are equal to zero over the whole computational grid.
In Figure \[Initcond\_2D\] we see the 2D initial conditions for the density distribution, showing a linearly increasing density from $\rho_{CH}=0.1$ up to $\rho_{CH}=0.5$ in the range $0.4\leq x \leq 0.6$. The initial density amplitude of the incoming wave has the same value along the whole $y$-axis. This initial 2D setup enables us to perform simulations of the wave propagation for different CH densities simultaneouosly. Hence, the differences of phase speed and amplitude values due to varying densities inside the CH can be compared immediately.
Morphology
==========
In Figure \[morphology\_1D\_part1\] and Figure \[morphology\_1D\_part2\] we have plotted the temporal evolution of the density distribution for five different CH densities, starting at the beginning of the simulation run at $t=0$ and ending at $t=0.5$. We can see overlayed vertical cuts through the $xz$-plane of our simulations at $y=0$, $y=0.25$, $y=0.5$, $y=0.75$ and $y=1$ at ten different time steps. We observe the temporal evolution of the incoming wave (hereafter named primary wave) and its interaction with the CHs of different density values. Moreover we can see the different behaviour of the reflected, transmitted and traversing waves (hereafter named secondary waves) due to varying density values inside the CH. We observe different kinds of stationary effects at the left CH boundary and also density depletions of varying depths, moving in the negative $x$-direction. In addition to that we find that the primary wave is able to push the left CH boundary in the direction of the primary wave’s propagation.
Primary Wave
------------
In Figure \[morphology\_1D\_part1\]a and Figure \[morphology\_1D\_part1\]b we can see how the primary wave is moving in the positive $x$-direction towards the left CH boundary. At the same time when the density amplitude starts decreasing we observe a broadening of the width of the wave that is accompanied by a steepening of the wave and a subsequent shock formation.
Secondary Waves
---------------
After the primary wave has reached the left CH boundary (see Figure \[morphology\_1D\_part1\]c) we find that the density amplitude quickly decreases when the wave starts traversing through the CH. The smallest density amplitude inside the CH can be seen for the case of an initial CH density of $\rho_{CH}=0.1$ (blue), whereas the largest wave amplitude is observed in the case of $\rho_{CH}=0.5$ (black). Furthermore, we observe immediate responses of the primary wave’s impact on the left CH boundary (see Figure \[morphology\_1D\_part1\]c and \[morphology\_1D\_part1\]d). First, one can see a stationary feature which appears as a stationary peak at $x\approx0.4$ in Figures \[morphology\_1D\_part1\]c-\[morphology\_1D\_part1\]e and Figure \[morphology\_1D\_part2\]a. The morphology of this stationary feature will be discussed in Section 3.3. Second, we observe a first reflective feature (seen at $x\approx0.37$ in Figures \[morphology\_1D\_part1\]c and \[morphology\_1D\_part1\]d), which is not able to move onward in the negative $x$-direction until the incoming wave has not completed the entry phase into the CH. Figure \[morphology\_1D\_part1\]e and Figures \[morphology\_1D\_part2\]a - \[morphology\_1D\_part2\]d show how this first reflection is finally moving toward the negative $x$-direction, where it is then difficult to distinguish from the background density. The first reflection is the same in all five cases of varying initial density inside the CH and it is located at the left side of the density depletions. The smaller the initial density value inside the CH, the smaller the minimum value of the density depletion, [[*i.e.*]{}]{} the stronger the density depletion.
In Figures \[morphology\_1D\_part1\]c - \[morphology\_1D\_part1\]e one can see how the waves are traversing through the CH with much lower density amplitude than that of the primary wave. We observe that the smaller the initial density value inside the CH, the smaller the density amplitude of the traversing wave and the faster the wave propagates through the CH. For a better comparison of the different traversing waves we zoom in the region $0.4\leq x\leq0.6$ in Figure \[morphology\_IN\_CH\_no0\], Figure \[morphology\_IN\_CH\_no1\] and Figure \[morphology\_IN\_CH\_no2\]. We choose the time interval from $t=0.22481$ to $t=0.47487$ which is the time interval where the traversing waves are moving back and forth inside the CH. Figure \[morphology\_IN\_CH\_no0\] shows that the waves are moving with approximately constant density amplitudes of $\rho=0.11$ (for $\rho_{CH}=0.1$), $\rho=0.25$ (for $\rho_{CH}=0.2$), $\rho=0.39$ (for $\rho_{CH}=0.3$), $\rho=0.52$ (for $\rho_{CH}=0.4$) and $\rho=0.65$ (for $\rho_{CH}=0.5$) towards the right CH boundary inside the CH. We saw in Figure \[morphology\_1D\_part2\] that at the time when the traversing waves reach the right CH boundary, one part of each wave is leaving the CH and is propagating onwards as a transmitted wave. In Figure \[morphology\_IN\_CH\_no1\] we find that another part of the traversing waves gets reflected at the right CH boundary inside the CH. When these second traversing waves, which are propagating in the negative $x$-direction now, reach the left CH boundary, again one part leaves the CH hole and this causes another stationary feature at the CH boundary outside of the CH (seen as sharp peaks in Figure \[morphology\_1D\_part2\]b - \[morphology\_1D\_part2\]d), this will be discussed in Section 3.3. Every one of these stationary features is followed by a wave that is moving in the negative $x$-direction (second reflection) while the stationary features can still be observed. The density ampitudes of this second reflection do not have a clear correlation with the initial density values inside the CH. A detailed analysis of the parameters of the second reflection will be performed in Section 4.2.
Due to the varying phase speeds inside the CH the traversing waves leave the CH at different times. Hence, the smaller the density value inside the CH, the earlier we can observe the transmitted wave propagating outside of the CH. After leaving the CH, all different transmitted waves keep moving onwards in the positive $x$-direction until the end of the simulation run at $t=0.5$ (see Figures \[morphology\_1D\_part2\]a - \[morphology\_1D\_part2\]e). One can see that the smaller the initial density inside the CH, the smaller the density amplitude of the transmitted wave (smallest density amplitude for the transmitted wave in case of $\rho_{CH}=0.1$, marked in blue; largest density amplitude for the transmitted wave in case of $\rho_{CH}=0.5$, marked in black).
Besides causing a second reflection, the second traversing waves get reflected inside the CH again and move a third time through the CH, now again in the positive $x$-direction (see Figure \[morphology\_IN\_CH\_no2\]). When this third traversing wave reaches the right CH boundary, it causes a kind of subwave inside the first transmitted wave, seen as a peak inside the already existing transmitted wave in Figure \[morphology\_IN\_CH\_no2\]. Since we can not see these additional peaks inside the transmitted waves very clearly in Figure \[morphology\_IN\_CH\_no2\], we zoom in the area of $0.6<x<0.9$. Figure \[zoom\_transmitted\_wave\] shows these peaks at $x\approx0.76$ (for $\rho_{CH}=0.1$) and at $x\approx0.685$ (for $\rho_{CH}=0.2$). This kind of second transmission moves together with the first transmission in the positive $x$-direction until the end of the simulation run at $t=0.5$ but it can only be seen in the cases of $\rho_{CH}=0.1$ (blue) and $\rho_{CH}=0.2$ (red).
Stationary Features
-------------------
We observe a first stationary feature at the left CH boundary, this appears as a stationary peak at $x\approx0.4$ in Figures \[morphology\_1D\_part1\]c-\[morphology\_1D\_part1\]e and Figure \[morphology\_1D\_part2\]a. This peak occurs in all five cases of different initial CH density but it can be seen most clearly for the case of $\rho_{CH}=0.5$ (black line). During the lifetime of these stationary features the rear part of the primary wave continues entering the CH and the traversing waves keep moving onwards inside the CH. Due to the plot resolution on the one hand and the time delay of this feature for the different cases of $\rho_{CH}$ on the other hand, the single peaks are hard to distinguish and detect in Figures \[morphology\_1D\_part1\]c and \[morphology\_1D\_part1\]d. Hence, we zoom in the area $0.3\leq x\leq 0.5$ for the time period in which this first stationary feature occurs, in order to be able to study the peak values and the lifetime of this feature for all cases of different initial density inside the CH. Figure \[zoom\_first\_stat\] shows that a stationary peak can be observed first in the case of $\rho_{CH}=0.1$ (blue line) at $x\approx0.4$, followed by the peaks in the cases $\rho_{CH}=0.2$, $\rho_{CH}=0.3$, $\rho_{CH}=0.4$ and $\rho_{CH}=0.5$. The density of all peak values decreases in time, starting from $t=0.22481$ and ending at $t=0.29467$. One can see that the smaller the initial density inside the CH, the larger the peak value of this first stationary feature. Moreover, we can observe that the peak values move slighty in the positive $x$-direction for all different values of $\rho_{CH}$.
In Figures \[morphology\_1D\_part2\]b, \[morphology\_1D\_part2\]c and \[morphology\_1D\_part2\]d we find a second stationary feature at the left CH boundary at about $x\approx0.43$. It occurs first for the case of $\rho_{CH}=0.1$ (blue), followed by the cases $\rho_{CH}=0.2$ (red), $\rho_{CH}=0.3$ (green), $\rho_{CH}=0.4$ (magenta) and $\rho_{CH}=0.5$ (black). In order to study the life time and the density peak values in detail we zoom in the region $0.3\leq x\leq 0.5$ between $t=0.31488$ and $t=0.44975$. In contrary to the first stationary feature, Figure \[zoom\_second\_stat\] shows that the smaller the initial density inside the CH, the larger the peak value of the second stationary feature. When we compare the time when the peaks show up with the time evolution of the traversing wave inside the CH, we see that the second stationary features appear at the time when the second traversing waves have reached the left CH boundary inside the CH again. We also find that the smaller the initial $\rho_{CH}$ inside the CH is, the longer the lifetime of the second stationary peak. The peaks of this feature remain observable while the second reflection is moving onward in the negative $x$-direction (see Figures \[morphology\_1D\_part2\]c, \[morphology\_1D\_part2\]d and \[morphology\_1D\_part2\]e).
Density Depletion
-----------------
In Figure \[morphology\_1D\_part1\]d we observe the beginning of the evolution of a density depletion at $t\approx0.39$, most clearly seen for the case $\rho_{CH}=0.1$ (blue) and located at the left side of the first stationary feature. This density depletion appears for all values of $\rho_{CH}$ but has different minimum values. It propagates in the negative $x$-direction, ahead of the second reflection. One can see that the smaller the value for $\rho_{CH}$, the smaller is the minimum value of the density depletion. One more time we zoom in the area of interest ($0.2\leq x\leq0.4$) to analyze and compare the different depletions in detail. Figure \[zoom\_density\_depletion\] shows the time evolution of the density depletions from $t=0.24967$ to $t=0.39931$.
2D Morphology
-------------
In Figure \[morphology\_rho\_2D\] we see the 2D temporal evolution of the density distribution for all cases of varying $\rho_{CH}$, starting at $y=0$ with $\rho_{CH}=0.1$ and increasing linearly to $\rho_{CH}=0.5$ at $y=1$. Figure \[morphology\_rho\_2D\]a shows the initial setup of the simulation run at $t=0$. In Figure \[morphology\_rho\_2D\]b one can see the primary wave shortly before the entry phase into the CH. What we observe in Figures \[morphology\_rho\_2D\]c and \[morphology\_rho\_2D\]d is how the wave enters the CH and starts traversing through the CH. We find that the smaller the value of $\rho_{CH}$, the faster the wave traverses through the CH. Figure \[morphology\_rho\_2D\]e shows that those waves which crossed a CH of low density already left the CH, while those which having entered a CH of a higher density value are still traversing through the CH. In Figure \[morphology\_rho\_2D\]f we see that the second stationary feature starts appearing at the left CH boundary, caused by the first traversing waves reaching the left CH boundary. Moreover, in Figures \[morphology\_rho\_2D\]g, \[morphology\_rho\_2D\]h and \[morphology\_rho\_2D\]i we can observe the evolution of the second stationary features for all cases of different $\rho_{CH}$ as well as the propagation of the density depletion and the second reflection in the negative $x$-direction. How the transmissive waves for all different values for $\rho_{CH}$ are moving forward in the positive $x$-direction can be seen in Figures \[morphology\_rho\_2D\]e - \[morphology\_rho\_2D\]i.
Figure \[evolution\_amplitude\] shows the temporal evolution of the density amplitude and its position for the traversing and the transmitted waves in two different cases of CH density ($\rho_{CH}=0.1$ marked in blue and $\rho_{CH}=0.3$ marked in red) with regard to a wave having no interaction with a CH (gray). Furthermore, we compare the final density distribution for $\rho_{CH}=0.1$ (blue) and $\rho_{CH}=0.3$ (red) at the end of the simulation run at $t=0.5$. One can see that the waves propagate faster through the CH ($0.4\leq x\leq0.6$) than the primary wave before entering the CH (gray for $0<t<0.2$). Morever, we can observe how the density amplitudes decrease when the wave is traversing through the CH and how they increase again after having left the CH. By comparing the density amplitudes inside the CH, we can see that the amplitude value is much smaller for the case $\rho_{CH}=0.1$ (blue) than in the case $\rho_{CH}=0.3$ (red).
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![Zoom into the area of the transmitted waves in the range $0.6\leq x\leq0.9$ for the cases $\rho_{CH}=0.1$ (blue), $\rho_{CH}=0.2$ (red), $\rho_{CH}=0.3$ (green), $\rho_{CH}=0.4$ (magenta) and $\rho_{CH}=0.5$ (black). []{data-label="zoom_transmitted_wave"}](paper2_transm_wave_zoom_resize.pdf){width="50.00000%"}
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Kinematics
==========
Primary Wave
------------
Figure \[Kin\_prim\_wave\] shows the temporal evolution for the peak values of the primary wave’s density, $\rho$, plasma flow velocity, $v_x$, phase speed, $v_w$, and magnetic field in the $z$-direction, $B_z$. In Figure \[Kin\_prim\_wave\]a we find that the density amplitude stays approximately constant at a value of $\rho\approx1.5$ until about $t=0.06$ and decreases subsequently to a density value of $\rho\approx1.4$ at $t=0.2$, the time at which the primary wave starts entering the CH. A similar decrease can be seen in Figures \[Kin\_prim\_wave\]d and \[Kin\_prim\_wave\]f , where we observe the plasma flow velocity and magnetic field component in the $z$-direction decreasing from $v_x\approx0.45$, $B_z\approx1.5$ at $t=0.05$ to $v_x\approx0.35$, $B_z\approx1.4$ at $t=0.2$. At the same time when the amplitude values of $\rho$, $v_x$ and $B_z$ start decreasing, we observe a broadening of the width of the wave, starting at $width_{wave}=0.08$ ($t=0$) and increasing to a value of $width_{wave}=0.13$ ($t=0.2$) (see Figure \[Kin\_prim\_wave\]c). Figure \[Kin\_prim\_wave\]b shows how the wave is propagating in the positive $x$-direction. In Figure \[Kin\_prim\_wave\]e it is evident that the phase speed of the primary wave decreases slightly until the beginning of the entry phase into the CH, i.e. $v_w=1.75$ (at $t=0.01$) decreases to $v_w=1.4$ (at $t=0.2$).
Secondary Waves
---------------
Figure \[Kin\_traver\_wave\] shows the temporal evolution of density, $\rho$, position of the amplitude, $Pos_{A}$, plasma flow velocity, $v_{x}$, phase speed, $v_{w}$, and magnetic field component in the $z$-direction, $B_{z}$, for the first traversing wave in every case of varying CH density, $\rho_{CH}$. In all five cases the wave is propagating with approximately constant amplitude in the positive $x$-direction (see Figues \[Kin\_traver\_wave\]a and \[Kin\_traver\_wave\]b), [[*i.e.*]{}]{} $\rho=0.11$ (for $\rho_{CH}=0.1$, blue), $\rho=0.25$ (for $\rho_{CH}=0.2$, red), $\rho=0.39$ (for $\rho_{CH}=0.3$, green), $\rho=0.52$ (for $\rho_{CH}=0.4$, magenta) and $\rho=0.65$ (for $\rho_{CH}=0.5$, black). Figure \[Kin\_traver\_wave\]e shows the values for the magnetic field component in $z$-direction. One can see that, like in the case of the density $\rho$, the amplitudes remain approximately constant and that the smaller the CH density, $\rho_{CH}$, the smaller the amplitude value of $B_{z}$. The tracking of the wave with $\rho_{CH}=0.5$ starts at a later time due to the different phase speeds of the traversing waves inside the CH. In Figures \[Kin\_traver\_wave\]c we observe approximately constant values for $v_x$, but in contrary to the density, $\rho$, and the magnetic field component in $z$-direction, $B_{z}$, the largest amplitudes can be seen in the case of $\rho_{CH}=0.1$ and the smallest ones for $\rho_{CH}=0.5$. The temporal evolution of the phase speed of the first traversing wave is shown in Figure \[Kin\_traver\_wave\]d. We find that the smaller the intial density inside the CH, the faster the wave propagates through the CH. In all five cases the phase speed decreases slightly until the wave leaves the CH, [[*i.e.*]{}]{} at $t=0.215$ we have $v_w\approx3.75$ (for $\rho_{CH}=0.1$), $v_w\approx2.5$ (for $\rho_{CH}=0.2$), $v_w\approx2.15$ (for $\rho_{CH}=0.3$) and $v_w\approx1.75$ (for $\rho_{CH}=0.4$). The speed tracking in the case of $\rho_{CH}=0.5$ starts a $t\approx0.222$ and supplies a value of $v_w\approx1.7$. The phase speed values decrease until $t=0.24$ to $v_w\approx3.2$ (for $\rho_{CH}=0.1$), $v_w\approx1.9$ (for $\rho_{CH}=0.2$), $v_w\approx1.9$ (for $\rho_{CH}=0.3$), $v_w\approx1.5$ (for $\rho_{CH}=0.4$) and $v_w\approx1.25$ (for $\rho_{CH}=0.5$). (Due to the very low amplitudes inside the CH one the one hand and the related tracking difficulties on the other hand, there will be no detailed kinematics study of the second and third traversing wave.)
The temporal evolution of the parameters of the transmitted waves is described in Figure \[Kin\_transm\_wave\]. In Figures \[Kin\_transm\_wave\]a, \[Kin\_transm\_wave\]b and \[Kin\_transm\_wave\]e, where one can see the amplitude values of $\rho$, $v_x$ and $B_z$, it is evident that the wave which was traversing through the CH in the case $\rho_{CH}=0.1$ (blue line) leaves the CH first, followed by the waves in the cases of $\rho_{CH}=0.2$ (red), $\rho_{CH}=0.3$ (green), $\rho_{CH}=0.4$ (magenta) and $\rho_{CH}=0.5$ (black). The density amplitude values of the transmitted waves start at $\rho=1.27$ (for $\rho_{CH}=0.1$ at $t\approx0.28$), $\rho=1.3$ (for $\rho_{CH}=0.2$ at $t\approx0.3$), $\rho=1.32$ (for $\rho_{CH}=0.3$ at $t\approx0.32$), $\rho=1.325$ (for $\rho_{CH}=0.4$ at $t\approx0.33$) and $\rho=1.325$ (for $\rho_{CH}=0.5$ at $t\approx0.337$) and decrease to $\rho=1.24$ (for $\rho_{CH}=0.1$), $\rho=1.28$ (for $\rho_{CH}=0.2$), $\rho=1.29$ (for $\rho_{CH}=0.3$), $\rho=1.3$ (for $\rho_{CH}=0.4$) and $\rho=1.305$ (for $\rho_{CH}=0.5$) at the end of the simulation run at $t=0.5$ (see Figure \[Kin\_transm\_wave\]a). Figure \[Kin\_transm\_wave\]c shows how the transmitted waves propagate in the positive $x$-direction in all five cases. The evolution of the phase speed of the transmitted waves is described in Figure \[Kin\_transm\_wave\]d. Here we can see that the values start at $v_w=1.21$ (for $\rho_{CH}=0.1$), $v_w=1.18$ (for $\rho_{CH}=0.2$), $v_w=1.17$ (for $\rho_{CH}=0.3$), $v_w=1.17$ (for $\rho_{CH}=0.4$) and $v_w=1.17$ (for $\rho_{CH}=0.5$) and decrease slightly in all five cases as the wave is moving further towards the positive $x$-direction.
Figure \[Kin\_first\_reflection\] describes the amplitude values of the first reflection. Due to a superposition, caused by a simultaneous entering of segments of the rear of the primary wave into the CH on the one hand and an already ongoing reflection of the the front segments of the wave on the other hand, this feature is not able to move in the negative $x$-direction until the primary wave has completed its entry phase into the CH. Hence, we will start describing the kinematics of this first reflection at $t\approx0.27$, when it starts moving in the negative $x$-direction.
As we can see in Figures \[morphology\_1D\_part2\]a - \[morphology\_1D\_part2\]e, the first reflection is the same in all cases of different $\rho_{CH}$. It moves from $x\approx0.3$ (seen in Figure \[morphology\_1D\_part2\]a) to $x\approx0.2$ (seen in Figure \[morphology\_1D\_part2\]d) and is located at the left side of the density depletions. Figure \[Kin\_first\_reflection\] describes the kinematics of this first reflection for all different $\rho_{CH}$. In Figure \[Kin\_first\_reflection\]a one can see that the amplitude density stays at an approximately constant value of about $\rho=1.0$ until $t\approx0.39$. At that time the first reflection approaches an area of oscillations that is caused by numerical effects (detailed description in @Piantschitsch2017). Here we can no longer get reasonable results for the first reflection. Similar to the density values of this first reflection the magnetic field component, $B_{z}$, and plasma flow velocity, $v_{x}$, stay approximately constant at values of $B_{z}=0.495$ or $v_{x}=0.001$ (see Figure \[Kin\_first\_reflection\]c and Figure \[Kin\_first\_reflection\]e). Figure \[Kin\_first\_reflection\]b shows how the reflection is moving in the negative $x$-direction. The temporal evolution of the phase speed of this first reflection is described in Figure \[Kin\_first\_reflection\]d. Here we observe that the value of the phase speed decreases from $v_{w}\approx-1.1$ to $v_{w}\approx-0.5$.
In Figure \[Kin\_second\_reflection\] we present the kinematic analysis of the second reflection. This reflection is caused by parts of the traversing wave leaving the CH at the left CH boundary at $t\approx0.36$. At the time we stop the simulation run at $t=0.5$ only the reflections for the cases $\rho_{CH}=0.1$, $\rho_{CH}=0.2$ and $\rho_{CH}=0.3$ have moved sufficiently far in the negative $x$-direction to compare their peak values. Figure \[Kin\_second\_reflection\] shows how the time at which the second reflection appears depends on the density inside the CH. In contrast to traversing and transmitted waves we do not have a linear correlation between the initial density values inside the CH and the amplitude values of the different reflection parameters. Figure \[Kin\_second\_reflection\]a shows that the density amplitude for the case $\rho_{CH}=0.2$ (red) is in fact larger than the density amplitude in the case $\rho_{CH}=0.1$ (blue). However, the density amplitude in the case $\rho_{CH}=0.3$ (green) is not larger than the one in the case $\rho_{CH}=0.2$ (red) but lies between the first two cases. A similar behaviour holds true for the plasma flow velocity, $v_x$, and the magnetic field component, $B_z$ (see Figures \[Kin\_second\_reflection\]b and \[Kin\_second\_reflection\]d). In Figure \[Kin\_second\_reflection\]c one can see how the second reflection is moving in the negative $x$-direction until the end of the simulation run at $t=0.5$.
Stationary Features
-------------------
The kinematics of the first stationary feature are described in Figure \[Kin\_first\_stat\]. In Figure \[Kin\_first\_stat\]a we can see that at about $t\approx0.22$ this feature occurs first in the case of $\rho_{CH}=0.5$ (black), starting with a density amplitude of $\rho\approx1.25$ and decreasing to $\rho\approx1.0$ at $t=0.36$. This density plot also shows that the appearance of the other density amplitudes follow one after each other: $\rho=1.23$ (for $\rho_{CH}=0.4$, magenta, at $t=0.22$), $\rho=1.2$ (for $\rho_{CH}=0.3$, green, at $t=0.225$), $\rho=1.18$ (for $\rho_{CH}=0.2$, red, at $t=0.23$) and $\rho=1.16$ (for $\rho_{CH}=0.1$, blue at $t=0.229$). Finally, these amplitude values decrease to $\rho\approx1.0$ (in all five cases of different $\rho_{CH}$). A similar decreasing behaviour can be observed for the magnetic field component in the $z$-direction, $B_z$, (see Figure \[Kin\_first\_stat\]e). The amplitude values of $B_z$ start at approximately the same values as the density amplitudes in Figure \[Kin\_first\_stat\]a and decrease also to a value of $B_{z}=1.0$ at $t=0.36$. Figures \[Kin\_first\_stat\]a and \[Kin\_first\_stat\]e show that the smaller the density inside the CH, the larger the wave’s amplitude values for density and magnetic field. The exact reverse behaviour can be observed for the plasma flow velocity, $v_{x}$, and phase speed, $v_{w}$, of this feature (see Figures \[Kin\_first\_stat\]b and \[Kin\_first\_stat\]d). Here it is evident that the smaller the density value inside the CH, the smaller the values for $v_{x}$ and $v_{w}$. Figure \[Kin\_first\_stat\]c shows that the first stationary feature is moving slightly in the positive $x$-direction in all five cases of different $\rho_{CH}$.
In Figure \[Kin\_second\_stat\] we present the kinematic analysis of the second stationary feature. Figures \[Kin\_second\_stat\]a and \[Kin\_second\_stat\]e show, in contrast to the first stationary feature, that the smaller the initial density inside the CH, $\rho_{CH}$, the larger the amplitude values for density, $\rho$, and magnetic field component, $B_z$. Another difference between the first and the second stationary features is the fact, that the second stationary feature is moving slightly in the negative $x$-direction (see Figure \[Kin\_second\_stat\]c). Since the movement of this feature is more or less only a small shift of its position to the left, plasma flow velocity $v_x$ and phase speed $v_w$ are very small too and finally decrease to a value of almost zero in all five cases of different $\rho_{CH}$ (see Figures \[Kin\_second\_stat\]b and \[Kin\_second\_stat\]d).
Density Depletion
-----------------
In Figure \[Kin\_density\_depletion\] we analyze the temporal evolution of the density depletion for all different cases of initial CH density. This feature occurs first for the case $\rho_{CH}=0.1$ (blue) at about $t=0.25$, followed by the density depletions for $\rho_{CH}=0.2$ (red), $\rho_{CH}=0.3$ (green), $\rho_{CH}=0.4$ (magenta) and $\rho_{CH}=0.5$ (black). The minimum density values of the depletion decrease from about $\rho=1.0$ (for all five cases) to $\rho=0.83.$ ($\rho_{CH}=0.1$, blue), $\rho=0.87$ ($\rho_{CH}=0.2$, red), $\rho=0.9$ ($\rho_{CH}=0.3$, green), $\rho=0.92$ ($\rho_{CH}=0.4$, magenta) and $\rho=0.94$ ($\rho_{CH}=0.5$, black) at $t=0.3$ and subsequently remain approximately constant at those values until the end of the run at $t=0.5$ (see Figure \[Kin\_density\_depletion\]a). An analogous behaviour to the density evolution can be found for the temporal evolution of the magnetic field component $B_{z}$ (see Figure \[Kin\_density\_depletion\]e). In Figure \[Kin\_density\_depletion\]b we find a decrease of the plasma flow velocity for all five cases of different intial CH density $\rho_{CH}$. At $t=0.25$ we find a value of $v_{x}\approx0.35$ which decreases down to $v_{x}\approx0.17$ (at $t=0.3$) in the case of $\rho_{CH}=0.1$ (blue). A similar decrease can be found for all the other cases: The density values $v_{x}\approx0.24$ (red, at $t\approx0.258$), $v_{x}\approx0.18$ (green, at $t\approx0.26$), $v_{x}\approx0.13$ (magenta, at $t\approx0.268$) and $v_{x}\approx.0.08$ (black, at $t\approx0.27$) decrease to $v_{x}\approx0.13.$ (red), $v_{x}\approx0.1$ (green), $v_{x}\approx0.07$ (magenta) and $v_{x}\approx0.05$ (black) and then remain at those values until $t=0.5$. Figure \[Kin\_density\_depletion\]c shows how all density depletions are moving towards the negative $x$-direction. Furthermore, we observe that the smaller the density inside the CH, the smaller the mean phase speed of the density depletion. (see Figure \[Kin\_density\_depletion\]d).
{width="\textwidth"}
![From top to bottom: Density, position of the amplitude, width of the wave, plasma flow velocity, phase velocity and magnetic field of the primary wave, from the beginning of the run ($t=0$) until the time when the wave is entering the CH ($t=0.2$). []{data-label="Kin_prim_wave"}](paper2_update_kin_prim_wave_resize.pdf){width="49.00000%"}
![From top to bottom: Temporal evolution of density, position of the amplitude, plasma flow velocity, phase velocity and magnetic field of the traversing wave for the cases $\rho_{CH}=0.1$ (blue), $\rho_{CH}=0.2$ (red), $\rho_{CH}=0.3$ (green), $\rho_{CH}=0.4$ (magenta) and $\rho_{CH}=0.5$ (black). Starting at about $t=0.215$, shortly after the primary wave has entered the CH and ending at $t=0.24$, when the traversing waves leave the CH.[]{data-label="Kin_traver_wave"}](paper2_kinem_trav_wave_correction_resize.pdf){width="49.00000%"}
![From top to bottom: Temporal evolution of density, plasma flow velocity, position of the amplitude, phase velocity and magnetic field of the transmitted wave for the cases $\rho_{CH}=0.1$ (blue), $\rho_{CH}=0.2$ (red), $\rho_{CH}=0.3$ (green), $\rho_{CH}=0.4$ (magenta) and $\rho_{CH}=0.5$ (black). Starting at about $t=0.26$, when the first transmitted wave (blue) occurs at the right CH boundary and ending at the end of the simulation run at $t=0.5$.[]{data-label="Kin_transm_wave"}](paper2_update_kin_transm_wave_resize.pdf){width="49.00000%"}
![From top to bottom: Temporal evolution of density, position of the amplitude, plasma flow velocity, phase velocity and magnetic field of the first reflection starting at ($t\approx0.27$) and ending at ($t=0.39$).[]{data-label="Kin_first_reflection"}](paper2_update_kin_1st_reflection_resize.pdf){width="49.00000%"}
![From top to bottom: Temporal evolution of density, plasma flow velocity, position of the amplitude and magnetic field of the second reflection for the cases $\rho_{CH}=0.1$ (blue), $\rho_{CH}=0.2$ (red) and $\rho_{CH}=0.3$ (green). Starting at about $t=0.35$, when the second reflection occurs for the case $\rho_{CH}=0.1$ (blue) and ending at the end of the simulation run at $t=0.5$.[]{data-label="Kin_second_reflection"}](paper2_update_kin_2nd_reflection_resize.pdf){width="49.00000%"}
![From top to bottom: Temporal evolution of density, plasma flow velocity, position of the amplitude, phase velocity and magnetic field of the first stationary feature for the cases $\rho_{CH}=0.1$ (blue), $\rho_{CH}=0.2$ (red), $\rho_{CH}=0.3$ (green), $\rho_{CH}=0.4$ (magenta) and $\rho_{CH}=0.5$ (black). Starting at about $t=0.22$, when this feature occurs first in case of $\rho_{CH}=0.5$ (black) and ending at $t\approx0.36$.[]{data-label="Kin_first_stat"}](paper2_update_kin_1st_stat_feature_resize.pdf){width="49.00000%"}
![From top to bottom: Temporal evolution of density, plasma flow velocity, position of the amplitude, phase velocity and magnetic field of the second stationary feature for the cases $\rho_{CH}=0.1$ (blue), $\rho_{CH}=0.2$ (red), $\rho_{CH}=0.3$ (green), $\rho_{CH}=0.4$ (magenta) and $\rho_{CH}=0.5$ (black). Starting at about $t=0.32$, when this feature occurs first in case of $\rho_{CH}=0.1$ (blue) and ending at the end of the simulation run at $t=0.5$.[]{data-label="Kin_second_stat"}](paper2_update_kin_2nd_stat_feature_resize.pdf){width="49.00000%"}
![From top to bottom: Temporal evolution of density, plasma flow velocity, position of the amplitude, phase velocity and magnetic field of the density depletion for the cases $\rho_{CH}=0.1$ (blue), $\rho_{CH}=0.2$ (red), $\rho_{CH}=0.3$ (green), $\rho_{CH}=0.4$ (magenta) and $\rho_{CH}=0.5$ (black). Starting at about $t=0.25$, when this feature occurs first in case of $\rho_{CH}=0.1$ (blue) and ending at the end of the simulation run at $t=0.5$.[]{data-label="Kin_density_depletion"}](paper2_update_kin_dens_depletion_resize.pdf){width="49.00000%"}
Discussion
==========
In @Piantschitsch2017 we showed that the impact of the incoming fast-mode MHD wave on the CH leads to effects like reflection, transmission and the formation of stationary fronts. In this paper we focus on how the CH density influences all these different features. We find that the CH density correlates with the peak values of the stationary features and the amplitudes of the secondary waves.
When we compare the first reflection and the first stationary feature with each other, we see that both effects are connected to a superposition of wave parts which are entering the CH and wave parts that were already reflected at the CH boundary. In detail this means, that segments of the rear of the primary wave are entering the CH while segments of the front of the primary wave have already been reflected at the left CH boundary but are prevented from moving in the negative $x$-direction due to the plasma flow associated with the primary wave. The significant difference between those two features is, that the first reflection is a moving feature and its parameters are the same for all five cases of different CH density. The first stationary feature on the other hand exhibits different amplitudes depending on the various CH densities. This means, further, that the first reflection is only caused by the immediate response of the primary’s wave impact on the CH boundary. The first stationary feature in contrary seems to be also affected by the different CH densities.
A comparison between the first and the second stationary feature shows that these effects depend on the initial CH density in the opposite manner. The amplitudes for the second stationary feature are larger, the smaller the initial CH density is. The density amplitudes of the first stationary feature in contrary are smaller, the smaller the initial CH density is. An explanation for this could be a combination of the effects of the traversing waves on the one hand and reflections inside the CH on the other hand in the case of the second stationary feature. We showed in the kinematics section that the smaller the initial CH density, the smaller the density amplitude of the traversing wave and the smaller the density amplitude of the transmitted wave. This consequently also means that in the case of low initial CH density a bigger part gets reflected inside the CH and leads finally to a larger peak value of the second stationary feature.
The second reflection exhibits notable properties since there is no linear correlation between its amplitudes and the initial CH density. This feature seems to be more complex since it combines effects of the traversing waves, their phase speed and reflections inside the CH.
During the analysis of the transmitted waves, we found an additional peak inside the wave, a kind of subwave which is moving with the transmitted wave in the positive $x$-direction. This phenomenon only occurs for the cases of $\rho_{CH}=0.1$ and $\rho_{CH}=0.2$. A reason for this is probably the limited runtime of the simulation, [[*i.e.*]{}]{} we expect to see those peaks in the transmitted waves for the other three cases as well for a longer runtime of the simulation. We found that these peaks occur when the third traversing wave reaches the right CH boundary.
When considering our simulation results we have to bear in mind that we are dealing with an idealized situation including many constraints, [[*e.g.*]{}]{} a homogeneous magnetic field, the fact that the pressure is equal to zero over the whole computational box, the assumption of a certain value for the initial wave amplitude and a simplified shape of the CH. Another thing we have to pay attention to is the fact that in our simulations we assume a certain width of the CH. We do not know so far how much a broader CH would influence the final phase speed of the traversing waves and hence the properties of the transmitted waves as well as the reflective features inside the CH.
In our simulations we observe a quite large density amplitude of the transmitted wave whereas in observations such transmitted waves are rarely found. Only in @Olmedo2012 for the first time a wave reported being transmitted through a CH. Hence, there are some aspects, like the intensity of the wave’s driver (solar flare or CME), the distance of the initial wave front to the CH, the shape and size of the CH and the magnetic field structure inside the CH, that we have to keep in mind when comparing observations with our simulations. More specifically, in our simulations we make sure that the amplitude of the incoming wave is large enough and that the distance to the CH is sufficiently small in order to guarantee a transmission through the CH. In observations, due to a possibly weak eruption or a large distance of the wave’s driver to the CH, this can not be guaranteed. Another issue is the shape of CH; in our simulations the wave is approaching exactely perpendicular to the CH at every point, whereas this is usually not the case in the observations. Moreover, the size of a CH can also be a reason for preventing a wave traversing through the whole CH. In our simulations we assume a homogenous magnetic field which does not reflect the actual magnetic field structure of a CH in the observations. The complexity of the magnetic field structure inside a CH may also be a cause for the wave not being transmitted through the CH due to, [[*e.g.*]{}]{} , dispersion of the wave on inhomogenities. We also have to be aware that our simulations are restricted to two dimensions, that is, the wave front is not capable of moving in the vertical direction as it would be the case in the observations.
Conclusions
===========
We present the results of a newly developed 2.5D MHD code performing simulations of a fast mode MHD wave interacting with CHs of different density and various Alfvén speed, respectively. In @Piantschitsch2017 we demonstrated that the impact of the incoming wave causes different effects like reflection, transmission and the formation of stationary fronts for the case of an initial density amplitude of $\rho=1.5$ and a fixed initial CH density of $\rho_{CH}=0.1$.
In this paper, we focus on comparing the properties of the different secondary waves and the stationary features with regard to various CH densities and different Alfvén speed, respectively. We observe that the CH density is correlated to the amplitude values of the secondary waves and the peak values of the stationary features. The main simulation results look as follows:
- For the first traversing wave we found that the smaller the initial CH density, the smaller the wave’s density amplitude and magnetic field component in $z$-direction, and the larger the amplitudes for phase speed and plasma flow velocity (see Figure \[morphology\_IN\_CH\_no0\] and Figure \[Kin\_traver\_wave\]). The crucial point is, that the different CH densities correspond to different Alfvén speeds inside the CH and hence to different phase speeds of the traversing waves.
- The analysis of the transmitted waves showed that the smaller the initial CH density, the smaller the amplitudes for density, magnetic field component in the $z$-direction and plasma flow velocity, and the larger the phase speed (see Figure \[zoom\_transmitted\_wave\] and Figure \[Kin\_transm\_wave\]).
- We observe a very weak dependence of the first reflection on the CH density with regard to the initial parameters we choose for our simulations. The reflection seems to be mostly driven by the impact of the incoming wave on the CH boundary (see Figure \[morphology\_1D\_part2\] and Figure \[Kin\_first\_reflection\]).
- The kinematic analysis of the second reflection has shown that we do not find a linear correlation between the initial CH density and the peak values for the different parameters of this feature like we have found for traversing and transmitted wave as well as for both stationary features (see Figure \[Kin\_second\_reflection\]).
- For the first stationary feature we have demonstrated that the smaller the initial CH density, the smaller the peak values of density and of magnetic field component in the $z$-direction. The stationary feature is moving slightly in the positive $x$-direction. (see Figure \[zoom\_first\_stat\] and Figure \[Kin\_first\_stat\])
- On the contrary, in the case of the second stationary feature we observe that the smaller the initial CH density, the larger the peak values of density and magnetic field component. This second stationary feature is moving slightly in the negative $x$-direction (see Figure \[zoom\_second\_stat\] and Figure \[Kin\_second\_stat\]).
- By analyzing the kinematics of the density depletion, we found that the smaller the initial CH density, the smaller the minimum density values of the depletion. Moreover, we find that the smaller the density values inside the CH, the larger the values of plasma flow velocity and phase speed (see Figure \[zoom\_density\_depletion\] and Figure \[Kin\_density\_depletion\]).
As already shown in @Piantschitsch2017, these findings strongly support the wave interpretation of large-scale disturbances in the corona. Firstly, effects like reflection and transmission can only be explained by a wave theory. We do not know of any other mechanism that would explain reflection or transmission of coronal waves. Secondly, the simulation results show that the interaction of an MHD wave and a CH is capable of forming stationary features, which were one of the main reasons for the development of a pseudo-wave theory.
We compared our simulation results to observations in @Kienreich_etal2012, where the authors observed reflected features which consist of a bright lane followed by a dark lane in base-difference images. These observations correspond to the first reflection and the density depletion in our simulation.
The authors gratefully acknowledge the helpful comments from the anonymous referee that have very much improved the quality of this paper. This work was supported by the Austrian Science Fund (FWF): P23618 and P27765. B.V. acknowledges financial support by the Croatian Science Foundation under the project 6212 „Solar and Stellar Variability“. I.P. is grateful to Ewan C. Dickson for the proof read of this manuscript. The authors gratefully acknowledge support from NAWI Graz.
|
---
abstract: 'We introduce a new kind of spontaneous four wave mixing process for the generation of photon pairs, in which the four waves involved counter-propagate in a guided-wave $\chi^{(3)}$ medium; we refer to this process as counter-propagating spontaneous four wave mixing (CP-SFWM). We show that for the simplest CP-SFWM source, in which all waves propagate in the same polarization and transverse mode and in which self- and cross-phase modulation effects are negligible, phasematching is attained *automatically* regardless of dispersion in the fiber or waveguide. Furthermore, we show that in two distinct versions of this source (both pumps pulsed, or one pump pulsed and the remaining one monochromatic), the two-photon state is *automatically* factorable provided that the length of the nonlinear medium exceeds a certain threshold, easily achievable in practice since this threshold length tends to be in the range of mm to cm. We also show that if one of the pumps approaches the monochromatic limit, and for a sufficient nonlinear medium length, the bandwidth of one of the two photons in a given pair may be reduced to the level of MHz, compatible with electronic transitions for the implementation of atom-photon interfaces, without the use of optical cavities.'
address:
- |
$^1$ Instituto de Ciencias Nucleares, Universidad Nacional Aut' onoma de M' exico,\
Apartado Postal 70-543, 04510 DF, M' exico
- '$^2$ Departamento de '' Optica, Centro de Investigaci'' on Cient'' ifica y de Educaci'' on Superior de Ensenada, Apartado Postal 360 Ensenada, BC 22860, M'' exico'
author:
- 'Jorge Monroy-Ruz$^1$, Karina Garay-Palmett$^2$[^1], and Alfred B. U’Ren$^1$'
bibliography:
- 'References.bib'
title: 'Counter-propagating spontaneous four wave mixing: photon-pair factorability and ultra-narrowband single photons'
---
\[sec.Intro\]Introduction
=========================
Photon pairs produced by spontaneous parametric processes have enabled many important advances in quantum-enhanced technologies such as quantum metrology [@Giovannetti11], quantum communications [@Gisin07] and quantum computation [@Kok07]. The processes of spontaneous parametric downconversion (SPDC) based on second-order non-linearities [@Burnham84] and of spontaneous four wave mixing (SFWM) based on third-order non-linearities [@Fiorentino02] are well-established as sources of photon pairs. The SFWM process, implemented in optical fibers [@Fulconis07; @Cohen09; @CruzDelgado14], has gained prominence as a viable alternative to SPDC with a number of distinct advantages including the elimination of losses associated with coupling of photon pairs into optical fibers, a greater scope for photon-pair engineering [@Garay07], as well as the possibility of an essentially unlimited interaction length in long optical fibers.
The development of photon-pair sources based on guided-wave non-linear optical media (fibers or waveguides) with tailored spatio-temporal properties is an ongoing field of research. On the one hand, it is well known that in order to herald a quantum-mechanically pure single photon from a photon-pair, it is essential that the two-photon quantum state be free from entanglement in all photonic degrees of freedom [@URen05]. While an appropriate combination of spectral and spatial filtering can render a two-photon state factorable, scalability to higher dimensions for protocols requiring multiple pure heralded single photons necessitates in practice photon-pair engineering so that filtering may be precluded. On the other hand, while photon-atom interfaces require single photons with both frequency and bandwidth matched to those of the atomic transition in question, photon-pair sources based on both SPDC and SFWM tend to be characterized by a bandwidth which is orders of magnitude larger than that of atomic transitions. In order to remedy this, one possibility is to resort to cavity-enhanced processes in which the nonlinear medium is placed inside a high-finesse optical cavity resulting in the emission of photon pairs in the (narrow) spectral modes supported by the cavity [@Ou99; @Kuklewicz06; @bao08; @Jeronimo10; @Fekete13; @Garay13].
In the spontaneous four wave mixing (SFWM) process, two pump photons are annihilated in a guided-wave $\chi^{(3)}$ medium, such as a fiber or waveguide, leading to the generation of signal and idler photon pairs in such a manner that energy and momentum are conserved. In all SFWM sources demonstrated to date, the four waves involved (the two pumps, the signal, and the idler) propagate in the same direction along the fiber or waveguide. In this paper we introduce a new kind of SFWM process, to the best of our knowledge not studied previously, in which the two pump waves are launched from opposite ends so that they counter-propagate in the non-linear medium. We refer to such a process as counter-propagating spontaneous four wave mixing, or CP-SFWM. In this process, one of the daugther photons, which we call signal, is emitted so that it backpropagates with respect to pump 1, while the conjugate idler photon backpropagates with respect to pump 2. Note that $\chi^{(2)}$-based processes have been studied in which the two generated photons counter-propagate leading to interesting spatio-temporal engineering possibilities [@Walton03; @torres04; @tsang2005; @christ2009; @caillet09; @boucher2014; @gatti2015; @corti2016]. Also note that classical implementations of four wave mixing (stimulated process) with counter-propagating fields have been previously proposed and demonstrated [@yariv1977; @jensen1978; @bian2004; @yan2011].
As we discuss below, the CP-SFWM process leads to some unique properties that distinguishes it from standard SFWM. First, regardless of the specific dispersion properties of the non-linear medium, phasematching is automatically attained for all conceivable single-mode fibers (or waveguides), as long as the four waves are characterized by the same dispersion relation, at generation frequencies ($\omega_s$ and $\omega_i$) that match those of the pumps ($\omega_1$ and $\omega_2$), according to $\omega_s=\omega_1$ and $\omega_i=\omega_2$. This symmetry is broken in the presence of self- and/or cross-phase modulation effects or if the four waves involve different polarizations and/or transverse modes leading to slight offsets from $\omega_s=\omega_1$ and $\omega_i=\omega_2$, thus facilitating experimental discrimination of the CP-SFWM photons from scattered pump photons. The automatic phasematching represents a considerable advantage as for any given optical fiber it becomes possible to freely choose the pump frequencies and thus also directly determine the generation frequencies, according to particular needs. Second, as we show in detail below, unlike the case of standard SFWM for which factorability can be accomplished under highly restrictive group velocity matching conditions, involving certain specific combinations of frequencies, in the case of CP-SFWM factorability is accomplished for any phasematched source design, as long as the nonlinear medium length exceeds a certain threshold. Automatic phasematching *and* automatic factorability indeed become a powerful combination in photon-pair source design. Third, we show below that when making one of the two pumps nearly monochromatic and if the nonlinear medium length exceeds a certain threshold, CP-SFWM also permits the generation of photon pairs for which one of the two photons can be characterized by an ultranarrow bandwidth, without resorting to the use of optical cavities.
\[Theory\]Theory of counterpropagating SFWM
============================================
While all conclusions reached in this paper could apply to both waveguide and fiber sources, henceforth we refer to the nonlinear medium as ‘fiber’ with the understanding that it could equally refer to a waveguide. Photon-pair generation experiments based on the process of spontaneous four wave mixing demonstrated to date involve four waves, i.e. pump 1, pump 2, signal, and idler, which propagate along the fiber in the same direction, see for example [@Fiorentino02; @Fulconis07; @Cohen09; @CruzDelgado14]. Here, we propose a SFWM scheme in which the pump fields counter-propagate, i.e. they are launched into the fiber from opposite ends. In such a SFWM interaction, a photon from the pump at frequency $\omega_1$ and travelling in the forward direction, together with and a photon from the pump at frequency $\omega_2$ travelling in the backward direction, are annihilated giving rise to the emission of a counterpropagating photon pair, which is a consequence of energy and momentum conservation constraints. The generated pair is comprised of a backward-propagating signal photon at frequency $\omega_s$ and a forward-propagating idler photon at frequency $\omega_i$. The described interaction is illustrated in figure \[FigEsquema\].
\[sec.State\]The two-photon state
---------------------------------
In this section we describe the two-photon state for the CP-SFWM process in a $\chi^{(3)}$ medium. We will initially write down expressions for the two-photon state which permit each of the four waves to propagate in different transverse and polarization modes, where $k_1(\omega)$, $k_2(\omega)$, $k_s(\omega)$, and $k_i(\omega)$ represent the frequency-dependent wavenumbers for each of the four waves: pump 1(1), pump 2(2), signal(s), and idler(i). Later in the paper we will concentrate our discussion on the case where all four waves are co-polarized and involve the same transverse mode so that $k_1(\omega)=k_2(\omega)=k_s(\omega)=k_i(\omega)\equiv k(\omega)$. Throughout this paper, while the pump $1$ and the idler waves are forward-propagating, the pump $2$ and signal waves are backward propagating; we adopt a sign convention for which all wavenumbers are positive, with explicit signs appearing in accordance to the direction of propagation.
We start from the interaction Hamiltonian governing SFWM processes, given by
$$\label{Hamilt}\hat{H}(t)=\frac{3}{4}\epsilon_o\chi^{(3)} \!\int\!\! d^3 \textbf{r} \hat{E}_{1}^{(+)}(\textbf{r},t)\hat{E}_{2}^{(+)}(\textbf{r},t)\hat{E}_{s}^{(-)}(\textbf{r},t)\hat{E}_{i}^{(-)}(\textbf{r},t),$$
where the integration is carried out over the portion of the nonlinear medium for which the pump fields are temporally and spatially overlapped, $\chi^{(3)}$ is the third-order nonlinear susceptibility, and $\epsilon_o$ is the vacuum electrical permittivity. In Eq. (\[Hamilt\]), the subscripts $^{(+)}/^{(-)}$ refer to the positive frequency / negative frequency parts of the electric field operators. In our analysis, we assume that the two pumps can be well-described by classical fields, i.e. no longer operators, of the form
$$\label{pump}\hat{E}_{\nu}^{(+)}(\textbf{r},t) \rightarrow A_{\nu}f_{\nu}\left(x,y\right)\int\!\!d\omega\alpha_{\nu_\pm}(\omega)\,\mbox{exp}\left[-i\left(\omega t\mp k(\omega)z\right)\right],$$
with $\nu=1,2$ for the two pumps and $A_{\nu}$ represents the field amplitude. $\alpha_{\nu_\pm}(\omega)$ is the spectral envelope (the meaning of the signs $\pm$ is defined below), and $f_{\nu}\left(x,y\right)$ is the transverse spatial field distribution, which is normalized so that $\int\!\!\int|f_{\nu}(x,y)|^2\,dxdy=1$ and is approximated to be frequency-independent within the pump bandwidth.
The quantized signal and idler fields are expressed as
$$\label{Ecuant}\hat{E}_{\mu}^{(+)}(\textbf{r},t)=i\sqrt{\delta k}f_{\mu}(x,y)\nonumber\sum_{k}\mbox{exp}\left[-i(\omega t\mp k(\omega)z)\right]\ell(\omega) \hat{a}_{\mu \pm}(k),$$
with $\mu=s,i$ and $\delta k=2\pi/L_Q$ the mode spacing, written in terms of the quantization length $L_Q$. Function $\ell(\omega)$ is given as follows
$$\label{ldek}\ell(\omega)=\sqrt{\frac{\hbar \omega}{\pi\epsilon_o n^2(\omega)}},$$
in terms of the (linear) refractive index of the nonlinear medium $n(\omega)$ and of Planck’s constant $\hbar$. In Eq. (\[Ecuant\]), $\hat{a}_{\mu \pm }(k)$ is the annihilation operator (the meaning of the signs $\pm$ is defined below), and $f_{\mu}(x,y)$ represents the transverse spatial distribution of the field, which is also normalized as the corresponding pump functions, and is assumed to be frequency-independent within the bandwidth of signal and idler modes.
Note that in equations (\[pump\]) and (\[Ecuant\]) the $-/+$ signs, in front of the propagation constant $k(\omega)$ and the corresponding subscripts $+/-$ in the annihilation operators and the pump spectral envelopes, indicate optical fields propagating along the fiber in the forward/backward directions.
Following a standard perturbative approach [@MandelWolf] and our treatment in reference [@Garay08], it can be shown that the two-photon state produced by CP-SFWM can be written as ${\ensuremath{\left|\Psi\right\rangle}}={\ensuremath{\left|0\right\rangle}}_s{\ensuremath{\left|0\right\rangle}}_i+\eta{\ensuremath{\left|\Psi\right\rangle}}_2$, in terms of the two-photon component
$$\label{2photonstate}
\label{state}{\ensuremath{\left|\Psi\right\rangle}}_2=\sum_{k_{s}}\sum_{k_{i}}\ell(k_{s})\ell(k_{i})F(k_{s},k_{i})\hat{a}^{\dagger}_{s-}(k_{s})\hat{a}^{\dagger}_{i+}(k_{i}){\ensuremath{\left|0\right\rangle}}_s{\ensuremath{\left|0\right\rangle}}_i,$$
and the constant $\eta$, which is related to the conversion efficiency and is given by
$$\label{eta}\eta=i(2\pi)\delta k\frac{3\epsilon_o \chi^{(3)}}{4\hbar}A_1A_2Lf_{eff},$$
where $L$ is the fiber length and $f_{eff}$ is the spatial overlap integral between the four fields given by
$$\label{overlap} f_{eff} =\int\! dx\!\int\! dyf_1(x,y)f_2(x,y)f^{\ast}_s(x,y)f^{\ast}_i(x,y).$$
In Eq. (\[state\]) $k_{s}\equiv k_s( \omega_{s} )$ is the propagation constant for the backward-propagating signal mode, and $k_{i}\equiv k_i( \omega_{i} )$ is the propagation constant for the forward-propagating idler mode; $\hat{a}^{\dagger}_{s-}(k)$ represents the creation operator for the backward-propagating signal mode, while $\hat{a}^{\dagger}_{i+}(k)$ represents the creation operator for the forward-propagating idler mode. $F(k_{s},k_{i})$ is the joint amplitude function, which can be written in terms of frequencies rather than wave numbers, in which case it is referred to as the joint spectral amplitude (JSA), and is expressed as $F(\omega_{s},\omega_{i})$.
In our analysis we first consider a source configuration in which both pumps are pulsed. In this case, the joint spectral amplitude function $F_{\it{P}}(\omega_s,\omega_i)$ can be shown to be given by
$$\label{JSAp}F_{\it{P}}(\omega_{s},\omega_{i})=\int d\omega\alpha_{1+}(\omega)\alpha_{2-}(\omega_{s}+\omega_{i}-\omega)\mbox{sinc}\left[\frac{L}{2}\Delta k\right]e^{i\frac{L}{2} \kappa}e^{i\omega \tau},$$
where $\alpha_{1+}(\omega)$ represents the pump spectral envelope for the forward-propagating pump, $\alpha_{2-}(\omega)$ represents the pump spectral envelope for the backward-propagating pump, and $\tau$ represents the time of arrival *difference* between the two pump pulses at their respective ends of the fiber. Note that $\tau$ can be controlled externally with a relative delay between the two pumps; in particular, $\tau=0$ implies that the pump pulses corresponding to pumps $1$ and $2$ arrive at the same time at the two ends of the fiber. Eq. (\[JSAp\]) is expressed in terms of the phase mismatch function $\Delta k\equiv\Delta k(\omega,\omega_s,\omega_i)$, and the function $\kappa\equiv \kappa(\omega,\omega_s,\omega_i)$ defined as
$$\begin{aligned}
\label{PMp}
\Delta k= k_1(\omega)- k_2(\omega_{s}+\omega_{i}-\omega)- k_s(\omega_{s})+k_i(\omega_{i})+\phi_{NL},\\
\kappa= k_1(\omega)+ k_2(\omega_{s}+\omega_{i}-\omega)+k_s(\omega_{s})+k_i(\omega_{i}),\end{aligned}$$
where $\phi_{NL}$ is a nonlinear phase shift derived from self-phase and cross-phase modulation (see below for further discussion and for expressions). Note that the energy conservation constraint is already included in the resulting joint amplitude.
Let us now consider a pumps configuration defined as the limit where the backward-propagating pump wave becomes monochromatic at frequency $\omega_{cw}$, in which case the corresponding electric field can be expressed as $E_{cw}^{(+)}(\textbf{r},t)=af_2\left(x,y\right)\mbox{exp}\left[-i\left(\omega_{cw} t+k(\omega_{cw})z\right)\right]$ with $a$ the field amplitude, while the forward-propagating pump remains broadband; we refer to this as the mixed pumps configuration. In this case, the JSA function becomes
$$\label{JSAm}F_{\it{M}}(\omega_{s},\omega_{i})=\alpha_{+}(\omega_{s}+\omega_{i}-\omega_{cw})\mbox{sinc}\left[\frac{L}{2}\Delta k_{\it{M}}\right]e^{i\frac{L}{2}\kappa_{\it{M}}},$$
where $\Delta k_{\it{M}}\equiv\Delta k_M(\omega_s,\omega_i)$ and $ \kappa_{\it{M}}\equiv\kappa_{\it{M}}(\omega_s,\omega_i)$ are defined as
$$\begin{aligned}
\label{PMm}
\Delta k_{\it{M}}= k_1(\omega_{s}+\omega_{i}-\omega_{cw})- k_2(\omega_{cw})- k_s(\omega_{s})+k_i(\omega_{i})+\phi_{NL},\\
\kappa_{\it{M}}= k_1(\omega_{s}+\omega_{i}-\omega_{cw})+ k_2(\omega_{cw})+k_s(\omega_{s})+k_i(\omega_{i}).\end{aligned}$$
![(a) Schematic of the CP-SFWM process. The gaussian-shaped pumps are represented in solid red (pump 1) and solid green (pump 2), while the generated CP-SFWM photons are indicated in dashed red (signal) and dashed green (idler); arrowheads indicate the directions of propagation for the four waves. (b) Energy level diagram of the process. (c) Phasematching diagram. The solid black lines represent the signal and idler frequencies that fulfil phasematching, as a function of $\omega_{1}$ and for $\omega_2$ fixed at the particular value $\omega_2 = (2\pi c)/0.532 \mu m$.[]{data-label="FigEsquema"}](Figure1.pdf){width="10cm"}
The nonlinear phase shift $\Phi_{NL}$, appearing in Eq. (\[PMp\]) (for the pulsed pumps case), can be been shown to be given as follows [@Garay07; @Garay2011r]
$$\begin{aligned}
\label{phiNL}
\Phi_{NL}=(\gamma_1-2\gamma_{21}-2\gamma_{s1}+2\gamma_{i1})
P_1 -(\gamma_2-2\gamma_{12}+2\gamma_{s2}-2\gamma_{i2}) P_2,\end{aligned}$$
where $P_1$ and $P_2$ represent the peak powers for pumps $1$ and $2$ (related, for pumps with Gaussian spectra, as $P_\nu=p_\nu \sigma_\nu / [\sqrt{2 \pi} R]$ with the average pump powers $p_\nu$, where $R$ is the repetition frequency and $\sigma_\nu$ is the corresponding bandwidth). The nonlinear phase shift $\Phi_{NL}$, appearing in Eq. (\[PMm\]) (for the mixed pumps case) is given by Eq. (\[phiNL\]), with the substitution $P_2 \rightarrow p_2$.
The coefficients $\gamma_{1}$ and $\gamma_{2}$ result from self-phase modulation (SPM) of the two pumps, and are given, with $\nu=1,2$ by
$$\label{gam1}
\gamma_{\nu}=\frac{3\chi^{(3)}\omega_{\nu}^0f^{\nu}_{eff}}{4\epsilon_0c^2n_{\nu}^2}.$$
In Eq. (\[gam1\]), the refractive index $n_{\nu}\equiv n(\omega_{\nu}^0)$ and the spatial overlap integral $f_{eff}^{\nu}\equiv \int\!\int\!dxdy|f_{\nu}(x,y)|^4$ (where the integral is carried out over the transverse dimensions of the fiber) are defined in terms of the carrier frequency $\omega_{\nu}^0$ for pump-mode $\nu$ [@Agrawal].
In contrast, coefficients $\gamma_{\mu\nu}$ ($\nu=1,2$ and $\mu=1,2,s,i$) correspond to the cross-phase modulation (CPM) contributions that result from the dependence of the refractive index experienced by wave $\mu$, with $\mu=1,2,s,i$, on the pump intensities ($\nu=1,2$; note that CPM of the signal/idler photons on the pumps as well as SPM of the signal and idler waves are small effects which we neglect). These coefficients are given by
$$\label{gam2}
\gamma_{\mu\nu}=\frac{3\chi^{(3)}\omega_{\mu}^0f^{\mu\nu}_{eff}}{4\epsilon_0c^2n_{\mu}n_{\nu}},$$
where $n_{\mu,\nu}\equiv n(\omega_{\mu,\nu}^0)$ is defined in terms of the central frequency $\omega_{\mu,\nu}^0$ for each of the four participating fields, and $f_{eff}^{\mu\nu}\equiv
\int\!\int\!dxdy|f_{\mu}(x,y)|^2|f_{\nu}(x,y)|^2$ is the two-mode spatial overlap integral (note that $f_{eff}^{\mu\nu}=f_{eff}^{\nu\mu}$).
\[sec.flux\]Expressions for the emitted flux
---------------------------------------------
In designing two-photon sources, it is helpful to be able to estimate the source brightness in terms of all relevant experimental parameters. Expressions for the emitted flux for co-propagating and co-polarized SFWM in single-mode fibers have been reported by us previously [@Garay08].
The number of photon pairs generated per second, or source brightness, is given by
$$\label{defbrightness}
\label{NumPhot} N = R\sum_k{\ensuremath{\left\langle\psi_2\right|}}\hat{a}_{s-}^{\dagger}(k)\hat{a}_{s-}(k){\ensuremath{\left|\psi_2\right\rangle}},$$
where ${\ensuremath{\left|\psi_2\right\rangle}}$ is defined in equation (\[state\]) and $R$ is the pump repetition rate (for the pulsed pump in the mixed pumps case, and assumed to be equal for both pumps in the pulsed pumps case); note that the brightness in Eq. (\[defbrightness\]) can likewise be expressed in terms of the idler annihilation operator. From this equation we have derived expressions for the number of photon pairs generated per second for the two pump configurations described above, which are represented by $N_{\it{P}}$ and $N_{\it{M}}$, respectively. In the analysis we assume that the pulsed pump fields have a gaussian spectral envelope
$$\label{envSpec}\alpha_\mu(\omega)=\frac{2^{1/4}}{\pi^{1/4}\sqrt\sigma_\mu}\,\exp
\left[-\frac{(\omega-\omega^0_\mu)^2}{\sigma_\mu^2}\right],$$
where $\omega^0_\mu$ and $\sigma_\mu$ (with $\mu=1,2$) are the central pump frequency and the pump bandwidth for the two pumps, respectively. Note that the function $\alpha_\mu(\omega)$ has been normalized so that $\int\!d\omega\! \mid\alpha_\mu(\omega)\mid^2=1$.
Substituting the two-photon state, Eq. (\[2photonstate\]), into Eq. (\[defbrightness\]) while appropriately turning sums into integrals in the limit $\delta k \rightarrow 0$, it can be shown that $N_{\it{P}}$ is given by
$$\label{NumPhot_P} N_{\it{P}}=\frac{2^5n_{1}n_{2}c^2L^2\gamma^2p_1p_2}{\pi^3\omega_{1}^0\omega_{2}^0\sigma_1\sigma_2R}\int\! d\omega_s\!\int\! d\omega_i\,h(\omega_s,\omega_i)\mid F_{\it{P}}(\omega_s,\omega_i)\mid^2,$$
where $n_1\equiv n_1(\omega_1^0)$ ($n_2\equiv n_2(\omega_2^0)$), $c$ is the speed of light in the vacuum, $L$ is the fiber length, $p_{\nu}$ and $\sigma_{\nu}$ (with $\nu=1,2$) are the average power and bandwidth of the two pumps, respectively, $F_{\it{P}}(\omega_s,\omega_i)$ is the JSA given in the equation (\[JSAp\]), and $h(\omega_s,\omega_i)$ is a function defined as
$$\label{funh} h(\omega_s,\omega_i)=\frac{\omega_sk'_s(\omega_s)}{n_s^2(\omega_s)}\frac{\omega_ik'_i(\omega_i)}{n_i^2(\omega_i)},$$
where $k'_{\mu}(\omega_{\mu})$ denotes the frequency derivate of the propagation constant $k_{\mu}(\omega_{\mu})$, and $\gamma$ is the SFWM nonlinear coefficient given by
$$\label{gamma} \gamma = \frac{3\chi^{(3)}\sqrt{\omega_{1}^0\omega_{2}^0}f_{eff}}{4\epsilon_0c^2n_{1}n_{2}},$$
in terms of the third-order nonlinear susceptibility, $\chi^{(3)}$, the electric permittivity of free space, $\epsilon_0$, and the spatial overlap integral $f_{eff}$ defined in Eq. (\[overlap\]). Note that in our analysis we have assumed that the transverse electric field distributions for the various fiber modes depend only weakly on the frequency, so that $\gamma$ has been regarded as a constant, and taken out of the integral, as in Eq. (\[NumPhot\_P\]).
Similarly, it can be shown that for the mixed pumps case the number of photon pairs generated per second, $N_{\it{M}}$, is given by
$$\label{NumPhot_M} N_{\it{M}}=\frac{2^{11/2}n_{1}n_{2}c^2L^2\gamma^2p_1p_2}{\pi^{3/2}\omega_{1}^0\omega_{2}\sigma}\int\! d\omega_s\!\int\! d\omega_i\,h(\omega_s,\omega_i)\mid F_{\it{M}}(\omega_s,\omega_i)\mid^2,$$
where $p_1$ and $\sigma$ represent the average power and bandwidth of the pulsed pump, respectively; $p_2$ is the power of the monochromatic pump wave; $F_{\it{M}}(\omega_s,\omega_i)$ is the JSA given in the equation (\[JSAm\]); $h(\omega_s,\omega_i)$ is given by equation (\[funh\]) and $\gamma$ is defined according to equation (\[gamma\]).
\[sec.PM\]Phasematching properties and SFWM-pump discrimination
---------------------------------------------------------------
In order for the CP-SFWM process to exist, linear momentum must be conserved which is equivalent to a phasematching condition $\Delta k =0$ for the pulsed pumps case, or $\Delta k_M=0$ for the mixed pumps case (note that since in general all four waves are polychromatic, these phasematching conditions are fulfilled exactly for specific frequencies regarded as “central” for each wave). It is straightforward to verify from Eqs. (\[PMp\]) and (\[PMm\]) that if all four waves propagate in the same transverse and polarization mode, then phasematching is always attained, provided that the nonlinear term $\phi_{NL}$ is negligible, at frequencies satisfying the following relationships: $\omega_1=\omega_{s}$ and $\omega_2=\omega_{i}$. This is a remarkable property of CP-SFWM: phasematching is fulfilled *automatically*, for an arbitrary single-mode fiber, using frequency non-degenerate pumps centered at $\omega_1$ and $\omega_2$ so as to generate a backward-propagating signal photon with frequency $\omega_{s}=\omega_1$ paired with a forward-propagating idler photon with frequency $\omega_{i}=\omega_2$. In other words, basic phasematching properties (i.e. the determination of emission SFWM frequencies as a function of pump frequencies), become decoupled from the fiber dispersion and are in fact identical for all conceivable single-mode fibers. Note that a particular case of the scenario above is that for which the pumps are frequency-degenerate which in fact leads to all four waves being frequency degenerate. Also, note that a possible source of noise in CP-SFWM is spontaneous Brillouin scattering of the pump fields, which would appear in the same frequencies and directions of propagation as the generated photons [@Agrawal].
In figure \[FigEsquema\](c) we have plotted the $\Delta k=0$ contour (solid black straight lines), i.e. the signal and idler frequencies which satisfy perfect phasematching, as a function of the pump frequency $\omega_1$, while $\omega_2$ remains fixed at $\omega_2=2\pi c/0.532\mu$m. Note that this diagram is “universal”, in the sense that it applies to all conceivable single-mode fibers. While a fixed pump 2 frequency leads to an equally fixed idler frequency, since $\omega_i=\omega_2$, there is a linear dependence between the remaining two frequencies, as $\omega_s=\omega_1$. Note also that the intersection of the two straight lines corresponds to the degenerate pumps case, for which $\omega_1=\omega_2=2\pi c/0.532\mu$m. In particular, in the figure \[FigEsquema\](c) the red vertical line corresponds to $\omega_1=2\pi c/0.820\mu$m, so that its intersection with the $\Delta k=0$ contour indicates that perfect phasematching occurs for $\omega_s=2\pi c/0.820\mu$m and $\omega_i=2\pi c/0.532\mu$m. A different choice of fixed pump $2$ frequency would simply lead to a vertically-displaced horizontal tuning curve for the idler photon. Let us emphasize that the automatic phasematching observed for CP-SFWM is *achromatic* in the sense that it is attained for any choice of $\omega_1$ and $\omega_2$ (leading to $\omega_s=\omega_1$ and $\omega_i=\omega_2)$, regardless of the specific underlying fiber dispersion. This opens a wealth of possibilities for the implementation of photon-pair sources in optical fibers.
Note that if the nonlinear term $\phi_{NL}$ is non-zero, the symmetry between each pair of counterpropagating pump and generated SFWM photon is broken; in principle, this could be useful in order to slightly offset the generation frequencies from the pump frequencies so as to simplify the experimental discrimination of signal and idler photons from scattered pump photons. Likewise, this symmetry can be broken with cross-polarized SFWM processes of the kind $xyxy$ or $xxyy$ in brirefringent fibers. However, for experimental conditions regarded as typical (conventional fibers and typical values of pump power and/or typical brifrefringence values) the resulting offset tends to be insufficient in practice for the effective discrimiantion between SFWM photon pairs and pump photons.
Another interesting possibility is for the four waves to propagate in different transverse modes (in a few-mode or multi-mode fiber), likewise leading to an offset of the generation frequencies from the pump frequencies. Thus, let us discuss the case of CP-SFWM implemented in a few-mode optical fiber [@Garay16; @Cruz2016] leading to an intermodal process; this means allowing some of the waves involved in the SFWM process to travel in higher-order transverse modes. As an example, if the forward propagating pump travels in the fundamental fiber mode and the signal photon travels in a certain higher-order mode $X$, while the backward-propagating pump mode travels in the same higher-order mode $X$ and the idler photon travels in the fundamental mode, then perfect phasematching will occur for signal and idler frequencies that are shifted from those of the pumps. Specifically, this will result in a signal photon with frequency $\omega_1+\delta$ and in an idler photon with frequency $\omega_2-\delta$, with a frequency offset $\delta$ which depends on the dispersion relations of the two modes; note that while the frequency offsets are equal (opposite in sign), the resulting wavelength offsets will differ between signal and idler. Importantly, as the order of mode $X$ increases, $\delta$ also increases. In table \[Tablamodes\] we summarize the emission wavelengths and the resulting wavelength offsets that result from intermodal CP-SFWM in a step-index fiber (with numerical aperture $NA=0.3$ and core radius $r=2\mu$m) that supports three higher-order modes, for the case in which the pump wavelengths are $820$nm (forward propagating pump) and $532$nm (backward propagating pump).
Fiber mode $X$ $\lambda_s$ (nm) $\Delta \lambda_s$(nm) $\lambda_i$ (nm) $\Delta \lambda_i$(nm)
---------------- ------------------ ------------------------ ------------------ ------------------------
$LP_{11}$ 816.1 -3.9 533.7 1.7
$LP_{21}$ 811.1 -8.9 535.8 3.8
$LP_{02}$ 809.7 -10.3 536.4 4.4
: Emission wavelengths ($\lambda_s$ and $\lambda_i$) and wavelength offsets ($\Delta \lambda_s$ and $\Delta \lambda_i$) for intermodal CP-SFWM, for different choices of excited mode $X$, in a few-mode step-index fiber with numerical aperture $NA=0.3$ and core radius $r=2\mu$m.[]{data-label="Tablamodes"}
We point out that while we have verified that the conclusions reached in this paper about factorability and ultra-narrowband single photon generation are unaffected by the use of the intermodal CP-SFWM process described above for signal/idler-pumps discrimination, for simplicity in the rest of the paper we concentrate on a CP-SFWM process which utilizes a single transverse mode.
\[sec.analytical\]Closed analytical expressions for the joint spectral amplitude and the emitted flux
-----------------------------------------------------------------------------------------------------
In this section we show that under certain approximations it becomes possible to derive analytical expressions, in closed form, for both the joint spectral amplitude and for the emitted flux. Specifically, these approximations involve: i) writing the propagation constant $k(\omega)$, for each of the four interacting fields, as a first-order Taylor expansion around the frequencies for which perfect phasematching is obtained, and ii) assuming that the function $h(\omega_s,\omega_i)$ (see eq. (\[funh\])) varies slowly within the spectral range of interest, so that we can regard it as a constant when evaluating the integrals in equations (\[NumPhot\_P\]) and (\[NumPhot\_M\]) in the section \[sec.flux\]. Note that these approximations are no longer valid for large spectral spreads of the signal and idler frequencies around the frequencies which yield perfect phasematching.
For the pulsed pumps case (assumed to be Gaussian in spectrum, see equation (\[envSpec\])), under the approximations mentioned above, and using the integral form of the $\mbox{sinc}$ function
$$\label{sincF} \mbox{sinc}(x)=\frac{1}{2}\int_{-1}^1d\xi e^{ix\xi},$$
the integral in equation (\[JSAp\]) can be carried out analytically resulting in the approximate expression for the joint spectral amplitude $ f^{lin}_{\it{P}}(\nu_s,\nu_i)=\alpha_{\it{P}}(\nu_s,\nu_i)\phi_{\it{P}}(\nu_s,\nu_i)$, where $\alpha_{\it{P}}(\nu_s,\nu_i)$ is determined by the two pump waves and is given by
$$\label{alpha_Pul} \alpha_{\it{P}}(\nu_s,\nu_i)=\mbox{exp}\left[-\frac{(\nu_s+\nu_i)^2}{\sigma_1^2+\sigma_2^2}\right],$$
while $\phi_{\it{P}}(\nu_s,\nu_i)$ is determined both by the pump waves and the properties of the fiber, and has the form
$$\label{phi_Pul}
\phi_{\it{P}}(x)=\mbox{exp}\left[-B^2x^2\right]\left[\mbox{erf}\left(\frac{1+\Lambda}{4B}+iBx\right) +\mbox{erf}\left(\frac{1-\Lambda}{4B}-iBx\right)\right],$$
given in terms of the variable $x$, and parameters $B$ and $\Lambda$, defined as
$$\begin{aligned}
\label{x} x=T_s\nu_s+T_i\nu_i,\\ \label{B}
B=\frac{\sqrt{\sigma_1^2+\sigma_2^2}}{t_{12}\sigma_1\sigma_2}, \\
\Lambda=\frac{1}{t_{12}}(2\tau +\tau_{12}),\end{aligned}$$
where $\mbox{erf(.)}$ denotes the error function, $\nu_{\mu} = \omega_{\mu}-\omega_{\mu}^0$ are detuning variables (${\mu}=s, i)$, and $T_s$, $T_i$, $t_{12}$, and $\tau_{12}$ are defined in tables \[Tpar\]; $\tau$ was defined in the context of Eq. (\[JSAp\]). Note that here $\omega_{\mu}^0$ (with $\mu=1,2,s,i$) represent the central frequencies of the four waves involved . The definitions provided in tables \[Tpar\] correspond to temporal variables; $\tau_{ij}$ terms represent transit time *differences* through the fiber between waves $i$ and $j$, while $t_{ij}$ terms represent transit time *sums* through the fiber between waves $i$ and $j$. In conventional fibers, with a length of a few cm, $t_{ij}$ is on the order of tenths of nanoseconds, while $\tau_{ij}$ is on the order of few picoseconds.
In the left-hand-side table we have shown the various temporal parameters which define the two-photon state in the general case where the four waves may involve different polarizations and transverse modes. In the right-hand-side table, we have specialized this to the case for which all four waves involve the same polarization and the same transverse mode.
$T_s=t_{2s}-\frac{\sigma_1^2}{\sigma_1^2+\sigma_2^2}t_{12}$ $T_{i}=\tau_{2i}-\frac{\sigma_1^2}{\sigma_1^2+\sigma_2^2}t_{12}$
------------------------------------------------------------- ------------------------------------------------------------------
$t_{12} = L(k^{\prime}_1+k^{\prime}_2)$ $\tau_{12}= L(k^{\prime}_1-k^{\prime}_2)$
$t_{1s} = L(k^{\prime}_1+k^{\prime}_s)$ $\tau_{1s} = L(k^{\prime}_1-k^{\prime}_s)$
$t_{1i} = L(k^{\prime}_1+k^{\prime}_i)$ $\tau_{1i} = L(k^{\prime}_1-k^{\prime}_i)$
$t_{2s} = L(k^{\prime}_2+k^{\prime}_s) $ $ \tau_{2s} = L(k^{\prime}_2-k^{\prime}_s)$
$t_{2i} = L(k^{\prime}_2+k^{\prime}_i) $ $ \tau_{2i} = L(k^{\prime}_2-k^{\prime}_i)$
: Temporal parameters in analytical expressions for two cases. Left: general case, for which the four waves could involve different polarizations and propagation modes, right: identical polarizations and propagation modes for all four waves; note that the definition $k^{\prime}_{\mu}\equiv k^{\prime}_{\mu}(\omega_{\mu}^0)$ is used throughout.[]{data-label="Tpar"}
$T_s=\frac{\sigma_2^2}{\sigma_1^2+\sigma_2^2}t_{12}$ $T_{i}=-\frac{\sigma_1^2}{\sigma_1^2+\sigma_2^2}t_{12}$
------------------------------------------------------ ---------------------------------------------------------
$t_{12} = L(k^{\prime}_1+k^{\prime}_2)$ $\tau_{12}= L(k^{\prime}_1-k^{\prime}_2)$
$t_{1s} = 2 L k^{\prime}_1 $ $\tau_{1s} =0$
$t_{1i} = t_{12}$ $\tau_{1i} = \tau_{12}$
$t_{2s} =t_{12} $ $ \tau_{2s} =-\tau_{12}$
$t_{2i} = 2 L k^{\prime}_2 $ $ \tau_{2i} = 0$
: Temporal parameters in analytical expressions for two cases. Left: general case, for which the four waves could involve different polarizations and propagation modes, right: identical polarizations and propagation modes for all four waves; note that the definition $k^{\prime}_{\mu}\equiv k^{\prime}_{\mu}(\omega_{\mu}^0)$ is used throughout.[]{data-label="Tpar"}
Note that for a sufficiently large time of arrival difference between the two pump pulses at the opposite fiber ends $\tau$, leading to the condition $\Lambda \gg 1$, the pump pulses overlap temporally outside the fiber and the process ceases to occur. If this time of arrival difference $\tau$ vanishes, the value of $\Lambda$ tends to be small since it is given by the ratio of the transit time *difference* through the fiber of the two pumps, divided by transit time *sum*; thus, often we may approximate $\Lambda \approx 0$.
Note that the assumptions (see first paragraph of this section) used for the derivation of the approximate expression for the joint spectral amplitude $f^{lin}_{\it{P}}(\nu_s,\nu_i)$ are no longer valid for sufficiently large signal and idler spectral spreads around the central SFWM frequencies $\omega_s^0$ and $\omega_i^0$. It is worth pointing out that for the specific source designs presented in this paper (see Figs. \[JSIsynthesis\] and \[JSIfactSyn\], below) these approximations are well justified: plots of the joint spectrum derived from the expression $|f^{lin}_{\it{P}}(\nu_s,\nu_i)|^2$ are in excellent agreement with plots derived from direct numerical integration, without resorting to approximations, according to Eq. (\[JSAp\]).
The same assumptions considered in the derivation of $ f^{lin}_{\it{P}}(\nu_s,\nu_i)$ can be applied in equation (\[NumPhot\_P\]) in order to get a closed analytical expression of the emission rate, which leads to
$$\label{N_analt_P} N_{\it{P}}^{lin}=\frac{2^5n_{1}n_{2}c^2\gamma^2p_1p_2h(\omega_s^0,\omega_i^0)}{R(k^{\prime}_1+k^{\prime}_2)( k^{\prime}_s+k^{\prime}_i)\omega_{1}^0\omega_{2}^0}\left[\mbox{erf}\left(\frac{1+\Lambda}{2\sqrt{2}B}\right) +\mbox{erf}\left(\frac{1-\Lambda}{2\sqrt{2}B}\right)\right].$$
From the above equation, using the property that the $\mbox{erf}(x)$ function saturates to a value of $1$ for $x \gtrsim 2$ (or to a value of $-1$ for $x \lesssim -2$), we may show that there exists an effective fiber length $L_{eff}$ given by
$$\label{Leff} L_{eff}=\frac{4\sqrt{2}\sqrt{\sigma_1^2+\sigma_2^2}}{(1+\Lambda)(k^{\prime}_1+k^{\prime}_2)\sigma_1\sigma_2}
= \frac{4\sqrt{2}\sqrt{\Delta t_1^2+\Delta t_2^2}}{(1+\Lambda)(k^{\prime}_1+k^{\prime}_2)},$$
where $\Delta t_1\equiv 1/ \sigma_1$ and $\Delta t_2\equiv 1/ \sigma_2$ represent the temporal durations for pump $1$ and pump $2$, respectively, with the property that increasing the fiber length beyond $L=L_{eff}$ does not lead to any further increase of the source brightness; thus, $L_{eff}$ corresponds to the maximum interaction length. Physically, $L_{eff}$ represents the length of fiber over which the two pumps overlap temporally. Note that making one of the two pumps approach the continuous wave (monochromatic) limit implies that the interaction length can increase without limit, which as is described below is helpful for the optimization of the source brightness.
Let us consider a case where both pumps have a non-zero bandwidth; we can then write $B=\sqrt{1+r}/(\sigma_1 t_{12})$, or $B = \sqrt{1+r}\Delta t_1 / t_{12}$, (with $r \equiv \sigma_1^2/\sigma_2^2$). Thus, if $\Delta t_1$ is much smaller than the sum of the transit times through the fiber of the two pump pulses, represented by $t_{12}$, then $B$ can be a small number. Essentially, a small $B$ implies that since the interaction length is much shorter than the fiber length, the two-photon state is free from any effects related to the air-fiber and fiber-air interfaces. In this $B\to0$ limit, which can always be reached through a combination of pulsed pumps with a sufficiently small pump duration together with a sufficiently long fiber, the phasematching function becomes $\phi_{\it{P}}(x)\to 2\mbox{exp}(-B^2x^2)$. This limit is interesting for applications where the suppression of the sinc-function sidelobes is beneficial, as is the case for the generation of very high-quality factorable states.
Let us now consider the case where the pump bandwidths are highly unbalanced. In particular, the condition $\sigma_1 \ll \sigma_2$ leads to $B \approx 1/(t_{12} \sigma_1)$, while similarly $\sigma_2 \ll \sigma_1$ leads to $B \approx 1/(t_{12} \sigma_2)$. In the limit where the smaller of the two bandwidths becomes very small, the value of $B$ becomes very large, in which case it may be shown that the phasematching function becomes $\phi_{\it{P}}(x)\to\mbox{sinc}(x/2)$. In practice, for values $B \gtrsim 1.0$, the phasematching function is already well described by a sinc function; in this regime, unlike for the small $B$ limit, the fiber edges play an essential role.
![Absolute value of the phasematching function \[see equation (\[phi\_Pul\])\] for different values of parameter $B$, with $\Lambda=-0.00685$ (corresponding to pumps at 532nm and 820nm with $\tau=0$, assuming a step-index fiber with core radius $r=1.5 \mu$m and numerical aperture $NA=0.13$).[]{data-label="PMfunc"}](Figure2.pdf){width="14cm"}
The behavior of the function $ |\phi_{\it{P}}(x)|$ as parameter $B$ varies is summarized in Fig. \[PMfunc\]. On the one hand, panel (a) illustrates the small $B$ limit (in this case with $B=0.01$), in which the function $|\phi_{\it{P}}(x)|$ becomes the Gaussian function $\mbox{exp}(-B^2x^2)$. On the other hand, panel (c) illustrates the large $B$ limit (in this case with $B=1$) in which the function $|\phi_{\it{P}}(x)|$ becomes $\mbox{sinc}(x/2)$. In both of these panels, the $|\phi_{\it{P}}(x)|$ function as given by Eq. (\[phi\_Pul\]) is plotted with a solid black line, while the Gaussian or sinc limiting behaviors are plotted with a dashed yellow line. It becomes evident that there is an excellent agreement between these. In panel (b) we show an intermediate case with $B=0.2$ for which $|\phi_{\it{P}}(x)|$ is not well described neither by a Gaussian nor by a sinc function.
Let us now analyze how the functions $\alpha_P(\nu_s,\nu_i)$ and $\phi_P(\nu_s,\nu_i)$ define the joint spectral intensity of CP-SFWM photon pairs. It is clear from equations (\[alpha\_Pul\]) and (\[phi\_Pul\]) that while $\alpha_P(\nu_s,\nu_i)$ is oriented at $-45^{\circ}$ in $\{\omega_s,\omega_i\}$ space with a width $\sqrt{\sigma_1^2+\sigma_2^2}$, the orientation and width of $\phi_{\it{P}}(\nu_s,\nu_i)$ depend, both, on pump and fiber parameters. The orientation angle of the function $\phi_{\it{P}}(\nu_s,\nu_i)$ is given by
$$\label{ang}
\theta_{si}=\arctan\left(-\frac{T_s}{T_i}\right)=\arctan \left(\frac{\sigma_2^2}{\sigma_1^2}\right),$$
with the last equality valid for the case where all four waves have the same polarization and transverse mode. Note that $\sigma_2^2/\sigma_1^2 \ge 0$, so that $\theta_{si}$ is constrained as $0\le\theta_{si}\le90^\circ$, i.e. the function $\phi_{\it{P}}(\nu_s,\nu_i)$ has contour curves with non-negative slope, including the two limiting cases of horizontal and vertical orientations. As regards the width of the function $\phi_{\it{P}}(\nu_s,\nu_i)$, it is helpful to consider separately the limiting cases for large $B$ where this function is well described by $\mbox{sinc}(x/2)$, and for small $B$ for which this function becomes $\exp(-B^2 x^2)$. In the first case, the width is inversely proportional to the fiber length $L$ (since both $T_s$ and $T_i$ in $x=T_s \nu_s+T_i \nu_i$ are linear in $L$). In the second case, the width no longer depends on $L$ (since $B$ is proportional to $L^{-1}$ while $x$ is proportional to $L$). Thus, as the fiber length is increased the width of the function $\phi_{\it{P}}(\nu_s,\nu_i)$ diminishes, eventually the shape turning Gaussian at which point the width and shape of the function $\phi_{\it{P}}(\nu_s,\nu_i)$ no longer responds to further increasing $L$. Thus, increasing $L$ beyond the length defined by non-zero temporal overlap between the two pump pulses, $L_{eff}$, has no effect *neither* on the flux *nor* on the joint spectral intensity.
![Spectral correlation properties of CP-SFWM two-photon states, assuming a step index fiber (core radius $r=1.5\mu$m and $NA=0.13$) with $L=1$cm. (a)-(d) Pulsed pumps case with $\sigma_1=0.01THz$ and $\sigma_2=0.03THz$. (e)-(h) Pulsed pump case with $\sigma_1=\sigma_2=0.01THz$. (i)-(l) Mixed pump case with $\sigma=0.01THz$. $\alpha(\omega_s,\omega_i)$ is the pump envelope function (given by equation (\[alpha\_Pul\]) for the pulsed case and by equation (\[pump\_analt\_M\]) for the mixed case). $\phi(\omega_s,\omega_i)$ is the phasematching function (given by equation (\[phi\_Pul\]) for the pulsed case and by equation (\[PM\_analt\_M\]) for the mixed case). $f^{lin}(\omega_s,\omega_i)$ represents the JSI under the linear $\Delta k$ approximation. $F(\omega_s,\omega_i)$ represents the JSI without resorting to approximations (calculated numerically from equation (\[JSAp\]) for the pulsed case and from equation (\[JSAm\]) for the mixed case).[]{data-label="JSIsynthesis"}](Figure3.pdf){width="14cm"}
The discussion of the previous paragraph is illustrated in figure \[JSIsynthesis\], in which we show the two-photon state obtained for a step-index fiber (with core radius $r=1.5 \mu$m and numerical aperture $NA=0.13$) with length $L=1$cm and two different pulsed pumps configurations: i) $\sigma_1=0.01THz$ and $\sigma_2=0.03THz$, panels (a)-(d), for which $B=1.07$; ii) $\sigma_1=\sigma_2=0.01$THz, panels (e)-(h), for which $B=1.43$. In this block of figures, the function $\alpha_P(\nu_s,\nu_i)$ is shown in the first column, the function $\phi_{\it{P}}(\nu_s,\nu_i)$ in the second column, the JSI $|\alpha_P(\nu_s,\nu_i)\phi_{\it{P}}(\nu_s,\nu_i)|^2$ in the third column, while the numerically-calculated JSI is shown in the fourth column. Note that while the third column corresponds to the analytical joint spectral intensity, defined as $|f^{lin}_{\it{P}}(\nu_s,\nu_i)|^2$, the fourth column was obtained by numerical integration of equation (\[JSAp\]) without resorting to the linear approximation of the $\Delta k$ function. It is evident that the approximate analytical results agree extremely well with the numerically-calculated ones. Also, consistent with Eq. (\[ang\]), while the orientation of the function $\phi_{\it{P}}(\nu_s,\nu_i)$ is $45^{\circ}$ for equal pump bandwidths, it approaches a vertical orientation for unequal pump banwidths $\sigma_2>\sigma_1$, and becomes fully vertical for $\sigma_2 \gg \sigma_1$. Similarly, (not shown in the figure), for $\sigma_2<\sigma_1$ the function $\phi_{\it{P}}(\nu_s,\nu_i)$ approaches a horizontal orientation while it becomes fully horizontal for $\sigma_2\ll\sigma_1$.
For highly unbalanced pump bandwidths, and in particular when one of the two pumps approaches the monochromatic limit, the interaction length between the two pump pulses increases, in principle, without limit. Such a mixed pumps configuration could have important implications for the ability to reach high emission rates.
Following a similar treatment as used above for the pulsed pumps case, it can be shown that the JSA function given in equation (\[JSAm\]) can be expressed, under the linear $\Delta k$ approximation, as $ f^{lin}_{\it{M}}(\nu_s,\nu_i)=\alpha_{\it{M}}(\nu_s,\nu_i)\phi_{\it{M}}(\nu_s,\nu_i)$, with the pump envelope function $\alpha_{\it{M}}(\nu_s,\nu_i)$ and the phasematching function $\phi_{\it{M}}(\nu_s,\nu_i)$ given by
$$\label{pump_analt_M} \alpha_{\it{M}}(\nu_s,\nu_i)=\mbox{exp}\left[-\frac{(\nu_s+\nu_i)^2}{\sigma^2}\right],$$
$$\label{PM_analt_M} \phi_{\it{M}}(\nu_s,\nu_i)=\mbox{sinc}\left[\frac{1}{2}(\tau_{1s}\nu_s+t_{1i}\nu_i)\right]\mbox{exp}\left[i t_{1s}\nu_s+i t_{1i}\nu_i\right],$$
where $\sigma$ is the bandwidth of the pulsed pump and $\tau_{1s}$, $t_{1s}$, and $t_{1i}$ are defined in table \[Tpar\]. Likewise, it can be demonstrated by integration of equation (\[NumPhot\_M\]), and under the linear phasemismatch approximation, that the emitted flux can be expressed as
$$\label{N_analt_M} N_{\it{M}}^{lin}=\frac{2^6n_{1}n_{2}c^2\gamma^2p_1p_2Lh(\omega_s^0,\omega_i^0)}{\omega_{1}^0\omega_{2}^0| k^{\prime}_s+k^{\prime}_i|},$$
where the dependence on the various experimental parameters of the emission rate appears explicitly. Particularly, it can be seen, as expected, that $N_{\it{M}}$ increases linearly with the fiber length, indicating that the interaction length is not capped as it is for the pulsed pumps configuration.
![Emitted flux as a function of the fibre length. (a)-(c) Pulsed pumps configurations. Vertical dashed lines indicate the effective length $L_{eff}$ (with the $L_{eff}$ values shown). (d) Mixed pumps configuration. The blue markers were obtained from numerical evaluation of equations (\[NumPhot\_P\]) and (\[NumPhot\_M\]) for the pulsed and mixed pumps configuration, respectively, while the black solid lines are the analytical results obtained under the linear phasematching approximation, according to Eqns. (\[N\_analt\_P\]) and (\[N\_analt\_M\]).[]{data-label="brightness"}](Figure4.pdf){width="16cm"}
In figures \[JSIsynthesis\](i) to \[JSIsynthesis\](l) we illustrate the two-photon state obtained for a CP-SFWM source in the mixed pumps configuration, for which we have assumed the same parameters as in figures \[JSIsynthesis\](e) to \[JSIsynthesis\](h), except that pump $2$ is now monochromatic ($\sigma_2\to 0$); in this case the contours of the phasematching function become horizontal. Note that for the mixed pumps configuration the dependence of the spectral envelope function and the phasematching function on the parameters of the source become decoupled; i.e., the width of $ \alpha_{\it{M}}(\nu_s,\nu_i)$ is proportional to the pulsed pump bandwidth (with an orientation at $-45^\circ$), while the width of $\phi_{\it{M}}(\nu_s,\nu_i)$ depends on fiber properties, including its length $L$ and dispersion (with a horizontal orientation) .
In Figure \[brightness\] we illustrate the behaviour of the source brightness as a function of the fiber length, for both the pulsed and mixed pumps cases. In panels (a)-(c) we show this behaviour for three different values of $\sigma_2$ (1 THz, 0.05 THz, and 0.005 THz, respectively), and for a fixed value $\sigma_1=1$ THz, where a vertical dashed line indicates the effective length $L_{eff}$; it may be appreciated that the brightness reaches a plateau at $L \approx L_{eff}$. In panel (d) we show the corresponding behaviour for the mixed pumps scheme, for $\sigma=1$THz; note that in this case the brightness grows linearly with $L$ without saturating to a fixed value.
\[fact\]Factorable two-photon states generation
===============================================
In this section we will show that when restricting our discussion to a co-polarized CP-SFWM process implemented in a fiber which supports a single transverse mode, a factorable state can be obtained for *any* phasematched configuration, with frequencies such that $\omega_1=\omega_s$ and $\omega_2=\omega_i$.
In a SFWM process for which all four waves propagate in the same polarization/transverse spatial mode, quantum entanglement can reside only in the spectral degree of freedom. Spectral correlation properties are then governed by the joint spectrum of the two-photon state, see equations (\[JSAp\]) and (\[JSAm\]). In order to facilitate the analysis we focus here on the analytical expressions of the JSA based on the linear approximation of $\Delta k$, which were introduced in section \[sec.analytical\], $f^{lin}_{\it{P}}(\nu_s,\nu_i)$ and $ f^{lin}_{\it{M}}(\nu_s,\nu_i)$ for the pulsed and mixed pumps configurations, respectively. In both cases, as discussed above, spectral correlations are determined by the relative orientation and spectral widths of the pump envelope and phasematching functions, see equations (\[alpha\_Pul\]) and (\[phi\_Pul\]) for the pulsed case, and equations (\[pump\_analt\_M\]) and (\[PM\_analt\_M\]) for the mixed case. The two-photon states becomes factorable if the JSA function is separable, i.e. if it can be written as $f(\omega_s,\omega_i)=S_s(\omega_s)I_i(\omega_i)$.
Let us consider the limit $B\rightarrow 0$, which as discussed in section \[sec.analytical\] can always be attained for a combination of sufficiently short pump pulses and for a sufficiently long fiber. In this case, the joint spectral intensity $I(\nu_s,\nu_i) \equiv |\alpha_P(\nu_s,\nu_i)\phi_P(\nu_s,\nu_i)|^2$ may be expressed as
$$\begin{aligned}
\label{JSAfactpuls}
I(\nu_s,\nu_i) &\propto \exp[-2B^2(T_s \nu_s+ T_i \nu_i)^2]\exp\left[-\frac{2(\nu_s+\nu_i)^2}{\sigma_1^2+\sigma_2^2}\right] \nonumber \\
&= \exp \left[ -\frac{2\nu_s^2}{\sigma_1^2} \right] \exp \left[ -\frac{2\nu_i^2}{\sigma_2^2} \right].\end{aligned}$$
Note that in order to write down the last equality, we have used the expressions for $B$, $T_s$, and $T_i$ valid for the case where all four waves propagate in the same polarization/transverse spatial mode (see table \[Tpar\]). This result, valid in the limit $B \rightarrow 0$, is remarkable on a number of fronts: i) the two-photon state is *automatically factorable*, in addition to the underlying phasematching condition being attained automatically, as already discussed in section \[sec.PM\], ii) the state becomes completely independent of fiber parameters and only depends on the two pumps, and iii) the bandwidth of the signal photon is identical to the pump $1$ bandwidth, while the bandwidth of the idler photon is identical to the pump $2$ bandwidth.
By direct plotting of the $\phi_P(x)$ function, we may verify that for $B \lesssim 0.14$ this function is essentially identical to $\exp(-B^2 x^2)$; this corresponds to the regime under which Eq. (\[JSAfactpuls\]) is valid. This leads to the following factorability condition,
$$\label{Lumbral}
L \gtrsim \frac{(0.14)^{-1}\sqrt{\sigma_1^2+\sigma_2^2}}{(k_1'+k_2')\sigma_1\sigma_2}=\frac{(0.14)^{-1}\sqrt{\Delta t_1^2+\Delta t_2^2}}{k_1'+k_2'}.$$
Eq. (\[Lumbral\]) provides a threshold fiber length (which decreases as the pump temporal durations are reduced) so that if the fiber length exceeds this threshold the two-photon state is always factorable. It is interesting to compare the factorability fiber length threshold (see eq. (\[Lumbral\])) with the maximum interaction length $L_{eff}$ (see Eq. (\[Leff\])). Note that these two expressions are essentially identical; indeed, if the fiber becomes longer than the distance over which the two pump pulses are temporally overlapped, two effects are observed: i) the brightness can no longer increase, and ii) edge effects related to the air-fused silica interfaces disappear. Thus, as $L$ is increased, the brightness plateaus at $L=L_{eff}$ and the state reaches the Gaussian factorable form as described by Eq. (\[JSAfactpuls\]). Note that for ps pumps the values of $L_{eff}$ tend to be in the range of mm to cm making this scheme for factorable photon-pair generation highly practical.
![Synthesis of the joint spectral intensity (obtained with the same fiber as assumed in Fig. 3) for: (a)-(d) Pulsed pumps case with $\sigma_1=0.01THz$ and $\sigma_2=0.03THz$, $L=0.12$m. (e)-(h) Pulsed pump case with $\sigma_1=\sigma_2=0.01THz$, $L=0.12$m. (i)-(l) Mixed pump case with $\sigma=0.01THz$, and $L=1$m. The specific fiber lengths considered here are longer than the threshold lengths in Eq. (\[Lumbral\]) and Eq. (\[LumbralCW\]) for the pulsed and mixed pumps configurations, respectively.[]{data-label="JSIfactSyn"}](Figure5.pdf){width="15cm"}
As one or both of the pump bandwidths are reduced, the effective length $L_{eff}$ increases without limit. Thus, in the limit where either $\sigma_1 \rightarrow 0$ and/or $\sigma_2 \rightarrow 0$, the interaction length can become arbitrarily large, in practice limited by the fiber length, and Eqns. (\[JSAfactpuls\]) and (\[Lumbral\]) derived above for the pulsed pumps case can no longer be applied. Thus, let us now consider the question of factorability in the mixed pumps case, for which pump $2$ is monochromatic, and pump $1$ has a certain non-zero bandwidth $\sigma$. In this case, the joint spectral intensity $I(\nu_s,\nu_i) \equiv |\alpha_M(\nu_s,\nu_i)\phi_M(\nu_s,\nu_i)|^2$ may be expressed as
$$\begin{aligned}
\label{JSIcwana}
I(\nu_s,\nu_i) &\propto \left(\mbox{sinc}\left[ \frac{1}{2}(\tau_{1s} \nu_s+ t_{1i} \nu_i )\right]\right)^2
\exp\left[-\frac{2(\nu_s+\nu_i)^2}{\sigma^2}\right] \nonumber \\
&= \left(\mbox{sinc}\left[ \frac{t_{12} \nu_i }{2}\right]\right)^2
\exp\left[-\frac{2(\nu_s+\nu_i)^2}{\sigma^2}\right].\end{aligned}$$
Note that in order to write down the last equality, we have used the expressions for $\tau_{1s}$ and $t_{1i}$ valid for the case where all four waves propagate in the same polarization / transverse spatial mode (see table \[Tpar\]). This JSI $I(\nu_s,\nu_i)$ then becomes factorable if the width of the sinc function, along $\nu_i$, is much less than the width of the exponential function, along $\nu_s+\nu_i$. With the help of the Gaussian approximation $\mbox{sinc}(x)\approx \exp(-\Gamma x^2)$ (with $\Gamma=0.193$), we then arrive at the following condition for factorability
$$\label{LumbralCW}
L \gg \frac {2 \Delta t}{\sqrt{\Gamma}(k_1'+k_2')},$$
which makes it clear that for a sufficiently long fiber, the two-photon state becomes factorable; note that in this equation $\Delta t \equiv \sigma^{-1}$. Note that because the sinc function depends only on the frequency $\nu_i$, the sidelobes associated with this function will run parallel to the $\nu_i$ axis and will not, therefore, introduce correlations (this observation also serves to justify the use of the Gaussian approximation). In this limit, we may set $\nu_i \rightarrow 0$ in the exponential term, so that the joint spectral intensity can be approximated as
$$\begin{aligned}
\label{jsiCWfact}
I(\nu_s,\nu_i)
\approx \left(\mbox{sinc}\left[ \frac{t_{12} \nu_i }{2}\right] \right)^2
\exp\left[-\frac{2\nu_s^2}{\sigma^2}\right].\end{aligned}$$
It is remarkable that for, both, the pulsed pumps and the mixed pumps configurations a factorable state can always be reached for a sufficiently long fiber. This behaviour is illustrated in figure \[JSIfactSyn\] in which, for the same pump configurations as in figure \[JSIsynthesis\], we show the synthesis of the joint spectral intensity for fiber lengths longer than the threshold lengths in Eq. (\[Lumbral\]) and Eq. (\[LumbralCW\]) for the pulsed and mixed pumps configurations, respectively. It is evident in this figure that the three source scenarios lead to factorable two-photon states. It is worth emphasizing that while in the case of standard (co-propagating) SFWM, factorability demands specific combinations of fiber length and pump bandwidth [@Garay07], for CP-SFWM the factorability conditions are considerably more relaxed and in fact *all* phasematched configurations can lead to a factorable state for a sufficient fiber length.
In order to quantify the degree of factorability of CP-SFWM photon pairs, we evaluate the heralded-single-photon state purity $p \equiv\mbox{Tr}(\hat{\rho}_s^2)=1/K$ in terms of the Schmidt number $K$, where $\hat{\rho}_s$ is the reduced density operator for the signal state [@URen05]. Thus, an ideal factorable two-photon state is related to an ideal single-photon purity $\mbox{Tr}(\hat{\rho}_s^2)=1$. In figure \[PurityVsSigma2\](a) we show the numerically-calculated purity as a function of $\sigma_2$, while $\sigma_1$ and $L$ remain fixed; results are shown for four different fiber lengths, as indicated, and $\lambda_{1}=0.820\mu$m, $\lambda_{2}=0.532\mu$m, and $\sigma_1=0.01THz$. Square markers in the figure correspond to the purity obtained for the mixed pump case, for which $\sigma_2\to0$, see equation (\[JSAm\]). Figure \[PurityVsSigma2\](b) shows the number of photon pairs emitted per second for the same parameters assumed in panel (a). Panels (c) and (d) are similar to (a) and (b), except for a larger value of the pump 1 bandwidth: $\sigma_1=1$THz. From these plots the following two behaviors as the fiber length is increased become apparent: i) the two-photon state becomes increasingly factorable, and ii) the source brightness reaches a plateau. In addition, increasing $\sigma_2$ leads to a reduced effective length $L_{eff}$, thus boosting the purity, for a given value of $L$. Note from Fig. \[PurityVsSigma2\] (b) and (d) that while the use of very short fibers would lead to the need for large pump bandwidths in order to attain factorability, with correspondingly larger self-/cross-phase modulation effects, there is no need in practice to use such short, e.g. sub-mm, fibers which are in addition comparatively more challenging to handle.
![(a) Purity versus $\sigma_2$ as a function of fiber length with $\sigma_1=0.01THz$. (b) Photon-pair emission rate vs $\sigma_2$ as a function of fiber length. The average power of the two pumps is 50mW. (c) and (d) Similar to (a) and (b), but with $\sigma_1=1THz$. (e)-(g) Joint spectral intensity for the $\sigma_2$ values indicated as e, f, g on panel (c) and $L=1$cm. For all three cases, the numerically evaluated purity reaches very close to unity.[]{data-label="PurityVsSigma2"}](Figure6.pdf){width="13cm"}
In Fig.\[PurityVsSigma2\](c) we have indicated with the letters e, f, and g three particular choices of parameters, which lead to the joint spectra shown in Fig.\[PurityVsSigma2\](e), (f), and (g). Note that for all of these three parameter choices, the two-photon state is essentially factorable.
As a final remark in this section, it is worth mentioning that in the intermodal CP-SFWM configuration, discussed at the end of section \[sec.PM\], the factorability of the two-photon state is preserved as compared with the case in which all interacting fields propagate in the fundamental mode, regardless of the higher-order fiber mode employed. The emission rate, however, may be compromised as the order of the excited mode used increases, due to a reduced overlap between the interacting modes.
\[fact\]Ultra-narrowband single-photon wavepacket generation
============================================================
Atom-photon interfaces rely on the ability of a single photon to be absorbed by a single atom; such interfaces involve matching both frequency and bandwidth of the single photons to the intended atomic transition in a given atomic species. While such electronic transitions typically have bandwidths in the region of MHz, the natural bandwidths of SPDC and SFWM sources tend to be many orders of magnitude greater. A possible solution is to place the nonlinear medium responsible for photon-pair generation inside a high-finesse cavity so as to restrict the emission bandwidth as needed, without adversely affecting the source brightness [@Garay13].
Let us observe from Eq. (\[JSAfactpuls\]) that in the pulsed pumps configuration, specifically in the regime $L>L_{eff}$ for which the JSI becomes factorable and fully Gaussian, the emission bandwidths are ‘inherited’ from the pumps: i.e $\sigma_s=\sigma_1$ and $\sigma_i=\sigma_2$. This is a reflection of the achromatic phasematching for which the fiber dispersion experienced by the signal and pump $1$, on the one hand, and by the idler and pump $2$, on the other hand, cancel each other out so that the two-photon state is determined exclusively by the pumps. As one or both of the pump bandwidths approach the monochromatic limit, the effective length $L_{eff}$ becomes infinite and the expression in Eq. (\[JSAfactpuls\]) for the two photon state is no longer valid.
Let us then consider the possibility of generating photon pairs in the mixed pumps configuration, for which at least one of the two pumps exhibits a very narrow bandwidth. For a sufficient fiber length (obeying Eq. (\[LumbralCW\])), the joint spectral intensity is given by Eq. (\[jsiCWfact\]). Let us observe that in this regime, the bandwidth of the signal photon ‘inherits’ the bandwidth of pump $1$, i.e. $\sigma_s=\sigma$, as occurs for the pulsed pumps case. However, note that the bandwidth of the idler photon $\sigma_i$ is determined not by the pumps but solely by fiber properties, and with the help of the Gaussian approximation, can be expressed as
$$\label{bandwI}
\sigma_i=\frac{2}{\sqrt{\Gamma} L (k_1'+k_2')}.$$
It is important to point out a key difference with respect to standard co-propagating SFWM. While for standard SFWM, the spectral properties are determined by reciprocal group velocity *difference* coefficients of the form $L(k_p'-k')$, for CP-SFWM the spectral properties are replaced by reciprocal group velocity *sum* coefficients of the form $L(k_1'+k_i')=L(k_1'+k_2')$. These reciprocal group velocity sum coefficients correspond to the sum of transit times for the pump 1 and pump 2 waves through the fiber, as opposed to transit time differences as appear in the case of standard SFWM. The fact that the sum coefficients tend to be orders of magnitude greater than the difference counterparts has a profound implication: because the idler bandwidth is inversely proportional to this reciprocal group velocity sum (difference) coefficient for counter-propagating (standard) SFWM, the resulting bandwidths are orders of magnitude smaller than for a comparable standard co-propagating source, as a direct consequence of the counter-propagating geometry. In practice this leads to the possibility of obtaining extremely small idler bandwidths for reasonable lengths of fiber.
Suppose that a bandwidth $\delta\omega$ is desired for the idler photon. We can then show from Eq. (\[bandwI\]) that the fiber length which guarantees such a bandwidth is given by
$$L=\frac{2}{\sqrt{\Gamma} \delta\omega}\frac{1}{ k_1'+k_2'}.$$
![Idler emission bandwidth (FWHM in intensity) (a) and purity (b) as function of fiber length $L$, obtained from CP-SFWM in the mixed pumps configuration. Results were evaluated assuming $\sigma=1$THz. Note from panel (b) that the two-photon state becomes essentially factorable for fiber lengths greater than the threshold length given in Eq. \[LumbralCW\], which in this case is around $0.5$mm. []{data-label="BandW"}](Figure7.pdf){width="12cm"}
In figure \[BandW\](a) we show results of the emission bandwidth $\sigma_i$ vs fiber length for a CP-SFWM source based on the mixed pumps scheme, with a pulsed pump of $0.42$nm bandwidth centered at $0.820\mu$m (compatible with a picosecond Ti:Sapphire laser), and a monochromatic pump at $0.532\mu$m. The red squares represent results obtained numerically from equation (\[JSAm\]), while the black solid line corresponds to those obtained analytically from equation (\[bandwI\]). As indicated in the figure, for fiber lengths longer than $\sim36$m, (idler) single-photon wavepackets with bandwidths narrower than $30$MHz can be generated. In contrast, the bandwidth of the signal photon essentially equals that of the pump, i.e. $1.18$THz (FWHM). In panel (b) of this figure we have shown the corresponding purity of a single idler photon vs fiber length, when heralded by the detection of a signal photon.
The source scheme described above is suitable for applications in which it suffices for only one of the two photons in each pair to be narrow-band. Note that if both pumps are monochromatic it becomes possible, for a sufficiently long fiber, to generate photon pairs characterized by ultra-narrowband signal *and* idler modes (this case has not been analyzed in detail in this paper).
As has been emphasised, an important feature of CP-SFWM is the resulting phasematching achromaticity. Thus, for a given fiber it becomes possible to tune the emission frequencies as controlled by the pump frequencies (with $\omega_s=\omega_1$ and $\omega_i=\omega_2$), while preserving the emission bandwidths. This is a significant advantage in designing two-photon state sources. For example, a source may be designed so that one of the emission modes corresponds to a specific atomic transition (in frequency and bandwidth), while the other is tuned to the telecommunications band [@Fekete13].
Conclusions
===========
In this paper we have described theoretically a new kind of spontaneous four wave mixing process, in which the two pump waves counter-propagate in the $\chi^{(3)}$ nonlinear medium, and in which the generated signal and idler photons likewise counter-propagate; we have referred to this process as counter-propagating spontaneous four wave mixing, or CP-SFWM. We have shown that in this process, phasematching is attained *automatically* regardless of the specific dispersion characteristics, leading to a signal frequency which equals the frequency of the pump wave travelling in the opposite direction, and likewise for the idler photon and the second pump wave. We have discussed that while a number of experimental aspects can slightly offset each of the generation frequencies with respect to the frequency of the corresponding pump wave, to aid discrimination of the SFWM photons from the pumps, the use of an intermodal CP-SFWM process seems to be the most practical alternative.
We have presented two versions of the CP-SFWM process: in the first, which we refer to as the pulsed pumps configuration, both pumps are assumed to be pulsed while in the second, which we refer to as the mixed pumps configuration, one pump is assumed pulsed and the remaining pump is assumed to be monochromatic. We have shown that in both of these cases, for an arbitrary phasematched source design, the state can always reach factorability for a sufficiently long fiber (or waveguide). Moreover, the threshold length for factorability tends to be in the range of a few mm to a few cm, making the resulting automatic phasematching *and* automatic factorability highly practical. We have also shown that in the mixed pumps configuration, the idler photon, which is emitted in counter-propagation to the monochromatic pump wave, can be made compatible in bandwidth with electronic transitions in atoms. The latter eliminates the need for optical cavities, and is a direct consequence of the counter-propagating geometry for which the emission bandwidths are governed by the transit time *sums* through the non-linear medium, rather than transit time *differences* as in the case of standard SFWM.
We point out that of the three properties discussed above, i.e. automatic phasematching, automatic factorability, and ultra-narrow single-photon bandwidths, at least the first two are amenable to integrated optics implementations, since they involve modest threshold lengths. We believe that this new type of spontaneous four wave mixing process, in which the waves involved counter-propagate in the non-linear medium, may prove useful in future implementations of fiber- or waveguide-based photon-pair sources with engineered spatio-temporal properties.
Acknowledgments
===============
This work was supported by CONACYT, México, PAPIIT (UNAM) grant number IN105915, and AFOSR grant FA9550-16-1-0458.
References
==========
[^1]: Corresponding author: [email protected]
|
---
abstract: 'Variability selection has been proposed as a powerful tool for identifying both low-luminosity AGN and those with unusual SEDs. However, a systematic study of sources selected in such a way has been lacking. In this paper, we present the multi-wavelength properties of the variability selected AGN in GOODS South. We demonstrate that variability selection indeed reliably identifies AGN, predominantly of low luminosity. We find contamination from stars as well as a very small sample of sources that show no sign of AGN activity, their number is consistent with the expected false positive rate. We also study the host galaxies and environments of the AGN in the sample. Disturbed host morphologies are relatively common. The host galaxies span a wide range in the level of ongoing star-formation. However, massive star-bursts are only present in the hosts of the most luminous AGN in the sample. There is no clear environmental preference for the AGN sample in general but we find that the most luminous AGN on average avoid dense regions while some low-luminosity AGN hosted by late-type galaxies are found near the centres of groups. AGN in our sample have closer nearest neighbours than the general galaxy population. We find no indications that major mergers are a dominant triggering process for the moderate to low luminosity AGN in this sample. The environments and host galaxy properties instead suggest secular processes, in particular tidal processes at first passage and minor mergers, as likely triggers for the objects studied. This study demonstrates the strength of variability selection for AGN and gives first hints at possibly triggering mechanisms for high-redshift low luminosity AGN.'
author:
- |
Carolin Villforth$^{1,2}$[^1], Vicki Sarajedini$^{1}$ and Anton Koekemoer$^{2}$\
$^{1}$Department of Astronomy, University of Florida, 32611 Gainesville, FL, USA\
$^{2}$Space Telescope Science Institute, 21218 Baltimore, MD, USA\
bibliography:
- 'VariableAGN.bib'
date: Accepted July 17th 2012
title: 'The SEDs, Host Galaxies and Environments of Variability selected AGN in GOODS-S'
---
\[firstpage\]
galaxies: active – galaxies: evolution – galaxies: structure
Introduction {#S:intro}
============
While our understanding of the physics of AGN has grown considerably over the last decades, an important questions about AGN remains unanswered until today: How are AGN triggered? This question has become of interest also for a wider field of astronomy research when it became clear that AGN likely play an important role in the evolution of galaxies. The masses of the central black holes in nearby galaxies correlate surprisingly well with the properties of their hosts [@gebhardt_relationship_2000; @graham_correlation_2001; @ferrarese_fundamental_2000; @novak_correlations_2006; @graham_expanded_2010] It has been argued that these correlations imply a causal connection between AGN and their hosts, or at least the galactic bulges of their hosts [e.g. @silk_quasars_1998]. However, other authors have argued that the data does not necessarily imply a casual connection [@peng_how_2007; @jahnke_non-causal_2010], but can be explained by progression to the mean during mergers throughout cosmic time. Also, the fact that both star formation and AGN activity peak around a redshift of two has been used as an argument for a causal connection between AGN and galaxies [e.g. @boyle_cosmological_1998; @merloni_tracing_2004].
Theoreticians have suggested several mechanisms in which AGN activity and galaxy evolution could be connected [e.g. @silk_quasars_1998; @sanders_luminous_1996; @hopkins_cosmological_2008]. Major mergers of galaxies have been the main mechanisms discussed [@sanders_luminous_1996; @silk_quasars_1998; @hopkins_characteristic_2009], in particular for the most luminous quasars. The evidence for major mergers as triggers for AGN activity has been mixed [e.g. @canalizo_quasi-stellar_2001; @hopkins_cosmological_2008; @kocevski_candels:_2011; @cisternas_bulk_2011; @schawinski_heavily_2012 and references therein] and latest studies seem to imply that earlier findings are mostly due to selection effects [@kocevski_candels:_2011; @cisternas_bulk_2011]. Recent studies have also emphasized that other mechanisms might play a role in triggering AGN activity in particular for low luminosity AGN and heavily obscured systems [@hopkins_characteristic_2009; @bournaud_black_2011]. These studies have emphasized that to properly understand AGN activity, one needs to study a wide range of AGN types, both in luminosity and obscuration.
Host galaxy studies commonly focus either on low redshift systems [e.g. @canalizo_quasi-stellar_2001; @veilleux_deep_2009] or high luminosity quasars at high redshift [e.g. @villforth_quasar_2008; @schramm_host_2008; @kocevski_candels:_2011]. A part of the parameter space that has been poorly explored are low luminosity AGN at high redshift, this is in particular due to the difficulty in identifying high-redshift low luminosity AGN.
Variability selection has been used as a powerful tool for selecting AGN, in particular due to the emergence of large surveys in the last decades [@schmidt_selecting_2010; @macleod_quasar_2011]. In particular, applying it to high-resolution HST data has been recognized as a valuable tool for selecting AGN at low luminosities up to high redshift [e.g. @sarajedini_v-band_2003; @villforth_new_2010; @sarajedini_variability_2011]. In a recent paper, @villforth_new_2010 published a catalogue of variability selected AGN in the GOODS Fields, containing a total of 155 variable sources, 88 of those located in GOODS-South. The objects found in this study are predominantly of Seyfert luminosities and have dominant host galaxies. In this paper, we aim to analyse the multi-wavelength properties of the 88 variability selected AGN candidates. We will summarize the properties of variability selected AGN and re-evaluate the issue of contamination from stars and false positives. We will also analyse this particular sample to asses questions concerning the likely triggering mechanisms and environments of lower luminosity AGN at higher redshift. This study will therefore explore a part of AGN parameter space that has been neglected so far and analyze the strength and weaknesses of variability selection.
The paper is structured as follows: Section 2 recaps the sample selection from @villforth_new_2010 and presents a short summary of data used in this study. Section 3 analyses the SEDs and general properties of the variability selected sources and discusses contamination by both stars and false positives. Host galaxies are analysed in Section 4. Section 5 discusses environments of variability selected AGN, followed by a discussion of the results in Section 6 and summary and conclusions in Section 7. The cosmology used is $H_{0}=70\textrm{km s}^{-1}\textrm{Mpc}^{-1},\Omega_{\Lambda}=0.7, \Omega_{m}=0.3$. Throughout the paper, we use AB magnitudes.
Data {#S:data_main}
====
Sample
------
In this study, we will use the variability selected sample from a previous study by the authors, details of the method used can be found in @villforth_new_2010, but the basic methods will be discussed here shortly.
Variability selection is performed on five epochs of GOODS data taken with the Advanced Camera for Surveys (ACS) Wide Field Channel (WFC) aboard HST in the F850LP (z) band. Data reduction is performed using MultiDrizzle [@koekemoer_multidrizzle:_2002; @koekemoer_candels:_2011]. The five epochs were separated by about 6 weeks in time, overall spanning about 7 months. An initial selection was performed to reject objects with SN $<$ 20 in the combined five epoch data. Additionally, only objects with good data in all five epochs were included.
The flux measurements were done by simple aperture photometry, tests were done with a range of aperture sizes, but finally an aperture radius of 0.36 " was chosen. Errors were determined using weight maps and corrected using the full sample. Visual inspection was performed to check for diffraction spikes and affected objects were discarded. The final sample was then selected using a $C$ statistic. A significance level of 99.99% was chosen to limit contamination by false positives, additionally a more rigorous ’clean’ catalogue (significance level 99.99%) which is expected to contain only one false positive is provided.
All sources where then cross-checked with a number of catalogues and databases to reject stars. After this, in the GOODS-S field 88 objects were variability selected, of which 14 where identified as stars, leaving 74 AGN candidates. We expect six sources to be false positives.
Short Description of Datasets Used {#S:data}
----------------------------------
In this study, we combine three different data sets covering the GOODS South field: the photometric redshift and multi-waveband optical-IR catalogue used in @dahlen_detailed_2010, the CDFS 4 Ms X-ray catalogue from @xue_chandra_2011 as well as the CDFS VLA 20cm and 6cm from @kellermann_vla_2008.
We use photometric redshifts and multi-wavelength data from @dahlen_detailed_2010. This dataset includes photometric data for the following bands: VLT/VIMOS (U-band), HST/ACS (F435W, F606W, F775W, and F850LP bands), VLT/ISAAC (J-, H-, and Ks-bands) as well as four Spitzer/IRAC channels (3.6, 4.5, 5.8, and 8.0$\mu$m).
The spatial resolution for the data at different wavelength are as follows: 0.11“ for the HST data, 0.35-0.65” for the ground-based IR data, 0.8“ for the U band data and 1.7-1.9” for the IRAC data. The resolution for the Chandra data is about 0.5“ with a typical spatial uncertainty of 0.4”. Finally, the VLA data have a typical resolution of about 3.5".
We also use the CDFS 4Ms catalogue from @xue_chandra_2011, this catalogue considerably improved the sensitivity from existing X-ray catalogues [@luo_chandra_2008] to $3.2 \times 10^{-17} \textrm{erg} \textrm{cm}^{-2} \textrm{s}^{-1} , 9.1 \times 10^{-18} \textrm{erg} \textrm{cm}^{-2} \textrm{s}^{-1} and 5.5 \times 10^{-17} \textrm{erg} \textrm{cm}^{-2} \textrm{s}^{-1}$ for the full, soft, and hard bands, respectively. Radio data at 1.4GhZ (20cm) and 4.8GhZ (6cm) is taken from @kellermann_vla_2008, with a rms per beam between 9-43 $\mu Jy$ for the 20cm band (lowest rms/beam is reached in the center of the field).
Both X-ray and radio catalogues are matched to the $z$-band variability selected catalogue in a simple radius matching technique. Matching radii correspond to the typical resolution of the data. The matching procedure is analysed to verify that there are no offsets between the coordinate systems. For this purpose, we analyse the distributions of distances in both right ascension and declination and check for offsets from zero. These offsets are then corrected for and the matching process is iterated. We find a small offset of about 0.1" in both right ascension and declination for the 4Ms data only. Correcting for it however does not significantly change the number of matches. No offset is found in the radio data matching.
Spectral Energy Distributions and General Properties of Variability Selected AGN {#S:sample}
================================================================================
With a 99.99% significance level, 88 variable objects were found in GOODS South, after rejecting stars, 74 variability selected AGN remain, of those sources, 21 had reported spectroscopic redshifts @villforth_new_2010. 21 additional sources have spectroscopic redshifts reported in the current paper, yielding a total of 42 sources with spectroscopic data. Three of the 88 initial sources do not show up in the multiwavelength catalogue and are thus discarded, leaving 71 objects studied here. All references for the spectroscopic redshifts can be found in Table \[T:AGN\] in the Appendix. Spectroscopic redshifts range from 0.0459 to as high as 3.7072. Adding the photometric redshift data from @dahlen_detailed_2010 gives us redshift estimates for the rest of the sources. The photometric redshifts range from 0.04 to 4.34. The mean and median of the redshift distribution are 0.94 and 0.6.
SED Fitting Procedure {#S:sed}
---------------------
We make use of the wide wavelength coverage in the optical-NIR available for GOODS South. As a comparison, we use the SWIRE Template Library [^2], which provides templates over a wide wavelength range for normal quiescent elliptical and spiral galaxies, starburst galaxies, quasars as well as mixed starburst and AGN objects. The templates are based on data from [@silva_modeling_1998; @gregg_reddest_2002; @berta_spatially-resolved_2003; @hatziminaoglou_sloan_2005; @polletta_chandra_2006]. Additionally, we use the Kurucz 1993 stellar models [@kurucz_model_1979] [^3]. SED Fitting is performed using a simply least square minimization technique. In a first step, stellar SEDs at a redshift of zero are fit to identify stellar contaminants, this is discussed in Section \[S:stars\]. AGN/Galaxy combination fits as well as typical errors, caveats, possible biases and sources of systematic errors are then discussed in Section \[S:sed\_agn\].
### Rejection of Stars {#S:stars}
Variable stars are a possible contaminant to the variability selected AGN sample. While the GOODS-South field is located far from the galactic plane, halo stars could still be present. It is also known that many stars show variability on time scales larger than a few hours, with a majority showing variability on the scale of a few percent on week and month time scales [e.g. @hartman_photometric_2011; @bryden_kepler_2011]. While this is generally shorter than typical variability timescales of AGN, which tend to be on the order of weeks instead of hours [e.g. @kelly_are_2009], stars will still show detectable variability on the month time-scales sampled by the data. Therefore, initially selecting against point sources is not useful since quasars are point sources.
In @villforth_new_2010, we identified 14 objects as stars through archive searches. Using SED Fitting, we additionally identify 10 objects as stars, usually of G-M type stars. Note that a majority of these stars were identified as low redshift early type galaxies in the photometric redshift catalogues due to their SED shape. Only the mid-IR data allows to reliably distinguish those objects from faint red galaxies.
In the following analysis, we will no longer include objects identified as stars through either SED fitting or through literature search. They are still listed in Table \[T:Stars\]. After flagging all stellar objects, a total of 61 non-stellar variability selected sources remain. We will not further consider objects identified as stars and continue with those 61 objects, which we will label AGN candidates.
### Combination Fits of Remaining AGN Candidates {#S:sed_agn}
For the sample of 61 AGN candidates, we perform mixed SED fits. This means that we fit a mixture of AGN and galaxy SEDs, we perform these fits for a range of different input SEDs for both the AGN and galaxy components.In total 6 different AGN SEDs and 11 different galaxy SEDs where used. For each of those combinations, we vary the fractional distribution of the galaxy and AGN in the $z$ band and fit the combined SED. The best fits for each combination are then compared and the overall best fit is selected. All results of the SED Fits are provided in Table \[T:AGN\] in the Appendix.
Out of the 61 objects, a clear best fit was found for 47 objects. For the other 14 objects, the fit was ambiguous. We could not determine the properties of those sources reliably. The sources with failed fits are predominantly faint, extended and show clear disturbed morphology. The failed fits might be due to several reasons. Due to the long time-span over which the data were taken, the AGN variability can cause excess scatter. Strong line emission could cause additional bumps in the SED and multi-wavelength photometry might be less reliable for sources with complex morphology due to the differences in resolution. Additionally, failed fits generally show erratic SEDs that might be indicative of an AGN dominated SED with strong variability. The failed fits will be discussed in more detail in Section \[S:failed\_fits\].
The SED fits contain several sources of errors. We will first qualitatively discuss different influences and then quantify these effects to derive expected errors on determined magnitudes:
- **Variability:** The AGN in this sample were selected through their variability. Due to the fact that the multi-wavelength data were taken over a timespan of several years [@dahlen_detailed_2010], the AGN variability causes additional scatter in the SED that are not accounted for in the photometric errors. This causes poorer fits than expected. Therefore SED fitting is especially difficult for sources that are AGN dominated, as well as for those with already large photometric errors (see also Section \[S:failed\_fits\]). A proper treatment of this effect is complicated since variability strength varies across the SED and photometric data is often taken over a wide time-span [@dahlen_detailed_2010]. This is beyond the scope of the current project. However, we do not expect this effect to cause systematic errors in our fits.
- **Size of template library:** For template fitting, there is a trade-off between using a large sets of templates and therefore being able to account for subtle differences in the SEDs against using a smaller subset, resulting in less accurate fits but more robust results. Due to the complexity of the SEDs, we decide for a limited template library. A larger number of templates will cause difficulties in finding global minima in the fit.
- **Unclear Redshift:** Two thirds of the AGN (42/61) in this sample have spectroscopic redshifts, leaving 19 objects for which we rely on photometric data. Some of the spectroscopic redshifts are of lower quality (i.e. rely on noisy spectra or single line detections). While the photometric redshifts of sources with significant galaxy contribution are relatively reliable (see Section \[S:redshift\]), some of the sources have very uncommon SEDs and we cannot find reliable redshift fits. While the majority of the sample has accurate enough redshifts to have little influence on the resulting fits, a few sources have SEDs for which we can not determine the proper redshift (see Section \[S:failed\_fits\] as well as Fig. \[F:Ind\_bumpy1\]).
Taking into account the factors mentioned above as well as typical uncertainties in the fit, the resulting errors in the AGN and galaxy magnitudes are typically 0.5-0.7 magnitudes for the non-dominant component and around 0.1 magnitudes for the dominant component. Errors in the percentage contribution are typically 1-5 per cent points.
To determine the actual luminosity of the AGN component, it is important to disentangle the contribution of galaxy and AGN. Fig. \[F:LP\_AGNMags\] shows the contribution of the AGN to the overall magnitude for all variability selected sources for which the fit succeeded. The AGN contribution can be as low as $\sim$5%. In the following, we will call sources with $>$90% AGN contribution in $z$ AGN dominated, those with $<$10% AGN contribution galaxy dominated and all others mixtures. Fig. \[F:SED\_General\] shows the magnitudes and redshifts of AGN dominated, mixture and galaxy dominated sources. Galaxy dominated sources are only present at very low redshifts where the small aperture used in @villforth_new_2010 only includes a small proportion of galaxy light and therefore makes the detection of variability of a faint source against a background galaxy easier. A majority of sources beyond redshift 0.5 are mixture objects.
![Percentage contribution of AGN in $z$ plotted against absolute $z$-band AGN magnitude. Only sources with successful fits that have not been identified as false positives are shown. Errors in the AGN magnitudes are between 0.1 and 0.5 mags and errors in the AGN contribution are $\leq$ 5 percent points.[]{data-label="F:LP_AGNMags"}](AGNMags.pdf){width="8cm"}
![ Observed $z$ band magnitudes versus redshift for all variability selected AGN by AGN contribution: black x: failed fit; red star: AGN dominated, green pentagons: mixture, blue circles: galaxy dominated. The magnitudes are shown for the entire object, i.e. contribution from the host galaxy is not subtracted off.[]{data-label="F:SED_General"}](LP_SED_lin.pdf){width="8cm"}
Redshift distribution of AGN candidates and accuracy of photometric redshift {#S:redshift}
----------------------------------------------------------------------------
Comparing the photometric redshifts of all GOODS South sources to the AGN candidates, we see that the variability selected sources are predominantly at rather low redshift. About 1% of all sources in the $z$ band catalogue are detected to be variable at a redshift below 0.5. The fraction of variability selected AGN candidates drops off at higher redshift. The lower signal-to-noise for objects at higher redshift makes it harder to detect variability for those sources. On the other hand, the fact that the spectrum is redshifted and therefore shorter wavelengths are sampled for higher redshift objects makes detection at higher redshifts more efficient since variability is stronger at shorter wavelengths [@vanden_berk_ensemble_2004; @trevese_variability-selected_2008]. Our results show that the drop in signal-to-noise dominates over the favourable change in observed rest-frame wavelength (see Fig. \[F:redshift\]). Since more stars were identified in this paper, we also show the redshift distribution with those objects removed (3rd panel from top). The overall trend does not change significantly after removing stars.
Interestingly, [@sarajedini_variability_2011] found an increase in the percentage of galaxies with varying nuclei with increasing redshift (their Figure 8). The primary reason for this appears to be due to the different wavelength at which the variability surveys were conducted. In the V-band, the distribution of galaxies is weighted towards lower redshifts when compared to the z-band survey (see Fig 8a in Sarajedini et al. as compared to the top panel in our Figure \[F:redshift\]). While the redshift distribution of V-band and z-band selected variables is similar, the galaxy distributions are not. Thus, the trend in the percentage of galaxies hosting variables as a function of redshift in the two surveys differs. The fact that the redshift distributions of the variable AGN samples are quite similar regardless of their parent populations is a testament to the robustness of using variability to identify low-luminosity AGN across a range of redshifts.
![The Distribution of photometric redshifts for all GOODS-S sources, all variability selected sources and variability selected AGN candidates only.[]{data-label="F:redshift"}](PhotZDistributions.pdf){width="8cm"}
One of the main reasons to use variability was to find lower luminosity AGN at higher redshift, the absolute magnitudes of our sample are shown in Fig. \[F:LP\_AGNMags\]. As we can see, the variability selected AGN are of moderate luminosity, a majority of the sources lie below the nominal upper limit for Seyfert Galaxies of $M=-23$ [@veron-cetty_catalogue_2006], showing that variability selection is capable of identifying low-luminosity AGN. The lack of luminous AGN is not entirely surprising since those sources are generally rare in pencil beam surveys such as GOODS due to their low number density.
In photometric redshift codes, galaxy templates are fitted to multi-waveband data. Since AGN spectra are distinct from galaxy spectra, there is concern that the code might be less accurate in determining redshifts. Fig. \[F:z\_accuracy\] shows a comparison between the photometric and spectroscopic redshift for the 42 objects with spectroscopic data. We see that in most cases the agreement is very good with a few strong outliers, all of which are later on found to have SED dominated by the AGN emission. Therefore, we caution that this finding should not be generalized to the general accuracy of photometric redshifts for AGN since the AGN in this sample are generally of low luminosity.
Inspecting the SEDs of the AGN candidates shows why this is the case: the stellar IR bump stays prominent in most lower luminosity AGN, leaving a Balmer break that allows to constrain the redshift. This is explored in more detail in the following sections . On the other hand, AGN dominated sources have SEDs that make it difficult or even impossible to derive redshifts. See for example the SED of the likely blazar in Figure \[F:Ind\_Blazar\].
![The relation between photometric and spectroscopic redshifts from @dahlen_detailed_2010. Small black dots show all objects from @dahlen_detailed_2010 and red circles show all variability selected AGN with spectroscopic redshifts, errorbars for variability selected AGN show the 68% confidence limits.[]{data-label="F:z_accuracy"}](PhotRedshift_1.pdf){width="8cm"}
Detailed Description of Variability Selected AGN SEDs {#S:subsamples}
-----------------------------------------------------
In this section, we will discuss the SEDs of variability selected AGN in detail and argue that there are clear signs of AGN activity in a vast majority of the variability selected AGN sample.
Properties of X-ray detected AGN {#S:xray}
--------------------------------
Next, we analyse the X-ray properties of the AGN candidates. Twenty variability selected AGN have X-ray detections, 14 were already detected in the previous 2Ms data [@luo_chandra_2008]. Thus six are only detected in the deeper 4Ms data.. The X-ray detected sources have a wide range of luminosities and range from moderate to high redshifts, see Fig. \[F:LP\]. It should also be noted that Fig. \[F:LP\] (as well as Figure \[F:LP\_detail\]) show the integrated observed $z$ band magnitudes of the sources, i.e. the galaxy contribution is not subtracted in these plots, and actual AGN magnitudes are fainter. Additionally, since we show the same filter for the whole redshift range, the restframe wavelengths covered are quite different.
Fig. \[F:Gamma\] shows the distribution of spectral indexes $\Gamma$ from @xue_chandra_2011 for the full 4 Ms sample as well as the variability selected AGN. The spectral indexes of the variability selected AGN are soft, ranging from -0.1 to 2.3. Typically the line between soft (unobscred) and hard (obscured) AGN is drawn at a $\Gamma = 1$, with larger $\Gamma$ values indicating softer spectra [@xue_chandra_2011]. Of our sample, only 3 sources lie below this line (J033217.06-274921.9, J033224.54-274010.4, J033228.30-274403.6). Of those three sources, two have low signal-to-noise ratios, making the derived spectral indices unreliable (J033224.54-274010.4 and J033228.30-274403.6). The other source (J033217.06-274921.9) appears to be obscured in the X-ray, it is unclear if the variability selection is due to some weak variability from obscured emission, or if the source shows a peculiar X-ray spectrum. Besides those sources, our sample includes a majority of AGN with soft spectral indexes, as expected from the fact that variability selection strongly favours unobscured sources. This is consistent with the result of @sarajedini_variability_2011 where V-band selected variables were found to have generally softer X-ray spectra based on hardness ratios.
The optical to X-ray ratio of AGN has been studied in detail and shows a rather small dispersion [@brandt_nature_2000; @strateva_color_2001]. Fig. \[F:alphaox\] shows the correlation between the 0.5-8kev X-ray flux and optical luminosity for both the AGN candidates and the general 4Ms catalogue, for sources without X-ray detection, we show the catalogue upper limits. The AGN candidates follow the general trend of the X-ray catalogues, however, they avoid the area of very X-ray dominated AGN. The AGN candidates cover almost the whole range in X-ray luminosities. X-ray detections are provided in Table \[T:AGN\] in the Appendix.
![The redshifts and observed frame $z$ magnitudes of all variability selected AGN, X-ray ands radio detections are overplotted. The magnitudes are shown for the entire object, i.e. contribution from the host galaxy is not subtracted off. The solid and dotted lines show the 68%/95% confidence regions for the objects with photometric redshifts.[]{data-label="F:LP"}](LP_xray.pdf){width="8cm"}
![The Distribution of $\Gamma$ for the full X-ray sample as well as the variability selected AGN (bottom) panel.[]{data-label="F:Gamma"}](GammaDistribution.pdf){width="8cm"}
![X-ray versus UV absolute luminosity for all X-ray sources (black dots) as well as X-ray detected variability selected AGN. Xray fluxes are full Chandra fluxes (0.5-8kev).[]{data-label="F:alphaox"}](AlphOX.pdf){width="8cm"}
Nine of the twenty x-ray detected AGN are morphologically dominated by the AGN. Eleven are clearly extended. Of the nine point source dominated objects, only two have clearly visible host galaxies. The absorption corrected X-ray luminosities for the point sources with X-ray detection range between $2.3 \times 10^{43} - 3.6 \times 10^{44}$ erg/s placing them just below the knee of the X-ray luminosity function [@aird_evolution_2010].
The eleven extended sources with X-ray detections have luminosities lower than those of the point sources, ranging between $3.4\times 10^{39} - 4.6 \times 10^{43}$ erg/s, ranging all the way from low luminosity to moderatly luminous AGN. This reaches down to the limits found for local ultra low luminosity AGN [@ho_nuclear_2008]. The morphologies of the extended X-ray detected AGN are varied: three late-type galaxies, three early-type galaxies, three disturbed systems as well as two clear disk-bulge mixtures. Two of the disks show clear central point sources, one shows a weak tidal tail, one of the irregular systems is classified as a train-wreck merger.
Out of the 20 X-ray detected sources, five show SEDs clearly dominated by the galaxy, with best fit SEDs having very low levels of star formation. Inspection of the ACS B-band imaging data for all objects with galaxy dominated SEDs show clear central compact sources.
Seven objects are classified as mixture objects, with AGN contributions in $z$ ranging from 20-50%. All of these show clear excess both bluewards of the Balmer break and in the IRAC bands with respect to galaxy templates. A representative example is shown in Fig. \[F:Ind\_XrayNice\].
![Multiwavelength SED of J033252.88-275119.8, overlaid the best fit mixture. Inset plots shows a $5\times5$ arcsec cut-out of the object in $z$.[]{data-label="F:Ind_XrayNice"}](Individual_ExampleXrayStamp.pdf){width="8cm"}
The five objects with quasar dominated SEDs are all point sources. Their SEDs are relatively flat spectra over a wide wavelength range. Redshifts for those objects are problematic. Several redshifts seem wrong even though they are spectroscopic. The featureless spectra make it difficult to determine proper redshifts.
One spectacular variability selected object with X-ray detection is J033228.30-274403.6. The object has a photometric redshift of 3.25. The SED is fit reasonably by either a Spiral or a Seyfert 1.8 SED. However, there are erratic jumps in the SED that indicate variability. The object is detected in the soft X-rays, but not the hard X-rays. The X-ray luminosity is $2.12\times10^{43}$ erg/s. The SED, as well as the image in $z$ is shown in Fig. \[F:Ind\_AwesomeSeyfert\]. The objects is relatively faint, extended and highly disturbed. This fascinating source shows the incredible power of variability selection.
![Multiwavelength SED of J033228.30-274403.6, overlaid the two best fit SED. Inset plots shows a $5\times5$ arcsec cut-out of the object in $z$.[]{data-label="F:Ind_AwesomeSeyfert"}](Individual_AwesomeSeyfertStamp.pdf){width="8cm"}
Properties of radio detected AGN {#S:radio}
--------------------------------
Radio emission is prominent in only about 10% of all AGN, however, in those radio-loud objects it can be significant. We therefore investigate the radio properties of the AGN candidates. Only five sources have radio detections. Of the five radio-detected variable AGN, three are radio loud, J033210.91-274414.9, J033217.14-274303.3 and J033239.47-275300.5. J033210.91-274414.9 and J033217.14-274303.3 are also detected in the X-ray. Morphologically, they are distinct, J033210.91-274414.9 and J033217.14-274303.3 are point sources, while J033239.47-275300.5 is located in a close pair of galaxies with a weak stellar bridge. Of the five radio detected AGN, two are low-redshift spirals and three are point-like objects. Three of the radio objects are also detected in the X-ray, two are compact sources and one is a low-redshift spiral galaxy. The SEDs of the radio detected sources are discussed in more detail below. As expected, radio selection only identifies a small subset of AGN. Radio detection are provided in Table \[T:AGN\] in the Appendix.
J033210.91-274414.9 has a spectroscopic redshift of 1.6 and a photometric redshift of 1.86. It is extremely radio-loud, with a value $log(R)\sim3.5$, it is also detected in the X-rays with $\Gamma=1.6$ and has an absorption corrected absolute luminosity of $2.38\times10^{44}$ erg/s, making it one of the most luminous objects in this sample. The object has an absolute magnitude in $z$ of -23.04. The entire SED is shown in Fig. \[F:Ind\_Blazar\]. The SED is rather peculiar. The optical-NIR spectrum is a power-law. Such shapes are usually common in blazars: highly beamed, extremely variable AGN [@urry_unified_1995]. The spectra of this objects shows broad, but relatively weak, emission lines. The radio spectrum has a slope of $\alpha=0.5$, putting it just on the border between Flat Spectrum Radio Quasars (FSRQs) and Steep Spectrum Radio Quasars (SSRQs). This suggests that the objects is likely a blazar.
![Multiwavelength SED of J033210.91-274414.9, overlaid several of the best fitting SEDs, redshifted to the spectroscopic redshift of 1.613 derived for this object. The best fitting SED was QSO1, followed by QSO2 and Mrk231. Inset plots shows a $5\times5$ arcsec cut-out of the object in $z$.[]{data-label="F:Ind_Blazar"}](Individual_BlazarStamp.pdf){width="8cm"}
J033217.14-274303.3 has a spectroscopic redshift of 0.566 and a photometric redshift of 0.49. It has a radio-loudness $log(R)\sim1.7$ and is also detected in the X-ray ($\Gamma=1.87$). The radio spectrum is rather flat ($\alpha=0.5$). An absolute magnitude in $z$ of -22 puts it below the limit for quasars. The object is clearly extended with a early-type morphology. The object shows a clear contribution from a stellar bump as well as a steep rise at 8$\mu m$, typical of many of the variability selected objects discussed in the following sections.
J033239.47-275300.5 does not have a spectroscopic redshift. The photometric redshift is 0.64. It is radio-loud with $log(R)\sim2.2$ and is not detected in the X-rays. With an $\alpha$ of 0.4, it is the radio-detected object with the flattest spectrum in our sample. The AGN is located in an elliptical galaxy that is located in a galaxy pair. The second galaxy is also spheroidal but more symmetric than the galaxy with the AGN. They seem to be connected by a faint stream, suggesting that they are actually undergoing a merger. The projected distance between the two galaxies is about 1.2 arcsec, which corresponds to approximately 8kpc at the redshift of the object. The SED is galaxy dominated. Apart from the extremely strong radio flux, there is no clear sign of AGN activity, however, we decide to keep this in the sample to due its radio-loudness.
Two of our objects (J033229.88-274424.4 and J033229.99-274404.8 ) are clearly extended spiral galaxies at very low redshift (both objects have a spectroscopic redshift of 0.076). They show clear extended spiral galaxy patterns as well as clear central point sources. Both objects have radii of about 2 arcseconds. Both objects are detected in radio but are radio-quiet ($R\sim2-3$). Given the fact that the AGN contribution to the overall flux in these objects is lower than 10%, this is a clear lower limit and the AGN itself is radio-loud in both objects. Both objects are very bright, with absolute $z$ magnitudes around 16.5, making them moderately bright spiral galaxies with $z$ band magnitudes around -21. One of the objects has spectroscopic data and shows a relatively featureless spectrum with only weak lines.
Due to the small contribution from the central point source, the X-ray emission is so weak that of those two object, only one (J033229.99-274404.8) is detected in the X-rays, with a low X-ray luminosity of only $1.3\times 10^{40}$ erg/s and $\Gamma=1.6$. The other object remains undetected, even in the 4Ms data. Interestingly, the two radio-detected spirals are at the same redshift and are only separated by a projected distance of 30kpc. It is unclear if they are gravitationally bound or if tidal forces might have contributed to AGN activity in both of those sources, this might fit with an emerging picture that close interaction without actual merging enhance AGN activity [e.g. @ellison_galaxy_2011; @koss_understanding_2012].
SEDs of AGN candidates not detected in X-ray or radio
-----------------------------------------------------
In the following sections, we will shortly discuss those sources not detected in X-ray or radio, we will start by identifying false positives and then continue on to different subsamples of AGN, divided by percentage AGN contribution.
### Likely False Positives {#S:contaminants}
While a large majority of objects in the sample show clear signs of AGN activity in their SEDs, in a few cases, the SEDs are very well described by a galaxy SED without other signs of AGN activity. In particular, there are six cases which might possibly not be AGN but false positive detections, we will shortly discuss their properties.
J033241.87-274651.1 is a ’tadpole’ type galaxy [see e.g. @straughn_tracing_2006 for a description of this class of galaxies]. The best fit SED is a Spiral galaxy SED, but the actual SED appears slightly reddened. While weak AGN activity in the ’head’ is not excluded, this object is probably a false positive.
J033219.86-274110.0 is entirely galaxy dominated, with a best-fit ’Sd’ SED. The rest-frame UV imaging data shows no clear central point source. The object appears to be undergoing a merger. It is very likely a false positive.
J033210.52-274628.9 appears to be somewhat reddened with respect to a normal S0 SED, there is no clear evidence of AGN activity in the SED. The object is very compact but resolved. It is likely a false positive.
J033246.37-274912.8 is reasonably well fit by a S0 SED, the morphology shows a clear mixture of bulge and disk components, there is no central point source visible in the rest-frame UV frames, this is likely a false positive.
J033209.57-274634.9 is reasonably well fit by a Sdm SED. There is a red excess that can be fit by including a QSO component, however, there is no UV excess. This might be explained by obscuration depressing the UV excess from star formation or possibly by invoking an UV-weak AGN. This source is possibly a false positive.
J033225.10-274403.2 is well fit by an S0 SED, it shows a clear disk+bulge morphology. It is at a redshift of 0.076 and very clearly extended. The UV excess in this source is extremely weak. This is likely a false positive.
We do caution that while those cases are likely false positives, these objects are marked in Table \[T:AGN\] in the Appendix. Note that all of these sources are in the ’normal’ catalogue of variability selected sources [@villforth_new_2010] that is expected to contain around six false positives in the GOODS-S field. Therefore this false positive detection rate is expected. We no longer consider false positives in the following discussion.
### Quasars {#S:quasars}
A total of ten sources have quasar dominated SEDs, four of which are X-ray detected, one source is detected in both radio and X-ray, leaving six sources with quasar dominated SEDs but lacking X-ray and radio detection. Redshifts for these sources are relatively unsecure.
### Seyferts {#S:seyferts}
As Seyferts, we label objects not detected in the X-rays but with SEDs classified as mixtures (the AGN contributes between 10-90% in the $z$ band). A total of 19 objects show mixture fits, of which nine are detected in neither X-ray nor radio. A representative example of this group is shown in Figure \[F:Ind\_Seyfert\]. Similar to X-ray detected AGN with mixture SEDs discussed in the previous Section, these sources show clear excesses both bluewards of the Balmer break as well as in the IR.
![Multi-wavelength SED of the Seyfert type object J033218.70-275149.3, overlaid the best fit mixture model. Inset plots shows a $5\times5$ arcsec cut-out of the object in $z$.[]{data-label="F:Ind_Seyfert"}](Individual_SeyfertStamp.pdf){width="8cm"}
### Ultra-low luminosity AGN {#S:faint_AGN}
As ultr-low-luminosity AGN, we label those sources for which the mixture fits indicate AGN contributions below 10%. 14 objects belong to this class, of which eight lack both X-ray and radio detection. In general, their SEDs resemble the SEDs of Seyfert type objects, showing excesses in both the UV and MIR. However, for those objects the excesses are notably weaker. The extra component needed to fit the SED is in general bluer than the QSO template SED, indicating a hotter accretion disk. This is expected for less massive black holes [e.g. @laor_massive_1989]. A representative example is shown in Fig. \[F:Ind\_ULLAGN\].
![The Ultra-low luminosity AGN J033205.40-274429.2, overlaid the best fit mixture model. Inset plots shows a $5\times5$ arcsec cut-out of the object in $z$.[]{data-label="F:Ind_ULLAGN"}](Individual_ULLAGNStamp.pdf){width="8cm"}
Failed Fits {#S:failed_fits}
-----------
The failed fit sources are well fit neither by a mixture model, single galaxy or AGN SED nor a stellar SED. 12 objects belong in this group. Their SEDs show erratic bumps that are indicative either of variability or strong line emission, failed fits objects show no clear SED features such as breaks that are reproduced by the templates. Figure \[F:Ind\_bumpy1\] shows a representative example of this group, apart from the poor fit, it is apparent that there is no clear stellar IR bump that it usually needed to provide a reasonable fit. Failed fit objects are generally faint but extremely compact, their featureless SEDs and erratic bumps indicate that they are highly variable and likely quasar dominated. Since no good fit is found for those objects, their redshifts are poorly constrained. Carefully modelling the variability across the wavelength regime in these sources might help recover the underlying SED, but this is beyond the scope of the current paper.
![SED of J033219.81-275300.9 with best fit mixture SED overlaid, the SED is clearly bumpy ad yields a poor fit for all templates. Inset plots shows a $5\times5$ arcsec cut-out of the object in $z$.[]{data-label="F:Ind_bumpy1"}](Individual_bumpy1Stamp.pdf){width="8cm"}
IRAC Colours {#S:irac}
------------
IRAC colour selection has commonly been used as a way to select AGN. Due to the fact that the dust in AGN is heated to a higher temperature than in normal starforming galaxies, their IR SEDs show considerable differences [e.g. @sanders_warm_1988; @lacy_obscured_2004; @stern_mid-infrared_2005; @donley_spitzers_2008]. Two different IRAC colour-colour plots have been suggested to separate AGN from star-forming galaxies: @lacy_obscured_2004 and @stern_mid-infrared_2005, both defined a wedge within which most objects should be AGN dominated. Figure \[F:Irac\] shows the colour-colour plots for AGN candidates as well as the full GOODS-S Sample. For the AGN dominated sources with IRAC detection, 80% are in the Lacy Wedge and 50% are inside the Stern wedge. For the Mixture objects 83%/54% are inside the wedge for Lacy and Stern respectively. For the galaxy dominated sources, those numbers drop to 12%/0% for Lacy and Stern respectively. For the objects with failed fits, 54%/ 18% would be flagged as possible AGN using the Lacy and Stern correction respectively. This shows that both selection methods recover moderately luminous AGN but are not suitable to detect the lowest luminosity AGN.
[@donley_spitzers_2008] have studied the reliability of different AGN IR selection techniques and found that contamination of normal star-forming galaxies is rather common in both the Lacy and Stern wedge, especially when low-luminosity sources are included. They showed that the colour-colour selection will include most actual AGN, but with a significant contamination. However, here we demonstrate that low-luminosity AGN can be located outside the Stern and Lacy wedges. We therefore argue that IR colour-colour selection misses most low luminosity systems in which the SED is dominated by the galaxy, as also noted in @donley_identifying_2012.
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Summary of SED Fitting Results
------------------------------
Here we will short summarize the results presented in this section.
SED fist were performed on 71 of the initial 88 sources found in GOODS-S by @villforth_new_2010, the rejected 14 objects being stars previously identified in the literature. SED fits revealed another ten stars, mostly of G-M type, listed in Table \[T:Stars\] in the Appendix but further disregarded in the analysis, this leaves 61 AGN candidates.
For the remaining 61 AGN candidates UV-IR AGN/Galaxy mixture SED fits were performed, six are likely false positives, in good agreement with the expected number of false positives in the sample. This leaves 55 confirmed AGN. 43 had successful fits, leaving 12 failed fits, discussed in Section \[S:failed\_fits\]. Twenty sources have X-ray detection, five have radio detection, three are detected in both X-ray and radio. Of the 43 sources with successful fits, ten are quasar dominated, 19 are mixture objects and 14 are galaxy dominated. Especially the low luminosity AGN would be missed by IRAC selection techniques.
In summary, we show that variability selection can reliably identify AGN far below the limits of even extremely deep X-ray data and not identified using commonly used IRAC colour-colour selection methods. SED fits where succesful for 43/61 sources, with the remaining SEDs showing clear non-stellar SEDs with clear bumps, indicating that they are indeed AGN.
The host galaxies of variability selected AGN {#S:hosts}
=============================================
Morphologies of variability selected AGN {#S:morph}
----------------------------------------
The properties of AGN host galaxies are of importance for understanding the triggering mechanisms of AGN. In this section, we will discuss the morphologies of the host galaxies in this sample. Morphological Classification is performed by eye. The classifications scheme is described in detail in the Appendix, and all morphological classifications are provided in Table \[T:AGN\]. In the following discussion we will distinguish between unresolved sources, disks, bulges, disk-bulge mixtures, mergers and unclassifiable.
Figure \[F:LP\_detail\] show the morphologies in the luminosity-redshift parameter space. Interaction is common, especially at the low luminosity end. At lower redshift, late-type morphologies dominate. No AGN is hosted by an early-type galaxy at redshifts higher than $\sim$1. Out of the 12 sources classified as late-type, eight show clear central compact sources, two sources show weak tidal tails indicating possible recent mergers. For five out of twelve late-type hosts, the orientation can be determined clearly, of those, four are face-on while only one is clearly edge on. While the numbers are too small to perform a detailed analysis, this might indicate that variability selection is more successful in face-on systems. This could either indicate that AGN are aligned with the host galaxy, or that absorption is stronger in edge-on systems.
Out of the mere five early-type galaxies, only two show very obvious central compact sources and one shows a close-by neighbour (J033239.47-275300.5, discussed in more detail in Section \[S:radio\]). Five objects are classified as disk-bulge mixture, of which one is clearly edge-on. All of those spheroids were best fit with an S0 galaxy template and show AGN contribution in $z$ of $\sim 5\%$. 18 objects are classified as interacting, of those, three show clear central compact cores, four are very clear ongoing mergers and five are classified as clumpy or train-wreck. Of the fourteen objects without a clear morphological classification, two show compact cores, two have nearby neighbours and one shows a weak tidal tail. Implications of these findings will be discussed in Section \[S:discussion\].
Red sequence and blue cloud: are variability selected AGN experiencing quenching of star formation? {#S:redblue}
---------------------------------------------------------------------------------------------------
![Star formation properties and morphology of variability selected AGN in a observed frame $z$ band magnitude redshift plot. Colours indicate star formation level: red: passive, green: moderately star forming; blue: strongly star-forming. Marker style show morphology: circles: disk galaxies; down triangles: elliptical; up triangle: disturbed morphology; pentagons: disk/bulge mixtures; cross: unclear (too faint); stars: point source dominated. The magnitudes are shown for the entire object, i.e. contribution from the host galaxy is not subtracted off. False positives are not shown.[]{data-label="F:LP_detail"}](LP_starformation_lin.pdf){width="8cm"}
One of the questions that is commonly discussed is if and how strongly AGN activity correlates with star formation in the host galaxy. To address this question, we show the level of ongoing star formation in the host galaxies as a function of redshift and luminosity. We divide the host galaxy SEDs into three groups: purely passive (no young stellar population), mildly star forming (contribution from unobscured young star forming regions) and starbursting (young stellar population dominant, considerable reddening). The results are shown in Figure \[F:LP\_detail\]. Starburst type SEDs are common only in high luminosity AGN, low luminosity AGN hosts are dominated by either mildly star-forming or passively evolving host galaxies. The implication of these finding will be discussed in Section \[S:discussion\].
![Optical Colours of variability selected AGN host galaxies from SED Fits compared to the full sample of galaxies. Colours indicate star formation level: red: passive, green: moderately star forming; blue: strongly star-forming. Symbols indicate morphology, as described earlier in Section \[S:morph\]: stars: point source, triangle: interacting; circles: disks; pentagons: elliptical; x: unclassified. Only sources with successful fits that have not been identified as false positives are shown.[]{data-label="F:redblue"}](RedSequenceBlueCloud.pdf){width="8cm"}
It has been noticed that galaxies often fall into two rather distinct groups: massive, passively evolving ’red and dead’ elliptical galaxies and star-forming, less massive spiral galaxies [e.g. @strateva_color_2001; @balogh_bimodal_2004; @faber_galaxy_2007]. These groups are known as the red sequence and blue cloud. Additionaly, there is a transitional phase of green valley galaxies in which the star formation is just turning off [e.g. @strateva_color_2001; @balogh_bimodal_2004; @faber_galaxy_2007]. It has been argued that the host galaxies of AGN should be located in the green valley since the AGN could be responsible for quenching the star formation [@silk_quasars_1998; @hopkins_cosmological_2008; @somerville_semi-analytic_2008]. Evidence for this scenario has been mixed [@cardamone_dust-corrected_2010; @schawinski_galaxy_2010]. We show the location of the host galaxies of our AGN with respect to the red sequence and blue cloud in Figure \[F:redblue\]. The AGN host galaxies are located in all areas of the colour-magnitude diagram. In the two low redshift bins, the hosts of AGN are found in the green valley as well as the red sequence. In the two higher redshift bins, the AGN host galaxies are located in the blue cloud as well as the green valley. V-band selected variable AGN hosts appear to be distributed over a range of colors as well, from the red sequence through the green valley and into the blue cloud [@sarajedini_variability_2011]. A slightly higher fraction of blue cloud hosts were found among the variability selected AGN when compared to those identified in X-ray surveys.
We caution that the technique used to determine host galaxy colours does not entirely remove the contamination from the central source and might therefore be biased. Properly accounting for this effect would require careful PSF Fitting in multiple bands which is beyond the scope of the current study. In the redshift range below $z<0.75$, the location of AGN host galaxies in the red sequence and blue cloud is comparable to the Seyfert host galaxies studies by [@schawinski_galaxy_2010]: the late-type hosts are predominantly located at the high-mass end and avoid the blue cloud.
Are more luminous AGN located in higher mass galaxies? {#S:AGNvsGalMass}
------------------------------------------------------
An important aspect regarding the properties of AGN hosts is the fact that more luminous AGN seem to be hosted by more luminous and therefore more massive galaxies [@hutchings_optical_1984; @bahcall_hubble_1997 e.g.]. We show the relation between galaxy and AGN magnitudes for our sample in Figure \[F:AGNVSGal\] (at restframe $\sim$ 1 $\mu$m), together with comparable data from the literature [@kotilainen_host_2006; @kotilainen_nuclear_2007; @hamilton_fundamental_2008]. Our data show that there is a clear correlation between the luminosities of AGN and their hosts. However, the luminosity of the AGN rises faster than the luminosity of the galaxy similar to findings in other studies [e.g. @hutchings_optical_1984; @bahcall_hubble_1997; @schramm_host_2008 and references therein]. This can be interpreted as a sign that higher mass black holes accrete at higher Eddington ratios. But it can also imply a tendency for black holes in more massive galaxies to be comparably overmassive, i.e. have higher $M_{BH}$ to $M_{galaxy}$ ratios. However, in a few cases, the galaxy luminosity is observed to rise faster at the high luminosity end [@schramm_host_2008]. Also, rare cases in which high luminosity AGN show no sign of host galaxy contribution are known [@magain_discovery_2005].
Selection effects play an important role in this context. Systems with very bright AGN and very faint hosts (lower right corner of Figure \[F:AGNVSGal\]), while not generally selected against, have SEDs clearly dominated by the AGN and therefore galaxy magnitudes are difficult to derive. On the other hand, systems with very high galaxy to AGN luminosity ratios (upper left corner of Figure \[F:AGNVSGal\]) are selected against because the AGN in such systems are evasive and will not be identified as AGN, even using variability selection. Another effect is that high-redshift AGN are generally more luminous [@croom_2df-sdss_2009], moving them towards the upper right corner. Selection effects also differ at higher redshifts: systems with high AGN-galaxy luminosity ratios are even harder to detect due to surface brightness dimming. Additionally, galaxy dominated systems are also harder to detect since low luminosity AGN are more difficult to identify at higher redshifts. This should results in a narrowing of the accessible parameter space towards higher redshift.
To analyse the correlation in more detail, we have performed a linear fit to the data for the full sample, as well as a number of subsamples. The fit for the entire sample shows a flat slope of $\sim 0.76$, similar to the slopes found in similar earlier studies [@hutchings_optical_1984; @bahcall_hubble_1997]. This therefore indicates higher Eddington ratios in more luminous AGN or comparably overmassive black holes in these systems. The latter might indicate that those AGN are in a later stage of their black hole growth.
Dividing the sample into subsamples according to the star formation level in the host, we find that the slope is significantly flatter for AGN with active star formation. This seems to indicate either that the tendency for higher mass black holes to show higher Eddington ratios is more strongly pronounced in actively star-forming systems, while it is weak for AGN with passive hosts. But it can also mean that star-forming systems are generally in a more advanced stage of black hole growth where the black hole is comparably more massive. This could indicate that those systems are nearing the end of their activity.
Dividing the sample into subsamples according to morphology, we find that there is no significant difference in the slope, but that there is a clear offset between the two datasets: AGN in interacting hosts having on average either higher Eddington ratios or larger $M_{BH} / M_{galaxy}$ ratios. It is however unclear if these findings are due to selection effects or if the trends found here reflect actual physical trends in the AGN population. Studying connections between host galaxies and AGN is complex, and it has been shown that carefully implementing selection effects can remove certain trends [e.g. @aird_primus:_2011].
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The environments of variability selected AGN {#S:environment}
============================================
In the context of AGN triggering, it is of importance to understand the environments of AGN of different types, since this can constrain suspected triggering mechanisms such as major or minor mergers, disk instabilities or inflow of gas.
To asses the environment of our AGN sample, we use the large-scale structure maps for GOODS-S from [@salimbeni_comprehensive_2009]. They used the catalogues of galaxies in GOODS-S with both spectroscopic and photometric redshifts to identify groups and clusters of galaxies. A total of twelve over-densities where found in GOODS-S, with redshifts between $\sim$0.6–2.5. These over-densities overall represent larger groups rather than clusters, with virial masses on the order of $10^{14} M_\odot$ and peak overdensities of 6–10. Fig. \[F:clusters\] shows the galaxy groups from [@salimbeni_comprehensive_2009] as well as all variability selected AGN in the same redshift regime.
It is noticeable that some sources that are located near the centres of over-densities have been classified as late-type or disk-bulge mixtures while interacting systems as well as luminous AGN are either located towards the edges of over-densities or in the field areas. This seems to run counter to previous findings and theoretical expectations [@hopkins_characteristic_2009; @strand_agn_2008; @hickox_host_2009 e.g.] that high luminosity systems show enhanced clustering on small scales while Seyferts do not. In particular, this is surprising in the light of the expected connection of quasars with major mergers and Seyferts with secular processes @hopkins_characteristic_2009. Generally, our sources do not seem to favour dense areas, somewhat consistent with finding that AGN are on average slightly less clustered than normal galaxies [e.g. @constantin_clustering_2006; @li_clustering_2006].
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Another hint towards triggering mechanisms are nearest neighbour distances. We measure the projected distance to the nearest neighbour for each object and compare the results for the AGN to those of the general population. The distributions differ significantly: variability selected sources show a clear excess of close-by neighbours (Fig. \[F:neighbors\]). The significance of the nearest neighbour excess depends on the accepted redshift mismatch used for matching a neighbour. (The values are: p=$4.5\times 10^{-5}$ for $\sigma_{z}=0.1$, p=0.001 for $\sigma_{z}=0.05$, and p=0.08 for $\sigma_{z}=0.01$ using 2 sample KS). The decreasing significance for stricter matching in redshift might indicate a spurious result. However, with a typical scatter in the photometric redshifts of about 0.04, the more strict matching criteria will likely miss actual close neighbours. We find that using the 0.05 results (corresponding to typical scatter in the photometric redshifts) the closest neighbours of AGN hosting galaxies are on average at a projected distance of about 72kpc, while those of the control sample are on average at a distance of about 101kpc (Fig. \[F:neighbors\]). The results argue for a possible enhancement of AGN activity through tidal interactions. There is no clear trend for galaxies with more close-by neighbours to preferentially have disturbed morphologies or star-burst SEDs. This might indicate that weak tidal interaction may trigger AGN activity without significantly disturbing the host. The implications of these finding will be discussed in more detail in Section \[S:discussion\].
![Cumulative histograms for nearest projected nearest neighbour distances for variable AGN and control sample respectively.[]{data-label="F:neighbors"}](NearestNeighbours.pdf){width="8cm"}
Discussion {#S:discussion}
==========
In this paper, we have shown that variability selection identifies low luminosity AGN up to moderately high redshift. We have also demonstrated that commonly used selection methods would have missed a large number of these objects. We have also clearly demonstrated that those sources are indeed AGN using SED fitting. In this section, we aim to discuss the implications of these findings for AGN triggering mechanisms.
Theoretical models propose a number of different possible triggering mechanisms for AGN. Triggering an AGN requires funnelling substantial amounts of cold gas to the vicinity of the black hole while loosing a large fraction of its angular momentum. Several mechanisms could be invoked to achieve this.
The most popular triggering mechanisms currently discussed is major (wet) mergers: during a major merger of two gas rich systems, a large reservoir of gas is available, gravitational torques allow cold gas to settle to the centre. This gas reservoirs can be used to fuel both a central starburst and an AGN. This scenario has been proposed early [@sanders_luminous_1996; @silk_quasars_1998] and has recently gained popularity due to its ability to alleviate problems in galaxy formation models [e.g. @hopkins_cosmological_2008; @somerville_semi-analytic_2008]. However, while early observations seemed to support this scenario [e.g. @canalizo_quasi-stellar_2001], recent studies seem to imply that it is not responsible for the triggering of a majority of AGN [@cisternas_bulk_2011; @cisternas_secular_2011; @kocevski_candels:_2011; @schawinski_heavily_2012].
Observable signatures for an AGN triggered in a recent merger are disturbed morphology, a recent star-burst as well as a possible lack of close-by neighbours. While disturbed morphologies are common in the host galaxies of the variability selected AGN, they are seldom accompanied by star-burst SEDs. Also, the fact that the closest neighbour of the AGN in this sample are at a smaller distance than for the general population might speak against such a scenario. Also, the host galaxies of the AGN in this sample do not preferentially populate the green valley, as is expected for many models [e.g. @hopkins_cosmological_2008]. However, since we did not attempt to compare the morphologies and star formation rates to a properly matched comparison sample, these results remain tentative. While we cannot exclude major mergers as a trigger for at least some of the AGN in this sample, we do not find evidence for major mergers being the dominant process of AGN triggering. Our findings agree with recent studies finding that while merger triggering is likely in a subset of AGN [@ramos_almeida_are_2011], it is not dominant in the general population [@kocevski_candels:_2011; @cisternas_bulk_2011; @cisternas_secular_2011; @schawinski_heavily_2012]. On the other hand, the large incidence of disturbed morphology in our sample are consistent with other findings showing that disturbed morphologies are common in low-luminosity AGN [@de_robertis_ccd_1998; @bennert_evidence_2008].
Minor mergers occur much more frequently than major mergers [@croton_many_2006; @somerville_semi-analytic_2008; @lotz_major_2011] but their signatures are more difficult to identify: disturbed morphologies last for shorter time in minor mergers and the morphologies deviate less from those of undisturbed galaxies [@lotz_major_2011]. Theoretical models suggest that minor mergers might indeed be important triggering mechanisms for lower luminosity AGN [e.g. @hopkins_characteristic_2009 and references therein]. Minor mergers can cause disturbed morphologies in the host [@lotz_major_2011] but are likely not sufficient to trigger major starbursts [@somerville_semi-analytic_2008; @hopkins_characteristic_2009]. This scenario agrees well with the findings that low luminosity AGN show disturbed morphology but no signs of starbursts. This also is broadly consistent with the properties of Seyfert host galaxies [@de_robertis_ccd_1998; @bennert_evidence_2008; @constantin_active_2008].
Other triggering mechanisms have also been suggested, including cold flows [@bournaud_black_2011], tidal triggering through first passage before a merger [@hopkins_cosmological_2008] and bars in spiral galaxies[e.g. @shlosman_bars_1989; @knapen_subarcsecond_2000; @alexander_what_2011]. Given the dominance of $z<1-2$ sources in our sample, cold flow triggering is unlikely for sources in this particular sample since cold flows are more common at higher redshifts [@dekel_galaxy_2006]. Also, the process proposed in [@bournaud_black_2011] will predominantly produced rather heavily obscured AGN, which are not part of our sample due to the selection technique in this study. Spiral bars are not addressed in this study, the resolution is not sufficient to clearly examine the presence of bars in this sample, and we will therefore not comment on this possibility. However, we do note that late-type galaxies only represent a small subset of our sample at low redshift.
A mechanisms that seems possible for our sample are triggering during first passage. We find an excess of close neighbours in our sample compared to the general galaxy population, but no clear difference in the general density. This is consistent with first passage triggering. Our findings are also consistent with other studies finding enhanced AGN fractions in close pairs [e.g. @ellison_galaxy_2011; @farina_study_2011; @koss_understanding_2012]. Interestingly, obscured low luminosity AGN are found to have more close neighbours than unobscured sources [@dultzin-hacyan_close_1999; @koulouridis_local_2006; @de_robertis_ccd_1998-1]. This might indicate that first passage triggering is connected with higher levels of obscuration. Given the preference of variability selection to find unobscured sources, the excess in close neighbours we find might therefore be a lower limit to the actual incidence of AGN with close neighbours.
While this seems somewhat cynical, there are several studies suggesting that AGN activity has no preference at all in environment [e.g. @miller_environment_2003; @mclure_cluster_2001; @wold_radio-quiet_2001] or host galaxy mass as well as stellar populations [@aird_primus:_2011]. These finding would imply a ’random’ triggering process that has no preference on galaxy properties or environment. We caution therefore that this study is hampered by a lack of comparison sample, other studies have demonstrated that well matched comparison samples are essential since some apparent trends can be solely due to selection effects [@kocevski_candels:_2011; @cisternas_bulk_2011; @cisternas_secular_2011; @aird_primus:_2011].
Our findings are inconsistent with major merger triggering in a majority of sources, especially at the low luminosity end. Our data suggest that minor mergers and triggering through first passage likely play a large role for lower luminosity AGN. A more detailed analysis of triggering mechanisms will require carefully matched control samples, this is however beyond the scope of the current study.
Summary and Conclusions {#S:summary}
=======================
In this paper we have presented the SEDs, host galaxy properties and environmental properties of variability selected AGN in GOODS-S. Our results can be summarized as follows:
- It has been demonstrated that variability selection can complement and expand other AGN selection techniques and expand them to much lower luminosities. Variability selection can reveal AGN too faint to be detected in even extremely deep X-ray exposures as well as AGN in which the galaxy emission is too dominant to reveal them as AGN in IR colour-colour plots. We have also shown that variability selection indeed identifies low-luminosity AGN and that contamination with normal galaxies is extremely small (6/88), consistent with expected false positive rates. This demonstrates that the method used in @villforth_new_2010 predicts expected false positive rates correctly.
- Careful SED analysis has revealed that a considerable number of red point-source in the variability selected sample turn out to be stars, in particular of types G-M. In total 24/88 objects from the initial sample turn out to be stars. This reveals the possibility of using variability selection also to find possible rare halo stars as well as the importance of more carefully checking point sources for further variability selection studies.
- Of the 61 AGN candidates for which SED fits were performed, six are likely false positives, in good agreement with the expected number of false positives in the sample. Of the 55 confirmed AGN, 43 had successful fits, leaving 12 failed fits. Twenty sources have X-ray detection, five have radio detection, three are detected in both X-ray and radio. Of the 43 sources with successful fits, ten are quasar dominated, 19 are mixture objects and 14 are galaxy dominated.
- SED Fits reveal that variability selection is suitable for selecting AGN that contribute as little as 10% to the overall emission in a given bands. Variability selection has revealed a considerable number of considerable sub-Seyfert AGN up to redshifts close to 3.
- Star formation in the hosts of AGN appears most prominent in the highest luminosity objects, while lower luminosity objects appear quiescent or moderately star-forming.
- The AGN hosts in these study do not have a preferred location with respect to the red sequence and blue cloud, rather the lower redshift low luminosity objects are located in the red sequence and partially the green valley, while higher redshift and higher luminosity objects tend to lie in the blue cloud and partially the green valley. It is not possible to determine for this sample if the redshift or luminosity are the deciding factor for this difference.
- Disturbed morphologies are common in variability selected AGN and dominate the sample at low AGN luminosities, elliptical galaxies as hosts are rare.
- We find that the luminosity of AGN and their hosts are generally correlated, but that the AGN luminosity rises faster than that of its host. These finding indicate that either the Eddington ratio or black hole mass to galaxy mass ratio might be higher in more luminous AGN. There are also tentative findings that this trend is stronger for AGN with ongoing starbursts and that AGN with disturbed morphologies on average have higher Eddington ratios or comparably overmassive black holes given their host galaxy mass.
- There is no clear evidence for AGN favouring either very high density or very low density environments, however, there is a trend for AGN near the cores of groups to be hosted by late-type galaxies, high luminosity AGN are only found in the field.
- Nearest neighbours are about 25% closer in the variability selected AGN sample than they are in the general population, arguing for tidal interactions as possible triggers for weak AGN activity.
- We find possible indications for different triggering mechanisms at the high and low luminosity end. Lower luminosity AGN in this sample appear to be connected to events which disturb the hosting galaxy but are not capable of triggering extreme star-bursts. We suggest minor mergers or tidal forces during first passage as possible triggering mechanisms. Higher luminosity AGN seem to be connected to events triggering strong starbursts, widely consistent with major mergers. However, since the AGN inhibits a careful analysis of the host galaxy morphology in those systems, it is unclear if they are connected to major mergers or other possible triggers. We would like to caution that we have not properly analysed selection effects. Further careful comparison with a control sample of non-AGN hosting galaxies could show if these findings are due to selection effects. This is however beyond the scope of the current paper.
This study therefore demonstrated the strength of variability selection, for the first time determined its relaibility in identifying low luminosity AGN that cannot be found and gave preliminary hints for the possible AGN triggering mechanisms in low luminosity high redshift AGN.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank the anonymous referee for a careful reading of the manuscript and constructive comments. We would like to acknowledge funding through National Science Foundation (NSF) Award AST-1009628 as well as HST grants AR-09935, GO-10134, GO-10530 and GO-11262.
Morphological Classifications {#A:morphology}
=============================
Detailed Morphological Classifications are provided in Table \[T:AGN\]. All Morphological classifications are performed by eye in the $z$ band, additionally, we also inspected other available HST bands (corresponding to approximately U,V and I bands) to verify our classification. The resolution of the data is 0.1 arcsec, corresponding to a physical scale between 0.2kpc (z=0.1) and 0.8kpc (z=3). It should be noted that by using the $z$ band as the prime band for inspection, we introduce possible biases since shorter and shorter rest-wavelengths are sampled at higher redshifts. In particular, this might cause us to overestimate the incidence of disturbed morphologies at higher redshifts.
Unresolved sources can either be classified as ’true unresolved’ (U), i.e. visual classification does not show signs of host galaxy contribution and ’weak host’ (UH) in which visual classification shows clear signs of host galaxy contribution. For those sources, additional flags can be set (see below).
For the extended sources, we have five main morphology classes as well as additional flags (see below). The five main are: late-type galaxies (D, clear disk, no strong bulge), early-type galaxies (E), ’spheroidal’: a clear disk-bulge component (S), interacting/disturbed (I), all other sources that are too faint for a clear classification are classified as unclear (X). The following flags are available additionally for all extended as well as unresolved with clear host sources: clear dominant core (C), nearby neighbour (N), edge-on late-type (E), face-on late-type (F), clear merger (M), weak tidal tails (T), multiple cores/clumpy/trainwreck (X).
Name RA (J2000) Dec (J2000) $mag_{z}$ z Morphology Type AGN Galaxy Ratio Xray? Radio?
--------------------- ------------ ------------- ----------- -------------- ------------ ------ -------- -------- ------- ------- -------- --
J033203.00-274213.6 53.01248 -27.7037872 25.16 2.66$^{p}$ EX Q BQSO1 Spi4 0.9 No No
J033203.01-274544.7 53.0125331 -27.7624232 25.12 0.54$^{p}$ ED G BQSO1 Spi4 0.1 No No
J033203.26-274530.3 53.013574 -27.7584257 22.72 0.04$^{p}$ ED G BQSO1 Spi4 0.1 No No
J033205.40-274429.2 53.0224983 -27.7414518 22.99 1.039$^{s}$ EDF G BQSO1 Spi4 0.05 No No
J033208.68-274508.0 53.036184 -27.7522317 22.71 1.296$^{s}$ EDC X - - - No No
J033209.57-274634.9 53.0398585 -27.7763722 22.97 0.5375$^{s}$ EEC FP - - - No No
J033209.58-274241.8 53.0399142 -27.7116218 23.44 0.681$^{s}$ EIC G BQSO1 Spi4 0.1 No No
J033209.80-274308.6 53.0408277 -27.7190613 23.15 2.48$^{p}$ EIM Q BQSO1 Ell2 0.95 No No
J033210.52-274628.9 53.0438334 -27.7747007 23.88 1.615$^{s}$ EXN FP - - - No No
J033210.91-274414.9 53.0454704 -27.7374846 22.34 1.613$^{s}$ UQXRH Q Mrk231 - 1.0 Yes Yes
J033211.02-274919.8 53.045918 -27.8221721 23.45 1.9431$^{s}$ EIMX M BQSO1 Spi4 0.3 No No
J033212.16-274408.8 53.0506774 -27.7357915 24.81 0.74$^{p}$ EX M BQSO1 Ell2 0.3 No No
J033213.21-274715.7 53.0550544 -27.7876915 24.19 0.523$^{s}$ EDEC G BQSO1 Ell2 0.1 No No
J033217.06-274921.9 53.0710682 -27.8227401 19.19 0.34$^{s}$ EDT G TQSO S0 0.1 Yes No
J033217.14-274303.3 53.0714326 -27.7175864 20.58 0.566$^{s}$ EE M TQSO S0 0.2 Yes Yes
J033217.72-274703.0 53.0738469 -27.7841607 23.63 1.607$^{s}$ EI Q BQSO1 Sdm 0.9 No No
J033218.24-275241.4 53.0760024 -27.8781606 24.26 2.801$^{s}$ UQX Q BQSO1 Sey18 0.9 Yes No
J033218.70-275149.3 53.0778965 -27.8637054 21.76 0.458$^{s}$ ESE M QSO1 Ell5 0.15 No No
J033218.81-274908.5 53.0783542 -27.8190409 23.88 1.128$^{s}$ EIX X - - - No No
J033218.84-274529.2 53.078519 -27.7581146 18.79 0.296$^{s}$ EE G TQSO Ell13 0.05 Yes No
J033219.81-275300.9 53.0825539 -27.8835727 24.50 3.706$^{s}$ EI X - - - No No
J033219.86-274110.0 53.0827492 -27.686119 23.40 1.542$^{s}$ EIM FP - - - No No
J033222.82-274518.4 53.0950956 -27.7550986 23.24 1.087$^{s}$ EXC M BQSO1 Ell2 0.3 No No
J033223.53-274707.5 53.0980434 -27.785425 23.26 0.69$^{p}$ EDCF M BQSO1 Ell2 0.2 No No
J033224.23-274129.5 53.1009433 -27.691518 24.89 4.34$^{p}$ EI Q BQSO1 Ell2 1.0 No No
J033224.54-274010.4 53.1022685 -27.6695645 21.91 0.96$^{p}$ EI X - - - Yes No
J033224.80-274617.9 53.1033449 -27.7716431 23.09 1.306$^{s}$ EX X - - - No No
J033225.10-274403.2 53.1045636 -27.7342138 17.76 0.076$^{s}$ ES FP QSO1 S0 0.0 No No
J033225.99-274142.9 53.1082952 -27.6952603 24.56 0.0459$^{s}$ EICN Q BQSO1 Spi4 0.95 No No
J033226.40-275532.4 53.109991 -27.9256529 24.44 1.77$^{p}$ EI X - - - No No
J033226.49-274035.5 53.1103938 -27.6765399 19.62 1.031$^{s}$ UQXHDCT Q QSO1 I19524 0.9 Yes No
J033227.01-274105.0 53.1125287 -27.6847238 19.03 0.734$^{s}$ UQ M QSO1 S0 0.75 Yes No
J033227.18-274416.5 53.113269 -27.73791 19.38 0.418$^{s}$ ES G TQSO S0 0.05 Yes No
J033227.51-275612.4 53.114644 -27.9367764 24.04 0.663$^{s}$ EXC X - - - No No
J033228.30-274403.6 53.1179209 -27.7343234 23.69 3.25$^{p}$ EI X - - - Yes No
J033229.88-274424.4 53.1244949 -27.7401248 16.46 0.076$^{s}$ EDCF G N6240 - - No Yes
J033229.98-274529.9 53.1249148 -27.7583013 21.11 1.215$^{s}$ UQ M TQSO I22491 0.5 Yes No
J033229.99-274404.8 53.1249588 -27.7346753 16.85 0.076$^{s}$ EDCF M Sey2 Spi4 0.3 Yes Yes
J033230.06-274523.5 53.1252547 -27.756535 21.82 0.955$^{s}$ EIX M TQSO S0 0.3 Yes No
J033230.22-274504.6 53.1258995 -27.7512749 21.51 0.738$^{s}$ UQ M BQSO1 Sc 0.5 Yes No
J033230.36-275133.2 53.1264805 -27.8592312 25.02 0.59$^{p}$ EX X - - - No No
J033232.04-274523.9 53.1334996 -27.7566329 23.44 0.43$^{p}$ EIX M BQSO1 Ell2 0.15 No No
J033232.49-275044.0 53.1353687 -27.8455431 23.44 0.41$^{p}$ EDC G QSO1 Ell2 0.1 No No
J033232.61-275316.7 53.1358826 -27.8879636 22.96 0.988$^{s}$ EIMC G BQSO1 Spi4 0.1 No No
J033232.67-274944.6 53.1361117 -27.829048 18.29 0.14$^{p}$ ES G BQSO1 S0 0.05 Yes No
J033233.68-274035.6 53.1403383 -27.6765489 24.97 0.81$^{p}$ EX X - - - No No
J033235.38-274704.3 53.1474321 -27.7845155 25.16 0.91$^{p}$ EX X - - - No No
J033236.92-275308.4 53.1538193 -27.885679 24.42 0.42$^{p}$ EXT M BQSO1 Ell2 0.3 No No
J033238.12-273944.8 53.1588307 -27.6624444 20.45 0.837$^{s}$ UQ Q BQSO1 I19524 0.95 Yes No
J033238.89-275406.7 53.1620545 -27.9018501 24.62 1.467$^{s}$ EXN M BQSO1 Ell2 0.8 No No
J033239.09-274601.8 53.1628593 -27.7671602 21.02 1.216$^{s}$ UQX M BQSO1 I19524 0.5 Yes No
J033239.47-275300.5 53.1644567 -27.8834689 20.55 0.64$^{p}$ EENC G - S0 0.0 No Yes
J033240.89-275449.2 53.1703678 -27.9136643 25.19 2.53$^{p}$ EX Q BQSO1 Sdm 0.9 No No
J033241.87-274651.1 53.1744478 -27.7808655 23.35 0.7$^{p}$ EI FP - Spi4 0.0 No No
J033243.24-274914.2 53.1801493 -27.8206046 22.70 1.906$^{s}$ UQX M TQSO I22491 0.5 Yes No
J033243.93-274351.1 53.1830401 -27.7308524 24.28 0.63$^{p}$ UQ M BQSO1 Ell2 0.5 No No
J033245.02-275207.7 53.1875887 -27.8688155 22.52 0.721$^{s}$ EDCT G Mrk231 S0 0.05 No No
J033246.37-274912.8 53.1932061 -27.8202112 21.95 0.683$^{s}$ ES FP TQSO S0 0.05 No No
J033247.98-274855.7 53.1999289 -27.8154702 20.57 0.233$^{s}$ EE X - - - Yes No
J033252.88-275119.8 53.2203537 -27.8555099 21.87 1.22$^{s}$ EDC M TQSO N6090 0.3 Yes No
J033253.44-275001.4 53.2226606 -27.8337103 24.35 1.245$^{s}$ EIX M BQSO1 Spi4 0.8 No No
\[T:AGN\]
Name Right Ascension (J2000) Declination (J2000) Spectral Type $mag_{z}$ \[AB\]
--------------------- ------------------------- --------------------- ---------------- ------------------
J033204.41-274635.5 53.018367 -27.7765167 M $^{a}$ 21.14
J033205.11-274317.5 53.021302 -27.7215411 M $^{a}$ 21.91
J033213.34-274210.5 53.0555659 -27.7029202 M $^{a}$ 23.84
J033215.16-274754.6 53.0631513 -27.7985117 K $^{a}$ 22.98
J033215.93-275329.3 53.0663591 -27.8914797 M2Iab $^{c}$ 19.86
J033216.34-274851.7 53.0681031 -27.8143745 M $^{a}$ 21.98
J033216.87-274916.7 53.0702817 -27.8212937 K $^{a}$ 23.40
J033220.80-275144.5 53.0866794 -27.8623513 M $^{a}$ 21.22
J033221.52-274358.7 53.0896509 -27.732984 M2Iab $^{d}$ 21.00
J033225.15-274933.3 53.1048118 -27.8259053 G0Iab $^{c}$ 21.61
J033227.86-275335.6 53.1160832 -27.8932186 M3III $^{e,i}$ 21.84
J033227.87-275335.9 53.1161233 -27.8933035 M3III $^{e,i}$ 21.63
J033228.45-274203.8 53.118527 -27.7010451 K $^{a}$ 24.23
J033231.80-275110.4 53.1324925 -27.8528853 M3III $^{d,i}$ 21.03
J033231.82-275110.6 53.1325905 -27.8529435 M3III $^{d,i}$ 21.06
J033231.94-274531.3 53.1330882 -27.7587052 G5V $^{f}$ 21.11
J033232.12-275636.8 53.1338346 -27.9435593 WD $^{a,b}$ 21.04
J033232.32-274316.4 53.1346778 -27.7212189 M $^{a}$ 22.11
J033237.93-274609.1 53.1580245 -27.769192 K $^{a,b}$ 19.96
J033241.05-275234.1 53.1710547 -27.8761468 G0V $^{c}$ 20.72
J033242.61-275453.8 53.1775348 -27.9149453 G8Iab $^{c}$ 20.82
J033244.10-275212.9 53.1837535 -27.8702564 K3III $^{d}$ 20.88
J033246.39-274820.1 53.1932909 -27.8055737 M2Iab $^{h}$ 21.16
J033247.53-275159.9 53.1980298 -27.8666421 G5V $^{a}$ 21.01
\[T:Stars\]
\[lastpage\]
[^1]: E-mail:[email protected]
[^2]: $http://www.iasf-milano.inaf.it/~polletta/templates/swire\_templates.html$
[^3]: Downloaded from http://www.stsci.edu/hst/observatory/cdbs/k93models.html
|
---
abstract: 'We study the ground state eigenvalues of Baxter’s $\Q$-operator for the eight-vertex model in a special case when it describes the off-critical deformation of the $\Delta=-\frac{1}{2}$ six-vertex model. We show that these eigenvalues satisfy a non-stationary Schrodinger equation with the time-dependent potential given by the Weierstrass elliptic $\Wp$-function where the modular parameter $\tau$ plays the role of (imaginary) time. In the scaling limit the equation transforms into a “non-stationary Mathieu equation” for the vacuum eigenvalues of the $\Q$-operators in the finite-volume massive sine-Gordon model at the super-symmetric point, which is closely related to the theory of dilute polymers on a cylinder and the Painlevé III equation.'
address: |
Department of Theoretical Physics,\
Research School of Physical Sciences and Engineering,\
Australian National University, Canberra, ACT 0200, Australia.
author:
- 'Vladimir V. Bazhanov and Vladimir V. Mangazeev'
title: 'Eight-vertex model and non-stationary Lamé equation'
---
.3 cm
Introduction and Summary {#sect:intro}
========================
The $\Q$-operators introduced by Baxter in his pioneering paper [@Bax72] on the eight-vertex model continue to reveal their exceptional properties in the theory of integrable quantum systems. These operators play a central role in the remarkable connection of Conformal Field Theory (CFT) with the spectral theory of the Schrödinger equation [@Vor92] discovered a few years ago [@DT99b]. As shown in [@DT99b; @BLZsdet], the vacuum eigenvalues of the $\Q$-operators [@BLZ97a] in CFT, with the central charge $c<1$, can be identified with the spectral determinants of certain one-dimensional stationary Schrödinger equations. Some further developments and applications of this connection can found in the recent review article [@DT03a].
Apart from a few exceptions, the vacuum eigenvalues of the $\Q$-operators (considered as functions of the spectral parameter) do not generally satisfy any ordinary second order differential equation themselves. One such exception is the case of the $c=0$ CFT where the these eigenvalues for particular Virasoro vacuum states are known to satisfy the Bessel differential equation [@Fen99]. Remarkably, a similar property holds for the lattice counterparts of these eigenvalues in the $\Delta=-\frac{1}{2}$ six-vertex model [@FSZ01; @Str01a] for the chain of an odd number of sites. In this paper we explain how this property is generalized for the corresponding cases of the lattice eight-vertex model and the massive finite volume sine-Gordon model with $N=2$ supersymmetry.
We study the eight-vertex model on a periodic chain of an odd length, $N=2n+1$, $n=0,1,2,\ldots\infty$. The eigenvalues of the transfer matrix of the model, $T(u)$, satisfy Baxter’s famous TQ-equation T(u)Q(u)=(u-)Q(u+2)+(u+)Q(u-2), \[TQ\] where $u$ is the spectral parameter, (u)=\_1\^N(u|q), q=e\^[[i]{} ]{}, [Im]{}>0,\[phi-def\] and $\vt_1(u\,|\,q)$ is the standard theta-function with the periods $\pi$ and $\pi \tau$ (we follow the notation of [@WW]). Here we consider a special case $\eta=\pi/3$, where the ground state eigenvalue is known [@Bax89; @Str01a] to have a very simple form for all (odd) $N$ T(u)=(u), =.\[T-simple\] The equation with this eigenvalue, $T(u)$, has two different solutions [@BLZ97a; @KLWZ97], $Q_\pm(u)\equiv
Q_\pm(u,q,n)$, which are entire functions of the variable $u$ and obey the following periodicity conditions [@Bax72; @McCoy2][^1] Q\_(u+)=(-1)\^n Q\_(u), Q\_(u+)=q\^[-N/2]{} e\^[-iNu]{} Q\_(u),Q\_(-u)=Q\_(u). \[Qper\] The above requirements uniquely determine $Q_\pm(u)$ to within a common $u$-independent normalization factor. For further references it is convenient rewrite the functional equation for $Q_\pm(u)$ in the form (u)Q\_(u)+(u+)Q\_(u+)+ (u+)Q\_(u+)=0.\[TQ1\] We show that the functions \_(u)\_(u,q,n)= Q\_(u,q,n),\[fpm\] which are meromorphic functions of the variable $u$ for any fixed values of $q$ and $n$, satisfy the non-stationary Schrödinger equation 6q(u,q,n)= {-+ 9 n (n+1)(3u|q\^3)+c(q,n)}(u,q,n), \[lame-pde\]where the modular parameter $\tau$ plays the role of (imaginary) time and the time-dependent potential is defined through the elliptic Weierstrass $\Wp$-function [@WW] (our function $\Wp(v\,|\,e^{i\pi\epsilon})$ has the periods $\pi$ and $\pi\epsilon$). The constant $c(q,n)$ appearing in is totally controlled by the normalization of $Q_\pm(u)$ and can be explicitly determined once this normalization is fixed (see Sec.3 below).
Equation is obviously related to the Lamé differential equation and could be naturally called the “non-stationary Lamé equation”. To our knowledge this equation[^2] (in fact, a more general equation) first explicitly appeared in [@EK94] as a particular case of the Knizhnik-Zamolodchikov-Bernard equation [@KZ84; @Ber88] for the one-point correlation function in the $\textsl{sl(2)}$-WZW-model on the torus. Here, we will not explore this and other [@Ols04] interesting connections of the equation leaving that for the future.
It is fairly trivial to show that the partial differential equation for the meromorphic functions $\f_\pm(u)$ is equivalent to the Baxter equation . Indeed, every solution of , with the required analytic properties in the variable $u$, implied by and , satisfies eq.. However, the very existence of exactly two solutions with these properties is by no means trivial and reflects some rather special features of eq. discussed below.
Equation has been discovered by virtue of a remarkable [*polynomial property*]{} of the eigenvalues $Q_\pm(u)$. Using a combination of analytical and numerical techniques we have explicitly solved the equation for all values of $n\le 10$. We have found that properly normalized eigenvalues $Q_\pm(u)$ could always be written as Q\_(u,q,n)=P\_[2n+1]{}\^[()]{}(\_3(|q\^[1/2]{}), \_4(|q\^[1/2]{}),), = -\^2,\[P-prop\] where $P_{2n+1}^{(\pm)}(\alpha,\beta,\g)$ are homogeneous polynomials of the degree $2n+1$ in the variables $\alpha$ and $\beta$, with coefficients being polynomials in the variable $\g$ with [*integer coefficients*]{}. Then we considered a class of linear second order partial differential equations in two variables $u$ and $q$ where the coefficients of the second order derivatives are independent of $n$ and all other coefficients are at most second degree polynomials in $n$. Eq. was then found as the only equation in this class satisfied by with all explicitly calculated polynomials with $n\le10$. It turned out that this equation uniquely defines two and only two such polynomials for every value $n=0,1,2,\ldots,\infty$. It would be interesting to clarify the combinatorial nature of these polynomials, given that the related $\Delta=-\frac{1}{2}$ six-vertex model is connected to various important enumeration problems [@SR2001; @BdGN].
In the scaling limit n, q0,t=8n q\^[3/2]{}=,\[scaling\] the functions essentially reduce to the ground state eigenvalues $\QQ_\pm(\t)\equiv\QQ_\pm(\t,t)$ of the ${\bf Q}$-operators of the restricted massive sine-Gordon model (at the so-called, super-symmetric point) on a cylinder of the spatial circumference $R$, where $t=M R$ and $M$ is the soliton mass. The equations and become \_()=\_(+2i)+\_(-2i), \_(+3i)=e\^ \_(),\_+()=\_-(-), where the variable $\t$ is defined as $u=\pi\tau/2-i\t/3$. With a suitable $t$-dependent normalization of $\QQ_\pm(\t)$ equation could be brought to a particularly simple form t\_(,t)={- t\^2 (2-1) } \_(,t). \[mathieu-pde\] With the same normalization the asymptotic behavior of $\QQ_\pm(\t)$ at large $\t$ is given by \_()&=&-t e\^+\_(t) +2(\_t\_(t)-t/8)e\^[-]{}\
&& +2(\^2\_t\_(t)-\_t\_(t)/t) e\^[-2]{} +O(e\^[-3]{}),+,\[asymp\] where $\Dcal_\pm(t)$ are the Fredholm determinants which previously appeared in connection with the calculation of the “supersymmetric index” and the problem of dilute polymers on a cylinder [@FS92; @FS94; @Zam94; @TW96]. Note, in particular, that the quantity F(t)=U(t), U(t)= , describes the free energy of a single incontractible polymer loop and satisfies the Painlevé III equation [@FS92] t U(t)=2 U(t). In this connection, it is useful to mention the other celebrated appearances of Painlevé transcendents in the theory of the two-dimensional Ising model [@BMW] and in the problem of isomonodromic transformations of the second order differential equations [@JMSM].
We did not attempt to make this paper self-contained. Detailed results are presented in [@BMfuture]. The main reference for the eight-vertex model and its commuting $\T$- and $\Q$-matrices is Baxter’s original paper [@Bax72]; Sec.2 is meant to be read in conjunction with this paper. The definitions of the $\T$- and $\Q$-operators in continuous quantum field theory are given in [@BLZ97a; @BLZ97b]. The connection of the $\Q$-operators with the problem of dilute polymers is explained in [@Fen99].
The eight-vertex model and TQ-relation {#eight}
======================================
We consider the eight-vertex model on the $N$-column square lattice with periodic boundary conditions and assume that $N$ is an odd integer $N=2n+1$. Following [@Bax72] we parameterize the Boltzmann weights $a$, $b$, $c$, $d$ of the eight-vertex model as[^3] &a= \_4(2|q\^2) \_4(u-|q\^2) \_1(u+|q\^2),&\
&b= \_4(2|q\^2) \_1(u-|q\^2) \_4(u+|q\^2), &\
&c= \_1(2|q\^2) \_4(u-|q\^2) \_4(u+|q\^2), &\[weights\]\
&d= \_1(2|q\^2) \_1(u-|q\^2) \_1(u+|q\^2), &and fix the normalization factor $\rho$ as =2 \_2(0|q)\^[-1]{} \_4(0|q\^2)\^[-1]{}.\[ei2\] It is convenient to introduce new variables $\g$ and $x$, where $\g$ is defined by (cf. [@Bax72]), == -\^2,\[gamma\] and $x$ satisfies the quadratic equation (-)\^2=-,\[x-def\] where we choose the following root x=, \_3(u)=\_3(|q\^[1/2]{}), \_4(u)=\_4(|q\^[1/2]{}).\[xz-def\]
The row-to-row transfer-matrices ${\bf T}(u)$ and the $\Q(u)$-matrices of the model form a commutative family; their eigenvalues satisfy the functional relation . In [@Bax72] Baxter explicitly constructed the matrix ${\bf Q}(u)$ provided that $\eta =\frac{\pi m}{2L}$ for integer $m$ and $L$. We only consider a special case when the weights satisfy the constraint [@Str01a] (a\^2+ab)(b\^2+ab)=(c\^2+ab)(d\^2+ab), which is equivalent to the condition $\eta=\pi/3$. In [@Str01a; @Str01c] it was conjectured that the largest eigenvalue of the transfer matrix (corresponding to the double-degenerate ground state of the model) has the simple form, , T(u)=(a+b)\^N=(u).\[tab\] Here we study the corresponding eigenvalues $Q_\pm(u)$ of the $\Q$-matrix. As noted in [@McCoy1], the method used in [@Bax72] for the construction of the $\Q$-matrix cannot be executed in its full strength for $\eta=\pi/3$, since some axillary $\Q$-matrix, $\Q_R(u)$, in [@Bax72] is not invertible in the full $2^N$-dimensional space of states of the model. The numerical results presented in Table 1 of [@McCoy1] make it natural to suggest that the rank of $\Q_R(u)$ in this case is given by the $N$-th Lucas number $((1+\sqrt{5})/2)^N+((1-\sqrt{5})/2)^N$ which coincides with the dimension of the space of states of the $N$-site hard hexagon model [@Bax80][^4]. This indicates that for $\eta=\pi/3$ the construction of [@Bax72] only provides a “restricted” $\Q^{(r)}(u)$-matrix which acts in some “RSOS-projected” subspace of the full space of states of the eight-vertex model. This interesting phenomenon certainly deserves special investigations in its own right; we have verified for several small values of $N$ that the ground state eigenvectors corresponding to belongs to this RSOS-projected subspace and that the eigenvalues of $\Q^{(r)}(u)$ satisfy . Below we will assume that this is true for all (odd) $N$.
After this brief review, let us now describe our main results. For $\eta=\pi/3$ the variable $\g\equiv\g(q)$, defined by , depends on $q$ only, while the variable $x\equiv x(u,q)$, defined by , depends on $u$ and $q$. Below it will be more convenient to use the combinations Q\_1(u)=(Q\_+(u)+Q\_-(u))/2,Q\_2(u)= (Q\_+(u)-Q\_-(u))/2. \[Q12-def\] which are simply related by the periodicity relation Q\_[1,2]{}(u+)=(-1)\^nQ\_[2,1]{}(u).\[Qper3\] Bearing in mind this simple relation we will only quote results for $Q_1(u)$, writing it as $Q^{(n)}_1(u)$ to indicate the $n$-dependence. We have found that all the eigenvalues $Q^{(n)}_1(u)$ can be written as Q\^[(n)]{}\_1(u)=[N]{}(q,n) \_3(u)\_4\^[2n]{}(u) ¶\_n(x,z), z=\^[-2]{},\[Q1\] where ${\mathcal N}(q,n)$ is an arbitrary normalization factor and $\P_n(x,z)$ are polynomials in $x,z$ of the degree $n$ in $x$, ¶\_n(x,z)=\_[k=0]{}\^n r\^[(n)]{}\_k(z)x\^k.\[P-def\] while $r^{(n)}_i(z)$, $i=0,\ldots,n$, are polynomials in $z$ with integer coefficients. The normalization of $\P_n(x,z)$ is fixed by the requirement $r^{(n)}_n(0)=1$. The polynomials $\P_n(x,z)$ are uniquely determined by the following partial differential equation in the variables $x$ and $z$, {A(x,z)\_x\^2 +B\_n(x,z)\_x +C\_n(x,z) + T(x,z)[\_z]{}} ¶\_n(x,z)=0,\[P-pde\] where A(x,z)&=&2x(1 + x - 3xz + x\^2z)(x + 4z - 6xz -3xz\^2 + 4x\^2z\^2),\
B\_n(x,z)&=&4(1+x-3xz+x\^2z)(x+3z-7xz+3x\^2z\^2)+\[gen8\]\
&&-1.0em +2nx(1-14z+21z\^2-8x\^3z\^3+3x\^2z(3z\^2+6z-1)-x(1-9z+23z\^2+9z\^3))\
C\_n(x,z)&=&n\[gen9\]\
T(x,z)&=&-2z(1-z)(1-9z)(1 + x - 3xz + x\^2z).\[gen10\] The first few polynomials $\P_n(x,z)$ read ¶\_0(x,z)=1, ¶\_1(x,z)=x+3,¶\_2(x,z)=x\^2(1+z)+5x(1+3z)+10,\[gen4\] ¶\_3(x,z)=x\^3(1+3z+4z\^2)+7x\^2(1+5z+18z\^2)+7x(3+19z+18z\^2)+35+21z.\[gen5\] In the next Section, we will prove that the eigenvalues $Q^{(n)}_1(u)$ given by (as well as all related eigenvalues $Q^{(n)}_2(u)$ and $Q^{(n)}_\pm(u)$ given by and ) automatically satisfy the functional relation , merely as a consequence of the defining property of $\P_n(x,z)$. Of course, for small values of $n$ this functional relation can be checked directly. For example, it is not very difficult to check it for Q\^[(0)]{}\_1(u)\~\_3(u),Q\^[(1)]{}\_1(u)\~\_3(u), by employing various identities for elliptic functions, however for $n\ge2$ this does not appear to be practical.
The polynomials $\P_n(x,z)$ can be effectively calculated with the following procedure. It is easy to see that leads to descending recurrence relations for coefficients in in the sense that each coefficient $r^{(n)}_k(z)$ with $k<n$ can be recursively calculated in terms of $r^{(n)}_m(z)$, with $m=k+1,\ldots,n$ and, therefore, can be eventually expressed through the coefficient $s_n(z)\equiv r^{(n)}_n(z)$ of the leading power of $x$. These leading coefficients, $s_n(z)$, $n=0,1,2\ldots\infty$, are uniquely determined by the following recurrence relation &2z(z-1)(9z-1)\^2\[s\_n(z)\]”\_z+2(3z-1)\^2(9z-1)\[s\_n(z)\]’\_z+&\[gen19\]\
&+8(2n+1)\^2- \[4(3n+1)(3n+2)+(9z-1)n(5n+3)\]=0,with the initial condition $s_0(z)=s_1(z)\equiv 1$. In particular, for $z=1/9$ (corresponding to $q=0$) this gives s\_0()=1, s\_[n+1]{}()= s\_n().\[gen20\] Currently, the polynomials $\P_n(x,z)$ have been calculated explicitly for $n\le50$; using the above procedure they can be easily calculated for higher values of $n$.
Non-stationary Lamé equation
============================
Evidently, the algebraic form of the partial differential equation given by is not very illuminating (even though it is quite useful for the analysis of polynomial solutions). Fortunately this equation has a much more elegant equivalent form given by the non-stationary Lamé equation discussed in the Introduction. The details of transformations between these two forms will be presented elsewhere [@BMfuture]. Noting the frustrating expressions for the coefficients in it is not surprising that these transformations turn out to be rather tedious. They involve many elliptic function identities, in particular, the algebraic properties of the ring of theta-constants and their $q$-derivatives [@Zud00] happened to be extremely useful.
Let us choose the normalization factor ${\mathcal N}(q,n)$ in as (-1)\^n2\^[(3n\^2+5n+1)/2]{}s\_n() [N]{}(q,n)= i\^n \^[-n]{}’\_1(0|q)\^[-n-1/3]{} {(\^2-1)\_4\^4(0|q)}\^ and define the functions $\f_\pm(u,q,n)$ as in where $Q_\pm(u)$ are given by , and . Then the equation takes the form where the constant term $c(q,n)$ is given by c(q,n)=-3n(n+1). By construction the functions $\f_\pm(u,q,n)$ are meromorphic functions of the variable $u$, which obey periodicity relations (u+2)=(u),(u+2)=q\^[-6]{}e\^[-6iu]{} (u),(-u)=(-1)\^[n+1]{}(u) \[f-per\] and have $(n+1)$-th order zeros at $u=k\pi+m\pi\tau$, where $k,m\in{\mathbb Z}$, i.e., (+k+m)=O(\^[n+1]{}), 0, k,m.\[psi-cond\] Let us show that for $n\ge1$, equation , restricted to a class of functions $\f(u)$ with such analytic properties, is equivalent to the functional relation . Any such solution of , $\f(u)$, could have either an $(n+1)$-th order zero or an $n$-th order pole in the variable $u$, at all points $u=(3k\pm1)\pi/3+m\pi\tau$, with $k,m\in{\mathbb Z}$, and these are the only points where $\f(u)$ could have poles. Consider the function (u)= (u)+(u+)+(u+),\[Phi1-def\] which also satisfies along with $\f(u)$. When $u=0$ the second and third terms in may have $n$-th order poles, however, they must cancel each other due to the last relation in . Thus $\Phi(u)$ could have at most $(n-1)$-th order pole at $u=0$ (the first term in obviously vanishes due to ). As noted above such pole is forbidden by eq. and thus $\Phi(u)$ has the $(n+1)$-th order zero at $u=0$. Similarly, one concludes that $\Phi(u)$ vanishes at all points $u=k\pi/3+m\pi\tau$, with $k,m\in{\mathbb Z}$, and, therefore, has at least $12(n+1)$ zeroes in the periodicity parallelogram (of the periods $2\pi$ and $2\pi\tau$) and no poles at all. However, for $n\ge1$ this contradicts , unless $\Phi(u)\equiv0$, which is equivalent to . The special case $n=0$ is considered in [@BMfuture].
Now consider various limiting forms of the equation . When $q\to0$, with $u$ and $n$ kept fixed, the functions $\f_\pm(u)$ reduce to their analogs for the 6-vertex model \_(u,q,n)=q\^[(d\_+)]{} \^[(6v)]{}\_(u,n)(1+O(q)),q0, where $d_\pm=(1\mp6)/36$ and eq. becomes {-+ -(d\_+)}\_\^[(6v)]{}(s/3,n)=0, which is simply related to eq.(13) of [@Str01a]. Taking now $n\to\infty$ and $s\sim i\log n$ one recovers the eigenvalues of the $\Q$-operators of ref. [@BLZ97a] corresponding to $p=\pm1/6$ vacuum states in the $c=0$ CFT Q\_\^[(CFT)]{}(+t)=e\^\_[n]{} \_\^[(6v)]{}((8n/t)-), which are known [@Fen99] to satisfy the Bessel differential equation {-\^2\_+\_+t\^2 e\^[2]{} +d\_}Q\_\^[(CFT)]{}(+t)=0.\[Bessel\]
In the most interesting scaling limit , the limiting functions \_(,t)=t\^[1/4]{}e\^[t\^2/16]{}e\^[/2]{} \_[n]{} n\^[-1/4]{}\_( /2-i/3,e\^[i]{}, n)|\_[=(8n/t)]{} \[Q-qft\]coincide with the eigenvalues [@Fen99] of the $\Q$-operators for special twisted vacuum states in the massive sine-Gordon at the supersymmetric point [@Zam94] (where the ground state energy of the model vanishes identically due to the supersymmetry). These eigenvalues satisfy the “non-stationary Mathieu equation” . Let us show that this equation completely determines the asymptotic expansion of these eigenvalues at large $\theta$, \_(,t)=-e\^+\_[k=0]{}\^ \_\^[(k)]{}(t) e\^[-k]{},+.\[asympt\] Consider the integral operator $\hat\Kcal(t)$ with the kernel (t|,’)=,u()=,\[intop\] which satisfies the following identity (t|,’)=te\^[-u()-u(’)+’]{}, where $\hat\Mcal_\t$ denotes the differential operator in the RHS of . Using this identity one can show that $\QQ_\pm(\t)$ satisfy the linear integral equation discovered in [@Fen99] \_(,t)=\_(t)e\^[-u()]{} \_[-]{}\^(t|,’) \_(’,t)d’\[int\], provided that the functions $\Dcal_\pm(t)$ coincide with the Fredholm determinants [@Zam94] \_(t)=[C]{}\_(1(t)), where ${\mathcal C}_\pm$ are numerical constants. Comparing with one concludes that $\Bcal_\pm^{(0)}(t)=\log\Dcal_\pm(t)$. Then eq. allows to express all higher coefficients in through the power $\Bcal_\pm^{(0)}(t)$ and its derivatives. The few first coefficients are shown in . Finally note, that in the limit $t\to0$, $\t\sim-\log t$, \_(,t)=t\^[d\_]{} Q\^[(CFT)]{}\_(+t) (1+O(t\^[4/3]{})), t0,\~-t, and the equation reduces to as it, of course, should[^5].
We expect that our results can be readily extended to elliptic generalizations of the other special lattice models [@DTS04], closely related to the $\eta=\pi/3$ six-vertex model. However, it would be more important to understand whether similar considerations could be applied for the eight-vertex with an arbitrary value of $\eta$ and whether the general scheme of correspondence between the $c<1$ CFT and ordinary differential equations, developed in [@DT99b; @BLZsdet; @BLZ03], can be extended to the massive field theory (sine-Gordon model) with the use of partial differential equations. The investigation of these questions is in progress.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors thank I.M. Krichever, S.L. Lukyanov, S.M. Sergeev, Yu.G. Stroganov, A.B. Zamolodchikov and Al.B. Zamolodchikov for stimulating discussions and valuable remarks.
[10]{}
Baxter, R. J. Partition function of the eight-vertex lattice model. Ann. Physics [**70**]{} (1972) 193–228.
Voros, A. Spectral zeta functions. In [*Zeta functions in geometry (Tokyo, 1990)*]{}, pages 327–358. Kinokuniya, Tokyo, 1992.
Dorey, P. and Tateo, R. Anharmonic oscillators, the thermodynamic [B]{}ethe ansatz and nonlinear integral equations. J. Phys. A [**32**]{} (1999) L419–L425. .
Bazhanov, V. V., Lukyanov, S. L., and Zamolodchikov, A. B. pectral determinants for [S]{}chrödinger equation and [Q]{}-operators of conformal field theory. J.Statist.Phys. [**102**]{} (2001) 567–576. .
Bazhanov, V. V., Lukyanov, S. L., and Zamolodchikov, A. B. Integrable structure of conformal field theory. [I]{}[I]{}. ${
Q}$-operator and [D]{}[D]{}[V]{} equation. Comm. Math. Phys. [**190**]{} (1997) 247–278. .
Dorey, P., Dunning, C., Millican-Slater, A., and Tateo, R. Differential equations and the Bethe ansatz. (2003). Proceedings of ICMP, 2003, Lisbon, \[[hep-th/0309054]{}\].
Fendley, P. Airy functions in the thermodynamic [B]{}ethe ansatz. Lett. Math. Phys. [**49**]{} (1999) 229–233.
Fridkin, V., Stroganov, Y., and Zagier, D. Finite size [$XXZ$]{} spin chain with anisotropy parameter [$\Delta=\frac12$]{}. J. Statist. Phys. [**102**]{} (2001) 781–794.
Stroganov, Y. The importance of being odd. J. Phys. A [**34**]{} (2001) L179–L185.
Whittaker, E. and Watson, G. . “Cambridge University Press”, Cambridge, 1996.
Baxter, R. J. Solving models in statistical mechanics. Adv. Stud. Pure Math. [**19**]{} (1989) 95–116.
Krichever, I., Lipan, O., Wiegmann, P., and Zabrodin, A. Quantum integrable models and discrete classical [H]{}irota equations. Comm. Math. Phys. [**188**]{} (1997) 267–304. .
Fabricius, K. and McCoy, B. M. New Developments in the Eight Vertex Model II. Chains of odd length. (2004). cond-mat/0410113.
Etingof, P. I. and Kirillov, Jr., A. A. Representations of affine [L]{}ie algebras, parabolic differential equations, and [L]{}amé functions. Duke Math. J. [**74**]{} (1994) 585–614.
Knizhnik, V. G. and Zamolodchikov, A. B. Current algebra and [W]{}ess-[Z]{}umino model in two dimensions. Nuclear Phys. B [**247**]{} (1984) 83–103.
Bernard, D. On the [W]{}ess-[Z]{}umino-[W]{}itten models on the torus. Nuclear Phys. B [**303**]{} (1988) 77–93.
Olshanetsky, M. Universality of [C]{}alogero-[M]{}oser model. .
Razumov, A. V. and Stroganov, Y. G. Spin chains and combinatorics. J. Phys. A [**34**]{} (2001) 3185–3190.
Batchelor, M. T., de Gier, J., and Nienhuis, B. The quantum symmetric [$XXZ$]{} chain at [$\Delta=-\frac 12$]{}, alternating-sign matrices and plane partitions. J. Phys. A [**34**]{} (2001) L265–L270.
Fendley, P. and Saleur, H. $N=2$ supersymmetry, Painleve III and exact scaling functions in $2$D polymers. Nuclear Phys. B [**388**]{} (1992) 609–626.
Cecotti, S., Fendley, P., Saleur, H., Intriligator, K., and Vafa, C. A new supersymmetric index. Nuclear Phys. B [**386**]{} (1992) 405–452.
Zamolodchikov, A. B. Painleve III and $2$D polymers. Nuclear Phys. B [**432**]{} (1994) 427–456.
Tracy, C. A. and Widom, H. Proofs of two conjectures related to the thermodynamic Bethe ansatz. Comm. Math. Phys. [**179**]{} (1996) 1–9.
Barouch, E., McCoy, B. M., and Wu, T. T. Zero-field susceptibility of the two-dimensional Ising model near $T_c$. Phys. Rev. Lett. [**31**]{} (1973) 1409–1411.
Jimbo, M., Miwa, T., Sato, M., and M[ô]{}ri, Y. Holonomic quantum fields. [T]{}he unanticipated link between deformation theory of differential equations and quantum fields. In [*Mathematical problems in theoretical physics (Proc. Internat. Conf. Math. Phys., Lausanne, 1979)*]{}, volume 116 of [*Lecture Notes in Phys.*]{}, pages 119–142. Springer, Berlin, 1980.
Bazhanov, V. V. and Mangazeev, V. V. Eight vertex model and partial differential equations. in preparation, 2004.
Bazhanov, V. V., Lukyanov, S. L., and Zamolodchikov, A. B. Quantum field theories in finite volume: excited state energies. Nuclear Phys. B [**489**]{} (1997) 487–531. .
Stroganov, Y. The [$8$]{}-vertex model with a special value of the crossing parameter and the related [$XYZ$]{} spin chain. In [*Integrable structures of exactly solvable two-dimensional models of quantum field theory (Kiev, 2000)*]{}, volume 35 of [*NATO Sci. Ser. II Math. Phys. Chem.*]{}, pages 315–319. Kluwer Acad. Publ., Dordrecht, 2001.
Fabricius, K. and McCoy, B. M. New developments in the eight vertex model. J. Statist. Phys. [**111**]{} (2003) 323–337.
Baxter, R. J. Hard hexagons: exact solution. J. Phys. A [**13**]{} (1980) L61–L70.
Sergeev, S. M. unpublished, 2004.
Zudilin, V. Thetanulls and differential equations. Sbornik: Mathematics. [**119**]{} (2000) 1827–1871.
Dorey, P., Suzuki, J., and Tateo, R. Finite lattice Bethe ansatz systems and the Heun equation. J. Phys. A [**37**]{} (2004) 2047–2061.
Bazhanov, V. V., Lukyanov, S. L., and Zamolodchikov, A. B. Higher-level eigenvalues of [Q]{}-operators and [S]{}chroedinger equation. Adv. Theor. Math. Phys. [**7**]{} (2003) 711–725.
[^1]: The factor $(-1)^n$ in and reflects our convention for labeling the eigenvalues for different $n$, which will be important in Sec. 3
[^2]: We are indebted to Prof. I.M.Krichever for informing us about the work [@EK94].
[^3]: We use the notation of [@WW] for theta-functions $\vt_k(u\,|\,q)$, $k=1,2,3,4$, of the periods $\pi$ and $\pi\tau$, $q=e^{i\pi\tau}$, $\im \tau>0$. The theta-functions $\HJ(v)$, $\TJ(v)$ of the nome $q_B$ used in [@Bax72] are given by $$q_B=q^2,\quad \HJ(v)=\vt_1(\frac{\pi v}{2\K_B}\,|\,\,q^2),\quad \TJ(v)=
\vt_4(\frac{\pi v}{2\K_B}\,|\,\,q^2),$$ where $\K_B$ is the complete elliptic integral of the first kind with the nome $q_B$.
[^4]: S.M.Sergeev noted [@Sergeev] that only a minor modification of the arguments of [@Bax72] leads to the matrix $\Q_R$ with the rank equal to $2^N$ for even $N$ and to $(2^N-2)$ for odd $N$.
[^5]: The normalization of integral operator has been fixed from the comparison of the $t\to0$ limit of the coefficients $\Bcal_\pm^{(1)}(t)$ with appropriate CFT results of [@BLZ97a]. This normalization corresponds to $\lambda=1/(4\pi)$ in eq.(3.1) of [@Zam94].
|
---
abstract: 'The redshift-space distortion (RSD) in the observed distribution of galaxies is known as a powerful probe of cosmology. Observations of large-scale RSD, caused by the coherent gravitational infall of galaxies, have given tight constraints on the linear growth rate of the large-scale structures in the universe. On the other hand, the small-scale RSD, caused by galaxy random motions inside clusters, has not been much used in cosmology, but also has cosmological information because universes with different cosmological parameters have different halo mass functions and virialized velocities. We focus on the projected correlation function $w(r_p)$ and the multipole moments $\xi_l$ on small scales ($1.4$ to $30\ h^{-1}\rm{Mpc}$). Using simulated galaxy samples generated from a physically motivated correspondence scheme in the Multiverse Simulation, we examine the dependence of the small-scale RSD on the cosmological matter density parameter $\Omega_m$, the satellite velocity bias with respect to MBPs, $b_v^s$, and the $\alpha$. We find that $\alpha=1.5$ gives an excellent fit to the $w(r_p)$ and $\xi_l$ measured from the SDSS-KIAS value added galaxy catalog. We also define the “strength” of Fingers-of-God as the ratio of the parallel and perpendicular size of the contour in the two-point correlation function set by a specific threshold value and show that the strength parameter helps constraining $(\Omega_m, b_v^s, \alpha)$ by breaking the degeneracy among them. The resulting parameter values from all measurements are $(\Omega_m,b_v^s)=(0.272\pm0.013,0.982\pm0.040)$, indicating a slight reduction of satellite galaxy velocity relative to the MBP. However, considering that the average MBP speed inside haloes is $0.94$ times the dark matter velocity dispersion, the main drivers behind the galaxy velocity bias are gravitational interactions, rather than baryonic effects.'
author:
- Motonari Tonegawa
- Changbom Park
- Yi Zheng
- 'Hyunbae Park (박현배)'
- 'Sungwook E. Hong (홍성욱)'
- Ho Seong Hwang
- 'Juhan Kim (김주한)'
title: 'Cosmological Information from the Small-scale Redshift Space Distortions'
---
[UTF8]{}[mj]{}
Introduction
============
The accelerated expansion of the Universe has been one of the most profound mysteries in astronomy and physics since observations have confirmed it through the redshift-distance relation of type Ia supernovae (@Riess1998 [@Perlmutter1999]). So far, the $\Lambda$CDM model gives the best description for these observations, although it involves several theoretical difficulties related to the smallness and fine-tuning of $\Lambda$, the exotic form of the energy in the Universe (@Frieman2008 [@Weinberg2013]). Another conceptual possibility for the apparent accelerated expansion is that the general relativity (GR), on which the $\Lambda$CDM model is built, may not be correct in cosmological scales. This idea gave rise to the modified gravity theories, which realized the same redshift-distance relation as that of $\Lambda$CDM model without relying on the dark energy but predict different gravitational growth history of the matter content of the Universe (@Joyce2016 [@Koyama2016]). Discriminating between the dark energy and modified gravity scenarios is essential to understand the origin and history of our Universe better.
space distortion (RSD) is a phenomenon that the distribution of galaxies is distorted from the one due to caused by galaxy peculiar motion [@Kaiser1987; @Hamilton1998]. It affects the statistical property of galaxy clustering such as the two-point correlation function (2pCF) and the power spectrum, making the line-of-sight direction a special one. As the galaxy peculiar velocity field is governed by the gravity law and background cosmological parameters, the anisotropy of the galaxy 2pCF is sensitive to the change of cosmological models, making RSD a powerful cosmological probe [@Weinberg2013]. Since the galaxy catalog used in an RSD analysis can also be used for such as large-scale structure topology [@Park2010; @Appleby2018], richness and size distributions of structures [@Hwang2016], baryon acoustic oscillations (BAO), and Alcock-Paczynski test [@Reid2012; @Li2016; @Sanchez2017], there have been a variety of galaxy redshift surveys (SDSS: @York2000; HectoMAP: @Geller2011; BOSS: @Dawson2013; 6dF: @Jones2005; WiggleZ: @Drinkwater2010; VIPERS: @Guzzo2014; FastSound: @Tonegawa2015; eBOSS: @Dawson2016). There are also further large upcoming surveys (PFS: @Takada2014; DESI: @DESI2016; WFIRST: @Spergel2015).
The large-scale RSD is caused by the infall motion of galaxies during the structure formation, and it has been detected by various redshift surveys, giving strong cosmological constraints on the growth rate of the large-scale structure $f=d\ln{D}/d\ln{a}$ [@Hawkins2003; @Guzzo2008; @Blake2011; @Beutler2012; @Samushia2012; @delaTorre2013; @Beutler2014; @Okumura2016; @Icaza-Lizaola2019], where $D$ is the growth factor and $a$ is the scale factor of the universe with $a=1$ at the present epoch. On the other hand, the small-scale RSD, called finger-of-god (FoG) effect, is caused by the orbital motion of galaxies inside galaxy groups and clusters. It has not been as much studied as the large-scale RSD but has rich cosmological information because different cosmological parameters lead to different halo mass functions and virialized velocities [@Marzke1995]. The difficulty in using the small-scale RSD lies in the theoretical prediction of the density and velocity field in highly non-linear scales. We cannot rely on the perturbation theory that is valid down to mildly non-linear regime [@Taruya2010], because the FoG effect takes place in the almost or completely relaxed objects. There attempts to (@Sheth1996 [@Juszkiewicz1998]; @Tinker2007; @Bianchi2016; @Kuruvilla2018). their models typically include free parameters which may depend on cosmological models and are not easy to derive from the first principles thus far. Nevertheless, because of the small statistical uncertainty, the use of the small-scale clustering will significantly enhance our constraining power from the limited size of observational data sets.
While the analytic prescription of the non-linear structure formation is a timely topic itself (@Tinker2007), $N$-body simulations can serve as an alternative small-scale cosmology studies (@DeRose2019). In this study, we use the Multiverse , the Horizon Run 4 Simulation [@Kim2015], and a physically motivated galaxy assigning scheme [@Hong2016] to the galaxy distributions in redshift space for different matter density parameters $\Omega_m$, satellite velocity bias parameters $b_v^s$, and merger time scale parameters, $\alpha$. The galaxy-halo correspondence in our model has more physical meaning than the halo occupation distribution (HOD) approach. @Reid2014 adopted the HOD approach and successfully explained the redshift-space clustering of the BOSS CMASS data, obtaining a $2.5\%$ constraint of the growth rate, which clearly proved the usefulness the small-scale clustering information. Although being the standard method to connect galaxies and haloes, the HOD has several issues to examine carefully. The HOD prescribes the probability of a halo of mass $M$ having $N$ galaxies, $P(N|M)$, with typically five parameters and specific functional forms. However, there is not a particular reason for the number of parameters and the functions. The number $N$ may also depend on secondary parameters, such as halo age and galaxy assembly history [@Wang2013; @Dorta2017; @Beltz-Mohrmann2019]. By contrast, our galaxy-halo corresponding scheme traces the merger tree and automatically places galaxies into subhaloes, which avoids the theoretical uncertainties. We constrain the matter density parameter $\Omega_m$ as well as the velocity bias parameter for satellite galaxies $b_v^s$ and the merger time scale parameter $\alpha$ by simultaneously matching the measurements of the projected correlation function and the multipole moments of the two-point correlation function between simulation and observation. We also define the “strength” of FoG and show that adding it helps us to constrain our model parameters more strongly.
The structure of this paper is as follows. In section \[section:data\], we describe the simulation and observational data that we use. In section \[section:measurements\], we measure the correlation functions and covariance matrix. We also quantify the FoG to extract cosmological information from the small-scale 2pCF. In section \[section:results\], we show our constraints on the parameters of our model, followed by discussions in section \[section:discussions\]. Finally we summarize our study in section \[section:conclusions\].
Data and Models {#section:data}
===============
The KIAS-VAGC catalog
---------------------
We use the KIAS as observational data. This catalog is based on the New York University Value-Added Galaxy Catalog (NYU-VAGC; @Blanton2005) as part of Sloan Digital Sky Survey Data Release 7 [@Abazajian2009], but supplements missing redshifts with other galaxy redshift catalogs for better redshift completeness. The KIAS-VAGC covers $\sim 8000\ {\rm deg^2}$ on the celestial plane and contains $593,514$ redshifts of the SDSS Main galaxies in $r$-band Petrosian magnitude of $10.0<m_r<17.6$. The supplementation increased the area with completeness higher than $0.97$, from $39.8\%$ to $54.3\%$. There are still missing redshifts even after this supplementation, which is mainly caused by the fiber collision effect and poor observing conditions. The fiber collision rate is estimated to be $\sim 5\%$, but lower in the overlapping regions. In the KIAS-VAGC catalog, these galaxies are marked and given redshifts of the nearest galaxy on the celestial plane.
We use the volume-limited sample “D5” with a redshift cut $0.025<z<0.10713$ and an $r$-band absolute magnitude cut as defined in [@Park2009]. The number density is $0.063 \ (h^{-1}\rm{Mpc})^{-3}$ and median redshift is $0.083$. Also, we restrict the sample to the largest area that satisfies $-65.0^\circ<\lambda<65.0^\circ$ and $-37.0^\circ<\eta<43.0^\circ$, where $\lambda$ and $\eta$ are the SDSS survey coordinates. The KIAS-VAGC also provides the survey mask, which indicates the spectroscopic completeness in each of $0.025\times0.025 \ {\rm deg^2}$ patch in the survey area. To avoid the bad observing condition and shot noise, we only use the region where the completeness is above 0.8. All galaxies in the valid region are assigned the weight as the inverse of the completeness. Some of the target galaxies are not allocated fibers to, due to the mechanical limitation about the minimum separation of two galaxies on the sky. This is called the fiber collision effect, occurring on small scales ($\sim0.1\ {\rm Mpc}$) and potentially weakens the FoG effect. If a spectroscopic target cannot be allotted a fiber, the redshift of the nearest neighbor galaxy is given to the galaxy. The validity of this approach will be discussed .
The random catalog is needed to measure the correlation function. Since we are using a volume-limited sub-sample, we make random catalogs as the uniform distribution in a comoving volume. The angular completeness mask is then applied after the conversion from $(X,Y,Z)$ to $(\lambda, \eta)$ to discard points on the region of ${\rm completeness}<0.8$. The size of the random catalog is $\sim 30$ times larger than the corresponding data.
The Multiverse Simulation
-------------------------
The Multiverse a collection of largevolume cosmological simulations with different cosmological parameters . There are five realizations that have different matter density $\Omega_m$ and the equation of state of the dark energy $w$: $(\Omega_m, w)=(0.21,-1.0)$, $(0.26,-1.0)$, $(0.31,-1.0)$, $(0.26,-0.5)$ , and $(0.26,-1.5)$, keeping $\Omega_m+\Omega_\Lambda=1$, while other parameters are fixed the Wilkinson Microwave Anisotropy Probe 5-year result [@Dunkley2009]: $\Omega_b=0.044$, $n=0.96$, $H_0=72 \ \rm{km\,s^{-1}\,Mpc^{-1}}$, and $\sigma_8=0.79$. We use the first three out of five because the change in $w$ will not be important for the clustering properties on the scales that we are interested in. The comoving box size is $1024^3\ (h^{-1}\rm{Mpc})^3$ and $2048^3$ dark matter (DM) particles are evolved inside, which leads to a particle mass of $9\times10^9 (\Omega_m/0.26)M_{\sun}$. The starting redshift is $z=99$, and $1980$ snapshots are saved until $z=0$. Haloes are identified through a friend-of-friend (FOF; @Davis1985) algorithm with a commonly-used linking length of $b=0.2$. The minimum number of particles to be qualified as a halo is $30$, which means the minimum halo mass of $2.7\times10^{11}(\Omega_m/0.26) M_{\sun} $. Haloes in each snapshot are searched for the most bound particle (MBP), which is located at the lowest gravitational potential. The merger tree is built by tracking the merger trajectories of .
Simulated galaxies are assigned to the DM haloes by the MBP-galaxy correspondence approach as described by @Hong2016. All MBPs marked in the merger tree are regarded as galaxy proxies and physical properties of MBPs such as mass, position, and velocity are allocated to the modeled galaxies. If a merger occurs, according to the of the haloes in the previous time step. Then, the satellites are monitored to determine their fates (i.e., escape from the gravitational potential of its host or tidally disrupted), according to the merger time scale of [@Jiang2008]: $$\label{equation:Jiang08}
\frac{t_{\rm merge}}{t_{\rm dyn}}=\frac{(0.94\epsilon^{0.60}+0.60)/0.86}{\ln{[1+(M_{\rm host}/M_{\rm sat})]}} \left (\frac{M_{\rm host}}{M_{\rm sat}} \right)^\alpha,$$ where $\epsilon$, $M_{\rm host}$ and $M_{\rm sat}$ are the circularity of the satellite’s orbit, mass of host and satellite haloes, and $t_{\rm dyn}$ is the dynamical timescale $$t_{\rm dyn}=\frac{R_{\rm vir}}{V_{\rm vir}},$$ with $R_{\rm vir}$ and $V_{\rm vir}$ being the virial radius and circular velocity, respectively. The $\alpha$ parameter the merger timescale of satellites. Increasing $\alpha$ increases the number of satellite galaxies and will enhance the overall amplitude of correlation functions as well as the FoG effect. Therefore, we use $\alpha=1.5$ in this study as a fiducial model, but also show some results and comparisons with other $\alpha$ values.
The Horizon Run 4 Simulation
----------------------------
The Horizon Run simulations [@Kim2009; @Kim2015] are large cosmological $N$-body simulations run by . To date, there are four realizations (Run 1, 2, 3, and 4) with different box sizes and particle numbers. The Horizon Run 4 (HR4) has evolved $6300^3$ particles with the mass of $3.0\times10^9 M_{\sun}$ in a $3150\ h^{-1}{\rm Mpc}$-long cubic box. HR4 is $27$ times larger than the Multiverse , allowing us to estimate the covariance matrix more accurately. Adopted cosmological parameters are the same as those of the case of the Multiverse . While we use the Multiverse to investigate the small-scale clustering property for different cosmological parameters, we use the HR4 simulation to calculate the covariance matrix and to test systematics including the fiber collision effect. The galaxy assignment was performed in an identical way to those of the Multiverse . To estimate the covariance matrix, we divide the HR4 simulation box into $5\times9\times5=405$ sub-cubes. The choice of the number is because of the geometry of the SDSS Main Galaxy survey volume, whose length in one dimension is longer than those of the other two. In each sub-cube, an origin is set and the galaxy positions are converted into $({\rm RA}, {\rm DEC}, z)$. Then, the RSD effect is applied using the line-of-sight velocity of galaxies (see the next subsection). We set $\alpha=1.5$ and the velocity bias parameter $b_v^s=1$ for the calculation of the covariance matrix. The covariance matrix may be a function of these parameters, but we will ignore it. The parameter fitting in section \[subsection:fitting\] is performed using the covariance matrix obtained here, and all error bars in the measurements of Figures \[figure:wrp\], \[figure:multipoles\], \[figure:FoG\], and \[figure:best\] are the square root of the diagonal elements of the covariance matrix.
The RSD and Velocity Bias {#subsection:vbias}
-------------------------
While the mock galaxy distribution is simulated in the real space, the observed galaxies clustering statistics come from the redshift-space distribution. Thus, we need to apply the RSD effect to the simulation data.
The redshift-space distortion alters the apparent galaxy position [@Hamilton1998]. As we take the third axis of the simulation as the line-of-sight direction, the positions are modified as $$x_3^g \mapsto x_3^g + v_3^g/aH$$ where ${\bf x^g} = (x_1^g,x_2^g,x_3^g)$ and ${\bf v^g}=(v_1^g,v_2^g,v_3^g)$ are the comoving position and velocity of a galaxy, and $H$ is the Hubble parameter at redshift $z$, respectively. The periodic boundary condition is applied if the modified position exceeds the boundary of the simulation box. For the first term of the right-hand side, we use the MBP positions as a proxy of galaxy positions.
Recently it has been argued, based on the observations and simulations [@Munari2013; @Wu2013; @Guo2015; @Ye2017], that the galaxy velocity distribution may not be the same as that of DM inside haloes. @Guo2015 found that the speed of satellite galaxies inside haloes was lower (typically $\sim 80\%$) than the velocity dispersion of the DM, $\sigma_v$. Possible origins of such discrepancy include statistical bias, dynamical friction, galaxy interactions, and hydrodynamic effects. Also, central galaxies may not be at rest at halo centers, with the velocity dispersion of $\sim 0.3\sigma_v$. Given that, we parameterize the satellite velocity bias by a single parameter $b_v^s$:
$$\label{equation:velocitybias}
{\bf v^g}-{\bf v^h}=b_v^s({\bf v^{MBP}}-{\bf v^h}),$$
where ${\bf v^{MBP}}$ and ${\bf v^h}$ are the MBP velocity of the galaxy and the host halo velocity, respectively. The host halo velocity is defined as the average velocity of the member particles. Equation (\[equation:velocitybias\]) means that, in the rest frame of the hosting halo, the velocity of the visible part of galaxies is different from that of the (represented by MBPs) by a factor of $b_v^s$. Note that our definition of the velocity bias is different from that of [@Guo2015]. They refer to $\alpha_v^s$ as the RMS velocity of satellites relative to the velocity dispersion of DM of their hosts: $\braket{|{\bf v^g}-{\bf v^h}|}=\alpha_v^s \sigma_v$ and use $\alpha_v^s$ to fit to the SDSS volume-limited sample. Therefore, their $\alpha_v^s$ includes all factors that cause the velocity difference between the baryonic component of galaxies and DM inside haloes. Some of the factors for the velocity bias are hydrodynamic, and others gravitational. Because the assignment approach naturally includes all gravitational effect, the difference between $|\bf{v^{\rm MBP}}-\bf{v^{h}}|$ and $\sigma_v$ will reflect these effects. As in Equation (\[equation:velocitybias\]), our $b_v^s$ is defined as the difference between the visible component and represented by MBPs inside halo; hence, $b_v^s$ will indicate only the baryonic effects that cause the velocity bias. Combining $\alpha_v^s$ and $b_v^s$ will tell us to what degree each origin contributes to the velocity bias. Figure \[figure:vbias\] shows the relation between $|\bf{v^{\rm MBP}}-\bf{v^{h}}|$ of centrals or satellites and $\sigma_v$ of DM haloes in the case of $\Omega_m=0.26$. The colored lines show central $68\%$ and $95\%$ percentile intervals, which are obtained by quantile regression using B-splines [@Ng2015]. The galaxy density is set to be similar to the observation data which we use. The median of $|\bf{v^{\rm MBP}}-\bf{v^{h}}|/\sigma_v$ is $0.94$ for satellite galaxies; the center of mass of satellite galaxies moves slightly slower than the velocity dispersion of DM inside the hosting halo.
One might wonder that the trajectories of MBPs and galaxies may diverge (i.e., the position of a galaxy in the next time step would be inconsistent with the corresponding MBP) if MBPs and galaxies have different velocities. Ideally, if the MBP represents the galaxy position and velocity correctly over cosmic time, $b_v$ has to be one. Our logic behind Equation (\[equation:velocitybias\]) is that we try to absorb the secondary effects which may cause the velocity difference between the $N$-body simulation and the real observation, in response to the results of previous studies. Although we expect that $b_v$ should be close to $1$ even if such effects are present, a significant deviation from $b_v\sim1$, if detected, would indicate an incompleteness of using $N$-body simulations to fit the observational data on the small scales.
Considering that the central galaxies have spent a relatively longer time inside clusters and should be better relaxed ($\bf{v^g} \sim \bf{v^{MBP}}$), the MBP velocity would be a good representative of the velocity of the baryonic part of the central galaxy. As seen in Figure \[figure:vbias\], the MBP velocity is in the range of $30\%$ to $50\%$ of $\sigma_v$. This compares with the estimate on $\alpha_v^s\sim0.3$ in @Guo2015. In an appendix, we will present the result obtained by modifying the MBP velocity for central galaxies. Also, note that the velocity bias can be a function of galaxy property such as age and mass, and studying the dependence of velocity bias in detail would help us to understand the dynamical aspects of the evolution of galaxies, but we will only use a single parameter $b_v^s$ in this work.
In summary, our model parameters are
- matter density parameter: $0.15<\Omega_m<0.37$.
- : $\alpha=1.5$
- satellite velocity bias: $0.3<b_v^s<1.7$.
Measurements {#section:measurements}
============
Multipole Moments of the Correlation Function
---------------------------------------------
As a statistical quantity of the redshift-space clustering, we use the multipole moments of the correlation function. First, the two-point correlation function is given by the Landy-Szalay estimator [@Landy1993], $$\xi(\mathbf{s}) = \frac{DD-2DR+RR}{RR},$$ where $DD$, $DR$, and $RR$ are the counts of galaxy-galaxy, galaxy-random, and random-random pairs, respectively. The vector $\mathbf{s}$ can be $\mathbf{s} = (r_p,r_\pi)$ or $(s, \mu)$, where $r_p$ and $r_\pi$ are the transverse and parallel components of the separation of galaxy pairs while $s=|\mathbf{s}|$ and $\mu=r_\pi/s$.
The multipole moments are calculated as $$\xi_l(s)=\frac{2l+1}{2}\int^{1}_{-1} \xi(s,\mu)L_l(\mu)d\mu$$ where $L_l(\mu)$ is the Legendre polynomial of $l$-th degree. Because the moments of odd numbers vanish due to the symmetry[^1] and higher order multipoles become less informative due to higher measurement noises, we use only $l=0$ (monopole), $l=2$ (quadrupole), and $l=4$ (hexadecapole): $L_0=1$, $L_2(\mu)=\frac{1}{2}(3\mu^2-1)$, and $L_4(\mu)=\frac{1}{8}(35\mu^4-30\mu^2+3)$. Because of the Kaiser effect, $\xi_0$ is enhanced and $\xi_2$ becomes negative in redshift space in scales larger than $\gtrsim 10 \ h^{-1}{\rm Mpc}$, while the opposite holds at the cluster scales. We use the bin size of $\Delta \mu=0.05$ and logarithmic bins from to $30 \ h^{-1}{\rm Mpc}$.
The Projected Correlation Function
----------------------------------
The projected correlation function is obtained by the integration along the line-of-sight, $$w(r_p) = \int^{r_{\pi,{\rm max}}}_{-r_{\pi,{\rm max}}} \xi(r_p,r_\pi)dr_\pi.$$ We set $r_{\pi,{\rm max}}=40 \ h^{-1}{\rm Mpc}$ and confirm that larger $r_{\pi,{\rm max}}$ hardly changes $w(r_p)$. The projected correlation function is a measure of clustering in real space, because the line-of-sight projection eliminates the RSD effect, whereas the multipole moments are the redshift-space quantities. It will be shown that using the projected correlation function can break the degeneracy between the ($\Omega_m$), the ($\alpha$) and the velocity bias ($b_v^s$) that are not fully broken by using multipoles only.
The FoG Ratio
-------------
Multipole moments of the correlation function are measures of the RSD effects, but they are also affected by the change of the overall clustering amplitude, which can vary due to the cosmic variance and other systematics. Thus, we try to extract a pure RSD information which is independent of the amplitude. We use a measure of the strength of the RSD effects as follows, $$\label{equation:FoG}
R_{|\xi=3}=\frac{r_{\pi{|\xi=3}}}{r_{p{|\xi=3}}},$$ which is the ratio of the separations along and across the line-of-sight from a point close to the origin to locations where the correlation function drops to $3$. The ratio for different threshold values can be defined likewise. The schematic image is given by Figure \[figure:image\_R\]. By taking a ratio, the cosmic variance in density fluctuations is expected to cancel out, giving a clean measurement of the strength of the FoG effect.
We calculate the correlation functions for the Multiverse and KIAS-VAGC catalogs, covering $0.1<r_p<30 \ h^{-1}{\rm Mpc}$ and $0.1<r_\pi<30 \ h^{-1}{\rm Mpc}$ with $15\times15$ logarithmic bins. Then, we take the fourth smallest bins ($\sim 0.4 \ h^{-1} {\rm Mpc}$), $\xi(0.4,r_\pi)$ and $\xi(r_p,0.4)$, to locate the point at which the correlation function becomes a certain threshold value. The scale $\sim 0.4 h^{-1} {\rm Mpc}$ is chosen to be sufficiently small to capture the FoG feature while keeping statistical uncertainty small with enough pair counts.
The Covariance Matrix
---------------------
The covariance matrix is necessary for evaluating the goodness of fit. We use the mock galaxy catalogs created from the HR4 data, which has a $3150^3 \ (h^{-1}{\rm Mpc})^3$ volume. Using the $405$ mock catalogs from HR4, we have found that the distributions of our observables follow the Gaussian distribution. For each data point, we compared the distribution of mock values to the Gaussian distribution of the same mean and variance using the Kolmogorov-Smirnov test for the null hypothesis of the mocks following Gaussian. The resulting $p$-values are $0.4$–$0.9$, indicating no evidence for non-Gaussian distributions. Therefore, we can use the standard $\chi^2$ statistics to evaluate the goodness of fit.
We adopt the $\chi^2$ statistics to constrain the model parameters, $$\label{equation:chi2}
\chi^2 = \left[{\mathbf X}^{\rm obs}-{\mathbf X}^{\rm th}({\mathbf \theta})\right]^T \mathbb{C}^{-1} \left[{\mathbf X}^{\rm obs}-{\mathbf X}^{\rm th}({\mathbf \theta})\right],$$ where ${\mathbf X}$ is the data vector, $\mathbb{C}$ is the covariance matrix corresponding to ${\mathbf X}$, and the superscripts represent the observation and the model prediction for parameters ${\mathbf \theta}=(\Omega_m, \alpha, b_v^s)$ respectively. For example, if we use $\xi_0$ and $\xi_2$ for the fitting, ${\mathbf X}$ will be a vector of $8\times2=16$ elements and be a $16\times16$ sized matrix. We apply the correction of [@Hartlap2007] to the covariance matrix to account for the underestimation of the covariance matrix due to the finite number of realizations. Because we use $405$ mock catalogs, the correction factor is $1.02$ to $1.10$, depending on the size of the data vector. Also, we multiply the covariance matrix by $(1+V_{\rm obs}/V_{\rm simu})=1.02$ to account for the uncertainty arising from the finite volume of the simulation box $V_{\rm simu}$ used to model the observation of volume $V_{\rm obs}$ [@Zheng2016].
Results {#section:results}
=======
The Correlation Functions
-------------------------
The projected correlation function is shown in Figure \[figure:wrp\]. Different colors correspond to different $\Omega_m$ while different line types to different $\alpha$. Because we fix the overall amplitude, $\sigma_8=0.79$, increasing $\Omega_m$ shifts the matter-radiation equality, resulting in weaker correlations at the scales which we are interested in. An increase in $\alpha$ enhances the overall amplitude due to the increased number of satellite galaxies. The change is more drastic on small scales than large scales, implying that the small-scale information is useful to discriminate different $\alpha$ scenarios, in turn giving a better constraint on $\Omega_m$. Also, it should be mentioned that the measurement error is small on smaller scales due to the larger number of pairs. While $\alpha=1.0$ and $2.0$ fail to reproduce the observation, the most probable value of $\alpha$ seems to be around $1.5$. Note that $w(r_p)$ does not depend on $b_v^s$ because $w(r_p)$ is a real-space quantity and not affected by RSD.
Figure \[figure:multipoles\] shows the dependence of multipole moments on $\Omega_m$ and $b_v^s$. Different panels are for different multipole moments. The dependence of multipoles on $\Omega_m$ is complicated. Both the Kaiser effect and FoG become stronger in a higher $\Omega_m$ universe [@Feldman2003; @Linder2005] and thus $\xi_0$ should be suppressed (enhanced) at small (large) scales, which is not the case at relatively large scales ($\sim 20 h^{-1}{\rm Mpc}$). This contradiction is caused by the weaker real-space clustering for higher $\Omega_m$ as we saw in Figure \[figure:wrp\], which is not fully compensated by the stronger Kaiser effect. For $\xi_2$, the positive (negative) sign indicates the elongated (squashed) feature. On larger scales, where the Kaiser effect dominates, $\xi_2<0$, and on small scales, where the FoG does, $\xi_2>0$. Again, a contradictory trend is seen in the middle panel of Figure \[figure:multipoles\], which we attribute to the overall amplitude of the real-space clustering. Another notable feature is the position of the peak of $\xi_2$. If we increase $b_v^s$, the peak shifts toward larger $s$. This is because large $b_v^s$ leads to a strong FoG effect, increasing the transition scale from the FoG to Kaiser effect. The position of the peak supports $b_v^s$ close to $1.0$.
The FoG Ratio
-------------
\[figure:FoG\]
Figure \[figure:FoG\] shows $R_{|\xi}$ for different $\Omega_m$ and $b_v^s$ as a function of the threshold value. $R_{|\xi}$ is smaller for lower thresholds because lower thresholds correspond to larger scales where the FoG effect is dominant and the Kaiser effect becomes effective, which reduce $R_{|\xi}$. A strong degeneracy between $\Omega_m$ and $b_v^s$ is seen. Also, the higher velocity bias means the higher galaxy motion inside clusters and thus the stronger FoG. Within the error bars of the observation, both of $(\Omega_m, b_v^s)=(0.21,1.0)$ and $(0.31,0.7)$ reproduce the observed FoG ratio reasonably well. If we had complete knowledge of $\alpha$ and $b_v^s$, the FoG ratios would give a constraint of $\Delta \Omega_m \sim0.02$ with our data. However, if we allow these to vary, using only the FoG ratio is insufficient to obtain a meaningful constraint on $\Omega_m$.
The Fitting {#subsection:fitting}
-----------
Next, let us see how well the fittings work. The peak position of the middle panel of Figure \[figure:multipoles\] tells us that $b_v^s\sim1.0$ will give the best fit. Then, we notice that $\Omega_m\sim0.26$ is preferred in the top panel by comparing the observation with the solid colored lines. Figure \[figure:contour\] shows the probability distribution of $\Omega_m$ and $b_v^s$ based on the $\chi^2$ statistics for $\alpha=1.5$. Because we only have three $\Omega_m$ realizations, any statistical quantity ($\xi_l$, $w(r_p)$, and $R_{|\xi}$) for other $\Omega_m$ is obtained by interpolation. Different lines correspond to what type(s) of information is(are) used to fit. The bold red contour is the combined result obtained by fitting to the monopole, quadruple, and hexadecapole moments and the projected 2pCF. The thin blue contour is from the FoG ratio and the bold blue is the combination of . Note that the projected 2pCF is the real-space quantity; therefore it cannot constrain $b_v^s$, but can indirectly contribute to determining $b_v^s$ better by constraining $\Omega_m$. All contours overlap one another in the $\alpha=1.5$ case. This means that $\alpha=1.5$ can explain all measurements simultaneously, supporting the validity of our modeling of the galaxy clustering. The best-fit values for the $\xi_0+\xi_2+\xi_4+w(r_p)$ case are $(\Omega_m,b_v^s)=(0.262\pm0.014,1.032\pm0.051)$. As seen in Figure \[figure:contour\], including $\xi_4$ and $w(r_p)$ does not change the best-fit within statistical uncertainty nor tighten the constraint significantly. However, note that $w(r_p)$ has given a good constraint on $\alpha$ as seen in \[figure:wrp\]. Adding the FoG ratio yields $(\Omega_m,b_v^s)=(0.272\pm0.013,0.982\pm0.040)$.
Figure \[figure:best\] gives a comparison between the observation and the best-fit models for $\alpha=1.5$ and $2.0$. The fitting procedure for $\alpha=2.0$ is identical to that for $\alpha=1.5$, except that we have used $\alpha=2.0$ in Equation (\[equation:Jiang08\]) to produce the simulated galaxies in the Multiverse Simulation. The top panel shows the projected 2pCF while the bottom the multipoles. Except for a weak deviation ($\sim 1\sigma$) of at $s>10\ h^{-1}\rm{Mpc}$, our $\alpha=1.5$ model reproduces $w(r_p), \xi_0$, and $\xi_2$ very well. A moderate deviation is seen at $s\sim 1\ h^{-1}\rm{Mpc}$ of $\xi_4$, which mainly contributes to our $\chi^2$ (see appendix \[section:chi2\] for detailed discussions). Also, we can see a good fit of $w(r_p)$ at $r_p<1\ h^{-1}\rm{Mpc}$ in spite of our fitting range, which also supports our galaxy model. Figure \[figure:xirppi\] shows the 2-D correlation function for the best models. We can see an excellent correspondence between the observation and our $\alpha=1.5$ model at small scales. We see some deviation beyond $10\ h^{-1}\rm{Mpc}$, but this will be within $1\sigma$ uncertainty as explained in Figure \[figure:best\].
Discussions {#section:discussions}
===========
Interpretation of $b_v^s$ and $\Omega_m$
----------------------------------------
As we have seen in the previous section, the inferred velocity bias is $b_v^s=0.982\pm0.040$ for $\alpha=1.5$. This value is slightly smaller than $1$, but the deviation is not significant for the size of the statistical error. We will discuss how this result compares with other studies. As we mentioned in §\[subsection:vbias\], our parameter $b_v^s$ is different from the definition used in the literature. The velocity bias is usually defined as the velocity of visible part of galaxies relative to the DM velocity dispersion inside their haloes. Thus, it is the multiplication of two factors: the velocity bias of baryonic component of galaxies (i.e., the observed galaxies) with respect to the whole galaxies (represented by MBPs), and MBPs’ velocity bias relative to the DM velocity dispersion. The sources of the galaxy velocity bias are also separated into two classes. The first one is related with the gravitational interactions such as dynamical friction, tidal stripping, and mergers. The other is baryonic effects including the star formation, radiative cooling, feedback from stars/supernovae/AGNs and the heat dissipation. Given the fact that $N$-body simulations implement all gravitational effects, the velocity of MBPs should reflect the velocity bias induced by gravitational interactions. Since we have defined $b_v^s$ as the ratio between the velocity of the baryonic component of galaxies and MBPs, $b_v^s$ will indicate the velocity bias generated by hydrodynamic effects. Figure \[figure:vbias\] shows that the median velocity of MBPs for satellite galaxies is $0.94$ of the DM velocity dispersion. As a result, the total velocity bias amounts to $0.94 \times 0.982=0.93$ and broadly consistent with the one obtained by @Guo2015. Our results imply that the satellite velocity bias is attributed more to dynamical effects than baryonic effects.
@Munari2013 ran $N$-body simulations with and without baryon cooling, star formation, and supernova and AGN feedbacks. Although it is not straightforward to compare their results with ours due to the different host halo mass focused on, their Figure 8 suggests $\sim10\%$ reduction of the velocity bias for the simulation with hydrodynamic effects. This is because the star formation and radiative emission cool galaxies down, forming a dense core and making galaxies resistant of tidal stripping. Tidal stripping selectively disrupts slow-moving galaxies leading to higher mean galaxy velocity. Thus, adding baryon cooling counteracts it and thereby reduces the averaged velocity. @Wu2013 investigated the effect of baryons on the galaxy velocity in their simulations, finding the reduction of galaxy velocity depending on the distance from the cluster center, from $30\%$ (close to halo center) to $0\%$ (at virial radius of host haloes). They suggest the reasoning for their result similarly to @Munari2013, but also mention the baryon dragging [@Puchwein2005]. Considering that the fraction of satellite galaxies enclosed within the innermost region around the halo center is subdominant [@Watson2012], the averaged reduction would be at most $10\%$. @Ye2017 found that the velocity bias depended on the ratio between the stellar mass and host halo mass, implying that the velocity bias is mostly caused by the dynamical effects. They argue that the dependence on stellar mass is a result of dynamical friction (a high-mass galaxy suffers from losing energy due to the two-body problem) and the dependence on host halo mass is related to the halo formation time (a high-mass halo is formed late, giving less time for dynamical effects to operate). All of these studies indicate that the velocity reduction caused by baryonic physics will be less significant than that caused by dynamical effects, which agrees with our results. @Ye2017 also found that the velocity bias was a complicated function of other physical quantities, including age and color. Future studies can include investigating such dependence, because knowing the detailed properties of the galaxy velocity bias would be useful for future surveys such as DESI and PFS, most of which apply color selections of galaxies to define the observation strategies. They would also push forward our understanding of the kinematic perspectives of the galaxy formation and evolution. For instance, we can classify galaxies by age, using the fitting technique, and measure the velocity bias through clustering measurements for each class.
Our constraint on $\Omega_m$ is $0.272\pm0.013$ when we use $\alpha=1.5$. The value is consistent with the WMAP5 result ($\Omega_m=0.26$; @Dunkley2009), but lower than that of the Planck ($\Omega_m=0.31$; @Planck2018). In our simulation, the normalization of the power spectrum is set to give $\sigma_8=0.79$, which is lower than the Planck results. Thus, the correlation functions of our simulations are systematically weaker. As we saw in Figures \[figure:wrp\] and \[figure:multipoles\], the correlation is stronger for lower $\Omega_m$, which explains our $\Omega_m$ consistent with the WMAP5 rather than the Planck. While we only run five simulations due to the large amount of resources required, efficient methods of searching parameter space using $N$-body simulations are being studied by several projects [@Nishimichi2018; @DeRose2019]. A more comprehensive study, including other cosmological parameters, would be beneficial.
The Usefulness of FoG Ratio
---------------------------
We have introduced a measure of the FoG strength as Equation (\[equation:FoG\]). As discussed in @Park2000 and @Tinker2007, taking a ratio removes the dependence on the overall amplitude of the real-space correlation function. The clustering amplitude depends on cosmological parameters such as $\Omega_m$, $\sigma_8$, and the linear growth rate $f$, which we usually wish to constrain, but also on unwanted factors including cosmic variance and some sort of systematic errors. The cosmological parameters inferred from only the multipole moments and projected 2pCF can be contaminated by the latter factors. On the other hand, the FoG ratio is free from these uncertainties after division if these factors are universal. As seen in the previous section, we should note that the constraining power is not strong because our FoG ratio uses the correlation function along $\mu=0$ and $1$ directions only.
The FoG ratio depends on cosmological parameters differently from the multipoles and projected 2pCF. Figure \[figure:contour\_2.0\] shows the probability distribution of $(\Omega_m,b_v^s)$ for $\alpha=2.0$, which is to be compared with Figure \[figure:contour\]. In each figure, combining $\xi_l$ and $w(r_p)$ gives preferred values of ($\Omega_m$, $b_v^s)$. However, the contour from the FoG ratio disagree with from the others in the $\alpha=2$ case, which supports $\alpha=1.5$. Noticeably, the contours from correlation functions and the FoG ratio shift toward different directions when the parameters are changed.
For the case of $\xi_l+w(r_p)$, increasing $\alpha$ results in lower $\Omega_m$ but higher $b_v^s$. The reason is as follows. There are more satellite galaxies when $\alpha$ is increased, leading to higher amplitude of the correlation function. On the other hand, increasing $\Omega_m$ decreases the amplitude of the galaxy 2pCF (see Figure \[figure:wrp\]). This is because we fix $\sigma_8$, which means that the integration of the correlation over all scales remains the same. Increasing $\Omega_m$ increases the amplitude on very large scales, thus decreasing the correlation at the scales of our interest. Therefore, $\alpha$ and $\Omega_m$ are anti-correlated. In contrast, the positive correlation between $\alpha$ and $b_v^s$ stems from the amplitudes of $\xi_l$. Considering the error bars of the observation, our constraints mainly come from $\sim1$–$3 \ h^{-1}\rm{Mpc}$. At such scales, increasing $b_v^s$ reduces $\xi_l$ due to the enhanced FoG effect, which compensates for the $\alpha$ increment. The degeneracy between the three parameters are the result of the fact that $\xi_l$ and $w(r_p)$ depend on the overall clustering amplitude, unlike the FoG ratio. Increasing both $\alpha$ and $b_v^s$, indeed, results in too strong FoG effect, which can be seen in Figure \[figure:xirppi\].
For the FoG ratio, on the other hand, an increase in $\alpha$ decreases both $\Omega_m$ and $b_v^s$. This behavior is easily understood. Increasing $\alpha$ increases the number of satellite galaxies, resulting in stronger FoG. In order to cancel this out, both $\Omega_m$ and $b_v^s$ need to be smaller. Since the FoG ratio does not depend on the overall clustering amplitude, how the parameters degenerate with one another is totally different from the 2pCFs. Although we have mainly used $\alpha=1.5$ in our study, the FoG ratio will tighten the constraints if we allow $\alpha$ to vary. Changing $\alpha$ alters the galaxy-halo connection, which is equivalent to changing the HOD parameters in that framework. Therefore, the FoG ratio would also help to constrain the parameters in the HOD approach. Another benefit of adding the FoG ratio would be that it gives a consistency check. In the absence of the cosmic variance and systematic errors, $w(r_p)$, $\xi_l$, and the FoG ratio overlap in the parameter space if the correct model is chosen. However, since $w(r_p)$ and $\xi_l$ are subject to the uncertainties of the clustering amplitude while the FoG ratio not, a discrepancy would be seen in the presence of systematic errors caused by the cosmic variance, observations, and data processing and analysis, even if the employed fitting model were sufficiently accurate.
Conclusions {#section:conclusions}
===========
The small-scale galaxy clustering can provide a wealth of information about the cosmological model and galaxy-halo connection, owing to the availability of precise measurements. In this study, we used the Multiverse and a physically motivated galaxy assignment scheme [@Hong2016] to study the small-scale redshift-space clustering. Specifically, we measured the projected correlation function $w(r_p)$ and the multipole moments $\xi_l(s)$ of the correlation function from $1.4$ to $30\ h^{-1}\rm{Mpc}$ to examine their dependence on the matter density parameter $\Omega_m$ and the merger time scale parameter $\alpha$. We also implemented the satellite velocity bias parameter $b_v^s$ to account for the possible velocity difference between galaxies and dark matter inside haloes [@Munari2013; @Wu2013; @Guo2015; @Ye2017]. We have measured the correlation functions of a volume-limited sample from the KIAS-VAGC catalog , which is based on the SDSS DR7 spectroscopic data to compare with those of the Multiverse Simulation. In the comparison, we have newly defined the strength of the FoG effect, $R_{|\xi}$, which is free from the change in the overall amplitude of the correlation function due to the cosmic variance and systematic errors. We have found that $\alpha=1.5$ reproduce the observation well, with $(\Omega_m,b_v^s)=(0.272\pm0.013,0.982\pm0.040)$. While our $b_v^s$ broadly agreed with previous observational and simulation studies [@Munari2013; @Wu2013; @Guo2015; @Ye2017], $\Omega_m$ was smaller than the Planck results [@Planck2018], which we attributed to the lower $\sigma_8$ that we assumed in the Multiverse simulations. Considering that the velocity of MBPs for satellite galaxies are $0.93$ of that of dark matter, the slow motions of galaxies relative to the dark matter velocity dispersion found by [@Guo2015] is mainly caused by dynamical effect rather than baryonic effects. The FoG ratio was found to be useful to break the degeneracy between the parameters and can be used to check the consistency of the fit obtained by $w(r_p)$ and $\xi_l(s)$.
We thank Christophe Pichon for fruitful discussions on this work and Junsup Shim for useful comments on the paper. HP acknowledges the support by World Premier International Research Center Initiative (WPI), MEXT, Japan. SEH was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education (2018R1A6A1A06024977).
Authors acknowledge the Korea Institute for Advanced Study for providing computing resources (KIAS Center for Advanced Computation Linux Cluster System). This work was supported by the Supercomputing Center/Korea Institute of Science and Technology Information, with supercomputing resources including technical support (KSC-2016-C3-0071) and the simulation data were transferred through a high-speed network provided by KREONET/GLORIAD.
Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the US Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS website is <http://www.sdss.org/>.
The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, Max Planck Institute for Astronomy (MPIA), the Max Planck Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the US Naval Observatory, and the University of Washington.
The $\chi^2$ statistics {#section:chi2}
=======================
Table \[table:chi2\] shows the minimum $\chi^2$ values and those per degree of freedom (d.o.f), obtained from Equation (\[equation:chi2\]) for different sets of measurements. The best-fit $\chi^2/{\rm d.o.f}$ is $1.67$ for the case where we use all measurements, which might be slightly high.
First, we used the covariance matrix estimated from the mock galaxy catalogs rather than jackknife resampling. We have also measured the covariance matrix using the jackknife method and found that the diagonal elements from the mocks are about half of those from the jackknife method, which means that the size of our error bars are smaller by a factor of $\sqrt{2}$. Because doubling the covariance matrix halves $\chi^2$, this partly describes the different $\chi^2$ values obtained by @Guo2015 and us. Then, why are the values of the covariance matrix from mock catalogs smaller than the jackknife resampling? One reason is related to the inherent feature of the jackknife; @Norberg2009 demonstrated that the jackknife returned the error bars accurately beyond $\sim 10\ h^{-1}\rm{Mpc}$ but significantly overestimated those below $10^{0.5}\ h^{-1}\rm{Mpc}$ for both $w(r_p)$ and $\xi(s)$, where our constraints mainly come from. Another possible reason is specific to our data; it includes the Sloan Great Wall [@Gott2005], which is centered at $z=0.08$. The unusually huge structure may lead to the large region-to-region variance, enhancing the error bars from the jackknife.
@Sinha2018 investigated the effect of the noise from the limited number of mocks on the resulting $\chi^2$. Due to the non-linearity of the inverse operation of the covariance matrix, this kind of noise enters into a $\chi^2$ analysis in an unpredictable manner. @Sinha2018 provided a solution to use the principal component analysis to extract some eigenvectors with large signal-to-noise ratios, and obtained smaller $\chi^2/{\rm d.o.f}$ values.
As obvious from Table \[table:chi2\], $\xi_4$ contributes hugely to the large $\chi^2$ that we obtain. However, Figure \[figure:contour\] shows that the inclusion of $\xi_4$ does not improve the constraining power. These facts might mean that the information of $\xi_4$ is already included in the combination of $\xi_0$ and $\xi_2$ or our model is insufficient to reproduce up to $\xi_4$. Improvements of our model can include allowing the central galaxy velocity bias parameter $b_v^c$ to change, but we will only try $b_v^c=0$ in the next appendix and leave the detailed analysis to future works.
\[table:chi2\]
The Model with Zero Central Galaxy Velocity Bias
================================================
While we have assumed $\mathbf{v}^g \sim \mathbf{v}^{\rm MBP}$ for central galaxies in the main text, we show the fitting result when the central velocity bias $b_v^c=0$, i.e., the central galaxies are rest at the center of haloes ($b_v^c$ is defined similarly to Equation (\[equation:velocitybias\])). Figure \[figure:contour\_ac0\] shows the $\chi^2$ contour obtained in the same manner as Figure \[figure:contour\]. Although the final constraint (blue bold line) is apparently consistent with that in the main text, it is simply a coincidence because each contour for $\xi_l$ does not overlap one another.
The shift of the FoG ratio (blue thin line) can be interpreted easily, owing to the fact that the FoG ratio is a pure measurement of FoG. The degree of the FoG effect is governed by the quadratic sum of the velocities of central and satellite galaxies. Therefore, $b_v^s$ has to be larger to compensate for nullifying central galaxy velocities inside haloes. On the other hand, understanding the shifts of the correlation functions is not straightforward. On small scales, the higher velocity bias takes the galaxy pairs to large separations in redshift space, reducing the amplitudes of $\xi_l$. Similarly to the FoG ratio, the decrease of the central galaxy velocity can be partly canceled out by the increase of the satellite velocity bias. However, not only the increase of $b_v^s$ but also the decrease in $\Omega_m$ can also increase the correlation amplitudes at such scales. Furthermore, the data points from larger scales have to be fit simultaneously, making the degeneracy of $(\Omega_m,b_v^c,b_v^s)$ complicated when we use the correlation functions.
Note that, only the diagonal elements of the covariance matrix is used to produce Figure \[figure:contour\_ac0\]. Due to the strong correlation and anticorrelation between each bin of $\xi_l$ and $w(r_p)$, the off-diagonal elements of $\mathbb{C}^{-1}$ are noisy. If the theoretical model were reasonably correct, the contributions of off-diagonal elements to $\left[{\mathbf X}^{\rm obs}-{\mathbf X}^{\rm th}({\mathbf \theta})\right]^T \mathbb{C}^{-1}
\left[{\mathbf X}^{\rm obs}-{\mathbf X}^{\rm th}({\mathbf \theta})\right]$ would not affect the best-fit values significantly because $|{\mathbf X}^{\rm obs}-{\mathbf X}^{\rm th}({\mathbf \theta})|$ is small. While we have confirmed that the best-fit values in the main text were stable even if we use only the diagonal components, we found that the blue bold line were afar from the rest contours when we used the full covariance matrix here. This will imply that $a_c=0$ is not a good model, probably giving unstable behaviors of the off-diagonal component of $\left[{\mathbf X}^{\rm obs}-{\mathbf X}^{\rm th}({\mathbf \theta})\right]^T \mathbb{C}^{-1}
\left[{\mathbf X}^{\rm obs}-{\mathbf X}^{\rm th}({\mathbf \theta})\right]$ caused by large $|{\mathbf X}^{\rm obs}-{\mathbf X}^{\rm th}({\mathbf \theta})|$, which supports previous studies that claim the existence of the central galaxy velocity bias.
The Fiber Collision Effect {#appendix:NNtest}
==========================
Due to the mechanical limitation of the SDSS spectroscopic instrument, when a galaxy pair is separated by an angular separation less than $55''$, only either one can be observed by a single run. This is called a fiber collision effect and leads to systematics in the correlation function measurements (@Zehavi2002 [@Guo2012]). Because the so-called nearest neighbor (NN) method is adopted in our study and it can cause systematic errors on the small scale measurements [@Reid2014], we have tested the validity of it by simulating the fiber collision effect using the HR4 simulation data. We use the concept of [@Guo2012] to model the fiber collision effect.[^2] The angular FoF grouping is performed to the objects on the celestial plane with an linking length of $55''$. Then, the objects are classified into three groups:
- D1: galaxies which are isolated
- D2: galaxies which collide with one close galaxy (typically doublets)
- D2’: galaxies which collide with more than one galaxy (typically the middle one of triplets)
We divide the HR4 galaxy catalog into $18$ boxes to create “flux-limited" samples which correspond to the parent photometric catalog of the KIAS-VAGC. Specifically, we select the heaviest galaxies within the distance range, which starts from $10\ {\rm Mpc}/h$ to $800\ {\rm Mpc}/h$ with a bin size of $10\ {\rm Mpc}/h$ to obtain the same number density. Because the fiber collision occurs in the parent photometric catalog before any redshift and luminosity cut, we require much more distant galaxies, which leads to the small number of realizations (18) compared to the ones for the covariance matrix (405).
We apply this classification to the KIAS-VAGC parent catalog to estimate the fraction of fiber-allocated galaxies as a function of the population (D1, D2, or D2’) and the number of plates covering the position of objects ($N_{\rm tile}$). Then, for each of the 18 realizations, the same classification code is run and “observed" galaxies are determined according to these fractions.
Then, we assign the nearest neighbor redshift to the “unobserved" galaxies, apply the redshift and mass cut, and measure the correlation functions. The comparison is given in Figure \[figure:fiber\_multipoles\] and \[figure:collision\]. We also perform another fiber collision correction method based on the pairwise-inverse-probability weights (PIP; [@Bianchi2017]). In this case, we repeat the selection process $1000$ times to create the logical array of length $N_{\rm bits}=1000$ for each galaxy, each elements of which is either 0 (unobserved) or 1 (observed). The correlation function is then measured using the pairwise weight given by Equation (14) of [@Bianchi2017].
The PIP scheme is accurate over almost all scales, as it is an unbiased way of correcting for the missing observations. The NN method is, however, still possible to use for our interested scales. This result is different from the argument of @Reid2014, but can be explained by the difference of minimum scale probed and the different collision scale in the comoving space ($<0.1 {\rm Mpc}/h$ and $<0.4 {\rm Mpc}/h$ for our and their works, respectively). While the higher redshift data such as BOSS and eBOSS would require the PIP method for small-scale clustering study, the NN method suffices for our study.
Abazajian, K. N., Adelman-McCarthy, J. K., Ag[ü]{}eros, M. A., et al. 2009, , 182, 543 Alam, S., Zhu, H., Croft, R. A. C., et al. 2017, , 470, 2822 Appleby, S., Chingangbam, P., Park, C., et al. 2018, , 863, 200 Beltz-Mohrmann, G. D., Berlind, A. A., & Szewciw, A. O. 2019, arXiv:1908.11448 Beutler, F., Blake, C., Colless, M., et al. 2012, , 423, 3430 Beutler, F., Saito, S., Seo, H.-J., et al. 2014, , 443, 1065 Bianchi, D., Percival, W. J., & Bel, J. 2016, , 463, 3783 Bianchi, D., & Percival, W. J. 2017, , 472, 1106 Blake, C., 2011, , 415, 2876 Blanton, M. R., Schlegel, D. J., Strauss, M. A., et al. 2005, , 129, 2562 Choi, Y.-Y., Han, D.-H., & Kim, S. S. 2010, Journal of Korean Astronomical Society, 43, 191 Davis, M., Efstathiou, G., Frenk, C. S., & White, S. D. M. 1985, , 292, 371 Dawson, K. S., Schlegel, D. J., Ahn, C. P., et al. 2013, , 145, 10 Dawson, K. S., Kneib, J.-P., Percival, W. J., et al. 2016, , 151, 44 DESI Collaboration, Aghamousa, A., Aguilar, J., et al. 2016, arXiv:1611.00036 de la Torre, S., Guzzo, L., Peacock, J. A., et al. 2013, , 557, A54 DeRose, J., Wechsler, R. H., Tinker, J. L., et al. 2019, , 875, 69 Drinkwater, M. J., Jurek, R. J., Blake, C., et al. 2010, , 401, 1429 Dunkley, J., Komatsu, E., Nolta, M. R., et al. 2009, , 180, 306 Feldman, H., Juszkiewicz, R., Ferreira, P., et al. 2003, , 596, L131 Frieman, J. A., Turner, M. S., & Huterer, D. 2008, , 46, 385 Geller, M. J., Diaferio, A., & Kurtz, M. J. 2011, , 142, 133 Gott, J. R., III, Juri[ć]{}, M., Schlegel, D., et al. 2005, , 624, 463 Guo, H., Zehavi, I., & Zheng, Z. 2012, , 756, 127 Guo, H., Zheng, Z., Zehavi, I., et al. 2015, , 453, 4368 Guo, H., Zheng, Z., Behroozi, P. S., et al. 2016, , 459, 3040 Guzzo, L., Pierleoni, M., Meneux, B., et al. 2008, , 451, 541 Guzzo, L., Scodeggio, M., Garilli, B., et al. 2014, , 566, A108 Hawkins, E., 2003, , 346, 78 Hamilton, A. J. S. 1998, The Evolving Universe, 231, 185 Hartlap, J., Simon, P., & Schneider, P. 2007, , 464, 399 Hong, S. E., Park, C., & Kim, J. 2016, , 823, 103 Hong, S., Jeong, D., Hwang, H. S., et al. 2020, , 493, 5972 Hwang, H. S., Geller, M. J., Park, C., et al. 2016, , 818, 173 Icaza-Lizaola, M., Vargas-Maga[ñ]{}a, M., Fromenteau, S., et al. 2019, arXiv:1909.07742 Jiang, C. Y., Jing, Y. P., Faltenbacher, A., Lin, W. P., & Li, C. 2008, , 675, 1095 Jones, D. H., Saunders, W., Read, M., & Colless, M. 2005, , 22, 277 Joyce, A., Lombriser, L., & Schmidt, F. 2016, Annual Review of Nuclear and Particle Science, 66, 95 Juszkiewicz, R., Fisher, K. B., & Szapudi, I. 1998, , 504, L1 Kaiser, N. 1987, , 227, 1 Kim, J., Park, C., Gott, J. R., III, & Dubinski, J. 2009, , 701, 1547 Kim, J., Park, C., L’Huillier, B., & Hong, S. E. 2015, Journal of Korean Astronomical Society, 48, 213 Koyama, K. 2016, Reports on Progress in Physics, 79, 046902 Kuruvilla, J., & Porciani, C. 2018, , 479, 2256 Landy, S. D., & Szalay, A. S. 1993, , 412, 64 Li, X.-D., Park, C., Sabiu, C. G., et al. 2016, , 832, 103 Linder, E. V. 2005, , 72, 043529 Marzke, R. O., Geller, M. J., da Costa, L. N., & Huchra, J. P. 1995, , 110, 477 Montero-Dorta, A. D., Perez, E., Prada, F., et al. 2017, https://arxiv.org/abs/1705.00013 Munari, E., Biviano, A., Borgani, S., Murante, G., & Fabjan, D. 2013, , 430, 2638 Ng, P. T., & Maechler, M. 2015, COBS: COnstrained B-Splines, ascl:1505.010 Nishimichi, T., Takada, M., Takahashi, R., et al. 2018, arXiv:1811.09504 Norberg, P., Baugh, C. M., Gazta[ñ]{}aga, E., & Croton, D. J. 2009, , 396, 19 Okumura, T., Hikage, C., Totani, T., et al. 2016, , 68, 38 Park, C. 2000, , 319, 573 Park, C., & Choi, Y.-Y. 2009, , 691, 1828 Park, C., & Kim, Y.-R. 2010, , 715, L185 Park, H., Park, C., Sabiu, C. G., et al. 2019, , 881, 146 Planck Collaboration, Aghanim, N., Akrami, Y., et al. 2018, arXiv:1807.06209 Perlmutter, S., Aldering, G., Goldhaber, G., et al. 1999, , 517, 565 Puchwein, E., Bartelmann, M., Dolag, K., & Meneghetti, M. 2005, , 442, 405 Reid, B. A., Samushia, L., White, M., et al. 2012, , 426, 2719 Reid, B. A., Seo, H.-J., Leauthaud, A., et al. 2014, , 444, 476 Riess, A. G., Filippenko, A. V., Challis, P., et al. 1998, , 116, 1009 Samushia, L, Percival, W. J & Raccanelli, A 2013, , 420, 2102 S[á]{}nchez, A. G., Scoccimarro, R., Crocce, M., et al. 2017, , 464, 1640 Sheth, R. K. 1996, , 279, 1310 Shin, J., Kim, J., Pichon, C., et al. 2017, , 843, 73 Sinha, M., Berlind, A. A., McBride, C. K., et al. 2018, , 478, 1042 Spergel, D. N., Bean, R., Dor[é]{}, O., et al. 2007, , 170, 377 Spergel, D., Gehrels, N., Baltay, C., et al. 2015, arXiv:1503.03757 Takada, M., Ellis, R. S., Chiba, M., et al. 2014, , 66, R1 Taruya, A., Nishimichi, T., & Saito, S. 2010, , 82, 063522 Tinker, J. L. 2007, , 374, 477 Tonegawa, M., Totani, T., Okada, H., et al. 2015, , 67, 81 Vikhlinin, A., Kravtsov, A. V., Burenin, R. A., et al. 2009, , 692, 1060 Wang, L., Weinmann, S. M., De Lucia, G., & Yang, X. 2013, , 433, 515 Watson, D. F., Berlind, A. A., McBride, C. K., et al. 2012, , 749, 83 Weinberg, D. H., Mortonson, M. J., Eisenstein, D. J., et al. 2013, , 530, 87 Wu, H.-Y., Hahn, O., Evrard, A. E., Wechsler, R. H., & Dolag, K. 2013, , 436, 460 Ye, J.-N., Guo, H., Zheng, Z., & Zehavi, I. 2017, , 841, 45 York, D. G., Adelman, J., Anderson, J. E., Jr., et al. 2000, , 120, 1579 Zehavi, I., Blanton, M. R., Frieman, J. A., et al. 2002, , 571, 172 Zehavi, I., Zheng, Z., Weinberg, D. H., et al. 2011, , 736, 59 Zheng, Z., & Guo, H. 2016, , 458, 4015
[^1]: The relativistic effect can cause asymmetry by breaking the symmetry along the line of sight [@Alam2017], but we do not consider it because the effect is much smaller than RSD.
[^2]: The public code is available at http://sdss4.shao.ac.cn/guoh/
|
---
abstract: 'Parametric down conversion (PDC) is widely interpreted in terms of photons, but, even among supporters of this interpretation, many properties of the photon pairs have been described as “mind-boggling" and even “absurd". In this article we argue that a classical description of the light field, taking account of its vacuum fluctuations, leads us to a consistent and rational description of all PDC phenomena. “Nonlocality" in quantum optics is simply an artifact of the Photon Concept. We also predict a new phenomenon, namely the appearance of a second, or satellite PDC rainbow. (This article will appear in the Proceedings of the Second Vigier Conference held in York University, Canada in August 1997. A somewhat more formal version has been submitted to Phys. Rev. Letters, and may be found at http://xxx.lanl.gov/abs /quant-ph/9711029.)'
author:
- |
Trevor W. Marshall\
Department of Mathematics, University of Manchester\
Manchester M13 9PL, U.K.
title: '[**The myth of the down converted photon**]{}'
---
Introduction
============
In an article for the last conference in this series[@vig] we gave a description of the Parametric Down Conversion (PDC) process based on the real vacuum electromagnetic-field fluctuations. We indicated that there was a serious unsolved problem, in that detectors must somehow subtract away these fluctuations; such a mechanism must come into play in order to explain the very low dark rates actually observed. We have since published a series of articles[@pdc1; @pdc2; @pdc3; @pdc4] in which a great variety of PDC phenomena have been analyzed using this description. Since we have been able to establish a formal parallel, through the Wigner representation, between the new (or rather the old!) field description and the presently dominant Photon Theory, it is clear that, [*once the reality of the zeropoint field has been accepted, there are no PDC phenomena which require photons.*]{} Furthermore we have made considerable progress on the subtraction problem[@pdc4]; all that is needed to explain the low dark rates of detectors is the recognition of their extremely large time windows (5ns is a very large number of light oscillations).
The approach of the above series of articles was a kind of compromise between the standard nonlocal theory of Quantum Optics, where the interaction of the various field modes is represented by a hamiltonian, and a fully maxwellian theory, which would be both local and causal. In this latter case the nonlinear crystal would be represented as a spatially localized current distribution, modified of course by the incoming electromagnetic field; the outgoing field would then be expressed as the retarded field radiated by this distribution. A preliminary attempt at such a theory was made[@magic], using first-order perturbation theory. However, we showed, in the above series of articles, that a calculation of the relevant counting rates, to lowest order, requires us to find the [*second*]{}-order perturbation corrections to the Wigner density, and the close formal parallel between these two theories means that the same considerations will apply to the maxwellian theory.
What is PDC?
============
It is necessary to pose this question, because, depending on the answer given, PDC may be described as either a local or a nonlocal phenomenon.
An example of the modern, nonlocal description is provided by Greenberger, Horne and Zeilinger[@ghz]. A nonlinear crystal, pumped by a laser at frequency $\omega_0$, produces conjugate pairs of signals, of frequency $\omega$ and $\omega_0-\omega$ (see Fig.1). Since light is supposed to consist of photons, this means that an incoming laser photon “down converts" into a pair of lower-energy photons. Naturally, since we know that $E=\hbar\omega$, that means energy is conserved in the PDC process, which must be very comforting. However, the above authors themselves refer to the PDC photon-counting statistics as “mind-boggling", and a more recent commentary[@zeil] even uses the term “absurd".
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There is an older description, which I suggest is more correct than the modern one. It had only a short life. Nonlinear optics was born in the late 1950s, with the invention of the laser. Up to about 1965, when Quantum Optics was born, the PDC process would have been depicted[@saleh; @yariv] by Fig.2; an incoming wave of frequency $\omega$ is down converted, by the pumped crystal, into an
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(159.33,44.67) (70.00,44.67)[(0,-1)[40.00]{}]{} (90.00,4.67)[(0,1)[40.00]{}]{} (0.00,24.67)[(1,0)[70.00]{}]{} (70.00,24.67)[(-5,1)[70.00]{}]{} (90.00,24.67)[(5,1)[69.33]{}]{} (159.33,10.67)[(-5,1)[69.33]{}]{} (31.33,19.67)[(0,0)\[cc\][laser]{}]{} (31.33,38.67)[(0,0)\[cc\][input$(\omega)$]{}]{} (122.33,38.34)[(0,0)\[cc\][signal$(\omega_0-\omega)$]{}]{} (122.00,10.00)[(0,0)\[cc\][idler$(\omega)$]{}]{} (80.00,24.67)[(0,0)\[cc\][[LINEAR]{}]{}]{} (80.00,31.67)[(0,0)\[cc\][[NON]{}]{}]{} (80.00,17.67)[(0,0)\[cc\][[CRYSTAL]{}]{}]{} (12.67,24.67)[(1,0)[7.67]{}]{} (14.67,35.67)[(4,-1)[1.00]{}]{} (119.33,18.67)[(4,-1)[1.00]{}]{} (119.67,30.34)[(4,1)[1.00]{}]{} (32.67,28.00)[(0,0)\[cc\][$\theta(\omega)$]{}]{}
outgoing signal of frequency $\omega_0-\omega$. The explanation of the frequency relationships lies in the multiplication, by the nonlinear crystal, of the two input amplitudes; we have no need of $\hbar$!
This process persists when the intensity of the input is reduced to zero, because all modes of the light field are still present in the vacuum, and the nonlinear crystal modifies vacuum modes in exactly the same way as it modifies input modes supplied by an experimenter. What we see emerging from the crystal is the familiar PDC rainbow. This is because the angle of incidence $\theta$, at which PDC occurs, is different for different frequencies on account of the variation of refractive index with frequency.
We depict the process of PDC from the vacuum in Fig.3, but note that this figure shows only two conjugate modes of the light field; a complete picture would show all frequencies participating in conjugate pairs, with varying angles of incidence. In contrast with Fig.2, where we showed only the one relevant input, we must now take account also of the conjugate input mode of the zeropoint, since the first mode itself has only the zeropoint amplitude.
The zeropoint inputs, denoted by interrupted lines in Fig.3, do not activate photodetectors, because the threshold of these devices is set precisely at the level of the zeropoint intensity, as discussed in Ref.[@pdc4].
=0.75mm
(159.33,44.67) (70.00,44.67)[(0,-1)[40.00]{}]{} (90.00,4.67)[(0,1)[40.00]{}]{} (0.00,24.67)[(1,0)[70.00]{}]{} (31.67,28.34)[(0,0)\[cc\][$\theta(\omega)$]{}]{} (90.00,24.67)[(5,1)[69.33]{}]{} (159.33,10.67)[(-5,1)[69.33]{}]{} (11.33,19.67)[(0,0)\[cc\][laser]{}]{} (31.33,38.67)[(0,0)\[cc\][input($\omega$)]{}]{} (122.33,38.34)[(0,0)\[cc\][+idler($\omega_0-\omega$)]{}]{} (122.00,10.00)[(0,0)\[cc\][idler($\omega$)]{}]{} (80.00,24.67)[(0,0)\[cc\][[LINEAR]{}]{}]{} (80.00,31.67)[(0,0)\[cc\][[NON]{}]{}]{} (80.00,17.67)[(0,0)\[cc\][[CRYSTAL]{}]{}]{} (12.67,24.67)[(1,0)[7.67]{}]{} (14.67,36.00)[(4,-1)[1.00]{}]{} (119.33,18.67)[(4,-1)[1.00]{}]{} (119.67,30.67)[(4,1)[1.00]{}]{} (70.00,24.67)[(-5,1)[42.33]{}]{} (15.67,35.67)[(-5,1)[15.67]{}]{} (0.00,38.13)[(0,0)[0.00]{}]{} (31.33,44.00)[(0,0)\[cc\][zeropoint]{}]{} (122.67,43.67)[(0,0)\[cc\][signal($\omega_0-\omega$)]{}]{} (122.00,5.33)[(0,0)\[cc\][+signal($\omega$)]{}]{} (0.00,10.67)[(4,1)[16.00]{}]{} (70.00,24.67)[(-5,-1)[39.67]{}]{} (31.33,13.33)[(0,0)\[cc\][zeropoint]{}]{} (31.33,9.00)[(0,0)\[cc\][input($\omega_0-\omega$)]{}]{} (33.33,21.67)[(0,0)\[cc\][$\theta(\omega_0-\omega)$]{}]{}
However, the two idlers have intensities above that of their corresponding inputs. Also there is no coherence between a signal and an idler of the same frequency, so their intensities are additive in both channels. Hence there are photoelectron counts in both of the outgoing channels of Fig.3.
The question we have posed in this section could be rephrased as “What is it that is down converted?". According to the thinking behind Fig.1, the laser photons are down converted, whereas according to Fig.3 it is the zeropoint modes; they undergo both down conversion, to give signals, and amplification, to give idlers.
Photon production rates in PDC
==============================
There is a small, but important difference between the maxwellian theory and the theory outlined in our Wigner series[@pdc1; @pdc2; @pdc3; @pdc4], though both of them could be said to be based on Fig.3. The Wigner series gave us the undulatory version of quantum optics, but its starting point is a hamiltonian which takes the creation of photon pairs as axiomatic. The maxwellian theory, whose details are given elsewhere[@puc1], starts from a nonlinear expression for the induced current and deduces a coupling between the field modes. This coupling is very similar, but not identical, to that deduced from the Wigner-based theory. As we have emphasized, there are no photons in the maxwellian theory, but if we translate the intensities of the outgoing signals in Fig.3 into photon terms, we obtain the result $$\frac{n_i(\omega)+n_s(\omega)}{n_i(\omega_0-\omega)+n_s(\omega_0-\omega)}=
\frac{\cos[\theta(\omega_0-\omega)]}
{\cos[\theta(\omega)]}\;. \label{pdcint}$$
So we conclude that [*the photon rate in a given channel is inversely proportional to the cosine of the rainbow angle*]{}. In the Photon Theory, the above ratio is one.
There seems little chance of finding out directly which of these theories is correct; the difference between the two ratios is small, since the rainbow angles are typically around 10 degrees, and it is not possible to measure at all accurately the efficiency of light detectors as a function of frequency. It is true that some of the experiments we have analysed, using the standard theory, in Refs.[@pdc1; @pdc2; @pdc3; @pdc4], have slightly different results in the present theory, for example the fringe visibility in the experiment of Zou, Wang and Mandel[@zwm]. Some details will be published shortly, but we can say that an experimental discrimination will be very difficult.
Parametric up conversion from the vacuum
========================================
There is, however, at least one prediction of the new theory which differs dramatically from the standard theory. An incident wave of frequency $\omega$, as well as being down converted by the pump to give a PDC signal of frequency $\omega_0-\omega$, may also be [*up converted to give a PUC signal*]{} of frequency $\omega_0+\omega$. We depict this phenomenon, which is well known[@saleh; @yariv] in classical nonlinear optics, in Fig.4.
=0.75mm
(159.33,86.80) (70.00,85.00)[(0,-1)[80.00]{}]{} (90.00,5.00)[(0,1)[80.00]{}]{} (0.00,45.00)[(1,0)[70.00]{}]{} (80.00,45.00)[(0,0)\[cc\][[LINEAR]{}]{}]{} (80.00,52.00)[(0,0)\[cc\][[NON]{}]{}]{} (80.00,38.00)[(0,0)\[cc\][[CRYSTAL]{}]{}]{} (12.67,45.00)[(1,0)[7.67]{}]{} (70.00,45.00)[(-5,3)[69.67]{}]{} (90.00,45.00)[(5,-3)[69.33]{}]{} (159.33,3.40)[(0,0)[0.00]{}]{} (90.00,45.00)[(6,-1)[69.33]{}]{} (49.00,51.00)[(0,0)\[cc\][$\theta_u(\omega)$]{}]{} (32.33,38.67)[(0,0)\[cc\][laser]{}]{} (33.67,73.67)[(0,0)\[cc\][input($\omega$)]{}]{} (131.67,43.33)[(0,0)\[cc\][signal($\omega_0+\omega$)]{}]{} (115.67,21.00)[(0,0)\[cc\][idler($\omega$)]{}]{} (120.00,27.00)[(3,-2)[1.00]{}]{} (121.67,39.67)[(4,-1)[1.00]{}]{} (27.67,70.33)[(3,-2)[1.00]{}]{}
Note that the angle of incidence, $\theta_u(\omega)$, at which PUC occurs is quite different from the PDC angle, which in Fig.2 was denoted simply $\theta(\omega)$, but which we should now call $\theta_d(\omega)$.
Now, following the same argument which led us from Fig.2 to Fig.3, we predict the phenomenon of PUC from the Vacuum, which we depict in Fig.5.
When we come to calculate the intensity of the PUC rainbow, there is an important difference from the PDC situation, because we find that the idler intensities are now less than the input zeropoint intensities. The signal intensities in both channels almost, but not quite, cancel this shortfall, so that the PUC intensities are only about 3 per cent of the PDC intensities, which may explain why nobody has yet observed them. Also, note that there is a detectable signal only in the lower-frequency channel, because the relation corresponding to eq.(\[pdcint\]) is $$\frac{n_i(\omega)+n_s(\omega)}{n_i(\omega_0+\omega)+n_s(\omega_0+\omega)}=
-\frac{\cos[\theta_u(\omega_0+\omega)]}
{\cos[\theta_u(\omega)]}\;,$$ which means that in one of the channels (actually the upper-frequency one), the total output intensity is less than the zeropoint, so nothing will be detected in this channel. My prediction therefore is that, as well as the main PDC rainbow $\theta_d(\omega)$, [*there is also a satellite rainbow*]{}, whose intensity is about 3 percent of the main one, at $\theta_u(\omega)$. An approximate calculation[@puc1] shows that $\theta_u(\omega)$ is about 2.5 times $\theta_d(\omega)$.
=0.75mm
(159.33,85.00) (70.00,85.00)[(0,-1)[80.00]{}]{} (90.00,5.00)[(0,1)[80.00]{}]{} (0.00,45.00)[(1,0)[70.00]{}]{} (80.00,45.00)[(0,0)\[cc\][[LINEAR]{}]{}]{} (80.00,52.00)[(0,0)\[cc\][[NON]{}]{}]{} (80.00,38.00)[(0,0)\[cc\][[CRYSTAL]{}]{}]{} (12.67,45.00)[(1,0)[7.67]{}]{} (90.00,45.00)[(5,-3)[69.33]{}]{} (159.33,3.40)[(0,0)[0.00]{}]{} (32.33,38.67)[(0,0)\[cc\][laser]{}]{} (33.67,73.67)[(0,0)\[cc\][input($\omega$)]{}]{} (131.67,43.33)[(0,0)\[cc\][+idler($\omega_0+\omega$)]{}]{} (115.67,21.00)[(0,0)\[cc\][idler($\omega$)]{}]{} (120.00,27.00)[(3,-2)[1.00]{}]{} (121.67,39.67)[(4,-1)[1.00]{}]{} (27.67,70.00)[(3,-2)[1.00]{}]{} (33.33,79.67)[(0,0)\[cc\][zeropoint]{}]{} (70.00,45.00)[(-6,1)[30.67]{}]{} (18.00,54.00)[(-6,1)[18.00]{}]{} (13.00,55.00)[(4,-1)[1.00]{}]{} (25.00,56.33)[(0,0)\[cc\][input($\omega_0+\omega$)]{}]{} (25.00,61.33)[(0,0)\[cc\][zeropoint]{}]{} (131.33,50.00)[(0,0)\[cc\][signal($\omega_0+\omega$)]{}]{} (115.67,14.67)[(0,0)\[cc\][+signal($\omega$)]{}]{} (70.00,45.00)[(-5,3)[27.00]{}]{} (28.67,69.33)[(-5,3)[22.33]{}]{} (90.00,45.00)[(6,-1)[32.33]{}]{} (137.33,36.33)[(6,-1)[21.33]{}]{}
Conclusion
==========
Our contribution to the previous conference in this series[@vig] was entitled “The myth of the photon”. The present article repeats this theme, but covers a narrower range of phenomena. This is because the local theory of nonlinear crystals is now very much more complete than the corresponding theory for atoms. In retrospect, the word “obsolete”, which we used in the previous article, for [*all*]{} photon theories, was excessively triumphalist. Of course one could argue that they became obsolete once their nonlocal nature was revealed, that is a quarter of a century ago, but there was nothing local on offer at that time. The claim we made, maybe prematurely, was based on having demonstrated, by the use of certain model theories with very limited fields of application, that local theories were, in all cases [*possible*]{}. Now we have passed into a new phase of the programme; we now have, for a very wide and growing area of investigation, [*a well defined alternative theory which makes certain new predictions*]{}. If and when such predictions are verified, I think that down-converted photons, for example those depicted in our Fig.1, will be very definitely obsolete.
[**Acknowledgement**]{}
I have had a lot of help with the ideas behind this article, and also in developing the argument, from Emilio Santos.
[99]{} T. W. Marshall and E. Santos, [*The myth of the photon*]{} in [*The Present Status of the Quantum Theory of Light*]{}, eds. S. Jeffers et al, (Kluwer, Dordrecht, 1997) pages 67–77. See also http://xxx.lanl.gov/abs /quant-ph/9711046. A. Casado, T. W. Marshall, and E. Santos, [*J. Opt. Soc. Am. B*]{}, [**14**]{}, 494–502 (1997). A. Casado, A. Fernandez Rueda, T. W. Marshall, R. Risco Delgado, and E. Santos, [*Phys.Rev.A*]{}, [**55**]{}, 3879–3890 (1997). A. Casado, A. Fernandez Rueda, T. W. Marshall, R. Risco Delgado, and E. Santos, [*Phys.Rev.A*]{}, [**56**]{}, 2477-2480 (1997) A. Casado, T. W. Marshall and E. Santos, [*J. Opt. Soc Am. B*]{} (awaiting publication). See also http://xxx.lanl.gov/abs /quant-ph/9711042. T. W. Marshall, [*Magical Photon or Real Zeropoint?*]{} in [*New Developments on Fundamental Problems in Quantum Physics*]{}, eds. M. Ferrero and A. van der Merwe (Kluwer, Dordrecht, 1997) D. M. Greenberger, M. A. Horne and A. Zeilinger, [*Phys. Today*]{}, [**46**]{} No.8, 22 (1993) D. Bouwmeester and A. Zeilinger, [*Nature*]{} [**388**]{}, 827-828 (1997) B.E.A.Saleh and M.C.Teich, [*Fundamentals of Photonics*]{}, (John Wiley, New York, 1991) Chap. 19 A. Yariv, [*Quantum Electronics*]{} (John Wiley, New York, 1989) Chaps. 16 and 17 T. W. Marshall, submitted to [*Phys. Rev. A*]{} See also http://xxx.lanl.gov/abs /quant-ph/9711030 L. J. Wang, X. Y. Zou and L. Mandel, [*Phys. Rev. A*]{}, [**44**]{}, 4614 (1991)
|
---
abstract: 'Using the cosmological smoothed particle hydrodynamical code [GADGET-3]{} we make a realistic assessment of the technique of using constant cumulative number density as a tracer of galaxy evolution at high redshift. We find that over a redshift range of $3\leq z \leq7$ one can [*on average*]{} track the growth of the stellar mass of a population of galaxies selected from the same cumulative number density bin to within $\sim 0.20$ dex. Over the stellar mass range we probe ($10^{10.39}\leq M_s/{M_\odot}\leq 10^{10.75}$ at $z =$ 3 and $10^{8.48}\leq M_s/{M_\odot}\leq 10^{9.55}$ at $z =$ 7) one can reduce this bias by selecting galaxies based on an evolving cumulative number density. We find the cumulative number density evolution exhibits a trend towards higher values which can be quantified by simple linear formulations going as $-0.10\Delta z$ for descendants and $0.12\Delta z$ for progenitors. Utilizing such an evolving cumulative number density increases the accuracy of descendant/progenitor tracking by a factor of $\sim2$. This result is in excellent agreement, within $0.10$ dex, with abundance matching results over the same redshift range. However, we find that our more realistic cosmological hydrodynamic simulations produce a much larger scatter in descendant/progenitor stellar masses than previous studies, particularly when tracking progenitors. This large scatter makes the application of either the constant cumulative number density or evolving cumulative number density technique limited to average stellar masses of populations only, as the diverse mass assembly histories caused by stochastic physical processes such as gas accretion, mergers, and star formation of individual galaxies will lead to a larger scatter in other physical properties such as metallicity and star-formation rate.'
author:
- 'Jason Jaacks$^1*$, Steven L. Finkelstein$^1$ & Kentaro Nagamine$^{2,3}$'
title: 'Connecting the Dots: Tracking Galaxy Evolution Using Constant Cumulative Number Density at $3\leq \lowercase{z} \leq 7$'
---
Introduction {#sec:intro}
============
Understanding how galaxies evolved from minute perturbations in the distant Universe into the diverse zoo of shapes and sizes we see today is one of the fundamental goals of modern astronomy. The current frontier lies at the edge of the observable universe, $\sim500\ {\rm Myr}$ after the Big Bang, and is primarily possible using the Wide Field Camera 3 (WFC3) instrument aboard the [*Hubble Space Telescope*]{} ([*HST*]{}). Using the Lyman-break technique and/or photometric redshifts to select candidate galaxies, programs such as CANDELS (PIs Faber & Ferguson), BoRG [@Trenti.etal:11], and HUDF09/UDF12 [@Bouwens.etal:11a; @Ellis:13], are able to identify galaxies down to rest-frame UV magnitudes of ${\rm M_{uv}}\sim-17.5$ [e.g., @Finkelstein.etal:12; @Finkelstein:14; @Trenti.etal:11; @Bouwens.etal:12b; @Ellis:13]. Two of these galaxies have been spectroscopically confirmed to be the earliest known galaxies to date with redshifts of $z$=7.51 [@Finkelstein:13] and $z$=7.73 [@Oesch:15]. With the help of the gravitational lensing effect of massive foreground galaxy clusters, campaigns such as CLASH [@Bouwens:14a] and the [*HST Frontier Fields*]{} will extend our understanding even further to $z\ge9$ and UV magnitudes as faint as ${\rm M_{uv}}\sim-13$, greatly increasing the dynamic range of the observed galaxy population for study.
From these surveys, fundamental properties such as stellar mass, age and star formation rate (SFR) can be derived by comparing the spectral energy distributions (SEDs) of galaxies to stellar population models. By selecting galaxies at different epochs (“snapshots” in time), we can in principle directly observe how galaxies evolve. However, tracing galaxies from one epoch to another has proven challenging, and frequently accompanied by misinterpretation. Previous studies have matched galaxies at different epochs by comparing samples selected to have similar physical tracers, such as at fixed UV luminosity [e.g., @Stark.etal:09]. This can be problematic as two galaxies, one at $z$=6 and the other at $z$=3, with similar UV luminosities could have dramatically different mass assembly histories depending on their individual star formation histories (SFHs), environments and/or merger histories.
A novel approach to this problem was suggested by @vandokkum:10 who tracked a population of galaxies through cosmic time ($z$=2 to $z$=0.1) selected to have a constant number density, using the observed cumulative galaxy stellar mass function (CSMF). The critical assumption this approach makes is that galaxies in the same number density bin will grow at a similar, smooth rate with a conserved rank order. Monte Carlo simulations were utilized to test the effects of mergers and starbursts, and it was concluded that only 1:1 mass-ratio mergers were able to break the rank order of galaxies.
The use of this technique was tested numerically by @leja:13 using the Millennium dark matter (DM) halo catalog [@Springel:05b] and semi-analytic models [@Guo:11]. They found that the actual mass growth tracked the mass growth inferred at a constant number density to within $40\%$ over a redshift range of $z$=3 to $z$=0. They note, however, that they did not reproduce the observed mass evolution of galaxies. This was attributed to the fact that the sub-grid physics (star formation, feedback, mergers) in the semi-analytic models utilized for their project might not be accurately capturing the physics of the observed galaxies.
@Papovich.etal:11 extended this technique to track the growth of observed galaxies at much higher redshift, from $z$=8 to $z$=3, concluding that galaxies exhibit a SFH that *increases* with time (contrary to previous assumptions of decaying SFHs, see also @Reddy:12 [@Finlator.etal:11; @Jaacks.etal:12b]). This result has critical implications for the estimates of physical properties of observed galaxies. To justify the use of this method at high-redshift, the authors used the DM halo catalogs from the Millennium simulations to track halo growth, making the assumption that halo growth directly tracks galaxy growth (i.e., $n_{\rm gal}(>L_{\rm uv})\approx n_{\rm gal}(>M_{s})\approx n_{\rm gal}(>M_{\rm vir})$). While they found that halo mass growth at a constant number density of $n=2\times10^{-4}\ {{\rm Mpc}^{-3}}$ was indeed smooth over the same time period, there was no consideration for the physics of formation and evolution of the galaxies themselves. A more recent numerical test at high-$z$ was performed by [@Behroozi:13] using the Bolshoi DM halo catalog [@Klypin:11] along with the merger tree and an abundance matching technique which assigns galaxies to halos based on their relative number densities. This study found that when comparing stellar mass evolution across redshift one must account for the fact the the number density of a given population is not constant to avoid mass estimate errors of $\sim0.3$ dex. While the technique used by @Behroozi:13 did extend to tracking descendants of $z$=6 galaxies, it did not explicitly include the details of fundamental physics involved in galaxy formation and evolution.
A contemporaneous work by @Torrey:15 utilized the Illustris [@Vogelsberger:14] hydrodynamical simulations to test the constant number density evolution tracking method at $0\leq z \leq 3$, and found that galaxies do not evolve at constant number densities due to the effects of merger events and scatter in individual galaxies growth rates. We will show that these results are very consistent to this work even though focused on different redshift epochs.
In this work we will use cosmological smoothed particle hydrodynamical code [GADGET-3]{}, where galaxies form in-situ, to test the technique of using constant cumulative number density as a tracer for tracking galaxy evolution at high-redshift (i.e. $3\leq z \leq 7$). Motivated by recent tests of this technique which rely the combination of dark-matter-only simulations with either abundance matching or semi-analytic models, we intend to capture the realistic mass assembly histories of galaxies in simulations with sophisticated baryonic physics. This work will be organized as follows: In Section \[sec:sim\] we will detail the simulations used in this exercise; Section \[sec:methods\] discusses the methods used for identifying and tracking galaxies in our simulations; Section \[sec:results\] presents our results; In Section \[sec:disc\] we present additional discussion, and we summarize in Section \[sec:sum\].
Simulations {#sec:sim}
===========
[ccccc]{} Box size & $N_{p}$ & $m_{DM}$ & $m_{\rm gas}$ & $\epsilon$\
(${h^{-1} {\rm Mpc}}$) & (DM, Gas) & ($h^{-1} {M_\odot}$) & ($h^{-1} {M_\odot}$) & (${h^{-1} {\rm kpc}}$)\
\
\[-1.5ex\] $100.0$ & $2 {\times} 800^{3}$ & $1.17 {\times} 10^{8}$ & $2.38 {\times} 10^{7}$ & $5.00$\
\[+1.0ex\]\
Simulation parameters used in this paper. The parameter $N_p$ is the number of gas and dark matter particles; $m_{\rm DM}$ and $m_{\rm gas}$ are the particle masses of dark matter and gas; $\epsilon$ is the comoving gravitational softening length. \[tbl:Sim\]
We use a modified version of the smoothed particle hydrodynamics (SPH) code GADGET-3 [originally described in @Springel:05]. Our code includes radiative cooling by H, He, and metals [@Choi:09], heating by a uniform UV background [UVB; @Faucher.etal:09], the UVB self-shielding effect [@Nagamine.etal:10], the initial power spectrum of , supernovae feedback, a sub-resolution multiphase ISM model [@Springel:03], Multicomponent Variable Velocity (MVV) wind model [@Choi:11a] and ${\rm H_2}$ regulated star formation [@Thompson:14]. Our current simulations do not include AGN feedback which is expected to contribute significantly at $z<3$ in massive galaxies. See @Thompson:14 and @Jaacks:13 for detailed comparisons of our simulations to both high and low redshift observations. In particular we find excellent agreement with both the observed $z$=4-7 UV luminosity functions in both faint-end slope ($\alpha$) and characteristic brightness ($M_{UV}^*$) found in @Finkelstein:14 and @Bouwens:15a, as well as the number densities ($\phi$) from the observed stellar mass functions in @Song:15 over the mass range studied here.
[cccccc]{} ID &$n$ (center) & $\log n$ & $\log M_s$ at $z$=7 & $\log M_s$ at $z$=3 & total \#\
& \[Mpc$^{-3}$\] & & \[${M_\odot}$\] & \[${M_\odot}$\] &\
\
\[-1.5ex\]
n1 & $2.00\times10^{-4}$ & $-3.70\pm0.20$ & $8.70^{+0.22}_{-0.23}$ &$10.38^{+0.10}_{-0.10}$ & $538$\
n2 & $8.10\times10^{-5}$ & $-4.10\pm0.20$ & $9.10^{+0.15}_{-0.18}$ &$10.54^{+0.08}_{-0.07}$ & $201$\
n3 & $2.87\times10^{-5}$ & $-4.50\pm0.20$ & $9.40^{+0.15}_{-0.13}$ &$10.69^{+0.06}_{-0.05}$ & $75$\
\[tbl:data\]
Our simulation is set up with $800^3$ particles for both gas and dark matter in a comoving box with sides of $100h^{-1}$ Mpc. We will refer to this run as N800L100 and complete simulation parameters are summarized in Table \[tbl:Sim\]. We use the 2LPTic [@Scoccimarro.etal:98] code which utilizes second-order Lagrangian perturbation theory for generating our initial conditions at $z$=99. The adopted cosmological parameters are consistent with the WMAP7 data [@Komatsu:09]: ${\Omega_{\rm m}}=0.26$, ${\Omega_{\Lambda}}=0.74$, ${\Omega_{\rm b}}=0.044$, $h=0.72$, $\sigma_{8}=0.80$, $n_{s} =0.96$.
Snapshots were recorded every $20\ {\rm Myr}$ starting at $z$=10 allowing us to more accurately track the growth of each galaxy, resulting in $72$ snapshots from $z$=7 to 3. Galaxies were identified in each snapshot using a variant of the SUBFIND algorithm [@Springel:01] which groups particles based on baryonic density peaks. A minimum of 32 particles (gas$+$stars) is required to be considered for grouping. However we require a minimum of $100$ baryonic particles and at least 20 star particles for the galaxies to be considered within our mass resolution.
It should be noted that our simulations assumed a Chabrier initial mass function (IMF) [@Chabrier:03] and when appropriate, data such as stellar masses which assumed a different IMF (e.g., @Salpeter:55) are converted for consistency using $M_{IMF}=M_{\rm Salpeter}/f_{IMF}$ where $f_{IMF}=1.6$ for Chabrier. [@Nagamine.etal:06; @Raue.etal:12].
Methods {#sec:methods}
=======
Galaxy selection {#subsec:select}
----------------
![Cumulative stellar mass function for $z=3, 4, 5, 6\ {\rm and}\ 7$ (solid blue lines, top to bottom). The horizontal solid, dashed and dashed-dot lines represent constant number density cuts n1, n2, n3 (see Table \[tbl:data\] for details), respectively. For each number density studied, the total bin width is shown by the horizontal orange, green and blue shaded areas. The gray shaded region denotes the approximate mass resolution for our simulation run, which corresponds to $\sim 100$ baryon particles. The mass value associated with the intersection of the CSMF and the constant number density cuts will be referred to as the [*inferred*]{} mass growth.[]{data-label="fig:cmf"}](fig1.pdf){width="1.10\columnwidth"}
Galaxies are selected based on symmetric number density bins as determined by the cumulative stellar mass function (CSMF, Figure \[fig:cmf\]) from our simulation at $z$=7. For our fiducial number density bin to select galaxies, we use $n=2.0\times10^{-4}$ \[Mpc$^{-3}$\], a choice which is motivated by its use in previous studies [e.g., @vandokkum:10; @Papovich.etal:11; @leja:13]. This number density can then be translated via the CSMF to a corresponding stellar mass of $M_s\approx 10^{8.7} {M_\odot}$ at $z$=7. We also explore lower number densities of $n=8.10\times10^{-5}$ \[Mpc$^{-3}$\] and $n=2.87\times10^{-5}$ \[Mpc$^{-3}$\] which have corresponding $z =$ 7 masses of $M_s\approx 10^{9.1} {M_\odot}$ and $M_s\approx 10^{9.4} {M_\odot}$, respectively. Throughout this work these number density bins will be referred to as n1, n2, and n3, respectively. Full details of each number density bin studied can be found in Table \[tbl:data\]. The choices of the values for n2 and n3 are motived by covering a wide range of masses within our simulation mass resolution, with little to no overlap between populations. We do not explore higher number density (lower mass) values due to the restrictions set by our simulation’s resolution (see gray shading in Figure \[fig:cmf\]). We also do not explore values lower than n3, as such a bin would have $<50$ tracked galaxies, leading to poor number statistics.
It should be noted that the symmetric number density bin leads to a slightly asymmetric mass bin (i.e. more low mass galaxies), depending on the shape of the CSMF at the selection redshift. However this choice allows for a constant number of galaxies at each redshift for a given number density, whereas a symmetric mass bin would lead to varying galaxy counts due to the changing shape of the CSMF.
Galaxy tracking {#sec:tracking}
---------------
The tracking of single galaxies across multiple snapshots is achieved through particle ID matching and the “most massive" descendant/progenitor algorithm. The galaxy at each subsequent snapshot which contains the highest number of matching particles from the original galaxy is considered to be its descendant/progenitor.
One downside to the “most massive” descendant/progenitor algorithm is that there are scenarios when the procedure is not completely reversible. For example, in merger events the progenitor path will choose the most massive of the two galaxies whereas the descendant path may have originated via the lower mass galaxy. To test the impact of this effect on our results we conducted a test with a random sample population of galaxies selected at $z$=7. This population was tracked to $z$=3 and the final $GIDs$ were used as a starting point for a progenitor track back to $z$=7. We found that $\sim10\%$ of the progenitor population differed from the original $z$=7 population. However, this difference only led to a median mass differential of $\sim0.02$ dex. Therefore, we conclude that the partial non-reversibility of the “most massive” algorithm utilized in this work has a negligible impact on our results.
{width="1.4\columnwidth"}
![[*Actual*]{} mass growth for progenitors, compared to the [*inferred*]{} mass growth from the CSMF. All symbols, lines and shading represent similar quantities as in Figure \[fig:m\_evo\_d\], though here for progenitors of $z =$ 3 galaxies. While again this technique appears to work on average, both the bias and the scatter of the constant number density tracking technique are larger when tracking progenitors than when tracking decendants.[]{data-label="fig:m_evo_p"}](fig3.pdf){width="1.1\columnwidth"}
While the algorithm for tracking galaxies in the simulation is simple, in practice this is a non-trivial exercise. All group-finders introduce some amount of uncertainty when associating particles with a galaxy, especially during merger events. Particles which were included in one snapshot can be excluded in the subsequent one, primarily those which are located more near the edges of the identified system. This uncertainty manifests itself as “noise” in the mass growth of each galaxy (see jagged gray lines in Figure. \[fig:m\_evo\_d\]). While this “noise” is aesthetically unpleasing, it does not affect the overall median growth trend.
Actual versus inferred mass growth
----------------------------------
Throughout this work we will refer to the [*actual*]{} mass growth and the [*inferred*]{} mass growth of the studied galaxy population. The [*inferred*]{} growth refers to the mass assembly of the ensemble galaxy population as derived from the CSMF by taking the mass associated with a constant number density cut at each redshift (e.g. the orange line intersection with CSMF in Figure \[fig:cmf\]). The [*actual*]{} mass growth is the mass assembly unique to each individual galaxy (gray lines in Figure. \[fig:m\_evo\_d\]) determined at each snapshot as described in Sec. \[sec:tracking\]. In each case the [*actual*]{} mass growth will be represented by the median of the distribution with the $1\sigma$ scatter represented by shading (e.g. orange circles, orange shade Figure. \[fig:m\_evo\_d\]).
Results {#sec:results}
=======
Descendants
-----------
{width="1.00\columnwidth"} {width="1.00\columnwidth"}
In Figure \[fig:m\_evo\_d\] we compare the inferred mass growth to the median of the actual mass growth for each galaxy in the n1, n2, and n3 bins. This comparison illustrates that the [*inferred*]{} growth tracks the [*actual*]{} mass growth on average, though there appears a systematic bias where the inferred mass growth is $\sim0.20$ dex higher (within a factor of $<2$ in mass) than the actual mass growth. This demonstrates that, on average, using constant cumulative number density to track the descendants of a population of galaxies through cosmic time can be an effective method of measuring stellar mass growth. This result is consistent with studies using the same techniques at lower redshifts [@vandokkum:10; @leja:13] and at high-$z$ [@Papovich.etal:11; @Behroozi:13]. We caution however that this is an [*on average*]{} result which has a large scatter ($\sim \pm0.30$ dex) which we will discuss in more detail in Section \[subsec:scatter\] and a systematic offset which suggest that evolution of the population must be considered (see Section \[sec:evo\]).
In Figure \[fig:m\_evo\_d\] we also compare to stellar mass growth estimates obtained through observations by @Salmon:15 and @Papovich.etal:11. This comparison demonstrates that our simulations, on average, are reproducing the observed growth of galaxies selected from a bin similar to our n1. As @Salmon:15 results were obtained using the evolving cumulative number density technique, it also serves as a good check that our simulations are reproducing the observed galaxy properties and evolution well.
Progenitors {#sec:progen}
-----------
We also use our simulations to test the effectiveness of the constant cumulative number density technique to track the *progenitors* of a population of galaxies selected at $z$=3. For the purposes of this study we are defining a galaxy progenitor to be it’s most massive progenitor. We acknowledge that in a hierarchical galaxy formation scenario each galaxy is the aggregate of many progenitors of varied masses. However, from an observational perspective, it is impossible to link multiple progenitors to a single descendant. Therefore we feel that our definition more closely approximates what can be observed. Figure \[fig:m\_evo\_p\] shows the mass assembly history of $z =$ 3 galaxy populations in the n1, n2, and n3 bins. As with the descendants, the constant cumulative number density technique, [*on average*]{}, reproduces the mass assembly history of the population of galaxies when compared to the [*inferred*]{} assembly, although with a larger bias of $0.30$ dex (factor $\sim2$ in mass).
However, when using this technique to track progenitors the scatter in the individual galaxy history increases substantially to $\sim \pm0.70$ dex at $z$=7 (Figure \[fig:m\_evo\_p\]), and is much higher than was seen when tracking descendants ($\sim 0.30$ dex). Further, we find that $\sim30\%$ of the galaxies in the original number density bin at $z$=3 do not exist at $z$=7, forming at various points between $z$=3 and $z$=7. This population of “late-forming” galaxies substantially increases the scatter in the [*actual*]{} assembly history and thus increases the offset between the [*actual*]{} and [*inferred*]{} mass growth. It should be noted that we drop the late-forming galaxies from the calculation of average mass growth prior to their formation redshift to avoid including objects with zero stellar mass. We include more discussion regarding the impact of this “late-forming” galaxy population in Section \[subsec:late\].
Evolving number density {#sec:evo}
-----------------------
The systematic offset between the [*actual*]{} and [*inferred*]{} mass growth suggests that the number density of the original population is evolving with redshift. This evolution can be seen in Figure \[fig:dens\_evo\] which shows the evolution of the number density as a function of redshift for both descendants (left) and progenitors (right), for each of our initial number density bins. The number density for each individual galaxy is obtained at each snapshot by relating its stellar mass to a number density via the CSMF. All populations show a trend of increasing number density as a function of cosmic time (with decreasing redshift for descendants, and with increasing redshift for progenitors). This demonstrates that the average [*actual*]{} mass growth of galaxies from each number density bin are growing slower than would be inferred from the CSMF (see offset between [*inferred*]{} and [*actual*]{} stellar mass growth in Figures \[fig:m\_evo\_d\] & \[fig:m\_evo\_p\]).
This trend is highly consistent with findings from @Behroozi:13 who utilized the Bolshoi [@Klypin:11] dark matter simulations and abundance matching (see Section \[subsec:abund\] for further details). In Figure \[fig:dens\_evo\] we compare our linear best fit (solid lines) to the results from the public code provided by @Behroozi:13; dashed lines. Our work shows excellent agreement (within $\sim 0.10$ dex) with the evolution generated by the @Behroozi:13 code over the studied redshift range.
Thus, in order to track galaxies more accurately through cosmic time one must utilize an [*evolving*]{} cumulative number density. By taking the average of the least square fit to the evolutionary trend for each population (see least square fits in Figure \[fig:dens\_evo\]) of descendants of $z$=7 galaxies ($10^{8.48}\leq M_s/{M_\odot}\leq 10^{9.55}$) we find, $$\label{eq:one}
\log n_f = \log n_i-0.10 \Delta z$$ Similarly the number density of $z$=3 ($10^{10.39}\leq M_s/{M_\odot}\leq 10^{10.75}$) progenitors can be traced utilizing, $$\label{eq:two}
\log n_f = \log n_i+0.12 \Delta z$$ We caution that these relations are only valid in the mass range and redshift range studied here ($3\leq z \leq7$). As pointed out in @Behroozi:13, the evolution changes at $z\leq3$, therefore equations \[eq:one\] & \[eq:two\] should not be extrapolated to lower redshifts. Note that the mass ranges referenced above are derived from the overall number density range (n1, n2, n3) at either $z=$7 for descendants or $z=$3 for progenitors (see Table \[tbl:data\]).
Completeness {#subsec:comp}
------------
![Completeness fraction, defined to be the fraction of tracked galaxies at each redshift which fall within the desired number density bin, keeping the bin width constant. The solid orange line represents the completeness fraction for the n1 bin. We also compare to @Papovich.etal:11 (gray circles), who define the completeness as the fraction of galaxies with a number density less than or equal to the value of the selection bin. Our direct comparison to the published data is the dashed orange line. The cyan solid and dashed lines represent the completeness fraction after applying the evolving cumulative number density, both using our definition of completeness, and that of Papovich et al. We find that even after the correction for the evolving number density, only $23\%$ of the $z=$7 population is recovered at $z=$3.[]{data-label="fig:comp"}](fig5.pdf){width="1.0\columnwidth"}
One method which has been used to quantify how well the constant cumulative number density method is tracking the evolution of the galaxy population is to examine the completeness fraction at each redshift, keeping the bin width constant. Here we define the completeness fraction to be the fraction of tracked simulated galaxies which fall within the desired number density bin at a given redshift. In Figure \[fig:comp\] we present the results of this study, where we find that only $\sim 13\%$ of $z$=7 descendants fall within the original number density bins at $z=3$. For clarity, we only show results for the n1 bin, however, the results for the n2 and n3 bins are quantitatively similar. We do not present the completeness fraction for progenitors for reasons which we discuss in Section \[subsec:scatter\].
We also compare our results to those found in @Papovich.etal:11. To make this comparison we need to modify our definition of completeness to match that used in Papovich et al. They define completeness as the fraction of galaxies with a number density of at most the value of the selection bin ($n(>M_s)$). Our direct comparison to this definition can be seen by the dashed orange line. This definition results in a completeness fraction of $\sim 35\%$, much lower than the @Papovich.etal:11 result of $\sim 55\%$. To obtain a similar result to the @Papovich.etal:11 study we would need to expand our target mass bin at $z$=3 to $\pm0.29$ dex.
Repeating both of the completeness calculations above, now applying the evolving cumulative number density leads to a completeness fraction of $\sim23\%$ as defined in this work and $\sim58\%$ for the @Papovich.etal:11 definition. This makes our completeness fraction nearly identical to the @Papovich.etal:11 result using their less restrictive definition. However, we see a modest increase of only $10\%$ at $z$=3 after applying the evolving cumulative number density when using our completeness definition. This discrepancy is likely due to the larger scatter between halo mass, galaxy mass and luminosity found in our simulations. The study by @Papovich.etal:11 did not include any of these sources of scatter in their estimates.
The results from the completeness fraction study are somewhat misleading as the width of the $z$=3 mass bin associated with the selected number density is smaller than the original $z$=7 mass bin by a factor of $\sim 2$ (see Table \[tbl:data\]). The reduced mass bin width is the result of the steepening of the CSMF over the redshift range studied, and it is the primary reason why we only see a modest completeness fraction increase of $10\%$ at $z$=3 when applying the evolving cumulative number density. If the mass bin width is fixed at the $z=$7 value of $\sim\pm 0.20$ dex (see Table \[tbl:data\]), the completeness fraction at $z=$3 rises to $\sim55\%$.
Scatter {#subsec:scatter}
-------
{width="1.5\columnwidth"}
![The same lines and symbols as Figure \[fig:dscatter\], only here for progenitors of $z =$ 3 galaxies tracked to $z =$ 7. The red histogram bar in the progenitor panel represents the population “late-forming” galaxies found in the progenitor study. Progenitors of $z=$3 galaxies have a much wider distribution in both mass and number density that seen from descendants of $z=$7 (see Figure \[fig:dscatter\]).[]{data-label="fig:pscatter"}](fig7.pdf){width="1.1\columnwidth"}
As demonstrated by the low completeness fraction, the majority of galaxies selected at $z$=7 are not falling within either the evolved or constant number density target bin at $z$=3. It is therefore useful to constrain where the majority of these descendants fall on the CSMF. Figures \[fig:dscatter\] and \[fig:pscatter\] shows the median number density evolution of simulated galaxies in our n1 bin plotted with the CSMF at each redshift for descendants and progenitors, respectively. These figures synthesize much of the data provided in this work as it also includes information regarding the mass and number density distribution (blue/orange histograms) for both descendants and progenitors.
For the n1 bin descendants, we find that $\sim 11\%$ of galaxies fall in the n1 bin at $z$=3, as discussed in Section \[subsec:comp\], and the remaining galaxies are distributed either at higher number densities ($\sim 67\%$), or at lower number densities ($\sim 22\%$) than the original number density bin. The overall scatter in stellar mass at $z$=3 is $\pm0.32$ dex, which corresponds to a number density scatter which has a total width of $\sim 1.2$ dex, given the $z =$ 3 CSMF. For progenitors of the n1 bin, we find that $\sim 13\%$ fall within the target $z =$ 7 number density bin with $\sim67\%$ at higher number densities, and $\sim21\%$ at lower number densities. This distribution leads to a stellar mass scatter of $\pm0.75$ dex at $z$=3, and a number density scatter with total width of $\sim1.1$ dex. It should be noted that the “late-forming” galaxies discussed in Section \[sec:progen\] are included in the fraction of galaxies which fall above the n1 bin. However, they are removed from the sample at their formation redshift. Therefore, they do not contribute to the median scatter (gray shaded area in Figure \[fig:pscatter\]) at $z=$3. The n2 and n3 density bins have quantitatively similar results.
It is interesting to consider the growth histories of galaxies which started in the selected number density bin at $z$=7, and end up outside the bin at $z$=3. We performed a test to determine the average growth rates of these populations of galaxies, calculating the average stellar mass at each time step of galaxies outside the bin (separately for those with both higher and lower number densities). We find that these *average* growth histories show that these galaxies initially grow too quickly or too slowly (moving to lower or higher number densities, respectively), jumping out of the selected number density bin, but after that, they grow at a relatively constant rate, similar to those galaxies inside the selected bin. Therefore, there is no systematic difference in the growth histories of these galaxies; rather, after they initially scatter out of the number density bin, they grow fairly similarly to the tracked population.
The fact that both the descendants and the progenitors have similar fractions of galaxies located in, above and below the original bin is misleading. As mentioned in Section \[subsec:comp\], the slope of the CSMF is steepening with decreasing redshift, resulting in a narrower number density bin at $z$=3 and a lower descendant completion fraction. This must also be considered when looking at progenitors as the bin is widening as we approach $z$=7, thus making it more likely that a galaxy will fall within the selection bin (i.e. higher completion fraction).
As demonstrated by the mass and number density histograms included in Figure \[fig:pscatter\], even using an evolving cumulative number density selection results in a wide scatter in both number density and mass for progenitors. We conclude that using an evolving cumulative number density will allow for the tracking of [*average*]{} stellar mass for progenitors, but that this method is an insufficient tracer of individual galaxy descendant/progenitor evolution in our simulations, as nearly any bin on the $z$=3 CSMF can be populated with galaxies that originated from nearly any point on the $z$=7 CSMF. We show further evidence of this scatter and its impact in Section \[subsec:disc\_scatter\].
Discussion {#sec:disc}
==========
Scatter {#subsec:disc_scatter}
-------
![The mass distribution of $z =$ 7 simulated galaxies, which were tracked as progenitors of $z =$ 3 galaxies, which were themselves tracked as descendants of $z =$ 7 galaxies in the n1 number density bin (effectively a “round-trip” tracking exercise). Here we show the stellar mass distribution for *all* galaxies selected in a mass bin at $z =$ 3 which contained 50% of the originally-tracked $z =$ 7 descendants (see Section \[subsec:comp\] for details). The gray shading indicates the original n1 bin width and the red bar represents the late-forming population found during progenitor tracking (see Section \[sec:progen\]). This “round trip” tracking exercise demonstrates that a given point on the $z$=3 CSMF can be populated by galaxies which have origins at nearly any point on the $z$=7 CSMF.[]{data-label="fig:50"}](fig8.pdf){width="1.00\columnwidth"}
To further explore the implications of the large scatter in descendants, an exercise was conducted to determine the mass bin width required at $z$=3 to obtain a completeness fraction, as defined in this work, of $50\%$. It was found that an increase of a factor of $\sim 3$ from $\pm0.10$ dex in stellar mass to $\pm0.29$ dex in stellar mass at $z$=3 is required to recover $50\%$ of the original $z$=7 selected population (see Table \[tbl:data\]). While a mass bin with a width of $\pm0.29$ dex may not seem extreme, as it is comparable to the observed mass error estimates at $z=3$, it will contain thousands of galaxies that were not part of the original $z$=7 density bin, which may have very different mass assembly histories. This can be see in Figure \[fig:50\] where the stellar mass distribution for a sample of $2175$ galaxies were selected from the aforementioned $50\%$ bin. The progenitors of this sample were then tracked back to $z$=7 for comparison to the original $z$=7 population. The blue histogram represents the $z =$ 7 stellar mass distribution of the $2175$ $z$=3 progenitors, and the gray shaded region indicates the mass width of the original n1 number density bin. This exercise effectively demonstrates that even a moderate widening of the target mass bin will lead to substantial contamination from a diverse population of galaxies. This exercise also shows that a large scatter makes the application of either the constant cumulative number density or evolving cumulative number density technique limited to average population masses only. The diverse mass assembly histories of individual galaxies can make associating other properties such as age, metallicity, star-formation rate, etc., untenable.
Late-forming galaxies {#subsec:late}
---------------------
The $\sim30\%$ of $z$=3 progenitor galaxies found in our simulation to form after $z$=7 have a significant impact on the results of the progenitor portion of this study. In particular, the scatter of the stellar mass and number density evolution is increased by a factor of $\sim2$. The high fraction, and significant impact of the “late-forming” population dictate that further discussion is warranted. Specifically, there is the possibility that these galaxies are artifacts of our simulations due to resolution limitations. The large volume of our simulation box is chosen so that we can best represent the dynamic range of observed high redshift galaxies. The consequence of this design is that we are not resolving low-mass halos with $M_h \lesssim 10^9 {M_\odot}$ at $z>$7. Therefore, the possibility exists that a theoretical simulation box with unlimited mass resolution would have resolved these “late-forming” galaxies at an earlier redshift.
In a $\Lambda$CDM Universe with a hierarchical structure formation scenario, it is possible that these galaxies would have then been merged into a higher mass galaxy by $z$=7, thus reducing the scatter. It is also possible that no mergers took place which would leave the galaxy in a lower mass bin (i.e. much higher on the CSMF). In this scenario the scatter would remain largely unchanged. To properly study this issue, a resolution test would be needed, which requires a simulation with a substantially higher particle count. Unfortunately, such a hydrodynamic simulation is beyond the scope and resource of this study.
There is also the possibility that the “late-forming” galaxies are very “real”, and representative of early galaxy formation. A simple calculation, exploring the growth rate required for a galaxy at $z$=5 with $M_s\approx10^{7.1}{M_\odot}$ to reach the target stellar mass of $M_s\approx10^{10.4}{M_\odot}$ by $z$=3, demonstrates that such a galaxy would require a star formation rate (SFR) of $\sim25\ {M_\odot}\ yr^{-1}$ over the $\sim 1$ Gyr. This SFR is reasonable, given that a typical observed $L_*$ galaxy has an SFR$\approx50\ {M_\odot}\ yr^{-1}$ at $z$=5 @Finkelstein:14 . This is without consideration for mergers, which could increase the stellar mass much more rapidly. Therefore, there is no physical motivation for us to definitively conclude that the “late-forming” galaxies are mere resolution artifacts.
We investigated the photometric properties of the putative “late-forming” galaxies to determine if it would be possible to detect $z$=3 observational signatures. More specifically, we examined the D4000 spectral indicator, which measures the relative strength of filters placed around the $4000\text{\AA}$ spectral features associated with Calcium H and K lines [@Bruzual:83]. The strength of this break is correlated with age, as younger, high SFR galaxies will produce an increase of UV photons at wavelengths less than $4000\text{\AA}$. The spectra of these galaxies will also contain less observable metal-line absorption, which will result in a decreased D4000 ratio. Utilizing the BC03 [@BC03] population synthesis models to generate spectra for each of our simulated galaxies, we found that there is no significant difference in the D4000 break for “late-forming” galaxies as compared to the remaining studied population. It is likely that this is the result of a combination of merger events and rapid star formation that leads to the effective mixing of older stellar populations with the newly formed populations making the $4000\text{\AA}$ an unreliable age indicator for our galaxy population.
On the origin of the evolving number density {#subsec:evo}
--------------------------------------------
When interpreting the results of this work, and other similar studies, it is critical to remember that number density is not a physical property of a galaxy (i.e. unlike stellar mass, SFR, metallicity, etc.). The number density of any individual galaxy is dependent directly on the distribution function from which it is taken, in this case the cumulative stellar mass function. As seen in Figure \[fig:cmf\], the CSMF is evolving with decreasing redshift towards more galaxies at higher masses, but also toward a stronger high-mass exponential cut-off. There are two primary forces driving this evolution of the CSMF over the studied redshift range: First, the stellar masses of existing galaxies are growing, which evolves the CSMF to toward higher masses. Second, new galaxies are forming and being included which, in a “bottom-up” hierarchical structure formation universe, will place them at higher number densities, effectively acting to steepen the exponential cutoff of the CSMF. We caution that this steepening is not related to the faint-end slope of the stellar mass function as our dynamic range does not include faint-end mass galaxies.
The steepening of the CSMF cutoff directly results in the evolution of the cumulative number density in our simulations. This can be clearly demonstrated by examining the rise in cumulative number density for a given stellar mass bin as the exponential cutoff becomes more pronounced (i.e. towards vertical). Therefore, a population of galaxies can have smoothly rising mass assembly histories, as seen in Figure \[fig:m\_evo\_d\], with no break in rank order, and still exhibit an evolving number density. This result is contradictory to the fundamental assumptions that are made in the constant cumulative number density technique, which has been shown to work at lower redshift [$0\leq z\leq3$; @vandokkum:10; @leja:13]. We do not however feel that our results invalidate the lower redshift results as the processes driving galaxy evolution are different during that more recent epoch (i.e. driven by stellar mass growth and not new galaxy formation, with an increased impact of feedback on massive galaxies).
Based on the above evolution argument, one would expect that the progenitor number density evolution would then trend towards lower number densities at higher redshifts. We do not see this trend. In fact we find the opposite, with progenitors evolving to higher number densities (Figure \[fig:pscatter\], bottom). This can be explained by the impact of the large “late-forming” galaxy population acting to lower the median stellar mass growth (Figure \[fig:m\_evo\_p\]), thus raising the cumulative number density.
It must be noted that the steepening of the CSMF in our simulations could also be the result of our mass resolution limitations as discussed in Section \[subsec:late\]. Low mass galaxies which are unresolved would fall at higher number densities, and could therefore steepen our $z=7$ CSMF, potentially eliminating the evolution seen in this work. Cosmological volume hydrodynamic simulations with sufficient dynamic range to test this effect are beyond the scope of this work based on the computational resources required.
Implications of scatter for abundance matching {#subsec:abund}
----------------------------------------------
The agreement with the @Behroozi:13 results is rather remarkable given the differing approaches taken to arrive at these conclusions. In the @Behroozi:13 work an abundance matching technique is employed. This technique converts the cumulative number density corresponding to a galaxy’s stellar mass at a given initial redshift to a dark matter halo mass with the same cumulative number density from the dark matter-only Bolshoi simulation halo catalog. The progenitors/descendants of these halos are then tracked within simulation using a similar ‘most-massive’ algorithm [@Behroozi:13c]. At the terminal redshift the mass of each halo is then converted back to a cumulative number density using the dark matter halo cumulative mass function [@Behroozi:13b].
At the core of the abundance matching technique is the fundamental assumption that halo mass accretion, whether it be through mergers or via the cosmic web, directly traces the galaxy stellar mass growth hosted within. This is a reasonable assumption given that we know that the baryonic matter in the Universe directly traces the underlying dark matter distribution [@Blanton:99]. It does not however account for the complicated physics of how this gas accretes onto galaxies and ultimately is converted into stars. This absence of fundamental physics is the primary motivation for this work.
The utilization of cosmological SPH simulations is the next logical step in testing these galaxy-tracking techniques, as they include both the underlying dark matter structure and the complicated hydrodynamic physics associated with the gas within (i.e. heating, cooling, chemistry). These simulations are also able to convert gas into stars via physically motivated sub-grid physics as well as account for the impact of star formation (i.e. supernova and stellar winds) on further star formation. This allows for galaxies to form in-situ with their respective dark matter halos, while computing the gas infall rate from cosmic filaments in a cosmological setting correctly.. Therefore we are able to directly track galaxy stellar mass growth without the need to make any assumptions regarding the relationship between the galaxy and the halo it resides within. The drawback to utilizing these types of high resolution SPH simulations is that they are computationally very expensive. This limits the dynamic mass range and redshift range we are able to study when compared to dark matter only simulations (i.e. Bolshoi, Millennium), which resolve lower mass structures. To probe this topic further, a combination of theoretical techniques is likely warranted.
Summary {#sec:sum}
=======
Using the cosmological smoothed particle hydrodynamical code [GADGET-3]{} we make a realistic assessment of the validity of the technique of using constant cumulative number density as a tracer of galaxy evolution at $3\leq z \leq7$. Our major conclusions are as follows:
- We find that, when utilizing the constant cumulative number density technique to track galaxies at $3\leq z \leq7$, an average stellar mass can be determined to within a factor of $<2$ for descendants and $\sim2$ for progenitors (Figure \[fig:m\_evo\_d\] & \[fig:m\_evo\_p\]).
- The systematic offset between the [*inferred*]{} mass growth and the [*actual*]{} mass growth suggest that the number density of both progenitors and descendants is evolving with redshift. This evolving cumulative number density is in very good agreement with @Behroozi:13 who find a similar evolution over the same redshift range (Figure \[fig:dens\_evo\]).
- On average, the cumulative number density for the stellar mass ranges studied in this work ($10^{8.48}\leq M_s/{M_\odot}\leq 10^{9.55}$) can be found using, $$\log n_f = \log n_i-0.10 \Delta z$$ for descendants, and, $$\log n_f = \log n_i+0.12 \Delta z$$ for progenitors.
- While the evolving cumulative number density is able to track the [*on average*]{} evolution of galaxies at $3\leq z \leq7$, we find that there is a large scatter of $\sim \pm 0.30$ dex for descendants at $z=$ 3, and $\sim \pm 0.70$ for progenitors at $z=$7 (Figures \[fig:m\_evo\_d\], \[fig:m\_evo\_p\], \[fig:dscatter\] & \[fig:pscatter\] ). Individual galaxies demonstrate diverse mass assembly histories due to the physical processes of gas accretion, quenching, mergers and star formation.
- When evaluating the completeness fraction, we find that only $\sim23\%$ of descendants fall within the original number density bin by $z=3$ (Figure \[fig:comp\]). This low fraction is due to the narrowing of the target mass bin which is caused by the steepening of the CSMF.
- Increasing the number density bin width at $z=$3 to improve the completeness fraction of $z=$7 descendants to $50\%$ leads the contamination of the original sample by thousands of galaxies which have origins outside of the targeted number density bin. Thus, significantly reducing the reliability of the evolving cumulative number density method.
The technique of utilizing an evolving cut of cumulative number density is a powerful tool in tracking the [*average*]{} mass growth of galaxies at $3\leq z \leq7$. This work demonstrates that galaxies in a given number density bin will have very diverse mass assembly histories even if they are of similar stellar mass. This diversity will reveal itself in the scatter of other physical properties of individual galaxies (i.e. metallicity, age, SFR, star formation history), thus caution should be used when applying this technique to study physical properties other than stellar mass.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank Volker Bromm, Peter Behroozi and Casey Papovich for stimulating conversations. We are grateful to V. Springel for allowing us to use the original version of [GADGET-3]{} code, on which the @Thompson:14 [@Jaacks:13] simulations are based. JDJ and SLF would like to acknowledge support from the University of Texas at Austin College of Natural Sciences. The simulations used in this paper were run and analyzed using pyGadgetReader [@pygr] on ‘Blue Waters’ at the National Center for Supercomputing Applications (NCSA). This research is part of the Blue Waters sustained petascale computing project, which is supported by the National Science Foundation (awards OCI-0725070 and ACI-1238993) and the state of Illinois. Blue Waters is a joint effort of the University of Illinois at Urbana-Champaign and NCSA. K.N. acknowledges the partial support by JSPS KAKENHI Grant Number 26247022.
[47]{} natexlab\#1[\#1]{}
, P. S., [Marchesini]{}, D., [Wechsler]{}, R. H., [Muzzin]{}, A., [Papovich]{}, C., & [Stefanon]{}, M. 2013, , 777, L10
, P. S., [Wechsler]{}, R. H., & [Conroy]{}, C. 2013, , 770, 57
, P. S., [Wechsler]{}, R. H., [Wu]{}, H.-Y., [Busha]{}, M. T., [Klypin]{}, A. A., & [Primack]{}, J. R. 2013, , 763, 18
, M., [Cen]{}, R., [Ostriker]{}, J. P., & [Strauss]{}, M. A. 1999, , 522, 590
, R. J., [Bradley]{}, L., [Zitrin]{}, A., [Coe]{}, D., [Franx]{}, M., [Zheng]{}, W., [Smit]{}, R., & [Host]{}, O. 2014, , 795, 126
, R. J., [Illingworth]{}, G. D., [Oesch]{}, P. A., [Labb[é]{}]{}, I., [Trenti]{}, M., [van Dokkum]{}, P., [Franx]{}, M., [Stiavelli]{}, M., [Carollo]{}, C. M., [Magee]{}, D., & [Gonzalez]{}, V. 2011, , 737, 90
, R. J., [Illingworth]{}, G. D., [Oesch]{}, P. A., [Trenti]{}, M., [Labb[é]{}]{}, I., [Bradley]{}, L., [Carollo]{}, M., & [van Dokkum]{}, P. G. 2015, , 803, 34
, R. J., [Illingworth]{}, G. D., [Oesch]{}, P. A., [Trenti]{}, M., [Labbe]{}, I., [Franx]{}, M., [Stiavelli]{}, M., & [Carollo]{}, C. M. 2012, , 752, L5
, G. & [Charlot]{}, S. 2003, , 344, 1000
, G. 1983, , 273, 105
, G. 2003, , 115, 763
, J. & [Nagamine]{}, K. 2009, , 395, 1776
—. 2011, , 410, 2579
, D. J. & [Hu]{}, W. 1999, , 511, 5
, R. S., [McLure]{}, R. J., [Dunlop]{}, J. S., [Robertson]{}, B. E., [Ono]{}, Y., [Schenker]{}, M. A., [Koekemoer]{}, A., & [Bowler]{}, R. A. A. 2013, , 763, L7
, C.-A., [Lidz]{}, A., [Zaldarriaga]{}, M., & [Hernquist]{}, L. 2009, , 703, 1416
, S. L., [Papovich]{}, C., [Dickinson]{}, M., [Song]{}, M., [Tilvi]{}, V., [Koekemoer]{}, A. M., [Finkelstein]{}, K. D., & [Mobasher]{}, B. 2013, , 502, 524
, S. L., [Papovich]{}, C., [Ryan]{}, R. E., [Pawlik]{}, A. H., [Dickinson]{}, M., [Ferguson]{}, H. C., [Finlator]{}, K., & [Koekemoer]{}, A. M. 2012, , 758, 93
, K., [Oppenheimer]{}, B. D., & [Dav[é]{}]{}, R. 2011, , 410, 1703
, Q., [White]{}, S., [Boylan-Kolchin]{}, M., [De Lucia]{}, G., [Kauffmann]{}, G., [Lemson]{}, G., [Li]{}, C., [Springel]{}, V., & [Weinmann]{}, S. 2011, , 413, 101
, J., [Nagamine]{}, K., & [Choi]{}, J. H. 2012, , 427, 403
, J., [Thompson]{}, R., & [Nagamine]{}, K. 2013, , 766, 94
, A. A., [Trujillo-Gomez]{}, S., & [Primack]{}, J. 2011, , 740, 102
, E., [Dunkley]{}, J., [Nolta]{}, M. R., [Bennett]{}, C. L., [Gold]{}, B., [Hinshaw]{}, G., [Jarosik]{}, N., [Larson]{}, D., & others. 2009, , 180, 330
, J., [van Dokkum]{}, P., & [Franx]{}, M. 2013, , 766, 33
, K., [Choi]{}, J.-H., & [Yajima]{}, H. 2010, , 725, L219
, K., [Ostriker]{}, J. P., [Fukugita]{}, M., & [Cen]{}, R. 2006, , 653, 881
, P. A., [van Dokkum]{}, P. G., [Illingworth]{}, G. D., [Bouwens]{}, R. J., [Momcheva]{}, I., [Holden]{}, B., [Roberts-Borsani]{}, G. W., & [Smit]{}, R. 2015, , 804, L30
, C., [Finkelstein]{}, S. L., [Ferguson]{}, H. C., [Lotz]{}, J. M., & [Giavalisco]{}, M. 2011, , 412, 1123
, M. & [Meyer]{}, M. 2012, , 426, 1097
, N. A., [Pettini]{}, M., [Steidel]{}, C. C., [Shapley]{}, A. E., [Erb]{}, D. K., & [Law]{}, D. R. 2012, , 754, 25
, B., [Papovich]{}, C., [Finkelstein]{}, S. L., [Tilvi]{}, V., [Finlator]{}, K., [Behroozi]{}, P., [Dahlen]{}, T., & [Dav[é]{}]{}, R. 2015, , 799, 183
, E. E. 1955, , 121, 161
, R. 1998, , 299, 1097
, M., [Finkelstein]{}, S. L., [Ashby]{}, M. L. N., [Grazian]{}, A., [Lu]{}, Y., [Papovich]{}, C., [Salmon]{}, B., & [Somerville]{}, R. S. 2015, arXiv:1507.05636
, V. 2005, , 364, 1105
, V. & [Hernquist]{}, L. 2003, , 339, 289
, V., [White]{}, S. D. M., [Jenkins]{}, A., [Frenk]{}, C. S., [Yoshida]{}, N., [Gao]{}, L., [Navarro]{}, J., & [Thacker]{}, R. 2005, , 435, 629
, V., [White]{}, S. D. M., [Tormen]{}, G., & [Kauffmann]{}, G. 2001, , 328, 726
, D. P., [Ellis]{}, R. S., [Bunker]{}, A., [Bundy]{}, K., [Targett]{}, T., [Benson]{}, A., & [Lacy]{}, M. 2009, , 697, 1493
, R. 2014, [pyGadgetReader: GADGET snapshot reader for python]{}, Astrophysics Source Code Library
, R., [Nagamine]{}, K., [Jaacks]{}, J., & [Choi]{}, J.-H. 2014, , 780, 145
, P., [Wellons]{}, S., [Machado]{}, F., [Griffen]{}, B., [Nelson]{}, D., [Rodriguez-Gomez]{}, V., [McKinnon]{}, R., & [Pillepich]{}, A. 2015, , 454, 2770
, M., [Bradley]{}, L. D., [Stiavelli]{}, M., [Oesch]{}, P., [Treu]{}, T., [Bouwens]{}, R. J., [Shull]{}, J. M., & [MacKenty]{}, J. W. 2011, , 727, L39
, P. G., [Whitaker]{}, K. E., [Brammer]{}, G., [Franx]{}, M., [Kriek]{}, M., [Labb[é]{}]{}, I., [Marchesini]{}, D., & [Quadri]{}, R. 2010, , 709, 1018
, M., [Genel]{}, S., [Springel]{}, V., [Torrey]{}, P., [Sijacki]{}, D., [Xu]{}, D., [Snyder]{}, G., [Nelson]{}, D., & [Hernquist]{}, L. 2014, , 444, 1518
|
---
author:
- |
Marián Kolesár\
Institute of Particle and Nuclear Physics, Faculty of Mathematics and Physics, Charles University in Prague, CZ-18000 Prague, Czech republic\
E-mail:
- |
Jiří Novotný\
Institute of Particle and Nuclear Physics, Faculty of Mathematics and Physics, Charles University in Prague, CZ-18000 Prague, Czech republic\
E-mail:
bibliography:
- 'Bibliography.bib'
title: 'Constraints on low energy QCD parameters from $\eta \to 3\pi$ and $\pi\pi$ scattering'
---
Introduction
============
Spontaneous breaking of chiral symmetry (SB$\chi$S) is a prominent feature of the QCD vacuum and thus its character has been under discussion for a long time [@Fuchs:1991cq; @DescotesGenon:1999uh]. The principal order parameters are the quark condensate and the pseudoscalar decay constant in the chiral limit[^1]
[ $$\hspace{0cm} \label{Sigma}
\Sigma(N_f) = -\langle\,0\,|\,\bar{q}q \,|\,0\,\rangle\,|_{m_q\to 0}\,,$$ ]{} [ $$\hspace{0cm} \label{F}
F(N_f) = F_P^a\,|_{m_q\to 0}\,,\quad i p_{\mu}\, F_P^a\ =\ \langle\,0\,|\,A_{\mu}^a\,|\,P\,\rangle,$$ ]{}
where $N_f$ is the number of quark flavors $q$ considered light and $m_q$ collectively denotes their masses. $A_{\mu}^a$ are the QCD axial vector currents, while $F_P^a$ the decay constants of the light pseudoscalar mesons $P$. The two flavor parameters are usually denoted as $\Sigma$ and $F$, while the three flavor ones as $\Sigma_0$ and $F_0$.
Chiral perturbation theory ($\chi$PT) [@Weinberg:1978kz; @Gasser:1983yg; @Gasser:1984gg] is constructed as a general low energy parametrization of QCD based on its symmetries and the discussed order parameters appear at the lowest order of the chiral expansion as low energy constants (LECs). Interactions of the light pseudoscalar meson octet, the pseudo-Goldstone bosons of the broken symmetry, directly depend on the pattern of SB$\chi$S and thus can provide information about the values of $\Sigma(N_f)$ and $F(N_f)$.
A convenient reparametrization of these order parameters, relating them to physical quantities connected with pion two point Green functions, can be introduced [@DescotesGenon:1999uh]
[ $$\hspace{0cm}
Z(N_f) = \frac{F(N_f)^2}{F_{\pi}^2},\quad
X(N_f) = \frac{2\hat{m}\,\Sigma(N_f)}{F_{\pi}^2M_{\pi}^2},$$ ]{}
where $\hat{m}\,$=$\,(m_u\,$+$\,m_d)/2$. Defined in this way, $X(N_f)$ and $Z(N_f)$ are limited to the range (0,1). $Z(N_f)$=0 would correspond to a restoration of chiral symmetry and $X(N_f)$=0 to a case with vanishing chiral condensate. Standard approach to chiral perturbation series tacitly assumes values of $X(N_f)$ and $Z(N_f)$ not much smaller than one, which means that the leading order terms should dominate the expansion.
Several recent results for the two and three flavor order parameters are listed in Tables \[tab1\] and \[tab2\], respectively. As can be seen, while the two flavor case is quite settled, the values of $X(2)$ and $Z(2)$ indeed being not much smaller than one, the situation in the three flavor sector is much less clear. Some analyses suggest a significant suppression of X(3) and/or Z(3) and thus a non-standard behavior of the spontaneously broken QCD vacuum. It can also be noted that the lattice averaging group FLAG [@Aoki:2016frl] does not report an average for the three flavor chiral order parameters.
[|c|c|c|c|]{}
------------------------------------------------------------------------
phenomenology & $Z(2)$ & $X(2)$\
------------------------------------------------------------------------
$\pi\pi$ scattering [@DescotesGenon:2001tn] & 0.89$\pm$0.03 & 0.81$\pm$0.07\
------------------------------------------------------------------------
lattice QCD & $Z(2)$ & $X(2)$\
------------------------------------------------------------------------
RBC/UKQCD+Re$\chi$PT [@Bernard:2012fw] & 0.86$\pm$0.01 & 0.89$\pm$0.01\
------------------------------------------------------------------------
FLAG2016 $N_f$=2 [@Aoki:2016frl] & 0.87$\pm$0.02 & 0.84$\pm$0.11\
------------------------------------------------------------------------
FLAG2016 $N_f$=2+1 [@Aoki:2016frl] & 0.88$\pm$0.01 & 0.86$\pm$0.03\
[|c|c|c|c|]{}
------------------------------------------------------------------------
phenomenology & $Z(3)$ & $X(3)$\
------------------------------------------------------------------------
NNLO $\chi$PT (BE14) [@Bijnens:2014lea] & 0.59 & 0.63\
------------------------------------------------------------------------
NNLO $\chi$PT (free fit) [@Bijnens:2014lea] & 0.48 & 0.45\
------------------------------------------------------------------------
NNLO $\chi$PT (“fit 10”) [@Amoros:2001cp] & 0.89 & 0.66\
------------------------------------------------------------------------
Re$\chi$PT $\pi\pi$+$\pi K$ [@DescotesGenon:2007ta] & $>$0.2 & $<$0.8\
------------------------------------------------------------------------
lattice QCD & $Z(3)$ & $X(3)$\
------------------------------------------------------------------------
RBC/UKQCD+Re$\chi$PT [@Bernard:2012ci] & 0.54$\pm$0.06 & 0.38$\pm$0.05\
------------------------------------------------------------------------
RBC/UKQCD+large $N_c$ [@Ecker:2013pba] & 0.91$\pm$0.08 &\
------------------------------------------------------------------------
MILC 09A [@Bazavov:2009fk] & 0.72$\pm$0.06 & 0.62$\pm$0.07\
Up, down and strange quark masses are other parameters with strong influence on low energy QCD physics. A commonly used reparametrization can be introduced
[ $$\hspace{0cm}
\hat{m} = \frac{m_u + m_d}{2},\quad
r = \frac{m_s}{\hat{m}},\quad
R = \frac{m_s-\hat{m}}{m_d-m_u}.$$ ]{}
The values for the light quark mass average and the strange to light quark mass ratio are well known from lattice QCD and QCD sum rules [@Aoki:2016frl; @Narison:2014vka]. On the other hand, as can be seen in Table \[tab3\], the isospin breaking parameter $R$, directly related to the light quark mass difference, has not been determined with such a high precision by these or other methods yet.
[|c|c|c|]{}
------------------------------------------------------------------------
phenomenology & $R$\
------------------------------------------------------------------------
Dashen’s theorem LO [@Bijnens:2007pr] & 44\
------------------------------------------------------------------------
Dashen’s theorem NNLO [@Bijnens:2007pr] & 37\
------------------------------------------------------------------------
$\eta\to3\pi$ NNLO $\chi$PT [@Bijnens:2007pr] & 41.3\
------------------------------------------------------------------------
$\eta\to3\pi$ dispersive [@Kampf:2011wr] & 37.7$\pm$2.2\
------------------------------------------------------------------------
$\eta\to3\pi$ dispersive [@Colangelo:2016jmc] & 34.2$\pm$2.2\
------------------------------------------------------------------------
$\eta\to3\pi$ dispersive [@Albaladejo:2017hhj] & 32.7$\pm$3.0\
------------------------------------------------------------------------
lattice QCD & $R$\
------------------------------------------------------------------------
FLAG2016 $N_f$=2 [@Aoki:2016frl] & 40.7$\pm$4.3\
------------------------------------------------------------------------
FLAG2016 $N_f$=2+1 [@Aoki:2016frl] & 35.7$\pm$2.6\
In this paper, we use a Bayesian approach in the framework of resummed chiral perturbation theory to extract information on the three flavor chiral condensate, chiral decay constant and the mass difference of the light quarks. Our experimental input are well known observables connected to $\eta$$\,\to\,$$3\pi$ decays and $\pi\pi$ scattering. We assume a reasonable convergence of Green functions connected to these observables and investigate the constraints this assumption can provide for the discussed parameters. The results presented here are a significant update on our initial reports [@Kolesar:2013ywa; @Kolesar:2014zra; @Kolesar:2016iyz]
In Section \[sect\_resummed\_chpt\] we shortly summarize our theoretical foundation. Section \[eta3pi\_decays\] discusses the $\eta$$\,\to\,$$3\pi$ decays, while Section \[Calculation\] provides an overview of our calculation of these processes. Section \[sect\_pipi\_scattering\] introduces $\pi\pi$ scattering into our analysis. The Bayesian statistical approach is reviewed in Section \[bayesian\_analysis\] and a discussion of our assumptions can be found in Section \[assumptions\]. Section \[subthreshold\_parameters\] is concerned with investigating the compatibility of our theoretical predictions with the $\pi\pi$ scattering data. We employ a $\chi^2$ based analysis in Section \[chi2\_analysis\] to evaluate the quality with which our theoretical predictions reconstruct the experimental data and use it to choose between several assumptions. Finally, in Section \[Results\], the main results of the Bayesian analysis are presented and compared with available literature. We conclude in Section \[Conclusions\].
Resummed $\chi$PT \[sect\_resummed\_chpt\]
==========================================
We use an alternative approach to chiral perturbation theory, dubbed resummed $\chi$PT (Re$\chi$PT) [@DescotesGenon:2003cg], which was developed in order to accommodate the possibility of irregular convergence of the chiral expansion. This is a typical scenario if the $X(3)$ and $Z(3)$ were indeed suppressed and the leading order was not dominant in the chiral expansion. In such a case one would have to be careful in the way how chiral expansion is defined and dealt with, as reshuffling of chiral orders could lead to unexpectedly large higher orders.
The procedure can be shortly summarized in the following way:
- The Standard $\chi$PT Lagrangian and power counting [@Weinberg:1978kz; @Gasser:1983yg; @Gasser:1984gg] is used. In particular, the quark masses $m_{q}$ are counted as $O(p^{2})$.
- Only expansions of quantities related linearly to Green functions of QCD currents are trusted, these are called *safe observables*. It is assumed that the next-to-next-to-leading and higher orders in these expansions are reasonably small, though not necessary negligible. Leading order terms do not need to be dominant.
- Calculations are performed explicitly to next-to-leading order, higher orders are included implicitly in *remainders*. The first step consists of performing the *strict chiral expansion* of the safe observables, which is understood as an expansion constructed in terms of the parameters of the chiral Lagrangian, while strictly respecting the chiral orders.
- In the next step, the strict expansion is modified in order to correct the location of the branching points of the non-analytical part of the amplitudes, which need to be placed in their physical position. This can be done either by means of a matching with a dispersive representation or by hand. The *bare expansion* is obtained.
- After that, an algebraically exact non-perturbative reparametrization of the bare expansion is performed. It is obtained by expressing the $O(p^{4})$ LECs $L_i$ in terms of physical values of experimentally well established safe observables - the pseudoscalar decay constants and masses. The procedure generates additional higher order remainders. We refer to these as *indirect* remainders.
- The physical amplitude and other relevant observables are then obtained as algebraically exact non-perturbative expressions in terms of the related safe observables and higher order remainders.
- The higher order remainders are not neglected, but estimated and treated as sources of error.
The hope for resummed $\chi$PT is that by carefully avoiding dangerous manipulations a better converging series can be obtained. The procedure also avoids the hard to control NNLO LECs by trading them for remainders with known chiral order.
In general, a chiral expansion of a safe observable is written in the following way:
[ $$\hspace{0cm} G = G^{(2)} + G^{(4)} + \Delta_G^{(6)} = G^{(2)} + G^{(4)} + G\,\delta_G^{(6)},$$ ]{}
where $\Delta_G^{(6)}= G\,\delta_G^{(6)}$ is the remainder which contains all terms starting with NNLO. The basic assumption of this paper is that $\delta_G^{(6)}\ll 1$ for our chosen observables. We will quantify this assumption later on.
$\eta\to 3\pi$ decays \[eta3pi\_decays\]
========================================
The $\eta$$\,\to\,$$3\pi$ isospin breaking decays have not been exploited for an extraction of the chiral order parameters so far, yet we argue there is valuable information to be had. The theory seems to converge slowly for the decays. One loop corrections were found to be very sizable [@Gasser:1984pr], the result for the decay width of the charged channel was 160$\pm$50 eV, compared to the current algebra prediction of 66 eV. However, the experimental value is still much larger, the current PDG value is [@PDG:2016xmw]
[ $$\hspace{0cm} \Gamma^+_\mathrm{exp} = 300 \pm 12 \ \mathrm{eV}.$$ ]{}
The latest experimental value for the neutral decay width is [@PDG:2016xmw]
[ $$\hspace{0cm} \Gamma^0_\mathrm{exp} = 428 \pm 17 \ \mathrm{eV}.$$ ]{}
Only the two loop $\chi$PT calculation [@Bijnens:2007pr] has succeeded to obtain a reasonable result for the widths.
As we have shown in [@Kolesar:2016jwe], we argue that the four point Green functions relevant for the $\eta$$\,\to\,$$3\pi$ amplitude (see (\[Green\_f\]) below) do not necessarily have large contributions beyond next-to-leading order and a reasonably small higher order remainder is not in contradiction with huge corrections to the decay widths. The widths do not seem to be sensitive to the details of the Dalitz plot distribution, but rather to the value of leading order parameters - the chiral decay constant, the chiral condensate and the difference of $u$ and $d$ quark masses, i.e. the magnitude of explicit isospin breaking. Moreover, access to the values of these quantities is not screened by EM effects, as it was shown that the electromagnetic corrections up to NLO are very small [@Baur:1995gc; @Ditsche:2008cq]. This is our motivation for our effort to extract information about the character of the QCD vacuum from this decay.
The Dalitz plot distributions are experimentally well known as well [@KLOE:2008ht; @KLOE:2010mj; @WASA:2014aks; @BESIII:2015cmz; @KLOE:2016qvh]. The usual parametrization of the square of the amplitude is defined as
[ $$\hspace{0cm}
|A(s,t;u)|^{2}=|A(s_{0},s_{0};s_{0})|^{2}
\left(1+ay+by^{2}+dx^{2}+fy^{3}+gx^{2}y+\ldots \right)$$ ]{}
in the charged decay channel and as
[ $$\hspace{0cm}
|\overline{A}(s,t;u)|^{2}=|A(s_{0},s_{0};s_{0})|^{2}
\left( 1+2\alpha z+\ldots \right)$$ ]{}
in the $\eta\to 3\pi^0$ case, where
[ $$\hspace{0cm}
x = -\frac{\sqrt{3}\left( t-u\right)}{2M_{\eta}(M_\eta-3M_\pi)}, \quad
y = -\frac{3\left(s-s_{0}\right)}{2M_{\eta }(M_\eta-3M_\pi)} \quad
z = x^{2}+y^{2}$$ ]{}
and $s_0=\frac{1}{3}(s+t+u)$ is the center of the Dalitz plot.
However, as we have discussed in detail in [@Kolesar:2016jwe], we have not found the convergence of the theory in the case of the slopes reliable enough to include all the experimentally measured Dalitz plot parameters into the analysis. To stay on the conservative side, we used the lowest order parameter $a$ in the charged channel only. The latest and most precise experimental value is [@KLOE:2016qvh]
[ $$\hspace{0cm} \label{a_KLOE_new} a = -1.095 \pm 0.004.$$ ]{}
Calculation \[Calculation\]
===========================
The details of the calculations with explicit formulas can be found in [@Kolesar:2016jwe]. We only summarize the basic steps here, which closely follow the procedure outlined in [@Kolesar:2008jr].
We start by expressing the charged decay amplitude in terms of 4-point Green functions $G_{ijkl}$, obtained from the generating functional of the QCD currents. We compute at first order in isospin breaking, the amplitude then takes the form
[ $$\hspace{0cm} \label{Green_f}
F_\pi^3F_{\eta}A(s,t;u)
= G_{+-83}-\varepsilon_{\pi}G_{+-33}+\varepsilon_{\eta}G_{+-88} + \Delta^{(6)}_{G_D},$$ ]{}
where $\Delta^{(6)}_{G_D}$ is the direct higher order remainder to the 4-point Green functions. The physical mixing angles to all chiral orders and first order in isospin breaking can be expressed in terms of quadratic mixing terms of the generating functional to NLO (${M}_{38}^{(4)}$ and $Z_{38}^{(4)}$) and related remainders $\Delta_{M_{38}}^{(6)}$ and $\Delta _{Z_{38}}^{(6)}$
[ $$\hspace{0cm}
\varepsilon_{\pi,\eta} = -\frac{F_{0}^{2}}{F_{\pi^0,\eta}^{2}}
\frac{({M}_{38}^{(4)}+\Delta_{M_{38}}^{(6)}) -
M_{\eta,\pi^0}^{2}(Z_{38}^{(4)}+\Delta _{Z_{38}}^{(6)})}
{M_\eta^2-M_{\pi^0}^2}.$$ ]{}
In this approximation the neutral decay amplitude can be straightforwardly obtained from the charged one using isospin symmetry and charge conjugation invariance
[ $$\hspace{0cm} \label{isospin rel}
\overline{A}(s,t;u) = -A(s,t;u)-A(t,s;u)-A(u,t;s).$$ ]{}
In accord with the method, $O(p^2)$ parameters appear inside loops in the strict chiral expansion, while physical quantities in outer legs. Such a strictly derived amplitude has an incorrect analytical structure due to the leading order masses in loops, cuts and poles being in unphysical positions. We have developed several ways to account for this in [@Kolesar:2008jr], explicit formulas for the $\eta$$\,\to\,$$3\pi$ case are listed in [@Kolesar:2016jwe]. The simplest approach is to exchange the LO masses in unitarity corrections and chiral logarithms for physical ones. A more sophisticated method is to use a dispersive representation for the unitarity part of the amplitude
[ $$\hspace{0cm} F_\pi^3 F_\eta A(s,t;u) = \mathcal{P}(s,t;u) + F_\pi^3 F_\eta \mathcal{U}(s,t;u) + O(p^{6}),$$ ]{}
where $\mathcal{U}(s,t;u)$ is the unitary part and $\mathcal{P}(s,t;u)$ is an unknown polynomial. This can be obtained by using the reconstruction theorem [@Zdrahal:2008bd], first used in [@Knecht:1995tr]. Then the two representations can be sewed together
[ $$\hspace{0cm} F_\pi^3 F_\eta A(s,t;u) = G_\mathrm{pol}(s,t;u) + F_\pi^3F_ \eta \mathcal{U}(s,t;u)
+ \Delta^{(6)}_{\mathcal{G}_D}$$ ]{}
where the polynomial part $G_\mathrm{pol}(s,t;u)$ is obtained from (\[Green\_f\]) by the sewing procedure.
However, there is an ambiguity in the derivation of $\mathcal{U}(s,t;u)$. When using the Cutkosky rule, there is a freedom in the way how to define the relation of the amplitude and the Green function of the entering sub-process at leading order. The most straightforward way is to keep the order by order relation
[ $$\hspace{0cm} \label{disp_Fp}
S^{(n)}(s,t;u)=\left( \prod_{i=1}^4F_{P_i}\right)^{-1}G^{(n)}(s,t;u).$$ ]{}
Such a definition has the advantage of satisfying perturbative unitarity. On the other hand, a suppression factor $F_0^4/(\prod_i F_{P_i}^2)$ appears in the loop functions, where $P_i$ are the pseudocalars running in the loop. Because we expect the unitarity correction in the case of the $\eta$$\,\to\,$$3\pi$ decays to be sizable, we decided to implement an alternative definition
[ $$\hspace{0cm} \label{disp_F0}
S^{(2)}(s,t;u) = F_0^{-4}G^{(2)}(s,t;u).$$ ]{}
No suppression factor occurs in this case. Both approaches are valid and they differ in the definition of the higher order remainder $\Delta^{(6)}_{\mathcal{G}_D}$. In our view, we should prefer such a definition where the higher orders are under better control. The approaches will be numerically compared in Section \[chi2\_analysis\].
The next step is the treatment of the LECs. As discussed above, the leading order ones, as well as quark masses, are expressed in terms of convenient parameters
[ $$\hspace{0cm} X=X(3),\ \ Z=Z(3),\ \ r,\ \ R.$$ ]{}
At next-to-leading order, the LECs $L_4$-$L_8$ are algebraically reparametrized in terms of pseudoscalar masses, decay constants and the free parameters $X$, $Z$ and $r$ using chiral expansions of two point Green functions, similarly to [@DescotesGenon:2003cg]. Because expansions are formally not truncated, each generates an unknown higher order remainder
[ $$\hspace{0cm} L_4,L_5,L_6,L_7,L_8\ \to\ \delta_{F_\pi},\delta_{F_K},\delta_{M_\pi},\delta_{M_\eta},\delta_{M_K}.$$ ]{}
We don’t have a similar procedure ready for $L_1$-$L_3$ at this point. Therefore we collect a set of standard $\chi$PT fits [@Bijnens:2014lea; @Amoros:2001cp; @Bijnens:2011tb; @Bijnens:1994ie] and by taking their mean and spread, while ignoring the much smaller reported error bars, we obtain an estimate of their influence
[ $$\hspace{0cm} L_1^r(M_\rho) = (0.57 \pm 0.18) \cdot 10^{-3}$$ ]{} [ $$\hspace{0cm} L_2^r(M_\rho) = (0.82 \pm 0.28) \cdot 10^{-3}$$ ]{} [ $$\hspace{0cm} L_3^r(M_\rho) = (-2.95 \pm 0.38) \cdot 10^{-3}$$ ]{}
These uncertainties enter our statistical analysis. However, as it is discussed in [@Kolesar:2016jwe], the results depend on the value of the constants $L_1$-$L_3$ only very weakly.
The $O(p^6)$ and higher order LECs, notorious for their abundance, are implicit in a relatively smaller number of higher order remainders. We have eight indirect remainders - three generated by the expansions of the pseudoscalar masses, three by the decay constants and two by the mixing angles. We expand the direct remainder to the 4-point Green functions around the center of the Dalitz plot $s_0=1/3(M_\eta^2$+$2M_{\pi^+}^2$+$M_{\pi^0}^2)$ to second order in Mandelstam variables
[ $$\hspace{0cm}
\Delta^{(6)}_{\mathcal{G}_D} = \Delta_A+\Delta_B(s-s_0) + \Delta_C(s-s_0)^2+\Delta_D [(t-s_0)^2+(u-s_0)^2]$$ ]{}
and thus get four derived direct remainders, two NLO ones ($\Delta_C,\Delta_D$) and two NNLO ones ($\Delta_A,\Delta_B$). We should note that in this approximation we completely miss the analytical structure of the amplitude at higher chiral orders, which includes $\pi\pi$ rescattering effects at NNLO and higher. A deeper discussion of the issue can be found in [@Kolesar:2016jwe]. We argue that because the experimental curvature of the Dalitz plot is very small [@KLOE:2008ht], the expansion to second order in the Mandelstam variables is sufficient for the purpose of calculating the decay widths and the lowest order Dalitz slope $a$. On the other hand, the theory seems unreliable for the higher order Dalitz parameters, especially in the case of $b$ and $\alpha$, thus we have chosen to avoid them in this analysis.
For the calculation of the decay widths, we need to numerically integrate over the kinematic phase space. In order to perform the computation efficiently enough, we are forced to expand the amplitude around the center of the Dalitz plot
[ $$\hspace{0cm}
F_\pi^3 F_\eta A(s,t;u) = A+B(s-s_0) + C(s-s_0)^2+D [(t-s_0)^2+(u-s_0)^2].$$ ]{}
The same argument as above hold here as well - the curvature of the Dalitz plot is tiny, therefore this is a very good approximation for the objective of calculating the decay widths.
$\pi\pi$ scattering \[sect\_pipi\_scattering\]
==============================================
In addition to the $\eta$$\,\to\,$$3\pi$ parameters discussed above, we employ $\pi\pi$ scattering in a very similar way to [@DescotesGenon:2003cg]. We use the two lowest order subthreshold parameters in the expansion of the polynomial part of the amplitude, $\alpha_{\pi\pi}$ and $\beta_{\pi\pi}$
[ $$\hspace{-0.5cm}
A_{\pi\pi}(s,t,u)\ =\ \frac{\alpha_{\pi\pi}}{F_\pi^2}\frac{ M_\pi^2}{3} +
\frac{\beta_{\pi\pi}}{F_\pi^2} \left(s-\frac{4 M_\pi^2}{3}\right) +
\frac{\lambda_1}{F_\pi^4} \left(s-2 M_\pi^2\right)^2 +
\frac{\lambda_2}{F_\pi^4} \left[(t-2 M_\pi^2)^2 + (u-2 M_\pi^2)^2 \right] +$$ ]{} [ $$\hspace{2cm} +\frac{\lambda_3}{F_\pi^6} \left(s-2 M_\pi^2\right)^3 +
\frac{\lambda_4}{F_\pi^6} \left[(t-2 M_\pi^2)^3 + (u-2 M_\pi^2)^3 \right] +
U_{\pi\pi}^{(4+6)}(s,t;u) + {\mathcal{O}}(p^8),$$ ]{}
where $U_{\pi\pi}^{(4+6)}(s,t;u)$ is the unitary part of the amplitude to NNLO, not given here explicitly. $\alpha_{\pi\pi}$ and $\beta_{\pi\pi}$ can be expressed as
$$\begin{aligned}
\alpha _{\pi \pi } &=&1+\frac{3r}{r+2}\epsilon (r)-\frac{2Yr}{r+2}\eta (r)+\frac{2(1-X)}{r+2}+\frac{4(1-Y)}{r+2} - \nonumber\\
&&-\frac{1}{2}Y^{2}\left( \frac{M_{\pi }}{4\pi F_{\pi }}\right) ^{2}\left(
\frac{r}{(r-1)(r+2)}\left( (r+2)\log \left( \frac{M_{\eta }^{2}}{M_{K}^{2}}\right) -(r-2)\log \left( \frac{M_{K}^{2}}{M_{\pi }^{2}}\right) \right) +\frac{7}{3}\right) - \nonumber\\
&&-\frac{6}{r+2}\left( \frac{r+1}{r-1}\delta _{M_{\pi }}-\left( \epsilon (r)+\frac{2}{r-1}\right) \delta _{M_{K}}\right)
-Y\frac{2r}{r+2}\left( \frac{r+1}{r-1}\delta _{F_{\pi }}-\left( \eta (r)+\frac{2}{r-1}\right) \delta _{F_{K}}\right) + \nonumber\\
&&+2Y\delta _{F_{\pi }}+\delta _{\alpha _{\pi \pi }}\end{aligned}$$
$$\begin{aligned}
\beta _{\pi \pi } &=&1+\frac{r\eta (r)}{r+2}+\frac{2(1-Z)}{r+2} + \nonumber\\
&&+\frac{1}{2}Y\left( \frac{M_{\pi }}{4\pi F_{\pi }}\right) ^{2}\left( \frac{r}{(r-1)(r+2)}\left( (2r+1)\log \left( \frac{M_{\eta }^{2}}{M_{K}^{2}}\right) +(4r+1)\log \left( \frac{M_{K}^{2}}{M_{\pi }^{2}}\right) \right)
-5\right) - \nonumber\\
&&-\frac{2}{r+2}\left( \frac{r+1}{r-1}\delta _{F_{\pi }}-\left( \eta (r)+\frac{2}{r-1}\right) \delta _{F_{K}}\right) +\delta _{\beta _{\pi\pi},}\end{aligned}$$
with
$$Y = \frac{X}{Z},\quad
\epsilon (r)=\frac{2}{r^{2}-1}\left( 2\frac{F_{K}^{2}M_{K}^{2}}{F_{\pi
}^{2}M_{\pi }^{2}}-r-1\right) ,\quad
\eta (r)=\frac{1}{r-1}\left( \frac{F_{K}^{2}}{F_{\pi }^{2}}-1\right).$$
We use the experimental values extracted in [@DescotesGenon:2001tn]
[ $$\hspace{0cm} \alpha_{\pi\pi}^\mathrm{exp} = 1.381 \pm 0.242, \quad \beta_{\pi\pi}^\mathrm{exp} = 1.081 \pm 0.023,\quad \rho_{\pi\pi} = -0.14.$$ ]{}
$\rho_{\pi\pi}$ is the correlation coefficient between the two parameters.
Bayesian statistical analysis \[bayesian\_analysis\]
====================================================
For the statistical analysis, we use an approach based on Bayes’ theorem [@DescotesGenon:2003cg]
[ $$\hspace{0cm} \label{Bayes}
P(X_i|\mathrm{data}) = \frac{P(\mathrm{data}|X_i)P(X_i)}{\int \mathrm{d}X_i\,P(\mathrm{data}|X_i)P(X_i)}\,,$$ ]{}
where $P(X_i|\mathrm{data})$ is the probability density of the parameters and remainders, denoted as $X_i$, having a specific value given the observed experimental data.
In the case of experimentally independent observables, $P(\mathrm{data}|X_i)$ is the known probability density of obtaining the observed values of the included observables $O_k$ in a set of experiments with uncertainties $\sigma_k$ under the assumption that the true values of $X_i$ are known
[ $$\hspace{0cm}
P(\mathrm{data}|X_i) = \prod_k\frac{1}{\sigma_k\sqrt{2\pi}}\,
\mathrm{exp}\left[-\frac{(\mathrm{O}_k^\mathrm{exp}-\mathrm{O}_k(X_i))^2}{2\sigma_k^2}\right].$$ ]{}
Our observables are the charged and neutral decay widths and the Dalitz slope $a$ of the $\eta$$\,\to\,$$3\pi$ decays and the $\pi\pi$ scattering subthreshold parameters $\alpha_{\pi\pi}$ and $\beta_{\pi\pi}$. However, the pair of $\pi\pi$ scattering observables cannot be treated as independent and we need to introduce a correlated probability function for this sector
[ $$\hspace{0cm}
P_{\pi\pi}(\mathrm{data}|X_i) = \frac{1}{2\pi}\sqrt{|C_{\pi\pi}|}\,
\mathrm{exp}\left(-\frac{1}{2}V_{\pi\pi}^T C_{\pi\pi} V_{\pi\pi}\right),$$ ]{}
where
[ $$\hspace{0cm}
V_{\pi\pi} = \left(
\begin{array}{c}
\alpha_{\pi\pi}^\mathrm{exp}-\alpha_{\pi\pi}\\
\beta_{\pi\pi}^\mathrm{exp}-\beta_{\pi\pi}
\end{array}\right), \quad
C_{\pi\pi} = \frac{1}{1-\rho_{\pi\pi}^2}\left(
\begin{array}{cc}
\frac{1}{\sigma_\alpha^2} & \frac{-\rho_{\pi\pi}}{\sigma_\alpha \sigma_\beta} \\
\frac{-\rho_{\pi\pi}}{\sigma_\alpha \sigma_\beta} & \frac{1}{\sigma_\alpha^2}
\end{array}\right).$$ ]{}
We resort to Monte Carlo sampling in order to perform the numerical integration in (\[Bayes\]). It turned out that the uncertainty of latest experimental measurement of the Dalitz parameter $a$ (\[a\_KLOE\_new\]) is so small that it is in fact a complication for performing the numerical integration. From this point of view the experimental error is negligible and therefore we can model the experimental distribution as a $\delta$ function rather then a normal distribution
[ $$\hspace{0cm}
\lim_{\sigma_a\to 0}\, \frac{1}{\sigma_a\sqrt{2\pi}}\,
\mathrm{exp}\left[-\frac{(a_\mathrm{exp}-a)^2}{2\sigma_a^2}\right] =
\delta(a_\mathrm{exp}-a).$$ ]{}
The experimental data thus become a constraint which can be solved algebraically. The solution is most straightforward in the case of the remainder $\delta_B$
[ $$\hspace{0cm} \label{delta_a}
\delta(a_\mathrm{exp}-a) = |K|\,\delta(1+K - \delta_B),$$ ]{}
where
[ $$\hspace{0cm}
K = \frac{4}{3}\frac{M_\eta(M_\eta-3 M_\pi)}{a_\mathrm{exp}}(1-\delta_A)
\mathrm{Re}\left(\frac{B}{A}\right).$$ ]{}
The remainder $\delta_B$ becomes fixed by this constraint and is thus no longer a source of uncertainty.
$P(X_i)$ in (\[Bayes\]) are the prior probability distributions of $X_i$. We use them to implement the theoretical uncertainties connected with our parameters and remainders. In this way we keep the theoretical assumptions explicit and under control. It also allows us to test various assumptions and formulate if-then statements as well as implement additional constraints (see below).
Assumptions \[assumptions\]
===========================
The following list summarizes the higher order remainders we need to deal with:
- $\eta$$\,\to\,$$3\pi$ direct remainders: $\delta_A$, $\delta_B$, $\delta_C$, $\delta_D$
- $\pi\pi$ scattering direct remainders: $\delta_{\alpha_{\pi\pi}}$, $\delta_{\beta_{\pi\pi}}$
- indirect remainders: $\delta_{M_\pi}$, $\delta_{F_\pi}$, $\delta_{M_K}$, $\delta_{F_K}$, $\delta_{M_\eta}$, $\delta_{F_\eta}$, $\delta_{M_{38}}$, $\delta_{Z_{38}}$
We use an estimate based on general arguments about the convergence of the chiral series [@DescotesGenon:2003cg]
[ $$\hspace{0cm} \label{Delta_G}\delta_G^{(4)}\ \approx\pm 0.3,\quad \delta_G^{(6)}\ \approx\pm 0.1,$$ ]{},
where $\delta_G^{(n)}$ collectively denotes all the remainders listed above, with the exception of $\delta_B$, which is fixed by the constraint (\[delta\_a\]) in the main analysis. We implement (\[Delta\_G\]) by using a normal distribution with $\mu\,$=0 and $\sigma\,$=0.3 or $\sigma\,$=0.1 for the NLO or NNLO remainders, respectively. The remainders are thus limited only statistically, not by any upper bound.
The remainders $\delta_{\alpha_{\pi\pi}}$ and $\delta_{\beta_{\pi\pi}}$, connected with $\pi\pi$ scattering, are defined in such a way that they are of order $O(\hat{m}m_s)$ instead of $O(m_s^2)$. This follows from the particular form of the chiral expansion of the subthreshold parameters $\alpha_{\pi\pi}$ and $\beta_{\pi\pi}$ in the $\hat{m}\to 0$ limit, see [@DescotesGenon:2003cg] for details. Therefore we can consider these remainders to be of the order
[ $$\hspace{0cm}
\delta_{\alpha_{\pi\pi}}\ \approx\pm 0.03,\quad \delta_{\beta_{\pi\pi}}\ \approx\pm 0.03.$$ ]{}
We assume the strange to light quark ratio $r$ to be known and use the lattice QCD average [@Aoki:2013ldr]
[ $$\hspace{0cm} r = 27.5 \pm 0.4.$$ ]{}
At last, we are left with three free parameters:
[ $$\hspace{0cm} X=X(3),\ Z=Z(3),\ R.$$ ]{}
These control the scenario of spontaneous breaking of chiral symmetry and isospin breaking in our results. In the case of $X$ and $Z$, we use a constraint from the so-called paramagnetic inequality [@DescotesGenon:1999uh] and assume these parameters to be in the range
[ $$\hspace{0cm} \label{prior1}0 < X < X(2),\quad 0< Z < Z(2).$$ ]{}
For the two flavor order parameters, we use the lattice QCD values [@Bernard:2012fw]. In addition, similarly to [@DescotesGenon:2003cg], we implement constraints following from
[ $$\hspace{0cm} \label{prior2}X(2),Z(2)>\,0, \qquad X/Z=Y<Y_\mathrm{MAX}.$$ ]{}
We use two approaches to deal with $R$. In the first one we assume it to be a known quantity. We use the $N_f$=2+1 lattice QCD average [@Aoki:2013ldr]
[ $$\hspace{0cm} R=35.8\pm2.6.$$ ]{}
Alternatively, we leave $R$ free, or more precisely, assume it to be in a wide range $R$$\in$(0,80).
Subthreshold parameters of $\pi\pi$ scattering \[subthreshold\_parameters\]
===========================================================================
As mentioned in Section \[eta3pi\_decays\], in [@Kolesar:2016jwe] we tested the compatibility of a reasonable convergence of the chiral series, laid out explicitly in the form of assumptions in the previous section, with the experimental data in the case of the $\eta\to 3\pi$ observables. This is an important first step in order to avoid using observables which are problematic from the theoretical or experimental point of view. Analogously, in this section we take a closer look at the subthreshold parameters $\alpha_{\pi\pi}$ and $\beta_{\pi\pi}$, which was not done in [@DescotesGenon:2003cg].
We numerically generated a large number of theoretical prediction for $\alpha_{\pi\pi}$ and $\beta_{\pi\pi}$, statistically distributed according to the assumptions described in Section \[assumptions\]. The parameter $Z$ was fixed in two scenarios ($Z$=0.5 and $Z$=0.9), while $X$ was varied in the full range $0<X<1$. Figure \[fig\_pipi\] displays the obtained theoretical distributions in comparison with the experimental data.
As can be seen, both parameters show a significant dependence on $X$, while $\beta_{\pi\pi}$ depends on $Z$ as well. In the case of $\beta_{\pi\pi}$, a broad range of the generated theoretical predictions is consistent with experimental data. Even though the observable seems sensitive to the values of the chiral order parameters, the amount of available information to be extracted might be limited by the experimental error, which is quite substantial.
The picture seems to be more tricky when considering $\alpha_{\pi\pi}$. The central experimental value is outside the 2$\sigma$ band of the theoretical distribution in the whole range of the free parameters. If taken at face value, possibly confirmed by more precise data, it would indicate a very small value of $X$, i.e. a vanishing chiral condensate. Such a scenario would be, however, inconsistent with any current determination of the order parameter (see Tab.\[tab2\]). [^2] The experimental error on the value of $\alpha$ is very large and our prior expectation therefore is that a large correction of the central value is very possible. It would be thus advisable to be cautious when interpreting the outcomes based on this data in the following.
Our conclusion with regard to the suitability of the subthreshold parameters of $\pi\pi$ scattering for the purpose of extraction of information about the chiral order parameters is hence twofold - both $\alpha_{\pi\pi}$ and $\beta_{\pi\pi}$ seem sensitive to the value of one or both of them, but at present available information is limited due to the quality od experimental data we have at hand.
\
$\chi^2$ based analysis \[chi2\_analysis\]
==========================================
In addition to the Bayesian analysis described in Section \[bayesian\_analysis\], the results of which will follow in Section \[Results\], we also perform a search for the minimum of the $\chi^2$ distribution in the Monte Carlo generated set of data points. The aim is to check the quality with which this set of theoretical predictions can reconstruct the experimental data, which is hard to quantify using the Bayesian method. This test also enables us to compare various theoretical approaches, e.g. the alternative ways the dispersive representation can be implemented.
Because we have much more free parameters than experimental inputs, we generally expect a well working theory to be able to reconstruct the experimental data very precisely with a large enough sample of generated data points. This means the minimum of $\chi^2/n$, where $n$ is the number of experimental observables employed, should be close to zero. On the other hand, a min.$\chi^2/n\approx 1$ means there is typically a 1$\sigma$ deviation between the best of the generated theoretical predictions and the experimental data, which might signal some tension and would not be fully satisfactory.
As the minimum of the $\chi^2$ distribution is subject to fluctuations stemming from the statistical nature of our procedure, we also report the number of points for which $\chi^2/n<1$. This metric then reveals how well the region of the parameter space where the experimental data lie is covered by the generated theoretical predictions. A reasonably high number should be considered a necessary condition for a well founded analysis, as it demonstrates that the theory has no obvious problem to reconstruct the experimental data and also that we have generated a large enough sample of points.
When comparing different theoretical approaches, one model can overlay the experimental region with fewer points than the other for several reason. It might be that the points originate from a less probable part of the theoretical distribution (e.g., the tail) which means that less likely values of the parameters and remainders are needed in order to reconstruct the experimental data. In that sense, the model is less probable to be true. The $\chi^2$ based analysis can thus form a basis for preference when choosing between alternative approaches. However, it is also possible that one model is simply more sensitive to the values of some of the parameters or remainders, which means the theoretical distribution has a larger spread. This is no reason to exclude to model, of course, and one should be aware of this possibility and check for it.
Let us first explore the issue of deciding between the two dispersive representations used to define the bare expansion of the $\eta\to 3\pi$ amplitude, as discussed in Section \[Calculation\]. For this purpose we numerically generated a set of $4-6\cdot10^6$ theoretical predictions and constructed a $\chi^2$ distribution. In order to check the consistency of the generated predictions and a complete set of $\eta\to 3\pi$ data, we have not used the constraint (\[delta\_a\]), but rather an experimental value
[ $$\hspace{0cm} a = -1.09 \pm 0.02.$$ ]{}
This is a slightly older measurement [@KLOE:2008ht] with large enough uncertainty for our purpose.
[|c|c|c|c|c|]{}
------------------------------------------------------------------------
free parameters & exp.data & disp.aproach & $\sqrt{\mathrm{min.}\chi^2/n}$ & $N(\chi^2/n<1)$\
------------------------------------------------------------------------
$X$,$Z$ & $\Gamma^+$,$\Gamma^0$,$a$ & (\[disp\_Fp\]) & 0.80 & 57\
------------------------------------------------------------------------
$X$,$Z$ & $\Gamma^+$,$\Gamma^0$,$a$ & (\[disp\_F0\]) & 0.59 & 1048\
------------------------------------------------------------------------
$Z$,$R$ & $\Gamma^+$,$\Gamma^0$,$a$ & (\[disp\_Fp\]) & 0.65 & 225\
------------------------------------------------------------------------
$Z$,$R$ & $\Gamma^+$,$\Gamma^0$,$a$ & (\[disp\_F0\]) & 0.46 & 1068\
As can be seen in Table \[tab4\], the approach (\[disp\_F0\]) is able to reconstruct the data more precisely, while the theoretical distributions have a very similar form. This conforms to our intuition from Section \[Calculation\] and thus, in what follows, we use the representation based on (\[disp\_F0\]) exclusively.
We use the PDG [@PDG:2014kda] as the source of input for the values of pseudoscalar masses and decay constants. Because the isospin breaking parameter $1/R$ carries the isospin symmetry breaking, we need to choose a value of these constants in the isospin limit. We use the averaged kaon mass and the well known decay constants of the charged pion and kaon. However, we found the situation to be quite subtle for the case of the pion mass. For the charged decay $\eta$$\,\to\,$$\pi^+\pi^-\pi^0$ it seems appropriate to use the averaged mass
[ $$\hspace{0cm} \overline{M_\pi}^2 = \frac{1}{3}\left(M_{\pi^0}^2+2 M_{\pi^+}^2\right),$$ ]{}
while in the case of the neutral channel $\eta$$\,\to\,$$3\pi^0$ to use the neutral pion mass $M_{\pi^0}$. However, in this approach the isposin relation (\[isospin rel\]) is not exactly fulfilled.
Alternatively, one could use the same pion mass for both channels, either the averaged or neutral one, and satisfy the relation (\[isospin rel\]). One could argue that the difference in the results for the decay widths should be very subtle, of the order $1/R$, and that certainly seems to be true. However, the ratio of the decay widths is known very precisely [@PDG:2016xmw]
[ $$\hspace{0cm} r_\Gamma\ =\ 1.43 \pm 0.02.$$ ]{}
This is an indication that even a slight difference in prediction of order $1/R$ might actually influence whether one can obtain an accurate enough prediction for both the decay widths at the same time.
As we have found out, this is really the case in the approach with an identical pion mass in both the charged and neutral channel amplitudes, where the predicted ratio $r_\Gamma$ comes out too high. This is reflected in both the minimum of $\chi^2/n$ not being quite close to zero and in the number of points for which $\chi^2/n<1$ to be substantially lower in our $\chi^2$ based test for any value of the parameters in the allowed range, as can be seen in Table \[tab5\]. Meanwhile, the form of the distributions do not change considerably, as might be expected when only the numerical value of an input parameters is changed.
[|c|c|c|c|c|]{}
------------------------------------------------------------------------
free parameters & exp.data & pion mass & $\sqrt{\mathrm{min.}\chi^2/n}$ & $N(\chi^2/n<1)$\
------------------------------------------------------------------------
$X$,$Z$ & $\Gamma^+$,$\Gamma^0$,$a$ & $\overline{M_\pi}$, $M_\pi^0$ & 0.59 & 1048\
------------------------------------------------------------------------
$X$,$Z$ & $\Gamma^+$,$\Gamma^0$,$a$ & $\overline{M_\pi}$ & 0.93 & 7\
------------------------------------------------------------------------
$X$,$Z$ & $\Gamma^+$,$\Gamma^0$,$a$ & $M_\pi^0$ & 0.92 & 3\
------------------------------------------------------------------------
$Z$,$R$ & $\Gamma^+$,$\Gamma^0$,$a$ & $\overline{M_\pi}$, $M_\pi^0$ & 0.46 & 1068\
------------------------------------------------------------------------
$Z$,$R$ & $\Gamma^+$,$\Gamma^0$,$a$ & $\overline{M_\pi}$ & 0.89 & 9\
------------------------------------------------------------------------
$Z$,$R$ & $\Gamma^+$,$\Gamma^0$,$a$ & $M_\pi^0$ & 0.93 & 3\
In other words, in this case the theory seems to have a harder time to reproduce the experimental data with the required precision. This might look surprising given the number of free parameters in the fit, but one has to realize that while the decay widths in the two channels are independent from the experimental point of view, they are very strongly correlated on the theoretical side given the relation (\[isospin rel\]). The experiment in fact shows that this relation is not precisely fulfilled in nature. This is the reason we have chosen to implement distinct values of the pion mass in the two channels for the main analysis, which is the same approach as was taken in [@Bijnens:2007pr].
Table \[tab6\] shows a summary of the $\chi^2$ based tests for our main analysis, as presented in the next section. This uses the dispersive representation (\[disp\_F0\]), distinct pion mass values for the two $\eta\to 3\pi$ decay modes and the constraint (\[delta\_a\]). The total number of generated data points is $2\cdot10^7$.
[|c|c|c|c|]{}
------------------------------------------------------------------------
free parameters & exp.data & $\sqrt{\mathrm{min.}\chi^2/n}$ & $N(\chi^2/n<1)$\
------------------------------------------------------------------------
$X$,$Z$ & $\Gamma^+$,$\Gamma^0$,$\delta(a-a_{exp})$ & 0.002 & 165874\
------------------------------------------------------------------------
$X$,$Z$ & $\Gamma^+$,$\Gamma^0$,$\delta(a-a_{exp})$,$\beta_{\pi\pi}$ & 0.03 & 104670\
------------------------------------------------------------------------
$X$,$Z$ & $\Gamma^+$,$\Gamma^0$,$\delta(a-a_{exp})$,$\alpha_{\pi\pi}$,$\beta_{\pi\pi}$ & 0.28 & 51278\
------------------------------------------------------------------------
$Z$,$R$ & $\Gamma^+$,$\Gamma^0$,$\delta(a-a_{exp})$ & 0.003 & 87034\
------------------------------------------------------------------------
$Z$,$R$ & $\Gamma^+$,$\Gamma^0$,$\delta(a-a_{exp})$,$\beta_{\pi\pi}$ & 0.02 & 40919\
------------------------------------------------------------------------
$Y$ & $\Gamma^+$,$\Gamma^0$,$\delta(a-a_{exp})$,$\alpha_{\pi\pi}$,$\beta_{\pi\pi}$ & 0.15 & 25041\
------------------------------------------------------------------------
$Y$ & $\Gamma^+$,$\Gamma^0$,$\delta(a-a_{exp})$ & 0.002 & 120130\
------------------------------------------------------------------------
$Y$ & $\Gamma^+$,$\Gamma^0$,$\delta(a-a_{exp})$,$\beta_{\pi\pi}$ & 0.02 & 62203\
------------------------------------------------------------------------
$Y$ & $\Gamma^+$,$\Gamma^0$,$\delta(a-a_{exp})$,$\alpha_{\pi\pi}$,$\beta_{\pi\pi}$ & 0.28 & 23428\
------------------------------------------------------------------------
$Y$,$R$ & $\Gamma^+$,$\Gamma^0$,$\delta(a-a_{exp})$ & 0.002 & 125724\
------------------------------------------------------------------------
$Y$,$R$ & $\Gamma^+$,$\Gamma^0$,$\delta(a-a_{exp})$,$\alpha_{\pi\pi}$,$\beta_{\pi\pi}$ & 0.18 & 20054\
As can be seen, the number of points seems sufficient. The coverage is better in the approach with $\eta\to 3\pi$ observables only. We can observe a drop in precision when $\alpha_{\pi\pi}$ is included, which correlates with the discussion in Section \[subthreshold\_parameters\]. As we will see, this corresponds to the fact that the signal in the used experimental value of $\alpha_{\pi\pi}$ is in some tension with the one contained in the $\eta\to3\pi$ data.
Results and discussion\[Results\]
=================================
In this section we present the outputs of the Bayesian analysis, i.e. the probability density functions $P(X_i|\mathrm{data})$, where $X_{i}\in \{X,Z,Y,R\}$ are the chiral symmetry breaking and explicit isospin breaking parameters, respectively and $\mathrm{data}$ represent a subset of the set $\left\{ \Gamma _{+},\Gamma _{0},a,\alpha _{\pi \pi
},\beta _{\pi \pi }\right\} $ of the $\eta \rightarrow 3\pi $ and $\pi \pi
\rightarrow \pi \pi $ observables.
The calculated probability density distributions $P(X_i|\mathrm{data})$ are depicted in Fig.\[fig\_X-Z\], Fig.\[fig\_R-Z\] and Fig.\[fig\_Y-R\]. The colors in these figures highlight the confidence regions of the parameter space with a particular confidence level [^3]. The corresponding one dimensional probability density functions for the parameters $X$, $Z$, $Y=X/Z$ and $R$ are obtained by integrating out all other free parameters. In particular, $$\begin{aligned}
P(Y|\mathrm{data}) &=&\int_{0}^{1}\mathrm{d}Z~Z~P(X,Z|\mathrm{data})|_{X\to ZY}.\end{aligned}$$The results are shown in Fig.\[fig\_1D\_R-fixed\], Fig.\[fig\_1D\_R-fixed\_pipi\], Fig.\[fig\_1D\_pipi\] and Fig.\[fig\_1D\_R-free\], along with the priors following from the assumptions (\[prior1\]) and (\[prior2\]). Error bars in these figures indicate an estimated error of the Monte Carlo integration, which is sufficiently low in all cases. We summarize some characteristics of these distributions in Tab.\[tab\_R-fixed\] and Tab.\[tab\_R-free\].
$\overline{x}$ $\sigma _{x} $ median $1\sigma$ C.L. $2\sigma $ C.L.
-------------------------------------------- ---------------- ---------------- -------- ----------------- -----------------
$P(X|\Gamma _{+},\Gamma _{0},a)$ 0.56 0.22 0.58 (0.31, 0.80) (0.11, 0.88)
$P(X|\Gamma _{+},\Gamma _{0},a,\pi \pi )$ 0.56 0.21 0.58 (0.32, 0.78) (0.13, 0.87)
$P(Z|\Gamma _{+},\Gamma _{0},a)$ 0.40 0.18 0.40 (0.22, 0.58) (0.08, 0.78)
$P(Z|\Gamma _{+},\Gamma _{0},a,\pi \pi ) $ 0.48 0.19 0.48 (0.28, 0.68) (0.11, 0.82)
$P(Y|\Gamma _{+},\Gamma _{0},a)$ 1.44 0.32 1.44 (1.11, 1.76) (0.78, 2.05)
$P(Y|\Gamma _{+},\Gamma _{0},a,\pi \pi )$ 1.20 0.30 1.20 (0.90, 1.50) (0.60, 1.80)
$P(Y|\alpha_{\pi\pi} )$ 0.55 0.42 0.48 (0, 0.72) (0, 1.38)
$P(Y|\alpha_{\pi\pi},\beta_{\pi\pi} )$ 0.53 0.38 0.49 (0, 0.70) (0, 1.25)
: Characteristics of obtained probability distributions, $R$ free.[]{data-label="tab_R-free"}
$\overline{x}$ $\sigma _{x} $ median $1\sigma $ C.L. $2\sigma $ C.L.
------------------------------------------- ---------------- ---------------- -------- ------------------ -----------------
$P(R|\Gamma _{+},\Gamma _{0},a)$ 43.0 13 41.8 (29.7, 56.2) (20.4, 72.2)
$P(R|\Gamma _{+},\Gamma _{0},a,\pi \pi )$ 34.4 11.6 32.7 (23.0, 45.9) (16.7, 62.0)
$P(X|\Gamma _{+},\Gamma _{0},a)$ 0.56 0.22 0.59 (0.31, 0.80) (0.10, 0.88)
$P(Z|\Gamma _{+},\Gamma _{0},a)$ 0.39 0.17 0.38 (0.21, 0.56) (0.08, 0.77)
$P(Y|\Gamma _{+},\Gamma _{0},a)$ 1.56 0.46 1.58 (1.09, 1.99) (0.57, 2.32)
: Characteristics of obtained probability distributions, $R$ free.[]{data-label="tab_R-free"}
[|c|c|c|c|]{}
------------------------------------------------------------------------
& $Z(3)$ & $L_4\cdot10^3$\
------------------------------------------------------------------------
RBC/UKQCD+large $N_c$ [@Ecker:2013pba] & 0.91$\pm$0.08 & -0.05$\pm$0.22\
------------------------------------------------------------------------
NNLO $\chi$PT (BE14) [@Bijnens:2014lea] & 0.59 & 0.3\
------------------------------------------------------------------------
NNLO $\chi$PT (free fit) [@Bijnens:2014lea] & 0.48 & 0.76$\pm$0.18\
------------------------------------------------------------------------
RBC/UKQCD+Re$\chi$PT [@Bernard:2012ci] & 0.54$\pm$0.06 & 1.06$\pm$0.29\
Let us first discuss the case with the fixed value of $R=35.8\pm 2.6$, which is the lattice QCD average [@Aoki:2013ldr] (see Section \[assumptions\]) and include only the $\eta \rightarrow 3\pi $ observables into the analysis. This is what we consider our main set of results.
As can be seen in the left panel of Fig.\[fig\_X-Z\], a large part of the parameter space can be excluded at 2$\sigma$ CL and the parameters $X$ and $Z$ appear to be quite strongly correlated. The latter statement can be made more quantitative by extracting the result for the ratio of the chiral order parameters $Y = X/Z = 2\hat{m}B_0/M_\pi^2$, see Fig.\[fig\_1D\_R-fixed\] and Tab.\[tab\_R-fixed\], for which we get:
[ $$\hspace{0cm} Y = X/Z = 1.44 \pm 0.32.$$ ]{}
We also obtain a higher bound for the three flavor chiral decay constant
[ $$\hspace{0cm} \label{Z_bound}Z < 0.78\quad \mathrm{(2\sigma\ CL)},$$ ]{}
which corresponds to
[ $$\hspace{0cm} F_0 < 81\ \mathrm{MeV}\quad \mathrm{(2\sigma\ CL)}.$$ ]{}
As can be confirmed from Fig.\[fig\_1D\_R-fixed\], a combination of two factors contribute to this result - the $\eta\to3\pi$ data disfavor high values of $Z$ and the prior (\[prior1\]), induced by the paramagnetic inequality, causes a sharp cutoff at $Z=Z(2)=0.86\pm0.01$.
As can be seen in Figure \[fig\_X-Z\], when assuming $R$=35.8$\pm$2.6, there is some tension with several of the previous determination of the chiral order parameters (Table \[tab2\]). The $\eta$$\,\to\,$$3\pi$ data seem to prefer a larger value for the ratio of the order parameters $Y = X/Z$ than recent $\chi$PT and lattice QCD fits. In addition, very large values of the chiral decay constant are excluded at 2$\sigma$ CL. and a relatively small value is favored. The uncertainties, however, are quite large.
Of course, these determinations are hardly compatible among themselves either and provide a very unclear picture. In our view, a reasonable guess could be that there are important differences in how NLO and NNLO low energy constants are treated in various works. In particular, one possible culprit could be the large $N_c$ suppressed LEC $L_4$, which is known to be anti-correlated with $F_0$ for a long time [@DescotesGenon:2000ct]. Indeed, this anti-correlation can be observed in the results we quote, see Tab.\[tab\_L4\].
In our approach, $L_4$ is reparametrized in terms of the remainders and free parameters, including $Z$. It can thus vary in a wide range, as we have shown in [@Kolesar:2016jwe]. As the $\eta\to 3\pi$ data seem to constrain $Z$ only mildly, as discussed above, we do not get significant information on $L_4$ either.
When concerning [@Bijnens:2014lea], both the main fit (BE14) and the free fit, which are based on the standard $\chi $PT at $O\left( p^{6}\right) $ and a large set of the experimental observables, are relatively close to the $2\sigma $ contour of our distribution. Note however, that the errors of these fits are not at our disposal. The two fits differ precisely in their treatment of $L_{4}$, as indicated in Tab.\[tab\_L4\]. If we roughly estimated the theoretical uncertainty of these fits as the distance of the corresponding points in the $(X,Z)$ plane, then these fits might look quite compatible with our distribution.
The apparent inconsistency with the result of [@Bernard:2012ci] is intriguing. It uses resummed $\chi$PT as well, paired with lattice data. One distinction is a different approach to the remainders - the authors use uniform distribution of the remainders inside a closed interval and thus a sharp cutoff. One could speculate that a normal distribution with unbounded tails, as we use, might lead to larger error bars.
The work [@Ecker:2013pba], which is based on a large $N_c$ motivated approximation of the standard $O\left( p^{6}\right) $ $\chi $PT calculations, used on lattice data, reports a very large value of the chiral decay constant and a very low value of $L_4$. This is consistent with the large $N_c$ picture assumed in this paper, but quite far away from other determinations and in tension with our limit (\[Z\_bound\]). In fact, a large part of the region covered by the fit [@Ecker:2013pba] is excluded by our prior for $Z$, namely by the constraint stemming from the paramagnetic inequality $Z<Z(2)$ (\[prior1\]).
Let us now add the $\pi \pi \rightarrow \pi \pi $ data into the analysis. As can be seen in the right panel of Fig.\[fig\_X-Z\] and Fig.\[fig\_1D\_R-fixed\_pipi\], the picture does not change appreciably when including the subthreshold parameters of $\pi\pi$ scattering. Though a bit disappointing, this outcome is not unexpected considering the significant errors connected with the experimental values of these observables and the weak constraints obtained in [@DescotesGenon:2003cg] and [@DescotesGenon:2007ta].
There is one difference, however, we can observe a slight shift of our probability distribution towards a lower ratio of order parameters $Y=X/Z$, as is confirmed by the mean $\overline{Y}=1.2$ ($P(Y|\Gamma _{+},\Gamma _{0},a,\pi \pi )$ in Tab.\[tab\_R-fixed\]). This could be interpreted as a move of our predictions in the direction of better compatibility with some of the available determinations (Tab.\[tab2\]). Here we have to be rather cautious though, because, as we have discussed in Sect.\[subthreshold\_parameters\] and Sect.\[chi2\_analysis\], we can expect some tension between the two sets of data. And indeed, this can be demonstrated when one compares the obtained probability distributions and confidence intervals for $Y$ from $\eta\to 3\pi$ (Fig.\[fig\_1D\_R-fixed\], $P(Y|\Gamma _{+},\Gamma _{0},a)$ in Tab.\[tab\_R-fixed\]) with one from $\pi\pi$ scattering alone (Fig.\[fig\_1D\_pipi\], $P(Y|\alpha_{\pi\pi},\beta_{\pi\pi} )$ in Tab.\[tab\_R-fixed\]), which are barely compatible. We can conclude that we need a more precise determination of the value of the $\pi\pi\to\pi\pi$ subthreshold parameters, especially for $\alpha_{\pi\pi}$, to be able to provide a more definite outcome and hopefully solve this puzzle of experimental data pointing in opposite directions.
The results with $R$ left as a free parameter are shown in Fig.\[fig\_R-Z\] and Fig.\[fig\_Y-R\]. The uncertainties are large and thus it’s hard to constrain $R$ without additional information on the chiral order parameters and the remainders. Even in this case a part of the parameter space can be excluded at 2$\sigma$ C.L. though. The obtained value for $R$ (Fig.\[fig\_1D\_R-free\], Tab.\[tab\_R-free\]) is compatible with available results (Table \[tab3\]).
We can also evaluate the obtained probability densities for $X$, $Z$ and $Y$ with $R$ left unconstrained (Tab.\[tab\_R-free\], Fig.\[fig\_1D\_R-free\]). Note that dismissing the very clear information on $R$ is not a reasonable assumption, we use it only as a test of robustness. And in this case, the results for $X$ and $Z$ seem almost independent on the value of $R$, which includes the obtained upper bound for the chiral decay constant (\[Z\_bound\]).
On the other hand, as Fig.\[fig\_Y-R\] shows, $R$ and the ratio $Y = X/Z$ seem to be quite strongly correlated. This also provides some additional insight into the large value of $Y$ obtained from $\eta\to 3\pi$ data when fixing $R$=35.8$\pm$2.6. Furthermore, as the $\pi\pi$ scattering data we use drag $Y$ down to lower values, one can observe a correlated shift in probability densities of $R$ to smaller numbers in Fig.\[fig\_Y-R\], Fig.\[fig\_1D\_R-free\] and Tab.\[tab\_R-free\].
Conclusions \[Conclusions\]
===========================
To summarize, we have used statistical methods in the framework of resummed chiral perturbation theory to generate large sets of theoretical predictions for $\eta\to 3\pi$ and $\pi\pi$ scattering observables, dependent on a variety of parameters and assumptions, and confronted them with experimental data.
We have developed a $\chi^2$ based analysis, which allowed us to form a basis for preference when choosing between alternative assumptions or models. In particular, it showed us that an approach using different pion masses for the charged and neutral decay channel observables in the isospin limit is significantly more consistent with data, despite violating the isospin relation, than using identical pion mass for both $\eta\to 3\pi$ decay modes.
For the main analysis, we have used Bayesian inference to obtain constraints on the values of three flavor chiral order parameters - the chiral decay constant $F_0$ and the chiral condensate $\Sigma_0$, which are connected with the spontaneous breaking of chiral symmetry in QCD.
When fixing the light quark difference by input from lattice QCD and using $\eta\to 3\pi$ observables only (the decay widths in both channels and the Dalitz plot parameter $a$), we could exclude a large part of the parameters space at 2$\sigma$ CL and have observed some correlation between the chiral order parameters. We have obtained an upper bound for the chiral decay constant, $F_0<81$MeV at 2$\sigma$ CL, and have extracted a fairly large value for the ratio of the order parameters $Y=2\hat{m}B_0/M_{\pi}^2$.
We have found some tension with several of the previous determination of the chiral order parameters, which, however, are neither very consistent with each other. The picture remains unclear, possibly stemming from differences in assumptions about the low energy constants, the large $N_c$ suppressed LEC $L_4$ being one candidate.
The picture have not changed appreciably when we performed a combined $\eta\to 3\pi$ and $\pi\pi\to\pi\pi$ analysis. However, we have observed some tension between $\eta\to 3\pi$ and $\pi\pi$ scattering data, which limited our ability to draw more definite conclusions. The possible source is the experimental error of the observables we used, the subthreshold parameters $\alpha_{\pi\pi}$ and $\beta_{\pi\pi}$. Specifically, the value of $\alpha_{\pi\pi}$ proved suspicious in our investigation, as it prefers a very low value for the chiral condensate, which is not very consistent with current expectations.
We have also tried to extract information on the difference of light quark masses, but the uncertainties proved to be very large. The result is consistent with available data though.
*Acknowledgment:* This work was supported by the Czech Science Foundation (grant no. GACR 15-18080S).
[^1]: We will abbreviate these to *chiral condensate* and *chiral decay constant* in the following
[^2]: A crash of the chiral condensate at three flavors would be also unexpected in the context of an SU(3) gauge theory with a varying number of light quark flavors, see e.g. [@Appelquist:2014zsa].
[^3]: The choice of such regions could be considered somewhat arbitrary, we constructed them by means of integrating the (discretized) probability densities according to decreasing probabilities starting from the maximal value until the desired confidence level was achieved.
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abstract: 'In this letter we examine a model recently proposed to produce phase synchronization \[K. Wood *et al*, Phys. Rev. Lett. [**96**]{}, 145701 (2006)\] and we show that the onset to synchronization corresponds to the emergence of an intermittent process that is non-Poisson and renewal at the same time. We argue that this makes the model appropriate for the physics of blinking quantum dots, and the dynamics of human brain as well.'
author:
- Simone Bianco$^1$
- 'Elvis Geneston$^{1}$'
- 'Paolo Grigolini$^{1,2,3}$'
- 'Massimiliano Ignaccolo$^{1}$'
title: Renewal Aging as Emerging Property of Phase Synchronization
---
Phase synchronization of coupled clocks (oscillators) is a growing field of research, which is fast developing from the seminal work of Winfree [@winfree] and Kuramoto [@kuramoto]. Using the model of coupled clocks of these authors it is possible to derive [@biosa] the Turing structure [@turing], which, in turn, triggered the research work in the field of diffusion reaction [@prigo]. Physiologists are using a clock as a representation of a single neuron. In the work of a neuron is a clock whose behavior is described by a chaotic attractor: the Rössler oscillator . The authors of and of have shown that coupled stochastic clocks can show cooperative (synchronized) behavior. Brain functions, such as a cognitive act, rest on the cooperative behavior of a collection of many neurons [@varela]. The authors of study the dynamic of the cooperation of a collection of many neurons by mapping the brain activity into a network. They find that the changes in the topology of the network describing the brain activity are driven by a non-Poisson renewal process operating in the non-ergodic regime [@nonergodic]. A collection of Blinking Quantum Dots (BQDs) [@bqd] has the same dynamical property as the brain activity. The non-Poisson non-ergodic character of the distribution of the sojourn times of the BQDs in the “light” and in the “dark” state is a well established property [@bqd]. The renewal character of the BQDs dynamic has been established only recently [@paradisi].
The similarity between the BQDs and the brain activity dynamics makes it plausible to search for a dynamic model accounting for both complex systems. In this Letter we show that a simplified version of the model of affords significant suggestions on how to realize this important purpose. The authors of use a system of coupled stochastic clocks, while we use a system coupled stochastic clocks. We shall prove that at the onset of phase synchronization the dynamics of the system has the same properties as the BQDs [@bqd; @paradisi] and brain dynamics [@eeg]. They are non-Poisson renewal processes operating in the non-ergodic regime.
We consider a Gibbs ensemble of systems with $N$ stochastic clocks, each of them coupled to $N_{c}$ clocks. We denote by $\mid$$1$$>$ and $\mid$$2$$>$ the two states of the clock, corresponding to the phases $\Phi$$=$$0$ and $\Phi$$=$$\pi$, respectively. The master equation for a single clock of a system of the Gibbs ensemble is $$\label{meq_first}
\left\{ \begin{array}{ll}
\frac{\displaystyle d}{\displaystyle dt} P_{1} = -g_{12} P_{1} + g_{21} P_{2} \\
\frac{\displaystyle d}{\displaystyle dt} P_{2} = -g_{21} P_{2} + g_{12} P_{1}
\end{array} .\right.$$ $P_{1}$ ($P_{2}$) is the probability of finding the clock in the state $\mid$$1$$>$ ($\mid$$2$$>$), and $g_{12}$ ($g_{21}$) is the rate of transitions from the state $\mid$$1$$>$ ($\mid$$2$$>$) to the state $\mid$$2$$>$ ($\mid$$1$$>$). The transition rates $g_{12}$ and $g_{21}$ are defined by means of the prescription of : $$\label{rate_definition}
g_{12(21)} = g \, \exp \left[K(\pi_{2(1)} - \pi_{1(2)})\right].$$ In Eq. (\[rate\_definition\]), $g$ is the unperturbed transition rate of a single clock, $K$$>$$0$ is the coupling constant and $\pi_{1}$ ($\pi_{2}$) is the fraction of the $N_{c}$ coupled clocks that are in the state $\mid$$1$$>$ ($\mid$$2$$>$). The authors of place their clocks in a d-dimensional lattice where only the nearest neighbors are coupled. Thus, every clock is coupled to $N_{c}$$=$$2d$ clocks. We adopt, instead, an all to all coupling: . With this choice, when , the mean field approximation [@stanley] $$\label{meanfieldapprox}
\pi_{1(2)}=P_{1(2)}$$ is an exact property. Thanks to the mean field approximation of Eq. (\[meanfieldapprox\]) and to the normalization condition $P_{1}$$+$$P_{2}$$=$$1$, the master equation Eq. (\[meq\_first\]) reduces to $$\label{mastereq_bigpi}
\frac{d\Pi(t)}{dt}\!=\!-2g \cosh(K \Pi) \Pi + 2g \sinh(K \Pi)\!=\!- \frac{\partial V(\Pi)}{\partial \Pi},$$ where and . Eq. (\[mastereq\_bigpi\]) describes the overdamped motion of a particle, whose position is $\Pi$, within the potential $V(\Pi)$ [@kramers]. Using Eq. (\[mastereq\_bigpi\]), we find that the potential $V(\Pi)$ is symmetric and the values of its minima depend only on the coupling constant $K$. Moreover, we find that there is a critical value $K_{c}$ of the coupling parameter $K$ $$\label{critical_value}
K = K_{c} = 1$$ such that: 1) If the potential $V(\Pi)$ has only one minimum for ; 2) if the potential $V(\Pi)$ is symmetric and has two minima separated by a barrier with the maximum centered in . As shown in Fig. \[figure1\], the value $\Pi_{\textrm{min}}$ and the height of the barrier V(0) are increasing function of the coupling constant $K$ [@potential]. In particular and when .
The time evolution of the variable $\Pi$ is determined by the minima and maxima of the potential $V(\Pi)$. Thus, two kinds of dynamical evolution are possible: 1) If , $\Pi(t)$ will reach, after a transient, an asymptotic value $\Pi(\infty)$$=$$0$ not depending on the initial conditions $\Pi(0)$; 2) if , $\Pi(t)$ will reach, after a transient, an asymptotic value $\Pi(\infty)$$=$$+$$\Pi_{\textrm{min}}$$\neq$$0$ ($\Pi(\infty)$$=$$-$$\Pi_{\textrm{min}}$$\neq$$0$) for an initial condition $\Pi(0)$$>$$0$ ($\Pi(0)$$<$$0$), while the initial condition $\Pi(0)$$=$$0$ will result in $\Pi(t)$$=$$0$ $\forall$$t$. In Fig. \[figure2\] we compare the minima $\pm\Pi_{\textrm{min}}$ of the potential $V(\Pi)$ and the numerical evaluation of $\Pi(\infty)$ for different values of the coupling constant $K$. Fig. \[figure2\] shows that a phase transition [@stanley] occurs at $K$$=$$K_{c}$$=$$1$.
For a single clock the condition $\Pi(\infty)$$=$$\pm$$\Pi_{\textrm{min}}$$=$$P_{1}$$-$$P_{2}$$\neq$$0$ corresponds to the statistical “preference” to be either in the state $\mid$$1$$>$ or $\mid$$2$$>$. This is a consequence of the fact that the transition rates $g_{12}$ and $g_{21}$ of Eq. (\[rate\_definition\]) are different if $K$$>$$K_{c}$. Plugging Eq. (\[meanfieldapprox\]) into Eq. (\[rate\_definition\]) and allowing $\Pi$ to reach its asymptotic value, we get $$\label{different_rates}
g_{12}=g \exp(-K \Pi(\infty)) \; \neq \; g_{21}=g \exp(K \Pi(\infty)).$$ Fig. \[figure3\] confirms the prediction of Eq. (\[different\_rates\]) showing that if $\Pi(\infty)$$=$$+$$\Pi_{\textrm{min}}$ ($\Pi(\infty)$$=$$-$$\Pi_{\textrm{min}}$) the single clock spends on average more time in the state $\mid$$1$$>$ ($\mid$$2$$>$).
The probability density function for the sojourn times in both the preferred and not-preferred state are exponential functions with different mean sojourn times.
Let us now explore the collective behavior of a single system of $N$ clocks under the all to all coupling condition. Following the authors of , we define the global clock variable $\xi(t)$ as $$\label{global_clock_1}
\xi(t) = \frac{\sum\limits_{j=1}^{N} \exp (i \; \Phi_{j}(t))}{N} = \frac{N_{1}(t)-N_{2}(t)}{N}.$$ The symbol $i$ indicates the imaginary unit, $\Phi_{j}$ is the phase of the j-th clock, $0$ ($\pi$) if the clock is in the state $\mid$$1$$>$ ($\mid$$2$$>$), and $N_{1}(t)$ ($N_{2}(t)$) is the number of clocks of the system in state $\mid$$1$$>$ ($\mid$$2$$>$) at a time $t$. When $N$$\rightarrow$$\infty$, the single system becomes a Gibbs ensemble on its own as all the clocks are identical, and, at the same time, the mean field approximation (Eq. (\[meanfieldapprox\])) becomes valid. In this case the master equation for the single system is the master equation Eq. (\[meq\_first\]), where $P_{1}$ ($P_{2}$) is now the probability of finding a clock of the system in the state $\mid$$1$$>$ ($\mid$$2$$>$). Finally, from Eq. (\[global\_clock\_1\]), in the limiting case $N$$=$$\infty$, we get $$\label{global_clock_2}
\xi(t)=P_{1}(t)-P_{2}(t)=\Pi(t).
%$\xi(t)$$=$$P_{1}(t)$$-$$P_{2}(t)$$=$$\Pi(t)$.$$ Thus, for a system with infinitely many clocks, the time behavior of the global clock variable $\xi$ is identical to that of the variable $\Pi$ of Eq. (\[mastereq\_bigpi\]): for $K$$>$$K_{c}$ (we exclude the initial unstable condition ) $$\label{global_clock_3}
\xi(\infty)=\pm\Pi_{\textrm{min}}\neq0 \:\:\:\;\;\;\textrm{if} \:\;\xi(0)\gtrless0.$$ The condition $\xi(\infty)$$\neq$$0$ of Eq. (\[global\_clock\_3\]) proves that a global phase synchronization occurs in the system at the onset of phase transition ($K$$>$$K_{c}$). Moreover, the time evolution of a single clock of the system is the one depicted by Fig. \[figure3\]: state $\mid$$1$$>$ ($\mid$$2$$>$) is statistically preferred if ().
Is the phase synchronization of Eq. (\[global\_clock\_3\]) present in a system with a finite number of clocks? From the definition of transition rate and of probability, it follows that $$\begin{aligned}
\label{rate_meaning} \left\{ \begin{array}{ll}
g_{12}P_{1} = \mathop {\lim } \limits_{N\to \infty} \frac{\displaystyle N_{1\mapsto2}}{\displaystyle N}\\
g_{21}P_{2} = \mathop {\lim } \limits_{N\to \infty} \frac{\displaystyle N_{2\mapsto1}}{\displaystyle N}
\end{array},\right.\end{aligned}$$ where $N_{1\mapsto2}$ ($N_{2\mapsto1}$) is the number of clocks that undergo a transition from state $\mid$$1$$>$ ($\mid$$2$$>$) to state $\mid$$2$$>$ ($\mid$$1$$>$) per unit of time, and $N$ is the number of clocks of the system. Using the law of large numbers [@linda], we get that, for any $N$ fitting the condition $\infty$$>$$N$$\gg$$1$, $$\begin{aligned}
\label{lawgrtnum} \left\{ \begin{array}{ll}
\frac{\displaystyle N_{1\mapsto2}}{\displaystyle N} = g_{12}P_{1} + \varepsilon_{12}P_{1}\\
\frac{\displaystyle N_{2\mapsto1}}{\displaystyle N} = g_{21}P_{2} + \varepsilon_{21}P_{2}\\
\end{array},\right.\end{aligned}$$ where $\epsilon_{12}$ and $\epsilon_{21}$ are fluctuating variables whose intensities are $\propto$$1$$/$$\sqrt(N)$. From Eqs. (\[rate\_meaning\]) and (\[lawgrtnum\]), we conclude that the master equation of a system with a finite number of clocks is equivalent to that of a system with an infinite number of clocks whose transition rates fluctuate: $$\label{meq_finitesystem}
\left\{ \begin{array}{ll}
\frac{\displaystyle d}{\displaystyle dt} P_{1} = -(g_{12}+\varepsilon_{12})P_{1} + (g_{21}+\varepsilon_{21})P_{2} \\
\frac{\displaystyle d}{\displaystyle dt} P_{2} = -(g_{21}+\varepsilon_{21})P_{2} + (g_{12}+\varepsilon_{12})P_{1}
\end{array} .\right.$$ If $\infty$$>$$N$$\gg$$1$, we can still consider the mean field approximation of Eq. (\[meanfieldapprox\]) to be valid. Using the master equation Eq. (\[meq\_finitesystem\]) and the normalization condition $P_{1}$$+$$P_{2}$$=$$1$, we get for the variable $\Pi$$=$$P_{1}$$-$$P_{2}$ the following equation of motion $$\label{mastereq_bigpi_fluct}
\frac{d\Pi(t)}{dt} = - \frac{\partial V(\Pi)}{\partial \Pi} - \eta(t) \Pi(t) + \theta (t).$$ The presence of the fluctuations $\eta$$=$$\epsilon_{12}+\epsilon_{21}$ and $\theta$$=$$\epsilon_{12}-\epsilon_{21}$ in Eq. (\[mastereq\_bigpi\_fluct\]) has the effect of triggering transitions from one well to the other of the potential $V(\Pi)$ of Fig. \[figure1\]. Thus, for a system with a finite number of clocks the phase synchronization of Eq. (\[global\_clock\_3\]) is not stable. The global clock variable $\xi$ of Eq.(\[global\_clock\_1\]) fluctuates (for $K$$>$$K_{c}$) between the two minima, $\pm$$\Pi_{\textrm{min}}$, of the potential $V(\Pi)$, as confirmed by Fig. \[figure4\]. The single clock follows the fluctuations of the global clock variable $\xi$, switching back and forth from the condition where the state $\mid$$1$$>$ is statistically preferred (time evolution described by the full line of Fig. \[figure3\]) to that where the state $\mid$$2$$>$ is (time evolution described by the dashed line of Fig. \[figure3\]).
The probability density functions of the sojourn times in the state $\xi$$>$$0$ or $\xi$$<$$0$ (Fig. \[figure4\]) are identical since the potential $V(\Pi)$ of Fig. \[figure1\] is symmetric. Thus, we denote them with the same symbol $\psi(\tau)$. Let us consider a condition where the coupling constant $K$ is close to the critical value . In this case, the height of the barrier $V(0)$ dividing the two wells of the potential $V(\Pi)$ of Fig. \[figure1\] is smaller than or comparable to the intensity of the fluctuations $\eta$ and $\theta$ of Eq. (\[mastereq\_bigpi\_fluct\]). Under this condition, we expect [@margolin] for an extended interval of sojourn times. This is exactly what we observe in Fig. \[figure5\], where the full line denotes the survival probability $\Psi(\tau)$, namely, the probability of observing a sojourn time larger than $\tau$ [@survivalprop]. For any fixed value of the unperturbed rate $g$ (Eq. (\[rate\_definition\])), the height $V(0)$ of the barrier dividing the two wells of the potential $V(\Pi)$ (Fig. \[figure1\]) increases as the value of the coupling parameter increases. Eventually, the height $V(0)$ will become much larger than the intensity of the fluctuations $\eta$ and $\theta$ (Eq. (\[mastereq\_bigpi\_fluct\])). As a consequence, the power law behavior ($\psi(\tau)$$\propto$$1/\tau^{1.5}$) of Fig. \[figure5\] disappears, the theoretical arguments of [@margolin] loses validity, and an exponential behavior emerges, as predicted by the Kramers theory [@kramers].
Finally, we show that the transition between the states $\xi$$>$$0$ and $\xi$$<$$0$ shown in Fig. \[figure4\] is a renewal process. For this purpose we use the aging experiment of Ref. [@paradisi]. We evaluate the survival probability $\Psi(t_{a},\tau)$ of age $t_{a}$. This is the probability of observing a sojourn time larger than $\tau$ if the observation starts at a time $t_{a}$ after a crossing from $\xi$$>$$0$ to $\xi$$<$$0$ or vice versa ($\Psi(0,\tau)$$=$$\Psi(\tau)$). We, then, compare $\Psi(t_{a},\tau)$ with the expected survival probability $\Psi_{r} (t_{a},\tau)$ of age $t_{a}$, evaluated according to the renewal theory [@paradisi]. If the process described by the survival probability $\Psi(\tau)$ is a renewal process. If $\Psi(\tau)$ is not an exponential function, $\Psi(t_{a},\tau)$ yields a slower decay than $\Psi(\tau)$, a condition that is denoted as “aging” [@paradisi]. Fig. \[figure5\] shows that the transition between the states $\xi$$>$$0$ and $\xi$$<$$0$ (Fig. \[figure4\]) is a renewal process and that there is aging. With increasing values of coupling parameter $K$ the renewal property is not lost, but the aging property is because the survival probability $\Psi(\tau)$ becomes an exponential function [@paradisi].
In conclusion, we have shown that, for a Gibbs ensemble of systems with $N$$=$$\infty$ coupled 2-state clocks, a phase transition occurs at the critical value of coupling parameter $K_{c}$$=$$1$ (Fig. \[figure2\]). The phase transition mirrors a statistical “preference” for a clock of one of the Gibbs ensemble systems to be in one of the two possible states (Eq.(\[different\_rates\])). For a single system with $N$$=$$\infty$ coupled 2-state clocks, the phase transition of Fig. \[figure2\] signals the onset of a global phase synchronization (Eqs. (\[global\_clock\_1\]), (\[global\_clock\_2\]) and (\[global\_clock\_3\])). If the number of clocks of the system is finite the phase synchronization is not stable. In this case, the variable $\xi$ (Eq. \[global\_clock\_1\]) describing the collective motion of the system (Fig. \[figure4\]) is characterized by non-Poisson intermittent behavior. At the onset of the phase transition (), the zero crossings of the global variable $\xi$ (Fig. \[figure4\]) define a series of events with the following properties: 1) The probability density function of inter events intervals (full line of Fig. \[figure5\]) has a non-Poisson non-ergodic character ( infinite mean inter event interval). 2) The sequence of inter events intervals satisfy the renewal aging condition (Fig. \[figure5\]). The properties 1) and 2) are observed properties of the events in both BQDs [@bqd; @paradisi; @margolin] and brain activity [@eeg], suggesting that a system consisting of a finite number of coupled clocks may be a good model for the dynamics of both processes.
The authors thankfully acknowledge Welch and ARO for financial support through Grant no. B-1577 and no. W911NF-05-1-0205, respectively.
[100]{} A. T. Winfree, *The Geometry of Biological Time*, Springer-Verlag, Berlin (1990). Y. Kuramoto, *Rhythms and Turbulence in Populations of Chemical Clocks* Physica A [**126**]{}, 128 (1981). G. Biosa, S. Bastianoni, and M. Rustici, Chem. Eur. J. [**12**]{}, 3430 (2006). A. M. Turing, Philos. Trans. R. Soc. London Ser. B [**327**]{}, 37 (1952). I. Prigogine and R. Lefever, J. Chem. Phys. [**48**]{}, 1695 (1967). C. J. Stam, Clinical Neurophysiology [**116**]{}, 2266 (2005). O. E. Rössler, Phys. Lett. [**57 A**]{}, 397 (1976). T. Prager, B. Naundorf, and L. Schimansky-Geier, Physica A [**325**]{}, 176 (2003). K. Wood, C. Van den Broeck, R. Kawai, and K. Lindenberg, Phys. Rev. Lett. [**96**]{}, 145701 (2006). F. Varela, J.-P.Lachaux, E. Rodriguez and J. Martinerie, Nature Review Neuroscience, [**2**]{}, 229 (2001). S. Bianco, P. Grigolini, M. Ignaccolo, M.Rider, M. Ross, P. Winsor, http://arxiv.org/abs/q-bio.NC/0610037 (submitted to PRE). G. Margolin and E. Barkai, Phys. Rev. Lett. [**94**]{} 080601 (2005); G. Bel and E. Barkai, Phys. Rev. Lett. [**94**]{}, 240602 (2005); G. Margolin and E. Barkai, J. Stat. Phys. [**122**]{}, 137 (2006); G. Bel and E. Barkai, Phys. Rev. E [**73**]{}, 016125 (2006). M. Nirmal, B.O. Dabbousi, M. G. Bawendi, J.J. Macklin, J. K. Trautman, T. D. Harris, L. E. Brus, Nature [**383**]{}, 802 (1996); M. Kuno, D.P. Fromm, S.T. Johmson, A. Gallagher and D.J. Nesbitt, Phys. Rev. B [**67**]{}, 125304 (2003); K.T. Shimizu, R.G. Neuhauser, C. A. Leatherdale, S. A. Empedocles, W.K. Woo and M. G. Bawendi, Phys. Rev. B [**63**]{}, 205316 (2001). S. Bianco, P. Grigolini, P. Paradisi, J. Chem. Phys. [**123**]{}, 174704 (2005). H. E. Stanley, *Introduction to Phase Transitions and Critical Phenomena*, Oxford University Press, Oxford (1971). H. A. Kramers, Physica [**7**]{}, 284 (1940). The arbitrary constant in the defintion of the potential $V(\Pi)$ is chosen to satisfy the condition . L. E. Reichl, *A Modern Course in Statistical Physics*, John Wiley, New York (1998). G. Margolin and E. Barkai, Phys. Rev. E [**72**]{}, 025101 (R) (2005). Note that $\psi(\tau)$$\propto$$1/\tau^{1.5}$$\Leftrightarrow$$\Psi(\tau)$$\propto$$1/\tau^{0.5}$.
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abstract: |
Random intersection graphs (RIGs) are an important random structure with applications in social networks, epidemic networks, blog readership, and wireless sensor networks. RIGs can be interpreted as a model for large randomly formed non-metric data sets. We analyze the component evolution in general RIGs, and give conditions on existence and uniqueness of the giant component. Our techniques generalize existing methods for analysis of component evolution: we analyze survival and extinction properties of a dependent, inhomogeneous Galton-Watson branching process on general RIGs. Our analysis relies on bounding the branching processes and inherits the fundamental concepts of the study of component evolution in Erdős-Rényi graphs. The major challenge comes from the underlying structure of RIGs, which involves its both the set of nodes and the set of attributes, as well as the set of different probabilities among the nodes and attributes.
**Keywords:** Random graphs, branching processes, probabilistic methods, random generation of combinatorial structures, stochastic processes in relation with random discrete structures.
author:
- 'Milan Bradonjić[^1], Aric Hagberg[^2], Nicolas W. Hengartner[^3], Allon G. Percus[^4]'
bibliography:
- 'intergraph.bib'
title: Component Evolution in General Random Intersection Graphs
---
Introduction {#sec.introduction}
============
Bipartite graphs, consisting of two sets of nodes with edges only connecting nodes in opposite sets, are a natural representation for many networks. A well-known example is a collaboration graph, where the two sets might be scientists and research papers, or actors and movies [@watts-1998-collective; @newman-2001-scientific]. Social networks can often be cast as bipartite graphs since they are built from sets of individuals connected to sets of attributes, such as membership of a club or organization, work colleagues, or fans of the same sports team. Simulations of epidemic spread in human populations are often performed on networks constructed from bipartite graphs of people and the locations they visit during a typical day [@eubank-2004-modelling]. Bipartite structure, of course, is hardly limited to social networks. The relation between nodes and keys in secure wireless communication, for examples, forms a bipartite network [@bloznelis-2009-component]. In general, bipartite graphs are well suited to the problem of classifying objects, where each object has a set of properties [@godehardt-2007-random]. However, modeling such classification networks remains a challenge. The well-studied Erdős-Rényi model, $G_{n,p}$, successfully used for average-case analysis of algorithm performance, does not satisfactorily represent many randomly formed social or collaboration networks. For example, $G_{n,p}$ does not capture the typical scale-free degree distribution of many real-world networks [@barabasi-1999-emergence]. More realistic degree distributions can be achieved by the configuration model [@newman-2001-random] or expected degree model [@chung-2002-average], but even those fail to capture common properties of social networks such as the high number of triangles (or cliques) and strong degree-degree correlation [@newman-2003-why; @albert-2002-statistical].
The most straightforward way of remedying these problems is to characterize each of the bipartite sets separately. One step in this direction is an extension of the configuration model that specifies degrees in both sets [@guillaume-2006-bipartite]. Another related approach is that of random intersection graphs (RIG), first introduced in [@singer-1995-thesis; @karonski-1999-random]. Any undirected graph can be represented as an intersection graph [@erdos-1966-representation]. The simplest version is the “uniform” RIG, $G(n,m,p)$, containing a set of $n$ nodes and a set of $m$ attributes, where any given node-attribute pair contains an edge with a fixed probability $p$, independently of other pairs. Two nodes in the graph are taken to be connected if and only if they are both connected to at least one common element in the attribute set. In our work, we study the more general RIG, $G(n,m,\boldsymbol{p})$ [@nikoletseas-2004-existence; @nikoletseas-2008-large], where the node-attribute edge probabilities are not given by a uniform value $p$ but rather by a set $\boldsymbol{p} = \{p_w\}_{w \in W}$: a node is attached to the attribute $w$, with probability $p_w$. This general model has only recently been developed and only a few results have obtained, such as expander properties, cover time, and the existence and efficient construction of large independent sets [@nikoletseas-2004-existence; @nikoletseas-2008-large; @spirakis-2009-expander].
In this paper, we analyze the evolution of components in general RIGs. Related results have previously been obtained for the uniform RIG [@behrisch-2007-component], and for two uniform cases of the RIG model where a specific overlap threshold controls the connectivity of the nodes, were analyzed in [@bloznelis-2009-component]. Our main contribution is a generalization of the component evolution on a general RIG. We provide stochastic bounds, by analyzing the stopping time of the branching process on general RIG, where the history of the process is directly dictated by the structure of the general RIG. The major challenge comes from the underlying structure of RIGs, which involves both the set of nodes and the set of attributes, as well as the set of different probabilities $\boldsymbol{p} = \{p_w\}_{w \in W}$.
Model and previous work
=======================
In this paper, we will consider the general intersection graph $G(n,m,\boldsymbol{p})$, introduced in [@nikoletseas-2004-existence; @nikoletseas-2008-large], with a set of probabilities $\boldsymbol{p} = \{p_w\}_{w \in W}$, where $p_w\in (0,1)$. We now formally define the model.
**Model.** There are two sets: the set of nodes $V = \{1,2,\dots,n\}$ and the set of attributes $W =
\{1,2,\dots,m\}$. For a given set of probabilities $\boldsymbol{p} = \{p_w\}_{w \in W}$, independently over all $(v,w) \in V \times W$ let \[eq.Avw\] A\_[v,w]{} := (p\_w). Every node $v \in V$ is assigned a random set of attributes $W(v) \subseteq W$ \[eq.Wv\] W(v) := {w W A\_[v,w]{} = 1}. The set of edges in $V$ is defined such that two different nodes $v_i,v_j \in V$ are connected if and only if \[eq.connectivity\] |W(v\_i) W(v\_j)| s, for a given integer $s \geq 1$. In our analysis, $p_w$ are not necessarily the same as in [@behrisch-2007-component; @bloznelis-2009-component] [^5], and for simplicity we fix $s=1$.
The component evolution of the uniform model $G(n,m,p)$ was analyzed by Behrisch in [@behrisch-2007-component], for the case when the scaling of nodes and attributes is $m=n^{\alpha}$, with $\alpha \ne 1$ and $p^2 m = c/n$. Theorem 1 in [@behrisch-2007-component] states that the size of the largest component $\Ncal(G(n,m,p))$ in RIG satisfies (i) $\Ncal(G(n,m,p)) \leq \frac{9}{(1-c^2)}\log n$, for $\al>1, c<1$, (ii) $\Ncal(G(n,m,p)) = (1+o(1))(1-\rho)n$, for $\al>1, c>1$, (iii) $\Ncal(G(n,m,p)) \leq \frac{10 \sqrt{c}}{(1-c^2)}\sqrt{\frac{n}{m}} \log m$, for $\al<1, c<1$, (iv) $\Ncal(G(n,m,p)) =(1+o(1))(1-\rho) \sqrt{c m n}$, for $\al<1, c>1$, where $\rho$ is the solution in $(0,1)$ of the equation $\rho = \exp(c(\rho-1))$.
The component evolution for the case $s \geq 1$ in the relation $|W(u) \cap W(v)| \geq s$ is considered in [@bloznelis-2009-component], where the following two RIG models are analyzed: (1) $G_s(n,m,d)$ model, where $\pr[W(v) = A] = {m \choose d }^{-1}$ for all $A \subseteq W$ on $d$ elements, for a given $d$; (2) $G'_s(n,m,p)$ model, where $\pr[W(v) = A] = p^{|A|}(1-p)^{m-|A|}$ for all $A \subseteq W$. In light of results of [@behrisch-2007-component], it has been shown in [@bloznelis-2009-component], that for $d = d(n), p=p(n), m=m(n), n = o(m)$, where $s$ is a fixed integer, and $d^{2s} \sim c m^s s!/n$, the largest component in $G_s(n,m,d)$ satisfies: (i) $\Ncal(G_{s}(n,m,d)) \leq \frac{9}{(1-c^2)}\log n$, for $c<1$, (ii) $\Ncal(G_{s}(n,m,d)) = (1+o(1))(1-\rho)n$, for $c>1$, in the case when $n \log n = o(m)$ for $s=1$ and $n = o(m^{s/(2s-1)})$ for $s\leq 2$. The same results for the giant component in $G_s(n,m,p)$ still hold for the case when $p^{2s} = cs! / m^s n$ and $n = o(m^{s/(2s-1)})$, see [@bloznelis-2009-component]. Both $G_s(n,m,d)$ and $G'_s(n,m,p)$ are special cases of a more general class studied in [@godehardt-2001-two], where the number of attributes of each node is assigned randomly as in the bipartite configuration model. That is, for a given probability distribution $(P_0, P_1, \dots, P_m)$, we have $\pr[|W(v)| = k] = P_k$ for all $0 \leq k \leq m$, and moreover given the size $k$, all of the sets $W(v)$ are equally probable, that is for any $A \subseteq W$, $\pr[W(v) = A : |W(v)| = k] = {m \choose k}^{-1}$. That is, we see that $G_s(n,m,d)$ is equivalent to the model of [@godehardt-2001-two] with the delta-distribution, where the probability of the $d$-th coordinate is $1$, while $G'_s(n,m,d)$ is equivalent to the model of [@godehardt-2001-two] with the $\bin(m,p)$ distribution. Comparing $G(n,m,\boldsymbol{p})$, the general RIG model, with the model of [@godehardt-2001-two], it follows that general RIG model does not perform a “uniform sampling”, as the model of [@godehardt-2001-two], since for a given set of probabilities $\{p_w\}_{w \in W}$, in general, $$\label{eq.non.unif.sampling}
\nonumber
\pr[W(v)=A \mid |W(v)| = k] = \frac{\pr[W(v)=A, |W(v)| = k]}{\pr[|W(v)| = k]} =
\frac{\prod_{\al \in A} p_{\al} \prod_{\bt \notin A} (1-p_{\bt})}{\sum_{A \subseteq W, |A|= k }\prod_{\al \in A} p_{\al} \prod_{\bt \notin A}(1-p_{\bt})} \neq {m \choose k}^{-1}. %\frac{1}{{m \choose k}}.$$ To complete the picture of previous work, in [@deijfen-2009-random], it was shown that when $n=m$ a set of probabilities $\boldsymbol{p} = \{p_w\}_{w \in W}$ can be chosen to tune the degree and clustering coefficient of the graph.
Let us now compare the general RIG $G(n,m,\boldsymbol{p})$ with the model of [@godehardt-2001-two]. Given the set of probabilities $\{p_w\}_{w \in W}$, $$\label{eq:prob.distr} P_k = \sum_{A \subseteq W: |A|= k } \pr[A] =
\sum_{A \subseteq W: |A|= k } \prod_{\al \in A} p_{\al} \prod_{\bt
\notin A} (1-p_{\bt}) = \prod_{i=1}^m(1-p_i) \prod_{\al \in A} \frac{p_{\al}}{1-p_{\al}}.$$ Moreover, $$\pr[W(v)=A : |W(v)| = k] = \frac{\pr[W(v)=A, |W(v)| = k]}{\pr[|W(v)| = k]} = \frac{\prod_{\al \in A} p_{\al} \prod_{\bt \notin A} (1-p_{\bt})}{\sum_{A \subseteq W: |A|= k }\prod_{\al \in A} p_{\al} \prod_{\bt \notin A}(1-p_{\bt})}.$$ It follows that for any set $\boldsymbol{p}$ in $G(n,m,\boldsymbol{p})$ there is a probability distribution $\mathcal{P} = (P_0, P_1, \dots, P_m)$. The other direction is not true, in general. For every $\mathcal{P}$, there is not a $\boldsymbol{p}$ as the following simple example shows. Let $\mathcal{P} = (1/2,1/4,1/4)$. Then from Eq. (\[eq:prob.distr\]) it follows that $p_{1/2} = (1 \pm i
\sqrt{7})/8$, or $p_{1/2} = (1 \mp i \sqrt{7})/8$), but $p_1,p_2$ are real numbers from $[0,1]$.
Mathematical preliminaries {#sec.math.prelim.}
==========================
In this paper, we analyze the component evolution of the general RIG structure. As we have already mentioned, the major challenge comes from the underlying structure of RIGs, which involves both the set of nodes and the set of attributes, as well as the set of different probabilities $\boldsymbol{p} = \{p_w\}_{w \in W}$.
Moreover, the edges in RIG are not independent. Hence, a RIG cannot be treated as an Erdős-Rényi random graph $G_{n,\hat p}$, with the edge probability $\hat p = 1 - \prod_{w \in W} (1-p_w^2)$. However, in [@fill-2000], the authors provide the comparison among $G_{n,\hat p}$ and $G(n,m,p)$, showing that for $m = n^\alpha$ and $\al > 6$, these two classes of graphs have asymptotically the same properties. In [@rybarczyk-2009], Rybarczyk has recently shown the equivalence of sharp threshold functions among $G_{n,\hat p}$ and $G_{n,m,p}$, when $m \geq n^3.$ In this work, we do not impose any constraints among $n$ and $m$, and we develop methods for the analysis of branching processes on RIGs, since the existing methods for the analysis of branching processes on $G_{n,p}$ do not apply.
We now briefly state the edge dependence. Consider three distinct nodes $v_i,v_j,v_k$ from $V$. Conditionally on the set $W(v_k)$, by the definition (\[eq.Wv\]), the sets $W(v_i) \cap W(v_k)$ and $W(v_j) \cap W(v_k)$ are mutually independent, which implies conditional independence of the events $\{v_i \sim v_k \mid W(v_k)\}, \{v_j \sim v_k \mid W(v_k)\}$, that is, \[eq:two-nodes-ind\] = . However, the latter does not imply independence of the events $\{v_i \sim v_k\}$ and $\{v_j \sim v_k\}$ since in general $$\begin{aligned}
\label{eq.rig.depend.}
\nonumber \pr [ v_i\sim v_k , v_j \sim v_k ] &=& {\mathbb E}[\pr [ v_i \sim v_k , v_j \sim v_k \mid W(v_k) ] \\
\nonumber &=& {\mathbb E} \left [\pr [ v_i \sim v_k \mid W(v_k) ] \pr [ v_j \sim v_k \mid W(v_k) ] \right ]\\
&\neq& {\mathbb P}[v_i \sim v_k]{\mathbb P}[v_j \sim v_k].\end{aligned}$$ Furthermore, the conditional pairwise independence (\[eq:two-nodes-ind\]) does not extend to three or more nodes. Indeed, conditionally on the set $W(v_k)$, the sets $W(v_i) \cap W(v_j), W(v_i) \cap W(v_k)$, and $W(v_j) \cap W(v_k)$ are not mutually independent, and hence neither are the events $\{v_i \sim v_j\}, \{v_i\sim v_k\}$, and $\{v_j \sim v_k\}$, that is, \[eg:tree-nodes-dep\] .
We now provide two identities, which we will use throughout this paper. For any $w \in W$, let $q_{w} := 1 - p_{w}$, and define $\prod_{\al \in \emptyset } q_{\al} = 1$.
\[claim:1\] For any node $u \in V$ and given set $A \subseteq W$, $$\label{eq:prob.A.empty}
\pr[W(u) \cap A = \emptyset | A ] = \prod_{\al \in A} (1-p_{\al}) = \prod_{\al \in A} q_{\al}.$$
Write = = \_[A]{} = \_[A]{} (1-p\_) = \_[A]{} q\_, which is the desired expression.
\[cl:AB\] For any node $u \in V$, and given sets $A \subseteq B \subseteq W$, $$\pr[W(u) \cap A = \emptyset, W(u) \cap B \neq \emptyset | A, B] =
\Big( \prod_{\al \in A} q_{\al} \Big) \Big( 1 - \prod_{\al \in B \setminus A} q_{\beta} \Big) \nonumber = \prod_{\al \in A} q_{\al} - \prod_{\bt \in B} q_{\bt}.$$
The sets $A$ and $B \setminus A$ are disjoint. The result follows from (\[eq:prob.A.empty\]).
It follows from (\[eq:prob.A.empty\]) that for any node $u,v \in V$, $\pr[u \sim v | W(v)] = 1 - \prod_{\al \in W(v)} q_{\al}$. Taking the expectation over $W(u)$ yields $$\label{eq.connection}
\pr[u \sim v ] = \sum_{W(u) \subseteq W} \pr[W(u)] \Big(1 - \prod_{\al \in W(v)} q_{\al}\Big) = 1 - \prod_{w \in W} (1-p_w^2).$$
It follows from (\[eq:prob.A.empty\]) that for any node $u,v \in V$, $$\pr[u \sim v | W(v)] = 1 - \prod_{\al \in W(v)} q_{\al}.$$ Taking the expectation over $W(u)$ yields $$\pr[u \sim v ] = \sum_{W(u) \subseteq W} \pr[W(u)] \Big(1 - \prod_{\al \in W(v)} q_{\al}\Big) = 1 - \prod_{w \in W} (1-p_w^2).$$
Auxiliary process on general random intersection graphs {#sec:branching.process}
=======================================================
Our analysis for the emergence of a giant component is inspired by the approach described in [@alon-2000-probabilistic]. The difficulty in analyzing the evolution of the stochastic process defined by equations (\[eq.Avw\]), (\[eq.Wv\]), and (\[eq.connectivity\]) resides in the fact that we need, at least in principle, to keep track of the temporal evolution of the sets of nodes and attributes being explored. This results in a process that is not Markovian. We construct an auxiliary process, which starts at an arbitrary node $v_0 \in V$, and reaches zero for the first time in a number of steps equal to the size of the component containing $v_0$. The process is algorithmically defined as follows.
**Auxiliary Process.** Let us denote by $V_t$ the cumulative set of nodes *visited* by time $t$, which we initialize to $V_0=\{v_0\}$, and set $W(v_0) = \{ v \not = v_0 : W(v) \cap W(v_0) \not = \emptyset \}$. Starting with $Y_0=1$, the process evolves as follows: For $t=1,2,3,\dots,n-1$ and $Y_t>0$, pick a node $v_t$ uniformly at random from the set $V \setminus V_{t-1}$ and update the set of visited nodes $V_t = V_{t-1} \cup \{v_t\}$. Denote by $W(v_t) = \{w \in W \mid A_{v_t,w} = 1\}$ the set of features associated to node $v_t$, and define $$Y_t = \left | \Big\{ v \in V \setminus V_t \mid W(v) \cap \cup_{\tau=0}^t W(v_\tau)\neq \emptyset \Big\}
\right |.$$ The random variable $Y_t$ counts the number of nodes outside the set of visited nodes $V_t$ that are connected to $V_t$. Following [@alon-2000-probabilistic], we call $Y_t$ the number of *alive* nodes at time $t$. We note that we do not need to keep track of the actual list of neighbors of $V_t$ $$\label{eq:neighbor-set}
\Big\{ v \in V \setminus V_t \mid W(v) \cap \cup_{\tau=0}^t W(v_\tau)\neq \emptyset \Big\},$$ as in [@alon-2000-probabilistic], because every node in $V \setminus V_t$ is equally likely to belong to the set (\[eq:neighbor-set\]). As a result, each time we need a random node from (\[eq:neighbor-set\]), we pick a node uniformly at random form $V \setminus V_t$.
To understand why this process is useful, notice that by time $t$, we know that the size of the component containing $v_0$ is at least as large as the number of visited nodes $V_t$ plus the number $Y_t$ of neighbors of $V_t$ not yet visited. Once the number $Y_t$ of neighbors connected to $V_t$ but not yet visited drops to zero, the size of $V_t$ is equal to the size of the component containing $v_0$. We formalize this last statement by introducing the stopping time $$\label{eq:stopping.time}
T(v_0) = \inf \{ t > 0: Y_t = 0 \},$$ whose value is $|C(v_0)|$.
Finally, our analysis of that process requires us to keep track of the history of the feature sets uncovered by the process \[eq:history\] \_t = {W(v\_0), W(v\_1), …, W(v\_t)}.
To define this process, we label at each time the nodes in $V$ as *alive*, *neutral*, or *dead*. Specifically, we initialize the process at time $t=0$ by labeling all the nodes in $V$ as neutral. Then we pick one node $v_0$ uniformly at random among the neutral node, and label all the nodes connected to $v_0$ as alive. Finally, label $v_0$ as dead. At each subsequent time $t \geq 1$, pick one node $v_t$ uniformly at random from the alive nodes and label all the neutral nodes connected to $v_t$ as alive, and $v_t$ as dead.
We describe this process in terms of the random variables $(N_t,Y_t,Z_t)$: $N_t$ the number of remaining neutral nodes at the end of iteration $t$, $Y_t$ the number of alive nodes at the end of iteration $t$, and $Z_t$ the number of neutral nodes that become alive in the course of iteration $t$. These random variables satisfy $Y_0 =1, N_0 = n-1$ and the recursion relation $$\begin{aligned}
%\label{eq:y0} Y_0 &\equiv& 1,\\
\label{eq:yt} Y_{t} &=& Y_{t-1} + Z_{t}- 1, \textrm{ for } t \geq 1,\\
\label{eq:ntsum} N_{t} &=& n - 1 - \sum_{\tau=1}^{t} Z_{\tau}, \textrm{ for } t \geq 1.\end{aligned}$$ Moreover $$\begin{aligned}
\label{eq:yty0}Y_t - Y_0 &=& \sum_{\tau=1}^{t} Z_{\tau} - t, \textrm{ for } t \geq 1,\\
\label{eq:nt} N_t &=& n - t - Y_{t}, \textrm{ for } t \geq 0.\end{aligned}$$ Define the stopping time $$\label{eq:stopping.time}
T(v_0) = \inf \{ t > 0: Y_t = 0 \}.$$ By construction of the process $\{Y_t\}$, the size of the connected component $C(v_0)$, containing $v_0$, is $T(v_0)$. Note that the previously defined process on a graph stops at the first $t$ for which $Y_t = 0$.
Process description in terms of random variable $Y_t$
-----------------------------------------------------
As in [@bloznelis-2009-component], we denote the cumulative feature set associated to the sequence of nodes $v_0, \dots, v_t$ from the auxiliary process by W\_[\[t\]]{} := \_[=0]{}\^[t]{} W(v\_). We will characterize the process $\{Y_t\}_{t \geq 0}$ in terms of the number $Z_t$ of newly discovered neighbors to $V_t$. The latter is directly related to the increment, defined by of the process $Y_t$ \[eq.increment.yt\] Z\_t = Y\_t - Y\_[t-1]{} + 1, where the term +1 reflects the fact that one node, $Y_{t-1}$ decreases by one when the node $v_t$ becomes a visited node at time $t$. The events that any given node, which is neither visited nor alive, becomes *alive* at time $t$ are conditionally independent given the history ${\mathcal H}_t$, since each event involves a different subsets of the indicator random variables $\{ A_{v,w} \}$. In light of Claim \[cl:AB\], the conditional probability that a node $u$ becomes alive at time $t$ is $$\begin{aligned}
\label{eq:rate}
\nonumber r_t &:=& \pr[u \sim v_t, u \not\sim v_{t-1}, u \not\sim v_{t-2}, \dots, u \not\sim v_0 | \Hcall_t ] \\
\nonumber &=& \pr[W(u) \cap W(v_t) \neq \emptyset, W(u) \cap W_{[t-1]} = \emptyset | \Hcall_t ]\\
\nonumber &=& \pr[W(u) \cap W(v_t) \neq \emptyset, W(u) \cap W_{[t-1]} = \emptyset | W(v_t), W_{[t-1]} ]\\
\nonumber &=& \prod_{\al \in W_{[t-1]}} q_{\al} - \prod_{\bt \in W_{[t]}} q_{\bt}\\
&=& \phi_{t-1} - \phi_{t},\end{aligned}$$ where we set $\phi_{t} := \prod_{\al \in W_{[t]}} q_{\al}$, and use the convention $W_{[-1]} = W(\emptyset) \equiv \emptyset$ and $\phi_{-1} \equiv 1$. Observe that the probability (\[eq:rate\]) does not depend on $u$. Hence the number of new alive nodes at time $t$ is, conditionally on the history $\Hcall_t $, a Binomial distributed random variable with parameters $r_t$ and \[eq:nt\] N\_t = n - t - Y\_[t]{}. Formally, $$\label{eq:zt+1}
Z_{t+1} | \Hcall_t \sim \mbox{Bin}(N_t,r_t).$$ This allows us to describe the distribution of $Y_t$ in the next lemma.
\[lm:nt\] For times $t \geq 1$, the number of alive nodes satisfies $$Y_t | \Hcall_{t-1} \sim \bin \Big(n-1, 1-\prod_{\tau = 0}^{t-1}(1-r_\tau) \Big) -t +1.$$
The proof of this lemma requires us to establish the following result first.
\[lm:binbin\] Let random variables $\Lm_1, \Lm_2$ satisfy: $\Lm_1 \sim \bin(m,\nu_1)$ and $\Lm_2 \textrm{ given } \Lm_1 \sim \bin(\Lm_1,\nu_2)$. Then marginally $\Lm_2 \sim \bin(m, \nu_1\nu_2)$ and $\Lm_1 - \Lm_2 \sim \bin(m, \nu_1 (1-\nu_2))$.
Let $U_1,\ldots,U_m$ and $V_1,\ldots,V_m$ be i.i.d. Uniform$(0,1)$ random variables. Writing \_1 \_[j=1]{}\^m [I]{}(U\_j \_1) \_2 | \_1 \_[k : U\_k < \_1]{} [I]{}(V\_k \_2), we have that \_2 \_[k=1]{}\^m [I]{}(U\_k \_1) [I]{}(V\_k \_2)\
\_[k=1]{}\^m [I]{}(U\_k \_1 \_2), from which the conclusion follows.
(Proof of Lemma \[lm:nt\]) We prove the assertion on the Lemma by induction in $t$. For $t=0$, $Y_0 = 1$ and $t=1$, $Y_1 = Z_1 ~\sim \bin(n-1,r_0)$. Hence, the Lemma is true for $t = 1$ and $t=0$. Assume that the assertion is true for some $t\geq 1$, $$%\label{eq:ind.eq.1}
%N_t | \Hcall_{t-1}\sim \bin\big(n-1, \prod_{\tau = 0}^{t-1}(1-r_\tau)\big).
Y_t | \Hcall_{t-1} \sim \bin \Big(n-1, 1-\prod_{\tau = 0}^{t-1}(1-r_\tau) \Big) -t +1.$$ From (\[eq:zt+1\]), we have $Z_{t+1} | \Hcall_t \sim \bin(N_t,r_t) = \bin(n-t -Y_t,r_t)$, Now, from (\[eq.increment.yt\]) and Lemma \[lm:binbin\], it follows $$%\label{eq:ind.eq.1}
Y_{t+1} | \Hcall_{t} \sim \bin \Big(n-1, 1-\prod_{\tau = 0}^{t}(1-r_\tau) \Big) - t.$$ Hence, by mathematical induction, the Lemma holds for any $t \geq 0$.
Auxiliary process on general random intersection graphs {#sec:mod.branching.process}
=======================================================
In the Section \[sec:branching.process\], we have described the process for the emergence of a giant component on general RIGs, inspired by the approach for $G_{n,p}$, described in [@alon-2000-probabilistic]. The difficulty of the underlying structure of general RIGs is given by the set of equations (\[eq.Avw\]), (\[eq.Wv\]), and (\[eq.connectivity\]), which protects us to further analyze the component evolution in a general RIG. We define an auxiliary (modified) process starting at an arbitrary node $v_0$, whose stopping time will give the size of the component $C(v_0)$ containing $v_0$. The algorithmic definition and correctness of the process follow.
**Algorithm 1** Set $V_{-1} := \emptyset$ and $\tilde Y_0 = 1$. At each time step $t \geq 0$, draw uniformly at random a node $v_t \in V \setminus V_{t-1}$, and update $V_t = V_{t-1} \cup \{ v_t \}$. Note that $|V_t| = t+1$, for any $t \geq 0$. Once $v_t$ is drawn, obtain its feature set $W(v_t) = \{w \in W \mid A_{v_t,w} = 1\}$, see (\[eq.Wv\]). Keep track of $\tilde Y_{t+1}$, which represents a number of nodes in $V \setminus V_t$ attached to $\cup_{\tau=0}^t W(v_\tau)$, that is, \[eq.yt\] Y\_[t+1]{} = | { v V V\_t W(v) \_[=0]{}\^t W(v\_)} |.
Define the stopping time $$\label{eq:aux.stopping.time}
\tilde T(v_0) = \inf \{ t > 0: \tilde Y_t = 0 \}.$$ Note that the previously defined process on a graph stops at the first $t$ for which $ \tilde Y_t = 0$.
By construction of the process $\{\tilde Y_t\}$, $\tilde T(v_0)$ is the size of the connected component $C(v_0)$, containing $v_0$.
Consider the processes on a general RIGs, defined in Section \[sec:mod.branching.process\] and in Section \[sec:branching.process\]. We argue by induction in $t$, that at every time step $t$ given the history, the distributions $\tilde Y_t | \Hcall_t \sim Y_t | \Hcall_t$, do not differ in these two processes, that is, the auxiliary process defined by **Algorithm 1** correctly estimates the size of $C(v_0)$.
Using (\[eq.yt\]), for simplicity, let us call the increment $\tilde Z_{t+1} := \tilde Y_{t+1} - \tilde Y_t + 1$.
By definition $\tilde Y_0 = 1$, that is, $\tilde Y_0 = Y_0$. Given the history $\Hcall_0 = \{W(v_0)\}$ we have $\tilde Z_1 = \tilde Y_1 $, that is, $\tilde Z_1 \mid \Hcall_0 = \tilde Y_1 \mid \Hcall_0 \sim \bin(n-1,r_0)$, which is identical as $Z_1 \mid \Hcall_0$.
Suppose now, that for the some time $t \geq 0$, for all $0 \leq \tau \leq t$, we have $\tilde Z_{\tau+1} | \Hcall_\tau \sim Z_{\tau+1} | \Hcall_\tau$, and $\tilde Y_{\tau+1} | \Hcall_\tau \sim Y_{\tau+1} | \Hcall_\tau$.
At the beginning of the iteration at time step $t$, given the history $\Hcall_{t-1}$, we have: (i) $t = |V_{t-1}|$ already visited nodes, (ii) $\tilde Y_t$ neighbors of all previously visited nodes belonging to $V_{t-1}$, excluding $V_{t-1}$, itself. From definition given in (\[eq.yt\]), notice that Z\_[t+1]{} \_t = Y\_[t+1]{} - Y\_t + 1 \_t \~(n- t - Y\_t,r\_t) \~(n- t - Y\_t,r\_t) \~Z\_[t+1]{} \_t, which concludes the proof.
Expectation and variance of $\phi_t$
------------------------------------
The history ${\mathcal H}_t$ embodies the evolution of how the features are discovered over time. It is insightful to recast that history in terms of the discovery times $\Gamma_w$ of each feature in $W$. Given any sequence of nodes $v_0,v_1,v_2,\ldots$, the probability that a given feature $w$ is first discovered at time $t < n$ is $$\begin{aligned}
{\mathbb P}[\Gamma_w = t ] & = &
{\mathbb P}[A_{v_t,w}=1,A_{v_{t-1},w}=0,\ldots,A_{{v_0},w}=0] \\
&= & p_w (1-p_w)^t.\end{aligned}$$ If a feature $w$ is not discovered by time $n-1$, we set $\Gamma_w = \infty$ and note that \[\_w = \] = (1-p\_w)\^n. From the independence of the random variables $A_{v,w}$, it follows that the discovery times $\{ \Gamma_w : w\in W \}$ are independent. We now focus on describing the distribution of $\phi_t = \prod_{\al \in W_{[t]}} q_{\al}$. For $t \geq 0$, we have \[eq:phi.geom.\] \_t = \_[W\_[\[t\]]{}]{} q\_= \_[j=0]{}\^t \_[s(v\_j) S\[j-1\]]{} q\_ \_[j=0]{}\^t \_[w W]{} q\_w\^[[I]{}(\_w=j)]{} = \_[w W]{} q\_w\^[ (\_w t)]{}. Using the fact that for a $B \sim \textrm{Bernoulli}(r)$, the expectation ${\mathbb E}[a^B] = 1-(1-a)r$, we can easily calculate the expectation of $\phi_t$ $$\begin{aligned}
\label{eq.phigeomexp}
\nonumber \E[\phi_t] &=& \E[ \prod_{w \in W} q_w^{ \ind(\Gamma_w \leq t)} ] = \prod_{w \in W} \Big(1- (1-q_w)\pr[\Gamma_w \leq t ]\Big)\\
&=& \prod_{w \in W} \Big(1 -(1-q_w)(1-q_w^{t+1}) \Big).\end{aligned}$$ The concentration of $\phi_0$ will be crucial for the analysis of the supercritical regime, Subsection \[sub.supercritical.regime\]. Hence, we here provide $\E[\phi_0]$ and $\E[\phi_0^2]$. From (\[eq.phigeomexp\]) it follows \[eq.phi.zero.exp.\] = \_[w W]{} (1 - p\_w\^2) = 1 - \_[w W]{} p\_w\^2 + ø(\_[w W]{} p\_w\^2). Moreover, from (\[eq:phi.geom.\]) it follows $$\begin{aligned}
\label{eq.phi.zero.sqaure.exp.}
\nonumber \E[\phi_0^2] &=& \E[ \prod_{w \in W} q_w^{ 2\ind(\Gamma_w \leq 0)} ] = \prod_{w \in W} \Big(1- (1-q_w^2)\pr[\Gamma_w = 0]\Big) = \prod_{w \in W} \Big(1- (1-q_w^2) p_w \Big) \\
%&=& \prod_{w \in W} \Big(1- 2 p_w^2 + p_w^3 \Big) = 1 - 2 \sum_{w \in W} p_w^2 + \sum_{w \in W} p_w^3 + \o(\sum_{w \in W} 2p_w^2 - p_w^3).
&=& \prod_{w \in W} \Big(1- 2 p_w^2 + p_w^3 \Big) = 1 - 2 \sum_{w \in W} p_w^2 + \o(\sum_{w \in W} p_w^2).\end{aligned}$$
Giant component {#sec:giant.component}
===============
With the process $\{Y_t\}_{t \geq 0}$ defined in the previous section, we analyze both the subcritical and supercritical regime of our random intersection graph by adapting the percolation based techniques to analyze Erdős-Rényi random graphs [@alon-2000-probabilistic]. The technical difficulty in analyzing that stopping time rests in the fact that the distribution of $Y_t$ depends on the history of the process, dictated by the structure of the general RIG. In the next two subsections, we will give conditions on non-existence, that is, on existence and uniqueness of the giant component in general RIGs.
Subcritical regime {#sub.subcritical.regime}
------------------
\[thm.sub.subcritical.regime\] Let \_[w W]{} p\_w\^3 = O(1/n\^2) p\_w = O(1/n) w. For any positive constant $c<1$, if $\sum_{w \in W} p_w^2 \leq c/n$, then all components in a general random intersection graph $G(n,m,\boldsymbol{p})$ are of order $O(\log n)$, with high probability[^6].
We generalize the techniques used in the proof for the sub-critical case in $G_{n,p}$ presented in [@alon-2000-probabilistic]. Let $T(v_0)$ be the stopping time define in (\[eq:stopping.time\]), for the process starting at node $v_0$ and note that $T(v_0)=|C(v_0)|$. We will bound the size of the largest component, and prove that under the conditions of the theorem, all components are of order $O(\log n)$, . For all $t \geq 0$, $$\begin{aligned}
\nonumber
{\mathbb P}[ T(v_0)>t ] & = & {\mathbb E} \left [ \pr\left[T(v_0)>t \mid \Hcall_t \right] \right ] \leq
{\mathbb E} \left [ \pr[Y_t>0 \mid \Hcall_t ] \right ]\\
&=& {\mathbb E} \left [ \pr[\bin(n-1,1 - \prod_{\tau=0}^{t-1} (1-r_\tau)) \geq t \mid \Hcall_t ] \right ].
\label{eq:prob.stopping.time}\end{aligned}$$ Bounding from above, which can easily be proven by induction in $t$ for $r_\tau \in [0,1]$, we have \[eq.rate.upper.bound\] 1 - \_[=0]{}\^[t-1]{} (1-r\_) \_[=0]{}\^[t-1]{} r\_= \_[=0]{}\^[t-1]{} (\_[-1]{}-\_) = 1 - \_[t-1]{}. By using stochastic ordering of the Binomial distribution, both in $n$ and in $\sum_{\tau=0}^{t-1} r_\tau$, and for any positive constant $\nu$, which is to be specified later, it follows \[eq:prtt\] && =\
&=&\
&& +\
&&\
&& + . Furthermore, using the fact that the event $\{ 1 - \phi_{t-1} < (1 - \nu) t/n \}$ is $\Hcall_t$-measurable, together with the stochastic ordering of the binomial distribution, we obtain . Taking the expectation with respect to the history ${\mathcal H}_t$ in (\[eq:prtt\]) yields $${\mathbb P}[T(v_0) > t] \leq \pr[\bin(n, (1-\nu) t/n) \geq t] + \pr[1 - \phi_{t-1} \geq (1-\nu)t/n ].$$ For $t = K_0 \log n$, where $K_0$ is a constant large enough and independent on the initial node $v_0$, the Chernoff bound ensures that $\pr[\bin(n, (1-\nu) t/n) \geq t] = o(1/n)$. To bound $\pr[1 - \phi_{t-1} \geq (1-\nu) t/n \mid \Hcall_t ]$, use (\[eq:phi.geom.\]) to obtain $$\begin{aligned}
\{ 1 - \phi_{t-1} \geq (1-\nu) t/n \} &=&\left \{ \prod_{w \in W} q_w^{ \ind(\Gamma_w \leq t)} \leq 1-\frac{(1-\nu) t}{n}
\right \}\\
&=& \left \{ \sum_{w \in W} \log \left ( \frac{1}{1-p_w} \right ) \ind(\Gamma_w \leq t) \geq -\log
\left ( 1-\frac{(1-\nu) t}{n} \right ) \right \}.\end{aligned}$$ Linearize $-\log(1-(1-\nu) t/n) = (1-\nu) t/n +o(t/n)$ and define the bounded auxiliary random variables $X_{t,w} = n \log(1/(1-p_w)) \ind(\Gamma_w \leq t)$. Direct calculations reveal that $$\begin{aligned}
\nonumber \E[X_{t,w}] &=& n \log \Big(\frac{1}{1-p_w} \Big) (1-q_w^t)
= n \Big( p_w + \o(p_w) \Big) \Big(1-(1-p_w)^t \Big)\\
&=& n \Big( p_w + \o(p_w) \Big) \Big(tp_w + \o(t p_w)) \Big)
= n t p_w^2 + \o \Big( n t p_w^2\Big), %\label{eq:bernstein}\end{aligned}$$ which implies \_[w W]{} = n t \_[w W]{} p\_w\^2 + ø( n t \_[w W]{} p\_w\^2). Thus under the stated condition that n \_[w W]{} p\_w\^2 c < 1, it follows that $0 < (1-c) t \leq t - \sum_{w \in W} \E[ X_{t,w}]$. In light of Bernstein’s inequality [@bernstein-1924], we bound $$\begin{aligned}
&& \pr[1 - \phi_{t-1} \geq (1-\nu) t/n] = {\mathbb P} \left [ \sum_{w \in W} X_{t,w} \geq (1-\nu) t \right ]
\leq {\mathbb P} \left [ \sum_{w \in W} \big( X_{t,w}-{\mathbb E}[X_{t,w}] \big) \geq (1-\nu - c) t \right ] \nonumber \\
&\leq&
\exp\Big( -\frac{ \frac{3}{2} ((1-\nu - c) t )^2}{3 \sum_{w \in W} \mbox{Var}[X_{t,w}] + n t\max_w \{p_w\} (1+\o(1)) } \Big). \label{eq:bernstein}\end{aligned}$$ Since $$\begin{aligned}
\nonumber \E[X_{t,w}^2] &=& \Big(n \log \Big(\frac{1}{1-p_w} \Big) \Big)^2 (1-q_w^t) = n^2 \Big( p_w + \o(p_w) \Big)^2 \Big(1-(1-p_w)^t \Big)\\
&=& n^2 \Big( p_w^2 + \o(p_w^2) \Big) \Big(tp_w + \o(t p_w)) \Big) = n^2 t p_w^3 + \o \Big( n^2 t \sum_{w \in W} p_w^3\Big), \end{aligned}$$ it follows that for some large constant $K_1>0$ \_[w W]{} \[X\_[t,w]{}\] \_[w W]{} = n\^2 t \_[w W]{} p\_w\^3 + ø( n\^2 t \_[w W]{} p\_w\^3) K\_1 t. Finally, the assumption of the theorem implies that there exists constant $K_2 > 0$ such that n \_[w W]{} p\_w K\_2. Substituting these bounds into (\[eq:bernstein\]) yields ( - t ), and taking $\nu \in (0,1-c)$ and $t=K_3 \log n$ for some constant $K_3$ large enough and not depending on the initial node $v_0$, we conclude that $\pr[1 - \phi_{t-1} \geq (1-\nu) t/n] = o(n^{-1})$, which in turn implies that taking constant $K_4 = \max \{ K_0,K_3\}$, ensures that = ø(1/n) for any initial node $v_0$. Finally, a union bound over the $n$ possible starting values $v_0$ implies that \[\_[v\_0 V]{} T(v\_0) > K\_4 n \] n ø(n\^[-1]{}) = o(1), which implies that all connected components in the random intersection are of size $O(\log n)$, .
**Remarks.** We now consider the conditions of the theorem. From the Cauchy-Schwarz inequality, we obtain $\Big(\sum_{w \in W} p_w^3\Big) \Big(\sum_{w \in W} p_w\Big) \geq \Big(\sum_{w \in W} p_w^2\Big)^2$. Moreover, given that $\sum_{w \in W} p_w^3 = O(1/n^2)$ and $p_w = O(1/n)$, it follows $\sum_{w \in W} p_w^2 = \Omega(\sqrt{m/n^3})$. Hence, for $\sum_{w \in W} p_w^2 = c/n$, when $c<1$, it follows $m = \Omega(n)$, which is consistent with the results in [@behrisch-2007-component] on the non-existence of a giant component in a uniform RIG.
Supercritical regime {#sub.supercritical.regime}
--------------------
We now turn to the study of the supercritical regime in which $\lim_{n \rightarrow \infty} n \sum_{w \in W} p_w^2 = c > 1$.
\[thm.super.subcritical.regime\] Let \_[w W]{} p\_w\^3 = o() p\_w = o(), w. For any constant $c>1$, if $\sum_{w \in W} p_w^2 \geq c/n$, then there exists a unique largest component in $G(n,m,\boldsymbol{p})$, of order $\Theta(n)$. Moreover, the size of the giant component is given by $n \zeta_c (1 + \o(1))$, where $\zeta_c$ is the solution in $(0,1)$ of the equation $1-e^{-c \zeta} = \zeta$, while all other components are of size $O(\log n)$.
**Remarks.** The conditions on $p_w$ and $\sum_w p_w^3$ are weaker than ones in the case of the sub-critical regime.
The proof proceeds as follows. The first step is to bound, both from above and below, the value $1 - \prod_{\tau = 0}^{t-1}(1-r_\tau)$ that governs the behavior the branching process $\{Y_t\}_{t \geq 0}$, see Lemma \[lm:nt\]. With the lower bound, we show the emergence with high probability of at least one giant component of size $\Theta(n)$. We use the upper bound to prove uniqueness of the giant component. Technically, we make use of these bounds to compare our branching process to branching processes arising in the study of Erdős-Réneyi random graphs.
We start by bounding $1 - \prod_{\tau = 0}^{t-1}(1-r_\tau)$. The upper bounds $\sum_{\tau=0}^{t-1} r_{\tau}$ has been previously established in (\[eq.rate.upper.bound\]). For the lower bound, we apply Jensen’s inequality to the function $\log (1-x)$ to get $$\begin{aligned}
\nonumber \log \prod_{\tau = 0}^{t-1} (1-r_\tau) &=& \sum_{\tau = 0}^{t-1} \log (1-r_\tau)
= \sum_{\tau = 0}^{t-1} \log \Big( 1- (\phi_{\tau-1} - \phi_{\tau}) \Big) \\
%\leq t \log\Big( 1- \frac{1}{t}\sum_{\tau=0}^{t-1}(\phi_{\tau-1} - \phi_{\tau})\Big) \\
%&=& t \log\Big( 1- \frac{\phi_{-1} - \phi_{t-1}}{t}\Big) = t \log\Big( 1- \frac{1 - \phi_{t-1}}{t}\Big).
&\leq& t \log\Big( 1- \frac{1}{t}\sum_{\tau=0}^{t-1}(\phi_{\tau-1} - \phi_{\tau})\Big) = t \log\Big( 1- \frac{1 - \phi_{t-1}}{t}\Big).\end{aligned}$$ In light of (\[eq:phi.geom.\]), $\phi_t$ is decreasing in $t$, and hence $$\label{eq:rate.sandwich}
1- \Big( 1- \frac{1 - \phi_{0}}{t}\Big)^t \leq 1- \Big( 1- \frac{1 - \phi_{t-1}}{t}\Big)^t \leq 1 - \prod_{\tau = 0}^{t-1} (1-r_\tau) \leq \sum_{\tau=0}^{t-1} r_{\tau} = 1 - \phi_{t-1}.$$ To further bound $1- \Big( 1- \frac{1 - \phi_{0}}{t}\Big)^t$, consider the function $f_t(x) = 1 - (1 - x/t)^t$ for $x$ in a neighborhood of the origin and $t \geq 1$. For any fixed $x$, $f_t(x)$ decreases to $1-e^{-x}$ as $t$ tends to infinity. The latter function is concave, and hence for all $x \leq \varepsilon$, x f\_t(x). Note that $(1-e^{-\varepsilon})/\varepsilon$ can be made arbitrary close to one by taking $\varepsilon$ small enough. Furthermore, $f_t(x)$ is increasing in $x$ for fixed $t$. From (\[eq:phi.geom.\]), $1-\phi_0 \leq 1- \phi_t$, hence $1- ( 1- \frac{1 - \phi_{0}}{t})^t \leq 1- ( 1- \frac{1 - \phi_{t-1}}{t})^t$. Looking closer at $1 - \phi_0$, from (\[eq.phi.zero.sqaure.exp.\]) and (\[eq.phi.zero.exp.\]), by using Chebyshev inequality, with $\sum_{w \in W} p_w^2 = c/n$, it follows that $\phi_0$, is concentrated around its mean $\E[\phi_0] = c/n$. That is, for any constant $\dl>0$, $\phi_0 \in ((1 - \delta)c/n, (1 + \dl)c/n)$, with probability $1 - o(1/n)$. We conclude that for any $\dl>0$ there is $\eps>0$ such that $(c - \dl)\frac{1 - e^{-\eps}}{\eps} > 1$, since constant $c>1$. Moreover, since $\lim_{\eps \to 0} \frac{1 - e^{-\eps}}{\eps} = 1$, by choosing $\eps$ sufficiently small, $\frac{1 - e^{-\eps}}{\eps}$ can be arbitrarily close to $1$. It follows that $1 - \prod_{\tau = 0}^{t-1} (1-r_\tau) > c'/n$, for some constant $c>c'>1$ arbitrarily close to $c$. Hence, the branching process on RIG is stochastically lower bounded by the $\bin(n-1, c'/n)$, which stochastically dominates a branching process on $G_{n,c'/n}$. Because $c^\prime > 1$, there exists a giant component of size $\Theta(n)$ in $G_{n,c'/n}$. This implies that the stopping of the branching process associated to $G_{n,c'/n}$ is $\Theta(n)$ with high probability, and so is the stopping time $T_{v}$ for some $v \in V$, which implies that there is a giant component in a general RIG, .
Let us look closer at the size of that giant component. From the representation (\[eq:phi.geom.\]) for $\phi_{t-1}$, consider the previously introduced random variables $X_{t,w} = n \log(1/(1-p_w)) \ind(\Gamma_w \leq t)$. Similarly, as in the proof of the Theorem \[thm.sub.subcritical.regime\], it follows that under the conditions of the theorem there is a positive constant $\dl>0$ such that $\sum_w X_{t,w}$ is concentrated within $(1\pm \delta)\sum_w\E[X_{t,w}] =
(1 \pm \delta)c/n$, with probability $1 - o(1)$. Hence, there exists $p^{+} = c^{+}/n$, for some constant $c^{+}>c>1$, such that $1 - \phi_{t-1} \leq 1 - (1 - p^{+})^t$, which is equivalent to $- \log \phi_{t-1} \leq t \log (1 - p^{+}) = tp^{+} + \o(tp^{+}) = t c^{+}/n + \o(t/n)$. Similarly, the concentration of $\phi_{t-1}$ implies that there exists $p^{-} = c^{-}/n$, with $c > c^{-}>1$, such that $1 - (1 - p^{-})^t \leq 1- ( 1- (1 - \phi_{t-1})/t)^t$, which implies that $- \log \phi_{t-1} \geq t \log (1 - p^{-}) = tp^{-} + \o(tp^{-}) = t c^{-}/n + \o(t/n)$. Combining the upper and lower bound, we conclude that with probability $1 - o(1)$, the rate of the branching process on RIG is bracketed by \[eq:rate.better.sandwich\] 1 - (1 - p\^[-]{})\^t 1 - \_[= 0]{}\^[t-1]{} (1-r\_) 1 - (1 - p\^[+]{})\^t. The stochastic dominance of the Binomial distribution together with (\[eq:rate.better.sandwich\]), implies \[eq:sandwich.stopping.time\] &&\
&& . In light of (\[eq:rate.better.sandwich\]), the branching process $\{Y_t\}_{t \geq 0}$ associated to a RIG is stochastically bounded from below and form above by the branching processes associated to $G_{n,p^{-}}$ and $G_{n,p^{+}}$, respectively (for the analysis on an Erdős-Rényi graph, see [@alon-2000-probabilistic]). Since both $c^{-}, c^{+}>1$, there exist giant components in both $G_{n,p^{-}}$ and $G_{n,p^{+}}$, **whp**.
In [@hofstad-notes], it has been shown that the giant components in $G_{n,\lm/n}$, for $\lm>1$, is unique and of size $\approx n \zl$, where $\zeta_\lambda$ is the unique solution from $(0,1)$ of the equation \[eq.tree.size\] 1-e\^[-]{} = . Moreover, the size of the giant component in $G_{n,\lm/n}$ satisfies the central limit theorem \[eq:clt.distr.\] (0,). From the definition of the stopping time, see (\[eq:prob.stopping.time\]), and since (\[eq:sandwich.stopping.time\]) and (\[eq:clt.distr.\]), it follows there is a giant component in a RIG, of size, at least, $n \zl (1 - \o(1))$, . Furthermore, the stopping times of the branching processes associated to $G_{n,p^{-}}$ and $G_{n,p^{+}}$ are approximately $\zeta n$, where $\zeta$ satisfy (\[eq.tree.size\]), with $\lm^{-} = n p^{-}$ and $\lm^{+} = n p^{+}$, respectively. These two stopping times are close to one another, which follows from analyzing the function $F(\zeta,c) = 1-\zeta - e^{-c\zeta}$, where $(\zeta,c)$ is the solution of $F(\zeta,c) = 0$, for given $c$. Since all partial derivatives of $F(\zeta,c)$ are continuous and bounded, the stopping times of the branching processes defined from $G_{n,p^{-}}$, $G_{n,p^{+}}$ are ‘close’ to the solution of (\[eq.tree.size\]), for $\lm=c$. From (\[eq:sandwich.stopping.time\]), the stopping time of a RIG is bounded by the stopping times on $G_{n,p^{-}}$, $G_{n,p^{+}}$. We conclude by proving that , the giant component of a RIG is unique by adapting the arguments in [@alon-2000-probabilistic] to our setting. Let us assume that there are at least two giant components in a RIG, with the sets of nodes $V_1, V_2 \subset V$. Let us create a new, independent ‘sprinkling’ $\widehat{\textrm{RIG}}$ on the top of our RIG, with the same sets of nodes and attributes, while $\hat{p}_w = p_w^\gamma$, for $\gamma>1$ to be defined later. Now, our object of interest is $\textrm{RIG}_{new} = \textrm{RIG} \cup \widehat{\textrm{RIG}}$. Let us consider all $\Theta(n^2)$ pairs $\{v_1,v_2\}$, where $v_1 \in V_1, v_2 \in V_2$, which are independent in $\widehat{\textrm{RIG}}$, (but not in RIG), hence the probability that two nodes $v_1, v_2 \in V$ are connected in $\widehat{\textrm{RIG}}$ is given by \[eq:edge.prob.in.righat\] 1 - \_w (1 - \_w\^2) = 1 - \_w (1 - p\_w\^[2]{}) = \_w p\_w\^[2]{} + ø(\_w p\_w\^[2]{}), which is true, since $\gamma>1$ and $p_w = O(1/n)$ for any $w$. Given that $\sum_w p_w^2 = c/n$, we choose $\gamma>1$ so that $\sum_w p_w^{2\gamma} = \omega(1/n^2)$. Now, by the Markov inequality, there is a pair $\{v_1,v_2\}$ such that $v_1$ is connected to $v_2$ in $\widehat{\textrm{RIG}}$, implying that $V_1, V_2$ are connected, **whp**, forming one connected component within $\textrm{RIG}_{new}$. From the previous analysis, it follows that this component is of size at least $2n \zl (1 - \dl)$ for any small constant $\dl>0$. On the other hand, the probabilities $p_w^{new}$ in $\textrm{RIG}_{new}$ satisfy p\_w\^[new]{}= 1 - (1-p\_w)(1-\_w) = p\_w + \_w(1-p\_w) = p\_w + p\_w\^(1-p\_w) = p\_w (1 + ø(1)), which is again true, since $\gamma>1$ and $p_w = O(1/n)$ for any $w$. Thus, \_[w W]{} (p\^[new]{}\_w)\^2 = \_[w W]{} p\_w\^2 + (\_[w W]{} p\_w\^[1+]{} (1 - p\_w)) = \_[w W]{} p\_w\^2(1+ø(1)) = c/n + o(1/n). Given that the stopping time on RIG is bounded by the stopping times on $G_{n,p^{-}}$, $G_{n,p^{+}}$, and from its continuity, it follows that the giant component in $\textrm{RIG}_{new}$ cannot be of size $2n \zl (1 - \dl)$, which is a contradiction. Thus, there is only one giant component in RIG, of size given by $n \zeta_c (1 + \o(1))$, where $\zeta_c$ satisfies (\[eq.tree.size\]), for $\lm = c$. Moreover, knowing behavior of $G_{n,p}$, from (\[eq:sandwich.stopping.time\]), it follows that all other components are of size $O(\log n)$.
Conclusion {#sec.conclusion}
==========
The analysis of random models for bipartite graphs is important for the study of social networks, or any network formed by associating nodes with shared attributes. In the random intersection graph (RIG) model, nodes have certain attributes with fixed probabilities. In this paper, we have considered the general RIG model, where these probabilities are represented by a set of probabilities $\boldsymbol{p} = \{p_w\}_{w \in W}$, where $p_w$ denotes the probability that a node is attached to the attribute $w$.
We have analyzed the evolution of components in general RIGs, giving conditions for existence and uniqueness of the giant component. We have done so by generalizing the branching process argument used to study the birth of the giant component in Erdős-Rényi graphs. We have considered a dependent, inhomogeneous Galton-Watson process, where the number of offspring follows a binomial distribution with a different number of nodes and different rate at each step during the evolution. The analysis of such a process is complicated by the dependence on its history, dictated by the structure of general RIGs. We have shown that in spite of this difficulty, it is possible to give stochastic bounds on the branching process, and that under certain conditions the giant component appears at the threshold $n \sum_{w \in W} p_w^2 = 1$, with probability tending to one, as the number of nodes tends to infinity.
Acknowledgments {#acknowledgments .unnumbered}
===============
Part of this work was funded by the Department of Energy at Los Alamos National Laboratory under contract DE-AC52-06NA25396 through the Laboratory-Directed Research and Development Program, and by the National Science Foundation grant CCF-0829945. Nicolas W. Hengartner was supported by DOE-LDRD 20080391ER.
[^1]: Theoretical Division, and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM 87545, USA, `[email protected]`,
[^2]: Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA, `[email protected]`,
[^3]: Statistical Sciences Group Los Alamos National Laboratory, NM 87545, USA, `[email protected]`,
[^4]: School of Mathematical Sciences Claremont Graduate University, Claremont, CA 91711, USA, `[email protected]`.
[^5]: Note that $p_w$’s do not sum up to $1$. Moreover, we can eliminate the cases $p_w = 0$ and $p_w = 1$. These two cases respectively correspond when none or all nodes $v$ are attached to the attribute $w$.
[^6]: We will use the notation “with high probability” and denote as , meaning with probability $1- o(1)$, as the number of nodes $n \to \infty$.
|
---
abstract: 'The ultrafast non-radiative relaxation of a molecular ensemble coupled to a cavity mode is considered theoretically and by real-time quantum dynamics. For equal coupling strength of single molecules to the cavity mode, the non-radiative relaxation rate from the upper to the lower polariton states is found to strongly depend on the number of coupled molecules. For $N>2$ molecules, the $N-1$ dark light-matter states between the two optically active polaritons feature true collective conical intersection crossings, whose location depends on the internal atomic coordinates of each molecule in the ensemble, and which contribute to the ultrafast non-radiative decay from the upper polariton. At least $N=3$ coupled molecules are necessary for cavity-induced collective conical intersections to exist and, for identical molecules, they constitute a special case of the Jahn-Teller effect.'
author:
- Oriol Vendrell
bibliography:
- 'cites.bib'
title: 'Collective conical intersections through light-matter coupling in a cavity'
---
The interaction of atoms and molecules with quantized light has the potential to open new routes towards manipulating their physical and chemical properties, and towards the development of hybrid matter-light systems with new attributes . Over the past few years, ground breaking experiments that realize the aforementioned scenario using, e.g., microcavities, have demonstrated the effective tuning of reaction rates and probabilities [@hut12:1624], energy transfer rates among different molecular species [@zho17:9034] and of molecular vibrations [@geo15:281; @geo16:153601]. A growing body of theoretical results has lead, among others, to propose mechanisms to suppress [@gal16:13841] and catalyze chemical processes [@gal17:136001] through a cavity mode, or modify the non-adiabatic dynamics of a single molecule strongly coupled to an electromagnetic mode [@kow16:2050].
Experimentally, it has been observed that photo-excitation of the upper polariton branch (UPB) in a coupled cavity-matter system is followed by population transfer to the lower polariton branch (LPB) before light emission from the UPB can take place [@col11:3691; @sch13:125]. Time-resolved measurements in hybrid organic dye-molecule systems indicate that population transfer from the UPB to the LPB occurs within a time-scale of tens to hundreds of femtoseconds [@sch13:125], orders of magnitude shorter than the radiative life-time of the molecular excited states in isolation. Theoretical predictions based on incoherent relaxation rates obtained by Fermi’s golden rule and based on modelling phonons coupled to the polaritonic excitation also predict relaxation rates of the order of tens to hundreds of femtoseconds [@agr03:85311; @sae18:arXiv].
Even though it is well established that the ultrafast relaxation rates of molecular polaritons arise from vibronic interactions of the participant molecules [@agr03:85311; @col11:3691; @rib18:6325; @sae18:arXiv], a microscopic, real-time description of such phenomena, which additionally sheds light onto the connection with standard descriptions of non-radiative phenomena in chemical systems, is still missing. In this work, the vibronic interactions leading to ultrafast non-radiative decay of a molecular ensemble coupled a single electromagnetic mode will be discussed theoretically and on the basis of numerically converged quantum dynamics simulations. It will be shown how the vibronic origin of the ultrafast relaxation is of similar nature as for isolated molecular excitations and related to the existence of collective conical intersections (CCI) among dark states, whose topological properties will be discussed and compared to their intramolecular counterparts.
The starting point is an ensemble of non-interacting diatomic molecules aligned with the polarization axis of the quantized light mode (also referred throughout the paper as cavity mode). The Hamiltonian for this system reads $$\begin{aligned}
\label{eq:ham1}
\hat{H} & = \hat{T}_{n} + \hat{H}_\mathsf{el} + \hat{H}_\mathsf{cav} +
\hat{H}_\mathsf{las},
\end{aligned}$$ where $\hat{T} = \sum_\kappa^N \hat{t}_n^{(\kappa)}$ is the sum of nuclear kinetic energy operators for each $\kappa$-th molecule, $\hat{H}_\mathsf{el} = \sum_\kappa^N \hat{h}_\mathsf{el}^{(\kappa)}$ is the sum of all other intramolecular Hamiltonian terms for each molecule $$\begin{aligned}
\label{eq:ham2}
\hat{h}_\mathsf{el}^{(\kappa)} & =
\hat{t}_{e}^{(\kappa)} +
\hat{v}_{ee}^{(\kappa)} +
\hat{v}_{en}^{(\kappa)} +
\hat{v}_{nn}^{(\kappa)},
\end{aligned}$$ $\hat{H}_\mathsf{cav}$ is the cavity and cavity-ensemble Hamiltonian, and $\hat{H}_\mathsf{las}$ describes the eventual coupling to an external laser field. The terms in Eq. (\[eq:ham2\]) correspond to the $\kappa$-th electronic kinetic energy and the Coulombic terms represent the electron-electron repulsion, electron-nuclei attraction and nuclei-nuclei repulsion, respectively. The cavity Hamiltonian is given by [@Faisal_1987; @gal15:41022; @fli17:3026; @ven18:55] $$\begin{aligned}
\label{eq:ham3}
\hat{H}_\mathsf{cav} & =
\hbar\omega_c\left(\frac{1}{2}+\hat{a}^\dagger\hat{a}\right) +
g\; \vec{\epsilon_c}\cdot\vec{\hat{D}}
\left( \hat{a}^\dagger + \hat{a} \right),
\end{aligned}$$ where $\omega_c$ is the angular frequency of the cavity mode, $\vec{\epsilon}_c$ is its polarization direction, and is the coupling strength between the cavity and the molecules where $V$ is the quantization volume. $\vec{D}=\sum_\kappa^N\vec{\mu}^{(\kappa)}$ is total dipole operator of the ensemble. In Eq. (\[eq:ham3\]), the quadratic dipole self-energy term is being neglected, which is only relevant at much higher coupling strengths than considered here. For further details see, e.g., Refs. [@Faisal_1987; @fli17:3026]. The coupling to an external laser field is introduced semiclassically in the length gauge and dipole approximation as $\hat{H}_\mathsf{las} = - \vec{E}(t)\vec{\hat{D}}$, where the electric field takes the form $\vec{E}(t)=\vec{\epsilon_L}A(t)\cos(\omega_L t)$. It is assumed for simplicity that external laser fields couple to the molecules and do not pump the cavity mode directly. In general this is not necessarily the case and direct coupling to the cavity mode can be easily introduced [@sae18:arXiv]. However, since strong coupling is assumed (cf. discussion below), this model choice has only observable consequences at times below the Rabi cycling period of the hybrid system, in the order of a few tens of femtoseconds, and is not of relevance for our discussion. As an illustrative molecular example we consider sodium iodide (NaI), whose ultrafast photo-dissociation dynamics in the first excited electronic state $^1A$ coupled to the ground state $^1X$ has been the subject of extensive experimental and theoretical investigations (See e.g., Ref. [@zew88:1645]), also in the context of cavity-induced chemistry [@kow16:2050; @ven18:55]. Details on the potential energy and transition dipole curves of NaI [@ven18:55] and a detailed account of the quantum dynamics numerical techniques employed in this work can be found elsewhere [@man92:3199; @bec00:1; @ven18:55]. Throughout this work the effective cavity-matter coupling is taken as $g/\omega_c$=0.01, where $g$ was defined around Eq. (\[eq:ham3\]) and can be seen as the rms vacuum electric field amplitude of the cavity mode [@har89:24]. This coupling strength is small compared to the single-molecule ultra-strong coupling regime, characterized by a Rabi splitting of the polaritonic energy levels (at zero detuning) $\hbar\omega_R=2 g\mu_{01}$ comparable to the transition energy. The collective Rabi splitting is given by [@tho92:1132], where $N$ is the number of coupled molecules and $\mu_{01}$ the transition dipole matrix element. For NaI at the Franck-Condon geometry this coupling strength results in $\hbar\Omega_R=0.13$ eV for $N=1$ and $\hbar\Omega_R=0.30$ eV for $N=5$. Rabi splittings of this order have been observed experimentally in micro-cavities with coupled organic dye molecules [@sch13:125; @zho17:9034].
We note here that for an ensemble of molecules featuring a dissociative excited state and coupled to a cavity mode, photo-dissociation can only occur in the LPB. This is because, as one of the molecules dissociates, it ceases to be resonant with the cavity mode, the lower polariton turns into a pure electronic excitation of that molecule and the excitation energy is not available anymore for neither the cavity nor the other molecules, which subsequently remain in their respective ground states (cf. Fig. (\[fig:pes\_spectra\]c)). For this reason, the rate of photo-dissociation directly corresponds to the rate of relaxation from the UPB to the LPB, which will be used below.
![Left panels: one-dimensional cuts along adiabatic polaritonic potential energy surfaces obtained by diagonalization of $\mathcal{H}_\mathsf{el-cav}^{[1]}(\mathbf{R})$ (cf. Eq. (\[eq:model\])). Right panels: absorption spectrum of the hybrid system (blue solid curves) and single-molecule photo-dissociation probability at the corresponding photon-energy (red dots). Isolated molecule (a), single coupled molecule (b) and ensemble of 5 molecules (c). Legends are explained in the main text. []{data-label="fig:pes_spectra"}](fig01.pdf){width="8.5cm"}
We consider first the absorption spectra of a single isolated NaI molecule and compare it afterwards to the absorption spectra of a molecular ensemble. Excitation of an isolated NaI molecule to its first singlet excited electronic state results immediately in ballistic dissociation of the nuclear wave packet on the corresponding potential energy surface. Therefore, the photo-absorption spectrum features a single absorption band of, in this case, full width at half maximum (fwhm) about 0.2 eV centered at an excitation energy of about 3.7 eV, which is seen in the right panel in Fig. (\[fig:pes\_spectra\]a). The absorption spectrum is computed from the dipole auto-correlation function , where and $|\Psi_0\rangle$ is the ground state of the system. The strongly dissociative potential energy surface responsible for the fast dissociation is shown in the left panel of Fig. (\[fig:pes\_spectra\]a), where the ground state potential energy surface shifted by the cavity photon energy is shown as a dashed curve for comparison.
The photo-dissociation dynamics is significantly changed when the molecules are coupled to the cavity mode. In the case of a single molecule coupled to the cavity, the absorption spectrum is shown in the right panel of Fig. (\[fig:pes\_spectra\]b). A broad absorption band is found in the energy region of the LPB and a set of discrete absorption peaks are seen in the region of the UPB. These markedly different spectral regions can be explained by the potential energy curves of the lower and upper polaritons obtained by diagonalization of the $\hat{H}_\mathsf{el}+\hat{H}_\mathsf{cav}$ Hamiltonian as a function of $R$, which correspond to dissociative and bound potentials, respectively, and are shown in the left panel of Fig. (\[fig:pes\_spectra\]b).
Population promoted to the LPB immediately dissociates as in the isolated molecule case. Conversely, the upper polariton features a sharp progression of long-lived vibrational excitations, which correspond, to a good approximation, to the vibrational energy levels on the ground electronic state potential energy surface shifted by the photon energy of the cavity mode. Clearly, there is no ballistic photo-dissociation from the upper polariton, as the LPB cannot be immediately reached. The collective behaviour of a molecular ensemble is now investigated by increasing the number of molecules coupled to the cavity mode from $N=1$ to $N=5$ while keeping the same single-molecule coupling strength as before. The absorption spectrum is shown in the right panel in Fig. (\[fig:pes\_spectra\]c). The LPB region features now a broad band with superimposed internal structure. The former is indicative of ballistic dissociation along one of the molecular coordinates, as in the single molecule case. The latter corresponds to bound-type dynamics along the other vibrational degrees of freedom [@hel81:368; @tannor-book]. These dynamics of the LPB are observed independently of the number of molecules in the ensemble for $N=2$ up to $N=5$ (not shown) and will not be discussed further. In the UPB, the linear absorption spectrum still features several peaks reminiscent of the vibrational progression of the single molecule case, but broader, which is indicative of decay from those states within the first few hundreds of femtoseconds after photo-excitation.
The relaxation dynamics of the UPB in real time is accessed by pumping the system with an external laser of pulse of duration 30 fs (fwhm), photon energy tuned to the upper polariton region (cf. absorption spectra) and a peak field amplitude of $5\cdot10^{-4}$ au. This relatively weak field ensures that in all cases light absorption takes place in the linear single photon regime. The time-dependent dissociation probability is defined as $P_\mathsf{dis}(t)=\langle\Psi(t)|\hat{\Theta}(R_1-R_d)|\Psi(t)\rangle$, where $\Theta(x)$ is the Heaviside step function and $R_d=15$ au. As was mentioned above, the relaxation dynamics from the upper to the lower polariton branches can be traced through the probability of photo-dissociation, which can only occur in the LPB.
Inspection of $P_\mathsf{dis}(t)$ in Fig (\[fig:diss\_prob\]) for different ensemble sizes pumped to the UPB indicates that the dissociation occurs progressively, as compared to ballistic dynamics in the LPB. Even though the individual coupling of each molecule to the cavity, the laser parameters, and the total amount of population pumped by the laser to the UPB are the same in all cases, the rate of dissociation and therefore the rate of relaxation significantly depends on the size of the ensemble.
![Single-molecule photo-dissociation probability $P_\mathsf{dis}(t)=\langle\Psi(t)|\Theta(R_1-R_b)|\Psi(t)\rangle$ as a function of time for an ensemble of different size interacting with a laser pulse of photon energy $\hbar\omega_L=3.9$ eV. $R_b=$ 15 au. The dashed curves correspond to fitted first order rate expressions.[]{data-label="fig:diss_prob"}](fig02.pdf){width="8.5cm"}
The calculated population curves in Fig (\[fig:diss\_prob\]) can be used to fit effective first order relaxation rate constants $\kappa_N$ to the LPB starting from the UPB. For $N=5$, $\kappa_5^{-1}\approx 350$ fs, whereas $\kappa_1^{-1}\approx 3800$ fs. The first order rates result in the superimposed dashed curves in Fig. (\[fig:diss\_prob\]), which however do not completely capture all features of the quantum dynamics at short times. This marked dependence of the decay rate on the number of molecules is not found in expressions based on Fermi’s golden rule and an assumed density of vibrational states [@agr03:85311; @sae18:arXiv], and hints at the participation of decay pathways involving non-adiabatic nuclear dynamics with coupling between the electronic-polaritonic states and the vibrational degrees of freedom.
In order to shed light onto the nature of the relaxation pathways and to understand the origin of these different kinds of dissociative dynamics, we direct our attention to the dark states found between the upper and lower polaritons. In the Tavis-Cummings model of an ensemble of two-level atoms coupled to a cavity, and in the zero-detuning case, all dark states are degenerate, appear at the average energy of the two bright polaritons and feature no dipole coupling to the ground state of the hybrid system [@tav67:714]. Their dark nature is clearly manifest in Fig. (\[fig:pes\_spectra\]c) by the practical lack of absorption between $3.65$ and $3.85$ eV in the linear absorption spectrum. When nuclear motion is present, the degeneracy of the dark states can be lifted through nuclear displacements that modulate the energy gap of the corresponding molecule. This leads to CCI among the dark polaritonic states, which are referred to as *collective* to emphasize the fact that their location in coordinate space depends on the internal coordinates of the different molecules in the ensemble. The existence of points of intersection among dark polaritonic states has been noted recently as well by Feist and collaborators [@fei17:205].
A cut through the potential energy surfaces obtained by diagonalization of the $\hat{H}_\mathsf{el}+\hat{H}_\mathsf{cav}$ Hamiltonian in the single-excitation space (SES) and for $N=5$ is shown in the left panel of Fig. (\[fig:pes\_spectra\]c). For five molecules, six polaritonic states are present in the SES, two of which are the bright polaritons and four of them are nominally dark. In the cut shown, $R_2=R_3=5.35$ au, $R_4=R_5=5.5$ au, and $R_1$ is scanned between $4$ and $7.5$ au. Two CCI, the origin of which will be discussed below, are seen along this PES cut at precisely $R_1=5.35$ and $R_1=5.5$, which is not fortuitous.
The matrix representation of the $\hat{H}_{\mathsf{el}} + \hat{H}_{\mathsf{cav}}$ operator in the basis of non-interacting cavity-ensemble states and in the SES results in the molecular Tavis-Cummings Hamiltonian [@support] $$\begin{aligned}
\label{eq:model}
\mathcal{H}_\mathsf{el-cav}^{[1]}=%(\mathbf{R}) =
\begin{pmatrix}
\hbar\omega_c & \gamma^{(1)}(R_1) & \gamma^{(2)}(R_2) & \gamma^{(3)}(R_3) & \cdots \\
\gamma^{(1)}(R_1) & \Delta^{(1)}(R_1) & 0 & 0 & \cdots \\
\gamma^{(2)}(R_2) & 0 & \Delta^{(2)}(R_2) & 0 & \cdots \\
\gamma^{(3)}(R_3) & 0 & 0 & \Delta^{(3)}(R_3) & \cdots \\
\vdots & \vdots & \vdots & \vdots & \ddots \\
\end{pmatrix},
\end{aligned}$$ where $\Delta^{(\kappa)}(R_\kappa)=V_1^{(\kappa)}-V_0^{(\kappa)}$ is the energy gap of the $\kappa$-th molecule and $\gamma^{(\kappa)}(R_\kappa)=g\mu_{01}^{(\kappa)}$ is the dipole coupling of the $\kappa$-th molecule to the cavity mode. Hamiltonian (\[eq:model\]) has the form of an arrowhead matrix, whose properties have been investigated in the contexts of applied mathematics [@wil65:algebra; @oe90:497] and molecular physics [@wal84:729]. The most important property of arrowhead matrices for our purposes is the fact that for every $m$ molecules with the same energy gap $\Delta$, there is an eigenvalue of multiplicity $m-1$ [@oe90:497]. Hence, for $m=2$ molecules with the same energy gap, there is one eigenstate of $\mathcal{H}_\mathsf{el-cav}^{[1]}$ at the corresponding energy value. In case $m=3$ molecules have the same energy gap, e.g., molecules $1$ to $3$, there are two degenerate eigenstates states at that energy resulting in a CCI of order two. For the case of identical molecules, this degeneracy is found in the one-dimensional space and it is lifted by displacements in the two-dimensional branching space that removes this equality. The local symmetry of the identical molecules giving rise to the CCI can be described in the permutation symmetry group $S_3$ [@bunker-book:symmetry], which among others is isomorph with the $D_3$ point group of an equilateral triangle [@koep10]. As is well known, molecules of $D_3$ and related symmetry point groups, e.g. $C_{3v}$, are Jahn-Teller active [@dom04; @koep10] and the degeneracy of electronic state energies is lifted linearly for displacements out of the highly symmetric atomic arrangement, resulting in conically intersecting potential energy surfaces. Therefore, cavity-induced CCI are, from a mathematical standpoint, analogous to the commonly encountered [@Tru03:32501] intra-molecular conical intersection case , including their divergent non-adiabatic coupling matrix elements at the regions of intersection [@support], where $|\psi_j\rangle$ are the eigenstates of $\mathcal{H}_\mathsf{el-cav}^{[1]}$. Generalising to larger numbers of molecules, the molecular Tavis-Cummings Hamiltonian features a CCI of order $m-1$ for any subset of $m$ molecules at molecular geometries that result in an equal energy gap for all molecules of the subset. As a final example, the polaritonic states resulting of four identical molecules with the same geometry can be classified according to the symmetry representations of the $S_4$ group, which is isomorph with the $T_d$ point group of the tetrahedron and which leads to triply degenerate potential energy crossings and to the $T_2
\otimes t_2$ Jahn-Teller effect [@koep10]. Summarizing, real-time wave packet simulations of a molecular ensemble coupled to a cavity mode show that the non-radiative energy relaxation rate from the upper to the lower polaritonic states is strongly dependent on the number of coupled molecules. This is in contrast to descriptions based on Fermi’s golden rule rates. Once the system reaches the manifold of nominally dark states, the vibronic relaxation dynamics proceeds through cavity-induced CCI, which mathematically are a consequence of the arrowhead-matrix form of the molecular Tavis-Cummings Hamiltonian in the SES. From a mechanistic perspective, the conical intersection topology funnels nuclear wave packets through it by first attracting them when in the upper part of the cone and then pushing them away once the probability amplitude appears on the lower electronic state [@dom04]. This mechanistic idea represents the cornerstone of ultrafast non-radiative relaxation in isolated molecular systems. In the context of polariton relaxation, CCI may be an important ingredient for fast localization and decay of collectively coupled excitations. Since only the energy gap is determinant for the existence of the CCI [@oe90:497], these are expected to be robust against molecular rotations or other external perturbations that modulate the off-diagonal coupling strength of individual molecules to the cavity mode, as well as to local interactions with an environment, which will lead only to displacements of the locus of intersection. The precise role of CCI in collective decay mechanisms, specially for larger ensembles and more complex molecules, remains to be further investigated.
I want to thank L.B. Madsen, L.S. Cederbaum, H.-D. Meyer and J. Feist for insightful discussions.
|
---
abstract: |
Let $ G $ be a connected, simply connected nilpotent Lie group and $ \Gamma < G $ a lattice. We prove that each ergodic diffeomorphism $ \phi(x\Gamma)=uA(x)\Gamma $ on the nilmanifold $ G/\Gamma $, where $ u\in G $ and $ A:G\to G $ is a unipotent automorphism satisfying $ A(\Gamma)=\Gamma $, enjoys the property of asymptotically orthogonal powers (AOP). Two consequences follow:
\(i) Sarnak’s conjecture on Möbius orthogonality holds in every uniquely ergodic model of an ergodic affine unipotent diffeomorphism;
\(ii) For ergodic affine unipotent diffeomorphisms themselves, the Möbius orthogonality holds on so called typical short interval: $$\frac1M\sum_{M\leq m<2M}\left|\frac1H\sum_{m\leq n<m+H}
f(\phi^n(x\Gamma)){\boldsymbol{\mu}}(n)\right|\to 0$$ as $ H\to\infty $ and $ H/M\to0 $ for each $ x\Gamma\in G/\Gamma $ and each $ f\in C(G/\Gamma) $.
In particular, the results in (i) and (ii) hold for ergodic nil-translations. Moreover, we prove that each nilsequence is orthogonal to the Möbius function ${\boldsymbol{\mu}}$ on a typical short interval.
We also study the problem of lifting of the AOP property to induced actions and derive some applications on uniform distribution.
address:
- |
Unité Mixte de Recherche CNRS 8524\
Unité de Formation et Recherche de Mathématiques\
Université de Lille\
F59655 Villeneuve d’Asq CEDEX\
FRANCE
- 'Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland'
author:
- 'L. Flaminio'
- 'K. Frczek'
- 'J. Kułaga-Przymus'
- 'M. Lemańczyk'
bibliography:
- 'aop\_biblio.bib'
title: Approximate orthogonality of powers for ergodic affine unipotent diffeomorphisms on nilmanifolds
---
[^1]
[^2]
Introduction
============
Following [@Sa], we say that a homeomorphism $T$ of a compact metric space $X$ is [*Möbius orthogonal*]{} if for each $ f\in C(X) $ and $ x\in X $, we have $$\frac1N\sum_{n\leq N}f(T^nx)\, {\boldsymbol{\mu}}(n)\to 0 ,$$ as $ N\to\infty $. Here $ {\boldsymbol{\mu}}:{{\mathbb{N}}}\to\C $ stands for the Möbius function defined by: $ {\boldsymbol{\mu}}(1)=1 $, $ {\boldsymbol{\mu}}(n)=(-1)^k $, if $ n=p_1p_2\cdots p_k $ for distinct prime numbers $ p_1,\ldots,p_k $, and $ {\boldsymbol{\mu}}(n)=0 $ otherwise. In 2010, P. Sarnak formulated the following conjecture, referred to as *Sarnak’s conjecture on Möbius orthogonality*:
> [Every zero entropy homeomorphism of a compact metric space is Möbius orthogonal.]{}
Since then, the Möbius orthogonality has been proved in many particular cases of zero entropy homeomorphisms: [@Ab-Ka-le], [@Ab-Le-Ru0], [@Bo], [@Bo-Sa-Zi], [@De-Dr-Mu], [@Do-Ka], [@Fe-Ku-Le-Ma], [@Fe-Ma], [@Gr-Ta], [@Kar], [@Mu], [@Pe].
Sarnak’s conjecture is mainly motivated by some open problems in number theory, for example, the famous Chowla conjecture on auto-correlations of the Möbius function implies Sarnak’s conjecture (see [@Ab-Ku-Le-Ru], [@Sa], [@Ta] for more details).
As stated above, Sarnak’s conjecture deals with topological dynamical systems. However, we may consider it by looking at all invariant measures for the homeomorphim $ T $. (By the variational principle, all such measures have measure-theoretic entropy zero.) Then we will deal with an ergodic theory problem considered in the class of measure-theoretic dynamical systems with zero measure-theoretic entropy.
Reversing the problem, we may start with an ergodic, zero entropy automorphism $ R $ of a probability standard Borel space $ (Z,{{\mathcal D}},\nu) $ and suppose that $ T $ is a *uniquely ergodic model of $ R $*, that is, $ T $ is a homeomorphism of a compact metric space $ X $ with a unique $ T $-invariant probability Borel measure $ \mu $ such that the measure-theoretic systems $ (Z,{{\mathcal D}},\nu,R) $ and $ (X,{{\mathcal B}}(X),\mu,T) $ are measure-theoretically isomorphic. We recall that, by the Jewett-Krieger theorem, each ergodic system $ R $ has many uniquely ergodic models, some of which may even be topologically mixing (see also [@Leh]). In such a setting it is natural to ask if there are some ergodic properties of $ R $ which are sufficient to guarantee that [**every**]{} uniquely ergodic model $ T $ of $ R $ will be Möbius orthogonal.
For example, irrational rotations are well-known to be Möbius orthogonal [@Da] but the fact that the Möbius orthogonality holds in all uniquely ergodic topological models of irrational rotations has been proved only recently in [@Ab-Le-Ru]. In [@Ab-Le-Ru], such a result has been achieved by introducing a new invariant of measure-theoretic isomorphism, namely, the AOP property (which, in particular, holds for all irrational rotations). The Möbius orthogonality holds true in every uniquely ergodic model of a measure-theoretic automorphism $ R $ satisfying the AOP property.
Let us recall briefly the definition of the AOP property (see Section \[joiAOP\], Definition \[defAOP\] and Remark \[r:weaktop\] for more details). An ergodic automorphism $ R $ of a probability standard Borel space $ (Z,{{\mathcal D}},\nu) $ is said to have [ *asymptotically orthogonal powers*]{} (AOP), or to satisfy the *AOP property*, if the Hausdorff distance of the set $ J(R^p,R^q) $ of joinings between $ R^p $ and $ R^q $ from the singleton $ \{\nu{\otimes}\nu\} $ goes to zero when $ p $ and $ q $ are different prime numbers tending to infinity.
The AOP property for the automorphism $ R $ implies that all non-zero powers of it are ergodic, i.e. $ R $ is [*totally ergodic*]{}.
Denoting by $ \mathscr{P} $ the set of prime numbers, a consequence of the AOP property is that in each uniquely ergodic model $ (X,T) $ of $ (Z,{{\mathcal D}},\nu,R) $, we have $$\label{intr1} \limsup_{\substack{p, q\to\infty\\
p,q\in\mathscr{P}\\
p\neq q}}\,\, \limsup_{K\to\infty}\left|\frac1{b_{K+1}} \sum_{k\leq
K}\sum_{b_k\leq n<b_{k+1}}f(T^{pn}x_k){\overline}{f(T^{qn}x_k)}\right|=0$$ for each increasing sequence of integers $ (b_k) $ with $ b_{k+1}-b_k\to\infty $, for each sequence of points $ (x_k)\subset X $ and a zero mean function $ f\in C(X) $. By the Kátai-Bourgain-Sarnak-Ziegler criterion[^3] ([@Bo-Sa-Zi], [@Ka]), we obtain $$\label{intr2} \frac1{b_{K+1}}\sum_{k\leq K}\left(\sum_{b_k\leq n<b_{k+1}}f
(T^nx_k){\boldsymbol{u}}(n)\right)\to0,\text{ when }K\to\infty,$$ for every multiplicative function $ {\boldsymbol{u}}:{{\mathbb{N}}}\to\C $ bounded by $ 1 $ and all $ (b_k) $, $ (x_k) $ and $ f $ as above. If we set $ x_k=x $ for all $ k\geq1 $ and ${\boldsymbol{u}}={\boldsymbol{\mu}}$, then means that the Möbius orthogonality holds for $T$. However, in many concrete situations, from , we can deduce the stronger conclusion $$\lim_{K\to \infty} \frac1{b_{K+1}}\sum_{k\leq K}\left|\sum_{b_k\leq
n<b_{k+1}}f(T^nx){\boldsymbol{u}}(n)\right|=0,$$ or, equivalently (see [@Ab-Le-Ru]), $$\frac1M\sum_{M\leq m<2M}\left|\frac1H\sum_{m\leq
n<m+H}f(T^nx){\boldsymbol{u}}(n)\right|,\text { when }H\to\infty,H/M\to0,$$ for each zero mean function $ f\in C(X) $, $ x\in X $ and $ {\boldsymbol{u}}$ as above.
The AOP property has been proved in [@Ab-Le-Ru] for so-called quasi-discrete spectrum automorphisms [@Ab]. This implies the Möbius orthogonality of all uniquely ergodic models of quasi-discrete spectrum automorphisms. Some uniquely ergodic models of such automorphisms are given by affine transformations on compact abelian (metric) groups. For [**these**]{} particular models the Möbius orthogonality has been proved earlier [@Gr-Ta], [@Li-Sa]. Recall that the affine transformations are examples of distal homeomorphisms.[^4] Another natural class of distal (uniquely ergodic) homeomorphisms is given by nil-translations and, more generally, affine unipotent diffeomorphisms of nilmanifolds. Recall that if $ G $ is a connected, simply connected, nilpotent Lie group, $ \Gamma < G $ is a lattice and $ u\in G $, then each rotation $ l_u(x\Gamma):=ux\Gamma $ acting on $ G/\Gamma $ is called a [*nil-translation by $ u $*]{}. More generally, we may consider *affine unipotent* diffeomorphisms on $ G/\Gamma $, that is maps of $ G/\Gamma $ of the form $ \phi(x\Gamma):=uA (x)\Gamma $, where $ A $ is a unipotent automorphism of $ G $ such that $ A(\Gamma)=\Gamma $ and $ u\in G $. For such maps, the Möbius orthogonality has been proved in [@Gr-Ta], where even some quantitative version of it has been proved. Liu and Sarnak in [@Li-Sa] suggest that perhaps instead of proving Sarnak’s conjecture, it will be easier first to prove the Möbius orthogonality in the distal case (see [@Ku-Le], [@Li-Sa] and [@Wang] for some results in this direction). Instead, we can ask if, assuming total ergodicity, we have the AOP property.
In the present paper, we confirm that the AOP approach may bring some fruits by proving the following.
\[ThA\] Let $ G $ be a connected, simply connected, nilpotent Lie group and $ \Gamma < G $ a lattice. Then, every ergodic affine unipotent diffeomorphism $ \phi:G/\Gamma\to G/\Gamma $ has the AOP property.
By the previous discussion, the following result is immediate:
\[CorB\]The Möbius orthogonality holds in every uniquely ergodic model of any affine unipotent diffeomorphism of a compact connected nilmanifold.
Moreover, the algebraic structure of the underlying space $ G/\Gamma $ will let us prove the following.
\[CorC\] Let $ G $ be a connected, simply connected, nilpotent Lie group and $ \Gamma < G $ a lattice. Let $ \phi:G/\Gamma\to G/\Gamma $ be an ergodic affine unipotent diffeomorphism. Then, for every $ x\in G$, for every zero mean function $ f\in C(G/\Gamma) $ and for every multiplicative function $ {\boldsymbol{u}}:{{\mathbb{N}}}\to\C $ bounded by $ 1 $, we have $$\label{eq:zbkr} \frac1M\sum_{M\leq m<2M}\left|\frac1H \sum_{m\leq
h<m+H}f(\phi^h(x\Gamma))\,{\boldsymbol{u}}(h)\right|\to 0$$ as $ H\to\infty $ and $ H/M\to0 $. For $ {\boldsymbol{u}}={\boldsymbol{\mu}}$ the result holds for arbitrary $ f\in C(G/\Gamma) $.
The property expressed by will be referred to as the [*Möbius orthogonality on typical short interval*]{}. One more consequence of Theorem \[ThA\] is the following sample of a result when this property takes place.
\[PropD\] Assume that $ P\in{{\mathbb{R}}}[x] $ is a non-zero degree polynomial with irrational leading coefficient. Then for all $ \gamma\in{{\mathbb{R}}}\setminus\{0\} $ and $\varrho\in{{\mathbb{R}}}$, we have $$\label{eq:zast1} \frac1M\sum_{M\leq m<2M}\left|\frac1H\sum_{m\leq
h<m+H} e^{2\pi iP([\gamma h+\varrho])}{\boldsymbol{\mu}}(h)\right|\longrightarrow 0$$ as $ H\to\infty $ and $ H/M\to0 $.
Recall that a sequence $(a_n)\subset\C$ is called a [*nilsequence*]{} if it is a uniform limit of sequences of the form $(f(l_u^n(x\Gamma)))$, where $G/\Gamma$ is a compact nilmanifold.[^5] Green and Tao [@Gr-Ta] proved that all nilsequences are orthogonal to ${\boldsymbol{\mu}}$ (their result is quantitative). We will prove the Möbius orthogonality on typical short interval:
\[TheoremE\] For each nilsequence $(a_n)\subset\C$, we have $$\frac1M\sum_{M\leq m<2M}\left|\frac1H\sum_{m\leq
h<m+H}a_h{\boldsymbol{\mu}}(h)\right|\to 0$$ when $H\to\infty$, $H/M\to0$.
As Leibman in [@MR2643713] proved that all polynomial multi-correlation sequences are nilsequences in the Weyl pseudo-metric (see Section \[pmc\]), we obtain the following result.
\[CorollaryF\]
For every automorphism $ T $ of a probability standard Borel space ${(X,{\mathcal B},\mu)}$, the polynomial multi-correlations functions are orthogonal to the Möbius function on a typical short interval, that is, for every $ g_i\in L^\infty(X,\mu) $, $ p_i\in{{\mathbb{Z}}}[x] $, $ i=1,\ldots,k $ ($k\geq1$), we have
$$\frac1M\sum_{M\leq m<2M}\left| \frac1H\sum_{m\leq h<m+H}
{\boldsymbol{\mu}}(h)\int_X g_1\circ T^{p_1(h)}\cdot\ldots\cdot g_k\circ
T^{p_k(h)}\,d\mu\right|\longrightarrow 0$$ when $ H\to\infty $ and $ H/M\to0 $.
In some classical cases (e.g. when $ G $ is $ H_3({{\mathbb{R}}}) $), it is well-known that nil-translations are time automorphisms of the suspension flows over affine automorphisms of the torus. Since affine automorphisms on tori enjoy the AOP property [@Ab-Le-Ru], a natural question arises whether the AOP property of an automorphism implies the AOP property of the suspension flow. This, in fact, motivates a more general question. The AOP property can be studied for actions of (general) abelian groups, see Section \[liftingAOPia\]. Passing from automorphisms (i.e.$ {{\mathbb{Z}}}$-actions) to their suspensions ($ {{\mathbb{R}}}$-actions) is a particular case of inducing [@Zi]. Our aim will be to prove the following result about the “relative” AOP property for induced actions.
\[PropositionG\] Let $ H $ be a closed cocompact subgroup of a locally compact second countable abelian group $ G $. Assume that $H$ has no non-trivial compact subgroups. Assume, moreover, that an $ H $-action $ {{\mathcal S}}$ on a probability standard Borel space ${(Y,{\mathcal C},\nu)}$ has the AOP property. Then, for each $ E,F\in L^2(Y\times G/H)\ominus L^2(G/H) $, we have $$\lim_{p\neq q, p,q\in\mathscr{P},p,q\to\infty}
\sup_{\kappa\in
J^e(\widetilde{{\mathcal S}}^{(p)},\widetilde{{\mathcal S}}^{(q)})}\left|\int_{Y\times
G/H\times Y\times G/H}E{\otimes}F\,d\kappa\right|=0$$ for the induced $ G $-action $ \widetilde{{{\mathcal S}}}, $ i.e. the induced $G$-action has the “relative” AOP property.
An application of this result for $ H=k{{\mathbb{Z}}}$ and $ G={{\mathbb{Z}}}$ yields the following:
\[CorollaryH\] \[c:km\] Assume that $ T $ is a uniquely ergodic homeomorphism of $ X $, with the unique invariant measure $ \mu $. Assume moreover, that $ (X,\mu,T) $ has the AOP property. Then, for each multiplicative function $ {\boldsymbol{u}}:{{\mathbb{N}}}\to\C $, $ |{\boldsymbol{u}}|\leq1 $, for each $ k\geq 1 $ and $ 0\leq j<k $, we have $$\label{in24} \frac1N\sum_{n\leq N}f(T^nx){\boldsymbol{u}}(kn+j)\to0$$ for each $ f\in C(X) $ of zero mean and each $ x\in X $. In particular, $$\label{in25} \frac1N\sum_{n\leq N}f(T^nx){\boldsymbol{\mu}}(kn+j)\to0.$$
\[CorollaryI\] Let $(a_n)$ be a sequence $\big(f(\phi^n(x\Gamma))\big)$, $\big(e^{2\pi iP([\gamma n+\varrho])}\big)$, an arbitrary nil-sequence or $\big(\int_Xg_1\circ T^{P_1(n)}\cdot\ldots\cdot g_r\circ
T^{p_r(n)}\,d\mu\big)$ as in Corollary \[CorC\], Proposition \[PropD\], Theorem \[TheoremE\] and Corollary \[CorollaryF\], respectively. Then, for each $k,j\in{{\mathbb{N}}}$, we have $$\label{eq:corI}
\frac1M\sum_{M\leq m<2M}\left|\sum_{m\leq n<m+H} a_n{\boldsymbol{\mu}}(kn+j)\right|\to 0$$ when $H\to\infty$, $H/M\to0$.
Finally, we will also give new examples of AOP flows with partly continuous singular spectra.
The second part of the paper has a form of an appendix in which we provide some more information on Lie groups, but also we elucidate the approach to prove the AOP property for nil-translations through Lie group apparatus as special properties of measure-theoretical distal automorphisms.
On the AOP property and the Möbius orthogonality
================================================
Joinings. The AOP property {#joiAOP}
--------------------------
Assume that $ T $ and $ S $ are ergodic automorphisms of probability standard Borel spaces $ {(X,{\mathcal B},\mu)}$ and $ {(Y,{\mathcal C},\nu)}$, respectively. A $ T\times S $-invariant probability measure $ \rho $ on $ (X\times Y,{{\mathcal B}}{\otimes}{{\mathcal C}}) $ is called a [*joining*]{} of $ T $ and $ S $ if the marginals of $ \rho $ on $ X $ and $ Y $ are equal to $ \mu $ and $ \nu $, respectively. The product measure $ \mu{\otimes}\nu $ is a joining of $ T $ and $ S $, called the *product joining*. In particular, the set $ J(T,S) $ of joinings of $ T $ and $ S $ is not empty. If $ \rho\in J(T,S) $ is ergodic for $ T\times S $, then $ \rho $ is called an [*ergodic joining*]{} and we write $ \rho\in J^e(T,S) $. The set $ J(T,S) $, endowed with the vague topology, is a closed simplex for the natural affine structure on the space of probability Borel measures on $ (X\times Y,{{\mathcal B}}{\otimes}{{\mathcal C}}) $. Then $ J^e(T,S) $ is the set of extremal points of $ J (T,S) $.
The automorphisms $ T $ and $ S $ are called [*disjoint*]{} if their only joining is the product joining $ \mu{\otimes}\nu $, (see [@Fu]); in this case we write $ T\perp S $.
All above notions have obvious generalizations to measure preserving actions of groups and semi-groups.
The following definition can be introduced for ergodic actions of more general groups (see Section \[liftingAOPia\]); later on, we shall consider the AOP property for flows.
Let $ L^2_0{(X,{\mathcal B},\mu)}$ stand for the subspace of zero mean function in $ L^2{(X,{\mathcal B},\mu)}$ and recall that $ \mathscr{P} $ stands for the set of prime natural numbers.
\[defAOP\] A totally ergodic automorphism $ T $ of a probability standard Borel space $ {(X,{\mathcal B},\mu)}$ has [*asymptotically orthogonal powers*]{} (AOP) if for each $ f,g\in L^2_0{(X,{\mathcal B},\mu)}$, we have $$\label{cond:aop} \limsup_{\substack{p,q\to\infty\\
p, q \in \mathscr{P}\\
p\neq q }}\sup_{\rho\in J^e(T^p,T^q)}\left|\int_{X\times X}f{\otimes}g\,d\rho\right|=0.$$ In this case, we also say that $ T $ has the AOP property.
Clearly, if the prime powers of $ T $ are pairwise disjoint, then $ T $ enjoys the AOP property. There are, however, other natural examples, see [@Ab-Le-Ru], [@Ku-Le]; in fact, it may even happen that all non-zero powers of automorphism having AOP are isomorphic.
\[r:weaktop\] By definition, $ \rho_n\to\rho $ for the vague topology if and only if we have $ \int_{X\times Y}f{\otimes}g\,d\rho_n\to \int_{X\times Y}f{\otimes}g\,d\rho $ for all $ f,g\in L^2_0{(X,{\mathcal B},\mu)}$. Then, since $ L^2_0{(X,{\mathcal B},\mu)}$ is separable, the vague topology on $ J(T,S) $ is metrizable. Then the AOP property for an automorphism $ T $ states that, when $ p $ and $ q $ are distinct large primes, all ergodic joinings (and hence all joinings) of $ T^p $ and $ T^q $ are uniformly close to product measure.
Moreover, in order to show the AOP property for $ T $, we only need to check the property for $ f,g $ belonging to a linearly dense subset in $ L^2_0{(X,{\mathcal B},\mu)}$.
Assume that automorphisms $ T $ and $ S $ have a [*common factor*]{}, i.e. there exists an ergodic automorphism $ R $ defined on a probability space $ {(Z,{\mathcal D},\kappa)}$ with equivariant factor maps $ \pi_{X,Z}:{(X,{\mathcal B},\mu)}\to{(Z,{\mathcal D},\kappa)}$ and $ \pi_{Y,Z}:{(Y,{\mathcal C},\nu)}\to{(Z,{\mathcal D},\kappa)}$. Let $$\mu=\int_{Z}\mu_z\,d\kappa(z),\;\nu=\int_Z\nu_z\,d\kappa(z)$$ be the disintegrations of the measures $ \mu $ and $ \nu $. Then every joining $ \rho\in J(R,R) $ has a natural extension to a joining $ \widehat{\rho}\in J(T,S) $ determined by[^6] $$\label{rie} \int_{X\times Y}f{\otimes}g\,d\widehat{\rho}=\int_{Z\times Z}\left
(\int_Xf\,d\mu_{z_1}\int_Y g\,d\nu_{z_2}\right)\,d\rho(z_1,z_2)$$ for all $ f,g\in L^\infty{(X,{\mathcal B},\mu)}$. The joining $ \widehat\rho $ is called the [*relatively independent extension*]{} of $ \rho $. The joining $ \widehat{\rho} $ need not be ergodic, even if $ \rho $ is; however, the image via $ \pi_{X,Z}\times\pi_{Y,Z} $ of almost every ergodic component of $ \widehat{\rho} $ is equal to $ \rho $.
A criterion for AOP and Möbius orthogonality on typical short intervals
-----------------------------------------------------------------------
Theorem \[ThA\] and Corollaries \[CorB\] and \[CorC\] will be proved by showing that affine unipotent diffeomorphisms of compact nilmanifolds satisfy the hypotheses of the following theorem.
\[thm:prop\] Let $ T:X\to X $ be a homeomorphism of a compact metric space. Assume that $ T $ is totally and uniquely ergodic for the $ T $-invariant probability Borel measure $ \mu $. Let $ \mathscr{C}\subset C(X)\cap L^2_0(X,\mu) $ be a set whose linear span is dense in $ L^2_0 (X,\mu) $.
- Assume that, for all $ f_1,f_2\in\mathscr{C} $ and for all but a finite number of pairs of distinct prime numbers $ (r,s) $, we have $ \rho(f_1\otimes \bar{f}_2)=0 $ for all ergodic joinings $ \rho $ of $ T^r $ and $ T^s $.
Then $ T $ satisfies the AOP property and the Möbius orthogonality holds in every uniquely ergodic model of $ (X,\mu,T) $.
- Assume further that for all $ f\in\mathscr{C} $ and all $ \omega\in \C $ with $ |\omega|=1 $ there exists a homeomorphism $ S:X\to X $ such that $ f(T^n(Sx))=\omega f(T^n x) $ for every $ x\in X $ and $ n\in{{\mathbb{Z}}}$.
Then, for every $ x\in X $, for every every zero mean $ f\in C(X) $ and every multiplicative $ {\boldsymbol{u}}:{{\mathbb{N}}}\to\C $ bounded by $ 1 $, we have $$\label{eq:zbkr1} \frac 1 M\sum_{M\leq m<2M}\left|\frac1H\sum_{m\leq
n<m+H}f(T^nx){\boldsymbol{u}}(n)\right| \longrightarrow 0$$ when $ H\to\infty $ and $ H/M\to 0 $. If $ {\boldsymbol{u}}={\boldsymbol{\mu}}$ the result holds for arbitrary $ f\in C(X) $.
In view of Remark \[r:weaktop\], since the set $ \mathscr{C} $ is linearly dense in $ L^2_0(X,\mu) $, Part (i) of the above theorem was proved in [@Ab-Le-Ru] (Theorem 2). So we only need to prove Part (ii).
\[rem:mrt\] If $ f $ is constant or more general $f(T^nx)=\exp(ina)$ for some $a\in{{\mathbb{R}}}$, then holds true, when $ {\boldsymbol{u}}$ is the Möbius function, by Theorem 1.7 in [@Ma-Ra-Ta] (see also the discussion preceding this theorem).
Suppose that the hypotheses of (i) and (ii) are satisfied. Let $ (b_k)_ {k\geq 1} $ be an increasing sequence of natural numbers such that $ b_{k+1}-b_k\to+\infty $.
Fix $ x\in X $, $ f\in \mathscr{C}\subset C(X)\cap L^2_0(X,\mu) $ and a multiplicative function $ {\boldsymbol{u}}:{{\mathbb{N}}}\to\C $ bounded by $ 1 $. For every $ k\geq 1 $, let $ \omega_k\in\C $ be the number of modulus $ 1 $ such that $$\Big|\sum_{b_k\leq n< b_{k+1}}f( T^nx){\boldsymbol{u}}(n)\Big|=
\omega_k\sum_{b_k\leq n< b_{k+1}}f( T^nx){\boldsymbol{u}}(n).$$ By hypothesis, there exists a homeomorphism $ S_k:X\to X $ such that $ f(T^n S_k x)=\omega_kf(T^nx) $ for every $ n\in{{\mathbb{Z}}}$. Let $ x_k:=S_k x $. Then $$\label{eq:convb0}
\Big|\sum_{b_k\leq n< b_{k+1}}f( T^nx){\boldsymbol{u}}(n)\Big|= \sum_{b_k\leq
n< b_{k+1}}f( T^nx_k){\boldsymbol{u}}(n).$$ By Theorem 3 in [@Ab-Le-Ru], the AOP property implies that for all zero mean $ f\in C(X) $, for all sequences $ (x_k)_{k\geq 1} $ in $ X $ and for all multiplicative functions $ {\boldsymbol{u}}:{{\mathbb{N}}}\to\C $ bounded by $ 1 $, we have $$\frac1{b_{K+1}}\sum_{k\leq K}\Big(\sum_{b_k\leq n<
b_{k+1}}f(T^nx_k){\boldsymbol{u}}(n)\Big) \to 0 \text{ when }K\to\infty.$$ In view of , it follows that $$\label{eq:convb1} \frac1{b_{K+1}}\sum_{k\leq K}\Big|\sum_{b_k\leq
n< b_{k+1}}f( T^nx){\boldsymbol{u}}(n)\Big|\to0\text{ when }K\to\infty.$$ As $ f \in \mathscr{C} $ was arbitrary, it follows that holds also for every $ f\in \Span\mathscr{C} $.
Let $ f $ be an arbitrary continuous function on $ X $ with zero mean. Since the space $ \Span\mathscr{C} $ is dense in $ L^2_0(X,\mu) $, for every $ \varepsilon>0 $ there exists $ f_\varepsilon\in \Span\mathscr{C} $ such that $ \int_{X}|f-f_\varepsilon|\,d\mu<\varepsilon $. Then $$\begin{aligned}
\frac1{b_{K+1}}\sum_{k\leq K} \Big|\sum_{b_k\leq n< b_{k+1}}f(T^nx){\boldsymbol{u}}(n)\Big| &\leq \frac1{b_{K+1}}\sum_{k\leq K} \Big|\sum_{b_k\leq n<
b_{k+1}}f_\varepsilon( T^nx){\boldsymbol{u}}(n)\Big|\\
&\quad+\frac1{b_{K+1}}\sum_{ n\leq b_{K+1}}\big|f(T^nx)-f_\varepsilon(T^nx)\big|.
\end{aligned}$$ Since $ T $ is uniquely ergodic and $ |f-f_\varepsilon| $ is continuous, we have $$\frac1{b_{K+1}}\sum_{ n\leq b_{K+1}}\big|f( T^nx)-f_\varepsilon(
T^nx)\big|\to \int_{X}|f-f_\varepsilon|\,d\mu<\varepsilon\text{ as
}K\to\infty.$$ It follows that $$\limsup_{K\to\infty}\frac1{b_{K+1}}\sum_{k\leq K}\Big|\sum_{b_k\leq
n< b_{k+1}}f( T^nx){\boldsymbol{u}}(n)\Big|\leq \varepsilon$$ for each $ \varepsilon>0 $ which proves for all continuous functions $ f $ with zero mean. According to [@Ab-Le-Ru] (see the proof of Theorem 5) this implies for every continuous function $ f $ with zero mean.
\[rem:riemint\] Let $ T $ be a uniquely ergodic homeomorphism of a compact metric space $ X $ and let $ \mu $ be the unique $ T $-invariant probability measure. Let $ {\boldsymbol{u}}$ be a multiplicative function bounded by $ 1 $. Suppose that the conclusion of Theorem \[thm:prop\] (ii) holds true: for each continuous functions $ f:X\to\C $ with zero mean and $ x\in X $, we have $$\label{eq:sumhm} A(f,M,H):=\frac1M\sum_{M\leq m<2M}\left|\frac1H\sum_
{m\leq n<m+H}f(T^nx){\boldsymbol{u}}(n)\right| \longrightarrow 0$$ as $ H\to\infty $ and $ H/M\to 0 $. Then the approximation argument used in the proof of Theorem \[thm:prop\] (ii) yields the validity of for a more general class of functions.
Let $ D(X) $ be the space of bounded measurable functions $ f:X\to\C $ such that the closure of the set of discontinuity point of $ f $ has zero measure. First note that every $ f\in D(X) $ satisfies the following equidistribution condition $$\label{eq:eqid} \frac{1}{N}\sum_{n\leq N}f(T^nx)\to\int_X f\,d\mu\text
{ for all }x\in X.$$ Indeed, since the closure $ F $ of the set of discontinuities of $ f $ has zero measure, for every $ \varepsilon>0 $ there exists an open set $ F\subset U $ such that $ \mu(\overline{U})<\varepsilon $. By Tietze’s extension theorem and the continuity of $ f $ restricted to $ X\setminus U $, there exists a continuous function $ f_\varepsilon:X\to\C $ such that $ f(x)=f_\varepsilon(x) $ for $ x\in X\setminus U $ and $ \|f_\varepsilon\|_{\sup}\leq \|f\|_{\sup} $. Then $$\begin{aligned}
\Big|\frac{1}{N}\sum_{n\leq N}f(T^nx)-\int_X f\,d\mu\Big|&\leq
\Big|\frac{1}{N}\sum_{n\leq N}f_\varepsilon(T^nx)-\int_X f_\varepsilon\,d\mu\Big|\\
&\quad+ 2\|f\|_{\sup}\Big(\frac{1}{N}\sum_{n\leq N}\chi_U(T^nx)+\mu
(U)\Big).
\end{aligned}$$ By unique ergodicity, $ \frac{1}{N}\sum_{n\leq N}\delta_{T^nx}\to\mu $ weakly. Therefore, $$\frac{1}{N}\sum_{n\leq N}f_\varepsilon(T^nx)\to\int_X
f_\varepsilon\,d\mu$$ and, by the regularity of $ \mu $, $$\limsup_{N\to\infty}\frac{1}{N}\sum_{n\leq N}\chi_U(T^nx)\leq
\limsup_{N\to\infty}\frac{1}{N}\sum_{n\leq N}\delta_{T^nx}(\overline
{U})\leq \mu(\overline{U})<\varepsilon.$$ It follows that $$\limsup_{N\to\infty}\Big|\frac{1}{N}\sum_{n\leq N}f(T^nx)-\int_X
f\,d\mu\Big|\leq 4\varepsilon\|f\|_{\sup}$$ for each $ \varepsilon>0 $ which proves .
Finally, for each $ f\in D(X) $ with zero mean and $ \varepsilon>0 $, we can find a continuous function $ f_\varepsilon $ with zero mean such that $ \int_{X}|f-f_\varepsilon|\,d\mu<\varepsilon $. Since $ |f-f_\varepsilon|\in D(X) $, by , we have $$\limsup_{N\to\infty}\frac{1}{N}\sum_{n\leq
N}|f(T^nx)-f_\varepsilon(T^nx)| <\varepsilon.$$
The quantity above bounds the asymptotical difference between $ A(f,M,H) $ and $ A(f_\varepsilon,M,H)$. Therefore holds for each $ f\in D(X) $ with zero mean and for arbitrary $ x\in X $.
AOP property for nil-translations
=================================
In this section we shall prove that ergodic nil-translations on compact connected nilmanifolds satisfy the hypotheses (i) and (ii) of the criterion provided by Theorem \[thm:prop\], thereby proving Theorem \[ThA\] and Corollaries \[CorB\] and \[CorC\] for $ \phi $ a nil-translation on a compact connected nilmanifold.
Background on nilpotent Lie groups {#sec:background}
----------------------------------
Let $ G $ be a connected simply connected $ k $-step nilpotent Lie group with Lie algebra $ \mathfrak g $, and let $ \Gamma $ be a lattice in $ G $. The quotient $ M=G/\Gamma $ is then a compact nilmanifold on which $ G $ acts on the left by translations. Denote by $ \lambda =\lambda_M $ the $ G $-invariant probability measure on $
M $ (locally given by a Haar measure of $ G $). Let $$\mathfrak g=\mathfrak g^{(1)}\supset\mathfrak g^{(2)}\supset\dots
\supset \mathfrak g^{(k)}\supset \mathfrak g^{(k+1)}=\{0\},\quad
\mathfrak g^{(i+1)}=[\mathfrak g,\mathfrak g^{(i)}], \ i=1,\dots, k,$$ be the descending central series of $ \mathfrak g $ (with $ \mathfrak
g^{(k)}\neq\{0\} $) and let $$G=G^{(1)}\supset G^{(2)}\supset\dots \supset G^{(k)}\supset G^{(k+1)}=\{e_G\},
\ G^{(i+1)}=[G,G^{(i)}], \ i=1,\dots, k,$$ be the corresponding series for $ G $. In this setting, there exists a strong Malcev basis through the filtration $ (\mathfrak g^{(i)})_ {i=1}^k
$ strongly based at the lattice $ \Gamma $, that is a basis $ X_1,
\dots , X_{\dim \mathfrak g} $ of $ \mathfrak g $ such that for an increasing sequence of integers $ 0=\ell_0< \ell_1 <\dots < \ell_k=\dim
\mathfrak g $ we have:
1. The elements $ X_{\ell_{i-1}+1}, X_{\ell_{i-1}+2}, \dots , X_{\dim
\mathfrak g} $ form a basis of $ \mathfrak g^{(i)} $;
2. For each $ i\in \{1, \dots, \dim \mathfrak g\} $ the elements $ X_
{i}, X_{i+1}, \dots , X_{\dim \mathfrak g} $ span an ideal of $
\mathfrak g $;
3. The lattice $ \Gamma $ is given by $$\Gamma=\big\{ \exp (n_1 X_1)\cdots \exp(n_{\dim \mathfrak g} X_{\dim
\mathfrak g})\mid n_i\in {{\mathbb{Z}}}, i=1,\dots, \dim \mathfrak g\big\}.$$
It will be convenient to set $ d_i=\ell_{i}-\ell_{i-1} $ for $ i=1,\ldots,k
$, so that $ d_1 $ denotes the dimension of the abelianized group $ G/
[G,G] $ and $ d_k $ is the dimension of the $ k $ derived group $ G^{(k)}
$, which, we recall, is included in the center $ Z(G) $ of $ G $.
Since, for all $ i=1,\dots , k-1 $, the group $ G^{(i+1)} $ is a closed normal subgroup of $ G $, we have natural epimorphisms $ \pi^ {(i)} \colon G\to G/G^{(i+1)} $. The group $ G^{(i+1)}\cap \Gamma $ is a lattice in $ G^{(i+1)} $ (Theorem 2.3 in [@Rag Corollary 1]); equivalently, $ G^{(i+1)}\Gamma $ is a closed subgroup of $ G $. Moreover, $\pi^ {(i)}(\Gamma)$ is a lattice in $G/G^{(i+1)}$ and $ M^{(i)}:=G/G^{(i+1)}\Gamma \approx (G/G^{(i+1)})/\pi^ {(i)}(\Gamma)$ is a nilmanifold. It follows that the short exact sequences $$0 \hookrightarrow G^{(i+1)} \hookrightarrow G \twoheadrightarrow G/G^
{(i+1)}\twoheadrightarrow 0, \qquad i=1,\dots,k-1,$$ induce $ G $-equivariant fiber bundles of nilmanifolds $$\label{eq:fibration} \pi^{(i)} \colon M=G/\Gamma \twoheadrightarrow
M^{(i)}=G/G^{(i+1)}\Gamma, \qquad i=1,\dots,k-1,$$ whose fibers are the orbits of $ G^{(i+1)} $ on $ G/\Gamma $, hence homeomorphic to the nilmanifolds $ G^{(i+1)} / (G^{(i+1)} \cap \Gamma)
$.
Two cases are of most interest for us. When $ i=1 $ the base nilmanifold $ M^{(1)}=G/[G,G]\Gamma $ is a torus of dimension $ d_1 $. As $ G/[G,G] $ is an abelian group, it is identified by the exponential map with (the additive group) $ \mathfrak g/\mathfrak g^{(2)}\approx {{\mathbb{R}}}^{d_1} $. Then the vectors $$\bar X_i = X_i \mod \mathfrak g^{(2)}, \qquad i=1,\dots, d_1,$$ form a set of generators of the lattice $ \pi^{(2)}(\Gamma)\approx\Gamma/
(G^{(2)}\cap\Gamma) $. We shall use an additive notation for the abelian group $ G/[G,G] $; thus we identify $ M^{(1)} $ with $ {{\mathbb{R}}}^{d_1}/{{\mathbb{Z}}}^
{d_1} $ by means of the basis $ \bar X_i $, ($ i=1,\dots ,d_1 $).
At the opposite extreme, when $ i=k-1 $, the group $ G^{(k)} $ is abelian, thus isomorphic to $ {{\mathbb{R}}}^{d_k} $. Let us consider the action of $ G^{(k)} $ by left translations on $
M=G/\Gamma $. As the stability group of any point in $ M $ is the subgroup $ G^{(k)}\cap \Gamma $, the action of $ G^{(k)} $ on $ M $ induces a free action of the torus group $ \mathbb T^{(k)}=G^{(k)}/ (G^
{(k)}\cap \Gamma) $ on $ M $. Ssnce the group $ \mathbb T^{(k)} $ acts transitively on the fibers of the fibration $ \pi^{(k-1)}\colon M\to M^
{(k-1)} $, the map $ \pi^{(k-1)} $ is a principal $ \mathbb T^{(k)} $-bundle.
It follows from the above that the Hilbert space $ L^2(M,\lambda) $ decomposes into a sum of $ G $-invariant orthogonal Hilbert subspaces $$L^2(M,\lambda)=\bigoplus_{\chi \in \widehat{ {\mathbb T}^{(k)}}} H_\chi,$$ where we have set, for each character $ \chi $ of the torus $ \mathbb
T^{(k)} $, $$\label{l2} H_\chi:=\{f\in L^2(M) \mid f(zx ) =\chi(z)f(x ), \,\forall
x \in M,\, \forall z\in G^{(k)}\}.$$ Let $ \lambda^{(k)} $ be the probability Haar measure on $ \mathbb T^{(k)} $. Denote by $ C_\chi $ the subspace of continuous functions in $
H_\chi $. The linear operator $ \mathcal F_\chi: L^2 (M,\lambda)\to
H_\chi $ given by $$\mathcal F_\chi(f)(x)=\int_{\mathbb T^{(k)}}\chi(z)f(z^{-1}x)\,d\lambda^
{(k)}(z)$$ is the orthogonal projector on $ H_\chi $. It follows that
\[lem:densc\] The space $C_\chi$ is dense in $H_\chi$.
Dynamics and joinings for nil-translations {#sec:dynamics-joinings}
------------------------------------------
For $ u\in G $, let $ l_u\colon M\to M $ be the left translation by $ u $. Then $ l_u $ is a measure preserving automorphism of $ (M,\lambda) $, which for simplicity we call a *nil-translation*.
By the normality of $ G^{(i+1)} $ in $ G $, the projection $ \pi^{(i)}
:M\to M^{(i)} $ intertwines the map $ l_u \colon M\to M $ with the nil-translation $ l_{\pi^{(i)}(u)} \colon M^{(i)}\to M^{(i)} $.
The celebrated theorem by Auslander, Green and Hahn (see [@Au-Gr-Ha]), states that the nil-translation $ l_u $ is ergodic (minimal and uniquely ergodic) if and only if the translation $ l_{\pi^ {(1)}(u)} $ by $ \pi^{(1)}(u) $ on the torus $ M^{(1)} $ is ergodic (minimal and uniquely ergodic). The latter condition is equivalent to saying that the rotation vector $ \alpha=(\alpha_1, \dots, \alpha_{d_1})\in {{\mathbb{R}}}^{d_1}/{{\mathbb{Z}}}^ {d_1}
$ defined by $$\pi^{(1)}(u) = \exp (\alpha_1 \bar X_1+ \dots+ \alpha_{d_1}\bar X_{d_1})$$ is irrational with respect to the lattice $ {{\mathbb{Z}}}^{d_1} $; equivalently, that the real numbers $ 1, \alpha_1', \dots, \alpha_{d_1}'
$ are linearly independent over $ {\mathbb{Q}}$, for any lift $ (\alpha_1',
\dots, \alpha_{d_1}') $ of $ \alpha $ to $ {{\mathbb{R}}}^{d_1} $. Using an additive notation we denote the translation $ l_{\pi^{(1)} (u)}
$ of the torus $ M^{(1)} $ by $ R_{\alpha} $, as it is uniquely determined by the rotation vector $ \alpha $. With a slight abuse of language, and coherently with the choice of using an additive notation for the torus group $ M^{(1)} $, we shall then say that the projection $ \pi^{(1)} $ intertwines the translation $ l_u $ with the rotation $ R_\alpha $.
Let $ l_u \colon M\to M $ be the nil-translation (not necessary ergodic with respect to $ \lambda $) by $u$ and let $ \mu $ be an $ l_u $-invariant ergodic probability measure on $ M $. Let us consider the stabilizer of the measure $ \mu $ $$\Lambda(\mu)=\{g\in G:(l_g)_*\mu=\mu\}.$$ Then $ \Lambda(\mu)<G $ is a closed subgroup of $ G $. The following celebrated theorem of Ratner [@Rat Theorem 1] (first proved independently, in the context of nilflows, by Starkov in [@St] and Lesigne in [@Les]) describes all such ergodic measures.
\[thm:ratner\] If $ \mu $ is a $ l_u $-invariant ergodic measure on $ M $ then there exists $ x\in M $ such that the orbit $
\Lambda(\mu) x\subset M $ is closed and it is the topological support of $ \mu $.
Let $ U\in \g $ and let $ \mu $ be a probability measure on $ M $ which is invariant and ergodic for the nilflow $ (l_{u^t})_{t\in {{\mathbb{R}}}}
$ ($ u^t=\exp(tU) $). Then the stabilizer $ \Lambda(\mu) $ is a connected group and the topological support of $ \mu $ is an orbit $
\Lambda(\mu) x\subset M $.
From now on, we fix an ergodic translation $ l_u $ of $ M $ (with respect to $ \lambda $) projecting to the corresponding ergodic rotation $ R_\alpha $ of the torus $ M^{(1)} $.
We now consider the group $ G\times G $ with the Lie algebra $
\mathfrak g \oplus \mathfrak g $, the lattice $ \Gamma\times\Gamma $ and the corresponding nilmanifold $ M\times M= (G\times G)/ (\Gamma\times
\Gamma) $ on which the group $ G\times G $ acts by left translations. We have natural projections $ p_1 $ and $ p_2 $ of $ M\times M $ onto $
M $, by selecting the first or the second coordinate of a point $ (x_1,x_2)\in
M\times M $, respectively.
Then we have $ (G\times G)^{(i)} = G^{(i)}\times G^{(i)} $, $ (\mathfrak
{g}\oplus\g)^{(i)}=\mathfrak g^{(i)}\oplus\mathfrak g^{(i)} $ for all $
i=1,\ldots, k $. Clearly $ M^{(i)} \times M^{(i)}= (G\times G)/(G\times
G)^{(i)}(\Gamma\times\Gamma) $ for all $ i=1,\ldots, k-1 $.
Let $ r,s\in {{\mathbb{N}}}$ be relatively prime numbers. Let $ \rho $ be an ergodic joining of the measure theoretical ergodic systems $ (M,l_ {u}^r,
\lambda) $ and $ (M,l_{u}^s, \lambda) $. We recall that, by definition, $ \rho $ is a measure on $ M\times M $, with marginals $ (p_i)_*\rho=\lambda
$, ($ i=1,2 $), which is invariant and ergodic for the product transformation $ l_{u}^r \times l_{u}^s $. The transformation $ l_u^r\times
l_u^s $ on $ M\times M $ is the left translation on $ M\times M $ by $
(u^r,u^s) $: $$l_u^r\times l_u^s (x_1,x_2)=(u^rx_1,u^sx_2), \quad \forall (x_1,x_2)\in
M\times M.$$ Then the projection map $ \pi^{(1)}\times\pi^{(1)} $ of $ M\times M $ onto the $ 2d_1 $-dimensional torus $ M^{(1)}\times M^{(1)}$ intertwines the map $ l_u^r\times l_u^s $ with the rotation $ R_ {\alpha}^r\times
R_{\alpha}^s $ of $ { M}^{(1)} \times{ M}^{(1)} $. Moreover, the image measure $ \rho^{(1)}:=(\pi^{(1)}\times\pi^{(1)})_*\rho $ is $ R_{\alpha}^r\times
R_{\alpha}^s $-invariant and ergodic.
\[lem:torus\] Let $R_\alpha$ a minimal rotation of the torus $d$-dimensional $\mathbb T^d$ and rotation vector $\alpha$. Let $r,s$ relatively prime integers and let $ \eta $ be an $ R_{\alpha}^r\times R_{\alpha}^s~
$-invariant and ergodic probability measure on $ \mathbb T^d \times
\mathbb T^d $. Then the stabilizer $ \Lambda(\eta)
$ of $\eta$ is the group $${\mathbb T}_{r,s}=\{(t_1,t_2)\in {\mathbb T}^d\times {\mathbb T}^d \mid st_1=rt_2\},$$ with Lie algebra $ \{(r v, s v)\mid v \in {{\mathbb{R}}}^d \} $.
As the map $ R_{\alpha}^r\times R_{\alpha}^s $ is the rotation $
R_{(r\alpha, s\alpha)} $ with rotation vector $ (r\alpha,s\alpha)
$ of the $2d$ dimensional torus $\mathbb T^d \times \mathbb T^d$, it preserves the orbits of the closed subgroup ${\mathbb T}_{r,s}$ that is, the cosets $${\mathbb T}_{r,s,c}=\{(t_1,t_2+c)\in {{\mathbb{T}}}^d \times {{\mathbb{T}}}^d \mid st_1=rt_2\},
\quad c\in {\mathbb T}^d.$$ We claim that the action of $ R_{(r\alpha, s\alpha)} $ on each coset $ {\mathbb T}_ {r,s,c} $ is minimal and uniquely ergodic. Indeed, let $ a , b \in {{\mathbb{Z}}}$ be such that $ ar+bs=1 $. The map $ I_c\colon {\mathbb T}_{r,s,c}\to {\mathbb T}^d $, given by $ I_c(t_1,t_2+c)=at_1+bt_2 $, is well defined. Its inverse is given by the formula $ t\in {{\mathbb{T}}}^d\mapsto (rt,st+c)\in {\mathbb T}_{r,s,c} $. It follows that the map $ I_c $ is a homeomorphism intertwining the action of $ R_{(r\alpha, s\alpha)} $ on $ {\mathbb T}_{r,s,c} $ with the action of $ R_\alpha $ on $ {\mathbb T}^d$; it also intertwines also the action of $ {\mathbb T}_{r,s}$ on $ {\mathbb T}_{r,s,c} $ with the action of ${\mathbb T}^d$ on itself. Since $ R_\alpha $ is minimal and uniquely ergodic, so is the action of $ R_{(r\alpha, s\alpha)}$ on $ {\mathbb T}_{r,s,c} $. Since $ \eta $ is an ergodic measure for $ R_{(r\alpha, s\alpha)}$, there exists a $c\in {\mathbb T}$ such that $ {\operatorname{supp}
}(\eta)={\mathbb T}_{r,s,c}={\mathbb T}_ {r,s}(0,c) $. It follows that $\Lambda (\eta)= {\mathbb T}_{r,s} $, which completes the proof.
Applying the above lemma to our setting we have:
\[lem:torus1\] Let $ \eta $ be an $ R_{\alpha}^r\times R_{\alpha}^s~
$-invariant and ergodic probability measure on $ M^{(1)}\times M^{ (1)} $. Then the Lie algebra of the stabilizer $ \Lambda(\eta) $ is t $$\mathfrak h_{r,s}:=\{(r \bar X, s \bar X) \mid \bar X \in
\mathfrak g/\mathfrak g^{(2)}\}\subset (\g\oplus \g)/(\g^{(2)}\oplus
\g^{(2)}).$$
Denote by $ H<G\times G $ the stabilizer of the joining $ \rho $ and let $ \mathfrak h\subset\g\oplus\g $ be its Lie algebra.
\[lem:rsh\] There exist elements $ X_1' , X_1'' , \dots, X_ {d_1}'
, X_{d_1}'' $ in $ \mathfrak g $ satisfying $$X'_i\equiv X_i''\equiv X_i \mod \mathfrak g^{(2)}, \text{ for all
} i=1,\dots, d_1$$ and such that $$\overline{Y}_i=(r X'_i, sX''_i)\in \mathfrak h.$$
Let us consider the image measure $ \rho^{(1)}:=(\pi^{(1)}\times\pi^
{(1)})_*\rho $ which is $ R_{\alpha}^r\times R_{\alpha}^s $-invariant and ergodic. By Theorem \[thm:ratner\], the topological support of the measure $ \rho $ is a closed coset $ H\bar x $ for some $ \bar x\in M\times M $. Then the topological support of $ \rho^{(1)} $ is $ (\pi^{(1)}\times\pi^
{(1)})(H) \bar t $ for some $ \bar t\in M^{(1)}\times M^{(1)} $.
Since $ \rho^{(1)} $ is $ R_{\alpha}^r\times R_{\alpha}^s $-invariant and ergodic, by Corollary \[lem:torus1\], the topological support of $ \rho^{(1)} $ is $
\Lambda(\rho^{(1)}) \bar t $ and the Lie algebra of $ \Lambda(\rho^
{(1)}) $ is $ \mathfrak h_{r,s} $. As the orbits $ (\pi^{(1)}\times\pi^
{(1)})(H) \bar t $ and $ \Lambda(\rho^{(1)}) \bar t $ are closed and equal, it follows that the Lie algebras of $ (\pi^{(1)}\times\pi^ { (1)})(H) $ and $ \Lambda(\rho^{(1)}) $ are the same and equal to $
\mathfrak h_{r,s} $. Hence the Lie algebra $ \mathfrak h $ of the Lie group $ H $ projects under the tangent map $ \mathrm d( \pi^{(1)}\times\pi^
{(1)}) $ onto the algebra $ \mathfrak h_{r,s}= \{ (r \bar X, s \bar
X) \mid \bar X \in \mathfrak g/\mathfrak g^{(2)}\} $. It follows that for every $ i=1,\ldots, d_1 $ there exists $ (X_i',X_i'')\in
\mathfrak h $ such that $$d( \pi^{(1)}\times\pi^{(1)})(X_i',X_i'')=(r\,d\pi^{(1)}(X_i),s\,d\pi^
{(1)}(X_i)),$$ which yields the required elements $ X_1' , X_1'' , \dots, X_ {d_1}'
, X_{d_1}'' \in \mathfrak g $.
\[lem:aop\_nil:1\] The Lie algebra $ \mathfrak h\subset\g\oplus\g
$ satisfies $$\{ (r^k Z, s^k Z )\mid Z\in \mathfrak g^{(k)}\} \subset \mathfrak
h \cap (\mathfrak g^{(k)}\oplus\mathfrak g^{(k)} ).$$ It follows that the group $ H $ contains the subgroup $$L{:=}\{ \exp(r^k Z, s^k Z)\mid Z\in\mathfrak g^{(k)}\} <G^{(k)}\times
G^{(k)}.$$
Let $ X_i',X_i'',\overline{Y}_i $ for $ i=1,\ldots, d_1 $ be given by Lemma \[lem:rsh\]. Consider the $ k $-fold Lie products of elements of the sets $ \overline S=\{ \overline Y_i \mid i=1, \dots ,d_1 \} $ and $ S=\{ X_i \mid i=1, \dots ,d_1 \} $; that is, for every $ (i_1,\dots,i_k)\in
\{1,\dots,d_1\}^k $, we set $$\overline S_{i_1,i_2,\dots,i_k}:=[\overline Y_{i_1},[\overline Y_{i_2},
\dots, [\overline Y_{i_{k-1}},\overline Y_{i_k}]\dots]]\in \g^{(k)}\oplus\g^
{(k)}$$ and $$S_{i_1,i_2,\dots,i_k}:=[X_{i_1},[X_{i_2}, \dots, [X_{i_{k-1}},X_{i_k}]\dots]]\in
\g^{(k)}.$$ Then, by definition and Lemma \[lem:aop\_nil:4\], we have $$\begin{aligned}
\overline S&_{i_1,i_2,\dots,i_k} =[(rX'_{i_1},sX_{i_1}'') ,[(rX'_{i_2},sX''_
{i_2}) , \dots, [(rX'_{i_{k-1}},sX''_{i_{k-1}}),(rX'_{i_{k}},sX''_
{i_{k}})]\dots]] \\
&=\big( [rX'_{i_1},[ rX'_{i_2}, \dots, [rX'_{i_{k-1}},rX'_{i_{k}}]\dots]],
[sX''_{i_1},[ sX''_{i_2}, \dots, [sX''_{i_{k-1}},sX''_{i_{k}}]\dots]]
\big)\\
&=\big( r^k \,[X_{i_1},[ X_{i_2}, \dots, [X_{i_{k-1}},X_{i_{k}}]\dots]]
, s^k\,[X_{i_1},[ X_{i_2}, \dots, [X_{i_{k-1}},X_{i_{k}}]\dots]]
\big)\\
&=\big( r^k S_{i_1,i_2,\dots,i_k} , s^k S_{i_1,i_2,\dots,i_k} \big).
\end{aligned}$$
Since $ X_1+\g^{(2)},\ldots,X_{d_1}+\g^{(2)} $ is a basis of $ \g/\g^
{(2)} $, by Lemma \[lem:aop\_nil:2\], the set $ \{X_1,\ldots,X_{d_1}\} $ generates the Lie algebra $ \g $. Therefore, in view of Lemma \[lem:aop\_nil:3\], the family of $ k $-fold products $ S_{i_1,i_2,\dots,i_k}~
$ spans $ \mathfrak g^{(k)} $.
By Lemma \[lem:rsh\], $ \overline Y_{i}\in\mathfrak h $ for every $ i=1,\ldots,
d_1 $. It follows that $$\big( r^k S_{i_1,i_2,\dots,i_k} , s^k S_{i_1,i_2,\dots,i_k} \big)=\overline
S_{i_1,i_2,\dots,i_k}\in\mathfrak h\cap (\mathfrak g^{(k)}\oplus\mathfrak
g^{(k)} )$$ for every $ (i_1,\dots,i_k)\in \{1,\dots,d_1\}^k $. Consequently, for all $ Z\in \mathfrak g^{(k)} $ we have $ (r^kZ,s^kZ)\in\mathfrak
h\cap (\mathfrak g^{(k)}\oplus\mathfrak g^{(k)} ) $, which completes the proof.
\[prop:aop\_nil:1\] Let $ \chi_1 $, $ \chi_2 $ be characters of the torus $ \mathbb T^{(k)} $ so that at least one is non-trivial. Then, if $ \chi_1^{r^k}\not= \chi_2^{s^k} $, for any $ f_i\in H_{\chi_i}
$, $ i=1,2 $, and any ergodic joining $ \rho $ of $ (M,l_{u}^r,
\lambda) $ and $ (M,l_{u}^s, \lambda) $, we have $$\rho(f_1\otimes \bar f_2)=0.$$
Recall there exists a closed subgroup $ H< G\times G $ such that the measure $ \rho $ is $ H $-invariant and $ H $ contains the subgroup $$L= \{ \exp(r^k Z, s^k Z)\mid Z\in\mathfrak g^{(k)}\} < G^{(k)}\times
G^{(k)}.$$ Since $ f_1\in H_{\chi_1} $ and $ f_2\in H_{\chi_2} $, for every $
(\exp(r^k Z), \exp(s^k Z))\in L $ and all $ ( x_1, x_2)\in
M \times M $ we have $$\begin{aligned}
f_1\otimes \bar f_2&((\exp(r^k Z), \exp(s^k Z))(x_1, x_2))=f_1(\exp
(r^k Z) x_1)\cdot \bar f_2(\exp(s^k Z) x_2)\\
& = \chi_1(\exp(r^k Z)) \overline{\chi_2(\exp(s^k Z))}\cdot f_1\otimes
\bar f_2(x_1, x_2)\\
& = (\chi_1^{r^k}\cdot\chi_2^{-s^k})(\exp(Z))\cdot f_1\otimes \bar
f_2(x_1, x_2).
\end{aligned}$$ As the probability measure $ \rho $ is invariant under the action of the group $ L<H $, it follows that $$\rho(f_1\otimes \bar f_2)=(\chi_1^{r^k}\cdot\chi_2^{-s^k})(z)\rho(f_1\otimes
\bar f_2)$$ for every $ z\in G^{(k)}/(G^{(k)}\cap \Gamma)=\mathbb T^{(k)} $. By assumption, the character $ \chi_1^{r^k}\cdot\chi_2^{-s^k} $ is non-trivial, so we can find $ z\in \mathbb T^{(k)} $ with $ (\chi_1^
{r^k}\cdot\chi_2^{-s^k})(z)\neq 1 $. This yields $ \rho(f_1\otimes
\bar f_2)=0 $, which completes the proof.
The following theorem shows that Theorem \[thm:prop\] applies to nil-translations. Consequently, Theorem \[ThA\] and Corollaries \[CorB\] and \[CorC\] are true for nil-translations.
\[thm:mainrot\] Let $ l_u $ be an ergodic nil-translation of a compact connected nilmanifold $ M=G/\Gamma $. Then there exists a set $ \mathscr{C}\subset C(M)\cap L^2_0(M,\lambda) $ whose span is dense in $ L^2_0(M,\lambda) $ such that:
- for any pair $ f_1,f_2\in\mathscr{C} $, for all but at most one pair of relatively prime natural numbers $ (r,s) $ we have $
\rho(f_1\otimes \bar{f}_2)=0 $ for all ergodic joinings $ \rho $ of $ l_u^r $ and $ l_u^s $;
- for every $ f\in\mathscr{C} $ and every $ \omega\in \C $ with $ |\omega|=1
$ there exists an element $ g\in G $ such that $ f(l_u^n (l_gx))=\omega f(l_u^n
x) $ for every $ x\in M $ and every $ n\in{{\mathbb{Z}}}$.
The proof is by induction on the class of nilpotency $ k $ of $G$.
If $ k=1 $ then $ M $ is the torus $ {{\mathbb{T}}}^{(1)} $ and $ H_\chi=\C\chi $ for every character $ \chi $ of $ {{\mathbb{T}}}^{(1)} $. Let $ \mathscr{C} $ be the set of nontrivial characters. Then $\mathscr{C}$ is linearly dense in $L^2_0(M,\lambda)$. Moreover, for every pair of nontrivial characters $ \chi_1 $, $ \chi_2 $ there is at most one pair of relatively prime natural numbers $ r,s $ such that $ \chi_1^r=\chi_2^s $. In view of Proposition \[prop:aop\_nil:1\], the hypotheses (i) of Theorem \[thm:prop\] are verified. Moreover, if $ \chi\in\mathscr {C} $ then for every $ z\in {{\mathbb{T}}}^{(1)} $ we have $ \chi(l_u^n(l_zx))=\chi (l_z(l_u^nx))=\chi(z)\chi(l_u^n x) $ which verifying the hypotheses (ii) of Theorem \[thm:prop\].
Suppose that for every ergodic nil-translation on any compact connected nilmanifold of class $ k-1 $ the required set of continuous functions does exist. Let us consider an ergodic nil-translation $ l_u $ on a compact connected nilmanifold $ M=G/\Gamma
$ of class $ k $. Then $ M^{(k-1)} $ is a compact connected nilmanifold of class $ k-1 $ and $ \pi^{(k-1)}\colon M\to M^{(k-1)}
$ intertwines $ l_u $ with the ergodic rotation $ l_{\pi^{(k-1)}u} $ on $ M^{(k-1)} $. Denote by $ \mathscr{C}^ {(k-1)} $ the subset of continuous functions on $ M^{(k-1)} $ derived from the induction hypothesis. Next we use $$L^2(M,\lambda)=\bigoplus_{\chi \in \widehat{ {\mathbb T}^{(k)}}} H_\chi.$$ If $ \chi=1 $ is the trivial character then $ H_1 $ consists of $ G^
{(k)} $-periodic functions and $ H_1 $ can be naturally identified with $ L^2(M^{(k-1)}) $. Since the identification is $ G $-equivariant, for functions from $ H_1 $ the dynamics given by the rotation $ l_u $ coincides with the dynamic of $ l_{\pi^{(k-1)}u} $ on $ M^{(k-1)} $. Let $$\mathscr{C}:=\mathscr{C}^{(k-1)}\cup\bigcup_{\chi \in \widehat{ {\mathbb
T}^{(k)}}\setminus\{1\}}C_{\chi}.$$ By Lemma \[lem:densc\] and by the induction hypothesis, the set $\mathscr{C}$ is linearly dense in the space $L_0^2(M)$.
First note that for all functions from the subset $ \mathscr{C}^{(k-1)}\subset
\mathscr{C} $ both properties (i) and (ii) follows from the induction hypothesis. To complete (i) we need to take $ f_1\in H_{\chi_1}
$ and $ f_2\in H_{\chi_2} $ with at least one non-trivial character $
\chi_1 $, $ \chi_2 $ of $ {{\mathbb{T}}}^{(k)} $. As there is at most one pair of relatively prime natural numbers $ r,s $ such that $ \chi_1^{r^k}=\chi_2^
{s^k} $, Proposition \[prop:aop\_nil:1\] implies (i).
Let $ f\in H_\chi $ for a non-trivial character $ \chi $. Since $
G^{(k)} $ is a subgroup of the center $ Z(G) $, the action $ z\in {{\mathbb{T}}}^{(k)}\mapsto l_z $ commutes with the nil-translation $ l_u $. It follows that $$f(l_u^n(l_zx))=f(l_z(l_u^nx))=\chi(z)f(l_u^n x).$$ The non-triviality of $ \chi $ completes the proof of (ii).
AOP for affine diffeomorphisms of nilmanifolds
==============================================
In this section we shall prove Theorem \[ThA\] and Corollaries \[CorB\] and \[CorC\] for zero entropy ergodic affine diffeomorphisms on compact connected nilmanifolds. Recall that each such affine diffeomorphism is unipotent, see [@MR0260975] and [@MR1758456]. In fact, we shall show that the hypotheses of Theorem \[thm:prop\] apply to such maps.
On affine diffeomorphisms of nilmanifolds
-----------------------------------------
For any Lie group $G$, an *affine diffeomorphism of $ G $* is a mapping of the form $ g
\mapsto u A(g) $, where $ u\in G $ and $ A $ is an automorphism of $ G
$.
Let $ l_u $ be the left translation on $ G $ by an element $ u\in G $. Then the above affine map is the composition $ l_u \circ A $. We shall however use the shortened notation $ u A $, whenever convenient.
Let $ M=G/\Gamma $ be a compact nilmanifold, with $ \Gamma $ a lattice in $ G $ and $G$ a connected, simply connected (nilpotent) Lie group. An affine diffeomorphism $ uA $ of $ G $ induces a quotient diffeomorphism of $ M $ if and only if $ A(\Gamma)=\Gamma $; an *affine diffeomorphism of $ M $* is a map of $ M $ that arises as such a quotient. For simplicity, whenever the condition $ A(\Gamma)=\Gamma $ is satisfied, the symbol $ \phi = uA $ (or $ \phi = l_u\circ A $) will denote both an affine diffeomorphism of $ G $ and the induced quotient affine diffeomorphism of $ M $.
We recall that the group $ \Aut(G) $ of automorphisms of $ G $ is identified, via the exponential map, with the group $ \Aut(\g) $ of automorphisms of the Lie algebra $ \g $ of $ G $; thus, for $ A\in \Aut(G)\approx\Aut (\g) $, we have $ \exp(A(X))= A(\exp X) $, for all $ X\in \g $. An affine diffeomorphism $ uA $ of $ M $ (or of $ G $) is *unipotent*, if $ A:\g\to\g $ is a unipotent automorphism. (Recall that an ergodic affine diffeomorphism of $ M $ has zero entropy if and only if it is unipotent.)In this case we can write $
A=\exp B $, with $ B:\g\to \g $ a nilpotent derivation of $\g$; by definition of derivation we have $$\label{eq:deriv} B([X,Y])=[BX,Y]+[X,BY]\ \text{ for }\ X,Y\in\g.$$
### Some rationality issues {#sec:some-rati-issu}
Let $\Gamma$ be a lattice in $G$. Then $\Gamma$ determines a rational structure on the Lie algebra $\g$. In fact, by Theorem 5.1.8 in [@Co-Gr], the vector space $ \g_\Gamma:={\mathbb{Q}}$-$ \Span(\log
\Gamma) $ is a Lie algebra over ${\mathbb{Q}}$ such that $
\g=\g_\Gamma\otimes_{\mathbb{Q}}{{\mathbb{R}}}$. Indeed, any strong Malcev basis strongly based on the lattice $ \Gamma $ is a $ {\mathbb{Q}}$-basis of $ \g_\Gamma $.
Recall that a subalgebra $ \mathfrak h\subset \g $ is *rational (with respect to $\Gamma$)* if $ (\mathfrak
h\cap \g_\Gamma)\otimes_{\mathbb{Q}}{{\mathbb{R}}}=\mathfrak h $, i.e. if $ \mathfrak h $ has a basis contained in $ \g_\Gamma $. For example, each $ \g^ {(i)} $ is a rational ideal with respect to any lattice of $G$.
By definition, a connected closed subgroup $ H=\exp(\mathfrak h) $ is *a rational subgroup of $G$* if the subalgebra $ \mathfrak h $ is rational.
In view of Theorem 5.1.11 in [@Co-Gr], a connected closed subgroup $H$ is a rational subgroup of $ G
$ if and only if the intersection subgroup $
H \cap \Gamma $ is a lattice in $ H $. If furthermore $H$ is normal in $G$, then $ G/\Gamma H $ is a compact nilmanifold; thus we obtain a smooth $ G $-equivariant factor map $
\pi_H:G/\Gamma \to G/\Gamma H $.
If $uA$ is an affine unipotent diffeomorphism of $M=G/\Gamma$, since $ A(\Gamma)=\Gamma $, we have $ A(\g_\Gamma)=\g_\Gamma
$. As the logarithm of a unipotent automorphims is a rational map, if $A=\exp B$ we have $ B(\g_\Gamma)\subset\g_\Gamma $. Hence $B$ is a rational endomorphism of $\g_\Gamma$.
Suspensions {#sec:danis-construction}
-----------
We consider the usual construction turning an affine diffeomorphism into a translation [@dani1977]. Let $ \phi=uA $ be a unipotent affine diffeomorphism of the nilmanifold $ M=G/\Gamma $ such that $ A\neq I $. Let $ B $ be the (nilpotent) derivation of $ \g $ such that $ A=\exp B $ and let $ \mathcal A= \{A^t\}_
{t\in{{\mathbb{R}}}}$, with $ A^t=\exp t B $, be the one-parameter subgroup of automorphisms of $ G $ generated by $B$. (We have a natural identification $A^t\in \mathcal A\mapsto t\in {{\mathbb{R}}}$.)
The semi-direct product $ \widetilde G=G\rtimes \mathcal A $ is a simply connected, connected subgroup of the affine diffeomophisms of $ G $ with the inherited product rule defined by $$\label{eq:semi-direct}
(g_1A^{t_1})\cdot (g_2A^{t_2})= g_1 A^{t_1}(g_2)\, A^{t_1+t_2}.$$ We regard $ G $ as a normal subgroup of $ \widetilde G $ via the inclusion map $ l\colon g \in G \mapsto l_g\in \widetilde G $. Thus we have a split exact sequence $$\label{eq:NilpotentAOP-06-14:1}
0 \to G \xhookrightarrow{l} \widetilde G \xrightarrow{\check p} {{\mathbb{R}}}\to 0$$ where $\check p (gA^{t}) = t$, for all $gA^{t}\in \widetilde G$.
The set $ \widetilde \Gamma=\{\gamma A^n\mid \gamma\in \Gamma, n\in {{\mathbb{Z}}}\} $ is a closed lattice subgroup of $ \widetilde G $; thus $ \widetilde M=\widetilde
G/\widetilde \Gamma $ is a compact nilmanifold. By formulas and , since $\check p(\widetilde \Gamma)={{\mathbb{Z}}}$, the map $\check p$ induces a fibration (in fact a $G$-bundle) $$\label{eq:NilpotentAOP-06-14:2}
p\colon g A^t\widetilde \Gamma \in \widetilde G/\widetilde \Gamma \mapsto t\in {{\mathbb{R}}}/{{\mathbb{Z}}}$$ with compact fibers $ p^{-1}\{t\}=\{gA^t\widetilde \Gamma\mid g\in G\}$ parametrized by $ t\in
{{\mathbb{R}}}/{{\mathbb{Z}}}$. Since $ gA^t\widetilde \Gamma= g' A^t\widetilde \Gamma$, for $t\in {{\mathbb{R}}}/{{\mathbb{Z}}}$ and $g,g'\in G$, if and only if $g A^t(\Gamma) = g' A^t(\Gamma)$ we identify $
p^{-1}\{t\}\approx G/A^t(\Gamma)$. In particular, $p^{-1}\{0\}\approx
G/\Gamma$, the identification being given by the embedding $ i\colon
G/\Gamma\hookrightarrow \widetilde G/\widetilde \Gamma $ defined by $ i(g\Gamma)=g \widetilde \Gamma $. Thus we shall consider $G/\Gamma$ as a subset of $\widehat G/\widetilde \Gamma $.
Let $ \widetilde \g $ be the Lie algebra of the group $ \widetilde G
$. Since this group is generated by the normal subgroup $G$ and by $\mathcal A=(\exp tB)_{t\in {{\mathbb{R}}}}$, the Lie algebra $ \widetilde \g $ is the semi-direct product $ \g\rtimes {{\mathbb{R}}}B $; in addition to the commutation rules of elements of Lie algebra $ \g $, we have the rule[^7] $$\label{eq:susplie}
[ B, X] = B(X), \quad \text { for all }X \in \g.$$ Let $\widetilde u= uA$. Since the derivation $B$ is nilpotent, the group $ \widetilde G $ is nilpotent. Thus the exponential map of $\widetilde G$ is a homeomorphism and we can write $ \widetilde u= \exp (B+v) $ for some $ v\in \g $. Indeed, since $\g$ is an ideal in $\widetilde \g$, by the Baker-Campbell-Hausdorff formula, setting $u=\exp v_1$, we have $u A= \exp v_1 \cdot \exp B = \exp (B +v)$ for some $v\in \g$. On the compact nilmanifold $ \widetilde M=\widetilde G/\widetilde \Gamma $ we consider the nil-translation $ l_{\widetilde u} $. Then $$\label{eq:elle_u}
l_{\widetilde u}(gA^s\widetilde \Gamma)= \widetilde u \cdot (gA^s)\widetilde \Gamma = u A(g)A^{s+1}
\widetilde \Gamma = u A(g)A^{s} \widetilde \Gamma,$$ where in the last line we used the observation that $ A^{-1} \in
\widetilde \Gamma $. It follows that the nil-translation $ l_{\widetilde u} $ preserves the fibres $ p^{-1}\{t\} $ of the fibration . By the formula , the homeomorphism $ i\colon G/\Gamma\to p^{-1}\{0\}\subset \widetilde G/\widetilde \Gamma $ intertwines the affine diffeomorphism $ uA :G/\Gamma \to G/\Gamma $ with the homeomorphism $ (l_{\widetilde u})_{|p^{-1}\{0\}}\colon p^{-1}\{0\}\to p^{-1}\{0\} $. We shall therefore identify these two maps.
Let $\{\widetilde u^t\}_{t\in {{\mathbb{R}}}} < \widetilde G $ be the one-parameter group defined by $ \widetilde u^t= \exp t (B+v)$. Again, the Baker-Campbell-Hausdorff formula $\exp t (B+v)\exp (-t B)=u_t\in G$ for every $t\in{{\mathbb{R}}}$. Therefore, $\widetilde u^t= u_t A^t$ for every $t\in{{\mathbb{R}}}$.
Denote by $ ({\widetilde \phi}_t)_{t\in {{\mathbb{R}}}} $ the nilflow on $ \widetilde
G/\widetilde \Gamma $ defined by the one-parameter group $\{\widetilde u^t\}_{t\in {{\mathbb{R}}}}$. By definition, $ {\widetilde \phi}_1=l_{\widetilde u} $ on $ \widetilde G/\widetilde \Gamma $ and $ p\circ {\widetilde \phi}_t=p+t \mod {{\mathbb{Z}}}$. From the invariance of the set $ p^{-1}\{0\}$ under the map $l_{\widetilde u} $ and since the first return time of any point in $p^{-1}\{0\} $ for the flow $ ({\widetilde \phi}_t)_ {t\in {{\mathbb{R}}}} $ is equal to $1$, we conclude that $p^{-1}\{0\} $ is a Poincaré section of the flow $ ({\widetilde \phi}_t)_
{t\in {{\mathbb{R}}}} $ (with constant return time 1). Moreover, $p^{-1}\{t\}= {\widetilde \phi}_t p^{-1}\{0\} $ for every $t\in{{\mathbb{R}}}/{{\mathbb{Z}}}$. As the map $ l_{\widetilde u} $ restricted to $ p^{-1}\{0\} $ is identified with $ \phi:M\to M $, we conclude, by standard arguments, that any $\phi$ invariant measure $\mu$ on $G/\Gamma$ defines a unique invariant measure $\widetilde \mu$ for the flow $(\widetilde \phi^t)$ on $\widetilde G/\widetilde \Gamma$ given by $$\widetilde \mu(f) = \int_{0}^{1} dt\int_{G/\Gamma} d\mu (x)\, f (\widetilde\phi^t
(i (x)))= \int_{0}^{1} dt\int_{p^{-1}\{t\}} d\widetilde \mu_t (y)\, f (y) .$$ where $\widetilde \mu_t$ is a probability measure on $p^{-1}\{t\}$ given by $\widetilde \mu_t=\widetilde\phi^t_*i_*\mu$. The above formula shows that the family of measures $(\widetilde \mu_t)_{t\in[0,1)}$ form the conditional measures of the measure $\widetilde \mu$ with respect to the projection $p$.
The measure preserving nilflow $(\widetilde G/\widetilde \Gamma, (\widetilde \phi^t), \widetilde \mu) $ is called the *suspension* of the measure preserving affine unipotent diffeomorphism $(G/\Gamma, \phi, \mu)$.
A simple application of these ideas is the following lemma.
\[lem:meas\] Let $\phi$ be a unipotent affine diffeomorphims of the nilmanifold $G/\Gamma$ preserving a measure $\mu$. Let $(\widetilde G/\widetilde \Gamma, (\widetilde \phi^t), \widetilde \mu)$ be the measure preserving nilflow suspension of $(G/\Gamma, \phi,\mu)$ and $Y$ the one-parameter group of $\widetilde G$ generating the flow. Let $\Lambda(\widetilde\mu)< \widetilde G $ be the stabilizer of $ \widetilde\mu $. Then $\Lambda(\widetilde\mu)= H \rtimes \{\exp t Y\}$ where $Y=B+v$ and $H$ is a closed subgroup of $G$ satisfying $\Lambda(\mu)_0 < H
<\Lambda(\mu)$, where $\Lambda(\mu)<G$ is the stabilizer of $ \mu $ and $\Lambda (\mu)_0$ denotes the connected component of the identity of $\Lambda (\mu)$.
As usual, we identify the fiber $p^{-1}(\{0\})$ of the fiber bundle $p\colon \widetilde G/\widetilde \Gamma \to {{\mathbb{R}}}/{{\mathbb{Z}}}$ with $G/\Gamma$ and the measure $\mu$ with a measure $\widetilde \mu_0$ supported on $p^{-1}(\{0\})\approx
G/\Gamma$.
The one-parameter group $ \{\exp t Y\}$ generating the flow $\widetilde \phi^t$ is clearly contained in $ \Lambda(\widetilde \mu)$; hence $H:= \Lambda(\widetilde \mu) \cap G$ is a normal closed subgroup of $ \Lambda(\widetilde \mu)$ and $\Lambda(\widetilde \mu) = H \rtimes \{\exp t Y\}$.
For each $h\in H$, the left translation $l_h: \widetilde G/\widetilde \Gamma \to \widetilde G/\widetilde \Gamma$ is a smooth diffeomorphism fibering over the circle ${{\mathbb{R}}}/{{\mathbb{Z}}}$, i.e. $p\circ l_h=p$ and $l_h(p^{-1}(\{t\}))=p^{-1}(\{t\})$ for $t\in{{\mathbb{R}}}/{{\mathbb{Z}}}$. Thus, for all $h\in
H$, from $(l_h)_*\widetilde\mu=\widetilde\mu$ we obtain that $(l_h)_*\widetilde\mu_t=\widetilde\mu_t$ for almost all $t\in {{\mathbb{R}}}/{{\mathbb{Z}}}$. Since the mapping $(h,t) \in H\times
{{\mathbb{R}}}/{{\mathbb{Z}}}\to (l_h)_*\widetilde\mu_t = (l_h)_*\phi^{t}_*\widetilde\mu_0$ is continuous, we have $(l_h)_*\widetilde\mu_t=\widetilde\mu_t$ for all $h\in H$ and *all* $t\in {{\mathbb{R}}}/{{\mathbb{Z}}}$. In particular, $$(l_h)_* \mu =(l_h)_* \widetilde \mu_0 = \widetilde \mu_0 =\mu .$$ This shows that $H$ is contained in the stabilizer $ \Lambda (\mu)$ of $\mu$.
The measure $\widetilde \mu_0$ on $\widetilde G/\widetilde \Gamma$ is preserved by the time one map $\widetilde \phi^1$, since this map restricted to $ p^{-1}\{0\} $ coincides with the affine map $\phi$. Hence, for all $h\in \Lambda
(\mu)$ and all $n\in {{\mathbb{Z}}}$, $( \widetilde \phi^n \circ l_h \circ \widetilde
\phi^{-n} )_*\widetilde \mu_0=\widetilde \mu_0$, that is $\exp (n Y) \Lambda (\mu) \exp (-n
Y) =\Lambda (\mu)$. Since the adjoint action of $\widetilde G$ on $\g$ is algebraic, we obtain the identity $$\exp (t Y) \Lambda (\mu)_0 \exp (-t
Y) =\Lambda (\mu)_0,$$ for all $t\in {{\mathbb{R}}}$. It follows that if $h\in \Lambda (\mu)_0$ then $\widetilde \mu_0$ is also $\widetilde \phi^{-t}\circ l_h\circ\widetilde \phi^{t}$-invariant for every $t\in{{\mathbb{R}}}$. Therefore, $(l_h)_*\widetilde\mu_t=\widetilde\mu_t$ for every $t\in{{\mathbb{R}}}$ and $(l_h)_*\widetilde\mu=\widetilde\mu$. Consequently, $\Lambda(\mu)_0 < H$.
### A bit of categorical thinking
Nilmanifolds are the objets of a category $\NilMan$ which we could formalise in the following way: the objects of this category are pairs $(G,\Gamma)$ with $G$ a connected, simply connected nilpotent Lie group and $\Gamma$ a lattice in $G$; a morphism $f$ from $(G,\Gamma)$ to $(G',\Gamma')$ is a smooth homomorphism $f\colon G\to G'$ such that $f(\Gamma) \subset\Gamma'$. Thus a morphism $f$ determines a smooth map $\bar f\colon G/\Gamma\to G'/\Gamma'$.
This category can enriched by adding new structures to an object $(G,\Gamma)$; for example, we may add an element $X\in \g$, (or, equivalently, the one-parameter group $\{\exp t X\}$, or the flow determined by $\{\exp t X\}$ on $G/\Gamma$). Such category will be called the *category of nilflows*.
Morphisms for these enriched categories are morphisms $\NilMan$ respecting the additional structures. As another example, we may consider the *category of measure preserving nilflows*, whose objects are quadruples consisting of a nilmanifold $(G,\Gamma)$, an element $X\in \g$ and a probability measure $\mu$ on $G/\Gamma$ invariant by the flow determined by $\{\exp t X\}$ on $G/\Gamma$. A morphism $f$ from the measure preserving nilflow $(G,\Gamma, X, \mu)$ to $(G',\Gamma', X', \mu')$ is a morphism of nilmanifolds $f\colon (G,\Gamma) \to (G',\Gamma')$ such that $f_* X=X'$ and $\bar f_* \mu=\mu'$, where, as before, $\bar f$ denotes the quotient map $\bar f\colon G/\Gamma\to G'/\Gamma'$.
The reader will be easily define in an analogous way the category $\Uaff$ of *measure preserving unipotent affine diffeomorphisms of nilmanifolds*.
The interest of these categories lies in the fact that we have already seen some functorial constructions.
The *abelianization functor $\Ab$* from the category of measure preserving nilflows to itself associates to each measure preserving nilflow $F=(G, \Gamma, X, \mu)$ the toral flow $\Ab(F) = (G/G^{(2)}, \Gamma/\Gamma^{(2)} , \bar X, \bar \mu)$ where $ \bar X = X + \g^{(2)}$ and $\bar \mu =\pi^{(2)}_* \mu$ is image of $\mu$ by the projection $\pi^{(2)}\colon G/\Gamma \to G/G^{(2)}\Gamma$ (see ). We define, for any homomorphism $f\colon G\to G_1$, the abelianized homomorphism $\Ab(f)$ as the induced homomorphism $G/G^{(2)}\to G_1/G_1^{(2)}$. It is routine to check that $\Ab$ is a well defined functor from the category of measure preserving nilflows to itself. The Auslander- Green-Hahn criterion states that the object $\Ab( F)$ is ergodic if and only if $F$ is ergodic.
The suspension construction we discussed above is in fact an isomorphism of categories. Let us define the category $\Nilover$ of *measure preserving nilflows fibering over the circle flow*. The objects of this category are quadruples $(\widetilde G,\widetilde \Gamma, p, Y, \widetilde \mu)$ where $p$ is a morphism from nilmanifolds $(\widetilde G,\widetilde \Gamma)$ to the nilmanifold $({{\mathbb{R}}},
{{\mathbb{Z}}})$, $Y$ is an element in $\widetilde \g$ is such that $p_* Y=d/dt$, and $\widetilde \mu$ is a $Y$-invariant measure on $\widetilde G/\widetilde \Gamma$. (For short, we write such an object as $(p, Y,\widetilde \mu)$ since $(\widetilde G,\widetilde \Gamma)$ is implied by $p$).Morphisms for this category are defined in the obvious way (this is in fact a slice category).
The suspension construction associates to each measure preserving unipotent affine diffeomorphisms of nilmanifolds $(G, \Gamma, uA , \mu)$ a nilmanifold $(\widetilde G, \widetilde \Gamma)$ and a morphism $\check p\colon (\widetilde G, \widetilde \Gamma) \to ({{\mathbb{R}}}, {{\mathbb{Z}}})$ inducing a fiber bundle $p \colon\widetilde G/\widetilde \Gamma \to {{\mathbb{R}}}/{{\mathbb{Z}}}$; it further defines a vector $Y=B+v = \log (uA)\in\widetilde \g $, satisfying $\pi_*Y = d/dt$ and a $Y$-invariant measure $\widetilde \mu $ on $\widetilde G/\widetilde \Gamma$. Thus to each object $(G, \Gamma, uA , \mu)$ in the category $\Uaff$ of measure preserving unipotent affine diffeomorphisms of nilmanifolds we have associated the object $(\pi,Y,\widetilde \mu) = \Susp (G, \Gamma, uA , \mu)$ in the category $\Nilover$ of measure preserving nilflows fibering over the circle flow.
Suppose $f \colon (G, \Gamma, uA , \mu) \to (G', \Gamma', u'A' ,
\mu')$ is a morphism of $\Uaff$. By definition $f$ is a homomorphism $f \colon G \to G'$ such that $f(\Gamma)\subset
\Gamma'$ and such that the induced map $\bar f \colon G/ \Gamma \to
G'/ \Gamma'$ is a morphism of the measure preserving dynamical systems $(G/\Gamma, uA , \mu) $ and $(G'/\Gamma', u'A' ,
\mu')$. Let $(\widetilde
G,\widetilde \Gamma, \check p, Y, \widetilde \mu)$ and $(\widetilde G',\widetilde \Gamma', \check p', Y', \widetilde
\mu')$ be the suspensions of the systems $(G, \Gamma, uA , \mu)$ and $(G', \Gamma', u'A' ,
\mu')$. By definition $\widetilde G = G \rtimes \{A^t\}$, $Y=\log
uA$ etc. Define $\widetilde f = \Susp (f)$ by setting
$$\widetilde f
\colon \widetilde
G \to \widetilde G',\quad \widetilde f (g A^t)= f(g) (A')^t \qquad
\text{for all }g\in G.$$ Since $A^t g A^{-t} = A^t(g)$ and $f\circ A=A'\circ f$, we have $$\begin{split}\widetilde f (A^t g A^{-t}) &= \widetilde f ( A^t(g) ) = f ( A^t(g) ) = (A')^t f(g)\\& = (A')^t f(g)
(A')^{-t} = \widetilde f (A^t) \widetilde f(g) \widetilde f (A^{-t});
\end{split}$$ thus $\widetilde f$ is a homomorphism. We leave to the reader the care of checking that $\widetilde f$ is a morphism of $\Nilover$, that is that $\widetilde f(\widetilde \Gamma) \subset
\widetilde \Gamma'$; that $p'\circ \widetilde f = p$; that $\widetilde f_* Y = Y'$; and that $\overline {\widetilde f}_* \widetilde \mu = \widetilde \mu '$.
We have showed that $\Susp$ is a functor. The functor $\Susp$ is in fact an isomorphism of categories. In fact given $(\check p ,Y,\widetilde \mu)$, where $\check p\colon (\widetilde G, \widetilde \Gamma) \to ({{\mathbb{R}}}, {{\mathbb{Z}}})$ induces a fiber bundle $p\colon \widetilde G/\widetilde \Gamma \to {{\mathbb{R}}}/{{\mathbb{Z}}}$, we recover $G$ as $\ker \check p$ and $\Gamma$ as $\widetilde \Gamma \cap G$; the affine map $uA$ is obtained as the first return map of the flow generated by $Y$ to the fiber $p^{-1}(\{0\})=G/\Gamma $; finally the measure measure $\mu$ is obtained as the conditional measure of $\mu$ on the fiber $p^{-1}(\{0\})= G/\Gamma$. If $F\colon (\check p ,Y,\widetilde \mu) \to
(\check p' ,Y',\widetilde \mu')$ is a morphism of $\Nilover$ from $ \check
p' \circ F = p$ we obtain that $ F(\ker p) \subset \ker p'$; thus $F$ restricted to $G:=\ker p$ is a homomorphism of $G$ into $G':=\ker
p'$. We let to the reader the verification that $F_{|G}$ is a morphism of $\Uaff$ from $ \Susp^{-1}(\check p ,Y,\widetilde \mu)$ to $\Susp ^{-1}(\check p' ,Y',\widetilde \mu')$,
Just as for the abelianization functor, the functor $\Susp$ has the property that the object $(G, \Gamma, uA , \mu)$ is ergodic if and only if the image $\Susp (G, \Gamma, uA , \mu)$ is ergodic.
For nil-translations the proof of the AOP property was based on the functorial properties of the abelianization. For affine unipotent maps it is based on the interplay of the two functors $\Ab$ and $\Susp$; more precisely on study of the functor $\Susp^{-1}\circ \Ab \circ \Susp$.
Invariant measures for unipotent affine diffeomorphisms
-------------------------------------------------------
The construction above gave a correspondence, which preserves ergodicity, between affine unipotent diffeomorphisms of nilmanifolds and nilflows. Criteria of ergodicity for these dynamical systems were given by Parry [@MR0260975] and by Hahn [@MR0155956; @MR0164001], for affine diffeomorphims and, as previouly mentioned, by Auslander, Green and Hahn for the nilflows. We shall exploit this correspondence and generalize it to non-ergodic measures. To simplify matters and to clarify a main point in the proof let us start examining the simplest case.
Assume $G={{\mathbb{R}}}^d$ and let $\phi=uA$ be an affine unipotent diffeomorphism of the a torus $ G/\Gamma $. The suspension of $ (G/\Gamma, \phi) $ yields a nilmanifold $ \widetilde G/\widetilde \Gamma $ and a flow $\widetilde \phi^t$. If $ A=\exp B $, the Lie algebra of $\widetilde G$ is $\widetilde \g = {{\mathbb{R}}}^n \oplus {{\mathbb{R}}}B$ with the only commutation relation $[B,X]=B(X)$ for any $X\in {{\mathbb{R}}}^n$ (here we identified $\g$ with ${{\mathbb{R}}}^n$). In particular the class of nilpotency of $\widetilde G$ is equal to the class of nilpotency of the endomorphism $B$.
The first derived subalgebra $[\widetilde \g, \widetilde \g] $ is therefore the subspace of $\g$ image of $B$, denoted $\mathfrak v$, and the group $[\widetilde G, \widetilde G] $ is the subgroup $V$ of $G$ generated by $\mathfrak v$. The subspace $\mathfrak v$ is rational because the automorphism of the torus $A$ maps the lattice $\Gamma$ to itself (hence it is an element of $\SL_{n}({{\mathbb{Z}}})$ is a suitable integral basis for $\Gamma$) and because the $\log$ of a unipotent automorphism is a rational map. Hence the orbits of the subgroup $V$ are closed and the space of orbits $G/V\Gamma$ is a lower dimensional quotient torus of the torus $G/\Gamma$.
The affine map $\phi=uA$ passes to the quotient torus $G/V\Gamma$, (since $\mathfrak v = B({{\mathbb{R}}}^n)$ and $A(\mathfrak v) = (\exp B) (\mathfrak v) \subset \mathfrak v$), yielding a quotient affine map $\bar \phi$ of the torus $G/V\Gamma$ onto itself. However, since $B$ is nilpotent, the automorphism induced by $A$ on the quotient torus $G/V\Gamma$ is the identity automorphism. It follows that the map $\bar \phi$ is a pure translation by the element $\bar u$, projection of $u$ in $G/V$.
Coming back to the suspended nilflow $(\widetilde G/\widetilde \Gamma, \widetilde \phi^t)$, we know by the Auslander-Green-Hahn criterion that this flow is ergodic if and only if the abelianized flow on the torus $\widetilde G/[\widetilde G, \widetilde G]\widetilde \Gamma$ is. Since $\widetilde G/[\widetilde G, \widetilde G]\widetilde \Gamma=\widetilde G/V\widetilde \Gamma$, the abelianized flow is the suspension of the translation $l_{\bar u}$ on the torus $G/V\Gamma$. (The latter assertion can be easily verified going through the steps of the construction of the suspension, but it is, in fact, a consequence of the functoriality of the construction.)
The reader will have no difficulty in generalizing the above discussion to the case where $G/\Gamma$ is a general connected nilmanifold. The conclusion is the following lemma.
\[lem:susp\] Let $\phi$ be a unipotent affine diffeomorphism of the nilmanifold $G/\Gamma$ preserving a probability measure $\mu$. Let $(\widetilde G/\widetilde \Gamma, (\widetilde \phi^t), \widetilde \mu)$ be the measure preserving nilflow suspension of $(G/\Gamma, \phi,\mu)$.
Let $F$ be a rational subgroup of $G$ normal in $ \widetilde G$ (hence normal in $G$). Then
1. The affine map $\phi$ project to an affine map $\phi_F$ on $G/F\Gamma$, preserving the measure $\mu_F$ image of $\mu$ by the quotient map $G/\Gamma \to G/F\Gamma$.
2. The suspension of the unipotent affine diffeomorphim $(G/F\Gamma, \phi_F,\mu_F)$ is the flow $(\widetilde G/F\widetilde \Gamma, (\widetilde{\phi_F}^t),\widetilde \mu_F)$, where $(\widetilde{\phi_F}^t)$, $\widetilde \mu_F$ are the images of $(\widetilde \phi^t)$, $\widetilde \mu$ by the quotient map $\widetilde
G/\widetilde \Gamma \to \widetilde
G/F\widetilde \Gamma$.
We remark that the above lemma applies to the case $F=[\widetilde G, \widetilde G^{(i)}]$. In fact we have:
\[lem:rationcomm\] The commutator $[\widetilde G, \widetilde G]$ is a rational subgroup of $G$ normal in $\widetilde G$. In fact $[\widetilde \g, \widetilde \g]= [\g, \g]+ B\g$ is a rational sub-algebra of $G$. Furthemore the descending central series $[\widetilde G, \widetilde G^{(i)}]$ and $[\widetilde \g, \widetilde \g^{(i)}]$ of $\widetilde G$ and of $\widetilde \g$ form a descending series of rational normal subgroups of $G$ and of rational normal ideals of $\g$ .
The equality $[\widetilde \g, \widetilde \g]= [\g, \g]+ B\g$ and the inclusion $[\widetilde G, \widetilde G]\triangleleft G$ are immediate consequences of the definition of suspended flow $(\widetilde G/\widetilde \Gamma, \widetilde \phi^t)$ and of the commutation relations . Since $[\widetilde \g, \widetilde \g]$ is a rational sub-algebra of $\widetilde \g$ the orbits of $[\widetilde G, \widetilde G]$ are closed in $\widetilde G/\widetilde \Gamma$. Thus the intersections of the $[\widetilde G, \widetilde G]$ orbits with $G/\Gamma\subset \widetilde G/\widetilde \Gamma$ are also closed. It follows that $[\widetilde G, \widetilde G]$ is a rational subgroup of $G$.
Alternatively we may argue as in paragraph \[sec:some-rati-issu\] that both $[\widetilde \g, \widetilde g]$ and $B(\g)$ are rational sub-algebras of $\widetilde \g$ and use the fact that the sum of rational sub-algebras is rational.
*Mutatis mutandis*, the same arguments apply to $[\widetilde \g, \widetilde
\g^{(i)}]$ and to $[\widetilde G, \widetilde G^{(i)}]$.
\[cor:susp\] Let $\Sigma= ( G, \Gamma, \phi, \mu)$ be a measure preserving unipotent affine diffeomorphism with $\phi =uA$. Let $(\widetilde G/\widetilde \Gamma, (\widetilde \phi^t), \widetilde \mu)$ be the measure preserving nilflow corresponding to the suspension $\widetilde \Sigma= \Susp (\Sigma)$.
1. The affine map $\phi$ projects to a translation $\phi_T$ on the torus $G/[\widetilde G, \widetilde G]\Gamma$ by the element $\bar u = u[\widetilde G, \widetilde G]$. The translation $\phi_T$ preserves the measure $\mu_T$ image of $\mu$ by the quotient map $G/\Gamma \to G/[\widetilde G, \widetilde G]\Gamma$.
2. The suspension of the toral translation $\Sigma_T:=( G/[\widetilde G,
\widetilde G]\Gamma, \phi_T,\mu_T)$ is the linear flow $(\widetilde G/[\widetilde G,
\widetilde G]\widetilde \Gamma, (\widetilde{\phi_T}^t),\widetilde \mu_T)$, where $ (\widetilde{\phi_T}^t)$, $\widetilde \mu_T$ are the images of $(\widetilde \phi^t)$, $\widetilde \mu$ by the quotient map $\widetilde
G/\widetilde \Gamma \to \widetilde
G/[\widetilde G, \widetilde G]\widetilde \Gamma$.
The nilmanifold $\widetilde G/[\widetilde G, \widetilde G]\widetilde \Gamma$ is clearly a torus. The nilmanifold $G/[\widetilde G, \widetilde G]\Gamma$ is a torus because it is a quotient of the torus $G/[\widetilde G,\widetilde G]\Gamma$. Thus the nilflow $(\widetilde{\phi_T}^t)$ is a linear flow. The affine unipotent map $\phi_T$ is indeed a translation on $G/[G,G]\Gamma$: in fact, since $B(\g) \subset [\widetilde\g,\widetilde\g]$ the derivation $B$ is trivial on $\g/ [\widetilde\g,\widetilde\g]$, which implies that the automorphims $A$ projects to the identity automorphism of $G/[\widetilde G, \widetilde
G]$.
The other statements of the corollary are proved in Lemmata \[lem:susp\] and \[lem:rationcomm\].
Observe that the nilflow $\widetilde \Sigma_T=(\widetilde G/[\widetilde G, \widetilde G]\widetilde \Gamma, (\widetilde{ \phi_T}^t),\widetilde \mu_T)$ is just the abelianization $\Ab(\widetilde \Sigma)$ of $\widetilde\Sigma=\Susp(\Sigma)$ and that $$\Sigma_T = \Susp^{-1}(\widetilde \Sigma_T)= \Susp^{-1}(\Ab(
\Susp(\Sigma))).$$
Thus the main content of the above corollary is the toral rotation $\Sigma_T$ is obtained from $\Sigma$ via the quotient morphism $G/\Gamma \to G/[\widetilde G, \widetilde G]\Gamma$.
Joinings of unipotent affine diffeomorphisms of nilmanifolds
------------------------------------------------------------
Let $ \phi=uA $, with $ u\in G $ and $ A\in \Aut(G) $, be an ergodic affine unipotent diffeomorphism of the connected nilmanifold $ G/\Gamma $ such that $ A\neq I $. Denote by $ \lambda $ the uniquely $ \phi $-invariant probability measure on $ G/\Gamma $, i.e. the Haar measure on $ G/\Gamma $.
Let $ r,s\in {{\mathbb{N}}}$ be relatively prime positive integers. Let us consider the product diffeomorphism $ \phi^r\times \phi^s $ on $ G^2/\Gamma^2=(G\times G)/(\Gamma\times \Gamma) $. The map $ \phi^r\times \phi^s$ is an affine unipotent diffeomorphism of $ G/\Gamma\times G/\Gamma$. Indeed, $ \phi^r\times \phi^s=u_{r,s}A_{r,s} $ with $ u_{r,s}=(u_r,u_s)\in G^2 $ and $ A_{r,s}=A^r\times A^s\in \Aut(G^2) $. Note that $ B_{r,s}:=\log A_ {r,s}=(rB,sB) $, with $B:=\log A$. Thus
Let $ \rho $ be an ergodic joining of the measure theoretical systems $ \Sigma_r= (G/\Gamma,\phi^r, \lambda) $ and $ \Sigma_s=(G/\Gamma,\phi^s, \lambda) $. By definition $ \rho $ is a measure on $ G/\Gamma\times G/\Gamma $ with marginals $ \lambda $ on each factor, invariant and ergodic for the transformation $ \phi^r\times \phi^s $. Thus we have an measure preserving unipotent affine diffeomorphism $\Sigma_{r,s} =(G^2, \Gamma^2, \phi^r\times \phi^s, \rho)$ and morphisms $q_i \colon \Sigma_{r,s} \to\Sigma_i$, $(i=r,s)$.
Let $\widetilde \Sigma_{r,s}=( \widetilde G_{r,s} , \widetilde \Gamma_{r,s} , p_{r,s},
Y_{r,s}, \widetilde \rho_{r,s})=\Susp ( \Sigma_{r,s})$. Here $ \widetilde G_{r,s} = G^2 \rtimes \{\exp t B_{r,s}\}$ and $Y_{r,s} = \log ( \phi^r\times \phi^s) = (rY, sY)$, with $Y=\log \phi=\log uA= B + v$, for some $v\in \g$. In the sequel we shall write $\widetilde \Sigma_{r,s}=( \widetilde G_{r,s}/\widetilde \Gamma_{r,s} , Y_{r,s}, \widetilde
\rho_{r,s})$, for short. Let $ \widetilde\g_{r,s} $ denote the Lie algebra of $ \widetilde G_{r,s} $. Then $ \widetilde\g_{r,s} = \g^2\oplus{{\mathbb{R}}}B_{r,s}$.
For $i=r,s$, let $\widetilde \Sigma_{i}=(\widetilde G_{i}, \widetilde \Gamma_{i}, p_i, Y_i , \lambda_i)$ be the suspensions of $\Sigma_i$ and denote by $\g_i$ the Lie algebra of $G_i$. Observe that $\lambda_r$ and $\lambda_s$ are the Haar measures on the corresponding nilmanifolds and that $Y_r=rY$ and $Y_s=sY$.
Note that, since $\widetilde\g_{r,s}$ is the semi-direct product of $\g^2 =\g\oplus \g$ and the span of the derivation $B_{r,s}= (rB,sB)$ of $\g^2$ we have $$\widetilde\g_{r,s}= (\g\oplus \g)\rtimes {{\mathbb{R}}}(rB,sB) = (\g \rtimes {{\mathbb{R}}}r B)
\oplus (\g \rtimes {{\mathbb{R}}}s B)= \widetilde \g_r \oplus\widetilde \g_s = \widetilde \g
\oplus \widetilde \g =\widetilde \g^2,$$ where $\widetilde\g= \g \rtimes {{\mathbb{R}}}B$. Hence $ \widetilde G_{r,s}=\widetilde G_r\times\widetilde G_s =\widetilde G\times\widetilde G$, with $\widetilde G = G \rtimes \{\exp tB\}$. Let $k$ be the class of nilpotency of the group $\widetilde G$. Then, for all $i=2, \dots, k$, we have: $$\begin{gathered}
\widetilde
\g^{(i)}_{r,s}=\widetilde\g_r^{(i)}\oplus\widetilde\g_s^{(i)}=\widetilde\g^{(i)}\oplus\widetilde\g^{(i)}\subset\g\oplus\g\intertext{and} \widetilde G^{(i)}_{r,s}=\widetilde G_r^{(i)}\times\widetilde
G_s^{(i)}=\widetilde G^{(i)}\times\widetilde G^{(i)}< G\times G.\end{gathered}$$ We have[^8]: $$\begin{tikzcd}[column sep=small]
& \Sigma_{r,s}=( (G/\Gamma)^2, \phi^r\times \phi^s , \rho)
\arrow{dl}[description]{q_r} \arrow{dr}[description]{q_s}
\arrow[dashed]{dd}[description]{\Susp}
& \\
\Sigma_{r}=(G/\Gamma, \phi^r ,
\lambda)\arrow[dashed]{dd}[description]{\Susp} &
& \Sigma_{s}=(G/\Gamma, \phi^s , \lambda)\arrow[dashed]{dd}[description]{\Susp}\\
&\widetilde \Sigma_{r,s}=( \widetilde G_{r,s}/\widetilde \Gamma_{r,s} , Y_{r,s} ,
\widetilde \rho_{r,s}) \arrow{dl}[description]{\widetilde q_r=\Susp( q_r)}
\arrow{dr}[description]{\widetilde q_s=\Susp( q_s)}\\
\widetilde \Sigma_{r}=(\widetilde G_{r}/ \widetilde \Gamma_{r}, rY , \lambda_r) & &
\widetilde \Sigma_{s}=(\widetilde G_{s}/\widetilde \Gamma_{s}, sY , \lambda_s)
\end{tikzcd}$$ By Lemma \[lem:meas\] we have:
\[lem:stabil\] Denote by $ \widetilde {\mathfrak h}_{r,s}\subset \widetilde\g_{r,s} $ and by $ {\mathfrak h}_{r,s} $ the Lie algebras of the stabilisers $\Lambda (\widetilde\rho_{r,s})<\widetilde G_{r,s} $ and $\Lambda (\rho_{r,s})< G^2 $ of the measures $\widetilde\rho_{r,s} $ and $\rho$. Then $$\label{eq:NilpotentAOP-06-14:3}
\widetilde {\mathfrak h}_{r,s} = {\mathfrak h}_{r,s} \rtimes {{\mathbb{R}}}Y_{r,s}.$$
By Corollary \[cor:susp\], applying the functor $\Ab'=\Susp^{-1}\circ \Ab\circ \Susp$ to the triangle $\Sigma_{r,s}$, $\Sigma_{r}$, $\Sigma_{s}$ we obtain a diagram $$\begin{tikzcd}[column sep=small]
& \Ab'(\Sigma_{r,s})=( T^2, l_{\bar u}^r\times l_{\bar u}^s ,
\rho') \arrow{dl}[description]{\Ab'(q_r)}
\arrow{dr}[description]{\Ab'(q_s)}
& \\
\Ab'(\Sigma_{r})=( T, l_{\bar u}^r , \lambda_{\mathbb T}) & &
\Ab'(\Sigma_{s})=( T, l_{\bar u}^s, \lambda_{\mathbb T})
\end{tikzcd}$$ where $ T$ is the torus $G/[\widetilde G,\widetilde G]\Gamma$, $\lambda_{ T}$ is the Lebesgue/Haar measure on $G/[\widetilde G,\widetilde G]\Gamma$, $\Ab'(q_i)$ are the projections of $ T^2= T\times T$ on the corresponding factors, $l_{\bar u}$ is the left translation by $\bar u=u [\widetilde G,\widetilde G]$ and $\rho'$ is the projection of the ergodic joining $\rho$ via the map $(G/\Gamma)^2\to (G/[\widetilde G,\widetilde G]\Gamma)^2$. In particular the measure $ \rho'$ is a joining of the ergodic and minimal toral rotations $ \Ab'(\Sigma_{r})$ and $\Ab'(\Sigma_{s})$.
Since the measure $\rho$ is ergodic, the measure $\rho'$, image of $\rho$ by the morphism $\Sigma_{r,s} \to \Ab'(\Sigma_{r,s})$ is an ergodic measure for the for the rotation $ l^r_{\bar u}\times l^s_{\bar u} $ of $\mathbb T^2$. By Lemma \[lem:torus\], the stabilizer $\Lambda(\rho')$ of the measure $\rho'$ is the connected group $$\Lambda(\rho')= \big\{ \exp(rX,sX)\mid X\in \operatorname
{Lie}(T)=\g/[\widetilde \g,\widetilde\g] \big\}.$$
Now we focus on the quadrangle of morphisms given by Corollary \[cor:susp\]. $$\begin{tikzcd}[column sep=small]
\Sigma_{r,s}=( (G/\Gamma)^2, \phi^r\times \phi^s , \rho) \arrow{r}
\arrow[dashed]{d}[description]{\Susp} & \Ab'(\Sigma_{r,s})=( T^2,
l_{\bar u}^r\times l_{\bar u}^s , \rho')
\\
\widetilde \Sigma_{r,s}=( \widetilde G_{r,s}/\widetilde \Gamma_{r,s} , Y_{r,s} , \widetilde
\rho_{r,s}) \arrow{r} & \Ab(\widetilde \Sigma_{r,s})=(\widetilde
G_{r,s}/[\widetilde G_{r,s}, \widetilde G_{r,s}]\widetilde \Gamma_{r,s},
Y'_{r,s}, \widetilde \rho_{r,s}')
\arrow[dashed]{u}[description]{\Susp^{-1}}
\end{tikzcd}$$
\[lem:aop\_nil:12\] The stabilizer $\Lambda( \widetilde \rho_{r,s}')$ of the measure $ \widetilde \rho_{r,s}'$, image of $\widetilde \rho_{r,s}$ by the morphism $\widetilde \Sigma_{r,s}\to \Ab(\widetilde \Sigma_{r,s})$, is the connected group $$\Lambda( \widetilde \rho_{r,s}')= \big\{ \exp(rX,sX)\mid X\in \widetilde
\g/[\widetilde \g,\widetilde\g] \big\}.$$
Since the toral flow $ \Ab( \Sigma_{r,s})$ is a suspension of the toral rotation $\Ab'(\Sigma_{r,s})$, it follows by elementary reasons of by Lemma \[lem:meas\] that we have $$\Lambda( \widetilde \rho_{r,s}')= \Lambda(\rho')\times \{\exp t
Y'_{r,s}\}= \big\{ \exp(rX,sX)\mid X\in \g/[\widetilde \g,\widetilde\g]
\big\}\times \{\exp t Y'_{r,s}\},$$ where $ Y'_{r,s} = Y_{r,s}\mod [\widetilde \g,\widetilde \g]^2 $. Thus, $ Y'_{r,s}= (r Y', s Y')$, with $Y'= Y \mod [\widetilde \g,\widetilde \g]$. Since the span of $\g$ and of $Y$ is the Lie algebra $\widetilde \g$, the statement follows.
By Lesigne-Starkov-Ratner Theorem \[thm:ratner\], since the measure $\widetilde\rho_{r,s}$ is ergodic, the stabilizer $ \Lambda (\widetilde\rho_{r,s})<\widetilde G_{r,s} $ of $\widetilde\rho_{r,s}$ is a connected closed subgroup of $\widetilde G_{r,s} $ such that the topological support of $ \widetilde\rho_{r,s} $ is a $ \Lambda (\widetilde\rho_{r,s} )$-orbit in $ \widetilde G_{r,s}/\widetilde \Gamma_ {r,s} $. Clearly the same properties hold for the stabilizer $\Lambda( \widetilde \rho_{r,s}')$ of the measure $ \widetilde \rho_{r,s}'$. Since the action of $\widetilde G_{r,s}/[\widetilde G_{r,s}, \widetilde G_{r,s}]$ on the torus $\widetilde G_{r,s}/[\widetilde G_{r,s}, \widetilde G_{r,s}]\widetilde \Gamma_{r,s}$ is locally free, we obtain:
\[lem:aop\_nil:120\] The projection of the stabilizer $ \Lambda (\widetilde\rho_{r,s})<\widetilde G_{r,s} $ onto $\widetilde G_{r,s}/[\widetilde G_{r,s}, \widetilde G_{r,s}]$ is the stabilizer $\Lambda( \widetilde \rho_{r,s}')$ of the measure $ \widetilde \rho_{r,s}'$.
\[lem:aop\_nil:121\] The Lie algebra $ \widetilde{\mathfrak h}_{r,s}\subset\widetilde\g\oplus\widetilde\g $ satisfies $$\{ (r^k Z, s^k Z )\mid Z\in \widetilde\g^{(k)}\} \subset \widetilde{\mathfrak
h}_{r,s} \cap (\widetilde\g^{(k)}\oplus\widetilde\g^{(k)} ).$$ It follows that the stabilizer $ \Lambda(\rho)<G\times G $ of the joining $ \rho $ contains the subgroup $$L{:=}\{ \exp(r^k Z, s^k Z)\mid Z\in\widetilde\g^{(k)}\} <\widetilde
G^{(k)}\times\widetilde G^{(k)}=\widetilde G^{(k)}_{r,s}.$$
The first statement follows from the previous two lemmata and from Lemma \[lem:aop\_nil:4\] as in the proof of Lemma \[lem:aop\_nil:1\]. The second is an application of Lemma \[lem:stabil\].
Recall that $ \widetilde G^{(k)}<G $ is a closed normal subgroup in the center of $ G $ such that its Lie algebra $ \widetilde\g^{(k)} $ is a rational ideal in $ \g $ include in the kernel of $ B:\g\to\g $. It follows that $ M^{(k-1)}:=G/\Gamma \widetilde G^{(k)} $ is a nilmanifold and $ {{\mathbb{T}}}^{(k)}:=\widetilde G^{(k)}/(\widetilde G^{(k)}\cap\Gamma) $ is a torus. The group $ {{\mathbb{T}}}^{(k)} $ acts on $ G/\Gamma $ by left translation. Thus, we have an orthogonal decomposition $$L^2(M,\lambda) = \bigoplus_{\chi \in \widehat{{{\mathbb{T}}}^{(k)}}} H_\chi$$ where, for each character $ \chi $ of the torus $ \mathbb T^{(k)} $ we set $$H_\chi= \big\{ f \in L^2(M,\lambda) \mid f(zx)= \chi (z)f(x),\, \forall
z\in \mathbb T^{(k)}, \,\forall x \in M\big\}.$$
\[prop:aop\_nil:2\] Let $ \chi_1 $, $ \chi_2 $ be characters of the torus $ \mathbb T^{(k)} $ so that at least one is non-trivial. Then, if $ \chi_1^{r^k}\not= \chi_2^{s^k} $, for any $ f_i\in H_{\chi_i} $, $ i=1,2 $, and any ergodic joining $ \rho $ of $ (M,\phi^r, \lambda) $ and $ (M,\phi^s, \lambda) $, we have $$\label{eq:zero} \rho(f_1\otimes \bar{f}_2)=0.$$ Moreover, if $ f\in H_\chi $ for nontrivial $ \chi $ then for every $ \omega\in \C $ with $ |\omega|=1 $ there exists $ g\in G $ such that $$f(\phi^n(l_gx))=\omega f(\phi^n x)\text{ for all }x\in M,\ n\in{{\mathbb{Z}}}.$$
Suppose that $ \chi_1^{r^k}\neq \chi_2^{s^k} $ and $ f_1\in H_{\chi_1} $ and $ f_2\in H_{\chi_2} $. Let $ \rho $ be any ergodic joining of $ (M,\phi^r,\lambda) $ and $ (M,\phi^s,\lambda) $. By Lemma \[lem:aop\_nil:121\], every element of the group $ L<\widetilde G^{(k)}\times \widetilde G^{(k)} $ stabilizes the measure $ \rho $. It follows that the rotation by $ (z^{r^k},z^{s^k})\in {{\mathbb{T}}}^{(k)}\times {{\mathbb{T}}}^{(k)} $ on $ M\times M $ preserves $ \rho $ for every $ z\in {{\mathbb{T}}}^{(k)} $. Therefore $$\rho(f_1\otimes
\bar{f}_2)=(\chi_1^{r^k}\chi_2^{-r^k})(z)\rho(f_1\otimes
\bar{f}_2),$$ which gives .
Since $ \widetilde\g^{(k)}\subset \ker B $, for every $ h\in \widetilde G^{(k)} $ we have $ A(h)=h $, so $$\phi(l_{h}g\Gamma)=uA(hg)\Gamma=l_{h}\phi(g\Gamma)\text{ for every
}g\Gamma\in M.$$ It follows that, if $ f\in H_\chi $ for nontrivial $ \chi $ then for every $ \omega\in \C $ with $ |\omega|=1 $ there exists $ g\in \widetilde G^{(k)} $ such that $$f(\phi^n(l_gx))=f(l_g\phi^n(x))=\chi(g) f(\phi^n x)=\omega
f(\phi^n x)\text{ for all }x\in M,\ n\in{{\mathbb{Z}}},$$ and the proof is complete.
Finally, proceeding by induction on $ k $ the clas of nilpotency of the group $\widetilde G$, we obtain the following result.
\[thm:mainaff\] Let $ \phi $ be an ergodic affine unipotent diffeomorphism of a compact connected nilmanifold $ M=G/\Gamma $. Then there exists a set $ \mathscr{C}\subset C(M) \cap L^2_0(M,\lambda) $ whose span is dense in $ L^2_0(M,\lambda) $ such that:
- for any pair $ f_1,f_2\in\mathscr{C} $, for all but at most one pair of relatively prime natural numbers $ (r,s) $ we have $ \rho(f_1\otimes \bar{f}_2)=0 $ for all ergodic joinings $ \rho $ of $ \phi^r $ and $ \phi^s $;
- for every $ f\in\mathscr{C} $ and any $ \omega \in \C $ with $ |\omega|=1 $ there exists $ g\in G $ such that we have $ f(\phi^n (l_gx))=\omega f(\phi^n x) $ for every $ x\in M $ and $ n\in{{\mathbb{Z}}}$.
First note that if $ \chi$ is the trivial character, then every element of $ H_\chi $ is invariant under the action of $ \widetilde G^{ (k)} $, hence it can be treated as an element of $ L^2(M^{(k-1)},\lambda) $. Furthermore, we can define the affine unipotent diffeomorphism $ \phi^ {(k-1)}:M^{(k-1)}\to M^{(k-1)} $ by $$\phi^{(k-1)}(g\Gamma\widetilde G^{(k)})=uA(g)\Gamma\widetilde G^{(k)}.$$ As $ A(\Gamma)=\Gamma $ and $ A(\widetilde G^{(k)})=\widetilde G^{(k)} $, this map is well defined and is a factor of $ \phi $ via the natural $ G $-equivariant projection $ \pi^{(k)}:G/\Gamma\to G/\Gamma \widetilde G^{(k)} $. It allows us to proceed by induction.
The inductive step is given by Proposition \[prop:aop\_nil:2\]. Note that Proposition \[prop:aop\_nil:2\] requires the non-triviality of the automorphism $ A $. Therefore induction must start from an affine diffeomorphism with trivial automorphism, so from a nil-translation. Thus the basis case of induction is derived from Theorem \[thm:mainrot\]. Finally, the construction of the set $ \mathscr {C} $ runs as in the proof of Theorem \[thm:mainrot\].
Theorem \[ThA\] as well as the Corollaries \[CorB\] and \[CorC\] are direct consequences of Theorem \[thm:mainaff\] and Theorem \[thm:prop\].
Nil-translations on non-connected nilmanifolds
----------------------------------------------
Suppose that $ G $ is a nilpotent Lie group and $ \Gamma< G $ its lattice. Then $ M=G/\Gamma $ is a compact nilmanifold which we assume to be non-connected. Let $ l_u $ for $ u\in G $ be a uniquely ergodic translation on $ M $. Denote by $ G_0<G $ the identity component of $ G $. The group $ G_0 $ is closed (and open) normal, and we may assume that it is simply connected. Moreover, $ M_0=(G_0\Gamma)/\Gamma\approx G_0/
(G_0\cap\Gamma) $ is a connected component of $ M $. Since $ l_u $ is minimal and $ M $ is compact, there is a minimal integer $ m>1 $ such that $ M $ is the disjoint union of clopen sets $ l^k_uM_0 $ for $ 0\leq k<m $ and $ l^m_uM_0=M_0
$. Moreover, $ l^m_u:M_0\to M_0 $ is a uniquely ergodic diffeomorphism which is a unipotent affine diffeomorphism of the compact connected nilmanifold $ M_0= G_0/(G_0\cap\Gamma) $. Indeed, as $ l^m_u
((G_0\Gamma)/\Gamma)=(G_0\Gamma)/\Gamma $, we have $ u^m\in G_0\Gamma $, and hence there are $ u_0\in G_0 $ and $ \gamma\in \Gamma $ such that $
u^m=u_0\gamma $. It follows that $ l_u^m=l_{u_0}\circ \Ad_{\gamma} $ on $ M_0 $, where $ l_{u_0} $ is a nil-translation on the nilmanifold $ G_0/(G_0\cap\Gamma) $ and $ \Ad_{\gamma}:G_0\to G_0 $ is a unipotent automorphism of $ G_0 $ with $
\Ad_{\gamma}(G_0\cap\Gamma)=
G_0\cap\Gamma
$. Therefore $ l^m_u:M_0\to M_0 $ is an ergodic unipotent affine diffeomorphism which will be denoted by $ \phi:M_0\to M_0 $
Let us consider the factor map $ p:M\to{{\mathbb{Z}}}/ m{{\mathbb{Z}}}$ defined by $$p(l_u^kx)=k\mod
m{{\mathbb{Z}}}, \quad \text{ if }~ k\in{{\mathbb{Z}}}\text{ and~} x\in M_0 .$$ Then the map $ p $ intertwines the nil-translation $ l_u $ on $ M $ with the rotation on $ {{\mathbb{Z}}}/ m{{\mathbb{Z}}}$ by $ 1 $. Denote by $ H^+ $ the space of $ L^2 (M,\lambda) $ of functions constant on fibers of $ p $ and by $ H^- $ its orthocomplement in $ L^2(M,\lambda) $. Then $ H^- $ coincides with the space of $ L^2 $-functions which have zero mean on each fiber $ M_k:=l^k_uM_0
$ for $ 0\leq k<m $.
\[prop:nonconnect\] Let $ l_u $ be an ergodic nil-translation of a compact nilmanifold $ M=G/\Gamma $ with $m$ connected components. Then there exists a set $
\mathscr{C}\subset C(M)\cap H^- $ whose span is dense in $ H^- $ such that:
- for any $ f_1,f_2\in\mathscr{C} $, for all but a finite number of pairs of distinct prime numbers $(r, s)$, we have $
\rho(f_1\otimes \bar{f}_2)=0 $ for all ergodic joinings $ \rho $ of $ l_u^r $ and $ l_u^s $;
- for every $ f\in\mathscr{C} $ and any $ \omega\in \C $ with $ |\omega|=1
$ there exists a homeomorphism $ S:M\to M $ such that $ f(l_u^n
(Sx))=\omega f(l_u^n x) $ for every $ x\in M $ and $ n\in{{\mathbb{Z}}}$.
Let us consider the ergodic unipotent affine diffeomorphism $ \phi:=l_ {u_0}\Ad_\gamma $ on $ M_0 $. By Theorem \[thm:mainaff\] applied to $ \phi $, there exists a set $ \mathscr{C}_0\subset C(M_0)\cap L^2_0(M_0) $ whose span is dense in $ L^2_0(M_0,\lambda) $ and which satisfies the conditions (i) and (ii) of that theorem. We consider $ \mathscr{C}_0$ as a set of continuous functions on $M$ by extending its elements as $0$ on $M\setminus M_0$. For every $ k\in {{\mathbb{Z}}}/ m{{\mathbb{Z}}}$ the map $ l_u^k: M\to M $ establishes a homeomorphism between $ M_0 $ and $ M_k $. Set $ \mathscr{C}_k= \{f\circ l_u^{-k}\mid f\in \mathscr{C}_0\}$. Elements of $ \mathscr{C}_k$ are continuous functions on $M$ supported on $ M_k $. Then $ \mathscr{C}:=\bigcup_ {k\in {{\mathbb{Z}}}/ m{{\mathbb{Z}}}}\mathscr{C}_k $ is a linearly dense set in $ H^- $.
Fix $ f\in \mathscr{C}_k $ for some $ 0\leq k<m $ and $ \omega\in\C
$ with $ |\omega|=1 $. Then $ f $ is supported on $ M_k $ and there exists $ h\in \mathscr{C}_0 $ such that the restriction of $
f $ to $ M_k $ is equal to $ h\circ l_u^{-k} $. By (ii) in Theorem \[thm:mainaff\], there exists $ g\in G_0 $ such that $ h(\phi^n (l_gx))=\omega
h(\phi^n x) $ for every $ x\in M_0 $ and $ n\in{{\mathbb{Z}}}$. Let $ S:M\to M
$ be the homeomorphism defined by $ Sx=l_u^{k'}\circ l_g\circ l_u^{-k'}x
$ if $ x\in M_{k'} $. Then $$\label{eq:dicon} f(l_u^n(Sx))=\omega f(l_u^nx)\text{ for all }n\in{{\mathbb{Z}}}\text
{ and }x\in M.$$ Indeed, let $ x\in M_{k'} $: if $ n\neq k-k'\mod m $ then both $
l_u^nx $ and $ l_u^n(Sx) $ do not belong to $ M_k $, and hence both sides of vanish. If $ n=k-k'+n' m $ for some $ n'\in{{\mathbb{Z}}}$ then $ x=l^{k'}_uy $ for some $ y\in M_0 $ and $$f(l_u^n(Sx))=h(l_u^{n+k'-k}l_{g}y)=h(\phi^{n'} l_{g}y)=\omega h(\phi^
{n'} y) =\omega h(l_u^{n+k'-k}y)=\omega f(l_u^nx).$$
Note that there exits only finitely many pairs of distinct prime numbers $(r, s)$ such that $m, r, s$ are not pairwise coprime. Therefore to prove we can assume that $(r,s)$ is a pair such that $m, r, s$ are pairwise coprime. Then both the nil-translations $
l_u^r $ and $ l_u^s $ are also uniquely ergodic.
Let $ \rho $ be an ergodic joining of $ l_u^r $ and $ l_u^s $. Choose any pair $ f_1,f_2 $ of functions in $ \mathscr{C} $. Suppose that $ f_1\in \mathscr{C}_{k_1} $ and $ f_2\in \mathscr {C}_{k_2}
$ for some $ 0\leq k_1,k_2<m $. Then, for $ i=1,2 $, $ f_i $ is supported on $ M_{k_i} $ and $ f_i= h_i\circ l_u^{-k_i} $ for some $ h_i\in
\mathscr{C}_0 $. Denote by $ \rho_{k_1,k_2} $ the restriction of $
\rho $ to the set $ M_k\times M_{k'} $. Then the measure $ (l_u^
{-k_1}\times l_u^{-k_2})_*\rho_{k_1,k_2} $ is supported on $ M_0\times
M_0 $ and coincides with the restriction of $ (l_u^ {-k_1}\times l_u^{-k_2})_*\rho
$ to $ M_0\times M_0 $. Then we have $$\rho(f_1\otimes \bar{f}_2)=(l_u^{-k_1}\times l_u^{-k_2})_*\rho_{k_1,k_2}
(h_1\otimes \bar{h}_2).$$ If the measure $ \rho_{k_1,k_2} $ vanishes then $
\rho(f_1\otimes \bar{f}_2)=0 $. Thus we may assume that $ \rho_{k_1,k_2} $ is not zero. As $ l_u^r $ and $ l_u^s $ are uniquely ergodic and $
l_u^{-k_1}\times l_u^{-k_2} $ commutes with $ l_u^{r}\times l_u^{s} $, the measure $ (l_u^{-k_1}\times l_u^{-k_2})_*\rho $ is also an ergodic joining of $ l_u^r $ and $ l_u^s $.
Since $ r,s,m $ are pairwise coprime and $ (l_u^{-k_1}\times l_u^ {-k_2})_*\rho
$ does not vanish on $ M_0\times M_0 $, the measure $ (l_u^{-k_1}\times
l_u^{-k_2})_*\rho $ is supported on $ \bigcup_{k=0}^{m-1}(l_u^r\times
l_u^s)^k(M_0\times M_0) $, where the elements of the union are pairwise disjoint and $ (l_u^r\times l_u^s)^m(M_0\times M_0)=M_0\times
M_0 $. It follows that the restriction $ (l_u^{-k_1}\times l_u^{-k_2})_*\rho_
{k_1,k_2} $ is an $ l_u^{rm}\times l_u^{sm} $-invariant ergodic measure on $ M_0\times M_0 $. By (i) in Theorem \[thm:mainaff\], it follows that $$\rho(f_1\otimes \bar{f}_2)=(l_u^{-k_1}\times l_u^{-k_2})_*\rho_{k_1,k_2}
(h_1\otimes \bar{h}_2)=0$$ for all but one pair $ (r,s) $, which completes the proof.
\[thm:nonconnect\] Let $ l_u $ be a nil-translation of the compact nilmanifold $ M=G/\Gamma $. Then $$\label{short1} \frac1M\sum_{M\leq m<2M}\left|\frac1H\sum_{m\leq
n<m+H}f(l_u^nx){\boldsymbol{\mu}}(n)\right|\to 0$$ when $ H\to\infty $, $ H/M\to0 $ for each $ x\in M $ and each $ f\in
C(M) $. If additionally $l_u$ is ergodic then hold for all $f\in D(M)$.
Assume that the nil-translation is ergodic. Denote by $ \mathcal{B}_{rat}\subset\mathcal{B} $ the maximal factor of $ l_u $ with the rational discrete spectrum. Then $ H^+=L^2 (M,\mathcal
{B}_{rat},\lambda) $. In view of (i) in Proposition \[prop:nonconnect\], the nil-translation satisfies the AOP property (i.e. ) for all functions $ f,g\in H^-=L^2
(M,\mathcal{B}_{rat},\lambda)^\perp $. Since the space $ H^- $ does not depend on the choice of the topological model of the nil-translation, similar arguments to those used in the proof of Theorem 3 in [@Ab-Le-Ru] show that for each zero mean $ f\in C(M)\cap H^- $, for any sequence $ (x_k)_{k\geq 1} $ in $ M $ and any $ (b_k)_ {k\geq
1} $ with $ b_{k+1}-b_k\to +\infty $ we have $$\frac1{b_{K+1}}\sum_{k\leq K}\Big(\sum_{b_k\leq n< b_{k+1}}f(l_u^nx_k){\boldsymbol{\mu}}(n)\Big)\to0\text{ when }K\to\infty.$$ Together with (ii) in Proposition \[prop:nonconnect\] this gives for every $ f\in C(M)\cap
H^- $. If $ f\in H^+ $ then the sequence $(f(l_u^nx))$ is periodic and the property follows from Theorem 1.7 in [@Ma-Ra-Ta] applied to the Möbius function (cf. Remark \[rem:mrt\]). If follows that holds for every $ f\in C(M) $, and finally, by Remark \[rem:riemint\], also for every $ f\in D(M) $.
We need now to consider the case where the nil-translation $ l_u $ is not ergodic. Then, for any $ x\in M $ denote by $ M_x $ its orbit closure. By [@Rat91], the restriction of $ l_u $ to $ M_x $ is topologically isomorphic to an ergodic nil-translation on a compact (not necessary connected) nilmanifold. Therefore, our claim is reduced to the ergodic case.
Polynomial type sequences
-------------------------
\[thm:gammaaffine\] Let $ \phi $ be an ergodic affine unipotent diffeomorphism of a compact connected nilmanifold $ M=G/\Gamma $. Then for all $ \gamma\in {{\mathbb{R}}}\setminus\{0\} $ and $\varrho\in{{\mathbb{R}}}$, we have $$\label{short2} \frac1M\sum_{M\leq m<2M}\left|\frac1H\sum_{m\leq
n<m+H}f(\phi^{[\gamma n+\varrho]}x){\boldsymbol{\mu}}(n)\right|\to 0$$ when $ H\to\infty $, $ H/M\to0 $ for each $ x\in M $ and each $ f\in
C(M) $.
Let $ \widetilde \Sigma=(\widetilde G, \widetilde \Gamma, p , ({\widetilde \phi}_t)_{t\in {{\mathbb{R}}}}, \widetilde \lambda) $ be the suspension of the ergodic affine unipotent diffeomorphism $\Sigma=(G,\Gamma, \phi,\lambda)$, with $\lambda$ and $\widetilde
\lambda$ the Haar measures on the corresponding nilmanifolds. The nilflow $\widetilde \Sigma$ is uniquely ergodic. As usual we identify $G/\Gamma$ with $p^{-1}\{0\}$. For each continuous function $f$ on $G/\Gamma$ let $\widetilde f$ be the unique function on $\widetilde
G/\widetilde \Gamma$ defined by the condition $$\widetilde f({\widetilde \phi}_t x ) = f(x), \quad \forall x \in G/\Gamma\approx p^{-1}\{0\},~
\forall t\in [0,1)\,,$$ or, equivalently, by the condition $$\label{eq:fractional}
\widetilde f({\widetilde \phi}_t x ) = f(\phi^{\lfloor t\rfloor}x), \quad \forall
x \in G/\Gamma\approx p^{-1}\{0\} ,~
\forall t\in {{\mathbb{R}}}\,.$$
Since the set of discontinuities of $ \widetilde{f} $ is contained in $p^{-1}\{0\}$, the function $\widetilde f$ belong so the class $D(\widetilde{M}) $ defined in Remark \[rem:riemint\].
The map $ {\widetilde \phi}_\gamma$ is a nil-translation $ l_{\widetilde {u}}
$ on $ \widetilde G/\widetilde \Gamma $ by an element $\widetilde u\in \widetilde G$; thus we shall consider two cases.
**Ergodic case:** Suppose that the nil-translation $ {\widetilde
\phi}_\gamma =l_{\widetilde {u}}$ on $ \widetilde G/\widetilde \Gamma $ is ergodic. Then ${\widetilde
\phi}_{\gamma n+\varrho}={\widetilde
\phi}_{\varrho}\circ l_{\widetilde {u}}^n $ and, by Theorem \[thm:nonconnect\] and formula , we have, for all $x\in G/\Gamma$, $$\begin{aligned}
\frac1M&\sum_{M\leq m<2M}\left|\frac1H\sum_{m\leq n<m+H}f(\phi^{[\gamma
n+\varrho]}x){\boldsymbol{\mu}}(n)\right|\\
&=\frac1M\sum_{M\leq m<2M}\left|\frac1H\sum_{m\leq n<m+H}\widetilde{f}\circ {\widetilde
\phi}_{\varrho}
(l_{\widetilde {u}}^n(x)){\boldsymbol{\mu}}(n)\right|\to 0.
\end{aligned}$$ **Non-ergodic case:** Assume that the nil-translation $ {\widetilde
\phi}_\gamma = l_{\widetilde {u}}$ on $ \widetilde G/\widetilde \Gamma $ is not ergodic. Then, by [@Rat91], for every $ x=g\Gamma\in G/\Gamma $ there exists a closed subgroup $ H<\widetilde{G} $ so that $ \widetilde{u}\in H $, the $ l_{\widetilde u}
$-orbit closure of $ \widetilde{x}:=g\widetilde \Gamma $ coincides with the orbit $ H\widetilde{x} $ and the restriction of $
l_{\widetilde{u}} $ to the sub-nilmanifold $ \widetilde W:=H\widetilde{x}\approx H/
(H\cap g\widetilde \Gamma g^{-1}) $ is uniquely ergodic. The sub-nilmanifold $ \widetilde W$ projects via $p$ to a sub-nilmanifold of ${{\mathbb{R}}}/{{\mathbb{Z}}}$, that is to a closed subgroup $W$ of ${{\mathbb{R}}}/{{\mathbb{Z}}}$. There are two possibilities: either $W$ is finite or $W={{\mathbb{R}}}/{{\mathbb{Z}}}$. In the first case the nilmanifold $ \widetilde W $ has a finite number of components and the restriction $ {\widetilde{f}\circ {\widetilde
\phi}_{\varrho}}_{| \widetilde W}$ of $ \widetilde{f} \circ {\widetilde
\phi}_{\varrho}$ to $\widetilde W$ is continuous. In the second case, the discontinuities of $ {\widetilde{f}\circ {\widetilde
\phi}_{\varrho}}_{| \widetilde W}$ belong to the lower dimensional sub-nilmanifold $\widetilde W \cap p^{-1} \{-\varrho\}$.
In summary, in both cases we have $ {\widetilde{f}\circ {\widetilde
\phi}_{\varrho}}_{| \widetilde W}\in D(\widetilde{W}) $ and it can be treated as a function on the compact nilmanifold $\widetilde{W}= H/
(H\cap g\widetilde \Gamma g^{-1}) $. Then, by Theorem \[thm:nonconnect\], we have $$\frac1M\sum_{M\leq m<2M}\left|\frac1H\sum_{m\leq n<m+H}\widetilde{f}\circ {\widetilde
\phi}_{\varrho}
(l_{\widetilde {u}}^n(x)(\widetilde{x})){\boldsymbol{\mu}}(n)\right|\to 0,$$ which completes the proof in the non-ergodic case.
Let $ P(x)=a_dx^d+\ldots+a_1x+a_0\in{{\mathbb{R}}}[x] $ have the leading coefficient $ a_d $ irrational. Let $ \alpha=a_d\cdot d! $ and let $
\phi\colon{{\mathbb{T}}}^d\to{{\mathbb{T}}}^d $ be given by $$\phi(x_1,x_2,\ldots, x_d)=(x_1+\alpha, x_1+x_2,\ldots, x_{d-1}+x_d).$$ Then $ \phi $ is an ergodic affine unipotent diffeomorphism of $ {{\mathbb{T}}}^d
$. Following [@Fu] (see also [@Ei-Wa]), we can now find $ x_1,\ldots,x_d\in{{\mathbb{T}}}$ so that $$P(n)=\binom{n}{d} \alpha+ \binom{n}{d-1}x_1+\ldots+nx_{d-1}+x_d\bmod
1\text{ for each }n\in{{\mathbb{Z}}}.$$ Moreover, notice that (mod 1) we have $${n\choose d}
\alpha+{n\choose d-1}x_1+\ldots+nx_{d-1}+x_d=f(\phi^n(x_1,\ldots,x_d)),$$ where $ f\colon {{\mathbb{T}}}^d\to{{\mathbb{T}}}$ is the projection on the last coordinate. Finally, the assertion of the theorem follows directly from Theorem \[thm:gammaaffine\] applied to the function $\exp(\imath f)$.
Nilsequences and polynomial multiple correlations {#pmc}
-------------------------------------------------
Following [@MR2138068] and [@MR2643713], recall that a bounded sequence $(c_n)\in \C^{{\mathbb{N}}}$ is called a *basic nilsequence* if there exist a compact but not necessarily connected nilmanifold $G/\Gamma$, a continuous function $f\in C(G/\Gamma)$, an element $u\in G$ and a point $x\in G/\Gamma$ such that $$c_n=f(l_{u}^n x), \qquad \forall n\in{{\mathbb{N}}},$$ where as usual $l_u$ denotes the nil-translation $G/\Gamma\ni x
\mapsto ux\in G/\Gamma$. A sequence $(d_n)\in \C^{{\mathbb{N}}}$ is called a *nilsequence*, if it is a uniform limit of basic nilsequences, i.e. for every $\varepsilon>0$, there exists a basic nil-sequence $(c_n)$ such that $$|c_n-d_n|<\varepsilon\text{ for all }n\in{{\mathbb{N}}}.$$
Theorem \[TheoremE\] follows directly from Theorem \[thm:nonconnect\].
Let $T$ be an automorphism of a probability standard Borel space $(X,{\mathcal B},\mu)$. Given, for all $i=1,\ldots,k$, functions $g_i\in L^\infty(X,\mu)$ and polynomials $p_i\in{{\mathbb{Z}}}[x]$, there exists a nilsequence $(d_n)$ such that $$\label{eq:leibman}
\limsup_{{N-M}\to\infty} \frac1{N-M}\sum_{n=M}^{N-1} \left|d_n-\int_X
g_1\circ T^{p_1(n)}\cdot\ldots\cdot g_k\circ T^{p_k(n)}\,d\mu\right|=0.$$
By Theorem \[TheoremE\], we have $$\frac1M\sum_{M\leq m<2M}\left| \frac1H\sum_{m\leq n<m+H} {\boldsymbol{\mu}}(n) d_n\right|\to 0$$ for a nilsequence $(d_n)$ satisfying the assertion of Leibman’s theorem. The statement follows now by an immediate application of the triangle inequality.
Lifting AOP to induced action
=============================
Induced actions {#ap-d}
---------------
In this section, we follow [@Ma], [@Zi]. Assume that $ G $ is a locally compact second countable (lcsc) group. Assume that $ H\subset G $ is a closed subgroup of $ G $. The quotient space $ G/H $ is lcsc for the quotient topology. Let $
\pi:G\to G/H $ be the canonical, continuous quotient map. Let $ \tau=
(\tau_g)_{g\in G} $ denote the $ G $-action on $ G/H $ by left translations: $ \tau_g(xH)= gxH $. The action $ \tau $ is continuous, transitive, hence ergodic for any $ G $-quasi-invariant Borel probability measure $ m $ on $ G/H $ (such measures always exist). We say that $ G/H $ is a finite volume space if $ G/H $ supports a $ G $-invariant probability measure $ m_{G/H} $,
Let $ {{\mathcal S}}=(S_h)_{h\in H} $ be an almost free Borel left action of $ H $ on a probability standard Borel space $ {(Y,{\mathcal C},\nu)}$. The group $ H $ acts on the right on the product $ Y\times G $, by $$(y,g)h = (S_{h^{-1}} y, gh)$$ Let $ (Y\times G)/H $ be the orbit space endowed with the quotient measurable structure. Since $ G/H $ is Hausdorff, since $ {(Y,{\mathcal C},\nu)}$ is a standard Borel space, and since the projection $ (Y\times G)/H \to
G/H $ measurable, the Borel space $ (Y\times G)/H $ is countably separated (i.e. the $ H $-action on $ Y\times G $ is smooth, in the sense of [@Zi Def. 2.1.9]).
The $ G $-action $ ( (\tau_{{{\mathcal S}}})_g)_{g\in G} $ on $ (Y\times G)/H $ is just given by left translation on the second factor: $$\label{eq:NilpotentAOP-14-06:1} (\tau_{{{\mathcal S}}})_g: (Y\times G)/H\to (Y\times
G)/H, \quad (\tau_{{{\mathcal S}}})_g(y,g_1)H=(y, gg_1)H$$ is called the *$ G $-action induced from $ {{\mathcal S}}$*.
If $ m $ is any $ G $-quasi-invariant Borel probability measure on $
G/H $, then we may define a Borel probability measure $ \nu \otimes_{\mathcal
S} m $ on $ (Y\times G)/H $, by setting, for any positive measurable function $ F $ on $ (Y\times G)/H $, $$\nu \otimes_{\mathcal S} m (F) = \int_{G/H} \left(\int_Y \widetilde
F(y,g) \, \D \nu(y)\right)\, \D m (gH)$$ where $ \widetilde F $ is the $ H $-invariant lift of $ F $ to $ Y\times
G $. The measure $ \nu \otimes_{\mathcal S} m $ is quasi-invariant for the $ G $-action induced from $ {{\mathcal S}}$ and it is ergodic if the action $ {{\mathcal S}}$ is ergodic on $ {(Y,{\mathcal C},\nu)}$. Furthermore, if $ G/H $ admits a $ G $-invariant probability measure $ m_{G/H} $, then $ \nu
\otimes_{\mathcal S} m_{G/H} $ is a $ G $-invariant Borel probability measure on $ (Y\times G)/H $.
Let $$\label{in2} s:G/H\to G,\text{ a Borel map, }\pi\circ s=
\operatorname{Id}
_{G/H}$$ be a (measurable) selector for $ \pi $. The map $ \theta $ defined on $ G\times G/H $ by setting $$\label{in3} \theta(g,xH)=s\big(gxH\big)^{-1}\, g \,s\big(xH\big).$$ takes its values in $ H $, since $$s\big(gxH\big) H= gxH, \quad \text{and}\quad g \,s\big(xH\big)= gxH.$$
\[l:in1\] The map $ \theta:G\times G/H\to H $ is a (left) cocycle for the $ G $-action $ \tau $.
We have $$\begin{split}
\theta(g_1g_2,xH)&=s(g_1g_2xH)^{-1} g_1g_2 s(xH)\\
&= s(g_1g_2xH)^{-1} g_1\,s(g_2 xH)^{-1}s(g_2 xH)\,g_2\,s(xH)\\
&= \theta(g_1,g_2xH)\,\theta(g_2,xH).\qedhere
\end{split}$$
The skew product $ G $-action $ \tau_{\theta,{{\mathcal S}}}=((\tau_{\theta,{{\mathcal S}}})_g)_
{g\in G} $, defined by $$\label{in5} (\tau_{\theta,{{\mathcal S}}})_g: Y\times G/H\to Y\times G/H,\qquad\quad
(\tau_{\theta,{{\mathcal S}}})_g(y, xH)=(S_{\theta(g,xH)}y,gxH)$$ is isomorphic to the $ G $-action $ \tau_{{{\mathcal S}}} $ on $ (Y\times G)/H $ induced from $ {{\mathcal S}}$, via the Borel isomorphism $ \Phi\colon Y\times
G/H\to (Y\times G)/H $ defined by $$\Phi( y, gH) = \big(S_{g^{-1}s(gH)}y, g\big)H = \big(y, s(gH)\big)H.$$
\[r:in1\] In the vocabulary of [@Le-Le], [@Le-Pa], the induced $ G $-action is a Rokhlin cocycle extension of the $ G $-action $ \tau $ through the $ H $-valued cocycle $ \theta $ and an ergodic $ H $-action $ {{\mathcal S}}$. Both descriptions and for the $ G $-action induced from $ {{\mathcal S}}$ have their advantages and disadvantages: the first is natural and intrinsic; the second, albeit depending of a arbitrary selector $ \theta $, yields an easier description of the measure $ \nu\otimes_{{{\mathcal S}}} m $ which is just $
\nu\otimes m $ on the space $ Y\times G/H $; the second definition also make apparent that $ (Y\times G)/H $ endowed with the measure $
\nu\otimes m $ is a standard Borel probability space.
The AOP property {#liftingAOPia}
----------------
\[ap-e\] Assume now that $ G $ is an abelian group without torsion elements, that $ H<G $ is closed and co-compact and let $ m_{G/H} $ be the $ G $-invariant probability measure on $ G/H $. Assume also that we have an ergodic $ H $-action $ {{\mathcal S}}=(S_h)_{h\in H} $ on $ {(Y,{\mathcal C},\nu)}$ and let $ p\in{{\mathbb{Z}}}$. Then $ p $ determines another $ H $-action $ {{\mathcal S}}^{(p)}=
(S^{(p)}_h)_{h\in H} $, where $$\label{in8} S^{(p)}_h(y)=(S_h)^p(y)=S_{h^p}(y);$$ indeed the map $ h\mapsto h^p $ is a group homomorphism.
Following [@Ab-Le-Ru], we now define:
An ergodic $ H $ action $ {{\mathcal S}}$ is said to have the AOP property if for each $ f,g\in L^2_0{(Y,{\mathcal C},\nu)}$ $$\label{in9} \lim_{p\neq q, p,q\in\mathscr{P},p,q\to\infty}
\sup_{\kappa\in J^e({{\mathcal S}}^{(p)},{{\mathcal S}}^{(q)})}\left|\int_{Y\times Y}f{\otimes}g\,d\kappa\right|=0.$$
If is satisfied for $ f,g $, then we say that $ f,g $ satisfy the AOP property.
\[r:in00\] (i) In order to talk about the AOP property, we need to know that the $ H $-actions are “totally” ergodic, that is that $ {{\mathcal S}}^{(p)} $ is ergodic for each prime $ p $. To see its meaning let us pass to the character group $ \widehat{H} $ and consider the endomorphism $ E_p: \chi\mapsto \chi^p $. We need to assume that the kernel of $ E_p $ is a set of measure zero for the maximal spectral type of $ {{\mathcal S}}$ on $ L^2_0{(Y,{\mathcal C},\nu)}$. If $ H $ equals $ {{\mathbb{Z}}}$ then it simply means that $ T $ has no roots of unity as eigenvalues. If $ \widehat{H} $ is torsion free, as for example it holds for $ H={{\mathbb{R}}}$, then each ergodic action is totally ergodic.
\(ii) If holds then it also holds if $ f\in L^2{(Y,{\mathcal C},\nu)}$ and $ g\in L^2_0{(Y,{\mathcal C},\nu)}$ (or vice versa); indeed, we write $ f=f_0+c $, with $ c=\int_Yf\,d\nu $ then $ \int_{Y\times Y}c{\otimes}g\,d\kappa=\int_Y g\,d\nu=0 $.
\(iii) If holds, then, in fact, we can replace the set of ergodic joinings $ J^e({{\mathcal S}}^
{(p)},{{\mathcal S}}^{(q)}) $ by $ J({{\mathcal S}}^{(p)},{{\mathcal S}}^{(q)}) $; indeed, $$\sup_{\kappa\in J^e({{\mathcal S}}^{(p)},{{\mathcal S}}^{(q)})}\left|\int_{Y\times Y}f{\otimes}g\,d\kappa\right|=\sup_{\kappa\in J({{\mathcal S}}^{(p)},{{\mathcal S}}^{(q)})}\left|\int_
{Y\times Y}f{\otimes}g\,d\kappa\right|$$ for each $ f,g\in L^2{(Y,{\mathcal C},\nu)}$.
\(iv) If $ L $ is a closed subgroup of $ H $ and the $ L $-subaction of $ {{\mathcal S}}$ has the AOP property, then the original action has the AOP property. Indeed, any ergodic invariant measure for $ S_{h^p}\times
S_{h^q} $, $ h\in H $ is an invariant measure for the subaction $ S_
{\ell^p}\times S_{\ell^q} $, $ \ell\in L $. Then use (iii).
We will constantly assume that $$\label{in1}\mbox{$H$ has no non-trivial compact subgroups.}\footnote{Equivalently, we assume that the dual group $\widehat{H}$ is connected. It may happen that $G$ has non-trivial compact subgroups.}$$ Since $H$ is cocompact, there is a relatively compact fundamental domain, that is, a subset $A\subset G$ such that $$\label{in2a}
\mbox{${\overline}{A}$ is compact, $hA\cap h'A=\emptyset$ whenever $h\neq h'$ and $\bigcup_{h\in H}hA=G$.}$$ We can always assume that (cf. ) $$\label{in2b}
s:H\to A\text{ and }s(H)=e.$$
Ergodic components for the G-action tau\^(p) times tau\^ (q) and regularity of cocycles
---------------------------------------------------------------------------------------
Recall $ \tau^{(p)}_g(xH)=g^pxH $. Assume that $ (p,q)=1 $ and let $ a,b\in{{\mathbb{Z}}}$ be so that $$ap+bq=1.$$ The following result is a more general version of Lemma 2.2.1 in [@Ku-Le] and Lemma \[lem:torus\]. Since the proof runs along similar lines, we omit it.
\[l:in2\]The ergodic components of $ \tau^{(p)}\times\tau^{(q)}=
(\tau^{(p)}_g\times\tau^{(q)}_g)_{g\in G} $ are of the form $${{\mathcal H}}_c:=\big\{(x_1H,x_2cH):\:x_1^qH=x_2^pH\big\},\,c\in G. \footnote{Note that since $G$ is abelian, $x^mH=(xH)^m$ in $G/H$ }$$ Moreover, the action of $ \tau^{(p)}\times\tau^{(q)} $ on $ {{\mathcal H}}_c $ is (topologically) [^9] isomorphic to $ \tau $.
We are now interested in the (infinite) skew product $ G $-action $
(\tau_\theta)^{(p)}\times(\tau_\theta)^{(q)} $, $$\begin{split}
&\big((\tau_\theta)^{(p)}\times(\tau_\theta)^{(q)}
\big)_g(x_1H,h_1,x_2H,h_2)=\\ &\qquad\qquad
\big(g^px_1H,\theta(g^p,x_1H)h_1,g^qx_2H, \theta(g^q,x_2H)h_2\big)
\end{split}$$ restricted to $ {{\mathcal H}}_c\times H\times H $ (up to a natural interchange of coordinates).
\[l:in3\]For each $ c\in G $, the map $ \widetilde{R}(x_1H,h_1,x_2cH,h_2):=
(x_1^ax_2^bH,h_1,h_2) $ establishes an isomorphism of $ (\tau_\theta)^
{(p)}\times(\tau_\theta)^{(q)}|_{{{\mathcal H}}_c} $ with $ \tau_{\theta^{(p,q)}}
$, where the cocycle $$\theta^{(p,q)}:G\times G/H\to H\times H,$$ is given by the formula $$\label{in11} \theta^{(p,q)}(g,yH):=\big(\theta(g^p,y^pH),\theta(g^q,y^qcH)\big).$$
First notice that indeed $ \theta^{(p,q)} $ is a cocycle, that is, we have $$\begin{aligned}
\theta^{(p,q)}(g_1g_2,yH)&=\big( \theta((g_1g_2)^p,y^pH),(\theta((g_1g_2)^q,y^qcH)\big)\\
&=
\big(\theta(g_1^p,y^pH),\theta(g_1^qy^qcH)\big)\cdot \big(\theta(g_2^p,g_1^py^pH),\theta
(g_2^q,g_1^qy^qcH)\big)\\
& =
\theta^{(p,q)}(g_1,yH)\cdot\theta^{(p,q)}(g_2,\tau_{g_1}(yH)).
\end{aligned}$$ Since $$\theta^{(p,q)}(g,x_1^ax_2^bH)=\big(\theta(g^p,x_1^{pa}x_2^{pb}H),
\theta(g^q,x_1^{qa}x_2^{qb}cH)\big)$$ and $$x_1^{pa}x_2^{pb}H=x_1^{pa}x_1^{qb}H=x_1H$$ with a similar observation concerning the second coordinate, the equivariance easily follows.
\[l:in4\] The cocycle $ \theta^{(p,q)} $ is regular.
Let $ J $ be an algebraic automorphism of $ H\times H $ given by the matrix $ \begin{bmatrix}
q & -p\\
a& b
\end{bmatrix}
$ [^10] and consider the cocycle $ J\circ \theta^{(p,q)}=:(\Psi_1,\Psi_2)
$, where (in view of ) $$\begin{aligned}
\Psi_1(g,yH)&=\left(\theta(g^p,y^pH)\right)^q\left(\theta(g^q,y^qcH)
\right)^{-p}\\
&=
\left(g^ps(g^py^pH)^{-1}s(y^pH)\right)^q \left( g^qs(g^qy^qcH)^{-1}s
(y^qcH)\right)^{-p}\\
&=
\left(s(g^py^pH)^{-1}s(y^pH)\right)^q \left(s(g^qy^qcH)^{-1}s(y^qcH)\right)^
{-p}.
\end{aligned}$$ In view of the values of the cocycle $ \Psi_1 $ belong to the set $ (A^{-1}A)^{-p+q} $ which is relatively compact. It follows by and Theorem 5.2 in [@Mo-Sch] that the cocycle $ \Psi_1 $ is a coboundary. Now, $$\begin{aligned}
\Psi_2(g,yH)&= \left(\theta(g^p,y^pH)\right)^a\left(\theta(g^q,y^qcH)
\right)^{b}\\
&=
g^{pa}\left(s(g^py^pH)^{-1}s(y^pH)\right)^a g^{qb}\left(s(g^qy^qcH)^
{-1}s(y^qcH)\right)^{b}\\
&=
g\left(s(g^py^pH)^{-1}s(y^pH)\right)^a \left(s(g^qy^qcH)^{-1}s(y^qcH)\right)^
{b}.
\end{aligned}$$ Note that for each $ h\in H $, we have $$\Psi_2(h,yH)=h.$$ It follows immediately that $ \tau_{\Psi_2} $ is transitive, [^11] whence $ \Psi_2 $ is an ergodic cocycle.
It follows that our cocycle $ J\circ \theta^{(p,q)} $ is cohomologous to a cocycle taking values in the group $ \{e\}\times H
$ and the cocycle is ergodic (the corresponding skew product is transitive). If we write $$J\circ \theta^{(p,q)}(g,yH)= \big(\eta(gyH)^{-1}\eta(yH),\Psi_2(g,yH)\big)$$ then $$\theta^{(p,q)}(g,yH)=
\big(\Psi_2(x,yH)^p \big(\eta(gyH)^{-1}\eta(yH)\big)^b, \Psi_2
(x,yH)^q\big(\eta(gyH)^{-1}\eta(yH)\big)^{-a}\big).$$ It follows that $ \theta^{(p,q)} $ is cohomologous to the cocycle $$(g,yH)\mapsto\left(\left(\Psi_2(g,yH)\right)^p,\left(\Psi_2(x,yH)\right)^q\right)$$ taking values in the subgroup $${{\mathcal F}}^{(p,q)}:=\big\{(h_1,h_2)\in H\times H:\: h_1^q=h_2^p\big\}=\big\{(h^p,h^q):\:h\in
H\big\}.$$ The latter cocycle is ergodic, [^12] which completes the proof.
Ergodic joinings of the G-action with
--------------------------------------
We are interested in a description of $
\tilde{{\lambda}}\in J^e((\tau_{\theta,{{\mathcal S}}})^{(p)},(\tau_ {\theta,{{\mathcal S}}})^{(q)})
$. Recall that $$\begin{aligned}
\left((\tau_{\theta,{{\mathcal S}}})^{(p)}\times (\tau_{\theta,{{\mathcal S}}})^{(q)}\right)_g
&\big((x_1H,y_1),(x_2H,y_2)\big)\\=&
\big(g^px_1H,S_{\theta(g^p,x_1H)}(y_1),g^qx_2H,S_{\theta(g^q,x_2H)}(y_2)\big).\end{aligned}$$ But $ \widetilde{{\lambda}}|_{G/H\times G/H}=:{\lambda}$ is an ergodic joining of $ \tau^{(p)} $ with $ \tau^{(q)} $, and by unique ergodicity, it is an ergodic component of the $ G $-action $ \tau^{(p)}\times \tau^{(q)} $. Hence we can pass to $ {{\mathcal H}}_c $ replacing $ (G/H\times G/H,{\lambda}) $ by $
(G/H,m_{G/H}) $, with $ c\in G/H $ in this notation implicit. It follows that we now consider the $ G $-action $ \tau_{\theta^{(p,q)},{{\mathcal S}}{\otimes}{{\mathcal S}}}
$ with an ergodic measure $ \widetilde{{\lambda}} $ (whose projection on the first and the second, and the first and the third coordinates are equal to $ m_{G/H}{\otimes}\nu $). Here, $ {{\mathcal S}}{\otimes}{{\mathcal S}}$ denotes the product $
H\times H $-action $ (S_h\times S_{h'})_{(h,h')}\in H\times H $. Moreover, $$\left(\tau_{\theta^{(p,q)},{{\mathcal S}}{\otimes}{{\mathcal S}}}\right)_g(xH,y_1,y_2)= \big(\tau_g(xH),S_
{\theta(g^p,x^pH)}(y_1),S_{\theta(g^p,x^pcH)}(y_2)\big).$$ But, by Lemma \[l:in4\], $ \theta^{(p,q)} $ is cohomologous to the cocycle $ (\Psi_2^p,\Psi_2^q)
$ taking values in $ {{\mathcal F}}^{(p,q)}\subset H\times H $, and the latter cocycle is ergodic. As a matter of fact: $$\theta^{(p,q)}\cdot\big(\eta(g,\cdot)^b, \eta(g,\cdot)^
{-a}\big)=(\eta^b,\eta^{-a}) \cdot\big(\Psi_2^p,\Psi_2^q\big).$$ This yields an isomorphism (an equivariant map) $$Id_{(S_{\eta^b},S_{\eta^{-a}})}\big(xH,y_1,y_2\big)= \big(xH,S_{\eta
(xH)^b}(y_1),S_{\eta(xH)^{-a}}(y_2)\big)$$ between the $ G $-actions $ \tau_{\theta^{(p,q)},{{\mathcal S}}{\otimes}{{\mathcal S}}} $ and $
\tau_{(\Psi_2^p,\Psi_2^q),{{\mathcal S}}^{(p)}\times{{\mathcal S}}^{(q)}} $.
Now, by Theorem 3 in [@Le-Me-Na], since $ (\Psi^p_2,\Psi_2^q) $ is ergodic, we obtain the following.
\[l:in5\] We have $$\label{in21}
\Big(\big(Id_{(S_{\eta^b},S_{\eta^{-a}})}\big)^{-1}\Big)_\ast (\widetilde
{{\lambda}})=m_{G/H}{\otimes}\kappa,$$ where $ \kappa\in J^e({{\mathcal S}}^{(p)},{{\mathcal S}}^{(q)}) $.
Proof of Proposition \[PropositionG\]
-------------------------------------
Before we start proving Proposition \[PropositionG\], let us make the following observation. Assume that $
{{\mathcal R}}=(R_g)_{g\in G} $ is a $ G $-action on $ (Z,\mu) $ and let $ (W,\xi)
$ be a probability standard Borel space. Assume that we have two Rokhlin cocycles: $$\Theta,\Psi:G\times Z\to {\rm Aut}(W,\xi)$$ and a measurable map $ \Sigma:Z\to {\rm Aut}(W,\xi) $. Assume that $
Id_{\Sigma} $ establishes an isomorphism of $ ({{\mathcal R}}_{\Psi},Z\times
W,\mu{\otimes}\xi) $ with $ ({{\mathcal R}}_{\Theta},Z\times W,\rho) $, where $ \rho|_Z=\mu
$. Assume that $ f\in L^1(Z,\mu) $, $ g\in L^1(Y,\xi) $, then $$\begin{aligned}
\int_{Z\times W}f(z)g(w)\,d\rho(z,w)&= \int_{Z\times W}(f{\otimes}g)\circ Id_{\Sigma} (z,w)\,d\mu(z)d\xi(w)
\\&=
\int_{Z}f(z)\left(\int_W g(\Sigma_z(w))\,d\xi(w)\right)d\mu(z)
\\&=
\int_{Z}f(z)\left(\int_W g(w)\,d\left((\Sigma_z)_\ast(\xi)\right)(w)\right)d\mu
(z).\end{aligned}$$ It follows that $$\label{in22} \left|\int_{Z\times W}f(z) g(z)\,d\rho(z,w)\right|\leq \|f\|_
{L^1(\mu)}\sup_{z\in Z}\left|\int_W g\,d\left(\left(\Sigma_z\right)_\ast
(\xi)\right)\right|.$$
Consider now the situation in which $ {{\mathcal R}}=\tau $, $ Z=G/H $, $\mu=m_{G/H}$, $ W=Y\times
Y $, $\xi=\kappa$: here $ {{\mathcal R}}_{\Theta}=\tau_{\theta^{(p,q)},{{\mathcal S}}\otimes{{\mathcal S}}} $ acts on $ G/H\times
Y\times Y $ and preserves $ \rho=\tilde{{\lambda}} $ (the parameter $ c $ of the ergodic component is implicit), and the other $ G $-action ${{\mathcal R}}_{\Psi}=
\tau_{(\Psi_2^p,\Psi_2^q),{{\mathcal S}}^{(p)}\times{{\mathcal S}}^ {(q)}} $ preserves $ m_{G/H}{\otimes}\kappa $. In view of , we consider the isomorphism $ Id_{(S_{\eta^b},S_{\eta^{-a}})} $ between $ \tau_{\theta^
{(p,q)},{{\mathcal S}}\otimes{{\mathcal S}}} $ and $ \tau_{(\Psi_2^p,\Psi_2^q),{{\mathcal S}}^ {(p)}\times{{\mathcal S}}^
{(q)}} $. Hence, in our notation, $$\Sigma_{xH}=S_{\eta(xH)^b}\times S_{\eta(xH)^{-a}}.$$ It follows that the fiber automorphisms are of the form $ S_{h^{b}}\times
S_{h^{-a}} $. Therefore, each fiber automorphism commute with the $ G
$-action $ {{\mathcal S}}^{(p)}\times {{\mathcal S}}^{(q)} $. It easily follows that $$\left(\Sigma_{xH}\right)_\ast \kappa\in J^e({{\mathcal S}}^{(p)},{{\mathcal S}}^{(q)})$$ and from , we obtain that for each $ f\in L^2(G/H,m_{G/H})
$ and $ F_1,F_2\in L^2(Y,\nu) $, we have $$\label{in23} \left|\int_{G/H\times Y\times Y}f{\otimes}F_1{\otimes}F_2\,d\widetilde
{{\lambda}}\right|\leq \|f\|_1\sup_{\kappa'\in J^e({{\mathcal S}}^{(p)},{{\mathcal S}}^{(q)})}\left|
\int_{Y\times Y}F_1{\otimes}F_2\,d\kappa'\right|.$$ To deduce the assertion, it is now enough to notice that, although on the LHS of , the constant $ c $ is implicit, it has no influence since if we consider (as we should) $ f_1{\otimes}F_1{\otimes}f_2{\otimes}F_2
$ as a member of $ L^2(G/H\times Y\times G/H\times Y,\widetilde{{\lambda}}) $ (with $ \widetilde{{\lambda}} $ an arbitrary ergodic joining of $ (\tau_ {\theta,{{\mathcal S}}})^
{(p)} $ and $ (\tau_{\theta,{{\mathcal S}}})^{(q)} $) then by the Schwarz inequality, the $ L^1(\widetilde{{\lambda}}) $-norm of $ f_1{\otimes}f_2 $ will be bounded by $ \|f_1\|_{L^2(m_{G/H})}\|f_2\|_{L^2(m_{G/H})} $, hence does not depend on $ \widetilde{{\lambda}} $. The result follows.
Examples and applications
-------------------------
Assume that we have an ergodic $ G $-action $ {{\mathcal S}}=(S_g)_{g\in G} $ on $ {(Y,{\mathcal C},\nu)}$. If $ H\subset
G $ is cocompact then, by [@Zi], the induced $ G $-action from the subaction $ (S_h)_{h\in H}
$ is isomorphic to $ {{\mathcal S}}\times \tau $ (by an isomorphism being the identity on the $ \tau $-coordinates).
It follows now from Proposition \[PropositionG\] that:[^13]
If $ {{\mathcal S}}=(S_g)_{g\in G} $ is as above and the subaction $ (S_h)_ {h\in
H} $ has the AOP property, then the $ G $-action $ {{\mathcal S}}\times\tau= (S_g\times\tau_g)_
{g\in G} $ on $ (Y\times G/H,\nu\otimes m_{G/H}) $ has the relative (with respect to $ \tau $) AOP property.
If we consider $ {{\mathbb{Z}}}\subset{{\mathbb{R}}}$, then a natural choice of the selector $ s:{{\mathbb{R}}}/Z\to{{\mathbb{R}}}$ satisfying , and is to set $ s(r):=\{r\} $ being the fractional part of $ r\in{{\mathbb{R}}}$. Then $$\theta(t,\{r\})=-\{t+\{r\}\}+t+\{r\}=[t+\{r\}]\in{{\mathbb{Z}}}.$$ Assume now that $ S $ is an ergodic automorphism of $ {(Y,{\mathcal C},\nu)}$. It follows that the induced $ {{\mathbb{R}}}$-action $
\widetilde{S} $ is given by $$\widetilde{S}_t(y,\{r\})=(S^{\lfloor t+\{r\}\rfloor}y,\{t+r\}),$$ hence, it is the standard [*suspension*]{} construction over $ S $.
Consider $ n{{\mathbb{Z}}}\subset{{\mathbb{Z}}}$. Then $ s:{{\mathbb{Z}}}/n{{\mathbb{Z}}}\to{{\mathbb{Z}}}$ is given by $ s(k+n{{\mathbb{Z}}})=k\text
{ mod }n $. Then, by writing $ m+k=tn+r $ with $ 0\leq r<n $, we have $$\theta(m,k\text{ mod }n)=-(m+(k\text{ mod }n))+m+(k\text{ mod }n)=-r+m+k=tn\in
n{{\mathbb{Z}}}.$$ If we are now given an ergodic $ n{{\mathbb{Z}}}$-action [^14] we can induce. Instead of doing this directly, first write the obvious automorphism $ n\ell=\ell
$ between $ n{{\mathbb{Z}}}$ and $ {{\mathbb{Z}}}$ and then rewrite the cocycle $ \theta $, which now becomes $$\theta(m,x)=\left\lfloor\frac{x+m}n\right\rfloor.$$ Since this is a $ {{\mathbb{Z}}}$-cocycle, it is entirely determined by the function $ \theta=\theta(1,\cdot) $ given by: $ \theta(x)=0 $ if $
x=0,\ldots,n-2 $ and $ \theta(n-1)=1 $. If now $ S $ is an ergodic automorphism of $ {(Y,{\mathcal C},\nu)}$ then its ($ n $-discrete) induced is given by $ \widetilde{S} $ acting on $ Y\times\{0,1,\ldots,n-1\} $ by the formula: $ \widetilde{S}(y,j)=(y,j+1) $ if $ j=0\ldots,n-2 $, and $
\widetilde{S}(y,n-1)=(Sy,0) $. Hence, we obtain the classical $ n $-discrete suspension.
The $ k $-discrete suspension $ \widetilde{T} $ of $ T $ is a uniquely ergodic homeomorphism of the space $ X\times\{0,1,\ldots,k-1\} $. Define $ F(z,i)=0 $ if $ i\neq j $ and $ z\in X $, and $ F(z,j)=f (z) $. Then $ F\perp L^2(\{0,1,\ldots,k-1\}) $. Now, by the relative AOP property, we have $$\frac1N\sum_{n\leq N}F(\widetilde{T}^n(x,0)){\boldsymbol{u}}(n)\to0.$$ It is however not hard to see that by the definition of $ F $, $$\frac1N\sum_{n\leq N}F(\widetilde{T}^n(x,0)){\boldsymbol{u}}(n)=\frac1N\sum_{n\leq
N,n=j\text{ mod }k}f(T^nx){\boldsymbol{u}}(n)$$ and the result follows.
\[l:podciag\] Let $ T $ be a uniquely ergodic homeomorphism of $ X $, with the unique invariant measure $ \mu $. Assume that $ (X,\mu,T) $ has the AOP property. Assume additionally that there exists a set $ \mathscr{C}\subset
C(X)\cap L^2_0(X,\mu) $ whose linear span is dense in $ L^2_0 (X,\mu) $ such that for all $ f\in\mathscr{C} $ and all $
\omega\in \C $ with $ |\omega|=1 $ there exists a homeomorphism $ S:X\to X $ such that $ f(T^n(Sx))=\omega f(T^n x) $ for every $ x\in X $ and $ n\in{{\mathbb{Z}}}$.
Let $\widetilde{T}:\widetilde{X}\to\widetilde{X}$ be a discrete suspension of $T$. Then for every $F\in C(\widetilde{X})$ and every $\widetilde{x}\in \widetilde{X}$ we have $$\label{eq:krotper:0}
\frac 1 M\sum_{M\leq m<2M}\left|\frac1H\sum_{m\leq
n<m+H}F(\widetilde{T}^n\widetilde{x}){\boldsymbol{\mu}}(n)\right| \longrightarrow 0$$ when $ H\to\infty $ and $ H/M\to 0 $.
Suppose that $\widetilde{T}$ the $k$-discrete suspension of $T$. Let us consider the subspace $H:=L^2(\widetilde{X})\ominus L^2(\{0,\ldots,k-1\})$. By Proposition \[PropositionG\], the subspace $H$ satisfies the AOP property.
For every $f\in \mathscr{C}$ and $0\leq j<k$ denote by $f_j:\widetilde{X}\to\C$ the continues function which vanishes on all level sets except $X\times\{j\}$, where is given by $f_j(x,j)=f(x)$. Functions of such form establish a linearly dense subset of $H$.
Suppose that $ S:X\to X $ is a homeomorphism such that $ f(T^n(Sx))=\omega f(T^n x) $ for every $ x\in X $ and $ n\in{{\mathbb{Z}}}$. Then for the homeomorphism $\widetilde{S}:\widetilde{X}\to\widetilde{X}$ given by $\widetilde{S}(x,l)=(Sx,l)$ we have $ f_j(\widetilde{T}^n(\widetilde{S}\widetilde{x}))=\omega f_j(\widetilde{T}^n \widetilde{x}) $ for every $ \widetilde{x}\in \widetilde{X} $ and $ n\in{{\mathbb{Z}}}$. Now the arguments used in the proof of Theorem \[thm:prop\] applied to the family of functions $\{f_j\}$ show that holds for every continuous function $F\in H$. If $F\in L^2(\{0,\ldots,k-1\})$ then the sequence $\big(F(\widetilde{T}^n\widetilde{x})\big)$ is periodic and follows from Remark \[rem:mrt\]. It follows that holds for every $F\in C(\widetilde{X})$.
Let $\widetilde{T}:\widetilde{X}\to\widetilde{X}$ be a homomorphism coming from Lemma \[l:podciag\]. Then for every $k\geq 1$, $0\leq j\leq k$, $F\in C(\widetilde{X})$ and $\widetilde{x}\in\widetilde{X}$ we have $$\label{eq:krotper:01}
\frac 1 M\sum_{M\leq m<2M}\left|\frac1H\sum_{m\leq
n<m+H}F(\widetilde{T}^n\widetilde{x}){\boldsymbol{\mu}}(kn+j)\right| \longrightarrow 0$$ when $ H\to\infty $ and $ H/M\to 0 $.
Following the proof of Corollary \[CorollaryH\] we consider the $ k $-discrete suspension $ \widetilde{\widetilde{T}} $ of $\widetilde{T}$. By Proposition 2.4 in [@Zi], $ \widetilde{\widetilde{T}} $ is also a discrete suspension of ${T}$.
For every $F\in C(\widetilde{X})$ define $\widetilde{F}(\widetilde{z},j)=F(\widetilde{z})$ and $\widetilde{F}(\widetilde{z},l)=0$ for $l\neq j$. Since $\widetilde{F}\in C(\widetilde{\widetilde{X}})$, by Lemma \[l:podciag\] applied to $ \widetilde{\widetilde{T}} $ as a discrete suspension of ${T}$, we have $$\frac 1 {kM}\sum_{kM\leq m<2kM}\left|\frac1{kH}\sum_{m\leq
n<m+kH}\widetilde{F}(\widetilde{\widetilde{T}}^n(\widetilde{x},0)){\boldsymbol{\mu}}(n)\right|\longrightarrow 0.$$ Moreover, $$\begin{aligned}
\frac 1 M\sum_{M\leq m<2M}&\left|\frac1H\sum_{m\leq
n<m+H}F(\widetilde{T}^n\widetilde{x}){\boldsymbol{\mu}}(kn+j)\right|\\&= k^2\frac 1 {kM}\sum_{kM\leq km<2kM}\left|\frac1{kH}\sum_{km\leq
n<km+kH}\widetilde{F}(\widetilde{\widetilde{T}}^n(\widetilde{x},0)){\boldsymbol{\mu}}(n)\right|\\&
\leq k^2\frac 1 {kM}\sum_{kM\leq m<2kM}\left|\frac1{kH}\sum_{m\leq
n<m+kH}\widetilde{F}(\widetilde{\widetilde{T}}^n(\widetilde{x},0)){\boldsymbol{\mu}}(n)\right|\longrightarrow 0,
\end{aligned}$$ which completes the proof.
First observe the it suffices to focus only on basic nilsequences, since all considered sequences are or are appropriately approximated by basic nil-sequences. Moreover, every nilsequence is of the form $\big(F(\widetilde{T}^n\widetilde{x})\big)$, where $T$ is a homeomorphism satisfying the assumption of Lemma \[l:podciag\] and $F\in C(\widetilde{X})$. Indeed, $T$ is an ergodic affine unipotent diffeomorphism on a compact connected nilmanifold. Therefore, by Lemma \[l:podciag\], the sequence $\big(F(\widetilde{T}^n\widetilde{x})\big)$ meets .
All examples of AOP actions that appeared so far in the paper have either purely discrete or, they have mixed spectrum: discrete and Lebesgue. We will now give examples of AOP flows that have mixed spectrum: discrete and continuous singular, making use of recent results from [@Ku-Le1].
Below, we use Remark \[r:in1\] and .
\[l:sre1\] Let $ T $ be an ergodic automorphism of probability a standard Borel space $ {(X,{\mathcal B},\mu)}$ and $ {\varphi}:X\to K $ a cocycle with values in an abelian lcsc group. Then the formula $$\widetilde{{\varphi}}(t,(x,s)):={\varphi}^{(\lfloor t+s\rfloor )}(x)$$ defines a cocycle for the $ {{\mathbb{R}}}$-action of the suspension $
\widetilde{T} $ of $ T $.
First notice that $$\widetilde{T}_t(x,s)=(T^{\lfloor t+s\rfloor }x,\{t+s\}),$$ whence $ \widetilde{T}_{t_1+t_2}(x,s)=\widetilde{T}_{t_2}(\widetilde
{T}_{t_1}(x,s)) $ implies $$\label{su1} \lfloor t_1+t_2+s\rfloor =\lfloor t_1+s\rfloor +\lfloor t_2+\{t_1+s\}\rfloor .$$ Now, we have $$\begin{aligned}
\widetilde{{\varphi}}(t_1+t_2,(x,s))&={\varphi}^{(\lfloor t_1+t_2+s\rfloor )}(x)\stackrel{\eqref
{su1}}{=}{\varphi}^{(\lfloor t_1+s\rfloor +\lfloor t_2+\{t_1+s\}\rfloor )}(x)\\
& =
{\varphi}^{(\lfloor t_1+s\rfloor )}(x)+{\varphi}^{(\lfloor
t_2+\{t_1+s\}\rfloor )}(T^{\lfloor t_1+s\rfloor }x)
\\&=
\widetilde{{\varphi}}(t_1,(x,s))+\widetilde{{\varphi}}(t_2,\widetilde{T}_{t_1}
(x,s)).
\end{aligned}$$
Assume now that $ K={{\mathbb{R}}}$, so $ {\varphi}:X\to{{\mathbb{R}}}$. We will now consider a special class of extensions of $ T $. Namely, let $ {{\mathcal S}}=(S_t) $ be an ergodic flow on a probability standard Borel space $ {(Y,{\mathcal C},\nu)}$. Then consider the (probability) space $ (X\times Y,{{\mathcal B}}{\otimes}{{\mathcal C}},\mu{\otimes}\nu) $ on which we consider the (measure-preserving) automorphism: $${T_{{\varphi},{{\mathcal S}}}}(x,y):=(Tx,S_{{\varphi}(x)}(y)).$$ Note that if $ {{\mathcal S}}$ is a continuous flow acting on $ Y $ compact and if $ {\varphi}$ is continuous then $ {T_{{\varphi},{{\mathcal S}}}}$ is a homeomorphism of $ X\times
Y $. Our special interest in this class of actions come from the following result.
\[t:kule\] For each $ {\varphi}:{{\mathbb{T}}}\to{{\mathbb{R}}}$ of class $ C^2 $, $ {\varphi}$ different from a trigopolynomial, there is $ \alpha $ irrational that if $ Tx=x+\alpha $ and $ {{\mathcal S}}$ is an arbitrary uniquely ergodic flow then $ {T_{{\varphi},{{\mathcal S}}}}$ has the AOP property. If $ {\varphi}(x)=x-\frac12 $ then for each $ \alpha $ with bounded partial quotients and each uniquely ergodic $ {{\mathcal S}}$ which has no non-trivial rational eigenvalues, $ {T_{{\varphi},{{\mathcal S}}}}$ has the AOP property.
It follows now from Theorem \[PropositionG\] that the suspension of the $ {T_{{\varphi},{{\mathcal S}}}}$ with the AOP property will also enjoy the same property (for flows and relatively). We will now try to say a little bit more on the structure of these suspensions (including a relationship with nilflows, see Proposition \[p:negK\]).
\[l:sre2\] We have (up to natural isomorphism) $ \widetilde{{T_{{\varphi},{{\mathcal S}}}}}=\widetilde
{T}_{\widetilde{{\varphi}},{{\mathcal S}}} $.
We have $$\begin{aligned}
\left(\widetilde{{T_{{\varphi},{{\mathcal S}}}}}\right)_t((x,y),s)&=\big(\left({T_{{\varphi},{{\mathcal S}}}}\right)^{\lfloor t+s\rfloor }
(x,y),\{t+s\}\big)\\& = \big(T^{\lfloor t+s\rfloor }x, S_{{\varphi}^{(\lfloor t+s\rfloor )}(x)}(y),\{t+s\}\big)
\\&=
\big(T^{\lfloor t+s\rfloor }x, \{t+s\}, S_{{\varphi}^{(\lfloor t+s\rfloor )}(x)}(y)\big)=\big(\widetilde{T}_t(x,s),S_
{\widetilde{{\varphi}}(t,(x,s))}(y)\big)
\\&= \left(\widetilde{T}_{\widetilde{{\varphi}},{{\mathcal S}}}\right)_t
((x,s),y).
\end{aligned}$$
By its definition, each suspension flow has the linear flow $ {\mathcal L}
$: $ L_tx=x+t $ on $ {{\mathbb{T}}}$, as its factor, [^15] so if we think about good ergodic properties of the suspension, it should be considered relatively to this factor $ {\mathcal L} $. For example, if $
T $ is weakly mixing then $ \widetilde{T} $ is relatively weakly mixing over $ {\mathcal L} $. Indeed, $$\begin{split}
\left(\widetilde T\times_{\mathcal L}\widetilde T\right)_t(x_1,x_2,s)&=
\big(T^{\lfloor t+s\rfloor }x_1,T^{\lfloor t+s\rfloor }x_2,\{s+t\}\big)\\&=
\big((T\times T)^{\lfloor t+s\rfloor }(x_1,x_2),\{s+t\}\big),
\end{split}$$ so the relative product is just the suspension over $ T\times T $. Similar calculation can be done if we want to compute some relative properties of $ \widetilde{T_{{\varphi},{{\mathcal S}}}}$ over $ \widetilde{T} $. Indeed, for the relative product we have: $$\begin{aligned}
&(\widetilde{T_{{\varphi},{{\mathcal S}}}})\times_{\widetilde T}(\widetilde{T_{{\varphi},{{\mathcal S}}}})
((x,y_1),(x,y_2),s)\\&\qquad=
\big(({T_{{\varphi},{{\mathcal S}}}})^{\lfloor t+s\rfloor }(x,y_1),({T_{{\varphi},{{\mathcal S}}}})^{\lfloor t+s\rfloor }(x,y_2)
,\{s+t\}\big)\\&\qquad
=
\big(\left({T_{{\varphi},{{\mathcal S}}}}\times_{T}{T_{{\varphi},{{\mathcal S}}}}\right)^{\lfloor t+s\rfloor }(x,y_1,y_2),\{s+t\}\big),\end{aligned}$$ so again the relative product of the suspension is the suspension of the relative product. Now, the examples of $ {T_{{\varphi},{{\mathcal S}}}}$ in Theorem \[t:kule\] are relatively weakly mixing over $ T $ whenever $ {{\mathcal S}}$ is weakly mixing. It follows that, we have the following.
\[c:kule\] Assume that $ T $ and $ {\varphi}$ are as in Theorem \[t:kule\]. Assume that $ {{\mathcal S}}$ is weakly mixing. Then $
\widetilde{T_{{\varphi},{{\mathcal S}}}}$ have the relative AOP property and they are relatively weakly mixing extensions of $ \widetilde{T} $. [^16]
The situation changes if on the fibers, instead of $ {{\mathcal S}}$ weakly mixing, we put a distal flow. For example, if we apply the above to $
{\varphi}(x)=x-\frac12 $ and $ S_t=x+t $, we obtain a nilflow (the Heisenberg case), but because of restrictions on $ {{\mathcal S}}$ (which in this case has rational eigenvalues), we cannot apply [@Ku-Le1] directly. Consider $ S_tx=x+\beta t $ with $ \beta $ irrational. Here we obtain that $ {T_{{\varphi},{{\mathcal S}}}}$ has the AOP property. The natural question arises whether in case of $ \beta $ irrational, we also obtain a nilflow. The answer is negative as the following result shows.
\[p:negK\] If $ {\varphi}(x)=x-\frac12 $ and $ S^{(\beta)}_t(x):=x+t\beta
$ with $ \beta\notin{\mathbb{Q}}$ then $ \widetilde{T_{{\varphi},{{\mathcal S}}}}$ has singular spectrum.
We have $ \widetilde{T}_{\widetilde{{\varphi}},{{\mathcal S}}^{(\beta)}}=\widetilde
{T}_{\beta\widetilde{{\varphi}},{{\mathcal S}}^{(1)}} $. Then $$\widetilde{T}_{\beta\widetilde{{\varphi}},{{\mathcal S}}^{(1)}}=\widetilde{T_{\beta{\varphi},{{\mathcal S}}^
{(1)}}}.$$ We have to now argue that the maximal spectral type of $ T_{\beta{\varphi},{{\mathcal S}}^
{(1)}} $ is singular. In view of [@Le-Pa3], to compute the maximal spectral type of $ T_{\beta{\varphi},{{\mathcal S}}^
{(1)}} $, we need to calculate the maximal spectral types of weighted unitary operators given by the multiples $ r\beta{\varphi}$, $ r\in{{\mathbb{R}}}$, and then integrate this against the maximal spectral type of $
{{\mathcal S}}^{(1)} $. The latter is simply a purely atomic measure whose atoms are in $ {{\mathbb{Z}}}$. This means that we are interested only in the weighted operators given by $ m\beta{\varphi}$ and they are all with singular spectrum by [@Iw-Le-Ma].
Appendix {#appendix .unnumbered}
========
Cocycles and group extensions
=============================
Assume that $ T $ is an ergodic automorphism of a probability standard Borel space $ {(X,{\mathcal B},\mu)}$. Let $ K $ be a compact (metric) abelian group. Each Borel function $
{\varphi}:X\to K $ is called a [*cocycle*]{}. Actually, $ {\varphi}$ determines $
{\varphi}^{(\,\cdot\,)}(\,\cdot\,):{{\mathbb{Z}}}\times X\to K $: $$\label{defcoc} {\varphi}^{(n)}(x)=\left\{ \begin{array}{ccc}
{\varphi}(x)+{\varphi}(Tx)+\ldots+ {\varphi}(T^{n-1}x)&\mbox{if}&n\geq1\\
0&\mbox{if}&n=0\\
-({\varphi}(T^{n}x)+\ldots+{\varphi}(T^{-1}x))& \mbox{if}&n<0,
\end{array}
\right.$$ satisfying the cocycle identity $ {\varphi}^{(m+n)}(x)={\varphi}^{(m)}(x)+{\varphi}^{(n)}
(T^mx) $ for all $ m,n\in{{\mathbb{Z}}}$ and $ \mu $-a.e. $ x\in X $. Having $ T $ and $ {\varphi}$, we can define the corresponding [*group extension*]{} $ T_{\varphi}$ of $ T $ by setting: $$T_{\varphi}:X\times K\to X\times K,\,T_{\varphi}(x,k)=(Tx,{\varphi}(x)+k).$$ Clearly $ T_{\varphi}$ is an automorphism of $ (X\times K,{{\mathcal B}}{\otimes}{{\mathcal B}}(K),\mu{\otimes}{\lambda}_K)
$, where $ {\lambda}_K $ stands for Haar measure on $ K $. Then $ T_{\varphi}$ is ergodic if and only if the only measurable solutions $ \xi:X\to{\mathbb{S}}^1
$ of the equations $$\chi\circ {\varphi}=\xi\circ T/\xi,\,\chi\in\widehat{K}$$ (i.e. $ \chi\circ{\varphi}$ is a [*coboundary*]{}) exist when $ \xi $ is a constant function and $ \chi={\mathbbm{1}}$ [@An].
Let $ \sigma_k:X\times K\to X\times K $ be defined by $ \sigma_k(x,k')=
(x,k'+k) $. Then $ \sigma_k $ ($ k\in K $) is an automorphism of $ (X\times
K,{{\mathcal B}}{\otimes}{{\mathcal B}}(K),\mu{\otimes}{\lambda}_K) $ and $ \sigma_k$ is an element of the centralizer $ C({T_{{\varphi}}}) $. Then, by a slight abuse of notation, we have $ K\subset C({T_{{\varphi}}}) $ and $ K $ is a compact (abelian) subgroup in the weak topology. The reciprocal is also true.
\[p:grext\] Assume that $ {\overline}{T} $ is an ergodic automorphism of a probability standard Borel space $ ({\overline}{X},{\overline}{{{\mathcal B}}},{\overline}{\mu}) $. Assume that $ K\subset C({\overline}{T}) $ is a compact abelian subgroup (we assume that $ K $ acts freely on $ X $) of the centalizer. Let $${\overline}{{\mathcal A}}:=\{{\overline}{A}\in{\overline}{{{\mathcal B}}}:\:k{\overline}A={\overline}A\text{ for each } k\in K\}.$$ Then $ {\overline}{{{\mathcal A}}} $ is an $ {\overline}{T} $-invariant $ \sigma $-algebra and $$K=\{{\overline}R\in C({\overline}T):\:{\overline}R{\overline}A={\overline}A\text{ for each }{\overline}{A}\in{\overline}{{{\mathcal A}}}\}.$$ If $ T={\overline}{T}|_{{\overline}{{\mathcal A}}} $ is the factor automorphism acting on the factor space $${(X,{\mathcal B},\mu)}:= ({\overline}{X}/{\overline}{{{\mathcal A}}},{\overline}{{{\mathcal A}}},{\overline}{\mu}|_
{{\overline}{{\mathcal A}}})$$ then there is a cocycle $ {\varphi}:X\to K $ such that $ {\overline}{T}
$ is isomorphic to $ {T_{{\varphi}}}$ (with an isomorphism being the identity on $ {\overline}{{\mathcal A}}$).
We recall how to define $ {\varphi}$. Let $ \pi:{\overline}X\to X $ be the factor map. Then the fibers $ \pi^{-1}(x) $ are copies of $ K $. Let $ \xi:X\to {\overline}X $ be a measurable selector (for $ \pi $). Then for each $ {\overline}x\in\pi^{-1}(x) $ there is a unique $ k_{{\overline}x}\in K
$ such that $ k_{{\overline}x}{\overline}x=\xi(x) $. Note that then $$k_{{\overline}x}({\overline}T {\overline}x)={\overline}T(k_{{\overline}x}{\overline}x)={\overline}T\xi(x),$$ whence if we define $ {\varphi}(x)\in K $ as the only element of $ K $ such that $ {\varphi}(x)({\overline}T\xi(x))=\xi(Tx) $ then the map $ {\overline}x\mapsto
(\pi({\overline}x),k_{{\overline}x}) $ establishes an isomorphism between $ {\overline}T $ and $ T_{\varphi}$.
Invariant measures for group extensions
=======================================
If $ T_{\varphi}$ is ergodic, then $ \mu{\otimes}{\lambda}_K $ is the only $ {T_{{\varphi}}}$-invariant probability measure $ \kappa $ whose projection $ \left(\pi_X\right)_\ast
(\kappa) $ is $ \mu $ [@Fu]. This result has a refinement as follows (see, e.g. [@Ke-Ne], [@Le-Me]). If $ {\varphi}:X\to K $ is not ergodic then there is a (unique) closed subgroup $ K'\subset K $, a cocycle $ {\varphi}':X\to K' $ and a Borel map $ j:X\to K $ such that $$\label{essgp1} \mbox{$ T_{{\varphi}'} $ is ergodic as an automorphism of~$
\big(X\times K',{{\mathcal B}}{\otimes}{{\mathcal B}}(K'),\mu{\otimes}{\lambda}_{K'}\big) $}$$ and $$\label{essgp2} {\varphi}(x)={\varphi}'(x)+j(Tx)-j(x)$$ for a measurable $ j:X\to K $ (i.e. $ {\varphi}$ is [*cohomologous*]{} to a cocycle $ {\varphi}' $ which is taking values in a smaller closed subgroup $
K' $ and $ T_{{\varphi}'} $ is ergodic). The group $ K' $ is called the [*group of essential values*]{} of $ {\varphi}$ and is denoted by $ E({\varphi}) $. Moreover, $$\label{essgp3} E({\varphi}')=E({\varphi})=\Lambda({\varphi})^\perp,$$ where $$\Lambda({\varphi}):=\big\{\chi\in \widehat{K}:\: \chi\circ{\varphi}\text{ is a
coboundary}\big\}.$$ In particular, $$\label{essgp4} \mbox{$ {T_{{\varphi}}}$ is ergodic if and only if~$ E({\varphi})=K $.}$$
We intend to describe the set $ {{\mathcal M}}(X\times K,{T_{{\varphi}}};\mu) $ of all $ {T_{{\varphi}}}$-invariant probability measures $ \kappa $ on $ X\times K $ that $ \left(\pi_X\right)_\ast
(\kappa)=\mu $. It is again a simplex (with its natural affine structure). The set of extremal points is equal to $ {{\mathcal M}}^e(X\times K,{T_{{\varphi}}};\mu)
$ of ergodic members of $ {{\mathcal M}}(X\times K,{T_{{\varphi}}};\mu) $. We have the following.
\[p:transl1\]
If $ \kappa\in {{\mathcal M}}^e(X\times K,{T_{{\varphi}}};\mu) $ then $
\left(\sigma_{k}\right)_\ast(\kappa)\in {{\mathcal M}}^e(X\times K,{T_{{\varphi}}};\mu) $ for each $ k\in K $. Moreover, if $ \kappa,\kappa'\in {{\mathcal M}}^e(X\times
K,{T_{{\varphi}}};\mu) $ then $ \left(\sigma_{k_0}\right)_\ast(\kappa)=\kappa' $ for some $ k_0\in K $.
It follows from Proposition \[p:transl1\] that to describe all ergodic members of $ {{\mathcal M}}(X\times
K,{T_{{\varphi}}};\mu) $, we need to describe just one. Let $ \kappa\in {{\mathcal M}}^e(X\times
K,{T_{{\varphi}}};\mu) $. Set
$$\operatorname{stab}(\kappa):=\{k\in K:\:\left(\sigma_k\right)_\ast(\kappa)=\kappa\}.$$
\[l:leme\] For each $ \kappa\in {{\mathcal M}}^e(X\times K,{T_{{\varphi}}};\mu) $, we have $ \operatorname{stab}(\kappa)=E({\varphi}) $.
By , $$\label{essgp33}{{\mathcal M}}\big(X\times E({\varphi}),T_{{\varphi}'};\mu\big)={{\mathcal M}}^{e}\big(X\times E({\varphi}),T_
{{\varphi}'};\mu\big)=\{\mu{\otimes}{\lambda}_{E({\varphi})}\}.$$ Moreover, by , the map $ \Theta:X\times E({\varphi})\to X\times
K $ given by $$\Theta(x,k')=(x,k'+j(x))$$ is equivariant, i.e. $ \Theta\circ T_{{\varphi}'}={T_{{\varphi}}}\circ \Theta $. It follows that $ \kappa:=\Theta_\ast(\mu{\otimes}{\lambda}_{E({\varphi})})\in {{\mathcal M}}^{e}(X\times
K,{T_{{\varphi}}};\mu) $. Finally, notice that if $ j:X\to K $ satisfies then, for each $ k\in K $, also $ j_k(x):=j(x)+k $ satisfies (moreover, each Borel $ {\overline}{j}:X\to K $ satisfying is of the form $ j_k $). Taking into account this and Proposition \[p:transl1\], we obtain the following.
\[p:leme\] If $ \kappa\in {{\mathcal M}}^e(X\times K,{T_{{\varphi}}};\mu) $ then there exists a Borel $ j:X\to K $ satisfying for which $$\kappa=\int_X \left(\sigma_{j(x)}\right)_\ast(\delta_x{\otimes}{\lambda}_{E({\varphi})})\,d\mu
(x).$$
If we take $ f\in L^2{(X,{\mathcal B},\mu)}$ and a character $ \chi\in\widehat{K} $ then, once $ \chi $ is non-trivial, the tensor $ f{\otimes}\chi $ has the zero mean for $ \mu{\otimes}{\lambda}_K $. We will now show that the zero mean phenomenon holds for many characters and ergodic members of $ {{\mathcal M}}(X\times
K,{T_{{\varphi}}};\mu) $.
\[c:leme\] Assume that $ f\in L^2{(X,{\mathcal B},\mu)}$. If $ \chi\notin\Lambda
({\varphi}) $ then $ \int_{X\times K}f{\otimes}\chi\,d\kappa=0 $ for each $
\kappa\in {{\mathcal M}}^e(X\times K,{T_{{\varphi}}};\mu) $.
By , $ \chi\notin\Lambda({\varphi}) $ if and only if $ \chi\notin
E({\varphi})^\perp $ and the latter is equivalent to saying that $ \chi|_{E
({\varphi})}\neq{\mathbbm{1}}$. By Proposition \[p:leme\], $$\label{fub}\kappa=\int_X \left(\sigma_{j(x)}\right)_\ast(\delta_x{\otimes}{\lambda}_
{E({\varphi})})\,d\mu(x).$$ But if $ \chi|_{E({\varphi})}\neq{\mathbbm{1}}$ then $ \int_{K}\chi\,d{\lambda}_{E({\varphi})}=0
$ and the same integral vanishes if we integrate against an arbitrary shift of $ {\lambda}_{E({\varphi})} $. The result now follows from .
Assume now that $ S $ is an ergodic automorphism of a probability standard Borel space $ {(Y,{\mathcal C},\nu)}$ and let $ \psi:Y\to L $ be a cocycle, where $ L $ is a compact (metric) abelian group. Assume that $ {T_{{\varphi}}}$ and $ S_\psi $ are ergodic and let $ \widetilde{\rho}\in J^e({T_{{\varphi}}},S_\psi)
$. Denote $ \rho=\widetilde{\rho}|_{X\times Y} $. Then $ \rho\in J^e(T,S)
$. We also have $$\widetilde{\rho}\in {{\mathcal M}}^e\big((X\times Y)\times (K\times L), (T\times S)_
{{\varphi}\times\psi};\rho\big),$$ where $ {\varphi}\times\psi:(X\times Y,\rho)\to K\times L $: $$({\varphi}\times\psi)(x,y)=({\varphi}(x),\psi(y))\text{ for }\rho-\text{a.e. }(x,y)\in
X\times Y.$$ In other words, $$\label{jge} J^e({T_{{\varphi}}},S_\psi)\subset {{\mathcal M}}^e\big((X\times Y)\times (K\times
L), (T\times S)_{{\varphi}\times\psi};\rho\big).$$
We can now take up a more general problem and study the form of $${\overline}{\rho}\in {{\mathcal M}}^e\big(((X\times Y)\times (K\times L), (T\times S)_{{\varphi}\times\psi};\rho\big)$$ (with $ \rho={\overline}\rho|_{X\times Y} $). This problem is equivalent to investigating the invariant measures for the group extensions $ (T\times S,\rho)_
{{\varphi}\times\psi} $ with $ \rho\in J^e(T,S) $. By denoting $ E_\rho ({\varphi}\times\psi)
$ the corresponding group of essential values, the problem is reduced to study the form of such groups.
\[l:rzut1\] If $ \rho\in J^e(T,S) $ and the cocycles $ {\varphi}$ and $ \psi $ considered over $ (T\times S,\rho) $ are ergodic then $
\pi_{K}(E_\rho({\varphi}\times\psi))=K $ and $ \pi_{L}(E_\rho({\varphi}\times\psi))=L
$.
It is not hard to see that $ E(\pi_K\circ ({\varphi}\times\psi))=\pi_K(E_\rho
({\varphi}\times\psi)) $ and since $ \pi_K\circ ({\varphi}\times\psi)={\varphi}$ is ergodic over $ (T\times S,\rho) $ by assumption, the result follows from .
\[r:uwdonil\] Although the problem of describing all elements of the set $ \big\{E_\rho({\varphi}\times \psi):\:\rho\in J^e(T,S)\big\} $ looks as a problem more general than the problem of description of the members of $ J^e({T_{{\varphi}}},S_\psi) $, in fact, quite often they are the same. We will see the equality of the two sets in when we study nil-cocycles in forthcoming sections.
A criterion to lift AOP to a group extension
============================================
Assume again that $ T $ is a totally ergodic automorphism of a probability standard Borel space $ {(X,{\mathcal B},\mu)}$. Let $ K $ be a compact (metric) abelian group and let $ {\varphi}:X\to K $ be a cocycle. We assume also that $ {T_{{\varphi}}}$ is also totally ergodic. [^17] We want to study the AOP property for $ {T_{{\varphi}}}$ (assuming that $ T $ enjoys it). To study this property, we will have to consider the set $ J^e\left(({T_{{\varphi}}})^r,({T_{{\varphi}}})^s\right) $ for $
r,s $ coprime. Note that $ ({T_{{\varphi}}})^r=(T^r)_{{\varphi}^{(r)}} $, so we are in the framework of the previous subsection in which $ T $, $
{\varphi}$, $ S $, $ \psi $ are $ T^r $, $ {\varphi}^{(r)} $, $ T^s $ and $ {\varphi}^{(s)} $, respectively.
Assume that $ K={{\mathbb{T}}}^d $ and identify characters of $ K $ with $ {{\mathbb{Z}}}^d $: given $ {\overline}m\in{{\mathbb{Z}}}^d $, we set $ \chi_{{\overline}m}({\overline}x)=e^{2\pi i\langle
{\overline}m,{\overline}x\rangle} $. Given $ k\geq1 $ and $ r,s $ coprime, let $$A_{k,r,s}:=\big\{({\overline}m,{\overline}n)\in{{\mathbb{Z}}}^d\times{{\mathbb{Z}}}^d:\:s^k{\overline}m=r^k{\overline}n\big\}\subset
\widehat{K}\times\widehat{K}.$$
\[p:critAOP\] Let $ T $ be a totally ergodic automorphism of a probability standard Borel space $ {(X,{\mathcal B},\mu)}$. Let $ {\varphi}:X\to {{\mathbb{T}}}^d $ be a cocycle so that $ {T_{{\varphi}}}$ is totally ergodic. Assume that for some $ k\geq1 $ and for all $ r,s $ coprime, we have $$\label{zawier} \Lambda_{({\varphi}^{(r)}\times{\varphi}^{(s)},\rho)}\subset A_
{k,r,s}\text{ for each }\rho\in J^e(T^r,T^s).$$ If $ T $ enjoys the AOP property, so does $ {T_{{\varphi}}}$.
Fix $ f,g\in L^2{(X,{\mathcal B},\mu)}$ and $ {\overline}m_0,{\overline}n_0\in{{\mathbb{Z}}}^d $ such that at least one is no-zero. It is enough to show that for all $ r,s $ coprime and sufficiently large, we have $$\label{pao1} \int f{\otimes}\chi_{{\overline}m_0}{\otimes}g{\otimes}\chi_{{\overline}n_0}\,d\widetilde
{\rho}=0\text{ for each }\widetilde{\rho}\in J^e\left({T_{{\varphi}}})^r,({T_{{\varphi}}})^s\right),$$ cf. Remark \[r:weaktop\]. By , it is sufficient that holds for each $ \widetilde{\rho}\in{{\mathcal M}}^e\big((X\times X)\times({{\mathbb{T}}}^d\times{{\mathbb{T}}}^d),
(T^r\times T^s)_{{\varphi}^{(r)}\times{\varphi}^{(s)}};\rho\big) $. By Corollary \[c:leme\], we only need to show that for all $ r,s $ coprime and sufficiently large, $ ({\overline}m_0,{\overline}n_0)\notin\Lambda_{{\varphi}^{(r)}\times{\varphi}^
{(s)}} $ for all $ \rho\in J^e(T^r,T^s) $. Hence, by assumption, it is sufficient to show that $ ({\overline}m_0,{\overline}n_0)\notin A_{k,r,s} $ for all $ r,s $ coprime and sufficiently large. This is however clear, if $ ({\overline}m_0,{\overline}n_0)\in A_{k,r,s} $ then $ s^k{\overline}m_0=r^k{\overline}n_0 $, whence the coordinates of $ {\overline}m_0 $ must be multiples of $
r^k $ and the coordinates of $ {\overline}n_0 $ must be multiples of $ s^k $ and this can hold only for finitely many $ r,s $. The result follows.
Since $ (r,s)=1 $, we have $ A_{k,r,s}=\{(r^k{\overline}j,s^k {\overline}j):\:{\overline}j\in{{\mathbb{Z}}}^d\}
$. Then, it follows that $$A_{k,r,s}^\perp=\big\{({\overline}x,{\overline}y)\in{{\mathbb{T}}}^d\times {{\mathbb{T}}}^d: r^k{\overline}x+s^k{\overline}y={\overline}0\big\}.$$
Using Proposition \[p:critAOP\], Lemma \[l:leme\] and , we obtain the following.
\[c:critAOP\] Let $ T $ be a totally ergodic automorphism of a probability standard Borel space $ {(X,{\mathcal B},\mu)}$. Let $ {\varphi}:X\to {{\mathbb{T}}}^d $ be a cocycle so that $ {T_{{\varphi}}}$ is totally ergodic. Assume that for some $ k\geq1 $ and for all $ r,s $ coprime, we have $$\label{zawier10} \operatorname{stab}(\widetilde{\rho})\supset A^\perp_{k,r,s}$$
for some $ \widetilde{\rho}\in{{\mathcal M}}^e\big((X\times X)\times({{\mathbb{T}}}^d\times{{\mathbb{T}}}^d),
(T^r\times T^s)_{{\varphi}^{(r)}\times{\varphi}^{(s)}};\rho\big) $ and for every $ \rho\in
J^e(T^r,T^s) $. If $ T $ enjoys the AOP property, so does $ {T_{{\varphi}}}$.
Some lemmata on nilpotent Lie algebras
======================================
In the sequel $ \g $ is a $ k $-step nilpotent Lie algebra and $$\mathfrak g=\mathfrak g^{(1)}\supset\mathfrak g^{(2)}\supset\dots
\supset \mathfrak g^{(k)}$$ its descending central series. By Lemma 1.1.1 in [@Co-Gr], $$[\g^{(i)},\g^{(j)}]\subset \g^{(i+j)}\quad\text{for all}\ i,j\geq 1.$$
\[lem:aop\_nil:2\] A set $ S=\{X_1, \dots , X_j\}\subset \g $ is a minimal set of generators for a nilpotent Lie algebra $ \mathfrak
g $ if and only if $ S[\mathfrak g,\mathfrak g] $ is a basis of the vector space $ \mathfrak g/[\mathfrak g,\mathfrak g] $.
The proof is by induction on $ k $. The statement is true for $ \g $ abelian ($ k=1 $). Suppose the statement true for all nilpotent Lie algebras of class of nilpotency $ \ell \le k $ and assume $ \g $ nilpotent of class of nilpotency $ k +1 $.
Let $ S\subset \g $ be a set such that $ S[\mathfrak g,\mathfrak g] $ is a basis of the vector space $ \mathfrak g/[\mathfrak g,\mathfrak
g] $. Then $ \g={{\langle}}S{{\rangle}}+ [\g,\g] $, where $ {{\langle}}S{{\rangle}}$ denotes the linear span of $ S $. It follows that $ \g^{(k+1)}= [\g, \g^ {(k)}]=
[{{\langle}}S{{\rangle}}, \g^{(k)}] $ as $ [[\g,\g],\g^{(k)}]\subset \g^{(k+2)}=0 $. Furthermore, the set $$(S\g^{(k+1)})\big[(\mathfrak g/\g^{(k+1)}),(\mathfrak g/\g^{ (k+1)})\big] ]\approx S[\g,\g]$$ is a basis of the vector space $ (\mathfrak
g/\g^{(k+1)})/\big[(\mathfrak g/\g^{(k+1)}),(\mathfrak g/\g^{ (k+1)})\big]\approx \g/[\g,\g] $. Since $ \g/\g^{(k+1)} $ is nilpotent of class of nilpotency $ k $, the induction hypothesis applies to the Lie algebra $ \g/\g^{(k+1)} $ and its subset $ S\g^ {(k+1)} $, so $ S\g^{(k+1)} $ is a minimal set of generators of $ \g/\g^{(k+1)}
$. Thus the Lie subalgebra $ \g_S $ generated by $ S $ projects onto $ \g/\g^{(k+1)} $ under the quotient mapping $ \g \mapsto \g/\g^
{(k+1)} $. Thus it suffices to show that $ S $ generates a set spanning $ \g^{(k+1)} $. Let $ T\subset \g_S $ be a finite set projecting to a spanning set of $ \g^{(k)}/\g^{(k+1)} $. Then $ \g^
{(k)}={{\langle}}T{{\rangle}}+\g^{(k+1)} $. By definition $ [{{\langle}}S{{\rangle}},{{\langle}}T{{\rangle}}]\subset \g_S
$. It follows that $$\g^{(k+1)}= [{{\langle}}S{{\rangle}}, \g^{(k)}]= [ {{\langle}}S{{\rangle}}, {{\langle}}T{{\rangle}}+\g^{(k+1)}]\subset [{{\langle}}S{{\rangle}},
{{\langle}}T{{\rangle}}] + \g^{(k+2)}= [{{\langle}}S{{\rangle}}, {{\langle}}T{{\rangle}}]\subset \g_S.$$
Finally observe that $ S $ is minimal, since any proper subset of $
S $ does not project to a basis of $ \g/[\g,\g] $, i.e. does not generate $ \g/[\g,\g] $. A fortiori it does not generate $ \g $.
Assume $ S $ is a minimal set of generators of $ \g $. Then $ S $ projects to a generating set of $ \g/[\g,\g] $, that is a finite spanning set for the vector space $ \g/[\g,\g] $. Let $ S_1\subset
S $ be a subset projecting to a basis of $ \g/[\g,\g] $. By the above $ S_1 $ generates $ \g $. By the minimality of $ S $ we have $ S=S_1 $, concluding the proof.
Let $ S=\{X_1,\dots, X_j\} $ be a minimal generating set for the $k$-step nilpotent Lie algebra $ \g $. For every $ (i_1,\dots,i_k)\in \{1,\dots,j\}^k
$ define the Lie $ k $-fold product $$S_{i_1,i_2,\dots,i_k}:=[X_{i_1},[X_{i_2}, \dots, [X_{i_{k-1}},X_{i_k}]\dots]]$$ and let $ V_k(S)\subset\g^{(k)} $ be the linear span of the set of $ k
$-fold products.
\[lem:aop\_nil:3\] Let $ \g $ be a $ k $-step nilpotent Lie algebra and $ S $ a minimal generating set for $ \g $. Then $ \g^{(k)}=
V_k(S) $.
The proof is by induction on $ k $. When $ \g $ is abelian the statement is obvious. Suppose that the Lemma is true for all nilpotent Lie algebras of class $ \ell <k $ and let $ \g $ be of class $ k $ and $ S $ be a minimal generating set for $ \g $. By the previous Lemma, the projection $ \bar S=\{\bar X_1,\dots, \bar X_j\}
$ of $ S $ into $ \g/\g^{(k)} $ is a set of generators for $ \g/\g^{ (k)} $. Then, by the induction hypothesis, the set of $ (k-1) $-fold products of the elements $ \bar S $ span $ \g^{(k-1)}/\g^{(k)} $. It follows that $ \g^{(k-1)}= \g^{(k)} + V_{k-1}(S) $. Since $ V_
{k-1}(S)\subset\g^{(k-1)} $, this gives $$\g^{(k)}=[\g,\g^{(k-1)}]= [{{\langle}}S{{\rangle}}+\g^{(2)}, \g^{(k)} + V_{k-1}(S) ]=
[{{\langle}}S{{\rangle}}, V_{k-1}(S) ]$$ The proof is concluded by the observations that $ [{{\langle}}S{{\rangle}}, V_{k-1}(S)
] \subset V_{k}(S) $ and that, as $ \g $ is of class $ k $, the opposite inclusion is true by definition of $ \g^{(k)} $.
\[lem:aop\_nil:4\] Let $ S=\{X_1,\dots, X_j\} $ and $ S'=\{X'_1,\dots,
X'_j\} $ be subsets of a nilpotent $ k $-step Lie algebra $ \g $ such that $ X_i=X'_i\mod [\g,\g] $ for all $ i=1,\dots,j $. Then for every $ (i_1,\dots,i_k)\in \{1,\dots,j\}^k $ we have $$S_{i_1,i_2,\dots,i_k}= S'_{i_1,i_2,\dots,i_k}$$
Again the proof is by induction on the class of nilpotency. The statement being trivially true in the abelian case, we assume that the Lemma is true for all nilpotent Lie algebras of class $ \ell <k $. Let $ \g $ be of class $ k $. Let $ X_i'=X_i+Y_i $ with $ Y_i\in [\g,\g]
$, for all $ i=1,\dots,j $. By the induction hypothesis, applied to the algebra $ \g/\g^{(k)} $ for all $ (i_2,\dots,i_k)\in \{1,\dots,j\}^
{k-1} $ , the elements $ S_{i_2,\dots,i_k} $ and $ S'_{i_2,\dots,i_k}
$ coincide modulo $ \g^{(k)} $, i.e. differ by an element $ \alpha_{i_2,\dots,i_k}\in
\g^{(k)} $. Thus we have $$\begin{aligned}
S'_{i_1, i_2,\dots,i_k}&= [X'_{i_1}, S'_{i_2,\dots,i_k}] = [X_{i_1}+
Y_{i_1}, S_{i_2,\dots,i_k} + \alpha_{i_2,\dots,i_k}]\\
& = [X_{i_1}, S_{i_2,\dots,i_k} ]=S_{i_1, i_2,\dots,i_k}
\end{aligned}$$ concluding the proof.
[^1]: L. Flaminio supported in part by the Labex CEMPI (ANR-11-LABX-07)
[^2]: M. Lemańczyk supported by Narodowe Centrum Nauki grant 2014/15/B/ST1/03736
[^3]: This criterion (in the form used in [@Ab-Le-Ru]) says that if $ (a_n)\subset\C $ is bounded and $ \limsup_{p\neq q\to\infty, p,q\in\mathscr{P}}\limsup_{N\to
\infty}\left|\frac1N\sum_ {n\leq N}a_{pn}{\overline}{a}_{q_n}\right|=0 $ then $ \frac1N\sum_{n\leq N}a_n{\boldsymbol{u}}(n)\to 0 $ for each multiplicative $ {\boldsymbol{u}}:{{\mathbb{N}}}\to\C $, $ |{\boldsymbol{u}}|\leq1 $.
We use this criterion by considering $ (a_n) $ given by $ (f(x_1),\ldots,f (T^{b_1-1}x_1),f(T^{b_1}x_2),\ldots) $.
[^4]: A homeomorphism $ T $ is distal if the orbit $ (T^nx,T^ny) $, $ n\in{{\mathbb{N}}}$, is bounded away from the diagonal in $ X\times X $ for each $ x\neq y $.
[^5]: $G/\Gamma$ need not be connected and $l_u$ need not be ergodic.
[^6]: Recall that, up to an abuse of notation, $ \int_Xf\,d\mu_{z_1}={{\mathbb{E}}}(f|\pi_{X,Z}^{-1}({{\mathcal D}}))(z_1) $ and similarly $ \int_Y g\,d\nu_{z_2}={{\mathbb{E}}}(g|\pi^{-1}_{Y,Z}({{\mathcal D}}))(z_2) $.
[^7]: In fact, from , $(\exp t B)\cdot \exp X\cdot (\exp -t B) = (\exp t B) (X)$ for every $t\in
{{\mathbb{R}}}$ and every $X\in \g$.
[^8]: Remark that $\widetilde \Sigma_{r,s}$ is not a joining of $\widetilde \Sigma_{r}$ and $\widetilde \Sigma_{s}$, since $\widetilde G_{r,s}/\widetilde \Gamma_{r,s} \not= \widetilde G_{r}/\widetilde \Gamma_{r} \times \widetilde
G_{s}/\widetilde \Gamma_{s}$. In fact $\dim \widetilde G_{r,s} = 2 \dim G +1 $ and $\dim \widetilde G_{r}=\dim \widetilde G_{s}= \dim G +1$; thus $\dim \widetilde G_{r,s} \not= \dim \widetilde G_{r}+\dim \widetilde G_{s}$.
[^9]: The action of $ \tau $ on $ G/H $ is uniquely ergodic and so is the action of $ \tau^{(p)}\times\tau^{(q)} $ on $ {{\mathcal H}}_c $.
[^10]: $ J(h_1,h_2)=(h_1^qh_2^{-p},h_1^ah_2^b) $. The inverse of it is given by $ \left[
\begin{array}{cc}
b & p\\
-a& q
\end{array}
\right] $.
[^11]: Indeed, we have $$\left(\tau_{\Psi_2}\right)_{hg}(H,e)=\left(\tau_{\Psi_2}\right)_h
\left(\tau_{\Psi_2}\right)_g(H,e)=
\left(\tau_{\Psi_2}\right)_h(gH,\Psi_2(g,H))=(gH,h\Psi_2(g,H)).$$
[^12]: The map $ (xH,(h_1,h_2))\mapsto (xH,h_1^ah_2^b) $ settles an isomorphism of $ \tau_{(\Psi_2^p,\Psi_2^q)} $ with $ \tau_{\Psi_2}
$.
[^13]: This should be compared with Remark \[r:in00\] (iv).
[^14]: This is of course a $ {{\mathbb{Z}}}$-action.
[^15]: The factor map is given by $ (x,s)\mapsto s $.
[^16]: Note that this is completely different case than nilflows which are distal extensions of of linear flows.
[^17]: This means that $ {T_{{\varphi}}}$ has no root of unity in its spectrum. This is equivalent to saying that for no $ n\geq2 $, no character $ \chi\in\widehat{K} $, we can solve the functional equation $ \chi\circ {\varphi}=e^{2\pi i/n}\xi\circ T/\xi $ for a measurable $ \xi:X\to{\mathbb{S}}^1 $ [@An].
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In a recent Letter [@Johnson; @and; @Quiroga], Johnson and Quiroga have obtained several interesting exact results for electrons with $1/r^2$ interactions in a two-dimensional quantum dot. A parabolic confining potential of the form ${\textstyle{1\over 2}} m \omega_0^2 r^2$ is assumed, and the system is subjected to a uniform perpendicular magnetic field of strength $B$. In particular, they have shown that there exists a collective “breathing” mode excitation with frequency $$\omega = 2 \Omega ,
\label{spectrum}$$ where $\Omega^2 \equiv \omega_0^2 + {\textstyle{1 \over 4}} \omega_{\rm c}^2 $, $\omega_{\rm c} \equiv e B /m c$ is the cyclotron frequency, and $m$ is the effective mass. The exact spectrum of the interacting electron system therefore contains an infinite ladder of energy levels at integer multiples of $2 \hbar \Omega$.
The purpose of this Comment is to point out that the quantum breathing mode excitation spectrum can also be obtained directly from the property of the many-particle Hamiltonian under a certain scale transformation, and in a manner that makes evident the special property of the inverse-square-law interaction for the quantum breathing mode. We shall work in the symmetric gauge and write the Hamiltonian as $ H = T + {\textstyle{1\over 2}} \omega_{\rm c} L_z + V + U, $ where $T \equiv \sum_n { p_n^2 / 2 m}$, with ${\bf p}_n$ the canonical momentum, $L_z \equiv \sum_n ({\bf r}_n \times {\bf p}_n) \cdot \hat z$ is the $z$ component of the canonical angular momentum, $ V \equiv \sum_n {\textstyle{1 \over 2}} m \Omega^2 r_n^2 $ is the effective field-dependent parabolic confining potential, and $$U \equiv \sum_{n<n'} {g \over | {\bf r}_n - {\bf r}_{n'} |^\alpha}$$ is any [*power-law*]{} electron-electron interaction.
We first note the properties of $H$ under a scale transformation $ O \rightarrow e^{i \lambda S} O e^{-i \lambda S}$ generated by $$S \equiv {1 \over 2} \sum_n ( {\bf r}_n \cdot {\bf p}_n
+ {\bf p}_n \cdot {\bf r}_n ).$$ This transformation performs a radial displacement of each coordinate by an amount proportional to its distance from the origin; that is, it generates a “breathing” motion. Under this transformation, $T \rightarrow T - 2 \lambda T$, $L_z \rightarrow L_z$, $V \rightarrow V + 2 \lambda V$, and $U \rightarrow U - \alpha \lambda U$, to first order in $\lambda$. Therefore, $$H \rightarrow
H + i \lambda [S,H]
= H - 2 \lambda T + 2 \lambda V - \alpha \lambda U .
\label{scaling of H}$$ Equation (\[scaling of H\]), in turn, may be regarded as an equation of motion for $S$. In fact, noting that $dV/dt = \Omega^2 S $, we obtain the operator equation of motion $${d^2 V\over d t^2} + (2+\alpha) \Omega^2 V =
\alpha \Omega^2 (H- {\textstyle{1\over 2}} \omega_{\rm c} L_z )
+ (2 - \alpha) \Omega^2 T,
\label{equation of motion}$$ which is the same as one would obtain classically.
The breathing mode of the corresponding classical system of point charges may be obtained from (\[equation of motion\]) by considering small oscillations about an equilibrium configuration, where the velocities are zero. Because $H$ and $L_z$ are constants of the motion, whereas the physical kinetic energy $T + {1\over 2} \omega_c L_z
+ \omega_c^2 V / 4 \Omega^2$ is zero to first order in the displacements, the [*classical*]{} breathing mode frequency is generally $\omega = \sqrt{(2 + \alpha) \omega_0^2 + \omega_c^2}.$ For example, the classical breathing frequency of electrons with Coulomb interactions ($\alpha = 1$) in a parabolic dot with no magnetic field is $\sqrt{3} \ \! \omega_0$, a result first obtained by Schweigert and Peeters [@Peeters].
Quantum zero-point motion, however, generally modifies the breathing mode and the other classical normal modes, by shifting their frequencies and by giving them a finite lifetime of the order of $a/R$, with $a$ denoting the Bohr radius and $R$ the radius of the droplet of charge in the dot. An exception occurs when $\alpha = 2$: In this case $V$ becomes an [*exact*]{} quantum collective coordinate with frequency (\[spectrum\]), independent of $g$ and $N$. The collective coordinate $V$ may be separated into a center-of-mass and relative-coordinate part, $V = V_{\rm cm} + V_{\rm rel}$. For $\alpha = 2$, it can be shown that each component separately satisifes a harmonic oscillator equation of motion of the form (\[equation of motion\]) with frequency $2 \Omega$. $V_{\rm rel}$ is the collective coordinate corresponding to the breathing mode discovered by Johnson and Quiroga [@Johnson; @and; @Quiroga].
This work was supported by the NSF Grants No. DMR-9403908 and DMR-9416906. We gratefully acknowledge the hospitality of the Condensed Matter Theory Group at Indiana University, where this work was initiated, and we thank Allan MacDonald for stimulating discussions, and Francois Peeters for first drawing our attention to the breathing mode in classical systems.
N. F. Johnson and L. Quiroga, Phys. Rev. Lett. [**74**]{}, 4277 (1995).
V. A. Schweigert and F. M. Peeters, Phys. Rev. B [**51**]{}, 7700 (1995); F. M. Peeters, V. A. Schweigert and V. M. Bedanov, Physica B [**212**]{}, 710 (1995).
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---
abstract: 'In this paper, we consider a two-hop amplify-and-forward (AF) relaying system, where the relay node is energy-constrained and harvests energy from the source node. In the literature, there are three main energy-harvesting (EH) protocols, namely, time-switching relaying (TSR), power-splitting (PS) relaying (PSR) and ideal relaying receiver (IRR). Unlike the existing studies, in this paper we consider $\alpha$-$\mu$ fading channels. In this respect, we derive accurate unified analytical expressions for the ergodic capacity for the aforementioned protocols over independent but not identically distributed (i.n.i.d) $\alpha$-$\mu$ fading channels. Three special cases of the $\alpha$-$\mu$ model, namely, Rayleigh, Nakagami-$m$ and Weibull fading channels were investigated. Our analysis is verified through numerical and simulation results. It is shown that finding the optimal value of the PS factor for the PSR protocol and the EH time fraction for the TSR protocol is a crucial step in achieving the best network performance.'
author:
- |
$\textrm{Galymzhan~Nauryzbayev}$, $\textrm{Khaled~M.~Rabie}^{\ddagger}$, $\textrm{Mohamed~Abdallah}^{*}$\
and $\textrm{Bamidele~Adebisi}^{\ddagger}$\
title: 'Ergodic Capacity Analysis of Wireless Powered AF Relaying Systems over $\alpha$-$\mu$ Fading Channels'
---
Ergodic capacity (EC), amplify-and-forward (AF) relaying, energy-harvesting (EH), $\alpha$-$\mu$ fading, wireless power transfer.
Introduction
============
power transfer has attracted significant research interest as a potential technology to prolong the life-time of wireless battery-powered devices [@K1][@GS2]. Exploiting radio-frequency (RF) signals for simultaneous delivery of information and power promises to be one of the main efficient techniques for wireless energy-harvesting (EH). In the literature, there are three known architectures for simultaneous wireless information and power transfer (SWIPT), namely, power-splitting (PS), time-switching (TS) and ideal relaying protocols [@L4].
Over the past few years, the performance of two-hop SWIPT systems has been widely analyzed. In two-hop SWIPT systems, the relay node scavenges the energy from the received RF signal which is then utilized to forward the source information to the destination node. The authors in [@L4], for instance, analyzed the performance of two-hop amplify-and-forward (AF) systems in Rayleigh fading channels. The analysis considered two relaying protocols, namely, time-switching relaying (TSR) and power-splitting relaying (PSR) protocols. In [@L5], the authors evaluated the outage probability for a two-hop decode-and-forward (DF) underlay cooperative cognitive network deploying the TSR and PSR protocols, again, in Rayleigh fading channels. Furthermore, accurate analytical expressions of the ergodic capacity (EC) and achievable throughput of a DF relaying system deploying the TSR and PSR protocols over Rayleigh fading channels were derived in [@L6]. The outage performance in two-hop AF and DF over log-normal fading channels is studied in [@khaled1] and [@L8], respectively. In addition, the authors in [@L4] and [@khaled2] focused on AF relaying systems with EH constraints for an ideal relaying receiver (IRR) protocol.
To the best of our knowledge, the performance of wireless EH two-hop AF relaying networks over $\alpha$-$\mu$ fading channels has not been evaluated before in the literature. Therefore, we derive new expressions for the EC in independent and not necessarily identically distributed (i.n.i.d.) $\alpha$-$\mu$ fading channels. It is worth to note that the $\alpha$-$\mu$ distribution can be regarded as a generalized model covering small-scale fading channels, namely, Rayleigh, Nakagami-$m$, Weibull, etc. The derived EC expression is unified in the sense that it considers three different relaying protocols, namely, TSR, PSR and IRR protocols. The obtained exact analytical expressions explain the behavior of these protocols under various parameters constituting various special cases of the $\alpha$-$\mu$ model, namely, Rayleigh, Nakagami-$m$ and Weibull fading channels. Our analysis is also validated by Monte Carlo simulations. Results reveal that the optimization of the EH TS and PS factors in the corresponding TSR and PSR protocols maximizes the achievable EC. Moreover, the IRR protocol achieves the best performance and the optimized PSR-based approach always outperforms the optimized TSR scheme.
The remainder of this paper is organized as follows. Section II defines the system model and its performance metric used in this paper. Sections III, IV and V derive new analytical expressions for the EC over i.n.i.d $\alpha$-$\mu$ fading channels for the TSR, PSR and IRR protocols, respectively. Section VI presents a discussion of numerical and simulation results. Finally, Section VII concludes the paper.
System and Channel Model
========================
Consider a two-hop AF wireless communication system as shown in Fig. 1. In this system, we assume that a source node $(S)$ transmits information to a destination node $(D)$ via energy-limited relay node $(R)$ operating in the AF mode and no direct link exists between $S$ and $D$. The relay node amplifies and then forwards the received signal to the destination. All nodes are equipped with a single antenna and operate in half-duplex mode. We assume that the relay has no external power supply (i.e., powered by energy-harvesting from source-transmitted signal). We also assume that the amount of power consumed by the relay during data processing is negligible. The source-to-relay ($S$-$to$-$R$) and relay-to-destination ($R$-$to$-$D$) links given by $h_1$ and $h_2$ are subject to quasi-static i.n.i.d. $\alpha$-$\mu$ fading. The $S$-$to$-$R$ and $R$-$to$-$D$ distances are denoted by $d_1$ and $d_2$, respectively; the corresponding path-loss exponents are given by $m_1$ and $m_2$. It is also assumed that the fading coefficients remain constant during a transmission block time $T$ but vary independently from one block to another.
Since we assumed that the channel envelope $r$ follows the $\alpha$-$\mu$ distribution, the probability density function (PDF) of the $i-$th hop is given by [@galym] $$\label{pdf_h_i}
f_{h_i}(r) = \frac{\alpha_i \mu_i^{\mu_i} r^{\alpha_i \mu_i - 1}}{\hat{r}^{\alpha_i\mu_i} \Gamma(\mu_i)} \exp\left( -\frac{\mu_i}{\hat{r}^{\alpha_i}} r^{\alpha_i} \right),$$ where $\alpha_i > 0$ is an arbitrary parameter, $\hat{r}$ indicates the $\alpha-$root mean value given by $\hat{r} = \sqrt[\alpha]{\mathbb{E}\left[r^{\alpha_i}\right]}$, $\mathbb{E}\left[\cdot\right]$ stands for the expectation operator and $\Gamma\left(\cdot\right)$ denotes the Gamma function defined as $\Gamma(s) = \int_{0}^{\infty} t^{s-1} e^{-t} dt $ [@gradstein]. Also, $\mu_i\ge \frac{1}{2}$ is the inverse of normalized variance of $r^{\alpha_i}$ given by $$\mu_i = \mathbb{E}\left[r^{\alpha_i}\right] / \left( \mathbb{E}\left[ r^{2\alpha_i} \right] - \mathbb{E}^2\left[r^{\alpha_i}\right] \right).$$ Note that the $\alpha$-$\mu$ distribution is the most appropriate fading distribution that can be used to describe small-scale fading channels, e.g., Rayleigh ($\alpha=2$, $\mu=1$), Nakagami$-m$ ($\mu$ is the fading parameter with $\alpha=2$), Weibull ($\alpha$ is the fading parameter with $\mu=1$) [@magableh].
Ergodic Capacity
----------------
We define the EC as $$\label{erg_capacity}
\mathbb{E}\left[C_D\right] = \frac{1}{2}\mathbb{E}\left[\log_2\left(1+\gamma_D\right)\right],$$ where $C_D$ and $\gamma_D$ indicate the capacity and signal-to-noise ratio (SNR) at the destination, respectively; the factor $\frac{1}{2}$ shows that the $S$-to-$D$ transmission requires two time slots.
Time-Switching Relaying
=======================
The principle of the TSR protocol is as follows. The time required for $S$-$to$-$D$ information transmission is given by $T$, and the time fraction designated for EH is given by $\eta T$ ($0\le\eta \le1$). The remaining time, $(1-\eta)T$, consists of two time slots to maintain the $S$-$to$-$R$ and $R$-$to$-$D$ transmissions.
The received signal at the relay can be written as [@khaled1] $$\label{relay_signal}
y_R(t) = \sqrt{\frac{P_S}{d_1^{m_1}}} h_1 s(t) + n_a(t),$$ where $P_S$ stands for the source transmit power. The information signal and the noise at the relay are denoted as $s(t)$, satisfying $\mathbb{E}\left[ |s(t)|^2 \right] = 1$, and $n_a(t)$, with variance $\sigma_a^2$, respectively. Accordingly, the energy harvested at the relay can be presented as $$\label{harv_energy}
E_H^{TSR} = \theta \eta T \left( \frac{P_S}{d_1^{m_1}} h_1^2 + \sigma_a^2 \right),$$ where the EH conversion efficiency $\theta$ $(0<\theta\le1)$ is mainly affected by the circuitry. During the $R$-$to$-$D$ transmission, the relay performs base-band processing and then amplifies the signal which can be written as $$\label{tx_relay signal}
s_R(t) = \sqrt{\frac{P_R P_S}{d_1^{m_1}}} G h_1 s(t) + \sqrt{P_R} G n_R(t),$$ where $P_R$ represents the relay transmit power, $G$ is the relay gain defined as $G = 1/\sqrt{\frac{P_S}{d_1^{m_1}} h_1^2 + \sigma_R^2}$ and $n_R(t) = n_a(t) + n_c(t)$ denotes the overall noise at the relay with $\sigma_R^2 = \sigma_a^2 + \sigma_c^2$, where $n_c(t)$ indicates the noise added by the information receiver. Hence, the received signal at the destination can be expressed by $$\begin{aligned}
\label{rx_dest_signal}
\hspace{-0.2cm}y_D(t) = \sqrt{\frac{P_R}{d_2^{m_2}}}G h_2 \left(\sqrt{\frac{P_S}{d_1^{m_1} }} h_1 s(t) + n_R(t) \right) + n_D(t),\end{aligned}$$ where $n_D(t)$ denotes the noise term at the destination with variance $\sigma_D^2$. Since the harvested energy relates to the relay transmit power as $P_R = E_H^{TSR} / \left( (1-\eta)T/2\right) $, it can be further rewritten using as $$\label{Pr_TSR}
P_R = \frac{2\theta \eta}{1-\eta} \left( \frac{P_S}{d_1^{m_1}} h_1^2 + \sigma_a^2 \right).$$
With this in mind, substituting into and after some algebraic manipulations, we can write the SNR at $D$ as $$\label{dest_snr}
\gamma_D = \frac{2 \theta \eta P_S h_1^2 h_2^2}{2\theta \eta h_2^2 d_1^{m_1} \sigma_R^2 + (1-\eta) d_1^{m_1} d_2^{m_2} \sigma_D^2}.$$
Now, to obtain an expression for EC for the TSR-based system, we first let $a_1 = 2 \theta \eta P_S,~a_2 = (1-\eta) d_1^{m_1} d_2^{m_2} \sigma_D^2,~a_3 = 2 \theta \eta d_1^{m_1} \sigma_R^2,~\mathcal{A} = a_1 X$, and $\mathcal{B} = a_2 \bar{Y},$ where $X = h_1^2$ and $\bar{Y} = h_2^{-2}$. Having these equations, the SNR $\gamma_D$ can be presented as $$\label{dest_snr1}
\gamma_D = \frac{\mathcal{A}}{\mathcal{B} + a_3}.$$
Using and , the TSR-based EC can be written as $$\mathbb{E}\left[ C_D \right] = \frac{1-\eta}{2} \mathbb{E} \left[ \log_2 \left( 1 + \frac{\mathcal{A}}{\mathcal{B} + a_3} \right)\right].$$ The term $(1 - \eta)$ implies that the data transmission occurs only within this time fraction while the remaining time is used for EH purposes.
To simplify the EC analysis, we adopt in this paper the following lemma [@khairi_lemma] $$\label{lemma}
\mathbb{E}\left[ \ln\left( 1 + \frac{u}{v} \right) \right] = \int_{0}^{\infty} \frac{1}{s} \left( \Phi_v(s) -\Phi_{v,u}(s)\right)ds,~\forall~u,v>0,$$ where $\Phi_v(s)$ is the moment generating function (MGF) of the random variable (RV) $v$. $\Phi_{v,u}(s)$ can be calculated as $\Phi_{v,u} = \Phi_v(s) \Phi_u(s)$ if $v$ and $u$ are independent.
Since $\mathcal{A}$ and $\mathcal{B}$ are independent, the EC at $D$ can be calculated using as $$\label{capacity_TSR}
\mathbb{E}\left[ C_D \right] = \frac{1-\eta}{2\ln (2)} \int_{0}^{\infty} \frac{1}{s}\left( 1- \Phi_{\mathcal{A}}(s) \right)\Phi_{\mathcal{B}+a_3}(s)ds,$$ where $\Phi_{\mathcal{A}}(s)$ and $\Phi_{\mathcal{B}+a_3}(s)$ indicate the MGFs of $\mathcal{A}$ and $\mathcal{B} + a_3$ given by $\Phi_{\mathcal{A}}(s) = \Phi_X(a_1 s)$ and $\Phi_{\mathcal{B}+a_3}(s) = \Phi_{\bar{Y}}(a_2 s)\exp\left( -a_3 s \right)$, respectively. Since $X$ and $\bar{Y}$ are drawn from the $\alpha$-$\mu$ distribution, we need to modify the PDF given in to meet the changes taken on the RVs.
Let $Z$ be a continuous RV with generic PDF $f(z)$ defined over the support $t_1 < z < t_2$ and $Q = g(Z)$ be an invertible function of $Z$ with inverse function $Z = \nu\left( Q \right)$. Then, using the change of variable method, the PDF of $Q$ can be expressed as $$\label{change}
f_{Q}(q) = f_{Z}\left( \nu(q) \right)|\nu'\left(q\right)|$$ defined over the support $g(t_1) < q < t(c_2)$.
Since we consider $X = h_1^2$ and $\bar{Y} = h_2^{-2}$, the corresponding PDFs can be rewritten using as $$\begin{aligned}
\label{pdf_h_1_2}
f_{X}(r) &= \frac{\alpha_1 \mu_1^{\mu_1} r^{\frac{\alpha_1 \mu_1}{2} - 1}}{2\hat{r}^{\alpha_1\mu_1} \Gamma(\mu_1)} \exp\left( -\frac{\mu_1}{\hat{r}^{\alpha_1}} r^{\frac{\alpha_1}{2}} \right),\\
\label{pdf_h_2_2}
f_{\bar{Y}}(r) &= \frac{\alpha_2 \mu_2^{\mu_2} r^{-\frac{\alpha_2 \mu_2}{2} - 1}}{2\hat{r}^{\alpha_2\mu_2} \Gamma(\mu_2)} \exp\left( -\frac{\mu_2}{\hat{r}^{\alpha_2}} r^{-\frac{\alpha_2}{2}} \right).\end{aligned}$$
The corresponding MGFs of the PDFs in and , which will be used later in the EC analysis for the various considered EH modes, can be calculated as $$\begin{aligned}
\label{MGF}
\Phi (s) = \int_{0}^{\infty}\exp\left( -s r \right) f(r) dr.\end{aligned}$$ Substituting and into , we can write the corresponding MGFs as follows $$\begin{aligned}
\label{MGF1}
\Phi_{X} &= \frac{\alpha_1 \mu_1^{\mu_{1}}}{2\hat{r}^{\alpha_1\mu_1} \Gamma(\mu_1)} \int_{0}^{\infty} r^{\frac{\alpha_1 \mu_1}{2} - 1} e^{-s r} e^{-\frac{\mu_1}{\hat{r}^{\alpha_1}} r^{\frac{\alpha_1}{2}}} dr,\\
\label{MGF2}
\Phi_{\bar{Y}}&= \frac{\alpha_2 \mu_2^{\mu_2}}{2\hat{r}^{\alpha_2\mu_2} \Gamma(\mu_2)} \int_{0}^{\infty} r^{-\frac{\alpha_2 \mu_2}{2} - 1} e^{-s r} e^{-\frac{\mu_2}{\hat{r}^{\alpha_2}} r^{-\frac{\alpha_2}{2}}} dr.\end{aligned}$$ These MGF integrals can be calculated in closed-form if the exponential functions in the integrands are represented in terms of Meijer’s G-functions as [@prudnikov Eq. (8.4.3.1)]. Using [@prudnikov Eq. (2.24.1.1) and (8.2.2.14)], and after some manipulations and integration steps, the MGFs of $X$ and $\bar{Y}$ are written as and at the top of the next page, where $l$ and $k$ are prime numbers such that $l/k = \alpha/2$ and $\Delta\left(\beta,\phi\right)=\left\lbrace \frac{\phi}{\beta}, \frac{\phi+1}{\beta}, \ldots, \frac{\phi+\beta-1}{\beta} \right\rbrace$. It is worth to note that the same derivation approach will be further applied for the other EH protocols. Finally, using these MGFs, we can obtain $\Phi_{\mathcal{A}}$ and $\Phi_{\mathcal{B}+a_3}$ given by and .
$$\label{mgf_h1}
\Phi_{X} (s) = \frac{\alpha_1 \mu_1^{\mu_1} k^{\frac{1}{2}} l^{\frac{\alpha_1 \mu_1 - 1}{2}} s^{-\frac{\alpha_1 \mu_1}{2}}}{2 \hat{r}^{\alpha_1 \mu_1} \Gamma(\mu_1) \left( 2\pi \right)^{\frac{l+k-2}{2}}}~G_{l,k}^{k,l}\left( \left.\left( \frac{\mu_1}{\hat{r}^{\alpha_1} k} \right)^{k} \left( \frac{l}{s} \right)^{l} \right\vert \begin{array}{c}
\Delta\left(l, 1 - \frac{\alpha_1 \mu_1}{2}\right)\\
\Delta\left(k, 0\right)
\end{array} \right)$$ $$\label{mgf_h2}
\Phi_{\bar{Y}} (s) = \frac{\alpha_2 \mu_2^{\mu_2} k^{\frac{1}{2}} l^{-\frac{\alpha_2 \mu_2 - 1}{2}} s^{\frac{\alpha_2 \mu_2}{2}}}{2 \hat{r}^{\alpha_2 \mu_2} \Gamma(\mu_2) \left( 2\pi \right)^{\frac{l+k-2}{2}}}~G_{k+l,0}^{0,k+l}\left( \left.\left( \frac{\mu_2 k}{\hat{r}^{\alpha_2}} \right)^{k} \left( \frac{l}{s} \right)^{l} \right\vert \begin{array}{c}
\Delta\left(l, 1 + \frac{\alpha_2 \mu_2}{2}\right), \Delta\left( k,0 \right)\\
\textendash
\end{array} \right)$$ $$\label{mgf_tsr1}
\Phi_{\mathcal{A}} (s) = \frac{\alpha_1 \mu_1^{\mu_1} k^{\frac{1}{2}} l^{\frac{\alpha_1 \mu_1 - 1}{2}} (2 \theta \eta P_S s)^{-\frac{\alpha_1 \mu_1}{2}}}{2 \hat{r}^{\alpha_1 \mu_1} \Gamma(\mu_1) \left( 2\pi \right)^{\frac{l+k-2}{2}}}~G_{l,k}^{k,l}\left( \left.\left( \frac{\mu_1}{\hat{r}^{\alpha_1} k} \right)^{k} \left( \frac{l}{2 \theta \eta P_S s} \right)^{l} \right\vert \begin{array}{c}
\Delta\left(l, 1 - \frac{\alpha_1 \mu_1}{2}\right)\\
\Delta\left(k, 0\right)
\end{array} \right)$$ $$\begin{gathered}
\label{mgf_tsr2}
\Phi_{\mathcal{B} + a_3} (s) = \frac{\alpha_2 \mu_2^{\mu_2} k^{\frac{1}{2}} l^{-\frac{\alpha_2 \mu_2 - 1}{2}} ((1-\eta) d_1^{m_1} d_2^{m_2} \sigma_D^2 s)^{\frac{\alpha_2 \mu_2}{2}}}{2 \hat{r}^{\alpha_2 \mu_2} \Gamma(\mu_2) \left( 2\pi \right)^{\frac{l+k-2}{2}}}~\exp\left(-2 \theta \eta d_1^{m_1} \sigma_R^2 s\right) \times \\
~G_{k+l,0}^{0,k+l}\left( \left.\left( \frac{\mu_2 k}{\hat{r}^{\alpha_2}} \right)^{k} \left( \frac{l}{(1-\eta) d_1^{m_1} d_2^{m_2} \sigma_D^2 s} \right)^{l} \right\vert \begin{array}{c}
\Delta\left(l, 1 + \frac{\alpha_2 \mu_2}{2}\right), \Delta\left( k,0 \right)\\
\textendash
\end{array} \right)
\end{gathered}$$
Power-Splitting Relaying
========================
The PSR protocol operates over the time period, $T$, formed by two identical parts to maintain the $S$-$to$-$R$ and $R$-$to$-$D$ transmission sessions. During the $S$-$to$-$R$ transmission, the relay utilizes a portion of the received signal power for EH, $\rho P_S$, while the rest of the power, $(1-\rho)P_S$, is dedicated for the $R$-$to$-$D$ transmission. Therefore, the received signal at the input of the energy harvester can be presented as $$\sqrt{\rho}y_{R}(t) = \sqrt{\frac{\rho P_S}{d_1^{m_1}}}h_1 s(t) + \sqrt{\rho}n_a(t).$$
The PSR-based harvested energy can be calculated as $$\label{EH_PSR}
E_H^{PSR} = \frac{\theta \rho T}{2}\left(\frac{P_S}{d_1^{m_1}} h_1^2 + \sigma_a^2\right).$$ This energy is used to amplify and forward the information to $D$. Thus, the relay transmit signal can be written as $$s_R(t) = \sqrt{\frac{(1-\rho)P_S P_R}{d_1^{m_1}}} G h_1 s(t) + \sqrt{P_R} G n_R(t),$$ where $G$ is the relay gain defined in this case as $G = 1/\sqrt{\frac{(1-\rho)P_S}{d_1^{m_1}}h_1^2 + \sigma_R^2}$ and $n_R(t) = \sqrt{1-\rho}n_a(t) + n_c(t)$. Hence, the signal at the destination can be written as $$\begin{aligned}
\label{yD_PSR}
\hspace{-0.1cm} y_{D}(t) = \sqrt{\frac{P_R}{d_2^{m_2}}}G h_2 \left(
\sqrt{\frac{(1-\rho)P_S }{d_1^{m_1}}} h_1 s(t)
+ n_R(t) \right) + n_D(t).\end{aligned}$$
The relay transmit power relates to the harvested energy as $P_R = \frac{2 E_H^{PSR}}{T}$ and therefore can be rewritten using as $$\label{Pr_PSR}
P_R = \theta\rho\left(\frac{P_S}{d_1^{m_1}} h_1^2 + \sigma_a^2\right).$$
Substituting into , and after some algebraic manipulations, we derive the SNR at $D$ for the PSR system as $$\label{snr_psr}
\gamma_D = \frac{\theta \rho (1-\rho)P_S h_1^2 h_2^2}{d_1^{m_1}\left(\theta \rho \sigma_c^2 h_2^2 + \theta \rho (1-\rho) \sigma_a^2 h_2^2 + (1-\rho) d_2^{m_2} \sigma_D^2 \right)}.$$
Below, to obtain an expression for the EC, first we let $b_1 = \theta \rho (1-\rho) P_S,~b_2 = (1-\rho)d_1^{m_1} d_2^{m_2} \sigma_D^2,~b_3 = \theta \rho d_1^{m_1} \sigma_c^2,~b_4 = \theta \rho (1 - \rho) d_1^{m_1} \sigma_a^2,~\mathcal{K} = b_1 X$ and $\mathcal{L} = b_2 \bar{Y}$. Using these definitions, we can rewrite as $$\label{snr_psr1}
\gamma_D = \frac{\mathcal{K}}{\mathcal{L} + b_3 + b_4}.$$
Substituting into , the EC can be calculated as $$\label{erg_capacity_psr}
\mathbb{E}\left[C_D\right] = \frac{1}{2}\mathbb{E}\left[\log_2\left(1+\frac{\mathcal{K}}{\mathcal{L} + b_3 + b_4}\right)\right],$$ which, using , can also be written as $$\label{erg_capacity_psr1}
\mathbb{E}\left[C_D\right] = \frac{1}{2\ln(2)}\int_{0}^{\infty}\frac{1}{s}\left(1 - \Phi_{\mathcal{K}}(s)\right) \Phi_{\mathcal{L} + b_3 + b_4}(s) ds,$$ where $\Phi_{\mathcal{K}}(s)$ and $\Phi_{\mathcal{L} + b_3 + b_4}(s)$ denote the MGFs of the RVs $\mathcal{K}$ and $(\mathcal{L}+b_3+b_4)$ given by $\Phi_{\mathcal{K}}(s) = \Phi_{X}(b_1 s)$ and $\Phi_{\mathcal{L} + b_3 + b_4}(s) = \Phi_{\bar{Y}}(b_2 s) \exp(-b_3 s) \exp(-b_4 s)$, respectively, and presented at the top of the next page. Now, substituting and into provides the PSR-based EC expression.
$$\label{mgf_psr1}
\Phi_{\mathcal{K}} (s) = \frac{\alpha_1 \mu_1^{\mu_1} k^{\frac{1}{2}} l^{\frac{\alpha_1 \mu_1 - 1}{2}} (\theta \rho (1-\rho) P_S s)^{-\frac{\alpha_1 \mu_1}{2}}}{2 \hat{r}^{\alpha_1 \mu_1} \Gamma(\mu_1) \left( 2\pi \right)^{\frac{l+k-2}{2}}}~G_{l,k}^{k,l}\left( \left.\left( \frac{\mu_1}{\hat{r}^{\alpha_1} k} \right)^{k} \left( \frac{l}{\theta \rho (1-\rho) P_S s} \right)^{l} \right\vert \begin{array}{c}
\Delta\left(l, 1 - \frac{\alpha_1 \mu_1}{2}\right)\\
\Delta\left(k, 0\right)
\end{array} \right)$$ $$\begin{gathered}
\label{mgf_psr2}
\Phi_{\mathcal{L} + b_3 + b_4} (s) = \frac{\alpha_2 \mu_2^{\mu_2} k^{\frac{1}{2}} l^{-\frac{\alpha_2 \mu_2 - 1}{2}} ((1-\rho)d_1^{m_1} d_2^{m_2} \sigma_D^2 s)^{\frac{\alpha_2 \mu_2}{2}}}{2 \hat{r}^{\alpha_2 \mu_2} \Gamma(\mu_2) \left( 2\pi \right)^{\frac{l+k-2}{2}}}~\exp\left(-\theta \rho d_1^{m_1} \sigma_c^2 s\right) \exp\left(-\theta \rho (1 - \rho) d_1^{m_1} \sigma_a^2 s\right) \times \\
~G_{k+l,0}^{0,k+l}\left( \left.\left( \frac{\mu_2 k}{\hat{r}^{\alpha_2}} \right)^{k} \left( \frac{l}{(1-\rho)d_1^{m_1} d_2^{m_2} \sigma_D^2 s} \right)^{l} \right\vert \begin{array}{c}
\Delta\left(l, 1 + \frac{\alpha_2 \mu_2}{2}, \Delta\left( k,0 \right)\right)\\
\textendash
\end{array} \right)
\end{gathered}$$
Ideal Relaying Receiver
=======================
Unlike the TSR and PSR systems, the IRR-based system has the capability to independently and concurrently process the information signal and harvest energy from the same received signal. Therefore, during the first time slot, the relay harvests energy and processes the information signal whereas in the second time slot the relay uses the harvested energy to amplify and then forward the signal to the destination node. Therefore, following the same procedure shown in Section IV, we get the SNR at $D$ as $$\label{snr_irr}
\gamma_D = \frac{\theta P_S h_1^2 h_2^2}{\theta d_1^{m_1} \sigma_R^2 h_2^2 + d_1^{m_1} d_2^{m_2} \sigma_D^2}.$$
Below, to obtain an expression for the EC over $\alpha$-$\mu$ fading channels, first we let $c_1 = \theta P_s,~c_2 = d_1^{m_1} d_2^{m_2} \sigma_D^2,~c_3 = \theta d_1^{m_1} \sigma_R^2,~\mathcal{E} = c_1 X$ and $\mathcal{F} = c_2 \bar{Y}$. Therefore, we can rewrite as $$\label{snr_irr1}
\gamma_D = \frac{\mathcal{E}}{\mathcal{F} + c_3}.$$
Similar to the previous sections, the EC can be given as $$\label{erg_capacity_iir}
\mathbb{E}\left[C_D\right] = \frac{1}{2}\mathbb{E}\left[\log_2\left(1 + \frac{\mathcal{E}}{\mathcal{F} + c_3}\right)\right].$$
Also, $$\label{erg_capacity_irr1}
\mathbb{E}\left[C_D\right] = \frac{1}{2\ln(2)}\int_{0}^{\infty}\frac{1}{s}\left(1 - \Phi_{\mathcal{E}}(s)\right) \Phi_{\mathcal{F} + c_3}(s) ds,$$ where $\Phi_{\mathcal{E}}(s)$ and $\Phi_{\mathcal{F} + c_3}(s)$ denote the MGFs of the RVs $\mathcal{E}$ and $(\mathcal{F}+c_3)$ given by $\Phi_{\mathcal{E}}(s) = \Phi_{X}(c_1 s)$ and $\Phi_{\mathcal{F} + c_3}(s) = \Phi_{\bar{Y}}(c_2 s) \exp(-c_3 s)$, respectively, shown at the top of the next page. Finally, substituting and into provides the EC for this system.
$$\label{mgf_irr1}
\Phi_{\mathcal{E}} (s) = \frac{\alpha_1 \mu_1^{\mu_1} k^{\frac{1}{2}} l^{\frac{\alpha_1 \mu_1 - 1}{2}} (\theta P_s s)^{-\frac{\alpha_1 \mu_1}{2}}}{2 \hat{r}^{\alpha_1 \mu_1} \Gamma(\mu_1) \left( 2\pi \right)^{\frac{l+k-2}{2}}}~G_{l,k}^{k,l}\left( \left.\left( \frac{\mu_1}{\hat{r}^{\alpha_1} k} \right)^{k} \left( \frac{l}{\theta P_s s} \right)^{l} \right\vert \begin{array}{c}
\Delta\left(l, 1 - \frac{\alpha_1 \mu_1}{2}\right)\\
\Delta\left(k, 0\right)
\end{array} \right)$$ $$\begin{gathered}
\label{mgf_irr2}
\Phi_{\mathcal{F} + c_3} (s) = \frac{\alpha_2 \mu_2^{\mu_2} k^{\frac{1}{2}} l^{-\frac{\alpha_2 \mu_2 - 1}{2}} (d_1^{m_1} d_2^{m_2} \sigma_D^2 s)^{\frac{\alpha_2 \mu_2}{2}}}{2 \hat{r}^{\alpha_2 \mu_2} \Gamma(\mu_2) \left( 2\pi \right)^{\frac{l+k-2}{2}}}~\exp\left(-\theta d_1^{m_1} \sigma_R^2 s\right) \times \\
~G_{k+l,0}^{0,k+l}\left( \left.\left( \frac{\mu_2 k}{\hat{r}^{\alpha_2}} \right)^{k} \left( \frac{l}{d_1^{m_1} d_2^{m_2} \sigma_D^2 s} \right)^{l} \right\vert \begin{array}{c}
\Delta\left(l, 1 + \frac{\alpha_2 \mu_2}{2}, \Delta\left( k,0 \right)\right)\\
\textendash
\end{array} \right)
\end{gathered}$$
Simulation Results
==================
This section presents numerical results for the EC expressions derived above. The adopted system parameters are as follows: $G=1$, $m_1 = m_2 = 2.7$ m, $\sigma_R = \sigma_D = 0.02$ W, $\sigma_a = \sigma_c = \sigma_R/2$. By assigning different values for the $\alpha$ and $\mu$ parameters, we derive the Rayleigh ($\alpha=2$ and $\mu=1$), Nakagami-$m$ ($\alpha=2$) and Weibull ($\mu=1$) fading coefficients.
Ergodic Capacity
----------------
Here, we study the impact of $\eta$ and $\rho$ on the EC for the TSR- and PSR-based systems. We consider the following network parameters: $d_1 = d_2 = 3$ m, $P_S = 1$ W and $\theta = \{0.5; 1\}$. Fig. \[results1\] illustrates some numerical and simulation results for the ECs as a function of $\rho$ and $\eta$ for different channel distributions derived from the $\alpha$-$\mu$ distribution. Eqs. and are used to obtain the analytical results for the TSR and PSR systems, respectively. For the case of the TSR protocol, no sufficient time is allocated for EH when $\eta$ is small and thus the relay scavenges only a small amount of power which correspondingly results in poor capacity. Being $\eta$ too large leads to the excessive amount of the harvested energy at the price of the time dedicated for information transmission which obviously results in poor capacity. The similar justification can be applied to $\rho$ for the case of the PSR protocol. It is worth to note that the selection of $\eta$ and $\rho$ mainly defines the performance of these protocols.
System Optimization
-------------------
Now, to assess the performance of the optimized PSR and TSR systems, first we find the optimal values of $\eta$ and $\rho$ given by $\eta^*$ and $\rho^*$ with $P_S = \{1; 5\}$ W and $\theta = 1$ for the corresponding PSR and TSR protocols by solving the following $d\left\lbrace \mathbb{E}\left[C_D\right] \right\rbrace / d\eta = 0$ and $d\left\lbrace \mathbb{E}\left[C_D\right] \right\rbrace / d\rho = 0$. It is worthwhile to mention that it is not easy to derive a closed-form solution for these equations; however, it can be calculated with software tools such as $Mathematica$.
Fig. \[results2\] presents the maximum achievable EC according to $\eta^*$ and $\rho^*$ versus the $R$-$to$-$D$ distance when the end-to-end distance is kept fixed at 10 m. It can be seen that the optimized PSR-based system always outperforms the optimized TSR protocol irrespective of the relay location. The IRR protocol provides the best performance among all the protocols under consideration. At $d_2 = 9$ m (when the relay node is located close to the source node), the optimized PSR-based relaying approaches the performance of that of the IRR system. Moreover, the lowest EC for the three systems is detected when the relay location is midway between the source and destination nodes. This occurs due to the fact that, EH in this region achieves its peak which considerably affects the time dedicated for information transmission and hence the overall EC.
Conclusion
==========
In this paper, we analyzed the performance of various EH relaying protocols over different i.n.i.d. $\alpha$-$\mu$ fading channels, namely, Rayleigh, Nakagami-$m$ and Weibull fadings. We derived accurate analytical expressions for the EC for the three EH systems, namely, TSR, PSR and IRR, which were validated with Monte Carlo simulations. The results showed that a proper choice of the TS and PS factors in the corresponding protocols is a key to acquire the best performance. In addition, the IRR-based system was shown to have the best performance and the optimized PSR-based approach always outperforms the optimized TSR scheme.
Acknowledgment
==============
This publication was made possible by NPRP grant number 9-077-2-036 from the Qatar National Research Fund (a member of Qatar Foundation). The statements made herein are solely the responsibility of the authors.
[99]{} X. Zhou, R. Zhang, and C. K. Ho, Wireless information and power transfer: Architecture design and rate-energy tradeoff, *IEEE Trans. Commun.*, vol. 61, pp. 47544767, Nov. 2013. K. Huang and E. Larsson, Simultaneous information and power transfer for broadband wireless systems, *IEEE Trans. Signal Process.*, vol. 61, pp. 59725986, Dec. 2013. L. R. Varshney, Transporting information and energy simultaneously, *IEEE Int. Symp. Inf. Theory (ISIT)*, pp. 16121616, Jul. 2008. D. Mishra, S. De, S. Jana, S. Basagni, K. Chowdhury, and W. Heinzelman, Smart RF energy harvesting communications: challenges and opportunities, *IEEE Commun. Mag.*, vol. 53, no. 4, pp. 7078, Apr. 2015. S. Arzykulov, G. Nauryzbayev, T. A. Tsiftsis, and M. Abdallah, Error Performance of Wireless Powered Cognitive Relay Networks with Interference Alignment, *in Proc. IEEE Int. Symp. Personal, Indoor, and Mobile Radio Commun. (PIMRC)*, 16, Oct. 2017. S. Arzykulov, G. Nauryzbayev, T. A. Tsiftsis, and M. Abdallah, On the Capacity of Wireless Powered Cognitive Relay Network with Interference Alignment, *in Proc. IEEE Global Commun. Conf. (Globecom)*, 16, Dec. 2017. A. A. Nasir, X. Zhou, S. Durrani, and R. A. Kennedy, Relaying protocols for wireless energy harvesting and information processing, *IEEE Trans. Wireless Commun.*, vol. 12, no. 7, pp. 36223636, Jul. 2013. P. N. Son and H. Y. Kong, Exact outage analysis of energy harvesting underlay cooperative cognitive networks, *IEICE Trans. Commun.*, vol. E98.B, no. 4, pp. 661672, Apr. 2015. A. A. Nasir, X. Zhou, S. Durrani, and R. A. Kennedy, Throughput and ergodic capacity of wireless energy harvesting based DF relaying network, *IEEE Int. Conf. Commun. (ICC)*, pp. 40664071, Jun. 2014. K. M. Rabie, A. Salem, E. Alsusa, and M. S. Alouini, Energy-harvesting in cooperative AF relaying networks over Log-normal fading channels, *IEEE Int. Conf. Commun. (ICC)*, pp. 17, May 2016. K. M. Rabie, B. Adebisi, and M. S. Alouini, Wireless power transfer in cooperative DF relaying networks with Log-normal fading, *IEEE Global Commun. Conf. (Globecom)*, pp. 16, Dec. 2016. A. Salem, K. Hamdi, and K. M. Rabie, Physical layer security with RF energy harvesting in AF Multi-Antenna Relaying Networks, *IEEE Trans. Commun.*, vol. 64, no. 7, pp. 30253038, Jul. 2016. S. Arzykulov, G. Nauryzbayev, and T. A. Tsiftsis, Underlay cognitive relaying system over $\alpha$-$\mu$ fading channels, *IEEE Commun. Lett.*, vol. 21, no. 1, pp. 216219, Jan. 2017. I. S. Gradshteyn and I. M. Ryzhik, *Table of Integrals, Series, and Products*, San Diego, CA, USA: Academic Press, 2007. A. M.Magableh and M. M. Matalgah, Moment generating function of the generalized $\alpha$-$\mu$ distribution with applications, *IEEE Commun. Lett.*, vol. 13, no. 6, pp. 411413, Jun. 2009. K. Hamdi, A useful lemma for capacity analysis of fading interference channels, *IEEE Trans. Commun.*, vol. 58, pp. 411416, Feb. 2010. A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, *Integrals and Series, Vol. 3: More Special Functions*, Gordon and Breach, New York, 1990.
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abstract: 'We report on the experimental observation of bunching dynamics with temporal cavity solitons in a continuously-driven passive fibre resonator. Specifically, we excite a large number of ultrafast cavity solitons with random temporal separations, and observe in real time how the initially random sequence self-organizes into regularly-spaced aggregates. To explain our experimental observations, we develop a simple theoretical model that allows long-range acoustically-induced interactions between a large number of temporal cavity solitons to be simulated. Significantly, results from our simulations are in excellent agreement with our experimental observations, strongly suggesting that the soliton bunching dynamics arise from forward Brillouin scattering. In addition to confirming prior theoretical analyses and unveiling a new cavity soliton self-organization phenomenon, our findings elucidate the manner in which sound interacts with large ensembles of ultrafast pulses of light.'
address: 'The Dodd-Walls Centre for Photonic and Quantum Technologies, and Physics Department, The University of Auckland, Private Bag 92019, Auckland 1142, New Zealand'
author:
- 'Miro Erkintalo, Kathy Luo, Jae K. Jang, Stéphane Coen, and Stuart G. Murdoch'
title: Bunching of temporal cavity solitons via forward Brillouin scattering
---
August 2015
Introduction
============
Temporal cavity solitons (CSs) are pulses of light that can persist in externally, coherently-driven passive nonlinear optical resonators [@wabnitz_suppression_1993; @leo_temporal_2010]. They are genuine solitons in that their shape does not evolve upon propagation: temporal broadening induced by chromatic dispersion is balanced by an optical nonlinearity [@kivshar_optical_2003; @agrawal_nonlinear_2006]. In addition, CSs have the ability to continuously extract energy from the coherent field driving the resonator so as to balance the power losses they suffer at each cavity roundtrip. This double balancing act makes CSs unique attracting states, and allows them to circulate indefinitely despite the absence of an amplifier or saturable absorber in the resonator. More generally, temporal CSs belong to the broader class of localized dissipative structures or dissipative solitons [@nicolis_self-organization_1977; @akhmediev_dissipative_2008; @purwins_dissipative_2010].
Because of the presence of the coherent driving beam, CSs are superimposed and phase-locked on a homogeneous background field filling the entire resonator [@firth_cavity_2002]. Consequently, they do not possess phase-rotation symmetry. For that reason, temporal CSs are fundamentally different from pulses in mode-locked lasers [@firth_temporal_2010]. For example, for the exact same system parameters, a passive driven nonlinear resonator can sustain at once an arbitrary number of CSs at arbitrary temporal positions. In other words, there are many different co-existing solutions that the intracavity field can assume and the system exhibits massive multi-stability (see also [@marconi_how_2014]). Moreover, each of the CSs can be individually addressed, which means that they can be turned on [@leo_temporal_2010] and off [@jang_writing_2015] and even temporally shifted with respect to each other [@jang_temporal_2015] without affecting adjacent pulses.
In terms of their identifying characteristics and dynamics, temporal CSs are similar to their spatial counterparts — self-localized beams of light that persist in coherently-driven *diffractive* nonlinear cavities [@ackemann_chapter_2009; @barland_cavity_2002]. In particular, both spatial and temporal CSs obey the paradigmatic mean-field Lugiato-Lefever equation (LLE) [@lugiato_spatial_1987]. But whilst spatial CSs have been extensively studied for more than two decades [@mcdonald_spatial_1990; @tlidi_localized_1994; @firth_optical_1996-1; @barland_cavity_2002; @lugiato_introduction_2003], research into temporal CSs only started in 2010, when they were first observed experimentally by Leo et al using an optical fibre ring resonator [@leo_temporal_2010]. Due to their unique characteristics, temporal CSs were identified as ideal candidates for bits in all-optical buffers, which stimulated many subsequent studies using similar fibre cavity designs [@leo_dynamics_2013; @jang_ultraweak_2013; @jang_observation_2014; @jang_temporal_2015]. In addition to macroscopic fibre cavities, temporal CSs have also recently attracted great interest in the context of microscopic Kerr resonators. In particular, it has been shown both theoretically [@coen_modeling_2013; @coen_universal_2013; @chembo_spatiotemporal_2013; @erkintalo_coherence_2014; @godey_stability_2014] and experimentally [@herr_temporal_2014; @brasch_photonic_2014] that temporal CSs are intimately linked to the formation of broadband Kerr frequency combs that have been observed in such devices [@delhaye_optical_2007; @kippenberg_microresonator-based_2011; @moss_new_2013].
Of course, a defining trait of solitons is concerned with the way they interact with each other [@zabusky_interaction_1965; @stegeman_optical_1999; @rotschild_long-range_2006], and CSs are no exceptions. Theoretical and experimental studies have revealed that CSs are connected to the surrounding background field through oscillatory tails and that adjacent CSs interact when their tails overlap and/or lock [@barashenkov_bifurcation_1998; @schapers_interaction_2000; @tlidi_interaction_2003; @bodeker_measuring_2004; @parra-rivas_dynamics_2014]. These interactions can induce rich dynamics in their own right, including the formation of bound states [@schapers_interaction_2000; @jang_controlled_2015], but are short range due to the exponentially decaying nature of the oscillatory tails. Experiments with temporal CSs in fibre-based cavities have however also revealed extremely long range interactions between solitons separated by hundreds of characteristic widths [@jang_ultraweak_2013]. These were found to be mediated by electrostriction [@boyd_nonlinear_2008], which causes temporal CSs to excite transverse acoustic waves in the fibre core and cladding. The acoustic waves give rise to refractive index changes through the acousto-optic effect, and long-range interactions ensue when a trailing CS overlaps with the perturbation created by a leading one [@jang_ultraweak_2013].
Long before temporal CSs were even observed, electrostriction-induced interactions were studied in the context of optical-fibre telecommunication systems [@smith_experimental_1989; @dianov_long-range_1992; @jaouen_transverse_2001], as well as in passively mode-locked fibre lasers [@grudinin_passive_1993; @pilipetskii_acoustic_1995; @grudinin_passive_1997]. In particular, Pilipetskii et al have numerically demonstrated that acoustic effects could be responsible for the bunching of pulses in fibre lasers [@pilipetskii_acoustic_1995]. Although experimental observations of pulse bunching abound in the ultrafast fibre laser literature [@grudinin_passive_1993; @tang_bound-soliton_2002; @zhao_passive_2009; @chouli_rains_2009], quantitative comparisons with the theory of acoustic interactions are hindered by the many competing effects that influence pulse dynamics in such lasers, including saturable absorption and gain depletion and recovery [@grudinin_passive_1997; @kutz_stabilized_1998; @korobko_long-range_2015]. Continuously-driven passive fibre cavities are void of these complications, and acoustic interactions of a pair of temporal CSs have been successfully modelled quantitatively [@jang_ultraweak_2013]. Moreover, because temporal CSs are phase-locked to the cavity driving beam, the acoustic interactions they experience are orders of magnitude weaker than in other systems [@jang_ultraweak_2013]. The pertinent dynamics can therefore be easily monitored in real time. Temporal CSs in coherently-driven passive fibre cavities thus appear as the ideal platform to explore electrostriction-mediated pulse interaction effects. So far, however, experiments with temporal CSs have only been performed with a small number of co-existing solitons [@jang_ultraweak_2013; @jang_temporal_2015]. Accordingly, no pulse bunching effects have yet been observed.
In this Article, we experimentally and theoretically investigate the acoustic interactions of a very large number of temporal CSs. Specifically, we excite a large number of randomly-spaced temporal CSs in a continuously-driven passive fibre cavity (hence based on a simple Kerr nonlinearity), and we examine their interaction dynamics in real time. We find that the initially random sequence of pulses self-organizes into regular bunches whose spacing agrees very well with the frequency of the acoustic modes that interact most efficiently with light in the fibre core. To quantitatively show that the bunching behaviour originates from acoustic effects, we develop a simple model that allows the full dynamical evolution to be simulated. Very good agreement is observed between simulations and experiments.
Experiment
==========
Experimental setup
------------------
Our experimental setup is similar to the one used in [@jang_ultraweak_2013], and is schematically illustrated in Fig. \[setup\]. The passive all-fibre cavity is 100-m long and constructed of standard telecommunications single-mode optical fibre (SMF28) that is closed on itself with a 90/10 fibre coupler. The cavity incorporates an optical isolator to prevent depletion of the driving beam by *backward* stimulated Brillouin scattering [@agrawal_nonlinear_2006], a wavelength-division multiplexer (WDM) to couple in addressing pulses used to “excite” the CSs (see below), and a 1% output coupler through which the intracavity CS dynamics can be monitored. Overall, the cavity has a finesse of $21.5$, corresponding to $29.2$% power losses per roundtrip.
The cavity is coherently driven with an ultra-narrow linewidth ($<1$ kHz) continuous-wave (cw) laser centred at 1550 nm wavelength, that is externally amplified to about 1 W with an erbium-doped fibre amplifier (EDFA). Amplified spontaneous emission noise is removed using a $0.6$-nm-wide bandpass filter (BPF) centred at 1550 nm before the field is injected into the cavity via the 90/10 fibre coupler. The light that is reflected off from the cavity is fed to a servo-controller that actuates the driving laser frequency so as to maintain the reflected power at a set level. In this way, the frequency of the driving laser follows any changes in the cavity resonances due, e.g., to environmental perturbations, ensuring that the phase detuning between the driving laser and the cavity is locked. This is an integral part of our experiment, as temporal CSs rely critically on phase-sensitive interactions with the driving field. Note that the cavity locking scheme employed here is more robust than that used in the first experimental observation of Leo et al [@leo_temporal_2010]; in our setup, CSs can routinely be sustained for several minutes or even hours. This stability is crucial to our study, since the acoustic CS interactions are so weak that very long measurement times are necessary to observe the full dynamics [@jang_ultraweak_2013].
To excite temporal CSs, we use the optical addressing technique introduced in [@leo_temporal_2010]. Specifically, ultrashort pulses from a 10 GHz mode-locked laser with a different wavelength (here, 1532 nm) than that of the driving field are launched into the cavity through the WDM. They then interact through cross-phase modulation with the intracavity cw background and each of them excites an independent CS. After one roundtrip, the addressing pulses exit the cavity through the WDM, and only the temporal CSs persist. The process is controlled by picking pulses from the mode-locked laser with a sequence of two intensity modulators. The first modulator is driven by a 10 GHz pattern generator synchronized to the mode-locked laser and selects the pattern of CSs to be excited. The second modulator is used as a gate to block the mode-locked laser beam after addressing is complete. Once CSs are excited, we monitor their dynamics in real time by recording the field at the cavity output using a fast $12.5$ GHz photodetector connected to a 40 GSa/s oscilloscope. Before detection, the output field passes through a narrow bandpass filter centred at 1551 nm, one nanometer away from the driving wavelength. This removes the cw background component of the CSs, improving the signal-to-noise ratio of the measurements [@leo_temporal_2010].
Experimental results
--------------------
In previous studies, we have examined configurations involving a small number of temporal CSs [@jang_ultraweak_2013; @jang_temporal_2015]. Here, in contrast, we are interested in studying the intracavity dynamics when a very large number of CSs co-exist. To this end, we start the experiment by exciting a densely-packed sequence of temporal CSs. This is achieved by programming a random sequence into the pattern generator driving the first modulator, in essence selecting a corresponding random series of pulses from the mode-locked addressing laser, while the second modulator is kept open for *several* cavity roundtrip times $t_\mathrm{R}$. Because the mode-locked laser repetition rate, the length of the random sequence, and the cavity free-spectral-range ($\mathrm{FSR} = 1/t_\mathrm{R}$) are not commensurate, the resultant temporal CS sequence is to a large extent random.
The curve in Fig. \[expr\](a) illustrates the result of the addressing process. It shows the temporal intensity profile of the intracavity field recorded by the oscilloscope at the beginning of the experiment, highlighting the presence in the cavity of a sequence of temporal CSs with essentially random spacing. Note that (i) for clarity we only show a small 50 ns-long segment of the full 480-ns roundtrip, and (ii) that the electronic bandwidth of our detectors prevents closely-spaced temporal CSs to be individually resolved. In this context, we remark that, for given parameters, all temporal CSs have identical characteristics (energy, duration, and peak power) [@leo_temporal_2010]. The different amplitudes observed in Fig. \[expr\](a) therefore simply represent bunches that contain different numbers of temporal CSs spaced by less than the detector 80 ps response time.
In Fig. \[expr\](c) we show a similar measurement but taken approximately 10 s after the temporal CSs were excited and allowed to freely interact. Here we see clearly that the CSs have formed almost regularly spaced aggregates, with an average separation of about $2.6$ ns. This bunching behaviour can be more readily appreciated from the false colour density plot in Fig. \[expr\](b), which maps the measured dynamical evolution of the CS field during the 10 s of free interaction. To form this plot, we have vertically concatenated 100 oscilloscope profiles \[like those shown in Figs \[expr\](a) and (c)\] measured at regular intervals (10 frames/s) so as to display how the intracavity pulse sequence evolves over time (top to bottom). We see clearly how the CSs exhibit complex interaction dynamics, with individual pulses gradually forming bunches. To the best of our knowledge, this represents the first direct experimental observation of pulse bunching dynamics in a fibre resonator. We also highlight that, as in [@jang_ultraweak_2013], the interactions are exceedingly weak. During the 10 s measurement shown in Fig. \[expr\], the temporal CSs complete about 20 million roundtrips (corresponding to 2 million kilometres of propagation length), yet their temporal separations only change by a few nanoseconds.
Theory
======
In this Section, we show theoretically that the bunching dynamics observed in the experiment described above can be quantitatively explained in terms of electrostriction-induced acoustic interactions. We first recount the basic mechanisms that underpin the interactions, and subsequently develop a simple model that allows the acoustic interactions of a large number of temporal CSs to be examined. Our approach is adapted from that developed by Pilipetskii et al to investigate acoustic interactions in passively mode-locked fibre lasers [@pilipetskii_acoustic_1995].
Acoustic soliton interactions
-----------------------------
Pulses of light travelling in optical fibres can excite, through electrostriction, transverse acoustic waves propagating (nearly) orthogonally to the fibre axis [@jaouen_transverse_2001; @boyd_nonlinear_2008], giving rise to refractive index perturbations that are left behind in the wake of the excitation pulses. The physical mechanism coincides with guided acoustic wave Brillouin scattering (also referred to as forward Brillouin scattering) that was first studied by Shelby et al in the context of cw fields [@shelby_resolved_1985; @shelby_guided_1985]. Dianov et al [@dianov_long-range_1992] were the first who suggested that this mechanism could explain long-range interpulse interactions previously observed in optical fibres [@smith_experimental_1989].
Figure \[acoustic\](a) is a plot of the temporal impulse response of the effective refractive index perturbation, $\delta n(\tau)$, generated through this process in the fibre core. The response was calculated following the approach of Dianov et al [@dianov_long-range_1992] and using parameters (in particular the CS energy) corresponding to our experiment (see Ref. [@jang_ultraweak_2013] for details). The perturbation is fairly weak but extends over tens of nanoseconds. The overall shape of the response is dominated by 1–3 ns wide spikes that are separated by about 21 ns, arising from successive acoustic reflections on the fibre cladding-jacket boundary. We must note that the temporal CSs in our experiment have a $\sim 3$ ps duration. The impulse response shown in Fig. \[acoustic\](a) is thus a fair representation of the refractive index perturbation induced by acoustic waves generated through electrostriction by an isolated temporal CS.
The refractive index perturbation shown in Fig. \[acoustic\](a) is continuously generated by a temporal CS as it propagates down the fibre at the speed of light, and exists as a spatially extended tail behind it. Due to its time-dependence, it can affect the group velocity of a trailing temporal CS, thus giving rise to long-range interactions. Specifically, if a temporal CS overlaps with a portion of the $\delta n(\tau)$ perturbation that has a negative (positive) gradient, the CS will speed up (slow down), leading to a time-domain drift towards the maxima of the refractive index change induced by the CSs leading it. For the case of two temporal CSs, the perturbation is simply given by the impulse response shown in Fig. \[acoustic\](a). Accordingly, the trailing CS will increase or decrease its separation from the leading one until it coincides with one of the maxima of the response shown in Fig. \[acoustic\](a) [@jang_ultraweak_2013].
When more than two temporal CSs are involved, the dynamical evolution of a particular one is affected by the superposition of refractive index perturbations induced by all the temporal CSs leading it. In general, this superposition can assume a very complex temporal profile. Pilipetskii et al have however numerically shown, in the context of passively mode-locked fibre lasers [@pilipetskii_acoustic_1995], that a large sequence of light pulses may spontaneously form bunches whose separations correspond to the acoustic frequency that interacts the most efficiently with light. To gauge whether this hypothesis is related to our experimental observations, we plot in Fig. \[acoustic\](b) the absolute value of the electrostrictive frequency response in our system, i.e., $|\mathcal{F}\left[\delta n(\tau)\right]|$, where $\mathcal{F}[\cdot]$ denotes Fourier transformation and $\delta
n(\tau)$ is the impulse response shown in Fig. \[acoustic\](a). As can be seen, maximum spectral amplitude is reached at a frequency of 370 MHz, and the corresponding $2.7$ ns period is in very good agreement with the $2.6$ ns bunch spacing observed in the experimental results of Fig. \[expr\]. This strongly suggests that the observed bunching dynamics is indeed due to acoustic interactions of the very large number of temporal CSs.
Simulation model
----------------
To establish quantitatively that electrostriction-induced interactions can explain our experimental observations, we have performed numerical simulations of the underlying dynamics. In Ref. [@jang_ultraweak_2013], a nonlinear partial differential equation was derived that was shown to accurately model the dynamics of temporal CSs and their acoustic interactions. Unfortunately, direct brute force simulations of that model is not computationally feasible here due to the very large number of temporal CSs and the extremely different timescales involved. We instead develop and use a simplified model that de-couples the soliton physics from the acoustic effects [@pilipetskii_acoustic_1995]. Specifically, we represent the entire CS sequence using only the temporal positions $\tau_i$ of the individual solitons ($i=1$, 2, 3, $\ldots$), and we examine how those positions evolve over time under the influence of acoustic waves generated by the corresponding CSs. To this end, we first need to quantitatively establish how the velocity of a CS is modified by a given refractive index perturbation. In this context, we note that temporal CSs in passive cavities react very differently to perturbations than pulses in mode-locked fibre lasers, and we therefore cannot simply use the approach of [@pilipetskii_acoustic_1995].
We start by considering the full partial-differential model of a Kerr cavity (a generalized LLE) that takes acoustic refractive index perturbations into account [@jang_ultraweak_2013]. Assuming the CSs act as Dirac-$\delta$ functions in exciting acoustic waves, the evolution of the intracavity field $E(t,\tau)$ can be written in dimensionless form as [@jang_ultraweak_2013]: $$\centering
\frac{\partial E(t,\tau)}{\partial t} = \left[ -1-i\Delta+i|E|^2+i\frac{\partial^2}{\partial\tau^2}\right]E + S +i\nu(\tau)E.
\label{eq:LL}$$ The normalization of this equation is the same as that used in the Supplementary Information of Ref. [@leo_temporal_2010]. The variable $t$ corresponds to the *slow* time of the resonator that describes evolution of the field envelope $E$ at the scale of a photon lifetime, whilst $\tau$ is a *fast* time describing the temporal profile of the field envelope. The first five terms on the right-hand side of Eq. (\[eq:LL\]) describe, respectively, the total cavity losses, phase detuning of the pump from resonance (with $\Delta$ the detuning coefficient), Kerr nonlinearity, anomalous group-velocity dispersion, and external driving (with $S$ the amplitude of the cw driving field).
The last term on the right-hand side of Eq. (\[eq:LL\]) describes the (normalized) acoustic-induced refractive index perturbation created by the temporal CSs present in the field $E(t,\tau)$. As can be seen, it amounts to introducing a time-dependent perturbation to the cavity detuning $\Delta$. Earlier studies of spatial CSs have revealed that detuning perturbations cause CSs to alter their velocities in proportion to the gradient of the perturbation [@maggipinto_cavity_2000; @caboche_cavity-soliton_2009]. To verify this behaviour, and also to find the proportionality constant for our parameters \[see caption of Fig. \[sim1\]\], we have numerically integrated Eq. (\[eq:LL\]) for a wide variety of different perturbation gradients. Specifically, we ran a set of simulations with detuning perturbations of the form $\nu(\tau) = A\tau$, where the gradients $A$ were chosen to have similar magnitudes to those arising from acoustic effects. For each value of $A$, we started the simulation with a single temporal CS centred at $\tau = \tau_i$, and we extracted the rate at which its temporal position drifts: $V =
\mathrm{d}\tau_i/\mathrm{d} t$.
Figure \[sim1\] shows results from these simulations. A linear relationship between the CS drift rate $V$ and the detuning gradient is evident. For our experimental conditions, we can thus approximate $V =
\mathrm{d}\tau_i/\mathrm{d}t \approx r\,\mathrm{d}\nu/\mathrm{d}\tau|_{\tau_i}$, with the proportionality constant $r\approx 1.43$. Transforming to dimensional units, we find that the CS temporal positions $\tau_i$ obey the following first-order ordinary differential equation $$\centering
\frac{\mathrm{d}\tau_i}{\mathrm{d}t}=r\frac{|\beta_2|L^2\mathcal{F}}{t_\mathrm{R}\lambda_0}
\left.\frac{\mathrm{d}n_{\mathrm{tot}}}{\mathrm{d}\tau}\right|_{\tau_i}.
\label{taui}$$ Here $\beta_2$ is the fibre group-velocity dispersion coefficient, $L$ is the cavity length, $\mathcal{F}$ is the cavity finesse, $t_\mathrm{R}$ is the cavity roundtrip-time, and $\lambda_0$ is the wavelength of the driving field. Finally, $n_{\mathrm{tot}}(\tau)$ corresponds to the total acoustic-induced refractive index perturbation existing in the cavity. It is given by $$n_{\mathrm{tot}}(\tau) = \sum_{j}\delta n(\tau-\tau_j) + \sum_{j}\delta n(t_\mathrm{R}+\tau-\tau_j).
\label{ntot}$$ where $\delta n(\tau)$ is the impulse response introduced above. Given the causal nature of $\delta n(\tau)$, the first term represents simply the superposition of the index perturbations induced by all CS present before time $\tau$. For single-pass propagation through an optical fibre, this term alone would appear. For a fibre cavity, one must however also take into account the periodic nature of the boundary conditions. Specifically, a temporal CS completing its $m^{th}$ roundtrip across the cavity may be affected by index perturbations induced by CSs during the $(m-1)^{th}$ roundtrip. This is accounted for by the second term in Eq. (\[ntot\]). Note that our cavity roundtrip time $t_\mathrm{R} = 480$ ns is much longer than the lifetime of the acoustic waves \[see Fig. \[acoustic\](a)\], and therefore only CS present at temporal positions behind time $\tau$ contribute to this term in practice. For the same reason, perturbations created more than one roundtrip earlier do not need to be considered.
Simulation results
------------------
Equations \[taui\] and \[ntot\] make it possible to efficiently simulate the acoustic interactions of an arbitrary sequence of temporal CSs. Figure \[sim2\] shows results from numerical integration using parameters corresponding to our experiment above \[and listed in the caption of Fig. \[sim2\]\]. Since extracting the precise random CS sequence that was excited in our experiment is difficult, we assume here that the cavity initially contains 2000 CSs whose temporal positions follow a uniform random distribution.
Figures \[sim2\](a) and (c) show 50-ns-long snapshots of the temporal intensity profiles of the CS sequences at the beginning and end of the simulation, respectively, while Fig. \[sim2\](b) reveals the full dynamics over 15 s, using the same representation as for the experimental data in Fig. \[expr\]. To facilitate visualization, and to mimic the temporal resolution of the oscilloscope, each CS is represented as a $\mathrm{sech}$ profile with 80 ps full-width at half maximum. The simulated CS interaction dynamics is clearly in excellent agreement with the experimental observations. In particular, we see that the initial random sequence \[Fig. \[sim2\](a)\] self-organizes into regularly-spaced bunches \[Fig. \[sim2\](c)\]. We note that the about 3 ns average bunch spacing observed at the output of our simulations is somewhat larger than the $2.6$ ns experimental figure. This discrepancy is attributed to an imperfect knowledge of the acoustic impulse response that was already identified in [@jang_ultraweak_2013]. Compounded by further uncertainties in the initial CS configuration and other experimental parameters, this could also explain the slightly different time-scale over which the bunching takes place in our experiment in comparison to simulations.
To further confirm our interpretation, we have plotted as a red dash-dotted line in Fig. \[acoustic\](c) the total refractive index perturbation $n_{\mathrm{tot}}(\tau)$ at the end of the simulation. This perturbation assumes an almost sinusoidal shape, and its approximately 3 ns period remarkably matches the simulated bunch spacing. It is also very clear from the dynamical evolution trajectories \[Fig. \[sim2\](b)\] that the temporal CS bunches experience an overall drift towards the perturbation maxima. However, even when continuing the simulations over much longer time scales, the bunches never reach the maxima, and always stay slightly offset from them \[this is already visible in Fig. \[sim2\](c)\]. In that way, the bunches eventually reach a quasi-stationary state in which they all drift with a non-zero near-constant velocity, chasing the maxima. Of course, the maxima themselves keep shifting in the same direction, as they are constantly re-formed by the drifting temporal CSs. This probably constitutes a general feature of this kind of interaction. Moreover, we suspect that the drift of the refractive index perturbation explains why the solitons in each bunch do not come arbitrarily close to each other — in our simulations, the CSs stay spaced by some tens of picoseconds within a bunch. If the refractive index perturbation was stationary, the CSs present within each bunch would meet at a maximum, where they would merge into one or annihilate [@jang_controlled_2015]. In that scenario, the bunches would progressively disappear, which is clearly not consistent with our experimental observations (see Fig. \[expr\]).
Conclusions
===========
To conclude, we have experimentally and theoretically studied the acoustic dynamics of a very large number of temporal CS in a coherently-driven passive fibre resonator. Our experiment reveals that the CSs exhibit complex interactions, resulting in the formation of almost regularly-spaced bunches, each made up of multiple CSs. To explain our observations, we have developed a simple theoretical framework that allows the electrostriction-induced long-range interactions of arbitrary temporal CS sequences to be simulated. Numerical results are in very good agreement with experimental observations, confirming that the observed bunching dynamics originate from the excitation of transverse acoustic waves. In addition to unveiling a new dynamical behaviour of temporal CS ensembles, our results quantitatively confirm the 1995 theoretical predictions of Pilipetskii et al concerning pulse bunch formation via electrostriction-induced interactions. We expect our result to greatly expand our understanding of temporal CSs and their interactions, as well as the manner in which sound interacts with long sequences of ultrafast pulses of light.
Acknowledgements {#acknowledgements .unnumbered}
================
We acknowledge support from the Finnish Cultural Foundation and the Marsden Fund of The Royal Society of New Zealand.
References {#references .unnumbered}
==========
[10]{}
Wabnitz S 1993 [*Opt. Lett.*]{} [**18**]{} 601–3
Leo F, Coen S, Kockaert P, Gorza S P, Emplit P and Haelterman M 2010 [ *Nature Photon.*]{} [**4**]{} 471–6
Kivshar Y S and Agrawal G 2003 [*Optical solitons: From fibers to photonic crystals*]{} (Amsterdam, Boston:Academic Press)
Agrawal G P 2006 [*Nonlinear fiber optics*]{} (Academic Press)
Nicolis G and Prigogine I 1977 [*Self-organization in nonequilibrium systems: From dissipative structures to order through fluctuations*]{} (New York:Wiley)
Akhmediev N N and Ankiewicz A (eds) 2008 [*Dissipative solitons: From optics to biology and medicine*]{} (Berlin Heidelberg:Springer)
Purwins H G, B[ö]{}deker H U and Amiranashvili S 2010 [*Adv. Phys.*]{} [**59**]{} 485–701
Firth W J and Weiss C O Feb 2002 [*Opt. Photonics News*]{} [**13**]{} 54–8
Firth W 2010 [*Nature Photon.*]{} [**4**]{} 415–7
Marconi M, Javaloyes J, Balle S and Giudici M 2014 [*Phys. Rev. Lett.*]{} [ **112**]{} 223901
Jang J K, Erkintalo M, Murdoch S G and Coen S 2015 [*arXiv*]{} 1501.05289
Jang J K, Erkintalo M, Coen S and Murdoch S G 2015 [*Nat. Commun.*]{} [**6**]{} 7370
Ackemann T, Firth W J and Oppo G L 2009 [*Adv. At. Mol. Opt. Phys.*]{} [**57**]{} 323–421
Barland S, Tredicce J R, Brambilla M, Lugiato L A, Balle S, Giudici M, Maggipinto T, Spinelli L, Tissoni G, Kn[ö]{}dl T, Miller M and J[ä]{}ger R 2002 [*Nature*]{} [**419**]{} 699–702
Lugiato L A and Lefever R 1987 [*Phys. Rev. Lett.*]{} [**58**]{} 2209–11
McDonald G S and Firth W J 1990 [*J. Opt. Soc. Am. B*]{} [**7**]{} 1328–35
Tlidi M, Mandel P and Lefever R 1994 [*Phys. Rev. Lett.*]{} [**73**]{} 640–3
Firth W J and Scroggie A J 1996 [*Phys. Rev. Lett.*]{} [**76**]{} 1623–6
Lugiato L A 2003 [*IEEE J. Quantum Elec.*]{} [**39**]{} 193–6
Leo F, Gelens L, Emplit P, Haelterman M and Coen S 2013 [*Opt. Express*]{} [ **21**]{} 9180–91
Jang J K, Erkintalo M, Murdoch S G and Coen S 2013 [*Nature Photon.*]{} [**7**]{} 657–63
Jang J K, Erkintalo M, Murdoch S G and Coen S 2014 [*Opt. Lett.*]{} [**39**]{} 5503–6
Coen S, Randle H G, Sylvestre T and Erkintalo M 2013 [*Opt. Lett.*]{} [**38**]{} 37–9
Coen S and Erkintalo M 2013 [*Opt. Lett.*]{} [**38**]{} 1790–2
Chembo Y K and Menyuk C R 2013 [*Phys. Rev. A*]{} [**87**]{} 053852
Erkintalo M and Coen S 2014 [*Opt. Lett.*]{} [**39**]{} 283–6
Godey C, Balakireva I V, Coillet A and Chembo Y K 2014 [*Phys. Rev. A*]{} [ **89**]{} 063814
Herr T, Brasch V, Jost J D, Wang C Y, Kondratiev N M, Gorodetsky M L and Kippenberg T J 2014 [*Nature Photon.*]{} [**8**]{} 145–52
Brasch V, Herr T, Geiselmann M, Lihachev G, Pfeiffer M H P, Gorodetsky M L and Kippenberg T J 2014 [*arXiv*]{} 1410.8598
Del’Haye P, Schliesser A, Arcizet O, Wilken T, Holzwarth R and Kippenberg T J 2007 [*Nature*]{} [**450**]{} 1214–7
Kippenberg T J, Holzwarth R and Diddams S A 2011 [*Science*]{} [**332**]{} 555–9
Moss D J, Morandotti R, Gaeta A L and Lipson M 2013 [*Nat Photon*]{} [**7**]{} 597–607
Zabusky N J and Kruskal M D 1965 [*Phys. Rev. Lett.*]{} [**15**]{} 240–3
Stegeman G I and Segev M 1999 [*Science*]{} [**286**]{} 1518–23
Rotschild C, Alfassi B, Cohen O and Segev M 2006 [*Nature Phys.*]{} [**2**]{} 769–74
Barashenkov I V, Smirnov Y S and Alexeeva N V 1998 [*Phys. Rev. E*]{} [**57**]{} 2350–64
Sch[ä]{}pers B, Feldmann M, Ackemann T and Lange W 2000 [*Phys. Rev. Lett.*]{} [**85**]{} 748–51
Tlidi M, Vladimirov A G and Mandel P 2003 [*IEEE J. Quantum Elec.*]{} [**39**]{} 216–26
B[ö]{}deker H U, Liehr A W, Frank T D, Friedrich R and Purwins H G 2004 [ *New J. Phys.*]{} [**6**]{} 62
Parra-Rivas P, Gomila D, Mat[í]{}as M A, Coen S and Gelens L 2014 [*Phys. Rev. A*]{} [**89**]{} 043813
Jang J K, Erkintalo M, Luo K, Oppo G L, Coen S and Murdoch S G 2015 [*arXiv*]{} 1504.07231
Boyd R W 2008 [*Nonlinear optics, third edition*]{} (Amsterdam Boston:Academic Press)
Smith K and Mollenauer L F 1989 [*Opt. Lett.*]{} [**14**]{} 1284–6
Dianov E M, Luchnikov A V, Pilipetskii A N and Prokhorov A M 1992 [*Appl. Phys. B*]{} [**54**]{} 175–80
Jaou[ë]{}n Y and du Mouza L 2001 [*Opt. Fib. Tech.*]{} [**7**]{} 141–69
Grudinin A, Richardson D and Payne D 1993 [*Electron. Lett.*]{} [**29**]{} 1860–1
Pilipetskii A N, Golovchenko E A and Menyuk C R 1995 [*Opt. Lett.*]{} [**20**]{} 907–9
Grudinin A B and Gray S 1997 [*J. Opt. Soc. Am. B*]{} [**14**]{} 144–54
Tang D Y, Zhao B, Shen D Y, Lu C, Man W S and Tam H Y 2002 [*Phys. Rev. A*]{} [**66**]{} 033806
Zhao L M, Tang D Y, Cheng T H, Lu C, Tam H Y, Fu X Q and Wen S C 2009 [*Opt. Quantum Elec.*]{} [**40**]{} 1053–64
Chouli S and Grelu P 2009 [*Opt. Express*]{} [**17**]{} 11776–81
Kutz J, Collings B, Bergman K and Knox W 1998 [*IEEE J. Quantum Elec.*]{} [ **34**]{} 1749–57
Korobko D A, Okhotnikov O G and Zolotovskii I O 2015 [*Opt. Lett.*]{} [**40**]{} 2862–5
Shelby R M, Levenson M D and Bayer P W 1985 [*Phys. Rev. Lett.*]{} [**54**]{} 939–42
Shelby R M, Levenson M D and Bayer P W 1985 [*Phys. Rev. B*]{} [**31**]{} 5244–52
Maggipinto T, Brambilla M, Harkness G K and Firth W J 2000 [*Phys. Rev. E*]{} [**62**]{} 8726–39
Caboche E, Barland S, Giudici M, Tredicce J, Tissoni G and Lugiato L A 2009 [*Phys. Rev. A*]{} [**80**]{} 053814
|
---
author:
- |
V. Antonelli$^{a,b}$, E. Torrente-Lujan$^{c}$\
[*$^{a}$ Dip. di Fisica, Università degli Studi di Milano, Milano, Italy\
$^{b}$ I.N.F.N., Sezione di Milano, Milano, Italy\
$^{c}$ Dept. de Fisica, Universidad de Murcia, Murcia, Spain\
*]{}
title: Statistically improved Analysis of Neutrino Oscillation Data with the latest KamLAND results
---
Introduction
============
Evidence of antineutrino disappearance in a beam of antineutrinos in the Kamland experiment has been presented [@klJun]. The analysis of previous experimental results on reactor physics and solar neutrinos [@klothers] in terms of neutrino oscillations has largely improved our knowledge of neutrino mixing. Thus, the solar neutrino data evidence prior to autumn of 2003 converged to two distinct allowed regions in parameter space, often referred to as LMAI (centered around the best-fit point of $\Delta m^2_{\odot}=7.1\times10^{-5} \eV^2$, $\tan^2\theta_{\odot}=0.47$) and LMAII (centered around $\Delta m^2_{\odot}=1.5\times 10^{-4} \eV^2$, $\tan^2\theta_{\odot}=0.48$). The inclusion of SNO phase II data eliminated the LMAII region at about $4\sigma$.
The KamLAND measurementes presented recently [@klJun] corresponds to the first about 317 days live time, they confirm the conclusions obtained from previous data and give a more stringent limit on the neutrino mass parameters. As we will see in this work, the new KamLAND information further excludes the LMAII region now at $\approx$ 5$\sigma$.
In addition to an increased statistics, a significant change in the KamLAND analysis technique is related to the fiducial volume definition. Whereas in the previous setup, events taking place at the outer edge of the nylon balloon were rejected [@klJun; @Eguchi:2002dm], the recent analysis adopts a more sophisticated coincidence-measurement technique to exclude unwanted backgrounds. Additionally, a better understanding of the fuel cycle on the reactors has lead to the collaboration to estimate the incoming neutrino flux with better accuracy: the estimated error on this initial flux $\phi_0\left(\overline{\nu}_{e}\right)$ is now of the order of 2%.
The aim of this work is to present a comprehensive updated analysis of all recent solar neutrino data including the KamLAND reactor-experiment results to determine the extent of the remaining viable region in the parameter space and to obtain favoured values for the neutrino physical parameters in a two-neutrino framework, and, with the inclusion of results coming from atmospheric oscillation evidence give an estimation of the elements of the three neutrino mass matrix. Some key analysis novelties are presented along this work, for example regarding the treatment of sistematic correlations in the analysis of the SNO spectrum (day+night) and a improved analysis of the KL data with an appropriate consideration of its low statistics data bins.
The structure of this paper is the following: in section [\[kamland\]]{} discuss our approach to the latest KamLAND results and older solar evidence. All solar neutrino experiments are discussed in section [\[solar\]]{}. We specially discuss the importance of the SNO data and the spectrum results. We then proceed, in section [\[analysis\]]{}, to explain the procedure adopted in our analysis with some emphasys on the special treatment needed by the KamLand and SNO spectra and in section [\[results\]]{} we present our results. Finally we summarize and conclude in section [\[summary\]]{}.
The KamLAND and solar evidence {#kamland}
==============================
\[simulations\]
Reactor anti-neutrinos with energies above 1.8 MeV produced in some 53 commercial reactors are detected in the KamLAND detector via the inverse $\beta$-decay reaction $\overline{\nu}_{e}+p\rightarrow n + e^{+}$. The mean reactor-detector distance and energy window of these $\overline{\nu}_{e}$ makes KamLAND an ideal testing ground for the LMA region of the $\nu_{\odot}$ parameter space. The first results published by the KamLAND collaboration eliminated all possible solutions to the solar neutrino problem (SNP) except the LMA region of the parameter space [@Eguchi:2002dm]. The sensitivity of this experiment to the $\Delta m^2$ parameter divided the previously whole LMA region into two distinct regions, the one relative to the smaller mass-squared difference being preferred by data [@postKamland].
In our analysis (see also Ref.[@torrente] for further details) we model the reactor incoming flux through a constant, time-averaged fuel composition for all of the commercial reactors within detectable distance of the Kamioka site, namely ${}^{235}\text{U}=56.3\% $, ${}^{238}\text{U}=7.9\% $, ${}^{239}\text{Pu}=30.1\% $, and ${}^{241}\text{Pu}=5.7\% $. We used the full cross-section including electron recoil corrections. We analyzed the data above threshold of $2.6 \MeV$, as the low-energy end of the spectrum had effectively no events. The information relative to the no-oscillation initial flux for the two periods can be extracted from fig. (1.a) of [@klJun]. We neglect all backgrounds, including geological background above 2.6 MeV. We use the resolutions published by the collaboration for the two different data sets, namely $\sigma\left(E\right) = 6.2\%/\sqrt{E}$ for the recent data (post upgrade) and $\sigma\left(E\right) = 7.3\%/\sqrt{E}$ for earlier data (pre-upgrade). The total systematic error is estimated at $6.5 \%$.
In order to use all the data available, we use a simple MC simulation to estimate an equivalent efficiency for the two pre-upgrade and post-upgrade phases. Finally, no matter effects are taken into consideration for the KamLAND data alone, as it was shown that, for this experiment, any asymmetry due to matter effects is negligible for $\Delta m^2$ of the order of $10 \times 10^{-5}$ eV$^2$.
Solar data {#solar}
----------
The most ponderous data present in our analysis come from the solar neutrino experiments. The experimental results are compared to an expected signal which we compute numerically by convoluting solar neutrino fluxes [@bpb2001], sun and earth oscillation probabilities, neutrino cross sections and detector energy response functions. We closely follow the same methods already well explained in previous works [@Aliani:2001zi; @Aliani:2001ba; @Aliani:2002ma; @Aliani:2002rv], we will mention here only a few aspects of this computation. We determine the neutrino oscillation probabilities using the standard methods found in literature [@torrente], as explained in detail in [@Aliani:2001zi] and in [@Aliani:2002ma]. We use a thoroughly numerical method to calculate the neutrino evolution equations in the presence of matter for all the parameter space. For the solar neutrino case the calculation is split in three steps, corresponding to the neutrino propagation inside the Sun, in the vacuum (where the propagation is computed analytically) and in the Earth. We average over the neutrino production point inside the Sun and we take the electron number density $n_e$ in the Sun by the BPB2001 model [@bpb2001]. The averaging over the annual variation of the orbit is also exactly performed. To take the Earth matter effects into account, we adopt a spherical model of the Earth density and chemical composition [@torrente]. The joining of the neutrino propagation in the three different regions is performed exactly using an evolution operator formalism [@torrente]. The final survival probabilities are obtained from the corresponding (non-pure) density matrices built from the evolution operators in each of these three regions.
In summary, double-binned day-night and zenith angle bins are computed in order to analyze the full SuperKamiokande data [@Smy:2002fs], whereas single-binned data is used for the SNO detector [@newSNO; @Poon:2001ee; @Ahmad:2002jz]. The global signals only are used for the radiochemical experiments Homestake [@homestake], SAGE [@sage1999; @sage], GallEx [@gallex] and GNO [@gno2000]. The next paragraphs are dedicated to a description of SNO component of the solar evidence.
\[NaCl\]
The Sudbury Neutrino Observatory (SNO) collaboration has presented data relative to the NaCl phase of the experiment [@newSNO; @newSNO03]. We remind that the addition of NaCl to a pure $\text{D}_{2}\text{O}$ detection medium has the effect of increasing the detector’s sensitivity to the neutral-current (NC) reactions within its fiducial volume. The NC detection efficiency has changed from a previous ’no-salt’ phase of approximately a factor three. This and other novelties have made it possible for the SNO collaboration to analyze their data without making use of the no-spectrum-distortion hypothesis. Furthermore, they have adopted a new ’event topology’ criterion [@newSNO03] to distinguish among the different channels within the detector. The SNO Collaboration has now devised a new data-analysis technique which relies on the topology of the three different events. The new parameter ($\beta_{\ell}$) relative to which they marginalize is known as the ’isotropy’ of the Cerenkov light distribution was used to separate the CC, ES and NC signals, something that was not possible in th previous two data sets. The measured fluxes are reported in [@newSNO].
The comparison of the new SNO results and the previous phase-II data can easily be made because the SNO collaboration has included in the recent paper results which were obtained following their previous method, along with the new unconstrained data. The new results are compatible with the previous ones. It seems that the overall effect of un-constraining the analysis is an increase in the measured fluxes, although the estimated total $\Phi_{\text{B}}$ has decreased relative to the previous best-fit value, leaving even less space for eventual sterile neutrinos. We make use of all day+night published data and refer the reader to section [\[analysis\]]{} for details. The procedure we used to introduce the SNO spectrum data is an extention of the one use for the SK spectrum analysis and is explained in the next section.
Our analysis {#analysis}
============
We use standard statistical techniques to test the non-oscillation hypothesis. Two different sets of analyses are possible with the present data on neutrino oscillations: 1) short-baseline reactor data, solar data including the SK spectrum and previous phase-I (CC only) SNO spectrum, phase-II SNO global result, combined with new the KamLAND spectrum and, 2) the previous set with the use of the phase-II SNO spectrum result and the new KamLAND data. In order to use all the SNO data, we consider the phase-I and phase-II results as two distinct but fully correlated experiments.
For the purpose of this analysis, a $\chi^{2}$ function is defined which is the the sum of the distinct contributions. The contribution of all solar neutrino experiments is summarized in the term: $$\chi^{2}_{sun}=\chi_{rad}+\chi_{SK}^{2}+\chi^2_{\text{SNO}}.$$ Where, the $\chi^{2}$ function for the global rates of the radiochemical experiments is as follows, $$\label{chi_radiochemical}
\chi^{2}_{\text{rad}}=\left(\mathbf{R}^{\text{th}}-\mathbf{R}^{\text{exp}}\right)^{T}\left(\sigma_{\text{sys}}+\sigma_{\text{stat}}\right)^{-1}\left(\mathbf{R}^{\text{th}}-\mathbf{R}^{\text{exp}}\right),$$ where $\mathbf{R}^{\text{th,exp}}$ are length-two vectors containing the theoretical (or experimental) signal-to-no-oscillation expectation for the chlorine and gallium experiments. Correlated systematic and uncorrelated statistical errors are considered in $\sigma_{\text{syst}}$ and $\sigma_{\text{stat}}$ respectively. Note that the parameter-dependent $R^{\text{th}}$ is an averaged day-night quantity, as the radiochemical experiments are not sensitive to day-night variations. For the next component We consider the double-binned SK spectrum comprising of 8 energy bins for a total of 6 night bins and one day bin. The $\chi^{2}$ is given by $$\label{chi_spectrum_sk}
\chi^2_{\text{SK}}= (\alpha\mathbf{R}^{\text{th}} -\mathbf{R}^{\text{exp}})^T
\left (\sigma^{2}_{\text{unc}} + \sigma^{2}_{\text{cor}}\right )^{-1}
(\alpha\mathbf{R}^{\text{th}}-\mathbf{R}^{\text{exp}}).$$ The covariance matrix $\sigma$ is a 4-rank tensor containing information relative to the statistical errors and energy and zenith-angle bin-correlated and uncorrelated uncertainties. Since the publication of the first SNO NC results, we have adopted their estimate of $\phi_{B}$ and incorporated the new parameter $\alpha$ in the $\chi^2$ representing the normalization with respect to this quantity. In determining our best-fit points, we minimize with respect to it. Note that the quantities $\mathbf{R}^{\text{exp}}$ and $\mathbf{R}^{\text{th}}$ contain the number of events normalized to the no-oscillation scenario.
We deal next with the SNO component. We present two different analysis of some of the phase II SNO data sets including total day/night quantities.
The first consideres the global signal alone, the second incorporates the total signal spectrum. We consider here the two SNO results as if coming from two independent experiments, but fully correlated. We use the backgrounds as listed in tables X of Ref.[@newSNO] and 1 of Ref.[@Ahmad:2002jz] for the phase-II data. The detector resolution is obtained from Refs. [@newSNO; @howto].
In the second case we make use of the spectral data. The spectrum used for our analusis is presented in table \[\[sno\_spectrum\]\]). The $\chi_{\text{SNO}}^{2}$ has the same formal expression as before, where it is understood that $\mathbf{R}^{\text{th,exp}}$ are now length 32 containing two 16-bin relative to the two SNO data sets. We consider the two fully correlated.
The main difficulty in using the total spectrum data lies in correctly estimating the, highly correlated, systematic error. By using the information contained in tables XIX and XX of Ref.[@newSNO], we have computed the influence of all the different sources of error on our response function considering the correlation/anti-correlation as presented in table 1 of [@howto]. The different backgrounds spectral correlations are included from table XXXIV of Ref.[@newSNO]. The procedure we used to introduce the SNO spectrum data is an extension of the one used for the SK spectrum analysis. For each point in the parameter space $\Delta m^2$, $\theta$ we start from a correlation matrix obtained by using the non-deformed spectrum assumption. We calculate, for each bin, the sum of the signals ES+NC+CC and we extract weights for each single contributions. After that, we compare our theoretical results with the ones given by the SNO collaboration and impose a 3-$\sigma$ cut. By using these zero-order weights and the correlation errors obtained by the SNO table, we reconstruct a correlation matrix. The correlation matrix is introduced into the $\chi^2$ analysis by adding a free parameter $\delta_{cor}$ which is determined in a minimization process togethed with the weights of the single $i=ES+NC+CC$ contributions to the signal:
$$\begin{aligned}
\chi^2_{SNO}&=&
\sum_{i}(\alpha R^{th}- R^{exp})^t
(\sigma^2_{unc}+\delta_{cor}\sigma^2_{cor})^{-1}
(\alpha R^{th}-R^{exp}) \\
&&+ \chi^2_{\alpha} + \chi^2_{\delta_{cor}},\end{aligned}$$
The full process is designed to be interated a number of times, in practise we obtain that after two iterations the process is convergent and give us the desired results.
The Kamland statistical analysis
--------------------------------
The total KamLAND contribution to the $\chi^2$ is defined as: $$\label{chi_kamland}
\chi^2_{\text{KL}} = \chi^2_{\text{KL,glob}}
+\chi^2_{\text{KL},\lambda}$$ where the global contribution is simply $$\label{chi_kamland_both}
\chi^2_{\text{KL,glob}}=
\frac{\left(R^{\text{th}}-R^{\text{exp}}\right)^2}{\sigma_{\text{stat+sys}}^2}.$$
The statistical consideration of the KL spectrum signal, $\chi^2_{KL,\lambda}$, is however worthy of special attention. Due to the fact that at high energy KamLAND observes a small number of events alternatives should be used instead the Gaussian approximation. This means among other things that the correlated systematic deviations cannot be introduced in a straightforward way. Due to these reasons, we use an alternative technique for the KamLAND data bins. A detailed account of some statistical considerations is presented in the Appendix.
It is possible to present an unified approach [@read88] to all the commonly used multinomial models (Pearson’s, log-likelihood among them) by defining a family of statistics $\chi^2(\lambda)$ for testing the fit of observed frequencies $R_i^{exp}$ to the expected ones $R_i^{th}$ [@read88]. All the statistics belonging to this family have similar well-behaved properties but however results as best fit parameters and exclusion regions may significantly depend on the use of one or another. Any decision as to which member of the family we should use to finally test the null hypothesis must depend on the type of the departure we wish to detect. The sensitivity of the statistic depends on how the defining function treats the large or small deviations.
Based on a comparative study it is recomended [@read88; @stat] to use $\chi^2(2/3)$ as a compromise candidate among the different test statistics optimizing diverse criteria as rate of convergence, sensitivity to the sample size and sensitivity to large or small bin deviations. The statistic corresponding to this value, the Read statistic, is the one used in this work: [$$\begin{aligned}
\chi^2_{KL}\left (\lambda=\frac{2}{3}\right )&=&
\frac{9}{5}
\sum_i R_i^{exp} \left ( \left( \frac{R_i^{exp}}{R_i^{th}}\right )^{2/3} - 1
\right )\\
&&
+\frac{2}{3} \left (R_i^{th}-R_i^{exp}\right ).\end{aligned}$$ ]{}
In the evaluation of $\chi^2_{\text{KL},\lambda}$ we use vectors that comprise therefore of 13 spectral points of width 0.425 MeV.
Results and Discussion. {#results}
=======================
To test a particular oscillation hypothesis against the parameters of the best fit (null hypothesis) and obtain allowed regions in parameter space we perform a minimisation of the full function $\chi^2$ with respect the oscillation and the rest of ancillary parameters. A given point in the oscillation parameter space is allowed if the globally subtracted quantity fulfills the condition $\Delta \chi^2=\chi^2 (\Delta m^2, \theta)-\chi_{\rm min}^2<\chi^2_n(CL)$. Where $\chi^2_(90\%,95\%,...)$ are the respective quantiles. In this way we obtain best fit mass differences and angles and joint exclusion regions. Additionally, we perform a second kind of analysis in order to obtain concrete values for the individual oscillation parameters and estimates for their uncertainties. We study the marginalised parameter constraints where the $\chi^2$ quantity is converted into likelihood using the expression ${\cal L}/{\cal L}_0=$ $e^{-(\chi^2-\chi_{min}^2)/2}$.
In table \[\[bestfitpoints\]\] we report the values of the mixing parameters $\Delta m^2_{\odot}$, $\tan^2\theta_{\odot}$, and the $\chi^2$ obtained from minimization and from the peak of marginal likelihood distribution.
The results are shown in Figs.\[fexclusions\] where we have generated acceptance contours in the $\Delta m^2$-$\tan^2\theta$ plane. In fig. \[\[fexclusions\]-(left)\] we show the exclusion plots for the solar, radiochemical + Cerenkov solar data and KamLAND with the global signal of the SNO phase-II data, whereas the right panel refers to the KamLAND spectrum, radiochemical + Cerenkov solar data and the SNO phase-II spectrum information. Contour lines correspond to the the allowed areas at 90, 95, 99 and 99.7% CL relative to the absolute minimum.
Thes normalized marginal likelihood, obtained from the integration of ${\cal L}$ for each of the variables, is plotted in Figs. (\[fmarginal\]) for each of the oscillation parameters $\Delta m^2$ and $\tan^2\theta$. Concrete values for the parameters are extracted by fitting one- or two-sided Gaussian distributions to any of the peaks (fits not showed in the plots). In both cases, for angle and the mass difference distributions the goodness of fit of the Gaussian fit to each individual peak is excellent (g.o.f $\sim 100\%$). The errors obtained from this method are assigned to the $\chi^2$ minimisation values. The central values are fully consistent and very similar to the values obtained from simple $\chi^2$ minimisation. Systematics variability of these results can come from the use of a different prior information or mixing parameterizations, however this variability or systematic error due to the procedure is small. We will again use the technique of marginal distributions in the next paragraphs to obtain an estimation of the individual elements of the neutrino mass matrix and their errors.
The main difference with previous analysis is a better resolution in parameter space. The previously two well separated solutions LMAI,LMAII have now completely disappeared. In particular the secondary region at larger mass differences (LMAII) is now completely excluded.
The introduction of the new KamLand data in general strongly diminishes the favored value for the mixing angle with respect to the KamLAND result alone [@Aliani:2002na]. The final value is more near to those values favored by the solar data alone than to the KamLAND ones. As an important consequence, the combined analysis of solar and KamLAND data concludes that maximal mixing is not favored at $\sim 4-5\sigma$. This conclusion is not supported by the antineutrino, earth-controlled, conceptually simpler KamLAND results alone. As we already pointed out in Ref.[@Aliani:2002na], this effect could be simply due to the present low KamLAND statistics or, more worrying, to some statistical artifact derived from the complexity of the analysis and of the heterogeneity of binned data involved.
An estimation of the neutrino mass matrix
-----------------------------------------
We proceed now to an estimation of the neutrino mass matrix in different aproximations. Our main objective is however to estimate how well the individual errors of the mass matrix can be extracted already at present by the existing experimental evidence. For this purpose we have applied similar arguments as those used before to obtain marginal distributions and errors for individual parameters from them.
The square of the neutrino mass matrix can be written in the flavour basis as $M^2=U M_D^2 U^\dagger $ where $M_D$ is diagonal and $U$ is an unitary (purely active oscillations are assumed) mixing matrix. Subtracting one of the diagonal entries we have $$M^2=m_1^2 I+M_0^2= m_1^2 I + U M_D^{\prime 2} U^\dagger,$$ where $I$ is the identity matrix. In this way we distinguish in the mass matrix a part, $M_0^2$, which affects and can be determined by oscillation experiments and another one, $m_1^2 I$, which does not. Evidently, the off-diagonal elements of the mass matrix are fully measurable by oscillation experiments.
First, we restric ourselves for the sake of simplicity to two neutrino oscillations, we have in this case [$$\begin{aligned}
M^2=m_1^2 I+M_0^2&=& m_1^2 I + \Delta m^2
\pmatrix{\sin^2\theta & \sin\theta \cos\theta \cr
\sin\theta \cos\theta & \cos^2\theta }
\label{eq1000}\end{aligned}$$ ]{} with $\Delta m^2=m_2^2-m_1^2$. The individual elements of the matrix $M_0$ can simply be estimated from the oscillation parameters obtained before. For example for $\tan^2\theta\sim 0.40$, $\Delta m^2\sim 7-8\times 10^{-5}\ eV^2$ we would obtain $(M_0^2)_{22}\sim 5-6\times 10^{-5}\ eV^2$.
Using again as likelihood function the quantity ${\cal L}/{\cal L}_0(\Delta m^2,\tan^2\theta) = $ $ e^{-(\chi^2-\chi_{min}^2)/2}$ we obtained the individual probability distributions for any of the elements of the matrix $M_0$. Average values and $1\sigma$ errors are obtained from two-sided Gaussian fits to these distributions. From this procedure we obtain: $$\begin{aligned}
M_0^2&=&
10^{-5}\ eV^2
\pmatrix{ 2.05^{+0.25}_{-0.26} & 3.12^{+0.25}_{-0.26}\cr
3.12^{+0.25}_{-0.26} & 4.50^{+0.51}_{-0.40}}.
\label{eq1001}\end{aligned}$$
One can go further supposing a concrete value for $m_1^2$ from elsewhere. If we take $m_1^2 >> \Delta m^2$ then we can directly write the mass matrix $$\begin{aligned}
M&=&m_1 I+\frac{1}{2 m_1} M_0^2.\end{aligned}$$ Supposing for example $m_1=1\ eV$, [$$\begin{aligned}
M&=& eV
\pmatrix{ 1.0+1.02^{+0.12}_{-0.12}\ 10^{-5}& 1.56^{+0.12}_{-0.13}\
10^{-5}\cr
1.56^{+0.12}_{-0.13}\ 10^{-5}& 1.0+2.25^{+0.25}_{-0.20}\ 10^{-5}}.\end{aligned}$$ ]{} this is the final two neutrino mass matrix which can be obtained from present oscillation evidence coming from solar and reactor neutrinos.
We obtain now an estimation of the three neutrino mass matrix. For this purpose we make the same reasoning as before and introduce the existing evidence of the individual values of the two additional angles and the square mass difference. Naturally knowledge of these parameters is still very poor and the elements of the final mass matrix will have much larger errors. First we substract a diagonal part and write the square mass matrix $M^2$ as: $$\begin{aligned}
M^2=m_2^2 I+M_0^2&=& m_2^2 I + \Delta m^2_{12} M_{12}^2
+\Delta m^2_{32} M_{32}^2\end{aligned}$$ with $\Delta m^2_{ij}=m_i^2-m_j^2$.
We write the mixing matrix as a product of three single rotations around each of the axis: $$U=u_{12}(\theta_{12})u_{23}(\theta_{23})u_{13}(\theta_{13}).$$ With this notation the matrices $M_{12},M_{32}$ are written $$M_{12}= (u_{23} u_{13})^t M_0^2 u_{23} u_{13},$$ $$M_{32}= (u_{23} u_{13})^t M_3 u_{23} u_{13}$$ where $M_3=Diag(0,0,1)$. The matrix $M_{32}$ does not depend on the angle $\theta_{12}$. The dependence on this angle is fully contained in $M_0$ which is the $3\times 3$ enlarged version of the $M_0$ matrix appearing in Eqs.(\[eq1000\],\[eq1001\]).
We take the best values for $\Delta m_{32}$ known at present (see for example Ref.[@neutrinos3] and references therein) and from CHOOZ evidence [@chooznew; @PaloVerde] the value of the $(13)$ angle: $ 1.3 \,\times\, 10^{-3}\,\mbox{eV}^2\,\leq\, |\Delta m^2_{atm}|\, \leq \,3.1\, \times\,10^{-3}\,\mbox{eV}^2,$ $ 0.90 \leq \sin^22\theta_{23} \leq 1.0,$ $\sin^2 \theta_{13} < 0.047,\ 90\%~{\rm C.L.}.$ With this values and for those values obtained previously for the $2\times 2$ $M_0^2 $ matrix (Eq.\[\[eq1000\],\[eq1001\]\]) we finally obtain an estimation for the three neutrino squared mass matrix ($10^{-5}\ eV^2$ units) : [$$\begin{aligned}
M_0^2&=&
\pmatrix{ 8.1\pm 6.5 & 8.4\pm 6.6 & -27.7\pm 28.0
\cr 8.4\pm 6.6 & 8.6\pm 6.9 & -27.5\pm 29.0
\cr -27.7\pm 28.0 & -27.6\pm 29.0 & 202.0\pm 60.0}.\end{aligned}$$ ]{}
One can go further supposing a concrete value for the free parameter $m_2^2$ from elsewhere. If we take $m_2^2 >> \Delta m^2$ then we can directly write the mass matrix $$\begin{aligned}
M&=&m_2 I+\frac{1}{2 m_2} \left (M_{21}^2+M_{32}^2\right ).\end{aligned}$$ Supposing for example $m_1=1\ eV$, we obtain ($eV$ units) [$$\begin{aligned}
M&=&
\pmatrix{ 1+
(4.0\pm 3.2)\ 10^{-5}& 4.2\pm 3.2\ 10^{-5} &-13.5\pm 14.0\ 10^{-5}
\cr
4.2\pm 3.2\ 10^{-5}& 1+(4.3\pm 3.5)\ 10^{-5} &-13.5\pm 14.5\ 10^{-5}
\cr
13.5\pm 14.0\ 10^{-5}& -13.5\pm 14.5\ 10^{-5} &1+(100.0\pm 30.0)\ 10^{-5}
}.\end{aligned}$$ ]{} this is now the final three neutrino mass matrix which can be obtained from present oscillation evidence coming from solar and reactor neutrinos.
Summary and Conclusions {#summary}
=======================
We have presented an up-to-date analysis including the recent KamLAND results, the SNO-phase II spectrum and all other solar neutrino data The active neutrino oscillations hypothesis has been confirmed, and the decoupling of the atmospheric $\Delta m^2$-solar $\Delta m^2$ justifies a 2-flavour analysis as the one presented here. This justification is even stronger if we have into account the large experimental disparity among solar, earth reactor and atmospheric evidence and the very much different accuracy which can be obtained in each of them for the parameters of the $\mu\tau$ and $e\mu$ neutrino sectors. Moreover, the consideration in the analysis of the atmospheric data would only slightly modify the best values and allowed regions for the parameters. These modifications would be well within the error bars of these paramters according to the present determination.
The results presented along this work show how due to the increased statistics, the inclusion of the new KamLAND data determines with good accuracy the value of $\Delta m^2_{\odot}$, clearly selecting the LMAI solution, and brings us to a new era of precision measurements in the solar neutrino parameter space [@kamlandTalk].
We have introduced in this work diverse novelties in the treatment of the SNO and Kl spectra. For the first one we have improved upon previous works in the full consideration of the sistematic correlations. For the KL spectrum we have studied the variability of the best fit results with respect the statistical method in use. We have shown that appreciable differences can be obtained. We believe that a careful study and proper statistical treatment of the KL evidence is needed. Significant differences on the values of the oscillation parameters can be obtained basically due to poor statistics. These apparition of these differences can be easily missed or obscured by analysis which include large quantities of diverse data without the needed care of the individual components.
We have obtained the allowed area in parameter space and individual values for $\Delta m^2$ and $\tan^2\theta$ with error estimation from the analysis of marginal likelihoods. We have shown that it is already possible to determine at present active two neutrino oscillation parameters with relatively good accuracy. In the framework of two active neutrino oscillations we obtain $$\Delta m^2= 8.20\pm 0.08\times 10^{-5} \eV^2,\quad \tan^2\theta=
0.50^{+0.11}_{-0.06}.$$ We estimate the individual elements of the two neutrino mass matrix, we show that individual elements of this matrix can be determined with an error $\sim 10 \% $ from present experimental evidence.
The use of the SNO phase-II spectrum in the data set has mainly two effects: 1) a slight reduction in the overall area of the exclusion plot and 2) a slight decrease in the best-fit $\Delta m^2_{\odot}$.
The decrease in the best-fit mass squared difference can be understood by the fact that by including the SNO spectrum, we increase the statistical relevance of solar neutrino data, which [*prefer smaller $\Delta m^2$*]{}. Furthermore, the oscillation pattern (whose information is contained in the spectrum) is more sensitive to $\Delta m^2$.
It is interesting to note that the KamLAND data alone still continue to predict, for both their analyses, a value of $\tan^2\theta$ smaller than the one obtained with the previous data, and significantly different from 1, consequently making the aesthetically pleasing bi-maximal-mixing models strongly disfavored. This result confirms what was already evident in the solar neutrino data analyses. Nevertheless, improvement on the determination of $\tan^2\theta$ is necessary and it is known that KamLAND is only slightly sensitive to this mixing parameter. The (lower) accuracy with which we determine the solar mixing angle is evident in the marginalized likelihood plots of fig. \[\[fmarginal\]\]. Planning of future super-beam experiments aimed at determining the $\theta_{13}$ and eventual CP violating phases relies on the most accurate estimation of all the mixing parameters [@terranova]. It is expected that future solar neutrino experiments, notably phase-II SNO (higher statistics, due to be made public soon) and eventually future low energy experiments, and phase-III SNO (with helium) will further restrict the allowed range of parameters.
Acknowledgments {#acknowledgments .unnumbered}
---------------
We would like to thank F. Terranova and M. Smy for usefull discussions. We acknowledge the financial support of the Italian MIUR, the Spanish CYCIT funding agencies and the CERN Theoretical Division. P.A. acknowledges funding from the Inter-University Attraction Pole (IUAP) “fundamental interactions”. The core of the numerical computations were done at the computer farm of the Università degli Studi di Milano, Italy.
[99]{} T. Araki [*et al.*]{} \[KamLAND Collaboration\], ArXiv:hep-ex/0406035.
A. B. Balantekin, V. Barger, D. Marfatia, S. Pakvasa and H. Yuksel, arXiv:hep-ph/0405019.\
L. Oberauer, Mod. Phys. Lett. A [**19**]{} (2004) 337 \[arXiv:hep-ph/0402162\]. K. Eguchi [*et al.*]{} \[KamLAND Collaboration\], Phys. Rev. Lett. [**90**]{}, 021802 (2003). A. B. Balantekin and H. Yuksel, J. Phys. G [**29**]{}, 665 (2003), P. Aliani, V. Antonelli, M. Picariello and E. Torrente-Lujan, Phys. Rev. D [**69**]{} (2004) 013005. H. Nunokawa, W. J. Teves and R. Zukanovich Funchal, Phys. Lett. B [**562**]{}, 28 (2003) J. N. Bahcall, M. C. Gonzalez-Garcia and C. Pena-Garay, JHEP [**0302**]{}, 009 (2003) A. Bandyopadhyay, S. Choubey, R. Gandhi, S. Goswami and D. P. Roy, Phys. Lett. B [**559**]{}, 121 (2003) M. Maltoni, T. Schwetz and J. W. Valle, Phys. Rev. D [**67**]{}, 093003 (2003) G. L. Fogli, E. Lisi, A. Marrone, D. Montanino, A. Palazzo and A. M. Rotunno, Phys. Rev. D [**67**]{}, 073002 (2003) V. Barger and D. Marfatia, Phys. Lett. B [**555**]{}, 144 (2003) P. C. de Holanda and A. Y. Smirnov, JCAP [**0302**]{}, 001 (2003) A. B. Balantekin and H. Yuksel, J. Phys. G [**29**]{}, 665 (2003). J. N. Bahcall, M. H. Pinsonneault and S. Basu, Astrophys. J. [**555**]{}, 990 (2001). P. Aliani, V. Antonelli, M. Picariello and E. Torrente-Lujan, New J. Phys. [**5**]{}, 2 (2003). P. Aliani, V. Antonelli, R. Ferrari, M. Picariello and E. Torrente-Lujan, Phys. Rev. D [**67**]{}, 013006 (2003). M. B. Smy, arXiv:hep-ex/0202020.
B. Aharmim [*et al.*]{} \[SNO Collaboration\], arXiv:nucl-ex/0502021.
S. N. Ahmed [*et al.*]{} \[SNO Collaboration\], Phys. Rev. Lett. [**92**]{} (2004) 181301 \[arXiv:nucl-ex/0309004\]. P. Aliani, V. Antonelli, M. Picariello and E. Torrente-Lujan, Nucl. Phys. B [**634**]{}, 393 (2002).
M.C. Gonzalez-Garcia et al. hep-ph/0410030. S. Petcov et al., hep-ph/0410283. M.C. Gonzalez-Garcia,F. Maltoni, Y. Smirnov, hep-ph/0408170. F. Maltoni, JWF Valle et al. hep-ph/0309130.
P. Aliani, V. Antonelli, R. Ferrari, M. Picariello and E. Torrente-Lujan, Phys. Rev. D [**67**]{}, 013006 (2003). P. Aliani, V. Antonelli, M. Picariello and E. Torrente-Lujan, Nucl. Phys. Proc. Suppl. [**110**]{}, 361 (2002) \[arXiv:hep-ph/0112101\]. P. Aliani, V. Antonelli, R. Ferrari, M. Picariello and E. Torrente-Lujan, arXiv:hep-ph/0205061.
P. Aliani, V. Antonelli, M. Picariello and E. Torrente-Lujan, Phys. Rev. D [**69**]{}, 013005 (2004) \[arXiv:hep-ph/0212212\].
A. W. Poon \[SNO Collaboration\], arXiv:nucl-ex/0110005.
P. Aliani, V. Antonelli, R. Ferrari, M. Picariello and E. Torrente-Lujan, reactor neutrino physics,” AIP Conf. Proc. [**655**]{} (2003) 103. E. Torrente-Lujan, Phys. Rev. D [**59**]{} (1999) 093006. E. Torrente-Lujan, Phys. Rev. D [**59**]{} (1999) 073001. E. Torrente-Lujan, Phys. Lett. B [**441**]{} (1998) 305. V. B. Semikoz and E. Torrente-Lujan, Nucl. Phys. B [**556**]{} (1999) 353. E. Torrente-Lujan, Phys. Lett. B [**494**]{} (2000) 255. E. Torrente-Lujan, arXiv:hep-ph/9902339. S. Khalil and E. Torrente-Lujan, J. Egyptian Math. Soc. [**9**]{}, 91 (2001)\[arXiv:hep-ph/0012203\].
Q. R. Ahmad [*et al.*]{} \[SNO Collaboration\], Phys. Rev. Lett. [**89**]{} (2002) 011301, R. Davis, Prog. Part. Nucl. Phys. 32 (1994) 13. B.T. Cleveland et al., (HOMESTAKE Coll.) . B.T. Cleveland et al., (HOMESTAKE Coll.) Astrophys. J. 496 (1998) 505-526.
J.N. Abdurashitov et al. (SAGE Coll.) Phys. Rev. Lett. 83(23) (1999)4686.
A.I. Abazov et al. (SAGE Coll.), . D.N. Abdurashitov et al. (SAGE Coll.), . J.N. Abdurashitov et al., (SAGE Coll.), Phys. Rev. [**C60**]{} (1999) 055801; astro-ph/9907131. J.N. Abdurashitov et al., (SAGE Coll.), ; astro-ph/9907113.
P. Anselmann et al., GALLEX Coll., . W. Hampel et al., GALLEX Coll., . T.A. Kirsten, Prog. Part. Nucl. Phys. 40 (1998) 85-99. W. Hampel et al., (GALLEX Coll.) . M. Cribier, . W. Hampel et al., (GALLEX Coll.) . W. Hampel et al., (GALLEX Coll.) . see also new data in `http://neutrino2004.in2p3.fr/`
M. Altmann et al. (GNO Coll.) Phys. Lett. B490 (2000) 16-26.
J. Boger [*et al.*]{} \[SNO Collaboration\], Nucl. Instrum. Meth. A [**449**]{}, 172 (2000) G.L. Fogli, E. Lisi, A. Palazzo, A.M. Rotunno, Phys. Rev. D [**67**]{}, 073001 (2003)
G. L. Fogli, E. Lisi, A. Marrone and A. Palazzo, Phys. Lett. B [**583**]{} (2004) 149 \[arXiv:hep-ph/0309100\]. “HOWTO use the SNO Salt Flux Results,” \[SNO Collaboration\], can be found at `http://www.sno.phy.queensu.ca`
N. Cressie and T. Read. “Multinomial Goodness-of-fit Tests.”, J. R. Statist. Soc. B, 46(3):440-464, 1984.
T. Read and N. Cressie, “Goodness-of-Fit Statistics for Discrete Multivariate Data.”, Springer Series in Statistics. Springer Verlag, New York, 1988.
N. Taneichi, Y. Sekiya, H. Imai, “On a normalizing transformation of Multinomial Goodness-of-Fit statistics”. Preprint. N. Taneichi, Y. Sekiya, A. Suzukawa. J. Japan Statist. Soc. Vol. 31 No. 2 2001 207-224.
A. Basu, S. Ray, C. Park, “Improved power in Multinomial Goodness-of-fit statistics”, J. Roy. Stat. Soc. Vol. 51, no. 3. pp. 381-393. G.S. Watson, Biometrics, 15,440-468. W.G. Cochran, “The $\chi^2$ test of goodness-of-fit”. Ann. Math. Statist., 23, 315-345.
see [`http://neutrino2004.in2p3.fr` for KamLAND transparencies and Global analysis (S. Goswami)]{}
V. Antonelli, M. Picariello, F. Terranova, E. Torrente-Lujan, work in progress
M. Apollonio et al. (CHOOZ coll.) ; M. Apollonio [*et al.*]{}, .
Y. F. Wang \[Palo Verde Collaboration\], Int. J. Mod. Phys. A [**16S1B**]{}, 739 (2001); F. Boehm [*et al.*]{}, Phys. Rev. D [**64**]{}, 112001 (2001).
[cc|cc]{} $T_{eff}$ (MeV)& Evnts./500 keV &$T_{eff}$ (MeV)& Evnts./500 keV\
5.5- 6.0 & 225 & 9.5-10.0 & 155\
6.0- 6.5 & 225 &10.0-10.5 & 80\
6.5- 7.0 & 220 &10.5-11.0 & 95\
7.0- 7.5 & 255 &11.0-11.5 & 55\
7.5- 8.0 & 235 &11.5-12.0 & 40\
8.0- 8.5 & 225 &12.5-13.0 & 15\
8.5- 9.0 & 155 &13.5-14.0 & 15\
9.0- 9.5 & 145 &14.5-15.0 & 5\
\
[cc|cc]{} $T_{eff}$ (MeV)& Evnts./500 keV &$T_{eff}$ (MeV)& Evnts./500 keV\
5.5- 6.0 & $840\pm 20$ & 9.5-10.0 & $180 \pm 10 $\
6.0- 6.5 & $785\pm 20$ &10.0-10.5 & $110 \pm 10 $\
6.5- 7.0 & $705\pm 20$ &10.5-11.0 & $110 \pm 10 $\
7.0- 7.5 & $680\pm 20$ &11.0-11.5 & $50 \pm 10 $\
7.5- 8.0 & $560\pm 15$ &11.5-12.0 & $40 \pm 10 $\
8.0- 8.5 & $470\pm 15$ &12.5-13.0 & $10 \pm 10 $\
8.5- 9.0 & $245\pm 10$ &13.5-14.0 & $10 \pm 5 $\
9.0- 9.5 & $205\pm 10$ &14.5-15.0 & $0 \pm 5 $\
\
$\Delta m^2 (\eV^2) $ $\tan^2\theta$
-------------------------------- ----------------------------------- ------------------------
[from $\chi^2$ minimization]{}
KL (Sp+Gl)+Solar + SNO (Sp) $7.89\times 10^{-5}$ $0.40$
KL (Sp+Gl)+Solar + SNO (Gl) $8.17\times 10^{-5}$ $0.40$
[from marginalization]{}
KL (Sp+Gl)+Solar + SNO (Sp) $8.2^{+0.9}_{-0.8}\times 10^{-5}$ $0.50^{+0.11}_{-0.06}$
-- --
-- --
-- --
-- --
|
---
abstract: 'In $U$-duality-manifest formulations, supergravity fields are packaged into covariant objects such as the generalized metric and $p$-form fields ${\mathcal A}_p^{I_p}$. While a parameterization of the generalized metric in terms of supergravity fields is known for $U$-duality groups $E_n$ with $n\leq 8$, a parameterization of ${\mathcal A}_p^{I_p}$ has not been fully determined. In this paper, we propose a systematic method to determine the parameterization of ${\mathcal A}_p^{I_p}$, which necessarily involves mixed-symmetry potentials. We also show how to systematically obtain the $T$- and $S$-duality transformation rules of the mixed-symmetry potentials entering the multiplet. As the simplest non-trivial application, we find the parameterization and the duality rules associated with the dual graviton. Additionally, we show that the 1-form field ${\mathcal A}_1^{I_1}$ can be regarded as the generalized graviphoton in the exceptional spacetime.'
---
**Exotic branes and mixed-symmetry potentials II: duality rules and exceptional $p$-form gauge fields**
[José J. Fernández-Melgarejo$^{a}$]{}[^1], [Yuho Sakatani$^{b}$]{}[^2], [Shozo Uehara$^{b}$]{}[^3]
${}^a$[*Departamento de Física, Universidad de Murcia,*]{}\
[*Campus de Espinardo, 30100 Murcia, Spain*]{}
${}^b$[*Department of Physics, Kyoto Prefectural University of Medicine,*]{}\
[*Kyoto 606-0823, Japan*]{}
Introduction
============
In the previous paper [@1907.07177], we have conducted a detailed survey of mixed-symmetry potentials in 11D and type II supergravities. By considering their reduction to $d$ dimensions, they yield various $p$-form fields ${\mathcal A}_p^{I_p}$, which transform covariantly under $E_n$ $U$-duality transformation ($n=11-d$). In the $U$-duality-covariant formulation of supergravity known as exceptional field theory (EFT) [@1308.1673; @1312.0614; @1312.4542; @1406.3348] (see [@hep-th/0307098; @hep-th/0312247; @hep-th/0406150; @0712.1795; @0901.1581; @1008.1763; @1110.3930; @1111.0459; @1208.5884] for earlier fundamental works), and the $U$-duality-manifest approaches to brane actions [@1009.4657; @1712.07115; @1712.10316; @1802.00442], the $p$-form fields ${\mathcal A}_p^{I_p}$ play an important role in providing $U$-duality-covariant descriptions. However, to make contact with the standard descriptions in supergravity and brane actions, explicit parameterizations of ${\mathcal A}_p^{I_p}$ are needed. In this paper, we propose a systematic method to determine the parameterization of ${\mathcal A}_p^{I_p}$ by utilizing the equivalence between M-theory (or type IIA theory) and type IIB theory. In our method, in addition to the parameterization of the $p$-form fields, the duality transformation rules of various potentials can also be obtained. As the first non-trivial example, we obtain the $T$- and $S$-duality rules for the dual graviton, equations – and , respectively.
In EFT, the fundamental fields are the generalized metric ${\mathcal M}_{IJ}$ and $p$-form fields ${\mathcal A}_p^{I_p}$, as well as certain auxiliary fields. For $E_n$ EFT with $n\leq 8$, the parameterization of the generalized metric has been determined in [@1111.0459; @1303.2035] by means of the bosonic fields in 11D supergravity. The parameterization in terms of type IIB supergravity has been determined in [@1405.7894; @1612.08738] for $E_n$ EFT with $n\leq 7$. They are nothing more than the two different parameterizations of the same object ${\mathcal M}_{IJ}$, and as was concretely realized in [@1701.07819], we can relate the two parameterizations through some redefinitions of fields. As was shown in [@1701.07819], by rewriting the M-theory fields in terms of type IIA fields, these field redefinitions are precisely the $T$-duality transformation rule. However, the analysis of [@1701.07819] is limited to the $E_n$ EFT with $n\leq 7$, where the generalized metric contains only the standard $p$-form potentials. In this paper, we extend their analysis to the case of $E_8$ EFT, and find the $T$-duality and $S$-duality rule for the dual graviton. This gives a non-trivial check of our duality rules for the dual graviton advertised in the first paragraph.
If we look at the explicit parameterization of the 1-form field ${\mathcal A}_1^{I}$, its first component ${\mathcal A}_1^{i}$ is the graviphoton. In 11D, the graviphoton is defined as ${\hat{A}}_\mu^i \equiv \bm{{\hat{g}}}_{\mu\nu}\, {\hat{g}}^{\nu i}$, by using the 11D inverse metric ${\hat{g}}^{\hat{M}\hat{N}}$ and the metric $\bm{{\hat{g}}}_{\mu\nu}$ in the external spacetime. In this paper, we propose that the 1-form field ${\mathcal A}_1^{I}$ can be regarded as a generalized graviphoton in the exceptional spacetime $$\begin{aligned}
{\mathcal A}_\mu^I = \bm{m}_{\mu\nu}\, {\mathcal M}^{\nu I} \,,\qquad
\bm{m} \equiv ({\mathcal M}^{\mu\nu})^{-1} \,,\end{aligned}$$ where ${\mathcal M}^{\hat{I}\hat{J}}$ ($\hat{I}=\{\mu,\,I\}$) is the inverse generalized metric in $E_{11}$ EFT. We also find that the parameterizations of the higher $p$-form fields ${\mathcal A}_p^{I_p}$ ($p\geq 2$) can be easily obtained from that of the 1-form ${\mathcal A}_1^{I}$ through a simple antisymmetrization of indices.
Parameterization of the 1-form ${\mathcal A}_1^{I}$ {#sec:1-form}
===================================================
In this section, we explain our method to determine the parameterizations of the 1-form ${\mathcal A}_1^{I}$. The index $I$ transforms in a fundamental representation of the $E_{n}$ algebra with Dynkin label $[1,0,\dotsc,0]$, known as the vector representation or the particle multiplet. Our approach relies on the existence of two equivalent descriptions of EFT by deleting different nodes of the $E_n$ Dynkin diagram; M-theory and type IIB theory (see [@1907.07177] and references therein for details): $$\begin{aligned}
\scalebox{0.7}{
\xygraph{
*\cir<6pt>{} ([]!{+(0,-.4)} {\alpha_1}) - [r]
*\cir<6pt>{} ([]!{+(0,-.4)} {\alpha_2}) - [r]
\cdots ([]!{+(0,0.8)} {\text{M-theory}}) ([]!{+(0,-.4)} {}) - [r]
*\cir<6pt>{} ([]!{+(0,-.4)} {\alpha_{n-4}}) - [r]
*\cir<6pt>{} ([]!{+(0,-.4)} {\alpha_{n-3}})
(
- [u] *{\scalebox{2}{$\times$}}*\cir<6pt>{} ([]!{+(.5,0)} {\alpha_{n}}),
- [r] *\cir<6pt>{} ([]!{+(0,-.4)} {\alpha_{n-2}})
- [r] *\cir<6pt>{} ([]!{+(0,-.4)} {\alpha_{n-1}})
)}\quad ,
\qquad\qquad
\xygraph{
*\cir<6pt>{} ([]!{+(0,-.4)} {\alpha_1}) - [r]
*\cir<6pt>{} ([]!{+(0,-.4)} {\alpha_2}) - [r]
\cdots ([]!{+(0,0.8)} {\text{Type IIB theory}}) ([]!{+(0,-.4)} {}) - [r]
*\cir<6pt>{} ([]!{+(0,-.4)} {\alpha_{n-4}}) - [r]
*\cir<6pt>{} ([]!{+(0,-.4)} {\alpha_{n-3}})
(
- [u] *\cir<6pt>{} ([]!{+(.5,0)} {\alpha_{n}}),
- [r] *{\scalebox{2}{$\times$}}*\cir<6pt>{} ([]!{+(0,-.4)} {\alpha_{n-2}})
- [r] *\cir<6pt>{} ([]!{+(0,-.4)} {\alpha_{n-1}})
)}\quad.
}
\label{eq:Dynkin-decomposition}\end{aligned}$$ As it is explained in the accompanying paper [@1907.07177], in terms of M-theory, the 1-form field ${\mathcal A}_1^{I}$ is decomposed into ${\text{SL}}(n)$ tensors as follows: $$\begin{aligned}
\bigl({\mathcal A}_\mu^{I}\bigr)= \bigl({\mathcal A}_\mu^i,\,\tfrac{{\mathcal A}_{\mu; i_1i_2}}{\sqrt{2!}},\,\tfrac{{\mathcal A}_{\mu; i_1\cdots i_5}}{\sqrt{5!}},\,\tfrac{{\mathcal A}_{\mu; i_1\cdots i_7,i}}{\sqrt{7!}},\cdots\bigr)\,,
\label{eq:M-decomp}\end{aligned}$$ where $i,j=d,\dotsc,9,{z}$ are indices of the fundamental representation of ${\text{SL}}(n)$. On the other hand, in terms of type IIB theory, the 1-form field is decomposed into ${\text{SL}}(n-1)\times{\text{SL}}(2)$ tensors as follows: $$\begin{aligned}
\bigl(\bm{{\mathcal A}}_\mu^{{\mathsf{I}}}\bigr) &= \bigl(\bm{{\mathcal A}}_\mu^{\mathsf{m}},\, \bm{{\mathcal A}}_{\mu; {\mathsf{m}}}^\alpha ,\,\tfrac{\bm{{\mathcal A}}_{\mu; {\mathsf{m}}_1{\mathsf{m}}_2{\mathsf{m}}_3}}{\sqrt{3!}},\,\tfrac{\bm{{\mathcal A}}_{\mu; {\mathsf{m}}_1\cdots {\mathsf{m}}_5}^\alpha}{\sqrt{5!}},\,\tfrac{\bm{{\mathcal A}}_{\mu; {\mathsf{m}}_1\cdots {\mathsf{m}}_6,{\mathsf{m}}}}{\sqrt{6!}}, \cdots\bigr) \,,
\label{eq:IIB-decomp}\end{aligned}$$ where $\alpha,\beta =1,2$ are the ${\text{SL}}(2)$ $S$-duality indices and ${\mathsf{m}},{\mathsf{n}}=d,\dotsc,9$ are indices of the fundamental representation of ${\text{SL}}(n-1)$. In order to stress the difference between the two parameterizations, we have denoted the 1-form in type IIB parameterization by $\bm{{\mathcal A}}_\mu^{{\mathsf{I}}}$. Although we know the tensor structures of each component, it is not obvious how to determine the explicit parameterization in terms of the standard supergravity fields, which is the main subject of this paper.
As demonstrated in [@1701.07819], the two decompositions and can be related by using the equivalence between M-theory on $T^2$ with coordinates $(x^\alpha)=(x^{y},\,x^{z})$ and type IIB theory on $S^1$ with a coordinate ${\mathsf{x}}^{\mathsf{y}}$: $$\begin{aligned}
\begin{gathered}
\xymatrix{
\text{M-theory/}T^2 \ar[d]_{\text{compactification on $x^{z}$}} \ar@{<->}[drrr]^{\text{our map}} & & & \\
\text{ Type IIA theory/$S^1$ } \ar@{<->}[rrr]_{\text{$T$-duality along $x^{y}$/${\mathsf{x}}^{\mathsf{y}}$}} & & & \text{ Type IIB theory/$S^1$ } .\\
}
\end{gathered}
\label{eq:our-map}\end{aligned}$$ Here, $x^{z}$ is a coordinate along the M-theory circle, and the coordinate $x^{y}$ in M-theory (or type IIA theory) is mapped to the coordinate ${\mathsf{x}}^{\mathsf{y}}$ in type IIB theory under the $T$-duality. By using the map, we can rewrite various quantities in M-theory in terms of type IIB supergravity.
Supergravity fields
-------------------
In order to discuss the parameterization, we will briefly explain the supergravity fields considered in this paper. We basically follow the convention of [@1701.07819].
#### 11D supergravity:
In 11D supergravity, we consider the following bosonic fields, $$\begin{aligned}
\{{\hat{g}}_{\hat{M}\hat{N}},\,{\hat{A}}_{\hat 3},\,{\hat{A}}_{\hat 6},\,{\hat{A}}_{\hat 8,\hat 1}\}\qquad \bigl(\hat{M},\hat{N}=0,\dotsc,9,{z}\bigr)\,. \end{aligned}$$ The standard potentials ${\hat{A}}_3$ and ${\hat{A}}_6$ couple to M2-brane and M5-brane, respectively, while the dual graviton ${\hat{A}}_{8,1}$ couples to the Kaluza–Klein monopole $6^1$ (sometimes called MKK) [@hep-th/9802199]. When we consider a compactification to $d$ dimensions, the 11D metric ${\hat{g}}_{\hat{M}\hat{N}}$ is decomposed as $$\begin{aligned}
({\hat{g}}_{\hat{M}\hat{N}}) = \begin{pmatrix}
\bm{{\hat{g}}}_{\mu\nu} +{\hat{A}}_\mu^k\,{\hat{G}}_{kl}\,{\hat{A}}_\nu^l & -{\hat{A}}_\mu^k\,{\hat{G}}_{kj} \\
-{\hat{G}}_{ik}\,{\hat{A}}_\nu^k & {\hat{G}}_{ij}
\end{pmatrix} \qquad (\mu,\nu=0,\dotsc,d-1),\end{aligned}$$ where we have defined the graviphoton as ${\hat{A}}_\mu^i \equiv -{\hat{g}}_{\mu k}\,{\hat{G}}^{ki} = \bm{{\hat{g}}}_{\mu\nu}\, {\hat{g}}^{\nu i}$.
#### Type IIA supergravity:
When we consider type IIA supergravity, we use the following standard 11D–10D map: $$\begin{aligned}
\begin{split}
&({\hat{g}}_{\hat{M}\hat{N}})\equiv \begin{pmatrix}
{\hat{g}}_{MN} & {\hat{g}}_{M{z}} \\ {\hat{g}}_{{z}N} & {\hat{g}}_{{z}{z}}
\end{pmatrix}
=\begin{pmatrix}
{\operatorname{e}^{-\frac{2}{3}\,{\Phi}}}\,{g}_{MN}+{\operatorname{e}^{\frac{4}{3}\,{\Phi}}}\,{\mathscr{C}}_M\, {\mathscr{C}}_N & {\operatorname{e}^{\frac{4}{3}\,{\Phi}}}\, {\mathscr{C}}_M \\ {\operatorname{e}^{\frac{4}{3}\,{\Phi}}}\, {\mathscr{C}}_N & {\operatorname{e}^{\frac{4}{3}\,{\Phi}}}
\end{pmatrix} \,,
\\
&{\hat{A}}_{\hat{3}} = {\mathscr{C}}_3 + {\mathscr{B}}_2\wedge {{\mathrm{d}}}x^{z}\,,\qquad
{\hat{A}}_{\hat{6}} = {\mathscr{B}}_6 + \bigl({\mathscr{C}}_5 - \tfrac{1}{2!}\, {\mathscr{C}}_3\wedge {\mathscr{B}}_2\bigr)\wedge {{\mathrm{d}}}x^{z}\,,
\end{split}
\label{eq:11D-10D}\end{aligned}$$ where we have added the hat to the subscript, like ${\hat{A}}_{\hat{p}}$, to stress that it is a $p$-form in 11D. In our convention, the dual graviton ${\hat{A}}_{\hat{8},\hat{1}}=\{{\hat{A}}_{\hat{8}, 1},\,{\hat{A}}_{\hat{8}, {z}}\}$ follows the 11D–10D map, $$\begin{aligned}
{\hat{A}}_{\hat{8}, 1} = {\mathscr{A}}_{8, 1} + {\mathscr{A}}_{7, 1}\wedge {{\mathrm{d}}}x^{{z}} \,, \qquad
{\hat{A}}_{\hat{8}, {z}} = {\mathscr{A}}_{8} + \bigl({\mathscr{C}}_7 - \tfrac{1}{3!}\,{\mathscr{C}}_3\wedge{\mathscr{B}}_2\wedge{\mathscr{B}}_2\bigr)\wedge{{\mathrm{d}}}x^{{z}} \,,\end{aligned}$$ where ${\hat{A}}_{\hat{8}, {z}}$ corresponds to $\hat{\tilde{N}}$ studied in [@hep-th/9802199]. The metric and the graviphoton are defined as $$\begin{aligned}
({g}_{MN}) = \begin{pmatrix}
\bm{{g}}_{\mu\nu} +{\mathscr{A}}_\mu^p\,{\mathscr{G}}_{pq}\,{\mathscr{A}}_\nu^q & -{\mathscr{A}}_\mu^p\,{\mathscr{G}}_{pn} \\
-{\mathscr{G}}_{mp}\,{\mathscr{A}}_\nu^p & {\mathscr{G}}_{mn}
\end{pmatrix},\qquad
{\mathscr{A}}_\mu^m \equiv \bm{{g}}_{\mu\nu}\,{g}^{\nu m} \,. \end{aligned}$$ Then we find the 11D–10D map for the graviphoton, $$\begin{aligned}
{\hat{A}}_\mu^m = {\mathscr{A}}_\mu^m \,, \qquad
{\hat{A}}_\mu^{z}= -\bigl({\mathscr{C}}_\mu + {\mathscr{A}}_\mu^p\,{\mathscr{C}}_p\bigr)\,.
\label{eq:11D-10D-photon}\end{aligned}$$
#### Type IIB supergravity:
In type IIB theory, in addition to the standard Einstein-frame metric ${\mathsf{g}}_{MN}$, we consider the following ${\text{SL}}(2)$ $S$-duality-covariant tensors, $$\begin{aligned}
&({\mathsf{m}}_{\alpha\beta}) \equiv {\operatorname{e}^{{\varphi}}} \begin{pmatrix} {\operatorname{e}^{-2\,{\varphi}}} +({\mathsf{C}}_0)^2 & {\mathsf{C}}_0 \\ {\mathsf{C}}_0 & 1 \end{pmatrix},\qquad
({\mathsf{A}}^\alpha_2) \equiv \begin{pmatrix} {\mathsf{B}}_2 \\\ -{\mathsf{C}}_2\end{pmatrix} ,
\\
&{\mathsf{A}}_4 \equiv {\mathsf{C}}_4 - \frac{1}{2}\,{\mathsf{C}}_2\wedge {\mathsf{B}}_2\,,\qquad
({\mathsf{A}}^\alpha_6) \equiv \begin{pmatrix} {\mathsf{C}}_6 - {\mathsf{C}}_4\wedge {\mathsf{B}}_2 +\frac{1}{3}\, {\mathsf{B}}_2\wedge {\mathsf{C}}_2 \wedge {\mathsf{B}}_2 \\ -\bigl({\mathsf{B}}_6 - {\mathsf{C}}_4\wedge {\mathsf{C}}_2 + \frac{1}{6}\,{\mathsf{B}}_2\wedge {\mathsf{C}}_2 \wedge {\mathsf{C}}_2\bigr) \end{pmatrix} .\end{aligned}$$ We also consider the dual graviton ${\mathsf{A}}_{7,1}$, whose behavior under duality transformations is to be determined. Upon compactification to $d$ dimensions, the graviphoton is introduced as $$\begin{aligned}
({\mathsf{g}}_{MN}) = \begin{pmatrix}
\bm{{\mathsf{g}}}_{\mu\nu} + {\mathsf{A}}_\mu^{\mathsf{p}}\, {\mathsf{G}}_{{\mathsf{p}}{\mathsf{q}}}\, {\mathsf{A}}_\nu^{\mathsf{q}}& - {\mathsf{A}}_\mu^{\mathsf{p}}\, {\mathsf{G}}_{{\mathsf{p}}{\mathsf{n}}} \\
-{\mathsf{G}}_{{\mathsf{m}}{\mathsf{p}}}\, {\mathsf{A}}_\nu^{\mathsf{p}}& {\mathsf{G}}_{{\mathsf{m}}{\mathsf{n}}}
\end{pmatrix} ,\qquad {\mathsf{A}}_\mu^{\mathsf{m}}\equiv \bm{{\mathsf{g}}}_{\mu\nu}\,{\mathsf{g}}^{\nu {\mathsf{m}}}\,. \end{aligned}$$
Strategy: Linear map
--------------------
Here, let us explain the detailed procedure, how to determine the parameterization of the 1-form in both the M-theory and type IIB languages, $$\begin{aligned}
({\mathcal A}_\mu^I) = {\footnotesize\begin{pmatrix}
{\mathcal A}_\mu^i \\
\frac{{\mathcal A}_{\mu; i_1i_2}}{\sqrt{2!}} \\
\frac{{\mathcal A}_{\mu; i_1\cdots i_5}}{\sqrt{5!}} \\
\frac{{\mathcal A}_{\mu; i_1\cdots i_7,i}}{\sqrt{7!}} \\
\vdots
\end{pmatrix}},\qquad\quad
(\bm{{\mathcal A}}_\mu^{\mathsf{I}}) = {\footnotesize\begin{pmatrix}
\bm{{\mathcal A}}_\mu^{\mathsf{m}}\\
\bm{{\mathcal A}}^\alpha_{\mu; {\mathsf{m}}} \\
\frac{\bm{{\mathcal A}}_{\mu; {\mathsf{m}}_1{\mathsf{m}}_2{\mathsf{m}}_3}}{\sqrt{3!}} \\
\frac{\bm{{\mathcal A}}^\alpha_{\mu; {\mathsf{m}}_1\cdots {\mathsf{m}}_5}}{\sqrt{5!}} \\
\tfrac{\bm{{\mathcal A}}_{\mu; {\mathsf{m}}_1\cdots {\mathsf{m}}_6,{\mathsf{m}}}}{\sqrt{6!}} \\
\vdots
\end{pmatrix}},
\label{eq:1-form-MB}\end{aligned}$$ where ellipses stand for the rest of components that complete the $U$-duality multiplet which, potentially, involve further mixed-symmetry potentials.
To determine the parameterization, we make the following modest assumptions:
- The M-theory fields ${\mathcal A}_{1; p,q,r,\cdots}$ and the type IIB fields $\bm{{\mathcal A}}^{\alpha_1\cdots\alpha_s}_{1; p,q,r,\cdots}$ are respectively parameterized by the following fields: $$\begin{aligned}
{2}
&\text{M-theory: }&\quad &\{{\hat{A}}_\mu^i ,\, {\hat{A}}_{\hat{3}},\, {\hat{A}}_{\hat{6}},\,{\hat{A}}_{\hat{8},\hat{1}},\,\dotsc \}\,,
\\
&\text{Type IIB theory: }&\quad &\{{\mathsf{A}}_\mu^{\mathsf{m}},\, {\mathsf{A}}^\alpha_2,\, {\mathsf{A}}_4,\, {\mathsf{A}}^\alpha_6,\,{\mathsf{A}}_{7,1},\,\dotsc \}\,.\end{aligned}$$
- The top form is normalized with weight one: $$\begin{aligned}
\begin{split}
{\mathcal A}_{\mu ; p,q,r,\dotsc} &= {\hat{A}}_{\mu p,q,r,\dotsc} + \text{(sum of products of potentials)} \,,
\\
\bm{{\mathcal A}}^{\alpha_1\cdots\alpha_s}_{\mu ; p,q,r,\dotsc} &= {\mathsf{A}}^{\alpha_1\cdots\alpha_s}_{\mu p,q,r,\dotsc} + \text{(sum of products of potentials)} \,.
\end{split}\end{aligned}$$
According to these, the first components of the 1-forms should be, respectively, $$\begin{aligned}
{\mathcal A}_\mu^i = {\hat{A}}_\mu^i\quad (\text{M-theory})\,,\qquad \bm{{\mathcal A}}_\mu^{\mathsf{m}}= {\mathsf{A}}_\mu^{\mathsf{m}}\quad (\text{type IIB})\,.
\label{eq:graviphoton-M-B}\end{aligned}$$ In the following, we explain the procedure to determine the components with higher level, which is based on [@1701.07819]. In order to utilize the map , we decompose the physical coordinates on the $n$-torus in M-theory as $(x^i)=(x^a,\,x^\alpha)$ ($a,b=1,\dotsc,n-2$) and those on the $(n-1)$-torus in type IIB theory as $({\mathsf{x}}^{\mathsf{m}})=({\mathsf{x}}^a,\,{\mathsf{x}}^{\mathsf{y}})$. Under the decomposition, the 1-form fields are decomposed into ${\text{SL}}(n-2)\times {\text{SL}}(2)$ tensors as follows: $$\begin{aligned}
{\footnotesize
({\mathcal A}_\mu^I) = \begin{pmatrix}
{\mathcal A}_\mu^a \\ {\mathcal A}_\mu^\alpha \\ \hline
\frac{{\mathcal A}_{\mu; a_1a_2}}{\sqrt{2!}} \\ {\mathcal A}_{\mu; a \alpha} \\
{\mathcal A}_{\mu; {y}{z}} \\ \hline \frac{{\mathcal A}_{\mu; a_1\cdots a_5}}{\sqrt{5!}} \\
\frac{{\mathcal A}_{\mu; a_1\cdots a_4\alpha}}{\sqrt{4!}} \\
\frac{{\mathcal A}_{\mu; a_1a_2a_3 {y}{z}}}{\sqrt{3!}} \\ \hline
\frac{{\mathcal A}_{\mu; a_1\cdots a_5 {y}{z},a}}{\sqrt{5!}} \\
\frac{{\mathcal A}_{\mu; a_1\cdots a_5 {y}{z},\alpha}}{\sqrt{5!}} \\
\vdots
\end{pmatrix},\qquad\quad
({\mathcal A}_\mu^{\mathsf{I}}) = \begin{pmatrix}
\bm{{\mathcal A}}_\mu^a \\
\bm{{\mathcal A}}_\mu^{{\mathsf{y}}} \\ \hline
\bm{{\mathcal A}}^\alpha_{\mu; a} \\ \bm{{\mathcal A}}^\alpha_{\mu; {\mathsf{y}}} \\ \hline
\frac{\bm{{\mathcal A}}_{\mu; a_1a_2a_3}}{\sqrt{3!}} \\ \frac{\bm{{\mathcal A}}_{\mu; a_1a_2 {\mathsf{y}}}}{\sqrt{2!}} \\ \hline
\frac{\bm{{\mathcal A}}^\alpha_{\mu; a_1\cdots a_5}}{\sqrt{5!}} \\
\frac{\bm{{\mathcal A}}^\alpha_{\mu; a_1\cdots a_4{\mathsf{y}}}}{\sqrt{4!}} \\ \hline
\frac{\bm{{\mathcal A}}_{\mu; a_1\cdots a_5{\mathsf{y}},a}}{\sqrt{5!}} \\
\frac{\bm{{\mathcal A}}_{\mu; a_1\cdots a_5{\mathsf{y}},{\mathsf{y}}}}{\sqrt{5!}} \\
\vdots
\end{pmatrix} ,
}
\label{eq:SL(n-2)xSL(2)}\end{aligned}$$ where toroidal directions (either compactified or $T$-dualized) are shown explicitly. In terms of the Dynkin diagram given in , in M-theory we have first performed the level decomposition associated with the node $\alpha_n$. Secondly, we have done the level decomposition associated with $\alpha_{n-2}$. On the other hand, in type IIB theory the order is reversed. In the end, we obtain the same decomposition. Indeed, the set of ${\text{SL}}(n-2)\times {\text{SL}}(2)$ tensors appearing in have the same structure. Then, we make the following identifications [@1701.07819]: $$\begin{aligned}
{\footnotesize
\begin{pmatrix}
{\mathcal A}_\mu^a \\ {\mathcal A}_\mu^\alpha \\ \hline
\frac{{\mathcal A}_{\mu a_1a_2}}{\sqrt{2!}} \\
{\mathcal A}_{\mu; a \alpha} \\
{\mathcal A}_{\mu; {y}{z}} \\ \hline \frac{{\mathcal A}_{\mu; a_1\cdots a_5}}{\sqrt{5!}} \\
\frac{{\mathcal A}_{\mu; a_1\cdots a_4\alpha}}{\sqrt{4!}} \\
\frac{{\mathcal A}_{\mu; a_1a_2a_3 {y}{z}}}{\sqrt{3!}} \\ \hline
\frac{{\mathcal A}_{\mu; a_1\cdots a_5 {y}{z},a}}{\sqrt{5!}} \\
\frac{{\mathcal A}_{\mu; a_1\cdots a_5 {y}{z},\alpha}}{\sqrt{5!}} \\
\vdots
\end{pmatrix}_{\text{M}}
=\
\begin{pmatrix}
\bm{{\mathcal A}}_\mu^a \\
\bm{{\mathcal A}}_{\mu; {\mathsf{y}}}^\alpha \\ \hline
\frac{\bm{{\mathcal A}}_{\mu; a_1a_2{\mathsf{y}}}}{\sqrt{2!}} \\
\bm{{\mathcal A}}_{\mu; a}^\beta \,\epsilon_{\beta\alpha} \\
\bm{{\mathcal A}}_\mu^{\mathsf{y}}\\ \hline
\frac{\bm{{\mathcal A}}_{\mu; a_1\cdots a_5{\mathsf{y}},{\mathsf{y}}}}{\sqrt{5!}} \\
\frac{\bm{{\mathcal A}}^\beta_{\mu; a_1\cdots a_4} \epsilon_{\beta\alpha}}{\sqrt{4!}} \\
\frac{\bm{{\mathcal A}}_{\mu; a_1a_2a_3}}{\sqrt{3!}} \\ \hline
\frac{\bm{{\mathcal A}}_{\mu; a_1\cdots a_5{\mathsf{y}},a}}{\sqrt{5!}} \\
\frac{\bm{{\mathcal A}}_{\mu; a_1\cdots a_5}^\beta\,\epsilon_{\beta\alpha}}{\sqrt{5!}} \\
\vdots
\end{pmatrix}_{\text{IIB}}
\ \text{or}\quad
\begin{pmatrix}
\bm{{\mathcal A}}_\mu^a \\
\bm{{\mathcal A}}_\mu^{{\mathsf{y}}} \\ \hline
\bm{{\mathcal A}}^\alpha_{\mu; a} \\ \bm{{\mathcal A}}^\alpha_{\mu; {\mathsf{y}}} \\ \hline
\frac{\bm{{\mathcal A}}_{\mu; a_1a_2a_3}}{\sqrt{3!}} \\ \frac{\bm{{\mathcal A}}_{\mu; a_1a_2 {\mathsf{y}}}}{\sqrt{2!}} \\ \hline
\frac{\bm{{\mathcal A}}^\alpha_{\mu; a_1\cdots a_5}}{\sqrt{5!}} \\
\frac{\bm{{\mathcal A}}^\alpha_{\mu; a_1\cdots a_4{\mathsf{y}}}}{\sqrt{4!}} \\ \hline
\frac{\bm{{\mathcal A}}_{\mu; a_1\cdots a_5{\mathsf{y}},a}}{\sqrt{5!}} \\
\frac{\bm{{\mathcal A}}_{\mu; a_1\cdots a_5{\mathsf{y}},{\mathsf{y}}}}{\sqrt{5!}} \\
\vdots
\end{pmatrix}_{\text{IIB}}
= \begin{pmatrix}
{\mathcal A}_\mu^a \\
{\mathcal A}_{\mu; {y}{z}} \\ \hline
\epsilon^{\alpha\beta}\,{\mathcal A}_{\mu; a\beta} \\
{\mathcal A}_\mu^\alpha \\ \hline
\frac{{\mathcal A}_{\mu; a_1a_2a_3{y}{z}}}{\sqrt{3!}} \\
\frac{{\mathcal A}_{\mu; a_1a_2}}{\sqrt{2!}} \\ \hline
\frac{\epsilon^{\alpha\beta}\,{\mathcal A}_{\mu; a_1\cdots a_5 {y}{z},\beta}}{\sqrt{5!}} \\
\frac{\epsilon^{\alpha\beta}\,{\mathcal A}_{\mu; a_1\cdots a_4\beta}}{\sqrt{4!}} \\ \hline
\frac{{\mathcal A}_{\mu; a_1\cdots a_5 {y}{z},a}}{\sqrt{5!}} \\
\frac{{\mathcal A}_{\mu; a_1\cdots a_5}}{\sqrt{5!}} \\
\vdots
\end{pmatrix}_{\text{M}} \,,
}
\label{eq:M-B-map}\end{aligned}$$ where we have defined $$\begin{aligned}
\epsilon \equiv (\epsilon^{\alpha\beta})\equiv(\epsilon_{\alpha\beta})\equiv \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}.\end{aligned}$$ We will refer to the set of linear relations established in as the *linear map*. Actually, by using a constant matrix $S^I{}_{\mathsf{J}}$, it can be rewritten as $$\begin{aligned}
{\mathcal A}_\mu^I = S^I{}_{\mathsf{J}}\, \bm{{\mathcal A}}_\mu^{\mathsf{J}}\,,\qquad \bm{{\mathcal A}}_\mu^{\mathsf{I}}= (S^{-1})^{\mathsf{I}}{}_J\,{\mathcal A}_\mu^J \,.
\label{eq:linear-map-A}\end{aligned}$$ We note that this identification has been originally proposed in [@hep-th/0402140] in the context of $E_{11}$.
Now, for simplicity, we assume the standard $T$-duality rule for the NS–NS fields[^4] $$\begin{aligned}
\begin{split}
{g}_{AB} &\overset{\text{A--B}}{=} {\mathsf{g}}_{AB} - \frac{{\mathsf{g}}_{A {\mathsf{y}}}\,{\mathsf{g}}_{B {\mathsf{y}}}-{\mathsf{B}}_{A {\mathsf{y}}}\,{\mathsf{B}}_{B {\mathsf{y}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}}\,,\qquad
{g}_{A {y}}\overset{\text{A--B}}{=}-\frac{{\mathsf{B}}_{A {\mathsf{y}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}}\,,\qquad
{g}_{{y}{y}}\overset{\text{A--B}}{=}\frac{1}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}}\,,
\\
{\mathscr{B}}_{AB} &\overset{\text{A--B}}{=} {\mathsf{B}}_{AB} - \frac{{\mathsf{B}}_{A{\mathsf{y}}}\,{\mathsf{g}}_{B{\mathsf{y}}}-{\mathsf{g}}_{A{\mathsf{y}}}\,{\mathsf{B}}_{B{\mathsf{y}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}}\,,\qquad
{\mathscr{B}}_{A{y}} \overset{\text{A--B}}{=} -\frac{{\mathsf{g}}_{A{\mathsf{y}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}} \,,
\end{split}
\label{eq:Buscher-NS}\end{aligned}$$ where $A,B=\{\mu,\,a\}$ (i.e. nine directions except the $T$-dual direction $x^{y}$ or ${\mathsf{x}}^{\mathsf{y}}$). From these, we obtain the $T$-duality rule for the graviphoton $$\begin{aligned}
{\mathscr{A}}_\mu^a \overset{\text{A--B}}{=} {\mathsf{A}}_\mu^a\,,\qquad
{\mathscr{A}}_\mu^{y}\overset{\text{A--B}}{=} {\mathsf{B}}_{\mu {\mathsf{y}}} + {\mathsf{A}}_\mu^{\mathsf{p}}\,{\mathsf{B}}_{{\mathsf{p}}{\mathsf{y}}} \,.
\label{eq:Buscher-graviphoton}\end{aligned}$$ By using the 11D–10D relation , the first rule gives $$\begin{aligned}
{\mathcal A}_\mu^a = {\hat{A}}_\mu^a \overset{\text{M--A}}{=} {\mathscr{A}}_\mu^a \overset{\text{A--B}}{=} {\mathsf{A}}_\mu^a = \bm{{\mathcal A}}_\mu^a \,,\end{aligned}$$ which is nothing but the first row of .
Detailed procedures
-------------------
We will continue this process by considering the index structure. The second components of the 1-form in the M-theory and the type IIB parameterization are generically expanded as $$\begin{aligned}
{\mathcal A}_{\mu; i_1i_2} = {\hat{A}}_{\mu i_1i_2} + c_1\, {\hat{A}}_\mu^k\,{\hat{A}}_{ki_1i_2} \,,\qquad
\bm{{\mathcal A}}_{\mu; {\mathsf{m}}}^\alpha = {\mathsf{A}}_{\mu {\mathsf{m}}}^\alpha + c_2\, {\mathsf{A}}_\mu^{\mathsf{p}}\,{\mathsf{A}}^\alpha_{{\mathsf{p}}{\mathsf{m}}} \,,\end{aligned}$$ where $c_1$ and $c_2$ are parameters to be determined. From ${\mathcal A}_\mu^\alpha \overset{\text{M--B}}{=} \bm{{\mathcal A}}_{\mu; {\mathsf{y}}}^\alpha$ in , we have $$\begin{aligned}
{\hat{A}}_\mu^\alpha = {\mathcal A}_\mu^\alpha \overset{\text{M--B}}{=} \bm{{\mathcal A}}_{\mu; {\mathsf{y}}}^\alpha = {\mathsf{A}}_{\mu{\mathsf{y}}}^\alpha + c_2\, {\mathsf{A}}_\mu^{\mathsf{p}}\,{\mathsf{A}}^\alpha_{{\mathsf{p}}{\mathsf{y}}}\,.
\label{eq:graviphoton-rule2}\end{aligned}$$ On the other hand, the second rule of and the 11D–10D relation gives $$\begin{aligned}
{\hat{A}}_\mu^{y}\overset{\text{M--B}}{=} {\mathscr{A}}_\mu^{y}\overset{\text{A--B}}{=} {\mathsf{B}}_{\mu {\mathsf{y}}} + {\mathsf{A}}_\mu^{\mathsf{p}}\,{\mathsf{B}}_{{\mathsf{p}}{\mathsf{y}}} \,,\end{aligned}$$ and by comparing this with the $\alpha={y}$ component of , we find $c_2=1$.
Similarly, the map ${\mathcal A}_{\mu; {y}{z}}\overset{\text{M--B}}{=}\bm{{\mathcal A}}_\mu^{\mathsf{y}}$ in gives $$\begin{aligned}
{\mathcal A}_{\mu; {y}{z}} = {\hat{A}}_{\mu {y}{z}} + c_1\, {\hat{A}}_\mu^a\,{\hat{A}}_{a{y}{z}} \overset{\text{M--B}}{=} \bm{{\mathcal A}}_\mu^{\mathsf{y}}\,.
\label{eq:c1-identification}\end{aligned}$$ On the other hand, the second line of and the 11D–10D relation give $$\begin{aligned}
{\hat{A}}_{AB{z}} &\overset{\text{M--B}}{=} {\mathsf{B}}_{AB} - \frac{{\mathsf{B}}_{A{\mathsf{y}}}\,{\mathsf{g}}_{B{\mathsf{y}}}-{\mathsf{g}}_{A{\mathsf{y}}}\,{\mathsf{B}}_{B{\mathsf{y}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}}\,,\qquad
{\hat{A}}_{A{y}{z}} \overset{\text{M--B}}{=} -\frac{{\mathsf{g}}_{A{\mathsf{y}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}} \,.
\label{eq:A3-rule}\end{aligned}$$ By substituting the second relation into the left-hand side of and using ${\mathsf{g}}_{\mu{\mathsf{y}}}=-({\mathsf{A}}_\mu^a\,{\mathsf{g}}_{a{\mathsf{y}}} + {\mathsf{A}}_\mu^{\mathsf{y}}\,{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}})$, we obtain $$\begin{aligned}
\frac{{\mathsf{A}}_\mu^a\,{\mathsf{g}}_{a{\mathsf{y}}} + {\mathsf{A}}_\mu^{\mathsf{y}}\,{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}} - c_1\,{\mathsf{A}}_\mu^a\,{\mathsf{g}}_{a{\mathsf{y}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}} \overset{\text{B--M}}{=} {\hat{A}}_{\mu {y}{z}} + c_1\, {\hat{A}}_\mu^a\,{\hat{A}}_{a{y}{z}} \overset{\text{M--B}}{=} \bm{{\mathcal A}}_\mu^{\mathsf{y}}={\mathsf{A}}_\mu^{\mathsf{y}}\,,\end{aligned}$$ which shows $c_1=1$. Thus, the parameterizations of ${\mathcal A}_{\mu; i_1i_2}$ and $\bm{{\mathcal A}}_{\mu; {\mathsf{m}}}^\alpha$ have determined as $$\begin{aligned}
{\mathcal A}_{\mu; ij} = {\hat{A}}_{\mu ij} + {\hat{A}}_\mu^k\,{\hat{A}}_{kij} \,,\qquad
\bm{{\mathcal A}}_{\mu; {\mathsf{m}}}^\alpha = {\mathsf{A}}_{\mu{\mathsf{m}}}^\alpha + {\mathsf{A}}_\mu^{\mathsf{p}}\,{\mathsf{A}}^\alpha_{{\mathsf{p}}{\mathsf{m}}} \,.\end{aligned}$$
In order to determine the parameterization of further components of the 1-forms, the $T$-duality rules are not enough and we need additional $T$-duality rules. To find the $T$-duality rules, we assume that
- The $T$-duality rules have the 9D covariance (in the nine directions $x^A$ orthogonal to the $T$-duality direction ${\mathsf{x}}^{\mathsf{y}}$).
- The metric appears in the $T$-duality rule only through the combination $\frac{{\mathsf{g}}_{A{\mathsf{y}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}}$ and the graviphoton does not appear explicitly.
By using these assumptions, we obtain the set of standard $T$-duality rules.
For example, from $\alpha={z}$ component of , we find $$\begin{aligned}
{\hat{A}}_\mu^{z}\overset{\text{M--B}}{=} -{\mathsf{C}}_{\mu{\mathsf{y}}} - {\mathsf{A}}_\mu^{\mathsf{p}}\,{\mathsf{C}}_{{\mathsf{p}}{\mathsf{y}}}\,.\end{aligned}$$ In terms of the type IIA field, this is equivalent to $$\begin{aligned}
{\mathscr{C}}_\mu + {\mathscr{A}}_\mu^a\,{\mathscr{C}}_a + {\mathscr{A}}_\mu^{y}\,{\mathscr{C}}_{y}\overset{\text{A--B}}{=} {\mathsf{C}}_{\mu{\mathsf{y}}} + {\mathsf{A}}_\mu^a\,{\mathsf{C}}_{a{\mathsf{y}}}\,,\end{aligned}$$ and by using the identity ${g}_{\mu {y}}=-({\mathscr{A}}_\mu^a\,{g}_{ay} + {\mathscr{A}}_\mu^{y}\,{g}_{{y}{y}})$, we obtain $$\begin{aligned}
\Bigl({\mathscr{C}}_\mu - \frac{{g}_{\mu {y}}}{{g}_{{y}{y}}}\,{\mathscr{C}}_{y}\Bigr) + {\mathscr{A}}_\mu^a\,\Bigl({\mathscr{C}}_a - \frac{{g}_{a{y}}}{{g}_{{y}{y}}}\,{\mathscr{C}}_{y}\Bigr) \overset{\text{A--B}}{=} {\mathsf{C}}_{\mu{\mathsf{y}}} + {\mathsf{A}}_\mu^a\,{\mathsf{C}}_{a{\mathsf{y}}}\,. \end{aligned}$$ From the assumption that the $T$-duality rule does not contain the graviphoton explicitly, this implies the standard $T$-duality rule, $$\begin{aligned}
{\mathsf{C}}_{A{\mathsf{y}}} \overset{\text{B-A}}{=} {\mathscr{C}}_A - \frac{{\mathscr{C}}_{y}\,{g}_{A{y}}}{{g}_{{y}{y}}} \,. \end{aligned}$$ or conversely, $$\begin{aligned}
{\mathscr{C}}_A \overset{\text{A--B}}{=} {\mathsf{C}}_{A{\mathsf{y}}} - {\mathsf{C}}_0\,{\mathsf{B}}_{A{\mathsf{y}}} \,, \end{aligned}$$ where we have employed the standard rule ${\mathscr{C}}_{y}\overset{\text{A--B}}{=} {\mathsf{C}}_0$.
Similarly, if we consider the linear map ${\mathcal A}_{\mu; a\alpha} \overset{\text{M--B}}{=} \bm{{\mathcal A}}_{\mu; a}^\beta\,\epsilon_{\beta\alpha}$ in , we find $$\begin{aligned}
{\hat{A}}_{\mu a\alpha} + {\hat{A}}_\mu^k\,{\hat{A}}_{k a\alpha} \overset{\text{M--B}}{=} \bigl({\mathsf{A}}_{\mu a}^\beta + {\mathsf{A}}_\mu^{\mathsf{p}}\,{\mathsf{A}}^\beta_{{\mathsf{p}}a}\bigr)\,\epsilon_{\beta\alpha} \,.
\label{eq:3-form-2-form}\end{aligned}$$ In particular, for $\alpha={y}$, we obtain a map between the type IIA/IIB fields, $$\begin{aligned}
{\mathscr{C}}_{\mu a{y}} + {\mathscr{A}}_\mu^b\,{\mathscr{C}}_{ba{y}} -\bigl({\mathscr{C}}_\mu + {\mathscr{A}}_\mu^m\,{\mathscr{C}}_m\bigr)\,{\mathscr{B}}_{a{y}} \overset{\text{A--B}}{=} {\mathsf{C}}_{\mu a} + {\mathsf{A}}_\mu^b\,{\mathsf{C}}_{ba} + {\mathsf{A}}_\mu^{\mathsf{y}}\,{\mathsf{C}}_{{\mathsf{y}}a} \,,\end{aligned}$$ and this is equivalent to $$\begin{aligned}
\begin{split}
&{\mathscr{C}}_{\mu a{y}} - {\mathscr{C}}_\mu \,{\mathscr{B}}_{a{y}} + \frac{{\mathscr{C}}_{y}\,{\mathscr{B}}_{a{y}}\, {g}_{\mu {y}}}{{g}_{{y}{y}}}
+ {\mathscr{A}}_\mu^b\,\Bigl({\mathscr{C}}_{ba{y}} - {\mathscr{C}}_b\,{\mathscr{B}}_{a{y}} + \frac{{\mathscr{C}}_{y}\,{\mathscr{B}}_{a{y}}\, {g}_{b{y}}}{{g}_{{y}{y}}} \Bigr)
\\
&\overset{\text{A--B}}{=}
{\mathsf{C}}_{\mu a} - \frac{{\mathsf{C}}_{{\mathsf{y}}a}\,{\mathsf{g}}_{\mu{\mathsf{y}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}}
+ {\mathsf{A}}_\mu^b\,\Bigl({\mathsf{C}}_{ba} - \frac{{\mathsf{C}}_{{\mathsf{y}}a}\,{\mathsf{g}}_{b{\mathsf{y}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}}\Bigr) \,.
\end{split}\end{aligned}$$ Then, we find the $T$-duality rule $$\begin{aligned}
{\mathscr{C}}_{AB{y}} \overset{\text{A--B}}{=} {\mathsf{C}}_{AB} -2\, \frac{{\mathsf{C}}_{[A|{\mathsf{y}}}\,{\mathsf{g}}_{|B]{\mathsf{y}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}} \,.\end{aligned}$$
#### Further steps\
{#further-steps .unnumbered}
We can further proceed by considering a general expansion of the ${\text{SL}}(2)$ singlet $\bm{{\mathcal A}}_{\mu; {\mathsf{m}}_1{\mathsf{m}}_2{\mathsf{m}}_3}$, $$\begin{aligned}
\bm{{\mathcal A}}_{\mu; m_1m_2m_3} &= {\mathsf{A}}_{\mu {\mathsf{m}}_1{\mathsf{m}}_2{\mathsf{m}}_3} + c_3\, \epsilon_{\alpha\beta}\,{\mathsf{A}}_{\mu[{\mathsf{m}}_1}^\alpha\,{\mathsf{A}}_{{\mathsf{m}}_2{\mathsf{m}}_3]}^\beta
{\nonumber}\\
&\quad + c_4\, {\mathsf{A}}_\mu^{\mathsf{p}}\,{\mathsf{A}}_{{\mathsf{p}}{\mathsf{m}}_1{\mathsf{m}}_2{\mathsf{m}}_3} + c_5\, \epsilon_{\alpha\beta}\,{\mathsf{A}}_\mu^{\mathsf{p}}\,{\mathsf{A}}_{{\mathsf{p}}[{\mathsf{m}}_1}^\alpha\,{\mathsf{A}}_{{\mathsf{m}}_2{\mathsf{m}}_3]}^\beta \,.\end{aligned}$$ Similarly, unknown $T$-duality rules can be also expanded by considering possible 9D covariant expressions with parameters. Then, the consistency with the linear map determines all of the parameters. In this manner, by using the linear map , we can find both the parameterization of ${\mathcal A}_{\mu}^{I}$ and $T$-duality rules for the gauge potentials one after another.
Results {#sec:1-form-results}
-------
By continuing the above procedure, we have determined the M-theory parameterization as $$\begin{aligned}
({\mathcal A}_\mu^{I})
= \begin{pmatrix}
{\mathcal A}_\mu^i\\
\frac{{\mathcal A}_{\mu; i_1i_2}}{\sqrt{2!}}\\
\frac{{\mathcal A}_{\mu; i_1\cdots i_5}}{\sqrt{5!}}\\
\frac{{\mathcal A}_{\mu; i_1\cdots i_7,i}}{\sqrt{7!}}\\
\vdots
\end{pmatrix}
= \begin{pmatrix}
{\hat{A}}_\mu^i \\
\frac{1}{\sqrt{2!}}\,\bigl({\hat{N}}_{\mu; i_1i_2}+{\hat{A}}_\mu^k\,{\hat{N}}_{k; i_1i_2}\bigr) \\
\frac{1}{\sqrt{5!}}\,\bigl({\hat{N}}_{\mu; i_1\cdots i_5}+{\hat{A}}_\mu^k\, {\hat{N}}_{k; i_1\cdots i_5} \bigr) \\
\frac{1}{\sqrt{7!}}\, \bigl({\hat{N}}_{\mu; i_1\cdots i_7,i} + {\hat{A}}_{\mu}^k\,{\hat{N}}_{k; i_1\cdots i_7,i}\bigr)\\
\vdots
\end{pmatrix}.
\label{eq:cA-M}\end{aligned}$$ Remarkably, the two tensors ${\hat{N}}_{\mu; p,q,r,\dotsc}$ and ${\hat{N}}_{k; p,q,r,\dotsc}$ in each row can be regarded as particular components of 11D-covariant tensors: $$\begin{aligned}
\begin{split}
{\hat{N}}_{\hat{M}_1;\hat{M}_2\hat{M}_3} &={\hat{A}}_{\hat{M}_1\hat{M}_2\hat{M}_3} \,,
\\
{\hat{N}}_{\hat{M}_1;\hat{M}_2\cdots \hat{M}_6} &= {\hat{A}}_{\hat{M}_1 \cdots \hat{M}_6} - 5\,{\hat{A}}_{\hat{M}_1 [\hat{M}_2\hat{M}_3}\, {\hat{A}}_{\hat{M}_4\hat{M}_5\hat{M}_6]}\,,
\\
{\hat{N}}_{\hat{M}_1;\hat{M}_2\cdots \hat{M}_8, \hat{N}}
&\simeq {\hat{A}}_{\hat{M}_1\cdots \hat{M}_8, \hat{N}} - 21\,\bigl({\hat{A}}_{\hat{M}_1[\hat{M}_2\cdots \hat{M}_6}\,{\hat{A}}_{\hat{M}_7\hat{M}_8]\hat{N}} -{\hat{A}}_{\hat{M}_1[\hat{M}_2\cdots \hat{M}_6}\,{\hat{A}}_{\hat{M}_7\hat{M}_8\hat{N}]} \bigr)
\\
&\quad +35\,{\hat{A}}_{\hat{M}_1[\hat{M}_2\hat{M}_3}\,{\hat{A}}_{\hat{M}_4\hat{M}_5\hat{M}_6}\,{\hat{A}}_{\hat{M}_7\hat{M}_8]\hat{N}} \,,
\end{split}
\label{eq:N-M}\end{aligned}$$ where the meaning of the equivalence $\simeq$ is explained below.
As discussed in [@1907.07177], for any mixed-symmetry potential, not all of the components couple to supersymmetric branes. For the dual graviton ${\mathcal A}_{\mu; i_1\cdots i_7, i}$, only the components satisfying $$\begin{aligned}
i\in \{i_1,\dotsc, i_7\} \,,
\label{eq:SUSY-rule}\end{aligned}$$ couples to supersymmetric branes. The components which do not couple to supersymmetric branes correspond to the $E_{11}$ roots $\alpha$ satisfying $\alpha\cdot\alpha<2$, and they are not connected to the standard $p$-form potentials under $T$-duality and $S$-duality. Since our procedure to determine the parameterization is based on $T$-duality and $S$-duality, it can only provide the parameterization of the components which couple to supersymmetric branes. In this sense, it is more honest to express the last equation of as $$\begin{aligned}
{\hat{N}}_{\hat{M}_1;\hat{M}_2\cdots \hat{M}_7x, x}
&= {\hat{A}}_{\hat{M}_1\cdots \hat{M}_7x, x} - 21\,{\hat{A}}_{\hat{M}_1[\hat{M}_2\cdots \hat{M}_6}\,{\hat{A}}_{\hat{M}_7x]x} +35\,{\hat{A}}_{\hat{M}_1[\hat{M}_2\hat{M}_3}\,{\hat{A}}_{\hat{M}_4\hat{M}_5\hat{M}_6}\,{\hat{A}}_{\hat{M}_7x]x}
{\nonumber}\\
&= {\hat{A}}_{\hat{M}_1\cdots \hat{M}_7x, x} - 15\,{\hat{A}}_{\hat{M}_1x[\hat{M}_2\cdots \hat{M}_5}\,{\hat{A}}_{\hat{M}_6\hat{M}_7]x}
{\nonumber}\\
&\quad -10\,{\hat{A}}_{\hat{M}_1x[\hat{M}_2}\,{\hat{A}}_{\hat{M}_3\hat{M}_4\hat{M}_5}\,{\hat{A}}_{\hat{M}_6\hat{M}_7]x}
+15\,{\hat{A}}_{\hat{M}_1[\hat{M}_2\hat{M}_3}\,{\hat{A}}_{\hat{M}_4\hat{M}_5|x|}\,{\hat{A}}_{\hat{M}_6\hat{M}_7]x} \,.\end{aligned}$$ In this paper, equalities that hold under the restriction are denoted by $\simeq$. The parameterizations of mixed-symmetry potentials which do not satisfy the restriction are not determined in this paper.
Now we turn to the results in type IIB theory. The parameterization takes the form $$\begin{aligned}
(\bm{{\mathcal A}}_\mu^{{\mathsf{I}}})
= \begin{pmatrix}
{\mathsf{A}}_\mu^{\mathsf{m}}\\
\bigl({\mathsf{N}}^\alpha_{\mu;{\mathsf{m}}} + {\mathsf{A}}_\mu^{\mathsf{p}}\, {\mathsf{N}}^\alpha_{{\mathsf{p}};{\mathsf{m}}}\bigr) \\
\frac{1}{\sqrt{3!}}\, \bigl({\mathsf{N}}_{\mu;{\mathsf{m}}_1{\mathsf{m}}_2{\mathsf{m}}_3} + {\mathsf{A}}_\mu^{\mathsf{p}}\,{\mathsf{N}}_{{\mathsf{p}};{\mathsf{m}}_1{\mathsf{m}}_2{\mathsf{m}}_3} \bigr) \\
\frac{1}{\sqrt{5!}}\,\bigl({\mathsf{N}}^\alpha_{\mu;{\mathsf{m}}_1\cdots {\mathsf{m}}_5} + {\mathsf{A}}_\mu^{\mathsf{p}}\,{\mathsf{N}}^\alpha_{{\mathsf{p}};{\mathsf{m}}_1\cdots {\mathsf{m}}_5} \bigr) \\
\frac{1}{\sqrt{6!}}\,\bigl({\mathsf{N}}_{\mu;{\mathsf{m}}_1\cdots {\mathsf{m}}_6,{\mathsf{m}}} + {\mathsf{A}}_{\mu}^{\mathsf{p}}\,{\mathsf{N}}_{{\mathsf{p}};{\mathsf{m}}_1\cdots {\mathsf{m}}_6,{\mathsf{m}}}\bigr) \\
\vdots
\end{pmatrix},
\label{eq:cA-B}\end{aligned}$$ where $$\begin{aligned}
\begin{split}
{\mathsf{N}}_{M_1;M_2}^\alpha &={\mathsf{A}}_{M_1M_2}^\alpha\,,
\\
{\mathsf{N}}_{M_1;M_2M_3M_4} &\equiv {\mathsf{A}}_{M_1\cdots M_4} - \frac{3}{2}\,\epsilon_{\gamma\delta}\,{\mathsf{A}}^\gamma_{M_1[M_2}\,{\mathsf{A}}^\delta_{M_3M_4]}
\\
&={\mathsf{C}}_{M_1\cdots M_4} - 3\,{\mathsf{C}}_{M_1[M_2}\,{\mathsf{B}}_{M_3M_4]}\,,
\\
{\mathsf{N}}^\alpha_{M_1;M_2\cdots M_6} &\equiv {\mathsf{A}}^\alpha_{M_1 \cdots M_6} + 5\,{\mathsf{A}}^\alpha_{M_1[M_2}\, {\mathsf{A}}_{M_3\cdots M_6]}
+ 5\,\epsilon_{\gamma\delta}\,{\mathsf{A}}^\gamma_{M_1[M_2}\,{\mathsf{A}}^\delta_{M_3M_4}\,{\mathsf{A}}^\alpha_{M_5M_6]}
\\
&= \begin{pmatrix}
{\mathsf{C}}_{M_1\cdots M_6} -10\,{\mathsf{C}}_{M_1 [M_2M_3M_4}\,{\mathsf{B}}_{M_5M_6]} + 15\,{\mathsf{C}}_{M_1 [M_2}\,{\mathsf{B}}_{M_3M_4}\,{\mathsf{B}}_{M_5M_6]}
\\
-\bigl({\mathsf{B}}_{M_1\cdots M_6} -10\,{\mathsf{C}}_{M_1 [M_2M_3M_4}\,{\mathsf{C}}_{M_5M_6]}\bigr)
\end{pmatrix},
\\
{\mathsf{N}}_{M_1;M_2\cdots M_7,N} &\simeq
{\mathsf{A}}_{M_1\cdots M_7,N}
+ 6\,{\mathsf{B}}_{M_1 [M_2\cdots M_6}\,{\mathsf{B}}_{M_7] N}
- 6\,{\mathsf{C}}_{M_1 [M_2}\,{\mathsf{C}}_{M_3 \cdots M_7] N}
\\
&\
-60\,{\mathsf{C}}_{M_1 [M_2 M_3 M_4}\,{\mathsf{C}}_{M_5 M_6}\, {\mathsf{B}}_{M_7]N}
+10\, {\mathsf{C}}_{M_1 [M_2M_3M_4}\, {\mathsf{C}}_{M_5M_6M_7]N}
\\
&\
+ \frac{45}{2}\, \bigl({\mathsf{B}}_{M_1 [M_2}\, {\mathsf{B}}_{M_3 M_4}\, {\mathsf{C}}_{M_5 M_6}\, {\mathsf{C}}_{M_7]N}
- {\mathsf{C}}_{M_1 [M_2}\, {\mathsf{C}}_{M_3 M_4}\, {\mathsf{B}}_{M_5 M_6}\, {\mathsf{B}}_{M_7]N}\bigr) \,.
\end{split}
\label{eq:N-B}\end{aligned}$$ The last component ${\mathsf{N}}_{1;6,1}$ is relatively long, and the $S$-duality invariance is not clear. However, this is because of the definition of the dual graviton ${\mathsf{A}}_{7,1}$. As we will see later \[in Eq. \], a certain redefinition of ${\mathsf{A}}_{7,1}$ makes the expression of ${\mathsf{N}}_{1;6,1}$ simpler.
$T$-duality rule {#sec:T-dual}
----------------
In addition to the parameterizations, we have obtained the $T$-duality rules as follows: $$\begin{aligned}
\begin{split}
{g}_{AB} &\overset{\text{A--B}}{=} {\mathsf{g}}_{AB} - \frac{{\mathsf{g}}_{A{\mathsf{y}}}\,{\mathsf{g}}_{B{\mathsf{y}}}-{\mathsf{B}}_{A{\mathsf{y}}}\,{\mathsf{B}}_{B{\mathsf{y}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}}\,,\qquad
{g}_{A {y}}\overset{\text{A--B}}{=}-\frac{{\mathsf{B}}_{A {\mathsf{y}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}}\,,\qquad
{g}_{{y}{y}}\overset{\text{A--B}}{=}\frac{1}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}}\,,
\\
{\mathscr{A}}_\mu^a &\overset{\text{A--B}}{=} {\mathsf{A}}_\mu^a\,,\qquad
{\mathscr{A}}_\mu^{y}\overset{\text{A--B}}{=} {\mathsf{B}}_{\mu{\mathsf{y}}} + {\mathsf{A}}_\mu^{{\mathsf{p}}}\,{\mathsf{B}}_{{\mathsf{p}}{\mathsf{y}}} \,,
\\
{\mathscr{B}}_{AB} &\overset{\text{A--B}}{=} {\mathsf{B}}_{AB} - \frac{{\mathsf{B}}_{A {\mathsf{y}}}\,{\mathsf{g}}_{B {\mathsf{y}}}-{\mathsf{g}}_{A{\mathsf{y}}}\,{\mathsf{B}}_{B{\mathsf{y}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}}\,,\qquad
{\mathscr{B}}_{A{y}} \overset{\text{A--B}}{=} -\frac{{\mathsf{g}}_{A{\mathsf{y}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}} \,, \qquad
{\operatorname{e}^{2{\Phi}}}\overset{\text{A--B}}{=} \frac{{\operatorname{e}^{2{\varphi}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}} \,,
\\
{\mathscr{C}}_{A_1\cdots A_{n-1}{y}}&\overset{\text{A--B}}{=} {\mathsf{C}}_{A_1\cdots A_{n-1}} - (n-1)\,\frac{{\mathsf{C}}_{[A_1\cdots A_{n-2}|{\mathsf{y}}|}\,{\mathsf{g}}_{A_{n-1}]{\mathsf{y}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}}\,,
\\
{\mathscr{C}}_{A_1\cdots A_n} &\overset{\text{A--B}}{=} {\mathsf{C}}_{A_1\cdots A_n{\mathsf{y}}} - n\, {\mathsf{C}}_{[A_1\cdots A_{n-1}}\, {\mathsf{B}}_{A_n]{\mathsf{y}}} - n\,(n-1)\,\frac{{\mathsf{C}}_{[A_1\cdots A_{n-2}|{\mathsf{y}}|}\, {\mathsf{B}}_{A_{n-1}|{\mathsf{y}}|}\,{\mathsf{g}}_{A_n]{\mathsf{y}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}}\,.
\end{split}
\label{eq:Busc-standard}\end{aligned}$$ For the 6-form potential $B_6$ and the dual graviton $A_{7, 1}$, we find $$\begin{aligned}
{\mathscr{B}}_{A_1\cdots A_5{y}}&\overset{\text{A--B}}{=} {\mathsf{B}}_{A_1\cdots A_5 {\mathsf{y}}} -5\,{\mathsf{A}}_{[A_1\cdots A_4}\,{\mathsf{C}}_{A_5]{\mathsf{y}}} -5\,{\mathsf{A}}_{[A_1A_2A_3|{\mathsf{y}}|}\,{\mathsf{C}}_{A_4A_5]}
{\nonumber}\\
&\quad\
- \frac{45}{2}\, {\mathsf{C}}_{[A_1A_2}\,{\mathsf{B}}_{A_3A_4}\,{\mathsf{C}}_{A_5]{\mathsf{y}}}
- \frac{15}{2}\, {\mathsf{C}}_{[A_1A_2}\,{\mathsf{C}}_{A_3A_4}\,{\mathsf{B}}_{A_5]{\mathsf{y}}}
{\nonumber}\\
&\quad\
- \frac{10\,{\mathsf{A}}_{[A_1\cdots A_3|{\mathsf{y}}|}\,{\mathsf{C}}_{A_4|{\mathsf{y}}|}{\mathsf{g}}_{A_5]{\mathsf{y}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}}
- \frac{15\,{\mathsf{C}}_{[A_1A_2}\, {\mathsf{B}}_{A_3|{\mathsf{y}}|}\,{\mathsf{C}}_{A_4|{\mathsf{y}}|}\,{\mathsf{g}}_{A_5]{\mathsf{y}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}} \,,
\label{eq:Busc-B6A-B6B}
\\
{\mathscr{B}}_{A_1\cdots A_6}&\overset{\text{A--B}}{=}
{\mathsf{A}}_{A_1 \cdots A_6 {\mathsf{y}}, {\mathsf{y}}}
-6 \, {\mathsf{B}}_{[A_1\cdots A_5|{\mathsf{y}}|}\, {\mathsf{B}}_{A_6] {\mathsf{y}}}
{\nonumber}\\
&\quad\
+30 \, {\mathsf{A}}_{[A_1 A_2 A_3|{\mathsf{y}}}\, {\mathsf{C}}_{|A_4 A_5}\, {\mathsf{B}}_{A_6] {\mathsf{y}}}
+30 \, {\mathsf{A}}_{[A_1 \cdots A_4}\, {\mathsf{C}}_{A_5|{\mathsf{y}}|}\, {\mathsf{B}}_{A_6] {\mathsf{y}}}
{\nonumber}\\
&\quad\
- \frac{315}{2} \, {\mathsf{B}}_{[A_1 A_2}\, {\mathsf{B}}_{A_3|{\mathsf{y}}|}\, {\mathsf{C}}_{A_4 A_5}\, {\mathsf{C}}_{A_6] {\mathsf{y}}}
+ \frac{60\, {\mathsf{A}}_{[A_1 A_2 A_3|{\mathsf{y}}}\, {\mathsf{B}}_{A_4|{\mathsf{y}}}\, {\mathsf{C}}_{A_5|{\mathsf{y}}}\, {\mathsf{g}}_{|A_6]{\mathsf{y}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}} \,,
\\
{\mathscr{A}}_{A_1\cdots A_6{y}, {y}}&\overset{\text{A--B}}{=} {\mathsf{B}}_{A_1\cdots A_6}
- 30\, {\mathsf{B}}_{[A_1A_2}\,{\mathsf{C}}_{A_3A_4}\, {\mathsf{C}}_{A_5A_6]}
- \frac{6\,{\mathsf{B}}_{[A_1\cdots A_5|{\mathsf{y}}}\,{\mathsf{g}}_{|A_6]{\mathsf{y}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}}
+ \frac{20\, {\mathsf{A}}_{[A_1A_2A_3|{\mathsf{y}}|}\, {\mathsf{C}}_{A_4A_5}\,{\mathsf{g}}_{A_6]{\mathsf{y}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}}
{\nonumber}\\
&\quad\
+ \frac{30\, {\mathsf{B}}_{[A_1|{\mathsf{y}}|}\, {\mathsf{C}}_{A_2A_3}\, {\mathsf{C}}_{A_4A_5}\,{\mathsf{g}}_{A_6]{\mathsf{y}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}}
+ \frac{150\, {\mathsf{B}}_{[A_1A_2}\,{\mathsf{C}}_{A_3A_4}\, {\mathsf{C}}_{A_5|{\mathsf{y}}|}\,{\mathsf{g}}_{A_6]{\mathsf{y}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}}\,,
\label{eq:Busc-A71A-B6B}
\\
{\mathscr{A}}_{A_1\cdots A_6{y}, B}&\overset{\text{A--B}}{\simeq} {\mathsf{A}}_{A_1\cdots A_6{\mathsf{y}}, B}
- 6\,{\mathsf{C}}_{[A_1\cdots A_5|{\mathsf{y}}|}\,{\mathsf{C}}_{A_6]B}
- 6\,{\mathsf{B}}_{[A_1\cdots A_5|B|}\,{\mathsf{B}}_{A_6]{\mathsf{y}}}
{\nonumber}\\
&\quad\
+ 10\,{\mathsf{C}}_{[A_1A_2A_3|B|}\,{\mathsf{C}}_{A_4A_5A_6]{\mathsf{y}}}
+ 20\,{\mathsf{C}}_{[A_1A_2A_3|{\mathsf{y}}|}\,{\mathsf{B}}_{A_4|B|}\,{\mathsf{C}}_{A_5A_6]}
{\nonumber}\\
&\quad\
+ 40\,{\mathsf{C}}_{[A_1A_2A_3|{\mathsf{y}}|}\,{\mathsf{B}}_{A_4A_5}\,{\mathsf{C}}_{A_6]B}
+ 30\,{\mathsf{B}}_{[A_1A_2}\,{\mathsf{C}}_{A_3A_4}\, {\mathsf{C}}_{A_5A_6]}\,{\mathsf{B}}_{B{\mathsf{y}}}
{\nonumber}\\
&\quad\
+\frac{45}{2}\,{\mathsf{B}}_{[A_1A_2}\,{\mathsf{C}}_{A_3A_4}\,\bigl(-{\mathsf{B}}_{A_5|{\mathsf{y}}|}\,{\mathsf{C}}_{A_6]B} + {\mathsf{C}}_{A_5|{\mathsf{y}}|}\,{\mathsf{B}}_{A_6]B}\bigr)
{\nonumber}\\
&\quad\
-\frac{{\mathsf{A}}_{A_1 \cdots A_6 {\mathsf{y}}, {\mathsf{y}}}\,{\mathsf{g}}_{B{\mathsf{y}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}}
+ \frac{6\, {\mathsf{C}}_{[A_1 \cdots A_5 |{\mathsf{y}}|}\, {\mathsf{C}}_{A_6]{\mathsf{y}}}\,{\mathsf{g}}_{B{\mathsf{y}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}}
-\frac{6\, {\mathsf{C}}_{B{\mathsf{y}}}\, {\mathsf{C}}_{[A_1 \cdots A_5 |{\mathsf{y}}|}\,{\mathsf{g}}_{|A_6]{\mathsf{y}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}}
{\nonumber}\\
&\quad\
+\frac{6 \, {\mathsf{B}}_{B{\mathsf{y}}}\, {\mathsf{B}}_{[A_1\cdots A_5|{\mathsf{y}}}\, {\mathsf{g}}_{|A_6]{\mathsf{y}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}}
-\frac{10\, {\mathsf{C}}_{[A_1 A_2| B{\mathsf{y}}}\, {\mathsf{C}}_{|A_3 A_4 A_5 |{\mathsf{y}}|}\,{\mathsf{g}}_{A_6] {\mathsf{y}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}}
{\nonumber}\\
&\quad\
+\frac{120\, {\mathsf{C}}_{[A_1 A_2 |B{\mathsf{y}}|}\, {\mathsf{B}}_{A_3 A_4}\, {\mathsf{C}}_{A_5|{\mathsf{y}}|}\, {\mathsf{g}}_{A_6]{\mathsf{y}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}}
-\frac{20\, {\mathsf{B}}_{B{\mathsf{y}}}\,{\mathsf{C}}_{[A_1 A_2 A_3|{\mathsf{y}}|}\, {\mathsf{C}}_{A_4 A_5}\,{\mathsf{g}}_{A_6]{\mathsf{y}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}}
{\nonumber}\\
&\quad\
+\frac{40\, {\mathsf{C}}_{[A_1 A_2 A_3|{\mathsf{y}}|}\, {\mathsf{B}}_{A_4|B|}\, {\mathsf{C}}_{A_5|{\mathsf{y}}|}\,{\mathsf{g}}_{A_6] {\mathsf{y}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}}
-\frac{120\, {\mathsf{B}}_{B{\mathsf{y}}}\,{\mathsf{B}}_{[A_1 A_2}\, {\mathsf{C}}_{A_3 A_4}\, {\mathsf{C}}_{A_5|{\mathsf{y}}}\,{\mathsf{g}}_{|A_6] {\mathsf{y}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}}
{\nonumber}\\
&\quad\
+\frac{45}{2}\,\frac{{\mathsf{B}}_{[A_1 A_2}\, {\mathsf{B}}_{A_3|{\mathsf{y}}|}\, {\mathsf{C}}_{A_4 A_5}\, {\mathsf{C}}_{A_6]{\mathsf{y}}}\, {\mathsf{g}}_{B{\mathsf{y}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}} \,.
\label{eq:Busc-A71A-A71B}\end{aligned}$$ The $T$-duality rules and coincide with the known results [@hep-th/9806169] (see Appendix A therein), for which the following identification of supergravity fields are needed: $$\begin{aligned}
\begin{pmatrix} \mathsf{g}_{\mu\nu} \\ B \\ C^{(1)} \\ C^{(3)} \\ C^{(5)} \\ \tilde{B} \\ N \end{pmatrix}_{\!\!\!{\text{(IIA)}\atop\text{\cite{hep-th/9806169}}}}
=\begin{pmatrix}
{g}_{MN}\\ {\mathscr{B}}_2\\ {\mathscr{C}}_1\\ {\mathscr{C}}_3\\ {\mathscr{C}}_5\\ - {\mathscr{B}}_6 \\ {\mathscr{A}}_{7,1} \end{pmatrix}_{\!\!\!{\text{(IIA)}\atop\text{here}}} ,\quad
\begin{pmatrix}
g_{\mu\nu}\\ {\mathcal B}\\ C^{(0)}\\ C^{(2)}\\ C^{(4)}\\ C^{(6)}\\ \tilde{{\mathcal B}} \\ {\mathcal N}\end{pmatrix}_{\!\!\!{\text{(IIB)}\atop\text{\cite{hep-th/9806169}}}}
= \begin{pmatrix}
{\mathsf{g}}_{MN}\\ {\mathsf{B}}_2\\ - {\mathsf{C}}_0 \\ - {\mathsf{C}}_2\\ - {\mathsf{A}}_4\\ -\bigl( {\mathsf{C}}_6-\tfrac{1}{4}\, {\mathsf{B}}_2\wedge {\mathsf{B}}_2\wedge {\mathsf{C}}_2\bigr) \\ -\bigl( {\mathsf{B}}_6-\tfrac{1}{4}\, {\mathsf{C}}_2\wedge {\mathsf{C}}_2\wedge {\mathsf{B}}_2\bigr) \\ {\mathsf{A}}_{7,1}
\end{pmatrix}_{{\!\!\!{\text{(IIB)}\atop\text{here}}}} .\end{aligned}$$ On the other hand, has been obtained in [@hep-th/9908094], where ${\mathsf{B}}_2=0$ and ${\mathsf{C}}_2=0$ are assumed. If we truncate ${\mathsf{B}}_2$ and ${\mathsf{C}}_2$, we have ${\mathsf{A}}_4={\mathsf{C}}_4$ and the $T$-duality rule reduces to $$\begin{aligned}
{\mathscr{A}}_{A_1\cdots A_6{y},B}&\overset{\text{A--B}}{\simeq} {\mathsf{A}}_{A_1\cdots A_6{\mathsf{y}},B} + 10\,{\mathsf{A}}_{[A_1A_2A_3|B|}\,{\mathsf{A}}_{A_4A_5A_6]{\mathsf{y}}}
{\nonumber}\\
&\quad\
-\frac{{\mathsf{A}}_{A_1 \cdots A_6 {\mathsf{y}}, {\mathsf{y}}}\,{\mathsf{g}}_{B{\mathsf{y}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}}
-\frac{10\, {\mathsf{A}}_{[A_1 A_2| B{\mathsf{y}}}\, {\mathsf{A}}_{|A_3 A_4 A_5 |{\mathsf{y}}|}\,{\mathsf{g}}_{A_6] {\mathsf{y}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}} \,.\end{aligned}$$ More explicitly, according to the restriction , the direction $x\equiv B$ must be contained in $\{A_1,\dotsc, A_6\}$ and by choosing $A_6=x$, we have $$\begin{aligned}
{\mathscr{A}}_{A_1\cdots A_5xy, x}&\overset{\text{A--B}}{=} {\mathsf{A}}_{A_1\cdots A_5x{\mathsf{y}}, x} + 5\,{\mathsf{A}}_{[A_1A_2A_3|x|}\,{\mathsf{A}}_{A_4A_5]x{\mathsf{y}}} -\frac{{\mathsf{A}}_{A_1 \cdots A_5x{\mathsf{y}}, {\mathsf{y}}}\,{\mathsf{g}}_{x{\mathsf{y}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}}
{\nonumber}\\
&\quad\
+\frac{5\, {\mathsf{A}}_{[A_1 A_2| x{\mathsf{y}}}\, {\mathsf{A}}_{|A_3 A_4|x{\mathsf{y}}|}\,{\mathsf{g}}_{A_5] {\mathsf{y}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}}
-\frac{5\, {\mathsf{A}}_{[A_1 A_2| x{\mathsf{y}}}\, {\mathsf{A}}_{|A_3 A_4 A_5]{\mathsf{y}}}\,{\mathsf{g}}_{x{\mathsf{y}}}}{3\,{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}} \,.\end{aligned}$$ If we denote $k\equiv \partial_x$ and $h\equiv \partial_{{\mathsf{y}}}$, and define $$\begin{aligned}
N^{(7)}_{M_1\cdots M_6}\equiv {\mathscr{A}}_{M_1\cdots M_7, x}\,,\quad
{\mathsf{N}}^{(7)}_{M_1\cdots M_6}\equiv {\mathsf{A}}_{M_1\cdots M_7, x}\,,\quad
{\mathcal N}^{(7)}_{M_1\cdots M_7}\equiv {\mathsf{A}}_{M_1\cdots M_7, {\mathsf{y}}}\,, \end{aligned}$$ the result of [@hep-th/9908094] \[see Eq. (5.13)\] is precisely reproduced, $$\begin{aligned}
(\iota_k N^{(7)})_{A_1\cdots A_5{y}}&\overset{\text{A--B}}{=} (\iota_k\iota_h {\mathsf{N}}^{(7)})_{A_1\cdots A_5} - 5\,(\iota_k {\mathsf{A}})_{[A_1A_2A_3}\,(\iota_k\iota_h {\mathsf{A}})_{A_4A_5]}
- \frac{(\iota_k\iota_h {\mathcal N}^{(7)})_{A_1 \cdots A_5}\,{\mathsf{g}}_{x{\mathsf{y}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}}
{\nonumber}\\
& -\frac{5\, (\iota_k\iota_h {\mathsf{A}})_{[A_1 A_2}\, (\iota_k\iota_h {\mathsf{A}})_{A_3 A_4}\,{\mathsf{g}}_{A_5] {\mathsf{y}}}}{{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}}
+ \frac{5\, (\iota_h {\mathsf{A}})_{[A_1 A_2 A_3}\,(\iota_k\iota_h {\mathsf{A}})_{A_4 A_5]}\, {\mathsf{g}}_{x{\mathsf{y}}}}{3\,{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}} \,.\end{aligned}$$
In the above computation, we have shown only $T$-duality transformations from type IIB to type IIA, but we can easily find the inverse map. The standard rules have the same form even for the map from type IIA to type IIB, $$\begin{aligned}
{\mathsf{g}}_{AB} &\overset{\text{B--A}}{=} {g}_{AB} - \frac{{g}_{A{y}}\,{g}_{B{y}}-{\mathscr{B}}_{A{y}}\,{\mathscr{B}}_{B{y}}}{{g}_{{y}{y}}}\,,\qquad
{\mathsf{g}}_{A {\mathsf{y}}}\overset{\text{B--A}}{=}-\frac{{\mathscr{B}}_{A {y}}}{{g}_{{y}{y}}}\,,\qquad
{\mathsf{g}}_{{\mathsf{y}}{\mathsf{y}}}\overset{\text{B--A}}{=}\frac{1}{{g}_{{y}{y}}}\,,
{\nonumber}\\
{\mathsf{A}}_\mu^a &\overset{\text{B--A}}{=} {\mathscr{A}}_\mu^a\,,\qquad
{\mathsf{A}}_\mu^{\mathsf{y}}\overset{\text{B--A}}{=} {\mathscr{B}}_{\mu {y}} + {\mathscr{A}}_\mu^{p}\,{\mathscr{B}}_{p{y}} \,,
{\nonumber}\\
{\mathsf{B}}_{AB} &\overset{\text{B--A}}{=} {\mathscr{B}}_{AB} - \frac{{\mathscr{B}}_{A {y}}\,{g}_{B {y}}-{g}_{A {y}}\,{\mathscr{B}}_{B {y}}}{{g}_{{y}{y}}}\,,\qquad
{\mathsf{B}}_{Ay} \overset{\text{B--A}}{=} -\frac{{g}_{A{y}}}{{g}_{{y}{y}}} \,, \qquad
{\operatorname{e}^{2{\varphi}}}\overset{\text{B--A}}{=} \frac{{\operatorname{e}^{2{\Phi}}}}{{g}_{{y}{y}}} \,,
{\nonumber}\\
{\mathsf{C}}_{A_1\cdots A_{n-1}{\mathsf{y}}}&\overset{\text{B--A}}{=} {\mathscr{C}}_{A_1\cdots A_{n-1}} - (n-1)\,\frac{{\mathscr{C}}_{[A_1\cdots A_{n-2}|{y}|}\,{g}_{A_{n-1}]{y}}}{{g}_{{y}{y}}}\,,
\\
{\mathsf{C}}_{A_1\cdots A_n} &\overset{\text{B--A}}{=} {\mathscr{C}}_{A_1\cdots A_n{y}} - n\, {\mathscr{C}}_{[A_1\cdots A_{n-1}}\, {\mathscr{B}}_{A_n]{y}} - n\,(n-1)\,\frac{{\mathscr{C}}_{[A_1\cdots A_{n-2}|{y}|}\, {\mathscr{B}}_{A_{n-1}|{y}|}\,{g}_{A_n]{y}}}{{g}_{{y}{y}}}\,.
{\nonumber}\end{aligned}$$ Regarding the 6-form potential and the dual graviton, the results are as follows: $$\begin{aligned}
{\mathsf{B}}_{A_1\cdots A_5{\mathsf{y}}}
&\overset{\text{B--A}}{=}{\mathscr{B}}_{A_1\cdots A_5{y}}
+5\,{\mathscr{C}}_{[A_1\cdots A_4|{y}|}\,{\mathscr{C}}_{A_5]}
+5\,{\mathscr{C}}_{[A_1A_2A_3}\,{\mathscr{C}}_{A_4A_5]{y}}
{\nonumber}\\
&\quad -\frac{15\,{\mathscr{C}}_{[A_1A_2|{y}|}\,{\mathscr{C}}_{A_3,A_4|{y}|}\,{g}_{A_5]{y}}}{{g}_{{y}{y}}}
-\frac{5\,{\mathscr{C}}_{{y}}\,{\mathscr{C}}_{[A_1\cdots A_4|{y}|}\,{g}_{A_5]{y}}}{{g}_{{y}{y}}} \,,
\\
{\mathsf{B}}_{A_1\cdots A_6}
&\overset{\text{B--A}}{=}
{\mathscr{A}}_{A_1\cdots A_6{y},{y}}
-6\,{\mathscr{B}}_{[A_1\cdots A_5{y}}\,{\mathscr{B}}_{A_6]{y}}
-30\,{\mathscr{C}}_{[A_1\cdots A_4|{y}|}\,{\mathscr{C}}_{A_5}\,{\mathscr{B}}_{A_6]{y}}
{\nonumber}\\
&\quad -10\,{\mathscr{C}}_{[A_1A_2A_3}\,{\mathscr{C}}_{A_4A_5|{y}|}\,{\mathscr{B}}_{A_6]{y}}
+30\,{\mathscr{C}}_{[A_1A_2|{y}|}\,{\mathscr{C}}_{A_3A_4|{y}|}\,{\mathscr{B}}_{A_5A_6]}
{\nonumber}\\
&\quad -\frac{30\,{\mathscr{C}}_{{y}}\,{\mathscr{C}}_{[A_1\cdots A_4|{y}|}\,{\mathscr{B}}_{A_5|{y}|}\,{g}_{A_6]{y}}}{{g}_{{y}{y}}}
-\frac{90\,{\mathscr{C}}_{[A_1A_2|{y}|}\,{\mathscr{C}}_{A_3A_4|{y}|}\,{\mathscr{B}}_{A_5|{y}|}\,{g}_{A_6]{y}}}{{g}_{{y}{y}}}\,,
\\
{\mathsf{A}}_{A_1\cdots A_6{\mathsf{y}},{\mathsf{y}}}
&\overset{\text{B--A}}{=} {\mathscr{B}}_{A_1\cdots A_6}
-\frac{6\,{\mathscr{B}}_{[A_1\cdots A_5|{y}|}\,{g}_{A_6]{y}}}{{g}_{{y}{y}}}
+\frac{45}{2}\, \frac{{\mathscr{C}}_{[A_1A_2|{y}|}\,{\mathscr{B}}_{A_3A_4}\,{\mathscr{C}}_{A_5}\,{g}_{A_6]{y}}}{{g}_{{y}{y}}}\,,
\\
{\mathsf{A}}_{A_1\cdots A_6{\mathsf{y}}, B}
&\overset{\text{B--A}}{\simeq}
{\mathscr{A}}_{A_1\cdots A_6{y}, B}
-6\,{\mathscr{B}}_{[A_1\cdots A_5|B|}\,{\mathscr{B}}_{A_6]{y}}
+6\,{\mathscr{C}}_{[A_1\cdots A_5}\,{\mathscr{C}}_{A_6] B{y}}
{\nonumber}\\
&\quad -\frac{15}{2} \,{\mathscr{C}}_{[A_1\cdots A_4|{y}|}\,{\mathscr{C}}_{A_5A_6]B}
+20\,{\mathscr{C}}_{[A_1A_2A_3}\,{\mathscr{C}}_{A_4A_5|B|}\,{\mathscr{B}}_{A_6]{y}}
{\nonumber}\\
&\quad -20\,{\mathscr{C}}_{[A_1A_2A_3}\,{\mathscr{C}}_{A_4A_5|{y}|}\,{\mathscr{B}}_{A_6]B}
-40\,{\mathscr{C}}_{[A_1A_2A_3}\,{\mathscr{C}}_{A_4|B{y}|}\,{\mathscr{B}}_{A_5A_6]}
{\nonumber}\\
&\quad -\frac{45}{2} \,{\mathscr{C}}_{[A_1A_2|{y}|}\,{\mathscr{C}}_{A_3}\,{\mathscr{B}}_{A_4|B|}\,{\mathscr{B}}_{A_5A_6]}
-\frac{6\,{\mathscr{A}}_{[A_1\cdots A_5|B{y}, {y}}\,{g}_{|A_6]{y}}}{{g}_{{y}{y}}}
\\
&\quad
+\frac{6\,{\mathscr{B}}_{B{y}}\,{\mathscr{B}}_{[A_1\cdots A_5|{y}|}\,{g}_{A_6]{y}}}{{g}_{{y}{y}}}
+\frac{15\,{\mathscr{C}}_{[A_1\cdots A_4|{y}|}\,{\mathscr{C}}_{A_5|B{y}|}\,{g}_{A_6]{y}}}{{g}_{{y}{y}}}
{\nonumber}\\
&\quad
+\frac{15}{2}\,\frac{{\mathscr{C}}_{[A_1\cdots A_4|{y}|}\,{\mathscr{C}}_{A_5A_6]{y}}\,{g}_{B{y}}}{{g}_{{y}{y}}}
-\frac{20\,{\mathscr{C}}_{[A_1A_2A_3}\,{\mathscr{C}}_{A_4A_5|{y}|}\,{\mathscr{B}}_{A_6]{y}}\,{g}_{B{y}}}{{g}_{{y}{y}}}
{\nonumber}\\
&\quad
-\frac{195}{8} \,\frac{{\mathscr{C}}_{[A_1A_2|{y}|}\,{\mathscr{C}}_{A_3A_4|{y}|}\,{\mathscr{B}}_{A_5A_6]}\,{g}_{B{y}}}{{g}_{{y}{y}}}
-\frac{45}{4} \,\frac{{\mathscr{C}}_{[A_1A_2|{y}|}\,{\mathscr{C}}_{A_3A_4|{y}|}\,{\mathscr{B}}_{A_5|B|}\,{g}_{A_6]{y}}}{{g}_{{y}{y}}}
{\nonumber}\\
&\quad
-\frac{45}{2} \,\frac{{\mathscr{B}}_{B{y}}\,{\mathscr{C}}_{[A_1A_2|{y}|}\,{\mathscr{C}}_{A_3}\,{\mathscr{B}}_{A_4A_5}\,{g}_{A_6]{y}}}{{g}_{{y}{y}}}
-\frac{45}{2} \,\frac{{\mathscr{C}}_{{y}}\,{\mathscr{C}}_{[A_1A_2|{y}|}\,{\mathscr{B}}_{A_3A_4}\,{\mathscr{B}}_{A_5|B|}\,{g}_{A_6]{y}}}{{g}_{{y}{y}}}\,.
{\nonumber}\end{aligned}$$
Now, let us comment more on the restriction rule. In the $T$-duality rule , we are assuming that $B$ is contained in $\{A_1,\cdots ,A_6\}$. When the restriction is removed, we expect the right-hand side of the $T$-duality rule is modified. In general, the components which do not satisfy the restriction is in the same orbit as the $(\alpha\neq\beta)$-component of the type IIB potential ${\mathscr{A}}^{\alpha\beta}_8$, which is electric-magnetic dual to the 0-form potential ${\mathsf{m}}_{\alpha\beta}$. Therefore, it will be possible that ${\mathscr{A}}^{\alpha\beta}_{A_1 \cdots A_6B{\mathsf{y}}}$ appears on the right-hand side of .
$S$-duality rule
----------------
The standard $S$-duality transformation rules are reproduced as follows: $$\begin{aligned}
\begin{split}
&{\mathsf{g}}'_{MN} ={\mathsf{g}}_{MN}\,,\quad {\mathsf{A}}'^{\mathsf{m}}_\mu = {\mathsf{A}}_\mu^{\mathsf{m}}\,,\quad
{\mathsf{C}}'_0 = - \frac{{\mathsf{C}}_0}{({\mathsf{C}}_0)^2+{\operatorname{e}^{-2{\varphi}}}}\,,\quad {\operatorname{e}^{-{\varphi}'}} = \frac{{\operatorname{e}^{-{\varphi}'}}}{({\mathsf{C}}_0)^2+{\operatorname{e}^{-2{\varphi}}}}\,,
\\
&{\mathsf{B}}'_2 =- {\mathsf{C}}_2\,,\quad {\mathsf{C}}'_2 = {\mathsf{B}}_2\,,\quad {\mathsf{C}}'_4 = {\mathsf{C}}_4 - {\mathsf{B}}_2\wedge {\mathsf{C}}_2 \,, \quad
{\mathsf{A}}'_4 = {\mathsf{A}}_4\,,
\\
&{\mathsf{C}}'_6 = - {\mathsf{B}}_6 + \frac{1}{2}\, {\mathsf{B}}_2\wedge {\mathsf{C}}_2\wedge {\mathsf{C}}_2 \,, \quad
{\mathsf{B}}'_6 = {\mathsf{C}}_6 - \frac{1}{2}\, {\mathsf{C}}_2\wedge {\mathsf{B}}_2\wedge {\mathsf{B}}_2\,,\quad
{\mathsf{A}}'^1_6 = - {\mathsf{A}}^2_6\,,\quad {\mathsf{A}}'^2_6 = {\mathsf{A}}^1_6\,.
\end{split}\end{aligned}$$ From the $S$-duality invariance of $\bm{{\mathcal A}}_{\mu; {\mathsf{m}}_1\cdots {\mathsf{m}}_6, {\mathsf{m}}}$, we also find $$\begin{aligned}
{\mathsf{A}}'_{M_1\cdots M_7,M}&\simeq {\mathsf{A}}_{M_1\cdots M_7,M} + 7\,\bigl({\mathsf{B}}_{[M_1\cdots M_6}\,{\mathsf{B}}_{M_7]M}-{\mathsf{B}}_{[M_1\cdots M_6}\,{\mathsf{B}}_{M_7M]}\bigr)
{\nonumber}\\
&\quad + 7\,\bigl({\mathsf{C}}_{[M_1\cdots M_6}\,{\mathsf{C}}_{M_7]M} - {\mathsf{C}}_{[M_1\cdots M_6}\,{\mathsf{C}}_{M_7M]} \bigr)
{\nonumber}\\
&\quad - \frac{105}{2}\,\bigl({\mathsf{A}}_{[M_1\cdots M_4}\,{\mathsf{B}}_{M_5M_6}\,{\mathsf{C}}_{M_7]M} - {\mathsf{A}}_{[M_1\cdots M_4}\,{\mathsf{C}}_{M_5M_6}\,{\mathsf{B}}_{M_7M]}\bigr)
{\nonumber}\\
&\quad +\frac{945}{4}\,\bigl({\mathsf{B}}_{[M_1M_2}\,{\mathsf{B}}_{M_3M_4}\,{\mathsf{C}}_{M_5M_6}\,{\mathsf{C}}_{M_7]M} -{\mathsf{B}}_{[M_1M_2}\,{\mathsf{B}}_{M_3M_4}\,{\mathsf{C}}_{M_5M_6}\,{\mathsf{C}}_{M_7M]} \bigr)\,.
\label{eq:dual-graviton-S-dual}\end{aligned}$$
Another approach based on the generalized metric {#sec:generalized-metric}
================================================
In this section, we discuss another derivation of the $T$-/$S$-duality transformation rule for the dual graviton, which is based on the generalized metric. We also explain another method to determine the parameterization of the 1-form ${\mathcal A}_\mu^I$.
In $d$ dimensions, scalar fields are packaged into $U$-duality-covariant object called the generalized metric, which are denoted as ${\mathcal M}_{IJ}$ and ${\mathsf{M}}_{{\mathsf{I}}{\mathsf{J}}}$ in M-theory and type IIB, respectively. The generalized vielbein, ${\mathcal E}^I{}_J$ and ${\mathsf{E}}^{\mathsf{I}}{}_{\mathsf{J}}$ respectively, are defined such that $$\begin{aligned}
{\mathcal M}_{IJ} \equiv \delta_{KL}\,{\mathcal E}^K{}_I \,{\mathcal E}^L{}_J \,, \qquad
{\mathsf{M}}_{{\mathsf{I}}{\mathsf{J}}} \equiv \delta_{{\mathsf{K}}{\mathsf{L}}}\,{\mathsf{E}}^{\mathsf{K}}{}_{\mathsf{I}}\,{\mathsf{E}}^{\mathsf{L}}{}_{\mathsf{J}}\,.
\label{eq:gen-metric}\end{aligned}$$ According to [@1111.0459], the generalized vielbein can be constructed as follows. We first consider the positive-root generators of the $E_n$ algebra, which are summarized as $$\begin{aligned}
\{E_{\bm{\alpha}}\} = \{K^i{}_j \ (i<j),\, R^{i_1i_2i_3} ,\, R^{i_1\cdots i_6} ,\, R^{i_1\cdots i_8,i} ,\cdots \bigr\}\,,\end{aligned}$$ in the M-theory parameterization and as $$\begin{aligned}
\{{\mathsf{E}}_{\bm{\alpha}}\} = \{{\mathsf{K}}^{\mathsf{m}}{}_{\mathsf{n}}\ ({\mathsf{m}}<{\mathsf{n}}),\, {\mathsf{R}}_{22},\,{\mathsf{R}}_\alpha^{{\mathsf{m}}_1{\mathsf{m}}_2},\,{\mathsf{R}}^{{\mathsf{m}}_1\cdots {\mathsf{m}}_4},\,{\mathsf{R}}^{{\mathsf{m}}_1\cdots {\mathsf{m}}_6}_\alpha,\,{\mathsf{R}}^{{\mathsf{m}}_1\cdots {\mathsf{m}}_7,{\mathsf{m}}},\cdots\}\,,\end{aligned}$$ in the type IIB parameterization. We also consider the Cartan generators, $$\begin{aligned}
\{H_k\} = \{K^d{}_d - K^{d+1}{}_{d+1},\, \dotsc,\, K^9{}_9 - K^{z}{}_{z},\, K^8{}_8 + K^9{}_9 + K^{z}{}_{z}+ \tfrac{1}{3}\,D \} \,,\end{aligned}$$ in the M-theory parameterization ($D\equiv K^i{}_i$) and $$\begin{aligned}
\{{\mathsf{H}}_{\mathsf{k}}\} = \{{\mathsf{K}}^d{}_d - {\mathsf{K}}^{d+1}{}_{d+1},\, \dotsc,\, {\mathsf{K}}^7{}_7 - {\mathsf{K}}^8{}_8,\, {\mathsf{K}}^8{}_8 + {\mathsf{K}}^9{}_9 -\tfrac{1}{4}\,{\mathsf{D}}-{\mathsf{R}}_{12},\, 2\,{\mathsf{R}}_{12},\, {\mathsf{K}}^8{}_8 - {\mathsf{K}}^9{}_9 \} \,,\end{aligned}$$ in the type IIB parameterization (${\mathsf{D}}\equiv K^{\mathsf{m}}{}_{\mathsf{m}}$). Then, we prepare the matrix representations of these generators in the vector representation. In the M-theory parameterization, the matrix representations have been obtained in [@1111.0459] for $n\leq 7$ and in [@1303.2035] for $n=8$. In the type IIB parameterization, they have been determined in [@1405.7894; @1612.08738] for $n\leq 7$. The results for $n=8$ are given in Appendix \[app:generators\]. Then, we define the generalized vielbein in the M-theory parameterization as $$\begin{aligned}
\begin{split}
&{\mathcal E}\equiv ({\mathcal E}^I{}_J) \equiv \hat{{\mathcal E}}\, L\,,\qquad \hat{{\mathcal E}}\equiv {\operatorname{e}^{h^k\,H_k}} {\operatorname{e}^{\sum_{i<j} h_i{}^j\,K^i{}_j}}\,,
\\
&L\equiv (L^I{}_J) \equiv {\operatorname{e}^{\frac{1}{3!}\,\bm{{\hat{A}}}_{i_1i_2i_3}\,R^{i_1i_2i_3}}} {\operatorname{e}^{\frac{1}{6!}\,\bm{{\hat{A}}}_{i_1\cdots i_6}\,R^{i_1\cdots i_6}}} {\operatorname{e}^{\frac{1}{8!}\,\bm{{\hat{A}}}_{i_1\cdots i_8,i}\,R^{i_1\cdots i_8,i}}} \cdots \,,
\end{split}
\label{eq:L-M}\end{aligned}$$ and the generalized vielbein in the type IIB parameterization as $$\begin{aligned}
\begin{split}
&{\mathsf{E}}\equiv ({\mathsf{E}}^{\mathsf{I}}{}_{\mathsf{J}}) \equiv \hat{{\mathsf{E}}}\, {\mathsf{L}}\,,\qquad \hat{{\mathsf{E}}}\equiv {\operatorname{e}^{{\mathsf{h}}^{\mathsf{k}}\,{\mathsf{H}}_{\mathsf{k}}}} {\operatorname{e}^{\sum_{{\mathsf{m}}<{\mathsf{n}}} {\mathsf{h}}_{\mathsf{m}}{}^{\mathsf{n}}\,{\mathsf{K}}^{\mathsf{m}}{}_{\mathsf{n}}}}\,,
\\
&{\mathsf{L}}\equiv ({\mathsf{L}}^{\mathsf{I}}{}_{\mathsf{J}}) \equiv {\operatorname{e}^{\frac{1}{2!}\,\bm{{\mathsf{A}}}^\alpha_{{\mathsf{m}}_1{\mathsf{m}}_2}\,{\mathsf{R}}_\alpha^{{\mathsf{m}}_1{\mathsf{m}}_2}}} {\operatorname{e}^{\frac{1}{4!}\,\bm{{\mathsf{A}}}_{{\mathsf{m}}_1\cdots{\mathsf{m}}_4}\,{\mathsf{R}}^{{\mathsf{m}}_1\cdots{\mathsf{m}}_4}}} {\operatorname{e}^{\frac{1}{6!}\,\bm{{\mathsf{A}}}^\alpha_{{\mathsf{m}}_1\cdots{\mathsf{m}}_6}\,{\mathsf{R}}_\alpha^{{\mathsf{m}}_1\cdots{\mathsf{m}}_6}}}{\operatorname{e}^{\frac{1}{7!}\,\bm{{\mathsf{A}}}_{{\mathsf{m}}_1\cdots{\mathsf{m}}_7,{\mathsf{m}}}\,{\mathsf{R}}^{{\mathsf{m}}_1\cdots{\mathsf{m}}_7,{\mathsf{m}}}}} \cdots \,.
\end{split}
\label{eq:L-B}\end{aligned}$$ The objects $\bm{{\hat{A}}}$ and $\bm{{\mathsf{A}}}$ are again the M-theory and type IIB fields respectively, expressed in a new basis. Namely, $\bm{{\hat{A}}}$ and $\bm{{\mathsf{A}}}$ are respectively related to ${\hat{A}}$ and ${\mathsf{A}}$ by field redefinitions, as we will show in this section. The ellipses in both parameterizations disappear for $n\leq 8$.
The generalized metrics are then expressed as $$\begin{aligned}
{\mathcal M}_{IJ} \equiv ({\mathcal E}^{{\mathpalette{\raisebox{\depth}{$\m@th\intercal$}}}\,{\mathcal E})_{IJ} = (L^{{\mathpalette{\raisebox{\depth}{$\m@th\intercal$}}}\,\hat{{\mathcal M}}\,L)_{IJ} \,,\qquad
{\mathsf{M}}_{{\mathsf{I}}{\mathsf{J}}} \equiv ({\mathsf{E}}^{{\mathpalette{\raisebox{\depth}{$\m@th\intercal$}}}\,{\mathsf{E}})_{{\mathsf{I}}{\mathsf{J}}} = ({\mathsf{L}}^{{\mathpalette{\raisebox{\depth}{$\m@th\intercal$}}}\,\hat{{\mathsf{M}}}\,{\mathsf{L}})_{{\mathsf{I}}{\mathsf{J}}} \,,\end{aligned}$$ where we have defined the untwisted metrics as $$\begin{aligned}
\hat{{\mathcal M}}_{IJ} \equiv (\hat{{\mathcal E}}^{{\mathpalette{\raisebox{\depth}{$\m@th\intercal$}}}\,\hat{{\mathcal E}})_{IJ}\,,\qquad
\hat{{\mathsf{M}}}_{{\mathsf{I}}{\mathsf{J}}} \equiv (\hat{{\mathsf{E}}}^{{\mathpalette{\raisebox{\depth}{$\m@th\intercal$}}}\,\hat{{\mathsf{E}}})_{{\mathsf{I}}{\mathsf{J}}} \,,\end{aligned}$$ which are parameterized by the supergravity fields $h^k$ (vielbein) and ${\mathsf{h}}^k$ (vielbein and ${\mathsf{m}}_{\alpha\beta}$). Explicitly, the untwisted metrics take the following form: $$\begin{aligned}
\hat{{\mathcal M}} &\equiv {\lvert {{\hat{G}}} \rvert}^{\frac{1}{n-2}}\,{\footnotesize
\begin{pmatrix}
{\hat{G}}_{ij} & 0 & 0 & 0 & \\
0 & {\hat{G}}^{i_1i_2, j_1j_2} & 0 & 0 & \cdots \\
0 & 0 & {\hat{G}}^{i_1\cdots i_5, j_1\cdots j_5} & 0 & \\
0 & 0 & 0 & {\hat{G}}^{i_1\cdots i_7, j_1\cdots j_7}\,{\hat{G}}^{ij} & \\
& \vdots & & & \ddots
\end{pmatrix}},
\\
\hat{{\mathsf{M}}} &\equiv {\lvert {{\mathsf{G}}} \rvert}^{\frac{1}{n-2}}{\arraycolsep=0.5mm {\footnotesize \begin{pmatrix}
{\mathsf{G}}_{{\mathsf{m}}{\mathsf{n}}} & 0 & 0 & 0 & 0 & \\
0 & {\mathsf{m}}_{\alpha\beta} \,{\mathsf{G}}^{{\mathsf{m}}{\mathsf{n}}} & 0 & 0 & 0 & \\
0 & 0 & {\mathsf{G}}^{{\mathsf{m}}_1{\mathsf{m}}_2{\mathsf{m}}_3, {\mathsf{n}}_1{\mathsf{n}}_2{\mathsf{n}}_3} & 0 & 0 & \cdots \\
0 & 0 & 0 & {\mathsf{m}}_{\alpha\beta} \,{\mathsf{G}}^{{\mathsf{m}}_1\cdots {\mathsf{m}}_5, {\mathsf{n}}_1\cdots {\mathsf{n}}_5} & 0 & \\
0 & 0 & 0 & 0 & {\mathsf{G}}^{{\mathsf{m}}_1\cdots{\mathsf{m}}_6, {\mathsf{n}}_1\cdots {\mathsf{n}}_6}\,{\mathsf{G}}^{{\mathsf{m}}{\mathsf{n}}} & \\
& & \vdots & & & \ddots
\end{pmatrix}}} ,\end{aligned}$$ where $$\begin{aligned}
\begin{split}
\begin{alignedat}{2}
{\hat{G}}^{i_1\cdots i_p, j_1\cdots j_p} &\equiv \delta^{j_1\cdots j_p}_{k_1\cdots k_p}\,{\hat{G}}^{i_1k_1}\cdots {\hat{G}}^{i_pk_p}\,,&\qquad
{\lvert {{\hat{G}}} \rvert}&\equiv \det({\hat{G}}_{ij})\,,
\\
{\mathsf{G}}^{{\mathsf{m}}_1\cdots {\mathsf{m}}_p, {\mathsf{n}}_1\cdots {\mathsf{n}}_p} &\equiv \delta^{{\mathsf{n}}_1\cdots {\mathsf{n}}_p}_{{\mathsf{q}}_1\cdots {\mathsf{q}}_p}\,{\mathsf{G}}^{{\mathsf{m}}_1{\mathsf{q}}_1}\cdots {\mathsf{G}}^{{\mathsf{m}}_p{\mathsf{q}}_p}\,,&\qquad
{\lvert {{\mathsf{G}}} \rvert}&\equiv \det({\mathsf{G}}_{{\mathsf{m}}{\mathsf{n}}})\,.
\end{alignedat}
\end{split}\end{aligned}$$ On the other hand, the twist matrices $L$ and ${\mathsf{L}}$ contain various gauge potentials, which can be computed by using the matrix representations of the $E_n$ generators given in Appendix \[app:generators\].
As we have introduced the parameterization of the generalized metrics, let us explain the procedure to obtain the duality rules, which has been proposed in [@1701.07819] for $n\leq 7$.
Linear map between generalized metrics
--------------------------------------
Here, we explain how to determine the duality transformation rules from the generalized metric. As we have discussed in Section \[sec:1-form\], in the M-theory and type IIB parameterizations, we are using different basis, which are related through the linear map . Accordingly, the generalized metrics in the two parameterizations are related as $$\begin{aligned}
{\mathsf{M}}_{{\mathsf{I}}{\mathsf{J}}} = (S^{{\mathpalette{\raisebox{\depth}{$\m@th\intercal$}}})_{\mathsf{I}}{}^K\,{\mathcal M}_{KL}\,S^L{}_{\mathsf{J}}\,.
\label{eq:linear-map-metric}\end{aligned}$$ The explicit form of $S^I{}_{\mathsf{J}}$ has been obtained in [@1701.07819] only for $n\leq 7$, and in this paper, we extend the result to be applicable to $n\leq 8$. Under the linear map, the generalized coordinates are transformed as $$\begin{aligned}
x^I = S^I{}_{\mathsf{J}}\, {\mathsf{x}}^{\mathsf{J}}\,,\qquad {\mathsf{x}}^{\mathsf{I}}= (S^{-1})^{\mathsf{I}}{}_J\,x^J \,.
\label{eq:E8-linear-map}\end{aligned}$$ Since the matrix size of $S^I{}_{\mathsf{J}}$ is very large ($21\times 21$), we show the linear map as follows: $$\begin{aligned}
{\footnotesize
\underbrace{\begin{pmatrix}
x^a \\[-0.2mm]
x^\alpha \\ \hline
\frac{y_{a_1a_2}}{\sqrt{2!}} \\[-0.2mm]
y_{a \alpha} \\[-0.2mm]
y_{{y}{z}} \\ \hline
\frac{y_{a_1\cdots a_5}}{\sqrt{5!}} \\[-0.2mm]
\frac{y_{a_1\cdots a_4\alpha}}{\sqrt{4!}} \\[-0.2mm]
\frac{y_{a_1a_2a_3 {y}{z}}}{\sqrt{3!}} \\ \hline
\frac{y_{a_1\cdots a_6\alpha,a}}{\sqrt{6!}} \\[-0.2mm]
\frac{y_{a_1\cdots a_6(\alpha,\beta)}}{\sqrt{6!}} \\[-0.2mm]
\frac{\epsilon^{\alpha\beta}\,y_{a_1\cdots a_6[\alpha,\beta]}}{\sqrt{2!\,6!}} \\[-0.2mm]
\frac{y_{a_1\cdots a_5{y}{z},a}}{\sqrt{5!}} \\[-0.2mm]
\frac{y_{a_1\cdots a_5 {y}{z},\alpha}}{\sqrt{5!}} \\ \hline
\frac{y_{a_1\cdots a_6{y}{z},b_1b_2b_3}}{\sqrt{6!\,3!}} \\[-0.2mm]
\frac{y_{a_1\cdots a_6{y}{z},b_1b_2\alpha}}{\sqrt{6!\,2!}} \\[-0.2mm]
\frac{y_{a_1\cdots a_6{y}{z},a{y}{z}}}{\sqrt{6!}} \\ \hline
\frac{y_{a_1\cdots a_6{y}{z},b_1\cdots b_6}}{\sqrt{6!\,6!}} \\[-0.2mm]
\frac{y_{a_1\cdots a_6{y}{z},b_1\cdots b_5\alpha}}{\sqrt{6!\,5!}} \\[-0.2mm]
\frac{y_{a_1\cdots a_6{y}{z},b_1\cdots b_4{y}{z}}}{\sqrt{6!\,4!}} \\ \hline
\frac{y_{a_1\cdots a_6{y}{z},b_1\cdots b_6{y}{z},a}}{\sqrt{6!\,6!}}\\[-0.2mm]
\frac{y_{a_1\cdots a_6{y}{z},b_1\cdots b_6{y}{z},\alpha}}{\sqrt{6!\,6!}}
\end{pmatrix}}_{x^I}
=
\underbrace{\begin{pmatrix}
{\mathsf{x}}^a \\[-0.5mm]
{\mathsf{y}}_{\mathsf{y}}^\alpha \\ \hline
\frac{{\mathsf{y}}_{a_1a_2{\mathsf{y}}}}{\sqrt{2!}} \\[-0.5mm]
{\mathsf{y}}_a^\beta\,\epsilon_{\beta\alpha} \\[-0.5mm]
{\mathsf{x}}^{\mathsf{y}}\\ \hline
\frac{{\mathsf{y}}_{a_1\cdots a_5{\mathsf{y}},{\mathsf{y}}}}{\sqrt{5!}} \\[-0.5mm]
\frac{{\mathsf{y}}^\beta_{a_1\cdots a_4}\,\epsilon_{\beta\alpha}}{\sqrt{4!}} \\[-0.5mm]
\frac{{\mathsf{y}}_{a_1a_2a_3}}{\sqrt{3!}} \\ \hline
\frac{{\mathsf{y}}^\beta_{a_1\cdots a_6{\mathsf{y}},a{\mathsf{y}}}\,\epsilon_{\beta\alpha}}{\sqrt{6!}} \\[-0.5mm]
\frac{\epsilon_{\alpha\gamma}\,\epsilon_{\beta\delta}\,{\mathsf{y}}^{\gamma\delta}_{a_1\cdots a_6{\mathsf{y}}}}{\sqrt{6!}} \\[-0.5mm]
{\mathsf{Y}}_{a_1\cdots a_6} \\[-0.5mm]
{\mathsf{Y}}_{a_1\cdots a_5,a} \\[-0.5mm]
\frac{{\mathsf{y}}_{a_1\cdots a_5}^\beta\,\epsilon_{\beta\alpha}}{\sqrt{5!}} \\ \hline
\frac{{\mathsf{y}}_{a_1\cdots a_6{\mathsf{y}},b_1b_2b_3{\mathsf{y}}}}{\sqrt{6!\,3!}} \\[-0.5mm]
\frac{{\mathsf{y}}^\gamma_{a_1\cdots a_6{\mathsf{y}},b_1b_2}\,\epsilon_{\beta\alpha}}{\sqrt{6!\,2!}} \\[-0.5mm]
\frac{{\mathsf{y}}_{a_1\cdots a_6,a}}{\sqrt{6!}} \\ \hline
\frac{{\mathsf{y}}_{a_1\cdots a_6{\mathsf{y}},b_1\cdots b_6{\mathsf{y}},{\mathsf{y}}}}{\sqrt{6!\,6!}} \\[-0.5mm]
\frac{{\mathsf{y}}^\beta_{a_1\cdots a_6{\mathsf{y}},b_1\cdots b_5{\mathsf{y}}}\,\epsilon_{\beta\alpha}}{\sqrt{6!\,5!}} \\[-0.5mm]
\frac{{\mathsf{y}}_{a_1\cdots a_6{\mathsf{y}},b_1\cdots b_4}}{\sqrt{6!\,4!}} \\ \hline
\frac{{\mathsf{y}}_{a_1\cdots a_6{\mathsf{y}},b_1\cdots b_6{\mathsf{y}},a}}{\sqrt{6!\,6!}} \\[-0.5mm]
\frac{{\mathsf{y}}^\beta_{a_1\cdots a_6{\mathsf{y}},b_1\cdots b_6}\,\epsilon_{\beta\alpha}}{\sqrt{6!\,6!}}
\end{pmatrix}}_{S^I{}_{\mathsf{I}}\,{\mathsf{x}}^{\mathsf{I}}}
\quad \Leftrightarrow \quad
\underbrace{\begin{pmatrix}
{\mathsf{x}}^a \\[-0.5mm] {\mathsf{x}}^{{\mathsf{y}}} \\ \hline
{\mathsf{y}}^\alpha_a \\[-0.5mm] {\mathsf{y}}^\alpha_{{\mathsf{y}}} \\ \hline
\frac{{\mathsf{y}}_{a_1a_2a_3}}{\sqrt{3!}} \\[-0.5mm]
\frac{{\mathsf{y}}_{a_1a_2 {\mathsf{y}}}}{\sqrt{2!}} \\ \hline
\frac{{\mathsf{y}}^\alpha_{a_1\cdots a_5}}{\sqrt{5!}} \\[-0.5mm]
\frac{{\mathsf{y}}^\alpha_{a_1\cdots a_4{\mathsf{y}}}}{\sqrt{4!}} \\ \hline
\frac{{\mathsf{y}}_{a_1\cdots a_6,a}}{\sqrt{6!}} \\[-0.5mm]
\frac{{\mathsf{y}}_{a_1\cdots a_6,{\mathsf{y}}}}{\sqrt{6!}} \\[-0.5mm]
\frac{{\mathsf{y}}_{a_1\cdots a_5{\mathsf{y}},a}}{\sqrt{5!}} \\[-0.5mm]
\frac{{\mathsf{y}}_{a_1\cdots a_5{\mathsf{y}},{\mathsf{y}}}}{\sqrt{5!}} \\ \hline
\frac{{\mathsf{y}}^{\alpha\beta}_{a_1\cdots a_6{\mathsf{y}}}}{\sqrt{6!}}\\ \hline
\frac{{\mathsf{y}}^\alpha_{a_1\cdots a_6{\mathsf{y}},b_1b_2}}{\sqrt{6!\,2!}}\\[-0.5mm]
\frac{{\mathsf{y}}^\alpha_{a_1\cdots a_6{\mathsf{y}},a {\mathsf{y}}}}{\sqrt{6!}}\\ \hline
\frac{{\mathsf{y}}_{a_1\cdots a_6{\mathsf{y}},b_1\cdots b_4}}{\sqrt{6!\,4!}}\\[-0.5mm]
\frac{{\mathsf{y}}_{a_1\cdots a_6{\mathsf{y}},b_1b_2b_3{\mathsf{y}}}}{\sqrt{6!\,3!}}\\ \hline
\frac{{\mathsf{y}}^\alpha_{a_1\cdots a_6{\mathsf{y}},b_1\cdots b_6}}{\sqrt{6!\,6!}}\\[-0.5mm]
\frac{{\mathsf{y}}^\alpha_{a_1\cdots a_6{\mathsf{y}},b_1\cdots b_5{\mathsf{y}}}}{\sqrt{6!\,5!}}\\ \hline
\frac{{\mathsf{y}}_{a_1\cdots a_6{\mathsf{y}},b_1\cdots b_6{\mathsf{y}},a}}{\sqrt{6!\,6!}}\\[-0.5mm]
\frac{{\mathsf{y}}_{a_1\cdots a_6{\mathsf{y}},b_1\cdots b_6{\mathsf{y}},{\mathsf{y}}}}{\sqrt{6!\,6!}}
\end{pmatrix}}_{{\mathsf{x}}^{\mathsf{I}}}
= \underbrace{\begin{pmatrix}
x^a \\[-0.5mm]
y_{{y}{z}} \\ \hline
\epsilon^{\alpha\beta}\,y_{a\beta} \\[-0.5mm]
x^\alpha \\ \hline
\frac{y_{a_1a_2a_3{y}{z}}}{\sqrt{3!}} \\[-0.5mm]
\frac{y_{a_1a_2}}{\sqrt{2!}} \\ \hline
\frac{\epsilon^{\alpha\beta}\,y_{a_1\cdots a_5 {y}{z},\beta}}{\sqrt{5!}} \\[-0.5mm]
\frac{\epsilon^{\alpha\beta}\,y_{a_1\cdots a_4\beta}}{\sqrt{4!}} \\ \hline
\frac{y_{a_1\cdots a_6{y}{z},a{y}{z}}}{\sqrt{6!}} \\[-0.5mm]
Y_{a_1\cdots a_6} \\[-0.5mm]
Y_{a_1\cdots a_5,a} \\[-0.5mm]
\frac{y_{a_1\cdots a_5}}{\sqrt{5!}} \\ \hline
\frac{\epsilon^{\alpha\gamma}\epsilon^{\beta\delta}\,y_{a_1\cdots a_6(\gamma,\delta)}}{\sqrt{6!}}\\ \hline
\frac{\epsilon^{\alpha\beta}\,y_{a_1\cdots a_6{y}{z},b_1b_2\beta}}{\sqrt{6!\,2!}} \\[-0.5mm]
\frac{\epsilon^{\alpha\beta}\,y_{a_1\cdots a_6\beta,a}}{\sqrt{6!}} \\ \hline
\frac{y_{a_1\cdots a_6{y}{z},b_1\cdots b_4{y}{z}}}{\sqrt{6!\,4!}} \\[-0.5mm]
\frac{y_{a_1\cdots a_6{y}{z},b_1b_2b_3}}{\sqrt{6!\,3!}} \\ \hline
\frac{\epsilon^{\alpha\beta}\,y_{a_1\cdots a_6{y}{z},b_1\cdots b_6{y}{z},\beta}}{\sqrt{6!\,6!}} \\[-0.5mm]
\frac{\epsilon^{\alpha\beta}\,y_{a_1\cdots a_6{y}{z},b_1\cdots b_5\beta}}{\sqrt{6!\,5!}} \\ \hline
\frac{y_{a_1\cdots a_6{y}{z},b_1\cdots b_6{y}{z},a}}{\sqrt{6!\,6!}} \\[-0.5mm]
\frac{y_{a_1\cdots a_6{y}{z},b_1\cdots b_6}}{\sqrt{6!\,6!}}
\end{pmatrix}}_{(S^{-1})^{\mathsf{I}}{}_I\,x^I},
}\end{aligned}$$ where $$\begin{aligned}
\begin{pmatrix}
{\mathsf{Y}}_{a_1\cdots a_6} \\ {\mathsf{Y}}_{a_1\cdots a_5,a}
\end{pmatrix}
&\equiv
\begin{pmatrix}
\frac{9\sqrt{2}+1}{14}\,\frac{{{\boldsymbol\delta}}^{b_1\cdots b_6}_{a_1\cdots a_6}}{\sqrt{6!\,6!}} & \frac{3\sqrt{2}-2}{28}\,\frac{{{\boldsymbol\delta}}^{b_1\cdots b_5b}_{a_1\cdots a_6}}{\sqrt{6!\,5!}} \\
\frac{3\sqrt{2}-2}{28}\,\frac{{{\boldsymbol\delta}}_{a_1\cdots a_5a}^{b_1\cdots b_6}}{\sqrt{5!\,6!}} & \frac{{{\boldsymbol\delta}}_{a_1\cdots a_5}^{b_1\cdots b_5}\,\delta_a^b -\frac{3\sqrt{2}+5}{28}\, {{\boldsymbol\delta}}_{a_1\cdots a_5a}^{b_1\cdots b_5b}}{\sqrt{5!\,5!}}
\end{pmatrix}
\begin{pmatrix}
\frac{{\mathsf{y}}_{b_1\cdots b_6,{\mathsf{y}}}}{\sqrt{6!}} \\
\frac{{\mathsf{y}}_{b_1\cdots b_5{\mathsf{y}},b}}{\sqrt{5!}}
\end{pmatrix} ,
\\
\begin{pmatrix}
Y_{a_1\cdots a_6} \\ Y_{a_1\cdots a_5,a}
\end{pmatrix}
&\equiv
\begin{pmatrix}
\frac{9\sqrt{2}+1}{14}\,\frac{{{\boldsymbol\delta}}^{b_1\cdots b_6}_{a_1\cdots a_6}}{\sqrt{6!\,6!}} & \frac{3\sqrt{2} -2}{28}\,\frac{{{\boldsymbol\delta}}^{b_1\cdots b_5b}_{a_1\cdots a_6}}{\sqrt{6!\,5!}} \\
\frac{3\sqrt{2}-2}{28}\,\frac{{{\boldsymbol\delta}}_{a_1\cdots a_5a}^{b_1\cdots b_6}}{\sqrt{5!\,6!}} & \frac{{{\boldsymbol\delta}}_{a_1\cdots a_5}^{b_1\cdots b_5}\,\delta_a^b -\frac{3\sqrt{2}+5}{28}\, {{\boldsymbol\delta}}_{a_1\cdots a_5a}^{b_1\cdots b_5b}}{\sqrt{5!\,5!}}
\end{pmatrix}
\begin{pmatrix}
\frac{\epsilon^{\alpha\beta}\,y_{b_1\cdots b_6[\alpha,\beta]}}{\sqrt{2!\,6!}} \\
\frac{y_{b_1\cdots b_5{y}{z},b}}{\sqrt{5!}}
\end{pmatrix}\end{aligned}$$ with ${{\boldsymbol\delta}}_{i_1\cdots i_n}^{j_1\cdots j_n}\equiv n!\,\delta_{i_1\cdots i_n}^{j_1\cdots j_n}$. The constant matrix $S^I{}_{\mathsf{J}}$ can be read off from the above map between the coordinates. We can check that the matrix $S^I{}_{\mathsf{I}}$ satisfies the property $$\begin{aligned}
S^I{}_{\mathsf{K}}\, (S^{\mathbb{T}})^{\mathsf{K}}{}_J = \delta^I_J\,,\qquad
(S^{\mathbb{T}})^{\mathsf{I}}{}_K\,S^K{}_{\mathsf{J}}= \delta^{\mathsf{I}}_{\mathsf{J}}\,,\end{aligned}$$ under the generalized transpose, which is defined for a matrix $A=(A^I{}_J)$ as $$\begin{aligned}
(A^{\mathbb{T}})^I{}_J \equiv \delta^{IK}\,(A^{{\mathpalette{\raisebox{\depth}{$\m@th\intercal$}}})_K{}^L \,\delta_{LJ}
\equiv \delta^{IK}\,A^L{}_K\,\delta_{LJ} \,,\end{aligned}$$ namely the standard matrix transpose ${}^{{\mathpalette{\raisebox{\depth}{$\m@th\intercal$}}}$ followed by a flip in the position of the indices. This property shows that the flat metric is preserved under the linear map, $$\begin{aligned}
\delta_{{\mathsf{I}}{\mathsf{J}}} = S^K{}_{\mathsf{I}}\, S^L{}_{\mathsf{J}}\, \delta_{KL} \,. \end{aligned}$$
Now, the constant matrix $S^I{}_{\mathsf{J}}$ has been completely determined and the relation connects the two parameterizations. By comparing both sides, we can express the M-theory fields in terms of the type IIB fields, and vice versa. In the case $n\leq 7$, the generalized metric does not contain the dual graviton, but it appears in $n\geq 8$ and here we consider the generalized metric in $E_8$ EFT.
Connection between two parameterization
---------------------------------------
By comparing the two parameterizations of the $E_8$ generalized metric, we find the following relation between the M-theory fields and type IIB fields: $$\begin{aligned}
\bigl({\hat{G}}_{ij}\bigr)
&\ \, = \ \begin{pmatrix}
{\hat{G}}_{ab} & {\hat{G}}_{a\beta} \\ {\hat{G}}_{\alpha b} & {\hat{G}}_{\alpha\beta}
\end{pmatrix}
{\nonumber}\\
&\overset{\text{M--B}}{=} {\operatorname{e}^{-\frac{2}{3}\,{\varphi}}}\,{\mathsf{G}}_{{\mathsf{y}}{\mathsf{y}}}^{1/3}
\begin{pmatrix}
\delta_a^c & -\bm{{\mathsf{A}}}_{a{\mathsf{y}}}^\gamma \\ 0 & \delta_\alpha^\gamma
\end{pmatrix}
\begin{pmatrix}
\frac{2\,{\mathsf{G}}_{c{\mathsf{y}}, d{\mathsf{y}}}}{{\mathsf{G}}_{{\mathsf{y}}{\mathsf{y}}}} & 0 \\ 0 & \frac{{\operatorname{e}^{{\varphi}}}}{{\mathsf{G}}_{{\mathsf{y}}{\mathsf{y}}}}\, {\mathsf{m}}_{\gamma\delta}
\end{pmatrix}
\begin{pmatrix}
\delta^d_b & 0 \\ -\bm{{\mathsf{A}}}_{b{\mathsf{y}}}^\delta & \delta^\delta_\beta
\end{pmatrix} \,,
\\
\bm{{\hat{A}}}_{a {y}{z}} &\overset{\text{M--B}}{=} \frac{{\mathsf{G}}_{a {\mathsf{y}}}}{{\mathsf{G}}_{{\mathsf{y}}{\mathsf{y}}}} \,,
\\
\bm{{\hat{A}}}_{ab \alpha} &\overset{\text{M--B}}{=} \Bigl(\bm{{\mathsf{A}}}^\beta_{ab} - \frac{2\,\bm{{\mathsf{A}}}^\beta_{[a|{\mathsf{y}}|}\,{\mathsf{G}}_{b]{\mathsf{y}}}}{{\mathsf{G}}_{{\mathsf{y}}{\mathsf{y}}}}\Bigr)\,\epsilon_{\beta\alpha}\,,
\\
\bm{{\hat{A}}}_{abc} &\overset{\text{M--B}}{=} \bm{{\mathsf{A}}}_{abc {\mathsf{y}}} -\frac{3}{2}\,\epsilon_{\gamma\delta}\,\bm{{\mathsf{A}}}^\gamma_{[ab}\,\bm{{\mathsf{A}}}^\delta_{c]{\mathsf{y}}}
- \frac{3\,\epsilon_{\gamma\delta}\,\bm{{\mathsf{A}}}^\gamma_{[a|{\mathsf{y}}|}\,\bm{{\mathsf{A}}}^\delta_{b|{\mathsf{y}}|}\,{\mathsf{G}}_{c]{\mathsf{y}}}}{{\mathsf{G}}_{{\mathsf{y}}{\mathsf{y}}}} \,,
\\
\bm{{\hat{A}}}_{a_1\cdots a_4 {y}{z}}
&\overset{\text{M--B}}{=} \bm{{\mathsf{A}}}_{a_1\cdots a_4}- \Bigl(\frac{2\,\bm{{\mathsf{A}}}_{[a_1a_2a_3|{\mathsf{y}}|}\, {\mathsf{G}}_{a_4]{\mathsf{y}}}}{{\mathsf{G}}_{{\mathsf{y}}{\mathsf{y}}}} + \frac{3\,\epsilon_{\gamma\delta}\,\bm{{\mathsf{A}}}^\gamma_{[a_1a_2}\,\bm{{\mathsf{A}}}^\delta_{a_3|{\mathsf{y}}|}\,{\mathsf{G}}_{a_4]{\mathsf{y}}}}{{\mathsf{G}}_{{\mathsf{y}}{\mathsf{y}}}} \Bigr)\,,
\\
\bm{{\hat{A}}}_{a_1\cdots a_5 \alpha}
&\overset{\text{M--B}}{=}
\Bigl(\bm{{\mathsf{A}}}_{a_1\cdots a_5{\mathsf{y}}}+ 5\,\bm{{\mathsf{A}}}_{[a_1a_2a_3|{\mathsf{y}}|}\,\bm{{\mathsf{A}}}^\beta_{a_4a_5]}
+\frac{5}{2}\, \epsilon_{\gamma\delta}\,\bm{{\mathsf{A}}}^\gamma_{[a_1a_2}\,\bm{{\mathsf{A}}}^\delta_{a_3|{\mathsf{y}}|}\,\bm{{\mathsf{A}}}^\beta_{a_4a_5]}
{\nonumber}\\
&\quad\ -\frac{10\,\bm{{\mathsf{A}}}_{[a_1a_2a_3|{\mathsf{y}}|}\,\bm{{\mathsf{A}}}^\beta_{a_4|{\mathsf{y}}|}{\mathsf{G}}_{a_5]{\mathsf{y}}}}{{\mathsf{G}}_{{\mathsf{y}}{\mathsf{y}}}}
-\frac{15\,\epsilon_{\gamma\delta}\,\bm{{\mathsf{A}}}^\gamma_{[a_1a_2}\,\bm{{\mathsf{A}}}^\delta_{a_3|{\mathsf{y}}|}\,\bm{{\mathsf{A}}}^\beta_{a_4|{\mathsf{y}}|}\,{\mathsf{G}}_{a_5]{\mathsf{y}}}}{{\mathsf{G}}_{{\mathsf{y}}{\mathsf{y}}}} \Bigr) \,\epsilon_{\beta\alpha}
\\
\bm{{\hat{A}}}_{a_1\cdots a_6}
&\overset{\text{M--B}}{=}
\bm{{\mathsf{A}}}_{a_1\cdots a_6{\mathsf{y}}, {\mathsf{y}}}
-15\,\epsilon_{\gamma\delta}\,\bm{{\mathsf{A}}}_{[a_1a_2a_3|{\mathsf{y}}|}\,\Bigl(\bm{{\mathsf{A}}}^\gamma_{a_4a_5}\,\bm{{\mathsf{A}}}^\delta_{a_6]{\mathsf{y}}}
+\frac{2\,\bm{{\mathsf{A}}}^\gamma_{a_4|{\mathsf{y}}|}\,\bm{{\mathsf{A}}}^\delta_{a_5|{\mathsf{y}}|}\,{\mathsf{G}}_{a_6]{\mathsf{y}}}}{{\mathsf{G}}_{{\mathsf{y}}{\mathsf{y}}}}\Bigr) \,,
\\
\bm{{\hat{A}}}_{a_1\cdots a_6 {y}{z}, \alpha}
&\overset{\text{M--B}}{=}
\biggl[\bm{{\mathsf{A}}}^\beta_{a_1\cdots a_6}
+\frac{\bm{{\mathsf{A}}}^\beta_{[a_1a_2}\,\bigl(20\,\bm{{\mathsf{A}}}_{a_3a_4a_5|{\mathsf{y}}|}+30\,\epsilon_{\gamma\delta}\,\bm{{\mathsf{A}}}^\gamma_{a_3a_4}\,\bm{{\mathsf{A}}}^\delta_{a_5|{\mathsf{y}}|}\bigr)\,{\mathsf{G}}_{a_6]{\mathsf{y}}}}{{\mathsf{G}}_{{\mathsf{y}}{\mathsf{y}}}} \biggr]\,\epsilon_{\beta\alpha} \,,
\\
\bm{{\hat{A}}}_{a_1\cdots a_6 {y}{z}, b}
&\overset{\text{M--B}}{=}
\bm{{\mathsf{A}}}_{a_1\cdots a_6{\mathsf{y}}, b}
-\frac{15}{2}\,\bm{{\mathsf{A}}}_{[a_1\cdots a_4}\, \bm{{\mathsf{A}}}_{a_5a_6] b{\mathsf{y}}}
-10\, \epsilon_{\alpha\beta}\, \bm{{\mathsf{A}}}_{[a_1a_2a_3|{\mathsf{y}}|}\, \bm{{\mathsf{A}}}^\alpha_{a_4a_5}\,\bm{{\mathsf{A}}}^\beta_{a_6]b}
{\nonumber}\\
&\quad +\frac{15}{2}\,\epsilon_{\alpha\beta}\,\epsilon_{\gamma\delta}\,\bm{{\mathsf{A}}}^\alpha_{[a_1a_2}\,\bm{{\mathsf{A}}}^\beta_{a_3|b|}\,\bm{{\mathsf{A}}}^\gamma_{a_4a_5}\,\bm{{\mathsf{A}}}^\delta_{a_6]{\mathsf{y}}}
{\nonumber}\\
&\quad +\frac{20 \,\bm{{\mathsf{A}}}_{[a_1a_2a_3|{\mathsf{y}}|}\, \bm{{\mathsf{A}}}_{a_4a_5 |b{\mathsf{y}}|}\, {\mathsf{G}}_{a_6]{\mathsf{y}}}}{{\mathsf{G}}_{{\mathsf{y}}{\mathsf{y}}}}
- \frac{20\,\epsilon_{\alpha\beta}\,\bm{{\mathsf{A}}}_{[a_1a_2a_3|{\mathsf{y}}|}\,\bm{{\mathsf{A}}}^\alpha_{a_4 |b|}\,\bm{{\mathsf{A}}}^\beta_{a_5|{\mathsf{y}}|}\, {\mathsf{G}}_{a_6]{\mathsf{y}}}}{{\mathsf{G}}_{{\mathsf{y}}{\mathsf{y}}}}
{\nonumber}\\
&\quad - \frac{10\,\epsilon_{\alpha\beta}\,\bm{{\mathsf{A}}}_{[a_1a_2a_3|{\mathsf{y}}|}\,\bm{{\mathsf{A}}}^\alpha_{a_4a_5}\,\bm{{\mathsf{A}}}^\beta_{|b{\mathsf{y}}|}\, {\mathsf{G}}_{a_6]{\mathsf{y}}}}{{\mathsf{G}}_{{\mathsf{y}}{\mathsf{y}}}}
+ \frac{30\,\epsilon_{\alpha\beta}\,\bm{{\mathsf{A}}}_{[a_1a_2|b{\mathsf{y}}|}\, \bm{{\mathsf{A}}}^\alpha_{a_3a_4}\,\bm{{\mathsf{A}}}^\beta_{a_5|{\mathsf{y}}|}\,{\mathsf{G}}_{a_6]{\mathsf{y}}}}{{\mathsf{G}}_{{\mathsf{y}}{\mathsf{y}}}}
{\nonumber}\\
&\quad +\frac{15\,\epsilon_{\alpha\beta}\,\epsilon_{\gamma\delta}\,\bigl(\bm{{\mathsf{A}}}^\alpha_{[a_1a_2}\,\bm{{\mathsf{A}}}^\beta_{a_3|b|}\,\bm{{\mathsf{A}}}^\gamma_{a_4|{\mathsf{y}}|}\,\bm{{\mathsf{A}}}^\delta_{a_5|{\mathsf{y}}|}
- \bm{{\mathsf{A}}}^\alpha_{[a_1a_2}\,\bm{{\mathsf{A}}}^\beta_{a_3|{\mathsf{y}}|}\,\bm{{\mathsf{A}}}^\gamma_{a_4a_5}\,\bm{{\mathsf{A}}}^\delta_{|b{\mathsf{y}}|}\bigr)\,{\mathsf{G}}_{a_6]{\mathsf{y}}}}{{\mathsf{G}}_{{\mathsf{y}}{\mathsf{y}}}} \,. \end{aligned}$$
By making identifications $$\begin{aligned}
\begin{split}
\bm{{\hat{A}}}_{\hat{M}_1\hat{M}_2\hat{M}_3} &= {\hat{A}}_{\hat{M}_1\hat{M}_2\hat{M}_3}\,,\qquad
\bm{{\hat{A}}}_{\hat{M}_1\cdots \hat{M}_6} = {\hat{A}}_{\hat{M}_1\cdots \hat{M}_6}\,,
\\
\bm{{\hat{A}}}_{\hat{M}_1\cdots \hat{M}_8, \hat{N}}
&\simeq {\hat{A}}_{\hat{M}_1\cdots \hat{M}_8, \hat{N}} - 28\,{\hat{A}}_{[\hat{M}_1\cdots \hat{M}_6}\,{\hat{A}}_{\hat{M}_7\hat{M}_8]\hat{N}} \,,
\end{split}
\label{eq:MA-bmMA}\end{aligned}$$ for M-theory fields and $$\begin{aligned}
\bm{{\mathsf{A}}}^\alpha_{M_1M_2} &= {\mathsf{A}}^\alpha_{M_1M_2}\,,\qquad
\bm{{\mathsf{A}}}_{M_1\cdots M_4} = {\mathsf{A}}_{M_1\cdots M_4} \,,\qquad
\bm{{\mathsf{A}}}^\alpha_{M_1\cdots M_6} = {\mathsf{A}}^\alpha_{M_1\cdots M_6}\,,
\\
\bm{{\mathsf{A}}}_{M_1\cdots M_7, N} &\simeq {\mathsf{A}}_{M_1\cdots M_7, N} + 7\,\bigl({\mathsf{B}}_{[M_1\cdots M_6}\,{\mathsf{B}}_{M_7]N} - {\mathsf{B}}_{[M_1\cdots M_6}\,{\mathsf{B}}_{M_7N]} \bigr)
{\nonumber}\\
&\quad +\frac{105}{4}\, \bigl[ {\mathsf{C}}_{[M_1\cdots M_4}\, \bigl({\mathsf{B}}_{M_5M_6}\,{\mathsf{C}}_{M_7]N}-3\,{\mathsf{C}}_{M_5M_6}\,{\mathsf{B}}_{M_7] N} \bigr)
{\nonumber}\\
&\quad\qquad\quad - {\mathsf{C}}_{[M_1\cdots M_4}\, \bigl({\mathsf{B}}_{M_5M_6}\,{\mathsf{C}}_{M_7N]} -3\,{\mathsf{C}}_{M_5M_6}\,{\mathsf{B}}_{M_7N]} \bigr) \bigr]
{\nonumber}\\
&\quad +\frac{315}{8}\,\bigl({\mathsf{C}}_{[M_1M_2}\,{\mathsf{C}}_{M_3M_4}\, {\mathsf{B}}_{M_5M_6}\,{\mathsf{B}}_{M_7]N} - {\mathsf{C}}_{[M_1M_2}\,{\mathsf{C}}_{M_3M_4}\, {\mathsf{B}}_{M_5M_6}\,{\mathsf{B}}_{M_7N]}\bigr) \,,
\label{eq:BA-bmBA}\end{aligned}$$ for type IIB fields, and by using the 11D–10D map, these relations are precisely the $T$-duality rules obtain in Section \[sec:T-dual\]. The $S$-duality rule for the new dual graviton is simply $$\begin{aligned}
\bm{{\mathsf{A}}}'_{M_1\cdots M_7, N} = \bm{{\mathsf{A}}}_{M_1\cdots M_7, N} \,,\end{aligned}$$ which is consistent with under the identification .
Note that, in order to obtain the duality rules for the higher mixed-symmetry potentials, we need to consider the $E_n$ generalized metric with $n\geq 9$.
1-form ${\mathcal A}_\mu^{I}$ as the generalized graviphoton
------------------------------------------------------------
In Section \[sec:1-form\], we found that the 1-form gauge field ${\mathcal A}_\mu^{I}$ has a simple structure in terms of the tensors ${\hat{N}}$ and ${\mathsf{N}}$, $$\begin{aligned}
{\mathcal A}_\mu^{I} = {\hat{N}}_\mu^{I} + \hat{A}_\mu^k\,{\hat{N}}_k^{I}\quad (\text{M-theory})\,,\qquad
\bm{{\mathcal A}}_\mu^{{\mathsf{I}}} = {\mathsf{N}}_\mu^{{\mathsf{I}}} + {\mathsf{A}}_\mu^{\mathsf{p}}\,{\mathsf{N}}_{\mathsf{p}}^{{\mathsf{I}}}\quad (\text{type IIB})\,.
\label{eq:Amu-structure}\end{aligned}$$ In fact, this combination has a clear origin. The basic idea is as follows.
#### Generalized graviphoton in DFT: {#generalized-graviphoton-in-dft .unnumbered}
In type IIB theory, the graviphoton is given by $$\begin{aligned}
{\mathsf{A}}_\mu^{\mathsf{m}}= \bm{{\mathsf{g}}}_{\mu\nu}\,{\mathsf{g}}^{\nu {\mathsf{m}}} \,. \end{aligned}$$ We can consider a generalization of this graviphoton in double field theory (DFT). In DFT, the inverse of the generalized metric has the form,[^5] $$\begin{aligned}
{\mathcal H}^{\hat{I}\hat{J}} = \begin{pmatrix} {\hat{g}}^{MN} & {\hat{g}}^{MK}\,\hat{B}_{KN} \\ -\hat{B}_{MK}\,{\hat{g}}^{KN} & ({\hat{g}}-\hat{B}\,{\hat{g}}^{-1}\,\hat{B})_{MN}
\end{pmatrix},\qquad
(x^{\hat{I}})\equiv (x^M,\,\tilde{x}_M)\,.\end{aligned}$$ We decompose the physical coordinates as $(x^M)=(x^\mu,\,x^m)$ and define the generalized coordinates for the compact directions as $(x^I)\equiv (x^m,\,\tilde{x}_m)$ ($m=1,\dotsc,n-1$). Then, we find $$\begin{aligned}
\begin{split}
{\mathcal H}^{\mu I} &= \bigl({\hat{g}}^{\mu m},\, {\hat{g}}^{\mu K}\,\hat{B}_{Km}\bigr) = \bigl({\hat{g}}^{\mu m},\, {\hat{g}}^{\mu\nu}\,\hat{B}_{\nu m} + {\hat{g}}^{\mu p}\,\hat{B}_{pm}\bigr)
\\
&= {\hat{g}}^{\mu\nu}\, \bigl(A_\nu^m,\, \hat{B}_{\nu m} + A_\nu^p \,\hat{B}_{pm}\bigr)\,.\end{split}\end{aligned}$$ This leads us to define the generalized graviphoton as $$\begin{aligned}
{\mathcal A}_\mu^I \equiv \bm{g}_{\mu\nu}\,{\mathcal H}^{\nu I} = \begin{pmatrix} A_\mu^m \\ \hat{B}_{\mu m} + A_\mu^p\,\hat{B}_{p m} \end{pmatrix} \qquad
\bigl[\bm{g}\equiv ({\hat{g}}^{\mu\nu})^{-1} \bigr]\,,\end{aligned}$$ which transforms covariantly under ${\text{O}}(n-1,n-1)$ transformations, and is sometimes used in the double sigma model (see for example [@hep-th/0406102; @1802.00442]). By using $$\begin{aligned}
N_N{}^I \equiv \begin{pmatrix} \delta_N^m \\ \hat{B}_{Nm} \end{pmatrix} ,\end{aligned}$$ we observe that the generalized graviphoton can be expressed as $$\begin{aligned}
{\mathcal A}_\mu^{I} = N_\mu{}^{I} + A_\mu^p\,N_p{}^{I}\,,\end{aligned}$$ which has the same structure as .
#### Generalized graviphoton in EFT: {#generalized-graviphoton-in-eft .unnumbered}
We now consider the case of EFT starting with the generalized metric ${\mathcal M}_{\hat{I}\hat{J}}$ in $E_{11}$ EFT. Denoting the inverse matrix of ${\mathcal M}^{\mu\nu}$ by $\bm{m}_{\mu\nu}$, we define the generalized graviphoton as $$\begin{aligned}
{\mathcal A}_\mu^I \equiv \bm{m}_{\mu\nu}\,{\mathcal M}^{\nu I} \,. \end{aligned}$$ In the following, we show that this ${\mathcal A}_\mu^I$ is precisely the 1-form considered in Section \[sec:1-form\]. To this end, let us recall that the generalized metric has the structure $$\begin{aligned}
{\mathcal M}_{IJ} = (L^{{\mathpalette{\raisebox{\depth}{$\m@th\intercal$}}}\,\hat{{\mathcal M}}\,L)_{IJ}\,,\qquad
{\mathcal M}^{IJ} = (L^{-1}\,\hat{{\mathcal M}}^{-1}\,L^{-{{\mathpalette{\raisebox{\depth}{$\m@th\intercal$}}}})^{IJ} \,.\end{aligned}$$ By using the fact that the matrix $L$ has a lower-triangular form, we find $$\begin{aligned}
{\mathcal M}^{\mu\nu} &= \hat{{\mathcal M}}^{\mu\nu} = {\lvert {{\hat{g}}} \rvert}^{\frac{1}{2}}\, \bm{g}^{\mu\nu} \equiv \bm{m}^{\mu\nu}\,,
\\
{\mathcal M}^{\mu I} &= (\hat{{\mathcal M}}^{-1}\,L^{-{{\mathpalette{\raisebox{\depth}{$\m@th\intercal$}}}})^{\mu J} = \hat{{\mathcal M}}^{\mu N}\,(L^{-{{\mathpalette{\raisebox{\depth}{$\m@th\intercal$}}}})_{N}{}^{J} = {\lvert {{\hat{g}}} \rvert}^{\frac{1}{2}} \, \bm{g}^{\mu\nu} \, \bigl[(L^{-{{\mathpalette{\raisebox{\depth}{$\m@th\intercal$}}}})_{\nu}{}^I + A_\nu^k\,(L^{-{{\mathpalette{\raisebox{\depth}{$\m@th\intercal$}}}})_{k}{}^I \bigr] \,,\end{aligned}$$ where we have used $$\begin{aligned}
(\hat{{\mathcal M}}^{\hat{M}\hat{N}}) = {\lvert {{\hat{g}}} \rvert}^{\frac{1}{2}} \begin{pmatrix} \delta^\mu_\rho & 0 \\ {\hat{A}}^i_\rho & \delta^i_k \end{pmatrix}
\begin{pmatrix} \bm{g}^{\rho\sigma} & 0 \\ 0 & {\hat{G}}^{kl} \end{pmatrix}
\begin{pmatrix} \delta_\sigma^\mu & {\hat{A}}_\sigma^j \\ 0 & \delta_l^j \end{pmatrix}.\end{aligned}$$ Then, we obtain $$\begin{aligned}
{\mathcal A}_\mu^I = \bm{m}_{\mu\nu}\, {\mathcal M}^{\nu I} = (L^{-{{\mathpalette{\raisebox{\depth}{$\m@th\intercal$}}}})_{\mu}{}^I + {\hat{A}}_\mu^k\,(L^{-{{\mathpalette{\raisebox{\depth}{$\m@th\intercal$}}}})_{k}{}^I\,. \end{aligned}$$
In order to show that this is the same as the 1-form considered in Section \[sec:1-form\], let us compute the explicit form of $(L^{-{{\mathpalette{\raisebox{\depth}{$\m@th\intercal$}}}})_{\mu}{}^I$ in M-theory/type IIB parameterizations. In the M-theory parameterization, $L$ is defined as and by using the matrix representations of the $E_{11}$ generators given in Appendix \[app:generators\], we obtain $$\begin{aligned}
(L^{-{{\mathpalette{\raisebox{\depth}{$\m@th\intercal$}}}})_{\hat{N}}{}^I \simeq
\begin{pmatrix}
\delta^i_{\hat{N}} \\
\frac{\bm{{\hat{A}}}_{\hat{N} i_1i_2}}{\sqrt{2!}}
\\
\frac{\bm{{\hat{A}}}_{\hat{N} i_1\cdots i_5}-5\,\bm{{\hat{A}}}_{\hat{N}[i_1i_2}\,\bm{{\hat{A}}}_{i_3i_4i_5]}}{\sqrt{5!}} \\
\frac{\bm{{\hat{A}}}_{\hat{N}i_1\cdots i_7,i} - 7\,\bm{{\hat{A}}}_{\hat{N}i[i_1\cdots i_4}\,\bm{{\hat{A}}}_{i_5i_6i_7]} +35\,\bm{{\hat{A}}}_{\hat{N}[i_1i_2}\,\bm{{\hat{A}}}_{i_3i_4i_5}\,\bm{{\hat{A}}}_{i_6i_7]i}}{\sqrt{7!}}
\\
\vdots
\end{pmatrix} ,
\label{eq:L-T-M}\end{aligned}$$ where $i\in \{i_1,\dotsc ,i_7\}$ has been assumed for the fourth row. By using the identification , $(L^{-{{\mathpalette{\raisebox{\depth}{$\m@th\intercal$}}}})_{\hat{N}}{}^I$ is precisely the same as ${\hat{N}}_{\hat{N}}{}^I$ given in and the generalized graviphoton is the same as the 1-form .
On the other hand, in the type IIB parameterization, ${\mathsf{L}}$ is defined as and we obtain $$\begin{aligned}
({\mathsf{L}}^{-{{\mathpalette{\raisebox{\depth}{$\m@th\intercal$}}}})_N{}^{{\mathsf{I}}} \simeq
\begin{pmatrix}
\delta_\mu^{\mathsf{m}}\\
\bm{{\mathsf{A}}}_{N{\mathsf{m}}}^\alpha
\\
\frac{\bm{{\mathsf{A}}}_{N{\mathsf{m}}_1{\mathsf{m}}_2{\mathsf{m}}_3} - \frac{3}{2}\,\epsilon_{\gamma\delta}\,\bm{{\mathsf{A}}}^\gamma_{N[{\mathsf{m}}_1}\,\bm{{\mathsf{A}}}^\delta_{{\mathsf{m}}_2{\mathsf{m}}_3]}}{\sqrt{3!}}
\\
\frac{\bm{{\mathsf{A}}}^\alpha_{N{\mathsf{m}}_1 \cdots {\mathsf{m}}_5} + 5\,\bm{{\mathsf{A}}}^\alpha_{N[{\mathsf{m}}_1}\, \bm{{\mathsf{A}}}_{{\mathsf{m}}_2\cdots {\mathsf{m}}_5]}
+ 5\,\epsilon_{\gamma\delta}\,\bm{{\mathsf{A}}}^\gamma_{N[{\mathsf{m}}_1}\,\bm{{\mathsf{A}}}^\delta_{{\mathsf{m}}_2{\mathsf{m}}_3}\,\bm{{\mathsf{A}}}^\alpha_{{\mathsf{m}}_4{\mathsf{m}}_5]}}{\sqrt{5!}}
\\
\frac{\Bigl[\genfrac{}{}{0pt}{1}{\bm{{\mathsf{A}}}_{N {\mathsf{m}}_1\cdots {\mathsf{m}}_6, {\mathsf{m}}} + \epsilon_{\gamma\delta}\, \bm{{\mathsf{A}}}^\gamma_{N{\mathsf{m}}}\,\bm{{\mathsf{A}}}^\delta_{{\mathsf{m}}_1\cdots {\mathsf{m}}_6}+10\,\bm{{\mathsf{A}}}_{N[{\mathsf{m}}_1{\mathsf{m}}_2{\mathsf{m}}_3}\,\bm{{\mathsf{A}}}_{{\mathsf{m}}_4{\mathsf{m}}_5{\mathsf{m}}_6]{\mathsf{m}}}~~~~~~~~~~~~~~}{-30\,\epsilon_{\gamma\delta}\,\bm{{\mathsf{A}}}^\gamma_{N[{\mathsf{m}}_1}\,\bm{{\mathsf{A}}}^\delta_{{\mathsf{m}}_2{\mathsf{m}}_3}\,\bm{{\mathsf{A}}}_{{\mathsf{m}}_4{\mathsf{m}}_5{\mathsf{m}}_6]{\mathsf{m}}}+\frac{15}{2}\,\epsilon_{\alpha\beta}\,\epsilon_{\gamma\delta}\,\bm{{\mathsf{A}}}^\alpha_{N[{\mathsf{m}}_1}\,\bm{{\mathsf{A}}}^\beta_{{\mathsf{m}}_2{\mathsf{m}}_3}\,\bm{{\mathsf{A}}}^\gamma_{{\mathsf{m}}_4{\mathsf{m}}_5}\,\bm{{\mathsf{A}}}^\delta_{{\mathsf{m}}_6]{\mathsf{m}}}}\Bigr]}{\sqrt{6!}}
\\
\vdots
\end{pmatrix},
\label{eq:L-T-B}\end{aligned}$$ where ${\mathsf{m}}\in \{{\mathsf{m}}_1,\dotsc, {\mathsf{m}}_6\}$ has been assumed for the fifth row. Again by using the identification and ${\mathsf{m}}\in \{{\mathsf{m}}_1,\dotsc, {\mathsf{m}}_6\}$, $({\mathsf{L}}^{-{{\mathpalette{\raisebox{\depth}{$\m@th\intercal$}}}})_{N}{}^{{\mathsf{I}}}$ matches with ${\mathsf{N}}_{N}{}^{\mathsf{I}}$ given in and the generalized graviphoton in the type IIB parameterization is precisely the 1-form .
Here, let us comment on the relation to the series of papers [@1403.7198; @1409.6314; @1412.2768]. The standard wave solution in 11D supergravity has the non-vanishing flux associated with the graviphoton ${\mathcal A}_\mu^i$. In [@1403.7198; @1409.6314; @1412.2768], the wave solution was embedded into EFT, which has non-vanishing ${\mathcal A}_\mu^I$. Then, by rotating the duality frames, various brane solutions were obtained in EFT. Particularly in [@1412.2768], the 1-form ${\mathcal A}_\mu^I$ was regarded as the graviphoton in the $(4+56)$-dimensional exceptional space. Since all of their brane solutions in EFT couples to the generalized graviphoton ${\mathcal A}_\mu^I$, branes were interpreted as a kind of generalized waves in the exceptional space. Although the explicit parameterization of ${\mathcal A}_\mu^I$ was not determined there, conceptually, their idea is closely related to the result obtained here.
Parameterization of ${\mathcal A}_p^{I_p}$ {#sec:p-forms}
==========================================
In this section, we study the parameterization of the higher $p$-form fields ${\mathcal A}_p^{I_p}$.
2-form ${\mathcal A}_2^{I_2}$ {#form-mathcal-a_2i_2 .unnumbered}
-----------------------------
The 2-form gauge field ${\mathcal A}_2^{I_2}$ transforms in the string multiplet, characterized by the Dynkin label $[0,\dotsc,0,1,0]$. It is decomposed as $$\begin{aligned}
{\footnotesize
{\mathcal A}_{\mu\nu}^{I_2}
= \begin{pmatrix}
{\hat{A}}_{[\mu\nu]; i} \\
\frac{1}{\sqrt{4!}}\,{\hat{A}}_{[\mu\nu]; i_1\cdots i_4} \\
\frac{1}{\sqrt{6!}}\,{\hat{A}}_{[\mu\nu]; i_1\cdots i_6,i} \\
\vdots
\end{pmatrix},\qquad
\bm{{\mathcal A}}_{\mu\nu}^{I_2}
= \begin{pmatrix}
{\mathsf{A}}^\alpha_{\mu\nu} \\
\frac{1}{\sqrt{2!}}\,{\mathsf{A}}_{[\mu\nu]; {\mathsf{m}}_1{\mathsf{m}}_2} \\
\frac{1}{\sqrt{4!}}\,{\mathsf{A}}^\alpha_{[\mu\nu]; {\mathsf{m}}_1\cdots {\mathsf{m}}_4} \\
\frac{1}{\sqrt{6!}}\,{\mathsf{A}}^\alpha_{[\mu\nu]; {\mathsf{m}}_1\cdots {\mathsf{m}}_5,{\mathsf{m}}} \\
\vdots
\end{pmatrix}, }\end{aligned}$$ and for example, the first component, in each parameterization, can be expanded as $$\begin{aligned}
{\hat{A}}_{[\mu\nu]; i} &= {\hat{A}}_{\mu\nu i} + m_1\, {\hat{A}}_{[\mu}^k\, {\hat{A}}_{\nu] ki} + m_2\, {\hat{A}}_{[\mu}^k\,{\hat{A}}_{\nu]}^l\, {\hat{A}}_{kl i} \,,
\\
{\mathsf{A}}^\alpha_{\mu\nu} &= {\mathsf{A}}^\alpha_{\mu\nu} + b_1\, {\mathsf{A}}_{[\mu}^{\mathsf{p}}\, {\mathsf{A}}^\alpha_{\nu] {\mathsf{p}}} + b_2\, {\mathsf{A}}_{[\mu}^{\mathsf{p}}\,{\mathsf{A}}_{\nu]}^{\mathsf{q}}\, {\mathsf{A}}^\alpha_{{\mathsf{p}}{\mathsf{q}}} \,,\end{aligned}$$ by introducing parameters $m_1,\,m_2,\,b_1$, and $b_2$. We already have the $T$-duality rules, and by following the same procedure as the 1-form, we can determine these parameters.
Repeating the procedure, we find the parameterization [$$\begin{aligned}
{\mathcal A}_{\mu\nu}^{I_2}
&= \begin{pmatrix}
{\hat{N}}_{[\mu;\nu] i} + {\hat{A}}_{[\mu|}^k\,{\hat{N}}_{k;|\nu]i}^{\phantom{k}} \\
\frac{1}{\sqrt{4!}}\bigl({\hat{N}}_{[\mu;\nu] i_1\cdots i_4} + {\hat{A}}_{[\mu|}^k\,{\hat{N}}_{k;|\nu]i_1\cdots i_4}^{\phantom{k}}\bigr) \\
\frac{1}{\sqrt{6!}}\, \bigl({\hat{N}}_{[\mu;\nu] i_1\cdots i_6, i} + {\hat{A}}_{[\mu|}^k\,{\hat{N}}_{k;|\nu] i_1\cdots i_6,i}\bigr)\\
\vdots
\end{pmatrix},
\\
\bm{{\mathcal A}}_{\mu\nu}^{I_2}
&= \begin{pmatrix}
{\mathsf{N}}^\alpha_{[\mu;\nu]} + {\mathsf{A}}_{[\mu|}^{\mathsf{p}}\,{\mathsf{N}}^\alpha_{{\mathsf{p}};|\nu]} \\
\frac{1}{\sqrt{2!}}\bigl({\mathsf{N}}_{[\mu;\nu] m_1m_2} + {\mathsf{A}}_{[\mu|}^{\mathsf{p}}\,{\mathsf{N}}_{{\mathsf{p}};|\nu]{\mathsf{m}}_1{\mathsf{m}}_2}^{\phantom{p}}\bigr) \\
\frac{1}{\sqrt{4!}}\bigl({\mathsf{N}}^\alpha_{[\mu;\nu] {\mathsf{m}}_1\cdots {\mathsf{m}}_4} + {\mathsf{A}}_{[\mu|}^{\mathsf{p}}\,{\mathsf{N}}^\alpha_{{\mathsf{p}};|\nu]{\mathsf{m}}_1\cdots {\mathsf{m}}_4}\bigr) \\
\frac{1}{\sqrt{5!}}\bigl({\mathsf{N}}_{[\mu;\nu] {\mathsf{m}}_1\cdots {\mathsf{m}}_5,{\mathsf{m}}} + {\mathsf{A}}_{[\mu|}^{\mathsf{p}}\,{\mathsf{N}}_{{\mathsf{p}};|\nu]{\mathsf{m}}_1\cdots {\mathsf{m}}_5,{\mathsf{m}}}\bigr) \\
\vdots
\end{pmatrix}.\end{aligned}$$]{}Interestingly, the tensors ${\hat{N}}$ and ${\mathsf{N}}$ are precisely the same as those defined in Section \[sec:1-form-results\]. The origin of this simple structure can be understood as follows.
For example, let us consider the map $$\begin{aligned}
{\hat{A}}_{\mu a\alpha} + {\hat{A}}_\mu^k\,{\hat{A}}_{ka\alpha} \overset{\text{M--B}}{=} \bigl({\mathsf{A}}_{\mu a}^\beta + {\mathsf{A}}_\mu^{\mathsf{p}}\,{\mathsf{A}}^\beta_{{\mathsf{p}}a}\bigr)\,\epsilon_{\beta\alpha} \,,\end{aligned}$$ in which both sides are connected through $T$-duality. However, the $T$-duality rule is 9D covariant, and even if we replace the index $a$ by the 9D index $A=(\mu,\,a)$, the above relation is still satisfied. Then, choosing $A=\nu$ and antisymmetrizing $\mu$ and $\nu$, we get $$\begin{aligned}
{\hat{A}}_{\mu\nu \alpha} + {\hat{A}}_{[\mu}^k\,{\hat{A}}_{|k|\nu] \alpha} \overset{\text{M--B}}{=} \bigl({\mathsf{A}}_{\mu\nu}^\beta + {\mathsf{A}}_{[\mu}^{\mathsf{p}}\,{\mathsf{A}}^\beta_{|{\mathsf{p}}|\nu]}\bigr)\,\epsilon_{\beta\alpha} \,,\end{aligned}$$ which connects the first row of ${\mathcal A}_{\mu\nu}^{I_2}$ and the first row of $\bm{{\mathcal A}}_{\mu\nu}^{I_2}$. In this manner, simply by replacing an internal index $a$ with an external index $\nu$ and acting the antisymmetrization, we obtain the parameterization of the 2-form from the result of the 1-form.
In the literature, several components of the 1-form and 2-form have been studied for example in [@1312.0614; @1506.01065]. By following the notation of [@1802.00442], their M-theory parameterization are $$\begin{aligned}
{\mathcal A}_\mu{}^m = A_\mu{}^m\,,\qquad
{\mathcal A}_{\mu mn} = \frac{\hat{C}_{\mu mn}{}^\beta - A_\mu{}^k\, \hat{C}_{k mn}{}^\beta}{\sqrt{2}} \,, \qquad
{\mathcal B}_{\mu\nu m} = \frac{\hat{C}_{\mu\nu m} - A_{[\mu}{}^k \,\hat{C}_{\nu] m k}}{\sqrt{10}}\,,\end{aligned}$$ while the type IIB parameterizations are $$\begin{aligned}
{\mathcal A}_\mu{}^i = A_\mu{}^i\,,\qquad
{\mathcal A}_{\mu i\alpha} = \epsilon_{\alpha\beta}\, \bigl(\hat{C}_{\mu i}{}^\beta - A_\mu{}^k\, \hat{C}_{k i}{}^\beta \bigr) \,, \qquad
{\mathcal B}_{\mu\nu}{}^\alpha = \frac{C_{\mu\nu}{}^\alpha - A_{[\mu}{}^j\,\hat{C}_{|j|\nu]}{}^\alpha}{\sqrt{10}}\,.\end{aligned}$$ By comparing, for example ${\mathcal A}_{\mu mn}$ with ${\mathcal B}_{\mu\nu m}$, we find that their results also follow the antisymmetrization rule and seem to be consistent with our results up to conventions.
$3$-form and higher $p$-form {#form-and-higher-p-form .unnumbered}
----------------------------
Similar to the case of the 2-form, a parameterization of a general $p$-form can be obtained by acting the antisymmetrization to that of the 1-form. In the case of 3-form, we obtain [$$\begin{aligned}
{\mathcal A}_{\mu\nu\rho}^{I_2}
&= \begin{pmatrix}
{\hat{N}}_{[\mu;\nu\rho]} + {\hat{A}}_{[\mu|}^k\,{\hat{N}}_{k;|\nu\rho]}^{\phantom{k}} \\
\frac{1}{\sqrt{3!}}\bigl({\hat{N}}_{[\mu;\nu\rho] i_1i_2i_3} + {\hat{A}}_{[\mu|}^k\,{\hat{N}}_{k;|\nu\rho]i_1i_2i_3}^{\phantom{k}}\bigr) \\
\frac{1}{\sqrt{5!}}\, \bigl({\hat{N}}_{[\mu;\nu\rho] i_1\cdots i_5, i} + {\hat{A}}_{[\mu|}^k\,{\hat{N}}_{k;|\nu\rho] i_1\cdots i_5,i}\bigr)\\
\vdots
\end{pmatrix},
\\
\bm{{\mathcal A}}_{\mu\nu\rho}^{I_2}
&= \begin{pmatrix}
{\mathsf{N}}_{[\mu;\nu\rho] {\mathsf{m}}} + {\mathsf{A}}_{[\mu|}^{\mathsf{p}}\,{\mathsf{N}}_{{\mathsf{p}};|\nu\rho]{\mathsf{m}}}^{\phantom{p}} \\
\frac{1}{\sqrt{3!}}\bigl({\mathsf{N}}^\alpha_{[\mu;\nu\rho] {\mathsf{m}}_1{\mathsf{m}}_2{\mathsf{m}}_3} + {\mathsf{A}}_{[\mu|}^{\mathsf{p}}\,{\mathsf{N}}^\alpha_{{\mathsf{p}};|\nu\rho] {\mathsf{m}}_1{\mathsf{m}}_2{\mathsf{m}}_3}\bigr) \\
\frac{1}{\sqrt{4!}}\bigl({\mathsf{N}}_{[\mu;\nu\rho] {\mathsf{m}}_1\cdots {\mathsf{m}}_4,{\mathsf{m}}} + {\mathsf{A}}_{[\mu|}^{\mathsf{p}}\,{\mathsf{N}}_{{\mathsf{p}};|\nu\rho]{\mathsf{m}}_1\cdots {\mathsf{m}}_4,{\mathsf{m}}}\bigr) \\
\vdots
\end{pmatrix}.\end{aligned}$$]{}Compared to the 2-form, the first component in type IIB side ${\mathsf{N}}^\alpha_{\mu;\nu}$ has disappeared because the number of indices is not enough to account for a 3-form. The 4-form is [$$\begin{aligned}
{\mathcal A}_{\mu_1\cdots\mu_4}^{I_2}
&= \begin{pmatrix}
\frac{1}{\sqrt{2!}}\bigl({\hat{N}}_{[\mu_1;\mu_2\cdots\mu_4] i_1i_2} + {\hat{A}}_{[\mu_1|}^k\,{\hat{N}}_{k;|\mu_2\mu_3\mu_4]i_1i_2}^{\phantom{k}}\bigr) \\
\frac{1}{\sqrt{4!}}\, \bigl({\hat{N}}_{[\mu_1;\mu_2\cdots\mu_4] i_1\cdots i_4, i} + {\hat{A}}_{[\mu_1|}^k\,{\hat{N}}_{k;|\mu_2\mu_3\mu_4] i_1\cdots i_4,i}\bigr)\\
\vdots
\end{pmatrix},
\\
\bm{{\mathcal A}}_{\mu_1\cdots\mu_4}^{I_2}
&= \begin{pmatrix}
{\mathsf{N}}_{[\mu_1;\mu_2\cdots\mu_4]} + {\mathsf{A}}_{[\mu_1|}^{\mathsf{p}}\,{\mathsf{N}}_{{\mathsf{p}};|\mu_2\mu_3\mu_4]}^{\phantom{p}} \\
\frac{1}{\sqrt{2!}}\bigl({\mathsf{N}}^\alpha_{[\mu_1;\mu_2\cdots\mu_4] {\mathsf{m}}_1{\mathsf{m}}_2} + {\mathsf{A}}_{[\mu_1|}^{\mathsf{p}}\,{\mathsf{N}}^\alpha_{{\mathsf{p}};|\mu_2\mu_3\mu_4] {\mathsf{m}}_1{\mathsf{m}}_2}\bigr) \\
\frac{1}{\sqrt{3!}}\bigl({\mathsf{N}}_{[\mu_1;\mu_2\cdots\mu_4]{\mathsf{m}}_1{\mathsf{m}}_2{\mathsf{m}}_3,{\mathsf{m}}} + {\mathsf{A}}_{[\mu_1|}^{\mathsf{p}}\,{\mathsf{N}}_{{\mathsf{p}};|\mu_2\mu_3\mu_4]{\mathsf{m}}_1{\mathsf{m}}_2{\mathsf{m}}_3,{\mathsf{m}}}\bigr) \\
\vdots
\end{pmatrix},\end{aligned}$$]{}and higher $p$-forms are also obtained in a similar manner.
We note that if there exist certain invariant tensor $f_{I_p I_q}{}^{I_r}$ with symmetry $f_{I_p I_q}{}^{I_r}=(-1)^{pq}\,f_{I_q I_p}{}^{I_r}$, we can redefine an $r$-form ${\mathcal A}^{I_r}_r$ as $$\begin{aligned}
{\mathcal A}^{I_r}_r \ \to \ {\mathcal A}'^{I_r}_r = {\mathcal A}^{I_r}_r + f_{I_p I_q}{}^{I_r}\,{\mathcal A}^{I_p}_p\wedge {\mathcal A}_q^{I_q} \,.
\label{eq:re-define-A}\end{aligned}$$ In such case, the $r$-form field is not unique and we cannot fix the parameterization unambiguously.
Summary and Discussion {#sec:conclusions}
======================
In this paper, we have proposed a systematic way to determine the parameterization of the $p$-form field ${\mathcal A}^{I_p}_p$. As a demonstration, we have determined how the dual graviton enters the $p$-form field. We have also determined the duality rules for the dual graviton, which have been partially studied in the literature. Our procedure is based on the (factorized) $T$-duality and $S$-duality transformations, which form a subgroup of the full $U$-duality group. Accordingly, our procedure cannot determine the contribution of the mixed-symmetry potentials which do not couple to any supersymmetric branes. However, we have provided another approach to determine the parameterization of ${\mathcal A}^{I_p}_p$. We have found that the 1-form field is precisely the generalized graviphoton ${\mathcal A}_\mu^I = \bm{m}_{\mu\nu}\, {\mathcal M}^{\nu I}$ defined by the $E_{11}$ generalized metric. By following the procedure of [@hep-th/0104081; @hep-th/0307098; @1009.2624], we can in principle determine the parameterization of the $E_{11}$ generalized metric level by level. We can then determine the full parameterization of the 1-form field. As we have shown, once the parameterization of the 1-form field is determined, we can easily obtain the parameterization of the $p$-form field by antisymmetrizing the indices.
As future directions, it is interesting to revisit the worldvolume actions of exotic brane. In the case of exotic branes, the Wess–Zumino term contains the mixed-symmetry potentials, but at present, the explicit forms of the brane actions are known for a few examples [@hep-th/9908094; @1309.2653; @1404.5442; @1601.05589]. A manifestly $U$-duality-covariant Wess–Zumino term, which employs the $p$-form fields ${\mathcal A}^{I_p}_p$, has been proposed in [@1009.4657] and it is important to clarify the connection to the results of [@hep-th/9908094; @1309.2653; @1404.5442; @1601.05589] by using the concrete parameterization of ${\mathcal A}^{I_p}_p$. It is also interesting to develop another $U$-duality-manifest approach to brane actions [@1607.04265; @1712.10316] (see also [@1712.07115; @1802.00442] for a similar approach).
It is also useful to study the duality transformation rules for more mixed-symmetry potentials beyond the dual graviton. By following the procedure proposed in this paper, it is a straightforward task to determine such duality rules. Recently, $T$-duality manifest formulation for mixed-symmetry potentials has been studied in detail in [@1903.05601], which aims to be more useful to determine the $T$-duality rules. Nevertheless, in order to consider the $S$-duality rule or the M-theory uplifts, our $U$-duality-based procedure would potentially prove more useful.
Acknowledgments {#acknowledgments .unnumbered}
---------------
The work of JJFM is supported by Plan Propio de Investigación of the University of Murcia R-957/2017 and Fundación Séneca (21257/PI/19 and 20949/PI/18). The work of YS is supported by JSPS KAKENHI Grant Numbers 18K13540 and 18H01214.
Notation {#app:notation}
========
In this appendix, we summarize the notation that has been used along this work to denote various fields corresponding to each theory and each dimension, as well as the different types of indices.
M-theory and type IIA/IIB theory are defined in $D$ dimensions, where $D=11,10$ respectively. Upon a dimensional reduction on a torus, we have a $d$-dimensional supergravity theory, with a global symmetry group $E_n$, where $n=D-d$. According to this, all the splittings of the M-theory and type IIB coordinates and the higher/lower-dimensional indices that have been used are shown in Figure \[fig:indices\]. The $D$-dimensional coordinates in M-theory and type IIB theory are denoted by $x^{\hat{M}}$ and ${\mathsf{x}}^M$, respectively.
In addition, indices for the $p$-form multiplet are denoted as $I_p$ in M-theory and as ${\mathsf{I}}_p$ in type IIB theory. In particular, for the 1-form, we denote $I\equiv I_1$ and ${\mathsf{I}}_1\equiv {\mathsf{I}}$. In type IIB theory, the index of the vector representation of the ${\text{SL}}(2)$ $S$-duality group is represented by $\alpha=1,2$.
$$\begin{aligned}
\hat M \left[
\phantom{\begin{array}{c}
\mbox{}\\\mbox{}\\[8pt]\mbox{}\\\mbox{}\\\mbox{}\\\mbox{}\\\mbox{}\\\mbox{}
\end{array}
}
\right.
\hspace{-12pt}
\begin{array}{l}
M \left[
\phantom{\begin{array}{c}
\mbox{}\\[8pt]\mbox{}\\\mbox{}\\\mbox{}\\\mbox{}\\\mbox{}\\\mbox{}
\end{array}
}
\right.
\\
\phantom{\mbox{}}
\end{array}
\hspace{-12pt}
\begin{array}{l}
A \left[
\phantom{\begin{array}{c}
\mbox{}\\\mbox{}\\\mbox{}\\\mbox{}\\\mbox{}\\\mbox{}
\end{array}
}
\right.
\\
\phantom{\mbox{}}
\\
\phantom{\mbox{}}
\end{array}
\hspace{-12pt}
\begin{array}{l}
\phantom{\mbox{}}
\\
\phantom{\mbox{}}
\\
\phantom{\mbox{}}
\\
i \left[
\phantom{\begin{array}{c}
\mbox{}\\[8pt]\mbox{}\\\mbox{}\\\mbox{}\\\mbox{}
\end{array}
}
\right.
\end{array}
\hspace{-12pt}
\begin{array}{l}
\phantom{\mbox{}}
\\
\phantom{\mbox{}}
\\
\phantom{\mbox{}}
\\
m \left[
\phantom{\begin{array}{c}
\mbox{}\\[8pt]\mbox{}\\\mbox{}\\\mbox{}
\end{array}
}
\right.
\\
\phantom{\mbox{}}
\end{array}
\hspace{-12pt}
\begin{array}{r}
\mu \left[
\phantom{\begin{array}{c}
\mbox{}\\\mbox{}\\[5pt]\mbox{}
\end{array}
}
\right.
\\[8pt]
a \left[
\phantom{\begin{array}{c}
\mbox{}\\\mbox{}\\\mbox{}
\end{array}
}
\right.
\\[8pt]
\alpha \left[
\phantom{\begin{array}{c}
\mbox{}\\\mbox{}
\end{array}
}
\right.
\end{array}
\hspace{-12pt}
\left(
\begin{array}{c}
x^0\\\vdots\\ x^{d-1}
\\
x^d\\\vdots\\ x^{8}
\\
x^y\\x^z
\end{array}
\right)
\hspace{-12pt}
\begin{array}{c}
\mbox{}\\\mbox{}\mbox{}\\\mbox{}\\\mbox{}\\\mbox{}\\[5pt] \mbox{}\\ \quad \leftarrow \ T \text{-duality} \ \rightarrow \quad \\S^1
\end{array}
\hspace{-12pt}
\begin{array}{r}
\left(
\begin{array}{c}
{\mathsf{x}}^0\\\vdots\\ {\mathsf{x}}^{d-1}
\\
{\mathsf{x}}^d\\\vdots\\ {\mathsf{x}}^{8}
\\
{\mathsf{x}}^{\mathsf{y}}\end{array}
\right)
\\
\phantom{\mbox{}}
\end{array}
\hspace{-12pt}
\begin{array}{l}
\left.
\phantom{\begin{array}{c}
\mbox{}\\\mbox{}\\\mbox{}
\end{array}
}
\right] \mu
\\
\left.
\phantom{\begin{array}{c}
\mbox{}\\\mbox{}\\\mbox{}
\end{array}
}
\right] a
\\
\phantom{\begin{array}{c}
\mbox{}\\\mbox{}
\end{array}
}
\end{array}
\hspace{-12pt}
\begin{array}{l}
\phantom{\mbox{}}
\\[8pt]
\phantom{\mbox{}}
\\
\phantom{\mbox{}}
\\
\left.
\phantom{\begin{array}{c}
\mbox{}\\\mbox{}\\\mbox{}\\\mbox{}
\end{array}
}
\right] {\mathsf{m}}\\
\phantom{\mbox{}}
\end{array}
\hspace{-12pt}
\begin{array}{l}
\left.
\phantom{\begin{array}{c}
\mbox{}\\\mbox{}\\\mbox{}\\\mbox{}\\\mbox{}\\\mbox{}
\end{array}
}
\right] A
\\
\phantom{\mbox{}}
\\
\phantom{\mbox{}}
\end{array}
\hspace{-12pt}
\begin{array}{l}
\left.
\phantom{\begin{array}{c}
\mbox{}\\[8pt]\mbox{}\\\mbox{}\\\mbox{}\\\mbox{}\\\mbox{}\\\mbox{}
\end{array}
}
\right] M
\\
\phantom{\mbox{}}
\end{array}\end{aligned}$$
Field M-theory Type IIB Type IIA
------------------------------------------------------ ------------------------ ------------------------------------------------ ---------------------------------------------------
$U$-duality-covariant $p$-form ${\mathcal A}_p^{I_p}$ $\bm{{\mathcal A}}_p^{{\mathsf{I}}_p}$ –
generalized metric ${\mathcal M}$ ${\mathsf{M}}$ –
generalized vielbein ${\mathcal E}$ ${\mathsf{E}}$ –
twist matrix $L$ ${\mathsf{L}}$ –
$D$-dim. metric ${\hat{g}}$ ${\mathsf{g}}$ $g$
$D$-dim. fields ${\hat{A}}$ ${\mathsf{A}}$, ${\mathsf{B}}$, ${\mathsf{C}}$ ${\mathscr{A}}$, ${\mathscr{B}}$, ${\mathscr{C}}$
$D$-dim. fields (Section \[sec:generalized-metric\]) $\bm{{\hat{A}}}$ $\bm{{\mathsf{A}}}$ –
spacetime metric $\bm{{\hat{g}}}$ $\bm{{\mathsf{g}}}$ $\bm{g}$
internal metric ${\hat{G}}$ ${\mathsf{G}}$ $G$
: *Summary of the fields that have been used along this work. While the first four lines correspond to $U$-duality multiplets, the rest correspond to standard supergravity fields. In the last two lines we show the $d$-dimensional fields that appear after compactification.* []{data-label="tab:fields"}
In Table \[tab:fields\], we summarize the notation that we have used to represent the fields of various theories. Fields transforming as $U$-duality multiplets are considered. Similarly, standard supergravity fields of M-theory and type II theories, and the lower-dimensional fields that arise after compactification are considered.
$E_n$ generators {#app:generators}
================
In this appendix, we show the explicit matrix representation of the $E_n$ generators in the vector representation. In the M-theory parameterization, our matrices are consistent with [@1303.2035]. Through the linear map from M-theory parameterization to type IIB parameterization, we find the matrix representations also in the type IIB parameterization, which is new.
Here, we show the results for $E_8$, but the $E_n$ generators with $n\leq 7$ can be easily obtained through a truncation. For example, an $E_8$ generator $R^{i_{1\cdots8},i}$ disappears in $E_7$ because the index $i$ ranges over seven directions and $i_{1\cdots8}$ automatically vanishes. Conversely, our $E_8$ generators can be understood as a truncation of the $E_{11}$ generators. In $E_{11}$, the matrix representation becomes infinite dimensional, but the first several blocks are the same as the $E_8$ generators. Accordingly, although we have computed the matrix $(L^{-{{\mathpalette{\raisebox{\depth}{$\m@th\intercal$}}}})$ in by using the $E_8$ generators, the first four rows do not change even if we use the matrix representation of the $E_{11}$ generators.[^6] In that sense, the results given in this appendix can be understood as a truncation of the $E_{11}$ generators.
M-theory parameterization
-------------------------
In the M-theory parameterization, the $E_n$ generators are decomposed as $$\begin{aligned}
\{T_{\hat{\alpha}}\} = \bigl\{ K^i{}_j,\, R_{i_{123}} ,\, R_{i_{1\cdots6}} ,\, R_{i_{1\cdots8},i} ,\, R^{i_{123}} ,\, R^{i_{1\cdots6}} ,\, R^{i_{1\cdots8},i} ,\,\cdots\bigr\}\,,\end{aligned}$$ where the ellipses disappear for $n\leq 8$. In this appendix, we may use a short-hand notation for the multiple indices, $$\begin{aligned}
(\cdots)_{i_{1\cdots p}} \equiv (\cdots)_{i_1\cdots i_p}\,.\end{aligned}$$ We also using a notation, $$\begin{aligned}
{{\boldsymbol\delta}}_{i_1\cdots i_n}^{j_1\cdots j_n}\equiv n!\,\delta_{i_1\cdots i_n}^{j_1\cdots j_n}\,. \end{aligned}$$ If we restrict to the case $n\leq 8$, they satisfy the following commutation relations:[^7] $$\begin{aligned}
\bigl[K^i{}_j,\, K^k{}_l \bigr]
&= \delta_j^k\,K^i{}_l - \delta_l^i\, K^k{}_j \,,
\\
\bigl[K^i{}_j,\, R^{k_1k_2k_3} \bigr]
&= \frac{1}{2!}\,{{\boldsymbol\delta}}_{jr_1r_2}^{k_1k_2k_3} \,R^{ir_1r_2}\,,
\\
\bigl[K^i{}_j,\, R^{k_1\cdots k_6} \bigr]
&= \frac{1}{5!}\,{{\boldsymbol\delta}}_{jr_1\cdots r_5}^{k_1\cdots k_6} \, R^{ir_1\cdots r_5}\,,
\\
\bigl[K^i{}_j,\,R^{k_1\cdots k_8,k}\bigr]
&= \frac{1}{7!}\,{{\boldsymbol\delta}}^{k_1\cdots k_8}_{jr_1\cdots r_7}\, R^{ir_1 \cdots r_7,k} + \delta_j^k\,R^{k_1\cdots k_8,i}\,,
\\
\bigl[K^i{}_j,\, R_{k_1k_2k_3} \bigr]
&= -\frac{1}{2!}\,{{\boldsymbol\delta}}_{k_1k_2k_3}^{ir_1r_2} \, R_{jr_1r_2}\,,
\\
\bigl[K^i{}_j,\, R_{k_1\cdots k_6} \bigr]
&= -\frac{1}{5!}\,{{\boldsymbol\delta}}^{ir_1\cdots r_5}_{k_1\cdots k_6}\, R_{jr_1\cdots r_5}\,,
\\
\bigl[K^i{}_j,\,R_{k_1\cdots k_8,k}\bigr]
&= -\frac{1}{7!}\,{{\boldsymbol\delta}}_{k_1\cdots k_8}^{ir_1\cdots r_7}\, R_{jr_1 \cdots r_7,k} - \delta_k^i\,R_{k_1\cdots k_8,j}\,,
\\
\bigl[R^{i_1i_2i_3},\, R^{j_1j_2j_3}\bigr]
&= - R^{i_1i_2i_3 j_1j_2j_3}\,,
\\
\bigl[R^{i_1i_2i_3},\, R^{j_1 \cdots j_6}\bigr]
&= - \frac{1}{5!}\, {{\boldsymbol\delta}}^{j_1 \cdots j_6}_{r_1\cdots r_5 s}\, R^{i_1i_2i_3 r_1\cdots r_5,s}\,,
\\
\bigl[ R^{i_1i_2i_3},\,R_{j_1j_2j_3} \bigr]
&= \frac{1}{2!}\,{{\boldsymbol\delta}}_{r_1r_2 s}^{i_1i_2i_3}\,{{\boldsymbol\delta}}_{j_1j_2j_3}^{r_1r_2 t}\, K^s{}_t - \frac{1}{3}\,{{\boldsymbol\delta}}_{j_1j_2j_3}^{i_1i_2i_3}\,\delta^t_s\, K^s{}_t \,,
\\
\bigl[ R^{i_1i_2i_3},\, R_{j_1\cdots j_6} \bigr]
&= \frac{1}{3!}\, {{\boldsymbol\delta}}_{j_1\cdots j_6}^{i_1i_2i_3 r_1r_2r_3}\,R_{r_1r_2r_3} \,,
\\
\bigl[R^{i_1i_2i_3},\, R_{j_1 \cdots j_8,j}\bigr]
&= \frac{1}{5!}\, {{\boldsymbol\delta}}^{i_1i_2i_3 r_1\cdots r_5}_{j_1 \cdots j_8}\, R_{r_1\cdots r_5 j} \,,
\\
\bigl[R^{i_1\cdots i_6},\, R_{j_1j_2j_3}\bigr]
&= \frac{1}{3!}\,{{\boldsymbol\delta}}^{i_1\cdots i_6}_{j_1j_2j_3 r_1r_2r_3}\,R^{r_1r_2r_3} \,,
\\
\bigl[R^{i_1\cdots i_6},\,R_{j_1\cdots j_6} \bigr]
&= \frac{1}{5!}\, {{\boldsymbol\delta}}^{i_1\cdots i_6}_{r_1\cdots r_5 s}\, {{\boldsymbol\delta}}_{j_1\cdots j_6}^{r_1\cdots r_5 t}\, K^s{}_t - \frac{2}{3}\,{{\boldsymbol\delta}}_{j_1\cdots j_6}^{i_1\cdots i_6}\,\delta^t_s \,K^s{}_t \,,
\\
\bigl[R^{i_1 \cdots i_6},\, R_{j_1 \cdots j_8,j}\bigr]
&= \frac{1}{2!}\,{{\boldsymbol\delta}}^{i_1 \cdots i_6 r_1r_2}_{j_1 \cdots j_8}\, R_{r_1r_2j} \,,
\\
\bigl[R^{i_1\cdots i_8,i},\, R_{j_1j_2j_3}\bigr]
&= \frac{1}{5!}\, {{\boldsymbol\delta}}^{i_1 \cdots i_8}_{j_1j_2j_3 r_1\cdots r_5}\, R^{r_1\cdots r_5 i} \,,
\\
\bigl[R^{i_1 \cdots i_8,i}, R_{j_1 \cdots j_6}\bigr]
&= \frac{1}{2!}\, {{\boldsymbol\delta}}^{i_1 \cdots i_8}_{j_1\cdots j_6 r_1r_2}\, R^{r_1r_2 i} \,,
\\
\bigl[R^{i_1 \cdots i_8,i}, R_{j_1 \cdots j_8,j}\bigr]
&= {{\boldsymbol\delta}}^{i_1\cdots i_8}_{j_1 \cdots j_8}\, K^i{}_j \,,
\\
\bigl[R_{i_1i_2i_3},\,R_{j_1j_2j_3}\bigr]
&= R_{i_1i_2i_3j_1j_2j_3}\,,
\\
\bigl[R_{i_1i_2i_3},\, R_{j_1 \cdots j_6}\bigr]
&= \frac{1}{5!}\, {{\boldsymbol\delta}}_{j_1\cdots j_6}^{r_1\cdots r_5 s}\, R_{i_1i_2i_3r_1\cdots r_5,s}\,. \end{aligned}$$ We note that our convention will be related to that of [@1303.2035] as follows: $$\begin{aligned}
\begin{array}{|c||c|c|c|c|c|c|c|} \hline
\text{Here} & K^i{}_j & R_{i_{123}} & R^{i_{123}} & R_{i_{1\cdots 6}} & R^{i_{1\cdots 6}} & R_{i_{1\cdots 8},i} & R^{i_{1\cdots 8},i}
\\ \hline
\text{\cite{1303.2035}} & K^i{}_j & R_{i_{123}} & R^{i_{123}} & 2\,R_{i_{1\cdots 6}} & -2\,R^{i_{1\cdots 6}} & 2 \,R_{i_{1\cdots 8},i} & 2\, R^{i_{1\cdots 8},i} \\ \hline
\end{array} \,.\end{aligned}$$
Now, we show the matrix representations of these generators in the vector representation. In the M-theory parameterization, the vector representation (for $n\leq 8$) is decomposed as $$\begin{aligned}
\{x^I\} &= \Bigl\{x^i,\,\tfrac{y_{i_{12}}}{\sqrt{2!}},\,\tfrac{y_{i_{1\cdots 5}}}{\sqrt{5!}},\,\tfrac{y_{i_{1\cdots 7},i}}{\sqrt{7!}},\,\tfrac{y_{i_{1\cdots 8}}}{\sqrt{8!}},\,\tfrac{y_{i_{1\cdots 8},k_{123}}}{\sqrt{8!\,3!}},\,\tfrac{y_{i_{1\cdots 8},k_{1\cdots 6}}}{\sqrt{8!\,6!}},\,\tfrac{y_{i_{1\cdots 8},k_{1\cdots 8},i}}{\sqrt{8!\,8!}} \Bigr\},\end{aligned}$$ where $y_{[i_{1\cdots 7},i]}=0$. In this paper, in order to reduce the matrix size, we have combined $y_{i_{1\cdots 7},i}$ and $y_{i_{1\cdots 8}}$, and our $y_{i_{1\cdots 7},i}$ do not satisfy $y_{[i_{1\cdots 7},i]}=0$. We then find that the following matrices $(T_{\hat{\alpha}})^I{}_J$ satisfy the above $E_8$ algebra: $$\begin{aligned}
K^{p}{}_{q} &\equiv {\footnotesize {\arraycolsep=0.2mm \underset{7\times7}{\mathrm{diag}}
\begin{pmatrix}
-\delta_{q}^i \delta_j^{p} \\
\frac{{{\boldsymbol\delta}}_{i_{12}}^{pr} {{\boldsymbol\delta}}_{qr}^{j_{12}}}{\sqrt{2!\,2!}} \\
\frac{{{\boldsymbol\delta}}_{i_{1\cdots5}}^{pr_{1\cdots4}} {{\boldsymbol\delta}}_{qr_{1\cdots4}}^{j_{1\cdots5}}}{4!\sqrt{5!\,5!}} \\
\frac{\frac{1}{6!}{{\boldsymbol\delta}}_{i_{1\cdots7}}^{pr_{1\cdots6}} {{\boldsymbol\delta}}_{qr_{1\cdots6}}^{j_{1\cdots 7}}\delta_i^j +{{\boldsymbol\delta}}_{i_{1\cdots7}}^{j_{1\cdots 7}} \delta_{i}^{p}\delta_{q}^j}{\sqrt{7!\,7!}} \\
\frac{\frac{1}{7!}{{\boldsymbol\delta}}_{i_{1\cdots8}}^{pr_{1\cdots7}} {{\boldsymbol\delta}}_{qr_{1\cdots7}}^{j_{1\cdots8}}{{\boldsymbol\delta}}_{k_{123}}^{l_{123}} +\frac{1}{2!}{{\boldsymbol\delta}}_{i_{1\cdots8}}^{j_{1\cdots8}} {{\boldsymbol\delta}}_{k_{123}}^{pr_{12}}{{\boldsymbol\delta}}_{qr_{12}}^{l_{123}}}{\sqrt{8!\,3!\,8!\,3!}} \\
\frac{\frac{1}{7!}{{\boldsymbol\delta}}_{i_{1\cdots8}}^{pr_{1\cdots7}} {{\boldsymbol\delta}}_{qr_{1\cdots7}}^{j_{1\cdots8}}{{\boldsymbol\delta}}_{k_{1\cdots6}}^{l_{1\cdots6}} +\frac{1}{5!}{{\boldsymbol\delta}}_{i_{1\cdots8}}^{j_{1\cdots8}} {{\boldsymbol\delta}}_{k_{1\cdots6}}^{pr_{1\cdots5}}{{\boldsymbol\delta}}_{qr_{1\cdots5}}^{l_{1\cdots6}}}{\sqrt{8!\,6!\,8!\,6!}} \\
\frac{\frac{1}{7!}{{\boldsymbol\delta}}_{i_{1\cdots8}}^{pr_{1\cdots7}} {{\boldsymbol\delta}}_{qr_{1\cdots7}}^{j_{1\cdots8}}{{\boldsymbol\delta}}_{k_{1\cdots8}}^{l_{1\cdots8}}\delta_i^j + \frac{1}{7!}{{\boldsymbol\delta}}_{i_{1\cdots8}}^{j_{1\cdots8}} {{\boldsymbol\delta}}_{k_{1\cdots8}}^{pr_{1\cdots7}}{{\boldsymbol\delta}}_{qr_{1\cdots7}}^{l_{1\cdots8}}\delta_i^j + {{\boldsymbol\delta}}_{i_{1\cdots8}}^{j_{1\cdots8}} {{\boldsymbol\delta}}_{k_{1\cdots8}}^{l_{1\cdots8}} \delta^{p}_i \delta_{q}^j}{\sqrt{8!\,8!\,8!\,8!}}
\end{pmatrix} - \frac{\delta_{q}^{p} \,\delta^I_J}{9-n}}} \,,
\\
R^{p_{123}} &\equiv {\footnotesize
{\arraycolsep=0mm \begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
\frac{-{{\boldsymbol\delta}}_{j i_{12}}^{p_{123}}}{\sqrt{2!}}\!\! & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & \frac{{{\boldsymbol\delta}}^{j_{12} p_{123}}_{i_{1\cdots5}}}{\sqrt{5!\,2!}} & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & \!\!\!\!\!\!\frac{\frac{1}{2!}{{\boldsymbol\delta}}^{j_{1\cdots5} r_{12}}_{i_{1\cdots 7}} {{\boldsymbol\delta}}^{p_{123}}_{r_{12}i} -\frac{1}{4} {{\boldsymbol\delta}}^{j_{1\cdots5} p_{123}}_{i_{1\cdots 7}i}}{\sqrt{7!\,5!}}\!\!\!\!\!\!\!\!\!\!\!\! & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\frac{\frac{1}{2!} {{\boldsymbol\delta}}^{j_{1\cdots7} r}_{i_{1\cdots8}} {{\boldsymbol\delta}}^{p_{123}}_{rs_{12}} {{\boldsymbol\delta}}^{s_{12}j}_{k_{123}}-\frac{1}{4}{{\boldsymbol\delta}}^{j_{1\cdots7}j}_{i_{1\cdots8}} {{\boldsymbol\delta}}^{p_{123}}_{k_{123}}}{\sqrt{8!\,3!\,7!}}\!\!\!\!\!\!\!\!\! & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & \!\!\!\!\!\!\!\!\frac{{{\boldsymbol\delta}}^{j_{1\cdots8}}_{i_{1\cdots8}} {{\boldsymbol\delta}}^{l_{123}p_{123}}_{k_{1\cdots6}}}{\sqrt{8!\,6!\,8!\,3!}}\!\!\!\! & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & \!\!\!\!\!\!\!\!\frac{{{\boldsymbol\delta}}^{j_{1\cdots8}}_{i_{1\cdots8}} {{\boldsymbol\delta}}^{l_{1\cdots6}r_{12}}_{k_{1\cdots8}} {{\boldsymbol\delta}}^{p_{123}}_{i r_{12}}}{2!\sqrt{8!\,8!\,8!\,6!}} & ~0~
\end{pmatrix} } } ,
\\
R_{p_{123}} &\equiv {\footnotesize
{\arraycolsep=-1.0mm \begin{pmatrix}
~0~ & ~~\frac{-{{\boldsymbol\delta}}^{i j_{12}}_{p_{123}}}{\sqrt{2!}}~ & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & \!\!\frac{{{\boldsymbol\delta}}_{i_{12} p_{123}}^{j_{1\cdots5}}}{\sqrt{2!\,5!}}\! & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & \frac{\frac{1}{2!}{{\boldsymbol\delta}}_{i_{1\cdots5} r_{12}}^{j_{1\cdots 7}} {{\boldsymbol\delta}}_{p_{123}}^{r_{12}j} -\frac{1}{4} {{\boldsymbol\delta}}_{i_{1\cdots5} p_{123}}^{j_{1\cdots 7}j}}{\sqrt{5!\,7!}}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
& 0 & 0 & 0 \\
0 & 0 & 0 & 0& \!\!\!\!\!\!\!\!\!\!\!\!\!\!\frac{\frac{1}{2!} {{\boldsymbol\delta}}_{i_{1\cdots7} r}^{j_{1\cdots8}} {{\boldsymbol\delta}}_{p_{123}}^{rs_{12}} {{\boldsymbol\delta}}_{s_{12}i}^{l_{123}}-\frac{1}{4}{{\boldsymbol\delta}}_{i_{1\cdots7}i}^{j_{1\cdots8}} {{\boldsymbol\delta}}_{p_{123}}^{l_{123}}}{\sqrt{7!\,8!\,3!}}\!\!\!\!\!\!\!\!\! & 0 & 0 \\
0 & 0 & 0 & 0& 0 & \!\!\frac{{{\boldsymbol\delta}}_{i_{1\cdots8}}^{j_{1\cdots8}} {{\boldsymbol\delta}}_{k_{123}p_{123}}^{l_{1\cdots6}}}{\sqrt{8!\,3!\,8!\,6!}}\!\! & 0 \\
0 & 0 & 0 & 0& 0 & 0 & \!\!\!\!\!\frac{{{\boldsymbol\delta}}_{i_{1\cdots8}}^{j_{1\cdots8}} {{\boldsymbol\delta}}_{k_{1\cdots6}r_{12}}^{l_{1\cdots8}} {{\boldsymbol\delta}}_{p_{123}}^{j r_{12}}}{2!\sqrt{8!\,6!\,8!\,8!}} \\
0 & 0 & 0 & 0& 0 & 0 & 0
\end{pmatrix} } }
{\nonumber}\\
&= (R^{p_{123}})^{\mathbb{T}}\,,
\\
R^{p_{1\cdots6}} &\equiv {\footnotesize {\arraycolsep=0mm \begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
\frac{{{\boldsymbol\delta}}_{i_{1\cdots5} j}^{p_{1\cdots6}}}{\sqrt{5!}} & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & \!\! \frac{{{\boldsymbol\delta}}^{j_{12}}_{ir} {{\boldsymbol\delta}}^{p_{1\cdots6} r}_{i_{1\cdots 7}} +\frac{1}{2} {{\boldsymbol\delta}}^{p_{1\cdots6} j_{12}}_{i_{1\cdots 7}i}}{\sqrt{7!\,2!}}\!\!\!\!\!\! & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & \!\!\!\!\!\!\!\!\frac{{{\boldsymbol\delta}}^{j_{1\cdots5} r_{123}}_{i_{1\cdots8}} {{\boldsymbol\delta}}^{p_{1\cdots6}}_{k_{123} r_{123}}}{3!\sqrt{8!\,3!\,5!}}\!\!\!\!\!\!\!\! & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & \!\!\!\!\!\!\!\!\frac{{{\boldsymbol\delta}}^{r j_{1\cdots7}}_{i_{1\cdots8}} {{\boldsymbol\delta}}^{p_{1\cdots6}j}_{k_{1\cdots6} r} + \frac{1}{2} {{\boldsymbol\delta}}^{j_{1\cdots7}j}_{i_{1\cdots8}} {{\boldsymbol\delta}}^{p_{1\cdots6}}_{k_{1\cdots6}}}{\sqrt{8!\,6!\,7!}}\!\!\!\!\!\!\!\! & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & \!\!\!\!\frac{-{{\boldsymbol\delta}}^{j_{1\cdots8}}_{i_{1\cdots8}} {{\boldsymbol\delta}}^{p_{1\cdots6} r_{12}}_{k_{1\cdots8}} {{\boldsymbol\delta}}^{l_{123}}_{i r_{12}}}{2!\sqrt{8!\,8!\,8!\,3!}} & ~0~ & ~0~ \end{pmatrix} } } ,
\\
R_{p_{1\cdots6}} &\equiv {\footnotesize {\arraycolsep=0mm \begin{pmatrix}
0 & ~0~ & ~\frac{{{\boldsymbol\delta}}^{j_{1\cdots5} i}_{p_{1\cdots6}}}{\sqrt{5!}}~ & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & \!\!\!\!\!\! \frac{{{\boldsymbol\delta}}_{i_{12}}^{jr} {{\boldsymbol\delta}}_{p_{1\cdots6} r}^{j_{1\cdots 7}} +\frac{1}{2} {{\boldsymbol\delta}}_{p_{1\cdots6} i_{12}}^{j_{1\cdots 7}j}}{\sqrt{2!\,7!}} \!\!\!\!\!\!\!\! & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & \frac{{{\boldsymbol\delta}}_{i_{1\cdots5} r_{123}}^{j_{1\cdots8}} {{\boldsymbol\delta}}_{p_{1\cdots6}}^{l_{123} r_{123}}}{3!\sqrt{5!\,8!\,3!}} & 0 & 0 \\
0 & 0 & 0 & 0& 0 & \!\!\!\!\!\!\!\! \frac{{{\boldsymbol\delta}}_{r i_{1\cdots7}}^{j_{1\cdots8}} {{\boldsymbol\delta}}_{p_{1\cdots6}i}^{l_{1\cdots6} r} + \frac{1}{2} {{\boldsymbol\delta}}_{i_{1\cdots7}i}^{j_{1\cdots8}} {{\boldsymbol\delta}}_{p_{1\cdots6}}^{l_{1\cdots6}}}{\sqrt{7!\,8!\,6!}} \!\!\!\!\!\!\!\!\!\! & 0 \\
0 & 0 & 0 & 0& 0 & 0 & \!\!\!\!\!\!\!\! \frac{-{{\boldsymbol\delta}}_{i_{1\cdots8}}^{j_{1\cdots8}} {{\boldsymbol\delta}}_{p_{1\cdots6} r_{12}}^{l_{1\cdots8}} {{\boldsymbol\delta}}_{k_{123}}^{j r_{12}}}{2!\sqrt{8!\,3!\,8!\,8!}} \\
0 & 0 & 0 & 0& 0 & 0 & 0 \\
0 & 0 & 0 & 0& 0 & 0 & 0 \end{pmatrix}}}
{\nonumber}\\
&= (R^{p_{1\cdots6}})^{\mathbb{T}}\,,
\\
R^{p_{1\cdots8},p} &\equiv {\footnotesize {\arraycolsep=0.5mm \begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 \\
\frac{{{\boldsymbol\delta}}_{i_{1\cdots 7}j}^{p_{1\cdots8}}\delta_i^p+\frac{1}{4} {{\boldsymbol\delta}}^{p_{1\cdots8}}_{i_{1\cdots 7}i}\delta^p_j}{\sqrt{7!}} & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & -\frac{{{\boldsymbol\delta}}_{k_{123}}^{j_{12}p}{{\boldsymbol\delta}}^{p_{1\cdots8}}_{i_{1\cdots8}}}{\sqrt{8!\,3!\,2!}} & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & \frac{{{\boldsymbol\delta}}^{j_{1\cdots5}p}_{k_{1\cdots6}}{{\boldsymbol\delta}}^{p_{1\cdots8}}_{i_{1\cdots8}}}{\sqrt{8!\,6!\,5!}} & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & \!\!\frac{{{\boldsymbol\delta}}^{j_{1\cdots7} p}_{i_{1\cdots8}}{{\boldsymbol\delta}}^{p_{1\cdots8}}_{k_{1\cdots8}}\delta^j_i + \frac{1}{4}{{\boldsymbol\delta}}^{j_{1\cdots7}j}_{i_{1\cdots8}}{{\boldsymbol\delta}}^{p_{1\cdots8}}_{k_{1\cdots8}}\delta^p_i}{\sqrt{8!\,8!\,7!}} & 0 & 0 & 0 \end{pmatrix}}} ,
\\
R_{p_{1\cdots8},p} &\equiv {\footnotesize {\arraycolsep=0.5mm \begin{pmatrix}
0 & 0 & 0 & \frac{{{\boldsymbol\delta}}^{j_{1\cdots 7}i}_{p_{1\cdots8}}\delta^j_p+\frac{1}{4} {{\boldsymbol\delta}}_{p_{1\cdots8}}^{j_{1\cdots 7}j}\delta_p^i}{\sqrt{7!}} & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & -\frac{{{\boldsymbol\delta}}^{l_{123}}_{i_{12}p}{{\boldsymbol\delta}}_{p_{1\cdots8}}^{j_{1\cdots8}}}{\sqrt{2!\,8!\,3!}} & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & \frac{{{\boldsymbol\delta}}_{i_{1\cdots5}p}^{l_{1\cdots6}}{{\boldsymbol\delta}}_{p_{1\cdots8}}^{j_{1\cdots8}}}{\sqrt{5!\,8!\,6!}} & 0 \\
0 & 0 & 0 & 0& 0 & 0 & \frac{{{\boldsymbol\delta}}_{i_{1\cdots7} p}^{j_{1\cdots8}}{{\boldsymbol\delta}}_{p_{1\cdots8}}^{l_{1\cdots8}}\delta_i^j + \frac{1}{4}{{\boldsymbol\delta}}_{i_{1\cdots7}i}^{j_{1\cdots8}}{{\boldsymbol\delta}}_{p_{1\cdots8}}^{l_{1\cdots8}}\delta_p^j}{\sqrt{7!\,8!\,8!}} \\
0 & 0 & 0 & 0& 0 & 0 & 0 \\
0 & 0 & 0 & 0& 0 & 0 & 0 \\
0 & 0 & 0 & 0& 0 & 0 & 0 \end{pmatrix}}}
{\nonumber}\\
&= (R^{p_{1\cdots8},p})^{\mathbb{T}}\,.\end{aligned}$$ We can identify the Cartan generators as $$\begin{aligned}
\{H_k\} = \{K^1{}_1 - K^2{}_2,\, \dotsc,\, K^7{}_7 - K^8{}_8,\, K^6{}_6 + K^7{}_7 + K^8{}_8 + \tfrac{1}{3}\,K^i{}_i \} \,,\end{aligned}$$ and the positive/negative simple-root generators are $$\begin{aligned}
\{E_k\} = \{K^1{}_2 ,\, \dotsc,\, K^7{}_8,\, R^{678} \} \,,\qquad
\{F_k\} = \{K^2{}_1 ,\, \dotsc,\, K^8{}_7,\, R_{678} \} \,. \end{aligned}$$ They satisfy the relations $$\begin{aligned}
\begin{split}
&[H_k,\,E_l]=A_{kl}\,E_l\,,\qquad
[H_k,\,F_l]=-A_{kl}\,F_l\,,\qquad
[E_k,\,F_l]= \delta_{kl}\,H_l\,,
\\
&\frac{1}{\alpha_n}\,{\text{tr}}(H_k\,H_l) = A_{kl}\,,\qquad
\frac{1}{\alpha_n}\,{\text{tr}}(E_k\,F_l) = \delta_{kl}\,,
\end{split}\end{aligned}$$ where $$\begin{aligned}
(A_{kl})= {\footnotesize\begin{pmatrix}
2 & -1 & & & & \\
-1 & 2 & \ddots & & & \\
& \ddots & \ddots & -1 & & -1 \\
& & -1 & 2 & -1 & 0 \\
& & & -1 & 2 & 0 \\
& & -1 & 0 & 0 & 2
\end{pmatrix}},\qquad\quad
\begin{array}{|c||c|c|c|c|c|} \hline
n&4&5&6&7&8 \\ \hline
\alpha_n&3&4&6&12&60 \\ \hline
\end{array} \,.\end{aligned}$$ The set of the positive/negative root generators can be obtained by taking commutators of the simple-root generators $E_k$/$F_k$, and they can be summarized as $$\begin{aligned}
\begin{split}
\{E_{\bm{\alpha}}\} &= \{K^i{}_j\ (i<j),\, R_{i_{123}},\,R_{i_{1\cdots 6}},\,R_{i_{1\cdots 8},i}\}\,,
\\
\{F_{\bm{\alpha}}\} &= \{K^i{}_j\ (i>j),\, R^{i_{123}},\,R^{i_{1\cdots 6}},\,R^{i_{1\cdots 8},i}\}\,.
\end{split}\end{aligned}$$
Type IIB parameterization
-------------------------
We can transform the $E_n$ generators of the M-theory parameterization into the type IIB parameterization by using the linear map . Namely, we act the following operation to the matrix representations of the generators $$\begin{aligned}
{\mathcal T}(T_{\hat{\alpha}})^{\mathsf{I}}{}_{\mathsf{J}}\equiv (S^{\mathbb{T}})^{\mathsf{I}}{}_K\, (T_{\hat{\alpha}})^K{}_L\, S^L{}_{\mathsf{J}}\,.\end{aligned}$$ Then, ${\mathcal T}(T_{\hat{\alpha}})^{\mathsf{I}}{}_{\mathsf{J}}$ is the matrix representations in the type IIB parameterization. The explicit form of the constant matrix $S^I{}_{\mathsf{J}}$ has been determined such that the algebra of the type IIB generators is closed. We also change the name of the generators such that the ${\text{SL}}(7)\times {\text{SL}}(2)$ symmetry is manifest. Concretely, we convert the non-positive-level generators of the M-theory parameterization into those of the type IIB parameterization as follows: $$\begin{aligned}
\begin{gathered}
\xymatrix@R-20pt{
& {\mathcal T}(K^a{}_b) \ar@{=}[r] & {\mathsf{K}}^a{}_b \ar@/^/[rd] & \\
{\mathcal T}(K^i{}_j) \ar@/^/[ru] \ar[r] \ar@/_/[rd] \ar@/_/[rdd] & {\mathcal T}\bigl(\tfrac{K^a{}_a-2\, K^\alpha{}_\alpha}{3}\bigr) \ar@{=}[r] & {\mathsf{K}}^{\mathsf{y}}{}_{\mathsf{y}}\ar[r] & {\mathsf{K}}^{\mathsf{m}}{}_{\mathsf{n}}\\
& {\mathcal T}\bigl(K^\alpha{}_\beta -\tfrac{1}{2}\,\delta^\alpha_\beta\,K^\gamma{}_\gamma \bigr) \ar@{=}[r] & \epsilon^{\alpha\gamma}\,{\mathsf{R}}_{\gamma\beta} \ar[r] & {\mathsf{R}}_{\alpha\beta} \\
& {\mathcal T}(K^\alpha{}_a) \ar@{=}[r] & {\mathsf{R}}^\alpha_{a{\mathsf{y}}} \ar@/^/[rd] & \\
& {\mathcal T}(R_{a_1{y}{z}}) \ar@{=}[r] & {\mathsf{K}}^{\mathsf{y}}{}_{a_1} \ar@/_/[ruuu] & {\mathsf{R}}^\alpha_{{\mathsf{m}}_1{\mathsf{m}}_2} \\
{\mathcal T}(R_{i_1i_2i_3}) \ar@/^/[ur] \ar[r] \ar@/_/[rd] & {\mathcal T}(R_{a_1a_2\alpha}) \ar@{=}[r] & \epsilon^{{\mathpalette{\raisebox{\depth}{$\m@th\intercal$}}}_{\alpha\beta}\,{\mathsf{R}}^\beta_{a_1a_2} \ar@/_/[ru] & \\
& {\mathcal T}(R_{a_1a_2a_3}) \ar@{=}[r] & {\mathsf{R}}_{a_1a_2a_3{\mathsf{y}}} \ar[r] & {\mathsf{R}}_{{\mathsf{m}}_1\cdots{\mathsf{m}}_4} \\
& {\mathcal T}(R_{a_1\cdots a_4{y}{z}}) \ar@{=}[r] & {\mathsf{R}}_{a_1\cdots a_4} \ar@/_/[ru] & \\
{\mathcal T}(R_{i_1\cdots i_6}) \ar@/^/[ur] \ar[r] \ar@/_/[rd] & {\mathcal T}(R_{a_1\cdots a_5\alpha}) \ar@{=}[r] & \epsilon^{{\mathpalette{\raisebox{\depth}{$\m@th\intercal$}}}_{\alpha\beta}\,{\mathsf{R}}^\beta_{a_1\cdots a_5{\mathsf{y}}} \ar[r] & {\mathsf{R}}^\alpha_{{\mathsf{m}}_1\cdots {\mathsf{m}}_6} \\
& {\mathcal T}(R_{a_1\cdots a_6}) \ar@{=}[r] & {\mathsf{R}}_{a_1\cdots a_6{\mathsf{y}},{\mathsf{y}}} \ar@/^/[rd] & \\
{\mathcal T}(R_{i_1\cdots i_8,i}) \ar[r] \ar@/_/[rd] & {\mathcal T}(R_{a_1\cdots a_6{y}{z},\alpha}) \ar@{=}[r] & \epsilon^{{\mathpalette{\raisebox{\depth}{$\m@th\intercal$}}}_{\alpha\beta}\,{\mathsf{R}}^\beta_{a_1\cdots a_6} \ar@/_/[ruu] & {\mathsf{R}}_{{\mathsf{m}}_1\cdots{\mathsf{m}}_7,{\mathsf{m}}} \\
& {\mathcal T}(R_{a_1\cdots a_6{y}{z},a}) \ar@{=}[r] & {\mathsf{R}}_{a_1\cdots a_6{\mathsf{y}},a} \ar@/_/[ru]&
}
\end{gathered}\end{aligned}$$ A similar map for the positive-level generators can be found by taking the generalized transpose; for example, $$\begin{aligned}
{\mathcal T}(R^{a_1a_2a_3}) = \bigl[{\mathcal T}(R_{a_1a_2a_3})\bigr]^{\mathbb{T}}= \bigl[{\mathsf{R}}_{a_1a_2a_3{\mathsf{y}}}\bigr]^{\mathbb{T}}= {\mathsf{R}}^{a_1a_2a_3{\mathsf{y}}}\,. \end{aligned}$$ We then obtain the $E_n$ generators ($n\leq 8$) in the type IIB parameterization, $$\begin{aligned}
\{{\mathsf{T}}_{\bm{\alpha}}\} = \{{\mathsf{K}}^{\mathsf{m}}{}_{\mathsf{n}},\, {\mathsf{R}}_{\alpha\beta},\,{\mathsf{R}}_\alpha^{{\mathsf{m}}_{12}},\,{\mathsf{R}}^{{\mathsf{m}}_{1\cdots 4}},\,{\mathsf{R}}^{{\mathsf{m}}_{1\cdots6}}_\alpha,\,{\mathsf{R}}^{{\mathsf{m}}_{1\cdots 7},{\mathsf{m}}}\,{\mathsf{R}}^\alpha_{{\mathsf{m}}_{12}},\,{\mathsf{R}}_{{\mathsf{m}}_{1\cdots 4}},\,{\mathsf{R}}_{{\mathsf{m}}_{1\cdots6}}^\alpha,\,{\mathsf{R}}_{{\mathsf{m}}_{1\cdots 7},{\mathsf{m}}}\}\,. \end{aligned}$$ By using the notations, $$\begin{aligned}
{{\boldsymbol\delta}}_{{\mathsf{m}}_1\cdots {\mathsf{m}}_n}^{{\mathsf{n}}_1\cdots {\mathsf{n}}_n}\equiv n!\,\delta_{{\mathsf{m}}_1\cdots {\mathsf{m}}_n}^{{\mathsf{n}}_1\cdots {\mathsf{n}}_n}\,,\qquad
(\cdots)_{{\mathsf{m}}_{1\cdots p}} \equiv (\cdots)_{{\mathsf{m}}_1\cdots {\mathsf{m}}_p}\,,\qquad
\delta^{\alpha\beta}_{\gamma\delta}\equiv \delta^{(\alpha}_{(\gamma}\delta^{\beta)}_{\delta)}\,,\end{aligned}$$ their matrix representations are found as follows: $$\begin{aligned}
{\mathsf{K}}^{{\mathsf{r}}}{}_{{\mathsf{s}}} &\equiv {\arraycolsep=0.2mm \underset{10\times10}{\mathrm{diag}}
\begin{pmatrix}
-\delta_{{\mathsf{s}}}^{\mathsf{m}}\delta_{\mathsf{n}}^{{\mathsf{r}}} \\[-1mm]
\delta^\alpha_\beta \delta_{{\mathsf{m}}}^{{\mathsf{r}}} \delta_{{\mathsf{s}}}^{{\mathsf{n}}} \\
\frac{\frac{1}{2!}{{\boldsymbol\delta}}_{{\mathsf{m}}_{123}}^{{\mathsf{r}}{\mathsf{t}}_{12}} {{\boldsymbol\delta}}_{{\mathsf{s}}{\mathsf{t}}_{12}}^{{\mathsf{n}}_{123}}}{\sqrt{3!\,3!}}\\[1mm]
\frac{\frac{1}{4!}\delta^\alpha_\beta {{\boldsymbol\delta}}_{{\mathsf{m}}_{1\cdots 5}}^{{\mathsf{r}}{\mathsf{t}}_{1\cdots 4}} {{\boldsymbol\delta}}_{{\mathsf{s}}{\mathsf{t}}_{1\cdots 4}}^{{\mathsf{n}}_{1\cdots 5}}}{\sqrt{5!\,5!}} \\[1mm]
\frac{\frac{1}{5!}{{\boldsymbol\delta}}_{{\mathsf{m}}_{1\cdots 6}}^{{\mathsf{r}}{\mathsf{t}}_{1\cdots 5}} {{\boldsymbol\delta}}_{{\mathsf{s}}{\mathsf{t}}_{1\cdots 5}}^{{\mathsf{n}}_{1\cdots 6}}\delta_{\mathsf{m}}^{\mathsf{n}}+{{\boldsymbol\delta}}_{{\mathsf{m}}_{1\cdots 6}}^{{\mathsf{n}}_{1\cdots 6}} \delta_{{\mathsf{m}}}^{{\mathsf{r}}}\delta_{{\mathsf{s}}}^{\mathsf{n}}}{\sqrt{6!\,6!}} \\[1mm]
\frac{\frac{1}{6!}\delta^{\alpha_{12}}_{\beta_{12}}{{\boldsymbol\delta}}_{{\mathsf{m}}_{1\cdots 7}}^{{\mathsf{r}}{\mathsf{t}}_{1\cdots 6}} {{\boldsymbol\delta}}_{{\mathsf{s}}{\mathsf{t}}_{1\cdots 6}}^{{\mathsf{n}}_{1\cdots 7}}}{\sqrt{7!\,7!}} \\[1mm]
\frac{\delta^\alpha_\beta(\frac{1}{6!}{{\boldsymbol\delta}}_{{\mathsf{m}}_{1\cdots 7}}^{{\mathsf{r}}{\mathsf{t}}_{1\cdots 6}} {{\boldsymbol\delta}}_{{\mathsf{s}}{\mathsf{t}}_{1\cdots 6}}^{{\mathsf{n}}_{1\cdots 7}}{{\boldsymbol\delta}}_{{\mathsf{p}}_{12}}^{{\mathsf{q}}_{12}} + {{\boldsymbol\delta}}_{{\mathsf{m}}_{1\cdots 7}}^{{\mathsf{n}}_{1\cdots 7}} {{\boldsymbol\delta}}_{{\mathsf{p}}_{12}}^{{\mathsf{r}}{\mathsf{t}}}{{\boldsymbol\delta}}_{{\mathsf{s}}{\mathsf{t}}}^{{\mathsf{q}}_{12}})}{\sqrt{7!\,2!\,7!\,2!}} \\[1mm]
\frac{\frac{1}{6!}{{\boldsymbol\delta}}_{{\mathsf{m}}_{1\cdots 7}}^{{\mathsf{r}}{\mathsf{t}}_{1\cdots 6}} {{\boldsymbol\delta}}_{{\mathsf{s}}{\mathsf{t}}_{1\cdots 6}}^{{\mathsf{n}}_{1\cdots 7}}{{\boldsymbol\delta}}_{{\mathsf{q}}_{1\cdots 4}}^{{\mathsf{p}}_{1\cdots 4}} +\frac{1}{3!}{{\boldsymbol\delta}}_{{\mathsf{m}}_{1\cdots 7}}^{{\mathsf{n}}_{1\cdots 7}} {{\boldsymbol\delta}}_{{\mathsf{p}}_{1\cdots 4}}^{{\mathsf{r}}{\mathsf{t}}_{123}}{{\boldsymbol\delta}}_{{\mathsf{s}}{\mathsf{t}}_{123}}^{{\mathsf{q}}_{1\cdots 4}}}{\sqrt{7!\,4!\,7!\,4!}} \\[1mm]
\frac{\delta^\alpha_\beta (\frac{1}{6!}{{\boldsymbol\delta}}_{{\mathsf{m}}_{1\cdots 7}}^{{\mathsf{r}}{\mathsf{t}}_{1\cdots 6}} {{\boldsymbol\delta}}_{{\mathsf{s}}{\mathsf{t}}_{1\cdots 6}}^{{\mathsf{n}}_{1\cdots 7}}{{\boldsymbol\delta}}_{{\mathsf{p}}_{1\cdots 6}}^{{\mathsf{q}}_{1\cdots 6}} + \frac{1}{5!}{{\boldsymbol\delta}}_{{\mathsf{m}}_{1\cdots 7}}^{{\mathsf{n}}_{1\cdots 7}} {{\boldsymbol\delta}}_{{\mathsf{p}}_{1\cdots 6}}^{{\mathsf{r}}{\mathsf{t}}_{1\cdots 5}}{{\boldsymbol\delta}}_{{\mathsf{s}}{\mathsf{t}}_{1\cdots 5}}^{{\mathsf{q}}_{1\cdots 6}})}{\sqrt{7!\,6!\,7!\,6!}} \\[1mm]
\frac{\frac{1}{6!}{{\boldsymbol\delta}}_{{\mathsf{m}}_{1\cdots 7}}^{{\mathsf{r}}{\mathsf{t}}_{1\cdots 6}} {{\boldsymbol\delta}}_{{\mathsf{s}}{\mathsf{t}}_{1\cdots 6}}^{{\mathsf{n}}_{1\cdots 7}}{{\boldsymbol\delta}}_{{\mathsf{p}}_{1\cdots 7}}^{{\mathsf{q}}_{1\cdots 7}}\delta_{\mathsf{m}}^{\mathsf{n}}+ \frac{1}{6!}{{\boldsymbol\delta}}_{{\mathsf{m}}_{1\cdots 7}}^{{\mathsf{n}}_{1\cdots 7}} {{\boldsymbol\delta}}_{{\mathsf{p}}_{1\cdots 7}}^{{\mathsf{r}}{\mathsf{t}}_{1\cdots 6}}{{\boldsymbol\delta}}_{{\mathsf{s}}{\mathsf{t}}_{1\cdots 6}}^{{\mathsf{q}}_{1\cdots 7}}\delta_{\mathsf{m}}^{\mathsf{n}}+ {{\boldsymbol\delta}}_{{\mathsf{m}}_{1\cdots 7}}^{{\mathsf{n}}_{1\cdots 7}} {{\boldsymbol\delta}}_{{\mathsf{p}}_{1\cdots 7}}^{{\mathsf{q}}_{1\cdots 7}} \delta^{{\mathsf{r}}}_{\mathsf{m}}\delta_{{\mathsf{s}}}^{\mathsf{n}}}{\sqrt{7!\,7!\,7!\,7!}}
\end{pmatrix} - \frac{\delta_{{\mathsf{s}}}^{{\mathsf{r}}}\,\delta^{\mathsf{I}}_{\mathsf{J}}}{9-n}} \,,
\\
{\mathsf{R}}_{\gamma\delta} &\equiv {\footnotesize {\arraycolsep=0.5mm
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & \delta^\alpha_{(\gamma} \epsilon_{\delta)\beta}\delta_{\mathsf{m}}^{\mathsf{n}}& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & \frac{\delta^\alpha_{(\gamma} \epsilon_{\delta)\beta}{{\boldsymbol\delta}}^{{\mathsf{n}}_{1\cdots 5}}_{{\mathsf{m}}_{1\cdots 5}}}{\sqrt{5!\,5!}} & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & \frac{({\mathsf{R}}_{\gamma\delta})^{\alpha_{12}}_{\beta_{12}}\,{{\boldsymbol\delta}}^{{\mathsf{n}}_{1\cdots 7}}_{{\mathsf{m}}_{1\cdots 7}}}{\sqrt{7!\,7!}} & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & \frac{\delta^\alpha_{(\gamma} \epsilon_{\delta)\beta}{{\boldsymbol\delta}}^{{\mathsf{n}}_{1\cdots 7}}_{{\mathsf{m}}_{1\cdots 7}}\,{{\boldsymbol\delta}}^{{\mathsf{q}}_{12}}_{{\mathsf{p}}_{12}}}{\sqrt{7!\,2!\,7!\,2!}} & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{\delta^\alpha_{(\gamma} \epsilon_{\delta)\beta}{{\boldsymbol\delta}}^{{\mathsf{n}}_{1\cdots 7}}_{{\mathsf{m}}_{1\cdots 7}}\,{{\boldsymbol\delta}}^{{\mathsf{q}}_{1\cdots 6}}_{{\mathsf{p}}_{1\cdots 6}}}{\sqrt{7!\,6!\,7!\,6!}} & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix}}}
{\nonumber}\\
&\quad \Bigl[({\mathsf{R}}_{\gamma\delta})^{\alpha_{12}}_{\beta_{12}} \equiv
\delta^{\alpha_1\alpha_2}_{\beta_1(\gamma}\,\epsilon_{\delta)\beta_2}
+ \delta^{\alpha_1\alpha_2}_{\beta_2(\gamma}\,\epsilon_{\delta)\beta_1}\Bigr] \,,
\\
{\mathsf{R}}_\gamma^{{\mathsf{r}}_{12}}
&\equiv {\footnotesize{\arraycolsep=0.2mm
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
\delta_\gamma^\alpha{{\boldsymbol\delta}}_{{\mathsf{m}}{\mathsf{n}}}^{{\mathsf{r}}_{12}}\!\!\!\! & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & \!\!\!\!\frac{\epsilon_{\beta\gamma}{{\boldsymbol\delta}}_{{\mathsf{m}}_{123}}^{{\mathsf{n}}{\mathsf{r}}_{12}}}{\sqrt{3!}}\!\!\!\! & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & \!\!\!\!\frac{\delta_\gamma^\alpha{{\boldsymbol\delta}}_{{\mathsf{m}}_{1\cdots 5}}^{{\mathsf{n}}_{123}{\mathsf{r}}_{12}}}{\sqrt{5!\,3!}}\!\!\!\! & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & \!\!\!\!\!\!\frac{\epsilon_{\beta\gamma} \Bigl[\genfrac{}{}{0pt}{1}{c_2\,{{\boldsymbol\delta}}^{{\mathsf{n}}_{1\cdots 5}{\mathsf{r}}_{12}}_{{\mathsf{m}}_{1\cdots 6}{\mathsf{m}}}}{+{{\boldsymbol\delta}}_{{\mathsf{m}}_{1\cdots 6}}^{{\mathsf{n}}_{1\cdots 5}{\mathsf{t}}} {{\boldsymbol\delta}}_{{\mathsf{m}}{\mathsf{t}}}^{{\mathsf{r}}_{12}}}\Bigr]}{\sqrt{6!\,5!}}\!\!\!\!\!\!\!\!\!\! & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & \!\!\!\!\frac{\delta^{\alpha_{12}}_{\beta\gamma}{{\boldsymbol\delta}}^{{\mathsf{n}}_{1\cdots 5}{\mathsf{r}}_{12}}_{{\mathsf{m}}_{1\cdots 7}}}{\sqrt{7!\,5!}}\!\!\!\! & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & \!\!\!\!\!\!\!\!\!\!\!\!\frac{\delta_\gamma^\alpha \Bigl[\genfrac{}{}{0pt}{1}{c_2\,{{\boldsymbol\delta}}_{{\mathsf{m}}_{1\cdots 7}}^{{\mathsf{n}}_{1\cdots 6}{\mathsf{n}}} {{\boldsymbol\delta}}_{{\mathsf{p}}_{12}}^{{\mathsf{r}}_{12}}}{+{{\boldsymbol\delta}}_{{\mathsf{m}}_{1\cdots 7}}^{{\mathsf{n}}_{1\cdots 6}{\mathsf{s}}} {{\boldsymbol\delta}}_{{\mathsf{s}}{\mathsf{t}}}^{{\mathsf{r}}_{12}} {{\boldsymbol\delta}}_{{\mathsf{p}}_{12}}^{{\mathsf{t}}{\mathsf{n}}}}\Bigr]}{\sqrt{7!\,2!\,6!}} & \frac{\delta_{(\beta_1}^\alpha \epsilon_{\beta_2)\gamma} {{\boldsymbol\delta}}_{{\mathsf{m}}_{1\cdots 7}}^{{\mathsf{n}}_{1\cdots 7}}{{\boldsymbol\delta}}_{{\mathsf{p}}_{12}}^{{\mathsf{r}}_{12}}}{\sqrt{7!\,2!\,7!}}\!\!\!\! & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \frac{\epsilon_{\beta\gamma}{{\boldsymbol\delta}}^{{\mathsf{n}}_{1\cdots 7}}_{{\mathsf{m}}_{1\cdots 7}}{{\boldsymbol\delta}}^{{\mathsf{q}}_{12}{\mathsf{r}}_{12}}_{{\mathsf{p}}_{1\cdots 4}}}{\sqrt{7!\,4!\,7!\,2!}}\!\!\!\!\!\!\!\! & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & \!\!\!\!\frac{\delta_\gamma^\alpha{{\boldsymbol\delta}}^{{\mathsf{n}}_{1\cdots 7}}_{{\mathsf{m}}_{1\cdots 7}}{{\boldsymbol\delta}}^{{\mathsf{p}}_1\cdots{\mathsf{q}}_4{\mathsf{r}}_{12}}_{{\mathsf{p}}_{1\cdots 6}}}{\sqrt{7!\,6!\,7!\,4!}}\!\!\!\!\!\!\!\!\!\!\!\! & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \!\!\!\!\!\!\!\!\!\frac{\epsilon_{\beta\gamma}{{\boldsymbol\delta}}^{{\mathsf{n}}_{1\cdots 7}}_{{\mathsf{m}}_{1\cdots 7}}{{\boldsymbol\delta}}^{{\mathsf{q}}_{1\cdots 6}{\mathsf{t}}}_{{\mathsf{p}}_{1\cdots 7}}{{\boldsymbol\delta}}^{{\mathsf{r}}_{12}}_{{\mathsf{m}}{\mathsf{t}}}}{\sqrt{7!\,7!\,7!\,6!}} & 0
\end{pmatrix}}}
{\nonumber}\\
&\quad \biggl[c_2 \equiv \frac{4+\sqrt{2}}{14}\biggr]\,,
\\
{\mathsf{R}}^\gamma_{{\mathsf{r}}_{12}}
&\equiv {\footnotesize{\arraycolsep=0.2mm
\begin{pmatrix}
0 & \,\delta^\gamma_\beta{{\boldsymbol\delta}}^{{\mathsf{n}}{\mathsf{m}}}_{{\mathsf{r}}_{12}}\!\! & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & \!\!\!\!\frac{\epsilon^{\alpha\gamma}{{\boldsymbol\delta}}^{{\mathsf{n}}_{123}}_{{\mathsf{m}}{\mathsf{r}}_{12}}}{\sqrt{3!}}\!\!\!\! & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & \!\!\frac{\delta^\gamma_\beta{{\boldsymbol\delta}}^{{\mathsf{n}}_{1\cdots 5}}_{{\mathsf{m}}_{123}{\mathsf{r}}_{12}}}{\sqrt{3!\,5!}}\!\!\!\!\!\!& 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & \!\!\!\!\frac{\epsilon^{\alpha\gamma} \Bigl[\genfrac{}{}{0pt}{1}{c_2\,{{\boldsymbol\delta}}_{{\mathsf{m}}_{1\cdots 5}{\mathsf{r}}_{12}}^{{\mathsf{n}}_{1\cdots 6}{\mathsf{n}}}}{+{{\boldsymbol\delta}}^{{\mathsf{n}}_{1\cdots 6}}_{{\mathsf{m}}_{1\cdots 5}{\mathsf{t}}} {{\boldsymbol\delta}}^{{\mathsf{n}}{\mathsf{t}}}_{{\mathsf{r}}_{12}}}\Bigr]}{\sqrt{5!\,6!}} & \frac{\delta_{\beta_{12}}^{\alpha\gamma}{{\boldsymbol\delta}}_{{\mathsf{m}}_{1\cdots 5}{\mathsf{r}}_{12}}^{{\mathsf{n}}_{1\cdots 7}}}{\sqrt{5!\,7!}}\!\!\!\!\!\!\!\!\! & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & \!\!\!\!\!\!\!\! \frac{\delta^\gamma_\beta \Bigl[\genfrac{}{}{0pt}{1}{c_2\,{{\boldsymbol\delta}}^{{\mathsf{n}}_{1\cdots 7}}_{{\mathsf{m}}_{1\cdots 6}{\mathsf{m}}} {{\boldsymbol\delta}}^{{\mathsf{q}}_{12}}_{{\mathsf{r}}_{12}}}{+{{\boldsymbol\delta}}^{{\mathsf{n}}_{1\cdots 7}}_{{\mathsf{m}}_{1\cdots 6}{\mathsf{s}}} {{\boldsymbol\delta}}^{{\mathsf{s}}{\mathsf{t}}}_{{\mathsf{r}}_{12}} {{\boldsymbol\delta}}^{{\mathsf{q}}_{12}}_{{\mathsf{t}}{\mathsf{m}}}}\Bigr]}{\sqrt{6!\,7!\,2!}}\!\!\!\!\!\!\!\! & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & \!\!\!\frac{\delta^{(\alpha_1}_\beta \epsilon^{\alpha_2)\gamma} {{\boldsymbol\delta}}^{{\mathsf{n}}_{1\cdots 7}}_{{\mathsf{m}}_{1\cdots 7}}{{\boldsymbol\delta}}^{{\mathsf{q}}_{12}}_{{\mathsf{r}}_{12}}}{\sqrt{7!\,7!\,2!}}\!\!\!\!\!\! & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & \!\!\!\!\!\!\!\!\!\frac{\epsilon^{\alpha\gamma}{{\boldsymbol\delta}}_{{\mathsf{m}}_{1\cdots 7}}^{{\mathsf{n}}_{1\cdots 7}}{{\boldsymbol\delta}}_{{\mathsf{p}}_{12}{\mathsf{r}}_{12}}^{{\mathsf{q}}_{1\cdots 4}}}{\sqrt{7!\,2!\,7!\,4!}}\!\!\!\! & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \!\!\!\!\!\!\!\!\!\frac{\delta^\gamma_\beta{{\boldsymbol\delta}}_{{\mathsf{m}}_{1\cdots 7}}^{{\mathsf{n}}_{1\cdots 7}}{{\boldsymbol\delta}}_{{\mathsf{p}}_1\cdots{\mathsf{p}}_4{\mathsf{r}}_{12}}^{{\mathsf{q}}_{1\cdots 6}}}{\sqrt{7!\,4!\,7!\,6!}}\!\!\!\!\!\!\!\!\! & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \!\!\!\!\!\!\!\!\!\!\!\frac{\epsilon^{\alpha\gamma}{{\boldsymbol\delta}}_{{\mathsf{m}}_{1\cdots 7}}^{{\mathsf{n}}_{1\cdots 7}}{{\boldsymbol\delta}}_{{\mathsf{p}}_{1\cdots 6}{\mathsf{t}}}^{{\mathsf{q}}_{1\cdots 7}}{{\boldsymbol\delta}}_{{\mathsf{r}}_{12}}^{{\mathsf{n}}{\mathsf{t}}}}{\sqrt{7!\,6!\,7!\,7!}}\!\! \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix}}}
{\nonumber}\\
&= ({\mathsf{R}}_\gamma^{{\mathsf{r}}_{12}})^{\mathbb{T}}\,,
\\
{\mathsf{R}}^{{\mathsf{r}}_{1\cdots 4}} &\equiv {\footnotesize {\arraycolsep=0.2mm
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
\frac{{{\boldsymbol\delta}}^{{\mathsf{r}}_{1\cdots 4}}_{{\mathsf{m}}_{123}{\mathsf{n}}}}{\sqrt{3!}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & \!\!\!\!\frac{-\delta_\beta^\alpha {{\boldsymbol\delta}}^{{\mathsf{n}}{\mathsf{r}}_{1\cdots 4}}_{{\mathsf{m}}_{1\cdots 5}}}{\sqrt{5!}}\!\!\!\! & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & \!\!\!\!\frac{\Bigl[\genfrac{}{}{0pt}{1}{\frac{1}{3!}{{\boldsymbol\delta}}^{{\mathsf{n}}_{123}{\mathsf{t}}_{123}}_{{\mathsf{m}}_{1\cdots6}}{{\boldsymbol\delta}}^{{\mathsf{r}}_{1\cdots4}}_{{\mathsf{m}}{\mathsf{t}}_{123}}}{+c_4\,{{\boldsymbol\delta}}^{{\mathsf{n}}_{123}{\mathsf{r}}_{1\cdots 4}}_{{\mathsf{m}}_{1\cdots6}{\mathsf{m}}}}\Bigr]}{\sqrt{6!\,3!}}\!\!\!\! & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & \!\!\!\!\!\!\!\!\!\!\!\!\frac{\delta_\beta^\alpha{{\boldsymbol\delta}}^{{\mathsf{n}}_{1\cdots 5}{\mathsf{t}}_{12}}_{{\mathsf{m}}_{1\cdots 7}}{{\boldsymbol\delta}}^{{\mathsf{r}}_{1\cdots 4}}_{{\mathsf{t}}_{12}{\mathsf{p}}_{12}}}{2!\sqrt{7!\,2!\,5!}}\!\!\!\!\!\!\!\!\!\!\!\! & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & \!\frac{\Bigl[\genfrac{}{}{0pt}{1}{\frac{1}{3!}{{\boldsymbol\delta}}^{{\mathsf{n}}_{1\cdots 6}{\mathsf{s}}}_{{\mathsf{m}}_{1\cdots 7}}{{\boldsymbol\delta}}^{{\mathsf{t}}_{123}{\mathsf{n}}}_{{\mathsf{p}}_{1\cdots 4}}{{\boldsymbol\delta}}^{{\mathsf{r}}_{1\cdots 4}}_{{\mathsf{s}}{\mathsf{t}}_{123}}}{+c_4\,{{\boldsymbol\delta}}^{{\mathsf{n}}_{1\cdots 6}{\mathsf{n}}}_{{\mathsf{m}}_{1\cdots 7}}{{\boldsymbol\delta}}^{{\mathsf{r}}_{1\cdots 4}}_{{\mathsf{p}}_{1\cdots 4}}}\Bigr]}{\sqrt{7!\,4!\,6!}} & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & \frac{-\delta_\beta^\alpha{{\boldsymbol\delta}}_{{\mathsf{m}}_{1\cdots 7}}^{{\mathsf{n}}_{1\cdots 7}}{{\boldsymbol\delta}}_{{\mathsf{p}}_{1\cdots 6}}^{{\mathsf{q}}_{12}{\mathsf{r}}_{1\cdots 4}}}{\sqrt{7!\,6!\,7!\,2!}}\!\!\!\!\!\!\!\! & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & \!\!\!\!\!\!\!\!\frac{{{\boldsymbol\delta}}_{{\mathsf{m}}_{1\cdots 7}}^{{\mathsf{n}}_{1\cdots 7}}{{\boldsymbol\delta}}_{{\mathsf{p}}{\mathsf{t}}_{123}}^{{\mathsf{q}}_{1\cdots 4}}{{\boldsymbol\delta}}_{{\mathsf{p}}_{1\cdots 7}}^{{\mathsf{t}}_{123}{\mathsf{r}}_{1\cdots 4}}}{3!\sqrt{7!\,7!\,7!\,4!}} & 0 & 0
\end{pmatrix}}}
{\nonumber}\\
&\quad \biggl[c_4\equiv \frac{4+\sqrt{2}}{7}\biggr]\,,
\\
{\mathsf{R}}_{{\mathsf{r}}_{1\cdots 4}} &\equiv {\footnotesize {\arraycolsep=0.2mm
\begin{pmatrix}
0 & 0 & \frac{{{\boldsymbol\delta}}_{{\mathsf{r}}_{1\cdots 4}}^{{\mathsf{n}}_{123}{\mathsf{m}}}}{\sqrt{3!}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & \!\!\!\frac{-\delta^\alpha_\beta {{\boldsymbol\delta}}_{{\mathsf{m}}{\mathsf{r}}_{1\cdots 4}}^{{\mathsf{n}}_{1\cdots 5}}}{\sqrt{5!}}\!\!\!\!\!\!\!\!\!\! & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & \frac{\Bigl[\genfrac{}{}{0pt}{1}{\frac{1}{3!}{{\boldsymbol\delta}}_{{\mathsf{m}}_{123}{\mathsf{t}}_{123}}^{{\mathsf{n}}_{1\cdots6}}{{\boldsymbol\delta}}_{{\mathsf{r}}_{1\cdots4}}^{{\mathsf{n}}{\mathsf{t}}_{123}}}{+c_4\,{{\boldsymbol\delta}}_{{\mathsf{m}}_{123}{\mathsf{r}}_{1\cdots 4}}^{{\mathsf{n}}_{1\cdots6}{\mathsf{n}}}}\Bigr]}{\sqrt{3!\,6!}} & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & \frac{\delta^\alpha_\beta{{\boldsymbol\delta}}_{{\mathsf{m}}_{1\cdots 5}{\mathsf{t}}_{12}}^{{\mathsf{n}}_{1\cdots 7}}{{\boldsymbol\delta}}_{{\mathsf{r}}_{1\cdots 4}}^{{\mathsf{t}}_{12}{\mathsf{q}}_{12}}}{2!\sqrt{5!\,7!\,2!}}\!\!\!\!\!\!\!\! & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & \!\!\!\!\!\!\!\!\!\frac{\Bigl[\genfrac{}{}{0pt}{1}{\frac{1}{3!}{{\boldsymbol\delta}}_{{\mathsf{m}}_{1\cdots 6}{\mathsf{s}}}^{{\mathsf{n}}_{1\cdots 7}}{{\boldsymbol\delta}}_{{\mathsf{t}}_{123}{\mathsf{m}}}^{{\mathsf{q}}_{1\cdots 4}}{{\boldsymbol\delta}}_{{\mathsf{r}}_{1\cdots 4}}^{{\mathsf{s}}{\mathsf{t}}_{123}}}{+c_4\,{{\boldsymbol\delta}}_{{\mathsf{m}}_{1\cdots 6}{\mathsf{m}}}^{{\mathsf{n}}_{1\cdots 7}}{{\boldsymbol\delta}}_{{\mathsf{r}}_{1\cdots 4}}^{{\mathsf{q}}_{1\cdots 4}}}\Bigr]}{\sqrt{6!\,7!\,4!}}\!\!\!\!\!\!\!\!\! & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \!\!\!\!\!\!\!\!\!\frac{-\delta^\alpha_\beta{{\boldsymbol\delta}}^{{\mathsf{n}}_{1\cdots 7}}_{{\mathsf{m}}_{1\cdots 7}}{{\boldsymbol\delta}}^{{\mathsf{q}}_{1\cdots 6}}_{{\mathsf{p}}_{12}{\mathsf{r}}_{1\cdots 4}}}{\sqrt{7!\,2!\,7!\,6!}}\!\!\!\!\!\!\! & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \!\!\!\!\!\frac{{{\boldsymbol\delta}}^{{\mathsf{n}}_{1\cdots 7}}_{{\mathsf{m}}_{1\cdots 7}}{{\boldsymbol\delta}}^{{\mathsf{q}}{\mathsf{t}}_{123}}_{{\mathsf{p}}_{1\cdots 4}}{{\boldsymbol\delta}}^{{\mathsf{q}}_{1\cdots 7}}_{{\mathsf{t}}_{123}{\mathsf{r}}_{1\cdots 4}}}{3!\sqrt{7!\,4!\,7!\,7!}} \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix}}}
{\nonumber}\\
&= ({\mathsf{R}}^{{\mathsf{r}}_{1\cdots 4}})^{\mathbb{T}}\,,
\\
{\mathsf{R}}_\gamma^{{\mathsf{r}}_{1\cdots 6}} &\equiv {\footnotesize {\arraycolsep=0.2mm
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
\frac{\delta_\gamma^\alpha{{\boldsymbol\delta}}^{{\mathsf{r}}_{1\cdots6}}_{{\mathsf{m}}_{1\cdots 5}{\mathsf{n}}}}{\sqrt{5!}}\!\!\!\! & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & \!\!\!\!\!\!\!\!\frac{\epsilon_{\beta\gamma}\Bigl[\genfrac{}{}{0pt}{1}{c_6\,{{\boldsymbol\delta}}^{{\mathsf{n}}{\mathsf{r}}_{1\cdots 6}}_{{\mathsf{m}}_{1\cdots 6}{\mathsf{m}}}}{-{{\boldsymbol\delta}}^{{\mathsf{r}}_{1\cdots 6}}_{{\mathsf{m}}_{1\cdots 6}}\delta_{\mathsf{m}}^{\mathsf{n}}}\Bigr]}{\sqrt{6!}}\!\!\!\!\!\!\!\! & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & \!\!\!\!\frac{-\delta^{\alpha_{12}}_{\beta\gamma}{{\boldsymbol\delta}}^{{\mathsf{n}}{\mathsf{r}}_{1\cdots 6}}_{{\mathsf{m}}_{1\cdots 7}}}{\sqrt{7!}}\!\!\!\! & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & \!\!\!\!\!\!\!\!\frac{-\delta_\gamma^\alpha{{\boldsymbol\delta}}^{{\mathsf{r}}_{1\cdots 6}{\mathsf{t}}}_{{\mathsf{m}}_{1\cdots 7}}{{\boldsymbol\delta}}^{{\mathsf{n}}_{123}}_{{\mathsf{p}}_{12}{\mathsf{t}}}}{\sqrt{7!\,2!\,3!}}\!\!\!\!\!\!\! & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & \!\!\!\!\!\!\!\!\frac{\epsilon_{\beta\gamma}{{\boldsymbol\delta}}^{{\mathsf{r}}_{1\cdots 6}{\mathsf{t}}}_{{\mathsf{m}}_{1\cdots 7}}{{\boldsymbol\delta}}^{{\mathsf{n}}_{1\cdots 5}}_{{\mathsf{p}}_{1\cdots 4}{\mathsf{t}}}}{\sqrt{7!\,4!\,5!}}\!\!\!\!\!\!\!\!\!\!\!\! & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & \!\!\!\!\!\!\!\!\frac{-\delta_\gamma^\alpha\Bigl[\genfrac{}{}{0pt}{1}{c_6\,{{\boldsymbol\delta}}^{{\mathsf{n}}_{1\cdots 6}{\mathsf{n}}}_{{\mathsf{m}}_{1\cdots 7}}{{\boldsymbol\delta}}^{{\mathsf{r}}_{1\cdots 6}}_{{\mathsf{p}}_{1\cdots 6}}}{-{{\boldsymbol\delta}}^{{\mathsf{n}}_{1\cdots 6}}_{{\mathsf{p}}_{1\cdots 6}}{{\boldsymbol\delta}}^{{\mathsf{n}}{\mathsf{r}}_{1\cdots 6}}_{{\mathsf{m}}_{1\cdots 7}}}\Bigr]}{\sqrt{7!\,6!\,6!}} & \frac{\delta^\alpha_{(\beta_1} \epsilon_{\beta_2)\gamma} {{\boldsymbol\delta}}^{{\mathsf{n}}_{1\cdots 7}}_{{\mathsf{m}}_{1\cdots 7}}{{\boldsymbol\delta}}^{{\mathsf{r}}_{1\cdots 6}}_{{\mathsf{p}}_{1\cdots 6}}}{\sqrt{7!\,7!\,6!}}\!\!\!\!\!\!\!\!\!\!\!\! & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\frac{\epsilon_{\beta\gamma}{{\boldsymbol\delta}}^{{\mathsf{n}}_{1\cdots 7}}_{{\mathsf{m}}_{1\cdots 7}}{{\boldsymbol\delta}}_{{\mathsf{p}}_{1\cdots 7}}^{{\mathsf{r}}_{1\cdots 6}{\mathsf{t}}}{{\boldsymbol\delta}}^{{\mathsf{q}}_{12}}_{{\mathsf{m}}{\mathsf{t}}}}{\sqrt{7!\,7!\,7!\,2!}} & 0 & 0 & 0
\end{pmatrix}}}
{\nonumber}\\
&\quad \Bigl[c_6 \equiv \frac{2-3\sqrt{2}}{14}\Bigr] \,,
\\
{\mathsf{R}}^\gamma_{{\mathsf{r}}_{1\cdots 6}} &\equiv {\footnotesize {\arraycolsep=0.2mm
\begin{pmatrix}
0 & 0 & 0 & \frac{\delta^\gamma_\beta{{\boldsymbol\delta}}_{{\mathsf{r}}_{1\cdots6}}^{{\mathsf{n}}_{1\cdots 5}{\mathsf{m}}}}{\sqrt{5!}}\!\! & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & \!\!\!\!\!\!\frac{\epsilon^{\alpha\gamma}\Bigl[\genfrac{}{}{0pt}{1}{c_6\,{{\boldsymbol\delta}}_{{\mathsf{m}}{\mathsf{r}}_{1\cdots 6}}^{{\mathsf{n}}_{1\cdots 6}{\mathsf{n}}}}{-{{\boldsymbol\delta}}_{{\mathsf{r}}_{1\cdots 6}}^{{\mathsf{n}}_{1\cdots 6}}\delta^{\mathsf{n}}_{\mathsf{m}}}\Bigr]}{\sqrt{6!}} & \frac{-\delta_{\beta_{12}}^{\alpha\gamma}{{\boldsymbol\delta}}_{{\mathsf{m}}{\mathsf{r}}_{1\cdots 6}}^{{\mathsf{n}}_{1\cdots 7}}}{\sqrt{7!}} & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & \!\!\!\!\!\!\!\!\!\!\!\!\frac{-\delta^\gamma_\beta{{\boldsymbol\delta}}_{{\mathsf{r}}_{1\cdots 6}{\mathsf{t}}}^{{\mathsf{n}}_{1\cdots 7}}{{\boldsymbol\delta}}_{{\mathsf{m}}_{123}}^{{\mathsf{q}}_{12}{\mathsf{t}}}}{\sqrt{3!\,7!\,2!}}\!\!\!\!\!\!\!\!\!\!\!\! & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{\epsilon^{\alpha\gamma}{{\boldsymbol\delta}}_{{\mathsf{r}}_{1\cdots 6}{\mathsf{t}}}^{{\mathsf{n}}_{1\cdots 7}}{{\boldsymbol\delta}}_{{\mathsf{m}}_{1\cdots 5}}^{{\mathsf{q}}_{1\cdots 4}{\mathsf{t}}}}{\sqrt{5!\,7!\,4!}}\!\!\!\!\!\!\!\! & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\frac{-\delta^\gamma_\beta\Bigl[\genfrac{}{}{0pt}{1}{c_6\,{{\boldsymbol\delta}}_{{\mathsf{m}}_{1\cdots 6}{\mathsf{m}}}^{{\mathsf{n}}_{1\cdots 7}}{{\boldsymbol\delta}}_{{\mathsf{r}}_{1\cdots 6}}^{{\mathsf{q}}_{1\cdots 6}}}{-{{\boldsymbol\delta}}_{{\mathsf{m}}_{1\cdots 6}}^{{\mathsf{q}}_{1\cdots 6}}{{\boldsymbol\delta}}_{{\mathsf{m}}{\mathsf{r}}_{1\cdots 6}}^{{\mathsf{n}}_{1\cdots 7}}}\Bigr]}{\sqrt{6!\,7!\,6!}}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \!\!\!\!\!\!\!\!\frac{\delta_\beta^{(\alpha_1} \epsilon^{\alpha_2)\gamma} {{\boldsymbol\delta}}_{{\mathsf{m}}_{1\cdots 7}}^{{\mathsf{n}}_{1\cdots 7}}{{\boldsymbol\delta}}_{{\mathsf{r}}_{1\cdots 6}}^{{\mathsf{q}}_{1\cdots 6}}}{\sqrt{7!\,7!\,6!}}\!\!\!\!\!\!\!\! & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \!\!\!\!\!\!\!\!\!\!\!\!\frac{\epsilon^{\alpha\gamma}{{\boldsymbol\delta}}_{{\mathsf{m}}_{1\cdots 7}}^{{\mathsf{n}}_{1\cdots 7}}{{\boldsymbol\delta}}^{{\mathsf{q}}_{1\cdots 7}}_{{\mathsf{r}}_{1\cdots 6}{\mathsf{t}}}{{\boldsymbol\delta}}_{{\mathsf{p}}_{12}}^{{\mathsf{n}}{\mathsf{t}}}}{\sqrt{7!\,2!\,7!\,7!}} \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix}}}
{\nonumber}\\
&= ({\mathsf{R}}_\gamma^{{\mathsf{r}}_{1\cdots 6}})^{\mathbb{T}}\,,
\\
{\mathsf{R}}^{{\mathsf{r}}_{1\cdots 7},{\mathsf{r}}} &\equiv {\footnotesize {\arraycolsep=0.2mm
\begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
\frac{\Bigl[\genfrac{}{}{0pt}{1}{c_{7,1}\,\delta_{\mathsf{n}}^{\mathsf{r}}{{\boldsymbol\delta}}^{{\mathsf{r}}_{1\cdots 7}}_{{\mathsf{m}}_{1\cdots 6}{\mathsf{m}}}}{-{{\boldsymbol\delta}}^{{\mathsf{r}}_{1\cdots 7}}_{{\mathsf{n}}{\mathsf{m}}_{1\cdots 6}}\delta^{\mathsf{r}}_{\mathsf{m}}}\Bigr]}{\sqrt{6!}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & \frac{\delta_\beta^\alpha {{\boldsymbol\delta}}^{{\mathsf{n}}{\mathsf{r}}}_{{\mathsf{p}}_{12}} {{\boldsymbol\delta}}^{{\mathsf{r}}_{1\cdots 7}}_{{\mathsf{m}}_{1\cdots 7}}}{\sqrt{7!\,2!}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & \frac{{{\boldsymbol\delta}}^{{\mathsf{n}}_{123}{\mathsf{r}}}_{{\mathsf{p}}_{1\cdots 4}} {{\boldsymbol\delta}}^{{\mathsf{r}}_{1\cdots 7}}_{{\mathsf{m}}_{1\cdots 7}}}{\sqrt{7!\,4!\,3!}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & \frac{\delta_\beta^\alpha {{\boldsymbol\delta}}^{{\mathsf{n}}_{1\cdots 5}{\mathsf{r}}}_{{\mathsf{p}}_{1\cdots 6}} {{\boldsymbol\delta}}^{{\mathsf{r}}_{1\cdots 7}}_{{\mathsf{m}}_{1\cdots 7}}}{\sqrt{7!\,6!\,5!}} & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & \!\frac{\Bigl[\genfrac{}{}{0pt}{1}{c_{7,1}\,{{\boldsymbol\delta}}^{{\mathsf{n}}_{1\cdots 6}{\mathsf{n}}}_{{\mathsf{m}}_{1\cdots 7}}{{\boldsymbol\delta}}^{{\mathsf{r}}_{1\cdots 7}}_{{\mathsf{p}}_{1\cdots 7}}\delta_{\mathsf{m}}^{\mathsf{r}}}{-{{\boldsymbol\delta}}^{{\mathsf{n}}_{1\cdots 6}{\mathsf{r}}}_{{\mathsf{m}}_{1\cdots 7}}{{\boldsymbol\delta}}^{{\mathsf{r}}_{1\cdots 7}}_{{\mathsf{p}}_{1\cdots 7}}\delta^{\mathsf{n}}_{\mathsf{m}}}\Bigr]}{\sqrt{7!\,7!\,6!}} & 0 & 0 & 0 & 0 & 0
\end{pmatrix}}}
{\nonumber}\\
&\quad \biggl[c_{7,1}\equiv \frac{4+\sqrt{2}}{7\sqrt{2}}\biggr] \,,
\\
{\mathsf{R}}_{{\mathsf{r}}_{1\cdots 7},{\mathsf{r}}} &\equiv {\footnotesize {\arraycolsep=0.2mm
\begin{pmatrix}
0 & 0 & 0 & 0 & \frac{\Bigl[\genfrac{}{}{0pt}{1}{c_{7,1}\,\delta^{\mathsf{m}}_{\mathsf{r}}{{\boldsymbol\delta}}_{{\mathsf{r}}_{1\cdots 7}}^{{\mathsf{n}}_{1\cdots 6}{\mathsf{n}}}}{-{{\boldsymbol\delta}}_{{\mathsf{r}}_{1\cdots 7}}^{{\mathsf{m}}{\mathsf{n}}_{1\cdots 6}}\delta_{\mathsf{r}}^{\mathsf{n}}}\Bigr]}{\sqrt{6!}} & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & \frac{\delta^\alpha_\beta {{\boldsymbol\delta}}_{{\mathsf{m}}{\mathsf{r}}}^{{\mathsf{q}}_{12}} {{\boldsymbol\delta}}_{{\mathsf{r}}_{1\cdots 7}}^{{\mathsf{n}}_{1\cdots 7}}}{\sqrt{7!\,2!}} & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{{{\boldsymbol\delta}}_{{\mathsf{m}}_{123}{\mathsf{r}}}^{{\mathsf{q}}_{1\cdots 4}} {{\boldsymbol\delta}}_{{\mathsf{r}}_{1\cdots 7}}^{{\mathsf{n}}_{1\cdots 7}}}{\sqrt{3!\,7!\,4!}} & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{\delta^\alpha_\beta {{\boldsymbol\delta}}_{{\mathsf{m}}_{1\cdots 5}{\mathsf{r}}}^{{\mathsf{q}}_{1\cdots 6}} {{\boldsymbol\delta}}_{{\mathsf{r}}_{1\cdots 7}}^{{\mathsf{n}}_{1\cdots 7}}}{\sqrt{5!\,7!\,6!}} & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \frac{\Bigl[\genfrac{}{}{0pt}{1}{c_{7,1}\,{{\boldsymbol\delta}}_{{\mathsf{m}}_{1\cdots 6}{\mathsf{m}}}^{{\mathsf{n}}_{1\cdots 7}}{{\boldsymbol\delta}}_{{\mathsf{r}}_{1\cdots 7}}^{{\mathsf{q}}_{1\cdots 7}}\delta^{\mathsf{n}}_{\mathsf{r}}}{-{{\boldsymbol\delta}}_{{\mathsf{m}}_{1\cdots 6}{\mathsf{r}}}^{{\mathsf{n}}_{1\cdots 7}}{{\boldsymbol\delta}}_{{\mathsf{r}}_{1\cdots 7}}^{{\mathsf{q}}_{1\cdots 7}}\delta_{\mathsf{m}}^{\mathsf{n}}}\Bigr]}{\sqrt{6!\,7!\,7!}} \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix}}}
{\nonumber}\\
&=({\mathsf{R}}^{{\mathsf{r}}_{1\cdots 7},{\mathsf{r}}})^{\mathbb{T}}\,.\end{aligned}$$ Again, we can identify the Cartan generators as $$\begin{aligned}
\{{\mathsf{H}}_{\mathsf{k}}\} = \{{\mathsf{K}}^d{}_d - {\mathsf{K}}^{d+1}{}_{d+1},\, \dotsc,\, {\mathsf{K}}^7{}_7 - {\mathsf{K}}^8{}_8,\, {\mathsf{K}}^8{}_8 + {\mathsf{K}}^9{}_9 -\tfrac{1}{4}\,{\mathsf{D}}-{\mathsf{R}}_{12},\, 2\,{\mathsf{R}}_{12},\, {\mathsf{K}}^8{}_8 - {\mathsf{K}}^9{}_9 \} \,,\end{aligned}$$ and the positive/negative simple-root generators are $$\begin{aligned}
\{{\mathsf{E}}_{\mathsf{k}}\} = \{{\mathsf{K}}^1{}_2 ,\, \dotsc,\, {\mathsf{K}}^5{}_6,\, {\mathsf{R}}_1^{67},\,{\mathsf{R}}_{22},\,{\mathsf{K}}^6{}_7 \} \,,\qquad
\{{\mathsf{F}}_{\mathsf{k}}\} = \{{\mathsf{K}}^2{}_1 ,\, \dotsc,\, {\mathsf{K}}^6{}_5,\, {\mathsf{R}}^1_{67},\,-{\mathsf{R}}^{11},\,{\mathsf{K}}^7{}_6 \} \,. \end{aligned}$$ The set of the positive/negative root generators can be summarized as $$\begin{aligned}
\begin{split}
\{{\mathsf{E}}_{\bm{\alpha}}\} &= \{{\mathsf{K}}^{\mathsf{m}}{}_{\mathsf{n}}\ ({\mathsf{m}}<{\mathsf{n}}),\, {\mathsf{R}}_{22},\,{\mathsf{R}}_\alpha^{{\mathsf{m}}_{12}},\,{\mathsf{R}}^{{\mathsf{m}}_{1\cdots 4}},\,{\mathsf{R}}^{{\mathsf{m}}_{1\cdots6}}_\alpha,\,{\mathsf{R}}^{{\mathsf{m}}_{1\cdots 7},{\mathsf{m}}}\}\,,
\\
\{{\mathsf{F}}_{\bm{\alpha}}\} &= \{{\mathsf{K}}^{\mathsf{m}}{}_{\mathsf{n}}\ ({\mathsf{m}}>{\mathsf{n}}),\, {\mathsf{R}}_{11},\,{\mathsf{R}}^\alpha_{{\mathsf{m}}_{12}},\,{\mathsf{R}}_{{\mathsf{m}}_{1\cdots 4}},\,{\mathsf{R}}_{{\mathsf{m}}_{1\cdots6}}^\alpha,\,{\mathsf{R}}_{{\mathsf{m}}_{1\cdots 7},{\mathsf{m}}}\}\,.
\end{split}\end{aligned}$$
We have checked that the obtained type IIB generators satisfy the following $E_8$ algebra: $$\begin{aligned}
\bigl[{\mathsf{K}}^{\mathsf{m}}{}_{\mathsf{n}},\, {\mathsf{K}}^{\mathsf{p}}{}_{\mathsf{q}}\bigr]
&= \delta_{\mathsf{n}}^{\mathsf{p}}\,{\mathsf{K}}^{\mathsf{m}}{}_{\mathsf{q}}- \delta_{\mathsf{q}}^{\mathsf{m}}\, {\mathsf{K}}^{\mathsf{p}}{}_{\mathsf{n}}\,,
\\
\bigl[{\mathsf{K}}^{\mathsf{m}}{}_{\mathsf{n}},\, {\mathsf{R}}^{{\mathsf{p}}_{12}}_\alpha \bigr]
&= {{\boldsymbol\delta}}_{{\mathsf{n}}{\mathsf{r}}}^{{\mathsf{p}}_{12}} \,{\mathsf{R}}^{{\mathsf{m}}{\mathsf{r}}}_\alpha\,,
\\
\bigl[{\mathsf{K}}^{\mathsf{m}}{}_{\mathsf{n}},\, {\mathsf{R}}^{{\mathsf{p}}_{1\cdots 4}} \bigr]
&= \frac{1}{3!}\,{{\boldsymbol\delta}}_{{\mathsf{n}}{\mathsf{r}}_{123}}^{{\mathsf{p}}_{1\cdots 4}} \, {\mathsf{R}}^{{\mathsf{m}}{\mathsf{r}}_{123}}\,,
\\
\bigl[{\mathsf{K}}^{\mathsf{m}}{}_{\mathsf{n}},\, {\mathsf{R}}^{{\mathsf{p}}_{1\cdots 6}}_\alpha \bigr]
&= \frac{1}{5!}\,{{\boldsymbol\delta}}_{{\mathsf{n}}{\mathsf{r}}_{1\cdots 5}}^{{\mathsf{p}}_{1\cdots 6}} \, {\mathsf{R}}^{{\mathsf{m}}{\mathsf{r}}_{1\cdots 5}}_\alpha\,,
\\
\bigl[{\mathsf{K}}^{\mathsf{m}}{}_{\mathsf{n}},\, {\mathsf{R}}^{{\mathsf{p}}_{1\cdots 7},{\mathsf{p}}} \bigr]
&= \frac{1}{6!}\,{{\boldsymbol\delta}}_{{\mathsf{n}}{\mathsf{r}}_{1\cdots 6}}^{{\mathsf{p}}_{1\cdots 7}} \, {\mathsf{R}}^{{\mathsf{m}}{\mathsf{r}}_{1\cdots 6},{\mathsf{p}}} + \delta_{\mathsf{n}}^{\mathsf{p}}\,{\mathsf{R}}^{{\mathsf{p}}_{1\cdots 7},{\mathsf{m}}}\,,
\\
\bigl[{\mathsf{K}}^{\mathsf{m}}{}_{\mathsf{n}},\, {\mathsf{R}}_{{\mathsf{p}}_{12}}^\alpha \bigr]
&= - {{\boldsymbol\delta}}^{{\mathsf{m}}{\mathsf{r}}}_{{\mathsf{p}}_{12}} \,{\mathsf{R}}_{{\mathsf{n}}{\mathsf{r}}}^\alpha\,,
\\
\bigl[{\mathsf{K}}^{\mathsf{m}}{}_{\mathsf{n}},\, {\mathsf{R}}_{{\mathsf{p}}_{1\cdots 4}} \bigr]
&= - \frac{1}{3!}\,{{\boldsymbol\delta}}^{{\mathsf{m}}{\mathsf{r}}_{123}}_{{\mathsf{p}}_{1\cdots 4}} \, {\mathsf{R}}_{{\mathsf{n}}{\mathsf{r}}_{123}}\,,
\\
\bigl[{\mathsf{K}}^{\mathsf{m}}{}_{\mathsf{n}},\, {\mathsf{R}}_{{\mathsf{p}}_{1\cdots 6}}^\alpha \bigr]
&= - \frac{1}{5!}\,{{\boldsymbol\delta}}^{{\mathsf{m}}{\mathsf{r}}_{1\cdots 5}}_{{\mathsf{p}}_{1\cdots 6}} \, {\mathsf{R}}_{{\mathsf{n}}{\mathsf{r}}_{1\cdots 5}}^\alpha\,,
\\
\bigl[{\mathsf{K}}^{\mathsf{m}}{}_{\mathsf{n}},\, {\mathsf{R}}_{{\mathsf{p}}_{1\cdots 7},{\mathsf{p}}} \bigr]
&= - \frac{1}{6!}\,{{\boldsymbol\delta}}^{{\mathsf{m}}{\mathsf{r}}_{1\cdots 6}}_{{\mathsf{p}}_{1\cdots 7}} \, {\mathsf{R}}_{{\mathsf{n}}{\mathsf{r}}_{1\cdots 6},{\mathsf{p}}} - \delta^{\mathsf{m}}_{\mathsf{p}}\,{\mathsf{R}}_{{\mathsf{p}}_{1\cdots 7},{\mathsf{n}}}\,,
\\
\bigl[{\mathsf{R}}_{\alpha\beta},\, {\mathsf{R}}_{\gamma\delta} \bigr]
&= \delta^\sigma_{(\alpha}\epsilon_{\beta)\gamma}\,{\mathsf{R}}_{\sigma\delta} + \delta^\sigma_{(\alpha}\epsilon_{\beta)\delta}\,{\mathsf{R}}_{\gamma\sigma}\,,
\\
\bigl[{\mathsf{R}}_{\alpha\beta},\, {\mathsf{R}}^{{\mathsf{m}}_{12}}_\gamma \bigr]
&= \delta^\sigma_{(\alpha}\epsilon_{\beta)\gamma}\,{\mathsf{R}}^{{\mathsf{m}}_{12}}_\sigma\,,
\\
\bigl[{\mathsf{R}}_{\alpha\beta},\, {\mathsf{R}}^{{\mathsf{m}}_{1\cdots 6}}_\gamma \bigr]
&= \delta^\sigma_{(\alpha}\epsilon_{\beta)\gamma}\,{\mathsf{R}}^{{\mathsf{m}}_{1\cdots 6}}_\sigma\,,
\\
\bigl[{\mathsf{R}}_{\alpha\beta},\, {\mathsf{R}}_{{\mathsf{m}}_{12}}^\gamma \bigr]
&= -\delta^\gamma_{(\alpha}\epsilon_{\beta)\sigma}\,{\mathsf{R}}_{{\mathsf{m}}_{12}}^\sigma\,,
\\
\bigl[{\mathsf{R}}_{\alpha\beta},\, {\mathsf{R}}_{{\mathsf{m}}_{1\cdots 6}}^\gamma \bigr]
&= -\delta^\gamma_{(\alpha}\epsilon_{\beta)\sigma}\,{\mathsf{R}}_{{\mathsf{m}}_{1\cdots 6}}^\sigma\,,
\\
\bigl[{\mathsf{R}}^{{\mathsf{m}}_{12}}_\alpha,\, {\mathsf{R}}^{{\mathsf{n}}_{12}}_\beta \bigr]
&= -\epsilon_{\alpha\beta}\,{\mathsf{R}}^{{\mathsf{m}}_{12}{\mathsf{n}}_{12}} \,,
\\
\bigl[{\mathsf{R}}^{{\mathsf{m}}_{12}}_\alpha,\, {\mathsf{R}}^{{\mathsf{n}}_{1\cdots 4}} \bigr]
&= {\mathsf{R}}_\alpha^{{\mathsf{m}}_{12}{\mathsf{n}}_{1\cdots 4}} \,,
\\
\bigl[{\mathsf{R}}^{{\mathsf{m}}_{12}}_\alpha,\, {\mathsf{R}}^{{\mathsf{n}}_{1\cdots 6}}_\beta \bigr]
&= -\frac{1}{5!}\,\epsilon_{\alpha\beta}\, {{\boldsymbol\delta}}^{{\mathsf{n}}_{1\cdots 6}}_{{\mathsf{r}}_{1\cdots 5}{\mathsf{s}}}\,{\mathsf{R}}^{{\mathsf{m}}_{12}{\mathsf{r}}_{1\cdots 4},{\mathsf{s}}} \,,
\\
\bigl[{\mathsf{R}}^{{\mathsf{m}}_{12}}_\alpha,\, {\mathsf{R}}_{{\mathsf{n}}_{12}}^\beta \bigr]
&= \delta_\alpha^\beta\,{{\boldsymbol\delta}}^{{\mathsf{m}}_{12}}_{{\mathsf{p}}{\mathsf{r}}}\,{{\boldsymbol\delta}}_{{\mathsf{n}}_{12}}^{{\mathsf{q}}{\mathsf{r}}}\, {\mathsf{K}}^{\mathsf{p}}{}_{\mathsf{q}}- \frac{1}{4}\,\delta_\alpha^\beta\,{{\boldsymbol\delta}}^{{\mathsf{m}}_{12}}_{{\mathsf{n}}_{12}}\,\delta_{\mathsf{p}}^{\mathsf{q}}\, {\mathsf{K}}^{\mathsf{p}}{}_{\mathsf{q}}- \epsilon^{\beta\gamma}\,{{\boldsymbol\delta}}^{{\mathsf{m}}_{12}}_{{\mathsf{n}}_{12}}\,{\mathsf{R}}_{\alpha\gamma}\,,
\\
\bigl[{\mathsf{R}}^{{\mathsf{m}}_{12}}_\alpha,\, {\mathsf{R}}_{{\mathsf{n}}_{1\cdots 4}} \bigr]
&= \frac{1}{2!}\,\epsilon_{\alpha\beta}\,{{\boldsymbol\delta}}^{{\mathsf{m}}_{12}{\mathsf{r}}_{12}}_{{\mathsf{n}}_{1\cdots 4}}\,{\mathsf{R}}^\beta_{{\mathsf{r}}_{12}} \,,
\\
\bigl[{\mathsf{R}}^{{\mathsf{m}}_{12}}_\alpha,\, {\mathsf{R}}_{{\mathsf{n}}_{1\cdots 6}}^\beta \bigr]
&= -\frac{1}{4!}\,\delta_\alpha^\beta\, {{\boldsymbol\delta}}_{{\mathsf{n}}_{1\cdots 6}}^{{\mathsf{m}}_{12}{\mathsf{r}}_{1\cdots 4}}\,{\mathsf{R}}_{{\mathsf{r}}_{1\cdots 4}} \,,
\\
\bigl[{\mathsf{R}}^{{\mathsf{m}}_{12}}_\alpha,\, {\mathsf{R}}_{{\mathsf{n}}_{1\cdots 7},{\mathsf{n}}} \bigr]
&= \frac{1}{5!}\,\epsilon_{\alpha\beta}\, {{\boldsymbol\delta}}_{{\mathsf{n}}_{1\cdots 7}}^{{\mathsf{m}}_{12}{\mathsf{r}}_{1\cdots 5}}\,{\mathsf{R}}^\beta_{{\mathsf{r}}_{1\cdots 5}{\mathsf{n}}} \,,
\\
\bigl[{\mathsf{R}}^{{\mathsf{m}}_{1\cdots 4}},\, {\mathsf{R}}^{{\mathsf{n}}_{1\cdots 4}} \bigr]
&= \frac{1}{3!}\,{{\boldsymbol\delta}}^{{\mathsf{n}}_{1\cdots 4}}_{{\mathsf{r}}_{123}{\mathsf{s}}}\,{\mathsf{R}}^{{\mathsf{m}}_{1\cdots 4}{\mathsf{r}}_{123},{\mathsf{s}}} \,,
\\
\bigl[{\mathsf{R}}^{{\mathsf{m}}_{1\cdots 4}},\, {\mathsf{R}}_{{\mathsf{n}}_{12}}^\alpha\bigr]
&= \frac{1}{2!}\,\epsilon^{\alpha\beta}\,{{\boldsymbol\delta}}_{{\mathsf{n}}_{12}{\mathsf{r}}_{12}}^{{\mathsf{m}}_{1\cdots 4}}\,{\mathsf{R}}_\beta^{{\mathsf{r}}_{12}} \,,
\\
\bigl[{\mathsf{R}}^{{\mathsf{m}}_{1\cdots 4}},\, {\mathsf{R}}_{{\mathsf{n}}_{1\cdots 4}} \bigr]
&= \frac{1}{3!}\,{{\boldsymbol\delta}}^{{\mathsf{m}}_{1\cdots 4}}_{{\mathsf{p}}{\mathsf{r}}_{123}}\,{{\boldsymbol\delta}}_{{\mathsf{n}}_{1\cdots 4}}^{{\mathsf{q}}{\mathsf{r}}_{123}}\, {\mathsf{K}}^{\mathsf{p}}{}_{\mathsf{q}}- \frac{1}{2}\, {{\boldsymbol\delta}}^{{\mathsf{m}}_{1\cdots 4}}_{{\mathsf{n}}_{1\cdots 4}}\,\delta_{\mathsf{p}}^{\mathsf{q}}\, {\mathsf{K}}^{\mathsf{p}}{}_{\mathsf{q}}\\
\bigl[{\mathsf{R}}^{{\mathsf{m}}_{1\cdots 4}},\, {\mathsf{R}}_{{\mathsf{n}}_{1\cdots 6}}^\alpha \bigr]
&= \frac{1}{2!}\,{{\boldsymbol\delta}}^{{\mathsf{m}}_{1\cdots 4}{\mathsf{r}}_{12}}_{{\mathsf{n}}_{1\cdots 6}}\, {\mathsf{R}}^\alpha_{{\mathsf{r}}_{12}}\,,
\\
\bigl[{\mathsf{R}}^{{\mathsf{m}}_{1\cdots 4}},\, {\mathsf{R}}_{{\mathsf{n}}_{1\cdots 7},{\mathsf{n}}} \bigr]
&= -\frac{1}{3!}\,{{\boldsymbol\delta}}^{{\mathsf{m}}_{1\cdots 4}{\mathsf{r}}_{123}}_{{\mathsf{n}}_{1\cdots 7}}\, {\mathsf{R}}_{{\mathsf{r}}_{123}{\mathsf{n}}}
\\
\bigl[{\mathsf{R}}^{{\mathsf{m}}_{1\cdots 6}}_\alpha,\, {\mathsf{R}}_{{\mathsf{n}}_{12}}^\beta \bigr]
&= -\frac{1}{4!}\,\delta^\beta_\alpha\, {{\boldsymbol\delta}}^{{\mathsf{m}}_{1\cdots 6}}_{{\mathsf{n}}_{12}{\mathsf{r}}_{1\cdots 4}}\,{\mathsf{R}}^{{\mathsf{r}}_{1\cdots 4}} \,,
\\
\bigl[{\mathsf{R}}^{{\mathsf{m}}_{1\cdots 6}}_\alpha,\, {\mathsf{R}}_{{\mathsf{n}}_{1\cdots 4}} \bigr]
&= \frac{1}{2!}\,{{\boldsymbol\delta}}_{{\mathsf{n}}_{1\cdots 4}{\mathsf{r}}_{12}}^{{\mathsf{m}}_{1\cdots 6}}\, {\mathsf{R}}_\alpha^{{\mathsf{r}}_{12}}\,,
\\
\bigl[{\mathsf{R}}^{{\mathsf{m}}_{1\cdots 6}}_\alpha,\, {\mathsf{R}}_{{\mathsf{n}}_{1\cdots 6}}^\beta \bigr]
&= \frac{1}{5!}\,\delta_\alpha^\beta\,{{\boldsymbol\delta}}^{{\mathsf{m}}_{1\cdots 6}}_{{\mathsf{p}}{\mathsf{r}}_{1\cdots 5}}\,{{\boldsymbol\delta}}_{{\mathsf{n}}_{1\cdots 6}}^{{\mathsf{q}}{\mathsf{r}}_{1\cdots 5}}\, {\mathsf{K}}^{\mathsf{p}}{}_{\mathsf{q}}- \frac{3}{4}\,\delta_\alpha^\beta\,{{\boldsymbol\delta}}^{{\mathsf{m}}_{1\cdots 6}}_{{\mathsf{n}}_{1\cdots 6}}\,\delta_{\mathsf{p}}^{\mathsf{q}}\, {\mathsf{K}}^{\mathsf{p}}{}_{\mathsf{q}}- \epsilon^{\beta\gamma}\,{{\boldsymbol\delta}}^{{\mathsf{m}}_{1\cdots 6}}_{{\mathsf{n}}_{1\cdots 6}}\,{\mathsf{R}}_{\alpha\gamma}\,,
\\
\bigl[{\mathsf{R}}^{{\mathsf{m}}_{1\cdots 6}}_\alpha,\, {\mathsf{R}}_{{\mathsf{n}}_{1\cdots 7},{\mathsf{n}}} \bigr]
&= -\epsilon_{\alpha\beta}\,{{\boldsymbol\delta}}^{{\mathsf{m}}_{1\cdots 6}{\mathsf{r}}}_{{\mathsf{n}}_{1\cdots 7}}\, {\mathsf{R}}^\beta_{{\mathsf{r}}{\mathsf{n}}}\,,
\\
\bigl[{\mathsf{R}}^{{\mathsf{m}}_{1\cdots 7},{\mathsf{m}}},\, {\mathsf{R}}_{{\mathsf{n}}_{12}}^\alpha \bigr]
&= \frac{1}{5!}\,\epsilon^{\alpha\beta}\, {{\boldsymbol\delta}}^{{\mathsf{m}}_{1\cdots 7}}_{{\mathsf{n}}_{12}{\mathsf{r}}_{1\cdots 5}}\,{\mathsf{R}}_\beta^{{\mathsf{r}}_{1\cdots 5}{\mathsf{n}}} \,,
\\
\bigl[{\mathsf{R}}^{{\mathsf{m}}_{1\cdots 7},{\mathsf{m}}},\, {\mathsf{R}}_{{\mathsf{n}}_{1\cdots 4}} \bigr]
&= -\frac{1}{3!}\,{{\boldsymbol\delta}}_{{\mathsf{n}}_{1\cdots 4}{\mathsf{r}}_{123}}^{{\mathsf{m}}_{1\cdots 7}}\, {\mathsf{R}}^{{\mathsf{r}}_{123}{\mathsf{m}}}
\\
\bigl[{\mathsf{R}}^{{\mathsf{m}}_{1\cdots 7},{\mathsf{m}}} ,\, {\mathsf{R}}_{{\mathsf{n}}_{1\cdots 6}}^\alpha\bigr]
&= -\epsilon_{\alpha\beta}\,{{\boldsymbol\delta}}_{{\mathsf{n}}_{1\cdots 6}{\mathsf{r}}}^{{\mathsf{m}}_{1\cdots 7}}\, {\mathsf{R}}_\beta^{{\mathsf{r}}{\mathsf{m}}}\,,
\\
\bigl[{\mathsf{R}}^{{\mathsf{m}}_{1\cdots 7},{\mathsf{m}}} ,\, {\mathsf{R}}_{{\mathsf{n}}_{1\cdots 7},{\mathsf{n}}}\bigr]
&= {{\boldsymbol\delta}}_{{\mathsf{n}}_{1\cdots 7}}^{{\mathsf{m}}_{1\cdots 7}}\, {\mathsf{K}}^{\mathsf{m}}{}_{\mathsf{n}}\,,
\\
\bigl[{\mathsf{R}}_{{\mathsf{m}}_{12}}^\alpha,\, {\mathsf{R}}_{{\mathsf{n}}_{12}}^\beta \bigr]
&= \epsilon^{\alpha\beta}\,{\mathsf{R}}_{{\mathsf{m}}_{12}{\mathsf{n}}_{12}} \,,
\\
\bigl[{\mathsf{R}}_{{\mathsf{m}}_{12}}^\alpha,\, {\mathsf{R}}_{{\mathsf{n}}_{1\cdots 4}} \bigr]
&= - {\mathsf{R}}_{{\mathsf{m}}_{12}{\mathsf{n}}_{1\cdots 4}}^\alpha \,,
\\
\bigl[{\mathsf{R}}_{{\mathsf{m}}_{12}}^\alpha,\, {\mathsf{R}}_{{\mathsf{n}}_{1\cdots 6}}^\beta \bigr]
&= \frac{1}{5!}\,\epsilon^{\alpha\beta}\, {{\boldsymbol\delta}}_{{\mathsf{n}}_{1\cdots 6}}^{{\mathsf{r}}_{1\cdots 5}{\mathsf{s}}}\,{\mathsf{R}}_{{\mathsf{m}}_{12}{\mathsf{r}}_{1\cdots 5},{\mathsf{s}}} \,,
\\
\bigl[{\mathsf{R}}_{{\mathsf{m}}_{1\cdots 4}},\, {\mathsf{R}}_{{\mathsf{n}}_{1\cdots 4}} \bigr]
&= -\frac{1}{3!}\,{{\boldsymbol\delta}}_{{\mathsf{n}}_{1\cdots 4}}^{{\mathsf{r}}_{123}{\mathsf{s}}}\,{\mathsf{R}}_{{\mathsf{m}}_{1\cdots 4}{\mathsf{r}}_{123},{\mathsf{s}}} \,.\end{aligned}$$
[99]{} J. J. Fernández-Melgarejo, Y. Sakatani and S. Uehara, “Exotic branes and mixed-symmetry potentials I: predictions from $E_{11}$ symmetry,” arXiv:1907.07177 \[hep-th\].
O. Hohm and H. Samtleben, “Exceptional Form of D=11 Supergravity,” Phys. Rev. Lett. [**111**]{}, 231601 (2013) \[arXiv:1308.1673 \[hep-th\]\].
O. Hohm and H. Samtleben, “Exceptional Field Theory I: $E_{6(6)}$ covariant Form of M-Theory and Type IIB,” Phys. Rev. D [**89**]{}, no. 6, 066016 (2014) \[arXiv:1312.0614 \[hep-th\]\].
O. Hohm and H. Samtleben, “Exceptional field theory. II. E$_{7(7)}$,” Phys. Rev. D [**89**]{}, 066017 (2014) \[arXiv:1312.4542 \[hep-th\]\].
O. Hohm and H. Samtleben, “Exceptional field theory. III. E$_{8(8)}$,” Phys. Rev. D [**90**]{}, 066002 (2014) \[arXiv:1406.3348 \[hep-th\]\].
P. C. West, “$E_{11}$, SL(32) and central charges,” Phys. Lett. B [**575**]{}, 333 (2003) \[hep-th/0307098\].
A. Kleinschmidt and P. C. West, “Representations of ${\cal G}^{+++}$ and the role of space-time,” JHEP [**0402**]{}, 033 (2004) \[hep-th/0312247\].
P. C. West, “$E_{11}$ origin of brane charges and U-duality multiplets,” JHEP [**0408**]{}, 052 (2004) \[hep-th/0406150\].
F. Riccioni and P. C. West, “$E_{11}$-extended spacetime and gauged supergravities,” JHEP [**0802**]{}, 039 (2008) \[arXiv:0712.1795 \[hep-th\]\].
C. Hillmann, “Generalized $E_{7(7)}$ coset dynamics and D=11 supergravity,” JHEP [**0903**]{}, 135 (2009) \[arXiv:0901.1581 \[hep-th\]\].
D. S. Berman and M. J. Perry, “Generalized Geometry and M theory,” JHEP [**1106**]{}, 074 (2011) \[arXiv:1008.1763 \[hep-th\]\].
D. S. Berman, H. Godazgar, M. Godazgar and M. J. Perry, “The Local symmetries of M-theory and their formulation in generalised geometry,” JHEP [**1201**]{}, 012 (2012) \[arXiv:1110.3930 \[hep-th\]\].
D. S. Berman, H. Godazgar, M. J. Perry and P. West, “Duality Invariant Actions and Generalised Geometry,” JHEP [**1202**]{}, 108 (2012) \[arXiv:1111.0459 \[hep-th\]\].
D. S. Berman, M. Cederwall, A. Kleinschmidt and D. C. Thompson, “The gauge structure of generalised diffeomorphisms,” JHEP [**1301**]{}, 064 (2013) \[arXiv:1208.5884 \[hep-th\]\].
E. A. Bergshoeff and F. Riccioni, “D-Brane Wess-Zumino Terms and U-Duality,” JHEP [**1011**]{}, 139 (2010) \[arXiv:1009.4657 \[hep-th\]\].
A. S. Arvanitakis and C. D. A. Blair, “Unifying Type-II Strings by Exceptional Groups,” Phys. Rev. Lett. [**120**]{}, no. 21, 211601 (2018) \[arXiv:1712.07115 \[hep-th\]\].
Y. Sakatani and S. Uehara, “Exceptional M-brane sigma models and $\eta$-symbols,” PTEP [**2018**]{}, no. 3, 033B05 (2018) \[arXiv:1712.10316 \[hep-th\]\].
A. S. Arvanitakis and C. D. A. Blair, “The Exceptional Sigma Model,” JHEP [**1804**]{}, 064 (2018) \[arXiv:1802.00442 \[hep-th\]\].
H. Godazgar, M. Godazgar and M. J. Perry, “$E_8$ duality and dual gravity,” JHEP [**1306**]{}, 044 (2013) \[arXiv:1303.2035 \[hep-th\]\].
A. G. Tumanov and P. West, “Generalised vielbeins and non-linear realisations,” JHEP [**1410**]{}, 009 (2014) \[arXiv:1405.7894 \[hep-th\]\].
K. Lee, S. J. Rey and Y. Sakatani, “Effective action for non-geometric fluxes duality covariant actions,” JHEP [**1707**]{}, 075 (2017) \[arXiv:1612.08738 \[hep-th\]\].
Y. Sakatani and S. Uehara, “Connecting M-theory and type IIB parameterizations in Exceptional Field Theory,” PTEP [**2017**]{}, no. 4, 043B05 (2017) \[arXiv:1701.07819 \[hep-th\]\].
E. Bergshoeff, E. Eyras and Y. Lozano, “The Massive Kaluza-Klein monopole,” Phys. Lett. B [**430**]{}, 77 (1998) \[hep-th/9802199\].
P. C. West, “The IIA, IIB and eleven-dimensional theories and their common $E_{11}$ origin,” Nucl. Phys. B [**693**]{}, 76 (2004) \[hep-th/0402140\].
E. Eyras, B. Janssen and Y. Lozano, “Five-branes, KK monopoles and T duality,” Nucl. Phys. B [**531**]{}, 275 (1998) \[hep-th/9806169\].
E. Eyras and Y. Lozano, “Exotic branes and nonperturbative seven-branes,” Nucl. Phys. B [**573**]{}, 735 (2000) \[hep-th/9908094\].
C. M. Hull, “A Geometry for non-geometric string backgrounds,” JHEP [**0510**]{}, 065 (2005) \[hep-th/0406102\].
J. Berkeley, D. S. Berman and F. J. Rudolph, “Strings and Branes are Waves,” JHEP [**1406**]{}, 006 (2014) \[arXiv:1403.7198 \[hep-th\]\].
D. S. Berman and F. J. Rudolph, “Branes are Waves and Monopoles,” JHEP [**1505**]{}, 015 (2015) \[arXiv:1409.6314 \[hep-th\]\].
D. S. Berman and F. J. Rudolph, “Strings, Branes and the Self-dual Solutions of Exceptional Field Theory,” JHEP [**1505**]{}, 130 (2015) \[arXiv:1412.2768 \[hep-th\]\].
A. Baguet, O. Hohm and H. Samtleben, “E$_{6(6)}$ Exceptional Field Theory: Review and Embedding of Type IIB,” PoS CORFU [**2014**]{}, 133 (2015) \[arXiv:1506.01065 \[hep-th\]\].
P. C. West, “$E_{11}$ and M theory,” Class. Quant. Grav. [**18**]{}, 4443 (2001) \[hep-th/0104081\].
P. West, “$E_{11}$, generalised space-time and IIA string theory,” Phys. Lett. B [**696**]{}, 403 (2011) \[arXiv:1009.2624 \[hep-th\]\].
A. Chatzistavrakidis, F. F. Gautason, G. Moutsopoulos and M. Zagermann, “Effective actions of nongeometric five-branes,” Phys. Rev. D [**89**]{}, no. 6, 066004 (2014) \[arXiv:1309.2653 \[hep-th\]\].
T. Kimura, S. Sasaki and M. Yata, “World-volume Effective Actions of Exotic Five-branes,” JHEP [**1407**]{}, 127 (2014) \[arXiv:1404.5442 \[hep-th\]\].
T. Kimura, S. Sasaki and M. Yata, “World-volume Effective Action of Exotic Five-brane in M-theory,” JHEP [**1602**]{}, 168 (2016) \[arXiv:1601.05589 \[hep-th\]\].
Y. Sakatani and S. Uehara, “Branes in Extended Spacetime: Brane Worldvolume Theory Based on Duality Symmetry,” Phys. Rev. Lett. [**117**]{}, no. 19, 191601 (2016) \[arXiv:1607.04265 \[hep-th\]\].
E. Bergshoeff, A. Kleinschmidt, E. T. Musaev and F. Riccioni, “The different faces of branes in Double Field Theory,” arXiv:1903.05601 \[hep-th\].
[^1]: E-mail address: `[email protected]`
[^2]: E-mail address: `[email protected]`
[^3]: E-mail address: `[email protected]`
[^4]: This assumption is not necessary in the approach discussed in Section \[sec:generalized-metric\].
[^5]: Note that the $B$-field in this paper has the opposite sign to the one usually used in DFT.
[^6]: To be more precise, in our matrix representations in M-theory, in the fourth row and below that, we have used Schouten-like identities; i.e. terms with antisymmetrized nine indices $(\cdots)_{[i_1\cdots i_7 i j]}$ has been dropped because they disappears automatically in $n\leq 8$. However, this does not affect the computation of $(L^{-{{\mathpalette{\raisebox{\depth}{$\m@th\intercal$}}}})$ in because the restriction rule $i\in \{i_1,\dotsc ,i_7\}$ has been assumed there and terms with the structure $(\cdots)_{[i_1\cdots i_7 i j]}$ disappear even for $n=11$. In this sense, can be understood as obtained from the $E_{11}$ generalized metric.
[^7]: If we consider the $E_{11}$ algebra, for example $\bigl[R^{i_1i_2i_3},\, R_{j_1 \cdots j_8,j}\bigr]$ needs to be modified as $\bigl[R^{i_1i_2i_3},\, R_{j_1 \cdots j_8,j}\bigr] = \frac{1}{5!}\, \bigl({{\boldsymbol\delta}}^{i_1i_2i_3 r_1\cdots r_5}_{j_1 \cdots j_8}\, R_{r_1\cdots r_5 j} - {{\boldsymbol\delta}}^{i_1i_2i_3 r_1\cdots r_5}_{[j_1 \cdots j_8|}\, R_{r_1\cdots r_5 |j]}\bigr)$. However, the second term on the right-hand side identically vanishes for $n\leq 8$. In this sense, the commutation relations shown here are valid only in the case $n\leq 8$.
|
---
abstract: |
The mathematical problem of the static storage optimisation is formulated and solved by means of a variational analysis. The solution obtained in implicit form is shedding light on the most important features of the optimal exercise strategy. We show how the solution depends on different constraint types including carry cost and cycling constraint. We investigate the relation between intrinsic and stochastic solutions. In particular we give another proof that the stochastic problem has a “bang-bang” optimal exercise strategy. We also show why the optimal stochastic exercise decision is always close to the intrinsic one.
In the second half we develop a perturbation analysis to solve the stochastic optimisation problem. The obtained approximate solution allows us to estimate the time value of the storage option. In particular we find an answer to rather academic question of asymptotic time value for the mean reversion parameter approaching zero or infinity. We also investigate the differences between swing and storage problems. The analytical results are compared with numerical valuations and found to be in a good agreement.
title: Storage option an Analytic approach
---
Dmitry Lesnik\
Introduction
============
The problem of storage optimisation is driven by the necessity of the storage owners to optimise their expenses and maximise a potential profit which can be gained by operating the storage. There are plenty of real world storage examples: Gas storage, Oil storage, Hydro power plant, Coal stock, etc. By following some clever strategy – buying the underlying commodity cheep, storing it, and selling as the prices go up – the storage owner can make a profit. Thus, there is a need for the exercise strategy optimisation – such an exercise rule, which allows to maximise the profit and minimise the risks.
One of the most popular ways of the numerical storage optimisation is the “dynamic programming” algorithm. It allows to treat both – intrinsic (when the prices are supposed to be frozen) as well as stochastic (when the prices are supposed to follow some stochastic process) problems. Usually the numerical solution provides the answers to the two most important questions: what is the optimal exercise decision now given the current state (prices and volume level of the storage), and what is the expected profit, provided we follow the optimal exercise strategy. The next level of sophistication is to provide the hedge strategy – a portfolio of derivatives (futures, options or any other financial instruments) which would minimise the financial risks. Of course, financial risk is only a feature of the stochastic problem, as the solution of the intrinsic problem is deterministic.
In this paper we to develop an analytical approach to the storage optimisation problem. In the Sec. \[sec:Deterministic\_problem\] we consider an intrinsic problem which deals with predefined deterministic price curve. Section \[sec:other\_constraint\_types\] considers different special cases of constraints. Sec. \[sec:stochastic\_problem\] is devoted to the stochastic problem, where we make use of perturbation theory to solve the stochastic problem, and derive an estimate of the stochastic time value. In the sections \[sec:example\_calculation\_storage\] and \[sec:swing\_option\] we make an example calculation of the time value of a simple storage and swing options and compare the results with numerical evaluation. We discuss the results in Sec \[sec:discussion\].
Problem formulation
===================
The storage problem can be formulated as follows. The storage option holder is given a right to store some amount of underlying (let it be for simplicity gas) in a (virtual) storage facility. At every time moment the option holder may “do nothing”, inject or release the gas from the storage. Every time the gas is injected into the storage it must be bought on the market. Likewise every time the gas is released from the storage, it is sold on the market. Since the market price of gas changes with time, this may lead to a non-trivial cash-flow. The injection and release process must satisfy some operational constraints (for instance maximum injection/release rates, storage capacity, etc.), specified as boundary conditions. Every exercise profile (trajectory in the time-volume space) yields a different profit. In this sense the profit becomes a functional on the exercise trajectory. The aim of the storage option holder is to maximise the profit by choosing an optimal injection/release strategy. The problem can thus be formulated in terms of variational analysis – an optimal trajectory is the one delivering maximum of the profit functional.
If the market prices are known in advance and never change, such a problem is deterministic. The possible profit is bound from above and from below, and thus there exists a trajectory such that no other trajectory yields higher profit. The maximal profit of the static problem is called “intrinsic value”. In the current section we investigate the deterministic problem by means of variational analysis.
Let the storage time span be $t\in[0, T_e]$. Let $F(t)$ be the market price of gas by the time $t$. The curve $F(t)$ is called the forward price curve, since it is observed on the market prior the time $t=0$ and contains the information about the prices of gas with delivery in the future. By the definition of the static problem the forward curve never changes, and hence it does not depend on the observation time.
Let $q(t)$ be the amount of gas in the storage by the time $t$. The curve $q(t)$ defines the exercise trajectory. The initial and terminal conditions are $$\begin{aligned}
&q(0)= Q_{start}\,; \\
&q(T_e)= Q_{end}\,.\end{aligned}$$
The cash flow (per unit time) resulting from trading the gas according to the exercise strategy $q(t)$ is given by $$\begin{aligned}
\label{eq:1.1}
- \dot q(t)\,F(t)\,.\end{aligned}$$ An additional cash flow results from taking into account the injection/release costs (operating costs). Let us designate $\gamma(\dot q(t))$ the operating costs per unit time. The terminal profit is given by the cumulative cash flow. We thus introduce the target functional $$\begin{aligned}
\label{eq:unperturbed_action}
S_0[q(t)] = -\int_0^{T_e} \Big[ \dot q(t)\,F(t) + \gamma(\dot q(t)) \Big]\,dt\,.\end{aligned}$$ The value of this functional on the trajectory $q(t)$ gives the storage value conditional on that trajectory. We have used here subscript $0$ to indicate the unmodified action integral. In the next section we will introduce a modified action integral $S[q(t)]$, which includes additional terms intended to enforce the operational constraints. Running ahead we notice that on any trajectory $q(t)$ allowed by constraints the values of the modified and unmodified functionals coincide.
Using “physical” terminology we introduce the (unmodified) Lagrangian $L_0$ corresponding to the target functional $S_0$ as $$\begin{aligned}
\label{eq:unperturbed_lagrangian}
L_0 = - \dot q(t)\,F(t) - \gamma(\dot q(t))\,.\end{aligned}$$
As mentioned above, the target functional is bound from above and from below, and hence there must exist such trajectories on which the functional achieves its maximum and minimum[^1]. We make use of variational analysis to search the extremal trajectory, i.e. the trajectory on which the first variation of the target functional vanishes. Once the extremal trajectory is found, one has to make sure that it delivers the maximum of the functional. It could be done by evaluating the second variation and checking its sign.
The operational constraints may differ for different storage option types. Below we consider some typical constraints, which can be classified as *local*. Local property implies that a constraint at time $t$ can be expressed in terms of state variables and their derivatives $q(t),\dot q(t), F(t),\dot F(t), ... $ at time $t$.
We consider the following two operational constraints:
1. The volume $q(t)$ is allowed to be within the interval $$\begin{aligned}
q(t) \in [Q_{min}(t),Q_{max}(t)]\,,
\end{aligned}$$ where the boundaries $Q_{min}(t),Q_{max}(t)$ are time dependent.
2. The injection/release rate $\dot q(t)$ is bounded to $$\begin{aligned}
\dot q(t) \in [r_{min}, r_{max}]\,.
\end{aligned}$$ Generally the maximal injection/release rates may depend on the time and volume: $r = r(t,q(t))$. Below we only consider the constant injection/release rates. The special case of volume dependent rates will be considered in the Sec. \[sec:volume\_dependend\_rates\].
One of the most common examples of *nonlocal* constraints is the so called *cycle* constraint. It can be formulated as follows. One introduces an *intake* cycle variable as $$\begin{aligned}
\label{eq:cycle_variable}
c(T) = \int_0^{T} \dot q(t)\, \theta( \dot q(t) )\,dt\,;\qquad\text{where} \quad
\theta(x) = \left\{
\begin{array}{ll}
1\,,\quad & x\geq 0 \\
0\,, & x < 0
\end{array}
\right.\end{aligned}$$ which has a meaning of total injected volume by the time $T$. The cycle constraint requires that the terminal value $c(T_e)$ does not exceed certain threshold $$\begin{aligned}
\label{eq:cycle_constrain}
c(T_e) \leq c_{max}\,.\end{aligned}$$ Similarly one introduces a *release* cycle constraint.
Most of the scope of this paper is not dealing with non-local constraints. However we will return to the cycle constraint below in the section \[sec:cycle\_constraint\].
Solution of deterministic problem {#sec:Deterministic_problem}
=================================
Penalty functions {#sec:2.1}
-----------------
To restrict $q(t)$ from going beyond the range $[Q_{min},Q_{max}]$, we can introduce a parametrised penalty function $-\phi[q(t),N_\phi]$ and add it to the unmodified Lagrangian (\[eq:unperturbed\_lagrangian\]) $$\begin{aligned}
L = L_0 -\phi(q) = -\dot qz - \gamma(\dot q) - \phi(q).\end{aligned}$$
The penalty function is any smooth function, which is in the limit $N_\phi\to \infty$ approaches zero within the interval $(Q_{min},Q_{max})$ and positive infinity otherwise. A particular example of this function could be $$\phi = \Big[a(q(t)-b)\Big]^{2N_\phi}\,, \qquad \text{with} \quad
a = \frac{2}{Q_{max}-Q_{min}}\,,\quad
b = \frac{Q_{max} + Q_{min}}{2}\,.$$
A similar penalty function $-\psi[\dot q(t),N_\psi]$ can be introduced to restrict $\dot q(t)$ from going beyond the interval $[r_{min}, r_{max}]$. In the limit $N_\psi\to\infty$ it approaches zero within $(r_{min}, r_{max})$ and plus infinity otherwise. The modified Lagrangian becomes $$\begin{aligned}
\label{eq:perturbed_lagrangian}
L = L_0 - \phi(q) - \psi(\dot q) = - \Big[ \dot q F + \phi(q) + \psi(\dot q) + \gamma(\dot q) \Big]\,.\end{aligned}$$ It corresponds to the following modified action integral: $$\begin{aligned}
\label{eq:perturbed_action}
S = - \int_0^{T_e} \Big[\dot q F + \phi(q) + \psi(\dot q) + \gamma(\dot q) \Big]\,dt\,.\end{aligned}$$
Euler-Lagrange equation
-----------------------
As we know from the variational analysis, the extremal trajectory of the integral $\int L(q, \dot q)\,dt$ satisfies the Euler-Lagrange equation $$\begin{aligned}
\frac{{\partial}L(q, \dot q)}{{\partial}q} = \frac{d}{dt}\frac{{\partial}L(q,\dot q)}{{\partial}\dot q}\,.\end{aligned}$$
Thus, for the modified Lagrangian we obtain the following equation $$\begin{aligned}
\label{eq:EL}
\phi'(q) = \frac{d}{dt}\Big[F + \psi'(\dot q) + \gamma'(\dot q) \Big]\,,\end{aligned}$$ which needs to be solved in the limit $N_\phi\to\infty,N_\psi\to\infty$.
There are two types of solution of the latter equation. One is obtained on the interval where the trajectory remains strictly within the boundaries $$q(t) \in (Q_{min}, Q_{max} ).$$ The second solution type is, when the trajectory lies on the boundary $$q(t) \equiv Q_{min}(t), \quad \text{ or } \quad q(t) \equiv Q_{max}(t)\,.$$ The general solution consists of pieces of the solutions of the types one and two. Let us consider each of these types separately.
Solution within the boundaries
------------------------------
First we consider the solution in the interval, where the trajectory does not touch the boundary. In the limit $N_\phi\to\infty$ the l.h.s. of Eq. (\[eq:EL\]) vanishes. We thus have $$\begin{aligned}
\label{eq:within_boundaries}
F(t) + \psi'(\dot q) + \gamma'(\dot q) = C\,,\end{aligned}$$ where $C$ is *constant* for the whole period of time, where the solution does not touch the boundary.
Let us consider one particular example of the operating cost function. It is rather typical for a gas storage facility that the operating cost is proportional to the amount of the gas released or injected into the storage. Again we parametrise it with $N_\gamma$ so that for finite $N_\gamma$ the function $\gamma(\dot q)$ is smooth, and in the limit $N_\gamma\to \infty$ it is piecewise linear $$\begin{aligned}
\gamma(\dot q) = \left\{
\begin{array}{cc}
\gamma_{inj} \, \dot q\,, \quad& \text{for } \dot q > 0\,;\\
-\gamma_{rel} \, \dot q\,, \quad& \text{for } \dot q < 0\,;
\end{array}
\right.\end{aligned}$$ where $\gamma_{inj} > 0$ and $\gamma_{rel}>0$.
The solution of Eq. (\[eq:within\_boundaries\]) is now straightforward to find graphically (see Fig. \[fig:1\]). In case $r_{min}<0, r_{max}>0$ we obtain: $$\begin{aligned}
\label{eq:solution}
\dot q(t) = \left\{
\begin{array}{ll}
r_{min}(t)\,, \qquad & F(t) > C+\gamma_{rel} \,; \\
0\,, & C-\gamma_{inj} < F(t) < C+\gamma_{rel} \,; \\
r_{max}(t)\,, \qquad & F(t) < C - \gamma_{inj} \,;
\end{array}
\right.\end{aligned}$$ We see that the value $C$ can be interpreted as a trigger price. If the price $F(t)$ is within the interval $[C-\gamma_{inj},\, C+\gamma_{rel}]$, the extremal trajectory is constant: $\dot q = 0$. If the price is below this interval, the volume $q(t)$ is increasing at maximal rate: $\dot q(t) = r_{max}(t)$. If the price is above the interval, the volume is decreasing at maximum rate: $\dot q(t) = r_{min}(t)$.
Solution on the boundary
------------------------
We consider only a boundary $Q_{min}(t)$ and $Q_{max}(t)$, that moves slower than the limit rate, i.e. $r_{min}\leq \dot Q_{min} \leq r_{max}$ and $r_{min}\leq \dot Q_{max} \leq r_{max}$. If the boundary moves faster, then no trajectory can lie on it.
On the boundary the function $\psi'(\dot q)$ vanishes, and the function $\phi'(q)$ takes a finite value (positive if $q = Q_{max}$ or negative if $q = Q_{min}$). Eq. (\[eq:EL\]) on the boundary becomes $$\begin{aligned}
\label{eq:on_the_boundary}
\phi'(q) = \frac{d}{dt} \Big[F(t)+\gamma'(\dot q)\Big]\,;\end{aligned}$$
There is another view on the solution on the boundary. We may consider our problem as a general variational problem with spacial boundary condition. A path lying on the boundary can not be varied and hence be a part of extremal. But a solution, that maximises the functional, can consist of parts lying on the boundary and *extremals* – pieces of trajectory satisfying the extremal condition (i.e., vanishing first variation).
Approaching the boundary
------------------------
An important statement can be made about the connection point between extremal and the boundary. From the variational analysis it is known that an extremal touches the boundary smoothly.
The extremal solution of our variational problem for large but finite $N_{\phi,\psi,\gamma}$ is a smooth function and must satisfy the condition of smooth connection between extremal and boundary. Let us consider a simplified problem (a generalisation is straightforward), where we suppose a constant boundary condition, and zero operating cost: $$Q_{max} = const,\quad Q_{min} = const,\quad \gamma(\dot q)\equiv 0\,.$$ In this case the extremal trajectory should touch the boundary at zero slope: $$\dot q(t^*) = 0\,,$$ where $t^*$ is the connection time. From Eq. (\[eq:solution\]) in the limit $\gamma_{rel}\to 0$ and $\gamma_{inj}\to 0$ we conclude that $$\begin{aligned}
F(t^*) = C\end{aligned}$$ Taking the operating cost again into consideration leads to the following cases:
1. The trajectory touches the lower boundary from the left: $$\begin{aligned}
\label{eq:bc1}
F(t^*) = C + \gamma_{rel}\,; \quad \dot F(t^*) \leq 0\,.
\end{aligned}$$
2. The trajectory touches the upper boundary from the left: $$\begin{aligned}
\label{eq:bc2}
F(t^*) = C - \gamma_{inj}\,; \quad \dot F(t^*) \geq 0\,.
\end{aligned}$$
3. The trajectory touches the lower boundary from the right: $$\begin{aligned}
\label{eq:bc3}
F(t^*) = C - \gamma_{inj}\,; \quad \dot F(t^*) \leq 0\,.
\end{aligned}$$
4. The trajectory touches the upper boundary from the right: $$\begin{aligned}
\label{eq:bc4}
F(t^*) = C + \gamma_{rel}\,; \quad \dot F(t^*) \geq 0\,.
\end{aligned}$$
These conditions must be satisfied for the time interval $t\in(0,T_e)$. The extremal trajectory on the times $t=0$ and $t=T_e$ does not have to satisfy these conditions.
The conditions (\[eq:bc1\]-\[eq:bc4\]) can be derived from a rule of thumb: if at the moment $t^*$ of boundary touch the boundary was virtually not there, the trajectory had to switch the mode anyway (like from “full release” to “do nothing” in case (\[eq:bc1\]), et.c.). The requirement to the price curve derivative follows from the simple observation: when the trajectory approaches the boundary, the forward curve should enter the dead zone, and when the trajectory leaves the boundary, the forward curve should exit the dead zone.
In the limit $N_{\phi,\psi,\gamma} \to \infty$ the solution may be non smooth any more. In particular the connection point $q(t^*)$ will be a point of the slope discontinuity.
It is worth emphasising that the constant $C$ may be different for different extremal pieces of trajectory separated by the boundary touch.
Conclusion
----------
Eq. (\[eq:solution\]) together with boundary conditions (\[eq:bc1\]-\[eq:bc4\]) and condition on the start and end volume $$\begin{aligned}
\label{eq:init_term_cond}
q(0) = Q_{start}\,;\qquad q(T_e) = Q_{end} = Q_{start} + \int_0^{T_e} \dot q \, dt\end{aligned}$$ provides an implicit solution for the problem. We summarise some properties of the solution:
1. The optimal exercise strategy is always bang-bang. If expressed in “industrial” language, the compressor either stands or works with a full power.
2. For each piece of trajectory, separated from others with a boundary touch, there is a trigger price $C$ and a “dead zone” $[C-\gamma_{inj},\, C+\gamma_{rel}]$. The volume in the storage is increasing at maximum rate if the price is below the zone, decreasing at maximum rate if the price is above the zone, and is constant if the price is in the zone. The width of the zone is defined by the operating costs.
3. If the trajectory touches the boundary in the time interval $t\in(0,T_e)$, it must satisfy one of the boundary conditions (\[eq:bc1\]-\[eq:bc4\]).
4. Generally a trajectory satisfying all the previous conditions is not unique. Any solution must obey all these conditions, but not any function obeying all conditions is a solution.
The solution found in the previous section maximises the modified target functional $S[q(t)]$. Let $\bar q(t)$ be the obtained solution. Now the optimal path $\bar q(t)$ can be substituted to the unmodified functional $S_0[\bar q(t)]$ to give the value of option. A natural question is whether the trajectory $\bar q(t)$, which is optimal for modified functional, is still optimal for the unmodified one.
The optimal trajectory $\bar q$ belongs to the broad class of all locally integrable functions $\mathcal{L}_1$. Let us introduce a class of trajectories $\mathcal{K}\subset\mathcal{L}_1$ satisfying all the constraints. By definition penalty functions vanish on $\mathcal{K}$: $$\phi(q)\equiv 0\,;\quad \psi(\dot q)\equiv 0\,,\quad \text{iff}\quad q\in\mathcal{K}\,.$$ Obviously the values of modified and unmodified functionals coincide on $\mathcal{K}$: $$S[q] = S_0[q]\,,\quad \forall q\in\mathcal{K}\,.$$
Next we show that the optimal trajectory $\bar q$ belongs to the class $\mathcal{K}$ and hence satisfies the constraints. Indeed, generally on some parts of the maximal trajectory $\psi'(\dot{\bar q}) \not = 0$ and $\phi'(\bar q) \not = 0$. According to Eqs.(\[eq:within\_boundaries\]) and (\[eq:on\_the\_boundary\]) they take finite values. One can show (we leave it without proof) that in the limit $N_{\phi,\psi}\to\infty$ everywhere where $\phi'$ (or $\psi'$) take finite value, the function $\phi$ (or $\psi$) vanishes. We conclude that on the extremal trajectory $$\begin{aligned}
\label{eq:zero}
\phi(\bar q) \equiv 0\,;\qquad
\psi(\dot{\bar q}) \equiv 0\,.\end{aligned}$$ and hence $\bar q \in\mathcal{K}$. Form this it follows that the values of modified and unmodified target functionals coincide on the maximal trajectory: $$S[\bar q] = S_0[\bar q]\,.$$
Last step is to prove that the trajectory $\bar q$ delivers the maximum of the unmodified functional. It is obvious from the following consideration. The trajectory $\bar q$ maximises the modified functional on the space $\mathcal{L}_1$ and hence it will also maximise this functional on the smaller space $\mathcal{K}\subset\mathcal{L}_1$. Since the values of both functionals coincide on $\mathcal{K}$, the trajectory $\bar q$ will also maximise the unmodified functional $S_0$ on the space of all trajectories satisfying constraints.
From this point we drop the subscript “0” for the action integral, unless we want to emphasise the difference between modified and unmodified target functions. We also drop the “bar” for the optimal trajectory $\bar q(t)$ everywhere where it does not lead to ambiguity.
Other constraint types {#sec:other_constraint_types}
----------------------
### Carry cost
Some storage contracts may include a “carry cost”, which is the cost of keeping the commodity in the storage. For instance an oil storage is subject to a heating cost (a fuel oil in a storage has to be kept warm at approximately $60^oC$).
The carry cost is specified as a time-dependent price $\gamma_c(t)$ per unit time per unit volume. Thus, the target function becomes $$\begin{aligned}
\label{eq:action_carry_cost}
S_0 = -\int_0^{T_e} \Big[ \dot q\,F(t) + \gamma(\dot q) + \gamma_c(t)\,q(t) \Big]\,dt\end{aligned}$$ Substituting the new Lagrangian into the Lagrange-Euler equation we obtain between boundaries: $$\begin{aligned}
\label{eq:within_boundaries_carry_cost}
F(t) + \psi'(\dot q) + \gamma'(\dot q) = C(t)\,\end{aligned}$$ where $$\begin{aligned}
\label{eq:trigger_price_carry_cost}
\frac{dC(t)}{dt} = \gamma_c(t)\,.\end{aligned}$$ Thus, the solution of the problem with the carry cost is absolutely the same as that without but the only difference: the trigger price is not a constant, rather a function of time satisfying Eq. (\[eq:trigger\_price\_carry\_cost\]).
### Solution with free terminal condition {#sec:free_vs_fixed_terminal_condition}
Sometimes the storage problem can be defined with a free terminal condition. For instance, one is allowed to leave an arbitrary amount of underlying in the storage, and for the remaining volume one gets a additional pay-off equivalent to selling that volume at some effective price $F_e$, usually referred to as a *final unit price*. We define the target functional $\tilde S$ as $$\begin{aligned}
\label{eq:2.18}
\tilde S[q(t)] = -\int_0^{T_e} [ \dot q(t)\,F(t) + \gamma(\dot q)]\,dt + q_{end}\,F_e\,.\end{aligned}$$ The optimisation problem is now to find an optimal exercise trajectory $q(t)\,,\ q_{end} = q(t_{end}) = q(T_e)$, satisfying operational constraints, that would maximise the target functions (\[eq:2.18\]). It is easy to see that any extremal trajectory of (\[eq:2.18\]) is also extremal of (\[eq:unperturbed\_action\]). Indeed, if the trajectory $\bar q(t)$ is extremal of (\[eq:2.18\]), the variation of the target functional vanishes on any small variation $\delta \bar q(t)$. Since there is no terminal boundary condition, the variation $\delta \bar q(t)$ may not vanish at the end point. But if the target function does not change on the whole class of allowed trajectories $q(t) = \bar q(t) + \delta\bar q(t)$, it also does not change on the sub-class of trajectories $q(t) = \bar q(t) + \delta \hat q(t)$ with fixed terminal value ($\delta \hat q(T_e) = 0$).
We conclude that to find a maximising trajectory for the functional (\[eq:2.18\]), we need first to find a solution for the functional (\[eq:unperturbed\_action\]) with fixed terminal conditions. Then, considering this solution as a function of terminal state $q_{end}$, we need to find a maximum of (\[eq:2.18\]) as a plain function of $q_{end}$.
As will be shown in the Sec. \[sec:3\], if the target functional (\[eq:unperturbed\_action\]) is considered as a function of the terminal state $q_{end}$, then its derivative with respect to $q_{end}$ is $$\frac{{\partial}S_0}{{\partial}q_{end}} = -C\,.$$ Thus, the target functional (\[eq:2.18\]), considered as a function of $q_{end}$, has a derivative $$\begin{aligned}
\label{eq:4.21}
\frac{{\partial}\tilde S}{{\partial}q_{end}} = F_e - C \,,\end{aligned}$$ where $C$ is the trigger price of the last part of trajectory. We conclude that the condition for a target functional with free terminal state to have maximum is $$\begin{aligned}
\label{eq:4.22}
C = F_e\,.\end{aligned}$$
In practice the storage option with free terminal state has two possibilities. One possibility is that the terminal state lies on the boundary $Q_{min}$ or $Q_{max}$. In this case the problem has a solution with fixed end. Another possibility is that the trajectory ends between boundaries $Q_{min} < q(T_e) < Q_{max}$. In this case the condition (\[eq:4.22\]) must be fulfilled.
If the storage option has no final unit price (price for the remaining volume), it is most likely to terminate with the lowest possible volume. Indeed, as we already know, the derivative of the storage option value with respect to the terminal state equals $-C$. If we neglect the operating costs, the trigger price can not be smaller than the smallest gas price. Hence, if the gas price is positive, then the derivative of the option value with respect to the terminal level is negative, and thus the optimal trajectory must terminate at the lowest possible level. Consequently the storage option can be considered as a problem with fixed terminal state (although it may not be fixed according to the contract).
An opposite example is a swing option. A typical swing option is a contract between a gas supplier and a trading company. The trader buys gas at some predefined price from the supplier and sells it on the market at the current market price. The spread between the market and supplier price becomes the effective gas price for the trader. The swing option allows the trader to take the gas from the supplier according to some flexible scheme (i.e. to decide, when to take and how much within certain constraints). Thus, the swing option can be formulated in terms of a storage option. However unlike the storage option, the effective price (the spread) in the case of swing option can be both positive and negative. Depending on the actual prices and on the contract constraints, it may happen that the terminal state does not lie on either lower or upper boundaries. In this case we deal with the problem with the free terminal condition. For this problem the trigger level equals zero $$\begin{aligned}
\label{eq:4.23}
C = 0\,,\end{aligned}$$ since there is no equivalent for the final unit price in the swing contract.
### Cycle constraint {#sec:cycle_constraint}
The cycle constraint (\[eq:cycle\_constrain\]) is formulated as a maximum allowed injected volume during the operation period $t\in[0,T_e]$. Instead of considering it in the form of inequality we can proceed as follows. We find a solution in two steps. On the first step we solve the problem without the cycle constraint and calculate the cycle variable on the optimal trajectory. If the cycle variable is below the threshold $c_{max}$, the obtained solution satisfies the cycle constraint. However if the solution violates the cycle constraint, we can reformulate the problem, imposing the “biting” cycle constraint $$\begin{aligned}
\label{eq:cycle_constrain_eq}
c(T_e) = \int_0^{T_e} \dot q(t)\, \theta( \dot q(t) )\,dt = c_{max}\end{aligned}$$ This condition allows to formulate our problem in terms of conditional variational extremum. It is solved by means of Lagrange multiplier. Namely we introduce a modified Lagrangian $$\begin{aligned}
\label{eq:modif_lagr_cycle}
L = - \Big[ \dot q F + \phi(q) + \psi(\dot q) + \gamma(\dot q) + \lambda\, \dot q(t)\, \theta( \dot q(t) ) \Big]\,,\end{aligned}$$ where $\lambda$ is an independent variable (Lagrange multiplier). The solution of the latter problem will contain the undefined coefficient $\lambda$, which has to be found from the additional equation (\[eq:cycle\_constrain\_eq\]).
Within the boundaries the Lagrange-Euler equation for the constrained Lagrangian becomes (compare to Eq. (\[eq:within\_boundaries\])) $$\begin{aligned}
C = F(t) + \psi'(\dot q) + \gamma'(\dot q) + \lambda \,\Big(\theta(\dot q) + \dot q\,\delta(\dot q) \Big) =
F(t) + \psi'(\dot q) + \gamma'(\dot q) + \lambda \,\theta(\dot q) \,.\end{aligned}$$ Solving this equation we find that the solution is similar to that without cycle constraint, but has the “dead zone” by $\lambda$ wider. Thus, the cycle constraint is equivalent to some additional operating costs. One can find a fictive additional release and/or injection costs (same for the whole trajectory), such that if added to the storage without cycle constraint, the resulting trajectory would be extremal for the Lagrangian with imposed cycle constraint. It plays no role if the additional cost is for injection, release or both, since it is only the width of the dead zone which matters. The only exception from this rule is if the optimal trajectory does not inject or release enough volume. For instance if the trajectory never releases gas, the additional release cost will not effect the trajectory and will not help to respect the intake cycle constraint. Practically one should add injection costs to fulfil the intake cycle constraint, and release costs to fulfil the release cycle constraint.
Generally the solution between boundary touches contains two free parameters – trigger price $C$ and cycle Lagrange multiplier $\lambda$. They can be equivalently expressed by upper and lower bounds of the dead zone. These parameters have to be found in such a way that the obtained solution satisfies boundary touch conditions (\[eq:bc1\]-\[eq:bc4\]), initial and terminal conditions (\[eq:init\_term\_cond\]) and the cycle constraint (\[eq:cycle\_constrain\_eq\]).
### Volume dependent injection/release rates {#sec:volume_dependend_rates}
If the maximal injection/release rates are volume dependent, the penalty function $\psi$ becomes an explicit function of volume: $\psi = \psi(q,\dot q)$. In this case the Euler-Lagrange equation within the boundaries becomes $$\begin{aligned}
\label{eq:4.25}
{\partial}_q \,\psi(q,\dot q) = \frac{d}{dt}\Big[F + {\partial}_{\dot q}\,\psi(q,\dot q) + \gamma'(\dot q) \Big]\,.\end{aligned}$$ The l.h.s. of this equation is an integrable implicit function of time. Designating $$\frac{dC(t)}{dt} = {\partial}_q \,\psi(q(\tau),\dot q(\tau)) \,;\qquad C(t) = C_0 + \int_0^t {\partial}_q \,\psi(q(\tau),\dot q(\tau)) \,d\tau$$ we obtain the same solution as in the case of volume independent rates with the difference that $C$ is not constant any longer, rather a function of time. We conclude that the optimal exercise strategy still preserves the “bang-bang” property. However is does not posses the constant trigger level. The function $C(t)$ can be reverse engineered from the optimal trajectory $q(t)$, provided the latter is been found.
Let $q(t)$ be the optimal trajectory. On this trajectory the derivative ${\partial}_{\dot q}\,\psi(q,\dot q)$ takes a finite value, which can be found from the equation $${\partial}_{\dot q}\,\psi(q(t),\dot q(t)) = C(t) -F(t) - \gamma'(\dot q(t))\,.$$ We can also find the relation between the derivatives ${\partial}_q\,\psi(q,\dot q)$ and ${\partial}_{\dot q}\,\psi(q,\dot q)$. Indeed, if the trajectory $q(t)$ lies for instance on the maximum injection rate boundary $\dot q = r_{max}(q)$, then the relation between $dq$ and $d\dot q$ is simply $d \dot q = r'_{max}(q)\,dq$. It’s easy to show that the gradient $\{{\partial}_q\,\psi(q,\dot q),{\partial}_{\dot q}\,\psi(q,\dot q)\}$ must be orthogonal to the vector $\{dq, d\dot q\}$. Hence $${\partial}_q\,\psi(q,\dot q) = r'_{max}(q)\,{\partial}_{\dot q}\,\psi(q,\dot q) \,.$$ We thus obtain the differential equation $$\frac{dC(t)}{dt} = r'(q(t)) \Big( C(t) -F(t) - \gamma'(\dot q(t)) \Big)\,. \qquad
r'(q(t)) = \left.\frac{dr(q)}{dq}\right|_{q = q(t)}\,.$$ Here $r(q) = r_{max}$ must be used for “injection” and $r_{min}$ for “release” parts of the trajectory. This equation allows to find the function $C(t)$ provided the optimal trajectory $q(t)$ is already known.
Dependency on initial and terminal conditions {#sec:3}
---------------------------------------------
The optimal path $q(t)$ and the target functional $S[q(t)]$ are calculated for fixed terminal point $\{T_e,Q_{end}\}$. If this point is slightly shifted in either direction – time or volume – the optimal path becomes different, and the target function as well. In this sense the target function can be viewed as a function of terminal point $S = S(q,t)$. In this section we are interested in the properties of this function.
From the standard variational analysis we have: $$\begin{aligned}
\frac{\partial S}{\partial q} = \frac{\partial L}{\partial \dot q}\,; \qquad
\frac{\partial S}{\partial t} = L - \dot q \frac{\partial L}{\partial \dot q}\,.\end{aligned}$$ We apply these equations to the perturbed action $S[q(t)]$. Making use of Eq. (\[eq:zero\]) we will find the derivatives of the unperturbed action $S_0[q(t)]$.
### Volume derivative {#sec:spacial_derivative}
Using the definition (\[eq:perturbed\_lagrangian\]) of the perturbed Lagrangian we find $$\frac{\partial S}{\partial q} = - \Big[ F + \psi'(\dot q) + \gamma'(\dot q) \Big]\,.$$ The spacial derivative makes sense only within the boundaries. Making use of Eq. (\[eq:within\_boundaries\]) and replacing $S$ with $S_0$ we finally find $$\begin{aligned}
\label{eq:spacial_derivative}
\frac{\partial S_0}{\partial q_{end}} = \frac{\partial S}{\partial q_{end}} = - C\,,\end{aligned}$$ where $C$ is the trigger level of the last part of the trajectory. If $C$ is positive, one can draw a conclusion, that the smaller the terminal state $q(T_e)$, the bigger is the target functional $S_0(T_e)$.
Derivative with respect to the initial condition $q_{start}$ is given by inverting the sign: $$\begin{aligned}
\frac{\partial S_0}{\partial q_{start}} = \frac{\partial S}{\partial q_{start}} = C\,,\end{aligned}$$ where $C$ is the trigger level of the initial part of the trajectory.
From this an important consequence follows, which we already mentioned in the section \[sec:free\_vs\_fixed\_terminal\_condition\]. If there is no final unit price and if the prices $F(t)$ are strictly positive, so is the trigger price $C$. As a result the option value has a negative derivative with respect to the terminal level. Hence the optimal trajectory will always take at the end the smallest possible value allowed by constraints. By other words, we know that the right end of the optimal trajectory will lie on the lower boundary. Thus, even if the contract allows the storage to end with some remaining volume inside (free terminal condition), the optimisation problem still can be considered as the one with fixed terminal condition.
This is not any longer the case in presence a positive final unit price is given or if the prices $F(t)$ can be negative. With the final unit price $F_e$ the option value derivative is given by Eq. (\[eq:4.21\]) and can be positive, negative or zero. Also extremely high injection/release costs may lead to vanishing trigger price and the option value derivative.
### Time derivative
From definition we find $$\begin{aligned}
\frac{\partial S}{\partial t}& =
\dot q\Big[\psi'(\dot q) + \gamma'(\dot q)\Big] -
\phi(q) - \psi(\dot q) - \gamma(\dot q) = \nonumber \\
& = \dot q\Big[\psi'(\dot q) + \gamma'(\dot q)\Big] - \gamma(\dot q)\,.\end{aligned}$$ **Within the boundaries** we make use of Eq. (\[eq:within\_boundaries\]). Replacing $S$ with $S_0$ one obtains $$\begin{aligned}
\frac{\partial S_0}{\partial t_{end}} = \frac{\partial S}{\partial t_{end}} = \dot q(t_{end}) \Big[C - F(t_{end}) \Big] - \gamma(\dot q)\,,\end{aligned}$$ where $\dot q(t_{end})$ is the solution (\[eq:solution\]) obtained for fixed terminal condition. In particular one can see that $\partial S_0/\partial t_{end} \geq 0$ (provided $r_{min}<0, r_{max}>0$).
**On the boundary** in the special case $\dot Q_{max} = \dot Q_{min} = 0$ one obtains $\dot q= 0, \gamma(\dot q)= 0$ and $$\begin{aligned}
\frac{\partial S_0}{\partial t_{end}} = 0\,.\end{aligned}$$ It is worth noting that the derivatives with respect to terminal (initial) boundary conditions is universal: it is independent on whether the optimal trajectory touches the boundary or not. To calculate the derivatives, one just needs to know the solution $q(t)$ and the trigger price $C$ at the end (beginning) of the trajectory. This also remains valid with imposed cycle constraint.
Stochastic problem {#sec:stochastic_problem}
==================
The following section is devoted to the stochastic optimisation problem. As a starting point of the stochastic approach we take the solution of the corresponding intrinsic problem, which is supposed to be known. Under some very generic assumptions about the price process we derive a stochastic differential equation which governs the evolution of the storage value in the rolling intrinsic approximation. This allows us to find the expectation value of the terminal option value, and hence to estimate the time value of the storage option.
In real world the prices are not static, rather evolve with time in a stochastic way. Every time we observe a new forward curve on the market, we can solve a new intrinsic problem, obtaining a new solution. The forward price $F$, as well as the optimal intrinsic strategy $\dot {\bar q}$ and trigger price $C$ become not only functions of “delivery” (or “maturity”) time $T$, but also functions of the “current” (or “observation”) time $t$: $$C = C(t)\,;\quad F = F(t,T)\,;\quad \dot {\bar q} = \dot {\bar q}(t,T)\,.$$ By other words, $F(t,T)$ is the forward price with delivery time $T$ observed on the market at time $t$. $C(t)$ and $\dot {\bar q}(t,T)$ are trigger price and optimal intrinsic strategy, calculated at time $t$ according to current observed forward price curve $F(t,T)$. Throughout the following section we designate $$r(t, T) \equiv \dot {\bar q}(t, T)$$ the optimal intrinsic exercise strategy on the observation time $t$.
By definition of the stochastic problem, the initial forward curve $F(0,T)$ is known. For any future observation time $t>0$ there is no deterministic forward curve. The evolution of the forward curve is supposed to satisfy some stochastic differential equation. It means that any possible curve $F(t,T)$ can be assigned a certain probability amplitude, and the solution of the problem has to be formulated in the probabilistic language.
#### Assumptions:
To make further analysis we make some simplifying assumptions.
- First we neglect the injection/release costs by setting $\gamma\equiv 0$.
- We also neglect the “surface effects” – the possible influence of the boundary touch on the optimal trajectory and on the storage option value. Under the specified assumptions for every observation time $t$ there exists a single trigger price $C(t)$ for the whole trajectory, and the optimal intrinsic strategy can be presented in the form $$\begin{aligned}
\label{eq:solution_for_dot_q}
r(t, T) = r_{min} + \Delta r\,\theta(C(t) - F(t,T))\,,\end{aligned}$$ where $\Delta r = r_{max} - r_{min}$, $\theta(x)$ is the Heaviside function and $C(t)$ is the intrinsic trigger price at time $t$. We also introduce the following notation: $$\begin{aligned}
r^{(1)}(t,T) = \Delta r\,\delta(C(t)-F(t,T))\,;\qquad
r^{(2)}(t,T) = \Delta r\,\delta'(C(t)-F(t,T))\,.\end{aligned}$$ Here $\delta(x) = \theta'(x)$ is the Dirac delta function. The derivative is taken with respect to the argument of the $\delta$-function. Notice the dimensionality: $$\begin{aligned}
r = \left[\frac{Q}{t}\right]\,; \qquad
r^{(1)} = \left[\frac{Q}{t\,\$}\right]\,; \qquad
r^{(2)} = \left[\frac{Q}{t\,\$^2}\right]\,.\end{aligned}$$
By neglecting the “surface effects” we restrict our consideration to the so called non-flexible storages. The storage flexibility can be characterised by a number of times $N_c$ the storage can be filled and emptied completely during the storage life time (maximum number of cycles). The storage is usually called non-flexible if $N_c \lesssim 1$. For $N_c \gg 1$ the storage is referred to as flexible. Obviously, the surface effects have much bigger impact on the flexible storages.
- As an additional assumption we believe that the prices $F(t,T)$ are always strictly positive. This allows us to consider the optimisation problem as a problem with fixed initial and terminal conditions (see Sec. [\[sec:spacial\_derivative\]]{}). This condition is applicable for the storage option and is not for the swing option. The swing option will be considered separately in the section \[sec:swing\_option\].
Formulation of the stochastic optimisation problem {#sec:stoch_opt_problem}
--------------------------------------------------
For the intrinsic problem one could determine an optimal exercise strategy $r(T)$ which has a meaning of amount (per unit time) of underlying which has to be injected/released from the storage and bought/sold on the market at the time $T$. It makes no difference if this amount is sold on the spot (i.e. exactly on the time $T$) or on the forward market prior the delivery time $T$, since the price of the underlying does not change.
This is not the case in the stochastic reality. Since the price for the delivery on time $T$ changes with the observation time $t$, it is essential to specify when exactly the underlying is traded. We thus distinguish between spot trades and forward trades. A spot trade is associated with immediate delivery, and the corresponding price is referred to as “spot price” $s(t) = F(t,T=t)$. A forward trade has a delivery in the future, and generally can be any kind of financial products, including forward contracts or options. The forward trades usually have a purpose of financial hedging, so we will refer to them as *hedge trades*. For simplicity we only consider linear hedge products, i.e. forward contracts. Thus for the stochastic problem at every time $t$ we can introduce a trade profile $h(t,T)$, which has the following meaning:
- The up-front value $h(t,t)$ is the exercise trade. The amount of underlying $h(t,t)\,dt$ has to be injected in the storage within the time $dt$, and the corresponding amount has to be purchased on the market. The purchased amount is a net amount purchased on the forward and spot markets for the delivery on $T=t$. We will refer to the exercise trade $h(t,t)$ as *prompt exercise*.
- The volume $h(t,T)$ for $T>t$ is a hedge position. The value $h(t,T)\,dT$ has a meaning of the total volume purchased on the forward market (i.e. aggregated volume of all forward trades performed during the time period $[0,t]$) for the delivery period $[T,T+dT]$.
For the stochastic optimisation problem there exists no optimal exercise trajectory, which could be computed prior or during the contract time. Hence it is not possible do define a target functional in the form of integral (\[eq:unperturbed\_action\]). Instead we have to formulate the aim of the stochastic optimisation problem as finding an optimal trading strategy $h(t,T)$, such that the expectation value of the total profit is maximised.
The trading profile can be used for exercise and hedge trades on the future market, giving rise to the cash flow at the observation time[^2]. Since the hedge profile is different for every time moment $t$, one has to update the hedge position continuously by buying and selling a proper amount of the underlying on the future and spot markets. Let $h(t,T)$ be the hedge position taken by the time $t$. This also includes the exercise volume $h(t,t)\,dt$, which has to be delivered during the time $dt$. After the infinitesimal time increment $dt$ we observe the new prices on the market, and find a new optimal hedge strategy $h(t+dt,T)$. The volume $$\delta h(t,T) = h(t+dt,T) - h(t,T)$$ has to be purchased on the market in order to update the hedge position. This volume includes the spot trade $\delta h(t,t)\,dt$ and hedge trades $\delta h(t,T)\,dT$ for $T>t$. Of the whole hedge profile $h(t,T)$ the only initial value $$\begin{aligned}
\label{eq:hedge_profile_exercise_part}
\left. h(t, T)\right|_{T=t}\,.\end{aligned}$$ is associated with the “physical” delivery of the underlying, and the rest is covered by the forward contracts.
Our aim now is to find for every observation time $t$ an optimal hedge profile $h(t,T)$ conditional on the market state $F(t,T)$ and current initial condition $q(t)$. In order to give a precise mathematical definition of the stochastic optimisation problem we need to define the stochastic profit function.
Let $h(0,T)$ be the initial hedge profile, calculated on the basis of the initial conditions $F(0,T)$ and $q(0) = Q_{start}$. The integral $$\begin{aligned}
\label{eq:4.2}
P_0 = - \int_0^{T_e} h(0,T) \, F(0,T)\,dt\,.\end{aligned}$$ gives the total cost of setting up the initial hedge profile. Let $h(t,T)$ be the hedge profile at time $t$. We introduce the profit function $P(t)$, which is defined as a cumulative cash flow from all hedge trades. The profit function satisfies the following equations: $$\begin{aligned}
\label{eq:5.5}
&P(0) = P_0 = -\int_0^{T_e} h(0,T)\,F(0,T)\,dT \\
\label{eq:5.6}
&dP(t) = - \int_t^{T_e} \delta h(t,T)\,
\Big( F(t,T) + \delta F(t,T) \Big)\,dT \,,\end{aligned}$$ where $\delta h$ and $\delta F$ are the increments of the optimal hedge strategy and forward curve within a time-interval $dt$. The stochastic optimisation problem can now be formulated as finding at every time $t$ an optimal hedge profile $h(t,T)$ such that the expectation value of the terminal profit $P(T_e)$ is maximised.
The defined profit function does not represent the total portfolio value, since it only reflects the cash-flow and does not take into account the value of the storage option itself. However this is the only part of the portfolio, which can be influenced by our trading strategy, and hence optimisation of the profit function is equivalent to the optimisation of the Profit and Loss.
Note that the hedge profile $h(t,T)$ for $T>t$ does not need to satisfy the storage constraints, since it is not associated with the physical delivery. The only physical part of the hedge profile is the prompt exercise trade $h(t,t)$ which has to obey the storage constraints.
Notice that the integral (\[eq:4.2\]) coincides with the target functional (\[eq:unperturbed\_action\]) (we have neglected the operating costs). As an example of a (non-optimal) trading strategy we can do the following. Let the initial hedge profile coincides with the initial intrinsic strategy $$h(0,T) = r(0,T)$$ In this case the initial value of the profit function coincides with the intrinsic value $P(0) = P_0 = S_0$. If we decided not to change the hedge profile with the time[^3] $$h(t,T) = h(0,T)$$ then the profit function would remain constant, giving rise to the terminal profit equal to the initial intrinsic value. Thus the stochastic profit is at least as big as the intrinsic value. In reality for every time $t$ one can almost always find a new hedge profile $h(t,T)$ such that the profit function increment $\delta P$ is positive. The increase of the terminal profit function in comparison with the intrinsic value is referred to as *time value*.
Running ahead it is also interesting to notice that only the prompt exercise decision $h(t,t)$ is relevant for the expectation value of the profit. The rest of the hedge profile $h(t,T)$ for $T>t$ only effects the distribution of the terminal profit as a stochastic value. If our model of the stochastic price process was exact, and if we neglected the market frictions, then it would be theoretically possible to find such a hedge profile, which would make the distribution of the terminal profit function arbitrarily narrow.
Rolling intrinsic strategy
--------------------------
The optimal intrinsic solution at time $t$ is a function $\dot q(T) = r(t,T)$, which maximises the deterministic functional $$\begin{aligned}
- \int_t^{T_e} r(t,T) \, F(t,T)\,dT\end{aligned}$$ given the initial condition $q(t)$ and the market state $F(t,T)$ at the time $t$, and subject to the storage constraints.
If at every time moment $t$ the hedge strategy $h(t,T)$ coincides with the current optimal intrinsic exercise $$h(t,T) = r(t,T)\,,$$ then the strategy is called *rolling intrinsic*.
In the Sec. \[sec:4.6.4\] we show that under very general assumptions the rolling intrinsic is very good approximation of the optimal stochastic exercise strategy. Based on this fact and taking into account that the hedge strategy does not effect the expected value of the profit function, one can conclude that if we follow the rolling intrinsic strategy, the expected value of the terminal profit must be equal the true option value. This also allows one to calculate the time value as an average cumulative cash flow from all hedge and exercise trades. In App. \[ap:A2\] we give a systematic proof of this fact.
Forward curve evolution model
-----------------------------
Let $F(t,T)$ for each time $T$ be a zero mean stochastic process governed by the stochastic differential equation in Ito representation $$\begin{aligned}
\label{eq:generic_price_process}
&\frac{dF(t,T)}{F(t,T)} = dM_t(T)\,,\end{aligned}$$ where $M_t(T)$ is a martingale process with respect to the variable $t$ parametrised by the variable $T$. In general for each future time $T$ the martingale processes $M_t(T)$ are different processes, which might or might not be correlated.
The correlation between processes $M(T)$ is given by $$\begin{aligned}
& {\left\langle}\, dM_t(T_1)\,dM_t(T_2)\,{\right\rangle}= \sigma^2(t,T_1,T_2)\,dt\,;\qquad
{\left\langle}\, dM_{t_1}(T_1)\,dM_{t_2}(T_2)\,{\right\rangle}= 0 \quad \text{for} \quad t_1 \neq t_2\,,\end{aligned}$$ where the function $\sigma(t,T_1,T_2)$ is supposed to be a decaying function of $|T_1-t|$, $|T_2-t|$, and $|T_2-T_1|$. In the stochastic calculus the product $dM_{t_1}(T_1)\,dM_{t_2}(T_2)$ can be considered as deterministic and requires no averaging.
Let us designate $$\begin{aligned}
L(t, T_1,T_2) = dF(t, T_1)\,dF(t, T_2) = F(t,T_1)\,F(t,T_2)\,\sigma^2(t,T_1,T_2)\,dt\,.\end{aligned}$$ Notice that $L$ is a stochastic variable, since it depends explicitly on the stochastic prices $F$.
We define the price process *correlation function* as $$\begin{aligned}
\Lambda(t,T_1,T_2) = \frac{1}{dt}\,{\left\langle}L(t,T_1,T_2) {\right\rangle}_{F} =
\sigma^2(t,T_1,T_2) \,{\left\langle}F(t,T_1)\,F(t,T_2) {\right\rangle}\,,\end{aligned}$$ where ${\left\langle}\cdot{\right\rangle}_{F}$ is an averaging over the stochastic prices $F$.
As a particular example we consider the following process $$\begin{aligned}
& dM_t(T) = \sigma_0\,e^{-\alpha(T-t)}dW_t(T)\,,\\
& {\left\langle}\,dM_t(T_1)\,dM_t(T_2)\,{\right\rangle}=
\sigma_0^2\,e^{-\alpha(T_1-t)}\,
e^{-\alpha(T_2-t)}\,
e^{-\beta\,|T_2-T_1|}\,dt\,.\end{aligned}$$ where $W_t$ is a standard Brownian motion. The price process has the following correlation function $$\begin{aligned}
\Lambda(t, T_1,T_2) =
\sigma_0^2\,e^{-\alpha(T_1-t)}\,
e^{-\alpha(T_2-t)}\,
e^{-\beta\,|T_2-T_1|}\, {\left\langle}F(t,T_1)\,F(t,T_2) {\right\rangle}\,.\end{aligned}$$ Note that the case $\beta=0$ corresponds to the standard one-factor forward curve model. In this case the whole curve is driven by a single Wiener process $W_t$.
To find the correlation ${\left\langle}F(t,T_1)\,F(t,T_2){\right\rangle}$ we need to integrate the price process (\[eq:generic\_price\_process\]). For one-factor model we obtain $$\begin{aligned}
&{\left\langle}F(t,T_1)\,F(t,T_2){\right\rangle}= F_0(T_1)\,F_0(T_2)\,l(t,T_1,T_2)\,, \qquad\text{where} \\
&l(t,T_1,T_2)= \exp\left[
\frac{\sigma_0^2}{2\,\alpha}\, e^{-\alpha\,(\tau_1+\tau_2)} \left(1 - e^{-2\,\alpha\,t } \right)
\right]\,; \quad \tau_1 = T_1-t\,;\quad \tau_2 = T_2-t\,. \nonumber\end{aligned}$$ Note also that after a relaxation time $\approx 1/\alpha$ the normalised correlation function $l$ becomes stationary, i.e. depending only on the time differences $\tau_1$ and $\tau_2$.
It can be easily shown that for small volatility $\sigma_0^2\,t \ll 1$ the function $l(t,T_1,T_2)\approx 1$, and thus for one-factor model we get $$\begin{aligned}
\Lambda(t, T_1,T_2) \approx
\sigma_0^2\,e^{-\alpha( \tau_1 + \tau_2 )} \,F_0(T_1)\,F_0(T_2)\,.\end{aligned}$$
Constraint surface
------------------
We apply a variational analysis technique to find the increments of the hedge position $\delta h(t,T)$ in the rolling intrinsic approximation, and to derive a stochastic differential equation for the storage option value evolution based on Eqs. (\[eq:5.5\]) and (\[eq:5.6\]).
When finding the functional derivatives, it is important to distinguish between the two following cases. Let ${\cal L}[\phi_1(t), \phi_2(t)]$ be a functional defined on the functions $\phi_1(t)$ and $\phi_2(t)$. If they are considered as independent variables, the corresponding functional derivatives $$\frac{\delta{\cal L}}{\delta\phi_1(t)} \qquad \text{and} \qquad \frac{\delta{\cal L}}{\delta\phi_2(t)}$$ are analogous to the partial derivatives of an ordinary function of two variables.
On the other hand, if the variables $\phi_1$ and $\phi_2$ are dependent, then the variation $$\delta{\cal L} = \frac{\delta{\cal L}}{\delta\phi_1(t)}\,\delta\phi_1(t) +
\frac{\delta{\cal L}}{\delta\phi_2(t)}\,\delta\phi_2(t)\,; \qquad
\delta\phi_2(t) = f(\delta\phi_1(t))$$ is analogous to a total derivative of an ordinary function.
Let us consider a functional ${\cal L}$ defined on the variables $t,C(t),F(t,T)$. If we consider a time evolution of the forward curve $F(t,T)$, then all three increments $dt, \delta C(t)$ and $\delta F(t,T)$ become dependent. The relation between these increments follows from the requirement that the terminal level must be preserved: $$\begin{aligned}
& \delta Q_{end} = 0\,; \qquad \text{where} \nonumber \\
\label{eq:constraint}
& Q_{end} = Q(t) + \int_t^{T_e} r(t,T)\,dT \,;\end{aligned}$$ The equation $\delta Q_{end} = 0$ for every time $t$ defines a surface in the space of $F(t,T)$ and $C(t)$, which restricts the possible simultaneous variations of the trigger price and forward curve. We say that the equation $\delta Q_{end} = 0$ defines a *constraint surface*. If we find a variation $\delta {\cal L}$ which takes into account the relation between $\delta t$, $\delta C(t)$ and $\delta F(t, T)$, we refer to this variation as a *variation on the constraint surface*.
Since the time-dependent increment of the forward curve $\delta F$ is driven by a Wiener-type stochastic equation of motion, we will treat this increment in a stochastic sense. In the framework of the stochastic calculus the square increment $\delta F^2$ can not be neglected if compared with the first order variation, since $$\delta F^2 \sim dt\,.$$ Hence the time-dependent variational derivatives must contain the second order terms alongside with the first order terms.
The details of derivation can be found in the Appendix \[ap:Contraint\_surface\]. After some algebra we finally obtain $$\begin{aligned}
\label{eq:dC_2_order}
&\delta C = \frac{1}{K}
\left( \int r^{(1)} \,\delta F\,dT -
\frac12 \int r^{(2)}\,\delta F^2 \,dT +
\frac{1}{K}\iint r^{(2)}_u\,r^{(1)}_v\,L_{uv} -
\frac{M}{2\,K^2}\iint r^{(1)}_u\,r^{(1)}_v\,L_{uv}
\right)\,; \\
& \delta C^2 =\frac{1}{K^2}\iint r^{(1)}_u\,r^{(1)}_v\,L_{uv}\,,\end{aligned}$$ where we have used the notation $$\begin{aligned}
& r^{(1)}(t,T) = \Delta r\, \delta(C(t)-F(t,T)) \,; && r^{(2)}(t,T) = \Delta r\, \delta'(C(t)-F(t,T))\,; \\
& K(t) = \int_t^{T_e} r^{(1)}(t,T)\,dT\,; && M(t) = \int_t^{T_e} r^{(2)}(t,T)\,dT\,; \\
& L_{uv} = \delta F(u) \, \delta F(v)\,. \end{aligned}$$ In particular with this notation we can write $$\begin{aligned}
\left[\int r^{(1)}\,\delta F\,dt\right]^2 =
\iint r^{(1)}(u)\,r^{(1)}(v)\, \delta F(u) \, \delta F(v) \,du\,dv =
\iint r^{(1)}_u\,r^{(1)}_v\,L_{uv}\,.\end{aligned}$$ Eq. (\[eq:dC\_2\_order\]) defines the constraint surface. It relates the increments $dt, \delta C(t)$ and $\delta F(t,T)$. Note that according to the forward curve evolution model $L_{uv}\sim\delta t$.
The first order variation of the trigger price on the constraint surface is then given by $$\begin{aligned}
\label{eq:dC_1_order}
& dC(t) = \frac{1}{K(t)} \int_t^{T_e} r^{(1)}(t,T) \,\delta F(T)\,dT \,.\end{aligned}$$
Stochastic trigger price {#sec:4.6.4}
------------------------
In the deterministic formulation of the storage problem, for each predefined forward curve $F(t,T)$ we can find an optimal exercise trajectory $\dot{\bar q}(T) = r(t,T)$. The value of the target function $S$ on the optimal trajectory at time $t=0$ is called “intrinsic value”. If the prices are stochastic, there exist no optimal exercise trajectory, and the solution of the stochastic problem can only provide an optimal prompt exercise $r(t,t)$ at every time $t$.
In this section we show that the optimal stochastic exercise is bang-bang, and that for every time moment $t$ there exist a stochastic trigger price $C_{st}(t)$. We also show that the intrinsic trigger price is a good approximation for the stochastic one, and derive the conditions for this approximation.
Let $F(t,T)$ be the forward curve observed on the time $t$. We define a *spot price* process $s(t)$ as $$\begin{aligned}
s(t) := F(t,t).\end{aligned}$$ Since the forward price curve evolves with time, the spot price $s(t) = F(t,t)$ generally does not follow the original forward curve ($F(t,t) \neq F(0,t)$ almost everywhere). The spot process $s(t)$ is uniquely defined by the forward price process, and thus to each possible spot price curve $s(t)$ one can assign an amplitude or, in case of discrete price and time space, a probability $p$.
Let us consider now a set of all possible spot price curves $s_i(t)$ each having corresponding probability $p_i$. Considering the spot price curve $s_i(t)$ as a deterministic function, we can solve a static optimisation problem for the target functional $$S_i = -\int_0^{T_e} \dot q(t)\,s_i(t)\,dt\,.$$ Thus, for every spot price curve $s_i(t)$ we can find a corresponding trigger price $C_i$, exercise strategy $r_i(t)$ and intrinsic value $S_i$. On each spot price path the trigger price is different, and the exercise strategy $r_i(t)$ is optimal only for that path.
Now we are looking for an optimal prompt exercise decision $\dot q(0)$ for the stochastic problem. Since we can only make one exercise decision at a time, this exercise decision will not be optimal for some spot price paths. If on the $i$th spot price path the optimal exercise is $r_i(0)\neq\dot q(0)$, the suboptimal exercise decision will lead to a loss of value on that path. We need to make such a choice of $\dot q$ that would minimise the losses on each path, for which the decision is not optimal.
During the time interval $dt$ the volume increment in the storage is given by $dq = \dot q\,dt$. Thus, on the $i$th spot price path the volume in the storage is by $(\dot q(0) - r_i(0))\,dt$ bigger than it had to be, if the exercise was optimal for that path. It leads to the change of the value on the $i$th path by $$\begin{aligned}
\label{eq:4.13}
\delta S_i = \Big( C_i - F(0,0) \Big)\Big( \dot q(0) - r_i(0)\Big)dt\end{aligned}$$ Indeed, the change of the value has two reasons – additional expenses $\delta S_{i1}$ of buying an additional volume, and increased value $\delta S_{i2}$ due to change in the actual storage volume. The price of additional volume is $$\delta S_{i1} = - F(0,0)\, \Big( \dot q(0) - r_i(0)\Big)\,dt.$$ The change of the value due to change of initial volume follows from the formula ${\partial}S/ {\partial}q_{start} = C$ (see Sec. \[sec:3\]): $$\delta S_{i2} = C_i \,\Big( \dot q(0) - r_i(0)\Big)\,dt\,.$$ Combining the latter two equations we obtain Eq. (\[eq:4.13\]). Now we average $\delta S$ over all paths: $$\begin{aligned}
{\left\langle}\delta S {\right\rangle}= \sum_i p_i \Big( C_i - F(0,0) \Big)
\Big( \dot q(0) - r_i(0)\Big)dt\,,\end{aligned}$$ where $p_i$ is the probability of the $i$th path. The optimal exercise decision $\dot q(0)$ should maximise the expected change of option value ${\left\langle}\delta S {\right\rangle}$. Deriving ${\left\langle}\delta S {\right\rangle}$ with respect to $\dot q(0)$ we obtain $$\begin{aligned}
\frac{{\partial}{\left\langle}\delta S {\right\rangle}}{{\partial}\dot q(0)} = \sum_i p_i\,\Big( C_i - F(0,0) \Big)dt =
\Big( {\left\langle}C {\right\rangle}- F(0,0)\Big) \,dt\,,\end{aligned}$$ where ${\left\langle}C {\right\rangle}= \sum_i\,p_i\,C_i$ is the trigger price averaged over all possible spot price processes. If the spot price $F(0,0)$ is bigger than the average trigger price ${\left\langle}C{\right\rangle}$, then the derivative is negative, and the optimal stochastic exercise is the smallest possible allowed by constraints. Similarly if $F(0,0)$ is smaller than the average trigger price, then the optimal exercise is the biggest possible allowed by constraints. Thus, the optimal prompt stochastic exercise is $$\begin{aligned}
\dot q(0) = \left\{
\begin{array}{ll}
r_{min}(t)\,, \qquad & F(t) > {\left\langle}C {\right\rangle}\,; \\
r_{max}(t)\,, \qquad & F(t) < {\left\langle}C {\right\rangle}\,;
\end{array}
\right.\end{aligned}$$ We see that the optimal stochastic exercise is bang-bang, and that the expected value of the intrinsic trigger price ${\left\langle}C {\right\rangle}$ can be interpreted as the stochastic trigger price $C_{st}$: $$\begin{aligned}
\label{eq:stochastic_trigger_price_1}
C_{st} = {\left\langle}C {\right\rangle}\,.\end{aligned}$$
Next we can show that under some assumptions the stochastic trigger price in the leading order of Taylor expansion equals the intrinsic trigger price. Indeed, let us designate $s_0(t)$ that spot price path which coincides with the initial forward curve at time $t= 0$: $$s_0(t) = F(0,t)\,.$$ The trigger price $C_0$ is then the intrinsic trigger price, calculated at time $t=0$ on the basis of the forward curve $F(0,t)$. All other paths $s_i(t)$ are different, and we designate $$\delta s_i(t) = s_i(t) - s_0(t)\,.$$ This difference is zero at the beginning of time period: $\delta s_i(0) = 0$ since all paths of the spot price process start in the same point $s_i(0) = F(0,0)$. At the end of the period the difference $\delta s_i(T_e)$ may be arbitrary large, but the majority of paths stay within the range $$\sqrt{{\langle\!\langle}\delta s^2(T_e) {\rangle\!\rangle}}$$ If $\delta s_i(t)$ remains small, then the difference in the trigger level $\delta C_i = C_i-C_0$ can be calculated by means of the first order Taylor expansion (\[eq:dC\_1\_order\]): $$\delta C_i \approx \frac{\Delta r}{K(0)} \int_0^{T_e} \delta(C_0-s_0(t))\,\delta s_i(t)\,dt\,.$$ Since the forward price process is a martingale, we conclude that ${\left\langle}\delta s_i(t) {\right\rangle}= 0$, and hence in the first order ${\left\langle}\delta C_i {\right\rangle}= 0$. Consequently $$\begin{aligned}
\label{eq:stochastic_trigger_price}
C_{st} = {\left\langle}C {\right\rangle}= C_0 + {\left\langle}\delta C {\right\rangle}\approx C_0\,,\end{aligned}$$ which means that the stochastic trigger price equals the intrinsic trigger price. Since any time moment $t$ during the exercise period can be considered as a starting point, the following statement holds: for any time $t$ the optimal stochastic exercise is bang-bang, and the stochastic trigger price $C_{st}(t)$ is approximately equal to the intrinsic trigger price $C_0(t)$, calculated at time $t$ on the basis of the deterministic forward curve $F(t,T)$.
Note that to derive the relation (\[eq:stochastic\_trigger\_price\_1\]) we did not use any approximation, and hence this relation is exact. For derivation of relation (\[eq:stochastic\_trigger\_price\]) we used the only assumption of small variation $$\begin{aligned}
\label{eq:condition_for_rolling intrinsic}
\frac{ \sqrt{{\langle\!\langle}\delta s^2(T_e) {\rangle\!\rangle}}}{s_0(T_e)}\ll 1\,.\end{aligned}$$ For a 1-factor mean reversion price process the relative spot price variation can be easily estimated: $$\frac{ \sqrt{{\langle\!\langle}\delta s^2(T_e) {\rangle\!\rangle}}}{s_0(T_e)} =
\sqrt{ \exp \left[ \frac{\sigma_0^2}{2\,\alpha}\,\Big(1- e^{-2\,\alpha\,T_e}\Big)\right]-1 }\,.$$ In particular, for $\alpha\,T_e\ll 1$ the condition for the intrinsic exercise approximation becomes $$\sqrt{ \exp(\sigma_0^2\,T_e) -1} \ll 1\,,$$ and for $\alpha\,T_e\gg 1$ we get the condition $$\sqrt{ \exp \left(\frac{\sigma_0^2}{2\,\alpha}\right) - 1} \ll 1\,.$$
Variation of the intrinsic target function
------------------------------------------
In this section we find a variation of the intrinsic target function on the constraint surface. We take into account the dependency between variations of the forward curve and the intrinsic trigger price, as in Eq. (\[eq:dC\_2\_order\]).
Below we omit the argument $t$ and use the notation $F(T)\equiv F(t,T)$. We write the intrinsic target function in the form $$\begin{aligned}
S(t) = -\int_t^{T_e} r(T)\,F(T) \, dT\,;\qquad \text{where} \quad
r(T) = r_{min} + \Delta r\,\theta(C - F(T))\,.\end{aligned}$$ Next we use the expansion $$\begin{aligned}
\delta S = \delta_C S + \delta_F S + \frac12( \delta_{FF} S + 2\,\delta_{CF} S + \delta_{CC} S)\,.\end{aligned}$$ The variational derivatives can be found easily: $$\begin{aligned}
& \delta_C S = \frac{{\partial}S}{{\partial}C}\,\delta C = -C\,K\,\,\delta C\,; \\
& \delta_F S = \int \frac{\delta S}{\delta F(T)}\,\delta F(T)\,dT =
\int \Big( C\,r^{(1)} - r \Big)\,\delta F\,dT\,;\end{aligned}$$ $$\begin{aligned}
& \delta_{FF} S = \iint \frac{\delta^2 S}{\delta F(u)\,
\delta F(v)}\,\delta F(u)\,\delta F(v)\,du\,dv =
\int \Big( 2\,r^{(1)} - r^{(2)}\,F \Big)\,L_{uu}\,du \,; \\
& \delta_{FC} S = \int \frac{\delta^2 S}{\delta F(T)\,{\partial}C}\,\delta F(T)\,dT\,\,\delta C =
\frac{1}{K} \iint \Big( r^{(2)}_u\,F_u - r^{(1)}_u \Big)\,r_v \,
L_{uv}\,\,du\,dv = \nonumber \\
& \hspace{6cm} =\frac{C}{K} \iint r^{(2)}_u\,r^{(1)}_v\,L_{uv}\,\,du\,dv \,;\\
& \delta_{CC} S = -\int r^{(2)}\,F \,dt\,\,\delta C^2 = - \frac{K + C\,M}{K^2}
\iint r^{(1)}_u\,r^{(1)}_v\,L_{uv}\,\,du\,dv\,;\end{aligned}$$ where $L_{uv} = \delta F(u)\,\delta F(v)$, $L_{uu} = \delta F^2(u)$.
Combining all variation terms we finally obtain: $$\begin{aligned}
\label{eq:deltaS}
\delta S = & -\int r(T) \,\delta F(T)\,dT + \frac12 \int r^{(1)}\, L_{uu} \,du -
\frac{1}{2\,K} \iint r^{(1)}_u\,r^{(1)}_v \,L_{uv}\, \,du\,dv \,.\end{aligned}$$ We can easily interpret the first term appearing in the latter expression. The variation $$-\int r(T)\,\delta F(T)\,dT$$ reflects the change of the value of the current hedge volume of underlying due to the changed forward prices, provided the hedge volume equals the optimal intrinsic exercise $r(T)$.
Next non-vanishing term is of the second order: $$\frac12 \int r^{(1)}(T)\,\delta F^2(T)\,dT =
\frac{\Delta r}{2} \int \delta(C-F(T))\,\delta F^2(T)\,dT\,.$$ Due to the delta-function under the integral this term is reduced to a sum over trigger times $\{T_i: F(T_i) = C \}$. This term appears due to the change of the perturbed trajectory $r(T)+\delta r(T)$. It can be shown (see Sec. (\[sec:delta\_r\])) that the perturbed trajectory almost everywhere coincides with the unperturbed one $r(T)$ except a few points, where the transition between injection and release takes place. These points are the trigger times. Change of the strategy on an infinitely small period of time around every trigger time leads to the second order correction of the target function.
It is worth noticing that the second order correction is always non-negative. This reflects the fact that the rolling intrinsic strategy leads to a non-negative drift of the storage value (see below).
The term $$\frac{1}{2\,K} \iint r^{(1)}_u\,r^{(1)}_v \,L_{uv}\, \,du\,dv$$ takes into account the cross-correlations between the forward price returns for different maturities.
### Interpretation in financial terms
In the financial calculus the derivatives of the target function with respect to the different parameters deserve a special attention. The most relevant of them for the portfolio management are usually designated by Greek letters, and traditionally called all together “Greeks”.
One of the most important derivatives is the derivative of the option value $V$ with respect to the underlying price $F$: $$\Delta = \frac{{\partial}V}{{\partial}F}$$ It is termed “delta” and can be used for constructing a delta-neutral portfolio – such a portfolio, which value does not change under infinitesimal changes of the price.
The second derivative of the option value with respect to the spot price is termed “gamma”: $$\Gamma = \frac{{\partial}^2 V}{{\partial}F^2} \,.$$ The gamma is a measure of the curvature of the target function with respect to the underlying price. It is also responsible for the time value of the option.
Combining first and second derivatives, the variation of the option value can be represented up to the second order as $$\begin{aligned}
dV = \Delta\,dF + \frac12\,\Gamma\,dF^2\,.\end{aligned}$$
Comparing the latter equation with Eq. (\[eq:deltaS\]) we conclude that the first order term in the expansion represents the option “delta” (within the intrinsic strategy approximation): $$\begin{aligned}
\Delta \sim - \int r(u)\,\delta F(u) \,du\,,\end{aligned}$$ and the second order term represents the storage option “gamma”: $$\begin{aligned}
\Gamma \sim \int r^{(1)}(u)\,L_{uu}\,du -
\frac{1}{K} \iint r^{(1)}(u)\,r^{(1)}(v) \, L_{uv} \,\,du\,dv\,.\end{aligned}$$
Variation of the intrinsic extremal trajectory {#sec:delta_r}
----------------------------------------------
Here we find a variation of the optimal intrinsic trajectory $r(T)$ on the constraint surface. We remind that the optimal intrinsic trajectory can be represented as $$r(T) = r_{min} + \Delta r\,\theta(C - F(T))\,,$$ where we have omitted the observation time $t$. Expanding to the second order in $F$ and $C$ we obtain: $$\begin{aligned}
\delta r(T) = \frac{{\partial}r(T)}{{\partial}F}\,\delta F(T)
+ \frac{{\partial}r(T)}{{\partial}C}\,dC
+ \frac12\,\frac{{\partial}^2 r(T)}{{\partial}F^2}\,\delta F^2(T)
+ \frac{{\partial}^2 r(T)}{{\partial}F\, {\partial}C}\,\delta F\, dC
+ \frac12\,\frac{{\partial}^2 r(T)}{{\partial}C^2}\, dC^2 \, .\end{aligned}$$ Evaluating the partial derivatives and making use of Eq. (\[eq:dC\_2\_order\]) we finally get $$\begin{aligned}
\label{eq:delta_r}
\delta r(T)& = -r^{(1)}\,\delta F + \frac12\,r^{(2)}\,L_{tt} -
\frac{r^{(2)}}{K}\,\delta F \int r^{(1)}_v\,\delta F_v +
\frac{r^{(2)}}{2\,K^2}\iint r^{(1)}_u\,r^{(1)}_v\,L_{uv} +\nonumber \\
& \frac{r^{(1)}}{K}\,\Big(
\int r^{(1)}_u\,\delta F_u
- \frac12 \int r^{(2)}_u\,L_{uu}
+ \frac{1}{K}\iint r^{(2)}_u\,r^{(1)}_v\,L_{uv}
- \frac{M}{2\,K^2}\,\iint r^{(1)}_u\,r^{(1)}_v\,L_{uv}
\Big).\end{aligned}$$
We see that $\delta r(T)$ has singularities at the trigger times $\{T_i:\ F(T_i) = C\}$. We need to give an interpretation to this fact, since the trajectory correction $\delta r(T)$ in this form violates the maximum injection/release constraints. We can show that the singular trajectory variation can be interpreted as a shift of trigger times $T_i$ by some value $\delta T$.
We can give an interpretation to the obtained singularity of $\delta r$, if we consider it under an integral $\int \delta r(T) \,\phi(T)\,dt$, where $\phi(T)$ is an arbitrary function of time. Approximately the $\delta-$function can be replaced by a step function with support $\delta t$ and height $\Delta r$. The duration $\delta T$ should be found to fit the integral value.
Let us consider $\delta r$ in a form $$\delta r(T) = k\,r^{(1)}(T) = k\, \Delta r\,\delta(C-F(T))\,.$$ Let $T^*$ be a trigger time. Then the integral around this time gives: $$\int_{T^*-\epsilon}^{T^*+\epsilon} \delta r(T) \, \phi(T)\,dT =
\frac{k\,\Delta r}{|\dot F(T^*)|}\,\phi(T^*)\,.$$ As stated above, the variation $\delta r(T)$ around the trigger time $T^*$ can be approximately replaced with a step function $\eta(T)$ with support $\delta T$ and height $\Delta r$. The width $\delta T$ can be estimated from simple consideration: $$\int_{T^*-\epsilon}^{T^*+\epsilon} \delta r(T) \,\phi(T)\,dT =
k\,\Delta r\,\frac{\phi(T^*)}{|\dot F(T^*)|} \approx
\Delta r\,\delta T\,\phi(T^*) \,.$$ We thus find for the absolute value of time shift: $$\begin{aligned}
\delta T \approx \left| \frac{k}{ \dot F(T^*)}\right|\,.\end{aligned}$$ Since the time $t^*$ just splits the regions of $\dot q = r_{min}$ and $\dot q = r_{max}$, the previous consideration has a meaning, that the singularity of $\delta r$ can be interpreted as shifting of the trigger time $T^*$ by the value $\delta T$ to the right or left. If $k > 0$, then the trigger time is always shifted in the direction of injection. If $k < 0$, then the trigger time is always shifted in the direction of release. We can represent the time shift, which takes the sign into account: $$\begin{aligned}
\label{eq:time_shift}
\delta T = - \frac{k}{ \dot F(T^*)}\,.\end{aligned}$$
An important property of $\delta r$ is that the integral $\int \delta r(T)\,dT$ must vanish. Indeed, from Eq. (\[eq:constraint\]) we conclude that $$\delta Q_{end} = \int \delta r(T)\,dT = 0\,.$$ Integrating Eq. (\[eq:delta\_r\]) and taking into account the definitions $K = \int r^{(1)}(T)\,dT$\
and $M = \int r^{(2)}(T)\,dT$ we find $$\begin{aligned}
& \int \delta r(T)\,dT = 0\,.\end{aligned}$$ This proves the consistency of the expansion formula (\[eq:delta\_r\]).
Dynamic hedge and time value
----------------------------
As a “delta” of a vanilla option we understand a derivative of the option value with respect to the price of the underlying. This derivative has a dimension of volume and has an obvious meaning: this is an amount of the underlying, whose value changes (up to the first order) to the same extent under the small variation of the price, as the option value itself.
The delta can be used for the delta-hedging. If the holder of the option takes a forward (hedge) position, which equals the delta in its amount, but has an opposite sign, then the combination of both – option and hedge position – has vanishing derivative with respect to the price. This means that for a very small variation of the price, i.e. for a very short period of time, the value of the portfolio remains constant (up to the first order). In this case we say that the portfolio is delta-neutral. Since the delta itself depends on the price, the position, which was delta-neutral at some time point, will not be so after a short period of time. To remain delta-neutral, one has to recalculate the delta and update the hedge position continuously. In an ideal situation, if the update of the hedge position can be done continuously, and if the market friction can be neglected, the value of the portfolio $\Pi$ (which includes the option value, the value of the hedge volume and cumulative cash flows) remains constant.
The similar logic applies to the storage option. We can identify such a volume of underlying, which we can use as a hedge position. But unlike the vanilla option the storage option has an array of maturity times, which can be thought of as an array of different hedge products, each having a different forward price $F_k = F(T_k)$. For every delivery time $T_k$ we can calculate a derivative ${\partial}V(F)/{\partial}F_k$ with respect to the forward price on that particular maturity. The delta in this case becomes an array: $$\begin{aligned}
\Delta_k = \frac{{\partial}V(F)}{{\partial}F_k}\,,\end{aligned}$$ which is a discrete analogue of a functional derivative $\Delta(T) = \delta V/\delta F(T)$.
At the beginning of the storage contract the portfolio consists only of the storage option. The initial portfolio value (sometimes referred to as “Profit and Loss” - P&L) equals the estimated option value. As the storage starts operating, the owner of the storage sells and buys the underlying and changes the hedge position. The P&L changes according to the actual cash-flows, option value and the value of the hedge position. At the end of the storage contract the terminal P&L consists only of the cumulative cash-flow, since no open hedge position is left after the end of the storage period. If the hedge position is optimal at any time, then the P&L remains constant, and the terminal P&L equals the initial one, i.e. the storage owner earns exactly the amount predicted by the stochastic storage model.
If the storage delta can not be calculated exactly, then the P&L becomes volatile, fluctuating around some expected value. Depending on the “quality” of the hedge position, the deviation of the terminal P&L from the expected value may become smaller or bigger.
Below we show that the expected value of the terminal P&L depends exceptionally on the prompt exercise trades (i.e. on the value of the delta at the current time $T=t$), whereas the width of the distribution of the terminal P&L depends on the “quality” of the hedge position. Whatever the hedge strategy is used, the correct average time value is obtained only if at every time moment the prompt exercise $\dot q(t)$ is optimal.
In the next section we consider the rolling intrinsic exercise strategy as one of the possible hedge strategies. The rolling intrinsic strategy has a number of remarkable properties. One important feature of the rolling intrinsic is that it provides an (almost everywhere) optimal exercise decisions, and hence guarantees that the expectation value of the P&L equals the true option value. Rolling intrinsic also allows to estimate the option time value by calculating the cumulative cash flows from the hedge and exercise trades.
### Rolling Intrinsic strategy
The deterministic consideration of the storage problem – the intrinsic value – gives a good first order estimate of the storage value. The intrinsic value can be computed for example by a dynamic programming algorithm, and has a remarkable property of the trigger price: the optimal intrinsic exercise strategy is bang-bang, each bang triggered by crossing the trigger level by the forward price curve.
The stochastic reality makes the true value of the storage option different, adding to it a time value. It appears due to additional profit opportunities on the price fluctuations. As discussed in the Sec. \[sec:4.6.4\], under some assumptions a good approximation for the “stochastic” trigger price is the intrinsic one. It means that an optimal prompt exercise decision at any time moment $t$ can be made by means of trigger level $C_{st}(t)$ for which the intrinsic level $C_0(t)$ is a good approximation.
In Sec. (\[sec:stoch\_opt\_problem\]) we have introduced the profit function $P$, which has a meaning of the cumulative cash flow from all hedge and exercise trades, and formulated the stochastic optimisation problem in terms of probabilistic maximisation of the terminal profit. We repeat the same argumentation here in more details in application to the rolling intrinsic strategy.
Let $F(t,T)$ be the forward price observed in the market at time $t$. Let also $r(t,T)= \dot q(t,T)$ be the intrinsic optimal exercise trajectory calculated on the forward price curve $F(t,T)$. The volume of future trading $r(0,T)$ can be interpreted as the intrinsic hedge volume. Prior the exercise period the whole volume $r(0,T)$ can be traded forward, providing the owner of the storage option the guaranteed profit $$P(0) = -\int_0^{T_e} r(0,T)\,F(0,T)\,dT\,,$$ which equals the intrinsic value. Although strictly speaking the cash-flow from the forward contracts takes place at maturity times, we simplify the picture by assigning all the future cash-flows to the time moment the forward (hedge) position is taken. Thus, we can say that the intrinsic value is locked at the very first time moment.
During the exercise period the breathing prices cause the change of optimal intrinsic strategy. If at time $t$ the price curve is $F(t,T)$, then at time $t+dt$ it becomes different $$F(t+dt,T) = F(t,T) + \delta F(t,T)\,,$$ leading to variation of optimal intrinsic strategy $$r(t+dt,T) = r(t,T) + \delta r(t,T)\,.$$ The volume $\delta r(t,T)$ can be traded continuously (dynamic hedge), leading to continuous change of the profit function: $$\begin{aligned}
\label{eq:delta_S}
\delta P(t) = -\int_t^{T_e} \delta r(t,T) \Big(F(t,T) + \delta F(t,T) \Big)\,dT\end{aligned}$$ corresponding to time period $dt$ (again we assign the cash-flows from all future trades to the current time moment $t$). The described strategy is called “rolling intrinsic”. It assumes that at every time moment the option holder takes the hedge position equal to the current optimal intrinsic strategy.
Since the trajectory $r+\delta r$ is optimal for the price curve $F+\delta F$, the following inequality must hold (we omit the limits of integration and observation time $t$) $$\begin{aligned}
& - \int \Big(r(T) + \delta r(T) \Big) \Big(F(T) + \delta F(T) \Big) \,dT \geq
- \int r(T) \, \Big(F(T) + \delta F(T) \Big) \,dT\,,\end{aligned}$$ from which follows $$\begin{aligned}
\label{eq:PnL_inequality}
\delta P(t) \geq 0\,.\end{aligned}$$ Consequently the dynamic hedge in the rolling intrinsic strategy can only increase the value of the profit function. In other words, the time value of the storage option is positive (non-negative).
Interestingly, the inequality (\[eq:PnL\_inequality\]) is only valid for rolling intrinsic strategy.
In the appendix \[ap:A2\] we show how the time value of the storage option can be calculated from the cumulative cash flows of a rolling intrinsic hedge and exercise trades. In the following sections we apply this technique to estimate the time value of the storage option.
### Meaning of the Dynamic Hedge
In this section we show that the hedge position is not relevant for the expected value of the storage option. The only prompt exercise trade, i.e. the spot trade, is crucial for the expected value of the terminal P&L. The hedge position only effects the distribution of the terminal profit.
Let us consider some delivery time $T\in[0,T_e]$. Let also at some observation time $t<T$ the optimal hedge position be $$h_{opt}(t,T)\,.$$ We do not specify here how to calculate the optimal hedge position $h_{opt}(t,T)$. We only believe that it could be calculated in some way. Suppose that the real hedge position, which is taken at time $t$ $$h(t,T)$$ is not necessarily equal the optimal one: $$h(t,T)\not = h_{opt}(t,T)$$ almost everywhere. On the other hand, we demand that at the beginning the hedge position is zero, and as the time approach the maturity, the hedge position has to be equal the optimal one: $$h(0,T) \equiv 0\,;\qquad h(T,T) \equiv h_{opt}(T,T) \qquad \text{for all } T \,.$$ The volume of the underlying associated with the hedge position for the delivery period $[T, T+\delta T]$ is $$h(t,T)\,\delta T\,.$$ Since the initial hedge position is everywhere zero, there is no initial cash-flow, associated with the hedge position. The hedge position changes with the time $t$. The increment of the hedge position $$dh(t,T) = \frac{{\partial}}{{\partial}t}h(t,T)\,dt$$ leads to the cash-flow $$dP(t,T) = - \Big(F(t,T)+dF(t,T)\Big)\,dh(t,T)\,\delta T\,,$$ where $F(t,T)$ is the forward price for the delivery on $T$, observed on $t$. The total cash-flow per delivery period $\delta T$ becomes $$\begin{aligned}
\label{eq:expected_cashflow}
\frac{P(T)}{\delta T} = -\int_0^T F(t,T)\,dh(t,T)-\int_0^T dF(t,T)\,dh(t,T) \,.\end{aligned}$$
Now we apply integration by parts, which In Ito calculus has to be slightly modified. Indeed, let $$g(t) = y(t)\,z(t)\,,$$ where $z(t)$ and $y(t)$ are correlated stochastic processes. We use the Ito formula to obtain $$\begin{aligned}
\label{eq:ItoChainRule}
dg = \frac{{\partial}g}{{\partial}y}\,dy + \frac{{\partial}g}{{\partial}z}\,dz
+ \frac12\,\frac{{\partial}^2 g}{{\partial}y^2}\, dy^2
+ \frac{{\partial}^2 g}{{\partial}y\,{\partial}z}\, dy\,dz
+ \frac12\,\frac{{\partial}^2 g}{{\partial}z^2}\, dz^2 = z\,dy + y\,dz + dy\,dz \,.\end{aligned}$$ Integrating from 1 to 2 we obtain the modified “integration by parts” formula: $$\begin{aligned}
\left. (y\,z) \right|_1^2 = \int_1^2 y\,dz + \int_1^2 z\,dy + \int_1^2 dy\,dz\,;\end{aligned}$$ Applying this formula to Eq. (\[eq:expected\_cashflow\]) we obtain $$\begin{aligned}
\frac{1}{\delta T} P(T) =
- \left. \Big( F(t,T)\,h(t,T)\Big) \right|_{t=0}^{t = T} +
\int_0^T h(t,T) \, dF(t,T) \,.\end{aligned}$$ Averaging the obtained expression over the price increments $dF$, and noticing that $${\left\langle}h(t,T) \, dF(t,T) {\right\rangle}= h(t,T) \, {\left\langle}dF(t,T) {\right\rangle}= 0\,,$$ the expected cash-flow becomes $$\begin{aligned}
\frac{1}{\delta T}{\left\langle}P(T){\right\rangle}&= - {\left\langle}F(T,T)\,h(T,T){\right\rangle}= - {\left\langle}F(T,T)\,h_{opt}(T,T){\right\rangle}\,.\end{aligned}$$ Thus, the expected cash-flow is independent on the hedge strategy $h(t,T)$ for $t<T$, and only depends on the prompt exercise trade $h(T,T)$.
From this an important consequence follows: the rolling intrinsic strategy is a good approximation of the exact optimal strategy. Indeed, as has been shown in Sec. \[sec:4.6.4\] the intrinsic prompt exercise is a very good approximation of the stochastic one, and hence, rolling intrinsic leads to a very good expected value of the terminal storage profit.
On the other hand, the intrinsic hedge profile may be not close to the optimal hedge profile, and thus it may lead to rather poor efficiency of hedge in terms of stabilising the profit function. We can estimate the “quality” of the intrinsic hedge strategy by calculating the standard deviation of the terminal value of the profit function.
Time Value
----------
For the rolling intrinsic strategy the increment of the target function is given by Eq. (\[eq:delta\_S\]). Averaging the increment $\delta P(t)$ and its square with respect to the price increment ${ {\delta F} }$ we obtain: $$\begin{aligned}
\label{eq:4.55}
&{\left\langle}\delta P {\right\rangle}_{ {\delta F} }= - \int_t^{T_e} {\left\langle}\delta r(T) {\right\rangle}_{{ {\delta F} }}\, F(T)\,dT -
\int_t^{T_e} {\left\langle}\delta r(T)\,\delta F(T) {\right\rangle}_{{ {\delta F} }}\,dT \,; \\
\label{eq:4.56}
& {\left\langle}\delta P^2 {\right\rangle}_{{ {\delta F} }} =
\iint_t^{T_e} {\left\langle}\delta r(T')\,\delta r(T''){\right\rangle}_{{ {\delta F} }}\,F(T')\,F(T'')\,dT'\,dT''\,,\end{aligned}$$ where ${\left\langle}\cdot {\right\rangle}_{dF}$ is the averaging over the stochastic increment $dF$.
Both ${\left\langle}\delta P {\right\rangle}_{\delta F}$ and ${\langle\!\langle}\delta P^2 {\rangle\!\rangle}_{\delta F}$ are proportional to $\delta t$. Designating $$\begin{aligned}
\mu_p(t) = \frac{{\left\langle}\delta P(t) {\right\rangle}_{{ {\delta F} }}}{\delta t}\,;\qquad
\sigma_p^2(t) = \frac{{\left\langle}\delta P^2(t) {\right\rangle}_{{ {\delta F} }}}{\delta t}\,,\end{aligned}$$ we can assign to $P(t)$ an effective stochastic process $$\begin{aligned}
dP(t) = \mu_p(t)\, dt + \sigma_p(t) \,dW_t\,,\end{aligned}$$ Notice that the average ${\left\langle}P {\right\rangle}_{\delta F}$ as well as $\mu_p$ and $\sigma_p$ are stochastic values, since they depend (implicitly or explicitly) on the stochastic price $F(T)$. The drift $\mu_p$ and volatility $\sigma_p$ averaged over the forward price will allow us to find the distribution parameters of the terminal profit $P(T_e)$. Indeed, the expectation value and the variance of the time value $V_T = P(T_e) - P_0$ are given by (see App. \[ap:A2\]) $$\begin{aligned}
&{\left\langle}V_T {\right\rangle}= \int_0^{T_e} \bar \mu_p(t)\,dt \\
&{\langle\!\langle}V_T^2 {\rangle\!\rangle}= \int_0^{T_e} \bar \sigma^2_p(t)\,dt\end{aligned}$$ where we have designated $$\bar\mu_p = {\left\langle}\mu_p {\right\rangle}_F\,;\qquad \bar\sigma_p^2 = {\left\langle}\sigma_p^2 {\right\rangle}_F\,.$$
Substituting $\delta r(T)$ into Eq. (\[eq:4.55\]) and performing integration we obtain $$\begin{aligned}
\label{eq:delta_P}
&{\left\langle}\delta P {\right\rangle}_{ {\delta F} }= \frac12 \int r^{(1)}(u)\,L_{uu} \,du -
\frac{1}{2\,K} \iint r^{(1)}_u\,r^{(1)}_v\,L_{uv}\,\,du\,dv \,;\end{aligned}$$ where the integration is performed from $t$ to $T_e$. This result could also be obtained from Eq. (\[eq:deltaS\]), since (see App. \[ap:A2\]) $$\begin{aligned}
\label{eq:dP=dS}
{\left\langle}\delta P {\right\rangle}_{ {\delta F} }= {\left\langle}\delta S {\right\rangle}_{ {\delta F} }\,.\end{aligned}$$
Next we observe that $r^{(1)}(T) = \Delta r\,\delta(C-F(T))$ as well as $r^{(2)}(T) = \Delta r\,\delta'(C-F(T))$ are singular functions, and the integrals of the type $\int r^{(1)}(T)\,f(T)\,dT$ and $\int r^{(2)}(T)\,f(T)\,dT$ are reduced to a summation over the set of trigger times $\{T_i: F(T_i) = C\}$. Making use of the identities (\[eq:A4\]) and (\[eq:A5\]) we obtain: $$\begin{aligned}
\label{eq:mu}
& \mu_p(t) = \frac{\Delta r}{2\,\delta t} \sum_i \frac{L_i(t)}{|\dot F_i(t)|} -
\frac{\Delta r^2}{2\,K(t)\,\delta t}\sum_{ij} \frac{L_{ij}(t)}{|\dot F_i(t)|\,|\dot F_j(t)|}\,,\end{aligned}$$ where $$\begin{aligned}
& L_{ij}(t) = L(t,T_i,T_j)\,;\quad L_i(t) \equiv L_{ii}(t)\,; \\
& \dot F_i(t) = \left. \frac{{\partial}}{{\partial}T} F(t,T)\right|_{T = T_i} \,;\\
& K(t) = \Delta r \sum_{k>t} \frac{1}{|\dot F_k(t)|}\,.\end{aligned}$$ This value is stochastic for $t>0$. Using the results of App. \[app:averaged\_integral\] and using the notation $$\begin{aligned}
& \Lambda_{ij}(t) = \frac{1}{\delta t}\,{\left\langle}L_{ij}{\right\rangle}\,; \qquad
\Lambda_i(t) := \Lambda_{ii}(t)\,;\end{aligned}$$ we finally obtain the average of $\mu_p$ with respect to the forward curve $F$: $$\begin{aligned}
\label{eq:av_mu_storage}
& \bar \mu_p(t) \approx \frac{\Delta r}{2} \sum_i \frac{ \Lambda_i(t)}{|\dot F_{0i}(t)|} -
\frac{\Delta r^2}{2\, K_0(t)}\sum_{ij} \frac{\Lambda_{ij}(t)}{|\dot F_{0i}(t)|\,|\dot F_{0j}(t)|}\,; \quad
\text{where}\quad K_0(t) = \Delta r \sum_{k>t} \frac{1}{|\dot F_{0k}(t)|}\,.\end{aligned}$$
Using the results from the App. \[app:averaged\_integral\] the latter summation formula can under some assumptions be replaced with the integration, which would allow an easier analytic approximation.
Example calculation {#sec:example_calculation_storage}
-------------------
We apply now the expansion formula to a simple toy example of a storage contract with one factor price process (with $\beta= 0$). The correlation function for this price process is given by $$\Lambda(t,T_1,T_2) = \sigma_0^2\,e^{-\alpha(T_1-t)}\,e^{-\alpha(T_2-t)}\,{\left\langle}F(t,T_1)\,F(t,T_2){\right\rangle}\approx\sigma_0^2\,e^{-\alpha(T_1-t)}\,e^{-\alpha(T_2-t)}\,F_0(T_1)\,F_0(T_2) \,.$$ As initial price condition we consider a periodic function $$F_0(T) = F_c + \Delta F\,\sin(\omega \,T)\,;\qquad \omega = \frac{\pi\,N}{T_e} = \frac{\pi}{\Delta T}\,,$$ where $T_e$ is the exercise period, $N$ is the number of trigger times and $\Delta T$ is the distance between the nearest trigger times. Let also $$C = F_c$$ be the initial trigger level. We obtain: $$\begin{aligned}
T_i = \frac{i\,T_e}{N}\,;\qquad
F_{0i} = F_c \,;\qquad
|\dot F_{0i} | = \Delta F\,\omega = \pi\,\frac{\Delta F}{\Delta T}\,; \qquad
\ddot F_i = 0\,.\end{aligned}$$
Next we substitute $$\begin{aligned}
& \Lambda_{ij} = \sigma_i\,\sigma_j\,F_{0i}\,F_{0j} = F_c^2\,\sigma_i\,\sigma_j\,;\qquad \text{where}\quad
\sigma_i = \sigma(t,T_i) = \sigma_0\,e^{-\alpha(T_i-t)}\,; \\
& \Lambda_i = F_c^2\,\sigma_i^2\,; \\
& K(t) = \frac{\Delta r}{\pi\,\Delta F}\sum_i \Delta T =
\frac{\Delta r}{\pi\,\Delta F}\,(T_e-t)\end{aligned}$$ into Eq. (\[eq:av\_mu\_storage\]) to obtain $$\begin{aligned}
\bar\mu_p(t) =& \frac{\Delta r\,F_c^2}{2\pi\,\Delta F}\,\sum_i \sigma_i^2\,\Delta T -
\frac{\Delta r^2\,F_c^2}{2\,\pi^2\,\Delta F^2\,K(t)}\,\sum_i \sigma_i\,\Delta T\,\sum_j \sigma_j\,\Delta T
\approx \nonumber \\
& \hspace{1cm} \approx \frac{\Delta r\,F_c^2\,\sigma_0^2}{2\,\pi\,\Delta F}
\left[\int_0^{T_e-t} e^{-2\,\alpha\,T}\,dT -
\frac{1}{T-t} \left( \int_0^{T_e-t} e^{-\alpha\,T}\,dT \right)^2 \right]\,.\end{aligned}$$ Performing integration we get: $$\begin{aligned}
\label{eq:mu_final}
\bar \mu_p(t) =&
\frac{\Delta r\,F_c^2\,\sigma_0^2}{2\,\pi\,\Delta F}\,
\left[ \frac{ 1 - e^{-2\,\alpha\,(T_e-t)} }{2\,\alpha} -
\frac{1 - 2\,e^{-\alpha\,(T_e-t)} + e^{-2\,\alpha\,(T_e-t)} }{\alpha^2\,(T_e-t)}\,
\right]\,.\end{aligned}$$
For the option time value we finally obtain $$\begin{aligned}
\label{eq:storage_time_value}
& V_T := {\left\langle}P(T_e){\right\rangle}- P_0 = \int _0^{T_e} \bar \mu_p(t) \,dt =
\frac{\Delta r\,F_c^2\,\sigma_0^2\,T_e^2}{8\,\pi\,\Delta F} \, \Phi(\alpha\,T_e)\,,\end{aligned}$$ where we have introduced a function $$\begin{aligned}
\Phi(x) = \frac{1}{x^2}\,\left(
e^{-2\,x} - 1 + 2\,x -4\,\gamma - 8\,\Gamma(0,x) + 4\,\Gamma(0,2\,x) + 4\,\ln\frac{2}{x}
\right)\,.\end{aligned}$$ Here $\gamma\approx 0.577$ is the Euler’s constant and $\Gamma(a,z)$ is incomplete gamma function $$\Gamma(a,z) = \int_z^\infty t^{a-1}\,e^{-t}\,dt\,.$$
### Time Value as a function of $\alpha$
The function $\Phi(x)$ is bell-shaped with $x^2$ asymptotics on the left wing as $x\to +0$ and $1/x$ asymptotics on the right wing as $x\to\infty$ (see Fig. \[fig:2\]). The function reaches it’s maximum approximately at $x^*\approx 5.04$ and reaches there the value $$\Phi(x^*) \approx 0.12$$ Thus, if we consider the time value $P_t$ as a function of the parameter $\alpha$ (the time $T_e$ is kept constant), then it reaches its maximum at $$\begin{aligned}
\alpha^* \approx \frac{5}{T_e}\,,\end{aligned}$$ and the maximum time value is $$\begin{aligned}
V_T^* \approx \frac{\Delta r\,F_c^2\,\sigma_0^2}{67\,\pi\,\Delta F}\,T_e^2\,.\end{aligned}$$
The function $\Phi(x)$ in the whole range can approximately be represented in the form $$\begin{aligned}
\Phi(x) \approx k(x)\,\frac{-5 + e^{-2\,x} + 2\,x + 4\,e^{-x}\,(1+x)}{x^2}\,,\end{aligned}$$ where $k(x)$ could roughly be considered as constant. More precisely, $k\approx 0.5$ to better fit the left wing of $\Phi$ and $k\approx 0.9$ to better fit the right wing. Around maximum the best fit is achieved with $k\approx 0.6$. In the whole range we can take $k\approx 0.9-0.4\,e^{-x/18}$.
This approximation results from the replacement $$\frac{1-2\,e^{-x} + e^{-2\,x} }{x} \approx x\,e^{-x}\,.$$ in Eq. (\[eq:mu\_final\]).
### Limiting case $\alpha\,T_e \ll 1$
For small $x$ the function $\Phi(x)$ has an asymptotics $$\Phi(x) \approx \frac{x^2}{12}\,,\qquad x\to 0\,.$$ Thus, for small $\alpha$ satisfying the inequality $\alpha \ll \frac{1}{T_e}$ the asymptotic time value becomes $$\begin{aligned}
V_T \approx \frac{\Delta r\,F_c^2\,\sigma_0^2}{96\,\pi\,\Delta F}\,\alpha^2\,T_e^4\qquad
\alpha\,T_e \to 0\,.\end{aligned}$$ We conclude that the storage option time value for $\alpha\to 0$ is proportional to $\alpha^2$. In particular, the time value vanishes for $\alpha=0$.
### Limiting case $\alpha\,T_e \gg 1$
For this asymptotics we obtain $$\begin{aligned}
\Phi(x) \approx \frac{-0.5 + 2\,x - 4\,\ln x}{x^2} \approx \frac{2}{x}\,.\end{aligned}$$ Thus, for big $\alpha$ satisfying the inequality $\alpha \gg \frac{1}{T_e}$ the asymptotic time value becomes $$\begin{aligned}
V_T \approx \frac{\Delta r\,F_c^2\,\sigma_0^2}{4\,\pi\,\Delta F}\,\frac{T_e}{\alpha}\,,\qquad
\alpha\,T_e \to \infty\,.\end{aligned}$$
### Comparison of analytical results with numerical simulation
The comparison of the analytical formula for the time value with numerical simulation can be seen in the Fig. \[fig:3\].
For this graph we have created a toy storage contract with deliberately reduced influence of the surface effects. To a large extent the difference can be explained by the time discretisation effect. The analytical formula is obtained in the continuous time approximation, whereas in the real life the time is discrete (for instance, gas delivery is usually nominated once a day, and hence the smallest time step is one day). The discrete time system has less trading opportunities than the continuous time system, and hence its exercise is “suboptimal” from the continuous time perspective. For this reason the discrete time system has slightly smaller time value than the continuous one.
Other sources of error are surface effects (boundary influence), higher order corrections of the exercise strategy and some approximations used to simplify the analytic expression.
Swing option {#sec:swing_option}
------------
In the previous sections we have considered the storage option problem with fixed terminal condition. This condition has a significant impact on the option value. Indeed, this is simply an additional constraint which restricts the possible strategies and hence can only decrease the option value. Although in most of the storage contracts there is no restriction on the possible terminal level, the problem still can be considered as having the fixed terminal state constraint. The reason for that is simple. The derivative of the storage value with respect to the terminal state is given by Eq. (\[eq:spacial\_derivative\]) $$\frac{{\partial}S}{{\partial}q_{end}} = -C$$ where $C$ is the trigger price of the last part of the trajectory. If the prices $F(T)$ are strictly positive, the trigger price can only be positive as well, since the forward curve should cross the trigger level at some point[^4]. Thus, it is always profitable to have at the end as little gas in the storage as possible within the constraints boundaries. It means that the optimal trajectory has to terminate at the lower boundary, and hence the problem can be considered as having the fixed terminal state. If we variate the forward curve $F\to F+\delta F$, this will lead to variation of the optimal strategy. However, the new strategy preserves the terminal level.
There are examples of the contracts for which this logic is not applicable. For instance, if the remaining gas can be sold at the fixed price $F_e$, then the derivative of the storage value with respect to the terminal state is given by Eq. (\[eq:4.21\]) $$\frac{{\partial}S}{{\partial}q_{end}} = F_e - C = 0$$ from which the condition (\[eq:4.22\]) $$C = F_e$$ follows. The vanishing derivative implies that the trajectory may terminate anywhere between the upper and lower boundaries. The variation of the forward curve would lead to an “arbitrary” variation of the optimal trajectory which does not preserve the terminal state.
Another more common example is a swing option. Usually a swing option is a gas supply contract. The option holder is given a right to purchase gas from a producer at some “contract” price. He would then sell the gas on the market at the “market” price. Thus, the effective price for the option holder is the spread between the “contract” and the “market” price. This spread can be positive or negative. A typical swing contract obliges the option holder to take some amount of gas within the contract period, however the option holder has some flexibility: the total taken amount can be anything within the constraints – minimal and maximal total taken volumes. The option holder also has some flexibility during the contract time to take bigger or smaller amount of gas on particular days, making maximum of the opportunities to earn on the spread. The swing contract can be easily formulated in terms of a storage, and hence possesses all features of a storage option. However the matter of fact that the spread between contract and market prices can be negative, makes the swing option having slightly different peculiarities.
The difference between the swing and storage option is rather relative. One can find examples of swing options having all the features of a storage option. If the swing option is deep in the money (i.e. the market prices are much higher than the contract prices), then the option holder is most likely to take the maximal allowed amount of gas, making the optimal trajectory to terminate at the lower boundary. Similarly, if the option is deeply out of the money, the optimal trajectory is very likely to terminate on the upper boundary. However if the swing option is at the money, it is quite probable that the total taken amount of gas will be between maximum and minimum. In this case the swing option problem becomes a problem with free terminal state.
This section is devoted to the problem with the free terminal condition. We will show that this problem has a different time value and derive the time value in a similar way it has been done for the problem with fixed terminal state.
The problem with free terminal state has an important feature. As it was shown in the Sec. (\[sec:free\_vs\_fixed\_terminal\_condition\]), the derivative of the option value with respect to the terminal volume must vanish. This leads to the condition on the trigger price $C= F_e$. Since there is no final unit price in the swing contract we conclude that for the problem with free terminal condition $$C = 0\,,$$ from which obviously the condition $$\delta C = 0$$ follows. These conditions will simplify the derivation of the option value evolution equation. In particular, the variation of the target function becomes (compare with Eq. (\[eq:deltaS\])) $$\begin{aligned}
\label{eq:deltaS_swing}
\delta S = & -\int r(T) \,\delta F(T)\,dT + \frac12 \int r^{(1)}\, L_{uu} \,du \,.\end{aligned}$$ The second term gives a non-vanishing part of the average option value increment. If we compare this formula with Eq. (\[eq:deltaS\]) we see that $\delta S$ of the problem with fixed terminal state contains an additional negative term. Hence the averaged growth rate of the option value of the problem with free terminal state is higher than that of the problem with fixed terminal state, as expected.
The evolution equation for the option time value becomes $$dP(t) = \mu_p(t)\,dt + \sigma_p(t)\,dW_t\,;$$ with $$\begin{aligned}
\mu_p(t) = \frac{{\left\langle}\delta S {\right\rangle}_{ {\delta F} }}{\delta t} = \frac{1}{2\,\delta t} \int r^{(1)}\, L_{uu} \,du\,.\end{aligned}$$
Repeating the calculation in the previous sections we finally obtain $$\begin{aligned}
\label{eq:av_mu_swing}
& \bar \mu_p(t) = \frac{\Delta r}{2} \sum_i \frac{\Lambda_i(t)}{|\dot F_{0i}(t)|}\,\end{aligned}$$
### Example calculation {#example-calculation}
Let the evolution of the spread be described by a Normal Brownian one-factor price process $$\begin{aligned}
\label{eq:brownian_motion}
dF(t,T) = \kappa(t,T)\,dW_t\,;\qquad \kappa(t,T) = \kappa_0\,e^{-\alpha (T-t)}\,;\end{aligned}$$ Note that a natural choice for the price process describing the evolution of gas forward curves is a geometric Brownian motion, whereas for the spreads evolution a plane Brownian motion is a more suitable choice. Unlike the volatility $\sigma$ of the geometric Brownian motion, the Normal volatility $\kappa$ includes additional dimension of price.
The correlation function of the price process (\[eq:brownian\_motion\]) becomes $$\begin{aligned}
\Lambda_{ij}(t) = \frac{1}{\delta t}\,{\left\langle}dF(t,T_i)\,dF(t,T_j){\right\rangle}= \kappa_0^2\,e^{-\alpha(T_i-t)}\,e^{-\alpha(T_j-t)}\,.\end{aligned}$$ Let the initial forward curve be given by $$\begin{aligned}
F_0(T) = \Delta F\,\sin(\omega \,T)\,;\qquad \omega = \frac{\pi\,N}{T_e} = \frac{\pi}{\Delta T}\,,\end{aligned}$$ Repeating the calculation similar to that in the previous section we obtain the swing option time value $$\begin{aligned}
\label{eq:swing_option_value}
V_T = \frac{\Delta r\,\kappa_0^2\,T_e^2}{8\,\pi\,\Delta F}\,\Phi(\alpha\,T_e)\,;\qquad\text{where} \quad
\Phi(x) = \frac{e^{-2\,x}+2\,x-1}{x^2}\,.\end{aligned}$$ This formula resembles the formula of the time value of a storage option (\[eq:storage\_time\_value\]), where the volatility $\kappa$ replaces the product $F_c\,\sigma_0$. The asymptotics of the swing option time value for the big mean reversion parameter coincides with that for the storage option: $$\begin{aligned}
V_T \approx \frac{\Delta r\,\kappa_0^2}{4\,\pi\,\Delta F}\,\frac{T_e}{\alpha}\,,\qquad \alpha\,T_e\to\infty\,.\end{aligned}$$ However in the limit of small mean reversion the swing option time value reveals different properties: $$\begin{aligned}
V_T \approx \frac{\Delta r\,\kappa_0^2\,T_e^2}{8\,\pi\,\Delta F}\,\Big(2-\frac{4}{3}\alpha\,T_e\Big)\,,\qquad \alpha\,T_e\to 0\,.\end{aligned}$$ For $\alpha=0$ the time value reaches its maximum $$\begin{aligned}
V_T^* = \frac{\Delta r\,\kappa_0^2\,T_e^2}{4\,\pi\,\Delta F}\end{aligned}$$ and on the entire range the time value is a decreasing function of $\alpha$ (See Fig. \[fig:4\]).
This result can be easily interpreted. The main driver of the storage option time value is the change of exercise strategy. Since any change of the exercise strategy must preserve the terminal volume level, the storage option becomes a zero sum game: any decision to inject more at some time moment must be compensated by the decision to release more at some other time moment. Hence the storage option is in some sense similar to a set of time spread options. The only way the time spread option can increase its value is an asynchronous movement of different parts of the forward curve. In the limit $\alpha\to 0$ the forward curve moves in a parallel way (in log scale), and hence does not lead to any change of the exercise strategy, i.e. to the increase of the time value.
On the contrary, the swing option is similar to a strip of european call options. Indeed, since there is no terminal level preservation condition, any change of the strategy at one (delivery) time has no impact on the rest of the trajectory. If for instance we hedge the exercise trajectory at the very first observation time moment $t=0$, then the change of the exercise decision for any time $T$ would make a profit. The pay-off of the exercised volume on the time $t$ will have a structure of the european vanilla option. The effective square volatility of the option with exercise time $T$ is given by $$\bar\varsigma^2 = \int_0^T \sigma^2(\tau,T)\,d\tau\,.$$ It’s easy to see that the biggest effective volatility $\bar\varsigma^2$ is achieved for the flat volatility term structure $\alpha=0$, thus leading to the highest possible time value of the swing option.
This has a strong impact on the way the price process for the forward curve evolution should be calibrated. Let us consider a price process in the form $$\begin{aligned}
dF(t,T) = \kappa_1(t,T)\,dW_t^{(1)} + \kappa_2(t,T)\,dW_t^{(2)}\,;\end{aligned}$$ where $dW_t^{(1)}$ and $dW_t^{(2)}$ are (possibly correlated) standard Brownian processes. Let also $\kappa_1$ and $\kappa_2$ be exponential functions, describing the short term and long term volatilities $$\kappa_1 = \kappa_{10}\,e^{-\alpha_1\,(T-t)}\,;\qquad \kappa_2 = \kappa_{20}\,e^{-\alpha_2\,(T-t)}\,.$$ Here the mean reversion parameter $\alpha_1$ is relatively big (compared to $1/T_e$), i.e. the term $\kappa_1$ describes the short term volatility. The second term $\kappa_2$ describes the long term volatility, which means $\alpha_2\ll 1/T_e$. The correlation function decouples into three terms $$\begin{aligned}
& \Lambda_{ij} = \frac{1}{dt}\,{\left\langle}dF(T_i)\,dF(T_j){\right\rangle}=
\Lambda_{ij}^{(1)} + \Lambda_{ij}^{(12)} + \Lambda_{ij}^{(2)}\,, \qquad \text{where} \\
& \Lambda_{ij}^{(1)} = {\left\langle}F(T_i)\,F(T_j){\right\rangle}\,\kappa_1(T_i)\,\kappa_1(T_j)\,; \\
& \Lambda_{ij}^{(2)} = {\left\langle}F(T_i)\,F(T_j){\right\rangle}\,\kappa_2(T_i)\,\kappa_2(T_j)\,; \\
& \Lambda_{ij}^{(12)} = \rho\,{\left\langle}F(T_i)\,F(T_j){\right\rangle}\,
( \kappa_1(T_i)\,\kappa_2(T_j) + \kappa_2(T_i)\,\kappa_1(T_j))\,.\end{aligned}$$ Since the option time value (both storage and swing) is a linear functional on the correlation function, the time value also splits into three components, which can be estimated independently.
For evaluation of the storage option, the second and third terms will give no contribution, since they correspond to the long term volatility ($\alpha\to 0$). Consequently for the storage option the second term in the r.h.s. of the price process can be neglected.
This is not the case for the swing option. Since both – long term and short term volatilities contribute to the option time value, we can not neglect the long term volatility in the price process.
Discussion {#sec:discussion}
==========
In Sec. \[sec:Deterministic\_problem\] we considered a deterministic storage problem and have fount its solution in an implicit form. The solution provided by Eq. (\[eq:solution\]) contains a free parameter – trigger price, which can be found from boundary conditions. The obtained solution reveals the key features of the intrinsic optimal exercise strategy.
- The optimal exercise is bang-bang. There is a “dead zone” around the trigger price. The optimal strategy is to inject at maximum rate if the price is below the zone, release at maximum rate if the price is above the dead zone and do nothing if the price is within the dead zone.
- The width of the dead zone is defined by the operating costs.
- The trigger price is constant for every piece of the trajectory where it does not sticks to the boundary. Different pieces of trajectory separated by the boundary touch may have different trigger level. If the trajectory touches the boundary in the interval $t\in(0,T_e)$, the conditions (\[eq:bc1\]-\[eq:bc4\]) must be satisfied.
We also considered an impact of different constraints on the solution. In particular we found that
- the cycle constraint is equivalent to additional injection/release costs.
- the carry cost preserves the bang-bang property, but makes the trigger price growing with the time at the rate of the carry cost.
- The volume dependent injection/release costs lead to a time-dependent trigger level, but preserve the bang-bang property.
- Fir the price of the underlying is strictly positive, and there is no final unit price, then the storage problem (both intrinsic and stochastic) can be considered with fixed terminal condition. If there is a positive final unit price, or if the prices of the underlying can be negative, then the optimisation problem becomes a problem with free terminal condition. This problem has an additional condition on the trigger level (\[eq:4.22\]) for storage or (\[eq:4.23\]) for swing option.
In Sec. \[sec:stochastic\_problem\] we developed a perturbation analysis to find the solution of the stochastic problem in rolling intrinsic approximation.
First important result obtained in this section finds the relation between the intrinsic and stochastic solutions. In particular we show that the stochastic optimal exercise has exactly the same bang-bang feature with a trigger level and a dead zone. The stochastic trigger level is given by the averaged intrinsic trigger price (Eq. (\[eq:stochastic\_trigger\_price\_1\])), where the average is taken over all realisations of the spot price process. Under the condition (\[eq:condition\_for\_rolling intrinsic\]) the average trigger price equals approximately the current intrinsic trigger price (conditional on the current initial condition and forward curve). This condition defines the range of applicability of the rolling intrinsic approximation.
Next we show that the hedge position has no impact on the expected option value. The only relevant part of the hedge strategy is the instant (prompt) exercise at the current time. This justifies the assumption that the intrinsic exercise profile as a good approximation of the optimal stochastic hedge profile, and generally, the rolling intrinsic strategy is a reasonable approximation for the optimal stochastic strategy.
By applying a variational analysis to the intrinsic solution we obtain the option greeks – delta and gamma. This allows us to find the option time value. Making use of the option gamma we derive a stochastic differential equation governing the evolution of the option value. The drift term (Eq. (\[eq:av\_mu\_storage\]) for storage and Eq. (\[eq:av\_mu\_swing\]) for swing option) can be integrated yielding the option time value.
We apply the obtained time value formula for a toy examples of the storage and swing contracts with a simple 1-factor price process. The storage option time value (Eq. (\[eq:storage\_time\_value\])) is a bell-shaped function of $\alpha$, approaching zero for $\alpha\to 0$ and $\alpha\to\infty$, and reaching its maximum at $\alpha\,T_e\approx 5$. The swing time value appears to be a decreasing function of $\alpha$ for the entire range, reaching the maximum for $\alpha=0$ and approaching zero for $\alpha\to\infty$.
The option time value appears to be a linear functional on the prices correlation function. From this fact we conclude that if the correlation function can be decoupled into independent components, the option time value can also be decoupled. This allows us to investigate effect of different components of the price process on the time value independently.
The difference in the limiting case $\alpha\to 0$ between swing and storage option time value has an important consequence. This limiting case corresponds to the “long term” volatility due to the flat shape of the correlation function in this limit. If we consider a price process, whose correlation function could be decoupled into short and long terms, the impact of the long term component can be investigated independently. We conclude that the long term volatility has a significant impact on the swing option time value, but has no impact on the storage option time value. This fact can also be used in the price process calibration procedure.
Last important comment concerns the applicability of the obtained results. Throughout the derivation process we used a number of simplifying approximations. In particular, the rolling intrinsic approximation is valid for sufficiently small volatility. For the derivation of the option time value we also have neglected the surface effects (possible influence of the storage being completely full or empty), the bid-offer and operating costs, and used rather simple time-independent option constraints. For the derivation of analytic formulas of the time value we also implicitly used an assumption about sufficient smoothness of the forward curve, which justified the use of Taylor expansion. Another significant approximation was made for the definition of the swing option, where we have allowed the option holder to take any possible amount of the underlying limited only by the maximum release rate.
All these approximations restrict the applicability of the obtained results. However all the qualitative results remain valid for much bigger range of parameters. And the main results can still be used for understanding the option value driving mechanisms and for building an intuition about the influence of different factors on the option value.
Suppose that we are considering a real storage and we are interested in the applicability of the option value formula. For instance, we find that the intrinsic strategy spends a significant time on the boundary. This means that we can’t neglect the surface effects, and the option value formula is not applicable directly. However, as the experiments show, all the major qualitative results still remain valid, and the option value formula could be corrected by some multiplier, which is weekly dependent on the storage parameters. Thus evaluating the storage for one set of parameters, we can easily predict how the value would change for a different set of parameters.
Another example of the restricted usability of the option value formula is the swing contract. A real swing contract usually has two global constraints – maximum and minimum volume which can be taken during the contract period. These constraints were not taken into account in the swing value formula, which allowed us to use the free terminal condition approximation. However, if the option is “at the money”, the probability is very high that the total taken volume will be between the maximum and minimum, and thus satisfy the requirement of the free terminal condition. Hence the swing formula remains a good approximation for the at-the-money option. The swing options which are either deep in- or out-of-the-money, are very likely to finish on either lower or upper boundary of the total taken volume. Such an option satisfies the condition with fixed terminal level, and hence a storage formula can be used to estimate the time value of such an option. An intermediate case, when the swing option is only slightly in- or out-of-the-money, neither storage nor swing formula become directly applicable. However, one can see that these two formulas allow a continuous transition, since they only differ by one term, which is not present in the swing formula. Thus the ultimate formula could be easily adjusted for the particular swing option.
[**Appendix**]{}
Derivation of Constraint Surface equation {#ap:Contraint_surface}
=========================================
To find the equation of the constraint surface we vary the equation \[eq:constraint\]. Since $F(t,T)$ follows a Wiener type stochastic process, so does the $C(t)$. Hence we find the variation of the $Q_{end}$ up to the second order in $\delta F$ and $\delta C$: $$\begin{aligned}
dQ_{end} = & \Big(\dot q - r(t)\Big)\,dt +
\frac{{\partial}Q_{end}}{{\partial}C}\, \delta C +
\int \frac{\delta Q_{end}}{\delta F(T)}\,\delta F(T)\, dT +
\delta C \int \frac{\delta^2 Q_{end}}{{\partial}C\,\delta F(T)}\,\delta F(T)\,dT + \nonumber \\
& \hspace{1cm} +
\frac12\,\frac{{\partial}^2 Q_{end}}{{\partial}C^2}\, \delta C^2 +
\frac{1}{2} \iint \frac{\delta^2 Q_{end}}{\delta F(u)\,\delta F(v)}\,
\delta F(u)\,\delta F(v)\, du\,dv = 0\,.\end{aligned}$$ On the optimal intrinsic trajectory $\dot q(t) = r(t,t)$, and the first term in the r.h.s. vanishes.
We introduce the notation: $$\begin{aligned}
& r^{(1)}(t,T) = \Delta r\, \delta(C(t)-F(t,T)) \,; \\
& r^{(2)}(t,T) = \Delta r\, \delta'(C(t)-F(t,T))\,; \\
& K(t) = \int_t^{T_e} r^{(1)}(t,T)\,dT\,; \\
& M(t) = \int_t^{T_e} r^{(2)}(t,T)\,dT\,,\end{aligned}$$ where $\delta(x)$ is the Dirac delta-function, and $\delta'(x)$ – its derivative. Making use of Eq. (\[eq:solution\_for\_dot\_q\]) and the notation above, we find $$\begin{aligned}
& \frac{{\partial}Q_{end}}{{\partial}C} = K(t) \,; \\
& \frac{{\partial}^2 Q_{end}}{{\partial}C^2} = M(t) \,; \\
& \frac{\delta Q_{end}}{\delta F(T)} = -r^{(1)}(t,T) \,; \\
& \frac{\delta^2 Q_{end}}{{\partial}C\,\delta F(T)} = -r^{(2)}(t,T) \,; \\
& \frac{\delta^2 Q_{end}}{\delta F(t_1)\,\delta F(t_2)} = r^{(2)}(t,t_1)\,\delta(t_1-t_2)\,.\end{aligned}$$ We thus obtain a quadratic equation on $\delta C$: $$\begin{aligned}
\label{eq:quadratic_eq_for_dC}
\delta C = &\frac{1}{K}
\left( \int_t^{T_e} r^{(1)} \,\delta F\,dT -
\frac12 \int_t^{T_e} r^{(2)}\,\delta F^2 \,dT +
\delta C \int_t^{T_e} r^{(2)} \,\delta F\,dT -
\frac{M}{2}\,\delta C^2
\right)\,.\end{aligned}$$ Here for simplicity we omitted arguments of the functions: $r=r(t,T)$, $\delta F = \delta F(t,T)$, $K = K(t)$, $M = M(t)$ and $\delta C = \delta C(t)$.
We search the solution in form of a series. As a zeroth order solution we take $$\delta C_0 = 0\,.$$ The first order solution becomes $$\delta C_1 = \frac{1}{K}
\left(
\int r^{(1)} \,\delta F\,dT -
\frac12 \int r^{(2)}\,\delta F^2 \,dT
\right)$$ Substituting $\delta C_1$ into the r.h.s. of Eq. (\[eq:quadratic\_eq\_for\_dC\]), and leaving only 1st and 2nd order terms we finally obtain $$\begin{aligned}
&\delta C = \frac{1}{K}
\left( \int r^{(1)} \,\delta F\,dT -
\frac12 \int r^{(2)}\,\delta F^2 \,dT +
\frac{1}{K}\iint r^{(2)}_u\,r^{(1)}_v\,L_{uv} -
\frac{M}{2\,K^2}\iint r^{(1)}_u\,r^{(1)}_v\,L_{uv}
\right)\,; \\
& \delta C^2 =\frac{1}{K^2}\iint r^{(1)}_u\,r^{(1)}_v\,L_{uv}\,,\end{aligned}$$ where we have used the notation $$L_{uv} = \delta F(u) \, \delta F(v)\,.$$
Portfolio dynamics and time value {#ap:A2}
=================================
In this section we show how the option time value can be obtained from the dynamics of the portfolio components in the rolling intrinsic strategy.
We start with summarising some basic facts from the option pricing theory. Let $F$ be a price of some underlying with delivery at time $T$, and let $f(t,F)$ be a price of an option expiring on $T$. Let the price be following a geometric Brownian motion (for simplicity we disregard the interest rate), which in Ito representation reads: $$\frac{dF}{F} = \sigma(t)\,dW_t\,.$$ Using Ito rule we can find a differential of the option price: $$df = \frac{{\partial}f}{{\partial}t}\,dt + \frac{{\partial}f}{{\partial}F}\,dF + \frac12 \frac{{\partial}^2 f}{{\partial}F^2}\,dF^2 =
\left(\frac{{\partial}f}{{\partial}t} + \frac12\, \sigma^2\,F^2 \,\frac{{\partial}^2 f}{{\partial}F^2}\right)dt + \frac{{\partial}f}{{\partial}F}\,dF\,.$$ According to the non-arbitrage condition average of this expression must vanish: $${\left\langle}df {\right\rangle}= 0\,.$$ Noticing that ${\left\langle}dF {\right\rangle}= 0$ we obtain $$\label{eq:theta_and_gamma}
\frac{{\partial}f}{{\partial}t}\,dt + \frac12\, \frac{{\partial}^2 f}{{\partial}F^2}\,dF^2 = 0\,.$$ or substituting $dF^2$ explicitly $$\label{eq:BS}
\frac{{\partial}f}{{\partial}t} + \frac12\, \sigma^2\,F^2 \,\frac{{\partial}^2 f}{{\partial}F^2} = 0\,.$$ This equation (up to the interest rate terms) is known as Black-Scholes equation. For us it is essential that the operator $$\hat B = \frac{{\partial}}{{\partial}t} + \frac12\, \sigma^2\,F^2 \,\frac{{\partial}^2 }{{\partial}F^2} = \hat\theta + \frac12\, \sigma^2\,F^2 \,\hat \Gamma\,,\qquad\text{where}\quad
\hat\theta = \frac{{\partial}}{{\partial}t}\,;\quad \hat \Gamma = \frac{{\partial}^2 }{{\partial}F^2}\,;$$ applied to any martingale process vanishes: $$\hat B\,f = 0\,.$$ Designating the operator of the full differential as $$\begin{aligned}
d = dt\,\hat B + dF\,\frac{{\partial}}{{\partial}F} =
dt\,\frac{{\partial}}{{\partial}t} + dt\,\frac12\, \sigma^2\,F^2 \,\frac{{\partial}^2 }{{\partial}F^2} + dF\,\frac{{\partial}}{{\partial}F}\end{aligned}$$ we find $$\begin{aligned}
\label{eq:df}
&df = dt\,\hat B\,f + dF\,\frac{{\partial}f}{{\partial}F} = \frac{{\partial}f}{{\partial}F}\,dF\,;\end{aligned}$$ Next we derive the expression for the differential of the option Delta $d\frac{{\partial}f}{{\partial}F}$. To do that, we notice first that the operators $d$ and ${\partial}/{\partial}F$ do not commute. Using the obvious commutation rule $$\left[d\,, \frac{{\partial}}{{\partial}F}\right] =
d\, \frac{{\partial}}{{\partial}F} - \frac{{\partial}}{{\partial}F}\,d = - dt \, \sigma^2\,F\,\frac{{\partial}^2 }{{\partial}F^2}$$ we find $$\begin{aligned}
\label{eq:d2f}
& d\,\frac{{\partial}f}{{\partial}F} = \frac{{\partial}}{{\partial}F} \,df - \sigma^2\,F\,\frac{{\partial}^2 f}{{\partial}F^2} \, dt =
\frac{{\partial}^2 f }{{\partial}F^2}\,dF - \sigma^2\,F\,\frac{{\partial}^2 f}{{\partial}F^2} \, dt \,;\end{aligned}$$
Dynamics of the portfolio components
------------------------------------
Let us consider a standard situation when an owner of an option hedges it with a combination of linear products. The full portfolio consists of three components: an option expiring on $T$, a hedge sub-portfolio, consisting of linear products, and a cash account. Let $\Pi(t)$ be the value of the full portfolio at the observation time $t$. Then we have $$\Pi(t) = f(t) + H(t) + P(t)\,,$$ where $f(t)$ is the option value, $H(t)$ is the value of the hedge portfolio, and $P(t)$ is the balance of the cash account.
The dynamics of the option value $f$ is given by Eq. (\[eq:df\]) $$df = \frac{{\partial}f}{{\partial}F}\,dF\,.$$
Next we find the dynamic equation for other portfolio components. Let us consider a hedge approach, when the hedge portfolio consists of some amount of underlying $h$, i.e. hedge portfolio contains only linear products. The value of the hedge portfolio equals the current price of the hedge volume: $$H = F\,h\,.$$ In a classical scheme we may think of the option $f$ as a vanilla option on some kind of shares. In this case the hedge consists of some amount of shares which is purchased prior the expiry date. The same logic can be applied to a commodity market. For instance $f$ could imply an option to purchase some amount of gas. In this case hedge consists of forward contracts with delivery on the option expiry date. Although strictly speaking, the cash-flow associated with the delivery on the forward contract can take place on a later stage, we assign the cash-flow to the current (observation) time $t$, and hence the value of the hedge portfolio reflects the total price of underlying, and not only of the forward contracts.
The hedge portfolio is a product of two stochastic variables. Using Eq. (\[eq:ItoChainRule\]) from the main text, we find $$\begin{aligned}
dH = h\,dF + (F + dF)\,dh.\end{aligned}$$ The latter equation can be formulated as *retarded action principle*. This principle follows from a physical meaning of the dynamic hedge: the change of the hedge volume takes place after the change of the price is observed, and the additional hedge volume $dh$ has to be purchased at the new price $F+dF$. Notice however that the retarded action principle does not require any special differentiation rule.
The dynamics of the cash-flow component is easy to find. The cash-flow appears due to the change of the hedge volume: $$\begin{aligned}
dP = -(F+dF)\,dh\,.\end{aligned}$$
The increment of the portfolio value is given by $$\begin{aligned}
d\Pi = df + dH + dP = \left( \frac{{\partial}f}{{\partial}F} + h \right)\,dF\,.\end{aligned}$$
First we notice that if we use the delta-hedge, i.e. if the hedge volume equals the option delta with the opposite sign $$h = - \frac{{\partial}f}{{\partial}F}\,,$$ then, as expected, the variance of the portfolio equals zero: $$d\Pi = df + dH + dP\equiv 0\,.$$ Next important observation can be made about the relation between hedge portfolio and cash-flow. Averaging the increments $dH$ and $dP$ with respect to the stochastic price increment $dF$ we obtain: $$\begin{aligned}
\label{eq:dH}
{\left\langle}dH {\right\rangle}_{dF} &= - {\left\langle}dP {\right\rangle}_{dF}\,,\end{aligned}$$ where ${\left\langle}\cdot{\right\rangle}_{dF}$ is the averaging over the price increment.
Exact delta-hedge
-----------------
Let us consider the case when the hedge volume equals the exact (negative) option delta $$h = -f' = -\frac{{\partial}f}{{\partial}F}\,;\qquad H = -F\,f'\,,$$ Using Eq. (\[eq:d2f\]) we get for the hedge increment: $$dH = -f'\,dF - F\,df' - dF\,df' = - (f' + F\,f'')\,dF\,.$$ Note that on average the hedge portfolio value and the cash account do not change: $${\left\langle}dH {\right\rangle}= 0\,; \qquad {\left\langle}dP {\right\rangle}= 0\,.$$
Intrinsic target function and time value {#app:timevaluefromtargetfunction}
----------------------------------------
Here we show how the time value of a vanilla option can be calculated from the intrinsic target function. Let $\Delta_{int}$ be the intrinsic option delta, i.e. the delta calculated at the option maturity time $$\Delta_{int}(F) = \frac{{\partial}f(F,T)}{{\partial}F}\,.$$ Similarly we define the intrinsic gamma $$\Gamma_{int}(F) = \frac{{\partial}^2 f(F,T)}{{\partial}F^2}\,.$$ The intrinsic target function $S_{int}$ is defined such that at every time moment it is equal to the intrinsic option value $$S_{int}(F) = f(F,T)\,.$$ The variation of the target function is given by $$\begin{aligned}
\delta S_{int} = \Delta_{int}\,\delta F + \frac12\, \Gamma_{int}\,\delta F^2\,.\end{aligned}$$ Note that the increment $\delta F$ is independent, and hence $${\left\langle}\delta S_{int} {\right\rangle}=\frac12\, {\left\langle}\Gamma_{int}\,\delta F^2 {\right\rangle}\,.$$
Integrating $\delta S_{int}$ over the time and averaging over the realisations of the stochastic price process, we get $$\begin{aligned}
\int_0^T {\left\langle}\delta S_{int} {\right\rangle}= {\left\langle}S_{int}(F(T)){\right\rangle}- S_{int}(F(0))\,.\end{aligned}$$ Next we observe that $S_{int}(F(0))$ is the intrinsic option value at time $t=0$, and the expectation ${\left\langle}S_{int}(F(T)){\right\rangle}$ is nothing else but the option value at time $t=0$, and thus the latter integral equals the option time value $V_T$: $$\begin{aligned}
V_T = \int_0^T {\left\langle}\delta S_{int} {\right\rangle}= \frac12\int_0^T {\left\langle}\Gamma_{int}\,\delta F^2 {\right\rangle}\,.\end{aligned}$$
Intrinsic delta-hedge of the storage option
-------------------------------------------
A storage option can be considered as a sequence of simple trades of buying and selling some amount of underlying. The logic of the derivation of a vanilla option time value can be applied to the storage option, if we consider it as a strip of (correlated) vanilla options with a pay-off $-\dot q(T)\,F(T)\,dT$, where $\dot q(T)\,dT$ is the volume purchased at time $T$ for the delivery period $dT$ at the spot price $F(T)$.
For every delivery time $T$ there is a forward price $F(t,T)$ which is as a function of the observation time $t$. Suppose we have an optimisation model, which calculates an intrinsic exercise profile $q_{int}(t,T)$ based on the current forward curve $F(t,T)$. Then $\dot q_{int}(t,T)$ is the volume of underlying (per delivery period $dT$) to be traded on time $T$. The value $$s(t,T) = -\dot q_{int}(t,T)\,F(t,T)\,;\qquad t<T$$ can be interpreted as a *target function density* for the trade on the delivery time $T$. The same function after expiry is constant $$s(t,T) = -\dot q_{int}(T,T)\,F(T,T)\,;\qquad t>T\,.$$ Now the storage option target function can be expressed simply as an integral of the target function density over the delivery time $$\begin{aligned}
\label{eq:full_intrinsic_target_function}
S(t) = \int_0^{T_e} s(t,T)\,dT =
-\int_0^t \dot q_{int}(T,T)\,F(T,T)\,dT - \int_t^{T_e} \dot q_{int}(t,T)\,F(t,T)\,dT\end{aligned}$$ The first integral in this expression represents the value of the closed trades, whereas the second integral represents the future profit calculated with the current forward curve and exercise profile. At the last time moment $t=T_e$ the second integral vanishes, and the target function equals the cumulative cash flow from the closed trades. Note that the cumulative cash flow only depends on the exercise volumes $\dot q_{int}(T,T)$, and not on the hedge volumes $\dot q_{int}(t,T)$ for $t<T$.
Now we apply the logic of the section \[app:timevaluefromtargetfunction\] in order to calculate the storage option time value. Obviously the target function calculated at the time $t=0$ yields the intrinsic storage option value $$S(0) = V_{int}\,.$$ The expectation value of the terminal target function value gives the true option value $${\left\langle}S(T_e) {\right\rangle}= V$$ (this statement relies on the fact that the intrinsic exercise is an optimal prompt exercise, which has been justified in the Sec. \[sec:4.6.4\]), and hence, the time value can be obtained as $$\begin{aligned}
V_T = \int_0^{T_e} {\left\langle}dS {\right\rangle}\,,\end{aligned}$$ where $dS$ is a full differential of the target function.
Some comment should be made regarding the calculation of the target function differential. First we split the target function into two parts: $$S = S_c + S_o\,,$$ where $$\begin{aligned}
S_c = -\int_0^t \dot q_{int}(T,T)\,F(T,T)\,dT\end{aligned}$$ represents the “closed” trades, and $$\begin{aligned}
S_o = - \int_t^{T_e} \dot q_{int}(t,T)\,F(t,T)\,dT\end{aligned}$$ represents the open trades, i.e. the remaining “future” option value. The definition of the target function in the main text coincides with the second “open” component of the target function defined above. However we can show that the open target function can also be used for the derivation of the time value.
The variation of the target function should be done with respect to the forward curve $F$ and time $t$. The target function $S$ is an explicit function of the forward curve and of the time, and the dependence on the time is due to the limits of the integrals. It is easy to see that the time variance of the closed component equals in absolute value the time variance of the open component but has an opposite sign: $$\frac{{\partial}S_c}{{\partial}t} + \frac{{\partial}S_o}{{\partial}t} = 0\,.$$ Thus the target function can be considered as an explicit function of the forward curve only. The full differential includes only the differentiation with respect to the forward price: $$dS = \int \frac{\delta S}{\delta F}\,dF(u)\,du +
\frac12\, \int \frac{\delta^2 S}{\delta F(u)\,\delta F(v)}\,dF(u)\,dF(v)\,du\,dv$$ The closed part of the target function is constant with respect to the forward curve variation (the variation of the forward curve impacts only the future values of the curve $F(t,T)$ for $T>t$), and hence $$\begin{aligned}
dS = \eth S_o\,,\end{aligned}$$ where we designated $\eth$ the variation which should be calculated for constant time $t$. Thus $$\begin{aligned}
V_T = \int_0^{T_e} {\left\langle}\eth S_o {\right\rangle}\,.\end{aligned}$$
It is interesting to note that for the storage option the hedge value coincides with the (negative) open target function: $$H(t) = \int_t^{T_e} \dot q_{int}(t,T)\,F(t,T)\,dT = -S_o\,,$$ and hence $$\begin{aligned}
{\left\langle}\eth S_o {\right\rangle}= -{\left\langle}\eth H {\right\rangle}= {\left\langle}dP {\right\rangle}\end{aligned}$$ where $P$ is the cumulative cash flow from the hedge trades. Note that this relation would not be valid for a vanilla option, since in general case the target function can not be expressed as $q_{int}\,F$.
### Cumulative cash flow
There is another way to see how the time value can be obtained from the cumulative cash flow. By definition the cash flow (resulting from the hedge trades) with delivery on $T$ per delivery interval $dT$ is given by $$dP_T(t,T) = - (F + dF)\,dh\,; \quad t<T\,; \qquad \text{where}\quad h = \dot q_{int}\,.$$ $$dP_T(t,T) = 0\,,\quad\text{for}\quad t>T\,.$$ The value $P_T$ has a meaning of the cash flow density over the delivery time. The total cash flow increment at time $t$ is related to $dP_T$ as $$dP(t) = \int_0^{T_e} dP_T(t,T)\,dT\,,$$ from which the total cash flow follows as $$\begin{aligned}
P(t) = \int_0^t d\tau \int_\tau^{T_e} dP_T(\tau,T)\,dT = \int_0^t d\tau \int_0^{T_e} dP_T(\tau,T)\,dT\,.\end{aligned}$$ Both $P_T$ and $P$ should be considered as functions of time $t$ with $T$ been a parameter.
Integrating $dP_T(t,T)$ from 0 to $T$ and noticing that $P_T(0) = 0$, we get the cumulative cash flow per delivery period $[T, T+dT]$: $$P_T(T) = \int_0^T dP_T = - \int_0^T (F + dF)\,dh\,.$$ Now we use the integration by parts, which in our case can be represented as $$\begin{aligned}
\left. (F\,h)\right|_0^T = \int_0^T h\,dF + \int_0^T F\,dh + \int_0^T dF\,dh\end{aligned}$$ For the averaged cumulative cash flow we obtain $${\left\langle}P_T(T) {\right\rangle}= F(0)\,h(0) - {\left\langle}F(T)\,h(T) {\right\rangle}=
F(0)\,\dot q_{int}(0) - {\left\langle}F(T)\,\dot q_{int}(T) {\right\rangle}\,.$$ The total average cash flow is given by integrating ${\left\langle}P_T{\right\rangle}$ over the delivery time: $${\left\langle}P(T_e) {\right\rangle}= \int_0^{T_e}{\left\langle}P_T(T){\right\rangle}\,dT = \int \dot q(0,T)\,F(0,T)\,dT - {\left\langle}\int \dot q(T,T)\,F(T,T)\,dT {\right\rangle}$$ Noticing that $$- \int \dot q(0,T)\,F(0,T)\,dT$$ equals the option intrinsic value, and $$-{\left\langle}\int \dot q(T,T)\,F(T,T)\,dT {\right\rangle}$$ equals the true option value, we conclude that $$\begin{aligned}
V_T = {\left\langle}P(T_e) {\right\rangle}= \int_0^{T_e} {\left\langle}dP {\right\rangle}\,.\end{aligned}$$
Averaging an integral with delta function {#app:averaged_integral}
=========================================
In this section we consider an averaging of the integrals of the type $$I(t) = \int_t^{T_e} \delta(C - F)\,\phi(T,F)\,dT\,,$$ where $F = F(t,T)$ is the forward curve – a stochastic function of observation and delivery times, and $\phi$ is an arbitrary function of $T$ and $F$. We will be interested in the average of the integral over the stochastic prices ${\left\langle}I(t){\right\rangle}_F$.
Due to the delta-function under the integral we can write $$I(t) = \int_t^{T_e} \delta(C - F)\,\phi(T,C)\,dT\,,$$
Introducing the set of trigger times $\{T_i: F(t,T_i) = C\}$ we can calculate the integral $I(t)$ for every observation time: $$\label{eq:summation3}
I(t) = \sum_{i>t} \frac{\phi_i}{|\dot F_i|}\,;\quad\text{where}\quad
\phi_i = \phi(T_i,C)\,;
\ F_i = C\,;
\ \dot F_i =\left. \frac{{\partial}F(t,T)}{{\partial}T}\right|_{T=T_i} \,;$$ This integral is a stochastic variable. Indeed, the process $F(t,T)$ is stochastic, and hence, the set of trigger times as well as the derivatives $\dot F_i$ are stochastic.
The average over the stochastic function $F$ is given by $$\label{eq:averaged_delta_integral_1}
{\left\langle}I(t) {\right\rangle}= \int_t^{T_e} {\left\langle}\delta(C - F){\right\rangle}\,\phi(T,C) \,dT$$ Here we have assumed that $C$ is constant.
The average of the delta function can be easily found using Fourier transformation: $$\label{eq:delta_fourier}
\delta(C-F) = \frac{1}{2\,\pi}\int e^{i\,\omega\,(C-F)}\,d\omega$$ Making use of the cumulant expansion and keeping only the two first cumulants (which, strictly speaking, is only exact for Gaussian variables) we find the average of the exponential: $${\left\langle}e^{i\,\omega\,(C-F)} {\right\rangle}\approx \exp\left(i\,\omega\,(C-F_0) - \frac12\,\omega^2\,\sigma^2 \right)\,,$$ where we used $${\left\langle}F {\right\rangle}= F_0\,;\quad {\langle\!\langle}F^2 {\rangle\!\rangle}= \sigma^2\,.$$ Substituting the averaged exponential into Eq. (\[eq:delta\_fourier\]) and performing integration we obtain $${\left\langle}\delta(C-F) {\right\rangle}= \frac{1}{\sigma\,\sqrt{2\,\pi}}\, \exp\left(
- \frac{(C-F_0)^2}{2\,\sigma^2}
\right)\,.$$
For the averaged integral we get: $$\label{eq:averaged_delta_integral_2}
{\left\langle}I(t) {\right\rangle}\approx \int_t^{T_e} \frac{1}{\sigma\,\sqrt{2\,\pi}}\, \exp\left(
- \frac{(C-F_0)^2}{2\,\sigma^2}
\right)\, \phi(T,C) \,dT\,.$$ Note that both $F_0$ and $\sigma$ are functions of $t$ and $T$: $$F_0 = F_0(t,T)\,;\quad \sigma = \sigma(t,T)$$ The exponent under the integral reaches its maximum at the trigger times $\{T_i\}$, when $F_0(T_i) = C$. Expanding the forward curve around this value $$F_0(T) \approx C + \dot F_0(T_i)\,(T - T_i)$$ we notice that for each trigger time averaged delta function becomes a Gaussian bell (as a function of time $T$) of the width $\sigma_t$, which can be found from an approximate relation $$\sigma_t(T_i) \approx \frac{\sigma(T_i)}{|\dot F_0(T_i)|}$$
slow $\phi$ approximation
-------------------------
Suppose that the trigger times are distributed rather uniformly on the time axis. Then we may speak about characteristic time distance between the trigger times. Let $\Delta T$ be the average trigger time step $$\Delta T \sim \overline{ ( T_{i+1} - T_i) }\,.$$ We say that function $\phi(T,C)$ is slow if it changes insignificantly within the average trigger time step: $$\frac{\dot \phi}{\phi} \ll \frac{1}{\Delta T}\,,\qquad \text{where}\quad
\dot \phi = \frac{{\partial}}{{\partial}T}\phi(T,C)\,.$$ Let us also suppose that the time derivative $\dot F_i$ can be considered as constant plus maybe a small random correction: $$|\dot F_i| = \dot F_0 + \epsilon_i\,,\qquad |\epsilon_i| \ll \dot F_0\,.$$ In this case we can replace the summation in Eq. (\[eq:summation3\]) with the integral: $$I(t) = \int_t^{T_e} \delta(C - F)\,\phi(T,F)\,dT = \sum_{i>t} \frac{\phi_i}{|\dot F_i|}\approx
\frac{1}{|\dot F_0|} \int_t^{T_e} \phi(T,C)\,dT$$ where we have assumed that the derivatives $|\dot F_0|$ can be considered as constant.
The same approximation can be used for the averaged integral (\[eq:averaged\_delta\_integral\_2\]). The biggest contribution to the integral is made by the intervals of the width $\sim 2\,\sigma_t$ around the trigger times $${\left\langle}I{\right\rangle}\approx \sum_i \int_{T_i-\sigma_t}^{T_i+\sigma_t} \frac{1}{\sigma\,\sqrt{2\,\pi}}\, \exp\left(
- \frac{(C-F_0)^2}{2\,\sigma^2}
\right)\, \phi(T,C) \,dT\,.$$
If the volatility of the forward price process is not too big, so that $$\sigma_t \lesssim \Delta T$$ then the function $\phi$ is slow with respect to the width of the Gaussian bell around the trigger times in the integral (\[eq:averaged\_delta\_integral\_2\]) $$\frac{\dot \phi}{\phi} \ll \frac{1}{\sigma_t}$$ In this case the function $\phi$ can be roughly considered as constant on every integration interval $(T_i - \sigma_t, T_i+\sigma_t)$. We thus obtain $${\left\langle}I(t){\right\rangle}\approx \sum_i \phi(T_i,C) \int_{T_i-\sigma_t}^{T_i+\sigma_t} \frac{1}{\sigma\,\sqrt{2\,\pi}}\, \exp\left(
- \frac{(C-F_0)^2}{2\,\sigma^2}
\right) \,dT\,.$$ The volatility $\sigma$ as a slow function of the delivery time and can be considered as constant under the integral. Substituting in the first order around the trigger price $F_0(T)\approx C + \dot F_0 \,(T-T_i)$ we can estimate the integral over one trigger time as $1/|\dot F_0(T_i)|$, and the averaged integral becomes $${\left\langle}I(t){\right\rangle}\approx \sum_i \frac{\phi(T_i,C)}{|\dot F_0(T_i)|}$$ This means that even for the averaged integral under the specified assumptions the same summation formula can be used as for the stochastic integral (\[eq:summation3\]).
If we demand that the function $\phi(T,C)$ is a slow function of time not only with respect to the characteristic time $\sigma_t$ but also with respect to the time interval $\Delta T$ $$\left|\frac{\phi(T_{i+1},C) - \phi(T_i,C)}{\phi(T_i,C)}\right| \ll 1$$ and if $|\dot F_0(T_i)|$ can be considered as constant, then we can roughly estimate $$\begin{aligned}
\label{eq:integral_approximation}
{\left\langle}I(t){\right\rangle}\approx \frac{1}{|\dot F_0|} \int \phi(T,C)\,dT\,.\end{aligned}$$ This approximation is used in the example calculation of the storage option time value.
Some formulas used in the main text {#ap:A1}
===================================
For the arbitrary sufficiently smooth functions $f(t)$ and $\varphi(t)$ the following identities are easy to prove: $$\begin{aligned}
\label{eq:A1}
&\int_{-\infty}^{\infty} \delta(\varphi(t))\,f(t)\,dt =
\sum \, \frac{f(t_i)}{|\dot \varphi(t_i)| } \,; \\
\label{eq:A2}
& \int_{-\infty}^{\infty} \delta'(\varphi(t))\,f(t)\,dt =
-\int \delta(\varphi(t))\, \frac{d}{dt} \left[ \frac{f(t)}{\dot \varphi(t)} \right] dt =
- \sum \, \frac{1}{|\dot \varphi(t_i)|}\, \frac{d}{dt}
\left[ \frac{f(t)}{\dot \varphi(t)} \right]_{t=t_i} \,; \\
& \int_{-\infty}^{\infty} \delta''(\varphi(t))\,f(t)\,dt =
\sum \, \frac{1}{|\dot \varphi(t_i)|}\,\frac{d}{dt}\, \frac{1}{\dot \varphi(t_i) }\,\frac{d}{dt}
\left[ \frac{f(t)}{\dot \varphi(t)} \right]_{t=t_i} \,;\end{aligned}$$ where the sum is over all points $\{t_i: \varphi(t_i)=0\}$.
In particular for the case $\varphi(t) = C-F(t)$ we have $$\begin{aligned}
\label{eq:A4}
&\int_{-\infty}^{\infty} \delta(C-F(t))\,f(t)\,dt =
\sum \, \frac{f(t_i)}{|\dot F(t_i)| } \,; \\
\label{eq:A5}
& \int_{-\infty}^{\infty} \delta'(C-F(t))\,f(t)\,dt =
\sum \, \frac{1}{|\dot F(t_i)| }\, \frac{d}{dt}
\left[ \frac{f(t)}{\dot F(t)} \right]_{t=t_i} \,;\end{aligned}$$
Making use the definitions from the main text (we omit the index $t$ where it is unambiguous) $$\begin{aligned}
&r(t,T) = \Delta r\,\delta(C(t) - F(t,T)) = \Delta r\,\delta(C - F(T))\,; \\
&r^{(1)}(T) = \Delta r\,\delta'(C - F(T))\,; \\
&r^{(2)}(T) = \Delta r\,\delta''(C - F(T))\,; \\
& K = \int_t^{T_e} r^{(1)}(T)\,dT = \Delta r\, \sum \frac{1}{|\dot F(T_i)| }\,; \\
& M = \int_t^{T_e} r^{(2)}(T)\,dT = -\Delta r\, \sum \frac{\ddot F(T_i)}{\dot F^2(T_i) }\,;\end{aligned}$$ it is easy to prove the following identities: $$\begin{aligned}
& \int r^{(1)}(u)\,F(u)\,du = C\,K\,; \\
& \int r^{(2)}(u)\,F(u)\,du = K + C\,M\,; \\
& \int r^{(2)}(u)\,F(u)\,f(u) \,du = \int r^{(1)}(u)\, f(u)\,du + C\int r^{(2)}(u)\,f(u)\,du\,; \\
& \int r^{(2)}(u)\,F(u)\,L(u,u)\,du = \int r^{(1)}(u)\, L(u,u)\,du + C\int r^{(2)}(u)\,L(u,u)\,du\,; \\
& \iint r^{(2)}(u)\,r^{(1)}(v)\,F(u)\,L(u,v)\,\,du\,dv = \iint r^{(1)}(u)\,r^{(1)}(v)\, L(u,v)\,\,du\,dv + \nonumber \\
& \hspace{6cm} + C\iint r^{(2)}(u)\,r^{(1)}(v)\,L(u,v)\,\,du\,dv\,,\end{aligned}$$ where $f(u)$ is an arbitrary sufficiently smooth function.
[^1]: We do not conduct a thorough analysis of existence and uniqueness of the solution of the variational problem. However we point out that the class of functions $q(t)$ on which the target functional is defined and could reach maximum should be rather broad. In particular it must include all continuous locally integrable functions. Other functions under the functional integral are allowed to be discontinuous. We will also use the concept of convergence of functions, which we will always understand as a weak convergence, i.e. $\psi_k\to\psi$ if $S[\psi_k]\to S[\psi]$.
[^2]: In reality the exact timing of the cash flow depends on the type of financial instruments used for hedging, and can be spread between observation and delivery time. However it plays no role for our analysis, and we agree to associate the cash-flow from the forward deals with the trading (observation) time $t$. Thus we agree that if the volume $V$ is purchased on the observation time $t$ with delivery on $T$, then the corresponding cash-flow $-V\,F(t,T)$ is associated with the observation time $t$.
[^3]: We can do so because the intrinsic profile satisfies the storage constraints, and hence we will be able to fulfil the contractual obligations.
[^4]: Strictly speaking, we can construct some extreme example, when the trigger price can be zero. For instance if the storage at the beginning is fool and release costs are so high that for some periods of time it is profitable rather to do nothing than to release gas. Here we do not consider such examples.
|
---
abstract: 'In this study, we consider the problem of variable selection and estimation in high-dimensional linear regression models when the complete data are not accessible, but only certain marginal information or summary statistics are available. This problem is motivated from the Genome-wide association studies (GWAS) that have been widely used to identify risk variants underlying complex human traits/diseases. With a large number of completed GWAS, statistical methods using summary statistics become more and more important because of restricted accessibility to individual-level data sets. Theoretically guaranteed methods are highly demanding to advance the statistical inference with a large amount of available marginal information. Here we propose an $\ell_1$ penalized approach, REMI, to estimate high dimensional regression coefficients with marginal information and external reference samples. We establish an upper bound on the error of the REMI estimator, which has the same order as that of the minimax error bound of Lasso with complete individual-level data. In particular, when marginal information is obtained from a large number of samples together with a small number of reference samples, REMI yields good estimation and prediction results, and outperforms the Lasso because the sample size of accessible individual-level data can be limited. Through simulation studies and real data analysis of the NFBC1966 GWAS data set, we demonstrate that REMI can be widely applicable. The developed R package and the codes to reproduce all the results are available at <https://github.com/gordonliu810822/REMI>'
author:
- Jian Huang
- Yuling Jiao
- Jin Liu
- Can Yang
title: 'REMI: Regression with marginal information and its application in genome-wide association studies'
---
Keywords: Genome-wide association studies, marginal information, high dimensional regression.
Introduction
============
High dimensional regression has been widely applied in various fields, such as medicine, biology, finance and marketing [@Hastie2009elements]. Consider the linear regression model that relates a response variable $Y$ to a vector of $p$ predictors $X=(X_1,\dots,X_p)^T$: $$\label{Reg1}
Y = \sum^p_{j=1} X_j \beta^*_j + \epsilon,$$ where $\bfbeta^* = (\beta^*_1,\dots,\beta^*_p)^T$ is the vector of regression coefficients and $\epsilon$ is the random error term with mean zero and noise level $\sigma_{\epsilon}^2.$ In most applications, the data set is comprised of an $n\times p$ matrix $\bfX$ for variables in $X$ and a vector $\bfy=(y_1,\dots,y_n)^T$ for response $Y$ collected from $n$ individuals. Given the individual-level data $\{\bfX,\bfy\}$, there exist convex [@Tibshirani:1996; @CandesTao:2007] and nonconvex [@FanLi:2001; @Zhang:2010a] penalized methods for estimating $\bfbeta^*$ with theoretical guarantee [@ZhaoYu:2006; @MeinshausenBuhlmann:2006; @ZhangHuang:2008; @BickelRitovTsybakov:2009; @ZhangZhang:2012], just name a few. Also see the monographs [@buhlmann2011statistics; @hastie2015statistical] and the references therein.
Motivated from the applications in human genetics, we consider the problem of estimating $\bfbeta^*$ when the individual-level data $\{\bfX,\bfy\}$ is not accessible but the marginal information is available, such as $\bfX_j^T \bfy$ and $\bfX^T_j \bfX_j$, $j = 1,\dots,p$, where $\bfX_j$ is the $j$-th column of $\bfX$. For this reason, we refer to our problem formulation as “gression with arginal nformation” (REMI). To make our formulation feasible, we also assume that information of the covariance structure of variables in $X$ can be estimated via a reference panel data set in the form of an $n_{\textrm{r}}\times p$ data matrix $\bfX_{\textrm{r}}$, where $n_{\textrm{r}}$ is the number of samples from the reference panel and $n_{\textrm{r}}\ll p$. A natural question arises: Without accessing the individual-level data, can we use marginal information together with the reference data $\bfX_{\textrm{r}}$ to estimate $\bfbeta^*$, assuming observations in $\bfX_{\textrm{r}}$ and $\bfX$ are from the same distribution? In particular, our problem arises in genome-wide association studies (GWAS), which have been conducted over the past decade to study the genetic basis of human complex phenotypes, including both quantitative traits and complex diseases [@hindorff2009potential; @welter2014nhgri; @visscher201710]. As of April, 2018, more than 59,000 unique phenotype-variant (typically Single Nucleotide Polymorphism, or SNP in short) associations have been reported in about 3,300 publications of GWAS (see the GWAS Catalog database <https://www.ebi.ac.uk/gwas/>). An important lesson from GWAS [@yang2010common; @visscher2012five; @visscher201710] is that complex phenotypes are highly polygenic, that is, they are often affected by many genetic variants with small effects. Well-known examples include human height [@wood2014defining], psychiatric disorders [@gratten2014large], as well as diabetes [@fuchsberger2016genetic]. Due to the polygenicity, variants with small effects largely remain undiscovered yet and large sample sizes are required in exploring genetic architectures of complex phenotypes. Researchers world-wide are forming large genomic consortia, such as the Genetic Investigation of ANthropometric Traits (GIANT) Consortium and the psychiatric genomic consortium (PGC), to maximize sample size, aiming at a deeper understanding of the genetic architecture of complex phenotypes.
Although much efforts have been made for data sharing, it is still very difficult for a research group to fully access the individual-level genotype data available in a consortium. For example, a core research group from the GAINT consortium reported that they can only access genotype data from about 44,000 individuals [@yang2015genome] while the total sample size is more than 250,000 for the consortium [@wood2014defining]. There are several reasons for the restricted access to the individual-level data. First, privacy protection is always a big concern in sharing individual-level genotype data. Second, it is often time-consuming to achieve an agreement on data-sharing among different research groups. Third, many practical issues arise in data transportation and storage. In contrast, summary statistics from GWAS are widely available through many public gateways [@editor2012ng], e.g., the download session at the GWAS Catalog <https://www.ebi.ac.uk/gwas/downloads/summary-statistics>. Because these summary statistics (e.g., estimated effect sizes, standard errors, and $z$-values) are often generated by simple linear regression analysis, summary statistics are essentially marginal information. To meet the great demand of data analysis in GWAS, various statistical methods have been proposed to utilize marginal information. Using a few hundreds of human genome data from the 1000 Genome Project as a reference panel, information on the correlation structure of genetic variants (typically using “linkage disequilibrium” in genetics, or LD for short) becomes available. This allows these methods to bypass the individual-level data but only use marginal information. Here we roughly divide these methods into three categories: (a) Methods for heritability estimation. Heritability of a phenotype quantifies the relative importance of genetics and environment to the phenotype [@visscher2008heritability]. When individual-level data is accessible, linear mixed model (LMM)-based approaches (e.g., GCTA [@yang2010common; @yang2011gcta]) are widely used for heritability estimation [@lee2011estimating]. In the absence of individual-level data, Bulik et al. [@bulik2015ld] first introduced the LD score regression, named LDScore, for heritability estimation only using summary statistics and the reference data from the 1000 Genome Project. Based on the minimal norm quadratic unbiased estimation criteria, Zhou [@zhou2016unified] proposed a novel method of moments, MQS, for variance component estimation with summary statistics. (b) Methods for association mapping. Heritability estimation provides a global measure which quantifies the overall contribution from genetic factors while association mapping is to localize genetic variants associated with a given phenotype. Recently, a few statistical methods have been proposed for association mapping based on summary statistics, including FGWAS [@pickrell2014joint], PAINTOR [@kichaev2014integrating], CAVIAR [@hormozdiari2014identifying], and CAVIARBF [@chen2015fine]. Although these methods are very useful for performing association mapping on summary statistics, they still have their limitations. On one hand, they adopted some ad-hoc ways to reduce computational cost. For example, to avoid a combinatorial search, FGWAS assumes that there is only one causal signal in an LD block and PAINTOR searches no more than two causal variants in its default setting. On the other hand, statistical analysis is oversimplified to overcome estimation difficulties. For example, the non-centrality parameter in PAINTOR and the variance components in CAVIAR and CAVIARBF are pre-fixed rather than adaptively estimated from data. (c) Methods for effect size estimation and risk prediction. Recently, Vilhj[á]{}lmsson et al. [@vilhjalmsson2015modeling] proposed a Bayesian method, LDpred, for effect size estimation and risk prediction by accounting for LD. Along this line, Hu et al. [@hu2017leveraging] further introduced AnnoPred to improve LDpred by incorporating functional information in human genome. However, neither LDpred or AnnonPred should be considered as a marginal-information-based method because it requires individual-level data as validation data for its parameter tuning.
Although the existing statistical methods have shown a good empirical performance in GWAS data analysis, there are a number of open questions on REMI. First, the sample size of the reference panel is often very small. For example, there are only about 370 samples from the 1000 Genome Project that can be used as reference for analyzing GWAS data in European ancestry. It remains unclear why such a small sample size is often good enough for exploring the correlation structure of a large number of variables (i.e., SNPs). Second, the theoretical properties of the existing methods for effect size estimation and prediction error are not clear. Third, the sampling-based algorithms are often time-consuming as they need to run thousands of Markov Chain Monte Carlo (MCMC) iterations [@zhu2016bayesian]. In this paper, we propose a unified framework to address the above open questions. The rest of this paper is organized as follows: In Section \[model\], we introduce our REMI model and discuss REMI in GWAS. In Section \[empirical\], we present an efficient coordinate descent algorithm following by discussion on some practical issues. In Section \[theory\], we establish the error bound and prediction error of the proposed method. In particular, our theoretical results explain why a small number of samples (i.e., $n_{{\textrm{r}}}$) from the reference panel can be good enough for effect size estimation and risk prediction. In Section \[emp\_results\], we show the results from both simulation studies and real data analysis.
The REMI model {#model}
==============
The REMI model {#the-remi-model}
--------------
$$\begin{aligned}
\label{lm1}\bfy = \bfX\bfbeta^* + \bfepsilon,\end{aligned}$$ where $\bfy$ is an $n\times1$ vector of responses from $n$ samples, $\bfX$ is an $n\times p$ design matrix whose rows are i.i.d. random vectors with mean zero and covariance matrix $\bfSigma$, $\bfbeta^*$ is a $p\times 1$ vector of underlying true effect sizes of $p$ variables, and $\bfepsilon$ is an $n\times 1$ vector of i.i.d error terms with mean 0 and variance $\sigma_{\bfepsilon}^2$. Without loss of generality, we assume that both $\bfy$ and the columns of $\bfX$ have been centered.
For the linear regression model , if the individual-level data $(\bfy, \bfX)$ is available, a basic approach for estimating $\bfbeta^*$ in high-dimensional settings is the Lasso [@Tibshirani:1996]. The Lasso estimator is given by $$\begin{aligned}
\label{regularized_reg}
\widehat{\bfbeta} = \arg\min_{\bfbeta} \frac{1}{n} \|\bfy - \bfX \bfbeta\|^2 + \lambda \|\bfbeta\|_1, $$ where $\|\cdot\|_1$ is the $\ell_1$ norm and $\lambda \ge 0 $ is a regularization parameter. In our problem, however, the individual-level data $\{\bfX, \bfy\}$ is not accessible. Hence, direct application of the Lasso is not feasible here. We note that several other important penalized methods have been proposed, including SCAD [@FanLi:2001] and MCP [@Zhang:2010a]. We will focus on the Lasso penalty below, although our proposed approach can also be based on the other penalties.
We now describe our proposed REMI model with the Lasso penalty. Rewrite (\[regularized\_reg\]) as $$\begin{aligned}
\label{regularized_reg2}
\widehat{\bfbeta} &= \arg\min_{\bfbeta} \frac{1}{n} (\bfbeta^T \bfX^T \bfX \bfbeta -2\bfbeta^T \bfX^T \bfy + \bfy^T\bfy) + \lambda \|\bfbeta\|_1 \nonumber\\
& = \arg\min_{\bfbeta} \bfbeta^T \bfX^T \bfX \bfbeta/n -2\bfbeta^T \bfX^T \bfy/n + \lambda \|\bfbeta\|_1,\end{aligned}$$ where the second term only involves the inner product of the optimization variable $\bfbeta$ and marginal information, say, $ \widetilde{\bfy} = \bfX^T \bfy/n$, which we assume is available. The difficulty comes from the first term, where $\bfX^T\bfX/n$ is unknown since $\bfX$ is not observed. Motivated by the application in GWAS, we assume that there exists a reference $n_{\textrm{r}} \times p$ data matrix $\bfX_{\textrm{r}}$, where the rows of $\bfX_{\textrm{r}}$ are i.i.d. and have the same distribution with covariance matrix $\bfSigma$ as the rows of $\bfX$. Therefore, both $\widehat\bfSigma = \bfX^T\bfX/n$ and $\widehat{\bfSigma}_{\textrm{r}} = \bfX_{\textrm{r}}^T\bfX_{\textrm{r}}/n_{\textrm{r}}$ can be viewed as estimators of $\bfSigma$. So we propose to solve the following optimization problem to estimate $\bfbeta^*$: $$\begin{aligned}
\label{remi_1}
\widehat{\bfbeta}^{\textrm{c}} = \arg\min_{\bfbeta} \bfbeta^T \bfX_{\textrm{r}}^T \bfX_{\textrm{r}} \bfbeta/n_{\textrm{r}} -2\bfbeta^T\widetilde{\bfy} + \lambda \|\bfbeta\|_1,\end{aligned}$$ where $\widehat{\bfbeta}^{\textrm{c}}$ denotes the estimator using the reference covariance matrix. Clearly, the above model (\[remi\_1\]) only uses the marginal correlation between $\bfX$ and $\bfy$, with the covariance matrix estimated by an external reference panel $\bfX_{\textrm{r}}$.
REMI in GWAS {#remi-gwas}
------------
In the context of GWAS, the available marginal information may not be $\widetilde{\bfy} = \bfX^T \bfy/n$ but summary statistics $\{\widehat{\beta}^{\textrm{m}}_{j}, \hat{s}^2_j\}_{j=1,\dots,p}$ from univariate linear regression: $$\begin{aligned}
\widehat{\beta}^{\textrm{m}}_{j} = (\bfX^T_j\bfX_j)^{-1}\bfX^T_j \bfy, \quad \hat{s}^2_j = (n\bfX^T_j\bfX_j)^{-1}(\bfy-\bfX_j \widehat{\beta}^{\textrm{m}}_{j})^T(\bfy-\bfX_j\widehat{\beta}^{\textrm{m}}_{j}),$$ where superscript $^{\textrm{m}}$ is used to denote marginal information. $\widehat{\beta}_{j}^{\textrm{m}}$ and $\hat{s}^2_j$ are the estimated effect size and its variance for SNP $j$, respectively. Due to the polygenicity of many complex phenotypes, the standard errors can be well approximated by $\hat{s}_j \approx \sqrt{(n\bfX^T_j\bfX_j)^{-1}\bfy^T\bfy}$ (Zhu and Stephens 2016). Let $\widehat{\bfbeta}^{\textrm{m}} = [\widehat{\beta}_{1}^{\textrm{m}}, \dots, \widehat{\beta}_{p}^{\textrm{m}}]^T$, $\hat{\bfs}^2 = [\hat{s}^2_1,...,\hat{s}^2_p]^T$ be the vectors collecting estimated effect sizes and estimated variance, respectively, and $\widehat\bfS$ be a $p\times p$ diagonal matrix with $\hat{s}_j$ being its $j$-th diagonal element. Further, we introduce a $p\times p$ diagonal matrix $\widehat{\bfD}=\mathrm{diag}(\hat{d}_j)$ with its $j$-th diagonal element being the sample standard deviation of $\bfX_j$, i.e., $\hat{d}_j = \sqrt{\frac{\bfX_j^T \bfX_j}{n}}$, and correlation matrix $\widehat{\bfR}=[\hat{r}_{jk}]\in\mathbb{R}^{p\times p}$ with $\hat{r}_{jk}=\frac{\bfX_{j}^T \bfX_{k}}{(\bfX_{j}^T \bfX_{j})^{1/2}(\bfX_{k}^T \bfX_{k})^{1/2}}$. Noticing that $\hat{d}_j^{2} \widehat{\beta}_{j}^{\textrm{m}} = \bfX^T_j\bfy/n$ and $n^2 \hat{d}^2_j \hat{s}^2_j \approx \bfy^T\bfy$, the REMI formulation (\[regularized\_reg2\]) becomes $$\begin{aligned}
\widehat{\bfbeta} &= \arg\min_{\bfbeta}\, \bfbeta^T \bfX^T \bfX \bfbeta/n -2\bfbeta^T \bfX^T \bfy/n + \lambda \|\bfbeta\|_1, \nonumber\\
&= \arg\min_{\bfbeta}\, \bfbeta^T \widehat{\bfD}\widehat{\bfR}\widehat{\bfD} \bfbeta -2\bfbeta^T \widehat{\bfD}^{2}\widehat{\bfbeta}^{\textrm{m}} + \lambda \|\bfbeta\|_1,\nonumber \\
&\approx\arg\min_{\bfbeta}\, \frac{\bfy^T\bfy}{n^2}\bfbeta^T \widehat{\bfS}^{-1}\widehat{\bfR}\widehat{\bfS}^{-1} \bfbeta -2 \frac{\bfy^T\bfy}{n^2}\bfbeta^T \widehat{\bfS}^{-2}\widehat{\bfbeta}^{\textrm{m}} + \lambda \|\bfbeta\|_1, \\
&=\arg\min_{\bfbeta}\, \bfbeta^T \widehat{\bfS}^{-1}\widehat{\bfR}\widehat{\bfS}^{-1} \bfbeta -2 \bfbeta^T \widehat{\bfS}^{-2}\widehat{\bfbeta}^{\textrm{m}} + \widetilde{\lambda} \|\bfbeta\|_1, \nonumber \\
\end{aligned}$$ where $\widetilde{\lambda} = \frac{n^2}{\bfy^T\bfy} \lambda$ and the approximation holds in the case of polygenicity. Because $\widetilde\lambda$ is a tuning parameter that scales $\lambda$ with a constant factor ($\frac{n^2}{\bfy^T\bfy}$), we slightly abuse $\lambda$ for $\widetilde{\lambda}$ and propose to solve the following optimization problem $$\begin{aligned}
\label{remi_2}
\widehat{\bfbeta}^{\textrm{r}} = \arg\min_{\bfbeta}\, L(\bfbeta) +\lambda \|\bfbeta\|_1, $$ where $L(\bfbeta)=\bfbeta^T \widehat{\bfS}^{-1}\widehat{\bfR}\widehat{\bfS}^{-1} \bfbeta -2 \bfbeta^T \widehat{\bfS}^{-2}\widehat{\bfbeta}^{\textrm{m}}$, and $\widehat{\bfbeta}^{\textrm{r}}$ denotes the estimates using correlation information. Similar to REMI (\[remi\_1\]) in which covariance matrix $\widehat\bfSigma=\bfX^T\bfX/n$ needs to be estimated, here correlation matrix $\widehat{\bfR}$ needs to be estimated by samples from the reference panel $\bfX_{\textrm{r}}$. We refer (\[remi\_1\]) as REMI-C and (\[remi\_2\]) as the REMI-R, respectively.
Algorithm and Practical Issues {#empirical}
==============================
Algorithm
---------
Here we adopt the widely used coordinate descent algorithm, which updates one parameter at a time, say $\widehat{\beta}^{\textrm{c}}_{j}$, keeping all other parameters fixed at their current values. Thus the sub-problem for parameter $\widehat{\beta}^{\textrm{c}}_{j}$ can be written as $$\begin{aligned}
\label{remi_1_j}
\widehat{\beta}^{\textrm{c}}_{j}(\lambda) = \arg\min_{{\beta}_{j}} \widehat{\sigma}_{jj} \beta_{j}^2 - 2\left(\widetilde{y}_j- \sum_{k\ne j} \widehat{\beta}_{k}^{\textrm{c}}{\widehat\sigma}_{jk} \right){\beta}_{j} + \lambda|\beta_{j}|,\end{aligned}$$ where $\widehat{\sigma}_{jk}$ is an element in $\widehat{\bfSigma}_{\textrm{r}} = [\widehat{\sigma}_{jk}]\in\mathbb{R}^{p\times p}$. An efficient path algorithm can be developed based on the warm start and some other tricks as described in [@friedman2010regularization]. In particular, we generate a sequence of $\bflambda = (\lambda_1,\dots,\lambda_D)$ equally spaced in logarithm with $\lambda_1 = \lambda_{\mathrm{max}}$ and $\lambda_D = \tau\lambda_{\mathrm{max}}$, where $\lambda_{\mathrm{max}}$ is the minimum $\lambda$ that shrinks all parameters to zero and $\tau$ is usually set to 0.05. For each $\lambda$, we use the solution of (\[remi\_1\]) from the last $\lambda$ value as warm start. The path algorithm is described in Algorithm \[alg2\].
\[alg2\]
*Output*: Solution path for $\widehat{\bfbeta}^{\textrm{c}}(\bflambda)$.\
Similar to REMI-C, an efficient coordinate descent algorithm can developed for solving REMI-R (\[remi\_2\]). The efficient path algorithm is given in Algorithm \[alg4\].
\[alg4\]
*Output*: Solution path $\widehat{\bfbeta}^{\textrm{r}} (\bflambda)$.\
Reference panel {#refpanel}
---------------
In REMI-R model (\[remi\_2\]), it involves the cohort-based estimated correlation matrix. Based on the nature of the correlation patterns of the SNPs, $\bfR$ can be approximated by a block diagonal matrix. Specifically, we first partition the whole genome into $L$ blocks ($L=1,703$ for European ancestry and $L=1,445$ for Asian ancestry, respectively [@berisa2016approximately]). Then we calculate empirical correlation matrix $\widehat{\bfR}_{\mathrm{emp}}$ for each LD-block. To ensure a stable numerical result, we apply a simple shrinkage estimator to obtain $\widehat{\bfR}^{\mathrm{r}} = \kappa \widehat{\bfR}_{\mathrm{emp}} + (1-\kappa) \bfI$ within each block [@schafer2005shrinkage], where we used $\kappa=0.9$ as default (the estimate of $\bfbeta^*$ is insensitive to $\kappa$ [@pasaniuc2014fast]). Thus, similar to [@zhu2016bayesian], REMIs and its individual-level-data counterpart will produce approximately the same inferential results within a region. After plugging $\widehat\bfS$ and $\widehat\bfR^{\mathrm{r}}$ in (\[remi\_2\]), we can use the coordinate-descent algorithm to obtain $\widehat{\bfbeta}^{\mathrm{r}}(\lambda)$ (Algorithm \[alg4\]).
Choice of regularization parameter $\lambda$
--------------------------------------------
The REMIs have one regularization parameter $\lambda$. Here we briefly show how to choose this parameter for REMI-R and it is straightforward to develop the same strategy for REMI-C. Similarly to the Lasso solver [@friedman2010regularization], we generate a sequence of $\bflambda$ from $\lambda_{\mathrm{max}}$ to $\tau\lambda_{\mathrm{max}}$, where $\lambda_{\mathrm{max}}$ is the minimum value of $\lambda$ that shrinks all parameters to zero and $\tau$ is pre-specified with the default value at 0.05. Note that $\lambda_{\mathrm{max}} = \max \big\{2\hat\beta_j^{\mathrm{m}}/\hat{s}^2_j\big\}_{j=1,\dots,p}$. We search for optimal value $\lambda$ value using BIC, $$\mathrm{BIC}(\lambda_l) = L(\widehat\bfbeta^{\mathrm{r}}(\lambda_l)) + \mathrm{log}(n)\mathrm{df}(\lambda_l).$$ Zou et al. [@zou2007degrees] showed that the number of nonzero coefficients is an unbiased estimate for the degrees of freedom of Lasso. We choose $\mathrm{df}(\lambda_l)$ to be the number of nonzero coefficients given $\lambda_l$. To make fair comparison of REMIs with Lasso, we also use BIC to select the regularization parameter when individual-level-data is accessible.
Theoretical properties {#theory}
======================
In this section, we give nonasymptotic bounds on the estimation error $\|\bfbeta^{*}_{\mathcal{A}}-\widehat{\bfbeta}^{\textrm{c}}\|$ and the prediction error $\| \bfX_\mathrm{new} \bfbeta^{*}_{\mathcal{A}}-\bfX_\mathrm{new} \widehat{\bfbeta}^{\textrm{c}}\|^2/n_{\mathrm{new}}$, where $ \mathcal{A} $ denotes index of significant entries of $\bfbeta^*$ and $\bfbeta^*_{\mathcal{A}}$ denote the vector supported on $\mathcal{A}$.
Since in real applications of genetic data the underlying signal is not exactly sparse but with many small components. Here, we assume that the target $\bfbeta^*$ is weak sparse, i.e., in addition to some significant components indexed by $ \mathcal{A} $, there may be many nonzero entries in $\bfbeta^*$ with very small magnitude, as indexed by $\mathcal{I} = \mathcal{A}^c$. Let $\bfbeta^*_{\mathcal{I}}$ be the vector supported on $ \mathcal{I}$. It is reasonable to assume that $$\label{weaksparse}
s = |\mathcal{A}| \leq n, \quad \|\bfbeta^*_{\mathcal{I}}\|_{\infty} \leq 2\sigma_{\epsilon}\sqrt{\frac{\log p}{n}},$$ since the signal whose magnitude smaller than this order is undetectable. Then $$\label{dec}
\widetilde{\bfy} = \widehat{\bfSigma} \bfbeta^* + \widetilde{\bfepsilon} = \widehat{\bfSigma} \bfbeta^*_{\mathcal{A}} + \widehat{\bfSigma} \bfbeta^*_{\mathcal{I}}+ \widetilde{\bfepsilon},$$ where, $\widetilde{\bfepsilon} = \bfX^T \bfepsilon/n.$ Let $C_1 \geq \|\textrm{diag}(\bfSigma)\|_{\infty}$, $C_2 \geq \max_{j=1,...p} \{\|\bfSigma_j\|_1\}$, $C_3\geq \|\bfbeta^*_{\mathcal{A}}\|_1,$ $C_4 \geq \|\bfbeta^*_{\mathcal{I}}\|_1.$ Recall the restrict eigenvalue [@Bickel:2009] of $\widehat{\bfSigma}_{\textrm{r}}$ is defined as $$\phi_{\widehat{\bfSigma}_{\textrm{r}}} = \min_{0\neq \bfv \in \mathcal{C}_{\mathcal{A},3}} \frac{\bfv^T\widehat{\bfSigma}_{\textrm{r}}\bfv}{\|\bfv\|_2^{2}},$$ where $$\mathcal{C}_{\mathcal{A},3}= \{\bfv \in \mathcal{R}^p: \|\bfv_{\mathcal{I}}\|_1 \leq 3\|\bfv_{\mathcal{A}}\|_1\}.$$
\[esterr\] Assume the rows of $\bfX$ and $\bfX_{\mathrm{r}}$ are i.i.d sub-Gaussian samples drawn from population with mean $\textbf{0}$ and covariance matrix $\bfSigma$, and $\widehat{\bfSigma}_{\mathrm{r}}$ satisfying restricted eigenvalue condition with $\phi_{\widehat{\bfSigma}_{\mathrm{r}}} \geq \phi_0 > 0,$ and the noise vector $\bfepsilon$ is mean zero sub-Gaussian with noise level $\sigma_{\epsilon}$, and $n\geq \frac{4}{C}\log p,$ $n_{\mathrm{r}}\geq \frac{4}{C}\log p.$ Take $\lambda\geq 2 \lambda_0 = 4(\frac{C_1(C_3+C_4)}{\sqrt{C}}\sqrt{\frac{\log p}{n}} + \frac{C_1+\sqrt{C}C_2}{\sqrt{C}}\sigma_{\epsilon}\sqrt{\frac{\log p}{ n}} + \frac{C_1C_3}{\sqrt{C}} \sqrt{\frac{\log p}{n_{\mathrm{r}}}})$.
\(i) With probability at least $1-3/p^2- 1/p^3$ we have $$\| \widehat{\bfbeta}^{\mathrm{c}} -\bfbeta^*_{\mathcal{A}}\| \leq \frac{6}{\phi_0}(\frac{C_1(C_3+C_4)}{\sqrt{C}}\sqrt{\frac{s\log p}{n}} + \frac{C_1+\sqrt{C}C_2}{\sqrt{C}}\sigma_{\epsilon}\sqrt{\frac{s\log p}{ n}} + \frac{C_1C_3}{\sqrt{C}} \sqrt{\frac{s\log p}{n_{\mathrm{r}}}}).$$
\(ii) Suppose we observe $\bfX_{\mathrm{new}} \in \mathcal{R}^{n_{\mathrm{new}}\times p}$, whose rows are sampled from the same distribution as that of $\bfX$’s. Then with probability at least $1-3/p^2- 1/p^3$, the prediction error satisfies $$\|\bfX_{\mathrm{new}} (\widehat{\bfbeta}^{\mathrm{c}} - \bfbeta^*_{\mathcal{A}})\|_2^2/n_{\mathrm{new}}\leq \mathcal{O}((\sigma_{\epsilon}\sqrt{\frac{s\log{p}}{n}} + \sqrt{\frac{s\log{p}}{n_{\mathrm{r}}}})^2) (1+ s^2 (\sqrt{\frac{\log p}{n_{\mathrm{r}}}}+ \sqrt{\frac{\log p}{n_{\mathrm{new}}}}) ).$$
The assumption that $\bfX_{\mathrm{r}}$ are i.i.d sub-Gaussian samples drawn from population with mean $\textbf{0}$ and covariance matrix $\bfSigma$ implies the restricted eigenvalue condition $\phi_{\widehat{\bfSigma}_{\mathrm{r}}} \geq \phi_0 $ holds for some positive $\phi_0$ with high probability as long as $n_{\mathrm{r}} \geq \mathcal{O}(s\log{p})$ [@VanPeter:2009; @Vershynin:2010; @HuangJiaoLuZhu:2017]. As shown in Theorem \[esterr\], with the help of reference panel, we can also get an accurate estimator by even if we only have marginal information in high-dimension setting as long as $n \geq \mathcal{O}(s\log{p})$ and $n_{\mathrm{r}} \geq \mathcal{O}(s\log{p})$. Furthermore, the estimation error of REMI model achieve the minimax optimal rate as that of the Lasso [@raskutti2011minimax] if the number of samples of reference penal $n_{\mathrm{r}}$ is at the same order of the number of individual-level samples $n$. Moreover if the magnitude of the significant entries larger than $\mathcal{O}(\sigma_{\epsilon} \sqrt{\frac{s\log{p}}{n}} + \sqrt{\frac{s\log{p}}{n_{\mathrm{r}}}} )$, the estimated support $\mathrm{supp}(\widehat{\bfbeta}^{\mathrm{c}})$ coincide with the true significant set $\mathcal{A}$.
Suppose we know observe $\bfX_{\mathrm{new}} \in \mathcal{R}^{n_{\mathrm{new}}\times p}$, whose rows are sampled from same distribution as that of $\bfX$’s. Then with high probability, prediction error satisfies $$\|\bfX_{\mathrm{new}} (\widehat{\bfbeta}^{\mathrm{c}} - \bfbeta^*)\|_2^2/n_{\mathrm{new}}\leq \mathcal{O}((\sigma_{\bfepsilon}\sqrt{\frac{s\log{p}}{n}} + \sqrt{\frac{s\log{p}}{n_{\mathrm{r}}}})^2) (1+ s^2 (\sqrt{\frac{\log p}{n_{\mathrm{r}}}}+ \sqrt{\frac{\log p}{n_{\mathrm{new}}}}) ).$$ Indeed, let $\widehat{\bfSigma}_{\mathrm{new}} = \bfX_{\mathrm{new}}^{T} \bfX_{\mathrm{new}}/n_{\mathrm{new}}.$ Then, $$\begin{aligned}
&\|\bfX_{\mathrm{new}} (\widehat{\bfbeta}^{\textrm{c}} - \bfbeta^*_{\mathcal{A}})\|_2^2/n_{\mathrm{new}} = \langle\widehat{\bfSigma}_{\mathrm{new}}(\widehat{\bfbeta}^{\textrm{c}} - \bfbeta^*_{\mathcal{A}}), \widehat{\bfbeta}^{\textrm{c}} - \bfbeta^*_{\mathcal{A}}\rangle \\
&= \langle\Delta, \widehat{\bfSigma}_{\mathrm{r}} \Delta \rangle + \langle\Delta, (\widehat{\bfSigma}_{\mathrm{new}} - \widehat{\bfSigma}_{\mathrm{r}}) \Delta \rangle\\
& \leq \frac{3}{2}\lambda \|\Delta_{\mathcal{A}}\|_1 + \|\Delta\|_1^2 \|\widehat{\bfSigma}_{\mathrm{new}} - \widehat{\bfSigma}_{\mathrm{r}}\|_{\infty}\\
&\leq \frac{3}{2}\lambda \|\Delta_{\mathcal{A}}\|_1 + \|\Delta\|_1^2 \frac{2C_1}{\sqrt{C}}(\sqrt{\frac{\log p}{n_{\mathrm{r}}}}+ \sqrt{\frac{\log p}{n_{\mathrm{new}}}})\\
& \leq \mathcal{O}(\sigma_{\bfepsilon}\sqrt{\frac{s\log{p}}{n}} + \sqrt{\frac{s\log{p}}{n_{\mathrm{r}}}}) \|\Delta_{\mathcal{A}}\|_2 + \mathcal{O}(\sqrt{\frac{\log p}{n_{\mathrm{r}}}}+ \sqrt{\frac{\log p}{n_{\mathrm{new}}}}) s^2 \|\Delta_{\mathcal{A}}\|_2^2\\
&\leq \mathcal{O}((\sigma_{\bfepsilon}\sqrt{\frac{s\log{p}}{n}} + \sqrt{\frac{s\log{p}}{n_{\mathrm{r}}}})^2) (1+ s^2 (\sqrt{\frac{\log p}{n_{\mathrm{r}}}}+ \sqrt{\frac{\log p}{n_{\mathrm{new}}}}) )\\
$$ where, the first inequality uses and Cauchy-Schwartz inequality, and the second one is due to Lemma \[linf\], and the third inequality follow from and Cauchy-Schwartz inequality, the fourth inequality uses Theorem \[esterr\].
Numerical Studies {#emp_results}
=================
Simulation studies {#simulation_study}
------------------
In simulation studies, we compare REMI-C (\[remi\_1\]), REMI-R (\[remi\_2\]) and Lasso using individual-level data. To avoid unrealistic LD pattern in simulation, we used the genotype data $\bfX$ from the GERA data set [@hoffmann2011next]. The GERA data set provided 657,184 genotyped SNPs for 62,313 European individuals. We performed strict quality control on data using PLINK [@purcell2007plink]. We excluded SNPs with a minor allele frequency less than 1$\%$, having missing values in more than 1$\%$ of the individuals or with a Hardy-Weinberg equilibrium $p$-value below 0.0001. Moreover, we removed one member of pairs with genetic relatedness larger than 0.05. Finally, there remained 53,940 samples for 550,482 SNPs. As individual-level-based analyses often suffer from limited sample sizes due to the restricted access of individual-level data, summary-level-based analyses may have advantages because the sample sizes are often much larger. To simulate this situation, we prefixed the sample size for individual-level-based analyses at $n_{\textrm{ind}} =3,000$. Specifically we randomly selected $n_{\textrm{ind}}$ samples from 53,940 individuals in the GERA data set to form the genotype matrix $\bfX \in \mathbb{R}^{n_{\textrm{ind}}\times p}$, where $p=19,865$ was the total number of the genotyped SNP on chromosomes 16, 17 and 18. Then the phenotype vector $\bfy$ was generated as $\bfy=\bfX\bfbeta^* + \bfepsilon$, where $\bfepsilon\sim \mathcal{N}(0,\sigma^2_{\epsilon})$ and the heritability ($h^2 = \frac{\mathrm{Var}(\bfX\bfbeta) }{\mathrm{Var}(\bfX\bfbeta^*) + \sigma^2_{\epsilon}}$) was controlled at 0.2, 0.3, 0.4 and 0.5. Here $\bfbeta^*$ was the vector of true effect size with sparsity $\alpha$, i.e., $\alpha\times p$ entries in $\bfbeta^*$ were nonzero and they were sampled from $\mathcal{N}(0,1)$. In our simulation study, we varied $\alpha$ in $\{0.001, 0.003, 0.005, 0.007, 0.01, 0.02\}$. With $\{\bfX,\bfy\}$ at hand, the standard Lasso can be applied, serving as a reference of individual-level data analysis.
To generate summary-level data, we varied sample size $n$ from 3,000 to 50,000. We generated individual-level data as we described above and then we ran simple linear regression on $\{\bfX_j,\bfy\}$, $j = 1,\dots,p$ to obtain $\{\widehat{\bfbeta}^{(m)},\hat{\bfs}^2\}$. After that, we pretended that we did not have individual-level data $\{\bfX,\bfy\}$ and then only used $\bfX^T\bfy$ and $\{\widehat{\bfbeta}^{(m)},\hat{\bfs}^2\}$ as the input for REMI-C and REMI-R, respectively. We used 379 European samples from the 1000 Genome Project data as the reference panel to estimate covariance matrix (REMI-C) or correlation matrix (REMI-R), as detailed in Section \[refpanel\]. For each replication, we held out 200 independent samples to evaluate prediction accuracy. In total, we summarized our results based on 50 replications for each setting.
We compared the performance of REMI-C, REMI-R and the Lasso using individual-level data in terms of variable selection and prediction. Specifically, we used partial area under the receiver operating characteristic (ROC) curve (partial AUC) for variable selection performance and the Pearson’s correlation coefficient between predicted and observed phenotypes for prediction performance. The results of this simulation study are shown in Figure \[fig\_selection\] and Figure \[fig\_pred\]. First, we can observe that the difference between REMI-C and REMI-R is nearly invisible. This justifies the approximation made in REMI-R. Second, when the sample size ($n$ = 3,000 or 5,000) in summary-level data is similar to that of individual-level data ($n_{\mathrm{ind}} = 3,000$), the performance of variable selection and prediction for REMIs (both REMI-C and REMI-R) is similar to that of Lasso. Third, REMIs gradually outperform the Lasso as the sample size increases from 5,000 to 50,000, for both variable selection and prediction. This clearly indicates that REMIs can have big advantages over the Lasso when the sample sizes of summary-level data become much larger.
![The comparison of variable selection performance of REMIs (REMI-R and REMI-C) for summary-statistics data with Lasso for individual-level data with sample size 3000. The sample size used to produce summary statistics was varied and denoted as $n\in\{3,000, 5,000, 10,000, 20,000, 50,000\}$. We used partial AUC to measure the variable selection performance. []{data-label="fig_selection"}](boxplot_pAUC.png){width="100.00000%"}
![Prediction accuracy of REMIs with summary-level data and the Lasso for individual-level data of sample size 3,000. The sample size used to simulate summary statistics was varied $n\in\{3,000, 5,000, 10,000, 20,000, 50,000\}$. Prediction accuracy is measured by the Pearson’s correlation coefficient between predicted and observed phenotypes.[]{data-label="fig_pred"}](boxplot_Prediction.png){width="100.00000%"}
Real data analysis {#realdata}
------------------
To demonstrate the utility of REMIs, we first compared Lasso and REMIs based on the GWAS data set from the Northern Finland Birth Cohorts program (NFBC1966) [@sabatti2009genome]. The NFBC1966 data set contains information for 5,402 individuals with a selected list of phenotypic data related to cardiovascular disease including high-density lipoprotein (HDL), low-density lipoprotein (LDL), total cholesterol (TC), triglycerides (TG), C-reactive protein (CRP), glucose, insulin, body mass index (BMI), systolic (SysBP) and diastolic (DiaBP) blood pressure. For each individual, 364,590 SNPs have been genotyped. We performed strict quality control on data using PLINK [@purcell2007plink]. We first excluded individuals having discrepancies between reported sex and sex determined from the X chromosome. We also excluded SNPs with a minor allele frequency less than 1$\%$, having missing values in more than 1$\%$ of the individuals or with a Hardy-Weinberg equilibrium $p$-value below 0.0001. In particular, we selected well-imputed variants from HapMap 3 reference panel [@international2010integrating]. After the strict quality control, 5,123 individuals with 310,975 SNPs in NFBC1966 were remaining for the further analysis. As we have the individual-level data, it is possible to run REMI-C, REMI-R and Lasso for all these traits. The solution paths using Lasso, REMI-R, and REMI-C for the ten metabolic traits in NFBC1966 data set are presented in Figures \[fig3\]. The dotted vertical bars in these two figures indicate the corresponding selected tuning parameters based on BIC. One can see that the differences among solution paths for the Lasso using individual-level data, REMI-C and REMI-R using summary statistics are very minor, which is consistent with results from our simulation studies in Section \[simulation\_study\].
In the released GWAS summary-level data sets, it is often the case that $\{\widehat{\bfbeta}^{(m)},\hat{\bfs}^2\}$ rather than the inner product $\bfX^T\bfy$ is made available. Therefore, we applied REMI-R to analyze summary statistics for ten GWASs of complex phenotypes. The source of the GWASs is given in Table \[Tab:01\]. Because the individuals of summary-level data sets were all from European ancestry, we used 379 European-ancestry samples in 1000 Genome Project [@10002012integrated] as a reference panel to estimate correlation matrix. Due to the quality of SNPs in the summary statistics, we restricted our analysis to a set of common and well-imputed variants from the HapMap 3 reference panel [@international2010integrating], which included 1,197,724 SNPs in total. Figure \[fig1\] shows the Manhattan plots of summary statistics for height (Ht) including $-\log_{10}$($p$-value), $|\widehat\bfbeta^{\textrm{m}}|$ and $|\widehat\bfbeta^{\textrm{r}}|$. The Manhattan plots of the absolute effect sizes from REMI-R for all other nine traits are shown in Figure \[fig2\].
Besides the effect size estimation, we evaluated prediction performance using 5,123 samples from the NFBC1966 [@sabatti2009genome]. To make a fair comparison with Lasso, we first split all 5,123 samples into ten folds. On the one hand, we applied REMI-R on the summary statistics for these lipid traits listed in Table \[Tab:01\]. Again, we used 379 European-ancestry samples from the 1000 Genome Project as a reference panel. For each of 10 folds in NFBC1966 data set, we calculated the predicted phenotypic values and evaluated the Pearson’s correlation coefficients between the predicted phenotypic values and the observed ones. On the other hand, we fitted the Lasso on the individual-level NFBC1966 data using the same ten-fold data for cross-validation. Specifically, we randomly selected nine folds of individual-level data as the training set to fit the Lasso, and evaluated prediction accuracy of the fitted model using the remaining one fold. Note that we used the same remaining fold to evaluate the prediction accuracy of the fitted REMI-R model. The prediction performance for REMI-R and Lasso is shown in Figure \[fig5\]. Clearly, the prediction performance of REMI-R outperforms the standard Lasso as the sample size in the summary statistics for these lipid traits are around 100,000 but the individual-level data contains only 5,123$\times 9/10$ samples. These real data results indicate the great advantage of REMI over the Lasso for risk prediction.
![Solution paths of Lasso, REMI-R, and REMI-C for HDL, LDL, TG, CRP TC, BMI, SysBP, DiaBP, Insulin and Glucose using the NFBC1966 data sets.[]{data-label="fig3"}](HDL_path.png "fig:"){width="48.00000%"} ![Solution paths of Lasso, REMI-R, and REMI-C for HDL, LDL, TG, CRP TC, BMI, SysBP, DiaBP, Insulin and Glucose using the NFBC1966 data sets.[]{data-label="fig3"}](LDL_path.png "fig:"){width="48.00000%"} ![Solution paths of Lasso, REMI-R, and REMI-C for HDL, LDL, TG, CRP TC, BMI, SysBP, DiaBP, Insulin and Glucose using the NFBC1966 data sets.[]{data-label="fig3"}](TG_path.png "fig:"){width="48.00000%"} ![Solution paths of Lasso, REMI-R, and REMI-C for HDL, LDL, TG, CRP TC, BMI, SysBP, DiaBP, Insulin and Glucose using the NFBC1966 data sets.[]{data-label="fig3"}](CRP_path.png "fig:"){width="48.00000%"} ![Solution paths of Lasso, REMI-R, and REMI-C for HDL, LDL, TG, CRP TC, BMI, SysBP, DiaBP, Insulin and Glucose using the NFBC1966 data sets.[]{data-label="fig3"}](TC_path.png "fig:"){width="48.00000%"} ![Solution paths of Lasso, REMI-R, and REMI-C for HDL, LDL, TG, CRP TC, BMI, SysBP, DiaBP, Insulin and Glucose using the NFBC1966 data sets.[]{data-label="fig3"}](BMI_path.png "fig:"){width="48.00000%"} ![Solution paths of Lasso, REMI-R, and REMI-C for HDL, LDL, TG, CRP TC, BMI, SysBP, DiaBP, Insulin and Glucose using the NFBC1966 data sets.[]{data-label="fig3"}](SysBP_path.png "fig:"){width="48.00000%"} ![Solution paths of Lasso, REMI-R, and REMI-C for HDL, LDL, TG, CRP TC, BMI, SysBP, DiaBP, Insulin and Glucose using the NFBC1966 data sets.[]{data-label="fig3"}](DiaBP_path.png "fig:"){width="48.00000%"} ![Solution paths of Lasso, REMI-R, and REMI-C for HDL, LDL, TG, CRP TC, BMI, SysBP, DiaBP, Insulin and Glucose using the NFBC1966 data sets.[]{data-label="fig3"}](Insulin_path.png "fig:"){width="48.00000%"} ![Solution paths of Lasso, REMI-R, and REMI-C for HDL, LDL, TG, CRP TC, BMI, SysBP, DiaBP, Insulin and Glucose using the NFBC1966 data sets.[]{data-label="fig3"}](Glucose_path.png "fig:"){width="48.00000%"}
![Manhattan plots of analysis result of human height for: -$\mathrm{log}_{10} p$-value, $|\widehat\bfbeta^{\textrm{m}}|$ from marginal analysis and $|\widehat\bfbeta^{\textrm{r}}|$ from REMI-R.[]{data-label="fig1"}](ssLasso_mht_height.png){width="100.00000%"}
![Manhattan plots of $|\widehat\bfbeta^{\textrm{r}}|$ from REMI-R for HDL, LDL, TC, TG, CKD-overall, T2D, CARDI, AD and RA.[]{data-label="fig2"}](ssLasso_mht.png){width="100.00000%"}
![Prediction accuracy (measured by the Pearson’s correlation coefficients) of REMI-R and the Lasso for HDL, LDL, TC and TG in the NFBC1966 data sets, where REMI-R was fitted using independent summary-level data and the Lasso was fitted using the individual-level data from NFBC1966. Their prediction accuracies were evaluated on 1/10 of the NFBC1966 data set holded out for testing.[]{data-label="fig5"}](boxplot_prediction_NFBC.png){width="100.00000%"}
ID YEAR Traits Sample Size SNPs Link
------------- ------ -------------------------------- ------------- --------- -----------------------------------------------------------------------------------------------
AD 2013 Alzheimer Disorder 54162 1149751 <http://www.pasteur-lille.fr/en/recherche/u744/igap/igap_download.php>
CARDI 2015 Coronary Artery Disease 817857 1197724 <http://www.cardiogramplusc4d.org/data-downloads/>
CKD-overall 2015 eGFRcrea in overall population 133715 984086 <https://www.nhlbi.nih.gov/research/intramural/researchers/ckdgen>
HDL 2013 High-Density-Lipid cholesterol 94272 992986 <http://csg.sph.umich.edu//abecasis/public/lipids2013/>
Ht 2014 Height 252778 827344 <http://portals.broadinstitute.org/collaboration/giant/index.php/GIANT_consortium_data_files>
LDL 2013 Low-Density-Lipid cholesterol 89851 990583 <http://csg.sph.umich.edu//abecasis/public/lipids2013/>
TC 2013 Total Cholesterol 94556 992889 <http://csg.sph.umich.edu//abecasis/public/lipids2013/>
TG 2013 Triglycerides 90974 990915 <http://csg.sph.umich.edu//abecasis/public/lipids2013/>
RA 2010 Rheumatoid Arthritis 25708 989551 <http://www.broadinstitute.org/ftp/pub/rheumatoid_arthritis/Stahl_etal_2010NG/>
T2D 2008 Type 2 Diabetes 63390 1061515 <http://diagram-consortium.org/downloads.html>
: GWAS data sets in our experiment[]{data-label="Tab:01"}
Discussion
==========
In this study, we proposed a novel approach for high-dimensional regression analysis when only marginal regression information and an external reference panel data set are available. Our work is motivated from combining information from multiple GWAS. To date, a large number of GWAS have been conducted to find genetic factors associated with complex traits. Due to the need for privacy protection and issues in data-sharing of individual-level data, it is important to be able to effectively make full use of the summary statistics from separate studies. In contrast to the limited sample size in individual-level data based GWAS analysis, a prominent feature of summary-level data analysis is that it can effectively make use of multiple data sets, which leads to a much larger combined sample size. Under mild conditions, we prove that the REMI estimator based on the marginal information and the reference penal achieves the minimax optimal rate estimation error under reasonable conditions. In particular, the requirement on the size of the reference panel data is quite mild, it is only in the order of the logarithm of the model dimension. Our theoretical result successfully explains why a relatively small reference sample can be good enough for accurate estimation and prediction in real applications. We have conducted comprehensive simulations and real data analysis to demonstrate the utility of REMI. The experimental results show that the performance of REMI can be very similar to the Lasso when the sample sizes of summary-level data and individual-level data are the same. In genetic analysis, summary-level data sets are much easier to access and their sample sizes are often orders of magnitude larger than that of individual-level data sets. Consequently, REMI can be superior to the existing methods requiring complete data by taking advantages of the larger sample sizes, as demonstrated in our real data example.
To date, a large number of GWAS have been conducted to study the genetic basis of complex traits. Due to privacy protection and issues in reaching agreement on data-sharing and in data transportation and storage, a variety of methods have been proposed to extract useful information using summary statistics and a relevant reference panel data. In contrast to the limited sample size in individual-level data based analysis, a prominent feature of summary-level data analysis is that it can effectively make use of the much larger sample size. However, the sample size of the reference panel data is usually very small, e.g. 379 European-ancestry samples in the 1000 Genome Project. These methods usually use 1000 Genome Project samples as a reference panel data. Despite the small size of a reference panel data (379 European-ancestry samples in 1000 Genome Project), The success of using summary statistics depends on incorporating a reference panel data, as which 1000 Genome Project samples are usually served.
Acknowledgment {#acknowledgment .unnumbered}
==============
This work was supported in part by grants No. 11501579 and NO. 61501389 from National Science Funding of China, grants NO. 22302815, NO. 12316116 and NO. 12301417 from the Hong Kong Research Grant Council, and Initiation Grant NO. IGN17SC02 from University Grants Committee, startup grant R9405 from The Hong Kong University of Science and Technology, Shenzhen Fundamental Research Fund under Grant No. QTD2015033114415450 and grant R-913-200-098-263 from Duke-NUS Graduate Medical School, and AcRF Tier 2 (MOE2016-T2-2-029) from Ministry of Education, Singapore.
Appendix {#appendix .unnumbered}
========
We recall some simple properties of subgaussian and subexponential random variables.
\[basic1\]( Lemma 2.7.7 of [@Vershynin:2016] and Remark 5.18 of [@Vershynin:2010].) Let $\xi_1,\xi_2$ be sub-Gaussian random variables with noise level $\|\xi_1\|_{\psi_2} \leq \sigma_{{_{\xi_1}}}$ and $\|\xi_2\|_{\psi_2}\leq \sigma_{{_{\xi_2}}}$, respectively. Then both $\xi_1\xi_2$ and $ \xi_1\xi_2- \mathbb{E}[\xi_1\xi_2]$ are sub-exponential random variables, and there exist an absolute constant $C > 0$ such that $ \|\xi_1\xi_2- \mathbb{E}[\xi_1\xi_2]\|_{\psi_1} \leq C \sigma_{{_{\xi_1}}}\sigma_{{_{\xi_2}}}.$
We state the Bernstein-type inequality for the sum of independent and mean 0 sub-exponential random random variables
\[vershynin5.17\](Corollary 5.17 of [@Vershynin:2010]) Let $\xi_1,...,\xi_m$ be independent centered sub-exponential random variables. Then for every $t>0$ one has $$\mathbb{P}[|\sum_{i = 1}^m \xi_i|/m \geq t] \leq 2 \exp (- C\min\{\frac{t^2}{K^2},\frac{t}{K}\}m),$$ where, $C$ is a absolute constant and $K = \max_{i = 1,..m}\{\|\xi_i\|_{\psi_1}\}$.
\[linf\]
Suppose the rows of $\bfX$ and $\bfX_{\mathrm{r}}$ are i.i.d sub-Gaussian samples drawn from population with mean $\textbf{0}$ and covariance matrix $\bfSigma$. Then, with probability at least $1-1/p^2$, we have $$\|\widehat{\bfSigma} - \bfSigma \|_{\infty} \leq \frac{2C_1}{\sqrt{C}}\sqrt{\frac{\log p}{n}},$$ and $$\|\widehat{\bfSigma}_{\mathrm{r}}- \bfSigma\|_{\infty} \leq \frac{2C_1}{\sqrt{C}}\sqrt{\frac{\log p}{n_{\mathrm{r}}}},$$ as long as $n> \frac{4}{C}\log p$ and $n_{{\mathrm{r}}}> \frac{4}{C}\log p$.
**Proof of Lemma \[linf\].** Since the proof of these two results are similar, we give one of them. Let $\bfx_{i}$ be the $i-$th row of $\bfX$, $i = 1,...n$, and $(\bfx_{i})_{j}$ denote the $j-$th entry of $\bfx_{i}$. Define $ G_{j,k}^{i} := (\bfx_{i})_{j}(\bfx_{i})_{k} - \mathbb{E}[(\bfx_{i})_{j}(\bfx_{i})_{k}]\in \mathcal{R}^{1}, i = 1,...,n, j= 1,...,p, k=1,..,p,$ which is sub-exponential with $\|G_{j,k}^{i}\|_{\psi_1} \leq C_1$ by Lemma \[basic1\]. Therefore, $$\begin{aligned}
\mathbb{P}[\|\widehat{\bfSigma} - \bfSigma \|_{\infty}\geq t] &= \mathbb{P}[\|\sum_{i=1}^{n} (\bfx_i^{T}\bfx_i - \mathbb{E}[\bfx_i^T\bfx_i])/n \|_{\infty}\geq t]\nonumber\\
&= \mathbb{P}[\bigcup_{j= 1,k=1}^{p,p}|\sum_{i = 1}^{n} G_{j,k}^{i}/n| \geq t ]\nonumber\\
&\leq \sum_{j= 1, k=1}^{p,p} \mathbb{P}[|\sum_{i = 1}^{n} G_{j,k}^{i}|/n \geq t] \nonumber\\
&\leq p^2 \exp (- C\min\{\frac{t^2}{C_{1}^2},\frac{t}{C_1}\}n) \label{eq1}\\
& \leq p^2 \exp (- C \frac{t^2}{C_{1}^2}n)\nonumber\end{aligned}$$ where the first inequality is due to union bound, and the second one follows from Lemma \[vershynin5.17\] and the last inequality is because of restricting $t\leq {C_1}$. Then Lemma \[linf\] follows from setting $t = \frac{2C_1}{\sqrt{C}}\sqrt{\frac{\log p}{ n}}$ and the assumption that $n> \frac{4}{C}\log p$. $\hfill\Box$
The following Lemma \[linfdif\] - Lemma \[wnoise\] are building blocks for proving Theorem \[esterr\].
\[linfdif\] Under the same assumption as Lemma \[linf\], we have $$\|(\widehat{\bfSigma} - \widehat{\bfSigma}_{\mathrm{r}})\bfbeta^*_{\mathcal{A}} \|_{\infty} \leq \frac{2C_1C_3}{\sqrt{C}}(\sqrt{\frac{\log p}{n}} +\sqrt{\frac{\log p}{n_{\mathrm{r}}}}),$$ holds with probability at least $1-2/p^2$.
**Proof of Lemma \[linfdif\].** $$\begin{aligned}
\|(\widehat{\bfSigma} - \widehat{\bfSigma}_{\textrm{r}})\bfbeta^*_{\mathcal{A}} \|_{\infty} &\leq \|(\widehat{\bfSigma} - \bfSigma)\bfbeta^*_{\mathcal{A}}\|_{\infty} + \|(\bfSigma - \widehat{\bfSigma}_{\textrm{r}})\bfbeta^*_{\mathcal{A}}\|_{\infty}\\
&\leq \|\widehat{\bfSigma} - \bfSigma\|_{\infty}\|\bfbeta^*_{\mathcal{A}}\|_{1} + \|\bfSigma - \widehat{\bfSigma}_{\textrm{r}}\|_{\infty}\|\bfbeta^*_{\mathcal{A}}\|_{1}\\
&\leq \frac{2C_1C_3}{\sqrt{C}}\sqrt{\frac{\log p}{n}} + \frac{2C_1C_3}{\sqrt{C}} \sqrt{\frac{\log p}{{n_{\textrm{r}}}}},\end{aligned}$$ where the first inequality is due to triangle inequality, and the second inequality follows from Cauchy-Schwartz inequality, and the last one holds with probability larger than $1-2/p^2$ due to Lemma \[linf\]. This finishes the proof of Lemma \[linfdif\]. $\hfill\Box$
\[noise\] Suppose the rows of $\bfX$ are i.i.d sub-Gaussian samples drawn from population with mean $\textbf{0}$ and covariance matrix $\bfSigma$, and the entries of noise $\bfepsilon$ are i.i.d centered sub-Gaussian with noise level $\sigma_{\epsilon}$. With probability at least $1-1/p^3$, we have $$\| \widetilde{\bfepsilon}\|_{\infty} < 2\sigma_{\epsilon} \frac{C_1}{\sqrt{C}}\sqrt{\frac{\log p}{ n}},$$ provided that $n\geq \frac{4\log p}{C}.$
**Proof of Lemma \[noise\].** We have, $$\begin{aligned}
& \mathbb{P}[\|\widetilde{\bfepsilon}\|_{\infty}< t]=\mathbb{P}[ \|\bfX^{T}\bfepsilon/n\|_{\infty}< t]\nonumber\\
& = 1- \mathbb{P}[ \|\bfX^{T}\bfepsilon/n\|_{\infty}\geq t]\nonumber\\
&= 1- \mathbb{P}[\bigcup_{j= 1}^{p}|\bfX_j^T\bfepsilon/n | \geq t]\nonumber\\
&\geq 1- \sum_{ j=1}^{p} \mathbb{P}[|\sum_{i = 1}^{n} (\bfX_j)_{i} \epsilon_i|/n \geq t] \nonumber\\
&\geq 1- p \exp (- C\min\{\frac{t^2}{C_{1}^2\sigma^2_{\epsilon}},\frac{t}{C_1\sigma_{\epsilon}}\}n)\nonumber\\
& \geq1- p \exp (- C\frac{t^2}{C_{1}^2\sigma^2_{\epsilon}}n)\nonumber\\
& \geq 1-1/p^3,\label{eq2}\end{aligned}$$ the first inequality is due to union bound, and the second one follows from Lemma \[basic1\] and Lemma \[vershynin5.17\], where we use $\|(\bfX_j)_i \epsilon_i -\mathbb{E}[(\bfX_j)_i \epsilon_i]\|_{\psi_1} \leq C \sigma_{\epsilon} C_1$, and the last two inequality follows from by setting $t = 2\sigma_{\epsilon} C_1\sqrt{\frac{\log p}{C n}}$ and the assumption that $n> \frac{4}{C}\log p$, i.e., with probability at least $1-1/p^3,$ we have $$\|\widetilde{\bfepsilon}\|_{\infty}\leq 2\sigma_{\epsilon} C_1\sqrt{\frac{\log p}{C n}}.$$ $\hfill\Box$
\[wnoise\] Under the same assumption as Lemma \[noise\], we have, $$\|\widehat{\bfSigma} \bfbeta^*_{\mathcal{I}}\|_{\infty}\leq \frac{2C_1C_4}{\sqrt{C}}\sqrt{\frac{\log p}{n}} + 2C_2 \sigma_{\epsilon} \sqrt{\frac{\log p}{n}}.$$
with probability larger than $1-1/p^2$. **Proof of Lemma \[wnoise\].** $$\begin{aligned}
&\|\widehat{\bfSigma} \bfbeta^*_{\mathcal{I}}\|_{\infty} \leq \|\bfSigma \bfbeta^*_{\mathcal{I}}\|_{\infty} +\|(\widehat{\bfSigma} -\bfSigma) \bfbeta^*_{\mathcal{I}}\|_{\infty} \\
&\leq \max_{j=1,...p}\{\|\bfSigma_j\|_1\} \|\bfbeta^*_{\mathcal{I}} \|_{\infty} +\|\widehat{\bfSigma} -\bfSigma\|_{\infty}\|\bfbeta^*_{\mathcal{I}} \|_{1} \\
& \leq 2C_2 \sigma_{\epsilon} \sqrt{\frac{\log p}{n}} + \frac{2C_1C_4}{\sqrt{C}}\sqrt{\frac{\log p}{n}},\end{aligned}$$ where first inequality is due to triangle inequality, and the second one follows from Cauchy-Schwartz inequality, the third inequality holds with probability larger than $1-1/p^2$ by using and Lemma \[linf\]. This completes the proof of Lemma \[wnoise\]. $\hfill\Box$
Now we are ready to prove Theorem \[esterr\].
**Proof of Theorem \[esterr\].** (i). Let $\Delta = \widehat{\bfbeta}^{\textrm{c}} - \bfbeta^*_{\mathcal{A}}$. Define the event $$\mathcal{E} = \{ 2\|(\widehat{\bfSigma} - \widehat{\bfSigma_{r}}) \bfbeta^*_{\mathcal{A}}+ \widehat{\bfSigma} \bfbeta^*_{\mathcal{I}} + \widetilde{\bfepsilon}\|_{\infty}\leq \lambda_0 \}.$$ The optimality of $\widehat{\bfbeta}^{\textrm{c}}$ implies that $$\begin{aligned}
\langle \widehat{\bfbeta}^{\textrm{c}} , \widehat{\bfSigma}_{\textrm{r}} \widehat{\bfbeta}^{\textrm{c}} \rangle -{2}\langle\widetilde{\bfy},\widehat{\bfbeta}^{\textrm{c}} \rangle + \lambda \|\widehat{\bfbeta}^{\textrm{c}} \|_{1}&\leq \langle\bfbeta^*_{\mathcal{A}},\widehat{\bfbeta}^{\textrm{c}}\bfbeta^*_{\mathcal{A}} \rangle -{2}\langle\widetilde{\bfy},\bfbeta^*_{\mathcal{A}} \rangle + \lambda \|\bfbeta^*_{\mathcal{A}}\|_{1},\nonumber\\
&\Downarrow (\textrm{eq1})\nonumber\\ \langle\widehat{\bfbeta}^{\textrm{c}}-\bfbeta^*_{\mathcal{A}}, \widehat{\bfSigma}_{\textrm{r}}(\widehat{\bfbeta}^{\textrm{c}}-\bfbeta^*_{\mathcal{A}}) \rangle + 2\langle\bfbeta^*_{\mathcal{A}},\widehat{\bfSigma}_{\textrm{r}}(\widehat{\bfbeta}^{\textrm{c}}-\bfbeta^*_{\mathcal{A}}) \rangle&+\lambda(\|\widehat{\bfbeta}^{\textrm{c}}{_\mathcal{A}}\|_1 + \|\widehat{\bfbeta}^{\textrm{c}}{_\mathcal{I}}\|_1)\leq
{2}\langle\widetilde{\bfy},\widehat{\bfbeta}^{\textrm{c}} - \bfbeta^*_{\mathcal{A}} \rangle
+\lambda\|\bfbeta^*_{\mathcal{A}}\|,\nonumber\\
& \Downarrow (\textrm{eq2})\nonumber\\
\langle\Delta,\widehat{\bfSigma}_{\textrm{r}}\Delta \rangle +\lambda\|\Delta_{\mathcal{I}}\|_1\leq
{2}\langle\widetilde{\bfy},\Delta \rangle & - 2\langle\widehat{\bfSigma}_{\textrm{r}}\bfbeta^*_{\mathcal{A}}, \Delta \rangle
+\lambda\|\bfbeta^*_{\mathcal{A}}\| - \lambda\|\widehat{\bfbeta}^{\textrm{c}}{_\mathcal{A}}\|_1,\nonumber\\
& \Downarrow (\textrm{eq3})\nonumber\\
\langle\Delta,\widehat{\bfSigma}_{\textrm{r}}\Delta \rangle +\lambda\|\Delta_{\mathcal{I}}\|_1 \leq
{2}\langle (\widehat{\bfSigma} & - \widehat{\bfSigma}_{\textrm{r}}) \bfbeta^*_{\mathcal{A}}+ \widehat{\bfSigma} \bfbeta^*_{\mathcal{I}}+ \widetilde{\bfepsilon},\Delta \rangle
+\lambda\|\Delta_{\mathcal{A}}\|,\nonumber\\
& \Downarrow (\textrm{eq4})\nonumber\\
\langle\Delta,\widehat{\bfSigma}_{\textrm{r}}\Delta \rangle +\lambda\|\Delta_{\mathcal{I}}\|_1 \leq
{2}\|(\widehat{\bfSigma} &- \widehat{\bfSigma}_{\textrm{r}}) \bfbeta^*_{\mathcal{A}}+ \widehat{\bfSigma} \bfbeta^*_{\mathcal{I}}+ \widetilde{\bfepsilon}\|_{\infty}\|\Delta\|_1
+\lambda\|\Delta_{\mathcal{A}}\|,\nonumber\\
& \Downarrow (\textrm{eq5})\nonumber\\
\langle\Delta,\widehat{\bfSigma}_{\textrm{r}}\Delta \rangle +\lambda\|\Delta_{\mathcal{I}}\|_1 &\leq
\frac{\lambda}{2}(\|\Delta_{\mathcal{A}}\|_1 + \|\Delta_{\mathcal{I}}\|_1)
+\lambda\|\Delta_{\mathcal{A}}\|,\nonumber\\
& \Downarrow (\textrm{eq6})\nonumber\\
\langle\Delta,\widehat{\bfSigma}_{\textrm{r}}\Delta \rangle +\frac{\lambda}{2}\|\Delta_{\mathcal{I}}\|_1&\leq
\frac{3}{2} \lambda\|\Delta_{\mathcal{A}}\|_1,\label{basic2}\end{aligned}$$ where, (eq1) and (eq2) and (eq6) are due to some algebra, and (eq3) follows from , and (eq4) uses Cauchy-Schwartz inequality, and (eq5) holds by conditioning $\mathcal{E}$ and the assumption $\lambda_0\leq \lambda/2.$ It follow from that $$\label{concd}
\Delta \in \mathcal{C}_{\mathcal{A},3}.$$ Then, by the restricted eigenvalue condition on $\widehat{\bfSigma}_{\textrm{r}}$ and we deduce, $$\phi_0 \|\Delta\|_2^2 \leq \langle\Delta,\widehat{\bfSigma}_{\textrm{r}}\Delta\rangle\leq \frac{3}{2} \lambda\|\Delta_{\mathcal{A}}\|_1\leq \frac{3}{2}\sqrt{s}\lambda\|\Delta\|_2,$$ i.e., $$\|\Delta\|_ 2 \leq \frac{3}{2\phi_0} \sqrt{s}\lambda\leq \frac{6}{\phi_0}(\frac{C_1(C_3+C_4)}{\sqrt{C}}\sqrt{\frac{s\log p}{n}} + \frac{C_1+\sqrt{C}C_2}{\sqrt{C}}\sigma_{\epsilon}\sqrt{\frac{s\log p}{ n}} + \frac{C_1C_3}{\sqrt{C}} \sqrt{\frac{s\log p}{n_{\textrm{r}}}}).$$ The above induction is conditioning on $\mathcal{E}.$ We need give a lower bound on $\mathbb{P}[\mathcal{E}]$. Indeed, $$\begin{aligned}
{2}\|(\widehat{\bfSigma} - \widehat{\bfSigma}_{\textrm{r}}) \bfbeta^*_{\mathcal{A}}+ \widehat{\bfSigma} \bfbeta^*_{\mathcal{I}}+ \tilde{\bfepsilon}\|_{\infty} &\leq {2}\|(\widehat{\bfSigma} - \widehat{\bfSigma}_{\textrm{r}}) \bfbeta^*_{\mathcal{A}}\|_{\infty} + {2}\|\widehat{\bfSigma} \bfbeta^*_{\mathcal{I}}\|_{\infty} + {2}\| \tilde{\bfepsilon}\|_{\infty}\\\end{aligned}$$ Then, it follows Lemma \[linfdif\] and Lemma \[noise\] and Lemma \[wnoise\] that $\mathbb{P}[\mathcal{E}] \geq 1- 3/p^2 - 1/p^3.$ This completes the first part of proof of Theorem \[esterr\].
(ii). Let $\widehat{\bfSigma}_{\mathrm{new}} = \bfX_{\mathrm{new}}^{T} \bfX_{\mathrm{new}}/n_{\mathrm{new}}.$ Then, $$\begin{aligned}
&\|\bfX_{\mathrm{new}} (\widehat{\bfbeta}^{\textrm{c}} - \bfbeta^*_{\mathcal{A}})\|_2^2/n_{\mathrm{new}} = \langle\widehat{\bfSigma}_{\mathrm{new}}(\widehat{\bfbeta}^{\textrm{c}} - \bfbeta^*_{\mathcal{A}}), \widehat{\bfbeta}^{\textrm{c}} - \bfbeta^*_{\mathcal{A}}\rangle \\
&= \langle\Delta, \widehat{\bfSigma}_{\mathrm{r}} \Delta \rangle + \langle\Delta, (\widehat{\bfSigma}_{\mathrm{new}} - \widehat{\bfSigma}_{\mathrm{r}}) \Delta \rangle\\
& \leq \frac{3}{2}\lambda \|\Delta_{\mathcal{A}}\|_1 + \|\Delta\|_1^2 \|\widehat{\bfSigma}_{\mathrm{new}} - \widehat{\bfSigma}_{\mathrm{r}}\|_{\infty}\\
&\leq \frac{3}{2}\lambda \|\Delta_{\mathcal{A}}\|_1 + \|\Delta\|_1^2 \frac{2C_1}{\sqrt{C}}(\sqrt{\frac{\log p}{n_{\mathrm{r}}}}+ \sqrt{\frac{\log p}{n_{\mathrm{new}}}})\\
& \leq \mathcal{O}(\sigma_{\epsilon}\sqrt{\frac{s\log{p}}{n}} + \sqrt{\frac{s\log{p}}{n_{\mathrm{r}}}}) \|\Delta_{\mathcal{A}}\|_2 + \mathcal{O}(\sqrt{\frac{\log p}{n_{\mathrm{r}}}}+ \sqrt{\frac{\log p}{n_{\mathrm{new}}}}) s^2 \|\Delta_{\mathcal{A}}\|_2^2\\
&\leq \mathcal{O}((\sigma_{\epsilon}\sqrt{\frac{s\log{p}}{n}} + \sqrt{\frac{s\log{p}}{n_{\mathrm{r}}}})^2) (1+ s^2 (\sqrt{\frac{\log p}{n_{\mathrm{r}}}}+ \sqrt{\frac{\log p}{n_{\mathrm{new}}}}) )\\
$$ where, the first inequality uses and Cauchy-Schwartz inequality, and the second one is due to Lemma \[linf\], and the third inequality follow from and Cauchy-Schwartz inequality, the fourth inequality uses Theorem \[esterr\]. This completes the second part of proof of Theorem \[esterr\]. $\hfill\Box$
References {#references .unnumbered}
==========
[10]{}
T. Berisa and J. K. Pickrell. Approximately independent linkage disequilibrium blocks in human populations. , 32(2):283, 2016.
P. J. Bickel, Y. Ritov, and A. B. Tsybakov. . , 37(4):1705–1732, 2009.
P. J. Bickel, Y. Ritov, and A. B. Tsybakov. Simultaneous analysis of lasso and dantzig selector. , pages 1705–1732, 2009.
P. B[ü]{}hlmann and S. Van De Geer. . Springer, 2011.
B. K. Bulik-Sullivan, P.-R. Loh, H. K. Finucane, S. Ripke, J. Yang, N. Patterson, M. J. Daly, A. L. Price, B. M. Neale, S. W. G. of the Psychiatric Genomics Consortium, et al. Ld score regression distinguishes confounding from polygenicity in genome-wide association studies. , 47(3):291–295, 2015.
E. Candes, T. Tao, et al. The dantzig selector: Statistical estimation when p is much larger than n. , 35(6):2313–2351, 2007.
W. Chen, B. R. Larrabee, I. G. Ovsyannikova, R. B. Kennedy, I. H. Haralambieva, G. A. Poland, and D. J. Schaid. Fine mapping causal variants with an approximate bayesian method using marginal test statistics. , 200(3):719–736, 2015.
. G. P. Consortium et al. An integrated map of genetic variation from 1,092 human genomes. , 491(7422):56, 2012.
I. H. . Consortium et al. Integrating common and rare genetic variation in diverse human populations. , 467(7311):52, 2010.
J. Fan and R. Li. Variable selection via nonconcave penalized likelihood and its oracle properties. , 96(456):1348–1360, 2001.
J. Friedman, T. Hastie, and R. Tibshirani. Regularization paths for generalized linear models via coordinate descent. , 33(1):1, 2010.
C. Fuchsberger, J. Flannick, T. M. Teslovich, A. Mahajan, V. Agarwala, K. J. Gaulton, C. Ma, P. Fontanillas, L. Moutsianas, D. J. McCarthy, et al. The genetic architecture of type 2 diabetes. , 2016.
J. Gratten, N. R. Wray, M. C. Keller, and P. M. Visscher. Large-scale genomics unveils the genetic architecture of psychiatric disorders. , 17(6):782–790, 2014.
T. Hastie, R. Tibshirani, and J. Friedman. . Springer, 2009.
T. Hastie, R. Tibshirani, and M. Wainwright. . CRC press, 2015.
L. A. Hindorff, P. Sethupathy, H. A. Junkins, E. M. Ramos, J. P. Mehta, F. S. Collins, and T. A. Manolio. Potential etiologic and functional implications of genome-wide association loci for human diseases and traits. , 106(23):9362–9367, 2009.
T. J. Hoffmann, M. N. Kvale, S. E. Hesselson, Y. Zhan, C. Aquino, Y. Cao, S. Cawley, E. Chung, S. Connell, J. Eshragh, et al. Next generation genome-wide association tool: design and coverage of a high-throughput european-optimized snp array. , 98(2):79–89, 2011.
F. Hormozdiari, E. Kostem, E. Y. Kang, B. Pasaniuc, and E. Eskin. Identifying causal variants at loci with multiple signals of association. , 198(2):497–508, 2014.
Y. Hu, Q. Lu, R. Powles, X. Yao, C. Yang, F. Fang, X. Xu, and H. Zhao. Leveraging functional annotations in genetic risk prediction for human complex diseases. , 13(6):e1005589, 2017.
J. Huang, Y. Jiao, X. Lu, and L. Zhu. Robust decoding from 1-bit compressive sampling with least squares. , 2017.
G. Kichaev, W.-Y. Yang, S. Lindstrom, F. Hormozdiari, E. Eskin, A. L. Price, P. Kraft, and B. Pasaniuc. Integrating functional data to prioritize causal variants in statistical fine-mapping studies. , 10(10):e1004722, 2014.
S. H. Lee, N. R. Wray, M. E. Goddard, and P. M. Visscher. Estimating missing heritability for disease from genome-wide association studies. , 88(3):294–305, 2011.
N. Meinshausen and P. B[ü]{}hlmann. High-dimensional graphs and variable selection with the lasso. , 34(3):1436–1462, 2006.
E. of Nature Genetics. Asking for more. , 44:733, 2012.
B. Pasaniuc, N. Zaitlen, H. Shi, G. Bhatia, A. Gusev, J. Pickrell, J. Hirschhorn, D. P. Strachan, N. Patterson, and A. L. Price. Fast and accurate imputation of summary statistics enhances evidence of functional enrichment. , 30(20):2906–2914, 2014.
J. K. Pickrell. Joint analysis of functional genomic data and genome-wide association studies of 18 human traits. , 94(4):559–573, 2014.
S. Purcell, B. Neale, K. Todd-Brown, L. Thomas, M. A. Ferreira, D. Bender, J. Maller, P. Sklar, P. I. De Bakker, M. J. Daly, et al. : a tool set for whole-genome association and population-based linkage analyses. , 81(3):559–575, 2007.
G. Raskutti, M. J. Wainwright, and B. Yu. Minimax rates of estimation for high-dimensional linear regression over $\ell\_q $-balls. , 57(10):6976–6994, 2011.
C. Sabatti, A.-L. Hartikainen, A. Pouta, S. Ripatti, J. Brodsky, C. G. Jones, N. A. Zaitlen, T. Varilo, M. Kaakinen, U. Sovio, et al. Genome-wide association analysis of metabolic traits in a birth cohort from a founder population. , 41(1):35, 2009.
J. Sch[ä]{}fer, K. Strimmer, et al. A shrinkage approach to large-scale covariance matrix estimation and implications for functional genomics. , 4(1):32, 2005.
R. Tibshirani. Regression shrinkage and selection via the lasso. , 58(1):267–288, 1996.
S. A. Van De Geer, P. B[ü]{}hlmann, et al. On the conditions used to prove oracle results for the lasso. , 3:1360–1392, 2009.
R. Vershynin. High dimensional probability.
R. Vershynin. Introduction to the non-asymptotic analysis of random matrices. , 2010.
B. J. Vilhj[á]{}lmsson, J. Yang, H. K. Finucane, A. Gusev, S. Lindstr[ö]{}m, S. Ripke, G. Genovese, P.-R. Loh, G. Bhatia, R. Do, et al. Modeling linkage disequilibrium increases accuracy of polygenic risk scores. , 97(4):576–592, 2015.
P. M. Visscher, M. A. Brown, M. I. McCarthy, and J. Yang. Five years of gwas discovery. , 90(1):7–24, 2012.
P. M. Visscher, W. G. Hill, and N. R. Wray. Heritability in the genomics era–concepts and misconceptions. , 9(4):255–266, 2008.
P. M. Visscher, N. R. Wray, Q. Zhang, P. Sklar, M. I. McCarthy, M. A. Brown, and J. Yang. . , 101(1):5–22, 2017.
D. Welter, J. MacArthur, J. Morales, T. Burdett, P. Hall, H. Junkins, A. Klemm, P. Flicek, T. Manolio, L. Hindorff, et al. . , 42(D1):D1001–D1006, 2014.
A. R. Wood, T. Esko, J. Yang, S. Vedantam, T. H. Pers, S. Gustafsson, A. Y. Chu, K. Estrada, J. Luan, Z. Kutalik, et al. Defining the role of common variation in the genomic and biological architecture of adult human height. , 46(11):1173–1186, 2014.
J. Yang, A. Bakshi, Z. Zhu, G. Hemani, A. A. Vinkhuyzen, I. M. Nolte, J. V. van Vliet-Ostaptchouk, H. Snieder, T. Esko, L. Milani, et al. Genome-wide genetic homogeneity between sexes and populations for human height and body mass index. , page ddv443, 2015.
J. Yang, B. Benyamin, B. P. McEvoy, S. Gordon, A. K. Henders, D. R. Nyholt, P. A. Madden, A. C. Heath, N. G. Martin, G. W. Montgomery, et al. Common snps explain a large proportion of the heritability for human height. , 42(7):565–569, 2010.
J. Yang, S. H. Lee, M. E. Goddard, and P. M. Visscher. . , 88(1):76–82, 2011.
C.-H. Zhang. Nearly unbiased variable selection under minimax concave penalty. , 38(2):894–942, 2010.
C.-H. Zhang and J. Huang. The sparsity and bias of the [LASSO]{} selection in high-dimensional linear regression. , 36(4):1567–1594, 2008.
C.-H. Zhang and T. Zhang. A general theory of concave regularization for high-dimensional sparse estimation problems. , 27(4):576–593, 2012.
P. Zhao and B. Yu. On model selection consistency of [L]{}asso. , 7:2541–2563, 2006.
X. Zhou. A unified framework for variance component estimation with summary statistics in genome-wide association studies. , page 042846, 2016.
X. Zhu and M. Stephens. Bayesian large-scale multiple regression with summary statistics from genome-wide association studies. , page 042457, 2016.
H. Zou, T. Hastie, R. Tibshirani, et al. On the degrees of freedom of the lasso. , 35(5):2173–2192, 2007.
|
---
abstract: 'We introduce and develop a new semi-algebraic proof system, called Stabbing Planes that is in the style of DPLL-based modern SAT solvers. As with DPLL, there is only one rule: the current polytope can be subdivided by branching on an inequality and its “integer negation.” That is, we can (nondeterministically choose) a hyperplane $a x \geq b$ with integer coefficients, which partitions the polytope into three pieces: the points in the polytope satisfying $a x \geq b$, the points satisfying $a x \leq b-1$, and the middle slab $b-1 < a x < b$. Since the middle slab contains no integer points it can be safely discarded, and the algorithm proceeds recursively on the other two branches. Each path terminates when the current polytope is empty, which is polynomial-time checkable. Among our results, we show somewhat surprisingly that Stabbing Planes can efficiently simulate Cutting Planes, and moreover, is strictly stronger than Cutting Planes under a reasonable conjecture. We prove linear lower bounds on the [*rank*]{} of Stabbing Planes refutations, by adapting a lifting argument in communication complexity.'
author:
- |
Paul Beame\
University of Washington\
- |
Noah Fleming[^1]\
University of Toronto\
- |
Russell Impagliazzo\
University of California, San Diego\
- |
Antonina Kolokolova[^2]\
Memorial University of Newfoundland\
- |
Denis Pankratov\
University of Toronto\
- |
\
University of Toronto\
- |
Robert Robere\
University of Toronto\
bibliography:
- 'paper.bib'
title: Stabbing Planes
---
Introduction {#sec:introduction}
============
Preliminaries {#sec:defsp}
=============
Motivating Example: ${\ensuremath{\mathsf{SP}}}$ Refutations of Tseitin Formulas {#sec:tseitin}
================================================================================
Simulation Theorems {#sec:simulations}
===================
Impossibility Results {#sec:impossibility}
=====================
Conclusions {#sec:conclusions}
===========
[^1]: Research supported by NSERC.
[^2]: Research supported by NSERC.
|
---
abstract: 'In this paper, we investigate effects of the minimal length on quantum tunnelling from spherically symmetric black holes using the Hamilton-Jacobi method incorporating the minimal length. We first derive the deformed Hamilton-Jacobi equations for scalars and fermions, both of which have the same expressions. The minimal length correction to the Hawking temperature is found to depend on the black hole’s mass and the mass and angular momentum of emitted particles. Finally, we calculate a Schwarzschild black hole’s luminosity and find the black hole evaporates to zero mass in infinite time.'
author:
- 'Benrong Mu$^{a,b}$'
- 'Peng Wang$^{b}$'
- 'Haitang Yang$^{b,c}$'
title: Minimal Length Effects on Tunnelling from Spherically Symmetric Black Holes
---
Introduction
============
The classical theory of black holes predicts that anything, including light, couldn’t escape from the black holes. However, Stephen Hawking first showed that quantum effects could allow black holes to emit particles. The formula of Hawking temperature was first given in the frame of quantum field theory[@Hawking:1974sw]. After that, various methods for deriving Hawking radiation have been proposed. Among them is a semiclassical method of modeling Hawking radiation as a tunneling effect proposed by Kraus and Wilczek[@Kraus:1994by; @Kraus:1994fj], which is known as the null geodesic method. Later, the tunneling behaviors of particles were investigated using the Hamilton-Jacobi method[@Srinivasan:1998ty; @Angheben:2005rm; @Kerner:2006vu]. Using the null geodesic method and the Hamilton-Jacobi method, much fruit has been achieved[@Hemming:2001we; @Medved:2002zj; @Vagenas:2001rm; @Arzano:2005rs; @Wu:2006pz; @Nadalini:2005xp; @Chatterjee:2007hc; @Akhmedova:2008dz; @Akhmedov:2008ru; @Akhmedova:2008au; @Banerjee:2008ry; @Singleton:2010gz]. The key point of the Hamilton-Jacobi method is using WKB approximation to calculate the imaginary part of the action for the tunneling process.
On the other hand, various theories of quantum gravity, such as string theory, loop quantum gravity and quantum geometry, predict the existence of a minimal length[@Townsend:1977xw; @Amati:1988tn; @Konishi:1989wk]. The generalized uncertainty principle(GUP)[@Kempf:1994su] is a simply way to realize this minimal length.
An effective model of the GUP in one dimensional quantum mechanics is given by[@Hossenfelder:2003jz; @Hassan:2002qk] $$L_{f}k(p)=\tanh\left( \frac{p}{M_{f}}\right) , \label{tanh1}$$$$L_{f}\omega(E)=\tanh\left( \frac{E}{M_{f}}\right) , \label{tanh2}$$ where the generators of the translations in space and time are the wave vector $k$ and the frequency $\omega$, $L_{f}$ is the minimal length, and $L_{f}M_{f}=\hbar$. The quantization in position representation $\hat{x}=x$ leads to $$k=-i\partial_{x},\text{ }\omega=i\partial_{t}.$$ Therefore, the low energy limit $p\ll M_{f}$ including order of $\frac{p^{3}}{M_{f}^{3}}$ gives $$\begin{aligned}
p & =-i\hbar\partial_{x}\left( 1-\frac{\hbar^{2}}{M_{f}^{2}}\partial
_{x}^{2}\right) ,\label{eq:momentum}\\
E & =i\hbar\partial_{t}\left( 1-\frac{\hbar^{2}}{M_{f}^{2}}\partial_{t}^{2}\right) , \label{eq:energy}$$ where we neglect the factor $\frac{1}{3}$. From eqn. $\left( \ref{tanh1}\right) $, it is noted that although one can increase $p$ arbitrarily, $k$ has an upper bound which is $\frac{1}{L_{f}}$. The upper bound on $k$ implies that that particles could not possess arbitrarily small Compton wavelengths $\lambda=2\pi/k$ and that there exists a minimal length $\sim L_{f}$. Furthermore, the deformed Klein-Gordon/Dirac equations incorporating eqn. $\left( \ref{eq:momentum}\right) $ and eqn. $\left( \ref{eq:energy}\right)
$ have already be obtained in [@Hossenfelder:2003jz], which will be briefly reviewed in section \[Sec:DHJ\]. So [@Hossenfelder:2003jz] provides a way to incorporate the minimal length with special relativity, a good starting point for studying Hawking radiation as tunnelling effect.
The black hole is a suitable venue to discuss the effects of quantum gravity. Incorporating GUP into black holes has been discussed in a lot of papers[@Ali:2012mt; @Majumder:2011bv; @Bina:2010ir; @Chen:2002tu; @Xiang:2009yq; @Kim:2007hf]. The thermodynamics of black holes has also been investigated in the framework of GUP[@Xiang:2009yq; @Kim:2007hf]. In [@Majhi:2013koa], a new form of GUP is introduced$$\begin{aligned}
p^{0} & =k^{0},\\
p^{i} & =k^{i}\left( 1-\alpha\mathbf{k}+2\alpha^{2}\mathbf{k}^{2}\right) ,\end{aligned}$$ where $p^{a}$ is the modified four momentum, $k^{a}$ is the usual four momentum and $\alpha$ is a small parameter. The modified velocity of photons, two-dimensional Klein-Gordon equation the emission spectrum due to the Unruh effect is obtained there. Recently, the GUP deformed Hamilton-Jacobi equation for fermions in curved spacetime have been introduced and the corrected Hawking temperatures have been derived[@Chen:2013pra; @Chen:2013tha; @Chen:2013ssa; @Chen:2014xsa; @liu:2014zxa]. The authors consider the GUP of form$$\begin{aligned}
x_{i} & =x_{0i,}\\
p_{i} & =p_{0i}\left( 1+\beta p^{2}\right) ,\end{aligned}$$ where $x_{0i,}$ and $p_{0i}$ satisfy the canonical commutation relations. Fermions’ tunnelling, black hole thermodynamics and the remnants are discussed there.
In this paper, we investigate scalars and fermions tunneling across the horizons of black holes using the deformed Hamilton-Jacobi method which incorporates the minimal length via eqn. $\left( \ref{eq:momentum}\right) $ and eqn. $\left( \ref{eq:energy}\right) $. Our calculation shows that the quantum gravity correction is related not only to the black hole’s mass but also to the mass and angular momentum of emitted particles.
The organization of this paper is as follows. In section \[Sec:DHJ\], from the modified fundamental commutation relation, we generalize the Hamilton-Jacobi in curved spacetime. In section \[Sec:PT\], incorporating GUP, we investigate the tunnelling of particles in the black holes. In section \[Sec:TBH\], we investigate how a Schwarzschild black hole evaporates in our model. Section \[Sec:Con\] is devoted to our conclusion. In this paper, we take Geometrized units $c=G=1$, where the Planck constant $\hbar$ is square of the Planck Mass $m_{p}$. We also assume that the emitted particles are neutral.
Deformed Hamilton-Jacobi Equations {#Sec:DHJ}
==================================
To be generic, we will consider a spherically symmetric background metric of the form $$ds^{2}=f\left( r\right) dt^{2}-\frac{dr^{2}}{f\left( r\right) }-r^{2}\left( d\theta^{2}+\sin^{2}\theta d\phi^{2}\right) ,
\label{Schwarzschild metric}$$ where where $f\left( r\right) $ has a simple zero at $r=r_{h}$ with $f^{\prime}\left( r_{h}\right) $ being finite and nonzero. The vanishing of $f\left( r\right) $ at point $r=r_{h}$ indicates the presence of an event horizon. In this section, we will first derive the deformed Klein-Gordon/Dirac equations in flat spacetime and then generalize them to the curved spacetime with the metric $\left( \ref{Schwarzschild metric}\right) $.
In the $\left( 3+1\right) $ dimensional flat spacetime, the relations between $\left( p_{i},E\right) $ and $\left( k_{i},\omega\right) $ can simply be generalized to$$\begin{aligned}
L_{f}k_{i}(p) & =\tanh\left( \frac{p_{i}}{M_{f}}\right)
,\label{eq:wavevector}\\
L_{f}\omega(E) & =\tanh\left( \frac{E}{M_{f}}\right) ,
\label{eq:frequency}$$ where, in the spherical coordinates, one has for $\vec{k}$$$\vec{k}=-i\left( \hat{r}\frac{\partial}{\partial r}+\frac{\hat{\theta}}{r}\frac{\partial}{\partial\theta}+\frac{\hat{\phi}}{r\sin\theta}\frac{\partial}{\partial\phi}\right) .$$ Expanding eqn. $\left( \ref{eq:wavevector}\right) $ and eqn. $\left(
\ref{eq:frequency}\right) $ for small arguments to the third order gives the energy and momentum operator in position representation$$\begin{gathered}
E=i\hbar\partial_{t}\left( 1-\frac{\hbar^{2}}{M_{f}^{2}}\partial_{t}^{2}\right) ,\\
\vec{p}=\frac{\hbar}{i}\left[ \hat{r}\left( \partial_{r}-\frac{\hbar
^{2}\partial_{r}^{3}}{M_{f}^{2}}\right) +\hat{\theta}\left( \frac
{\partial_{\theta}}{r}-\frac{\hbar^{2}\partial_{\theta}^{3}}{r^{3}M_{f}^{2}}\right) +\hat{\phi}\left( \frac{\partial_{\phi}}{r\sin\theta}-\frac
{\hbar^{2}\partial_{\phi}^{3}}{r^{3}\sin^{3}\theta M_{f}^{2}}\right) \right]
,\end{gathered}$$ where we also omit the factor $\frac{1}{3}$. Substituting the above energy and momentum operators into the energy-momentum relation, the deformed Klein-Gordon equation satisfied by the scalar field with the mass $m$ is $$E^{2}\phi=p^{2}\phi+m^{2}\phi, \label{eq:KGFD}$$ where $p^{2}=\vec{p}\cdot\vec{p}.$ Making the ansatz for $\phi$ $$\phi=\exp\left( \frac{iI}{\hbar}\right) ,$$ and substituting it into eqn. $\left( \ref{eq:KGFD}\right) $, one expands eqn. $\left( \ref{eq:KGFD}\right) $ in powers of $\hbar$ and then finds that the lowest order gives the deformed scalar Hamilton-Jacobi equation in the flat spacetime $$\begin{gathered}
\left( \partial_{t}I\right) ^{2}\left( 1+\frac{2\left( \partial
_{t}I\right) ^{2}}{M_{f}^{2}}\right) -\left( \partial_{r}I\right)
^{2}\left( 1+\frac{2\left( \partial_{r}I\right) ^{2}}{M_{f}^{2}}\right)
-\frac{\left( \partial_{\theta}I\right) ^{2}}{r^{2}}\left( 1+\frac{2\left(
\partial_{\theta}I\right) ^{2}}{r^{2}M_{f}^{2}}\right) \nonumber\\
-\frac{\left( \partial_{\phi}I\right) ^{2}}{r^{2}\sin^{2}\theta}\left(
1+\frac{2\left( \partial_{\phi}I\right) ^{2}}{r^{2}\sin^{2}\theta M_{f}^{2}}\right) =m^{2}, \label{eq:HJFD}$$ which is truncated at $\mathcal{O}\left( \frac{1}{M_{f}^{2}}\right) $.
Similarly, the deformed Dirac equation for a spin-$1/2$ fermion with the mass $m$ takes the form as$$\left( \gamma_{0}E+\vec{\gamma}\cdot\vec{p}-m\right) \psi=0,
\label{eq:DiracFD}$$ where $\left\{ \gamma_{0},\gamma_{0}\right\} =2$, $\left\{ \gamma
_{a},\gamma_{b}\right\} =-2\delta_{ab}$, and $\left\{ \gamma_{0},\gamma
_{a}\right\} =0$ with the Latin index $a$ running over $r,\theta,$and $\phi$. Multiplying $\left( \gamma_{0}E+\vec{\gamma}\cdot\vec{p}+m\right) $ by eqn. $\left( \ref{eq:DiracFD}\right) $ and using the gamma matrices anticommutation relations, the deformed Dirac equation can be written as$$E^{2}\psi=\left( p^{2}+m^{2}\right) \psi-\frac{\left[ \gamma_{a},\gamma
_{b}\right] }{2}p_{a}p_{b}\psi. \label{eq:DiracFDM}$$ To obtain the Hamilton-Jacobi equation for the fermion, the ansatz for $\psi$ takes the form of $$\psi=\exp\left( \frac{iI}{\hbar}\right) v, \label{eq:fermionansatzD}$$ where $v$ is a vector function of the spacetime. Substituting eqn. $\left(
\ref{eq:fermionansatzD}\right) $ into eqn. $\left( \ref{eq:DiracFDM}\right)
$ and noting that the second term on RHS of eqn. $\left( \ref{eq:DiracFDM}\right) $ does not contribute to the lowest order of $\hbar$, we find the deformed Hamilton-Jacobi equation for a fermion is the same as the deformed one for a scalar with the same mass, namely eqn. $\left( \ref{eq:HJFD}\right) $. Note that one can use the deformed Maxwell’s equations obtained in [@Hossenfelder:2003jz] to get the deformed Hamilton-Jacobi equation for a vector boson. However, for simplicity we just stop here.
In order to generalize the deformed Hamilton-Jacobi equation, eqn. $\left(
\ref{eq:HJFD}\right) $, to the curved spacetime with the metric $\left(
\ref{Schwarzschild metric}\right) $, we first consider the Hamilton-Jacobi equation without GUP modifications. In ref. [@Benrong:2014woa], we show that the unmodified Hamilton-Jacobi equation in curved spacetime with $ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}$ is$$g^{\mu\nu}\partial_{\mu}I\partial_{\nu}I-m^{2}=0.$$ Therefore, the unmodified Hamilton-Jacobi equation in the metric $\left(
\ref{Schwarzschild metric}\right) $ becomes $$\frac{\left( \partial_{t}I\right) ^{2}}{f\left( r\right) }-f\left(
r\right) \left( \partial_{r}I\right) ^{2}-\frac{\left( \partial_{\theta
}I\right) ^{2}}{r^{2}}-\frac{\left( \partial_{\phi}I\right) ^{2}}{r^{2}\sin^{2}\theta}=m^{2}. \label{eq:HJC}$$ On the other hand, the unmodified Hamilton-Jacobi equation in flat spacetime can be obtained from eqn. $\left( \ref{eq:HJFD}\right) $ by taking $M_{f}\rightarrow\infty$,$$\left( \partial_{t}I\right) ^{2}-\left( \partial_{r}I\right) ^{2}-\frac{\left( \partial_{\theta}I\right) ^{2}}{r^{2}}-\frac{\left(
\partial_{\phi}I\right) ^{2}}{r^{2}\sin^{2}\theta}=m^{2}. \label{eq:HJFF}$$ Comparing eqn. $\left( \ref{eq:HJC}\right) $ with eqn. $\left(
\ref{eq:HJFF}\right) $, one finds that the Hamilton-Jacobi equation in the metric $\left( \ref{Schwarzschild metric}\right) $ can be obtained from the one in flat spacetime by making replacements $\partial_{r}I\rightarrow
\sqrt{f\left( r\right) }\partial_{r}I$ and $\partial_{t}I\rightarrow
\frac{\partial_{t}I}{\sqrt{f\left( r\right) }}$ in the no GUP modifications scenario. Therefore, by making replacements $\partial_{r}I\rightarrow
\sqrt{f\left( r\right) }\partial_{r}I$ and $\partial_{t}I\rightarrow
\frac{\partial_{t}I}{\sqrt{f\left( r\right) }}$, the deformed Hamilton-Jacobi equation in flat spacetime, eqn. $\left( \ref{eq:HJFD}\right) $, leads to the deformed Hamilton-Jacobi equation in the metric $\left( \ref{Schwarzschild metric}\right) $, which is to $\mathcal{O}\left(
\frac{1}{M_{f}^{2}}\right) ,$ $$\begin{gathered}
\frac{\left( \partial_{t}I\right) ^{2}}{f\left( r\right) }\left(
1+\frac{2\left( \partial_{t}I\right) ^{2}}{f\left( r\right) M_{f}^{2}}\right) -f\left( r\right) \left( \partial_{r}I\right) ^{2}\left(
1+\frac{2f\left( r\right) \left( \partial_{r}I\right) ^{2}}{M_{f}^{2}}\right) -\frac{\left( \partial_{\theta}I\right) ^{2}}{r^{2}}\left(
1+\frac{2\left( \partial_{\theta}I\right) ^{2}}{r^{2}M_{f}^{2}}\right)
\nonumber\\
-\frac{\left( \partial_{\phi}I\right) ^{2}}{r^{2}\sin^{2}\theta}\left(
1+\frac{2\left( \partial_{\phi}I\right) ^{2}}{r^{2}\sin^{2}\theta M_{f}^{2}}\right) =m^{2}. \label{eq:HJCG}$$
Quantum Tunnelling {#Sec:PT}
==================
In this section, we investigate the particles’ tunneling at the event horizon $r=r_{h}$ of the metric $\left( \ref{Schwarzschild metric}\right) $ where GUP is taken into account. Since the metric $\left(
\ref{Schwarzschild metric}\right) $ does not depend on $t$ and $\phi$, $\partial_{t}$ and $\partial_{\phi}$ are killing vectors. Taking into account the Killing vectors of the background spacetime, we can employ the following ansatz for the action $$I=-\omega t+W\left( r,\theta\right) +p_{\phi}\phi, \label{eq:Ianzatz}$$ where $\omega$ and $p_{\phi}$ are constants and they are the energy and the $z$-component of angular momentum of emitted particles, respectively. Inserting eqn. $(\ref{eq:Ianzatz})$ into eqn. $(\ref{eq:HJCG}),$ we find that the deformed Hamilton-Jacobi equation becomes $$p_{r}^{2}\left( 1+\frac{2f\left( r\right) p_{r}^{2}}{M_{f}^{2}}\right)
=\frac{1}{f^{2}\left( r\right) }\left[ \omega^{2}\left( 1+\frac
{2\omega^{2}}{f\left( r\right) M_{f}^{2}}\right) -f\left( r\right)
\left( m^{2}+\lambda\right) \right] , \label{eq:pr}$$ where $p_{r}=\partial_{r}W$, $p_{\theta}=\partial_{\theta}W$ and $$\lambda=\frac{p_{\theta}^{2}}{r^{2}}\left( 1+\frac{2p_{\theta}^{2}}{r^{2}M_{f}^{2}}\right) +\frac{p_{\phi}^{2}}{r^{2}\sin^{2}\theta}\left(
1+\frac{2p_{\phi}^{2}}{r^{2}\sin^{2}\theta M_{f}^{2}}\right) .$$ Since the magnitude of the angular momentum of the particle $L$ can be expressed in terms of $p_{\theta}$ and $p_{\phi},$ $$L^{2}=p_{\theta}^{2}+\frac{p_{\phi}^{2}}{\sin^{2}\theta},$$ one can rewrite $\lambda$ as$$\lambda=\frac{L^{2}}{r^{2}}+\mathcal{O}\left( \frac{1}{M_{f}^{2}}\right) .
\label{eq:lamda}$$ Solving eqn. $(\ref{eq:pr})$ for $p_{r}$ to $\mathcal{O}\left( \frac{1}{M_{f}^{2}}\right) $gives $$\begin{aligned}
\partial_{r}W_{\mp} & =\pm\frac{1}{f\left( r\right) }\sqrt{\omega
^{2}\left( 1+\frac{2\omega^{2}}{f\left( r\right) M_{f}^{2}}\right)
-f\left( r\right) \left( m^{2}+\lambda\right) }\times\nonumber\\
& \sqrt{1-\frac{2}{M_{f}^{2}f\left( r\right) }\left[ \omega^{2}\left(
1+\frac{2\omega^{2}}{f\left( r\right) M_{f}^{2}}\right) -f\left( r\right)
\left( m^{2}+\lambda\right) \right] }, \label{eq:dW}$$ where +/$-$ represent the outgoing/ingoing solutions. In order to get the imaginary part of $W_{\pm}$, we need to find residue of the RHS of eqn. $(\ref{eq:dW})$ at $r=r_{h}$ by expanding the RHS in a Laurent series with respect to $r$ at $r=r_{h}$. We then rewrite eqn. $(\ref{eq:dW})$ as $$\begin{aligned}
\partial_{r}W_{\mp} & =\pm\frac{1}{f\left( r\right) ^{\frac{5}{2}}}\sqrt{\omega^{2}\left( f\left( r\right) +\frac{2\omega^{2}}{M_{f}^{2}}\right) -f^{2}\left( r\right) \left( m^{2}+\lambda\right) }\times\nonumber\\
& \sqrt{f^{2}\left( r\right) -\frac{2}{M_{f}^{2}}\left[ \omega^{2}\left(
f\left( r\right) +\frac{2\omega^{2}}{M_{f}^{2}}\right) -f^{2}\left(
r\right) \left( m^{2}+\lambda\right) \right] }.\end{aligned}$$ Using $f\left( r\right) =f^{\prime}\left( r_{h}\right) \left(
r-r_{h}\right) +\frac{f^{\prime\prime}\left( r_{h}\right) }{2}\left(
r-r_{h}\right) ^{2}+\mathcal{O}\left( \left( r-r_{h}\right) ^{3}\right)
$, one can single out the $\frac{1}{r-r_{h}}$ term of the Laurent series $$\partial_{r}W_{\mp}\sim\frac{\pm a_{-1}}{r-r_{h}},$$ where we have $$a_{-1}=\frac{\omega}{f^{\prime}\left( r_{h}\right) }\left[ 1+\frac{2}{M_{f}^{2}}\left( m^{2}+\frac{L^{2}}{r_{h}^{2}}\right) \right]
+\mathcal{O}\left( \frac{1}{M_{f}^{4}}\right) .$$ Using the residue theory for semi circles, we obtain for the imaginary part of $W_{\pm}$ to $\mathcal{O}\left( \frac{1}{M_{f}^{2}}\right) $ $$\operatorname{Im}W_{\pm}=\pm\frac{\pi\omega}{f^{\prime}\left( r_{h}\right)
}\left[ 1+\frac{2}{M_{f}^{2}}\left( m^{2}+\frac{L^{2}}{r_{h}^{2}}\right)
\right] .$$
However, when one tries to calculate the tunneling rate $\Gamma$ from $\operatorname{Im}W_{\pm}$, there is so called factor-two problem[@Akhmedov:2006pg]. One way to solve the “factor two problem” is introducing a temporal contribution[@Akhmedova:2008au; @Akhmedova:2008dz; @Akhmedov:2008ru; @Chowdhury:2006sk; @Akhmedov:2006un]. To consider an invariance under canonical transformations, we also follow the recent work[@Akhmedova:2008au; @Akhmedova:2008dz; @Akhmedov:2008ru; @Chowdhury:2006sk; @Akhmedov:2006un] and adopt the expression ${\displaystyle\oint}
p_{r}dr=\int p_{r}^{+}dr-\int p_{r}^{-}dr$ for the spatial contribution to $\Gamma$. The spatial and temporal contributions to $\Gamma$ are given as follows.
**Spatial Contribution:** The spatial part contribution comes from the imaginary part of $W\left( r\right) $. Thus, the spatial part contribution is proportional to $$\begin{aligned}
& exp\left[ -\frac{1}{\hbar}\operatorname{Im}\oint p_{r}dr\right]
\nonumber\\
& =exp\left[ -\frac{1}{\hbar}\operatorname{Im}\left( \int p_{r}^{+}dr-\int
p_{r}^{-}dr\right) \right] \nonumber\\
& =exp\left\{ -\frac{2\pi\omega}{f^{\prime}\left( r_{h}\right) }\left[
1+\frac{2}{M_{f}^{2}}\left( m^{2}+\frac{L^{2}}{r_{h}^{2}}\right) \right]
\right\}\end{aligned}$$
**Temporal Contribution:** As pointed in Ref. [@Akhmedov:2008ru; @Akhmedov:2006pg; @Chowdhury:2006sk; @Akhmedov:2006un], the temporal part contribution comes from the “rotation” which connects the interior region and the exterior region of the black hole. Thus, the imaginary contribution from the temporal part when crossing the horizon is $\operatorname{Im}\left( \omega\Delta t^{out,in}\right) =\omega\frac{\pi
}{2\kappa}$, where $\kappa=\frac{f^{\prime}\left( r_{h}\right) }{2}$ is the surface gravity at the event horizon. Then the total temporal contribution for a round trip is $$\operatorname{Im}\left( \omega\Delta t\right) =\frac{2\pi\omega}{f^{\prime
}\left( r_{h}\right) }.$$ Therefore, the tunnelling rate of the particle crossing the horizon is $$\begin{aligned}
\Gamma & \propto exp\left[ -\frac{1}{\hbar}\left( \operatorname{Im}\left(
\omega\Delta t\right) +\operatorname{Im}\oint p_{r}dr\right) \right]
\nonumber\\
& =exp\left\{ -\frac{4\pi\omega}{f^{\prime}\left( r_{h}\right) }\left[
1+\frac{1}{M_{f}^{2}}\left( m^{2}+\frac{L^{2}}{r_{h}^{2}}\right) \right]
\right\} . \label{eq:Gamma}$$ This is the expression of Boltzmann factor with an effective temperature $$T=\frac{f^{\prime}\left( r_{h}\right) }{4\pi}\frac{\hbar}{1+\frac{1}{M_{f}^{2}}\left( m^{2}+\frac{L^{2}}{r_{h}^{2}}\right) }, \label{HKT1}$$ where $T_{0}=\frac{\hbar f^{\prime}\left( r_{h}\right) }{4\pi}$ is the original Hawking temperature.
For the standard Hawing radiation, all particles very close to the horizon are effectively massless on account of infinite blueshift. Thus, the conformal invariance of the horizon make Hawing temperatures of all particles the same. The mass, angular momentum and identity of the particles are only relevant when they escape the potential barrier. However, if quantum gravity effects are considered, behaviors of particles near the horizon could be different. For example, if we send a wave packet which is governed by a subluminal dispersion relation, backwards in time toward the horizon, it reaches a minimum distance of approach, then reverse direction and propagate back away from the horizon, instead of getting unlimited blueshift toward the horizon [@Unruh:1994je; @Corley:1996ar]. Thus, quantum gravity effects might make fermions and scalars experience different (effective) Hawking temperatures. However, our result shows that in our model, the tunnelling rates of fermions and scalars depend on their masses and angular momentums, but independent of the identities of the particles, to $\mathcal{O}\left( \frac{1}{M_{f}^{2}}\right) $. In other words, effective Hawking temperatures of fermions and scalars are the same to $\mathcal{O}\left( \frac{1}{M_{f}^{2}}\right) $ in our model as long as their masses, energies and angular momentums are the same.
Thermodynamics of Black Holes {#Sec:TBH}
=============================
For simplicity, we consider the Schwarzschild metric with $f\left( r\right)
=1-\frac{2M}{r}$ with the black hole’s mass, $M$. The event horizon of the Schwarzschild black hole is $r_{h}=2M$. In this section, we work with massless particles. Near the horizon of the the black hole, angular momentum of the particle $L\sim pr_{h}\sim\omega r_{h}$. Thus, one can rewrite $T$$$T\sim\frac{T_{0}}{1+\frac{2\omega^{2}}{M_{f}^{2}}}, \label{eq:Temp}$$ where $T_{0}=\frac{\hbar}{8\pi M}$ for the Schwarzschild black hole. As reported in [@AmelinoCamelia:2005ik], the authors obtained the relation $\omega\gtrsim\frac{\hbar}{\delta x}$ between the energy of a particle and its position uncertainty in the framework of GUP. Near the horizon of the the Schwarzschild black hole, the position uncertainty of a particle will be of the order of the Schwarzschild radius of the black hole [@Bekenstein:1973ur] $\delta x\sim r_{h}$. Thus, one finds for $T$$$T\sim\frac{T_{0}}{1+\frac{m_{p}^{4}}{2M^{2}M_{f}^{2}}}, \label{eq: Temp}$$ where we use $\hbar=m_{p}^{2}$. Using the first law of the black hole thermodynamics, we find the corrected black hole entropy is$$\begin{aligned}
S & =\int\frac{dM}{T}\nonumber\\
& \sim\frac{A}{4m_{p}^{2}}+\frac{4\pi m_{p}^{2}}{M_{f}^{2}}\ln\left(
\frac{A}{16\pi}\right) , \label{eq:entropy}$$ where $A=4\pi r_{h}^{2}=16\pi M^{2}$ is the area of the horizon. The logarithmic term in eqn. $\left( \ref{eq:entropy}\right) $ is the well known correction from quantum gravity to the classical Bekenstein-Hawking entropy, which have appeared in different studies of GUP modified thermodynamics of black holes[@Bina:2010ir; @Chen:2002tu; @Xiang:2009yq; @Majumder:2011xg; @Nozari:2012nf; @Banerjee:2010sd; @Cavaglia:2004jw; @Myung:2006qr; @Nouicer:2007jg; @Majumder:afa]. In general, the entropy for the Schwarzschild black hole of mass $M$ in four spacetime dimensions can be written in form of$$S=\frac{A}{4}+\sigma\ln\left( \frac{A}{16\pi}\right) +\mathcal{O}\left(
\frac{M_{f}^{2}}{A}\right) , \label{eq:EntropyExpansion}$$ where $\sigma=\frac{2M_{f}^{2}}{M^{2}}$ in our paper. Neglecting the terms $\mathcal{O}\left( \frac{M_{f}^{2}}{A}\right) $ in eqn. $\left(
\ref{eq:EntropyExpansion}\right) \,$, there could be three scenarios depending on the sign of $\sigma$.
1. $\sigma=0:$ This case is just the standard Hawking radiation. The black holes evaporate completely in finite time.
2. $\sigma<0:$ The entropy $S$ as function of mass develops a minimum at some value of $M_{\min}$. This predicts the existence of black hole remnants. Furthermore, the black holes stop evaporating in finite time. This is what happened in [@Bina:2010ir; @Chen:2002tu; @Xiang:2009yq; @Majumder:2011xg; @Nozari:2012nf; @Banerjee:2010sd; @Cavaglia:2004jw; @Myung:2006qr; @Nouicer:2007jg; @Majumder:afa], which is consistent with the existence of a minimal length.
3. $\sigma>0:$ This is a subtle case. In the remaining of the section, we will investigate how the black holes evaporates in our model.
For particles emitted in a wave mode labelled by energy $\omega$ and $L,$ we find from eqn. $\left( \ref{eq:Gamma}\right) $ that [@Hartle:1976tp]$$\begin{aligned}
& \left( \text{Probability for a black hole to emit a particle in this
mode}\right) \\
& =\exp\left( -\frac{\omega}{T}\right) \times(\text{Probability for a black
hole to absorb a particle in the same mode}),\end{aligned}$$ where $T$ is given by eqn. $\left( \ref{HKT1}\right) $. Neglecting back-reaction, detailed balance condition requires that the ratio of the probability of having $N$ particles in a particular mode with $\omega$ and $L$ to the probability of having $N-1$ particles in the same mode is $\exp\left(
-\frac{\omega}{T}\right) .$ One then follows the standard textbook procedure to get the average number $n_{\omega,L}$ in the mode$$n_{\omega,L}=n\left( \frac{\omega}{T}\right) ,$$ where we define$$n\left( x\right) =\frac{1}{\exp x-\left( -1\right) ^{\epsilon}},$$ and $\epsilon=0$ for bosons and $\epsilon=1$ for fermions. In [@Page:1976df], counting the number of modes per frequency interval with periodic boundary conditions in a large container around the black hole, Page related the expected number emitted per mode $n_{\omega,L}$ to the average emission rate per frequency interval $\frac{dn_{\omega,L}}{dt}$ by$$\frac{dn_{\omega,L}}{dt}=n_{\omega,L}\frac{d\omega}{2\pi\hbar}.
\label{eq:DnDt}$$ Following the Page’s argument, we find that in our model $$\frac{dn_{\omega,L}}{dt}=n_{\omega,L}\frac{\partial\omega}{\partial p_{r}}\frac{dp_{r}}{2\pi\hbar}=n_{\omega,L}\frac{d\omega}{2\pi\hbar},
\label{eq:DnDt-MDR}$$ where $\frac{\partial\omega}{\partial p_{r}}$ is the radial velocity of the particle and the number of modes between the wavevector interval $\left(
p_{r},p_{r}+dp_{r}\right) $ is $\frac{dp_{r}}{2\pi\hbar}$ where $p_{r}=\partial_{r}I$ is the radial wavevector. Since each particle carries off the energy $\omega$, the total luminosity is obtained from $\frac
{dn_{\omega,L}}{dt}$ by multiplying by the energy $\omega$ and summing up over all energy $\omega$ and $L$, $$L={\displaystyle\sum\limits_{l=0}}
\left( 2l+1\right) \int\omega n_{\omega,l}\frac{d\omega}{2\pi\hbar},$$ where $L^{2}=\left( l+1\right) l\hbar^{2}$ and the degeneracy for $l$ is $\left( 2l+1\right) $. However, some of the radiation emitted by the horizon might not be able to reach the asymptotic region. We need to consider the greybody factor $\left\vert T_{i}\left( \omega\right) \right\vert ^{2}$, where $T_{i}\left( \omega\right) $ represents the transmission coefficient of the black hole barrier which in general can depend on the energy $\omega$ and angular momentum $l$ of the particle. Taking the greybody factor into account, we find for the total luminosity$$L={\displaystyle\sum\limits_{l=0}}
\left( 2l+1\right) \int\left\vert T_{i}\left( \omega\right) \right\vert
^{2}\omega n_{\omega,l}\frac{d\omega}{2\pi\hbar}.$$ Usually, one needs to solve the exact wave equations for $\left\vert
T_{i}\left( \omega\right) \right\vert ^{2},$ which is very complicated. On the other hand, one can use the geometric optics approximation to estimate $\left\vert T_{i}\left( \omega\right) \right\vert ^{2}$. In the geometric optics approximation, we assume $\omega\gg M$ and high energy waves will be absorbed unless they are aimed away from the black hole. Hence we have $\left\vert T_{i}\left( \omega\right) \right\vert ^{2}=1$ for all the classically allowed energy $\omega$ and angular momentum $l$ of the particle. In this approximation, the Schwarzschild black hole is just like a black sphere of radius $R=3^{3/2}M$ [@Wald:1984rg], which puts an upper bound on $l\left( l+1\right) \hbar^{2},$$$l\left( l+1\right) \hbar^{2}\leqslant27M^{2}\omega^{2}.$$ Note that we neglect possible modifications from GUP to the bound since we are interested in the GUP effects near the horizon. Thus, the luminosity is$$\begin{aligned}
L & =\int_{0}^{\infty}\frac{\omega d\omega}{2\pi\hbar^{3}}\int_{0}^{27M^{2}\omega^{2}}n\left[ \frac{\omega}{T_{0}}\left( 1+\frac{1}{M_{f}^{2}}\frac{l\left( l+1\right) \hbar^{2}}{2M^{2}}\right) \right] d\left[
l\left( l+1\right) \hbar^{2}\right] \nonumber\\
& =\frac{T_{0}^{4}M^{2}}{2\pi\hbar^{3}}\int_{0}^{\infty}u^{3}du\int_{0}^{27}n\left[ u\left( 1+a^{2}ux\right) \right] dx, \label{eq:integral}$$ where we define $u=\frac{\omega}{T_{0}}$, $y=\frac{l\left( l+1\right)
\hbar^{2}}{M^{2}\omega^{2}}$ and $a=\frac{T_{0}}{\sqrt{2}M_{f}}=\frac
{m_{p}^{2}}{8\sqrt{2}\pi M_{f}M}$. For $M\gg\frac{m_{p}^{2}}{8\sqrt{2}\pi
M_{f}}$, we have $a\ll1$ and hence the luminosity is$$L\approx\frac{27}{32\pi^{2}\hbar^{3}}T_{0}^{4}A\int_{0}^{\infty}u^{3}n\left(
u\right) du,$$ which is just the Stefan’s law for black holes. Therefore for large black holes, they evaporate in almost the same way as in Case 1 until $M\sim
\frac{m_{p}^{2}}{8\sqrt{2}\pi M_{f}}$, when the term $a^{2}ux$ starts to dominate in eqn. $\left( \ref{eq:integral}\right) $. Then the luminosity is approximated by $$\begin{aligned}
L & \sim\frac{T_{0}^{4}M^{2}}{2\pi\hbar^{3}}\int_{0}^{\infty}u^{3}du\int
_{0}^{27}n\left( a^{2}u^{2}x\right) dx\nonumber\\
& =\frac{2M^{2}M_{f}^{4}}{\pi m_{p}^{6}}\int_{0}^{\infty}v^{3}dv\int_{0}^{27}n\left( v^{2}x\right) dx,\end{aligned}$$ where $v=au$. Not worrying about exact numerical factors, one has for the evaporation rate $$\frac{dM}{dt}=-L\sim-A\frac{M^{2}}{M_{f}^{2}}, \label{eq:EvaporationRate}$$ where $A>0$ is a constant. Solving eqn. $\left( \ref{eq:EvaporationRate}\right) $ for $M$ gives $M\sim\frac{M_{f}^{2}}{At}$. The evaporation rate considerably slows down when black holes’ mass $M\sim\frac{m_{p}^{2}}{8\sqrt{2}\pi M_{f}}$. The black hole then evaporates to zero mass in infinite time. However, the GUP predicts the existence of a minimal length. It would make much more sense if there are black hole remnants in the GUP models. How can we reconcile the contradiction? When we write down the deformed Hamilton-Jacobi equation, eqn. $\left( \ref{eq:HJCG}\right) $, we neglect terms higher than $\mathcal{O}\left( \frac{1}{M_{f}^{2}}\right) $. However, when $M\sim\frac{m_{p}^{2}}{8\sqrt{2}\pi M_{f}}$, our effective approach starts breaking down since contributions from higher order terms become the same important as these from terms $\mathcal{O}\left( \frac{1}{M_{f}^{2}}\right) $. Thus, one then has to include these higher order contributions. For example, if there are higher order corrections to eqn. $\left( \ref{eq:EvaporationRate}\right) $ as in $$\frac{dM}{dt}\sim-A\frac{M^{2}}{M_{f}^{2}}+B\frac{M^{3}}{M_{f}^{3}},$$ where $B>0$, one can easily see that there exists a minimum mass $M_{\min}\sim
M_{f}\sqrt{\frac{A}{B}}$ for black holes. In another word, the $\mathcal{O}\left( \frac{1}{M_{f}^{2}}\right) $ terms used in our paper are not sufficient enough to produce black hole remnants predicted by the GUP. To do so, one needs to resort to higher order terms if the full theory is not available. It is also noted that when at late stage of a black hole with $a\gtrsim1\,$, eqn. $\left( \ref{eq:Temp}\right) $ becomes $$T\sim\frac{T_{0}}{\frac{m_{p}^{4}}{2M^{2}M_{f}^{2}}}\sim M,$$ which means the tempreture of the black hole goes to zero as the mass goes to zero.
The GUP is closely related to noncommutative geometry. In fact, when the GUP is investigated in more than one dimension, a noncommutative geometric generalization of position space always appears naturally[@Kempf:1994su]. On the other hand, quantum black hole physics has been studied in the noncommutative geometry[@Nicolini:2008aj]. In [@Banerjee:2008gc], a noncommutative black hole’s entropy received a logarithmic correction with $\sigma<0$. However, [@Banerjee:2008du; @Spallucci:2009zz] showed that the corrections to a noncommutative schwarzschild black hole’s entropy might not involve any logarithmic terms. In either case, the tempreture of the noncommutative schwarzschild black hole reaches zero in finite time with remnants left.
Conclusion {#Sec:Con}
===========
In this paper, incorporating effects of the minimal length, we derived the deformed Hamilton-Jacobi Equations for both scalars and fermions in curved spacetime based on the modified fundamental commutation relations. We investigated the particles’ tunneling in the background of a spherically symmetric black holes. In this spacetime configurations, we showed that the corrected Hawking temperature is not only determined by the properties of the black holes, but also dependent on the angular momentum and mass of the emitted particles. Finally, we studied how a Schwarzschild black hole evaporates in our model. We found the black hole evaporates to zero mass in infinite time.
**Acknowledgements.**
We would like to acknowledge useful discussions with Y. He, Z. Sun and H. W. Wu. This work is supported in part by NSFC (Grant No. 11005016, 11175039 and 11375121) and SYSTF (Grant No. 2012JQ0039).
[99]{} S. W. Hawking, “Particle Creation by Black Holes,” Commun. Math. Phys. [**43**]{}, 199 (1975) \[Erratum-ibid. [**46**]{}, 206 (1976)\].
P. Kraus and F. Wilczek, “Selfinteraction correction to black hole radiance,” Nucl. Phys. B [**433**]{}, 403 (1995) \[gr-qc/9408003\].
P. Kraus and F. Wilczek, “Effect of selfinteraction on charged black hole radiance,” Nucl. Phys. B [**437**]{}, 231 (1995) \[hep-th/9411219\].
K. Srinivasan and T. Padmanabhan, “Particle production and complex path analysis,” Phys. Rev. D [**60**]{}, 024007 (1999) \[gr-qc/9812028\].
M. Angheben, M. Nadalini, L. Vanzo and S. Zerbini, “Hawking radiation as tunneling for extremal and rotating black holes,” JHEP [**0505**]{}, 014 (2005) \[hep-th/0503081\].
R. Kerner and R. B. Mann, “Tunnelling, temperature and Taub-NUT black holes,” Phys. Rev. D [**73**]{}, 104010 (2006) \[gr-qc/0603019\].
S. Hemming and E. Keski-Vakkuri, “The Spectrum of strings on BTZ black holes and spectral flow in the SL(2,R) WZW model,” Nucl. Phys. B [**626**]{}, 363 (2002) \[hep-th/0110252\]. A. J. M. Medved, “Radiation via tunneling from a de Sitter cosmological horizon,” Phys. Rev. D [**66**]{}, 124009 (2002) \[hep-th/0207247\].
E. C. Vagenas, “Semiclassical corrections to the Bekenstein-Hawking entropy of the BTZ black hole via selfgravitation,” Phys. Lett. B [**533**]{}, 302 (2002) \[hep-th/0109108\].
M. Arzano, A. J. M. Medved and E. C. Vagenas, “Hawking radiation as tunneling through the quantum horizon,” JHEP [**0509**]{}, 037 (2005) \[hep-th/0505266\]. S. Q. Wu and Q. Q. Jiang, “Remarks on Hawking radiation as tunneling from the BTZ black holes,” JHEP [**0603**]{}, 079 (2006) \[hep-th/0602033\].
M. Nadalini, L. Vanzo and S. Zerbini, “Hawking radiation as tunneling: The D dimensional rotating case,” J. Phys. A [**39**]{}, 6601 (2006) \[hep-th/0511250\].
B. Chatterjee, A. Ghosh and P. Mitra, “Tunnelling from black holes in the Hamilton Jacobi approach,” Phys. Lett. B [**661**]{}, 307 (2008) \[arXiv:0704.1746 \[hep-th\]\]. V. Akhmedova, T. Pilling, A. de Gill and D. Singleton, “Temporal contribution to gravitational WKB-like calculations,” Phys. Lett. B [**666**]{}, 269 (2008) \[arXiv:0804.2289 \[hep-th\]\].
E. T. Akhmedov, T. Pilling and D. Singleton, “Subtleties in the quasi-classical calculation of Hawking radiation,” Int. J. Mod. Phys. D [**17**]{}, 2453 (2008) \[arXiv:0805.2653 \[gr-qc\]\]. V. Akhmedova, T. Pilling, A. de Gill and D. Singleton, “Comments on anomaly versus WKB/tunneling methods for calculating Unruh radiation,” Phys. Lett. B [**673**]{}, 227 (2009) \[arXiv:0808.3413 \[hep-th\]\]. R. Banerjee and B. R. Majhi, “Quantum Tunneling and Back Reaction,” Phys. Lett. B [**662**]{}, 62 (2008) \[arXiv:0801.0200 \[hep-th\]\].
D. Singleton, E. C. Vagenas, T. Zhu and J. R. Ren, “Insights and possible resolution to the information loss paradox via the tunneling picture,” JHEP [**1008**]{}, 089 (2010) \[Erratum-ibid. [**1101**]{}, 021 (2011)\] \[arXiv:1005.3778 \[gr-qc\]\].
P. K. Townsend, “Small Scale Structure of Space-Time as the Origin of the Gravitational Constant,” Phys. Rev. D [**15**]{}, 2795 (1977). D. Amati, M. Ciafaloni and G. Veneziano, “Can Space-Time Be Probed Below the String Size?,” Phys. Lett. B [**216**]{}, 41 (1989). K. Konishi, G. Paffuti and P. Provero, “Minimum Physical Length and the Generalized Uncertainty Principle in String Theory,” Phys. Lett. B [**234**]{}, 276 (1990).
A. Kempf, G. Mangano and R. B. Mann, “Hilbert space representation of the minimal length uncertainty relation,” Phys. Rev. D [**52**]{}, 1108 (1995) \[hep-th/9412167\].
S. Hossenfelder, M. Bleicher, S. Hofmann, J. Ruppert, S. Scherer and H. Stoecker, “Collider signatures in the Planck regime,” Phys. Lett. B [**575**]{}, 85 (2003) \[hep-th/0305262\].
S. F. Hassan and M. S. Sloth, “TransPlanckian effects in inflationary cosmology and the modified uncertainty principle,” Nucl. Phys. B [**674**]{}, 434 (2003) \[hep-th/0204110\]. A. F. Ali, “No Existence of Black Holes at LHC Due to Minimal Length in Quantum Gravity,” JHEP [**1209**]{}, 067 (2012) \[arXiv:1208.6584 \[hep-th\]\]. B. Majumder, “Quantum Black Hole and the Modified Uncertainty Principle,” Phys. Lett. B [**701**]{}, 384 (2011) \[arXiv:1105.5314 \[gr-qc\]\]. A. Bina, S. Jalalzadeh and A. Moslehi, “Quantum Black Hole in the Generalized Uncertainty Principle Framework,” Phys. Rev. D [**81**]{}, 023528 (2010) \[arXiv:1001.0861 \[gr-qc\]\].
P. Chen and R. J. Adler,“Black hole remnants and dark matter,” Nucl. Phys. Proc. Suppl. [**124**]{}, 103 (2003) \[gr-qc/0205106\]. L. Xiang and X. Q. Wen, “Black hole thermodynamics with generalized uncertainty principle,” JHEP [**0910**]{}, 046 (2009) \[arXiv:0901.0603 \[gr-qc\]\]. W. Kim, E. J. Son and M. Yoon, “Thermodynamics of a black hole based on a generalized uncertainty principle,” JHEP [**0801**]{}, 035 (2008) \[arXiv:0711.0786 \[gr-qc\]\]. B. R. Majhi and E. C. Vagenas, “Modified Dispersion Relation, Photon’s Velocity, and Unruh Effect,” Phys. Lett. B [**725**]{}, 477 (2013) \[arXiv:1307.4195\].
D. Chen, H. Wu and H. Yang, “Fermion’s tunnelling with effects of quantum gravity,” Adv. High Energy Phys. [**2013**]{}, 432412 (2013) \[arXiv:1305.7104 \[gr-qc\]\]. D. Chen, H. Wu and H. Yang, “Observing remnants by fermions’ tunneling,” JCAP [**1403**]{}, 036 (2014) \[arXiv:1307.0172 \[gr-qc\]\]. D. Y. Chen, Q. Q. Jiang, P. Wang and H. Yang, “Remnants, fermions‘ tunnelling and effects of quantum gravity,’’ JHEP [**1311**]{}, 176 (2013) \[arXiv:1312.3781 \[hep-th\]\]. D. Chen and Z. Li, “Remarks on Remnants by Fermions’ Tunnelling from Black Strings,” Adv. High Energy Phys. [**2014**]{}, 620157 (2014) \[arXiv:1404.6375 \[hep-th\]\]. Z. Y. liu and J. R. Ren, “Fermions tunnelling with quantum gravity correction,” arXiv:1403.7307 \[gr-qc\].
M. Benrong, P. Wang and H. Yang, “Deformed Hamilton-Jacobi Method in Covariant Quantum Gravity Effective Models,” arXiv:1408.5055 \[gr-qc\]. E. T. Akhmedov, V. Akhmedova and D. Singleton, “Hawking temperature in the tunneling picture,” Phys. Lett. B [**642**]{}, 124 (2006) \[hep-th/0608098\].
B. D. Chowdhury, “Problems with Tunneling of Thin Shells from Black Holes,” Pramana [**70**]{}, 593 (2008) \[Pramana [**70**]{}, 3 (2008)\] \[hep-th/0605197\]. E. T. Akhmedov, V. Akhmedova, T. Pilling and D. Singleton, “Thermal radiation of various gravitational backgrounds,” Int. J. Mod. Phys. A [**22**]{}, 1705 (2007) \[hep-th/0605137\]. W. G. Unruh, “Sonic analog of black holes and the effects of high frequencies on black hole evaporation,” Phys. Rev. D [**51**]{}, 2827 (1995). S. Corley and T. Jacobson, “Hawking spectrum and high frequency dispersion,” Phys. Rev. D [**54**]{}, 1568 (1996) \[hep-th/9601073\]. G. Amelino-Camelia, M. Arzano, Y. Ling and G. Mandanici, “Black-hole thermodynamics with modified dispersion relations and generalized uncertainty principles,” Class. Quant. Grav. [**23**]{}, 2585 (2006) \[gr-qc/0506110\]. J. D. Bekenstein, “Black holes and entropy,” Phys. Rev. D [**7**]{}, 2333 (1973). B. Majumder, “Black Hole Entropy and the Modified Uncertainty Principle: A heuristic analysis,” Phys. Lett. B [**703**]{}, 402 (2011) \[arXiv:1106.0715 \[gr-qc\]\]. K. Nozari and S. Saghafi, “Natural Cutoffs and Quantum Tunneling from Black Hole Horizon,” JHEP [**1211**]{}, 005 (2012) \[arXiv:1206.5621 \[hep-th\]\].
R. Banerjee and S. Ghosh, “Generalised Uncertainty Principle, Remnant Mass and Singularity Problem in Black Hole Thermodynamics,” Phys. Lett. B [**688**]{}, 224 (2010) \[arXiv:1002.2302 \[gr-qc\]\]. M. Cavaglia and S. Das, “How classical are TeV scale black holes?,” Class. Quant. Grav. [**21**]{}, 4511 (2004) \[hep-th/0404050\].
Y. S. Myung, Y. W. Kim and Y. J. Park, “Black hole thermodynamics with generalized uncertainty principle,” Phys. Lett. B [**645**]{}, 393 (2007) \[gr-qc/0609031\]. K. Nouicer, “Quantum-corrected black hole thermodynamics to all orders in the Planck length,” Phys. Lett. B [**646**]{}, 63 (2007) \[arXiv:0704.1261 \[gr-qc\]\]. B. Majumder, “Black Hole Entropy with minimal length in Tunneling formalism,” Gen. Rel. Grav. [**11**]{}, 2403 (2013) \[arXiv:1212.6591 \[gr-qc\]\]. J. B. Hartle and S. W. Hawking, “Path Integral Derivation of Black Hole Radiance,” Phys. Rev. D [**13**]{}, 2188 (1976). D. N. Page, “Particle Emission Rates from a Black Hole: Massless Particles from an Uncharged, Nonrotating Hole,” Phys. Rev. D [**13**]{}, 198 (1976).
R. M. Wald, “General Relativity,” Chicago, Usa: Univ. Pr. ( 1984) 491p
P. Nicolini, “Noncommutative Black Holes, The Final Appeal To Quantum Gravity: A Review,” Int. J. Mod. Phys. A [**24**]{}, 1229 (2009) \[arXiv:0807.1939 \[hep-th\]\]. R. Banerjee, B. R. Majhi and S. Samanta, “Noncommutative Black Hole Thermodynamics,” Phys. Rev. D [**77**]{}, 124035 (2008) \[arXiv:0801.3583 \[hep-th\]\].
R. Banerjee, B. R. Majhi and S. K. Modak, “Noncommutative Schwarzschild Black Hole and Area Law,” Class. Quant. Grav. [**26**]{}, 085010 (2009) \[arXiv:0802.2176 \[hep-th\]\]. E. Spallucci, A. Smailagic and P. Nicolini, “Non-commutative geometry inspired higher-dimensional charged black holes,” Phys. Lett. B [**670**]{}, 449 (2009) \[arXiv:0801.3519 \[hep-th\]\].
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---
abstract: |
[ A routing $R$ of a given connected graph $G$ of order $n$ is a collection of $n(n-1)$ simple paths connecting every ordered pair of vertices of $G$. The vertex-forwarding index $\xi(G,R)$ of $G$ with respect to $R$ is defined as the maximum number of paths in $R$ passing through any vertex of $G$. The vertex-forwarding index $\xi(G)$ of $G$ is defined as the minimum $\xi(G,R)$ over all routing $R$’s of $G$. Similarly, the edge-forwarding index $ \pi(G,R)$ of $G$ with respect to $R$ is the maximum number of paths in $R$ passing through any edge of $G$. The edge-forwarding index $\pi(G)$ of $G$ is the minimum $\pi(G,R)$ over all routing $R$’s of $G$. The vertex-forwarding index or the edge-forwarding index corresponds to the maximum load of the graph. Therefore, it is important to find routings minimizing these indices and thus has received much research attention in the past ten years and more. In this paper we survey some known results on these forwarding indices, further research problems and several conjectures.]{}
[**Keywords**]{}:Vertex-forwarding index, Edge-forwarding index, Routing
[**AMS Subject Classification**]{}: 05C40
author:
- |
[Jun-Ming Xu[^1]]{}\
[School of Mathematical Sciences]{}\
[University of Science and Technology of China]{}\
[Hefei, Anhui, 230026, China]{}\
[E-mail addresses: [email protected]]{}\
\
[Min Xu ]{}\
[School of Mathematical Sciences]{}\
[Beijing Normal University]{}\
[Beijing, 100875, China]{}\
[E-mail addresses: [email protected]]{}
title: |
**The Forwarding Indices\
of Graphs – a Survey [^2]**
---
Introduction
=============
In a communication network, the message delivery system must find a route along which to send each message from its source to its destination. The time required to send a message along the fixed route is approximately dominated by the message processing time at either end-vertex, intermediate vertices on the fixed route relay messages without doing any extensive processing. Metaphorically speaking, the intermediate vertices pass on the message without having to open its envelope. Thus, to a first approximation, the time required to send a message along a fixed route is independent of the length of the route. Such a simple forwarding function can be built into fast special-purpose hardware, yielding the desired high overall network performance.
For a fully connected network, this issue is trivial since every pair of processors has direct communication in such a network. However, in general, it is not this situation. The network designer must specify a set of routes for each pair $(x,y)$ of vertices in advance, indicating a fixed route which carries the data transmitted from the message source $x$ to the destination $y$. Such a choice of routes is called a routing.
We follow [@x03] for graph-theoretical terminology and notation not defined here. A graph $G=(V,E)$ always means a simple and connected graph, where $V=V(G)$ is the vertex-set and $E=E(G)$ is the edge-set of $G$. It is well known that the underlying topology of a communication network can be modelled by a connected graph $G=(V,E)$, where $V$ is the set of processors and $E$ is the set of communication links in the network.
Let $G$ be a connected graph of order $n$. A routing $R$ in $G$ is a set of $n(n-1)$ fixed paths for all ordered pairs $(x,y)$ of vertices of $G$. The path $R(x,y)$ specified by $R$ carries the data transmitted from the source $x$ to the destination $y$. A routing $R$ in $G$ is said to be minimal, denoted by $R_m$, if each of the paths specified by $R$ is shortest; $R$ is said to be symmetric or bidirectional, if for all vertices $x$ and $y$, path $R(y,x)$ is the reverse of the path $R(x,y)$ specified by $R$; $R$ is said to be consistent, if for any two vertices $x$ and $y$, and for each vertex $z$ belonging to the path $R(x,y)$ specified by $R$, the path $R(x,y)$ is the concatenation of the paths $R(x,z)$ and $R(z,y)$.
It is possible that the fixed paths specified by a given routing $R$ going through some vertex are too many, which means that the routing $R$ loads the vertex too much. Load of any vertex is limited by capacity of the vertex, for otherwise it would affect efficiency of transmission, even result in malfunction of the network.
It seems quite natural that a “good" routing should not load any vertex too much, in the sense that not too many paths specified by the routing should go through it. In order to measure the load of a vertex, Chung, Coffiman, Reiman and Simon [@ccrs87] proposed the notion of the forwarding index.
Let $G$ be a graph with a give routing $R$ and $x$ be a vertex of $G$. The load of $x$ with respect to $R$, denoted by $\xi_x(G,R)$, is defined as the number of the paths specified by $R$ going through $x$. The parameter $$\xi(G,R)=\max\{\xi_x(G,R):\ x\in V(G)\}$$ is called the forwarding index of $(G,R)$, and the parameter $$\xi(G)=\min\{\xi(G,R):\ \forall\ R\},\quad
\xi_m(G)=\min\{\xi(G,R_m):\ \forall\ R_m\}$$ is called the forwarding index of $G$.
Similar problems are studied for edges by Heydemann, Meyer and Sotteau [@hms89]. The load of an edge $e$ with respect to $R$, denoted by $\pi_e(G,R)$, is defined as the number of the paths specified by $R$ which go through it. The edge-forwarding index of $(G,R)$, denoted by $\pi(G,R)$, is the maximum number of paths specified by $R$ going through any edge of $G$, i.e., $$\pi(G,R)=\max\{\pi_e(G,R):\ e\in E(G)\};$$ and the edge-forwarding index of $G$ is defined as $$\pi(G)=\min\{\pi(G,R):\ \forall\ R\},\quad
\pi_m(G)=\min\{\pi(G,R_m):\ \forall\ R_m\}.$$
Clearly, $\xi(G)\le \xi_m(G)$ and $\pi(G)\le \xi_m(G)$. The equality however does not always holds.
The original research of the forwarding indices is motivated by the problem of maximizing network capacity [@ccrs87]. Maximizing network capacity clearly reduces to minimizing vertex-forwarding index or edge-forwarding index of a routing. Thus, whether or not the network capacity could be fully used will depend on the choice of a routing. Beyond a doubt, a “good" routing should have a small vertex-forwarding index and edge-forwarding index. Thus it becomes very significant, theoretically and practically, to compute the vertex-forwarding index and the edge-forwarding index of a given graph and has received much attention the recent ten years and more.
Generally, computing the forwarding index of a graph is very difficult. In this paper, we survey some known results on these forwarding indices, further research problems, several conjectures, difficulty and relations to other topics in graph theory.
Since forwarding indices were first defined for a graph, that is, an undirected graph [@ccrs87], most of the results in the literature are given for graphs instead of digraphs, but they can be easily extended to digraphs. Nevertheless, we give here most of the results for graphs as they appear in the literature.
Basic Problems and Results
==========================
[**2.1.NP-completeness**]{}
Chung, Coffiman, Reiman and Simon [@ccrs87] asked whether the problem of computing the forwarding index of a graph is an NP-complete problem. Following [@gj79], we state this problem as follows.
Forwarding Index Problem,
: A graph $G$ and an integer $k$.
: $\xi(G)\le k$?
Heydemann, Meyer, Sotteau and Opatrný [@hmso94] first showed that Problem 2.1 is NP-complete for graphs of diameter at least $4$ when the routings considered are restricted shortest, consistent and symmetric; a P-problem for graphs of diameter $2$ when the routings considered are restricted to be shortest. Saad [@s93] proved that Problem 2.1 is NP-complete for for general routings even if the diameter of the graph is $2$. However, Problem 2.1 has not yet been solved for graphs of $3$ when the routings considered are restricted to be shortest be minimal and/or, consistent and/or symmetric.
The same problem was also suggested by Heydemann, Meyer and Sotteau [@hms89].
Edge-Forwarding Index Problem,
: A graph $G$ and an integer $k$.
: $\pi(G)\le k$?
Heydemann, Meyer, Sotteau and Opatrný [@hmso94] showed that Problem 2.2 is NP-complete for graphs of diameter at least $3$ when the routings considered are restricted to be minimal, consistent and symmetric; a P-problem for graphs of diameter $2$ when the routings considered are restricted to be minimal.
[**2.2.Basic Bounds and Relations**]{}
For a given connected graph $G$ of order $n$, set $$A(G)=\frac {1}{n}\sum\limits_{u\in V} \left(\sum\limits_{v\in
V\setminus\{u\}}(d_{G}(u,v)-1)\right),$$ and $$B(G)=\frac{1}{\varepsilon}\sum\limits_{(u,\,v)\in V\times
V}d_{G}(u,v).$$
The following bounds of $\xi (G)$ and $\pi(G)$ were first established by Chung, Coffiman, Reiman and Simon [@ccrs87] and Heydemann, Meyer and Sotteau [@hms89], respectively.
(Chung [*et al*]{} [@ccrs87]) Let $G$ be a connected graph of order $n$. Then $$\label{e1} A(G) \le \xi(G)\le (n-1)(n-2),$$ and the equality $\xi_{G}=\xi_m(G)=A(G)$ is true if and only if there exists a minimal routing in $G$ which induces the same load on every vertex. The graph that attains this upper bound is a star $K_{1,n-1}$.
(Heydemann [*et al*]{} [@hms89]) Let $G$ be a connected graph of order $n$. Then $$\label{e2} B(G)\le \pi(G) \le \left\lfloor \frac 12n^2\right\rfloor,$$ and the equality $\pi(G)=\pi_m(G)=B(G)$ is true if and only if there exists a minimal routing in $G$ which induces the same load on every edge. The graph that attains this upper bound is a complete bipartite graph $K_{\frac n2, \frac n2}$.
Recently, Xu [*et al*]{}. [@xzdy07] have showed the star $K_{1,n-1}$ is a unique graph that attains the upper bound in (\[e1\]).
Note that the upper bound given in (\[e2\]) can be attained. Give a characterization of graphs whose vertex- or edge-forwarding indices attain the upper bound in (\[e2\]).
Although the two concepts of vertex- and edge-forwarding index are similar, no interesting relationships is known between them except the following trivial inequalities.
(Heydemann [*et al*]{} [@hms89]) For any connected undirected graph $G$ of order $n$, maximum degree $\Delta$, minimum degree $\delta$,
(a) $2\xi(G)+2(n-1)\le \Delta\pi(G)$;
(b) $\pi(G)\le \xi(G)+2(n-1)$;
(c) $\pi_m(G)\le \xi_m(G)+2(n-\delta)$.
The inequality in (a) is also valid for minimal routings.
No nontrivial graph is found for which the forwarding indices hold one of the above equalities. Thus, it is necessary to further investigate the relationships between $\pi(G)$ and $\xi(G)$ or between $\pi_m(G)$ and $\xi_m(G)$.
For a graph $G$ and its line graph $L(G)$, investigate the relationships between $\xi(G)$ and $\pi(L(G))$ or between $\xi_m(G)$ and $\pi_m(L(G))$.
[**2.3.Optimal Graphs**]{}
A graph $G$ is said to be vertex-optimal if $\pi(G)=A(G)$, and edge-optimal if $\xi(G)=B(G)$. Note that if $R_m$ is a routing of $G$ such that $\pi(G,R_m)=A(G)$, then $$\label{e3} \xi(G)=\sum\limits_{y\in V}d(G;x,y)-(n-1), \quad \forall \ x\in V.$$ Heydemann [*et al*]{} [@hms89] showed that the equality (3) is valid for any Cayley graph.
Let $G$ be a connected Cayley graph with order $n$. Then $$\label{e4} \xi (G)=\xi_m(G)=\sum\limits_{y\in V}d(G;x,y)-(n-1), \quad
\forall\ x\in V.$$
From Theorem 2.8, Cayley graphs are vertex-optimal. Some results and problems on the forwarding indices of vertex-transitive or Cayley graphs, an excellent survey on this subject has been given by Heydemann [@h97].
Heydemann [*et al*]{} [@hmso94] have constructed a class of graphs for which the vertex-forwarding index is not given by a minimal consistent routing. Thus, they suggested the following problems worthy of being considered.
(Heydemann [*et al*]{} [@hmso94]) For which graph or digraph $G$ does there exist a minimal consistent routing $R$ such that $\xi_m(G)=\xi(G,R)$ or a consistent routing $R$ such that $\xi(G)=\xi(G,R)$?
Heydemann et al [@hmso94] have ever conjectured that in any vertex-transitive graph $G$, there exists a minimal routing $R_m$ in which the equality (4) holds.
The conjecture has attracted many researchers for ten years and more without a complete success until 2002. Shim, Širáň and Žerovnik [@ssz02] disproved this conjecture by constructing an infinite family of counterexamples, that is, $K_p\oplus P(10,2)$ for any $q\not\equiv0$(mod $3$), where $P(10,2)$ is the generalized Petersen graph and the symbol $\oplus$ denotes the strong product.
Gauyacq [@g971; @g97; @gmr98] defined a class of quasi-Cayley graphs, a new class of vertex-transitive graphs, which contain Cayley graphs, and are vertex-optimal. Solé [@so94] constructed an infinite family of graphs, the so-called orbital regular graphs, which are edge-optimal. We state the results of Gauyacq and Solé as the following theorem.
Any quasi-Cayley graph is vertex-optimal, and any orbital regular graph is edge-optimal.
However, we have not yet known whether a quasi-Cayley graph is edge-optimal and not known whether an orbital regular graph is vertex-optimal. Thus, we suggest to investigate the following problem.
Investigate whether a quasi-Cayley graph is edge-optimal and an orbital regular graph is vertex-optimal.
Considering $\pi(K_2\times K_p)$ for $p\ge 3$, Heydemann [*et al*]{} [@hms89] found that the equality (4) is not valid for $\pi(G)$, and proposed the following conjecture.
(Heydemann et al [@hms89]) For any distance-transitive graph $G=(V,E)$, there exists a minimal routing $R_m$ for which, $$\pi(G)=\pi (G,R_m)=\left\lceil\frac
{n}{\varepsilon}\sum\limits_{y\in V}d(G;x,y)\right\rceil, \quad
\forall\ x\in V.$$
[**Conjecture 2.13**]{}(Heydemann [*et al*]{} [@hms89]) For any distance-transitive graph $G=(V,E)$, there exists a minimal routing in which we have both
(a) the load of all vertices is the same, and then, $$\xi(G)=\xi_m(G)=\sum\limits_{y\in V}d(G;x,y)-(n-1), \quad \forall\
x\in V.$$
(b) the load of all the edges is almost the same (difference of at most one) and then, $$\pi(G)=\pi_m(G)=\left\lceil\frac {n}{\varepsilon}\sum\limits_{y\in
V}d(G;x,y)\right\rceil, \quad \forall\ x\in V.$$
[**2.4.For Cartesian Product Graphs**]{}
The cartesian product can preserve many desirable properties of the factor graphs. A number of important graph-theoretic parameters, such as degree, diameter and connectivity, can be easily calculated from the factor graphs. In particular, the cartesian product of vertex-transitive (resp. Cayley) graphs is still a vertex-transitive (resp. Cayley) graph (see Section 2.3 in [@x01]). Since quasi-Cayley graphs are vertex-transitive, the cartesian product of quasi-Cayley graphs is still a quasi-Cayley graph. Thus, determining the forwarding indices of the cartesian product graphs is of interest. Heydemann [*et al*]{} [@hms89] obtained the following results first.
Let $G$ and $G'$ be two connected graphs with order $n$ and $n'$, respectively. Then
(a) $\xi(G\times G')\le n\ \xi(G')+n'\ \xi(G)+(n-1)(n'-1)$;
(b) $\pi(G\times G')\le \max\{n\pi(G'),\ n'\pi(G)\}.$
The inequalities are also valid for minimal routings. Moreover, the equality in (a) holds if both $G$ and $G'$ are Cayley digraphs.
Recently, Xu [*et al*]{} [@xxh06] have considered the cartesian product of $k$ graphs and obtained the following results.
Let $G=G_1\times G_2\times
\cdots \times G_k$. Then
(a) $G$ is vertex-optimal and if $G_i$ is vertex-optimal for every $i=1,2,\cdots,k$ then $$\xi(G)= \sum\limits_{i=1}^k n_1n_2\cdots
n_{i-1}(\xi_i-1)n_{i+1}\cdots n_k+(k-1)n_1n_2\cdots n_k+1;$$
(b) $G$ is edge-optimal and if $G_i$ is edge-optimal for every $i=1,2,\cdots,k$ then $$\pi(G)= \max\limits_{1\le i\le k}
\{n_1n_2\cdots n_{i-1}\pi_in_{i+1}\cdots n_k\}.$$
By Theorem 2.15 and Theorem 2.10, the cartesian product of quasi-Cayley graphs is vertex-optimal and the cartesian product of orbital regular graphs is edge-optimal.
Connectivity Constraint
=======================
In this section, we survey the known results of the forwarding indices of $k$-connected or $k$-edge-connected graphs.
[**3.1.$\kappa$-connected Graphs**]{}
If $G$ is a $2$-connected graph of order $n$, then
(a) $\xi (G)\le \frac 12\,(n-2)(n-3)$, this bound is best possible in view of $K_{2,n-2}$ (Heydemann [*et al*]{} [@hms89]);
(b) $\xi_m(G)\le n^2-7n+12$ for $n\ge 6$ and diameter $2$, this bound is best possible since it is reached for a wheel of order $n$ minus one edge with both ends of degree $3$ (Heydemann [*et al*]{} [@hms89]);
(c) $\xi_m(G)\le n^2-7n+12$ for $n\ge 7$ (Heydemann [*et al*]{} [@hmos92]);
(d) $\pi(G)\le \lfloor\frac 14\,n^2\rfloor$ and this bound is best possible in view of the cycle $C_n$ (Heydemann [*et al*]{} [@hmos92]).
(Heydemann [*et al*]{} [@hmos92]) $\xi(G)\le n^2-(2k+1)n+2k$ for any $k$-connected graph $G$ of order $n$ with $k\ge 3$ and $n\ge
8k-10$.
Heydemann [*et al*]{} [@hmos92] proposed the following research problem.
Find the best upper bound $f(n,k), g(n,k), h(n,k)$ and $s(n,k)$ such that for any $k$-connected graph $G$ of order $n$ with $k\ge 2$, $\xi(G)\le
f(n,k), \xi_m(G)$ $\le g(n,k), \pi(G)\le h(n,k)$ and $\pi_m(G)\le
s(n,k)$ for $n$ large enough compared to $k$.
(de la Vega and Manoussakis [@fm92]) For any integer $k\ge 1$,
(a) $f(n,k)\le (n-1) \left\lceil\frac 1k\,(n-k-1)\right\rceil$;
(b) $g(n,k)\le \frac 12n^2-(k-1)n+\frac 38(k-1)^2$ if $n$ is substantially larger than $k$;
(c) $h(n,k)\le n\left\lceil\frac 1k\, (n-k-1)\right\rceil$.
Recently, Zhou [*et al*]{}. [@zxx08] have improved the upper bounds of $f(n,k)$ and $h(n,k)$ in Theorem 3.4 as follows.
If $G$ is a $k$-connected graph of order $n$ with the maximum degree $\Delta$, then $\xi (G)\le
(n-1)\lceil(n-k-1)/k\rceil-(n-\Delta-1)$ and $\pi (G)\le
n\lceil(n-k-1)/k\rceil-(n-\Delta)$.
(de la Vega and Manoussakis [@fm92]) For any positive integer $k$,
(a) $f(n,k)\le\lceil\frac 1k\,(n-k)(n-k-1)\rceil$ for $n\ge 2k\ge
2$, which would be best possible in view of the complete bipartite graph $K_{k,n-k}$;
(b) there exists a function $q(k)$ such that if $n\ge q(k)$, then $g(n,k)\le \frac 12n^2-(k-1)n-\frac 32k^2+k+\frac 72$;
(c) $h(n,k)\le \lceil\frac {n^2}{2k}\rceil$ for $n\ge 2k\ge 2$, which would be best possible in view of the graph obtained from two complete graphs $K_m$ plus a matching $e_1,e_2,\cdots,e_k$ between them, $m\ge k$.
It can be easily verified that the conjecture (a) and (c) are true for $k=1$ and $k=2$. Recently, Zhou [*et al*]{}. [@zxx08] have proved that Conjecture 3.6 (a) is true for $k=3$, that is,
If $G$ is a $3$-regular and $3$-connected graph of order $n\ge 4$. Then $\xi(G)\le \lceil(n-3)(n-4)/3\rceil$.
[**3.2.$\lambda$-edge-connected Graphs**]{}
If $G$ is a $2$-edge-connected graph of order $n$, then
(a) $\pi_m(G)\le \lfloor\frac 12\,n^2-n+\frac 12\rfloor$ (Heydemann [*et al*]{} [@hms89; @hmos92]);
(b) $\pi(G)\le \lfloor\frac 14\,n^2\rfloor$ (Cai [@cai90]).
Heydemann [*et al*]{} [@hms89] conjectured that for any $\lambda$-edge-connected graph $G$ of order $n$, $\pi(G)\le
\lfloor\frac 12\,n^2-(\lambda-1)n+\frac 12\, (\lambda-1)^2
\rfloor$. Latter, Heydemann, Meyer, Opatrný and Sotteau [@hmos92] gave a counterexample and proposed the following conjecture.
for any $\lambda$-edge-connected graph $G$ of order $n$ with $\lambda\ge 3$ and $n\ge 3\lambda+3$, $$\pi_m(G)=\max\left\{\left\lceil\frac
{n^2}2\right\rceil-n-2(\lambda-1)^2,\ \left\lfloor\frac
{n^2}2\right\rfloor-2n+5\lambda-\frac 32(\lambda^2+1)\right\}.$$
The same problem as the ones in Problem 3.3 can be considered for $\lambda$-edge-connected graphs.
Find the best upper bound $f'(n,\lambda), g'(n,\lambda), h'(n,\lambda)$ and $s'(n,\lambda)$ such that for any $\lambda$-connected graph $G$ of order $n$ with $\lambda\ge 2$, $\xi(G)\le f'(n,\lambda),\xi_m(G)$ $\le
g'(n,\lambda), \pi(G)\le h'(n,\lambda)$ and $\pi_m(G)\le
s'(n,\lambda)$ for $n$ large enough compared to $\lambda$.
The following theorem is the only result we have known as far on this problem.
(de la Vega and Manoussakis [@fm92]) For any integer $\lambda\ge 3$, $$g'(n,\lambda)=\left\lceil\frac
{n^2}2\right\rceil-n-2(\lambda-1)^2\ {\rm for}\
n\ge\max\left\{3\lambda+3,\ \frac 12(\lambda+1)^2\right\}.$$
[**3.3.Strongly Connected Digraphs**]{}
It is clear that the notion of the forwarding indices can be similarly defined for digraphs. Many general results, such as Theorem 2.3 and Theorem 2.4 are valid for digraphs. Manoussakis and Tuza [@mt962] consider the forwarding index of strongly $k$-connected digraphs and obtained the following result similar to Theorem 2.4.
Let $D=(V,E)$ be a strong digraph of order $n$. Then
\(1) $B(D)\le \pi(D)\le \pi_m(D)\le (n-1)(n-2)+1$, and
\(2) The equalities $\pi(D)=\pi_m(D)=B(D)$ are true if and only if there exists a minimal routing in $D$ which induces the same load on every edge.
In addition to validity of Theorem 3.2 for digraphs, they obtained the following results.
Let $D$ be a $k$-connected digraph of order $n\ge 3$, and $k\ge 1$. Then
(a) $\pi (D)\le (n-1)\lceil\frac 1k(n-k-1)\rceil+1$;
(b) $\xi_m(D)\le n^2-(2k+1)n+2k$ for $n\ge 2k+1$;
(c) $\pi_m(D)\le n^2-(3K+2)n+4k+3$ for $n\ge 4k-1$.
Degree Constraint
=================
Although Saad [@s93] showed that for any graph determining the forwarding index problem is NP-complete, yet many authors are interested in the forwarding indices of a graph. Specially, it is still of interest to determine the exact value of the forwarding index with some graph-theoretical parameters. For example, Chung [*et al*]{} [@ccrs87], Bouabdallah and Sotteau [@bs93] proposed to determine the minimum forwarding indices of $(n,\Delta)$-graphs that has order $n$ and maximum degree at most $\Delta$. Given $\Delta$ and $n$, let $$\begin{array}{rl}
&\xi_{\Delta,n}=\min\{\xi(G):\ |V(G)|=n, \Delta(G)=\Delta\},\\
&\pi_{\Delta,n}=\min\{\pi(G):\ |V(G)|=n, \Delta(G)=\Delta\}.
\end{array}$$
[**4.1.Problems and Trivial Cases**]{}
(Chung [*et al*]{} [@ccrs87]) Given $\Delta\ge 2$ and $n\ge 4$, determine $\xi_{\Delta,n}$, and exhibit an $(n,\Delta)$-graph $G$ and $R$ of $G$ for which $\xi(G,R)=\xi_{\Delta,n}$.
(Bouabdallah and Sotteau [@bs93]) Given $\Delta\ge 2$ and $n\ge 4$, determine $\pi_{\Delta,n}$, and exhibit an $(n,\Delta)$-graph $G$ and $R$ of $G$ for which $\pi(G,R)=\pi_{\Delta,n}$.
For $\Delta \ge n-1$, we can fully connect a graph, i.e., $G$ is a complete graph. In this case any routing $R$ can be composed only of single-edge paths so that the minimum $\xi=0$ and $\pi=2$ is achieved, that is, $\xi_{\Delta,n}=\xi(G,R)=0$ and $\pi(G,R)=\pi_{\Delta,n}=2$ for $\Delta\ge n-1$.
For $\Delta =2$ the only connected graph fully utilizing the degree constraint is easily seen to be a cycle. Because of the simplicity of cycles, the for vertex-forwarding index problem can be solve completely for $\Delta =2$.
For all $n\ge 3$, (a) $\xi_{2,n}=\xi(C_n,R_m)=\left\lfloor\frac
14(n-1)^2\right\rfloor$;
(b) $\pi_{2,n}=\pi(C_n,R_m)=\left\lfloor\frac
14n^2\right\rfloor$.
[**4.2.Results on $\xi_{\Delta,\,n}$**]{} $\Delta\ge 3$
Problem 4.1 was solved for $n\le 15$ or any $n$ and $\Delta$ with $\frac 13\,(n+4)\le \Delta\le n-1$ by Heydemann et al [@hms88].
(Heydemann [*et al*]{} [@hms88])
(a) if $n$ is even or $n$ odd and $\Delta $ even, $\xi_{\Delta,\,n}=n-1-\Delta$ for $\Delta\ge \frac 13\,(n+1)$ or for $n=12$ or $13$ and $\Delta =4$;
(b) if $n$ and $\Delta $ are odd, $\xi_{\Delta,\,n}=n-\Delta$ for $\Delta\ge \frac 13\,(n+4)$ or for $n=13$ and $\Delta =5$.
Problem 4.1 has not been completely solved for $\Delta<\frac 13\,(n+4)$.
(Heydemann [*et al*]{} [@hms88]) For any $n$ and $\Delta$,
(a) $\xi_{n-2p-1,\,n}=2p$ for any $n$ and $p$ such that $n\ge
3p+2$;
(b) $\xi_{2p+1,\,n}=n-2p-1$ for any odd $n$ such that $2p+1\le
n\le 6p-1$;
(c) $\xi_{2p,\,n}=n-2p-1$ for any $n$ and $p$ such that $2p+1\le
n\le 6p-1$ and $p\ge 3$;
(d) $\xi_{\Delta,\,n}\ge n-1-\Delta$;
(e) $\xi_{\Delta,\,n}=n-1-\Delta\Longrightarrow$ every $(n,\Delta)$-graph $G$ such that $\xi(G)=\xi_{\Delta,\,n}$ is $\Delta$-regular and diameter $2$;
(f) $\xi_{\Delta,\,n}\ge n-\Delta$ if $n$ and $\Delta$ are odd.
An asymptotic result on $\xi_{\Delta,\,n}$ has been given by Chung [*et al*]{} [@ccrs87].
For any given $\Delta \ge 3$, $$[1+o(1)]n\log_{\Delta-1}n\le \xi_{\Delta,\,n}\le
\left[3+O\left({\displaystyle}\frac
1{\log\Delta}\right)\right]n\log_{\Delta}n,$$ where the upper bound holds for $\Delta\ge 6$.
[**4.3.Results on $\pi_{\Delta,\,n}$**]{} $\Delta\ge 3$
Similar to Theorem 4.5, Bouabdallah and Sotteau [@bs93] obtained the following result on $\pi_{\Delta,\,n}$
For any $n$ and $\Delta\ge 3$,
(a) $\pi_{\Delta,\,n}\ge \lceil\frac {4(n-1)}{\Delta}\rceil-2$;
(b) $\pi_{\Delta,\,n}\ge \lceil\frac
{4(n-1)}{\Delta}\rceil-2\Longrightarrow$ every $(n,\Delta)$-graph $G$ such that $\pi(G)=\pi_{\Delta,\,n}$ is $\Delta$-regular and diameter $2$ and $G$ has a minimal routing for which the load of all edges is the same;
(c) $\pi_{\Delta,\,n}\ge \lceil\frac {4n-2)}{\Delta}\rceil-2$ if $n$ and $\Delta$ are odd;
(d) $\pi_{\Delta,\,n}\le \pi_{\Delta',\,n}$ for any $n$ and $\Delta'\le n-1$ with $\Delta'<\Delta$.
Problem 4.3 was solved for $n\le 15$ by Bouabdallah and Sotteau [@bs93], who also obtained $\pi_{n-2, n}=3$ for any $n\ge 6$, $n\ne 7$ and $\pi_{n-2, n}=4$ for any $n=4,5,7$. Recently, Xu [*et al*]{} [@xhx04] have determined $\pi_{n-2p-1,\,n}=8$ if $3p+\lceil\frac 13\, p \rceil +1 \leq n \leq 3p+\lceil\frac 35\,p
\rceil$ and $\ge 2$. The authors in the two [@bs93] and [@xhx04] have completely determined $\pi_{n-2p-1, n}$ for $n\ge
4p$ and $p\ge1$ except a little gap.
For any $p\geq 1$, $$\pi_{n-2p-1, n}=\left\{
\begin{array}{ll}
3, \ & {\rm if}\ n\geq 10p+1;\\
4, \ & {\rm if}\ 6p+1 \leq n < 10p+1;\\
5, \ & {\rm if}\ 4p+2\lceil\frac 13\, p \rceil +1 \leq n \leq 6p\,;\\
6, \ & {\rm if}\ 4p+1 \leq n \leq 4p+\lceil \frac 23\,p\rceil.
\end{array}\right.$$
Note the value of $\pi_{n-2p-1,\,n}$ has not been determined for $4p+\lceil \frac 23\,p \rceil+1\le n \le 4p+2\lceil\frac 13\,p
\rceil$. However, these two numbers are different only when $p=3k+1$. Thus, we proposed the following conjecture.
For any $p\geq 1$, $\pi_{n-2p-1, n}=5$ if $4p+\lceil\frac 23\,p \rceil+1\le n \le
4p+2\lceil\frac 13\, p \rceil$.
(Xu [*et al*]{} [@xxh05])For any $p\geq 1$, we have $$\pi_{n-2p,\,n}=\left\{
\begin{array}{ll}
3, \ & {\rm if}\ n\geq 10p-2\ {\rm or}\ n=10p-4;\\
4, \ & {\rm if}\ 6p+1 \leq n < 10p-4\ or\ n=10p-3;\\
6, \ & {\rm if}\ 4p+1 \leq n \leq 4p+\lceil \frac 13\,(2p-1)
\rceil-2.
\end{array}\right.$$
An asymptotic result on $\pi_{\Delta,\,n}$ has been given by Heydemann [*et al*]{} [@hms89].
For any given $\Delta \ge 3$, $$\left[\frac 2{\Delta}+o(1)\right]\,\,n\,\log_{\Delta-1}n\le
\pi_{\Delta,\,n}\le 24\frac{\log_2(\Delta-1)}{\Delta}\, n
\log_{\Delta-1}n,
$$ where the upper bound holds for $\Delta\ge 6$.
[**4.4.General Results Subject to Degree and Diameter**]{}
(Xu [*et al*]{}. [@xzdy07]) For any connected graph $G$ of order $n$ and maximum degree $\Delta$, $$\xi(G)\le (n-1)(n-2)-\left(2n-2-\Delta\left\lfloor 1+
\frac{n-1}{\Delta}\right\rfloor\right)\left\lfloor\frac{n-1}{\Delta}
\right\rfloor.$$
Considering a special case of $\Delta=n-1$ in Theorem 4.12, we obtain the upper bound in (\[e1\]) immediately.
(Heydemann [*et al*]{} [@hms89]) Let $G$ be a graph of order $n$, maximum degree $\Delta$ and diameter $d$,
(a) $\xi(G)\le \xi_m(G)\le (n-1)(n-2)-2(\varepsilon(G)-\Delta)$,
(b) $\xi(G)\le \xi_m(G)\le n^2-3n-\lfloor\frac
12\,d\rfloor^2-\lceil\frac 12\,d\rceil^2+d+2$.
(Heydemann [*et al*]{} [@hms89]) If $G$ is a graph of order $n$ and diameter $2$ with no end vertex, then $\pi_m(G)\le 2n-4$.
Manoussakis and Tuza [@mt962] obtained some upper bounds on the forwarding indies for digraphs subject to degree constraints.
Let $D$ be a strongly connected digraph of order $n$ and minimum degree $\delta$. Then
(a) $\xi_m(D)\le n^2-(\delta+2)n+\delta+1$;
(b) $\pi_m(D)\le \max\{n^2-3n\delta+2\delta^2+\delta,\
n^2-(2\delta+3)n+\delta^2+4\delta+3\}$ if $n$ is sufficient large compared to $\delta$.
Considering the minimum degree $\delta$ rather the maximum degree $\Delta$, we can propose an analogy of $\xi_{\Delta,n}$ and $\pi_{\Delta,n}$ as follows. Given $\Delta$ and $n$, let $$\begin{array}{rl}
&\xi_{\delta,n}=\min\{\xi(G):\ |V(G)|=n, \delta(G)=\delta\},\\
&\pi_{\delta,n}=\min\{\pi(G):\ |V(G)|=n, \delta(G)=\delta\}.
\end{array}$$
However, the problem determining $\xi_{\delta,n}$ and $\pi_{\delta,n}$ is simple.
(Xu [*et al*]{}. [@xzdy07])For any $n$ and $\delta$ with $n>\delta\ge 1$, $$\xi_{\delta,n}=\left\lceil\frac{2(n-1-\delta)}{\delta}\right\rceil\quad
{\rm and}\
\pi_{\delta,n}=\left\lceil\frac{2(n-1)}{\delta}\right\rceil.$$
Difficulty, Methods and Relations to Other Topics
=================================================
[**5.1.Difficulty of Determining the Forwarding Indices**]{}
As we have stated in Subsection 2.1, the problem of computing the forwarding indices of a general graph is an NP-complete problem. Also, for a given graph $G$, determining its forwarding indices $\xi(G)$ and $\pi(G)$ is also very difficult.
The first difficulty is designing a routing $R$ such that $\xi_x(G,R)$ for any $x\in V(G)$ or $\pi_x(G,R)$ for any $e\in E(G)$ can be conveniently computed. An ideal routing can be found by the current algorithms for finding shortest paths. However, in general, it is not always the case that the forwarding indices of a graph can be obtained by a minimum routing.
For example, consider the wheel $W_7$ of order seven. The hub $x$, other vertices $0,1,\cdots, 5$. A minimum and bidirectional routing $R_m$ is defined as follows. $$\left\{
\begin{array}{ll}
R_m(i,i+2)=R(i+2,i)=(i,i+1,i+2)({\rm mod}\ 6), & i=0,1,\cdots,5;\\
R_m(i,i+3)=R(i+3,i)=(i,x,i+3)({\rm mod}\ 6), & i=0,1,2;\\
{\rm direct\ edge}, & {\rm otherwise}.
\end{array}\right.$$ Then, $$\xi_x(W_7,R_m)=6,\ \xi_i(W_7,R_m)=2,\ i=0,1,\cdots,5.$$ Thus, we have $\xi(W_7,R_m)=6$.
However, if we define a routing $R$ that is the same as the minimum routing $R_m$ except for $R(2,5)=(2,1,0,5),\ R(5,2)=(5,4,3,2)$. Then the routing $R$ is not minimum. We have $$\xi_x(W_7,R)=4,\ \tau_2(W_7,R)=\tau_5(W_7,R)=2,\quad
\xi_i(W_7,R)=3,\ i=0,1,3,4.$$ Thus, we have $\xi(W_7,R)=4<6=\xi(W_7,R_m)$.
The second difficulty is that the forwarding indices are always attained by a bidirectional routing. For example, for the hypercube $Q_n$ $(n\ge 2)$, $\xi(Q_n)=n2^{n-1}-(2^n-1)=2^{n-1}(n-2)+1$. Since $2^{n-1}(n-2)+1$ is odd, $\xi(Q_n)$ can not be attained by a bidirectional routing.
[**5.2.Methods of Determining the Forwarding Indices**]{}
To the knowledge of the author, one of the actual methods of determining forwarding index is to compute the sum of all pairs of vertices according to (\[e4\]) for some Cayley graphs. In fact, the forwarding indices of many Cayley graphs are determined by using (\[e4\]), for example, the folded cube [@hxx05], the augmented cube [@xx06] and so on, list in the next section.
Although Cayley graphs, one class of vertex-transitive graphs, are of hight symmetry, it is not always easy to compute the distance from a fixed vertex to all other vertices for some Cayley graphs. For example, The $n$-dimensional cube-connected cycle $CCC_n$, constructed from $Q_n$ by replacing each of its vertices with a cycle $C_n=(0,1,\ldots,n-1)$ of length $n$, is a Cayley graph proved by Carlsson [*et al.*]{} [@ccsw85]. Until now, one has not yet determined exactly its sum of all pairs of vertices, and so only can give its forwarding indices asymptotically (see, Shahrokhi and Székely [@ss01], and Yan [*et al*]{}. [@yxy08]).
Unfortunately, for the edge-forwarding index, there is no an analogy of (\[e4\]). But the lower bound of $\pi(G)$ given in (\[e2\]) is useful. One may design a routing $R$ such that $\pi (G,R)$ attains this lower bound. For example, the edge-forwarding indices of the folded cube [@hxx05] and the augmented cube [@xx06] are determined by this method.
Making use of results on the Cartesian product is one of methods determining forwarding indices. Using Theorem 2.15, Xu [*et al*]{} [@xxh06] determined the vertex-forwarding indices and the edge-forwarding indices for the generalized hypercube $Q(d_1,d_2,\ldots,d_n)$, the undirected toroidal mesh $C(d_1,d_2,\ldots, d_n)$, the directed toroidal mesh $\overrightarrow C(d_1,d_2,\ldots$, $d_n)$, all of which can be regarded as the Cartesian products.
[**5.3.Relations to Other Topics**]{}
To the knowledge of the author, until now one has not yet find an approximation algorithm with good performance ratio for finding routings of general graphs. However, de le Vega and Manoussakis [@fm94] showed that the problem of determining the value of the forwarding index (respectively, the forwarding index of minimal routings) is an instance of the multicommodity flow problem (respectively, flow with multipliers). Since many very good heuristics or approximation algorithms are known for these flow problems [@a78; @h69; @l76], it follows from these results that all of these algorithms can be used for calculating the forwarding index.
Teranishi [@t02], Laplacian spectra and invariants of graphs.
Forwarding Indices of Some Graphs
=================================
The forwarding indices of some very particular graphs have been determined. We list all main results that we are interested in and have known, all of which not noted can be found in Heydemann et at [@hms89] or determined easily.
For a complete graph $K_n$, $\xi(K_n)=0$ and $\pi(K_n)=2$.
For a star $K_{1,\,n-1}$,$\xi(K_{1,\,n-1})=(n-1)(n-2)$ and $\pi(K_{1,\,n-1})=2(n-1)$.
For a path $P_n$, $\xi(P_n)=2\left\lfloor\frac
12n\right\rfloor\left(\left\lceil\frac 12n\right\rceil-1\right)$ and $\pi(P_n)=2\left\lfloor\frac 12n\right\rfloor\left\lceil\frac
12n\right\rceil$.
For the complete bipartite $K_{m,\,n}$ $(m\ge n)$, $\xi_m(K_{m,\,n})=\xi(K_{m,\,n})= \lceil\frac
{m(m-1)}n\rceil$ and $\pi_m(K_{m,1})=2m$ and if $2\le n\le m$, $$\left\lceil\frac {2m(m-1)+2n(n-1)}{mn}\right\rceil+2\le
\pi_m(K_{m,\,n})\le \left\lceil\frac {m-1}{n}\right\rceil.$$ In particular, $$\pi_m(K_{n,\,n})=\pi_m(K_{n,\,n})=\left\{\begin{array}{ll} 4, &\
{\rm for}\ n=2;\\
5, &\ {\rm for}\ n=3,\ 4;\\ 6, &\ {\rm for}\ n\ge
5;\end{array}\right.$$
For a directed cycle $C_d$ $(d\ge 3)$,$\xi(C_d)=\frac 12(d-1)(d-2)$. For an undirected cycle $C_d$ $(d\ge 3)$, $\xi_m=\xi(C_d)=\left\lfloor\frac
14(d-1)^2\right\rfloor$ and $\pi_m=\pi(C_d)=\left\lfloor\frac
14d^2\right\rfloor$.
The $n$-dimensional undirected toroidal mesh $C(d_1,d_2,\cdots, d_n)$ is defined as cartesian product $C_{d_1}\times C_{d_2}\times\cdots \times C_{d_n}$ of $n$ undirected cycles $C_{d_1},C_{d_2},\cdots$, $C_{d_n}$ of order $d_1,d_2,\cdots,
d_n$, $d_i\ge 3$ for $i=1,2,\cdots,n$. The $C(d,d,\cdots,d)$, denoted by $C_n(d)$, is called a $d$-ary $n$-cube a generalized $n$-cube. Xu [*et al*]{} [@xxh06] determined that $$\begin{aligned}
\xi(C(d_1,d_2,\cdots, d_n)) & = & \sum_{i=1}^{n}d_1d_2\cdots
d_{i-1}\left\lfloor \frac {d_i^2}4\right\rfloor d_{i+1}\cdots
d_n-d_1d_2\cdots d_n +1;\\
\pi(C(d_1,d_2,\cdots, d_n)) & = & \max\limits_{1\le i\le n}
\left\{d_1d_2\cdots d_{i-1}\left\lfloor\frac {d_i^2}4\right\rfloor
d_{i+1}\cdots d_n\right\}.\end{aligned}$$ In particular, $$\xi(C_n(d))=nd^{n-1}\left\lfloor \frac 14
d^2\right\rfloor-(d^n-1), \quad {\rm and}\quad
\pi(C_n(d))=d^{n-1}\left\lfloor \frac 14 d^2\right\rfloor.$$ The last result was obtained by Heydemann et al [@hms89].
The $n$-dimensional directed toroidal mesh $\overrightarrow C(d_1,d_2,\cdots, d_n)$ is defined as the cartesian product $\overrightarrow C_{d_1}\times \overrightarrow
C_{d_2}\times\cdots \times \overrightarrow C_{d_n}$ of $n$ directed cycles $\overrightarrow C_{d_1},\overrightarrow C_{d_2},\cdots,
\overrightarrow C_{d_n}$ of order $d_1,d_2,\cdots, d_n$, $d_i\ge 3$ for each $i=1,2,\cdots,n$. Set $\overrightarrow
C_n(d)=\overrightarrow C(d,d,\cdots,d)$. Xu [*et al*]{} [@xxh06] determined that $$\begin{aligned}
\xi(\overrightarrow{C}(d_1,d_2,\cdots, d_n)) & = &
\frac 12\left(\sum_{i=1}^n(d_i-3)\right)d_1d_2\cdots d_n+(n-1)d_1d_2\cdots d_n +1;\\
\pi(\overrightarrow{C}(d_1,d_2,\cdots, d_n)) & = & \frac 12
\max_{1\le i\le n} \{d_1\cdots d_{i-1}d_i(d_i-1)d_{i+1}\cdots
d_n\}.\end{aligned}$$ In particular, $$\xi(\overrightarrow{C}_n(d))=\frac n2\,d^n(d-1)-d^n+1,\quad {\rm
and}\quad \pi(\overrightarrow{C}_n(d))=\frac 12\,d^n(d-1).$$
The $n$-dimensional generalized hypercube, denoted by $Q(d_1,d_2,\cdots,d_n)$, where $d_i\ge 2$ is an integer for each $i=1,2,\cdots,n$, is defined as the cartesian products $K_{d_1}\times K_{d_2}\times \cdots \times K_{d_n}$. If $d_1=d_2=\cdots =d_n=d\ge 2$, then $Q(d,d,\cdots,d)$ is called the $d$-ary $n$-dimensional cube, denoted by $Q_n(d)$. It is clear that $Q_n(2)$ is $Q_n$. Xu [*et al*]{} [@xxh06] determined that $$\begin{aligned}
\xi(Q(d_1,d_2,\cdots,d_n)) &=& -\sum\limits_{i=1}^n d_1d_2\cdots
d_{i-1}d_{i+1}\cdots d_n+(n-1)d_1d_2\cdots d_n+1,\\
\pi(Q(d_1,d_2,\cdots,d_n)) &=& \max\limits_{1\le i\le n}
\{d_1d_2\cdots d_{i-1}2d_{i+1}\cdots d_n\}.\end{aligned}$$ In particular, $$\xi(Q_n(d))= ((d-1)n-d)d^{n-1}+1,\quad {\rm and}\quad \pi(Q_n(d))=
2d^{n-1}.$$ For the $n$-dimensional hypercube $Q_n$, $$\xi(Q_n)=(n-2)2^{n-1}+1\quad {\rm and}\quad
\pi(Q_n)=2^n.$$ The last result was obtained by Heydemann et al [@hms89].
For the crossed cube $CQ_n$ $(n\ge 2)$,$\pi (CQ_n)=\pi_m(CQ_n)=2^n$ decided by Chang [*et al*]{} [@csh00]. However, $\xi (CQ_n)$ has not been determined so far.
For the folded cube $FQ_n$, decided by Hou [*et al*]{} [@hxx05], $$\xi(FQ_n)=\xi_m(FQ_n)=(n-1)2^{n-1}+1-\frac{n+1}{2}{n\choose
\lceil\frac n2\rceil},$$ $$\pi(FQ_n)=\pi_m(FQ_n)=2^n-{n\choose
\lceil\frac n2\rceil}.$$
For the augmented cube $AQ_n$ proposed by Choudum and Sunitha [@cs02], Xu and Xu [@xx06] showed that $$\xi(AQ_n)=\frac{2^n}{9}+\frac{(-1)^{n+1}}{9}+\frac{n2^n}{3}-2^n+1,$$ and $$\pi(AQ_n)=2^{n-1}.$$
For the cube-connected cycle $CCC(n)$ and the $k$-dimensional wrapped butterfly $WBF_k(n)$, Yan, Xu, and Yang [@yxy08], Shahrokhi and Székely [@ss01], determined $$\xi (CCC_n) =\frac{7}{4} n^2 2^n (1 - o(1)),$$ $$\pi(CCC(n))=\pi_m (CCC(n))=\frac 54n^22^n(1-o(1)),$$ $$\pi(WBF(n))=\pi_m (WBF_2(n))=\frac 54n^22^{n-1}(1+o(1)).$$ Hou, Xu and Xu [@hxx09] determined $$\xi(WB_k(n))=\frac{3n(n-1)}2k^n-\frac{n(k^n-1)}{k-1}+1.$$ For the star graph $S_n$, Gauyacq [@g97] obtained that $$2(n-1)!(n-1)+\lceil 2\alpha\rceil\le \pi(S_n)\le
2(n-1)!(n-1)+2\lceil \alpha\rceil,$$ where $\alpha=(n-2)!\sum\limits_{i=2}^{n-1}\frac{n-i}i$.
For the complete-transposition graph $CT_n$, Gauyacq [@g97] obtained that $$2(n-2)!(2n-3)-\lfloor 2\beta\rfloor\le \pi(CT_n)\le
2(n-2)!(2n-3)-2\lfloor \beta\rfloor,$$ where $\beta=2(n-2)!\sum\limits_{i=3}^{n}\frac{1}i$.
For the undirected de Bruijn graph $UB(d,\,n)$ and Kautz graph $UK(d,\,n)$, the upper bounds of their vertex-forwarding indices and edge-forwarding indices have been, respectively, given as follows.
$$\xi(UB(d,\,n))\le (n-1)d^n,\quad \xi (UK(d,\,n))\le (n-1)d^n,$$ $$\pi(UB(d,\,n))\le 2nd^{n-1},\quad \pi (UK(d,\,n))\le
2(n-1)d^{n-2}(d+1).$$
[s2]{}
A. Assad, Multicommodity network flows - a survey. [*Networks*]{}, [**8**]{} (1978), 37-91.
A. Bouabdallah, and D. Sotteau, On the edge-forwarding index problem for small graphs. [*Networks*]{}, [**23**]{}(4) (1993), 249-255.
M.-C. Cai, Edge-forwarding indices of $2$-edge-connected graphs. [*J. Xinjiang Univ. Natur. Sci.*]{}, [**7**]{} (4) (1990), 11-13.
G. E. Carlsson, J. E. Cruthirds, H. B. Sexton and C. G. Wright, Interconnection networks based on a generalization of cube-connected cycles, IEEE Transactions on computers, 34(8) (1985), 769-772.
C.-P. Chang, T.-Y. Sung, and L.-H. Hsu, Edge congestion and topological properties of crossed cubes. [*IEEE Trans. Parallel and Distributed Systems*]{}, [**11**]{}(1) (2000), 64-80.
S. A. Choudum and V. Sunitha, Augmented cubes, [*Networks*]{}, [**40**]{} (2) (2002), 71-84.
F. R. K. Chung, E. G. Jr. Coffman, M. I. Reiman, and B. Simon, The forwarding index of communication networks. [*IEEE Trans. Inform. Theory,*]{} [**33**]{}(2) (1987), 224-232.
W. F. de la Vega, and L. M. Gordones, The forwarding indices of random graphs. [*Random Structures Algorithms*]{}, [**3**]{} (1992), no. 1, 107-116.
W. F. de la Vega, and Y. Manoussakis, The forwarding index of communication networks with given connectivity. [*Discrete Appl. Math.*]{}, [**37/38**]{} (1992), 147-155.
W. F. de le Vega, and Y. Manoussakis, Computation of the forwarding index via flows: a note. [*Networks*]{}, [**24**]{} (5) (1994), 273-276.
M. R. Garey, and D. S. Johnson, [*Computers and Intractability: A Guide to Theory of NP-Completeness*]{}. W. H. Freeman, San Francisco, 1979.
G. Gauyacq, On quasi-Cayley graphs. [*Discrete Applied Mathematics*]{}, [**77**]{} (1997), 43-58.
G. Gauyacq, Edge-forwarding index of star graphs and other Cayley graphs. [*Discrete Appl. Math.*]{}, [**80**]{} (2-3) (1997), 149-160.
G. Gauyacq, C. Micheneau, and A. Raspaud, Routing in recursive circulant graphs: edge forwarding index and Hamiltonian decomposition. [*Graph-theoretic concepts in computer science*]{} (Smolenice Castle, 1998), 227-241, [*Lecture Notes in Comput. Sci.*]{}, [**1517**]{}, Springer, Berlin, 1998.
M.-C. Heydemann, Cayley graphs and interconnection networks. In [*Graph Symmetry*]{} (G. Hahn and G. Sabidussi, eds.), Kluwer Academic Publishers, Netherlands, 1997, 167-224.
M.-C. Heydemann, J.-C. Meyer, and D. Sotteau, On the-forwarding index problem for small graphs. Eleventh British Combinatorial Conference (London, 1987), [*Ars Combin.*]{}, [**25**]{} (1988), 253-266.
M.-C. Heydemann, J.-C. Meyer, and D. Sotteau, On the-forwarding index of networks. [*Discrete Appl. Math.*]{}, [**23**]{}(2) (1989), 103-123.
M.-C. Heydemann, J.-C. Meyer, J. Opatrný, and D. Sotteau, Forwarding indices of $k$-connected graphs. [*Discrete Appl. Math.*]{}, [**37/38**]{} (1992), 287-296.
M.-C. Heydemann, J.-C. Meyer, J. Opatrný, and D. Sotteau, Realizable values of the forwarding index. Proceedings of the Twentieth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, FL, 1989). [*Congressus Numerantium*]{}, [**74**]{} (1990), 163-172.
M.-C. Heydemann, J.-C. Meyer, D. Sotteau, and J. Opatrný, Forwarding indices of consistent routings and their complexity. [*Networks*]{}, [**24**]{} (2) (1994), 75-82.
X.-M. Hou, M. Xu, and J.-M. Xu, Forwarding indices of folded $n$-cubes. [*Discrete Applied Math*]{}., [**145**]{} (3) (2005), 490-492.
X.-M. Hou, J.-M. Xu, and M. Xu, The forwarding index of wrapped btterfly networks. [*Networks*]{}, [**53**]{} (4) (2009), 329-333.
T. C. Hu, [*Integer Programming and Network Flows*]{}. Addison Wesley, Reading, MA, 1969.
E. L. Lawler, [*Combinatorial optimization: Networks and Matroids*]{}, Holt, Rinehart and Winston, New York, 1976.
Y. Manoussakis, and Z. Tuza, Optimal routings in communication networks with linearly bounded forwarding index. [*Networks*]{}, [**28**]{} (4) (1996), 177-180.
Y. Manoussakis, and Z. Tuza, The forwarding index of directed networks. [*Discrete Appl. Math.*]{}, [**68**]{} (3)(1996), 279-291.
R. Saad, Complexity of the-forwarding index problem. [*SIAM J. Discrete Math.*]{}, [**6**]{}(3) (1993), 418-427.
S. Shim, J. Širáň, and J. Žerovnik, Counterexamples to the uniform shortest path routing conjecture for vertex-transitive graphs. [*Discrete Applied Math.*]{}, [**119**]{} (2002), 281-286.
F. Shahrokhi, and L. A. Székely, Constructing integral uniform flows in symmetric networks with application to the edge-forwarding index problem. International Workshop on Graph-Theoretic Concepts in Computer Science (Smolenice Castle, 1998). [*Discrete Appl. Math*]{}., [**108**]{} (1-2) (2001), 175-191.
P. Solé, The edge-forwarding index of orbital regular graphs. [*Discrete Mathematics*]{}, [**130**]{} (1994), 171-176.
Y. Teranishi, Laplacian spectra and invariants of graphs. [*Discrete Mathematics*]{}, [**257**]{} (2002), 183-189.
Y. Wang, X.-G. Fang and D. F. Hsu, On the edge-forwarding indices of Frobenius graphs. [*Acta Mathematica Sinica*]{}, [**22**]{}(6) (2006), 1735-1744.
J.-M. Xu, [*Toplogical Structure and Analysis of Interconnection Networks*]{}. Kluwer Academic Publishers, Dordrecht/Boston/London, 2001.
J.-M. Xu, [*Thoery and Application of Graphs*]{}, Kluwer Academic Publishers, Dordrecht/Boston/London, 2003.
J.-M. Xu, M. Xu, and X.-M. Hou, Forwarding indices of Cartesian product graphs. [*Taiwan J. Math*]{}., [**10**]{} (5) (2006), 1305-1315.
M. Xu, X.-M. Hou, and J.-M. Xu, The proof of a conjecture of Bouabdallah and Sotteau. [*Networks*]{}, [**44**]{} (4) (2004), 292-296.
M. Xu, and J.-M. Xu, The forwarding indices of augmented cubes. [*Information Processing Letters*]{}, [**101**]{} (5) (2007), 185-189.
M. Xu, J.-M. Xu, and X.-M. Hou, On edge-forwarding index of graphs with degree restriction. [*J. China Univ. Sci. Tech*]{}., [**35**]{}(6) (2005), 732-737.
M. Xu, J.-M. Xu, and L. Sun, The forwarding index of the circulant networks. [*Journal of Mathematics*]{}. [**27**]{} (6) (2007), 623-629.
J.-M. Xu, T. Zhou, Y. Du, and J. Yan, A new upper bound on forwarding index of graphs. [*Ars Combinatoria*]{}, [**83**]{} (2007), 289-293.
J. Yan, J.-M. Xu, and C. Yang, Forwarding index of cube-connected cycles. [*Discrete Applied Mathematics*]{}, [**157**]{} (1) (2009), 1-7.
M.-J. Zhou, M. Xu, J.-M. Xu, and T. Zhou, Forwarding indices of $3$-connected and $3$-regular graphs. [*J. China Univ. Sci. Tech*]{}., [**38**]{}(5) (2008), 456-459,495.
[^1]: Corresponding author: [email protected] (J.-M. Xu)
[^2]: The work was supported partially by NNSF of China (No. 11071233).
|
---
abstract: 'We constrain the form of the primordial power spectrum using Wilkinson Microwave Anisotropy Probe (WMAP) 3-year cosmic microwave background (CMB) data (+ other high resolution CMB experiments) in addition to complementary large-scale structure (LSS) data: 2dF, SDSS, Ly-$\alpha$ forest and luminous red galaxy (LRG) data from the SDSS catalogue. We extend the work of the WMAP team to that of a fully Bayesian approach whereby we compute the comparative Bayesian evidence in addition to parameter estimates for a collection of seven models: (i) a scale invariant Harrison-Zel’dovich (H-Z) spectrum; (ii) a power-law; (iii) a running spectral index; (iv) a broken spectrum; (v) a power-law with an abrupt cutoff on large-scales; (vi) a reconstruction of the spectrum in eight bins in wavenumber; and (vii) a spectrum resulting from a cosmological model proposed by @Doran (L-D). Using a basic dataset of WMAP3 + other CMB + 2dF + SDSS our analysis confirms that a scale-invariant spectrum is disfavoured by between 0.7 and 1.7 units of log evidence (depending on priors chosen) when compared with a power-law tilt. Moreover a running spectrum is now significantly preferred, but only when using the most constraining set of priors. The addition of Ly-$\alpha$ and LRG data independently both suggest much lower values of the running index than with basic dataset alone and interestingly the inclusion of Ly-$\alpha$ significantly disfavours a running parameterisation by more than a unit in log evidence. Overall the highest evidences, over all datasets, were obtained with a power law spectrum containing a cutoff with a significant log evidence difference of roughly 2 units. The natural tilt and exponential cutoff present in the L-D spectrum is found to be favoured decisively by a log evidence difference of over 5 units, but only for a limited study within the best-fit concordance cosmology.'
date: 'Accepted —. Received —; in original form '
title: 'WMAP 3-year primordial power spectrum'
---
\[firstpage\]
cosmological parameters – cosmology:observations – cosmology:theory – cosmic microwave background – large-scale structure
Introduction
============
The recent release of 3-year Wilkinson Microwave Anisotropy Probe (WMAP3; @WMAP3) data have provided precise measurements of temperature fluctuations in the cosmic microwave background (CMB). The accepted inflationary paradigm suggests that a primordial spectrum of almost scale-invariant density fluctuations produced during inflation went on to produce the observed structure in the CMB and that seen on large-scales in the current distribution of matter. Now, for the first time a purely scale invariant primordial spectrum is ruled out at $1\sigma$ (@SpergelII; @Parkinson) in favour of a ‘tilted’ spectrum with $n <
1$. The WMAP team have already attempted limited constraints on the form of the spectrum and @Parkinson have conducted a model selection study to ascertain the necessity of a tilt in the spectrum with the new data. In this paper we extend both studies to a suite of models covering a wide variety of possibilities based on both physical and observational grounds. We use a fully Bayesian approach to determine the model parameters and comparative evidence to ascertain which model the data actually prefers.
Our previous paper [@Bridges] \[Bridges06\] used WMAP 1-year data (WMAP1; @WMAP1) to constrain the same set of models. These generalisations were motivated principally by observations of a decrement in power on large-scales from WMAP1 and a tilting spectrum on small-scales from high resolution experiments such as the Arcminute Cosmology Bolometer Array (ACBAR; @ACBAR), the Very Small Array (VSA ; @VSA) and the Cosmic Background Imager (CBI; @CBI). With two more years observing time and improved treatment of systematic errors the decrement in power on large scales is now somewhat reduced, yet still evident in WMAP3 and is now constrained almost to the cosmic variance limit while the tilting spectrum on small-scales is now seen even without the aid of high-resolution small scale experiments, due to tighter constraints on the second acoustic peak. On physical grounds we test a broken spectrum caused perhaps by double field inflation [@barriga] and a spectrum predicted by @Doran (L-D) naturally incorporating an exponential cutoff in power on large scales by considering the evolution of closed universes out of a big bang singularity, with a novel boundary condition that restricts the total conformal time available in the universe. We also aim to reconstruct the spectrum in a number of bins in wavenumber $k$.
Model Selection Framework
=========================
Bayesian model selection is now well established within the community as a reliable means of appropriately determining the most efficient parameterisation for a model, penalising any unecessary complication (@Jaffe, @Drell, @John, @Bridle, @McLachlan, @Slosar, @Saini, @Marshall, @Niarchou, @Basset, @Mukherjee, @Trotta, @Beltran, Bridges06). Recently much progress has been made in improving the speed and accuracy of evidence results [@Parkinson] by implementing the method of @Skilling known as *nested sampling*. In this paper we will employ our own implementation of this method [@Shaw] to evaluate the evidence and use standard Markov Chain Monte Carlo (MCMC) to make parameter constraints.
Markov Chain Monte Carlo sampling {#MCMCsec}
---------------------------------
A Bayesian analysis provides a coherent approach to estimating the values of the parameters, $\mathbf{\Theta}$, and their errors and a method for determining which model, $M$, best describes the data, $\mathbf{D}$. Bayes theorem states that $$P(\mathbf{\Theta}|\mathbf{D}, M) =
\frac{P(\mathbf{D}|\mathbf{\Theta},
M)P(\mathbf{\Theta}|M)}{P(\mathbf{D}|M)},$$ where $P(\mathbf{\Theta}|\mathbf{D}, M)$ is the posterior, $P(\mathbf{D}|\mathbf{\Theta}, M)$ the likelihood, $P(\mathbf{\Theta}|M)$ the prior, and $P(\mathbf{D}|M)$ the Bayesian evidence. Conventionally, the result of a Bayesian parameter estimation is the posterior probability distribution given by the product of the likelihood and prior. In addition however, the posterior distribution may be used to evaluate the Bayesian evidence for the model under consideration.
We will employ a MCMC sampling procedure to explore the posterior distribution using an adapted version of the [cosmoMC]{} package [@cosmomc] with four CMB datasets; WMAP3, ACBAR the VSA and CBI. We also include the 2dF Galaxy Redshift Survey [@2dF], the Sloan Digital Sky Survey [@sloan] and the Hubble Space Telescope (HST) key project [@HST]. These set of experiments comprise dataset I. Additionally we include two datasets which cover different scales and probe independent sources. The Ly-$\alpha$ forest (@McDonaldI; @McDonaldII) (which combined with dataset I makes up dataset II) comprises cosmological absorption by neutral hydrogen observed in quasar spectra in the intergalactic medium. It probes fluctuation scales that are small ($\sim$ Mpc) in comparison to the other datasets used at redshifts between 2-4 so that primordial information has not been erased by non-linear evolution. It thus provides a very useful complementary observation when constraining the form of the primordial spectrum. Previous authors (@Viel; @Seljak) have already examined this dataset in conjunction with WMAP3 and others and found that most of the interesting results observed by @SpergelII, namely lowered scalar amplitude and a non-vanishing running index can be removed when Ly-$\alpha$ is included. Observations of luminous red galaxies (LRG) @TegmarkII (when combined with dataset I becomes dataset III) consists of $>$ 46,000 galaxies taken from the full SDSS catalogue which represent a highly uniform galaxy sample containing only luminous early-types over the entire redshift range studied constituting an excellent tracer of large scale structure. @TegmarkII recently detected baryonic acoustic oscillations in the matter power spectrum extracted from this dataset, providing a welcome confirmation of early universe physics in large scale structure data.
In addition to the primordial spectrum parameters, we parameterise each model using the following five cosmological parameters; the physical baryon density $\Omega_b
h^2$; the physical cold dark matter density $\Omega_{c} h^2$; the curvature density $\Omega_k$; the Hubble parameter $h$ ($H_0 = h \times100 \mbox{kms}^{-1}$) and the redshift of re-ionisation $z_{re}$.
Bayesian evidence and nested sampling {#bayes}
-------------------------------------
The Bayesian evidence is the average likelihood over the entire prior parameter space of the model: $$\int\int{\mathcal{L}(\Theta_{\rm{C}},\Theta_{\rm{B}})P(\Theta_{\rm{C}})P(\Theta_{\rm{B}})}d^N\Theta_{\rm{C}} d^M\Theta_{\rm{B}},
\label{equation:evidence}$$ where $N$ and $M$ are the number of cosmological and Bianchi parameters respectively. Those models having large areas of prior parameter space with high likelihoods will produce high evidence values and *vice versa*. This effectively penalises models with excessively large parameter spaces, thus naturally incorporating Ockam’s razor.
The method of nested sampling is capable of much higher accuracy than previous methods such as thermodynamic integration (see e.g. @Beltran, Bridges06). due to a computationally more effecient mapping of the integral in Eqn. \[equation:evidence\] to a single dimension by a suitable re-parameterisation in terms of the prior *mass* $X$. This mass can be divided into elements $dX = \pi(\mathbf{\Theta})d^N \mathbf{\Theta}$ which can be combined in any order to give say $$X(\lambda) = \int_{\mathcal{L\left(\mathbf{\Theta}\right) > \lambda}} \pi(\mathbf{\Theta}) d^N
\mathbf{\Theta},$$ the prior mass covering all likelihoods above the iso-likelihood curve $\mathcal{L} = \lambda$. We also require the function $\mathcal{L}(X)$ to be a singular decreasing function (which is trivially satisfied for most posteriors) so that using sampled points we can estimate the evidence via the integral: $$\mathcal{Z}=\int_0^1{\mathcal{L}(X)}dX.
\label{equation:nested}$$ Via this method we can obtain evidences with an accuracy 10 times higher than previous methods for the same number of likelihood evaluations.
A standard scenario in Bayesian model selection would require the computation of evidences for two models A and B. The difference of log-evidences $\ln
\mathcal{Z}_A - \ln \mathcal{Z}_B$, also called the Bayes factor then quantifies how well A may fit the data when compared with model B. @Jeffreys provides a scale on which we can make qualitative conclusions based on this difference: $\Delta\mbox{ln} \mathcal{Z} < 1$ is not significant, $1 < \Delta\mbox{ln} \mathcal{Z} < 2.5$ significant, $2.5 < \Delta\mbox{ln}
\mathcal{Z} < 5$ strong and $\Delta\mbox{ln} \mathcal{Z} > 5$ decisive.
Primordial Power Spectrum Parameterisation
==========================================
H-Z, power-law and running spectra {#h_z}
----------------------------------
The early Universe as observed in the CMB is highly homogeneous on large scales suggesting that any primordial spectrum of density fluctuations should be close to scale invariant. The H-Z spectrum is described by an amplitude $A$ for which we assume a uniform prior of $[15,55]\times 10^{-8}$. Slow-roll inflation, given an exponential potential, predicts a slightly ‘tilted’ power-law spectrum parameterised as: $$P(k)=A \left(\frac{k}{k_0}\right)^{n-1},
\label{single_index}$$ where the spectral index should be close to unity; we assume a uniform prior on $n$ of $[0.5,1.5]$; $k_0$ is the pivot scale (set to 0.05 Mpc$^{-1}$) of which $n$ and the amplitude $A$ are functions. For a generic inflationary potential we should also account for any scale dependence of $n(k)$ called *running*, so that to first order: $$P(k) = A
\left(\frac{k}{k_0}\right)^{n-1+(1/2)\ln (k/k_0)(dn/d\ln k)},$$ where $dn/d\ln k$ is the running parameter $n_{run}$; for which we assume a uniform prior of $[-0.15,0.15]$.
The inclusion of WMAP3 in dataset I now places impressively tight constraints on all three models (see Fig. \[figure1\]). The unexpected reduction in the value of the optical depth to reionisation, to $\tau \sim 0.09$ has had the effect of reducing the overall amplitude of the power spectrum, due to the well known $\tau$-$A$ degeneracy. This effect is noticeable in all cases but most particularly so in the H-Z spectrum in Fig. \[figure1\] (c). For the first time the single spectral index model now exhibits a constraint, to $1\sigma$, of $n_s=0.95 \pm 0.02$ (see Fig. \[figure1\] (b)) excluding the possibility of a scale-invariant spectrum at this confidence level. Furthermore, a pure power-law (with $n_{run} = 0$) is also excluded at the $1\sigma$ level with $n_{run} = -0.038 \pm 0.030$ (Fig. \[figure1\] (a)).
\
\
The addition of Ly-$\alpha$ data further increases constraints on all spectral parameters (see Fig. \[figure2\]), particularly so on spectral running. While a scale invariant spectrum is also ruled out with this dataset ($n_s = 0.96 \pm 0.02$) a running spectrum is not preferred, with a constraint on $n_{run} = 0.015 \pm 0.015$ representing a doubling in accuracy over dataset I. A further tension exists between the amplitude of fluctuations as found using dataset I alone and when combined with Ly-$\alpha$, the latter preferring a much larger value of $\sigma_8$ and thus higher scalar fluctuation amplitude. [@Seljak] estimates this deviation to be at the $2\sigma$ level and treat it as a normal statistical fluctuation and *not* a sign of some unaccounted systematic flaw in either dataset. Since two independent analyses of WMAP3 + Ly-$\alpha$ and WMAP3 + LRG both suggest no significant running, a conservative conclusion is that the running observed with dataset I is simply a statistical anomaly albeit at close to 2$\sigma$.
\
\
Large scale cutoff
------------------
Confirmation by WMAP3 of the large-scale decrement in power at $\ell \sim 2$, close to the cosmic variance limit, supports the possibility of a spectrum with some form of cutoff. We reexamine the case of a sharp cutoff as did the WMAP team, parameterising the scale at which the power drops to zero, $k_c$ with a choice of prior $[0.0,0.0006]$ Mpc$^{-1}$ which was made to limit the study to regions up to $\ell \sim 6$ at which point an appreciable cutoff is no longer observed. $$P(k) = \left\{ \begin{array}{ll}
0,& \mbox{$k < k_c$}\\
A \left(\frac{k}{k_0} \right)^{n-1},& \mbox{$k \geq k_c$}\end{array}\right.$$ Although this model $P(k)$ is not continuous, as the physical spectrum should be, it does give an upper limit on the average cutoff scale, which is useful when comparing for instance the L-D spectrum which does predict a form for the cutoff.
On small scales this spectrum behaves just as the single-index power law and so constraints on $A$ and $n$ are similar (see Fig. \[figure3\]). Our WMAP1 constraint on $k_c$ showed a non-zero likelihood for a cutoff at $k=0$ i.e. no cutoff. This likelihood is marginally lower with WMAP3 presumably due to the higher amplitude of the $\ell = 3$ multipole which was more heavily suppressed in WMAP1 observations. The peak in the likelihood encouragingly remains at around $k_c=2.8\times 10^{-4}$ Mpc$^{-1}$. The scale of this cutoff is much larger than anything probed by either Ly-$\alpha$ (in dataset II) or LRG (in dataset III), so neither dataset improves on this scale constraint (see Fig. \[cutoff\_lya\_lrg\_figure\]).
Broken Spectrum
---------------
Multiple field inflation would produce a symmetry breaking phase transition in the early universe causing the mass of the inflaton field to change suddenly, momentarily violating the slow-roll conditions [@Adams]. The resultant primordial spectrum would be roughly scale invariant initially, followed by a sudden break lasting roughly 1 $e$-fold before returning to scale invariance. Because the slow-roll conditions are violated it is not trivial to calculate the form of the break, however a robust expectation is that it will be sharp as the field undergoing the phase transition evolves exponentially fast to its minimum [@barriga]. We parameterise the spectrum as: $$P(k) = \left\{ \begin{array}{ll}
A,& \mbox{$k \leq k_s$}\\
Ck^{\alpha-1}, & \mbox{$k_s < k \leq k_e$}\\
B, & \mbox{$k > k_e$} \end{array}\right.,$$ where the values of $C$ and $\alpha$ are chosen to ensure continuity. Four power spectrum parameters were varied in this model: the ratio of amplitudes before and after the break $A/B$ with prior $[0.3,7.2]$; $k_s$ indicating the start of the break with prior $[0.01,0.1]$ Mpc$^{-1}$; $\ln(k_e/k_s)$ to constrain the length of the break with prior $[0,4]$ and normalisation $A$ with prior $[14.9,54.6]\times 10^{-8}$. These represent a very conservative set of priors that allow the model a large degree of freedom in both the position of the break (which could occur anywhere from $k \sim 0.01$, well above any possible large scale cutoff) and the form, which could be extended, so as to mimic a tilted spectrum or occur as a sharp drop. In addition a prior that $k_e$ could not exceed 0.1 Mpc$^{-1}$ was imposed so that only a region well covered by the datasets used was explored.
In this parameterisation a scale invariant spectrum would have an amplitude ratio $A/B = 1$ and a large value of $\ln(k_e/k_s)$. Neither our WMAP1 analysis nor dataset I suggest this to be a plausible explanation (see Fig. \[figure4\]) with a distinct drop in amplitude ($A/B \sim
1.2$) starting on scales below $k_s \sim 0.01$ Mpc$^{-1}$. The bimodal distribution in $\ln
(k_e/k_s)$ vs. $k_s$ observed with WMAP1 is preserved, though less pronounced, with dataset I. This implies a preference for both a sharp break at a single scale of $k\approx 0.04$ Mpc$^{-1}$ and an extended break begining at $k\approx 0.01$. A drop in power is clearly a crude way of approximating a tilted spectrum, so the extended break was an expected result. The sharp break, could be indicative of a some early universe physics or it could simply be as @Recon suggest in their WMAP1 analysis, an artifact of the inclusion of the large scale structure datasets. The transition scale lies close to the point at which large scale structure data becomes statistically significant. Below this scale the WMAP data would disfavour a break and above it large scale structure data would. The inclusion of both Ly-$\alpha$ and LRG (see Fig. \[figure5\]) data shifts the start of the break to larger scales (most pronounced with LRG), but simultaneously lowers the amplitude of the break, thus producing a more gradual extended slope, providing a better approximation to a tilted spectrum. Thus, our results do not conclusively suggest that a break does exist in the primordial spectrum, rather they seem only to confirm the need for a spectral tilt.
\
\
\
\
Power spectrum reconstruction
-----------------------------
Many previous attempts have been made to reconstruct the primordial spectrum directly from data: @Wang; @Recon; @Souradeep; @SouradeepII; @Silk; @Steen; Bridges06. Most recently the WMAP team [@SpergelII] attempted to reconstruct it using a set of amplitude bins in $k$ using WMAP3 data alone. The method suffers from the natural side effect of imposing correlations between neighbouring bins and broadening other constraints. Therefore searching for features by this method is unreliable and difficult. However from a model selection viewpoint it does allow full freedom for the data to decide how many parameters are required of the model. Our parameterisation linearly interpolates between eight amplitude bins $a_n$ in $k$ on large scales between 0.0001 and 0.11 Mpc$^{-1}$ parameterised logarithmically with $k_{i+1} = 2.75k_i$ so that $$P(k) = \left\{ \begin{array}{ll} \frac{(k_{i+1} - k)a_i + (k-k_i)a_{i+1}}{k_{i+1}-k_i},&
\mbox{$k_i <k < k_{i+1}$}\\ a_n,& \mbox{$k \geq k_n$}\end{array}\right.$$ As expected an obvious tilt is discernible in our dataset I reconstruction (Fig. \[figure6\]) the mean bin amplitudes deviating significantly from the best fit scale invariant spectrum at high $k$, not observable in our WMAP1 analysis. Though, within 1$\sigma$ limits a H-Z spectrum can still be fitted to both sets of data, however it would require a lower amplitude to accommodate the values at large $k$. No cutoff is observed as was hinted at in our WMAP1 analysis; however uncertainty at this scale is dominated by high cosmic-variance, making constraints inherently difficult. We find the addition of LRG data in dataset III provides little further constraint beyond that obtained from dataset I alone (see Fig. \[figure7\]). The effect of including Ly-$\alpha$ data in dataset II is however, marked. Firstly the overall amplitude is lifted, owing to the larger $\sigma_8$ required by Ly-$\alpha$. Furthermore little obvious tilt is discernible in full agreement with our analysis in Sec. \[h\_z\]. Encouragingly similar features are seen in all three spectra such as the peaks at $0.003$ Mpc$^{-1}$ and $0.05$ Mpc$^{-1}$.
![Reconstruction of the primordial power spectrum in 8 bands of $k$ (with $1\sigma$ errors) for dataset I (red) and our WMAP1 (Bridges06) analysis (yellow) compared with the best fit 1-year (dotted) and 3-year (filled) H-Z spectrum.[]{data-label="figure6"}](wmap3_binned.ps){width="0.9\linewidth"}
Closed universe Inflation {#doran}
-------------------------
@Doran arrived at a novel model spectrum by considering a boundary condition that restricts the total conformal time available in the Universe, and requires a closed geometry. The resultant predicted perturbation spectrum encouragingly contains an exponential cutoff (as previously suggested phenomenologically by @efstathiou) at low $k$ that yields a corresponding deficit in power in the CMB power spectrum. The shape of the derived spectrum for a cosmology defined by $\Omega_0 = 1.04, \Omega_b h^2 =
0.0224, h = 0.6, \Omega_{cdm} h^2 = 0.110$ was parameterised by the function: $$P(k) = A (1-0.023y)^2(1-\exp(-(y+0.93)/0.47))^2,$$ where $y=\ln\left(\frac{k}{H_0/100}\times 3 \times 10^3\right)> -0.93$. However we do know this form to be fairly stable to changes in cosmology, particularly in the position of the cutoff. We therefore analyse this spectrum using dataset I, varying only the amplitude within the best fit ‘concordance’ cosmology found for the single index model in Sec. \[h\_z\]. As expected the amplitude is reduced when compared with our WMAP1 analysis ($A = 27.1 \pm 0.2$ Mpc$^{-1}$) due to the revised WMAP3 value for $\tau$. At small scales the spectrum then lies roughly in line with a tilted spectrum with $n_s \sim 0.96$ (see Fig. \[figure8\]). For comparison the plot also shows the other best fit spectra (including the bimodality in the broken spectrum).
![L-D spectrum (solid-red) shown with the WMAP3 best fitting H-Z (solid-black), single-index with a cutoff (dotted-orange), without (dotted-blue), a running index (dashed-pink) and broken (dotted-black).[]{data-label="figure8"}](wmap3_model_comparison_arrow.eps){width="0.9\linewidth"}
Model Selection
---------------
We shall now turn to the fundamental inference. Which of the models considered best describes the current data? As with our WMAP1 analysis we perform the analysis in two stages to accommodate the L-D spectrum: the first, within the full parameter space including all models bar L-D, while the second was carried out within the fixed, best-fit single-index cosmology (as determined using dataset I) including L-D. While this artificial division may seem flawed it should be remembered that within this particular fixed cosmology the only spectrum that is favoured is that of the single-index model itself, any other models should only be penalised. We chose two sets of priors (see Table \[WMAP3\_table1\]) on $n_s$ and $n_{run}$, the wider of which provides a very conservative range given the constraints available from current data. The results have been normalised to the single-index power law spectrum with wide priors.
--------------------------- ----------------
$A_s(\times 10^{-8})$ \[15,55\]
$n_s$ (wide priors) \[0.5,1.5\]
$n_s$ (narrow priors) \[0.8,1.2\]
$n_{run}$ (wide priors) $[-0.15,0.15]$
$n_{run}$ (narrow priors) $[-0.1,0.05]$
--------------------------- ----------------
: Priors placed on the principal primordial spectral parameters.
\[WMAP3\_table1\]
### Full cosmological parameter space exploration {#full}
Our corresponding analysis of WMAP1 data was unable to make any conclusive model selection; with statistical uncertainty dominating results. With WMAP3 data and the method of nested sampling however we have arrived at a cusp in cosmological model selection. With dataset I, Table \[WMAP3\_table2\] confirms the disfavouring of a scale-invariant spectrum that we saw in the parameter constraints in Sec. \[h\_z\], though not at a decisive level according to the Jeffrey’s scale. As expected from the parameter constraints a running spectrum is preferred, but only at a significant level if narrow priors are chosen on $n_s$ and $n_{run}$. Large scale power is suppressed in WMAP3 as it was in WMAP1, with little improvement in uncertainty: both datasets are almost cosmic-variance limited on these scales so a cutoff is preferred by a significant margin of over 2 units in log-evidence. The evidence in favour of a broken spectrum is likely only due to its mimicking of a tilted spectrum as opposed to modelling of an abrupt break.
Inclusion of Ly-$\alpha$ in dataset II confirms the conclusions drawn in Section \[h\_z\], showing a significantly reduced evidence in favour of spectral running (with narrow priors), from a +1.2 with dataset I to -1.3, a significant evidence difference between the two datasets. Unsurprisingly all three data combinations exhibit significant evidences in favour of a spectral cutoff, due presumably in each case to the decrement in WMAP3 data alone.
**Model** datset I + Ly-$\alpha$ + LRG
---------------------------------------------- ------------------ ----------------- ------------------
H-Z $-0.7$ $\pm$ 0.3 $-0.1$$\pm$ 0.3 $-0.4$ $\pm$ 0.3
Single-Index (wide priors) +0.0 $\pm$ 0.3 +0.0$\pm$ 0.3 +0.0 $\pm$ 0.3
Single-Index (narrow priors) +1.0 $\pm$ 0.3 +0.2$\pm$ 0.3 +0.7 $\pm$ 0.3
Running (wide priors) $-2.9$ $\pm$ 0.3 $-1.6$$\pm$ 0.3 $-1.8$ $\pm$ 0.3
Running (narrow priors on $n_s$) +0.4 $\pm$ 0.3 $-0.7$$\pm$0.3 +1.7 $\pm$ 0.3
Running (narrow priors on $n_{run}$ & $n_s$) +1.2 $\pm$ 0.3 $-1.3$$\pm$ 0.3 +1.0 $\pm$ 0.3
Cutoff +2.3 $\pm$ 0.3 +1.7$\pm$ 0.3 +2.9 $\pm$ 0.3
Barriga +1.0 $\pm$ 0.3 +1.2$\pm$ 0.3 +0.9 $\pm$ 0.3
\[WMAP3\_table2\]
### Primordial parameter space exploration
The large preference in favour of a spectral cutoff and increasingly tight constraints on the universal geometry being marginally closed (a fact that is particularly reinforced when examining the LRG data @TegmarkII), suggests that the L-D spectrum may provide a very good fit to current data. For the remainder of this analysis we will examine the L-D spectrum using dataset I only.
As with the WMAP1 analysis, dataset I, also prefers the L-D spectrum, now by a slightly larger log-evidence (see Table \[WMAP3\_table3\].), which according to Jeffreys’ scale now constitutes a decisive model selection. In Bridges06 we concluded that the ‘significant’ model detection was due to the form of the spectrum on large scales i.e. its exponential cutoff. However on small scales it behaves much like a tilted spectrum with slight running, a form we now know fits the data very well. Could it be that these small scale features are driving this model selection?
To test this hypothesis we have analysed three *hybrid* spectra, where we have divided the L-D spectrum about $k\approx 0.008$ Mpc$^{-1}$ (denoted by the arrow in Fig. \[figure8\]) which corresponds loosely with the angular scale ($l=8$) where a cutoff ceases to be observed. We have then spliced both sections, about this point, with various single-index spectra at large or small scales. Model A combines a single-index spectrum below $k = 0.0008$ Mpc$^{-1}$ with L-D thereafter; Model B: an exponential cutoff from L-D with a fixed single-index model ($n= 0.94$) for $k>0.0008$ and Model C: as for B but with a *varying* single-index model.
The results (see Table \[WMAP3\_table3\]) bear out our assertion; models A and B have roughly identical log-evidence values to the original L-D, demonstrating the data to be essentially indifferent to the presence of a cutoff. In comparison, model C is typically just as good as a power law spectrum (i.e. an evidence close to 0). This suggests that the L-D spectrum is attractive to the data as it naturally incorporates a tilt without the need to parameterise it. However the tilt present in the L-D spectrum ($n_s \sim 0.96$) does coincide well with the best fit values obtained in Sec. \[h\_z\] for a power law spectrum (which are fairly invariant among the datasets I, II or III). This fact coupled with a significant evidence preference for a large scale spectral cutoff suggests the L-D spectrum does provide a uniquely good fit to current data.
**Model** WMAP1 (Bridges06) dataset I
-------------- ------------------- ----------------
Single-Index +0.0 $\pm$ 0.5 +0.0 $\pm$ 0.2
L-D +4.1 $\pm$ 0.5 +5.2 $\pm$ 0.2
L-D (A) – +5.0 $\pm$ 0.2
L-D (B) – +5.2 $\pm$ 0.2
L-D (C) – +0.9 $\pm$ 0.2
: Differences of log-evidences (primordial parameters only) for all models with respect to the single-index models preferred cosmology: $\Omega_0 = 1.035, \Omega_b h^2 = 0.0221, h = 0.58, \Omega_{cdm} h^2 = 0.112$, with our previous WMAP1 results for comparison.
\[WMAP3\_table3\]
Conclusions
===========
A scale-invariant spectrum is now largely disfavoured by the dataset I with a spectral index $n_s=0.95 \pm 0.02$ deviating by at least 2$\sigma$ from $n_s=0$. Moreover a running spectrum ($n_{run} = -0.038 \pm 0.030$) is now significantly preferred but only using the most constraining prior. The addition of Ly-$\alpha$ forest data improves all constraints but does not alter the preferred spectral tilt greatly. It does however, along with LRG data, suggest a significantly smaller running index ($n_{run} = -0.015 \pm 0.015$, $n_{run} = 0.01 \pm 0.05$). This tension has previously been analysed by @Seljak who conclude that such discrepancies, even though at the 2$\sigma$ level are consistent with normal statistical fluctuations between datasets. A power law spectrum with a cutoff provides the best evidence fit in our full parameter space study with a significant evidence ratio of roughly 2 units across all three datasets. The similarity of this cutoff model with the L-D spectrum suggests the latter should also provide a very good fit. This is indeed borne out with decisively large evidence ratios within our limited primordial-only analysis.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was conducted in cooperation with SGI/Intel utilising the Altix 3700 supercomputer (UK National Cosmology Supercomputer) at DAMTP Cambridge supported by HEFCE and PPARC. We thank S. Rankin and V. Treviso for their computational assistance. MB was supported by a Benefactors Scholarship at St. John’s College, Cambridge and an Isaac Newton Studentship.
[99]{}
Abazajian K., et al., 2003, Astrophys. J., 126, 2081 Adams J.A., Ross G.G. & Sarkar S., 1997, Nucl. Phys. B. 503, 405 Barriga J., Gaztanaga E., Santos M.G., Sarkar S., 2001, MNRAS, 324, 977 Basset B.A., Corasaniti P.S., Kunz, M., Astrophys. J. Lett., 2004, 617, L1 Beltran M., Garcia-Bellido J., Lesgourgues J., Liddle A., Slosar A., 2005, Phys. Rev. D, 71, 063532 Bennett C.L., et al., 2003, Astrophys. J. Suppl., 148, 1 Bridges M., Lasenby A.N., Hobson, M.P., 2006, MNRAS, 369, 1123 Bridle S., Lewis A., Weller J., Efstathiou G., 2003, MNRAS, 342, L72 Contaldi C.R., Peloso M., Kofman L., Linde A., 2003, J. Cosmol. Astropart. Phys., 7, 2 Dickinson C. et al., 2004, MNRAS, 353, 732 Drell P.S., Loredo T.J., Wasserman, I., 2000, Astrophys. J., 530, 593 Efstathiou G., 2003, MNRAS, 346, 26 Freedman W.L., et al., 2001, Astrophys. J., 553, 47 Hannestad S., 2004, J. Cosmol. Astropart. Phys., 4, 2 Hinshaw G. et.al., 2007, Astrophys. J. Suppl., 170, 288 Hobson M.P., Bridle S.L., Lahav O., 2002, MNRAS, 335, 377 Hobson M.P., McLachlan C., 2003, MNRAS, 338, 765 Jaffe A., 1996, Astrophys. J., 471, 24 Jeffreys H., 1961, *Theory of Probability*, 3rd ed., Oxford University Press John M.V., Narlikar J.V., 2002, Phys. Rev. D, 65, 043506 Kuo C.L. et al., 2004, Ap. J., 600, 32 Lasenby A.N., Doran, C., 2005, Phys.Rev. D 71, 063502 Lewis A. and Bridle S., 2002, Phys. Rev. D, 66, 103511 Marshall P.J., Hobson M.P., Slosar A., 2003, MNRAS, 346, 489
McDonald P., et al., 2006, Astrophys. J. Suppl., 163, 80-109 McDonald P., et al., 2005, Astrophys. J., 635, 761-783
Mukherjee, P. Parkinson D. Liddle, A., 2006, Astrophys. J., 638, L51-L54 Mukherjee P. & Wang Y., 2003, Astrophys. J., 593, 38 Niarchou A., Jaffe A., Pogosian L., 2004, Phys.Rev. D 69 063515 Percival W.J. et al., 2001, MNRAS, 327, 1297 Readhead A.C.S. et al., 2004, Astrophys. J.,, 609, 498–512 Parkinson D., Mukherjee P., Liddle A.R., 2006, Phys. Rev. D., 73, 123523 Saini T.D., Weller J., Bridle S.L., 2004, MNRAS, 348, 603
Seljak U., Slosar A., McDonald P., 2006, J. Cosmol. Astropart. Phys., 10, 14
Skilling J., 2004, Bayesian Inference and Maximum Entropy Methods in Science and Engineering, Ed. R. Fisher et al., American Inst. Phys. conf. proc., 735, 395 Slosar A. et al., 2003, MNRAS, 341, L29 Shafieloo, A., Souradeep, T., 2004, Phys. Rev. D, 70, 043523
Shaw J.R., Bridges M., Hobson M.P., 2007, MNRAS, 378, 1365-1370 Sinha, R., Souradeep, T., 2006, Phys. Rev. D., 74, 043518 Spergel D.N. et al., 2003, Astrophys. J. Suppl., 148, 175 Spergel D.N. et al., 2007, Astrophys. J. Suppl., 170, 377
Tegmark M. et al., 2006,Phys. Rev. D, 74, 123507
Tocchini-Valentini D., Douspis M., Silk J., 2005, MNRAS, 359, 31 Trotta R., 2007, MNRAS, 378, 72 Viel M., Haehnelt M.G., Lewis A., 2006, MNRAS, 370, L51
Wang Y., 1994, Phys. Rev. D, 50, 6135
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---
abstract: 'The self-duality equations on a Riemann surface arise as dimensional reduction of self-dual Yang-Mills equations. Hitchin had showed that the moduli space ${\mathcal M}$ of solutions of the self-duality equations on a compact Riemann surface of genus $g >1$ has a hyperKähler structure. In particular ${\mathcal M}$ is a symplectic manifold. In this paper we elaborate on one of the symplectic structures, the details of which is missing in Hitchin’s paper. Next we apply Quillen’s determinant line bundle construction to show that ${\mathcal M}$ admits a prequantum line bundle. The Quillen curvature is shown to be proportional to the symplectic form mentioned above. We do it in two ways, one of them is a bit unnatural (published in R.O.M.P.) and a second way which is more natural.'
author:
- Rukmini Dey
title: 'Geometric quantization of the moduli space of the Self-duality equations on a Riemann surface '
---
Keywords: Geometric quantization, Quillen determinant line bundle, moment map.
Introduction
============
Geometric prequantization is a construction of a Hilbert space, namely the square integrable sections of a prequantum line bundle on a symplectic manifold $({\mathcal M}, \Omega)$ and a correspondence between functions on ${\mathcal M}$ (the classical observables are functions on the phase space ${\mathcal M}$) and operators on the Hilbert space such that the Poisson bracket of two functions corresponds to the commutator of the operators. The latter is ensured by the fact that the curvature of the prequantum line bundle is precisely the symplectic form $\Omega$ [@Wo]. Let $f \in C^{\infty}({\mathcal M})$. Let $X_f$ be the vector field defined by $\Omega(X_f, \cdot) = - df$. Let $\theta$ be the symplectic potential corresponding to $\Omega$. Then we can define the operator corresponding to the function $f$ to be $\hat{f} = -i h[X_f - \frac{i}{h}\theta(X_f)] + f$. Then if $f_1, f_2 \in C^{\infty}({\mathcal M})$ and $f_3 = \{ f_1, f_2 \}$, Poisson bracket of the two, then $[\hat{f}_1, \hat{f}_2] = -i h \hat{f}_3$.
A relevant example would be geometric quantization of the moduli space of flat connections. The moduli space of flat connections of a principal $G$-bundle on a Riemann surface has been quantized by Witten by a construction of the determinant line bundle of the Cauchy-Riemann operator, namely, ${\mathcal L}
= \wedge^{\rm{top}} ({\rm Ker} \bar{\partial}_A)^{*} \otimes \wedge^{\rm{top}}({\rm Coker}
\bar{\partial}_A)$, [@Wi], [@ADW]. It carries the Quillen metric such that the canonical unitary connection has a curvature form which coincides with the natural Kähler form on the moduli space of flat connections on vector bundles over the Riemann surface of a given rank [@Q].
Inspired by [@ADW], applying Quillen’s determinant line bundle construction we construct prequantum line bundles on the moduli space of solutions to the vortex equations [@D1] and the moduli space the self-duality equations over a Riemann surface which is a hyperKähler manifold [@H]. In this paper we quantize one of the symplectic forms in two ways. In [@D] we show the quantization of the full hyperKähler structure.
The self-duality equations on a Riemann surface arise from dimensional reduction of self-dual Yang-Mills equations from $4$ to $2$ dimensions [@H]. They have been studied extensively in [@H]. They are as follows. Let $M$ be a compact Riemann surface of genus $g>1$ and let $P$ be a principal unitary $U(n)$-bundle over $M$. Let $A$ be a unitary connection on $P$, i.e. $A = A^{(1,0)} + A^{(0,1)}$ such that $A^{(1,0)} = -A^{(0,1)*}$, where $*$ denotes conjugate transpose [@GH], [@K]. Let $\Phi$ be a complex Higgs field, $\Phi \in {\mathcal H} = \Omega^{1,0}(M; {\rm ad} P \otimes {{\ensuremath{\mathbb{C}}}})$. The pair $(A, \Phi)$ will be said to satisfy the self-duality equations if $$(1)\rm{\;\;\;\;\;}F = -[\Phi, \Phi^*],$$ $$(2)\rm{\;\;\;\;\;}d^{\prime\prime}_A \Phi = 0.$$ Here $\Phi^* = \phi^* d \bar{z}$ where $\phi^*$ is taking conjugate transpose of the matrix of $\phi$. Let the solution space to $(1)-(2)$ be denoted by $S$. There is a gauge group acting on the space of $(A, \Phi)$ which leave the equations invariant. If $g$ is an $U(n)$ gauge transformation then $(A_1, \Phi_1)$ and $(A_2, \Phi_2)$ are gauge equivalent if $d_{A_2} g = g d_{A_1}$ and $\Phi_2 g = g \Phi_1$ [@H], page 69. Taking the quotient by the gauge group of the solution space to $(1)$ and $(2)$ gives the moduli space of solutions to these equations and is denoted by ${\mathcal M}$. Hitchin shows that there is a natural metric on the moduli space ${\mathcal M}$ and further proves that the metric is hyperKähler [@H].
In the next section, we will elaborate on the natural metric and one of the symplectic forms, explicit mention of which is missing in [@H]. In the section after that we will construct a determinant line bundle on the moduli space ${\mathcal M}$ such that its Quillen curvature is precisely this symplectic form. This will put us in the context of geometric quantization.
We will do this in two different ways – the first one using $\overline{\partial} + \overline{A_0^{(1,0)}}
+ \overline{\Phi^{(1,0)}}$ which gauge transforms like $\bar{g} (\overline{\partial} + \overline{A_0^{(1,0)}}
+ \overline{\Phi^{(1,0)}}\bar{g}^{-1}.$ This is a bit unnatural since the action is with $\bar{g}$. (This approach was published in R.O.M.P.)
In the next section we will do it using $\bar{\partial} + A_0^{(0,1)} + \Phi^{(0,1)}$ which gauge transforms like $g(\bar{\partial} + A_0^{(0,1)} + \Phi^{(0,1)})g^{-1}$ which is more natural.
In the end we discuss the holomorphicity of the prequantum line bundle w.r..t the first complex structure.
In the paper where we construct prequantum line bundles for the full hyperKähler structure of the moduli space [@D] we mention the second approach to quantizing the first symplectic form.
Papers which may be of interest in this context are [@BDr], [@BDr1], [@H1], [@S] [@Wi1]. These papers use algebraic geometry and algebraic topology and may provide alternative methods to quantizing the hyperKähler system. Our method, in contrast, is very elementary and we explictly construct the prequantum line bundles. The only machinery we use is Quillen’s construction of the determinant line bundle, [@Q].
Note: After writing the paper the author found Kapustin and Witten’s paper [@KW] where they have applied Beilinson and Drinfeld’s quantization of the Hitchin system to study the geometric Langlands programme.
Symplectic structure of the moduli space
========================================
This section is an elaboration of what is implicit in [@H]. Following the ideas in [@H] we give a proof that $\Omega$ below is a symplectic form on ${\mathcal M}$.
Let the configuration space be defined as ${\mathcal C} = \{ (A, \Phi)|
A \in {\mathcal A}, \Phi \in {\mathcal H} \}$ where ${\mathcal A}$ is the space of unitary connections on $P$ and ${\mathcal H} = \Omega^{(1,0)}(M, {\rm ad} P \otimes {{\ensuremath{\mathbb{C}}}})$ is the space of Higgs field. Unitary connections satisfy $A = A^{(1,0)} + A^{(0,1)}$ where $A^{(1,0)*} = -A^{(0,1)}$, where $*$ is conjugate transpose.
Let us define a metric on the complex configuration space $$g ((\alpha, \gamma^{(1,0)}), (\beta, \delta^{(1,0)})) =
- {\rm Tr} \int_M ( \alpha \wedge *_1 \beta)
-2 {\rm Im} {\rm Tr} \int_M ( \gamma^{(1,0)} \wedge *_2 \delta^{(1,0){\rm tr}})$$ where $\alpha, \beta \in T_A {\mathcal A} = \{ \alpha \in \Omega^1(M, {\rm ad} P) |
\alpha^{(1,0)*} = - \alpha^{(0,1)} \}$ and $\gamma^{(1,0)}, \delta^{(1,0)} \in T_{\Phi}{\mathcal H} =
\Omega^{1,0}(M; {\rm ad} P \otimes {{\ensuremath{\mathbb{C}}}}).$ Here the superscript ${\rm tr} $ stands for tranpose in the Lie algebra of $U(n)$, $*_1$ denotes the Hodge star taking $dx$ forms to $dy$ forms and $dy$ forms to $-dx$ forms (i.e. $*_1 (\eta dz) = -i \eta dz$ and $*_1(\eta d \bar{z}) = i \eta d \bar{z}$) and $*_2$ denotes the operation (another Hodge star), such that $*_2(\eta dz) = \bar{\eta} d \bar{z}$ and $*_2(\eta d \bar{z}) = -\bar{\eta} d z$.
We check that this coincides with the metric on the moduli space ${\mathcal M}$ given by [@H], page 79 and page 88. Hitchin identifies $A$ with its $A^{(0,1)}$ part. (Since it is a unitary connection, $A^{(1,0)}$ is determined by $A^{(0,1)}$ by the formula $A^{(1,0)} =- A^{(0,1)*}$). This is different from our point of view where in the connection part we keep both $A^{(1,0)}$ and $A^{(0,1)}$ even though they are related. On $ T_{(A, \Phi)} {\mathcal C}
= T_A {\mathcal A} \times T_{\Phi} {\mathcal H}$ which is $\Omega^{(0,1)}(M, {\rm ad} P \otimes {{\ensuremath{\mathbb{C}}}}) \times
\Omega^{(1,0)}(M, {\rm ad} P \otimes {{\ensuremath{\mathbb{C}}}})$ for him, Hitchin defines a metric $g_1$ such that $g_1((\alpha^{(0,1)},\gamma^{(1,0)}), (\alpha^{(0,1)}, \gamma^{(1,0)}))
= 2i {\rm Tr} \int_M (\alpha^{(0,1)*} \wedge \alpha^{(0,1)})
+ 2i {\rm Tr} \int_M (\gamma^{(1,0)} \wedge \gamma^{(1,0)*})$. $*$ denotes conjugate transpose as usual. Let $ \gamma^{(1,0)}= c dz$, where $c$ is a matrix.
On $T_{(A, \Phi)} {\mathcal C}$, our metric $$\begin{aligned}
& & g((\alpha, \gamma^{(1,0)}), (\alpha, \gamma^{(1,0)})) \\
&=& - {\rm Tr} \int_M (\alpha \wedge *_1 \alpha)
- 2 {\rm Im} {\rm Tr} \int_M ( \gamma^{(1,0)} \wedge *_2 \gamma^{(1,0){\rm tr}})\\
&=& - {\rm Tr} \int_M (\alpha^{(1,0)} + \alpha^{(0,1)})
\wedge (-i \alpha^{(1,0)} + i \alpha^{(0,1)}) \\
& & -2 {\rm Im} {\rm Tr} \int_M (c dz \wedge c^* d \bar{z}) \\
&=& 2i {\rm Tr} \int_M (\alpha^{(0,1)} \wedge \alpha^{(1,0)})
-2 Im \int_M (-2i){\rm Tr} (c c^*) dx \wedge dy \\
&=& -2i {\rm Tr} \int_M (\alpha^{(0,1)} \wedge \alpha^{(0,1)*})
+ 4 \int_M Re ({\rm Tr} (c c^*)) dx \wedge dy \\
&=& 2i {\rm Tr} \int_M (\alpha^{(0,1)*} \wedge \alpha^{(0,1)})
+ 2i \int_M (-2i) {\rm Tr} (c c^*) . dx \wedge dy \\
&=& 2i {\rm Tr} \int_M (\alpha^{(0,1)*} \wedge \alpha^{(0,1)})
+ 2i \int_M {\rm Tr} (c c^*) dz \wedge d \bar{z} \\
&=& 2i {\rm Tr} \int_M (\alpha^{(0,1)*} \wedge \alpha^{(0,1)})
+ 2i {\rm Tr} \int_M \gamma^{(1,0)} \wedge \gamma^{(1,0)*} \\\end{aligned}$$ where we have used the fact that $\alpha^{(1,0)} = -\alpha^{(0,1)*}$ and that ${\rm Tr} (cc^*)$ is real. Thus we get the same metric as Hitchin does.
The symmetry as $\alpha$ is interchanged with $\beta$ and $\gamma$ is interchanged with $\delta$ is as follows. In the first term $\alpha \wedge *_1 \beta = (\alpha^{(1,0)} + \alpha^{(0,1)})
\wedge ( *_1 \beta^{(1,0)} + *_1 \beta^{(0,1)})
= i (\alpha^{(1,0)} \wedge \beta^{(0,1)}) - i (\alpha^{(0,1)}
\wedge \beta^{(1,0)})$. It is easy to check that $\beta \wedge *_1 \alpha$ is also exactly the same. In the second term, first note that ${\rm Re} {\rm Tr} (AB^*) = {\rm Re} {\rm Tr} (BA^*)$ for matrices $A, B$. But we have matrix valued one forms. Thus if $\gamma^{(1,0)} = A d z, \delta^{(1,0)} = B d z$, $A,B$ are matrices, then $*_2 \delta^{(1,0){\rm tr}} = B^* d \bar{z}$. Then the second term in the metric is the integral of $-2 {\rm Im} {\rm Tr} (A dz \wedge B^* d \bar{z}) =
-2{\rm Im} ({\rm Tr} (AB^*) dz \wedge d \bar{z}) = -2 {\rm Im} ({\rm Tr} (AB^*) . (-2i) dx \wedge dy) = -2 . {\rm Re} ({\rm Tr} (AB^*)). (-2) dx \wedge dy $ Interchanging $\gamma$ with $\delta$ amounts to interchanging $A$ and $B$ which doesnot change the term.
There is an almost complex structure on ${\mathcal C}$, namely, ${\mathcal I} = \left[ \begin{array}{cc}
*_1 & 0 \\
0 & i
\end{array} \right]. $ Thus ${\mathcal I}(\beta, \delta^{(0,1)}) =
(*_1 \beta, i \delta^{(0,1)})$.
(If one identifies $T_{A} {\mathcal A}$ with $\Omega^{(0,1)} (M, {\rm ad }P \otimes {{\ensuremath{\mathbb{C}}}})$ as Hitchin does, then ${\mathcal I} = \left[ \begin{array}{cc}
i & 0 \\
0 & i
\end{array} \right]$ i.e., ${\mathcal I}(\beta^{(0,1)}, \delta^{(0,1)}) =
(i\beta^{(0,1)}, i \delta^{(0,1)})$. This is the viewpoint we will take in a paper sequel to this one. The metric and the symplectic forms will remain the same.)
We can define a symplectic form $$(3)\rm{\;\;\;}\Omega ((\alpha, \gamma^{(1,0)}), (\beta, \delta^{(1,0)}))
= g ((\alpha, \gamma^{(1,0)}), {\mathcal I}(\beta, \delta^{(1,0)}))$$ Then $$\begin{aligned}
\Omega ((\alpha, \gamma^{(1,0)}), (\beta, \delta^{(1,0)}))
&=& {\rm Tr} \int_M (\alpha \wedge \beta)
-2 {\rm Im} {\rm Tr} \int_M (\gamma^{(1,0)} \wedge *_2(i\delta^{(1,0){\rm tr}})) \\
&=& {\rm Tr} \int_M (\alpha \wedge \beta)
-2 {\rm Im} {\rm Tr} \int_M ((-i) \gamma^{(1,0)} \wedge *_2(\delta^{(1,0){\rm tr}})) \\
&=& {\rm Tr} \int_M (\alpha \wedge \beta)
+ 2 {\rm Re} {\rm Tr} \int_M (\gamma^{(1,0)} \wedge \delta^{(1,0)*}) \\
&=& {\rm Tr} \int_M (\alpha \wedge \beta)
- {\rm Tr} \int_M (\gamma \wedge \delta)\end{aligned}$$ This is because $*_1^2=-1$, $*_2(i \delta^{(1,0){\rm tr} })
= -i *_2 (\delta^{(1,0){\rm tr}})$, ${\rm Re} (-iz) = {\rm Im} z$ and finally $*_2(\delta^{(1,0){\rm tr}}) = \delta^{(1,0)*}$. Note that the first term of this symplectic form appears also in [@AB], page 587.
We will show by a moment map construction that this form descends to a symplectic form on the moduli space ${\mathcal M}$.
We need to find out the vector field generated by the action of the gauge group on ${\mathcal C}$ and hence on $S$ the solution space to $(1)-(2)$. Let $g = e^{ \epsilon \zeta}$, $\zeta \in u(n), $ the Lie algebra of the gauge group, i.e. $\zeta^* = - \zeta$. Under the action of the gauge group connection $A \rightarrow A_g = gA g^{-1} + g dg^{-1} $ and $\Phi \rightarrow \Phi_g = g \Phi g^{-1} $. (The action on ${\mathcal A}$ can be derived as follows: $(d + A_g) = g (d + A)g^{-1}$, by [@H]. Thus, $(d + A_g) s = g d(g^{-1}s)+ g A g^{-1} s
= ds + g d(g^{-1})s + g A g^{-1} s. $ Thus $A_g = g A g^{-1} + g d g^{-1}$ ). Taking $\epsilon$ to be very small, we write $g = 1+ \epsilon \zeta$ and $ g^{-1} = 1 - \epsilon \zeta$ upto first order in $\epsilon$. Thus $A \rightarrow A - \epsilon ( d \zeta - [\zeta, A])$ Similarly, $\Phi \rightarrow \Phi + \epsilon [ \zeta, \Phi] $. Thus on ${\mathcal C}$ there is a vector field generated by the gauge action given by $X_{\zeta} = (X_1, X_2) = (-( d \zeta - [\zeta, A]),[ \zeta, \Phi] )$.
Define a moment map $\mu: {\mathcal C} \rightarrow u(n)^*$ as follows: $$\mu(A, \Phi) = (F(A) + [\Phi, \Phi^*]).$$ Given $\zeta$ as before, define the Hamiltonian to be $$H_{\zeta} = {\rm Tr} \int_M (F(A) + [\Phi, \Phi^*])\zeta.$$ Now $F(A) = d A + A \wedge A$. Thus $F^{\prime} = lim_{t \rightarrow 0} \frac{F(A + t \beta ) - F(A)}{t} = d \beta + [\beta, A]$ where $\beta \in T_{A} {\mathcal A}$. (Note that for ${\rm ad} P$ valued $p$, $q$ forms $[\omega^{p}, \omega^{q}] =
(-1)^{pq+1}[\omega^q, \omega^p]$, [@AB], page 546.) Let $$h_{\zeta} = {\rm Tr} \int_M F(A) \zeta = {\rm Tr}
\int_M \zeta F(A)$$ (since ${\rm Tr} (AB)={\rm Tr} (BA)$). Now for $u, v, w \in \Omega^* (M, {\rm ad} P)$, $[u,v] \wedge w = u \wedge [v,w],$ [@AB], page 546. Thus $ {\rm Tr} (\zeta \wedge [\beta, A] )
= {\rm Tr} ([\zeta, \beta] \wedge A)
= {\rm Tr} (-A \wedge [\zeta, \beta])
= {\rm Tr} (-[A, \zeta] \wedge \beta)
= {\rm Tr}([\zeta, A] \wedge \beta).$ Thus if $\beta \in T_{A} {\mathcal A}= \Omega^{1}(M, {\rm ad} P),$ $$\begin{aligned}
d h_{\zeta}(\beta) &=& {\rm Tr} \int_M \zeta ( d\beta + [\beta, A]) \\
&=& {\rm Tr} \int_M (- d \zeta \wedge \beta + \zeta [\beta, A])\\
&=& {\rm Tr} \int_M (-(d \zeta - [\zeta, A]) \wedge \beta)\\
&=& {\rm Tr} \int_M (X_1 \wedge \beta). \rm{\;\;\;\;\;\;\;\;\;\;\;\;\;(4)}\end{aligned}$$
Let $$f_{\zeta}(\Phi) = {\rm Tr} \int_M ([\Phi, \Phi^*] \zeta),$$ and $\delta^{(1,0)} \in \Omega^{(1,0)}(M, {\rm ad} P \otimes {{{\ensuremath{\mathbb{C}}}}}),$ the tangent space to ${\mathcal H}$. Let $\delta^{(1,0)} = e dz$, $\Phi = \phi d z$, $\Phi^* = \phi^* d \bar{z}$, $\delta^{(1,0)*} = e^* d \bar{z}$, where $e$ and $\phi$ are matrices.
$$\begin{aligned}
d f_{\zeta}(\delta^{(1,0)}) &=& {\rm Tr} \int_M ([\delta^{(1,0)}, \Phi^*] \zeta
+ [ \Phi, \delta^{(1,0)*}] \zeta)\\
&=& 2 {\rm Re} {\rm Tr} \int_M ([\zeta, \Phi] \wedge \delta^{(1,0)*}) \\
&=& 2 {\rm Re} {\rm Tr} \int_M (X_2 \wedge \delta^{(1,0)*}. \rm{\;\;\;\;\;\;\;\;\;(5)}\end{aligned}$$
This follows from the fact that $$\begin{aligned}
2 {\rm Re} ({\rm Tr} [\zeta, \Phi] \wedge \delta^{(1,0)*}) &=& 2 {\rm Re} ({\rm Tr} ([\zeta, \phi] e^* )d z \wedge d {\bar z}) \\
&=& 2 {\rm Re} ({\rm Tr} [\zeta, \phi] e^* (-2i d x \wedge dy )) \\
&=& 4 {\rm Im} {\rm Tr} ([\zeta, \phi] e^*) d x \wedge dy \end{aligned}$$ Now, $$\begin{aligned}
{\rm Im} {\rm Tr} ([\zeta , \phi] e^*) &=&
{\rm Tr} ([\zeta,\phi] e^* - e [\zeta ,\phi]^*)/2i \\
&=& {\rm Tr}([\zeta ,\phi] e^* + e [\phi^*, \zeta])/2i \\
&=& {\rm Tr}([\phi, e^*]\zeta + [e, \phi^*]\zeta)/2i.\end{aligned}$$ Here we have used the fact that $\zeta^* = - \zeta$ since $\zeta \in u(n)$ and ${\rm Tr} ([A,B]C)= {\rm Tr} ([B,C]A)$.
Thus, $$\begin{aligned}
4 {\rm Im} {\rm Tr} ([\zeta, \phi] e^*) d x \wedge dy &=& {\rm Tr} ([\phi, e^*] \zeta + [e, \phi^*] \zeta) dz \wedge d \bar{z}\\
&=& {\rm Tr} ([\Phi, \delta^{(1,0)*}] \zeta + [\delta^{(1,0)}, \Phi^*] \zeta)\end{aligned}$$
Thus $(5)$ follows.
From $(4)$ and $(5)$ it follows that $$\begin{aligned}
dH_{\zeta} ((\beta, \delta^{(1,0)})) &=& dh_{\zeta}(\beta) + df_{\zeta}(\delta^{(1,0)})\\
&=& {\rm Tr} \int_M (X_1 \wedge \beta)
+ 2 {\rm Re} {\rm Tr} \int_M (X_2 \wedge \delta^{(1,0)*}) \\
&=& \Omega(X_{\zeta}, (\beta, \delta^{(1,0)}))\end{aligned}$$ Therefore, $$dH_{\zeta}(Y) = \Omega(X_{\zeta}, Y)$$ Thus the gauge group action on ${\mathcal C}$ is Hamiltonian and arises from a moment map.
$\Omega$ is a symplectic form on ${\mathcal M}$.
We saw that equation $(1)$ is a moment map $\mu = 0$.
About equation $(2)$ it is easy to check that ${\mathcal I}X$ satisfies the linearization of equation $(2)$ iff $X$ satisfies it. (This basically is as follows. Eq$(2)$ is $(\bar{\partial} + A^{(0,1)}) \Phi = 0$. Its linearization is $\bar{\partial} \gamma^{(1,0)} + \alpha^{(0,1)}
\wedge \Phi + A^{(0,1)} \wedge \gamma^{(1,0)} = 0$, with $X= (\alpha, \gamma^{(1,0)}) \in T_{(A, \Phi)} {\mathcal C}$ as before. ${\mathcal I} X = ((-i \alpha^{(1,0)}, i \alpha^{(0,1)}), i \gamma^{(1,0)})$. Since in ${\mathcal I} X$ both $\alpha^{(0,1)}$ and $\gamma^{(1,0)}$ just get multiplied by $i$, linearization of eq$(2)$ is satisfied by ${\mathcal I} X$ iff it is satisfied by $X$).
If $S$ be the solution space to equation $(1)$ and $(2)$ and $X \in T_p S$ then ${\mathcal I}X \in T_p S$ iff $X \in T_{p}S$ is orthogonal to the gauge orbit $O_p = G \cdot p$. The reason is as follows. We let $X_{\zeta} \in T_p O_p,$ $g( X_{\zeta}, X) = \Omega (X_{\zeta}, {\mathcal I} X) =
{\rm Tr} \int_M \zeta \cdot d \mu ({\mathcal I}X), $ and therefore ${\mathcal I}X$ satisfies the linearization of equation $(1)$ iff $d \mu ({\mathcal I}X) = 0$ iff $g(X_{\zeta}, X) = 0$ for all $\zeta$. Similarly, $g(X_{\zeta}, {\mathcal I}X) = - \Omega(X_{\zeta}, X)
= - {\rm Tr} \int_M \zeta d \mu(X) $. Thus $X \in T_p S$ implies ${\mathcal I}X $ is orthogonal to $X_{\zeta}$ for all $\zeta$. Thus $X \in T_p S$ and $X$ orthogonal to $X_{\zeta}$ implies that ${\mathcal I}X \in T_p S$ and ${\mathcal I}X$ is orthogonal to $X_{\zeta}$.
Now we are ready to show that ${\mathcal M} $ has a natural symplectic structure and an almost complex structure compatible with the symplectic form $\Omega $ and the metric $g$.
First we show that the almost complex structure descends to ${\mathcal M}$. Then using this and the symplectic quotient construction we will show that $\Omega$ gives a symplectic structure on ${\mathcal M}$. To show that ${\mathcal I}$ descends as an almost complex structure we let ${\rm pr}: S \rightarrow S/G =
{\mathcal M}$ be the projection map and set $[p] = {\rm pr} (p)$. Then we can naturally identify $T_{[p]} {\mathcal M} $ with the quotient space $T_p S / T_p O_p, $ where $ O_p = G
\cdot p $ is the gauge orbit. Using the metric $g$ on $S$ we can realize $T_{[p]} {\mathcal M}$ as a subspace in $T_p S$ orthogonal to $T_p O_p$. Then by what is said before, this subspace is invariant under ${\mathcal I}$. Thus $I_{[p]} ={\mathcal I} |_{T_p (O_p )^{\perp}}$, gives the desired almost complex structure. This construction does not depend on the choice of $p$ since ${\mathcal I}$ is $G$-invariant.
The symplectic structure $\Omega$ descends to $\mu^{-1}(0) / G$, (by what we said before and by the Marsden-Wienstein symplectic quotient construction [@GS], [@H]) since the leaves of the characteristic foliation are the gauge orbits. Now, as a $2$-form $\Omega$ descends to ${\mathcal M}$ and so does the metric $g$. We check that equation $(2)$ does not give rise to new degeneracy of $\Omega$ (i.e. the only degeneracy of $\Omega$ is due to $(1)$ but along gauge orbits). Thus $\Omega $ is symplectic on ${\mathcal M}$.
Prequantum line bundles
=======================
A very clear description of the determinant line bundle can be found in [@BF] and [@Q]. Here we mention the formula for the Quillen curvature of the determinant line bundle $\wedge^{{\rm top}}({\rm Ker} \bar{\partial}_A)^{*}
\otimes \wedge^{{\rm top}}({\rm Coker} \bar{\partial}_A) = {\rm det}(\bar{\partial}_A),$ where $\bar{\partial}_A = \bar{\partial} + A^{(0,1)}$ , given the canonical unitary connection $\nabla_Q$, induced by the Quillen metric [@Q]. Namely, recall that the affine space ${\mathcal A}$ (notation as in [@Q]) is an infinite-dimensional Kähler manifold. Here each connection is identified with its $(0,1)$ part. Since the total connection is unitary (i.e. of the form $A= A^{(1,0)} + A^{(0,1)}$, where $A^{(1,0)} = -A^{(0,1)*}$) this identification is easy. In fact, for every $A \in {\mathcal A}$, $T_A^{\prime} ({\mathcal A}) = \Omega^{0,1} (M, {\rm ad} P)$ and the corresponding Kähler form is given by $$F(\alpha, \beta) = {\rm Re} {\rm Tr} \int_M (\alpha^{(0,1)} \wedge
\beta^{(0,1)*}) = - {\rm Re} {\rm Tr} \int_M (\alpha^{(0,1)} \wedge
\beta^{(1,0)})$$ where $\alpha^{(0,1)}, \beta^{(0,1)} \in \Omega^{0,1} (M, {\rm ad} P)$, and $\beta^{(1,0)} = - \beta^{(0,1)*}.$ It is skew symmetric if you interchange $\alpha^{(0,1)} = A d \bar{z}$ and $\beta^{(0,1)} = B d \bar{z}$ (follows from the fact that ${\rm Im} ({\rm Tr} (A B^*)) = - {\rm Im} ({\rm Tr} (B A^*))$ for matrices $A$ and $B$, using once again $ d \bar{z} \wedge d z $ is imaginary). Let $\alpha = \alpha^{(0,1)} + \alpha^{(1,0)}$, $\beta =\beta^{(0,1)} +
\beta^{(1,0)}$. It is clear from the fact that $ \alpha^{(1,0)} = -\alpha^{(0,1)*} $ and $\beta^{(1,0)} = -\beta^{(0,1)*}$ that $$F(\alpha, \beta) = \frac{-1}{2}
{\rm Tr} \int_M \alpha \wedge \beta.$$ Here we have used the fact that $$\begin{aligned}
& & 2 {\rm Re} {\rm Tr} \int_M \alpha^{(0,1)} \wedge \beta^{(1,0)}\\
&=& {\rm Tr} \int_M \alpha^{(0,1)} \wedge \beta^{(1,0)} + {\rm Tr} \int_M \overline{\alpha^{(0,1)}} \wedge \overline{\beta^{(1,0)}}\\
&=& {\rm Tr} \int_M \alpha^{(0,1)} \wedge \beta^{(1,0)} + {\rm Tr} \int_M (-\overline{\alpha^{(0,1)\rm tr}}) \wedge (-\overline{\beta^{(1,0)\rm tr}}) \\
&=& {\rm Tr} \int_M \alpha^{(0,1)} \wedge \beta^{(1,0)} + {\rm Tr} \int_M \alpha^{(1,0)} \wedge \beta^{(0,1)} \\
&=& {\rm Tr} \int_M \alpha \wedge \beta \end{aligned}$$
Then one has $ {\mathcal F}(\nabla_Q) = \frac{i}{\pi} F. $
Prequantization of the moduli space ${\mathcal M}$
--------------------------------------------------
In this section we show that ${\mathcal M}$ admits a prequantum line bundle, i.e.a line bundle whose curvature is the symplectic form $\Omega$.
The moduli space ${\mathcal M}$ of solutions to $(1)$ and $(2)$ admits a prequantum line bundle $P$ whose Quillen curvature ${\mathcal F} = \frac{i}{\pi} \Omega$ where $\Omega$ is the natural symplectic form on ${\mathcal M}$ as in $(3)$.
First we note that to the connection $A$ we can add any one form and still obtain a derivative operator. To a connection $A_0$ whose gauge equivalence class is fixed, we will add $\Phi$ to obtain new connections which will appear in the Cauchy-Riemann derivative operators.
[**Definitions:**]{} Let us denote by ${\mathcal L} =
{\rm det} (\bar{\partial}+ A^{0,1})$ a determinant bundle on ${\mathcal A}$.
Let ${\mathcal R} = {\rm det} (\overline{\partial} + \overline{A_0^{(1,0)}}
+ \overline{\Phi^{(1,0)}})$ where $A_0$ is a connection whose gauge equivalence class is fixed, i.e. $A_0$ is allowed to change only in the gauge direction. The ${\mathcal R}$ is a line bundle on ${\mathcal C}$ defined such that the fiber on $(A,\Phi)$ is that of ${\rm det} (\overline{\partial} + \overline{A_0^{(1,0)}}
+ \overline{\Phi^{(1,0)}})$, but the fiber over gauge equivalent $(A_g, \Phi_g)$ is that of ${\rm det}(\bar{g}(\overline{\partial} + \overline{A_0^{(1,0)}}
+ \overline{\Phi^{(1,0)}})\bar{g}^{-1})$. This is because the gauge group ${\mathcal G}$ acts on all of ${\mathcal A}$ simultaneously, so that when $A \rightarrow A_g$, $A_0 \rightarrow A_{0g}$.
(In the next section we have a better approach where we donot have to deal with $\bar{g}$).
Let ${\mathcal P} = {\mathcal L}^{-2} \otimes
{\mathcal R}^2$ denote a line bundle over ${\mathcal C}$. We will show that this line bundle is well defined on ${\mathcal M}$ and has Quillen curvature a constant multiple of the symplectic form $\Omega$.
${\mathcal P}$ is a well-defined line bundle over ${\mathcal M} \subset {\mathcal C}/{\mathcal G}$, where ${\mathcal G}$ is the gauge group.
First consider the Cauchy-Riemann operator $ D= \bar{\partial} + A^{(0,1)}$. Under gauge transformation $D=\bar{\partial} + A^{(0,1)}
\rightarrow D_g= g(\bar{\partial} + A^{(0,1)})g^{-1} $. We can show that the operators $D$ and $D_g$ have isomorphic kernel and cokernel and their corresponding Laplacians have the same spectrum and the eigenspaces are of the same dimension. Let $\Delta$ denote the Laplacian corresponding to $D$ and $\Delta_g$ that corresponding to $D_g$. Then $\Delta_g = g \Delta g^{-1}$. Thus the isomorphism of eigenspaces is $s \rightarrow g s$. Thus when one identifies $\wedge^{{\rm top} }({\rm Ker} D)^* \otimes \wedge^{{\rm top} }
({\rm Coker} D)$ with $\wedge^{{\rm top}}(K^a(\Delta))^* \otimes \wedge^{{\rm top}}
(D(K^a(\Delta)))$ where $K^a(\Delta)$ is the direct sum of eigenspaces of the operator $\Delta$ of eigenvalues $< a$, over the open subset $U^a = \{A | a \notin {\rm Spec} \Delta \}$ of the affine space ${\mathcal A}$ (see [@BF], [@Q] for more details), there is an isomorphism of the fibers as $D \rightarrow D_g$. Thus one can identify $$\wedge^{{\rm top}}(K^a(\Delta))^* \otimes \wedge^{{\rm top}}
(D(K^{a}(\Delta))) \equiv
\wedge^{{\rm top}}(K^a(\Delta_g))^* \otimes \wedge^{{\rm top}}
(D(K^{a}(\Delta_g))).$$ By extending this definition from $U^a$ to $V^a = \{(A, \Phi)| a \notin {\rm Spec} \Delta \}$, an open subset of ${\mathcal C}$, we can define the fiber over the quotient space ${\mathcal C}/{\mathcal G}$ to be the equivalence class of this fiber.
Similarly one can deal with the other case of ${\rm det} (\overline{\partial} + \overline{A_0^{(1,0)}} +
\overline{\Phi^{(1,0)}})$, because under gauge transformation, $\overline{\partial} + \overline{A_0^{(1,0)}} +
\overline{\Phi^{(1,0)}} \rightarrow
\bar{g}(\overline{\partial} + \overline{A_0^{(1,0)}} +
\overline{\Phi^{(1,0)}})\bar{g}^{-1}$. Let $([A], [\Phi]) \in {\mathcal C}/{\mathcal G},$ where $[A], [\Phi]$ are gauge equivalence classes of $A, \Phi$, respectively. Then associated to the equivalence class $([A], [\Phi])$ in the base space, there is an equivalence class of fibers coming from the identifications of ${\rm det}(\overline{\partial} + \overline{A_0^{(1,0)}} +
\overline{\Phi^{(1,0)}})$ with ${\rm det} (\bar{g}(\overline{\partial} + \overline{A_0^{(1,0)}} +
\overline{\Phi^{(1,0)}})\bar{g}^{-1})$ as mentioned in the previous case. Note that in this case, the equivalence class of the fiber above $([A], [\Phi])$ is changing as $[\Phi]$ is changing. It is unaffected by change in $[A]$, since $[A_0]$ is not changing.
This way one can prove that ${\mathcal P}$ is well defined on ${\mathcal C}/{\mathcal G}$. Then we restrict it to ${\mathcal M} \subset {\mathcal C}/{\mathcal G}$.
[**Curvature and symplectic form:**]{}
Recall $\alpha \in \Omega ^{1}(M, {\rm ad} P)$ has the decomposition $\alpha = \alpha^{(1,0)} + \alpha^{(0,1)}$, where $\alpha^{(1,0)}= - \alpha^{(0,1)*}$ . Similar decomposition holds for $\beta, \gamma, \delta \in \Omega^{1}(M, {\rm ad} P)$.
Let $p = (A, \Phi) \in S$ where $S$ is the space of solutions to Hitchin equations $(1)$ and $(2)$. Let $X, Y \in T_{[p]}{\mathcal M}$. We write $X =(\alpha, \gamma)$ and $Y=(\beta, \delta)$, where $\alpha^{(0,1)}, \beta^{(0,1)} \in T_{A} ({\mathcal A}^{(0,1)}) = \Omega^{(0,1)}(M, {\rm ad} P \otimes {{\ensuremath{\mathbb{C}}}})$ and $\gamma^{(1,0)}, \delta^{(1,0)} \in T_{\Phi} {\mathcal H} = \Omega^{(1,0)}(M,
{\rm ad} P \otimes {{\ensuremath{\mathbb{C}}}})$. Since $T_{[p]}{\mathcal M}$ can be identified with a subspace in $T_p S$ orthogonal to $T_p O_p$ (the tangent space to the gauge orbit) then $X,Y$ can be said to satisfy (a) $X, Y \in T_p S$ i.e. they satisfy linearization of $(1)$ and $(2)$ and (b) $X, Y$ are orthogonal to $T_p O_p $, the tangent space to the gauge orbit.
Let ${\mathcal F}_{{\mathcal L}^{-2}}$, $ {\mathcal F}_{{\mathcal R}^2}$, denote the Quillen curvatures of the determinant line bundles ${\mathcal L}^{-2}$, ${\mathcal R}^2$, respectively. Then, $$\begin{aligned}
{\mathcal F}_{{\mathcal L}^{-2}}((\alpha, \gamma^{(1,0)}), (\beta, \delta^{(1,0)}))
&=& -2 {\mathcal F}_{{\mathcal L}}((\alpha, \gamma^{(1,0)}), (\beta, \delta^{(1,0)}))\\
&=& - 2 \frac{i}{\pi} {\rm Re} {\rm Tr}\int_M (\alpha^{(0,1)}
\wedge \beta^{(0,1)*}) \\
&=& \frac{i}{\pi} {\rm Tr} \int_M \alpha \wedge \beta\end{aligned}$$
$$\begin{aligned}
{\mathcal F}_{{\mathcal R}^2}((\alpha, \gamma^{(1,0)}),
(\beta, \delta^{(1,0)}))
&=& 2{\mathcal F}_{{\mathcal R}}((\alpha, \gamma^{(1,0)}),
(\beta, \delta^{(1,0)})) \\
&=& 2\frac{i}{\pi} {\rm Re} {\rm Tr} \int_M \overline{\gamma^{(1,0)}} \wedge \overline{\delta^{(1,0)*}}\\
&=& 2\frac{i}{\pi} {\rm Re} {\rm Tr} \int_M \gamma^{(1,0)} \wedge
\delta^{(1,0)*} \\
&=& - \frac{i}{\pi}{\rm Tr} \int_M \gamma \wedge \delta\end{aligned}$$
Note: $\overline{\gamma^{(1,0)}}$ and $\overline{\delta^{(1,0)*}}$ contributes because of the term $\overline{\Phi^{(1,0)}}$ in ${\mathcal R}$. $\alpha$, $\beta$ donot contribute to this curvature because in the definition of ${\mathcal R}$ the gauge equivalence class of $A_0$ is fixed.
The Quillen curvature of ${\mathcal P} = {\mathcal L}^{-2} \otimes {\mathcal R}^2$ on ${\mathcal M}$ is $\frac{i}{\pi} \Omega$.
It is easy to check that ${\mathcal F}_{{\mathcal L}^{-2}}
+ {\mathcal F}_{{\mathcal R}^2}
= \frac{i}{\pi} \Omega$.
These lemmas prove the theorem $(3.1)$.
A more natural approach for the quantization
============================================
We define $\Phi^{(0,1)}= -\Phi^{(1,0)*}$. We call the old $\Phi$ in the Hitchin equations by $\Phi^{(1,0)}$.
Let us denote by ${\mathcal L} =
{\rm det} (\bar{\partial}+ A^{0,1})$ a determinant bundle on ${\mathcal A}$.
Let ${\mathcal R} = {\rm det} (\bar{\partial} + A_0^{(0,1)}
+ \Phi^{(0,1)})$ where $A_0$ is a connection whose gauge equivalence class is fixed, i.e. $A_0$ is allowed to change only in the gauge direction.
Let ${\mathcal P} = {\mathcal L}^{-2} \otimes
{\mathcal R}^2$ denote a line bundle over ${\mathcal C} = {\mathcal A} \times {\mathcal H}$.
(This combination will give the prequantum line bundle corresponding to $\Omega$).
${\mathcal P}$ is a well-defined line bundle over ${\mathcal M} \subset {\mathcal C}/{\mathcal G}$, where ${\mathcal G}$ is the gauge group.
Let us consider the Cauchy-Riemann operator $ D= \bar{\partial} + A_0^{(0,1)} + \Phi^{(0,1)}$ which appears in ${\mathcal R}$. The other case is analogous. Under gauge transformation $D=\bar{\partial} + A_0^{(0,1)} + \Phi^{(0,1)}
\rightarrow D_g= g(\bar{\partial} + A_0^{(0,1)} + \Phi^{(0,1)})g^{-1} $ since it is the $(0,1)$ part of the connection operator $d + A_0 + \Phi$ which transforms in the same way. We can show that the operators $D$ and $D_g$ have isomorphic kernel and cokernel and their corresponding Laplacians have the same spectrum and the eigenspaces are of the same dimension. Let $\Delta$ denote the Laplacian corresponding to $D$ and $\Delta_g$ that corresponding to $D_g$.The Laplacian is $\Delta = \tilde{D} D$ where $\tilde{D} = \partial + A_0^{(1,0)} + \Phi^{(1,0)}$, where recall $A_0^{(1,0)*} = -A_0^{(0,1)}$ and $\Phi^{(1,0)*} = - \Phi^{(0,1)}$. Note that $\tilde{D} \rightarrow \tilde{D}_g = g \tilde{D} g^{-1}$ under gauge transformation since it is the $(1,0)$ part of the connection operator $ d+ A_0+ \Phi$ which transforms in the same way. Thus $\Delta_g = g \Delta g^{-1}$. Thus the isomorphism of eigenspaces of $\Delta$ and $\Delta_g$ is $s \rightarrow g s$. Thus when one identifies ${\rm det} D$= $\wedge^{{\rm top} }({\rm Ker} D)^* \otimes \wedge^{{\rm top} }
({\rm Coker} D)$ with $\wedge^{{\rm top}}(K^a(\Delta))^* \otimes \wedge^{{\rm top}}
(D(K^a(\Delta)))$ where $K^a(\Delta)$ is the direct sum of eigenspaces of the operator $\Delta$ of eigenvalues $< a$, over the open subset $U^a = \{(A^{(0,1)} , \Phi^{(0,1)}) | a \notin {\rm Spec} \Delta \}$ of ${\mathcal C}$ (see [@BF], [@Q] for more details), there is an isomorphism of the fibers as $D \rightarrow D_g$. Thus one can identify $$\wedge^{{\rm top}}(K^a(\Delta))^* \otimes \wedge^{{\rm top}}
(D(K^{a}(\Delta))) \equiv
\wedge^{{\rm top}}(K^a(\Delta_g))^* \otimes \wedge^{{\rm top}}
(D(K^{a}(\Delta_g))).$$ We can define the fiber over the quotient space $U^a/{\mathcal G}$ to be the equivalence class of this fiber. Covering ${\mathcal C}$ by open sets of the type $U^a$ enables us to define it on ${\mathcal C}/{\mathcal G}$. Then we restrict it to the moduli space ${\mathcal M} \subset
{\mathcal C}/{\mathcal G}$.
${\mathcal L}$ also descends to the moduli space in the same spirit.
[**Curvatures and symplectic forms**]{}
Recall $\alpha \in \Omega ^{1}(M, {\rm ad} P)$ has the decomposition $\alpha = \alpha^{(1,0)} + \alpha^{(0,1)}$, where $\alpha^{(1,0)}= - \alpha^{(0,1)*}$ . Similar decomposition holds for $\beta, \gamma, \delta \in \Omega^{1}(M, {\rm ad} P)$.
Let $p = (A, \Phi) \in S$ where $S$ is the space of solutions to Hitchin equations $(1)$ and $(2)$. Let $X, Y \in T_{[p]}{\mathcal M}$. We write $X =(\alpha, \gamma)$ and $Y=(\beta, \delta)$, where $\alpha^{(0,1)}, \beta^{(0,1)} \in T_{A} ({\mathcal A}^{(0,1)}) = \Omega^{(0,1)}(M, {\rm ad} P \otimes {{\ensuremath{\mathbb{C}}}})$ and $\gamma^{(1,0)}, \delta^{(1,0)} \in T_{\Phi} {\mathcal H} = \Omega^{(1,0)}(M,
{\rm ad} P \otimes {{\ensuremath{\mathbb{C}}}})$. Since $T_{[p]}{\mathcal M}$ can be identified with a subspace in $T_p S$ orthogonal to $T_p O_p$ (the tangent space to the gauge orbit) then $X,Y$ can be said to satisfy (a) $X, Y \in T_p S$ i.e. they satisfy linearization of $(1)$ and $(2)$ and (b) $X, Y$ are orthogonal to $T_p O_p $, the tangent space to the gauge orbit.
Let ${\mathcal F}_{{\mathcal L}^{-2}}$, $ {\mathcal F}_{{\mathcal R}^2}$, denote the Quillen curvatures of the determinant line bundles ${\mathcal L}^{-2}$, ${\mathcal R}^2$, respectively. Then, $$\begin{aligned}
{\mathcal F}_{{\mathcal L}^{-2}}((\alpha, \gamma), (\beta, \delta))
&=& -2 {\mathcal F}_{{\mathcal L}}((\alpha, \gamma), (\beta, \delta))\\
&=& - 2 \frac{i}{\pi} {\rm Re} {\rm Tr}\int_M (\alpha^{(0,1)}
\wedge \beta^{(0,1)*}) \\
&=& \frac{i}{\pi} {\rm Tr} \int_M \alpha \wedge \beta\end{aligned}$$
(Since there is no $\Phi$-term in ${\mathcal L}$, $\gamma$ and $\delta$ donot contribute).
$$\begin{aligned}
{\mathcal F}_{{\mathcal R}^2}((\alpha, \gamma),
(\beta, \delta))
&=& 2{\mathcal F}_{{\mathcal R}}((\alpha, \gamma),
(\beta, \delta)) \\
&=& 2\frac{i}{\pi} {\rm Re} {\rm Tr} \int_M \gamma^{(0,1)} \wedge \delta^{(0,1)*}\\
&=& -2\frac{i}{\pi} {\rm Re} {\rm Tr} \int_M \gamma^{(0,1)} \wedge \delta^{(1,0)}\\
&=& -2\frac{i}{\pi} {\rm Re} {\rm Tr} \int_M \overline{(-\gamma^{(0,1)tr})} \wedge \overline{(-\delta^{(1,0)tr)})}\\
&=&-2\frac{i}{\pi} {\rm Re} {\rm Tr} \int_M \gamma^{(1,0)} \wedge \delta^{(0,1)}\\
&=& 2\frac{i}{\pi} {\rm Re} {\rm Tr} \int_M \gamma^{(1,0)} \wedge
\delta^{(1,0)*} \\
&=& -\frac{i}{\pi} {\rm Tr} \int_M \gamma \wedge \delta\end{aligned}$$
Note: $\gamma^{(0,1)}$ and $\delta^{(0,1)*}$ contributes because of the term $\Phi^{(0,1)}$ in ${\mathcal R}$. $\alpha$, $\beta$ donot contribute to this curvature because in the definition of ${\mathcal R}$ the gauge equivalence class of $A_0$ is fixed.
It is easy to check that the curvature of ${\mathcal P}$ is $${\mathcal F}_{{\mathcal L}^{-2}}
+ {\mathcal F}_{{\mathcal R}^2}
= \frac{i}{\pi} \Omega.$$
Recall that $${\mathcal I} (\alpha^{(0,1)}) = i \alpha^{(0,1)},$$ $${\mathcal I}(\gamma^{(1,0)}) = i \gamma^{(1,0)},$$ $${\mathcal I}(\alpha^{(1,0)}) = -i \alpha^{(1,0)},$$ $${\mathcal I} (\gamma^{(0,1)}) = -i \gamma^{(0,1)}$$.
Thus w.r.t. ${\mathcal I},$ $A^{(0,1)}$ is holomorphic and $\Phi^{(0,1)}$ is antiholomorphic. But in ${\mathcal P}^{-1} = {\mathcal L}^2 \otimes {\mathcal R}^{-2}$ has the $A^{(0,1)}$-term as it is and the $\Phi^{(0,1)}$-term in the inverse bundle. Thus ${\mathcal P}^{-1}$ is ${\mathcal I}$-holomorphic.
Thus we have the following:
${\mathcal P}^{-1}$ is a holomorphic prequantum line bundle on ${\mathcal M}$ with curvature $-\frac{i}{\pi} \Omega$.
[**Polarization:**]{} We can take ${\mathcal I}$-holomorphic sections of ${\mathcal P}^{-1}$ as our Hilbert space.
[99]{} M.F. Atiyah, R. Bott: The Yang-Mills equations over Riemann surfaces; Phil.Trans.R.Soc.Lond.A 308, 523-615 (1982). S. Axelrod, S. Della Pietra, E. Witten: Geometric Quantization of Chern-Simons Gauge Theory; J.Differential Geom. 33 no3, 787-902, (1991). A.A. Beilinson, V.G. Drinfeld: Quantization of Hitchin’s fibration and Langlands program; Algebraic and geometric methods in mathematical physics (Kaciveli, 1993),3–7, Math. Phys. Stud., 19, Kluwer Acad. Publ., Dordrecht, 1996. A. Beilinson, V. Drinfeld: Quantization of Hitchin’s integrable system and Hecke eigensheaves; preprint (1995), http://www.math.uchicago.edu/ arinkin/langlands/ J.M. Bismut, D.S. Freed: The Analysis of Elliptic Families.I. Metrics and Connections on Determinant Bundles; Commun.Math.Phys, 106, 159-176 (1986). R. Dey: Geometric quantization of a hyperKähler structure in the Self-dual Yang-Mills on a Riemann surface, preprint, math-ph/0605027. R. Dey: Geometric quantization of the moduli space of the vortex equations on a Riemann surface, preprint, math-ph/0605025. P. Griffiths, J. Harris: Principles of Algebraic Geometry; John Wiley and sons, Inc., (1994). V. Guillemin, S. Sternberg: Symplectic Techniques in Physics; Cambridge University Press, Cambridge,(1984). N.J. Hitchin: The Self-duality Equations on a Riemann surface, Proc.London Math.Soc.(3)55,59-126, (1987). N.J.Hitchin: Flat Connections and Geometric Quantization; Comm Math Phys 131 (1990) pp.347-380 A. Kapustin, E. Witten: Electric-Magnetic Duality and the Geometric Langlands program; hep-th/0604151 S. Kobayashi: Differential Geometry of Complex Vector Bundles; Iwanami Shoten, Publishers and Princeton University Press, Princeton (1987). D. Quillen: Determinants of Cauchy-Riemann Operators over a Riemann surface; Functional Analysis and Its Application, 19, 31-34, (1985). N. M. Romao: Quantum Chern-Simons vortices on a sphere; J.Math. Phys., 42, Issue 8, (2001) 3445-3469. C.T. Simpson: Higgs bundles and local systems; Publ.Math. I.H.E.S., 75, 5-95 (1992).69. E. Witten: Quantum Field Theory and Jones polynomial; Commun.Math.Phys. 121:351, (1989). E. Witten: Quantization Of Chern-Simons Gauge Theory With Complex Gauge Group; Commun.Math.Phys.137:29-66,1991 N.M.J. Woodhouse: Geometric Quantziation; Claredon press, Oxford, (1991)
School of Mathematics, Harish Chandra Research Institute, Jhusi, Allahabad, 211019, India. email: [email protected]
|
---
abstract: 'By projecting onto complex optical mode profiles, it is possible to estimate arbitrarily small separations between objects with quantum-limited precision, free of uncertainty arising from overlapping intensity profiles. Here we extend these techniques to the time-frequency domain using mode-selective sum-frequency generation with shaped ultrafast pulses. We experimentally resolve temporal and spectral separations between incoherent mixtures of single-photon level signals ten times smaller than their optical bandwidths with a ten-fold improvement in precision over the intensity-only Cramér-Rao bound.'
author:
- 'J. M. Donohue'
- 'V. Ansari'
- 'J. Řeháček'
- 'Z. Hradil'
- 'B. Stoklasa'
- 'M. Paúr'
- 'L. L. Sánchez-Soto'
- 'C. Silberhorn'
bibliography:
- 'Resolution.bib'
title: 'Quantum-limited time-frequency estimation through mode-selective photon measurement'
---
*Introduction.—* The time-honored Cramér-Rao lower bound (CRLB) [@Cramer:1946aa; @Rao:1945aa] is credibly the most appropriate tool to address the resolution limits for incoherent imaging, as highlighted in recent years [@Farrell:1966aa; @Orhaug:1969aa; @Helstrom:1970aa; @Helstrom:1970ab; @Zmuidzinas:2003aa; @Holmes:2013aa; @Motka:2016aa]. This is especially pertinent when photon shot noise is the dominant noise source (as in, for example, astronomical observations) and a statistical treatment of resolution is indispensable. Nonetheless, in spite of these compelling results, Cramér-Rao resolution limits did not demand a great deal of attention until recent works examined microscopy limitations from a photon-counting perspective [@Bettens:1999aa; @Ram:2006aa; @Chao:2016aa]. The chief idea can be formalized through the Fisher information [@Fisher:1925aa], which quantifies the amount of information gained per photon detection and is directly associated to the CRLB. For direct intensity imaging, the Fisher information drops to zero for object separations smaller than the spread of the optical field. This precipitous drop, named Rayleigh’s curse, limits the usefulness of photon counting for metrology.
This line of questioning cleared the way for a fresh reexamination of the problem by Tsang and coworkers [@Tsang:2016aa; @Nair:2016aa; @Nair:2016ab; @Tsang:2017aa; @Tsang:2018aa]. Surprisingly, when one calculates the quantum Fisher information [@Petz:2011aa] (i.e., optimized over all the possible quantum measurements), the associated quantum CRLB maintains a fairly constant value for any separation of the sources. This shows the potential for parameter estimation of distributions with precision unaffected by Rayleigh’s curse. The key behind these techniques is phase-sensitive measurement in mode bases other than intensity [@Lupo:2016aa; @Rehacek:2017aa]. This has been experimentally demonstrated for spatially separated objects by holographic mode projection [@Paur:2016aa], heterodyne detection [@Yang:2016aa; @Yang:2017aa], and parity-sensitive interferometers [@Tham:2016aa]. As the strong analogy between the space-momentum and time-frequency descriptions of light has already provided valuable insights and useful techniques such as temporal imaging [@Bennett:1994aa], it is worthwhile to consider the advantages these techniques can offer when adapted to different domains.
In this Letter, we show that mode-selective measurement can be harnessed to estimate separations in time and frequency well below the spread of the source light. In analogy to the Rayleigh limit in space, this allows us to overcome the Taylor criterion in measuring spectral separations [@Juvells:2006aa], which states that the minimum resolvable separation of the spectral maxima is equal to the half-maximum width. We experimentally realize this enhancement in both time and frequency estimation settings by projectively measuring Hermite-Gauss time-frequency modes using sum-frequency generation with shaped ultrafast pulses in group-velocity engineered nonlinear waveguides. We explicitly demonstrate precision below the intensity-only CRLB, establishing mode-selective measurement as a valuable tool for pushing metrological limits in multiple physical domains.
Quantum analysis of time-frequency metrological problems has already provided a plethora of useful tools. In particular, quantum advantages can be realized in time-of-flight measurement and synchronization by exploiting entanglement [@Giovannetti:2011aa], squeezing [@Lamine:2008aa], and bunching [@Lyons:2017aa], and considering quantum techniques and analysis has inspired classical techniques that outperform their pre-existing counterparts [@Kaltenbaek:2008aa]. Additionally, reductions in the standard quantum limit have been noted using homodyne techniques with shaped local oscillators in higher-order Hermite-Gaussian modes [@Jian:2012aa; @Thiel:2017aa]. Here, we show that quantum-inspired metrology finds application in measuring incoherent source superpositions with either time or frequency offsets. This form of frequency estimation has natural applications in, for example, measuring nearly degenerate atomic and stellar spectral lines, particularly after undergoing inhomogenous broadening. Precision time measurements find natural applications in time-of-flight ranging and in probing ultrafast system dynamics.
*Quantum-limited measurements.—* We formalize the parameter estimation problem under consideration in analogously to the spatial case [@Tsang:2016aa; @Paur:2016aa]. Two mutually incoherent (or phase-randomized) light sources with equal intensities emit at optical frequencies $\nu_0\pm\frac{\textgoth{s:}_\nu}{2}$. We assume that the central frequency $\nu_0$ is well-known and that the remaining quantity of interest is the spectral separation, $\textgoth{s:}_\nu$. If the sources have non-negligible spectral bandwidth, the optical spectrum $I(\nu, \textgoth{s:}_\nu)$ of the incoherent mixture as measured on a spectrometer will be $$I(\nu,\textgoth{s:}_\nu)=\frac{1}{2} \left(\left|\psi\left (\nu+ \frac{\textgoth{s:}_\nu}{2} \right )\right|^2+\left|\psi\left(\nu-\frac{\textgoth{s:}_\nu}{2}\right)\right|^2\right),$$ where $\psi(\nu)$ is the spectral amplitude shape. For specificity, we focus on the case of Gaussian spectral amplitudes (frequency-domain point-spread functions) with root-mean-square (RMS) widths $\sigma_\nu$, such that $$\psi\left(\nu\pm\frac{\textgoth{s:}_\nu}{2}\right)=\frac{1}{(2\pi\sigma_\nu^2)^\frac{1}{4}} \exp\left[-\frac{\left(\nu-\nu_0\pm\frac{\textgoth{s:}_\nu}{2}\right)^2}{4\sigma_\nu^2}\right].\label{eq:gaussian}$$
The standard method of estimating the spectral separation $\textgoth{s:}_\nu$ in the low-luminescence (i.e. photon counting) regime would be to measure the spectral intensity $I(\nu,\textgoth{s:}_\nu)$ on a spectrometer, such as a Fabry-Pérot interferometer or grating-based spectrograph, and use a fitting or deconvolution algorithm on the integrated photon counts. We quantify the amount of information in-principle available to estimate $\textgoth{s:}_\nu$ with $N$ detected photons (i.e., standard intensity detection) via the Fisher information $\mathcal{F}_\mathrm{std}$, given by $$\mathcal{F}_\mathrm{std}= N\int^\infty_{-\infty}{\mathrm{d}}\nu\,\frac{1}{I(\nu,\textgoth{s:}_\nu)}\left [ \frac{\partial I(\nu,\textgoth{s:}_\nu)}{\partial \textgoth{s:}_\nu}\right ]^2.$$ The Fisher information quantifies how sensitive the measured quantity $I(\nu,\textgoth{s:}_\nu)$ is to changes in the variable $\textgoth{s:}_\nu$, and can be used to construct the CRLB as $\mathrm{Var}(\hat{\textgoth{s:}}_\nu)\geq1/\mathcal{F}_\mathrm{std}$ [@Motka:2016aa], which defines the absolute minimum mean-squared error (variance) of the estimated separation, $\hat{\textgoth{s:}}_\nu$. For large separations, ${\textgoth{s:}_\nu\gg\sigma_\nu}$, the standard Fisher information is constant, providing a Cramér-Rao bounded variance of $\mathrm{Var}(\hat{\textgoth{s:}}_\nu)\geq(4\sigma_\nu^2)/N$. However, when $\textgoth{s:}_\nu\sim\sigma_\nu$, the CRLB bound grows dramatically, diverging as $\textgoth{s:}_\nu/\sigma_\nu$ approaches zero. This behavior is known as Rayleigh’s curse in the spatial domain, and is sometimes rephrased as the Taylor criterion in spectral measurements. Note that the exact same “curse” applies to estimating incoherent time separations, $\textgoth{s:}_t$, between two pulsed sources through direct timing measurement, for example with autocorrelation or streak-camera techniques [@bradley1971direct].
The curse can be lifted by performing phase- or parity-sensitive measurements, even though the source fields themselves have no coherent phase relationship. An optimal measurement basis is always provided by the partial derivatives of the amplitude point-spread function [@Rehacek:2017aa]. For the Gaussian point-spread function as in Eq. , the optimal measurement is then the Hermite-Gaussian basis [@Tsang:2016aa; @Paur:2016aa; @Rehacek:2017aa]. For separations $\textgoth{s:}_\nu\lesssim\sigma_\nu$, $\textgoth{s:}_\nu$ can be optimally estimated with only projections onto the first two Hermite-Gauss modes, expressed as $$\begin{split} \phi_{\hgzero}(\nu) &= \frac{1}{(2\pi\sigma_\nu^2)^\frac{1}{4}}\exp\left[-\frac{(\nu-\nu_0)^2}{4\sigma_\nu^2}\right] \\ \phi_{\hgone}(\nu) &= \frac{(\nu-\nu_0)}{(2\pi\sigma_\nu^6)^\frac{1}{4}}\exp\left[-\frac{(\nu-\nu_0)^2}{4\sigma_\nu^2}\right] .\end{split}$$ If projective measurements onto these modes can be realized, the estimator $\hat{\textgoth{s:}}_\nu$ has curse-free performance, with $\mathrm{Var}(\hat{\textgoth{s:}}_\nu)\geq(4\sigma_\nu^2)/N$ for arbitrarily small values of $\textgoth{s:}_\nu$. This value agrees exactly with the absolute quantum limit derived from the quantum Fisher information [@Tsang:2016aa]. To include estimation of the centroid, extend the technique to large separations $\textgoth{s:}_\nu\gg\sigma_\nu$, or in cases with unequal-intensity emission, higher-order mode projections may be used [@Rehacek:2017ab].
*Time-frequency mode selection.—* The key experimental requirement to enable this advantage is mode-selective projective measurement in the time-frequency domain. We implement such measurements using a technique known as the quantum pulse gate [@eckstein2011quantum; @manurkar2016multidimensional; @reddy2017engineering; @ansari2017temporal; @ansari2018tailoring], a sum-frequency process where a weak input signal is mixed with a spectrally shaped pump pulse to create an upconverted signal in a long nonlinear waveguide. To implement a quantum pulse gate, the input signal and pump pulses must have matched group velocities and the walkoff between the input and upconverted signals must be longer than the length of the input pulses. If these conditions are met, the probability of an upconversion event in the low-efficiency regime given an input spectral amplitude $\psi(\nu)$ and a pump amplitude $\alpha(\nu)$ can be expressed simply as [@eckstein2011quantum; @ansari2018tailoring] $$P_\alpha\propto\left|\int{\mathrm{d}}\nu\,\alpha(-\nu)\psi(\nu)\right|^2.\label{eq:QPGproj}$$ Measuring the upconverted pulse power thereby corresponds to a projective measurement on the broadband time-frequency mode defined by the shape of the pump pulse, $\alpha^*(-\nu)$ [@ansari2017temporal]. By counting photons in the upconverted mode while projecting on either the fundamental Gaussian mode or the first-order Hermite-Gaussian mode, we can easily construct an estimator by taking their ratio.
The mode selectivity of the quantum pulse gate is limited by the group-velocity walkoff between the input and upconverted signals, which we define by the walkoff parameter $\Delta=\frac{L}{2}\left(\frac{1}{u_{{\mathrm{in}}}}-\frac{1}{u_{{\mathrm{out}}}}\right)$, where $u_j$ is the group velocity and $L$ is the length of the nonlinear interaction. This walkoff defines the phasematching conditions of the interaction, imposing a RMS bandwidth of the upconverted light of ${\sigma_\mathrm{PM}\approx \frac{0.18}{\Delta}}$. When the input and pump are significantly broader than the phasematching bandwidth and the side lobes arising from the sinc-shaped phasematching curve are filtered out, the ratio of the lowest order Hermite-Gaussian projections is given by $$\frac{P_{\hgone}}{P_{\hgzero}}\approx \frac{\sigma_\mathrm{PM}^2}{4\sigma_\nu^2}+\frac{\textgoth{s:}_t^2}{16\sigma_t^2}+\frac{\textgoth{s:}_\nu^2}{16\sigma_\nu^2},\label{eq:QPGestimator}$$ where $\sigma_t$ and $\sigma_\nu$ are the RMS widths of the measurement pulse’s temporal and spectral profiles; a derivation of this result is presented in the appendix. If the signal is properly aligned in one of the two degrees of freedom ($\textgoth{s:}_\nu=0$ or $\textgoth{s:}_t=0$) and $\frac{\textgoth{s:}_\nu}{\sigma_\nu}\gg\frac{\sigma_\mathrm{PM}}{\sigma_\nu}$ or $\frac{\textgoth{s:}_t}{\sigma_t}\gg\frac{\sigma_\mathrm{PM}}{\sigma_\nu}$, Eq. shows that the square-root of the ratio of projection probability for the first two Hermite-Gauss modes can be used as an exact estimator for the separation between the signals. For separations small enough that the finite phasematching bandwidth cannot be completely neglected, Eq. can still be inverted to construct a estimator $\hat{\textgoth{s:}}_\nu$ or $\hat{\textgoth{s:}}_t$, although with slightly reduced precision relative to the quantum limit. As the phasematching bandwidth can be much smaller than the input bandwidths [@allgaier2017bandwidth], the precision of this method can be considerably finer than the broad bandwidth or temporal durations of the pulses being interrogated.
![**Experimental setup.** We carve signal pulses with shifting center frequencies and time delays from an attenuated broadband Ti:Sapphire OPO pulse at 1540 nm using a commercial telecommunications pulse shaper. We shape pump pulses at 875 nm into Hermite-Gaussian shapes using a 4f-line with a spatial light modulator (SLM). We then mix the pump and signal pulses in a PPLN waveguide, separate the sum-frequency signal with a 4f bandpass filter (BP), and count photons using an avalanche photodiode gated by a clock pulse from the Ti:Sa.[]{data-label="fig:expsetup"}](figures/fig_expsetup.pdf){width="1\columnwidth"}
[figures/plots\_raw\_Specinset.pdf]{}[(0,90) [(a)]{}]{}
\
[figures/plots\_raw\_Time.pdf]{}[(0,90) [(b)]{}]{}
*Experiment.—* In our experimental apparatus, sketched in Fig. \[fig:expsetup\], we generate shaped input signal and pump pulses from a Ti:Sa laser and optical parameter oscillator (OPO) with a repetition rate of 80 MHz. The strong pump pulses at 875 nm are shaped into Hermite-Gauss modes with a bandwidth of 1.3 nm full-width at half-maximum (FWHM) using a 4f-line with a spatial light modulator (SLM) at the focal plane, with approximately 2 mW coupled into the quantum pulse gate.
To create frequency- and time-shifted pulses, we carve Gaussian signals with intensity RMS widths of ${\sigma_\nu=182\pm2~\mathrm{GHz}}$ from the approximately $3$-THz FWHM emission of the OPO using a commercial pulse shaper (Finisar 4000S). Frequency shifts are imparted straightforwardly by carving different parts of the OPO spectrum, while time shifts are imparted by programming linear spectral phases with the pulse shaper. The width of the pulses in time was measured to be $\sigma_t={387\pm13~\mathrm{fs}}$ using the quantum pulse gate as an autocorrelator. Neutral density filters were used to attenuate the shaped pulses to approximately 1.1 photons per pulse coupled into the measurement waveguide. The incoherence of the time- and frequency-separated mixtures was assured by switching between positive and negative shifts and mixing the measured results.
To quantum pulse gate was realized by combing the shaped input and pump pulses on a dichroic mirror and coupling them into a 17-mm-long and 7-$\mu$m-wide periodically poled lithium niobate (PPLN) waveguide with a poling period of 4.4 $\mu$m and single-mode propagation at 1540 nm. The spectra of the upconverted light at 558 nm was cleaned with a 4f line to remove phasematching side lobes, resulting in an upconverted bandwidth of $\sigma_{\mathrm{PM}}=28$ GHz, a factor of six smaller than the input light. The internal upconversion efficiency, measured as the depletion of the transmitted signal for the Gaussian projection with zero offset, was approximately 18%. To reduce background noise due to detector dark counts, the upconverted signal was measured with an avalanche photodiode in coincidence with a clock pulse from the Ti:Sa sampled down by a factor of 50, resulting in an effective experimental repetition rate of 1.6 MHz.
[figures/plots\_vars\_mt\_time.pdf]{}[(6.5,98) [(b)]{}]{}
[figures/plots\_vars\_mt\_spec.pdf]{}[(1,98) [(a)]{}]{}
*Results and discussion.—* Twenty separations ranging from $0-2\sigma$ were programmed in both time and frequency during the experiment and each setting was measured 60 times. In addition to controlling the separation, the pulse shaper was also used to attenuate the weak input signal to 100%, 50%, or 25% of its original intensity, to demonstrate the lack of any bias due to background noise. The uncorrected estimator $\hat{\textgoth{s:}}=4\sqrt{P_{\hgone}/P_{\hgzero}}$ from Eq. is shown in Fig. \[fig:rawdata\]. The estimator is seen to reach the expected linear behavior for separations on the same order as the RMS widths, but the imperfect mode selectivity causes small, predictable deviations for very small separations. The observed extinction ratio between the first and zeroth order Hermite-Gauss mode when no separation is found to be ${-10\log_{10}(P_{\hgone}/P_{\hgzero})}={(22.9\pm0.3)}$ dB, corresponding to a minimum estimator value of $\hat{\textgoth{s:}}_{\mathrm{min}}=0.144\pm0.005$.
To construct an unbiased estimator resilient to the imperfect selectivity of our device, we use calibration data from projections onto the first three Hermite-Gauss modes to perform measurement tomography of our technique. Details on the tomography techniques are presented in the appendix. To demonstrate the precision of our technique, in Fig. \[fig:qlim\] we show the variance of the calibrated estimator $\mathrm{Var}(\hat{\textgoth{s:}})$ for both time and frequency measurements while varying the total number of detection events, alongside the standard and quantum CRLBs. The variance is above the quantum limit (in red), owing to mode-selectivity limitations and instabilities. However, it remains below the intensity-only bound $1/\mathcal{F}_{\mathrm{std}}$ for separations well below the point-spread function widths, with an improvement in precision by factor of as high as ten for small separations.
The above results clearly demonstrate that mode-selective time-frequency measurement can be exploited for precision parameter estimation problems where intensity measurements fail. Notably, the absolute time and frequency scales accessible are not strongly dependent on the scale of the measurement pulses, but rather the material properties, namely the phasematching bandwidth $\sigma_\mathrm{PM}$. In our realization, this corresponded to time and frequency scales of 200 fs and 100 GHz, respectively. The accessible time and frequency scales could be improved either along with the conversion efficiency by increasing the nonlinear interaction length, or at the expense of the detection rate through narrowband filtering of the upconverted signal. Alternative methods based on mode-selective atomic or solid-state Raman memories could provide greater sensitivity, particularly in the frequency domain [@fisher2016frequency; @munns2017temporal]. Techniques based on homodyne detection can also provide the necessary mode selectivity in the time-frequency domain [@Lamine:2008aa; @polycarpou2012adaptive; @roslund2014wavelength].
We have demonstrated that parameter estimation in the time-frequency domain can benefit greatly from quantum-inspired techniques and analysis. By exploiting time-frequency mode-selective measurement enabled by waveguided nonlinear interactions, we have shown that sub-pulse-width separations can be estimated with precision below the standard CRLB. By adapting these techniques to different scales, this method could find immediate practical use in atomic and stellar spectral characterization and time-of-flight imaging. Future work will explore different mode-selective systems to adapt to specific tangible metrological problems and apply higher-order projections to multi-parameter estimation protocols.
*Acknowledgements* - We thank K. Bonsma-Fisher, O. Di Matteo, J. Gil-López, M. Allgaier, and B. Brecht for fruitful discussions. This research has received funding from the European Union’s (EU) Horizon 2020 research and innovation program under Grant Agreement No. 665148, the Grant Agency of the Czech Republic (Grant No. 18-04291S), Palacký University (Grant No. IGA-PrF-2018-003), and the Spanish MINECO (Grant FIS2015-67963-P). J.M.D. gratefully acknowledges support from Natural Sciences and Engineering Resource Council of Canada.
Estimating separations with a quantum pulse gate {#sec:QPGestimator}
================================================
In this section, we derive Eq. of the main text, which provides the separation estimators $\hat{\textgoth{s:}}_\nu$ and $\hat{\textgoth{s:}}_t$ for measurements made with a quantum pulse gate. The quantum pulse gate operation relies on a large discrepancy between the group velocities of the input and upconverted signals, which manifests in the energy picture as a much larger input acceptance bandwidth than output signal bandwidth. In practice, this discrepancy is of course finite, which places limitations on the achievable time and frequency resolutions. Here, starting from the basic nonlinear interaction, we outline these limitations.
In our treatment herein, we assume that the three-field interaction takes place inside a single-mode $\chi^{(2)}$ waveguide, such that we may neglect the spatial modes involved. We also assume that we are working in the low-efficiency regime, such that a first-order approach is sufficient [@reddy2017engineering; @ansari2018tailoring]. We label the modes as the input (“1”), the QPG pump (“2”), and upconverted output (“3”), and group the central frequencies into the variables as $\tilde\nu=\nu-\nu_0$. In this case, the upconverted spectral amplitude $\gamma(\tilde\nu_3)$ is related to the spectral amplitude of the QPG pump $\alpha(\tilde\nu_2)$ and the input signal $\psi(\tilde\nu_1)$ as $$\gamma(\tilde\nu_3)=\theta\int{\mathrm{d}}\tilde\nu_1\,H(\tilde\nu_1,\tilde\nu_3)\alpha(\tilde\nu_3-\tilde\nu_1)\psi(\tilde\nu_1),\label{eq:threewavemixing}$$ where energy conservation $\tilde\nu_2=\tilde\nu_3-\tilde\nu_1$ has been accounted for, $\theta$ is a coupling constant representing factors such as the material nonlinearity, and $H(\tilde\nu_1,\tilde\nu_3)$ is the phasematching function, characterized by the relationships between the wavenumbers $k_j(\tilde\nu_j)=\frac{2\pi\nu_jn_j(\tilde\nu_j)}{c}$ of the interacting fields.
If process is phasematched at the central frequencies through periodic poling and chromatic dispersion within each field can be neglected, the phasematching function for an interaction length $L$ can be expressed as $$H(\tilde\nu_1,\tilde\nu_3)\propto L\,\mathrm{sinc}\left(\frac{L\left[(k'_1-k'_2)\tilde\nu_1-(k'_3-k'_2)\tilde\nu_3\right]}{2}\right)$$ where $k'_j=\frac{\partial k_j}{\partial\nu_j}\big\rvert_{\nu_{0,j}}=\frac{1}{2\pi u_{j}}$ is inversely proportional to the group velocity $u_{j}$. If the input signal and QPG pump are group-velocity matched $k'_1=k'_2$, the phasematching function simplifies to a function of only the output frequency $\tilde\nu_3$. If we use a bandpass filter to remove the side lobes of the sinc function, we can approximate the phasematching function as a Gaussian, $$H(\tilde\nu_3)\approx L\,e^{-\eta\frac{(L(k'_3-k'_1)\tilde\nu_3)^2}{4}}\vcentcolon= L\,e^{-\frac{\tilde\nu_3^2}{4\sigma_{\mathrm{PM}}^2}},$$ where $\sigma_{\mathrm{PM}}$ is the RMS phasematching bandwidth and $\eta\approx0.193$.
We assume that the input signal wavefunction is a Gaussian pulse with some offset $\delta\nu$ from the perfectly phasematched frequencies and a small time delay $\delta t$ relative to the QPG pump pulse, which we express as $$\psi(\tilde\nu_1)=\frac{1}{(2\pi\sigma_\nu^2)^\frac{1}{4}}\exp\left[-\frac{(\tilde\nu+\delta\nu)^2}{4\sigma_\nu^2}-i2\pi\tilde\nu\delta t\right].$$ Note that the RMS width of the pulse in time is $\sigma_t=1/(4\pi\sigma_\nu)$. The QPG pump pulse is shaped to the first two Hermite-Gauss temporal modes with bandwidth $\sigma_2$, given by $$\begin{split} \alpha_{\hgzero}(\tilde\nu_2) &= \frac{1}{(2\pi\sigma_2^2)^\frac{1}{4}}\exp\left[-\frac{\tilde\nu_2^2}{4\sigma_2^2}\right] \\ \alpha_{\hgone}(\tilde\nu_2) &= \frac{\tilde\nu_2}{(2\pi\sigma_2^6)^\frac{1}{4}}\exp\left[-\frac{\tilde\nu_2^2}{4\sigma_2^2}\right] .\end{split}$$ Substituting these and the phasematching function into Eq. and finding the relative upconversion probability as $P=\int{\mathrm{d}}\nu_3\,|\gamma(\tilde\nu_3)|^2$, the ratio of the upconversion probabilities for the first two modes is found to be $$\begin{array}{ccl} \frac{P_{\hgone}}{P_{\hgzero}}&=& \sigma_2^2\left[\frac{\sigma_\nu^2+16\pi^2\delta t^2\sigma_\nu^2+\sigma_2^2}{(\sigma_\nu^2+\sigma_2^2)^2}+\frac{\delta\nu^2-\sigma_\nu^2-\sigma_2^2-\sigma_{\mathrm{PM}}^2}{(\sigma_\nu^2+\sigma_2^2+\sigma_{\mathrm{PM}}^2)^2}\right]\\ &\overset{\sigma_2=\sigma_\nu}{=}& \frac{\sigma_\mathrm{PM}^2}{2(2\sigma_\nu^2+\sigma_\mathrm{PM}^2)} + 4\pi^2\delta t^2\sigma_\nu^2+\frac{\delta\nu^2\sigma_\nu^2}{2(2\sigma_\nu^2+\sigma_\mathrm{PM}^2)^2}\\ &\overset{\sigma_\nu^2\gg\sigma_\mathrm{PM}^2}{\approx}&\frac{\sigma_\mathrm{PM}^2}{4\sigma_\nu^2} + \frac{\delta t^2}{4\sigma_t^2} + \frac{\delta\nu^2}{4\sigma_\nu^2}.\label{eq:QPGestimatorDerivation}\end{array}$$ To get from the first line to the second, we have set the bandwidth of the QPG pump to be equal to the input signal, ensuring that the two pulses have matched temporal-mode bases. To get from the second line to the third, we have assumed that the phasematching bandwidth is narrower than the input pulses, such that ${2\sigma_\nu^2+\sigma_\mathrm{PM}^2\approx2\sigma_\nu^2}$. Since $P_{\hgzero}$ and $P_{\hgone}$ are both symmetric functions of $\delta\nu$ or $\delta t$, Eq. holds for incoherent mixtures of positive and negative shifts, and Eq. can be retrieved by substituting $\delta\nu\mapsto\frac{\textgoth{s:}_\nu}{2}$ and $\delta t\mapsto \frac{\textgoth{s:}_t}{2}$. It is apparent that the minimum resolvable shift will be on the order of $\sigma_\mathrm{PM}$ in frequency and $\frac{\sigma_\mathrm{PM}}{\sigma_\nu}\sigma_t$ in time, and that any misalignment in frequency or time will adversely effect the resolution of measurements in the other setting.
Measurement tomography methods {#sec:MeasTomo}
==============================
In this section, we describe the measurement tomography method used to retrieve an accurate separation estimator from the directly measured data. To characterize the device, we implement projections onto the first three Hermite-Gauss modes, where “ideal measurement” can be described by projections of the input signal on the three lowest-order Hermite-Gauss modes HG$_0$, HG$_1$, and HG$_2$. We denote $q_j(\textgoth{s:})$ the probability of the $j$th measurement output given the true separation is $\textgoth{s:}$. For a Gaussian point-spread function (PSF) of width $\sigma$, this probability reads $$\label{basis}
q_j(\textgoth{s:})=\frac{1}{k!}\left(\frac{\textgoth{s:}}{4\sigma}\right)^{2j} e^{-\left(\frac{\textgoth{s:}}{4\sigma}\right)^2},\qquad j=0,1,2\,.$$ Due to unavoidable imperfections, the actual detection probabilities $p_j(\textgoth{s:})$ differ slightly from $q_j(\textgoth{s:})$ and the measurement device needs to be characterized before using. Assuming the setup works well, that is, the differences between the actual and target distributions $p_j(\textgoth{s:})$ and $q_j(\textgoth{s:})$ are small, we expand the former using the latter as a basis as follows $$p_j(\textgoth{s:})=\sum\limits_{k=0}^M c_{jk}\, q_k(\textgoth{s:}),\qquad j=0,1,2\,.$$ Having repeatedly measured a set of known separations $\textgoth{s:}=\{\textgoth{s:}_1,\textgoth{s:}_2,\ldots,\textgoth{s:}_N\}$, the probabilities $p_j$ can be estimated by the corresponding relative frequencies $f_j=\langle n_j\rangle/\sum_j \langle n_j\rangle$. Denoting further $f^j_\alpha= f_j(\textgoth{s:}_\alpha)$, $q_{k\alpha}= q_k(\textgoth{s:}_\alpha)$, and $c^j_k=c_{jk}$, we obtain three sets of linear equations to be solved for the set of unknown detection coefficients $c^j_k$ $$f^j_k=\sum_k q_{k\alpha} c_k^j,\qquad j=0,1,2\,.$$ The pseudo-inverse can be used to obtain standard solutions minimizing the $L_2$ norm, $$\mathbf{c}^j=Q^{+} \mathbf{f}^j,\qquad j=0,1,2\,.$$ It turns out just a few ($M\approx 4$) basis functions in Eq. are required to observe excellent fits of the detected relative frequencies $f_j$ in terms of the corresponding theoretical models $p_j$ for all measured separations in the region of interest $\textgoth{s:}\in [0,2]$.
We next proceed to the parameter estimation step using our characterized measurement. Each measurement returns a three numbers, $n_0$, $n_1$, and $n_2$. Assuming Poissonian statistics, the separation is estimated by maximizing the log-likelihood $$\hat{\textgoth{s:}}=\arg\max\limits_{\textgoth{s:}}\left\{ \sum_j n_j \log\left[\frac{p_j(\textgoth{s:})}{\sum_{j'} p_{j'}(\textgoth{s:})}\right]\right\}\,$$ subject to $\hat{\textgoth{s:}}\ge 0$ using a suitable optimization tool. Finally, for every true separation we calculate the statistics of the estimates and compare the measurement errors to the relevant classical and quantum resolution limits.
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abstract: 'The ultraviolet spectra of all “weak emission line central stars of planetary nebulae” (WELS) with available IUE data is analyzed. We found that the WELS can be divided in three different groups regarding their UV: (1) Strong P-Cygni profiles (mainly in C IV 1549); (2) Weak P-Cygni features and (3) Absence of P-Cygni profiles. We have measured wind terminal velocities for all objects presenting P-Cygni profiles in N V 1238 and/or C IV 1549. The results obtained were compared to the UV data of the two prototype stars of the \[WC\]-PG 1159 class, namely, A30 and A78. They indicate that WELS are distinct from the \[WC\]-PG 1159 stars, in contrast to previous claims in the literature. In order to gain a better understanding about the WELS, we clearly need to determine their physical parameters and chemical abundances. First non LTE expanding atmosphere models (using the CMFGEN code) for the UV and optical spectra of the star Hen 2-12 are presented.'
author:
- 'W. L. F. Marcolino'
title: 'WELS - Ultraviolet Spectra and Expanding Atmosphere Models'
---
Introduction:
=============
Hydrogen deficient central stars of planetary nebulae (CSPN) are generally divided in three main groups[^1]: \[WR\], PG 1159 and \[WC\]-PG 1159 stars. However, besides these classes, there is the “weak emission line stars” (WELS or \[WELS\]) group, which was introduced in the extensive observational study presented by Tylenda et al. (1993).
Although a considerably advance in the understanding of the origin and evolution of \[WR\], \[WC\]-PG 1159 and PG 1159 stars have been achieved in the last decade (see e.g. Werner & Herwig 2006), the evolutionary status of the WELS remains an open question. Motivated by the fact that only a few studies have been done focusing these objects, and that most of them were done in the optical part of the spectrum, we have investigated the UV spectra of all WELS with available IUE data. Moreover, we have applied the non LTE expanding atmosphere code CMFGEN to model the UV and optical spectra of one WELS, namely, Hen 2-12.
The UV Spectra of the WELS:
===========================
![Terminal wind velocities of hydrogen deficient CSPN (from Marcolino et al. 2007).[]{data-label="vinfres"}](f1){width="270pt" height="220pt"}
We have used the MAST database to retrieve all the data available for the WELS from the IUE satellite. The total number of objects currently known is about 50, but only about 40% of this population was observed by the IUE. For all the details regarding the UV analysis, we refer the reader to the complete results published in Marcolino et al. (2007).
Instead of presenting a homogeneous set of features, we found that the WELS could be divided in three different groups: 1) Strong P-Cygni profiles (mainly in C IV 1549); (2) Weak P-Cygni feature in C IV 1549 and (3) Absence of P-Cygni profiles. In the case of group (3), the lines are very intense and most likely of nebular origin.
We have compared the UV spectra of the WELS with the UV spectra of the two prototypes of the \[WC\]-PG 1159 class: A30 and A78. These two objects present simultaneous intense P-Cygni emissions of N V 1238, O V 1371 and C IV 1549. In contrast, in the WELS, the O V 1371 line is weak or absent. The same is true for N V 1238 in some objects. It is also conspicuous that the P-Cygni profiles in A30 and A78 present a broader absorption than in the WELS spectra. These characteristics indicate that the WELS are not \[WC\]-PG 1159 stars, in contrast with previous claims in the literature (see Parthasarathy et al. 1998) on the basis of optical spectra. However, the situation is not clear for the stars NGC 6543, NGC 6567, and NGC 6572. They present simultaneous P-Cygni emissions of N V 1238, O V 1371 and C IV 1549, but considerably lower terminal velocities than the ones found for A30 and A78 (see Marcolino et al. 2007 for more details).
Wind Terminal Velocities:
-------------------------
After the spectral comparison between the WELS and the \[WC\]-PG 1159 stars A30 and A78, we have derived wind terminal velocities ($v_\infty$) for all objects presenting P-Cygni profiles in N V 1238 and/or C IV 1549. As most of the data available are of low resolution, we have used the calibration provided by Prinja (1994). For spectra of high resolution, the standard procedure to measure $v_\infty$ was used. Our results are displayed in Fig. \[vinfres\], along with data for \[WCE\], \[WCL\] and PG 1159 stars, obtained from Koesterke (2001). As it is clear from the figure, the $v_\infty$ distribution for the WELS stand mainly between the early and late type \[WR\] stars distribution. Moreover, the WELS have much lower terminal velocities than the \[WC\]-PG 1159 and PG 1159 stars.
First models for Hen 2-12:
==========================
![CMFGEN model (blue/dark grey) and observed spectra of the star Hen 2-12 (red/light grey). Optical and UV resolutions are 2Å and 6Å.[]{data-label="fitcmfgen"}](f2){width="430pt" height="430pt"}
A way to cast light in the evolutionary status of the WELS, is through the modeling of their spectra. In this manner, we can obtain their physical parameters and chemical abundances. Thereafter, their place in the H-R diagram could be estimated and a better comparison to other hydrogen deficient CSPN could be made. Our first efforts in this direction have been made by the analysis of the star Hen 2-12. We used the non LTE expanding atmosphere code of Hillier & Miller (1998; CMFGEN) to model the spectrum of this star from the UV to the optical. Our fit is shown in Fig. \[fitcmfgen\]. Several lines are not reproduced, since they are nebular emissions (they are indicated by “neb” in the figure).
The physical parameters derived from the fit are: $\dot{M} = 3.5 \times 10^{-9}M_{\sun}$ yr$^{-1}$ (clumped model, f=0.1); $v_\infty = 1350$ km s$^{-1}$; $R_{*} = 0.43R_{\sun}$; $T_{*} = 74$kK. A chemical abundance of $\beta _{C} = 70$, $\beta _{He} = 29$, $\beta _{O} = 0.02$ and $\beta _{N} = 0.05$, (% by mass) was adopted. We can see from Fig. \[fitcmfgen\] that the fit is not perfect. Some C IV absorptions are predicted in the optical, and are not observed. Despite these problems, the fit provides valuable informations. The weak absorption lines left to the blend in 4650Å, are found to be due to N V. Furthermore, the lack of model emissions in the 4650Å blend, suggest that these emissions could have a nebular origin. More test models are currently being computed.
Conclusions:
============
We have investigated the UV spectra of all WELS with available IUE data. We have found that they can be divided in three different groups: (1) Strong P-Cygni profiles (mainly in C IV 1549); (2) Weak P-Cygni features and (3) Absence of P-Cygni profiles. We have derived terminal velocities for all objects presenting P-Cygni profiles in N V 1238 and/or C IV 1549 in a homogeneous way. Both the $v_\infty$ measurements and the UV characteristics observed indicate that the WELS are not \[WC\]-PG 1159 stars.
We have presented first non LTE expanding atmosphere models for the star Hen 2-12. Optical line fits are presented for the first time. Although interesting results could be obtained, in order to make an efficient comparison to the \[WR\] and PG 1159 classes, and to gain insight to the evolutionary status of the WELS, we clearly need an analysis of a large sample.
Many thanks to the LOC and the hospitality of Klaus Werner and Thomas Rauch at the Tübingen meeting.
Hillier, D. J., & Miller, D. L., 1998, ApJ, 496, 407 Koesterke, L., 2001, Ap&SS, 275, 41 Marcolino, W. L. F., de Araujo, F. X., Junior, H. M. B., & Duarte, E. S., 2007, AJ, 134, 1380 Parthasarathy, M., Acker, A., & Stenholm, B., 1998, A&A, 329, L9 Prinja, R. K., 1994, A&A, 289, 221 Tylenda, R., Acker, A., & Stenholm, B., 1993, A&AS, 102, 595 Werner, K., & Herwig, F., 2006, PASP, 118, 183
[^1]: Here, we neglect other type of hydrogen deficient objects such as R Coronae Borealis (RCB) and O(He) stars
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abstract: 'We study numerically stabilization against ionization of a fully correlated two-electron model atom in an intense laser pulse. We concentrate on two frequency regimes: very high frequency, where the photon energy exceeds both, the ionization potential of the outer [*and*]{} the inner electron, and an intermediate frequency where, from a “single active electron”-point of view the outer electron is expected to stabilize but the inner one is not. Our results reveal that correlation reduces stabilization when compared to results from single active electron-calculations. However, despite this destabilizing effect of electron correlation we still observe a decreasing ionization probability within a certain intensity domain in the high-frequency case. We compare our results from the fully correlated simulations with those from simpler, approximate models. This is useful for future work on “real” more-than-one electron atoms, not yet accessible to numerical [*ab initio*]{} methods.'
address: |
Theoretical Quantum Electronics (TQE)[@www], Darmstadt University of Technology,\
Hochschulstr. 4A, D-64289 Darmstadt, Germany
author:
- 'D. Bauer and F. Ceccherini'
title: |
Electron correlation vs. stabilization: A two-electron model atom\
in an intense laser pulse
---
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Introduction
============
The advent of high intensity lasers led to an increasing interest in non-perturbative studies of atomic systems interacting with intense laser light (see, e.g., [@prot] for a review). One of the most frequently revisited topics during the last fifteen years was [*stabilization*]{} of atoms (or ions) against ionization in intense laser light, i.e., for increasing laser intensity the ionization rate [*decreases*]{}. This kind of stabilization was predicted by Gersten and Mittleman already in 1975 [@mittle]. Experimentally, stabilization of highly excited atoms has been reported [@boer] whereas measuring stabilization of atoms initially in the ground state is hard to achieve. This is due to the fact that, in order to see stabilization, the laser photon energy has to exceed the ionization potential. Unfortunately, there are not yet high intensity lasers available delivering such energetic photons. Therefore most of the studies in this field are of analytical or numerical nature: “high-frequency theory” [@gavrila], Floquet calculations [@doerr; @faisal], the numerical treatment of 1D model atoms, quantum [@su; @grobe; @grobe_ii; @vivi] and classical [@rosen], as well as in two-color laser fields [@cheng], 2D atoms in arbitrary polarized laser light [@patel], and full 3D hydrogen [@kul]. Of particular interest is whether the atom survives in a “real” laser pulse up to intensities where stabilization sets in, or whether it already ionizes almost 100% during the rise time of the pulse [@lamb]. In other words: is the atom able to pass through the “death valley” of ionization before arriving at the “magic mountain” of stabilization? There are also several papers where the authors came to the conclusion that stabilization does not exist at all (see [@fring; @geltman], and references therein).
In this paper we focus on how the electron correlation in a two-electron model atom affects the probability for stabilization, i.e., the probability that the model atom remains neutral after the pulse has passed by. For two frequency regimes we compare the results from the fully correlated calculation with approximate models like “single active electron” or time-dependent density functional theory. The purpose of these studies is, on one hand, to gain a qualitative picture of the stabilization mechanism in a more-than-one electron atom, and, on the other hand, testing approximate methods before applying them to 3D many-electron atoms where accurate, full [*ab initio*]{} studies are not possible with current days computers. To our knowledge only a few other numerical studies of correlated two-electron systems in the stabilization regime are reported in the literature so far [@grobe_iii; @lewen; @volkova].
The model atom {#modelatom}
==============
We study a model helium atom where both electrons are allowed to move in one dimension only, but with the electron-electron correlation fully taken into account. This leads to a two-dimensional time-dependent Schrödinger equation (TDSE) = (t) with the Hamiltonian (t)= ( \_1 + A(t))\^2 + ( \_2 + A(t))\^2 - - + . \[full\_hamil\] Here, the laser pulse is coupled to the atom in dipole approximation through the vector potential $A(t)$. $\xoperator_i$ and $\poperator_i$ ($i=1,2$) are the electrons’ coordinates and canonical momenta, respectively. We use atomic units (a.u.) throughout this paper. The regularization parameter $\epsilon$ was chosen $0.49$ which yielded, on our numerical grid, ionization potentials similar to real helium (0.9 a.u.for the first electron, and 2 a.u. for the second one).
The electric field $E(t)=-\partial_t A(t)$ was a trapezoidal pulse with a rising edge over 5 optical cycles, 5 cycles of constant amplitude $\Edach$, and a down-ramp over, again, 5 cycles. We started all our simulations with the field-free ground state $\ket{\Psi(0)}$. The wavefunction $\Psi(x_1,x_2,t)=\braket{x_1 x_2}{\Psi(t)}$ was propagated in time using an unconditionally stable, explicit “grid hopping” algorithm [@raedt]. Non-vanishing probability amplitude $\Psi(x_1,x_2)$ near the grid boundary was removed through an imaginary potential. The numerical grid was always several times (at least 10 times) larger than the excursion length =/\^2= / of a classical electron oscillating in the laser field of frequency $\omega$ and electric field amplitude $\Edach$ (vector potential amplitude $\Adach$). During time propagation we monitored the amount of probability density $\vert\Psi\vert^2$ inside a box $x_1,x_2\in[-5,+5]$. After the pulse is over, the density inside this box can be interpreted as the “survival” probability of the helium atom to remain neutral [@box_comment].
To analyze the results obtained with this fully correlated model atom we compare with several simplified models. Among those, the “single active electron” (SAE) approximation is the simplest one. There, one assumes that an [ *inner*]{} and an [*outer*]{} electron respond independently to the laser field. The inner electron “feels” the bare nucleus ($Z_i=2$, hydrogen-like). The outer one sees an effective nuclear charge, to be adjusted in such a way that the correct ionization potential (0.9 a.u.) is obtained. In our numerical model this was the case for $Z_o=1.1$. Thus, in the SAE approximation, we solved two independent TDSEs with no dynamic correlation at all, $$\begin{aligned}
\imagi\partial_t\Psi_i(x,t) &=& \left(\halb( -\imagi\partial_x + A(t))^2
- \frac{Z_i}{\sqrt{x^2 + \epsilon}}\right) \Psi_i(x,t), \\
\imagi\partial_t\Psi_o(x,t) &=& \left(\halb( -\imagi\partial_x + A(t))^2
- \frac{Z_o}{\sqrt{x^2 + \epsilon}}\right) \Psi_o(x,t).\end{aligned}$$
In order to incorporate correlation in a first step one can introduce a Hartree-type potential into the Hamiltonian for the inner electron, $$\begin{aligned}
\imagi\partial_t\Psi_i(x,t) &=& \left(\halb( -\imagi\partial_x + A(t))^2
- \frac{Z_i}{\sqrt{x^2 + \epsilon}} + \int \frac{\vert
\Psi_o(x',t)\vert^2}{\sqrt{(x-x')^2 + \epsilon}} \diff x'
\right) \Psi_i(x,t), \label{iso_i} \\
\imagi\partial_t\Psi_o(x,t) &=& \left(\halb( -\imagi\partial_x + A(t))^2
- \frac{Z_o}{\sqrt{x^2 + \epsilon}}\right) \Psi_o(x,t). \label{iso_ii}\end{aligned}$$ In this approximation, the inner electron feels the bare nuclear potential [*and*]{} the outer electron. Therefore, we call this model “inner sees outer” (ISO) approximation. It was utilized in Ref. [@watson] to study non-sequential ionization (NSI). In the ground state, the Hartree-potential leads to a screening of the bare nuclear charge. Thus, energetically the two electrons are almost equivalent in the beginning, though we labelled them “inner” and “outer” in Eqs. , . However, during the interaction with the laser field one of the electrons might become the outer one. We will also consider the opposite point of view where the outer electron sees the inner one (“outer sees inner”, OSI). In this case we have to deal with the system of TDSEs $$\begin{aligned}
\imagi\partial_t\Psi_i(x,t) &=& \left(\halb( -\imagi\partial_x + A(t))^2
- \frac{Z}{\sqrt{x^2 + \epsilon}}
\right) \Psi_i(x,t), \label{osi_i} \\
\imagi\partial_t\Psi_o(x,t) &=& \left(\halb( -\imagi\partial_x + A(t))^2
- \frac{Z}{\sqrt{x^2 + \epsilon}}+ \int \frac{\vert
\Psi_i(x',t)\vert^2}{\sqrt{(x-x')^2 + \epsilon}} \diff x'\right) \Psi_o(x,t). \label{osi_ii}\end{aligned}$$ with $Z=2$.
Finally, another way to study our model system is to apply time-dependent density functional theory (TDDFT) (see [@gross] for an overview) in local density approximation, leading to the (nonlinear) TDSE for [*one*]{} Kohn-Sham orbital $\Phi(x,t)$, \_t(x,t) = (( -\_x + A(t))\^2 - + x’ ) (x,t). \[dft\] The total electron probability density is given by $n(x,t)=2\vert\Phi(x,t)\vert^2$. Since, strictly speaking, $\Phi(x,t)$ cannot be interpreted as a physically meaningful single electron orbital it is not easy to deduce single electron quantities (such as single ionization for instance). On the other hand, the somewhat arbitrary distinction between an inner and an outer electron is avoided in TDDFT.
High-frequency results
======================
Single active electron-approximation {#sae_section}
------------------------------------
In this Section we want to compare the results from the fully correlated model atom with those from the corresponding SAE calculations. First, we focus on the high-frequency regime where the laser frequency exceeds both ionization potentials, $\omega = \pi > \energy_i >\energy_o$. From an SAE-point of view we expect both electrons to stabilize, especially the outer one since the frequency is 3.5 times larger than the ionization potential $\energy_o=0.9$. In Fig. \[fig\_i\] the amount of probability density inside the box $x_1,x_2\in[-5,+5]$ (PDIB) vs. time for $\alphadach=0.5,1.0,1.5$ and $2.0$ is shown for the inner and the outer electron. First we observe the expected result that the outer electron is more stabilized than the inner one: the amount of PDIB after the pulse has gone is greater for the outer electron. However, qualitatively the set of curves are quite similar. For the two higher $\alphadach$-values ($1.5$, drawn dashed, and $2.0$, drawn dashed-dotted) one observes that the curves bend sharply after the up-ramping at $t=5$ cycles, i.e., for both electrons ionization is much slower during the constant, intense part of the pulse. In fact, ionization happens almost exclusively during the rampings. We also note that the slight decrease of the PDIB during the constant part of the $\alphadach=1.5$ and $\alphadach=2.0$-pulses is linear in time, in contrast to tunneling ionization where we have an exponential dependence (in the case of a constant ionization rate). Approximately two cycles after the down-ramping at $t=10$ cycles, ionization starts to increase again. Finally, after the pulse is over (at $t=15$ cycles) the amount of PDIB remains stationary. Secondly, we observe that there is obviously no monotonous behavior of stabilization with increasing intensity (or $\alphadach$). Ionization is higher for $\alphadach=1.0$ (dotted curves) than for $\alphadach=0.5$ (solid curves). For $\alphadach=1.5$ (dashed curves) ionization starts to decrease, i.e., we are entering the stabilization domain at that point. Therefore the so-called “death valley” seems to be located around $\alphadach=1$ for [*both*]{} electrons in our model atom.
In Fig. \[fig\_ii\] the stabilization probability for both electrons is shown vs. the excursion $\alphadach$. The quiver amplitude $\alphadach$ is related to the laser intensity $I=\Edach^2$ through $\alphadach=I^{1/2}\omega^{-2}$. The stabilization probability of the inner electron exhibits an oscillatory behavior. The “death valley” is located at $\alphadach\approx 1$, followed by a maximum at $\alphadach\approx 4$. For higher intensity ionization increases again up to a stabilization minimum around $\alphadach\approx 7$. Stabilization of the inner electron recovers till the next maximum at $\alphadach
\approx 9$. The second maximum is below the first. Thus we observe an overall decrease of stabilization with increasing intensity. This is even more pronounced in the stabilization probability for the outer electron where the oscillations are less visible. The “death valley” for both electrons is located at $\alphadach\approx 1$ while the maxima are at different positions. The oscillatory character of the stabilization probability in 1D systems has been observed by other authors as well [@yao; @mill; @su_ii]. In contrast to our results an overall increase of stabilization with increasing intensity was found in [@yao; @mill]. This might be due to the fact that we are looking at the ionization [*probability*]{} after the pulse is over while in analytical papers often the ionization [*rate*]{} is discussed. In the former, ionization during the up and down-ramps is taken into account while in the latter it is commonly not.
The probability for our He model atom to survive as neutral He after the pulse is over is, in SAE approximation, simply the product of the probabilities for each electron to remain bound. In Fig. \[fig\_ii\] the corresponding curve is indicated by i$\cdot$o. The result from the fully correlated system is also shown (drawn dotted, indicated with ‘corr’). We infer that, especially for $\alphadach < 5$ the stabilization probability is strongly overestimated by the SAE treatment. We could argue that [*if*]{} the system stabilizes it will probably stabilize in such a way that the correlation energy is minimized. In that case it sounds more reasonable to take the square of the SAE stabilization probability for the [*inner*]{} electron. In what follows we will refer to this viewpoint as “independent electron” (IE) model since it follows from crossing out the correlation term in the Hamiltonian . The result is included in Fig. \[fig\_ii\], labelled i$\cdot $i. The IE-curve seems to oscillate around the fully correlated one, especially for $\alphadach>6$. From this result we conclude that, compared to the IE-model with two equivalent inner electrons, electron correlation washes out oscillations in the stabilization probability and, therefore, can stabilize as well as destabilize, depending on the intensity (for a given pulse shape and frequency).
To discuss that further we look at the time-averaged Kramers-Henneberger potential (TAKHP), i.e., we transform to the frame of reference where the quivering electron is at rest but the nuclear potential oscillates, and average over one cycle, V\_[KH]{}\^[corr]{}(x\_1,x\_2)=\_[i=1]{}\^2 \_0\^[2/]{} t + .\[av\_KH\] For sufficiently high frequencies this is the leading term in a perturbation series in $\omega^{-1}$ [@gavrila]. In the correlation term no $\alpha(t)$ appears since the interparticle distance is not affected by the KH transformation. We calculated numerically the TAKHP. The result is shown in Fig. \[fig\_iii\] for $\alpha(t)=\alphadach \sin\omega t$ with $\alphadach=5$. For comparison the TAKHP with the correlation term neglected is also shown (corresponding to the IE model). With correlation, there are two minima near $x_1=\alphadach, x_2=-\alphadach$ and $x_1=-\alphadach, x_2=\alphadach$ whereas without correlation there are two more, energetically equivalent minima at $x_1=x_2=\alphadach$ and $x_1=x_2=-\alphadach$. However, if we assume that the fully correlated system manages it [*somehow*]{} to occupy the ground state of $V_{KH}^{corr}$, the correlation energy will be small (for not too small $\alphadach$) since the interparticle distance is $2\alphadach$. The higher $\alphadach$ the lower the correlation energy. We believe that this is the physical reason that, for increasing $\alphadach$, the agreement of the IE results with the fully correlated ones becomes quite good (although the latter do not exhibit an oscillating stabilization probability). Our viewpoint is further supported by examining the probability density of the fully correlated system during the pulse. In Fig. \[fig\_iv\] $\vert\Psi(x_1,x_2)\vert^2$ is shown for $\omega=\pi$, $\alphadach=4.0$ at $t=7.5$ cycles, i.e., in the middle of the constant part of the trapezoidal pulse. We clearly observe [*dichotomy*]{}, i.e., two probability density peaks at the classical turning points, well known from one-electron systems [@gavrila]. Due to electron correlation we do not observe four peaks. Instead the peaks at $x_1=x_2=\pm 4$ are suppressed, in accordance with our discussion of the TAKHPs in Fig. \[fig\_iii\]. Therefore the correlation energy is rather small since the distance between the two peaks in the $x_1x_2$-plane is $\sqrt{8}\alphadach$. In the work by Mittleman [@mittle_ii] such multi-electron “dichotomized” bound states are calculated.
Time-dependent density functional theory {#tddft_section}
----------------------------------------
In Fig. \[fig\_v\] our results from the TDDFT calculations are presented. Although the Kohn-Sham orbital $\Phi(x,t)$ is an auxillary entity that has, in a rigorous sense, no physical meaning, we take it as an approximation to a single electron orbital. If we do this, $\vert\Phi(x,t)\vert^2\cdot \vert\Phi(x,t)\vert^2$, integrated over the region $-5 < x < 5$ after the pulse is over, is our TDDFT stabilization probability. We see that for $\alphadach<1.5$ the agreement between TDDFT and correct result is very good. The difference between TDDFT and IE (indicated by i$\cdot$i again) is a direct measure of correlation effects since both models differ by the Hartree-term in the Hamiltonian only. Up to $\alphadach\approx
5.5$ electron correlation suppresses stabilization compared to the IE approximation. In that region TDDFT agrees better with the full result. As mentioned before, for higher $\alphadach$ the IE curve oscillates around the correct result and therefore it comes occasionly to a very good agreement with the exact result. Also the TDDFT result agrees very well with the fully correlated curve for $\alphadach\geq 7$. In summary we can say that the TDDFT result is in good agreement with the exact, fully correlated stabilization probability. Both have their maximum around $\alphadach\approx 4$ and the “death valley” is also at the right position. For higher $\alphadach$ the agreement seems to become even better.
“Inner sees outer” and “outer sees inner”-approximation
-------------------------------------------------------
In order to explain non-sequential ionization (NSI; see, e.g., Ref. [@prot] for an overview) it is essential to incorporate electron correlation (see [@becker] for a very recent paper, and references therein). For that purpose Watson [*et al.*]{} [@watson] added a Hartree-type potential to the TDSE for the inner electron (see Eq.\[iso\_i\], ISO). By doing this the double ionization yield is greatly enhanced, in accordance with experimental results [@walker]. The question we address in this Section is whether this method is applicable to stabilization as well. We will also study the opposite procedure, i.e., where the outer electron feels the inner one (see Eqs. \[osi\_i\] and \[osi\_ii\], OSI). From the discussions on the SAE approximation above we can expect that ISO will probably not agree very well with the exact results since the assumption that the outer electron sees a static, effective nuclear charge is not valid. In Fig. \[fig\_vi\] we see that, for low $\alphadach$ ISO ($\bigtriangleup$) behaves like SAE (the i$\cdot$o-curve) while OSI ($\Diamond$) is similar to IE (the curve indicated by i$\cdot$i). In ISO approximation for $\alphadach>3$ the electron correlation obviously causes strong ionization, compared to the SAE result. Especially during the down-ramping, when probability density of the outer electron moves from the turning points $\pm\alphadach$ back toward the nucleus, ionization of the inner electron is enhanced. For $\alphadach>4.5$ the ISO curve even drops below the exact result (indicated by ‘corr’). In OSI approximation the stabilization probability is also underestimated for $\alphadach>4.5$. In summary we can say, that for $\alphadach<2$, OSI is in very good agreement with the correct result while ISO is not, due to the inappropriate assumption of the outer electron feeling just a static effective nuclear charge. However, for higher $\alphadach$ ISO and OSI tend to underestimate the stabilization probability while TDDFT does not (see Section \[tddft\_section\] and Fig. \[fig\_v\]).
Intermediate frequency results
==============================
In this Section we discuss the stabilization probability in the intermediate frequency regime $\energy_o < \omega < \energy_i$ where, according a single active electron point-of-view, the outer electron should stabilize while for the inner one ionization is more likely. In Fig. \[fig\_vii\] we compare the result from the fully correlated calculation with those from the SAE treatment. In SAE approximation, the outer electron is more stable than the inner one in the region $1.5 < \alphadach <
8.5$. For the inner electron no clear stabilization maximum is visible. For the outer electron the maximum is at $\alphadach\approx 6$, i.e., it is shifted toward higher $\alphadach$ compared to the high-frequency case. Both, i$\cdot$o and i$\cdot$i underestimates ionization, especially for low $\alphadach$ in the “death valley”-region. Electron correlation obviously enhances ionization. For lower frequencies this is the well-known effect of NSI ([@becker] and references therein). Although in the fully correlated result we observe a stabilization probability maximum around $\alphadach\approx 7.5$ the absolute value is below $0.04$, and, in our opinion, it makes no sense to talk about real “stabilization” in that case. As in the high-frequency-case we observe an overall decrease of the stabilization probability for very high $\alphadach$-values.
In Fig. \[fig\_viii\] we compare the result from the fully correlated model atom with the corresponding ones from the ISO, OSI, and TDDFT runs where electron correlation is included approximately. Let us first focus on the $\alphadach$-region “left from the death valley”, i.e., $\alphadach\leq
1$. There we observe that ISO is nearest to the correct result while OSI and TDDFT underestimates ionization. This is quite understandable within the present knowledge of how NSI works: the inner electron needs to interact with laser field [*and*]{} the outer electron in order to become free. Obviously this is best accounted for in the ISO approximation. For most $\alphadach$ TDDFT lies between the ISO and the OSI result. This is also quite clear since in TDDFT both correlated electrons are treated on an equal footing (one Kohn-Sham-orbital only) whereas in ISO (OSI) the inner (outer) electron feels the outer (inner) partner through Coulomb correlation, but not vice verse. However, all these approximations still overestimate the stabilization probability, at least in the interesting $\alphadach$-regime where the stabilization probability rises at all (i.e., for $ 2 < \alphadach < 7.5$).
To summarize this Section we can say that in order to achieve stabilization of our two-electron model atom it is necessary to choose a laser frequency that exceeds [*all*]{} ionization potentials. For an intermediate frequency the outer electron cannot stabilize owing to correlation. The SAE picture is not appropriate and even ISO, OSI, or TDDFT where electron correlation is included approximately fail.
Discussion and summary
======================
In this paper we studied how the electron correlation in a two-electron model atom affects the probability for stabilization. We found clear stabilization only for frequencies that exceed [*both*]{} ionization potentials. Although for the intermediate frequency we did not find a monotonous increase of the ionization probability with increasing intensity we prefer not calling this effect stabilization since, on an absolute scale, its probability was very small. In all cases electron correlation reduced the stabilization probability compared to the SAE picture. In the high-frequency case the two electrons behave more like two independent [*inner*]{} electrons. Similar results were obtained by Grobe and Eberly [@grobe_iii] for a H$^-$ model-ion. Lewenstein [*et al.*]{} [@lewen] performed classical calculations for a model-atom similar to ours which also showed that “dichotomized” two-electron states are dynamically accessible.
The agreement of the exact numerical result with TDDFT in the high-frequency case was quite good while in the intermediate frequency regime stabilization was overestimated by [*all*]{} approximate techniques (ISO, OSI, and TDDFT).
It is well-known that a slow time-scale in the stabilization dynamics is introduced owing to floppings between states in the time-averaged Kramers-Henneberger potential [@gavrila; @vivi]. It can be easily imagined that these slow floppings are affected by electron correlation because, e.g., the merging of the two dichotomous peaks into a single one is suppressed then. But even without correlation the results for the stabilization probability are quite sensitive to rise time and pulse duration of the laser field since it strongly depends on which Kramers-Henneberger states are mainly occupied at the time instant when the laser pulse ends. To avoid these additional complications in the interpretation of the numerical results we chose a rather short laser pulse duration so that low-frequency Rabi-floppings do not play a role. Therefore, in the high-frequency studies we just observed the two dichotomous peaks building up (as depicted in Fig. \[fig\_iv\]) but no peak-merging during the constant part of our trapezoidal laser pulse.
Finally, we would like to comment on the reduced dimensionality of our two-electron model atom. We also performed calculations with “real”, i.e., three-dimensional (3D) hydrogen-like ions in the stabilization regime. It seems to be the case that in 3D stabilization is less pronounced. Moreover, the oscillatory character is less visible, i.e., we observe a single stabilization maximum followed by a rather monotonous increase of the ionization probability. The difference of 1D models and 3D hydrogen was also studied in Ref. [@mill]. The effect of electron correlation in 3D stabilization will be the subject of a future paper [@cecch].
Acknowledgment {#acknowledgment .unnumbered}
==============
Fruitful discussions with Prof. P. Mulser are gratefully acknowledged. This work was supported in part by the European Commission through the TMR Network SILASI (Superintense Laser Pulse-Solid Interaction), No.ERBFMRX-CT96-0043, and by the Deutsche Forschungsgemeinschaft under Contract No. MU 682/3-1.
[99]{} http://www.physik.tu-darmstadt.de/tqe/ M. Protopapas, C. H. Keitel, and P. L. Knight, Rep. Progr. Phys. [**60**]{}, 389 (1997). J. I. Gersten and M. H. Mittleman, Phys. Rev. A [ **11**]{}, 1103 (1975). M. P. de Boer, J. H. Hoogenraad, R. B. Vrijen, L. D. Noordam, and H. G. Muller, Phys. Rev. Lett. [**71**]{}, 3263 (1993); M. P. de Boer, J. H. Hoogenraad, R. B. Vrijen, and L. D. Noordam, Phys. Rev. A [**50**]{}, 4133 (1994); N. J. van Druten, R. C. Constantinescu, J. M. Schins, H. Nieuwenhuize, and H. G. Muller, Phys. Rev. A [**55**]{}, 622 (1997). M. Gavrila in: [*Atoms in Intense Laser Fields*]{} ed. by M. Gavrila (Academic, New York, 1992), p. 435, and references therein. Martin Dörr, R. M. Potvliege, Daniel Proulx, and Robin Shakeshaft, Phys. Rev. A [**43**]{}, 3729 (1991). F. H. M. Faisal, L. Dimou, H.-J. Stiemke, M. Nurhuda, Journ. of Nonl. Opt. Phys. and Materials [**4**]{}, 701 (1995). Q. Su, J. H. Eberly, and J. Javanainen, Phys. Rev. Lett., 862 (1990). R. Grobe and J. H. Eberly, Phys. Rev. Lett. [ **68**]{},2905 (1992). R. Grobe and J. H. Eberly, Phys. Rev. A [**47**]{}, R1605 (1993). R. M. A. Vivirito and P. L. Knight, J. Phys. B: At. Mol. Opt. Phys. [**28**]{}, 4357 (1995). A. T. Rosenberger, C. C. Sung, S. D. Pethel, and C. M. Bowden, Phys. Rev. A [**56**]{}, 2459 (1997). Taiwang Chen, Jie Liu, and Shigang Chen, Phys. Rev. A [ **59**]{}, 1451 (1999). A. Patel, M. Protopapas, D. G. Lappas, and P. L. Knight, Phys. Rev. A [**58**]{}, R2652 (1998). Kenneth C. Kulander, Kenneth J. Schafer, and Jeffrey L. Krause, Phys. Rev. Lett. [**66**]{}, 2601 (1991). P. Lambropoulos, Phys. Rev. Lett. [**55**]{}, 2141 (1985). A. Fring, V. Kostrykin, and R. Schrader, J. Phys. B: At. Mol. Opt. Phys. [**29**]{}, 5651 (1996). S. Geltman, J. Phys. B: At. Mol. Opt. Phys. [**32**]{}, 853 (1999). R. Grobe and J. H. Eberly, Phys. Rev. A [**47**]{}, R1605 (1993). Maciej Lewenstein, Kazimierz Rzażewski, and Pascal Salières, in [*Super-Intense Laser-Atom Physics*]{}, ed. by B. Piraux et al., (Plenum, New York, 1993), p. 425. E. A. Volkova, A. M. Popov, and O. V. Tikhonova, JETP [**87**]{}, 875 (1998); \[Zh. Eksp. Teor. Fiz. [**114**]{}, 1618 (1998)\] . Hans DeRaedt, Comp. Phys. Rep. [**7**]{}, 1 (1987). The atom might be in an excited state after the pulse is over. In order to check this and allow for autoionization we continued the calculation for another 10 atomic time units (for $\omega=\pi$) or 20 a.u. (for $\omega=\pi/2$). J. B. Watson, A. Sanpera, D. G. Lappas, P. L. Knight, and K. Burnett, Phys. Rev. Lett. [**78**]{}, 1884 (1997). E. K. U. Gross, J. F. Dobson, and M. Petersilka, in [ *Topics in Current Chemistry, Vol. 181*]{} (Springer, Berlin, 1996), p. 81. Guanhua Yao and Shih-I Chu, Phys. Rev. A [**45**]{}, 6735 (1992). Thomas Millack, J. Phys. B: At. Mol. Opt. Phys. [**26**]{}, 4777 (1993). Q. Su, B. P. Irving, C. W. Johnson, and J. H. Eberly, J. Phys. B: At. Mol. Opt. Phys. [**29**]{}, 5755 (1996). Marvin H. Mittleman, Phys. Rev. A [**42**]{}, 5645 (1990). A. Becker and F. H. M. Faisal, Phys. Rev. A [**59**]{}, R1742 (1999). B. Walker, B. Sheehy, L. F. DiMauro, P. Agostini, K. J. Schafer, and K. C. Kulander, Phys. Rev. Lett. [**73**]{}, 1227 (1994). F. Ceccherini, D. Bauer, and P. Mulser, to be submitted.
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abstract: 'Starting from a general operator representation in the time-frequency domain, this paper addresses the problem of approximating linear operators by operators that are diagonal or band-diagonal with respect to Gabor frames. A characterization of operators that can be realized as Gabor multipliers is given and necessary conditions for the existence of (Hilbert-Schmidt) optimal Gabor multiplier approximations are discussed and an efficient method for the calculation of an operator’s best approximation by a Gabor multiplier is derived. The spreading function of Gabor multipliers yields new error estimates for these approximations. Generalizations (multiple Gabor multipliers) are introduced for better approximation of overspread operators. The Riesz property of the projection operators involved in generalized Gabor multipliers is characterized, and a method for obtaining an operator’s best approximation by a multiple Gabor multiplier is suggested. Finally, it is shown that in certain situations, generalized Gabor multipliers reduce to a finite sum of regular Gabor multipliers with adapted windows.'
address:
- 'Acoustics Research Institute, Austrian Academy of Science, Wohllebengasse 12-14, A-1040 Vienna, Austria'
- 'Laboratoire d’Analyse, Topologie et Probabilités, Centre de Mathématique et d’Informatique, 39 rue Joliot-Curie, 13453 Marseille cedex 13, France'
author:
- Monika Dörfler
- Bruno Torrésani
title: 'Representation of operators in the time-frequency domain and generalized [G]{}abor multipliers '
---
[^1]
[^2]
Introduction {#se:intro}
============
The goal of time-frequency analysis is to provide efficient representations for functions or distributions in terms of decompositions such as $${ f =
\sum_{\lambda\in\Lambda} \langle f,g_\lambda \rangle h_\lambda\ .
}$$ Here, $f$ is expanded as a weighted sum of [*atoms*]{} $ h_\lambda$ well localized in both time and frequency domains. The time-frequency coefficients $ \langle f,g_ \lambda \rangle$ characterize the function under investigation, and a [*synthesis map*]{} usually allows the reconstruction of the original function $f$.\
Concrete applications can be found mostly in signal analysis and processing (see [@Carmona98practical; @Daubechies92ten; @Mallat98wavelet] and references therein), but recent works in different areas such as numerical analysis may also be mentioned (see for example [@fest98; @fest03] and references therein).
Time-frequency analysis of operators, originating in the work on communication channels of Bello [@be63], Kailath [@ka62] and Zadeh [@za61], has enjoyed increasing interest during the last few years, [@hlma02; @kopf06; @pfwa05-1; @baporive05]. Efficient time-frequency operator representation is a challenging task, and often the intuitively appealing approach of operator approximation by modification of the time-frequency coefficients before reconstruction is the method of choice. If the modification of the coefficients is confined to be multiplicative, this approach leads to the model of time-frequency multipliers, as discussed in Section \[se:mult\]. The class of operators that may be well represented by time-frequency multipliers depends on the choice of the parameters involved and is restricted to operators performing only small time-shifts or modulations.
The work in this paper is inspired by a general operator representation in the time-frequency domain via a *twisted convolution*. It turns out, that this representation, respecting the underlying structure of the Heisenberg group, has an interesting connection to the so-called spreading function representation of operators. An operator’s spreading function comprises the amount of time-shifts and modulations, i.e. of time-frequency-shifts, effected by the operator. Its investigation is hence decisive in the study of time-frequency multipliers and their generalizations. Although no direct discretization of the continuous representation by an operator’s spreading function is possible, the twisted convolution turns out to play an important role in the generalizations of time-frequency multipliers. In the main section of this article, we introduce a general model for [*m*ultiple Gabor multipliers (MGM)]{}, which uses several synthesis windows simultaneously. Thus, by jointly adapting the respective masks, more general operators may be well-represented than by regular Gabor multipliers. Specifying to a separable mask in the modification of time-frequency coefficients within MGM, as well as a specific sampling lattice for the synthesis windows, it turns out, that the MGM reduces to one or the sum of a finite number of regular Gabor multipliers with adapted synthesis windows.\
For the sake of generality, most statements are given in a *Gelfand-triple*, rather than a pure Hilbert space setting. This choice bears several advantages. First of all, many important operators and signals may not be described in a Hilbert-space setting, starting from simple operators as the identity. Furthermore, by using distributions, continuous and discrete concepts may be considered together. Finally, the Gelfand-triple setting often allows for short-cut proofs of statements formulated in a general context.
This paper is organized as follows. The next section gives a review of the time-frequency plane and the corresponding continuous and discrete transforms. We then introduce the concept of Gelfand triples, which will allow us to consider operators beyond the Hilbert-Schmidt framework. The section closes with the important statement on operator-representation in the time-frequency domain via twisted convolution with an operator’s spreading function. Section \[Se:tfmult\] introduces time-frequency multipliers and gives a criterion for their ability to approximate linear operators. A fast method for the calculation of an operator’s best approximation by a Gabor multiplier in Hilbert-Schmidt sense is suggested. Section 3 introduces generalizations of Gabor multipliers. The operators in the construction of MGM are investigated and a criterion for their Riesz basis property in the space of Hilbert-Schmidt operators is given. We mention some connections to classical Gabor frames. A numerical example concludes the discussion of general MGM. In the final section, TST (twisted spline type) spreading functions are introduced. It is shown, that under certain conditions, a MGM reduces to a regular Gabor multiplier with an adapted window or a finite sum of regular multipliers with the same mask and adapted windows.
Operators from the Time-frequency point of view {#se:TFop}
===============================================
Whenever one is interested in time-localized frequency information in a signal or operator, one is naturally led to the notion of the time-frequeny plane, which, in turn, is closely related to the Weyl-Heisenberg group.
Preliminaries: the time-frequency plane
---------------------------------------
The starting point of our operator analysis is the so-called [*spreading function operator representation*]{}. This operator representation expresses linear operators as a sum (in a sense to be specified below) of time-frequency shifts $\pi(b,\nu) = M_\nu T_b$. Here, the translation and modulation operators are defined as $$T_b f(t) = f(t-b)\ , \quad M_\nu f(t) = e^{2i\pi\nu t}f(t)\ ,
\quad f\in \mathbf{L}^2(\RR)\ .$$ These (unitary) operators generate a group, called the Weyl-Heisenberg group $$\label{fo:WH.group}
\HH = \left\{(b,\nu,\varphi)\in\RR\times\RR\times [0,1[\right\}\ ,$$ with group multiplication $$(b,\nu,\varphi)(b',\nu',\varphi') = (b+b',\nu+\nu',\varphi+\varphi'-\nu'b)\ .$$ The specific quotient space $\PP= \HH / [0,1]$ of the Weyl-Heisenberg group is called [*phase space*]{}, or [*time-frequency plane*]{}, which plays a central role in the subsequent analysis. Details on the Weyl-Heisenberg group and the time-frequency plane may be found in [@Folland89harmonic; @Schempp86harmonic]. In the current article, we shall limit ourselves to the basic irreducible unitary representation of $\HH$ on $\mathbf{L}^2(\RR)$, denoted by $\pi^o$, and defined by $$\pi^o(b,\nu,\varphi) = e^{2i\pi\varphi}M_\nu T_b\ .$$ By $\pi(b,\nu) = \pi^o(b,\nu,0)$ we denote the restriction to the phase space. We refer to [@Dorfler07spreading] or [@gr01 Chapter 9] for a more detailed analysis of this quotient operation.
The left-regular (and right-regular) representation generally plays a central role in group representation theory. By unimodularity of the Weyl-Heisenberg group, its left and right regular representations coincide. We thus focus on the left-regular one, acting on $\mathbf{L}^2(\HH)$ and defined by $$\label{fo:left.regular}
\big[L(b',\nu',\varphi')F\big](b,\nu, \varphi) =
F(b-b',\nu-\nu',\varphi-\varphi' +b'(\nu-\nu'))\ .$$ Denote by $\mu$ the Haar measure. Given $F,G\in \mathbf{L}^2(\HH,d\mu)$, the associated (left) convolution product is the bounded function $F*G$, given by $$(F * G)(b,\nu,\varphi) = \int_\HH F(h)
\big[L(b,\nu,\varphi)G\big](h)\, d\mu(h).$$ After quotienting out the phase term, this yields the [*twisted convolution*]{} on $\mathbf{L}^2(\PP)$: $$\label{fo:twisted.conv}
(F\natural G) (b,\nu) = \int\bds\int\bds
F(b',\nu') G(b-b',\nu-\nu') e^{-2i\pi b'(\nu-\nu')}\,db'd\nu'\ .$$ The twisted convolution, which admits a nice interpretation in terms of group Plancherel theory [@Dorfler07spreading] is non-commutative (which reflects the non-Abelianess of $\HH$) but associative. It satisfies the usual Young inequalities, but is in some sense nicer than the usual convolution, since $\mathbf{L}^2(\RR^2) \natural \mathbf{L}^2(\RR^2) \subset
\mathbf{L}^2(\RR^2)$ (see [@Folland89harmonic] for details).
As explained in [@Grossmann85transforms; @Grossmann86transforms] (see also [@Fuhr05abstract] for a review), the representation $\pi^o$ is unitarily equivalent to a subrepresentation of the left regular representation. The representation coefficient is given by a variant of the [*short time Fourier transform*]{} (STFT), which we define next.
Let $g\in\mathbf{L}^2(\RR)$, $g\ne 0$. The STFT of any $f\in\mathbf{L}^2(\RR)$ is the function on the phase space $\PP$ defined by $${\mathscr{V}}_g f(b,\nu) = \langle f,\pi(b,\nu)g\rangle = \int\bds f(t) \overline{g}(t-b)
e^{-2i\pi\nu t}\,dt\ .$$
This STFT is obtained by quotienting out $[0,1]$ in the group transform $${\mathscr{V}}_g^o f(b,\nu,\varphi) = \langle f,\pi^o(b,\nu,\varphi)g\rangle\ .$$ The integral transform ${\mathscr{V}}_g^o$ intertwines $L$ and $\pi^o$, i.e. $L(h) {\mathscr{V}}_g^o = {\mathscr{V}}_g^o\pi^o(h)$ for all $h\in\HH$. The latter relation still holds true (up to a phase factor) when $\pi^o$ and ${\mathscr{V}}_g^o$ are replaced with $\pi$ and ${\mathscr{V}}_g$ respectively.
It follows from the general theory of square-integrable representations that for any $g\in\mathbf{L}^2(\RR)$, $g\ne 0$, the transform ${\mathscr{V}}_g^o$ is (a multiple of) an isometry $\mathbf{L}^2(\RR)\to
\mathbf{L}^2(\PP)$, and thus left invertible by the adjoint transform (up to a constant factor). More precisely, given $h\in\mathbf{L}^2(\RR)$ such that $\langle g,h\rangle\ne 0$, one has for all $f\in\mathbf{L}^2(\RR)$ $$\label{fo:STFT.inv}
f = \frac1{\langle h,g\rangle}\,
\int_\PP {\mathscr{V}}_gf(b,\nu)\,\pi(b,\nu) h\,dbd\nu\ .$$ We refer to [@Carmona98practical; @gr01] for more details on the STFT and signal processing applications.
The STFT, being a continuous transform, is not well adapted for numerical calculations, and, for practical issues, is replaced by the Gabor transform, which is a sampled version of it. To fix notation, we outline some steps of the Gabor frame theory and refer to [@Daubechies92ten; @gr01] for a detailed account.
\[Def:GabT\] Given $g\in\mathbf{L}^2(\RR)$ and two constants $b_0,\nu_0\in\RR^+$, the corresponding Gabor transform associates with any $f\in\mathbf{L}^2(\RR)$ the sequence of Gabor coefficients $${\mathscr{V}}_gf(mb_0,n\nu_0) = \langle f,M_{n\nu_0} T_{mb_0} g\rangle
= \langle f,g_{mn}\rangle\ ,$$ where the functions $
g_{mn}=M_{n\nu_0} T_{mb_0} g
$ are the Gabor atoms associated to $g$ and the lattice constants $b_0,\nu_0$.
Whenever the Gabor atoms associated to $g$ and the given lattice $\Lambda = b_0 \mathbb{Z}\times \nu_0\mathbb{Z}$ form a frame,[^3] the Gabor transform is left invertible, and there exists $h\in\mathbf{L}^2(\RR)$ such that any $f\in\mathbf{L}^2(\RR)$ may be expanded as $$f = \sum_{m,n} {\mathscr{V}}_gf(mb_0,n\nu_0)h_{mn}\ .$$
The Gelfand triple $( S_0, \mathbf{L}^2, S_0')$ {#Sec: Gelftrip}
-----------------------------------------------
We next set up a framework for the exact description of operators we are interested in. In fact, by their property of being compact operators, the Hilbert space of Hilbert-Schmidt operators turns out to be far too restrictive to contain most operators of practical interest, starting from the identity. Although the classical triple $({\mathscr{S}}, \mathbf{L}^2, {\mathscr{S}}')$ might seem to be the appropriate choice of generalization, we prefer to resort to the Gelfand triple $( S_0, \mathbf{L}^2, S_0')$, which has proved to be more adapted to a time-frequency environment. Additionally, the Banach space property of $S_0$ guarantees a technically less elaborate account.
\[defS0\] Let ${\mathscr{S}}(\mathbb{R})$ denote the Schwartz class. Fix a non-zero “window” function $\varphi \in {\mathscr{S}}(\mathbb{R} )$. The space $S_0 (\mathbb{R} ) $ is given by $$S_0 (\mathbb{R} ) = \{ f\in \mathbf{L}^2(\mathbb{R}) :
\| f\|_{S_0}:= \|\mathcal{V}_{\varphi}
f\|_{L^1(\mathbb{R}^2)}<\infty\}.$$
The following proposition summarizes some properties of $S_0(\mathbb{R} )$ and its dual, the distribution space $S_0'(\mathbb{R} )$.
\[Pro:S0\] $S_0(\mathbb{R} )$ is a Banach space and densely embedded in $\mathbf{L}^2(\mathbb{R}) $. The definition of $S_0(\mathbb{R})$ is independent of the window $\varphi \in
{\mathscr{S}}(\mathbb{R} ) $, and different choices of $\varphi \in {\mathscr{S}}(\RR) $ yield equivalent norms on $S_0(\mathbb{R} )$.
By duality, $\mathbf{L}^2(\mathbb{R}) $ is densely and weak$^{\ast}$-continuously embedded in $S_0'(\mathbb{R} )$ and can also be characterized by the norm $\| f\|_{S_0'} =
\|\mathcal{V}_{\varphi} f\|_{\mathbf{L}^{\infty}}$.
In other words, the three spaces $(S_0(\mathbb{R} ),
\mathbf{L}^2(\mathbb{R} ) , S_0'(\mathbb{R} ))$ Represent a special case of a *Gelfand triple* [@gelfand-vilenkin] or Rigged Hilbert space. For a proof, equivalent characterizations, and more results on $S_0$ we refer to [@feichtinger81; @FZ98a; @feichtinger-kozek98].
Via an isomorphism between integral kernels in the Banach spaces $S_0, S_0'$ and the operator spaces of bounded operators $S_0'\mapsto S_0$ and $S_0\mapsto S_0'$, we obtain, together with the Hilbert space of Hilbert-Schmidt operators, a Gelfand triple of operator spaces, as follows. We denote by ${\mathscr{B}}$ the family of operators that are bounded $S_0'\to S_0$ and by ${\mathscr{B}}'$ the family of operators that are bounded $S_0\to S_0'$. [We have the following correspondence between these operator classes and their integral kernels $\kappa$: ]{} $$H\in ({\mathscr{B}}, {\mathscr{H}}, {\mathscr{B}}') \longleftrightarrow \kappa_H \in
(S_0(\mathbb{R} ), \mathbf{L}^2(\mathbb{R} ) , S_0'(\mathbb{R} ))\ .$$ We will make use of the principle of *unitary Gelfand triple isomorphisms*, described for the Gelfand triples just introduced in [@feichtinger-kozek98]. The basic idea is the extension of a unitary isomorphism between $\mathbf{L}^2$-spaces to isomorphisms between the spaces forming the Gelfand triple. In fact, it may be shown, that it suffices to verify unitarity of a given isomorphic operator on the (dense) subspace $S_0$ in order to obtain a unitary Gelfand triple isomorphism, see [@feichtinger-kozek98 Corollary 7.3.4]. The most prominent examples for a unitary Gelfand triple isomorphism are the Fourier transform and the partial Fourier transform. For all further details on the Gelfand triples just introduced, we again refer to [@feichtinger-kozek98],
only mentioning here, that one important reason for investigating operator representations on the level of Gelfand triples instead of just a Hilbert space framework is the fact, that $S_0'$ contains distributions such as the Dirac functionals, Shah distributions or just pure frequencies and ${\mathscr{B}}'$ contains operators of great importance in signal processing, e.g. convolution, the identity or just time-frequency shifts.
Subsequently, we will usually assume that the analysis and synthesis windows $g,h$ are in $S_0$. This is a rather mild condition, which has almost become the canonical choice in Gabor analysis, for many good reasons. Among others, this choice guarantees a beautiful correspondence between the $\ell^p$-spaces and corresponding modulation space [@gr01]. In the $\ell^2$-case this means, that the sequence of Gabor atoms generated from time frequency translates of an $S_0$ window on an arbitrary lattice $\Lambda$ is automatically a Bessel sequence (in such a case, the window is termed “Bessel atom”), which is not true for general $\mathbf{L}^2$-windows.\
The Banach spaces $S_0$ and $S_0'$ may also be interpreted as Wiener amalgam spaces [@FZ98a Section 3.2.2]. These time-frequency homogeneous spaces are defined as follows. Let $\cF\mathbf{L}^1$ denote the Fourier image of integrable functions and let a compact function $\phi\in\cF\mathbf{L}^1 (\RR )$ with $\sum_{n\in\ZZ} \phi(x-n ) \equiv 1$ be given. Then, for $\mathbf{ X} (\RR ) = \cF\mathbf{L}^1(\RR )$ or $\mathbf{ X }(\RR ) = \mathbf{C}(\RR )$, i.e. the space of continuous functions on $\RR $, or any of the Lesbesgue spaces, we define, for $p\in [1,\infty )$, with the usual modification for $p=\infty$: $$\label{fo:amalgDef}
\mathbf{W}(\mathbf{ X} ,\ell^p ) = \big\{f \in \mathbf{ X}_{loc}:\|f\|_{\mathbf{W}(\mathbf{ X} ,\ell^p )} = \big(\sum_{n\in\ZZ}\|f T_n\phi\|_{\mathbf{X}}^p\big)^{1/p}<\infty\big\}$$ Now, $S_0 = \mathbf{W} (\cF\mathbf{L}^1,\ell^1 )$ and $S_0' = \mathbf{W} (\cF\mathbf{L}^{\infty},\ell^{\infty} )$, see [@FZ98a Section 3.2.2].\
The spreading function representation and its connections to the STFT {#Se:sprep}
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The so-called [*spreading function*]{} representation, closely related to the integrated Schrödinger representation [@gr01 Section 9.2], expresses operators in $({\mathscr{B}}, {\mathscr{H}}, {\mathscr{B}}')$ as a sum of time-frequency shifts. More precisely, one has (see [@gr01 Chapter 9]):
\[th:SFrep\] Let $H\in ({\mathscr{B}},{\mathscr{H}},{\mathscr{B}}')$; then there exists a spreading function $\eta_H$ in $(S_0(\RR^2),\mathbf{L}^2(\RR^2), S_0'(\RR^2))$ such that $$\label{fo:SFrep}
H = \int\bds\int\bds \eta_H(b,\nu) \pi(b,\nu)\, dbd\nu\ .$$ For $H\in\cH$, the correspondence $H\leftrightarrow\eta_H$ is isometric, i.e. $\|H\|_{\mathscr{H}}= \|\eta_H\|_{\mathbf{L}^2(\PP)}$.
[For $H\in{\mathscr{B}}$, the decomposition given in is absolutely convergent, whereas, for $H\in{\mathscr{B}}'$, it holds in the weak sense of bilinear forms on $S_0$.\
]{} When $\eta_H\in {\mathbf L}^2(\PP)$, $H$ is a Hilbert-Schmidt operator, and the above integral is defined as a Bochner integral.
The spreading function is intimately related to the integral kernel $\kappa = \kappa_H$ of $H$ via $$\label{fo:SF2Ker2SF}
\hspace{-0.8cm}\eta_H(b,\nu)=\int\bds \kappa_H(t,t-b) e^{-2i\pi\nu t}\, dt
\mbox{ and }
\kappa_H(t,s)= \int\bds \eta_H(t-s,\nu) e^{2i\pi\nu t}\,d\nu$$ As a consequence, for $\kappa_H\in (S_0,\mathbf{L}^2, S_0')$, we also have $\eta_H\in (S_0,\mathbf{L}^2, S_0')$. In particular, this leads to the following expression for a weak evaluation of Gelfand triple operators: $$\label{fo:OpEval1}
\langle K,L\rangle_{({\mathscr{B}}, {\mathscr{H}}, {\mathscr{B}}')} = \langle \kappa (K ) ,\kappa (L) \rangle_{(S_0,\mathbf{L}^2,S_0' )} = \langle \eta(K ) ,\eta (L) \rangle_{(S_0,\mathbf{L}^2,S_0' )}$$ For $L = g\otimes f^{\ast}$, i.e. the tensor product with kernel $\kappa (s,t) = g(s)\overline{f}(t)$ and spreading function $\eta (b,\nu ) = {\mathscr{V}}_g f(b,\nu ) $, we thus have: $$\label{fo:OpEval2}
\langle K,g\otimes f^{\ast}\rangle_{({\mathscr{B}}, {\mathscr{H}}, {\mathscr{B}}')} = \langle \kappa (K ) ,\kappa (g\otimes f^{\ast}) \rangle_{(S_0,\mathbf{L}^2,S_0' )} = \langle \eta(K ) ,{\mathscr{V}}_g f \rangle_{(S_0,\mathbf{L}^2,S_0' )}$$ Let us also mention that the spreading function is related to the operator’s Kohn-Nirenberg symbol via a *symplectic Fourier transform*, which we define for later reference.
\[SympFT\] The symplectic Fourier transform is formally defined by $$\cF_s F(t,\xi ) = \int_{\mathbb{P}}F(b,\nu )e^{-2\pi i (b \xi - t\nu )}dbd\nu$$
The symplectic Fourier transform is a self-inverse unitary automorphism of the Gelfand triple $(S_0,\mathbf{L}^2, S_0' )$. We will make use of the following relation.
\[Le:FSSTFT\]Assume that $f_1,f_2,g_1,g_2\in \mathbf{L}^2 (\mathbb{R})$. Then $$\cF_s ({\mathscr{V}}_{g_1}f_1 \overline{{\mathscr{V}}_{g_2} f_2})(x,\omega) = ({\mathscr{V}}_{f_1}f_2 \overline{{\mathscr{V}}_{g_2}g_1})(x,\omega ).$$
The analoguous statement for the conventional (Cartesian) Fourier transform reads $\cF ({\mathscr{V}}_{g_1}f_1 \overline{{\mathscr{V}}_{g_2} f_2})(x,\omega) = ({\mathscr{V}}_{f_1}f_2 \overline{{\mathscr{V}}_{g_2}g_1})(-\omega,x )$ and has been shown in [@gr03-2 Lemma 2.3.2]. The fact, that $$[\cF_s F](x,\omega) = \hat{F}(-\omega,x )$$ completes the argument.\
Recall that the spreading function of the product of operators corresponds to the twisted convolution of the operators’ spreading function. Assume $K_1$ in $({\mathscr{B}}, {\mathscr{H}}, {\mathscr{B}}') $ and $K_2$ in $({\mathscr{B}}', {\mathscr{H}}, {\mathscr{B}}) $ then
$$\label{fo:opTwist}
\eta (K_2\cdot K_1) = \eta(K_2)\natural \eta(K_1).$$
The spreading function representation of operators provides an interesting time-frequency implementation for operators, stated in the following proposition. It turns out to be closely connected to the tools described in the previous section, in particular twisted convolution and STFT.
\[prop:TwistRep\] Let $H$ be in $({\mathscr{B}}, {\mathscr{H}}, {\mathscr{B}}') $, and let $\eta = \eta_H$ be its spreading function in $(S_0(\RR^2),\mathbf{L}^2(\RR^2), S_0'(\RR^2)$. Let $g\in S_0(\RR)$, then the STFT of $Hf$ is given by a twisted convolution of $\eta_H$ and ${\mathscr{V}}_g f$: $${\mathscr{V}}_g Hf (z) = (\eta_H\natural {\mathscr{V}}_g f)(z).$$
By (\[fo:SFrep\]), we may write \_g Hf (z’) &=&Hf, (z’) g\
&=&\_H(z)(z) f, (z’ )gdz\[SFrepeq1\]\
&=&\_H(z)f, (z)\^(z’ )gdz\
&=&\_H(z)e\^[-2i b(’-)]{} f, (z’-z )gdz\
&=&\_H(z)\_g f(z’-z) e\^[-2i b(’-)]{} dz = (\_H\_g f)(z’)Note that $S_0$ is time-frequency shift-invariant, so $\pi (z) g$ is in $S_0$ for all $z$. Hence, the expression in is well-defined.\
If $f\in\mathbf{L}^2(\RR)$ and $H\in{\mathscr{H}}$, then ${\mathscr{V}}_gf$, ${\mathscr{V}}_g Hf$ and $\eta_H \in\mathbf{L}^2(\RR^2)$, which is in accordance with the fact that $\mathbf{L}^2\natural \mathbf{L}^2\subseteq\mathbf{L}^2$.\
If $f\in S_0(\RR)$, then $H$ may be in ${\mathscr{B}}'$, such that $\eta_H \in S_0'(\RR^2)$, hence $Hf\in S_0'(\RR )$. Hence, we have ${\mathscr{V}}_gf\in S_0(\RR^2)$ and ${\mathscr{V}}_g Hf\in S_0'(\RR^2)$. This leads to the inclusion $S_0'\natural \mathbf{W}(\mathbf{C},\ell^1)\subseteq \mathbf{L}^\infty$, which may easily be verified directly.\
On the other hand, if $f\in S_0'(\RR)$ and $H$ in ${\mathscr{B}}$, such that $\eta_H \in S_0(\RR^2)$, then $Hf$ is in $S_0(\RR)$. Hence, ${\mathscr{V}}_gf\in S_0'(\RR^2)$ and ${\mathscr{V}}_g Hf\in S_0(\RR^2)$. Here, this leads to the conclusion that we have, for $f\in S_0'(\RR)$: $$\label{eq:twistS0}
S_0\natural {\mathscr{V}}_g f\subseteq S_0 .$$\
Although it is known that ${\mathscr{V}}_g f$ is not only $\mathbf{L}^{\infty} (\RR^2 )$, but also in the Amalgam space $\mathbf{W}(\mathcal{F}\mathbf{L}^1, \ell^{\infty})$ for $f\in S_0'(\RR)$ and $g\in S_0(\RR)$,[@FZ98a], it is not clear, whether also holds for functions $F\in \mathbf{W}(\mathcal{F}\mathbf{L}^1, \ell^{\infty})$, which are not in the range of $S_0' (\RR )$ under ${\mathscr{V}}_g$. This and other interesting open questions concerning the twisted convolution of function spaces are currently under investigation[^4].
As a consequence of the last proposition, $H$ may be realized as a twisted convolution in the time-frequency domain: $$\label{fo:TwistRep}
Hf = \int\bds\int\bds \left(\eta_H\natural {\mathscr{V}}_gf \right)(b,\nu)
M_\nu T_b h\,dbd\nu\ \mbox{ for all }
f\in (S_0 , \mathbf{L}^2, S_0').$$
Notice that Proposition \[prop:TwistRep\] implies that the range of ${\mathscr{V}}_g$ is invariant under left twisted convolution. Notice also that this is no longer true if the left twisted convolution is replaced with the right twisted convolution. Indeed, in such a case, one has $${\mathscr{V}}_g f \natural \eta_H = {\mathscr{V}}_{H^* g}f\ .$$ Hence, one has the following simple rule: left twisted convolution on the STFT amounts to acting on the analyzed function $f$, while right twisted convolution on the STFT amounts to acting on the analysis window $g$. It is worth noticing that in such a case, applying ${\mathscr{V}}_g^*$ to ${\mathscr{V}}_gf \natural \eta_H$ yields the analyzed function $f$, up to some (possibly vanishing) constant factor.
As an illustrating example, let $g,h\in S_0$ be such that $\langle g,h\rangle = 1$ and consider the oblique projection $P: f\mapsto \langle f,g\rangle h$. The spreading function of this operator is given by ${\mathscr{V}}_g h$, and we have ${\mathscr{V}}_{\varphi} Pf (z) = \langle f,g\rangle \langle h,\pi (z) \varphi\rangle$. By virtue of the inversion formula for the STFT, which may be written as $\langle f,g\rangle h = \int \langle h , \pi (z) g\rangle \pi (z) f dz$, we obtain: $${\mathscr{V}}_{\varphi} Pf (z) = \int\langle h,\pi (z) g\rangle\langle \pi (z) f, \pi (z') \varphi\rangle dz = {\mathscr{V}}_g h\natural {\mathscr{V}}_{\varphi}f.$$ By completely analogous reasoning, we obtain the converse formula, if the operator is applied to the analysing window: $${\mathscr{V}}_{P\varphi}f = \langle f, \pi (z) P\varphi\rangle = {\mathscr{V}}_{\varphi}f\natural {\mathscr{V}}_g h .$$
Notice also that twisted convolution in the phase space is associated with the true translation structure. Indeed, time-frequency shifts take the form of twisted convolutions with a Dirac distribution on $\PP$: $$\delta_{b_0,\nu_0}\natural {\mathscr{V}}_g f = {\mathscr{V}}_g M_{\nu_0} T_{b_0} f\ .$$ This corresponds to the usage of engineers, who “adjust the phases” after shifting STFT coefficients [@dols1; @phavoc66].
Time-Frequency multipliers {#Se:tfmult}
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\[se:mult\] Section \[Se:sprep\] has shown the close connection between the spreading function representation of Hilbert-Schmidt operators and the short time Fourier transform. However, the twisted convolution representation is generally of poor practical interest in the continuous case, because it does not discretize well. Even in the finite case, it relies on the full STFT on $\CC^N$, which represents vectors with $N^2$ STFT coefficients, which may be far too large in practice, and sub-sampling is not possible in a straightforward way.
Time-frequency (in particular Gabor) multipliers represent a valuable alternative for time-frequency operator representation (see [@Feichtinger02first; @Hlawatsch03linear] and references therein for reviews). We analyze below the connections between these representations and the spreading function, and point out some limitations, before turning to generalizations.
### Definitions and main properties
Let $g,h\in S_0(\RR)$ be such that $\langle g,h\rangle =1$, let $\bm\in \mathbf{L}^\infty(\RR^2)$, and define the STFT multiplier $\MM_{\bm;g,h}$ by $$\MM_{\bm;g,h} f =
\int_\PP \mathbf{m}(b,\nu) {\mathscr{V}}_gf(b,\nu)\, \pi(b,\nu) h\, dbd\nu ,$$ This defines a bounded operator on $( S_0(\RR), \mathbf{L}^2(\RR), S_0'(\RR))$.
Similarly, given lattice constants $b_0,\nu_0\in\RR^+$, set $\pi_{mn} = \pi(mb_0,n\nu_0) = M_{n\nu_0}T_{mb_0}$. Then, for $\bm\in\ell^\infty(\ZZ^2)$, the corresponding Gabor multiplier is defined as $$\label{fo:GMdef}
\MM_{\bm;g,h}^G f = \sum_{m=-\infty}^\infty
\sum_{n=-\infty}^\infty \bm(m,n)
{\mathscr{V}}_gf(mb_0,n\nu_0)\, \pi_{mn} h\ .$$ Note that Gabor multipliers may be interpreted as STFT multipliers with multiplier $\bm$ in $S_0'$. In fact, in this case, $\bm$ is simply a sum of weighted Dirac impulses on the sampling lattice.
The definition of time-frequency multipliers can of course be given for $g,h\in \mathbf{L}^2 (\mathbb{R})$, many nice properties only apply with additional assumptions on the windows. Abstract properties of such multipliers have been studied extensively, and we refer to [@Feichtinger02first] for a review. One may show for example that, whenever the windows $g$ and $h$ are at least in $S_0$, if $\bm$ belongs to $\mathbf{L}^2(\PP)$ (or $\ell^2(\ZZ^2)$) then the corresponding multiplier is a Hilbert-Schmidt operator and maps $S_0'(\mathbb{R})$ to $\mathbf{L}^2 (\mathbb{R})$.
The spreading function of time-frequency multipliers may be computed explicitly.
The spreading function of the STFT multiplier $\MM_{\bm;g,h}$ is given by $$\label{fo:STFTMult.spreading}
\eta_{\MM_{\bm;g,h}} (b,\nu) = {\mathscr{M}}(b,\nu) {\mathscr{V}}_g h(b,\nu)\ ,$$ where ${\mathscr{M}}$ is the symplectic Fourier transform of the transfer function $\bm$ $${\mathscr{M}}(t,\xi) = \int_\PP \bm(b,\nu) e^{2i\pi(\nu t - \xi b)}\, dbd\nu\ .$$ Specifying to the Gabor multiplier $\MM_{\bm;g,h}^G$, we see the same expression for the spreading function, however, in this case, ${\mathscr{M}}= {\mathscr{M}}^{(d)}$ is the $(\nu_0\inv,b_0\inv)$-periodic symplectic Fourier transform of the discrete transfer function $\bm$ $$\label{fo:GabMult.spreading}
{\mathscr{M}}^{(d)}(t,\xi) = \sum_{m=-\infty}^\infty\sum_{n=-\infty}^\infty
\bm (m,n) e^{2i\pi(n\nu_0 t - mb_0\xi)}\ .$$
For $f\in S_0'$ and $\varphi\in S_0$, we may write $$\langle \MM_{\bm;g,h} f,\varphi \rangle =
\langle \bm, \overline{{\mathscr{V}}_g f}\cdot {\mathscr{V}}_h\varphi\rangle\ ,$$ where the right-hand side inner product has to be interpreted as an integral or infinite sum, respectively. By Lemma \[Le:FSSTFT\], applying the symplectic Fourier transform, we obtain $$\langle \MM_{\bm;g,h} f,\varphi \rangle
= \langle {\mathscr{M}}, {\mathscr{V}}_{\varphi} f\cdot\overline{{\mathscr{V}}_g h} \rangle
= \langle {\mathscr{M}}\cdot{\mathscr{V}}_g h, {\mathscr{V}}_{\varphi} f\rangle\ .$$ By calling on , this proofs . By virtue of that fact that for $g, h \in S_0$, ${\mathscr{V}}_g h$ is certainly in $\mathbf{L}^1 (\mathbb{P})$ and even in the Wiener Amalgam Space $W(C, \mathbf{L}^1)$, hence in particular continuous, the expressions for the spreading function given in the lemma are always well-defined. The symplectic Fourier transform is a Gelfand triple isomorphism of $(S_0,\mathbf{L}^2, S_0')$, i.e., $\bm \in (S_0,\mathbf{L}^2, S_0') \Longleftrightarrow {\mathscr{M}}\in (S_0,\mathbf{L}^2, S_0')$. Hence, ${\mathscr{M}}\cdot{\mathscr{V}}_g h\in (S_0,\mathbf{L}^2, S_0')$, which is in accordance with the fact, that for $\bm$ in $(\ell^1,\ell^2,\ell^{\infty})$, i.e., $(S_0,\mathbf{L}^2, S_0') (\ZZ^2)$, the kernel of the resulting operator (and hence its spreading function), is in $(S_0,\mathbf{L}^2, S_0')$, see [@Feichtinger02first] for details.
All expressions derived so far are easily generalized to Gabor frames for $\mathbb{R}^d$ associated to arbitrary lattices $\Lambda\subset\RR^{2d}$. In such situations, the spreading function takes a similar form, and involves some discrete symplectic Fourier transform of the transfer function $\bm$, which is in that case a $\Lambda^\circ$-periodic function, $\Lambda^\circ$ being the adjoint lattice of $\Lambda$, see Definition \[Def:AdLatt\] in Section \[se:multmult\].
Notice that as a consequence of Theorem \[th:SFrep\], one has the following “intertwining property” $${\mathscr{V}}_g \MM f = ({\mathscr{M}}_{\bm;g,h}\,{\mathscr{V}}_gh)\natural{\mathscr{V}}_gf\ .$$
\[rem:aaaaa\] It is clear from the above calculations that a general Hilbert-Schmidt operator may not be well represented by a TF-multiplier. For example, let us assume that the analysis and synthesis windows have been chosen, and let $\eta$ be the spreading function of the operator under consideration.
- In the STFT case, if the analysis and synthesis windows are fixed, the decay of the spreading function has to be fast enough (at least as fast as the decay of ${\mathscr{V}}_gh$) to ensure the boundedness of the quotient ${\mathscr{M}}=\eta/{\mathscr{V}}_gh$. [Such considerations have led to the introduction of the notion of [*underspread*]{} operators [@Kozek96matched] whose spreading function is compactly supported in a domain of small enough area.]{} A more precise definition of underspread operators will be given below.
- In the Gabor case, the periodicity of ${\mathscr{M}}^{(d)}$ imposes extra constraints on the spreading function $\eta$. In particular, the shape of the support of the spreading function must influence the choice of optimal parameters for the approximation by a Gabor multiplier, i.e. the shape of the window as well as the lattice parameters. The following numerical example indicates the direction for the choice of parameters in the approximation of operators by Gabor multipliers.
\[Ex1\] Consider two operators $OP1$ and $OP2$ with spreading functions as shown in Figure \[FI1\]. The values of the spreading functions are random and real, uniformly distributed in $[-0.5,0.5]$.\
Operator 1 has a spreading function with smaller support on the time-axis, which means that the corresponding operator exhibits time-shifts across smaller intervals than Operator 2, whose spreading function is, on the other hand, less extended in frequency. The effect in the opposite direction is, obviously, reverse. These characteristics are illustrated by applying the operators to a sinusoid with frequency $1$ and a Dirac impulse at $-1$, respectively.\
Next, we realize approximation[^5] by Gabor multipliers with two fixed pairs of lattice constants: $b_1 = 2, \nu_1 = 8$ and $b_2 = 8, \nu_2 = 2$. Furthermore, the windows are Gaussian windows varying from wide $(j = 0)$ to narrow $(j = 100)$.Thus, $j$ corresponds to the concentration of the window, in other words, $j$ is the reciprocal of the standard deviation. Now the approximation quality is investigated. The results are shown in the lower plots of Figure \[FI1\], where the left subplot shows the approximation quality for operator $OP1$ for $b_1,\nu_1$ (solid) and $b_2,\nu_2$ (dashed), while the right hand subplot gives the corresponding results for operator 2. The error is measured by $err = \|OP-APP\|_{{\mathscr{H}}}/\|OP+APP\|_{{\mathscr{H}}}$. Here, $APP$ denotes the approximation operator and the norm is the operator norm. The results show that, as expected, the “adapted” choice of time-frequency parameters leads to more favorable approximation quality. Here, the adapted choice of $b$ and $\nu$ mimics the shape of the support of the spreading function according to formula and the periodicity of ${\mathscr{M}}^{(d)}$. In brief, if the operator realizes frequency-shifts in a wider range, we will need more sampling-points in frequency and vice-versa. It is also visible, that the shape of the window has considerable influence on the approximation quality.
The previous example shows, that the parameters in the approximation by Gabor multipliers must be carefully chosen. Let us point out that the approximation quality achieved in the experiment described in Example \[Ex1\] is not satisfactory, especially when the time- and frequency shift parameters are not well adapted. Operators with a spreading function that is not well-concentrated around $0$, i.e. “overspread operators”, don’t seem to be well-represented by a Gabor multiplier even with high redundancy (the redundancy used in the example is $8$). Moreover, a realistic operator will have a spreading function with a much more complex shape. The next section will give some more details on approximation by Gabor multipliers before generalizations, which allow for approximation of more complex operators, are suggested.
### Approximation by Gabor multipliers
The possibility of approximating operators by Gabor multipliers in Hilbert-Schmidt sense depends on the properties of the rank one operators associated with time-frequency shifted copies of the analysis and synthesis windows.
Let $g,h\in S_0(\RR )$ be such that $\langle g,h\rangle = 1$. Let $\lambda =
(b_1,\nu_1)\in\PP$, and consider the rank one operator (oblique projection) $P_\lambda$ defined by $$P_\lambda f = (g_\lambda^{\ast}\otimes h_\lambda) f =
\langle f,g_\lambda\rangle h_\lambda\ ,\quad
f\in(S_0(\RR),\mathbf{L}^2(\RR),S_0'(\RR))\ .$$ Direct calculations show that the kernel of $P_\lambda$ is given by $$\kappa_{P_\lambda} (t,s) = \overline{g}_\lambda(s) h_\lambda(t)\ ,$$ and its spreading function reads $$\eta_{P_\lambda} (b,\nu) = e^{2i\pi(\nu_1 b-b_1\nu)} {\mathscr{V}}_gh(b,\nu)\ .$$
The following result characterizes the situations for which time-frequency rank one operators form a Riesz sequence, in which case the best approximation by a Hilbert-Schmidt operator is well-defined. This result first appeared in [@Feichtinger02wavelet]. Here, we give a slightly different version, which is obtained from the original statement by applying Poisson summation formula. This result was also given in [@bepf06] for general full-rank lattices in $\mathbb{R}^d$.
\[Prop:Ucond\] Let $g,h\in\mathbf{L}^2(\RR)$, with $\langle g,h\rangle\ne 0$, let $b_0,\nu_0\in\RR^+$, and set $$\cU(t,\xi) = \sum_{k,\ell=-\infty}^\infty
\left|{\mathscr{V}}_gh\left(t+\frac{k}{\nu_0},\xi + \frac\ell{b_0}\right)\right|^2\ .$$ The family $\{P_{mb_0,n\nu_0},\,m,n\in\ZZ\}$ is a Riesz sequence in $\cH$ if and only if there exist real constants $0<A\le B<\infty$ such that $$\label{fo:Ucond}
0 < A\le \cU(t,\xi)\le B<\infty\ \mbox{ a.e. on } [0,\nu_0\inv[\times [0,b_0\inv[ .$$ We call this condition the $\cU$ condition.
It turns out, that the approximation of a given operator via a standard minimization process yields an expression, which is only well-defined if the $\cU$ condition (\[fo:Ucond\]) holds.
\[th:gabmult.app\] Assume that ${\mathscr{V}}_gh$ and $b_0,\nu_0\in\RR^+$ are such that the $\cU$ condition (\[fo:Ucond\]) is fulfilled. Then the best Gabor multiplier approximation (in Hilbert-Schmidt sense) of $H\in{\mathscr{H}}$ is defined by the time-frequency transfer function $\bm$ whose discrete symplectic Fourier transform reads $$\label{Eq:MBestGM}
{\mathscr{M}}(b,\nu) =
\frac{\sum_{k,\ell=-\infty}^\infty
\overline{{\mathscr{V}}_gh}\left(b+k/\nu_0,\nu + \ell/b_0\right)
\eta_H\left(b+k/\nu_0,\nu + \ell/b_0\right)}
{\sum_{k,\ell=-\infty}^\infty
\left|{\mathscr{V}}_gh\left(b+k/\nu_0,\nu + \ell/b_0\right)\right|^2}$$
Let us denote as before by $\square$ the rectangle $\square = [0,\nu_0\inv[\times [0,b_0\inv[$, and set ${\mathscr{V}}={\mathscr{V}}_gh$ for simplicity of notation. First, notice that if $\eta_H\in L^2(\PP)$, then the function $(b,\nu)\in\square\to \sum_{k,\ell} |\eta_H(b+k/\nu_0,\nu+\ell/b_0)|^2$ is in $L^2(\square)$, and is therefore well defined almost everywhere in $\square$. Thus, by Cauchy-Schwarz inequality, the numerator in is well-defined a.e.\
The Hilbert-Schmidt optimization is equivalent to the problem $$\min_{{\mathscr{M}}\in {\mathbf L}^2(\square)} \|\eta_H -{\mathscr{M}}{\mathscr{V}}\|^2\ .$$ The latter squared norm may be written as $$\begin{aligned}
\|\eta_H\! -\!{\mathscr{M}}{\mathscr{V}}\|^2 &=&\!\!
\int\bds\int\bds \left|\eta_H(b,\nu) -{\mathscr{M}}(b,\nu){\mathscr{V}}(b,\nu)\right|^2\,dbd\nu\\
&=&\!\!\!\! \sum_{k,\ell=-\infty}^\infty\!\!\int\!\!\!\int_\square
\left|\eta_H(b+k/\nu_0,\nu+\ell/b_0)-{\mathscr{M}}(b,\nu){\mathscr{V}}(b+k/\nu_0,\nu+\ell/b_0)\right|^2
\,dbd\nu\\
&=&\!\! \int\!\!\!\int_\square \bigg[\sum_{k,\ell}
|\eta_H(b+k/\nu_0,\nu+\ell/b_0)|^2\\
&&\hphantom{aa}- 2 \Re\bigg( \overline{{\mathscr{M}}}(b,\nu) \sum_{k,\ell}
\eta_H(b+k/\nu_0,\nu+\ell/b_0)\overline{{\mathscr{V}}}(b+k/\nu_0,\nu+\ell/b_0)\bigg)\\
&&\hphantom{aa} +
|{\mathscr{M}}(b,\nu)|^2 \sum_{k,\ell} |{{\mathscr{V}}}(b+k/\nu_0,\nu+\ell/b_0)|^2
\bigg]db d\nu\end{aligned}$$ From this expression, the Euler-Lagrange equations may be obtained, which read $${\mathscr{M}}(b,\nu) \sum_{k,\ell} |{{\mathscr{V}}}(b+k/\nu_0,\nu+\ell/b_0)|^2
= \sum_{k,\ell}
\eta_H(b+k/\nu_0,\nu+\ell/b_0)\overline{{\mathscr{V}}}(b+k/\nu_0,\nu+\ell/b_0)\ ,$$ and the result follows.\
We next derive an error estimate for the approximation. Let us set, for $(b,\nu)\in\square$, $$\cE(b,\nu) = \frac{\left|\sum_{k,\ell}\eta_H(b+k/\nu_0,\nu+\ell/b_0)
\overline{{\mathscr{V}}}(b+k/\nu_0,\nu+\ell/b_0)\right|^2}
{\sum_{k,\ell}|\eta_H(b+k/\nu_0,\nu+\ell/b_0)|^2\,\cU(b,\nu)}\ .$$
\[Cor:ErrEst1\] With the above notation, we obtain the estimate $$\|H-\MM_\bm\|^2_{\mathscr{H}}\le \|\eta_H\|^2\,\|1-\cE\|_\infty$$for the best approximation of $H$ by a Gabor multiplier $\MM_\bm$ according to .
Set $\Gamma_H(b,\nu) = \sum_{k,\ell}|\eta_H(b+k/\nu_0,\nu+\ell/b_0)|^2$. Replacing the expression for ${\mathscr{M}}$ obtained in into the error term, we have $$\begin{aligned}
\|H-\MM_\bm\|^2_{\mathscr{H}}&=& \int\int_\square \left[\Gamma_H(b,\nu)
- |{\mathscr{M}}(b,\nu)|^2\cU(b,\nu)\right]\,dbd\nu\\
&=&\int\int_\square \Gamma_H(b,\nu)\left[1-\cE(b,\nu)\right]\,dbd\nu\\
&\le& \|\eta_H\|^2\,\|1-\cE\|_\infty
\ ,\end{aligned}$$ where we have used the fact that $\|\Gamma_H\|_{{\mathbf L}^2(\square)}
= \|\eta_H\|_{{\mathbf L}^2(\PP)}$. Clearly, Cauchy-Schwarz inequality gives $|\cE(b,\nu)|\le 1$ on $\square$, with equality if and only if there exists a function $\phi$ such that $\eta = \phi{\mathscr{V}}$, i.e. if and only if $H$ is a multiplier with the prescribed window functions. Hence we obtain $$\label{fo:GM.error}
\|H-\MM_\bm\|^2_{\mathscr{H}}\le \|H\|_{\mathscr{H}}^2\,
\left[1-\mathop{{\mathrm ess}\inf}\limits_{(b,\nu)\in\square} \cE(b,\nu)\right]
\ .$$
$\cE(b,\nu)$ essentially represents the cosine of the angle between vectors $\{\eta_H(b+k/\nu_0,\nu+\ell/b_0),\,k,\ell\in\ZZ\}$ and $\{{\mathscr{V}}(b+k/\nu_0,\nu+\ell/b_0),\,k,\ell\in\ZZ\}$. In other words, the closer to colinear these vectors, the better the approximation.
An example for operators which are poorly represented by this class of multipliers are those with a spreading function that is not “well-concentrated”. These are, in technical terms, overspread operators. The underspread/overspread terminology seems to originate from the context of time-varying multipath wave propagation channels [@Ken69]. However, different definitions exist in the literature. Here, we give the definition used in [@ko97-1]. Note that underspread operators have recently found renewed interest [@ko97; @bokoma02; @hlma98; @kopf06; @pfwa05-1].
Consider an operator $K$ with compactly supported spreading function: $$\supp(\eta_K )\subseteq Q(t_0,\xi_0), \mbox{ where }
Q(t_0,\xi_0):= [-t_0,t_0]\times [-\xi_0,\xi_0].$$ Then, $K$ is called underspread, if $t_0\xi_0<1/4$.
Most generally, operators which are *not* underspread, will be called overspread.\
It is generally known, that an operator must be underspread in order to be well-approximated by Gabor multipliers. Formula enables us to make this statement more precise.
Consider an underspread operator $H$. Then, for $(b_0,\nu_0)$ such that $1/b_0>2\xi_0$ and $1/\nu_0>2t_0$, it is possible to find a Gabor frame $\{g_{mn},m,n\in\ZZ\}$, with lattice constants $(b_0,\nu_0)$, and dual window $h$. Then, the symplectic Fourier transform ${\mathscr{M}}$ of the time-frequency transfer function of the best Gabor multiplier takes the form $${\mathscr{M}}= \frac{\eta_H\overline{{\mathscr{V}}_gh}}{\cU}\ ,$$ and the approximation error can be bounded by $$\left\|\eta_H-{\mathscr{M}}{\mathscr{V}}_gh\right\|^2 \le \|\eta_H\|^2
\mathop{{\mathrm ess}\sup}\limits_{(b,\nu)\in\square}
\left[ 1 - \frac{|{\mathscr{V}}_gh(b,\nu)|^2}{\cU(b,\nu)}\right]$$
$$\begin{aligned}
\left\|\eta_H-{\mathscr{M}}{\mathscr{V}}_gh\right\|^2 &=&
\int\int_\square \left[|\eta(b,\nu)|^2
-\frac{\left|\eta(b,\nu){\mathscr{V}}_gh(b,\nu)\right|^2}{\cU(b,\nu)}\right]\,dbd\nu\\
&\le& \|\eta_H\|^2 \mathop{{\mathrm ess}\sup}\limits_{(b,\nu)\in\square}
\left[ 1 - \frac{|{\mathscr{V}}_gh(b,\nu)|^2}{\cU(b,\nu)}\right]\end{aligned}$$\
The estimate in the last corollary shows, that approximation quality is a joint property of window and lattice, which is in accordance with the results of Example \[Ex1\].
Note that, although technically only defined for Hilbert-Schmidt operators, the approximation by Gabor multipliers can formally be extended to operators from ${\mathscr{B}}'$, see [@Feichtinger02first Section 5.8]. Also, the expression given in is well-defined in $S_0'$ whenever $\eta_H$ is at least in $S_0'$. However, for non-Hilbert-Schmidt operators, it is not clear, in which sense the resulting Gabor multiplier represents the original operator. The following example shows that at least in some cases, the result is however the intuitively expected one.
Consider the operator $\pi (\lambda )$, i.e. a time-frequency shift. Although this operator is clearly not a Hilbert-Schmidt operator, we may consider its approximation by a Gabor-multiplier according to . First note that the spreading function of the time-frequency shift $\pi (\lambda )= \pi (b_1,\nu_1)$ is given by $\eta_{\pi} = \delta (b-b_1)\cdot \delta (\nu -\nu_1)$. Then, we have $$\begin{aligned}
{\mathscr{M}}(b,\nu ) &= &
\sum_{k,l}\frac{\overline{{\mathscr{V}}_g h}(b+k/\nu_0, \nu+l/b_0 )\delta(b-b_1+k/\nu_0, \nu-\nu_1+l/b_0)}{\sum_{k',l'}|{\mathscr{V}}_g h(b+k'/\nu_0, \nu+l'/b_0 )|^2}\\
&=&
\overline{{\mathscr{V}}_g h }(b_1,\nu_1)\sum_{k,l}\frac{\delta(b-b_1+k/\nu_0, \nu-\nu_1+l/b_0)}{\sum_{k',l'}
|{\mathscr{V}}_g h(b_1+\frac{k'-k}{\nu_0},\nu_1+\frac{l'-l}{b_0})|^2}\\
&=&
\frac{\overline{{\mathscr{V}}_g h }(b_1,\nu_1)}{\cU(b_1,\nu_1)}
\sum_{k,l}\delta(b-b_1+k/\nu_0, \nu-\nu_1+l/b_0)\end{aligned}$$
Hence, from the inverse (discrete) symplectic Fourier transform we obtain: $$\begin{aligned}
\bm (m,n)& = &
\frac{\overline{{\mathscr{V}}_g h }(b_1,\nu_1)}{\cU(b_1,\nu_1)}
\int_0^{\frac{1}{\nu_0}}\int_0^{\frac{1}{b_0}}
\sum_{k,l}\delta(b-b_1+k/\nu_0, \nu-\nu_1+l/b_0)\\
&&
\qquad\qquad\qquad\qquad\qquad\times
e^{-2\pi i (mb_0 \nu -n\nu_0 b)}\,dbd\nu\\
&=&
\frac{\overline{{\mathscr{V}}_g h }(b_1,\nu_1)}{\cU(b_1,\nu_1)}\
e^{-2\pi i (mb_0 \nu_1 -n\nu_0 b_1)}\end{aligned}$$ As expected, the absolute value of the mask is constant and the phase depends on the displacement of $\lambda$ from the origin. [ This confirms the key role played by the phase of the mask of a Gabor multiplier.]{} Specializing to $\lambda = 0$, we obtain a constant mask and thus, if $h$ is a dual window of $g$ with respect to $\Lambda = b_0\mathbb{Z}\times \nu_0\mathbb{Z}$, up to a constant factor, the identity.
Generalizations: multiple Gabor multipliers and TST spreading functions {#se:multmult}
=======================================================================
In the last section it has become clear that most operators are not well represented as a STFT or Gabor multiplier.
Guided by the desire to extend the good approximation quality that Gabor multipliers warrant for underspread operators to the class of their overspread counterparts, we introduce generalized TF-multipliers. The basic idea is to allow for an extended scheme in the synthesis part of the operator: instead of using just one window $h$, we suggest the use of a set of windows $\{h^{(j)}\}$ in order to obtain the class of [*Multiple Gabor Multipliers*]{} (MGM for short).
Let $g\in S_0 (\mathbb{R})$ and a family of reconstruction windows $h^{(j)}\in S_0 (\mathbb{R})$, $j\in \mathcal{J}$, as well as corresponding masks $\bm_j\in\ell^{\infty}$ be given. Operators of the form $$\label{fo:multimulti}
\MM = \sum_{j\in\cJ} \MM^G_{\bm_j;g,h^{(j)}}$$ will be called [*Multiple Gabor Multipliers*]{} (MGM for short).
Note, that we need to impose additional assumptions in order to obtain a well-defined operator. For example, we may assume $\sum_j sup_{\lambda} |m_j (\lambda )| = C< \infty$ and $\max_j \|h^j\|_{S_0} = C <\infty$, which guarantees a bounded operator on $(S_0,\mathbf{L}^2,S_0')$. This follows easily from the boundedness of a Gabor multiplier under the condition that $\bm$ is $\ell^{\infty}$. Conditions for function space membership of MGMs are easily derived in analogy to the Gabor multiplier case. For example, if $\sum_j\sum_{\lambda} |m_j(\lambda )|^2<\infty$, we obtain a Hilbert-Schmidt operator, similarly, trace-class membership follows from an analogous $\ell^1$-condition.
As a starting point, we give the (trivial) generalization of the spreading function of a MGM as a sum of the spreading functions corresponding to the single Gabor multipliers involved.
The spreading function of a MGM is (formally) given by $$\label{fo:MGM.spreading}
\eta_{\MM_{\bm_j;g,h^{(j)}}}^G (b,\nu) =
\sum_{j\in\cJ}{\mathscr{M}}^{(j)}(b,\nu) {\mathscr{V}}_g h^{(j)}(b,\nu)\ ,$$ where the $(\nu_0\inv,b_0\inv)$-periodic functions ${\mathscr{M}}^{(j)}$ are the symplectic Fourier transforms of the transfer functions $\bm_j$.
Note that the issue of convergence for the series defining $\eta_{\MM_{\bm_j;g,h^{(j)}}}^G (b,\nu)$ will not be discussed, as in practice $|\mathcal{J}|$ will usually be finite. Let us just mention that by assuming $\bm_j\in\ell^2(\mathbb{Z}^3 )$, i.e., the Hilbert-Schmidt case with an additional $\ell^2$-condition for the masks in the general model, we have $\|\eta_{\MM_{\bm_j;g,h^{(j)}}}^G \|_2 = C\|\bm\|_2$.\
It is immediately obvious that this new model gives much more freedom in generating overspread operators. However, in order to obtain structural results, we will have to impose further specifications.\
Before doing so, we will state a generalization of Proposition \[Prop:Ucond\] to the more general situation of the family of projection operators $P^j_\lambda$ defined by $$P^j_\lambda f = (g_\lambda^{\ast}\otimes h^j_\lambda) f =
\langle f,g_\lambda\rangle h^j_\lambda , \mbox{ where }
\lambda\in\Lambda, j\in\mathcal{J}, \ |\mathcal{J}|<\infty .$$ Note that these projection operators are the building blocks for the MGM. The following theorem characterizes their Riesz property.
\[Prop:GenUcond\] Let $g,h^j\in\mathbf{L}^2(\RR)$, $j\in \mathcal{J}, \ |\mathcal{J}|<\infty$, with $\langle g,h^j\rangle\neq 0$, let $b_0,\nu_0\in\RR^+$, and let the matrix $\Gamma(b,\nu)$ be defined by $$\label{EqU1}
\Gamma(b,\nu)_{jj'} =
\sum_{k,\ell}\overline{{\mathscr{V}}_gh^{(j)}}(b+k/\nu_0,\nu+\ell/b_0)
{\mathscr{V}}_gh^{(j')}(b+k/\nu_0,\nu+\ell/b_0)$$ a.e. on $\square = [0,\nu_0^{-1}[\times [0,b_0^{-1}[$. Then the family of projection operators $\{P^j_\lambda, j\in \mathbb{Z}, \lambda\in\Lambda\}$ is a Riesz sequence in $\cH$ if and only if $\Gamma$ is invertible a.e.\
Alternatively, the Riesz basis property is characterized by invertibility of the matrix $U$ defined as $$\label{EqU2}
\cU^{jj'}(t,\xi)
= \sum_{k,l} U^{jj'}(kb_0,l\nu_0 ) e^{-2\pi i (l\nu_0 t-kb_0\xi)}$$ a.e. on the fundamental domain of $\Lambda$.
Recall that the family $\{P^j_\lambda, j\in \mathbb{Z}, \lambda\in\Lambda\}$ is a Riesz basis for its closed linear span if there exist constants $0<A,B<\infty$ such that $$A\|c\|_2^2\leq
\left\|\sum_{\lambda}\sum_j c^j_{\lambda} P^j_{\lambda}\right\|_{\HS}^2\leq
B\|c\|_2^2$$ for all finite sequences $c$ defined on $(\Lambda\times \mathcal{J})$. We have $$\begin{aligned}
\left\|\sum_{\lambda}\sum_j c^j_{\lambda} P^j_{\lambda}\right\|_{\HS}^2
&=& \left\langle \sum_{\lambda}\sum_j c^j_{\lambda} P^j_{\lambda} ,
\sum_{\mu}\sum_{j'}c^{j'}_{\mu} P^{j'}_{\mu}\right\rangle\\
&=&
\sum_{\lambda\mu}\sum_{jj'} c^j_{\lambda}\overline{c^{j'}_{\mu}}
\overline{\langle g_{\lambda},g_{\mu} \rangle}
\langle h^j_{\lambda},h_{\mu}^{j'} \rangle.\\\end{aligned}$$ Hence, by setting $U^{jj'} (b,\nu ) = [\overline{{\mathscr{V}}_g g}\cdot {\mathscr{V}}_{h^j}h^{j'}](b,\nu )$, we may write $$\begin{aligned}
\left\|\sum_{\lambda}\sum_j c^j_{\lambda} P^j_{\lambda}\right\|_{\HS}^2
&=&
\sum_{\lambda\mu}\sum_{jj'} c^j_{\lambda}\overline{c^{j'}_{\mu}}
U^{jj'} (\mu - \lambda )\\
&=&
\sum_{\mu}\sum_{jj'}\overline{c^{j'}_{\mu}}(c^j\ast U^{jj'})(\mu)\\
&=&
b_0\nu_0\sum_{jj'}\left\langle \cU^{jj'}\cdot\cC^j,
\cC^{j'}\right\rangle_{\mathbf{L^2}(\square )},\end{aligned}$$ where $\cU^{jj'}(t,\xi) = \sum_{k,l} U^{jj'}(kb_0,l\nu_0 )
e^{-2\pi i (l\nu_0 t-kb_0\xi)}$ is the discrete symplectic Fourier transform of $U^{jj'}$, and, analogously, $\cC^j$ is the discrete symplectic Fourier transform of the sequence $c^j$, defined on $\Lambda$, for each $j$. Hence, these are $\nu_0^{-1}\times b_0^{-1}$-periodic functions. The last equation can be rewritten as $$\begin{aligned}
\left\|\sum_{\lambda}\sum_j c^j_{\lambda} P^j_{\lambda}\right\|_2^2
&=&
b_0\nu_0\sum_{jj'}\int_{\square} \cU^{jj'}(t,\xi )\cdot\cC^j(t,\xi )
\overline{\cC^{j'}}(t,\xi )dtd\xi\\
&=&
\left\langle \cU\cdot \cC,\cC\right\rangle_{\mathbf{L^2}(\square )
\times\ell^2(\mathbb{Z})},\end{aligned}$$ where $\cU$ is the matrix with entries $\cU^{jj'}$. Note that this proves statement by positivity of the operator $\cU$.\
In order to obtain the condition for $\Gamma$ given in , first note that $$\cF_s ( \overline{{\mathscr{V}}_g g}\cdot {\mathscr{V}}_{h^j}h^{j'})(\lambda)
= ({\mathscr{V}}_g h^{j'}\cdot \overline{{\mathscr{V}}_g h^j})(\lambda)$$ by applying Lemma \[Le:FSSTFT\]. Furthermore, $F:={\mathscr{V}}_g h^{j'}\cdot \overline{{\mathscr{V}}_g h^j}$ is always in $\mathbf{L}^1$ for $g,h^j\in \mathbf{L}^2$. We may therefore look at the Fourier coefficients of its $\Lambda^{\circ}$-periodization, with $\lambda = (mb_0 ,n\nu_0 )$: $$\begin{aligned}
\cF_s^{-1} (P_{\Lambda^{\circ}}F) (\lambda)
&=& \int_{\square}( \sum_{k,\ell}F(b+\frac{k}{\nu_0},\nu+\frac{\ell}{b_0}))
e^{2\pi i (b n\nu_0-\nu m b_0)}dbd\nu\\
&=&\int_{\square}( \sum_{k,\ell}F(b+\frac{k}{\nu_0},\nu+\frac{\ell}{b_0}))
e^{2\pi i ((b+\frac{k}{\nu_0})n\nu_0-(\nu+\frac{\ell}{b_0})m b_0 )}dbd\nu\\
&=&\int_{\mathbb{R}^2}F(b,\nu) e^{2\pi i (bn\nu_0-\nu mb_0)}dbd\nu
= \cF_s^{-1} ({\mathscr{V}}_g h^{j'}\cdot \overline{{\mathscr{V}}_g h^j})(\lambda).\end{aligned}$$ Hence, we may apply the Poisson summation formula, with convergence in $\mathbf{L^2}(\square )$, to obtain: $$\begin{aligned}
P_{\Lambda^{\circ}}({\mathscr{V}}_g h^{j'}\cdot \overline{{\mathscr{V}}_g h^j}) (b,\nu)
&=&
b_0\nu_0\sum_{k,l}\cF_s^{-1} ({\mathscr{V}}_g h^{j'}\cdot
\overline{{\mathscr{V}}_g h^j})(kb_0,l\nu_0)e^{-2\pi i (b\ell\nu_0 -\nu kb_0)}\\
&=&
b_0\nu_0\sum_{k,l} \overline{{\mathscr{V}}_g g}\cdot {\mathscr{V}}_{h^j}h^{j'}
(kb_0,l\nu_0)e^{-2\pi i (b\ell\nu_0 -\nu kb_0)}.\end{aligned}$$We conclude that $$\left\|\sum_j\sum_{\lambda\in\Lambda} c_{\lambda}^j P_{\lambda}^j\right\|_2^2
= b_0\nu_0\int\Gamma(b,\nu)_{jj'}C^j(b,\nu ) \overline{C^{j'} (b,\nu )}dbd\nu,$$ and the Riesz basis property is equivalent to the invertibility of $\Gamma$.\
In the sequel, the discrete symplectic Fourier transforms of $\bm_j$ will be denoted by ${\mathscr{M}}_j$, and the vector with ${\mathscr{M}}_j$ as coordinates will be denoted by ${\mathscr{M}}$. We then obtain an expression for the best multiplier in analogy to the Gabor multiplier case discussed in Theorem \[th:gabmult.app\].
\[prop:multgabmult.app\] Let $g\in S_0(\RR)$ and $h^{(j)}\in S_0(\RR)$, $j\in\mathcal{J}$ be such that for almost all $b,\nu$, the matrix $\Gamma(b,\nu)$ defined in is invertible a.e. on $\square = [0,\nu_0^{-1}[\times [0,b_0^{-1}[$.
Let $H\in ({\mathscr{B}},{\mathscr{H}},{\mathscr{B}}')$ be an operator with spreading function $\eta\in (S_0, \mathbf{L}^2, S_0')$. Then the functions ${\mathscr{M}}_j$ yielding approximation of the form (\[fo:multimulti\]) may be obtained as $$\label{fo:tototo0}
{\mathscr{M}}= \Gamma^{-1}\cdot \mathcal{B}\ ,$$ where $\mathcal B$ is the vector whose entries read $$\cB_{j_0}(b,\nu)
= \sum_{k,\ell}\eta(b+k/\nu_0,\nu+\ell/b_0)
\overline{{\mathscr{V}}}_g h^{j_0}(b+k/\nu_0,\nu+\ell/b_0).
\label{fo:tototo}$$ For operators in ${\mathscr{H}}$ the obtained approximation is optimal in Hilbert-Schmidt sense.
The proof follows the lines of the Gabor multiplier case. The optimal approximation of the form (\[fo:multimulti\]), when it exists, is obtained by minimizing $$\big\|\eta - \sum_j {\mathscr{M}}_j{\mathscr{V}}_j\big\|^2
= \sum_{k,\ell}\int_\square \big|\eta(b+k/\nu_0,\nu+\ell/b_0)
- \sum_j {\mathscr{M}}_j(b,\nu){\mathscr{V}}_j(b+k/\nu_0,\nu+\ell/b_0)\big|^2\,dbd\nu$$ where one has set ${\mathscr{V}}_j = {\mathscr{V}}_gh^{(j)}$. Setting to zero the Gâteaux derivative with respect to $\overline{{\mathscr{M}}}_{j_0}$, we obtain the corresponding variational equation $$\sum_j{\mathscr{M}}_j(b,\nu)
\sum_{k,\ell}{\mathscr{V}}_j(b+k/\nu_0,\nu+\ell/b_0)
\overline{{\mathscr{V}}}_{j_0}(b+k/\nu_0,\nu+\ell/b_0) = \cB_{j_0}(b,\nu)\ ,$$ where $\cB_j(b,\nu)$ are as defined in (\[fo:tototo\]). Provided that the $\Gamma(b,\nu)$ matrices are invertible for almost all $b,\nu$, this implies that the functions ${\mathscr{M}}_j$ for approximation of the form (\[fo:multimulti\]) may indeed be obtained as in (\[fo:tototo0\]). .
In a next step, we are going to discern two basic approaches:\
(a) $m_j (\lambda) = m(\mu,\lambda)$, i.e. the synthesis windows are time-frequency shifted versions (on a lattice) of a single synthesis window: [$h_j = \pi (\mu_j ) h$, $\mu_j\in\Lambda_1$.]{}\
(b) $m_j (\lambda ) = m_1(\lambda)m_2(j)$, i.e. a separable multiplier function. If we set $h^{(j)} (t) = \pi(b_j,\nu_j)h(t)$ then this approach leads to what will be called TST spreading functions in Section \[Se:TSTSF\].\
In both cases we will be especially interested in the situation in which the $h^j$ are given as time-frequency shifted versions of a single synthesis window on the adjoint lattice $\Lambda^{\circ}$.
\[Def:AdLatt\] For a given lattice $\Lambda= b_0\mathbb{Z}\times \nu_0\mathbb{Z}$ the adjoint lattice is given by $\Lambda^{\circ}=\frac{1}{\nu_0}\mathbb{Z}\times \frac{1}{b_0}\mathbb{Z}$.
Note that the adjoint lattice is the dual lattice $\Lambda^{\perp}$ with respect to the symplectic character.\
Varying the multiplier: MGM with synthesis windows on the lattice
-----------------------------------------------------------------
We fix the synthesis windows $h_j$ to be time-frequency translates of a fixed window function, i.e. $$\label{fo:glou}
h^{(j)} (t) = \pi(b_j,\nu_j)h(t) = e^{2i\pi\nu_j t}h(t-b_j)\ .$$ We may turn our attention to the projection operators associated to the (Gabor) families $(g,\Lambda_1)$ and $(h,\Lambda_2)$. Note that it has been shown by Benedetto and Pfander [@bepf06] that the family of projection operators $\{P_{\lambda},\,\lambda\in\Lambda\}$, as discussed in Section \[se:mult\] either forms a Riesz basis or not a frame (for its closed linear span). The next corollary shows that, on the other hand, if we use the extended family of projection operators $\{P_{\lambda,\mu}$, $(\lambda,\mu) \in\Lambda_1\times\Lambda_2\}$, where $P_{\lambda,\mu}f = \langle f,\pi (\lambda )g\rangle\pi (\mu ) h$, we obtain a frame of operators for the space of Hilbert-Schmidt operators, whenever $(g,\Lambda_1)$ and $(h,\Lambda_2)$ are Gabor frames. This corollary is a special case of Theorem 4.1 in [@ba08a] and Proposition 3.2 in [@ba08b].
\[Co:ProdFram\] Let two Gabor frames $(g,\Lambda_1)$ and $(h,\Lambda_2)$ be given. Then the family of projection operators $\{P_{\lambda,\mu}$, $(\lambda,\mu) \in\Lambda_1\times\Lambda_2\}$ form a frame of operators in $\cH$ and any Hilbert-Schmidt operator $H$ may be expanded as $$H = \sum_{\lambda\in\Lambda_1, \mu\in\Lambda_2}\mathbf{c}(\lambda,\mu) P_{\lambda,\mu} .$$ The coefficients are given by $\mathbf{c}(\lambda,\mu) = \langle H, (P_{\lambda,\mu})^{\ast}\rangle = \langle H \pi(\mu)h,\pi(\lambda)g\rangle$.
Note that an analogous statement holds for Riesz sequences. Very recently, it has been shown [@bo08], that the converse of Corollary \[Co:ProdFram\] holds true for both frames and Riesz bases, i.e. the family of projection operators $\{P_{\lambda,\mu}$, $(\lambda,\mu) \in\Lambda_1\times\Lambda_2\}$ is a frame (a Riesz basis) for $\cH $ if and only if the two generating sequences form a frame (a Riesz basis) for $\mathbf{L}^2 (\RR)$. In particular, this leads to the conclusion, that the characterization of Riesz sequences given in Proposition \[Prop:GenUcond\] also yields a characterization of frames for $\mathbf{L}^2 (\RR)$ - it is well known, that $(g_{\lambda}, \lambda\in\Lambda)$ form a Gabor frame if and only if $(g_{\mu}, \mu\in\Lambda^{\circ})$ form a Riesz sequence. We can draw two conclusions.
\[Co:ProdFram2\]Let $g\in S_0 (\RR )$ and a lattice $\Lambda= b_0 \mathbb{Z}\times\nu_0\mathbb{Z}$ be given.
- The Gabor family $\{g_{\lambda}, \lambda\in\Lambda\} $ forms a frame for $\mathbf{L}^2(\RR )$ if and only if the matrix $$\begin{aligned}
\Gamma_{mn,m'n'}(b,\nu)
&=& \sum_{k,\ell}
\exp\big(-2i\pi [m/\nu_0(\nu+\ell\nu_0 -n/b_0)- m'/\nu_0(\nu+\ell\nu_0 -n'/b_0)]\big)
\\
&&\hphantom{aaaaaa}\times\overline{{\mathscr{V}}_g g}(b-m/\nu_0+kb_0,\nu-n/b_0+\ell\nu_0)\\
&&\hphantom{aaaaaa}\times{{\mathscr{V}}_g g}(b-m'/\nu_0+kb_0,\nu-n'/b_0+\ell\nu_0)\end{aligned}$$is, a.e. on $\square$, invertible on $\ell^2$.\
- In addition, we may state the following “Balian-Low Theorem for the tensor products of Gabor frames”:\
A family of projection operators given by $\{P_{\lambda,\mu} = g_\lambda^{\ast}\otimes g_\mu$, $(\lambda,\mu) \in\Lambda\times\Lambda^{\circ}\}$ forms a frame for the space of Hilbert-Schmidt operators on $\mathbf{L}^2(\RR )$ if and only if it forms a Riesz basis. Hence, in this case, $g$ cannot be in $S_0 (\RR )$.
Statement (a) is easily obtained from by observing that $${\mathscr{V}}_g \pi(mb_0,n\nu_0 )g(b,\nu) = e^{-2i\pi mb_0,(\nu-n\nu_0)}{\mathscr{V}}_gh(b-mb_0,\nu-n\nu_0)\ .$$ We then have that $\{P_{\lambda,\mu} = g_\lambda^{\ast}\otimes g_\mu$, $(\lambda,\mu) \in\Lambda^{\circ}\times\Lambda^{\circ}\}$ forms a Riesz basis in $\cH$ if and only if $\Gamma_{mn,m'n'}(b,\nu) $ is invertible. By the converse of Corollary \[Co:ProdFram\], this is equivalent to the Riesz property of $g_{\mu}, \mu\in\Lambda^{\circ} $, which, in turn, is equivalent to the frame property of $\{g_{\lambda}, \lambda\in\Lambda\} $ by Ron-Shen duality, see, e.g. [@gr01].\
To see (b), note that in this case $P_{\lambda,\mu}$ is a frame for $\cH$ $\Leftrightarrow$ $(g_\lambda,\Lambda)$ and $(g_\mu,\Lambda^{\circ})$ form a frame $\Leftrightarrow$ $(g_\lambda,\Lambda)$ and $(g_\mu,\Lambda^{\circ})$ form a Riesz basis $\Leftrightarrow$ $\{P_{\lambda,\mu} = g_\lambda^{\ast}\otimes g_\mu$, $(\lambda,\mu) \in\Lambda\times\Lambda^{\circ}\}$ is a Riesz basis for $\cH$. Furthermore, by the classical Balian-Low theorem, if a Gabor system is an $\mathbf{L}^2$-frame and at the same time a Riesz sequence (hence an $\mathbf{L}^2$-Riesz basis), then the generating window $g$ cannot be in $S_0$[^6]. .
We may next ask, when the projection operators form a Riesz sequence, if the reconstruction windows are TF-shifted versions of a single window $h$ on the adjoint lattice of $\Lambda = b_0 \mathbb{Z}\times \nu_0 \mathbb{Z}$. In fact, in this case, the matrix $\Gamma$ turns out to enjoy quite a simple form. To fix some notation, let $$\begin{aligned}
\cA_{mn}(b,\nu) =& \sum_{k,\ell}e^{2i\pi m[\nu-\ell/\nu_0]}\
\overline{{\mathscr{V}}_g h}(b-k/\nu_0,\nu-\ell/b_0)\\
\qquad\qquad\times & \
{{\mathscr{V}}_g h}(b-(k-m)/\nu_0,\nu-(\ell-n)/b_0), \end{aligned}$$ and introduce the right twisted convolution operator $$K_\cA^\natural(b,\nu): {\mathscr{M}}(b,\nu)\to {\mathscr{M}}(b,\nu)\natural \cA(b,\nu).$$
Let $g,h \in S_0$ as well as $b_0,\nu_0$ be given. Furthermore, let $h^{(j)} = \pi(\frac{m}{\nu_0}, \frac{n}{b_0})h$. Then the variational equations read $$\label{eq:DefA}
{\mathscr{M}}(b,\nu) \natural \cA(b,\nu) = \cB(b,\nu)\ .$$ Hence, if for all $b,\nu\in\RR^2$, the discrete right twisted convolution operator $K_\cA^\natural$ is invertible, then the family $P_{\lambda,\mu} = g_\lambda^{\ast}\otimes g_\mu$, $(\lambda,\mu) \in\Lambda\times\Lambda^{\circ}$ forms a Riesz sequence and the best MGM approximation of an Hilbert-Schmidt operator with spreading function $\eta$ is given by the family of transfer functions $${\mathscr{M}}_{mn}(b,\nu) = \left[(K_A^\natural(b,\nu))\inv \cB(b,\nu)\right]_{mn}\ ,$$ where $\cB$ is given in (\[fo:tototo\]).
As in the proof of Corollary \[Co:ProdFram2\], we may derive the given form of $\Gamma$ by direct calculation, achieving the final form by noting that for $\Lambda= b_0\mathbb{Z}\times\nu_0\mathbb{Z}$, the adjoint lattice is given by $\Lambda^{\circ}=\frac{1}{\nu_0}\mathbb{Z}\times\frac{1}{b_0}\mathbb{Z}$..\
We close this section with some results of numerical experiments testing the approximation quality of MGMs for slowly time-varying systems. .
We study the approximation of a (slowly) time-varying operator. The operator has been generated by perturbing a time-invariant operator. The spreading function is shown in the upper display of Figure \[FI2\]. The signal length is $32$, time- and frequency-parameters are $b_0 = 4$ and $\nu_0 = 4$, such that the redundancy of the Gabor frame used in the MGM approximation is $2$. The approximation is then realized in several steps for two different schemes. Scheme 1 adds three synthesis window corresponding to a frequency-shift by $4$, a time-shift by $4$ and a time-frequency-shift by $(4,4)$. The first step $1$ calculates the regular Gabor multiplier approximation. Step $2$ adds one (only frequency-shift) and so on. The rank of the resulting operator families is $64$, $128$ for both step 2 and step 3 (adding either time- or frequency-shift) and $256$ (time-shifted, frequency-shifted and time-frequency-shifted window added). The resulting approximation-errors are given by the solid line in the lower display of Figure \[FI2\].\
Scheme 2 considers synthesis windows shifted in time and frequency on the sub-lattice generated by $a = 8, b = 8$, the resulting families having rank $64$, $256$ and $576$. Here, we only plot the results for the case corresponding to three and eight additional synthesis windows, respectively. The results are given by the dotted line.\
For comparison, an approximation with a regular Gabor multiplier with redundancy $8$, i.e. an approximation family of rank $256$, has been performed. The approximation error for this situation is the diamond in the middle of the display.
It is easy to see that, depending on the behavior of the spreading function, different schemes perform advantageously for a certain redundancy. Note that for scheme 1, the best MGM with the same rank as the regular Gabor multiplier performs better than the latter. In the case of the present operator, scheme 2 performs the “wrong” time-frequency shifts on the synthesis windows in order to capture important characteristics of the operator. However, in a different setting, this scheme might be favorable (e.g. if an echo with a longer delay is present).
The example shows, that the choice of an appropriate sampling scheme for the synthesis windows is extremely important in order to achieve a good and efficient approximation by MGM. An optimal sampling scheme depends on the analysis window’s STFT, the lattice used in the analysis and on the behavior of the operator’s spreading function, which reflects the amount of delay and Doppler-shift created by the operator. Additionally, structural properties of the family of projections operators used in the approximation, based on the results in this section, have to be exploited to achieve numerical efficiency. An algorithm for optimization of these parameters is currently under development.[^7]
Varying the synthesis window: TST spreading functions {#Se:TSTSF}
-----------------------------------------------------
We next turn to the special case of separable functions $\bm_j (\lambda ) = m_1(\lambda)m_2(j)$ for the mask in the definition of MGMs. In this case the resulting operator is of the form $$\MM f = \sum_{\lambda} m_1(\lambda )\rho (\lambda )
(\sum_j m_2 (j) ( g^{\ast}\otimes h^j))(f)
=\sum_{\lambda} m_1(\lambda )\rho (\lambda )\PP_m f\ ,$$ where $\PP_m f = \sum_j m_2 (j) \langle f, g\rangle h^j$ and $\rho (\lambda )$ denotes a tensor product of time-frequency shifts: $$\rho (\lambda ) H := \pi(\lambda )H\pi^{\ast}(\lambda ).$$ Hence, the spreading function of $\MM$ is given by $$\eta_{\MM} = {\mathscr{M}}\cdot \eta_{\PP_m},$$ where ${\mathscr{M}}$ is the discrete symplectic FT of $m_1$. If the reconstruction windows are given by $h^j = \pi (\mu^j )h$, $\mu_j = (b_j,\nu_j )$, this becomes $$\eta_{\MM} = {\mathscr{M}}\cdot\sum_j m_2(j) {\mathscr{V}}_g h(\lambda -\mu_j )
e^{-2\pi i (\nu -\nu_j )b_j}\ .$$ Motivated by this result, we introduce the following definition.
Let $\phi$ be a given function from the function spaces $(S_0(\RR^2),
\mathbf{L}^2(\RR^2),S_0'(\RR^2))$ and let $b_1,\nu_1$ denote positive numbers. Let $\mathbf{ \alpha}$ be in $\ell^1(\mathbb{Z}^2)$. A spreading function $\eta=\eta_H$ of $H\in ({\mathscr{B}},{\mathscr{H}}, {\mathscr{B}}')$, that may be written as $$\label{fo:TST}
\eta(b,\nu) = \sum_{k,\ell} \alpha_{k\ell} \phi(b-kb_1,\nu-\ell\nu_1)
e^{-2i\pi(\nu-\ell\nu_1)kb_1}$$ will be called [*Twisted Spline Type*]{} function (TST for short).
By $\alpha$ in $\ell^1$, the series defining $\eta$ is absolutely convergent in $(S_0,\mathbf{L}^2,S_0' )$. For $\ell^2$-sequences $\alpha$, we obtain an $\mathbf{L}^2$-function $\eta$ for $\phi\in\mathbf{L}^2$.
TST functions are nothing but spline type functions (following the terminology introduced in [@Feichtinger02wavelet]), in which usual (Euclidean) translations are replaced with the natural (i.e. $\HH$-covariant) translations on the phase space $\PP$. In fact, by writing $\alpha (b,\nu ) = \sum_{k,\ell} \alpha_{k\ell} \delta(b-kb_1,\nu-\ell\nu_1)$, the TST spreading function may be written as a twisted convolution: $\eta (b,\nu ) = \alpha\natural \phi$. This leads to the following property of operators associated with TST spreading functions.
An operator $H\in ({\mathscr{B}},{\mathscr{H}}, {\mathscr{B}}')$ possesses a TST spreading function $\eta\in (S_0, \mathbf{L}^2,S_0')$ as in (\[fo:TST\]) if and only if it is of the form $$H_{\eta} = \sum_{k,\ell} \alpha_{k\ell} \pi(kb_1,\ell\nu_1) H_\phi\ ,$$ where $H_\phi $ is the linear operator with spreading function $\phi$.
The proof consists of a straight-forward computation which may be spared by noting that we have, by : $$H_{\eta} =H_{\alpha\natural \phi}= H_{\alpha}\cdot H_{\phi}
= \sum_{k,\ell} \alpha_{k\ell} \pi(kb_1,\ell\nu_1) H_{\phi}\ .$$
As before, we are particularly interested in the situation of the synthesis windows being given by time-frequency shifted versions of a single window: $h^j = \pi(\mu_j ) h$. In a next step we note, that under the condition $\pi(\lambda ) \pi(\mu_j ) = \pi(\mu_j )\pi(\lambda )$, i.e., $\mu_j\in\Lambda^{\circ}$, the MGM with separable multiplier results in a TST spreading function with a Gabor multiplier as basic operator $H_\phi$.
\[Le:MGM\_TST\] Assume that a MGM $\mathbb{M}$ with multiplier $m_j( \lambda ) = m_1 (\lambda ) m_2(j)$ is given. If the synthesis windows $h^j$ are given by $h^j = \pi (\mu _j ) h$, with $\mu_j\in\Lambda^{\circ}$, then $$\mathbb{M} = \sum_j m_2 (j) \pi (\mu_j ) \mathbb{M}^G_{m_1;g,h},$$ i.e., here, the operator $H_{\phi}$ is given by a regular Gabor multiplier with mask $m_1$ and synthesis window $h$.
Comparing the expression in the previous lemma to the expression $\mathbb{M} = \sum_{\lambda} m_1(\lambda )\rho (\lambda )\PP_m $ for the same operator, we note that in this situation, the operator may either be interpreted as a (weighted) sum of Gabor multipliers or as a Gabor multiplier with a generalized projection operator $\mathbb{P}_{\bm}$ in the synthesis process. In this situation, we may ask, whether the family of generalized projection operators, $\{\rho (\lambda )\mathbb{P}_{\bm}\}_{\lambda\in\Lambda}$ form a frame or Riesz basis for their linear span. In fact, if $\bm$ is in $\ell^1$ and $g,h\in S_0$, this question is easily answered by generalizing the result proved in [@bepf06 Theorem 3.2]. Here, $\{\rho (\lambda )\mathbb{P}_{\bm}\}_{\lambda\in\Lambda}$ is either a Riesz basis or not a frame for its closed linear span. Furthermore, there exists $r>0$ such that $\{\rho (\alpha\lambda )\mathbb{P}_{\bm}\}_{\lambda\in\Lambda}$ is a Riesz basis for its closed linear span whenever $\alpha>r$.
In generalizing the result of Lemma \[Le:MGM\_TST\], it is a natural next step to assume that the basic function $\phi$ entering in the composition of $\eta$ is the spreading function of a Gabor multiplier (at least in an approximate sense). According to the discussion of Section \[se:mult\], this essentially means that $\phi$ is sufficiently well concentrated in the time-frequency domain.\
(In the sequel we will write $\pi_{mn}$ for $\pi (mb_0,n\nu_0)$ whenever the applicable lattice constants are sufficiently clear from the context.)\
Hence, we assume that a Gabor multiplier $H_\phi$, as defined in is given. We may formally compute $$\begin{aligned}
\label{eq:TSTGabmul}
Hf &=& \sum_{k,\ell} \alpha_{k\ell} \pi_{k\ell}\sum_{m,n} \bm(m,n)
{\mathscr{V}}_gf(mb_0,n\nu_0) \pi (mb_0,n\nu_0) h\\\nonumber
&=& \sum_{m,n} \bm(m,n){\mathscr{V}}_gf(mb_0,n\nu_0)
\sum_{k,\ell} \alpha_{k\ell}\pi(kb_1,\ell\nu_1)\pi (mb_0,n\nu_0) h\end{aligned}$$ Based on this expression, one may pursue two different choices of the sampling-points $(kb_1, \ell\nu_1)$. First, in extension of the result given in Lemma \[Le:MGM\_TST\], we assume that the sampling points are associated to the adjoint lattice $\Lambda^{\circ} = \frac{1}{\nu_0}\mathbb{Z}\times \frac{1}{b_0}\mathbb{Z}$ of $\Lambda = b_0\mathbb{Z}\times \nu_0\mathbb{Z}$. The second choice of sampling points on the original lattice leads to a construction as introduced in [@Dorfler07spreading] as Gabor twisters and will not be further discussed in the present contribution.\
The following theorem extends the result given in Lemma \[Le:MGM\_TST\] to the case in which the sampling points in the TST expansion are chosen from a lattice containing the adjoint lattice. It turns out that the TST spreading function then leads to a representation as a sum of Gabor multipliers.
\[Th:TSTfacts\] Let $b_0,\nu_0\in\RR^+$ generate the time-frequency lattice $\Lambda$, and let $\Lambda^\circ$ denote the adjoint lattice. Let $g,h\in S_0(\RR)$ denote respectively Gabor analysis and synthesis windows, such that the $\cU$ condition (\[fo:Ucond\]) is fulfilled. Let $H$ denote the operator in $({\mathscr{B}}, {\mathscr{H}}, {\mathscr{B}}')$ defined by the twisted spline type spreading function $\eta$ as in (\[fo:TST\]), with $b_1,\nu_1\in\RR^+$.
1. Assume that $b_1$ and $\nu_1$ are multiple of the dual lattice constants. Then $H$ is a Gabor multiplier, with analysis window $g$, synthesis window $$\gamma =\sum_{k,\ell} \alpha_{k\ell}\pi(kb_1,\ell\nu_1) h\ ,$$ and transfer function $$\label{fo:TST.transfer.function}
\bm(m,n) = b_0\nu_0\int_\square {\mathscr{M}}(b,\nu) e^{-2i\pi(n\nu_0b - mb_0\nu)}\, db
d\nu\ ,$$ with $\square$ the fundamental domain of the adjoint lattice $\Lambda^\circ$, and $$\label{fo:TST.transfer.function2}
{\mathscr{M}}(b,\nu) = \frac{\sum_{k,\ell=-\infty}^\infty
\overline{{\mathscr{V}}_gh}\left(b+k/\nu_0,\nu + \ell/b_0\right)
\phi\left(b+k/\nu_0,\nu + \ell/b_0\right)}
{\sum_{k,\ell=-\infty}^\infty
\left|{\mathscr{V}}_gh\left(b+k/\nu_0,\nu + \ell/b_0\right)\right|^2}$$
2. Assume that the lattice generated by $b_1$ and $\nu_1$ contains the adjoint lattice: $$\label{fo:super.dual.lattice}
b_1 = \frac1{p\nu_0}\ ,\qquad \nu_1 = \frac1{qb_0}\ .$$ Then $H$ may be written as a finite sum of Gabor multipliers $$\label{fo:super.lattice.approx}
Hf = \sum_{i=1}^p \sum_{j=1}^q \bigg(
\sum_{m\equiv i\,[{ \rm mod}\,p]}\ \sum_{n\equiv j\,[{ \rm mod}\,q]}
\bm(m,n){\mathscr{V}}_gf(mb_0,n\nu_0) \pi_{mn}\bigg) \gamma_{ij}\ ,$$ with at most $p\cdot q$ different synthesis windows $\gamma_{ij}$ and the transfer function given in (\[fo:TST.transfer.function\]) and (\[fo:TST.transfer.function2\]).
\
Let us formally compute $$\begin{aligned}
Hf &=& \sum_{m,n} \bm(m,n){\mathscr{V}}_gf(mb_0,n\nu_0)
\sum_{k,\ell} \alpha_{k\ell}\pi(kb_1,\ell\nu_1)\pi_{mn} h\\
&=& \sum_{m,n} \bm(m,n){\mathscr{V}}_gf(mb_0,n\nu_0) \pi_{mn} \gamma_{mn}\ ,\end{aligned}$$ where $$\nonumber
\gamma_{mn} = \sum_{k,\ell} \alpha_{k\ell}
e^{2i\pi [knb_0\nu_1 - \ell m \nu_0 b_1]} \pi(kb_1,\ell\nu_1) h$$ Now observe that if $(b_1,\nu_1)\in\Lambda^\circ$, one obviously has $$\gamma_{mn} = \sum_{k,\ell} \alpha_{k\ell}\pi(kb_1,\ell\nu_1) h =
\gamma_{00}\ , \mbox{ for } (m,n)\in\ZZ^2\ ,$$ i.e. the above expression for $Hf$ involves a single synthesis window $\gamma=\gamma_{00}$. Therefore, in this case, $H$ takes the form of a standard Gabor multiplier, with fixed time-frequency transfer function, and a synthesis window prespribed by the coefficients in the TST expansion. This proves the first part of the theorem.
Let us now assume that the TST expansion of the spreading function is finer than the one prescribed by the lattice $\Lambda^\circ$, but nevertheless the lattice $\Lambda_1 = \ZZ b_1\times\ZZ\nu_1$ contains $\Lambda^\circ$. In other words, there exist positive integers $p,q$ such that holds.\
We then have $$\gamma_{mn} = \sum_{k,\ell} \alpha_{k\ell}
e^{2i\pi [\frac{knp-lmq}{pq} ]} \pi(kb_1,\ell\nu_1) h$$ and it is readily seen that there are at most $pq$ different synthesis windows $\gamma_{ij}$, $$\label{fo:super.synth.win}
\gamma_{ij} = \gamma_{m\,[{\rm mod}\,p],n\,[{\rm mod}\,q]}\ ,
i = 1,\ldots , p; \ j = 1,\ldots , q .$$ The operator $H$ may hence be written as a sum of Gabor multipliers, with one prescribed time-frequency transfer function, which is sub-sampled on several sub-lattices of the lattice $\Lambda$: $$\Lambda_{ij} = (p b_0\cdot\mathbb{Z}+i\cdot b_0)\times
(q \nu_0\cdot\mathbb{Z}+j\cdot \nu_0),\,
i = 0,\ldots, p-1; j = 0, \ldots , q-1,$$ and a single synthesis window per sub-lattice as given in . The resulting expression for $H$ is hence as given in .\
The expression for the transfer function is derived in analogy to the case discussed in Section \[se:mult\]. .
Let us observe that in this approximation, the time-frequency transfer function $\bm$ is completely characterized by the function $\phi$ used in the TST expansion. The choice of $\phi$ therefore imposes a fixed mask for the multipliers that come into play in equation (\[fo:super.lattice.approx\]).
We first assume, that for a given primal lattice $\Lambda =
b_0\mathbb{Z}\times \nu_0\mathbb{Z}$, the representation of a spreading function $\eta$ is given by $5$ building blocks: $$\eta (b,\nu ) =\sum_{k = -1}^1 \alpha_{k0} \phi (b
-\frac{k}{\nu_0},\nu )+\sum_{\ell = -1}^1 \alpha_{0\ell} \phi (b ,\nu
-\frac{\ell}{b_0}).$$ In this case, we obtain a single Gabor multiplier with synthesis window $$\gamma_{00} = \sum_{k = -1}^1 \alpha_{k0} \pi (\frac{k}{\nu_0},0)h
+\sum_{\ell = -1}^1 \alpha_{0\ell} \pi (0, \frac{\ell}{b_0})h\ .$$ [If we add the windows $ \phi (b \pm\frac{1}{2\nu_0},\nu
\pm\frac{1}{2b_0})$ to the representation of $\eta$, we are now dealing with the finer lattice $\Lambda =
\frac{1}{2\nu_0}\mathbb{Z}\times\frac{1}{2b_0} \mathbb{Z}$ and we obtain the sum of $4$ Gabor multipliers with the following synthesis windows: $$\begin{aligned}
\gamma_{00} &=& \sum_{k = -1}^1 \alpha_{k0}
\pi\! \left(\frac{k}{2\nu_0},0\right)h
+\sum_{\ell = -1}^1 \alpha_{0\ell}
\pi\! \left(0, \frac{\ell}{2b_0}\right)h, \\
\gamma_{01} &= &\sum_{k = -1}^1 \alpha_{k0}e^{\pi ik}
\pi\!\left(\frac{k}{2\nu_0},0\right)h +
\sum_{\ell = -1}^1 \alpha_{0\ell}
\pi\! \left(0,\frac{\ell}{2b_0}\right)h, \\
\gamma_{10} &=& \sum_{k = -1}^1 \alpha_{k0}
\pi\!\left(\frac{k}{2\nu_0},0\right)h
+\sum_{\ell = -1}^1e^{\pi i\ell}\alpha_{0\ell}
\pi\! \left(0, \frac{\ell}{2b_0}\right)h\ ,\\
\gamma_{11} &=& \sum_{k = -1}^1 \alpha_{k0}e^{\pi ik}
\pi\!\left(\frac{k}{2\nu_0},0\right)h +
\sum_{\ell = -1}^1 \alpha_{0\ell}e^{-\pi i\ell}
\pi\! \left(0,\frac{\ell}{2b_0}\right)h\ ,\end{aligned}$$ and corresponding lattices: $\Lambda_{00} = 2\mathbb{Z}b_0\times 2\mathbb{Z}\nu_0$, $\Lambda_{01} = 2\mathbb{Z}b_0\times (2\mathbb{Z}+1)\nu_0$, $\Lambda_{10} = (2\mathbb{Z} +1)b_0\times 2\mathbb{Z}\nu_0$, and $\Lambda_{11} = (2\mathbb{Z}+1)b_0\times (2\mathbb{Z}+1)\nu_0$. ]{}
It is important to note, that in both cases described in Theorem \[Th:TSTfacts\] as well as the above example, the transfer function $\bm$ can be calculated as the best approximation by a regular Gabor multiplier - a procedure which may be efficiently realized using . Fast algorithms for this exist in the literature, see [@fehakr04], however, the method derived in Section \[Se:tfmult\] appears to be faster.
Conclusions and Perspectives
============================
Starting from an operator representation in the continuous time-frequency domain via a twisted convolution, we have introduced generalizations of conventional time-frequency multipliers in order to overcome the restrictions of this model in the approximation of general operators. The model of multiple Gabor multipliers in principle allows the representation of any given linear operator. However, in oder to achieve computational efficiency as well as insight in the operator’s characteristics, the parameters used in the model must be carefully chosen. An algorithm choosing the optimal sampling points for the family of synthesis windows, based on the spreading function, is the topic of ongoing research. On the other hand, the model of twisted spline type functions allows the approximation of a given spreading function and results in an adapted window or family of windows. By refining the sampling lattice in the TST approximation, a rather wide class of operators should be well-represented. The practicality of this approach has to be shown in the context of operators of practical relevance. All the results given in this work will also be applied in the context of estimation rather than approximation.\
As a further step of generalization, frame types other than Gabor frames may be considered. Surprisingly little is known about wavelet frame multipliers, hence it will be interesting to generalize the achieved results to the affine group.
[10]{}
A.Bourouihiya. The tensor product of frames. , 7(1):65–76, Jan. 2008.
R. M. [B]{}alan, S. [R]{}ickard, H. V. [P]{}oor, and S. [V]{}erdú. Canonical [T]{}ime-[F]{}requency, [T]{}ime-[S]{}cale, and [F]{}requency-[S]{}cale representations of time-varying channels. , 5(2):197–226, 2005.
P. [B]{}alazs. ilbert-[S]{}chmidt [O]{}perators and [F]{}rames - [C]{}lassification, [B]{}est [A]{}pproximation. , 6(2):315 – 330, [M]{}arch 2008.
P. Balazs. Matrix-representation of operators using frames. , 7(1):39–54, Jan. 2008.
P. [B]{}ello. haracterization of [R]{}andomly [T]{}ime-[V]{}ariant [L]{}inear [C]{}hannels. , 11:360–393, 1963.
J. J. [B]{}enedetto and G. E. [P]{}fander. rame expansions for [G]{}abor multipliers. , 20(1):26–40, 2006.
H. [B]{}[ö]{}lcskei, R. [K]{}oetter, and S. [M]{}allik. oding and modulation for underspread fading channels. In [*[I]{}[E]{}[E]{}[E]{} [I]{}nternational [S]{}ymposium on [I]{}nformation [T]{}heory ([I]{}[S]{}[I]{}[T]{}) 2002*]{}, page 358. [E]{}[T]{}[H]{}-[Z]{}[ü]{}rich, jun 2002.
R. Carmona, W. Hwang, and B. Torrésani. , volume 9 of [*Wavelet Analysis and its Applications*]{}. Academic Press, San Diego, 1998.
I. Daubechies. . SIAM, Philadelphia, PA, 1992.
M. Dolson. The phase vocoder: a tutorial. , 10(4):11–27, 1986.
M. Dörfler and B. Torrésani. Spreading function representation of operators and [G]{}abor multiplier approximation. In [*Sampling Theory and Applications (SAMPTA’07), Thessaloniki, June 2007*]{}, 2007.
H. Feichtinger and G. Zimmermann. A [B]{}anach space of test functions for [G]{}abor analysis. In H. Feichtinger and T. Strohmer, editors, [*[G]{}abor Analysis and Algorithms: Theory and Applications*]{}, pages 123–170. [Birkhäuser]{}, Boston, 1998. Chap. 3.
H. G. Feichtinger. On a new [S]{}egal algebra. , 92(4):269–289, 1981.
H. G. Feichtinger. Spline type spaces in [G]{}abor analysis. In D. Zhou, editor, [*Wavelet analysis: twenty years’ developments*]{}, Singapore, 2002. World Scientific.
H. G. [F]{}eichtinger, M. [H]{}ampejs, and G. [K]{}racher. pproximation of matrices by [G]{}abor multipliers. , 11(11):883– 886, 2004.
H. G. Feichtinger and W. Kozek. Quantization of [T]{}[F]{} lattice-invariant operators on elementary [L]{}[C]{}[A]{} groups. In [*Gabor analysis and algorithms*]{}, pages 233–266. Birkhäuser Boston, Boston, MA, 1998.
H. G. Feichtinger and K. Nowak. A first survey of [G]{}abor multipliers. In H. G. Feichtinger and T. Strohmer, editors, [*Advances in Gabor Analysis*]{}, Boston, 2002. Birkhauser.
H. G. [F]{}eichtinger and T. [S]{}trohmer. irkh[ä]{}user, 1998.
H. G. [F]{}eichtinger and T. [S]{}trohmer. . irkh[ä]{}user, 2003.
J. Flanagan and R. M. Golden. Phase vocoder. , 45:1493 – 1509, 1966.
G. Folland. . Princeton University Press, Princeton, NJ, 1989.
H. Führ. . Number 1863 in Lecture Notes in Mathematics. Springer Verlag, Berlin; Heidelberg; New York, NY, 2005.
I. M. Gel[’]{}fand and N. Y. Vilenkin. . Academic Press \[Harcourt Brace Jovanovich Publishers\], New York, 1964 \[1977\]. Applications of harmonic analysis. Translated from the Russian by Amiel Feinstein.
K. [G]{}r[ö]{}chenig. . ppl. [N]{}umer. [H]{}armon. [A]{}nal. [B]{}irkh[ä]{}user [B]{}oston, 2001.
K. [G]{}r[ö]{}chenig. ncertainty principles for time-frequency representations. In H. [F]{}eichtinger and T. [S]{}trohmer, editors, [*[A]{}dvances in [G]{}abor [A]{}nalysis*]{}, pages 11–30. [B]{}irkh[ä]{}user [B]{}oston, 2003.
A. Grossmann, J. Morlet, and T. Paul. Transforms associated to square integrable group representations [I]{}: General results. , 26:2473–2479, 1985.
A. Grossmann, J. Morlet, and T. Paul. Transforms associated to square integrable group representations [II]{}: Examples. , 45:293, 1986.
F. Hlawatsch and G. Matz. Linear time-frequency filters. In B. Boashash, editor, [*Time-Frequency Signal Analysis and Processing: A Comprehensive Reference*]{}, page 466:475, Oxford (UK), 2003. Elsevier.
T. [K]{}ailath. easurements on time-variant communication channels. , 8(5):229– 236, 1962.
R. Kennedy. . Wiley, New York, 1969.
W. Kozek. . PhD thesis, NuHAG, University of Vienna, 1996.
W. [K]{}ozek. daptation of [W]{}eyl-[H]{}eisenberg frames to underspread environments. In H. [F]{}eichtinger and T. [S]{}trohmer, editors, [*[G]{}abor [A]{}nalysis and [A]{}lgorithms: [T]{}heory and [A]{}pplications*]{}, pages 323–352. [B]{}irkh[ä]{}user [B]{}oston, 1997.
W. [K]{}ozek. n the transfer function calculus for underspread [L]{}[T]{}[V]{} channels. , 45(1):219–223, [J]{}anuary 1997.
W. [K]{}ozek and G. E. [P]{}fander. dentification of operators with bandlimited symbols. , 37(3):867–888, 2006.
S. Mallat. . Academic Press, 1998.
G. [M]{}atz and F. [H]{}lawatsch. ime-frequency transfer function calculus (symbolic calculus) of linear time-varying systems (linear operators) based on a generalized underspread theory. , 39(8):4041–4070, 1998.
G. [M]{}atz and F. [H]{}lawatsch. Time-frequency characterization of random time-varying channels. In H. [F]{}eichtinger and T. [S]{}trohmer, editors, [*Time Frequency Signal Analysis and Processing: A Comprehensive Reference*]{}, pages 410 – 419. Prentice Hall, Oxford, 2002.
G. E. [P]{}fander and D. F. [W]{}alnut. perator [I]{}dentification and [F]{}eichtinger’s algebra. , 5(2):183–200, 2006.
W. Schempp. , volume 147 of [*Pitman Series*]{}. J. Wiley, New York, 1986.
L. A. [Z]{}adeh. ime-varying networks, [I]{}. In [*[P]{}roceedings of the [I]{}[R]{}[E]{}*]{}, volume 49, pages 1488–1502. others, [O]{}ctober 1961.
[^1]: The first author has been supported by the WWTF project MA07-025.
[^2]:
[^3]: The operator $$S_{g }f =
\sum_{m,n\in\mathbb{Z}}\langle f, M_{mb_0}T_{n\nu_0}g\rangle
M_{mb_0}T_{n\nu_0}g$$ is the [*frame operator*]{} corresponding to $g$ and the lattice defined by $(b_0, \nu_0)$. If $S_g $ is invertible on $L^2 (\mathbb{R} )$, the family of time-frequency shifted atoms $M_{mb_0}T_{n\nu_0}g$, $m,n\in\mathbb{Z}$, is a *Gabor frame* for $L^2 (\mathbb{R} )$.
[^4]: H. Feichtinger and F. Luef. Twisted convolution properties for [W]{}iener amalgam spaces. [*[I]{}n preparation*]{}, 2008.
[^5]: Best approximation is realized in Hilbert Schmidt sense, see the next section for details.
[^6]: More precisely, $g$ cannot even be in the space of continuous functions in the Wiener space $W(\mathbb{R})$, see [@gr01 Theorem 8.4.1]
[^7]: P. Balazs, M. Dörfler, F. Jaillet and B. Torrésani. An optimized sampling scheme for generalized Gabor multipliers. [*[I]{}n preparation*]{}, 2008.
|
---
abstract: 'Multicopters are becoming increasingly important in both civil and military fields. Currently, most multicopter propulsion systems are designed by experience and trial-and-error experiments, which are costly and ineffective. This paper proposes a simple and practical method to help designers find the optimal propulsion system according to the given design requirements. First, the modeling methods for four basic components of the propulsion system including propellers, motors, electric speed controls, and batteries are studied respectively. Secondly, the whole optimization design problem is simplified and decoupled into several sub-problems. By solving these sub-problems, the optimal parameters of each component can be obtained respectively. Finally, based on the obtained optimal component parameters, the optimal product of each component can be quickly located and determined from the corresponding database. Experiments and statistical analyses demonstrate the effectiveness of the proposed method. The proposed method is fast and practical that it has been successfully applied to a web server to provide online optimization design service for users.'
author:
- 'Xunhua Dai, Quan Quan, Jinrui Ren and Kai-Yuan Cai [^1]'
bibliography:
- 'IEEEabrv.bib'
title: An Analytical Design Optimization Method for Electric Propulsion Systems of Multicopter UAVs with Desired Hovering Endurance
---
Multicopter, Design optimization, Propulsion system, UAV.
Introduction
============
During recent years, multicopter Unmanned Aerial Vehicles (UAVs) are becoming increasingly popular in both civil and military fields [@doyle2013avian; @tauro2015large] including aerial photography, plant protection, package delivery and other fields. Limited by the battery technology, the flight time (hovering endurance) of multicopters is still too short for most applications. Since the performance and efficiency of a multicopter directly depend on the propulsion system, the design optimization for multicopter propulsion systems is urgently needed to increase the flight time.
The design optimization problem studied in this paper is to find the optimal combination of propulsion system components to satisfy the given hovering endurance requirement, and the obtained propulsion system should have smaller weight and higher efficiency as possible. A typical multicopter propulsion system usually consists of four basic components including the propeller, BrushLess Direct-Current (BLDC) motor, Electronic Speed Control (ESC) and Lithium Polymer (LiPo) battery [@quan2017introduction]. Traditional methods to determine a propulsion system are usually based on the experience and trial-and-error experiments. Considering that there are thousands of component products on the market, it is a costly and time-consuming work for the traditional design methods. Meanwhile, in the whole design process of a multicopter system, the propulsion system need to be repeatedly modified according to the actual controlled system until all the performance requirements and safety requirements are satisfied. According to [@oktay2013simultaneous; @oktay2016simultaneous], a more efficient way is to simultaneously design the body system (including the propulsion system) and the control system subject to the optimal objective and additional constraints. Therefore, the automatic design and optimization technologies for propulsion systems will be beneficial for reducing the prototyping needs for the whole multicopter system, and minimizing development and manufacturing cost. For such reasons, this paper proposes a simple, practical and automatic design optimization method to help designers quickly find the optimal propulsion system according to the given design requirements.
In our previous work [@Shi2017], based on the mathematical modeling methods for the components of propulsion systems, a practical method is proposed to estimate the flight performance of multicopters according to the given propulsion system parameters. In fact, the study in this paper is the reverse process of our previous work, namely estimating the optimal propulsion system parameters according to the given design requirements. This problem is more complicated and difficult because the number of design requirements is much less than the number (more than 15) of the propulsion system parameters.
There are many studies on the mathematical modeling [@Harrington2011; @Ramana2013; @Mccrink2015], the efficiency analysis [@Lawrence2005; @Stepaniak2009], and the performance estimation [@Shi2017; @Bershadsky2016a] of multicopter propulsion systems. To our best knowledge, there are few studies on the design optimization of multicopter propulsion systems. Most of them adopt numerical methods (fixed-wing aircraft [@Lundstrom2009] and multicopters [@Bershadsky2016a]) to search and traverse all the possible propulsion system combinations in the database based on the proposed cost functions. In [@Magnussen2014; @Magnussen2015], the multicopter optimization problem is described as a mixed integer linear program, and solved with the Cplex optimizer. However, these numerical methods have following problems: i) a large and well-covered product database is required for a better optimization effect; ii) the calculation speed is slow when there are large numbers of products in the database because the amount of product combinations is huge (the algorithm complexity is $O(n^{4})$, where $n$ is the number of the database products).
In order to solve the above problems, this paper proposes an analytical method to estimate the optimal parameters of the propulsion system components. First, the modeling methods for each component of the propulsion system are studied respectively to describe the problem with mathematical expressions. Secondly, the whole problem is simplified and decoupled into several small problems. By solving these sub-problems, the optimal parameters of each component can be obtained respectively. Finally, based on the obtained parameters, selection algorithms are proposed to determine the optimal combination of the propeller, the motor, the ESC and the battery products from their corresponding databases.
The contributions of this paper are as follows: i) an analytical method to solve the design optimization problem of multicopter propulsion systems is proposed for the first time; ii) the conclusion obtained through the theoretical analysis has a guiding significance for the multicopter design; iii) compared to the numerical traversal methods, the proposed method reduces the algorithm complexity from $O(n^{4})$ to $O(n)$, which is faster and more efficient for practical applications.
The paper is organized as follows. *Section\[Sec-2\]* gives a comprehensive analysis of the design optimization problem to divide it into twelve sub-problems. In *Section\[Sec-3\]*, the modeling methods for each component of the propulsion system are studied the describe the sub-problems with mathematical expressions. In *Section\[Sec-4\]*, the sub-problems are solved respectively to obtain the optimal components of the desired propulsion system. In *Section\[Sec-6\]*, statistical analyses and experiments are performed to verify the proposed method. In the end, *Section\[Sec-7\]* presents the conclusions.
Problem Formulation {#Sec-2}
===================
Design Requirements of Propulsion Systems {#Sec2-Des}
-----------------------------------------
The role of a propulsion system is to continuously generate the desired thrust within the desired time of endurance for a multicopter. The design requirements for a propulsion system are usually described by the following parameters: i) the number of the propulsion units $n_{\text{p}}$; ii) the hovering thrust of a single propeller $T_{\text{hover}}$ (unit: N) under the hovering mode when the multicopter stays fixed in the air; iii) the maximum thrust of a single propeller $T_{\text{max}}$ (unit: N) under the full-throttle mode when the autopilot gives the maximum throttle signal; iv) the nominal flight altitude $h_{\text{hover}}$ (unit: m); iv) the flight time $t_{\text{hover}}$ (unit: min) under the hovering mode.
This paper only focuses on studying the design optimization of propulsion systems with assuming $T_{\text{hover}}$ and $T_{\text{max}}$ are known parameters. Although the propulsion system parameters $T_{\text{hover}}$ and $T_{\text{max}}$ are usually not directly available, according to our previous research [@Shi2017], $T_{\text{hover}},T_{\text{max}}$ can be obtained by giving the aerodynamic coefficients, airframe parameters and the kinematic performance requirements. For example, for common multicopters, the hovering thrust $T_{\text{hover}}$ can be obtained by the total weight of the multicopter $G_{\text{total}}$ (unit: N) as $$T_{\text{hover}}=\frac{G_{\text{total}}}{n_{\text{p}}}.\label{eq:T0}$$ The kinematic performance of a multicopter is directly determined by the thrust ratio $\gamma\in\left(0,1\right)$ which is defined as $$\gamma\triangleq\frac{T_{\text{hover}}}{T_{\text{max}}}.\label{Eq31T0TpmaxThr}$$ where thrust ratio $\gamma$ describes the remaining thrust for the acceleration movement, which further determines the maximum forward speed and the wind resistance ability of a multicopter. Therefore, designers should estimate the desired $\gamma$ (usually $\gamma=0.5$ is selected for common multicopters) according to the kinematic performance requirements of the multicopter. Then, the desired full-throttle thrust $T_{\text{max}}$ of the propulsion system can be obtained through Eqs.(\[eq:T0\])(\[Eq31T0TpmaxThr\]).
Component Parameters
--------------------
The ultimate goal of the design optimization problem is to select the optimal products from four component databases with component parameters listed in Table\[tab1\], where $\Theta_{\text{p}}$, $\Theta_{\text{m}}$, $\Theta_{\text{e}}$ and $\Theta_{\text{b}}$ represent the parameter sets for propellers, motors, ESCs and batteries. In order to ensure commonality of the method, all component parameters in Table\[tab1\] are the basic parameters that can be easily found in the product description pages. The detailed introduction of each parameter in Table\[tab1\] can also be found in [@quan2017introduction pp. 31-46].
[|c|>p[0.38]{}|]{} Items & Parameters [\
]{} Propeller & $\Theta_{\text{p}}\triangleq${Diameter $D_{\text{p}}$ (m), Pitch Angle $\varphi_{\text{p}}$ (rad), Blade Number $B_{\text{p}}$}[\
]{} Motor & $\Theta_{\text{m}}\triangleq${Nominal Maximum Voltage $U_{\text{mMax}}$ (V), Nominal Maximum Current $I_{\text{mMax}}$ (A), KV Value $K_{\text{V}}$ (RPM/V), No-load Current $I_{\text{m0}}$ (A), Resistance $R_{\text{m}}$ ($\Omega$)}[\
]{} ESC & $\Theta_{\text{e}}\triangleq${Nominal Maximum Voltage $U_{\text{eMax}}$ (V), Nominal Maximum Current $I_{\text{eMax}}$ (A), Resistance $R_{\text{e}}$ ($\Omega$)}; [\
]{} Battery & $\Theta_{\text{b}}\triangleq${Nominal Voltage $U_{\text{b}}$ (V), Maximum Discharge Rate $K_{\text{b}}$ (A), Capacity $C_{\text{b}}$ (mAh), Resistance $R_{\text{b}}$ ($\Omega$)}; [\
]{}
The propeller pitch angle $\varphi_{\text{p}}$ (unit: rad) in Table\[tab1\] is defined according to the propeller diameter ${D_{\text{{p}}}}$ (unit: m) and the propeller pitch ${H_{\text{{p}}}}$ (unit: m) as $$\varphi_{\text{p}}\triangleq{\arctan\frac{{H_{\text{{p}}}}}{{\pi{D_{\text{{p}}}}}}}\label{Eq00-pitch}$$ where ${H_{\text{{p}}}}$ and ${D_{\text{{p}}}}$ are usually contained in the propeller model name.
The most commonly used unit for the battery voltage $U_{\text{b}}$, as well as the motor voltage $U_{\text{mMax}}$ and the ESC voltage $U_{\text{eMax}}$, is “S, which denotes the number of battery cells in series. For LiPo batteries, the voltage changes from 4.2V to 3.7V as the battery capacity decreases from full to empty, and the average voltage 4.0V is adopted for the unit conversion. For example, $U_{\text{b}}=12\text{\,S}=48\text{\,V}.$
Optimization Constraints {#subsec:OptiConstr}
------------------------
### Requirement Constraints
According to [@Shi2017], $t_{\text{hover}}$ and $T_{\text{max}}$ can be estimated by parameters $n_{\text{p}},T_{\text{hover}}$,$\Theta_{\text{p}},\Theta_{\text{m}},\Theta_{\text{e}},\Theta_{\text{b}}$. Therefore, two equality constraints can be obtained according to the design requirements in *Section\[Sec2-Des\]*.
### Safety Constraints
The electric components should work within their allowed operating conditions to prevent from being burnt out. Therefore, a series of inequality constraints can be obtained for electric components including the battery, the ESC and the motor of the propulsion system. The detailed inequality expressions will be introduced in the later sections.
### Product Statistical Constraints
In order to make sure that the obtained solution is meaningful and practical, the product statistical features should be considered. Otherwise, it is very possible that there is no product in reality matching with the obtained component parameters. The product features can be described by equality constraints based on the statistical models of the products in the database. In practice, products with different material, operating principle and processing technology may have different product features, for example, LiPo batteries and Ni-MH batteries. Therefore, different statistical models should be obtained for different types of products, and designers should select the required product type to the following optimization process to improve the precision and practicability of the obtained result.
Optimization Problem
--------------------
In practice, there are two methods to increase the flight time of a multicopter: i) decrease the total weight of the multicopter to allocate more free weight for the battery capacity; ii) increase the efficiency of the multicopter propulsion system to decrease the required battery capacity. For a typical multicopter, according to the weight statistical model in [@Bershadsky2016a], the propulsion system weight is the main source (usually more than 70%) of the multicopter weight, and the weight of the battery is further the main source (usually more than 60%) of the propulsion system weight. In fact, for a propulsion system, the maximum efficiency usually means the minimum battery weight (capacity). Therefore, the above two methods essentially share the same optimization objective, namely minimizing the weight of the propulsion system.
Assuming that the total weight of the propulsion system is defined as $G_{\text{sys}}$ (unit: N), the optimization objective of the design optimization can be described as
$$\min_{\Theta_{\text{p}},\Theta_{\text{m}},\Theta_{\text{e}},\Theta_{\text{b}}}G_{\text{sys}}.\label{eq:OptiEqua}$$
Along with the constraints in *Section\[subsec:OptiConstr\]*, the optimal solutions for the parameters of each component can be obtained from Eq.(\[eq:OptiEqua\]).
Problem Decomposition {#subsec:ProbDeco}
---------------------
According to [@Shi2017], the detailed mathematical expressions for Eq.(\[eq:OptiEqua\]) are very complex because there are 15 parameters listed in Table\[tab1\] that need to be optimized through solving complex nonlinear equations. As a result, decomposition and simplification are required to solve this problem.
### Weight Decomposition
The total weight of the propulsion system $G_{\text{sys}}$ is determined by the weight of each component as $$G_{\text{sys}}=n_{\text{p}}\left(G_{\text{p}}+G_{\text{m}}+G_{\text{e}}\right)+G_{\text{b}}\label{eq:Gsys}$$ where $G_{\text{p}},G_{\text{m}},G_{\text{e}},G_{\text{b}}$ (unit: N) denote the weight of the propeller, the motor, the ESC and the battery respectively. Therefore, based on the idea of the greedy algorithm, the optimization objective of minimizing $G_{\text{sys}}$ can be decomposed into four sub-problems of minimizing the weight of each component as $$\min G_{\text{sys}}\Rightarrow\min G_{\text{p}},\min G_{\text{m}},\min G_{\text{e}},\min G_{\text{b}}.\label{eq:WeightDecom}$$ As mentioned above, the battery weight $G_{\text{b}}$ is the most important factor for $G_{\text{sys}}$, and $G_{\text{b}}$ directly depends on the battery capacity $C_{\text{b}}$. Since, according to the analysis in [@quan2017introduction], the battery capacity $C_{\text{b}}$ depends on the efficiency of each component, the optimization objective of minimizing $G_{\text{b}}$ can also be decomposed into four sub-problems of maximizing the efficiency of each component as $$\min G_{\text{b}}\Rightarrow\max\eta_{\text{p}},\max\eta_{\text{m}},\max\eta_{\text{e}},\max\eta_{\text{b}}.\label{eq:EffDecom}$$ where $\eta_{\text{p}},\eta_{\text{m}},\eta_{\text{e}},\eta_{\text{b}}$ denote the efficiency of the propeller, the motor, the ESC and the battery respectively.
![Optimization objective decomposition diagram.[]{data-label="Fig02Decom"}](Fig04){width="40.00000%"}
Thus, by solving the eight sub-problems in Eqs.(\[eq:WeightDecom\])(\[eq:EffDecom\]), the optimal solutions for the parameters in Table\[tab1\] can be obtained with the results presented in Fig.\[Fig02Decom\], where the obtained optimal solutions are marked with a subscript “Opt”. Then, the optimal parameter sets for the propeller, the motor, the ESC and the battery can be determined and represented by $\Theta_{\text{pOpt}}$,$\Theta_{\text{mOpt}}$, $\Theta_{\text{eOpt}}$, and $\Theta_{\text{bOpt}}$ respectively.
### Product Selection Decomposition
The ultimate goal of the design optimization of the propulsion system is to determine the optimal propeller, motor, ESC and propeller products from their corresponding databases according to the obtained optimal parameter sets $\Theta_{\text{pOpt}},\Theta_{\text{mOpt}},\Theta_{\text{eOpt}},\Theta_{\text{bOpt}}$. This problem can also be divided into four sub-problems. Through solving the four sub-problems, the parameter sets of the obtained products are represented by $\Theta_{\text{pOpt}}^{\ast}$, $\Theta_{\text{pOpt}}^{\ast}$, $\Theta_{\text{pOpt}}^{\ast}$ and $\Theta_{\text{pOpt}}^{\ast}$.
Solving Procedures {#Sec2-End}
------------------
Through the above decomposition procedures, the whole design optimization problem can finally be simplified, decoupled and divided into twelve sub-problems. However, since there are argument-dependent relationships among the sub-problems, the solving sequence should be well-arranged. For instance, the propeller parameters $B_{\text{p}},\varphi_{\text{p}}$ are required for the motor optimization, and the optimal propeller diameter $D_{\text{p}}$ depends on the obtained motor parameters. As a result, the propeller and motor should be treated as a whole during the solving procedures.
In this paper, the twelve sub-problems will be solved separately by twelve steps with the solving sequence shown in Fig.\[Fig02-0\]. Fig.\[Fig02-0\] also presents the inputs, outputs, and parameter dependency relationships of each sub-problem. The detailed solving methods of each step will be introduced in *Section\[Sec-4\]* with twelve subsections.
![Solving procedures of the propulsion system optimization problem[]{data-label="Fig02-0"}](Fig02-0){width="40.00000%"}
Propulsion System Modeling {#Sec-3}
==========================
The whole propulsion system can be modeled by the equivalent circuit [@Shi2017] as shown in Fig.\[Fig02\], with which the optimization sub-problems can be described by mathematical expressions.
![Equivalent circuit model of the whole propulsion system[]{data-label="Fig02"}](Fig02){width="45.00000%"}
Propeller Modeling
------------------
### Propeller Aerodynamic Model
According to [@merchant2006propeller], the thrust force $T$ (unit: N) and torque $M$ (unit: N$\cdot$m) of fixed-pitch propellers can be obtained through equations $$\left\{ \begin{array}{c}
T=C_{\text{T}}\rho\left(\frac{N}{60}\right)^{2}D_{\text{p}}^{4}\\
M=C_{\text{M}}\rho\left(\frac{N}{60}\right)^{2}D_{\text{p}}^{5}
\end{array}\right.\label{Eq02PropTorque}$$ where $N$ (unit: RPM) is the propeller revolving speed, $\rho$ is the local air density (unit: kg/m$^{3}$), $C_{\text{T}}$ is the propeller thrust coefficient, $C_{\text{M}}$ is the propeller torque coefficient, and $D_{\text{p}}$ (unit: m, from ${\Theta_{\text{p}}}$) is the propeller diameter.
The air density $\rho$ is determined by the local temperature $T_{\text{t}}$ (unit: $^{\circ}$C) and the air pressure which is further determined by the altitude $h_{\text{hover}}$ (unit: m). According to the international standard atmosphere model [@cavcar2000international] $$\rho=f_{\rho}\left(h_{\text{hover}}\right)\triangleq\frac{273}{(273+T_{\text{t}})}(1-0.0065\frac{h_{\text{hover}}}{273+T_{\text{t}}})^{5.2561}\rho_{\text{0}}\label{Eq03AirDen}$$ where the standard air density $\rho_{\text{0}}=1.293$kg/m$^{3}$ ($^{\circ}$C, 273K).
The propeller coefficients $C_{\text{T}}$ and $C_{\text{M}}$ can be modeled by using the blade element theory as presented in [@Shi2017][@Mccrink2015]. A simplified form is introduced here as $$\left\{ \begin{array}{lll}
C_{\text{T}} & =f_{C_{\text{T}}}\left(B_{\text{p}},D_{\text{p}},\varphi_{\text{p}}\right) & \triangleq k_{\text{t0}}B_{\text{p}}\varphi_{\text{p}}\\
C_{\text{M}} & =f_{C_{\text{M}}}\left(B_{\text{p}},D_{\text{p}},\varphi_{\text{p}}\right) & \triangleq k_{\text{m0}}B_{\text{p}}^{2}\left(k_{\text{m1}}+k_{\text{m2}}\varphi_{\text{p}}^{2}\right)
\end{array}\right.\label{Eq04Cm}$$ where $k_{\text{t0}},k_{\text{m0}},k_{\text{m1}},k_{\text{m2}}$ are constant parameters determined by the shapes and aerodynamic characteristics of the propeller blades, and they can be obtained through the propeller model in [@Shi2017] as $$\begin{array}{llcl}
k_{\text{t0}} & \triangleq\frac{0.25{\pi^{3}}\lambda{\zeta^{2}}{K_{0}}\varepsilon}{{\pi A+{K_{0}}}}, & k_{\text{m0}} & \triangleq\frac{1}{{8A}}{\pi^{2}}\lambda{\zeta^{2}}\\
k_{\text{m1}} & \triangleq C_{{\rm {fd}}}, & k_{\text{m2}} & \triangleq\frac{\pi AK_{0}^{2}\varepsilon}{e{{\left({\pi A+{K_{0}}}\right)}^{2}}}
\end{array}\label{eq:CTCMParamete}$$ where the detailed definitions of the internal parameters of Eq.(\[eq:CTCMParamete\]) can be found in [@Shi2017]. Note that $k_{\text{t0}},k_{\text{m0}},k_{\text{m1}},k_{\text{m2}}$ may slightly vary with the difference of types, material, and technology of propellers. Based on the propeller data from T-MOTOR website [@TMotor2017], general parameters $k_{\text{t0}},k_{\text{m0}},k_{\text{m1}},k_{\text{m2}}$ for the carbon fiber propellers are given by $$\begin{array}{ll}
k_{\text{t0}}=0.323, & k_{\text{m0}}=0.0432\\
k_{\text{m1}}=0.01, & k_{\text{m2}}=0.9
\end{array}.\label{Eq04KCM}$$
### Propeller Efficiency Objective Function
Similar to the lift-drag ratio for airfoils, a widely used efficiency index to describe the efficiency of propellers is $\eta_{\text{T/M}}$, which is defined as the ratio between the thrust coefficient $C_{\text{T}}$ and the torque coefficient $C_{\text{M}}$ as $$\eta_{\text{T/M}}\triangleq\frac{C_{\text{T}}}{C_{\text{M}}}=\frac{k_{\text{t0}}\varphi_{\text{p}}}{k_{\text{m0}}B_{\text{p}}\left(k_{\text{m1}}+k_{\text{m2}}\varphi_{\text{p}}^{2}\right)}\label{Eq05EffTM}$$ where $\eta_{\text{T/M}}$ only depends on the aerodynamic design of the blade shape, which is convenient for manufacturers to improve the aerodynamic efficiency. Moreover, a higher $\eta_{\text{T/M}}$ means a smaller torque for generating the same thrust. Since $\eta_{\text{T/M}}$ is adopted by most of the manufacturers, this paper will use $\eta_{\text{T/M}}$ as the propeller efficiency objective function to obtain the optimal $\varphi_{\text{pOpt}}$ and $B_{\text{pOpt}}$.
### Propeller Weight Objective Function
Through analyzing the propeller products on the market, the propeller weight $G_{\text{p}}$ can be described by a statistical model that depends on the diameter $D_{\text{p}}$ and the blade number $B_{\text{p}}$ as $$G_{\text{p}}=f_{G_{\text{p}}}\left(B_{\text{p}},D_{\text{p}}\right)\label{eq:Gp}$$ where $f_{G_{\text{p}}}\left(\cdot\right)$ is an increasing function of $D_{\text{p}}$ and $B_{\text{p}}$. Therefore, the minimum propeller weight $G_{\text{p}}$ requires that both $D_{\text{p}}$ and $B_{\text{p}}$ should be chosen as small as possible, which is described as $$\min G_{\text{p}}\Rightarrow\min B_{\text{p}},\min D_{\text{p}}.\label{eq:minGp}$$
Motor Modeling
--------------
### Motor Circuit Model
The equivalent circuit of a BLDC motor has been presented in Fig.\[Fig02\], where $U_{\text{m}}$ (unit: V) is the motor equivalent voltage and $I_{\text{m}}$ (unit: A) is the motor equivalent current. According to [@Shi2017; @chapman2005electric], $U_{\text{m}}$ and ${I_{\text{{m}}}}$ can be obtained through $$\left\{ \begin{array}{lll}
{I_{\text{{m}}}} & = & \frac{\pi{M{K_{\text{{V}}}}{U_{\text{{m0}}}}}}{{30({U_{\text{{m0}}}}-{I_{\text{{m0}}}}{R_{\text{{m}}}})}}+{I_{\text{{m0}}}}\\
{U_{\text{{m}}}} & = & {I_{\text{{m}}}R_{\text{{m}}}}+\frac{{{U_{\text{{m0}}}}-{I_{\text{{m0}}}}{R_{\text{{m}}}}}}{{{K_{\text{{V}}}}{U_{\text{{m0}}}}}}N
\end{array}\right.\label{Eq05Um}$$ where $M$ (unit: N$\cdot$m) is the output torque of the motor which equals to the propeller toque in Eq.(\[Eq02PropTorque\]), $N$ is the motor rotating speed which equals to the propeller rotating speed in Eq.(\[Eq02PropTorque\]). The current and voltage measured under no-load (no propeller) tests are called the no-load current $I_{\text{{m0}}}$ (unit: A) and the no-load voltage $U_{\text{{m0}}}$ (unit: V), where ${U_{\text{{m0}}}}$ is a constant value defined by manufacturers (usually ${U_{\text{{m0}}}=10}$V). Note that, the nominal motor resistance $R_{\text{m0}}$ on the product description is usually not accurate enough, so it is recommended to obtain the actual resistance $R_{\text{m}}$ according to the test data of the full-throttle current $I_{\text{{m}}}^{*}$ and speed $N^{*}$, where a correction expression is derived from Eq.(\[Eq05Um\]) as $R_{\text{m}}$$\approx\left(U_{\text{{b}}}-N^{*}/K_{\text{V}}\right)/I_{\text{{m}}}^{*}\approx2\sim3R_{\text{m0}}$ according to our experimental results.
### Motor Constraints
The actual operation voltage of the motor depends on the battery voltage $U_{\text{b}}$ instead of the motor nominal maximum voltage (NMV) $U_{\text{mMax}}$. The motor NMV $U_{\text{mMax}}$ defines the range of the battery voltage $U_{\text{b}}$ that can ensure the motor work safely, which is described as $U_{\text{b}}\leq U_{\text{mMax}}$. According to [@Shi2017], to prevent the motor from burnout, the motor equivalent voltage $U$$_{\text{m}}$ and current $I_{\text{m}}$ satisfy the constraints that $$\begin{array}{l}
U_{\text{m}}\leq U_{\text{m\ensuremath{\sigma_{\text{max}}}}}=U_{\text{b}}\leq U_{\text{mMax}}\\
I_{\text{m}}\leq I_{\text{m\ensuremath{\sigma_{\text{max}}}}}\leq I_{\text{mMax}}
\end{array}\label{eq:MotConstraint}$$ where $U_{\text{m\ensuremath{\sigma_{\text{max}}}}}$ and $I_{\text{m\ensuremath{\sigma_{\text{max}}}}}$ are the motor voltage and current under the full-throttle mode ($\sigma_{\text{max}}=1$).
By letting the motor work under the maximum limit condition as $$U_{\text{m}}={U_{\text{{mMax}}}}\text{, }I_{\text{m}}=I_{\text{mMax}}.\label{eq:MotorConstraint}$$ Then, the maximum rotating speed ${N}_{\max}$ (unit: RPM) and torque ${M}_{\max}$ (unit: N$\cdot$m) of the motor can be obtained by combining the motor model in Eq.(\[Eq05Um\]) $$\left\{ \begin{array}{l}
{N}_{\max}=f_{{N}_{\max}}\left(\Theta_{\text{m}}\right)\triangleq\frac{{\left({U_{\text{{mMax}}}-{R_{\text{m}}}{I_{\text{mMax}}}}\right){K_{\text{V}}}{U_{\text{m0}}}}}{\left({{U_{\text{m0}}}-{I_{\text{m0}}}{R_{\text{m}}}}\right)}\\
{M}_{\max}=f_{{M}_{\max}}\left(\Theta_{\text{m}}\right)\triangleq\frac{{30\left({{I_{\text{mMax}}}-{I_{\text{m0}}}}\right)({U_{\text{m0}}}-{I_{\text{m0}}}{R_{\text{m}}})}}{\pi{K}_{\text{V}}{U_{\text{m0}}}}
\end{array}\right..\label{Eq18Mmax}$$ Moreover, according to the propeller model in Eq.(\[Eq02PropTorque\]), the theoretical maximum thrust ${T_{\text{pMax}}}$ (unit: N) can be obtained as $$T_{\text{pMax}}=\frac{{{C}_{\text{T}}}}{{{C}_{\text{M}}}}\frac{M_{\max}^{4/5}{{\rho}^{1/5}}{{C}_{\text{M}}^{1/5}{N}_{\max}^{2/5}}}{{{60}^{2/5}}}.\label{Eq19TpMax1}$$ To satisfy the maneuverability requirement, the theoretical maximum thrust range $\left[0,T_{\text{pMax}}\right]$ of the propulsion system should cover the required thrust range $\left[0,T_{\text{max}}\right]$, which means the following constraint should also be satisfied $$T_{\text{pMax}}\geq T_{\text{max}}.\label{eq:TmaxTpMaxEq}$$
![Statistical relationship between the motor maximum input power $U_{\text{{mMax}}}\cdot{I}_{\text{mMax}}$ and propeller maximum thrust $T_{\text{pMax}}$. The testing data come from the motor products website [@TMotor2017].[]{data-label="Fig05ThustWattVolt-1"}](Fig05-1){width="40.00000%"}
According to the statistical results in Fig.\[Fig05ThustWattVolt-1\], there is an equality constraint between the maximum input power ${U_{\text{{mMax}}}{I}_{\text{mMax}}}$ and the theoretical maximum thrust ${T_{\text{pMax}}}$ for motor products as $$\frac{{T_{\text{pMax}}}}{{U_{\text{{mMax}}}{I}_{\text{mMax}}}}\approx G_{\text{WConst}}\label{Eq37GWconst}$$ where $G_{\text{WConst}}$ (unit: N/W) is a constant coefficient that reflects the technological process and product quality of products. According to the curve fitting result in Fig.\[Fig05ThustWattVolt-1\], the coefficient for the tested motors is $G_{\text{WConst}}\approx0.0624$.
### Motor Efficiency Objective Function
The motor power efficiency $\eta_{\text{m}}$ is defined as $$\eta_{\text{m}}\triangleq\frac{P_{\text{p}}}{P_{\text{m}}}=\frac{M\frac{2\pi N}{60}}{U_{\text{m}}I_{\text{m}}}.$$ According to Eq.(\[Eq05Um\]), the propeller torque $M$ and rotating speed $N$ can be described by $U_{\text{m}}\text{ and }I_{\text{m}}$, which yields that $$\begin{array}{c}
\eta_{\text{m}}=\left(1-\frac{I_{\text{m}}}{{U_{\text{{m}}}}}{R_{\text{{m}}}}\right)\left(1-\frac{1}{I_{\text{m}}}I_{\text{m0}}\right)\end{array}.\label{eq:EqEffmOtor}$$
It can be observed from Eq.(\[eq:EqEffmOtor\]) that the motor efficiency $\eta_{\text{m}}$ has negative correlations with $R_{\text{{m}}}$ and $I_{\text{m0}}$. Therefore, $R_{\text{{m}}}$ and $I_{\text{m0}}$ should be chosen as small as possible for the maximum motor efficiency, which is described as $$\max\eta_{\text{m}}\Rightarrow\min R_{\text{{m}}},\min I_{\text{m0}}.\label{eq:EffM}$$
### Weight Optimization Objective Function
Through analyzing the motor products on the market, the motor weight $G_{\text{m}}$ can be described by a statistical model depending on the motor nominal maximum voltage (NMV) $U_{\text{{mMax}}}$, the Nominal Maximum Current (NMC) $I_{\text{mMax}}$, and the KV value $K_{\text{V}}$ as $$G_{\text{m}}=f_{G_{\text{m}}}\left(U_{\text{mMax}},I_{\text{mMax}},K_{\text{V}}\right).\label{eq:GmOrgin}$$ According to Eqs.(\[Eq19TpMax1\])(\[Eq37GWconst\]), $I_{\text{mMax}}$ and $K_{\text{V}}$ can be described by $U_{\text{mMax}}$ and $T_{\text{pMax}}$. Therefore, Eq.(\[eq:GmOrgin\]) can be rewritten into the following form $$G_{\text{m}}=f_{G_{\text{m}}}^{\prime}\left(U_{\text{mMax}},T_{\text{pMax}}\right).\label{eq:Gm}$$ Thus, by combining the constraints in Eqs.(\[eq:MotConstraint\])(\[eq:TmaxTpMaxEq\]), the motor weight optimization problem can be written into $$\begin{array}{c}
\underset{U_{\text{mMax}},T_{\text{pMax}}}{\min}f_{G_{\text{m}}}^{\prime}\left(U_{\text{mMax}},T_{\text{pMax}}\right)\\
\text{s.t. }U_{\text{b}}\leq U_{\text{mMax}},T_{\text{max}}\leq T_{\text{pMax}}
\end{array}.\label{eq:minGm}$$
In practice, the motor weight $G_{\text{m}}$ has a positive correlation with $U_{\text{mMax}}\text{ and }T_{\text{pMax}}$. Therefore, the minimum motor weight $G_{\text{m}}$ requires that the $U_{\text{mMax}}$ and $T_{\text{pMax}}$ should both be chosen as small as possible, which is described as $$\min G_{\text{m}}\Rightarrow\min U_{\text{mMax}},\min T_{\text{pMax}}.\label{eq:minGp-1}$$ Thus, solving Eq.(\[eq:minGm\]) gives that $$U_{\text{b}}=U_{\text{mMax}},T_{\text{max}}=T_{\text{pMax}}\label{eq:UbTmax}$$ where $U_{\text{mMax}}$ should be chosen as small as possible.
ESC Modeling
------------
### ESC Circuit Model
After receiving the throttle signal $\sigma{\in}\left[0,1\right]$ from the flight controller, the ESC converts the direct-current power of the battery to the PWM-modulated voltage $\sigma U_{\text{{e}}}$ for the BLDC motor without speed feedback. Then, the motor rotating speed is determined by both the motor and propeller models. Since the propeller model is nonlinear, the motor speed is not in proportion to the input throttle signal. According to the ESC equivalent circuit in Fig.\[Fig02\], the ESC current $I_{\text{e}}$ (unit: A) and voltage $U_{\text{e}}$ (unit: V) are given by $$\begin{array}{cl}
\sigma U_{\text{{e}}} & =U_{\text{{m}}}+I_{\text{{m}}}R_{\text{e}}\\
I_{\text{{e}}} & ={\sigma I_{\text{{m}}}}
\end{array}.\label{Eq06Ie}$$
### ESC Efficiency Objective Function
According to Eq.(\[Eq06Ie\]), the power efficiency of the ESC can be obtained as $$\begin{array}{lll}
\eta_{\text{e}} & \triangleq & \frac{U_{\text{m}}I_{\text{m}}}{U_{\text{e}}I_{\text{e}}}=\frac{1}{1+\frac{I_{\text{m}}}{U_{\text{m}}}R_{\text{e}}}\end{array}$$ which shows that the ESC efficiency $\eta_{\text{e}}$ increases as the resistance $R_{\text{e}}$ decreases, which yields that $$\max\eta_{\text{e}}\Rightarrow\min R_{\text{e}}.\label{eq:EffE}$$
### ESC Weight Objective Function
Through analyzing the ESC products on the market, the ESC weight $G_{\text{e}}$ can be described by a statistical model depending on the ESC nominal maximum voltage (NMV) $U_{\text{{eMax}}}$, the Nominal Maximum Current (NMC) $I_{\text{eMax}}$ as $$G_{\text{e}}=f_{G_{\text{e}}}\left(U_{\text{eMax}},I_{\text{eMax}}\right)\label{eq:GeOrgin}$$ where $f_{G_{\text{e}}}\left(\cdot\right)$ is in positive correlation with $U_{\text{eMax}}\text{ and }I_{\text{eMax}}$. Therefore, to minimize $G_{\text{e}}$, the ESC parameters $U_{\text{eMax}},I_{\text{eMax}}$ should be chosen as small as possible, which is described as
$$\min G_{\text{e}}\Rightarrow\min U_{\text{eMax}},\min I_{\text{eMax}}.\label{eq:GeMin}$$
Battery Modeling
----------------
### Battery Circuit Model
The battery is used to provide energy to drive the motor through ESC. The most commonly-used type battery is the LiPo battery because of the superior performance and low price. According to Fig.\[Fig02\], the battery model is given by $$U_{\text{b}}=U_{\text{\ensuremath{\text{e}}}}+{I}_{\text{b}}{R}_{\text{b}}\label{Eq08Ue}$$ where ${U}_{\text{b}}$ (unit: V) is the nominal battery voltage, and ${I}_{\text{b}}$ (unit: A) is the output current.
Assuming that the number of the propulsion unit on a multicopter is $n_{\text{p}}$, the battery current is given by $${I_{\text{{b}}}=n_{\text{p}}I_{\text{{e}}}+I_{\text{other}}}\label{Eq09Ob}$$ where $I_{\text{other}}$ (unit: A) is the current from other devices on the multicopter such as the flight controller and the camera. Usually, according to [@quan2017introduction], it can be assumed that $I_{\text{other}}\approx0.5\text{\,A}$ if there is only a flight controller on the multicopter.
According to [@Shi2017], the battery discharge time $t_{\text{discharge}}$ (unit: min) is determined by the battery capacity $C_{\text{b}}$ and the discharge current $I_{\text{b}}$ $$t_{\text{discharge}}\approx\frac{0.85C_{\text{b}}}{I_{\text{b}}}\cdot\frac{60}{1000}\label{eq:tdischarge}$$ where the coefficient 0.85 denotes a 15% remaining capacity to avoid over discharge. Note that the endurance computation equations in Eqs.(\[Eq08Ue\])-(\[eq:tdischarge\]) are simplified to reduce the computation time. They can be replaced by more accurate and nonlinear methods as presented in [@donateo2017design; @donateo2017new] to increase the precision of the battery optimization result.
### Battery Constraints
The Maximum Discharge Rate (MDR) $K_{\text{b}}$ (unit: mA/mAh or marked with symbol “C”) of the battery is defined as $$K_{\text{b}}=\frac{1000I_{\text{bMax}}}{C_{\text{b}}}$$ where $I_{\text{bMax}}$ (unit: A) is the maximum discharge current that the battery can withstand. Since the battery should be able to work safely under the full-throttle mode of the motor, the maximum discharge current $I_{\text{bMax}}$ should satisfy $$I_{\text{bMax}}\geq n_{\text{p}}I_{\text{e\ensuremath{\sigma_{\text{max}}}}}+I_{\text{other}}=n_{\text{p}}I_{\text{{mMax}}}+I_{\text{other}}\label{eq:Kbbb}$$ which yields that $$K_{\text{b}}\geq\frac{1000\left(n_{\text{p}}I_{\text{{mMax}}}+I_{\text{other}}\right)}{C_{\text{b}}}.\label{eq:KbConstraint}$$
### Battery Efficiency Objective Function
According to Eqs.(\[Eq08Ue\])(\[Eq09Ob\]), the battery efficiency $\eta_{\text{b}}$ can be written into $$\begin{array}{cc}
\eta_{\text{b}} & \triangleq\frac{n_{\text{p}}U_{\text{e}}I_{\text{e}}}{U_{\text{b}}I_{\text{b}}}=\left(1-\frac{I_{\text{b}}}{U_{\text{b}}}R_{\text{b}}\right)\left(1-\frac{I_{\text{other}}}{I_{\text{b}}}\right)\end{array}\label{Eq09BEFF}$$ which shows that the battery efficiency $\eta_{\text{b}}$ increases as the resistance $R_{\text{b}}$ decreases, which is described as
$$\max\eta_{\text{b}}\Rightarrow\min R_{\text{b}}.\label{eq:Effb}$$
### Battery Weight Objective Function
According to the definition of the battery power density $\rho_{\text{b}}$ (unit: Wh/kg), the battery weight can be described as $$G_{\text{b}}=\frac{C_{\text{b}}U_{\text{b}}}{1000g\rho_{\text{b}}}\label{eq:Gb}$$ where $g=9.8\text{\,m/s\ensuremath{^{2}}}$ is the acceleration of gravity. Limited by the battery technology, the power density $\rho_{\text{b}}$ for a specific battery type is statistically close to a constant value. For instance, $\rho_{\text{b}}\text{\ensuremath{\approx}}140\text{\,Wh/kg}$ for LiPo batteries. Moreover, according to the statistical results, the battery weight is positively correlated with $K_{\text{b}}$. Therefore, to minimize $G_{\text{b}}$, the battery parameters $C_{\text{b}}$, $U_{\text{b}}$ and $K_{\text{b}}$should be chosen as small as possible, which is described as
$$\min G_{\text{b}}\Rightarrow\min U_{\text{b}},\min K_{\text{b}},\min C_{\text{b}}.$$
Design Optimization {#Sec-4}
===================
Step 1: Propeller Efficiency Optimization {#Sec-DesOpProp}
-----------------------------------------
### Optimal Blade Number $B_{\text{pOpt}}$
According to Eq.(\[Eq05EffTM\]), if only the blade number $B_{\text{p}}$ is considered, the propeller thrust efficiency $\eta_{\text{T/M}}$ can be simplified into the following form $$\eta_{\text{T/M}}\propto\frac{1}{B_{\text{p}}}\label{Eqw14-2Bp}$$ where the symbol “$\propto$” means “in proportion to”. Eq.(\[Eqw14-2Bp\]) indicates that $\eta_{\text{T/M}}$ monotonically decreases as ${B_{\text{{p}}}}$ increases. Therefore, to maximize $\eta_{\text{T/M}}$, the blade number ${B_{\text{{p}}}}$ should be chosen as small as possible. Moreover, according to the weight optimization principle in Eq.(\[eq:minGp\]), the blade number should also be chosen as small as possible to minimize the propeller weight $G_{\text{p}}$. Considering that the blade number should satisfy the constraint that ${B_{\text{{p}}}\geq2}$, the optimal blade number ${B_{\text{{pOpt}}}}$ should be chosen as $${B_{\text{{pOpt}}}=2}.\label{Eq14BpOpt}$$
### Optimal Pitch Angle $\varphi_{\text{pOpt}}$
According to Eq.(\[Eq05EffTM\]), if only the pitch angle $\varphi_{\text{p}}$ is considered, $\eta_{\text{T/M}}$ can be simplified into the form as $$\eta_{\text{T/M}}\propto\frac{1}{k_{\text{m1}}\varphi_{\text{p}}^{-1}+k_{\text{m2}}\varphi_{\text{p}}}.\label{EqfH14EffFHD}$$ It is easy to obtain from Eq.(\[EqfH14EffFHD\]) that $\eta_{\text{T/M}}$ first increases then decreases as $\varphi_{\text{p}}$ increases. Therefore, the optimal pitch angle $\varphi_{\text{pOpt}}$ should be able to maximize $\eta_{\text{T/M}}$, which yields that $$\varphi_{\text{pOpt}}=\sqrt{\frac{k_{\text{m1}}}{k_{\text{m2}}}}.\label{Eq15DHOpt0}$$
Noteworthy, if it happens in some cases that there are few propeller products in database with pitch angles close to the obtained $\varphi_{\text{pOpt}}$, then the mean pitch angle $\overline{\varphi}_{\text{p}}$ can be chosen as the optimal pitch angle $\varphi_{\text{pOpt}}$=$\overline{\varphi}_{\text{p}}$ to ensure the method can find a proper propeller product.
Step 2: Motor Weight Optimization {#subsec:motorWeight}
---------------------------------
### Optimal Motor NMV $U_{\text{mMaxOpt}}$
![Statistical relationship between the motor NMV $U_{\text{{mMax}}}$ and the propeller maximum thrust $T_{\text{pMax}}$. The testing data come from the motor products website [@TMotor2017].[]{data-label="Fig05ThustWattVolt"}](Fig05){width="40.00000%"}
According to statistical results in Fig.\[Fig05ThustWattVolt\], the relationship between ${U_{\text{{mMax}}}}$ and ${T_{\text{pMax}}}$ can be described as a piecewise function $$f{_{{U_{\text{{mMax}}}}}}\left({T_{\text{pMax}}}\right){\triangleq}\left\{ \begin{array}{cc}
U_{1}, & {0}<{T_{\text{pMax}}}\leq{T}_{1}\\
U_{2}, & {T}_{1}<{T_{\text{pMax}}}\leq{T}_{2}\\
\cdots & \cdots
\end{array}\right.\label{Eq36UmMaxFunc}$$ where $f{_{{U_{\text{{mMax}}}}}}\left(\cdot\right)$ is obtained with the principle of minimizing $U_{\text{{mMax}}}$ for a given $T_{\text{pMax}}$.
Thus, by substituting $T_{\text{pMax}}=T_{\text{max}}$ into Eq.(\[Eq36UmMaxFunc\]), the optimal motor NMV $U_{\text{{mMaxOpt}}}$ can be obtained as $${U_{\text{{mMaxOpt}}}=f{_{{U_{\text{{mMax}}}}}}\left({T_{\text{max}}}\right)}.\label{Eq36UmMax}$$
### Optimal NMC $I_{\text{{mMaxOpt}}}$
According to Eq.(\[Eq37GWconst\]), there is a function relationship between $T_{\text{pMax}}$, $I_{\text{{mMax}}}$ and $U_{\text{{mMax}}}$. Therefore, by substituting $U_{\text{{mMax}}}=U_{\text{{mMaxOpt}}}$ and $T_{\text{pMax}}=T_{\text{max}}$ into Eq.(\[Eq37GWconst\]), the optimal motor NMC $I_{\text{{mMaxOpt}}}$ can be obtained as $${I_{\text{{mMaxOpt}}}=\frac{{T_{\text{max}}}}{G_{\text{WConst}}U_{\text{{mMaxOpt}}}}}.\label{Eq37ImMaX1}$$
### Optimal KV Value ${K}_{\text{VOpt}}$
Since the motor no-load current ${I_{\text{m0}}}$ and resistance $R_{\text{m}}$ are both very small in practice, it is reasonable to assume that $${I_{\text{m0}}}\approx0\text{ and }{R_{\text{m}}\approx0}.\label{Eq33ImRm}$$ Substituting Eq.(\[Eq33ImRm\]) into Eq.(\[Eq18Mmax\]) gives $$\begin{array}{l}
M_{\max}\approx\frac{30{{I}_{\text{mMax}}}}{\pi{{K}_{\text{V}}}}\\
N_{\max}\approx{U_{\text{{mMax}}}{K}_{\text{V}}}
\end{array}.\label{Eq34MmNm}$$ Thus, Eq.(\[Eq19TpMax1\]) can be simplified into the following form $${T_{\text{pMax}}\approx{k}_{\text{tm}}}\cdot{{\left(\frac{{{I}_{\text{mMax}}^{2}U_{\text{{mMax}}}}}{{{K}_{\text{V}}}}\right)}^{2/5}}\label{EQ35IUK}$$ where ${{k}_{\text{tm}}}$ is defined as $$k_{\text{tm}}=f_{k_{\text{tm}}}\left(\text{\ensuremath{B_{\text{p}},\varphi_{\text{p}},h_{\text{hover}}}}\right)\triangleq\sqrt[5]{k_{\text{c}}\frac{255\rho C_{\text{T}}^{5}}{\pi^{4}C_{\text{M}}^{4}}}\label{Eq36Km}$$ where $\rho=f_{\rho}\left(h_{\text{hover}}\right),$ $C_{\text{T}}=f_{C_{\text{T}}}\left(B_{\text{p}},D_{\text{p}},\varphi_{\text{p}}\right)$ and $C_{\text{M}}=f_{C_{\text{M}}}\left(B_{\text{p}},D_{\text{p}},\varphi_{\text{p}}\right)$ are defined in Eqs.(\[Eq03AirDen\])(\[Eq04Cm\]), and $k_{\text{c}}$ is a constant correction coefficient to compensate for the neglected factors including $I_{\text{m0}}$, $R_{\text{m}}$. According to the statistical analysis, the correction coefficient can be set as $k_{\text{c}}$$\approx0.82$$.$
Finally, by substituting $U_{\text{{mMaxOpt}}},$ $I_{\text{mMaxOpt}}$, $T_{\text{max}}$, $B_{\text{pOpt}},$ $\varphi_{\text{pOpt}}$ and $h_{\text{hover}}$ into Eqs.(\[EQ35IUK\])(\[Eq36Km\]), the expression for the optimal KV value ${{K}_{\text{VOpt}}}$ is obtained as $${K}_{\text{VOpt}}=f_{{k}_{\text{tm}}}^{5/2}\left(\text{\ensuremath{B_{\text{pOpt}},\varphi_{\text{pOpt}},h_{\text{hover}}}}\right)\frac{{{I}_{\text{mMaxOpt}}^{2}U_{\text{{mMaxOpt}}}}}{{T_{\text{max}}^{5/2}}}.\label{Eq37KVOpt}$$
Step 3: Motor Efficiency Optimization
-------------------------------------
### Optimal Motor Resistance ${R_{\text{{m}Opt}}}$ and No-load Current $I_{\text{m0Opt}}$
In practice, $R_{\text{{m}}}$ and $I_{\text{m0}}$ should satisfy the constraint that $$R_{\text{{m}}}>0\text{ and }I_{\text{m0}}>0.\label{eq:mRE}$$ By combining Eq.(\[eq:EffM\]) and Eq.(\[eq:mRE\]), the optimal motor resistance ${R_{\text{{m}Opt}}}$ and no-load current $I_{\text{m0Opt}}$ are marked with $${R_{\text{{m}Opt}}=0}\text{ and }I_{\text{m0Opt}}=0$$ which denote that the motor resistance and no-load current should be chosen as close to zero as possible.
Note that, in addition to parameters $R_{\text{{m}}}$,$I_{\text{m0}}$, the motor efficiency $\eta_{\text{m}}$ in Eq. (\[eq:EqEffmOtor\]) is also determined by the motor working state ($U_{\text{m}}$,$I_{\text{m}}$) which further depends on factors including the throttle, rotating speed, propeller parameters, other motor parameters, air density, etc. According to our experimental results, a larger motor with a larger propeller will have a higher efficiency for generating the same thrust. Therefore, only maximizing the motor efficiency $\eta_{\text{m}}$ to solve all the motor parameters may not obtain the desired result because the size and weight will be very large. Thus, this paper finds the optimal propulsion system by considering both the efficiency and the weight to ensure the practicability of the obtained results.
Step 4: Optimal Motor Product {#SecOptMotSel}
-----------------------------
Although the optimal motor parameters $\Theta_{\text{mOpt}}\triangleq\{U_{\text{mMaxOpt}}$, $I_{\text{mMaxOpt}}$, $K_{\text{VOpt}}$, $R_{\text{mOpt}}$, $I_{\text{m0Opt}}\}$ have been obtained through the above procedures, it is still difficult to determine a corresponding product from the database. For example, $U_{\text{mMax}}$ of motor products are usually given by discrete form like 20A, 30A, 40A, while the obtained solutions are usually given with continuous form like $I_{\text{mMaxOpt}}=33.5$A. To solve this problem, a method is proposed to determine the optimal motor product according to $\Theta_{\text{mOpt}}$. For simplicity, the parameter set of the obtained motor product is represented by $\Theta_{\text{mOpt}}^{\ast}\triangleq\{U_{\text{mMaxOpt}}^{\ast}$, $I_{\text{mMaxOpt}}^{\ast}$, $K_{\text{VOpt}}^{\ast}$, $R_{\text{mOpt}}^{\ast}$, $I_{\text{m0Opt}}^{\ast}\}$.
There are two selection principles for the optimal motor product:
\(i) The product should be selected by comparing with every parameter of $\Theta_{\text{mOpt}}$ in a proper sequence. Through the statistical analysis of the motor products on the market, a comparison sequence is given by considering the influence of each parameter on the motor weight as $$U_{\text{mMaxOpt}}^{*}\text{, }K_{\text{VOpt}}^{*}\text{, }I_{\text{mMaxOpt}}^{*}\text{, }R_{\text{{mOpt}}}^{*}\text{, }{I_{\text{m0Opt}}^{*}}.\label{eq:secPrin}$$
\(ii) When comparing one parameter, on the premise of ensuring safety requirements, the product should be chosen equal to or close to the corresponding parameter in $\Theta_{\text{mOpt}}$. For example, if $I_{\text{mMaxOpt}}=33.5$A and the available current options are 20A, 30A, 40A, then it should be chosen that $I_{\text{mMaxOpt}}^{\ast}=40$A for some safety margin. The safety constraints for the selection of motor products are given by $$\begin{array}{l}
U_{\text{mMaxOpt}}^{\ast}\geq U_{\text{mMaxOpt}},\,I_{\text{mMaxOpt}}^{\ast}\geq I_{\text{mMaxOpt}}\end{array}.\label{EqMotorSeq}$$
Step 5: Propeller Weight Optimization
-------------------------------------
### Optimal Diameter $D_{\text{pOpt}}$
According to Eq.(\[eq:TmaxTpMaxEq\]), the following constraint equation should be satisfied $$T_{\text{pMax}}=C_{\text{T}}\rho\left(\frac{N_{\text{max}}}{60}\right)^{2}D_{\text{p}}^{4}\geq T_{\text{max}}$$ which yields that $$D_{\text{p}}\geq\sqrt[4]{\frac{60^{2}T_{\text{max}}}{C_{\text{T}}\rho N_{\text{max}}^{2}}}.\label{eq:MinDia}$$
According to the optimization objective in Eq.(\[eq:minGp\]), the optimal diameter should be chosen as the minimum diameter under constraint in Eq.(\[eq:MinDia\]). Therefore, the optimal diameter $D_{\text{pOpt}}$ can be obtained by combining Eqs.(\[Eq03AirDen\])(\[Eq04Cm\])(\[Eq18Mmax\]) with the parameters $h_{\text{hover}}$, $D_{\text{pOpt}}$, $B_{\text{pOpt}}$ and $\Theta_{\text{mOpt}}^{*}$ as $$\begin{array}{cl}
D_{\text{pOpt}} & =\sqrt[4]{\frac{60^{2}T_{\text{max}}}{\rho C_{\text{T}}N_{\text{max}}^{2}}}=\sqrt[4]{\frac{60^{2}T_{\text{pMax}}}{\rho C_{\text{T}}N_{\text{max}}^{2}}}=\sqrt[5]{{\frac{{M}_{\max}}{{\rho{C_{\text{M}}}{{\left({\frac{{N}_{\max}}{{60}}}\right)}^{2}}}}}}\\
& =\sqrt[5]{{\frac{3600f_{{M}_{\max}}\left(\Theta_{\text{mOpt}}^{*}\right)}{f_{\rho}\left(h_{\text{hover}}\right)f_{C_{\text{M}}}\left(B_{\text{pOpt}},D_{\text{pOpt}}\right)f_{N_{\text{max}}}^{2}\left(\Theta_{\text{mOpt}}^{*}\right)}}}.
\end{array}\label{Eq19DpOpt}$$
Since the propeller pitch $H_{\text{{p}}}$ is more convenient to select a propeller product, according to the definition of the pitch angle $\varphi_{\text{p}}$ in Eq.(\[Eq00-pitch\]), the optimal propeller pitch $H_{\text{{pOpt}}}$ is given by $$H_{\text{{pOpt}}}=\pi\cdot D_{\text{pOpt}}\cdot\tan\varphi_{\text{pOpt}}.$$
Step 6: Optimal Propeller Product {#SecOpProp}
---------------------------------
With the obtained parameters $\Theta_{\text{pOpt}}=\left\{ B_{\text{pOpt}},\varphi_{\text{p}},D_{\text{pOpt}}\right\} $, the optimal propeller product can be determined by searching the propeller product database. The parameter set of the obtained optimal propeller product is represented by $\Theta_{\text{pOpt}}^{\ast}=\left\{ B_{\text{pOpt}}^{*},\varphi_{\text{p}}^{*},D_{\text{pOpt}}^{*}\right\} $.
Similar to the selection principles in *Section\[SecOptMotSel\]*, ** the optimal propeller product should be determined from the propeller database by comparing the parameters in the sequence $B_{\text{pOpt}}^{*}$,$\varphi_{\text{p}}^{*}$,$D_{\text{pOpt}}^{*}$, and each parameter in $\Theta_{\text{pOpt}}^{\ast}$ should be chosen equal or close to the corresponding parameter in $\Theta_{\text{pOpt}}$ with satisfying the safety constraints $$\begin{array}{c}
B_{\text{pOpt}}^{*}=B_{\text{pOpt}},\,D_{\text{pOpt}}^{*}\leq D_{\text{pOpt}}\end{array}.\label{eq:DpOpt}$$
Step 7: ESC Weight Optimization
-------------------------------
### Optimal ESC NMV $U_{\text{eMaxOpt}}$ and NMI $I_{\text{eMaxOpt}}$
Since the ESC and the motor are connected in series, their voltage and current should match with each other to ensure proper operations. For the safety, the ESC NMV $U_{\text{eMax}}$ and NMI $I_{\text{eMax}}$ should be able to support the maximum voltage and current from the motor, which means $$\begin{array}{c}
U_{\text{eMax}}\geq U_{\text{mMaxOpt}},\,I_{\text{eMax}}\geq I_{\text{mMaxOpt}}\end{array}.\label{eq:GeCons}$$ By combining Eq.(\[eq:GeMin\]) and Eq.(\[eq:GeCons\]), the optimal $U_{\text{eMaxOpt}}$ and $I_{\text{eMaxOpt}}$ are given by $$\begin{array}{c}
U_{\text{eMaxOpt}}=U_{\text{mMaxOpt}},\,I_{\text{eMaxOpt}}=I_{\text{mMaxOpt}}\end{array}.\label{eq:OptESC}$$
Step 8: ESC Efficiency Optimization
-----------------------------------
### Optimal ESC Resistance $R_{\text{eOpt}}$
According to Eq.(\[eq:EffE\]), the optimal ESC resistance $R_{\text{eOpt}}$ should be chosen as small as possible for the maximum ESC efficiency $\eta_{\text{e}}$. Since $R_{\text{e}}>0$ in practice, the optimal ESC resistance $R_{\text{eOpt}}$ is marked with $$R_{\text{eOpt}}=0$$ which denotes that the ESC resistance should be chosen as close to zero as possible.
Step 9: Optimal ESC Product
---------------------------
With the obtained parameters $U_{\text{eMaxOpt}},I_{\text{eMaxOpt}},R_{\text{eOpt}}$, the optimal ESC product can be determined by searching the ESC product database. The parameter set of the obtained optimal ESC product is represented by $\Theta_{\text{eOpt}}^{\ast}=\left\{ U_{\text{eMaxOpt}}^{*},I_{\text{eMaxOpt}}^{*},R_{\text{eOpt}}^{*}\right\} $.
Similar to the selection principles in *Section\[SecOptMotSel\]*, the optimal ESC product should be determined from the ESC database by comparing the parameters in the sequence $U_{\text{eMaxOpt}}^{*},I_{\text{eMaxOpt}}^{*},R_{\text{eOpt}}^{*}$, and each parameter in $\Theta_{\text{eOpt}}^{\ast}$ should be chosen equal or close to the corresponding parameter in $\Theta_{\text{eOpt}}$ with satisfying the constraints $$\begin{array}{c}
U_{\text{eMaxOpt}}^{\ast}\geq U_{\text{eMaxOpt}},\,I_{\text{eMaxOpt}}^{\ast}\geq I_{\text{eMaxOpt}}\end{array}.$$
Step 10: Battery Efficiency Optimization
----------------------------------------
### Optimal Battery Resistance ${R}_{\text{bOpt}}$
According to Eq.(\[eq:Effb\]), $R_{\text{b}}$ should be chosen as small as possible for the maximum battery efficiency $\eta_{\text{b}}$. considering that $R_{\text{b}}>0$ in practice, the optimal battery resistance ${R}_{\text{bOpt}}$ is marked with $${R}_{\text{bOpt}}=0$$ which denotes that the battery resistance should be chosen as close to zero as possible.
Step 11: Battery Weight Optimization
------------------------------------
### Optimal Battery Nominal Voltage $U_{\text{bOpt}}$
As analyzed in *Section\[subsec:motorWeight\]*, the actual working voltage of the motor is determined by the battery voltage, and the constraint in Eq.(\[eq:UbTmax\]) should be satisfied to make sure the motor has the minimum weight. Therefore, after the optimal motor NMV $U_{\text{mMaxOpt}}$ is determined, the optimal battery voltage $U_{\text{bOpt}}$ is also determined as $$U_{\text{bOpt}}=U_{\text{mMaxOpt}}.\label{eq:UbOpt}$$
### Optimal Battery Capacity $C_{\text{bOpt}}$
After the above procedures, the propeller, motor, ESC and battery of the propulsion system both have the maximum efficiency, which means the battery has the minimum current $I_{\text{b0}}$ under the hovering mode with propeller thrust $T_{\text{hover}}$. According to [@Shi2017], with knowing the parameters $\Theta_{\text{pOpt}}^{*},\Theta_{\text{mOpt}}^{*},\Theta_{\text{eOpt}}^{*},U_{\text{bOpt}},T_{\text{hover}},n_{\text{p}},h_{\text{hover}}$, the battery current $I_{\text{b0}}$ can be estimated through the propulsion system equivalent circuit in Fig.\[Fig02\]. By substituting the desired propeller thrust $T_{\text{hover}}$ into the propeller, motor, ESC and battery models in Eqs.(\[Eq02PropTorque\])(\[Eq05Um\])(\[Eq06Ie\])(\[Eq08Ue\]) successively, the battery discharge current $I_{\text{b0}}$ can be obtained. Then, the optimal battery capacity $C_{\text{bOpt}}$ can be obtained by substituting $t_{\text{discharge}}=t_{\text{hover}}$ into Eq.(\[eq:tdischarge\]), which yields that $$C_{\text{bOpt}}=\frac{t_{\text{hover}}I_{\text{b0}}}{0.85}\cdot\frac{1000}{60}\label{EqCbOpt}$$
### Optimal
The battery MDR $K_{\text{b}}$ should be chosen as small as possible within the constraint in Eq.(\[eq:KbConstraint\]). Therefore, by substituting the obtained $U_{\text{mMaxOpt}}$ and $C_{\text{bOpt}}$ into Eq.(\[eq:KbConstraint\]), the optimal battery MDR K$_{\text{bOpt}}$ is given by $$K_{\text{b}}=\frac{1000\left(n_{\text{p}}I_{\text{{mMaxOpt}}}+I_{\text{other}}\right)}{C_{\text{bOpt}}}.\label{eq:KBop}$$
Step 12: Optimal Battery Product
--------------------------------
With the obtained parameters $\Theta_{\text{bOpt}}=\left\{ U_{\text{bOpt}},K_{\text{bOpt}},C_{\text{bOpt}},R_{\text{bOpt}}\right\} $, the optimal battery product can be determined by searching the battery product database. The parameter set of the obtained optimal battery product is represented by $\Theta_{\text{bOpt}}^{*}=\left\{ U_{\text{bOpt}}^{*},K_{\text{bOpt}}^{*},C_{\text{bOpt}}^{*},R_{\text{bOpt}}^{*}\right\} $.
Similar to the selection principles in *Section\[SecOptMotSel\]*, the optimal battery product should be determined from the battery database by comparing the parameters in the sequence $U_{\text{bOpt}}^{*},K_{\text{bOpt}}^{*},C_{\text{bOpt}}^{*},R_{\text{bOpt}}^{*}$, and each parameter in $\Theta_{\text{bOpt}}^{\ast}$ should be chosen equal or close to the corresponding parameter in $\Theta_{\text{bOpt}}$ with satisfying the constraints $$\begin{array}{l}
U_{\text{bOpt}}^{\ast}=U_{\text{bOpt}},\,K_{\text{bOpt}}^{\ast}\geq K_{\text{bOpt}}\end{array}.$$ Noteworthy, the battery voltage $U_{\text{bOpt}}^{\ast}$ should satisfy the constraint that $U_{\text{bOpt}}^{\ast}=U_{\text{bOpt}}=U_{\text{mMaxOpt}}$ to ensure that the motor can work under the desired voltage.
In practice, designers have to build a battery pack to satisfy the above design requirements by connecting small battery cells in series or parallel. According to [@quan2017introduction pp. 46], by combining battery cells in series, a higher voltage can be obtained, with capacity unchanged. On the other hand, by combining battery cells in parallel, larger capacity and discharge current can be obtained, with voltage unchanged.
Experiments and Verification {#Sec-6}
============================
Statistical Model Verification {#Sec-6-1}
------------------------------
Comprehensive statistical analyses for the products of propellers, motors, ESCs and batteries on the market are performed to verify the weight statistical functions $f_{G_{\text{p}}}\left(\cdot\right)$, $f_{G_{\text{m}}}\left(\cdot\right)$, $f_{G_{\text{e}}}\left(\cdot\right)$ and $f_{G_{\text{b}}}\left(\cdot\right)$ in Eqs.(\[eq:Gp\])(\[eq:GmOrgin\])(\[eq:GeOrgin\])(\[eq:Gb\]) respectively. Some typical results are presented in Fig.\[Fig09-0\], where the products come from four most well-known manufacturers (APC, T-MOTOR, Hobbywing, Gens ACE). Fig.\[Fig09-0\] shows the relationship between the weight and the parameters of each component. The statistical results are consistent with the analysis results in *Section*\[Sec-4\].
![Weight statistical results for the propulsion system components.[]{data-label="Fig09-0"}](Fig09-0){width="45.00000%"}
Optimization Method Verification {#SecPropDiam-1}
--------------------------------
According to Eq.(\[Eq15DHOpt0\]), the optimal pitch angle for T-MOTOR series propellers (parameters are listed in Eq.(\[Eq04KCM\])) is obtained as $\varphi_{\text{pOpt}}=0.1054$. For the comparative validation, a series of multicopter propellers from the website [@TMotor2017] are listed in Table\[Table2TProp\]. It can be observed from Table\[Table2TProp\] that the statistical diameter/pitch ratio result is $D_{\text{p}}/H_{\text{p}}\approx3$ ($\varphi_{\text{p}}\approx0.1065$), which is in good agreement with the theoretical result $\varphi_{\text{pOpt}}=0.1054$. The comparison between the calculation value and the statistical value shows that the pitch angle optimization method is effective to find the optimal pitch angle adopted by manufacturers. Meanwhile, the propeller test data from the UIUC website [@UIUC2017] also show that the obtained pitch angle $\varphi_{\text{pOpt}}$ can guarantee high efficient in increasing the thrust and decreasing the torque.
Propeller P12x4 P14x4.8 P15x5 P16x5.4
----------------------------- --------- ---------- ---------- ----------
$D_{\text{p}}/H_{\text{p}}$ 3 2.92 3 2.963
Propeller P18x6.1 $\cdots$ G26x8.5 G28x9.2
$D_{\text{p}}/H_{\text{p}}$ 2.951 $\cdots$ 3.06 3.04
Propeller G30x10 G32x11 G34x11.5 G40x13.1
$D_{\text{p}}/H_{\text{p}}$ 3 2.91 2.96 3.05
: Diameter/pitch ratio of carbon fiber propellers from the website [@TMotor2017].[]{data-label="Table2TProp"}
The motor T-MOTOR U11 KV90 is adopted as an example to verify the proposed optimization method. The calibrated motor parameters of U11 KV90 are listed below
$$\begin{array}{c}
K_{\text{V}}=90\text{RPM/V},\text{ }U_{\text{mMax}}=48\text{V},\text{ }I_{\text{mMax}}=36\text{A},\\
U_{\text{m0}}=10\text{V},\text{ }I_{\text{m0}}=0.7\text{A},\text{ }R_{\text{m}}=0.3\Omega
\end{array}\label{eq:U11Param-1}$$
and the corresponding experiment results from [@TMotor2017] are listed in Table\[Tab4PropData\].
[|c|>p[0.04]{}|>p[0.05]{}|>p[0.035]{}|c|>p[0.05]{}|>p[0.04]{}|]{} Prop. & Current (A) & Power (W) & Thrust (N) & RPM & Torque (N$\cdot$m) & Tempe. ($^{\circ}$C) [\
]{} 27x8.8CF & 24.6 & 1180.8 & 81.4 & 3782 & 2.623 & 58.5[\
]{} 28x9.2CF & 28.3 & 1358.4 & 91.3 & 3696 & 3.068 & 66.5[\
]{} 29x9.5CF & 31.9 & 1531.2 & 98.8 & 3602 & 3.41 & 78.5[\
]{} 30x10.5CF & 36.3 & 1742.4 & 106.8 & 3503 & 3.846 & HOT$!$[\
]{}
From the perspective of theoretical calculation, the optimal diameter of motor U11 KV90 can be obtained by substituting the motor parameters in Eq.(\[eq:U11Param-1\]) into Eq.(\[Eq19DpOpt\]), where the obtained result is $D_{\text{pOpt}}=29.7$inches. Therefore, according to the constraint in Eq.(\[eq:DpOpt\]), the propeller diameter should be chosen as ${D_{\text{pOpt}}^{\ast}}=29$inches because the motor will overheat if the propeller diameter is larger than $D_{\text{pOpt}}$.
It can be observed from the experimental results in the last two rows of Table\[Tab4PropData\] that the motor temperature becomes overheated when the propeller diameter changes from 29 inches to 30 inches. Therefore, the optimal propeller diameter obtained from experiments should be 29inches, which agrees with the theoretical optimal solution ${D_{\text{pOpt}}^{\ast}}=29$inches.
Design Optimization Example
---------------------------
As an example, assume that the given design task is to select an optimal propulsion system for a multicopter ($n_{\text{p}}=4$) whose total weight $G_{\text{total}}=196\text{\,N}$ (20 kg) and flight time is $t_{\text{hover}}=17$min (flight altitude $h_{\text{hover}}=50\,\text{m}$). The component databases are composed of the ESC, BLDC motor, caber fiber propeller products from T-MOTOR website [@TMotor2017], and the LiPo battery products from GENS ACE website [@ACE2017].
The key calculation results of the design procedures are listed as follows.
i\) The thrust requirements for the propulsion system are obtained from Eqs.(\[eq:T0\])(\[Eq31T0TpmaxThr\]) as $T_{\text{hover}}=49$N and $T_{\text{max}}=98$N, where $\gamma=0.5$ is adopted here.
ii\) The optimal propeller efficiency parameters are obtained from Eqs.(\[Eq14BpOpt\])(\[Eq15DHOpt0\]) as $B_{\text{pOpt}}=2$, $\varphi_{\text{pOpt}}=0.1065$rad. Then, with statistical models in Fig.\[Fig05ThustWattVolt-1\] and Fig.\[Fig05ThustWattVolt\], the optimal motor parameters are obtained from Eqs.(\[Eq36UmMax\])(\[Eq37ImMaX1\])(\[Eq37KVOpt\]) as $U_{\text{mMaxOpt}}=48$V, ${{I}_{\text{mMaxOpt}}}=34$A, ${{K}_{\text{VOpt}}}=91$RPM/V. Therefore, by searching products from the T-MOTOR website according to principles in Eqs.(\[eq:secPrin\])(\[EqMotorSeq\]), the optimal motor is determined as U11 KV90.
iii\) With the blade parameters in Eq.(\[Eq04KCM\]), the optimal propeller diameter can be obtained from Eq.(\[Eq19DpOpt\]) as $D_{\text{pOpt}}=0.7468\text{\,m}$, and the optimal propeller product is selected as 29x9.5CF 2-blade. In the same way, the ESC parameters can also be obtained from Eq.(\[eq:OptESC\]) as $U_{\text{eMaxOpt}}=48$V and ${{I}_{\text{eMaxOpt}}}=34$A, and the optimal ESC product is selected as FLAME 60A HV.
iv\) For the battery, the optimal parameters are obtained from Eqs.(\[eq:UbOpt\])(\[EqCbOpt\])(\[eq:KBop\]) as $U_{\text{bOpt}}=48$V, $C_{\text{bOpt}}=16000$mAh and $K_{\text{bOpt}}=10$C. The optimal battery selected from ACE website is TATTU LiPo 6S 15C 16000mAh $\times$ $2$.
The obtained result has been verified by several multicopter designers. Experiments show that a quadcopter with the designed propulsion system can efficiently meet the desired design requirements. If a larger motor (like T-MOTOR U13) is selected, then a smaller propeller has to be chosen for generating the same full-throttle thrust $T_{\text{pMax}}$, which reduces the motor efficiency because the optimal operating condition cannot be reached. As a result, the obtained propulsion system is heavier than the optimal result according to our experiments. If a smaller motor is selected, then a larger propeller has to be chosen, which results in exceeding the safety current of the motor. Therefore, the obtained propulsion system is optimal within the given database.
Method Application
------------------
If the same optimization problem is solved by brute force searching methods to traverse all combinations and evaluate the performance of each combination, the time consumption will be far longer than the proposed method. For example, it takes about 100ms for our evaluation method in [@Shi2017] to calculate the performance of each propulsion combination. Assuming that the numbers of products in the propeller, motor, ESC and battery databases all equal to $n$, then it will take about $T\left(n\right)\approx C_{n}^{4}=O(n^{4})$ to traverse all products in the databases. By comparison, the computation amount of the proposed method to traverse 4 component databases is 4$n$. Since there are 15 parameters in Table\[tab1\], it is easy to verify that the total computation amount of the proposed method is approximate to $T\left(n\right)\approx15\cdot4n=O(n)$, which is much faster than the brute force searching methods.
The optimization algorithm proposed in this paper is adopted as a sub-function in our online toolbox (URL: ) to estimate the optimal propulsion system with giving the multicopter total weight. The program is fast enough to be finished within 30 ms by using a web server with low configuration (single-core CPU and 1GB of RAM). The feedback results from the users show that the optimization results are effective and practical for the multicopter design.
Conclusions {#Sec-7}
===========
In this paper, the precise modeling methods for the propeller, ESC, motor and battery are studied respectively to solve the optimization problem for the propulsion system of multicopters. Then, the key parameters of each component are estimated through mathematical derivations to make sure that the obtained propulsion system has the maximum efficiency. Experiments and feedback of the website demonstrate the effectiveness of the proposed method. The propulsion system is the most important part of a multicopter, and its design optimization method will be conducive to the fast, optimal and automatic design of the whole multicopter system or other types of aircraft systems. The theoretical analysis can be further used to directly maximize the endurance of all kinds of UAVs, which is interesting for future research.
[Xunhua Dai]{} received the B.S. and M.S. degrees from Beihang University, Beijing, China, in 2013, and 2016, respectively. Currently, he is a Ph.D. candidate of School of Automation Science and Electrical Engineering at Beihang University, Beijing, China. His main research interests are reliable flight control, model-based design and design optimization of UAVs.
[Quan Quan]{} received the B.S. and Ph.D. degrees from Beihang University, Beijing, China, in 2004, and 2010, respectively. He has been an Associate Professor with Beihang University since 2013, where he is currently with the School of Automation Science and Electrical Engineering. His research interest covers reliable flight control, vision-based navigation, repetitive learning control, and time-delay systems.
[Jinrui Ren]{} received the B.S. degree from Northwestern Polytechnical University, Xian, China, in 2014. Currently, She is a Ph.D. candidate of School of Automation Science and Electrical Engineering at Beihang University, Beijing, China. Her main research interests include nonlinear control, flight control, and aerial refueling.
[Kai-Yuan Cai]{} received the B.S., M.S., and Ph.D. degrees from Beihang University, Beijing, China, in 1984, 1987, and 1991, respectively. He has been a full professor at Beihang University since 1995. He is a Cheung Kong Scholar (chair professor), jointly appointed by the Ministry of Education of China and the Li Ka Shing Foundation of Hong Kong in 1999. His main research interests include software testing, software reliability, reliable flight control, and software cybernetics.
[^1]: The authors are with School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China.
|
---
abstract: 'This paper explores the relationship between the existence of an exact embedded Lagrangian filling for a Legendrian knot in the standard contact $\rr^3$ and the hierarchy of positive, strongly quasi-positive, and quasi-positive knots. On one hand, results of Eliashberg and especially Boileau and Orevkov show that every Legendrian knot with an exact, embedded Lagrangian filling is quasi-positive. On the other hand, we show that if a knot type is positive, then it has a Legendrian representative with an exact embedded Lagrangian filling. Further, we produce examples that show that strong quasi-positivity and fillability are independent conditions.'
address:
- 'Boston College, Chestnut Hill, MA 02467'
- 'Haverford College, Haverford, PA 19041'
author:
- Kyle Hayden
- 'Joshua M. Sabloff'
bibliography:
- 'main.bib'
title: Positive Knots and Lagrangian Fillability
---
Introduction {#sec:intro}
============
Properly embedded Lagrangian submanifolds of $B^4$ whose boundaries are Legendrian links in $S^3$, called [[**fillings**]{}]{} of the Legendrian links, are of interest in a variety of fields: in smooth knot theory, Lagrangian fillings minimize the slice genus of a link [@chantraine]; in Legendrian knot theory, Lagrangian fillings induce augmentations of the Legendrian Contact Homology DGA [@ekholm:lagr-cob; @ehk:leg-knot-lagr-cob; @rizell:lifting]; and Lagrangian fillings can even be used to answer questions about complex algebraic curves [@polyfillability]. These considerations motivate the following question:
> *Which smooth knot types have Legendrian representatives with Lagrangian fillings?*
Such a smooth knot type is termed [[**fillable**]{}]{}. In analyzing this question, we work in the equivalent setting of Legendrian links in the standard contact $\rr^3$ and Lagrangian fillings in the symplectization $\rr \times \rr^3$. We further require the Lagrangian fillings to be exact, orientable, embedded, and collared, i.e. equal to $\rr \times \Lambda$ outside a compact set.
Initial progress on the question above indicates a close relationship to the hierarchy of positivity in smooth knot theory. To describe the hierarchy, let $BP$ be the set of braid positive knot types, $P$ be the set of positive knot types, $SQP$ be the set of strongly quasi-positive knot types, and $QP$ be the set of quasi-positive knot types. The following relationships are well known (see [@hedden:pos; @rudolph:survey; @stoimenow:pos], for example): $$\label{eq:hierarchy}
BP \subsetneq P \subsetneq SQP \subsetneq QP.$$
The first main result of this paper delineates a sufficient condition for a smooth knot type to be fillable:
\[thm:pos-filling\] All positive knots are fillable.
To make progress towards a necessary condition, we begin by noting that quasi-positivity is necessary for fillability. To see this, first note that a result of Eliashberg [@yasha:lagr-cyl] can be extended to show that a Lagrangian filling of a Legendrian knot may be perturbed to a symplectic filling of a transverse knot (see [@polyfillability] for more details). From there, use the following result of Boileau and Orevkov [@bo:qp]: if a smooth knot type has a transverse representative with a symplectic filling, then it is quasi-positive.
As we shall see in Section \[sec:sqp-qp\], however, not all quasi-positive knots are fillable. Further, the intermediate condition of strong quasi-positivity is independent of fillability:
\[thm:sqp-bad\] There exists a fillable knot that is quasi-positive, but not strongly quasi-positive, and there exists non-fillable knot that is strongly quasi-positive.
Since this theorem shows that strong quasi-positivity is not relevant to fillability, we must seek an alternative condition to characterize fillable knots. Based on the results above and a survey of quasi-positive knots up to ten crossings,[^1] we make the following conjecture:
\[conj\] A smooth knot type $K$ is fillable if and only if it is quasi-positive and the HOMFLY bound on the maximum Thurston-Bennequin number of $K$ is sharp.
One implication of the conjecture is true: if a smooth knot is fillable, then, as discussed above, it is quasi-positive. Further, the Legendrian contact homology DGA of the fillable Legendrian representative has an augmentation [@ekholm:lagr-cob; @rizell:lifting], hence a graded ruling [@fuchs-ishk; @rulings], and hence the HOMFLY bound is sharp [@rutherford:kauffman].
The remainder of the paper is organized as follows: after reviewing background on the various notions of positivity touched on above, on rulings of Legendrian knots, and on the construction of Lagrangian cobordisms via handle attachment in Section \[sec:background\], we prove Theorem \[thm:pos-filling\] in Section \[sec:pos-fillable\] and Theorem \[thm:sqp-bad\] in Section \[sec:sqp-qp\].
Acknowledgements {#acknowledgements .unnumbered}
----------------
We thank Matt Hedden for several stimulating discussions about the material in this paper. The authors were partially supported by NSF grant DMS-0909273.
Background {#sec:background}
==========
In this section, we review positivity and (strong) quasi-positivity of smooth knots, rulings of Legendrian knots, and various constructions of Lagrangian fillings. We assume that the reader is familiar with basic notions of Legendrian knot theory as discussed, for example, in Etnyre’s survey [@etnyre:knot-intro]. In particular, we assume familiarity with front diagrams, the classical invariants, and the HOMFLY bound on the Thurston-Bennequin number.
Notions of Positivity {#ssec:gh}
---------------------
The notion of positivity is simple to define: an oriented link is [[**positive**]{}]{} if it has a projection in which all crossings are positive. The jump to quasi-positivity requires the concept of positive bands in a braid, which generalize positive crossings. Denote the standard generators of the braid group by $\sigma_i$. A positive band is a braid word of the form $w \sigma_i w^{-1}$, where $w$ is any word in the braid group. The more restricted notion of a positive embedded band, denoted $\sigma_{i,j}$, is a positive band of the form $$\label{eq:sigmaindices} \sigma_{i,j} \equiv (\sigma_i \cdots \sigma_{j-2}) \sigma_{j-1}(\sigma_i \cdots \sigma_{j-2})^{-1} , \qquad 1 \leq i < j \leq n.$$ We define a (strongly) quasi-positive braid to be a product of positive (embedded) bands, and we say that an oriented link is [[**(strongly) quasipositive**]{}]{} if it is the closure of such a braid.
The relationship between these notions of positivity — in particular, the inclusions displayed in Equation — has been treated extensively in the literature; see [@hedden:pos; @rudolph:survey; @stoimenow:pos] for surveys.
Rulings of Legendrian Links {#ssec:rulings}
---------------------------
A ruling is a combinatorial object associated to the front diagram of a Legendrian link. Hereafter, assume that all Legendrian links have been perturbed so that the cusps and crossings in their front diagrams have distinct $x$-coordinates. We define a **ruling** of such a front to be a one-to-one correspondence between left and right cusps, together with pairs of paths (called [[**companions**]{}]{}) in the front joining corresponding cusps. The companion paths must satisfy the following conditions:
1. paired paths intersect only at the cusps they join; and
2. unpaired paths intersect only at crossings.
In particular, the $x$-coordinate of each path is strictly monotonic and the paths cover the front. Further, since the companion paths meet only at the cusps, each pair of companions bounds a [[**ruling disk**]{}]{} in the plane.
To refine the notion of a ruling, we refer to the paths incident to a crossing of the front diagram as [[**crossing paths**]{}]{}. At a crossing, either the two crossing paths pass through each other or one path lies entirely above the other. In the latter case, we say that the ruling is [[**switched**]{}]{} at the crossing. We call a ruling [[**normal**]{}]{} if, at each switched crossing, the crossing paths have one of the configurations shown in Figure \[fig:normality\]. We can rephrase this definition in terms of ruling disks: a ruling is normal if, near a switch, the interiors of the disks involved in the switch are either nested or disjoint. If all switches occur at positive crossings, then we say that the ruling is [[**oriented**]{}]{}.[^2]
![The two leftmost configurations of crossing and companion strands are allowed in a normal ruling (as is the reflection of the center configuration through a horizontal line), but the rightmost configuration is not allowed.[]{data-label="fig:normality"}](all-switches.pdf){width=".55\linewidth"}
The set of all oriented normal rulings of a Legendrian trefoil appear in Figure \[fig:trefoil\].
![The three normal rulings of a Legendrian trefoil.[]{data-label="fig:trefoil"}](trefoil-rulings.pdf){width=".8\linewidth"}
The proof of Theorem \[thm:pos-filling\] relies on rulings that switch at every crossing. In such a ruling, ruling disks are either (globally) nested or disjoint, and hence such a ruling must be normal. This type of ruling is equivalent to an “admissible 0-resolution” of a front, as studied by Ng in [@lenny:khovanov]. A 0-resolution of a front is the diagram obtained by “smoothing” crossings as in Figure \[fig:0-resolution\]. The aforementioned paper shows that alternating links have Legendrian representatives with normal — but not necessarily oriented — rulings that switch at each crossing. In Section \[sec:pos-fillable\], below, we will revisit a result of Tanaka which implies that positive links also admit *oriented* normal rulings of this form.
Lagrangian Handle Attachment {#ssec:cobord}
----------------------------
In this subsection, we describe constructions for Lagrangian fillings of Legendrian links in the standard contact $\rr^3$. A Lagrangian submanifold $L$ of the symplectization is a **Lagrangian cobordism** between the Legendrian submanifolds ${\ensuremath{\Lambda}}_\pm \subset \rr^3$ if there exists a pair of real numbers $T_- < T_+$ such that $$\begin{aligned}
L \cap \big((-\infty,T_-] \times \rr^3 \big) &= (-\infty,T_-] \times {\ensuremath{\Lambda}}_-, \text{ and}\\
L \cap \big([T_+,\infty) \times \rr^3 \big) &= [T_+,\infty) \times {\ensuremath{\Lambda}}_+.\end{aligned}$$
As defined in the introduction, a Lagrangian filling of a Legendrian link is a cobordism from the empty set to the given link. Note that stacking one Lagrangian cobordism on top of another results in a new Lagrangian cobordism.
The following tool allows us to construct Lagrangian fillings with a prescribed Legendrian link as their boundary. It has been adapted from Theorem 4.2 of [@bst:construct], though this result appears in slightly different forms in the work of Ekholm-Honda-Kálman [@ehk:leg-knot-lagr-cob] and of Rizell [@rizell:surgery].
\[thm:construct\] If two Legendrian links ${\ensuremath{\Lambda}}_{-}$ and ${\ensuremath{\Lambda}}_{+}$ in the standard contact $\rr^{3}$ are related by any of the following three moves, then there exists an exact, embedded, orientable, and collared Lagrangian cobordism from ${\ensuremath{\Lambda}}_{-}$ to ${\ensuremath{\Lambda}}_{+}$.
Isotopy
: ${\ensuremath{\Lambda}}_{-}$ and ${\ensuremath{\Lambda}}_{+}$ are Legendrian isotopic.
$\boldsymbol{0}$-Handle
: The front of ${\ensuremath{\Lambda}}_{+}$ is the same as that of ${\ensuremath{\Lambda}}_{-}$ except for the addition of a disjoint Legendrian unknot as in the left side of Figure \[fig:handle\].
$\boldsymbol{1}$-Handle
: The fronts of ${\ensuremath{\Lambda}}_{\pm}$ are related as in the right side of Figure \[fig:handle\].
![Diagram moves corresponding to attaching a 0-handle and an oriented 1-handle.[]{data-label="fig:handle"}](3d-surgery.pdf){width=".6\linewidth"}
We will call a filling [[**decomposable**]{}]{} if it can be split into cobordisms arising as in Theorem \[thm:construct\]. Hereafter, Theorem \[thm:construct\] will be our primary means for producing links with Lagrangian fillings.
Positive Knots are Fillable {#sec:pos-fillable}
===========================
In this section, we prove Theorem \[thm:pos-filling\]. As mentioned above, the key tool in the proof is the existence of an oriented ruling in which all crossings are switched. The following lemma, essentially due to Tanaka [@tanaka:max-tb-pos], shows that we may assume the existence of such a ruling for a positive link.
\[lem:pos-ruling\] Every positive link admits a Legendrian representative whose front diagram carries an oriented normal ruling in which all crossings are switched.
In [@tanaka:max-tb-pos], Tanaka extends an earlier result of Yokota [@yokota:pos-poly] to show that every positive link has a Legendrian representative ${\ensuremath{\Lambda}}$ with the following properties:
1. \[it:crossing\] Every crossing has both strands oriented to the left (in particular, every crossing is positive); and
2. \[it:0-res\] The 0-resolution of ${\ensuremath{\Lambda}}$ consists of disjoint (but possibly nested) closed curves, each of which contains exactly one left cusp and one right cusp.
By (\[it:crossing\]), the smoothing of each crossing in the 0-resolution respects the orientation of ${\ensuremath{\Lambda}}$. Thus, the strands which connect left and right cusps in the 0-resolution correspond to ruling paths in an oriented normal ruling that switches at each crossing.
We now prove that all positive links are fillable.
$1$-handles \[b\] at 762 1470 Isotopy \[b\] at 762 600 ![At the top are diagrams of ${\ensuremath{\Lambda}}_+$ near the crossing constructed in the proof of Theorem \[thm:pos-filling\]. Diagrams of ${\ensuremath{\Lambda}}_-$ appear at the bottom, and passing from bottom to top is a sequence of moves defining a Lagrangian cobordism from ${\ensuremath{\Lambda}}_-$ up to ${\ensuremath{\Lambda}}_+$.[]{data-label="fig:allowedcrossings"}](Figures/AllowedCrossings3 "fig:"){width=".6\linewidth"}
By Lemma \[lem:pos-ruling\], it suffices to consider a fixed front ${\ensuremath{\Lambda}}_+$ that admits an oriented normal ruling in which all crossings are switched. The proof proceeds by induction on the number of crossings. If ${\ensuremath{\Lambda}}_+$ has no crossings, then it must be a disjoint union of maximal Thurston-Bennequin unknots. Such a link is fillable by the $0$-handle construction in Theorem \[thm:construct\].
For the inductive step, suppose that every Legendrian whose front diagram has fewer crossings than ${\ensuremath{\Lambda}}_+$ and that admits an oriented normal ruling in which all crossings are switched is fillable. We begin by showing that the ruling of ${\ensuremath{\Lambda}}_+$ must have a switch with a neighborhood equivalent to one of the topmost diagrams in Figure \[fig:allowedcrossings\] (up to an overall reversal of orientation or reflection through the horizontal). First, each switch occurs at a positive crossing, so the crossing paths must have the same horizontal direction. Second, companion paths are oriented in the opposite direction to the crossing paths, so the two companion strands must also have the same horizontal direction. Finally, because the ruling switches at each crossing, the ruling disks are either nested or disjoint. Any switch along the boundary of an innermost ruling disk will have the desired form.
Next, we use the crossing found above to construct a Legendrian link ${\ensuremath{\Lambda}}_-$ with the following properties:
1. The fronts of ${\ensuremath{\Lambda}}_+$ and ${\ensuremath{\Lambda}}_-$ are identical outside of a neighborhood of the crossing;
2. Near the crossing, the front of ${\ensuremath{\Lambda}}_-$ is of one of the forms depicted at the bottom of Figure \[fig:allowedcrossings\];
3. ${\ensuremath{\Lambda}}_-$ has an oriented normal ruling with all crossings switched; and
4. There exists a Lagrangian cobordism from ${\ensuremath{\Lambda}}_-$ to ${\ensuremath{\Lambda}}_+$.
Only the last condition needs some verification, which is carried out in Figure \[fig:allowedcrossings\].
Clearly, the Legendrian ${\ensuremath{\Lambda}}_-$ satisfies the inductive hypothesis and has one less crossing than ${\ensuremath{\Lambda}}_+$. It follows that ${\ensuremath{\Lambda}}_-$ is fillable, and hence, by condition (4), that ${\ensuremath{\Lambda}}_+$ is fillable as well. The theorem follows.
We say that an oriented ruling of a front has an [[**unlinked resolution**]{}]{} if the result of performing $0$-resolutions (see Figure \[fig:0-resolution\]) at all switches of the ruling results in an unlink, all of whose components are maximal $tb$ unknots. Using this language, we see that the proof above actually gives a stronger result than Theorem \[thm:pos-filling\]: a smooth knot type is fillable if it has a Legendrian representative whose front has an oriented ruling with an unlinked resolution.
(Strong) Quasi-positivity and Fillability {#sec:sqp-qp}
=========================================
We end this paper by providing the examples necessary to prove Theorem \[thm:sqp-bad\]. The first example proves the first part of the theorem, namely that there exists a fillable quasi-positive knot that is not strongly quasi-positive.
As shown in Figure \[fig:9-46\], the mirror of the $9_{46}$ knot is fillable, and hence quasi-positive. On the other hand, since this knot is a non-trivial slice knot, its slice genus differs from its Seifert genus and hence it is not strongly quasi-positive [@rudolph Prop. 2].
$\emptyset$ \[b\] at 165 5 $0$-handles \[l\] at 175 65 Isotopy \[l\] at 175 185 Isotopy \[l\] at 175 385 $1$-handle \[l\] at 175 585 ![The slice knot $m(9_{46})$ is fillable.[]{data-label="fig:9-46"}](Figures/m9-46-construction "fig:"){width=".4\linewidth"}
The example for the second part of the theorem — that there exists a non-fillable strongly quasi-positive knot — is somewhat more complicated. It was originally brought to light by Stoimenow [@stoimenow:pos Example 3] in the context of disproving a conjecture of Fiedler on an upper bound for the minimal degree of the Jones polynomial of a (quasi-positive) knot in terms of a band representation.
Let $K$ be the closure of the strongly quasi-positive $4$-braid $$\sigma_1^2 (\sigma_1 \sigma_2 \sigma_1^{-1}) \sigma_2 \sigma_1 \sigma_3 (\sigma_1 \sigma_2 \sigma_1^{-1}) \sigma_2 (\sigma_2 \sigma_3 \sigma_2^{-1}) (\sigma_1 \sigma_2 \sigma_1^{-1}) (\sigma_2 \sigma_3 \sigma_2^{-1}).$$
Suppose, for the sake of contradiction, that $K$ has a fillable Legendrian representative $\Lambda$. Denote the maximum Euler characteristic of a smooth slicing surface for $K$ by $\chi_4(K)$. We may easily compute from Rudolph’s formula [@rudolph:qp-obstruction Section 3] that $\chi_4 (K) = -7$. Since $\Lambda$ is fillable, Theorem 1.3 of [@chantraine] implies that $tb(\Lambda) = - \chi_4(K) = 7$ and that this is the maximal Thurston-Bennequin invariant for $K$, $\overline{tb}(K)$.
On the other hand, we compute[^3] that the degree in the framing variable of the HOMFLY polynomial of $K$ is $-10$. Thus, the HOMFLY bound on the Thurston-Bennequin number would be $\overline{tb}(K) \leq 9$. Clearly, this bound is not sharp. As argued after Conjecture \[conj\], however, if $K$ were fillable, then the HOMFLY bound would be sharp. We must conclude, then, that $K$ is not fillable.
[^1]: We used [@baader:slice-gordian] for a list of quasi-positive knots, Morrison’s code in the `KnotTheory` package [@knottheory] and Morton and Short’s `C++` program [@morton-short:homfly-program; @morton-short:homfly-article] to compute the HOMFLY polynomials, `Gridlink` [@gridlink] to find Legendrian representatives of quasi-positive knots, and our own constructions of Lagrangian fillings.
[^2]: In the literature, such a ruling is also called [[**$2$-graded**]{}]{}.
[^3]: Again, the computation was performed using Morrison’s code in the `KnotTheory` package [@knottheory] and Morton and Short’s `C++` program [@morton-short:homfly-program; @morton-short:homfly-article].
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